src/ZF/Constructible/Rec_Separation.thy
 author paulson Wed Jul 24 17:59:12 2002 +0200 (2002-07-24) changeset 13418 7c0ba9dba978 parent 13409 d4ea094c650e child 13422 af9bc8d87a75 permissions -rw-r--r--
tweaks, aiming towards relativization of "satisfies"
```     1 header{*Separation for Facts About Recursion*}
```
```     2
```
```     3 theory Rec_Separation = Separation + Datatype_absolute:
```
```     4
```
```     5 text{*This theory proves all instances needed for locales @{text
```
```     6 "M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*}
```
```     7
```
```     8 lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
```
```     9 by simp
```
```    10
```
```    11 subsection{*The Locale @{text "M_trancl"}*}
```
```    12
```
```    13 subsubsection{*Separation for Reflexive/Transitive Closure*}
```
```    14
```
```    15 text{*First, The Defining Formula*}
```
```    16
```
```    17 (* "rtran_closure_mem(M,A,r,p) ==
```
```    18       \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
```
```    19        omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
```
```    20        (\<exists>f[M]. typed_function(M,n',A,f) &
```
```    21 	(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
```
```    22 	  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
```
```    23 	(\<forall>j[M]. j\<in>n -->
```
```    24 	  (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
```
```    25 	    fun_apply(M,f,j,fj) & successor(M,j,sj) &
```
```    26 	    fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
```
```    27 constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
```
```    28  "rtran_closure_mem_fm(A,r,p) ==
```
```    29    Exists(Exists(Exists(
```
```    30     And(omega_fm(2),
```
```    31      And(Member(1,2),
```
```    32       And(succ_fm(1,0),
```
```    33        Exists(And(typed_function_fm(1, A#+4, 0),
```
```    34 	And(Exists(Exists(Exists(
```
```    35 	      And(pair_fm(2,1,p#+7),
```
```    36 	       And(empty_fm(0),
```
```    37 		And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
```
```    38 	    Forall(Implies(Member(0,3),
```
```    39 	     Exists(Exists(Exists(Exists(
```
```    40 	      And(fun_apply_fm(5,4,3),
```
```    41 	       And(succ_fm(4,2),
```
```    42 		And(fun_apply_fm(5,2,1),
```
```    43 		 And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
```
```    44
```
```    45
```
```    46 lemma rtran_closure_mem_type [TC]:
```
```    47  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
```
```    48 by (simp add: rtran_closure_mem_fm_def)
```
```    49
```
```    50 lemma arity_rtran_closure_mem_fm [simp]:
```
```    51      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```    52       ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```    53 by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```    54
```
```    55 lemma sats_rtran_closure_mem_fm [simp]:
```
```    56    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```    57     ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
```
```    58         rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```    59 by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
```
```    60
```
```    61 lemma rtran_closure_mem_iff_sats:
```
```    62       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```    63           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```    64        ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
```
```    65 by (simp add: sats_rtran_closure_mem_fm)
```
```    66
```
```    67 theorem rtran_closure_mem_reflection:
```
```    68      "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
```
```    69                \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
```
```    70 apply (simp only: rtran_closure_mem_def setclass_simps)
```
```    71 apply (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    72 done
```
```    73
```
```    74 text{*Separation for @{term "rtrancl(r)"}.*}
```
```    75 lemma rtrancl_separation:
```
```    76      "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
```
```    77 apply (rule separation_CollectI)
```
```    78 apply (rule_tac A="{r,A,z}" in subset_LsetE, blast )
```
```    79 apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
```
```    80 apply (drule subset_Lset_ltD, assumption)
```
```    81 apply (erule reflection_imp_L_separation)
```
```    82   apply (simp_all add: lt_Ord2)
```
```    83 apply (rule DPow_LsetI)
```
```    84 apply (rename_tac u)
```
```    85 apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
```
```    86 apply (rule sep_rules | simp)+
```
```    87 done
```
```    88
```
```    89
```
```    90 subsubsection{*Reflexive/Transitive Closure, Internalized*}
```
```    91
```
```    92 (*  "rtran_closure(M,r,s) ==
```
```    93         \<forall>A[M]. is_field(M,r,A) -->
```
```    94  	 (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
```
```    95 constdefs rtran_closure_fm :: "[i,i]=>i"
```
```    96  "rtran_closure_fm(r,s) ==
```
```    97    Forall(Implies(field_fm(succ(r),0),
```
```    98                   Forall(Iff(Member(0,succ(succ(s))),
```
```    99                              rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
```
```   100
```
```   101 lemma rtran_closure_type [TC]:
```
```   102      "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
```
```   103 by (simp add: rtran_closure_fm_def)
```
```   104
```
```   105 lemma arity_rtran_closure_fm [simp]:
```
```   106      "[| x \<in> nat; y \<in> nat |]
```
```   107       ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   108 by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   109
```
```   110 lemma sats_rtran_closure_fm [simp]:
```
```   111    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   112     ==> sats(A, rtran_closure_fm(x,y), env) <->
```
```   113         rtran_closure(**A, nth(x,env), nth(y,env))"
```
```   114 by (simp add: rtran_closure_fm_def rtran_closure_def)
```
```   115
```
```   116 lemma rtran_closure_iff_sats:
```
```   117       "[| nth(i,env) = x; nth(j,env) = y;
```
```   118           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   119        ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
```
```   120 by simp
```
```   121
```
```   122 theorem rtran_closure_reflection:
```
```   123      "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
```
```   124                \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
```
```   125 apply (simp only: rtran_closure_def setclass_simps)
```
```   126 apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
```
```   127 done
```
```   128
```
```   129
```
```   130 subsubsection{*Transitive Closure of a Relation, Internalized*}
```
```   131
```
```   132 (*  "tran_closure(M,r,t) ==
```
```   133          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
```
```   134 constdefs tran_closure_fm :: "[i,i]=>i"
```
```   135  "tran_closure_fm(r,s) ==
```
```   136    Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
```
```   137
```
```   138 lemma tran_closure_type [TC]:
```
```   139      "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
```
```   140 by (simp add: tran_closure_fm_def)
```
```   141
```
```   142 lemma arity_tran_closure_fm [simp]:
```
```   143      "[| x \<in> nat; y \<in> nat |]
```
```   144       ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   145 by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   146
```
```   147 lemma sats_tran_closure_fm [simp]:
```
```   148    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   149     ==> sats(A, tran_closure_fm(x,y), env) <->
```
```   150         tran_closure(**A, nth(x,env), nth(y,env))"
```
```   151 by (simp add: tran_closure_fm_def tran_closure_def)
```
```   152
```
```   153 lemma tran_closure_iff_sats:
```
```   154       "[| nth(i,env) = x; nth(j,env) = y;
```
```   155           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   156        ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
```
```   157 by simp
```
```   158
```
```   159 theorem tran_closure_reflection:
```
```   160      "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
```
```   161                \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
```
```   162 apply (simp only: tran_closure_def setclass_simps)
```
```   163 apply (intro FOL_reflections function_reflections
```
```   164              rtran_closure_reflection composition_reflection)
```
```   165 done
```
```   166
```
```   167
```
```   168 subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
```
```   169
```
```   170 lemma wellfounded_trancl_reflects:
```
```   171   "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   172 	         w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
```
```   173    \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
```
```   174        w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
```
```   175        wx \<in> rp]"
```
```   176 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   177           tran_closure_reflection)
```
```   178
```
```   179
```
```   180 lemma wellfounded_trancl_separation:
```
```   181 	 "[| L(r); L(Z) |] ==>
```
```   182 	  separation (L, \<lambda>x.
