src/ZF/Constructible/Rec_Separation.thy
 author wenzelm Mon Jul 29 18:07:53 2002 +0200 (2002-07-29) changeset 13429 2232810416fc parent 13428 99e52e78eb65 child 13434 78b93a667c01 permissions -rw-r--r--
tuned;
```     1
```
```     2 header {*Separation for Facts About Recursion*}
```
```     3
```
```     4 theory Rec_Separation = Separation + Datatype_absolute:
```
```     5
```
```     6 text{*This theory proves all instances needed for locales @{text
```
```     7 "M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*}
```
```     8
```
```     9 lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
```
```    10 by simp
```
```    11
```
```    12 subsection{*The Locale @{text "M_trancl"}*}
```
```    13
```
```    14 subsubsection{*Separation for Reflexive/Transitive Closure*}
```
```    15
```
```    16 text{*First, The Defining Formula*}
```
```    17
```
```    18 (* "rtran_closure_mem(M,A,r,p) ==
```
```    19       \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
```
```    20        omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
```
```    21        (\<exists>f[M]. typed_function(M,n',A,f) &
```
```    22         (\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
```
```    23           fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
```
```    24         (\<forall>j[M]. j\<in>n -->
```
```    25           (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
```
```    26             fun_apply(M,f,j,fj) & successor(M,j,sj) &
```
```    27             fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
```
```    28 constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
```
```    29  "rtran_closure_mem_fm(A,r,p) ==
```
```    30    Exists(Exists(Exists(
```
```    31     And(omega_fm(2),
```
```    32      And(Member(1,2),
```
```    33       And(succ_fm(1,0),
```
```    34        Exists(And(typed_function_fm(1, A#+4, 0),
```
```    35         And(Exists(Exists(Exists(
```
```    36               And(pair_fm(2,1,p#+7),
```
```    37                And(empty_fm(0),
```
```    38                 And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
```
```    39             Forall(Implies(Member(0,3),
```
```    40              Exists(Exists(Exists(Exists(
```
```    41               And(fun_apply_fm(5,4,3),
```
```    42                And(succ_fm(4,2),
```
```    43                 And(fun_apply_fm(5,2,1),
```
```    44                  And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
```
```    45
```
```    46
```
```    47 lemma rtran_closure_mem_type [TC]:
```
```    48  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
```
```    49 by (simp add: rtran_closure_mem_fm_def)
```
```    50
```
```    51 lemma arity_rtran_closure_mem_fm [simp]:
```
```    52      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```    53       ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```    54 by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```    55
```
```    56 lemma sats_rtran_closure_mem_fm [simp]:
```
```    57    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```    58     ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
```
```    59         rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```    60 by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
```
```    61
```
```    62 lemma rtran_closure_mem_iff_sats:
```
```    63       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```    64           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```    65        ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
```
```    66 by (simp add: sats_rtran_closure_mem_fm)
```
```    67
```
```    68 theorem rtran_closure_mem_reflection:
```
```    69      "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
```
```    70                \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
```
```    71 apply (simp only: rtran_closure_mem_def setclass_simps)
```
```    72 apply (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    73 done
```
```    74
```
```    75 text{*Separation for @{term "rtrancl(r)"}.*}
```
```    76 lemma rtrancl_separation:
```
```    77      "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
```
```    78 apply (rule separation_CollectI)
```
```    79 apply (rule_tac A="{r,A,z}" in subset_LsetE, blast )
```
```    80 apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
```
```    81 apply (drule subset_Lset_ltD, assumption)
```
```    82 apply (erule reflection_imp_L_separation)
```
```    83   apply (simp_all add: lt_Ord2)
```
```    84 apply (rule DPow_LsetI)
```
```    85 apply (rename_tac u)
```
```    86 apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
```
```    87 apply (rule sep_rules | simp)+
```
```    88 done
```
```    89
```
```    90
```
```    91 subsubsection{*Reflexive/Transitive Closure, Internalized*}
```
```    92
```
```    93 (*  "rtran_closure(M,r,s) ==
```
```    94         \<forall>A[M]. is_field(M,r,A) -->
```
```    95          (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
```
```    96 constdefs rtran_closure_fm :: "[i,i]=>i"
```
```    97  "rtran_closure_fm(r,s) ==
```
```    98    Forall(Implies(field_fm(succ(r),0),
```
```    99                   Forall(Iff(Member(0,succ(succ(s))),
```
```   100                              rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
```
```   101
```
```   102 lemma rtran_closure_type [TC]:
```
```   103      "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
```
```   104 by (simp add: rtran_closure_fm_def)
```
```   105
```
```   106 lemma arity_rtran_closure_fm [simp]:
```
```   107      "[| x \<in> nat; y \<in> nat |]
```
```   108       ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   109 by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   110
```
```   111 lemma sats_rtran_closure_fm [simp]:
```
```   112    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   113     ==> sats(A, rtran_closure_fm(x,y), env) <->
```
```   114         rtran_closure(**A, nth(x,env), nth(y,env))"
```
```   115 by (simp add: rtran_closure_fm_def rtran_closure_def)
```
```   116
```
```   117 lemma rtran_closure_iff_sats:
```
```   118       "[| nth(i,env) = x; nth(j,env) = y;
```
```   119           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   120        ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
```
```   121 by simp
```
```   122
```
```   123 theorem rtran_closure_reflection:
```
```   124      "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
```
```   125                \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
```
```   126 apply (simp only: rtran_closure_def setclass_simps)
```
```   127 apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
```
```   128 done
```
```   129
```
```   130
```
```   131 subsubsection{*Transitive Closure of a Relation, Internalized*}
```
```   132
```
```   133 (*  "tran_closure(M,r,t) ==
```
```   134          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
```
```   135 constdefs tran_closure_fm :: "[i,i]=>i"
```
```   136  "tran_closure_fm(r,s) ==
```
```   137    Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
```
```   138
```
```   139 lemma tran_closure_type [TC]:
```
```   140      "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
```
```   141 by (simp add: tran_closure_fm_def)
```
```   142
```
```   143 lemma arity_tran_closure_fm [simp]:
```
```   144      "[| x \<in> nat; y \<in> nat |]
```
```   145       ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   146 by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   147
```
```   148 lemma sats_tran_closure_fm [simp]:
```
```   149    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   150     ==> sats(A, tran_closure_fm(x,y), env) <->
```
```   151         tran_closure(**A, nth(x,env), nth(y,env))"
```
```   152 by (simp add: tran_closure_fm_def tran_closure_def)
```
```   153
```
```   154 lemma tran_closure_iff_sats:
```
```   155       "[| nth(i,env) = x; nth(j,env) = y;
```
```   156           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   157        ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
```
```   158 by simp
```
```   159
```
```   160 theorem tran_closure_reflection:
```
```   161      "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
```
```   162                \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
```
```   163 apply (simp only: tran_closure_def setclass_simps)
```
```   164 apply (intro FOL_reflections function_reflections
```
```   165              rtran_closure_reflection composition_reflection)
```
```   166 done
```
```   167
```
```   168
```
```   169 subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
```
```   170
```
```   171 lemma wellfounded_trancl_reflects:
```
```   172   "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   173                  w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
```
```   174    \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
```
```   175        w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
```
```   176        wx \<in> rp]"
```
```   177 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   178           tran_closure_reflection)
```
```   179
```
```   180
```
```   181 lemma wellfounded_trancl_separation:
```
```   182          "[| L(r); L(Z) |] ==>
```
```   183           separation (L, \<lambda>x.
