src/ZF/Constructible/Rec_Separation.thy
 author paulson Mon Aug 12 18:01:44 2002 +0200 (2002-08-12) changeset 13493 5aa68c051725 parent 13441 d6d620639243 child 13496 6f0c57def6d5 permissions -rw-r--r--
Lots of new results concerning recursive datatypes, towards absoluteness of
"satisfies"
```     1 (*  Title:      ZF/Constructible/Rec_Separation.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   2002  University of Cambridge
```
```     5
```
```     6 FIXME: define nth_fm and prove its "sats" theorem
```
```     7 *)
```
```     8
```
```     9 header {*Separation for Facts About Recursion*}
```
```    10
```
```    11 theory Rec_Separation = Separation + Datatype_absolute:
```
```    12
```
```    13 text{*This theory proves all instances needed for locales @{text
```
```    14 "M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*}
```
```    15
```
```    16 lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
```
```    17 by simp
```
```    18
```
```    19
```
```    20 subsection{*The Locale @{text "M_trancl"}*}
```
```    21
```
```    22 subsubsection{*Separation for Reflexive/Transitive Closure*}
```
```    23
```
```    24 text{*First, The Defining Formula*}
```
```    25
```
```    26 (* "rtran_closure_mem(M,A,r,p) ==
```
```    27       \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
```
```    28        omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
```
```    29        (\<exists>f[M]. typed_function(M,n',A,f) &
```
```    30         (\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
```
```    31           fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
```
```    32         (\<forall>j[M]. j\<in>n -->
```
```    33           (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
```
```    34             fun_apply(M,f,j,fj) & successor(M,j,sj) &
```
```    35             fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
```
```    36 constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
```
```    37  "rtran_closure_mem_fm(A,r,p) ==
```
```    38    Exists(Exists(Exists(
```
```    39     And(omega_fm(2),
```
```    40      And(Member(1,2),
```
```    41       And(succ_fm(1,0),
```
```    42        Exists(And(typed_function_fm(1, A#+4, 0),
```
```    43         And(Exists(Exists(Exists(
```
```    44               And(pair_fm(2,1,p#+7),
```
```    45                And(empty_fm(0),
```
```    46                 And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
```
```    47             Forall(Implies(Member(0,3),
```
```    48              Exists(Exists(Exists(Exists(
```
```    49               And(fun_apply_fm(5,4,3),
```
```    50                And(succ_fm(4,2),
```
```    51                 And(fun_apply_fm(5,2,1),
```
```    52                  And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
```
```    53
```
```    54
```
```    55 lemma rtran_closure_mem_type [TC]:
```
```    56  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
```
```    57 by (simp add: rtran_closure_mem_fm_def)
```
```    58
```
```    59 lemma arity_rtran_closure_mem_fm [simp]:
```
```    60      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```    61       ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```    62 by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```    63
```
```    64 lemma sats_rtran_closure_mem_fm [simp]:
```
```    65    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```    66     ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
```
```    67         rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```    68 by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
```
```    69
```
```    70 lemma rtran_closure_mem_iff_sats:
```
```    71       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```    72           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```    73        ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
```
```    74 by (simp add: sats_rtran_closure_mem_fm)
```
```    75
```
```    76 theorem rtran_closure_mem_reflection:
```
```    77      "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
```
```    78                \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
```
```    79 apply (simp only: rtran_closure_mem_def setclass_simps)
```
```    80 apply (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    81 done
```
```    82
```
```    83 text{*Separation for @{term "rtrancl(r)"}.*}
```
```    84 lemma rtrancl_separation:
```
```    85      "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
```
```    86 apply (rule separation_CollectI)
```
```    87 apply (rule_tac A="{r,A,z}" in subset_LsetE, blast )
```
```    88 apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
```
```    89 apply (drule subset_Lset_ltD, assumption)
```
```    90 apply (erule reflection_imp_L_separation)
```
```    91   apply (simp_all add: lt_Ord2)
```
```    92 apply (rule DPow_LsetI)
```
```    93 apply (rename_tac u)
```
```    94 apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
```
```    95 apply (rule sep_rules | simp)+
```
```    96 done
```
```    97
```
```    98
```
```    99 subsubsection{*Reflexive/Transitive Closure, Internalized*}
```
```   100
```
```   101 (*  "rtran_closure(M,r,s) ==
```
```   102         \<forall>A[M]. is_field(M,r,A) -->
```
```   103          (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
```
```   104 constdefs rtran_closure_fm :: "[i,i]=>i"
```
```   105  "rtran_closure_fm(r,s) ==
```
```   106    Forall(Implies(field_fm(succ(r),0),
```
```   107                   Forall(Iff(Member(0,succ(succ(s))),
```
```   108                              rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
```
```   109
```
```   110 lemma rtran_closure_type [TC]:
```
```   111      "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
```
```   112 by (simp add: rtran_closure_fm_def)
```
```   113
```
```   114 lemma arity_rtran_closure_fm [simp]:
```
```   115      "[| x \<in> nat; y \<in> nat |]
```
```   116       ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   117 by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   118
```
```   119 lemma sats_rtran_closure_fm [simp]:
```
```   120    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   121     ==> sats(A, rtran_closure_fm(x,y), env) <->
```
```   122         rtran_closure(**A, nth(x,env), nth(y,env))"
```
```   123 by (simp add: rtran_closure_fm_def rtran_closure_def)
```
```   124
```
```   125 lemma rtran_closure_iff_sats:
```
```   126       "[| nth(i,env) = x; nth(j,env) = y;
```
```   127           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   128        ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
```
```   129 by simp
```
```   130
```
```   131 theorem rtran_closure_reflection:
```
```   132      "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
```
```   133                \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