```
```   183 	      \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   184 	       w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
```
```   185 apply (rule separation_CollectI)
```
```   186 apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast )
```
```   187 apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
```
```   188 apply (drule subset_Lset_ltD, assumption)
```
```   189 apply (erule reflection_imp_L_separation)
```
```   190   apply (simp_all add: lt_Ord2)
```
```   191 apply (rule DPow_LsetI)
```
```   192 apply (rename_tac u)
```
```   193 apply (rule bex_iff_sats conj_iff_sats)+
```
```   194 apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats)
```
```   195 apply (rule sep_rules tran_closure_iff_sats | simp)+
```
```   196 done
```
```   197
```
```   198
```
```   199 subsubsection{*Instantiating the locale @{text M_trancl}*}
```
```   200 ML
```
```   201 {*
```
```   202 val rtrancl_separation = thm "rtrancl_separation";
```
```   203 val wellfounded_trancl_separation = thm "wellfounded_trancl_separation";
```
```   204
```
```   205
```
```   206 val m_trancl = [rtrancl_separation, wellfounded_trancl_separation];
```
```   207
```
```   208 fun trancl_L th =
```
```   209     kill_flex_triv_prems (m_trancl MRS (axioms_L th));
```
```   210
```
```   211 bind_thm ("iterates_abs", trancl_L (thm "M_trancl.iterates_abs"));
```
```   212 bind_thm ("rtran_closure_rtrancl", trancl_L (thm "M_trancl.rtran_closure_rtrancl"));
```
```   213 bind_thm ("rtrancl_closed", trancl_L (thm "M_trancl.rtrancl_closed"));
```
```   214 bind_thm ("rtrancl_abs", trancl_L (thm "M_trancl.rtrancl_abs"));
```
```   215 bind_thm ("trancl_closed", trancl_L (thm "M_trancl.trancl_closed"));
```
```   216 bind_thm ("trancl_abs", trancl_L (thm "M_trancl.trancl_abs"));
```
```   217 bind_thm ("wellfounded_on_trancl", trancl_L (thm "M_trancl.wellfounded_on_trancl"));
```
```   218 bind_thm ("wellfounded_trancl", trancl_L (thm "M_trancl.wellfounded_trancl"));
```
```   219 bind_thm ("wfrec_relativize", trancl_L (thm "M_trancl.wfrec_relativize"));
```
```   220 bind_thm ("trans_wfrec_relativize", trancl_L (thm "M_trancl.trans_wfrec_relativize"));
```
```   221 bind_thm ("trans_wfrec_abs", trancl_L (thm "M_trancl.trans_wfrec_abs"));
```
```   222 bind_thm ("trans_eq_pair_wfrec_iff", trancl_L (thm "M_trancl.trans_eq_pair_wfrec_iff"));
```
```   223 bind_thm ("eq_pair_wfrec_iff", trancl_L (thm "M_trancl.eq_pair_wfrec_iff"));
```
```   224 *}
```
```   225
```
```   226 declare rtrancl_closed [intro,simp]
```
```   227 declare rtrancl_abs [simp]
```
```   228 declare trancl_closed [intro,simp]
```
```   229 declare trancl_abs [simp]
```
```   230
```
```   231
```
```   232 subsection{*Well-Founded Recursion!*}
```
```   233
```
```   234 (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
```
```   235    "M_is_recfun(M,MH,r,a,f) ==
```
```   236      \<forall>z[M]. z \<in> f <->
```
```   237             5      4       3       2       1           0
```
```   238             (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
```
```   239 	       pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
```
```   240                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
```
```   241                xa \<in> r & MH(x, f_r_sx, y))"
```
```   242 *)
```
```   243
```
```   244 constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i"
```
```   245  "is_recfun_fm(p,r,a,f) ==
```
```   246    Forall(Iff(Member(0,succ(f)),
```
```   247     Exists(Exists(Exists(Exists(Exists(Exists(
```
```   248      And(pair_fm(5,4,6),
```
```   249       And(pair_fm(5,a#+7,3),
```
```   250        And(upair_fm(5,5,2),
```
```   251         And(pre_image_fm(r#+7,2,1),
```
```   252          And(restriction_fm(f#+7,1,0),
```
```   253           And(Member(3,r#+7), p(5,0,4)))))))))))))))"
```
```   254
```
```   255
```
```   256 lemma is_recfun_type_0:
```
```   257      "[| !!x y z. [| x \<in> nat; y \<in> nat; z \<in> nat |] ==> p(x,y,z) \<in> formula;
```
```   258          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   259       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   260 apply (unfold is_recfun_fm_def)
```
```   261 (*FIXME: FIND OUT why simp loops!*)
```
```   262 apply typecheck
```
```   263 by simp
```
```   264
```
```   265 lemma is_recfun_type [TC]:
```
```   266      "[| p(5,0,4) \<in> formula;
```
```   267          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   268       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   269 by (simp add: is_recfun_fm_def)
```
```   270
```
```   271 lemma arity_is_recfun_fm [simp]:
```
```   272      "[| arity(p(5,0,4)) le 8; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   273       ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   274 apply (frule lt_nat_in_nat, simp)
```
```   275 apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] )
```
```   276 apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1])
```
```   277 apply (rule le_imp_subset)
```
```   278 apply (erule le_trans, simp)
```
```   279 apply (simp add: succ_Un_distrib [symmetric] Un_ac)
```
```   280 done
```
```   281
```
```   282 lemma sats_is_recfun_fm:
```
```   283   assumes MH_iff_sats:
```
```   284       "!!x y z env.
```
```   285 	 [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   286 	 ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)"
```
```   287   shows
```
```   288       "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   289        ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
```
```   290            M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
```
```   291 by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
```
```   292
```
```   293 lemma is_recfun_iff_sats:
```
```   294   "[| (!!x y z env. [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   295                     ==> MH(nth(x,env), nth(y,env), nth(z,env)) <->
```
```   296                         sats(A, p(x,y,z), env));
```
```   297       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   298       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   299    ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
```
```   300 by (simp add: sats_is_recfun_fm [of A MH])
```
```   301
```
```   302 theorem is_recfun_reflection:
```
```   303   assumes MH_reflection:
```
```   304     "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
```
```   305                      \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
```
```   306   shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L), f(x), g(x), h(x)),
```
```   307                \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
```
```   308 apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
```
```   309 apply (intro FOL_reflections function_reflections
```
```   310              restriction_reflection MH_reflection)
```
```   311 done
```
```   312
```
```   313 text{*Currently, @{text sats}-theorems for higher-order operators don't seem
```
```   314 useful.  Reflection theorems do work, though.  This one avoids the repetition
```
```   315 of the @{text MH}-term.*}
```
```   316 theorem is_wfrec_reflection:
```
```   317   assumes MH_reflection:
```
```   318     "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
```
```   319                      \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
```
```   320   shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L), f(x), g(x), h(x)),
```
```   321                \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
```
```   322 apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
```
```   323 apply (intro FOL_reflections MH_reflection is_recfun_reflection)
```
```   324 done
```
```   325
```
```   326 subsection{*The Locale @{text "M_wfrank"}*}
```
```   327
```
```   328 subsubsection{*Separation for @{term "wfrank"}*}
```
```   329
```
```   330 lemma wfrank_Reflects:
```
```   331  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   332               ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
```
```   333       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   334          ~ (\<exists>f \<in> Lset(i).