```
```   184               \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   185                w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
```
```   186 apply (rule separation_CollectI)
```
```   187 apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast )
```
```   188 apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
```
```   189 apply (drule subset_Lset_ltD, assumption)
```
```   190 apply (erule reflection_imp_L_separation)
```
```   191   apply (simp_all add: lt_Ord2)
```
```   192 apply (rule DPow_LsetI)
```
```   193 apply (rename_tac u)
```
```   194 apply (rule bex_iff_sats conj_iff_sats)+
```
```   195 apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats)
```
```   196 apply (rule sep_rules tran_closure_iff_sats | simp)+
```
```   197 done
```
```   198
```
```   199
```
```   200 subsubsection{*Instantiating the locale @{text M_trancl}*}
```
```   201
```
```   202 theorem M_trancl_L: "PROP M_trancl(L)"
```
```   203   apply (rule M_trancl.intro)
```
```   204     apply (rule M_axioms.axioms [OF M_axioms_L])+
```
```   205   apply (rule M_trancl_axioms.intro)
```
```   206    apply (assumption | rule
```
```   207      rtrancl_separation wellfounded_trancl_separation)+
```
```   208   done
```
```   209
```
```   210 lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
```
```   211   and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
```
```   212   and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L]
```
```   213   and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L]
```
```   214   and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L]
```
```   215   and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L]
```
```   216   and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L]
```
```   217   and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L]
```
```   218   and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L]
```
```   219   and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L]
```
```   220   and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L]
```
```   221   and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L]
```
```   222   and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L]
```
```   223
```
```   224 declare rtrancl_closed [intro,simp]
```
```   225 declare rtrancl_abs [simp]
```
```   226 declare trancl_closed [intro,simp]
```
```   227 declare trancl_abs [simp]
```
```   228
```
```   229
```
```   230 subsection{*Well-Founded Recursion!*}
```
```   231
```
```   232 (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
```
```   233    "M_is_recfun(M,MH,r,a,f) ==
```
```   234      \<forall>z[M]. z \<in> f <->
```
```   235             5      4       3       2       1           0
```
```   236             (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
```
```   237                pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
```
```   238                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
```
```   239                xa \<in> r & MH(x, f_r_sx, y))"
```
```   240 *)
```
```   241
```
```   242 constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i"
```
```   243  "is_recfun_fm(p,r,a,f) ==
```
```   244    Forall(Iff(Member(0,succ(f)),
```
```   245     Exists(Exists(Exists(Exists(Exists(Exists(
```
```   246      And(pair_fm(5,4,6),
```
```   247       And(pair_fm(5,a#+7,3),
```
```   248        And(upair_fm(5,5,2),
```
```   249         And(pre_image_fm(r#+7,2,1),
```
```   250          And(restriction_fm(f#+7,1,0),
```
```   251           And(Member(3,r#+7), p(5,0,4)))))))))))))))"
```
```   252
```
```   253
```
```   254 lemma is_recfun_type_0:
```
```   255      "[| !!x y z. [| x \<in> nat; y \<in> nat; z \<in> nat |] ==> p(x,y,z) \<in> formula;
```
```   256          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   257       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   258 apply (unfold is_recfun_fm_def)
```
```   259 (*FIXME: FIND OUT why simp loops!*)
```
```   260 apply typecheck
```
```   261 by simp
```
```   262
```
```   263 lemma is_recfun_type [TC]:
```
```   264      "[| p(5,0,4) \<in> formula;
```
```   265          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   266       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   267 by (simp add: is_recfun_fm_def)
```
```   268
```
```   269 lemma arity_is_recfun_fm [simp]:
```
```   270      "[| arity(p(5,0,4)) le 8; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   271       ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   272 apply (frule lt_nat_in_nat, simp)
```
```   273 apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] )
```
```   274 apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1])
```
```   275 apply (rule le_imp_subset)
```
```   276 apply (erule le_trans, simp)
```
```   277 apply (simp add: succ_Un_distrib [symmetric] Un_ac)
```
```   278 done
```
```   279
```
```   280 lemma sats_is_recfun_fm:
```
```   281   assumes MH_iff_sats:
```
```   282       "!!x y z env.
```
```   283          [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   284          ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)"
```
```   285   shows
```
```   286       "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   287        ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
```
```   288            M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
```
```   289 by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
```
```   290
```
```   291 lemma is_recfun_iff_sats:
```
```   292   "[| (!!x y z env. [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   293                     ==> MH(nth(x,env), nth(y,env), nth(z,env)) <->
```
```   294                         sats(A, p(x,y,z), env));
```
```   295       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   296       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   297    ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
```
```   298 by (simp add: sats_is_recfun_fm [of A MH])
```
```   299
```
```   300 theorem is_recfun_reflection:
```
```   301   assumes MH_reflection:
```
```   302     "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
```
```   303                      \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
```
```   304   shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L), f(x), g(x), h(x)),
```
```   305                \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
```
```   306 apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
```
```   307 apply (intro FOL_reflections function_reflections
```
```   308              restriction_reflection MH_reflection)
```
```   309 done
```
```   310
```
```   311 text{*Currently, @{text sats}-theorems for higher-order operators don't seem
```
```   312 useful.  Reflection theorems do work, though.  This one avoids the repetition
```
```   313 of the @{text MH}-term.*}
```
```   314 theorem is_wfrec_reflection:
```
```   315   assumes MH_reflection:
```
```   316     "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
```
```   317                      \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
```
```   318   shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L), f(x), g(x), h(x)),
```
```   319                \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
```
```   320 apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
```
```   321 apply (intro FOL_reflections MH_reflection is_recfun_reflection)
```
```   322 done
```
```   323
```
```   324 subsection{*The Locale @{text "M_wfrank"}*}
```
```   325
```
```   326 subsubsection{*Separation for @{term "wfrank"}*}
```
```   327
```
```   328 lemma wfrank_Reflects:
```
```   329  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   330               ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
```
```   331       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   332          ~ (\<exists>f \<in> Lset(i).