```
```   335             M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
```
```   336                         rplus, x, f))]"
```
```   337 by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
```
```   338
```
```   339 lemma wfrank_separation:
```
```   340      "L(r) ==>
```
```   341       separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   342          ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
```
```   343 apply (rule separation_CollectI)
```
```   344 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   345 apply (rule ReflectsE [OF wfrank_Reflects], assumption)
```
```   346 apply (drule subset_Lset_ltD, assumption)
```
```   347 apply (erule reflection_imp_L_separation)
```
```   348   apply (simp_all add: lt_Ord2, clarify)
```
```   349 apply (rule DPow_LsetI)
```
```   350 apply (rename_tac u)
```
```   351 apply (rule ball_iff_sats imp_iff_sats)+
```
```   352 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   353 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   354 done
```
```   355
```
```   356
```
```   357 subsubsection{*Replacement for @{term "wfrank"}*}
```
```   358
```
```   359 lemma wfrank_replacement_Reflects:
```
```   360  "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
```
```   361         (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   362          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   363                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   364                         is_range(L,f,y))),
```
```   365  \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
```
```   366       (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   367        (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
```
```   368          M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
```
```   369          is_range(**Lset(i),f,y)))]"
```
```   370 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   371              is_recfun_reflection tran_closure_reflection)
```
```   372
```
```   373
```
```   374 lemma wfrank_strong_replacement:
```
```   375      "L(r) ==>
```
```   376       strong_replacement(L, \<lambda>x z.
```
```   377          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   378          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   379                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   380                         is_range(L,f,y)))"
```
```   381 apply (rule strong_replacementI)
```
```   382 apply (rule rallI)
```
```   383 apply (rename_tac B)
```
```   384 apply (rule separation_CollectI)
```
```   385 apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
```
```   386 apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
```
```   387 apply (drule subset_Lset_ltD, assumption)
```
```   388 apply (erule reflection_imp_L_separation)
```
```   389   apply (simp_all add: lt_Ord2)
```
```   390 apply (rule DPow_LsetI)
```
```   391 apply (rename_tac u)
```
```   392 apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
```
```   393 apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
```
```   394 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   395 done
```
```   396
```
```   397
```
```   398 subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
```
```   399
```
```   400 lemma Ord_wfrank_Reflects:
```
```   401  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   402           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   403              is_range(L,f,rangef) -->
```
```   404              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   405              ordinal(L,rangef)),
```
```   406       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   407           ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
```
```   408              is_range(**Lset(i),f,rangef) -->
```
```   409              M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
```
```   410                          rplus, x, f) -->
```
```   411              ordinal(**Lset(i),rangef))]"
```
```   412 by (intro FOL_reflections function_reflections is_recfun_reflection
```
```   413           tran_closure_reflection ordinal_reflection)
```
```   414
```
```   415 lemma  Ord_wfrank_separation:
```
```   416      "L(r) ==>
```
```   417       separation (L, \<lambda>x.
```
```   418          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   419           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   420              is_range(L,f,rangef) -->
```
```   421              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   422              ordinal(L,rangef)))"
```
```   423 apply (rule separation_CollectI)
```
```   424 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   425 apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
```
```   426 apply (drule subset_Lset_ltD, assumption)
```
```   427 apply (erule reflection_imp_L_separation)
```
```   428   apply (simp_all add: lt_Ord2, clarify)
```
```   429 apply (rule DPow_LsetI)
```
```   430 apply (rename_tac u)
```
```   431 apply (rule ball_iff_sats imp_iff_sats)+
```
```   432 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   433 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   434 done
```
```   435
```
```   436
```
```   437 subsubsection{*Instantiating the locale @{text M_wfrank}*}
```
```   438 ML
```
```   439 {*
```
```   440 val wfrank_separation = thm "wfrank_separation";
```
```   441 val wfrank_strong_replacement = thm "wfrank_strong_replacement";
```
```   442 val Ord_wfrank_separation = thm "Ord_wfrank_separation";
```
```   443
```
```   444 val m_wfrank =
```
```   445     [wfrank_separation, wfrank_strong_replacement, Ord_wfrank_separation];
```
```   446
```
```   447 fun wfrank_L th =
```
```   448     kill_flex_triv_prems (m_wfrank MRS (trancl_L th));
```
```   449
```
```   450
```
```   451
```
```   452 bind_thm ("iterates_closed", wfrank_L (thm "M_wfrank.iterates_closed"));
```
```   453 bind_thm ("exists_wfrank", wfrank_L (thm "M_wfrank.exists_wfrank"));
```
```   454 bind_thm ("M_wellfoundedrank", wfrank_L (thm "M_wfrank.M_wellfoundedrank"));
```
```   455 bind_thm ("Ord_wfrank_range", wfrank_L (thm "M_wfrank.Ord_wfrank_range"));
```
```   456 bind_thm ("Ord_range_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_range_wellfoundedrank"));
```
```   457 bind_thm ("function_wellfoundedrank", wfrank_L (thm "M_wfrank.function_wellfoundedrank"));
```
```   458 bind_thm ("domain_wellfoundedrank", wfrank_L (thm "M_wfrank.domain_wellfoundedrank"));
```
```   459 bind_thm ("wellfoundedrank_type", wfrank_L (thm "M_wfrank.wellfoundedrank_type"));
```
```   460 bind_thm ("Ord_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_wellfoundedrank"));
```
```   461 bind_thm ("wellfoundedrank_eq", wfrank_L (thm "M_wfrank.wellfoundedrank_eq"));
```
```   462 bind_thm ("wellfoundedrank_lt", wfrank_L (thm "M_wfrank.wellfoundedrank_lt"));
```
```   463 bind_thm ("wellfounded_imp_subset_rvimage", wfrank_L (thm "M_wfrank.wellfounded_imp_subset_rvimage"));
```
```   464 bind_thm ("wellfounded_imp_wf", wfrank_L (thm "M_wfrank.wellfounded_imp_wf"));
```
```   465 bind_thm ("wellfounded_on_imp_wf_on", wfrank_L (thm "M_wfrank.wellfounded_on_imp_wf_on"));
```
```   466 bind_thm ("wf_abs", wfrank_L (thm "M_wfrank.wf_abs"));
```
```   467 bind_thm ("wf_on_abs", wfrank_L (thm "M_wfrank.wf_on_abs"));
```
```   468 bind_thm ("wfrec_replacement_iff", wfrank_L (thm "M_wfrank.wfrec_replacement_iff"));
```
```   469 bind_thm ("trans_wfrec_closed", wfrank_L (thm "M_wfrank.trans_wfrec_closed"));
```
```   470 bind_thm ("wfrec_closed", wfrank_L (thm "M_wfrank.wfrec_closed"));
```
```   471 *}
```
```   472
```
```   473 declare iterates_closed [intro,simp]
```
```   474 declare Ord_wfrank_range [rule_format]
```
```   475 declare wf_abs [simp]
```
```   476 declare wf_on_abs [simp]
```
```   477
```
```   478
```
```   479 subsection{*For Datatypes*}
```
```   480
```
```   481 subsubsection{*Binary Products, Internalized*}
```
```   482
```
```   483 constdefs cartprod_fm :: "[i,i,i]=>i"
```
```   484 (* "cartprod(M,A,B,z) ==
```
```   485 	\<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
```
```   486     "cartprod_fm(A,B,z) ==
```
```   487        Forall(Iff(Member(0,succ(z)),
```
```   488                   Exists(And(Member(0,succ(succ(A))),
```
```   489                          Exists(And(Member(0,succ(succ(succ(B)))),
```
```   490                                     pair_fm(1,0,2)))))))"
```
```   491
```
```   492 lemma cartprod_type [TC]:
```
```   493      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
```
```   494 by (simp add: cartprod_fm_def)
```
```   495
```
```   496 lemma arity_cartprod_fm [simp]:
```
```   497      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   498       ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   499 by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   500
```
```   501 lemma sats_cartprod_fm [simp]:
```
```   502    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   503     ==> sats(A, cartprod_fm(x,y,z), env) <->
```
```   504         cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   505 by (simp add: cartprod_fm_def cartprod_def)
```
```   506
```
```   507 lemma cartprod_iff_sats:
```
```   508       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   509           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   510        ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
```
```   511 by (simp add: sats_cartprod_fm)
```
```   512
```
```   513 theorem cartprod_reflection:
```
```   514      "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
```
```   515                \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
```
```   516 apply (simp only: cartprod_def setclass_simps)
```
```   517 apply (intro FOL_reflections pair_reflection)
```
```   518 done
```
```   519
```
```   520
```
```   521 subsubsection{*Binary Sums, Internalized*}
```
```   522
```
```   523 (* "is_sum(M,A,B,Z) ==
```
```   524        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
```
```   525          3      2       1        0
```
```   526        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
```
```   527        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
```
```   528 constdefs sum_fm :: "[i,i,i]=>i"
```
```   529     "sum_fm(A,B,Z) ==
```
```   530        Exists(Exists(Exists(Exists(
```
```   531 	And(number1_fm(2),
```
```   532             And(cartprod_fm(2,A#+4,3),
```
```   533                 And(upair_fm(2,2,1),
```
```   534                     And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
```
```   535
```
```   536 lemma sum_type [TC]:
```
```   537      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
```
```   538 by (simp add: sum_fm_def)
```
```   539
```
```   540 lemma arity_sum_fm [simp]:
```
```   541      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   542       ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   543 by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   544
```
```   545 lemma sats_sum_fm [simp]:
```
```   546    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   547     ==> sats(A, sum_fm(x,y,z), env) <->
```
```   548         is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   549 by (simp add: sum_fm_def is_sum_def)
```
```   550
```
```   551 lemma sum_iff_sats:
```
```   552       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   553           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   554        ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
```
```   555 by simp
```
```   556
```
```   557 theorem sum_reflection:
```
```   558      "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
```
```   559                \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
```
```   560 apply (simp only: is_sum_def setclass_simps)
```
```   561 apply (intro FOL_reflections function_reflections cartprod_reflection)
```
```   562 done
```
```   563
```
```   564
```
```   565 subsubsection{*The Operator @{term quasinat}*}
```
```   566
```
```   567 (* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
```
```   568 constdefs quasinat_fm :: "i=>i"
```
```   569     "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
```
```   570
```
```   571 lemma quasinat_type [TC]:
```
```   572      "x \<in> nat ==> quasinat_fm(x) \<in> formula"
```
```   573 by (simp add: quasinat_fm_def)
```
```   574
```
```   575 lemma arity_quasinat_fm [simp]:
```
```   576      "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
```
```   577 by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   578
```
```   579 lemma sats_quasinat_fm [simp]:
```
```   580    "[| x \<in> nat; env \<in> list(A)|]
```
```   581     ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
```
```   582 by (simp add: quasinat_fm_def is_quasinat_def)
```
```   583
```
```   584 lemma quasinat_iff_sats:
```
```   585       "[| nth(i,env) = x; nth(j,env) = y;
```
```   586           i \<in> nat; env \<in> list(A)|]
```
```   587        ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
```
```   588 by simp
```
```   589
```
```   590 theorem quasinat_reflection:
```
```   591      "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
```
```   592                \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
```
```   593 apply (simp only: is_quasinat_def setclass_simps)
```
```   594 apply (intro FOL_reflections function_reflections)
```
```   595 done
```
```   596
```
```   597
```
```   598 subsubsection{*The Operator @{term is_nat_case}*}
```
```   599
```
```   600 (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
```
```   601     "is_nat_case(M, a, is_b, k, z) ==
```
```   602        (empty(M,k) --> z=a) &
```
```   603        (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
```
```   604        (is_quasinat(M,k) | empty(M,z))" *)
```
```   605 text{*The formula @{term is_b} has free variables 1 and 0.*}
```
```   606 constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i"
```
```   607  "is_nat_case_fm(a,is_b,k,z) ==
```
```   608     And(Implies(empty_fm(k), Equal(z,a)),
```
```   609         And(Forall(Implies(succ_fm(0,succ(k)),
```
```   610                    Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))),
```
```   611             Or(quasinat_fm(k), empty_fm(z))))"
```
```   612
```
```   613 lemma is_nat_case_type [TC]:
```
```   614      "[| is_b(1,0) \<in> formula;
```
```   615          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   616       ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
```
```   617 by (simp add: is_nat_case_fm_def)
```
```   618
```
```   619 lemma arity_is_nat_case_fm [simp]:
```
```   620      "[| is_b(1,0) \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   621       ==> arity(is_nat_case_fm(x,is_b,y,z)) =
```
```   622           succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(is_b(1,0)) #- 2)"
```
```   623 apply (subgoal_tac "arity(is_b(1,0)) \<in> nat")
```
```   624 apply typecheck
```
```   625 (*FIXME: could nat_diff_split work?*)
```
```   626 apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat
```
```   627                  succ_Un_distrib [symmetric] Un_ac
```
```   628                  split: split_nat_case)
```
```   629 done
```
```   630
```
```   631 lemma sats_is_nat_case_fm:
```
```   632   assumes is_b_iff_sats:
```
```   633       "!!a b. [| a \<in> A; b \<in> A|]
```
```   634               ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))"
```
```   635   shows
```
```   636       "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   637        ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
```
```   638            is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
```
```   639 apply (frule lt_length_in_nat, assumption)
```
```   640 apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
```
```   641 done
```
```   642
```
```   643 lemma is_nat_case_iff_sats:
```
```   644   "[| (!!a b. [| a \<in> A; b \<in> A|]
```
```   645               ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env))));
```
```   646       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   647       i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   648    ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
```
```   649 by (simp add: sats_is_nat_case_fm [of A is_b])
```
```   650
```
```   651
```
```   652 text{*The second argument of @{term is_b} gives it direct access to @{term x},
```
```   653   which is essential for handling free variable references.  Without this
```
```   654   argument, we cannot prove reflection for @{term iterates_MH}.*}
```
```   655 theorem is_nat_case_reflection:
```
```   656   assumes is_b_reflection:
```
```   657     "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
```
```   658                      \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
```
```   659   shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
```
```   660                \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