```
```   333             M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
```
```   334                         rplus, x, f))]"
```
```   335 by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
```
```   336
```
```   337 lemma wfrank_separation:
```
```   338      "L(r) ==>
```
```   339       separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   340          ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
```
```   341 apply (rule separation_CollectI)
```
```   342 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   343 apply (rule ReflectsE [OF wfrank_Reflects], assumption)
```
```   344 apply (drule subset_Lset_ltD, assumption)
```
```   345 apply (erule reflection_imp_L_separation)
```
```   346   apply (simp_all add: lt_Ord2, clarify)
```
```   347 apply (rule DPow_LsetI)
```
```   348 apply (rename_tac u)
```
```   349 apply (rule ball_iff_sats imp_iff_sats)+
```
```   350 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   351 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   352 done
```
```   353
```
```   354
```
```   355 subsubsection{*Replacement for @{term "wfrank"}*}
```
```   356
```
```   357 lemma wfrank_replacement_Reflects:
```
```   358  "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
```
```   359         (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   360          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   361                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   362                         is_range(L,f,y))),
```
```   363  \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
```
```   364       (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   365        (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
```
```   366          M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
```
```   367          is_range(**Lset(i),f,y)))]"
```
```   368 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   369              is_recfun_reflection tran_closure_reflection)
```
```   370
```
```   371
```
```   372 lemma wfrank_strong_replacement:
```
```   373      "L(r) ==>
```
```   374       strong_replacement(L, \<lambda>x z.
```
```   375          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   376          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   377                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   378                         is_range(L,f,y)))"
```
```   379 apply (rule strong_replacementI)
```
```   380 apply (rule rallI)
```
```   381 apply (rename_tac B)
```
```   382 apply (rule separation_CollectI)
```
```   383 apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
```
```   384 apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
```
```   385 apply (drule subset_Lset_ltD, assumption)
```
```   386 apply (erule reflection_imp_L_separation)
```
```   387   apply (simp_all add: lt_Ord2)
```
```   388 apply (rule DPow_LsetI)
```
```   389 apply (rename_tac u)
```
```   390 apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
```
```   391 apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
```
```   392 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   393 done
```
```   394
```
```   395
```
```   396 subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
```
```   397
```
```   398 lemma Ord_wfrank_Reflects:
```
```   399  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   400           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   401              is_range(L,f,rangef) -->
```
```   402              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   403              ordinal(L,rangef)),
```
```   404       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   405           ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
```
```   406              is_range(**Lset(i),f,rangef) -->
```
```   407              M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
```
```   408                          rplus, x, f) -->
```
```   409              ordinal(**Lset(i),rangef))]"
```
```   410 by (intro FOL_reflections function_reflections is_recfun_reflection
```
```   411           tran_closure_reflection ordinal_reflection)
```
```   412
```
```   413 lemma  Ord_wfrank_separation:
```
```   414      "L(r) ==>
```
```   415       separation (L, \<lambda>x.
```
```   416          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   417           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   418              is_range(L,f,rangef) -->
```
```   419              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   420              ordinal(L,rangef)))"
```
```   421 apply (rule separation_CollectI)
```
```   422 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   423 apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
```
```   424 apply (drule subset_Lset_ltD, assumption)
```
```   425 apply (erule reflection_imp_L_separation)
```
```   426   apply (simp_all add: lt_Ord2, clarify)
```
```   427 apply (rule DPow_LsetI)
```
```   428 apply (rename_tac u)
```
```   429 apply (rule ball_iff_sats imp_iff_sats)+
```
```   430 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   431 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   432 done
```
```   433
```
```   434
```
```   435 subsubsection{*Instantiating the locale @{text M_wfrank}*}
```
```   436
```
```   437 theorem M_wfrank_L: "PROP M_wfrank(L)"
```
```   438   apply (rule M_wfrank.intro)
```
```   439      apply (rule M_trancl.axioms [OF M_trancl_L])+
```
```   440   apply (rule M_wfrank_axioms.intro)
```
```   441    apply (assumption | rule
```
```   442      wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
```
```   443   done
```
```   444
```
```   445 lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L]
```
```   446   and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
```
```   447   and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
```
```   448   and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
```
```   449   and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
```
```   450   and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
```
```   451   and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
```
```   452   and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
```
```   453   and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
```
```   454   and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
```
```   455   and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
```
```   456   and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
```
```   457   and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
```
```   458   and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
```
```   459   and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
```
```   460   and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
```
```   461   and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L]
```
```   462   and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L]
```
```   463   and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L]
```
```   464
```
```   465 declare iterates_closed [intro,simp]
```
```   466 declare Ord_wfrank_range [rule_format]
```
```   467 declare wf_abs [simp]
```
```   468 declare wf_on_abs [simp]
```
```   469
```
```   470
```
```   471 subsection{*For Datatypes*}
```
```   472
```
```   473 subsubsection{*Binary Products, Internalized*}
```
```   474
```
```   475 constdefs cartprod_fm :: "[i,i,i]=>i"
```
```   476 (* "cartprod(M,A,B,z) ==
```
```   477         \<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
```
```   478     "cartprod_fm(A,B,z) ==
```
```   479        Forall(Iff(Member(0,succ(z)),
```
```   480                   Exists(And(Member(0,succ(succ(A))),
```
```   481                          Exists(And(Member(0,succ(succ(succ(B)))),
```
```   482                                     pair_fm(1,0,2)))))))"
```
```   483
```
```   484 lemma cartprod_type [TC]:
```
```   485      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
```
```   486 by (simp add: cartprod_fm_def)
```
```   487
```
```   488 lemma arity_cartprod_fm [simp]:
```
```   489      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   490       ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   491 by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   492
```
```   493 lemma sats_cartprod_fm [simp]:
```
```   494    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   495     ==> sats(A, cartprod_fm(x,y,z), env) <->
```
```   496         cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   497 by (simp add: cartprod_fm_def cartprod_def)
```
```   498
```
```   499 lemma cartprod_iff_sats:
```
```   500       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   501           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   502        ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
```
```   503 by (simp add: sats_cartprod_fm)
```
```   504
```
```   505 theorem cartprod_reflection:
```
```   506      "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
```
```   507                \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
```
```   508 apply (simp only: cartprod_def setclass_simps)
```
```   509 apply (intro FOL_reflections pair_reflection)
```
```   510 done
```
```   511
```
```   512
```
```   513 subsubsection{*Binary Sums, Internalized*}
```
```   514
```
```   515 (* "is_sum(M,A,B,Z) ==
```
```   516        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
```
```   517          3      2       1        0
```
```   518        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
```
```   519        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
```
```   520 constdefs sum_fm :: "[i,i,i]=>i"
```
```   521     "sum_fm(A,B,Z) ==
```
```   522        Exists(Exists(Exists(Exists(
```
```   523         And(number1_fm(2),
```
```   524             And(cartprod_fm(2,A#+4,3),
```
```   525                 And(upair_fm(2,2,1),
```
```   526                     And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
```
```   527
```
```   528 lemma sum_type [TC]:
```
```   529      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
```
```   530 by (simp add: sum_fm_def)
```
```   531
```
```   532 lemma arity_sum_fm [simp]:
```
```   533      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   534       ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   535 by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   536
```
```   537 lemma sats_sum_fm [simp]:
```
```   538    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   539     ==> sats(A, sum_fm(x,y,z), env) <->
```
```   540         is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   541 by (simp add: sum_fm_def is_sum_def)
```
```   542
```
```   543 lemma sum_iff_sats:
```
```   544       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   545           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   546        ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
```
```   547 by simp
```
```   548
```
```   549 theorem sum_reflection:
```
```   550      "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
```
```   551                \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
```
```   552 apply (simp only: is_sum_def setclass_simps)
```
```   553 apply (intro FOL_reflections function_reflections cartprod_reflection)
```
```   554 done
```
```   555
```
```   556
```
```   557 subsubsection{*The Operator @{term quasinat}*}
```
```   558
```
```   559 (* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
```
```   560 constdefs quasinat_fm :: "i=>i"
```
```   561     "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
```
```   562
```
```   563 lemma quasinat_type [TC]:
```
```   564      "x \<in> nat ==> quasinat_fm(x) \<in> formula"
```
```   565 by (simp add: quasinat_fm_def)
```
```   566
```
```   567 lemma arity_quasinat_fm [simp]:
```
```   568      "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
```
```   569 by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   570
```
```   571 lemma sats_quasinat_fm [simp]:
```
```   572    "[| x \<in> nat; env \<in> list(A)|]
```
```   573     ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
```
```   574 by (simp add: quasinat_fm_def is_quasinat_def)
```
```   575
```
```   576 lemma quasinat_iff_sats:
```
```   577       "[| nth(i,env) = x; nth(j,env) = y;
```
```   578           i \<in> nat; env \<in> list(A)|]
```
```   579        ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
```
```   580 by simp
```
```   581
```
```   582 theorem quasinat_reflection:
```
```   583      "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
```
```   584                \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
```
```   585 apply (simp only: is_quasinat_def setclass_simps)
```
```   586 apply (intro FOL_reflections function_reflections)
```
```   587 done
```
```   588
```
```   589
```
```   590 subsubsection{*The Operator @{term is_nat_case}*}
```
```   591
```
```   592 (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
```
```   593     "is_nat_case(M, a, is_b, k, z) ==
```
```   594        (empty(M,k) --> z=a) &
```
```   595        (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
```
```   596        (is_quasinat(M,k) | empty(M,z))" *)
```
```   597 text{*The formula @{term is_b} has free variables 1 and 0.*}
```
```   598 constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i"
```
```   599  "is_nat_case_fm(a,is_b,k,z) ==
```
```   600     And(Implies(empty_fm(k), Equal(z,a)),
```
```   601         And(Forall(Implies(succ_fm(0,succ(k)),
```
```   602                    Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))),
```
```   603             Or(quasinat_fm(k), empty_fm(z))))"
```
```   604
```
```   605 lemma is_nat_case_type [TC]:
```
```   606      "[| is_b(1,0) \<in> formula;
```
```   607          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   608       ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
```
```   609 by (simp add: is_nat_case_fm_def)
```
```   610
```
```   611 lemma arity_is_nat_case_fm [simp]:
```
```   612      "[| is_b(1,0) \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   613       ==> arity(is_nat_case_fm(x,is_b,y,z)) =
```
```   614           succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(is_b(1,0)) #- 2)"
```
```   615 apply (subgoal_tac "arity(is_b(1,0)) \<in> nat")
```
```   616 apply typecheck
```
```   617 (*FIXME: could nat_diff_split work?*)
```
```   618 apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat
```
```   619                  succ_Un_distrib [symmetric] Un_ac
```
```   620                  split: split_nat_case)
```
```   621 done
```
```   622
```
```   623 lemma sats_is_nat_case_fm:
```
```   624   assumes is_b_iff_sats:
```
```   625       "!!a b. [| a \<in> A; b \<in> A|]
```
```   626               ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))"
```
```   627   shows
```
```   628       "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   629        ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
```
```   630            is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
```
```   631 apply (frule lt_length_in_nat, assumption)
```
```   632 apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
```
```   633 done
```
```   634
```
```   635 lemma is_nat_case_iff_sats:
```
```   636   "[| (!!a b. [| a \<in> A; b \<in> A|]
```
```   637               ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env))));
```
```   638       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   639       i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   640    ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
```
```   641 by (simp add: sats_is_nat_case_fm [of A is_b])
```
```   642
```
```   643
```
```   644 text{*The second argument of @{term is_b} gives it direct access to @{term x},
```
```   645   which is essential for handling free variable references.  Without this
```
```   646   argument, we cannot prove reflection for @{term iterates_MH}.*}
```
```   647 theorem is_nat_case_reflection:
```
```   648   assumes is_b_reflection:
```
```   649     "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
```
```   650                      \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
```
```   651   shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
```
```   652                \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
```
```   653 apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
```