```
```   661 apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
```
```   662 apply (intro FOL_reflections function_reflections
```
```   663              restriction_reflection is_b_reflection quasinat_reflection)
```
```   664 done
```
```   665
```
```   666
```
```   667
```
```   668 subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
```
```   669
```
```   670 (*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   671    "iterates_MH(M,isF,v,n,g,z) ==
```
```   672         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   673                     n, z)" *)
```
```   674 constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i"
```
```   675  "iterates_MH_fm(isF,v,n,g,z) ==
```
```   676     is_nat_case_fm(v,
```
```   677       \<lambda>m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0),
```
```   678                      Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))),
```
```   679       n, z)"
```
```   680
```
```   681 lemma iterates_MH_type [TC]:
```
```   682      "[| p(1,0) \<in> formula;
```
```   683          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   684       ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
```
```   685 by (simp add: iterates_MH_fm_def)
```
```   686
```
```   687
```
```   688 lemma arity_iterates_MH_fm [simp]:
```
```   689      "[| p(1,0) \<in> formula;
```
```   690          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   691       ==> arity(iterates_MH_fm(p,v,x,y,z)) =
```
```   692           succ(v) \<union> succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(p(1,0)) #- 4)"
```
```   693 apply (subgoal_tac "arity(p(1,0)) \<in> nat")
```
```   694 apply typecheck
```
```   695 apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac
```
```   696             split: split_nat_case, clarify)
```
```   697 apply (rename_tac i j)
```
```   698 apply (drule eq_succ_imp_eq_m1, simp)
```
```   699 apply (drule eq_succ_imp_eq_m1, simp)
```
```   700 apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left)
```
```   701 done
```
```   702
```
```   703 lemma sats_iterates_MH_fm:
```
```   704   assumes is_F_iff_sats:
```
```   705       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   706               ==> is_F(a,b) <->
```
```   707                   sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
```
```   708   shows
```
```   709       "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   710        ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
```
```   711            iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
```
```   712 by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm
```
```   713               is_F_iff_sats [symmetric])
```
```   714
```
```   715 lemma iterates_MH_iff_sats:
```
```   716   "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   717               ==> is_F(a,b) <->
```
```   718                   sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env))))));
```
```   719       nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   720       i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   721    ==> iterates_MH(**A, is_F, v, x, y, z) <->
```
```   722        sats(A, iterates_MH_fm(p,i',i,j,k), env)"
```
```   723 apply (rule iff_sym)
```
```   724 apply (rule iff_trans)
```
```   725 apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all)
```
```   726 done
```
```   727
```
```   728 theorem iterates_MH_reflection:
```
```   729   assumes p_reflection:
```
```   730     "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
```
```   731                      \<lambda>i x. p(**Lset(i), f(x), g(x))]"
```
```   732  shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
```
```   733                \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
```
```   734 apply (simp (no_asm_use) only: iterates_MH_def)
```
```   735 txt{*Must be careful: simplifying with @{text setclass_simps} above would
```
```   736      change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
```
```   737      it would no longer match rule @{text is_nat_case_reflection}. *}
```
```   738 apply (rule is_nat_case_reflection)
```
```   739 apply (simp (no_asm_use) only: setclass_simps)
```
```   740 apply (intro FOL_reflections function_reflections is_nat_case_reflection
```
```   741              restriction_reflection p_reflection)
```
```   742 done
```
```   743
```
```   744
```
```   745
```
```   746 subsection{*@{term L} is Closed Under the Operator @{term list}*}
```
```   747
```
```   748 subsubsection{*The List Functor, Internalized*}
```
```   749
```
```   750 constdefs list_functor_fm :: "[i,i,i]=>i"
```
```   751 (* "is_list_functor(M,A,X,Z) ==
```
```   752         \<exists>n1[M]. \<exists>AX[M].
```
```   753          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
```
```   754     "list_functor_fm(A,X,Z) ==
```
```   755        Exists(Exists(
```
```   756 	And(number1_fm(1),
```
```   757             And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
```
```   758
```
```   759 lemma list_functor_type [TC]:
```
```   760      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
```
```   761 by (simp add: list_functor_fm_def)
```
```   762
```
```   763 lemma arity_list_functor_fm [simp]:
```
```   764      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   765       ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   766 by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   767
```
```   768 lemma sats_list_functor_fm [simp]:
```
```   769    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   770     ==> sats(A, list_functor_fm(x,y,z), env) <->
```
```   771         is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   772 by (simp add: list_functor_fm_def is_list_functor_def)
```
```   773
```
```   774 lemma list_functor_iff_sats:
```
```   775   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   776       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   777    ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
```
```   778 by simp
```
```   779
```
```   780 theorem list_functor_reflection:
```
```   781      "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
```
```   782                \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
```
```   783 apply (simp only: is_list_functor_def setclass_simps)
```
```   784 apply (intro FOL_reflections number1_reflection
```
```   785              cartprod_reflection sum_reflection)
```
```   786 done
```
```   787
```
```   788
```
```   789 subsubsection{*Instances of Replacement for Lists*}
```
```   790
```
```   791 lemma list_replacement1_Reflects:
```
```   792  "REFLECTS
```
```   793    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   794          is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
```
```   795     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   796          is_wfrec(**Lset(i),
```
```   797                   iterates_MH(**Lset(i),
```
```   798                           is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
```
```   799 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   800           iterates_MH_reflection list_functor_reflection)
```
```   801
```
```   802 lemma list_replacement1:
```
```   803    "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
```
```   804 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   805 apply (rule strong_replacementI)
```
```   806 apply (rule rallI)
```
```   807 apply (rename_tac B)
```
```   808 apply (rule separation_CollectI)
```
```   809 apply (insert nonempty)
```
```   810 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   811 apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   812 apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
```
```   813 apply (drule subset_Lset_ltD, assumption)
```
```   814 apply (erule reflection_imp_L_separation)
```
```   815   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   816 apply (elim conjE)
```
```   817 apply (rule DPow_LsetI)
```
```   818 apply (rename_tac v)
```
```   819 apply (rule bex_iff_sats conj_iff_sats)+
```
```   820 apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   821 apply (rule sep_rules | simp)+
```
```   822 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```   823 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   824 apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
```
```   825 done
```
```   826
```
```   827
```
```   828 lemma list_replacement2_Reflects:
```
```   829  "REFLECTS