```   654 apply (intro FOL_reflections function_reflections
```
```   655              restriction_reflection is_b_reflection quasinat_reflection)
```
```   656 done
```
```   657
```
```   658
```
```   659
```
```   660 subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
```
```   661
```
```   662 (*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   663    "iterates_MH(M,isF,v,n,g,z) ==
```
```   664         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   665                     n, z)" *)
```
```   666 constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i"
```
```   667  "iterates_MH_fm(isF,v,n,g,z) ==
```
```   668     is_nat_case_fm(v,
```
```   669       \<lambda>m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0),
```
```   670                      Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))),
```
```   671       n, z)"
```
```   672
```
```   673 lemma iterates_MH_type [TC]:
```
```   674      "[| p(1,0) \<in> formula;
```
```   675          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   676       ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
```
```   677 by (simp add: iterates_MH_fm_def)
```
```   678
```
```   679
```
```   680 lemma arity_iterates_MH_fm [simp]:
```
```   681      "[| p(1,0) \<in> formula;
```
```   682          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   683       ==> arity(iterates_MH_fm(p,v,x,y,z)) =
```
```   684           succ(v) \<union> succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(p(1,0)) #- 4)"
```
```   685 apply (subgoal_tac "arity(p(1,0)) \<in> nat")
```
```   686 apply typecheck
```
```   687 apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac
```
```   688             split: split_nat_case, clarify)
```
```   689 apply (rename_tac i j)
```
```   690 apply (drule eq_succ_imp_eq_m1, simp)
```
```   691 apply (drule eq_succ_imp_eq_m1, simp)
```
```   692 apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left)
```
```   693 done
```
```   694
```
```   695 lemma sats_iterates_MH_fm:
```
```   696   assumes is_F_iff_sats:
```
```   697       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   698               ==> is_F(a,b) <->
```
```   699                   sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
```
```   700   shows
```
```   701       "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   702        ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
```
```   703            iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
```
```   704 by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm
```
```   705               is_F_iff_sats [symmetric])
```
```   706
```
```   707 lemma iterates_MH_iff_sats:
```
```   708   "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   709               ==> is_F(a,b) <->
```
```   710                   sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env))))));
```
```   711       nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   712       i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   713    ==> iterates_MH(**A, is_F, v, x, y, z) <->
```
```   714        sats(A, iterates_MH_fm(p,i',i,j,k), env)"
```
```   715 apply (rule iff_sym)
```
```   716 apply (rule iff_trans)
```
```   717 apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all)
```
```   718 done
```
```   719
```
```   720 theorem iterates_MH_reflection:
```
```   721   assumes p_reflection:
```
```   722     "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
```
```   723                      \<lambda>i x. p(**Lset(i), f(x), g(x))]"
```
```   724  shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
```
```   725                \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
```
```   726 apply (simp (no_asm_use) only: iterates_MH_def)
```
```   727 txt{*Must be careful: simplifying with @{text setclass_simps} above would
```
```   728      change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
```
```   729      it would no longer match rule @{text is_nat_case_reflection}. *}
```
```   730 apply (rule is_nat_case_reflection)
```
```   731 apply (simp (no_asm_use) only: setclass_simps)
```
```   732 apply (intro FOL_reflections function_reflections is_nat_case_reflection
```
```   733              restriction_reflection p_reflection)
```
```   734 done
```
```   735
```
```   736
```
```   737
```
```   738 subsection{*@{term L} is Closed Under the Operator @{term list}*}
```
```   739
```
```   740 subsubsection{*The List Functor, Internalized*}
```
```   741
```
```   742 constdefs list_functor_fm :: "[i,i,i]=>i"
```
```   743 (* "is_list_functor(M,A,X,Z) ==
```
```   744         \<exists>n1[M]. \<exists>AX[M].
```
```   745          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
```
```   746     "list_functor_fm(A,X,Z) ==
```
```   747        Exists(Exists(
```
```   748         And(number1_fm(1),
```
```   749             And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
```
```   750
```
```   751 lemma list_functor_type [TC]:
```
```   752      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
```
```   753 by (simp add: list_functor_fm_def)
```
```   754
```
```   755 lemma arity_list_functor_fm [simp]:
```
```   756      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   757       ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   758 by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   759
```
```   760 lemma sats_list_functor_fm [simp]:
```
```   761    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   762     ==> sats(A, list_functor_fm(x,y,z), env) <->
```
```   763         is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   764 by (simp add: list_functor_fm_def is_list_functor_def)
```
```   765
```
```   766 lemma list_functor_iff_sats:
```
```   767   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   768       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   769    ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
```
```   770 by simp
```
```   771
```
```   772 theorem list_functor_reflection:
```
```   773      "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
```
```   774                \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
```
```   775 apply (simp only: is_list_functor_def setclass_simps)
```
```   776 apply (intro FOL_reflections number1_reflection
```
```   777              cartprod_reflection sum_reflection)
```
```   778 done
```
```   779
```
```   780
```
```   781 subsubsection{*Instances of Replacement for Lists*}
```
```   782
```
```   783 lemma list_replacement1_Reflects:
```
```   784  "REFLECTS
```
```   785    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   786          is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
```
```   787     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   788          is_wfrec(**Lset(i),
```
```   789                   iterates_MH(**Lset(i),
```
```   790                           is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
```
```   791 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   792           iterates_MH_reflection list_functor_reflection)
```
```   793
```
```   794 lemma list_replacement1:
```
```   795    "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
```
```   796 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   797 apply (rule strong_replacementI)
```
```   798 apply (rule rallI)
```
```   799 apply (rename_tac B)
```
```   800 apply (rule separation_CollectI)
```
```   801 apply (insert nonempty)
```
```   802 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   803 apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   804 apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
```
```   805 apply (drule subset_Lset_ltD, assumption)
```
```   806 apply (erule reflection_imp_L_separation)
```
```   807   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   808 apply (elim conjE)
```
```   809 apply (rule DPow_LsetI)
```
```   810 apply (rename_tac v)
```
```   811 apply (rule bex_iff_sats conj_iff_sats)+
```
```   812 apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   813 apply (rule sep_rules | simp)+
```
```   814 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```   815 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   816 apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
```