```
```   830    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   831          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   832            is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
```
```   833                               msn, u, x)),
```
```   834     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   835          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   836           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   837            is_wfrec (**Lset(i),
```
```   838                  iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
```
```   839                      msn, u, x))]"
```
```   840 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   841           iterates_MH_reflection list_functor_reflection)
```
```   842
```
```   843
```
```   844 lemma list_replacement2:
```
```   845    "L(A) ==> strong_replacement(L,
```
```   846          \<lambda>n y. n\<in>nat &
```
```   847                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   848                is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
```
```   849                         msn, n, y)))"
```
```   850 apply (rule strong_replacementI)
```
```   851 apply (rule rallI)
```
```   852 apply (rename_tac B)
```
```   853 apply (rule separation_CollectI)
```
```   854 apply (insert nonempty)
```
```   855 apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
```
```   856 apply (blast intro: L_nat)
```
```   857 apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
```
```   858 apply (drule subset_Lset_ltD, assumption)
```
```   859 apply (erule reflection_imp_L_separation)
```
```   860   apply (simp_all add: lt_Ord2)
```
```   861 apply (rule DPow_LsetI)
```
```   862 apply (rename_tac v)
```
```   863 apply (rule bex_iff_sats conj_iff_sats)+
```
```   864 apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
```
```   865 apply (rule sep_rules | simp)+
```
```   866 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   867 apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
```
```   868 done
```
```   869
```
```   870
```
```   871 subsection{*@{term L} is Closed Under the Operator @{term formula}*}
```
```   872
```
```   873 subsubsection{*The Formula Functor, Internalized*}
```
```   874
```
```   875 constdefs formula_functor_fm :: "[i,i]=>i"
```
```   876 (*     "is_formula_functor(M,X,Z) ==
```
```   877         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
```
```   878            4           3               2       1       0
```
```   879           omega(M,nat') & cartprod(M,nat',nat',natnat) &
```
```   880           is_sum(M,natnat,natnat,natnatsum) &
```
```   881           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
```
```   882           is_sum(M,natnatsum,X3,Z)" *)
```
```   883     "formula_functor_fm(X,Z) ==
```
```   884        Exists(Exists(Exists(Exists(Exists(
```
```   885 	And(omega_fm(4),
```
```   886          And(cartprod_fm(4,4,3),
```
```   887           And(sum_fm(3,3,2),
```
```   888            And(cartprod_fm(X#+5,X#+5,1),
```
```   889             And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
```
```   890
```
```   891 lemma formula_functor_type [TC]:
```
```   892      "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
```
```   893 by (simp add: formula_functor_fm_def)
```
```   894
```
```   895 lemma sats_formula_functor_fm [simp]:
```
```   896    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   897     ==> sats(A, formula_functor_fm(x,y), env) <->
```
```   898         is_formula_functor(**A, nth(x,env), nth(y,env))"
```
```   899 by (simp add: formula_functor_fm_def is_formula_functor_def)
```
```   900
```
```   901 lemma formula_functor_iff_sats:
```
```   902   "[| nth(i,env) = x; nth(j,env) = y;
```
```   903       i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   904    ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
```
```   905 by simp
```
```   906
```
```   907 theorem formula_functor_reflection:
```
```   908      "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
```
```   909                \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
```
```   910 apply (simp only: is_formula_functor_def setclass_simps)
```
```   911 apply (intro FOL_reflections omega_reflection
```
```   912              cartprod_reflection sum_reflection)
```
```   913 done
```
```   914
```
```   915 subsubsection{*Instances of Replacement for Formulas*}
```
```   916
```
```   917 lemma formula_replacement1_Reflects:
```
```   918  "REFLECTS
```
```   919    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   920          is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
```
```   921     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   922          is_wfrec(**Lset(i),
```
```   923                   iterates_MH(**Lset(i),
```
```   924                           is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
```
```   925 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   926           iterates_MH_reflection formula_functor_reflection)
```
```   927
```
```   928 lemma formula_replacement1:
```
```   929    "iterates_replacement(L, is_formula_functor(L), 0)"
```
```   930 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   931 apply (rule strong_replacementI)
```
```   932 apply (rule rallI)
```
```   933 apply (rename_tac B)
```
```   934 apply (rule separation_CollectI)
```
```   935 apply (insert nonempty)
```
```   936 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   937 apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   938 apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
```
```   939 apply (drule subset_Lset_ltD, assumption)
```
```   940 apply (erule reflection_imp_L_separation)
```
```   941   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   942 apply (rule DPow_LsetI)
```
```   943 apply (rename_tac v)
```
```   944 apply (rule bex_iff_sats conj_iff_sats)+
```
```   945 apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   946 apply (rule sep_rules | simp)+
```
```   947 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```   948 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   949 apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
```
```   950 txt{*SLOW: like 40 seconds!*}
```
```   951 done
```
```   952
```
```   953 lemma formula_replacement2_Reflects:
```
```   954  "REFLECTS
```
```   955    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   956          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   957            is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
```
```   958                               msn, u, x)),
```
```   959     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   960          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   961           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   962            is_wfrec (**Lset(i),
```
```   963                  iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
```
```   964                      msn, u, x))]"
```
```   965 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   966           iterates_MH_reflection formula_functor_reflection)
```
```   967
```
```   968
```
```   969 lemma formula_replacement2:
```
```   970    "strong_replacement(L,
```
```   971          \<lambda>n y. n\<in>nat &
```
```   972                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   973                is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
```
```   974                         msn, n, y)))"
```
```   975 apply (rule strong_replacementI)
```
```   976 apply (rule rallI)
```
```   977 apply (rename_tac B)
```
```   978 apply (rule separation_CollectI)
```
```   979 apply (insert nonempty)
```
```   980 apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
```
```   981 apply (blast intro: L_nat)
```
```   982 apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
```
```   983 apply (drule subset_Lset_ltD, assumption)
```
```   984 apply (erule reflection_imp_L_separation)
```
```   985   apply (simp_all add: lt_Ord2)
```
```   986 apply (rule DPow_LsetI)
```
```   987 apply (rename_tac v)
```
```   988 apply (rule bex_iff_sats conj_iff_sats)+
```
```   989 apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
```
```   990 apply (rule sep_rules | simp)+
```
```   991 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   992 apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
```
```   993 done
```
```   994
```