```   817 done
```
```   818
```
```   819
```
```   820 lemma list_replacement2_Reflects:
```
```   821  "REFLECTS
```
```   822    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   823          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   824            is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
```
```   825                               msn, u, x)),
```
```   826     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   827          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   828           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   829            is_wfrec (**Lset(i),
```
```   830                  iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
```
```   831                      msn, u, x))]"
```
```   832 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   833           iterates_MH_reflection list_functor_reflection)
```
```   834
```
```   835
```
```   836 lemma list_replacement2:
```
```   837    "L(A) ==> strong_replacement(L,
```
```   838          \<lambda>n y. n\<in>nat &
```
```   839                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   840                is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
```
```   841                         msn, n, y)))"
```
```   842 apply (rule strong_replacementI)
```
```   843 apply (rule rallI)
```
```   844 apply (rename_tac B)
```
```   845 apply (rule separation_CollectI)
```
```   846 apply (insert nonempty)
```
```   847 apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
```
```   848 apply (blast intro: L_nat)
```
```   849 apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
```
```   850 apply (drule subset_Lset_ltD, assumption)
```
```   851 apply (erule reflection_imp_L_separation)
```
```   852   apply (simp_all add: lt_Ord2)
```
```   853 apply (rule DPow_LsetI)
```
```   854 apply (rename_tac v)
```
```   855 apply (rule bex_iff_sats conj_iff_sats)+
```
```   856 apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
```
```   857 apply (rule sep_rules | simp)+
```
```   858 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   859 apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
```
```   860 done
```
```   861
```
```   862
```
```   863 subsection{*@{term L} is Closed Under the Operator @{term formula}*}
```
```   864
```
```   865 subsubsection{*The Formula Functor, Internalized*}
```
```   866
```
```   867 constdefs formula_functor_fm :: "[i,i]=>i"
```
```   868 (*     "is_formula_functor(M,X,Z) ==
```
```   869         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
```
```   870            4           3               2       1       0
```
```   871           omega(M,nat') & cartprod(M,nat',nat',natnat) &
```
```   872           is_sum(M,natnat,natnat,natnatsum) &
```
```   873           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
```
```   874           is_sum(M,natnatsum,X3,Z)" *)
```
```   875     "formula_functor_fm(X,Z) ==
```
```   876        Exists(Exists(Exists(Exists(Exists(
```
```   877         And(omega_fm(4),
```
```   878          And(cartprod_fm(4,4,3),
```
```   879           And(sum_fm(3,3,2),
```
```   880            And(cartprod_fm(X#+5,X#+5,1),
```
```   881             And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
```
```   882
```
```   883 lemma formula_functor_type [TC]:
```
```   884      "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
```
```   885 by (simp add: formula_functor_fm_def)
```
```   886
```
```   887 lemma sats_formula_functor_fm [simp]:
```
```   888    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   889     ==> sats(A, formula_functor_fm(x,y), env) <->
```
```   890         is_formula_functor(**A, nth(x,env), nth(y,env))"
```
```   891 by (simp add: formula_functor_fm_def is_formula_functor_def)
```
```   892
```
```   893 lemma formula_functor_iff_sats:
```
```   894   "[| nth(i,env) = x; nth(j,env) = y;
```
```   895       i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   896    ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
```
```   897 by simp
```
```   898
```
```   899 theorem formula_functor_reflection:
```
```   900      "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
```
```   901                \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
```
```   902 apply (simp only: is_formula_functor_def setclass_simps)
```
```   903 apply (intro FOL_reflections omega_reflection
```
```   904              cartprod_reflection sum_reflection)
```
```   905 done
```
```   906
```
```   907 subsubsection{*Instances of Replacement for Formulas*}
```
```   908
```
```   909 lemma formula_replacement1_Reflects:
```
```   910  "REFLECTS
```
```   911    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   912          is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
```
```   913     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   914          is_wfrec(**Lset(i),
```
```   915                   iterates_MH(**Lset(i),
```
```   916                           is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
```
```   917 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   918           iterates_MH_reflection formula_functor_reflection)
```
```   919
```
```   920 lemma formula_replacement1:
```
```   921    "iterates_replacement(L, is_formula_functor(L), 0)"
```
```   922 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   923 apply (rule strong_replacementI)
```
```   924 apply (rule rallI)
```
```   925 apply (rename_tac B)
```
```   926 apply (rule separation_CollectI)
```
```   927 apply (insert nonempty)
```
```   928 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   929 apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   930 apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
```
```   931 apply (drule subset_Lset_ltD, assumption)
```
```   932 apply (erule reflection_imp_L_separation)
```
```   933   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   934 apply (rule DPow_LsetI)
```
```   935 apply (rename_tac v)
```
```   936 apply (rule bex_iff_sats conj_iff_sats)+
```
```   937 apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   938 apply (rule sep_rules | simp)+
```
```   939 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```   940 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   941 apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
```
```   942 txt{*SLOW: like 40 seconds!*}
```
```   943 done
```
```   944
```
```   945 lemma formula_replacement2_Reflects:
```
```   946  "REFLECTS
```
```   947    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   948          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   949            is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
```
```   950                               msn, u, x)),
```
```   951     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   952          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   953           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   954            is_wfrec (**Lset(i),
```
```   955                  iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
```
```   956                      msn, u, x))]"
```
```   957 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   958           iterates_MH_reflection formula_functor_reflection)
```
```   959
```
```   960
```
```   961 lemma formula_replacement2:
```
```   962    "strong_replacement(L,
```
```   963          \<lambda>n y. n\<in>nat &
```
```   964                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   965                is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
```
```   966                         msn, n, y)))"
```
```   967 apply (rule strong_replacementI)
```
```   968 apply (rule rallI)
```
```   969 apply (rename_tac B)
```
```   970 apply (rule separation_CollectI)
```
```   971 apply (insert nonempty)
```
```   972 apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
```
```   973 apply (blast intro: L_nat)
```
```   974 apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
```
```   975 apply (drule subset_Lset_ltD, assumption)
```
```   976 apply (erule reflection_imp_L_separation)
```
```   977   apply (simp_all add: lt_Ord2)
```
```   978 apply (rule DPow_LsetI)
```
```   979 apply (rename_tac v)
```