```   995 text{*NB The proofs for type @{term formula} are virtually identical to those
```
```   996 for @{term "list(A)"}.  It was a cut-and-paste job! *}
```
```   997
```
```   998
```
```   999 subsubsection{*Instantiating the locale @{text M_datatypes}*}
```
```  1000 ML
```
```  1001 {*
```
```  1002 val list_replacement1 = thm "list_replacement1";
```
```  1003 val list_replacement2 = thm "list_replacement2";
```
```  1004 val formula_replacement1 = thm "formula_replacement1";
```
```  1005 val formula_replacement2 = thm "formula_replacement2";
```
```  1006
```
```  1007 val m_datatypes = [list_replacement1, list_replacement2,
```
```  1008                    formula_replacement1, formula_replacement2];
```
```  1009
```
```  1010 fun datatypes_L th =
```
```  1011     kill_flex_triv_prems (m_datatypes MRS (wfrank_L th));
```
```  1012
```
```  1013 bind_thm ("list_closed", datatypes_L (thm "M_datatypes.list_closed"));
```
```  1014 bind_thm ("formula_closed", datatypes_L (thm "M_datatypes.formula_closed"));
```
```  1015 bind_thm ("list_abs", datatypes_L (thm "M_datatypes.list_abs"));
```
```  1016 bind_thm ("formula_abs", datatypes_L (thm "M_datatypes.formula_abs"));
```
```  1017 *}
```
```  1018
```
```  1019 declare list_closed [intro,simp]
```
```  1020 declare formula_closed [intro,simp]
```
```  1021 declare list_abs [intro,simp]
```
```  1022 declare formula_abs [intro,simp]
```
```  1023
```
```  1024
```
```  1025
```
```  1026 subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
```
```  1027
```
```  1028 subsubsection{*Instances of Replacement for @{term eclose}*}
```
```  1029
```
```  1030 lemma eclose_replacement1_Reflects:
```
```  1031  "REFLECTS
```
```  1032    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1033          is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
```
```  1034     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1035          is_wfrec(**Lset(i),
```
```  1036                   iterates_MH(**Lset(i), big_union(**Lset(i)), A),
```
```  1037                   memsn, u, y))]"
```
```  1038 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1039           iterates_MH_reflection)
```
```  1040
```
```  1041 lemma eclose_replacement1:
```
```  1042    "L(A) ==> iterates_replacement(L, big_union(L), A)"
```
```  1043 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1044 apply (rule strong_replacementI)
```
```  1045 apply (rule rallI)
```
```  1046 apply (rename_tac B)
```
```  1047 apply (rule separation_CollectI)
```
```  1048 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1049 apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1050 apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
```
```  1051 apply (drule subset_Lset_ltD, assumption)
```
```  1052 apply (erule reflection_imp_L_separation)
```
```  1053   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1054 apply (elim conjE)
```
```  1055 apply (rule DPow_LsetI)
```
```  1056 apply (rename_tac v)
```
```  1057 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1058 apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
```
```  1059 apply (rule sep_rules | simp)+
```
```  1060 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```  1061 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1062 apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1063 done
```
```  1064
```
```  1065
```
```  1066 lemma eclose_replacement2_Reflects:
```
```  1067  "REFLECTS
```
```  1068    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```  1069          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```  1070            is_wfrec (L, iterates_MH (L, big_union(L), A),
```
```  1071                               msn, u, x)),
```
```  1072     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```  1073          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```  1074           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```  1075            is_wfrec (**Lset(i),
```
```  1076                  iterates_MH (**Lset(i), big_union(**Lset(i)), A),
```
```  1077                      msn, u, x))]"
```
```  1078 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1079           iterates_MH_reflection)
```
```  1080
```
```  1081
```
```  1082 lemma eclose_replacement2:
```
```  1083    "L(A) ==> strong_replacement(L,
```
```  1084          \<lambda>n y. n\<in>nat &
```
```  1085                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```  1086                is_wfrec(L, iterates_MH(L,big_union(L), A),
```
```  1087                         msn, n, y)))"
```
```  1088 apply (rule strong_replacementI)
```
```  1089 apply (rule rallI)
```
```  1090 apply (rename_tac B)
```
```  1091 apply (rule separation_CollectI)
```
```  1092 apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
```
```  1093 apply (blast intro: L_nat)
```
```  1094 apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
```
```  1095 apply (drule subset_Lset_ltD, assumption)
```
```  1096 apply (erule reflection_imp_L_separation)
```
```  1097   apply (simp_all add: lt_Ord2)
```
```  1098 apply (rule DPow_LsetI)
```
```  1099 apply (rename_tac v)
```
```  1100 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1101 apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
```
```  1102 apply (rule sep_rules | simp)+
```
```  1103 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1104 apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1105 done
```
```  1106
```
```  1107
```
```  1108 subsubsection{*Instantiating the locale @{text M_eclose}*}
```
```  1109 ML
```
```  1110 {*
```
```  1111 val eclose_replacement1 = thm "eclose_replacement1";
```
```  1112 val eclose_replacement2 = thm "eclose_replacement2";
```
```  1113
```
```  1114 val m_eclose = [eclose_replacement1, eclose_replacement2];
```
```  1115
```
```  1116 fun eclose_L th =
```
```  1117     kill_flex_triv_prems (m_eclose MRS (datatypes_L th));
```
```  1118
```
```  1119 bind_thm ("eclose_closed", eclose_L (thm "M_eclose.eclose_closed"));
```
```  1120 bind_thm ("eclose_abs", eclose_L (thm "M_eclose.eclose_abs"));
```
```  1121 *}
```
```  1122
```
```  1123 declare eclose_closed [intro,simp]
```
```  1124 declare eclose_abs [intro,simp]
```
```  1125
```
```  1126
```
```  1127 subsection{*Internalized Forms of Data Structuring Operators*}
```
```  1128
```
```  1129 subsubsection{*The Formula @{term is_Inl}, Internalized*}
```
```  1130
```
```  1131 (*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
```
```  1132 constdefs Inl_fm :: "[i,i]=>i"
```
```  1133     "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1134
```
```  1135 lemma Inl_type [TC]:
```
```  1136      "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
```
```  1137 by (simp add: Inl_fm_def)
```
```  1138
```
```  1139 lemma sats_Inl_fm [simp]:
```
```  1140    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1141     ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
```
```  1142 by (simp add: Inl_fm_def is_Inl_def)
```
```  1143
```
```  1144 lemma Inl_iff_sats:
```
```  1145       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1146           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1147        ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
```
```  1148 by simp
```
```  1149
```
```  1150 theorem Inl_reflection:
```
```  1151      "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
```
```  1152                \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
```
```  1153 apply (simp only: is_Inl_def setclass_simps)
```
```  1154 apply (intro FOL_reflections function_reflections)
```
```  1155 done
```
```  1156
```
```  1157
```
```  1158 subsubsection{*The Formula @{term is_Inr}, Internalized*}
```
```  1159
```
```  1160 (*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
```
```  1161 constdefs Inr_fm :: "[i,i]=>i"
```
```  1162     "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1163
```
```  1164 lemma Inr_type [TC]:
```
```  1165      "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
```
```  1166 by (simp add: Inr_fm_def)
```
```  1167
```
```  1168 lemma sats_Inr_fm [simp]:
```
```  1169    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1170     ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
```
```  1171 by (simp add: Inr_fm_def is_Inr_def)
```
```  1172
```
```  1173 lemma Inr_iff_sats:
```
```  1174       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1175           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1176        ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
```
```  1177 by simp
```
```  1178
```
```  1179 theorem Inr_reflection:
```
```  1180      "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
```
```  1181                \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
```
```  1182 apply (simp only: is_Inr_def setclass_simps)
```
```  1183 apply (intro FOL_reflections function_reflections)
```
```  1184 done
```
```  1185
```
```  1186
```
```  1187 subsubsection{*The Formula @{term is_Nil}, Internalized*}
```
```  1188
```
```  1189 (* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
```
```  1190
```
```  1191 constdefs Nil_fm :: "i=>i"
```
```  1192     "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
```
```  1193
```
```  1194 lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
```
```  1195 by (simp add: Nil_fm_def)
```
```  1196
```
```  1197 lemma sats_Nil_fm [simp]:
```
```  1198    "[| x \<in> nat; env \<in> list(A)|]
```
```  1199     ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
```
```  1200 by (simp add: Nil_fm_def is_Nil_def)
```
```  1201
```
```  1202 lemma Nil_iff_sats:
```
```  1203       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1204        ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
```
```  1205 by simp
```
```  1206
```
```  1207 theorem Nil_reflection:
```
```  1208      "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
```
```  1209                \<lambda>i x. is_Nil(**Lset(i),f(x))]"
```
```  1210 apply (simp only: is_Nil_def setclass_simps)
```
```  1211 apply (intro FOL_reflections function_reflections Inl_reflection)
```
```  1212 done
```
```  1213
```
```  1214
```
```  1215 subsubsection{*The Formula @{term is_Nil}, Internalized*}
```
```  1216
```
```  1217
```
```  1218 (*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
```
```  1219 constdefs Cons_fm :: "[i,i,i]=>i"
```
```  1220     "Cons_fm(a,l,Z) ==
```
```  1221        Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
```
```  1222
```
```  1223 lemma Cons_type [TC]:
```
```  1224      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
```
```  1225 by (simp add: Cons_fm_def)
```
```  1226
```
```  1227 lemma sats_Cons_fm [simp]:
```
```  1228    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1229     ==> sats(A, Cons_fm(x,y,z), env) <->
```
```  1230        is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```  1231 by (simp add: Cons_fm_def is_Cons_def)
```
```  1232
```
```  1233 lemma Cons_iff_sats:
```
```  1234       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```  1235           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1236        ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
```
```  1237 by simp
```
```  1238
```
```  1239 theorem Cons_reflection:
```
```  1240      "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
```
```  1241                \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
```
```  1242 apply (simp only: is_Cons_def setclass_simps)
```
```  1243 apply (intro FOL_reflections pair_reflection Inr_reflection)
```
```  1244 done
```
```  1245
```
```  1246 subsubsection{*The Formula @{term is_quasilist}, Internalized*}
```
```  1247
```
```  1248 (* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
```
```  1249
```
```  1250 constdefs quasilist_fm :: "i=>i"
```
```  1251     "quasilist_fm(x) ==
```
```  1252        Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
```
```  1253
```
```  1254 lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
```
```  1255 by (simp add: quasilist_fm_def)
```
```  1256
```
```  1257 lemma sats_quasilist_fm [simp]:
```
```  1258    "[| x \<in> nat; env \<in> list(A)|]
```
```  1259     ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
```
```  1260 by (simp add: quasilist_fm_def is_quasilist_def)
```
```  1261
```
```  1262 lemma quasilist_iff_sats:
```
```  1263       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1264        ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
```
```  1265 by simp
```
```  1266
```
```  1267 theorem quasilist_reflection:
```
```  1268      "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
```
```  1269                \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
```
```  1270 apply (simp only: is_quasilist_def setclass_simps)
```
```  1271 apply (intro FOL_reflections Nil_reflection Cons_reflection)
```
```  1272 done
```
```  1273
```
```  1274
```
```  1275 subsection{*Absoluteness for the Function @{term nth}*}
```
```  1276
```
```  1277
```
```  1278 subsubsection{*The Formula @{term is_tl}, Internalized*}
```
```  1279
```
```  1280 (*     "is_tl(M,xs,T) ==
```
```  1281        (is_Nil(M,xs) --> T=xs) &
```
```  1282        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
```
```  1283        (is_quasilist(M,xs) | empty(M,T))" *)
```
```  1284 constdefs tl_fm :: "[i,i]=>i"
```
```  1285     "tl_fm(xs,T) ==
```
```  1286        And(Implies(Nil_fm(xs), Equal(T,xs)),
```
```  1287            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
```
```  1288                Or(quasilist_fm(xs), empty_fm(T))))"
```
```  1289
```
```  1290 lemma tl_type [TC]:
```
```  1291      "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
```
```  1292 by (simp add: tl_fm_def)
```
```  1293
```
```  1294 lemma sats_tl_fm [simp]:
```
```  1295    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1296     ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
```
```  1297 by (simp add: tl_fm_def is_tl_def)
```
```  1298
```
```  1299 lemma tl_iff_sats:
```
```  1300       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1301           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1302        ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
```
```  1303 by simp
```
```  1304
```
```  1305 theorem tl_reflection:
```
```  1306      "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
```
```  1307                \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
```
```  1308 apply (simp only: is_tl_def setclass_simps)
```
```  1309 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1310              quasilist_reflection empty_reflection)
```
```  1311 done
```
```  1312
```
```  1313
```
```  1314 subsubsection{*An Instance of Replacement for @{term nth}*}
```
```  1315
```
```  1316 lemma nth_replacement_Reflects:
```
```  1317  "REFLECTS
```
```  1318    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1319          is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
```
```  1320     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1321          is_wfrec(**Lset(i),
```
```  1322                   iterates_MH(**Lset(i),
```
```  1323                           is_tl(**Lset(i)), z), memsn, u, y))]"
```
```  1324 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1325           iterates_MH_reflection list_functor_reflection tl_reflection)
```
```  1326
```
```  1327 lemma nth_replacement:
```
```  1328    "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
```
```  1329 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1330 apply (rule strong_replacementI)
```
```  1331 apply (rule rallI)
```
```  1332 apply (rule separation_CollectI)
```
```  1333 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1334 apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1335 apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
```
```  1336 apply (drule subset_Lset_ltD, assumption)
```
```  1337 apply (erule reflection_imp_L_separation)
```
```  1338   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1339 apply (elim conjE)
```
```  1340 apply (rule DPow_LsetI)
```
```  1341 apply (rename_tac v)
```
```  1342 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1343 apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
```
```  1344 apply (rule sep_rules | simp)+
```
```  1345 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1346 apply (rule sep_rules quasinat_iff_sats tl_iff_sats | simp)+
```
```  1347 done
```
```  1348
```
```  1349 ML
```
```  1350 {*
```
```  1351 bind_thm ("nth_abs_lemma", datatypes_L (thm "M_datatypes.nth_abs"));
```
```  1352 *}
```
```  1353
```
```  1354 text{*Instantiating theorem @{text nth_abs} for @{term L}*}
```
```  1355 lemma nth_abs [simp]:
```
```  1356      "[|L(A); n \<in> nat; l \<in> list(A); L(Z)|]
```
```  1357       ==> is_nth(L,n,l,Z) <-> Z = nth(n,l)"
```
```  1358 apply (rule nth_abs_lemma)
```
```  1359 apply (blast intro: nth_replacement transL list_closed, assumption+)
```
```  1360 done
```
```  1361
```
```  1362 end
```