```   980 apply (rule bex_iff_sats conj_iff_sats)+
```
```   981 apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
```
```   982 apply (rule sep_rules | simp)+
```
```   983 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```   984 apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
```
```   985 done
```
```   986
```
```   987 text{*NB The proofs for type @{term formula} are virtually identical to those
```
```   988 for @{term "list(A)"}.  It was a cut-and-paste job! *}
```
```   989
```
```   990
```
```   991 subsection{*Internalized Forms of Data Structuring Operators*}
```
```   992
```
```   993 subsubsection{*The Formula @{term is_Inl}, Internalized*}
```
```   994
```
```   995 (*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
```
```   996 constdefs Inl_fm :: "[i,i]=>i"
```
```   997     "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```   998
```
```   999 lemma Inl_type [TC]:
```
```  1000      "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
```
```  1001 by (simp add: Inl_fm_def)
```
```  1002
```
```  1003 lemma sats_Inl_fm [simp]:
```
```  1004    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1005     ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
```
```  1006 by (simp add: Inl_fm_def is_Inl_def)
```
```  1007
```
```  1008 lemma Inl_iff_sats:
```
```  1009       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1010           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1011        ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
```
```  1012 by simp
```
```  1013
```
```  1014 theorem Inl_reflection:
```
```  1015      "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
```
```  1016                \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
```
```  1017 apply (simp only: is_Inl_def setclass_simps)
```
```  1018 apply (intro FOL_reflections function_reflections)
```
```  1019 done
```
```  1020
```
```  1021
```
```  1022 subsubsection{*The Formula @{term is_Inr}, Internalized*}
```
```  1023
```
```  1024 (*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
```
```  1025 constdefs Inr_fm :: "[i,i]=>i"
```
```  1026     "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1027
```
```  1028 lemma Inr_type [TC]:
```
```  1029      "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
```
```  1030 by (simp add: Inr_fm_def)
```
```  1031
```
```  1032 lemma sats_Inr_fm [simp]:
```
```  1033    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1034     ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
```
```  1035 by (simp add: Inr_fm_def is_Inr_def)
```
```  1036
```
```  1037 lemma Inr_iff_sats:
```
```  1038       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1039           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1040        ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
```
```  1041 by simp
```
```  1042
```
```  1043 theorem Inr_reflection:
```
```  1044      "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
```
```  1045                \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
```
```  1046 apply (simp only: is_Inr_def setclass_simps)
```
```  1047 apply (intro FOL_reflections function_reflections)
```
```  1048 done
```
```  1049
```
```  1050
```
```  1051 subsubsection{*The Formula @{term is_Nil}, Internalized*}
```
```  1052
```
```  1053 (* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
```
```  1054
```
```  1055 constdefs Nil_fm :: "i=>i"
```
```  1056     "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
```
```  1057
```
```  1058 lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
```
```  1059 by (simp add: Nil_fm_def)
```
```  1060
```
```  1061 lemma sats_Nil_fm [simp]:
```
```  1062    "[| x \<in> nat; env \<in> list(A)|]
```
```  1063     ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
```
```  1064 by (simp add: Nil_fm_def is_Nil_def)
```
```  1065
```
```  1066 lemma Nil_iff_sats:
```
```  1067       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1068        ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
```
```  1069 by simp
```
```  1070
```
```  1071 theorem Nil_reflection:
```
```  1072      "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
```
```  1073                \<lambda>i x. is_Nil(**Lset(i),f(x))]"
```
```  1074 apply (simp only: is_Nil_def setclass_simps)
```
```  1075 apply (intro FOL_reflections function_reflections Inl_reflection)
```
```  1076 done
```
```  1077
```
```  1078
```
```  1079 subsubsection{*The Formula @{term is_Cons}, Internalized*}
```
```  1080
```
```  1081
```
```  1082 (*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
```
```  1083 constdefs Cons_fm :: "[i,i,i]=>i"
```
```  1084     "Cons_fm(a,l,Z) ==
```
```  1085        Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
```
```  1086
```
```  1087 lemma Cons_type [TC]:
```
```  1088      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
```
```  1089 by (simp add: Cons_fm_def)
```
```  1090
```
```  1091 lemma sats_Cons_fm [simp]:
```
```  1092    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1093     ==> sats(A, Cons_fm(x,y,z), env) <->
```
```  1094        is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```  1095 by (simp add: Cons_fm_def is_Cons_def)
```
```  1096
```
```  1097 lemma Cons_iff_sats:
```
```  1098       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```  1099           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1100        ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
```
```  1101 by simp
```
```  1102
```
```  1103 theorem Cons_reflection:
```
```  1104      "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
```
```  1105                \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
```
```  1106 apply (simp only: is_Cons_def setclass_simps)
```
```  1107 apply (intro FOL_reflections pair_reflection Inr_reflection)
```
```  1108 done
```
```  1109
```
```  1110 subsubsection{*The Formula @{term is_quasilist}, Internalized*}
```
```  1111
```
```  1112 (* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
```
```  1113
```
```  1114 constdefs quasilist_fm :: "i=>i"
```
```  1115     "quasilist_fm(x) ==
```
```  1116        Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
```
```  1117
```
```  1118 lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
```
```  1119 by (simp add: quasilist_fm_def)
```
```  1120
```
```  1121 lemma sats_quasilist_fm [simp]:
```
```  1122    "[| x \<in> nat; env \<in> list(A)|]
```
```  1123     ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
```
```  1124 by (simp add: quasilist_fm_def is_quasilist_def)
```
```  1125
```
```  1126 lemma quasilist_iff_sats:
```
```  1127       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1128        ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
```
```  1129 by simp
```
```  1130
```
```  1131 theorem quasilist_reflection:
```
```  1132      "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
```
```  1133                \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
```
```  1134 apply (simp only: is_quasilist_def setclass_simps)
```
```  1135 apply (intro FOL_reflections Nil_reflection Cons_reflection)
```
```  1136 done
```
```  1137
```
```  1138
```
```  1139 subsection{*Absoluteness for the Function @{term nth}*}
```
```  1140
```
```  1141
```
```  1142 subsubsection{*The Formula @{term is_tl}, Internalized*}
```
```  1143
```
```  1144 (*     "is_tl(M,xs,T) ==
```
```  1145        (is_Nil(M,xs) --> T=xs) &
```
```  1146        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
```
```  1147        (is_quasilist(M,xs) | empty(M,T))" *)
```
```  1148 constdefs tl_fm :: "[i,i]=>i"
```
```  1149     "tl_fm(xs,T) ==
```
```  1150        And(Implies(Nil_fm(xs), Equal(T,xs)),
```
```  1151            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
```
```  1152                Or(quasilist_fm(xs), empty_fm(T))))"
```
```  1153
```
```  1154 lemma tl_type [TC]:
```
```  1155      "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
```
```  1156 by (simp add: tl_fm_def)
```
```  1157
```
```  1158 lemma sats_tl_fm [simp]:
```
```  1159    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1160     ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
```
```  1161 by (simp add: tl_fm_def is_tl_def)
```
```  1162
```
```  1163 lemma tl_iff_sats:
```
```  1164       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1165           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1166        ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
```
```  1167 by simp
```
```  1168
```
```  1169 theorem tl_reflection:
```
```  1170      "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
```
```  1171                \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
```
```  1172 apply (simp only: is_tl_def setclass_simps)
```
```  1173 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1174              quasilist_reflection empty_reflection)
```
```  1175 done
```
```  1176
```
```  1177
```
```  1178 subsubsection{*An Instance of Replacement for @{term nth}*}
```
```  1179
```
```  1180 lemma nth_replacement_Reflects:
```
```  1181  "REFLECTS
```
```  1182    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1183          is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
```
```  1184     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1185          is_wfrec(**Lset(i),
```
```  1186                   iterates_MH(**Lset(i),
```
```  1187                           is_tl(**Lset(i)), z), memsn, u, y))]"
```
```  1188 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1189           iterates_MH_reflection list_functor_reflection tl_reflection)
```
```  1190
```
```  1191 lemma nth_replacement:
```
```  1192    "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
```
```  1193 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1194 apply (rule strong_replacementI)
```
```  1195 apply (rule rallI)
```
```  1196 apply (rule separation_CollectI)
```
```  1197 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1198 apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1199 apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
```
```  1200 apply (drule subset_Lset_ltD, assumption)
```
```  1201 apply (erule reflection_imp_L_separation)
```
```  1202   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1203 apply (elim conjE)
```
```  1204 apply (rule DPow_LsetI)
```
```  1205 apply (rename_tac v)
```
```  1206 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1207 apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
```
```  1208 apply (rule sep_rules | simp)+
```
```  1209 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1210 apply (rule sep_rules quasinat_iff_sats tl_iff_sats | simp)+
```
```  1211 done
```
```  1212
```
```  1213
```
```  1214
```
```  1215 subsubsection{*Instantiating the locale @{text M_datatypes}*}
```
```  1216
```
```  1217 theorem M_datatypes_L: "PROP M_datatypes(L)"
```
```  1218   apply (rule M_datatypes.intro)
```
```  1219       apply (rule M_wfrank.axioms [OF M_wfrank_L])+
```
```  1220   apply (rule M_datatypes_axioms.intro)
```
```  1221       apply (assumption | rule
```
```  1222         list_replacement1 list_replacement2
```
```  1223         formula_replacement1 formula_replacement2
```
```  1224         nth_replacement)+
```
```  1225   done
```
```  1226
```
```  1227 lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
```
```  1228   and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
```
```  1229   and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
```
```  1230   and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
```
```  1231   and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
```
```  1232
```
```  1233 declare list_closed [intro,simp]
```
```  1234 declare formula_closed [intro,simp]
```
```  1235 declare list_abs [simp]
```
```  1236 declare formula_abs [simp]
```
```  1237 declare nth_abs [simp]
```
```  1238
```
```  1239
```
```  1240 subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
```
```  1241
```
```  1242 subsubsection{*Instances of Replacement for @{term eclose}*}
```
```  1243
```
```  1244 lemma eclose_replacement1_Reflects:
```
```  1245  "REFLECTS
```
```  1246    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1247          is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
```
```  1248     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1249          is_wfrec(**Lset(i),
```
```  1250                   iterates_MH(**Lset(i), big_union(**Lset(i)), A),
```
```  1251                   memsn, u, y))]"
```
```  1252 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1253           iterates_MH_reflection)
```
```  1254
```
```  1255 lemma eclose_replacement1:
```
```  1256    "L(A) ==> iterates_replacement(L, big_union(L), A)"
```
```  1257 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1258 apply (rule strong_replacementI)
```
```  1259 apply (rule rallI)
```
```  1260 apply (rename_tac B)
```
```  1261 apply (rule separation_CollectI)
```
```  1262 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1263 apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1264 apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
```
```  1265 apply (drule subset_Lset_ltD, assumption)
```
```  1266 apply (erule reflection_imp_L_separation)
```
```  1267   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1268 apply (elim conjE)
```
```  1269 apply (rule DPow_LsetI)
```
```  1270 apply (rename_tac v)
```
```  1271 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1272 apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
```
```  1273 apply (rule sep_rules | simp)+
```
```  1274 txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
```
```  1275 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1276 apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1277 done
```
```  1278
```
```  1279
```
```  1280 lemma eclose_replacement2_Reflects:
```
```  1281  "REFLECTS
```
```  1282    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```  1283          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```  1284            is_wfrec (L, iterates_MH (L, big_union(L), A),
```
```  1285                               msn, u, x)),
```
```  1286     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```  1287          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```  1288           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```  1289            is_wfrec (**Lset(i),
```
```  1290                  iterates_MH (**Lset(i), big_union(**Lset(i)), A),
```
```  1291                      msn, u, x))]"
```
```  1292 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1293           iterates_MH_reflection)
```
```  1294
```
```  1295
```
```  1296 lemma eclose_replacement2:
```
```  1297    "L(A) ==> strong_replacement(L,
```
```  1298          \<lambda>n y. n\<in>nat &
```
```  1299                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```  1300                is_wfrec(L, iterates_MH(L,big_union(L), A),
```
```  1301                         msn, n, y)))"
```
```  1302 apply (rule strong_replacementI)
```
```  1303 apply (rule rallI)
```
```  1304 apply (rename_tac B)
```
```  1305 apply (rule separation_CollectI)
```
```  1306 apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
```
```  1307 apply (blast intro: L_nat)
```
```  1308 apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
```
```  1309 apply (drule subset_Lset_ltD, assumption)
```
```  1310 apply (erule reflection_imp_L_separation)
```
```  1311   apply (simp_all add: lt_Ord2)
```
```  1312 apply (rule DPow_LsetI)
```
```  1313 apply (rename_tac v)
```
```  1314 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1315 apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
```
```  1316 apply (rule sep_rules | simp)+
```
```  1317 apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
```
```  1318 apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1319 done
```
```  1320
```
```  1321
```
```  1322 subsubsection{*Instantiating the locale @{text M_eclose}*}
```
```  1323
```
```  1324 theorem M_eclose_L: "PROP M_eclose(L)"
```
```  1325   apply (rule M_eclose.intro)
```
```  1326        apply (rule M_datatypes.axioms [OF M_datatypes_L])+
```
```  1327   apply (rule M_eclose_axioms.intro)
```
```  1328    apply (assumption | rule eclose_replacement1 eclose_replacement2)+
```
```  1329   done
```
```  1330
```
```  1331 lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
```
```  1332   and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
```
```  1333
```
```  1334 end
```