```
```   134 apply (simp only: rtran_closure_def setclass_simps)
```
```   135 apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
```
```   136 done
```
```   137
```
```   138
```
```   139 subsubsection{*Transitive Closure of a Relation, Internalized*}
```
```   140
```
```   141 (*  "tran_closure(M,r,t) ==
```
```   142          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
```
```   143 constdefs tran_closure_fm :: "[i,i]=>i"
```
```   144  "tran_closure_fm(r,s) ==
```
```   145    Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
```
```   146
```
```   147 lemma tran_closure_type [TC]:
```
```   148      "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
```
```   149 by (simp add: tran_closure_fm_def)
```
```   150
```
```   151 lemma arity_tran_closure_fm [simp]:
```
```   152      "[| x \<in> nat; y \<in> nat |]
```
```   153       ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   154 by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   155
```
```   156 lemma sats_tran_closure_fm [simp]:
```
```   157    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   158     ==> sats(A, tran_closure_fm(x,y), env) <->
```
```   159         tran_closure(**A, nth(x,env), nth(y,env))"
```
```   160 by (simp add: tran_closure_fm_def tran_closure_def)
```
```   161
```
```   162 lemma tran_closure_iff_sats:
```
```   163       "[| nth(i,env) = x; nth(j,env) = y;
```
```   164           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   165        ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
```
```   166 by simp
```
```   167
```
```   168 theorem tran_closure_reflection:
```
```   169      "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
```
```   170                \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
```
```   171 apply (simp only: tran_closure_def setclass_simps)
```
```   172 apply (intro FOL_reflections function_reflections
```
```   173              rtran_closure_reflection composition_reflection)
```
```   174 done
```
```   175
```
```   176
```
```   177 subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
```
```   178
```
```   179 lemma wellfounded_trancl_reflects:
```
```   180   "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   181                  w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
```
```   182    \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
```
```   183        w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
```
```   184        wx \<in> rp]"
```
```   185 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   186           tran_closure_reflection)
```
```   187
```
```   188
```
```   189 lemma wellfounded_trancl_separation:
```
```   190          "[| L(r); L(Z) |] ==>
```
```   191           separation (L, \<lambda>x.
```
```   192               \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   193                w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
```
```   194 apply (rule separation_CollectI)
```
```   195 apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast )
```
```   196 apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
```
```   197 apply (drule subset_Lset_ltD, assumption)
```
```   198 apply (erule reflection_imp_L_separation)
```
```   199   apply (simp_all add: lt_Ord2)
```
```   200 apply (rule DPow_LsetI)
```
```   201 apply (rename_tac u)
```
```   202 apply (rule bex_iff_sats conj_iff_sats)+
```
```   203 apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats)
```
```   204 apply (rule sep_rules tran_closure_iff_sats | simp)+
```
```   205 done
```
```   206
```
```   207
```
```   208 subsubsection{*Instantiating the locale @{text M_trancl}*}
```
```   209
```
```   210 lemma M_trancl_axioms_L: "M_trancl_axioms(L)"
```
```   211   apply (rule M_trancl_axioms.intro)
```
```   212    apply (assumption | rule rtrancl_separation wellfounded_trancl_separation)+
```
```   213   done
```
```   214
```
```   215 theorem M_trancl_L: "PROP M_trancl(L)"
```
```   216 by (rule M_trancl.intro
```
```   217          [OF M_triv_axioms_L M_axioms_axioms_L M_trancl_axioms_L])
```
```   218
```
```   219 lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
```
```   220   and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
```
```   221   and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L]
```
```   222   and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L]
```
```   223   and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L]
```
```   224   and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L]
```
```   225   and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L]
```
```   226   and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L]
```
```   227   and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L]
```
```   228   and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L]
```
```   229   and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L]
```
```   230   and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L]
```
```   231   and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L]
```
```   232
```
```   233 declare rtrancl_closed [intro,simp]
```
```   234 declare rtrancl_abs [simp]
```
```   235 declare trancl_closed [intro,simp]
```
```   236 declare trancl_abs [simp]
```
```   237
```
```   238
```
```   239 subsection{*Well-Founded Recursion!*}
```
```   240
```
```   241
```
```   242 text{*Alternative definition, minimizing nesting of quantifiers around MH*}
```
```   243 lemma M_is_recfun_iff:
```
```   244    "M_is_recfun(M,MH,r,a,f) <->
```
```   245     (\<forall>z[M]. z \<in> f <->
```
```   246      (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M].
```
```   247              MH(x, f_r_sx, y) & pair(M,x,y,z) &
```
```   248              (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M].
```
```   249                 pair(M,x,a,xa) & upair(M,x,x,sx) &
```
```   250                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
```
```   251                xa \<in> r)))"
```
```   252 apply (simp add: M_is_recfun_def)
```
```   253 apply (rule rall_cong, blast)
```
```   254 done
```
```   255
```
```   256
```
```   257 (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
```
```   258    "M_is_recfun(M,MH,r,a,f) ==
```
```   259      \<forall>z[M]. z \<in> f <->
```
```   260                2      1           0
```
```   261 new def     (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M].
```
```   262              MH(x, f_r_sx, y) & pair(M,x,y,z) &
```
```   263              (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M].
```
```   264                 pair(M,x,a,xa) & upair(M,x,x,sx) &
```
```   265                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
```
```   266                xa \<in> r)"
```
```   267 *)
```
```   268
```
```   269 text{*The three arguments of @{term p} are always 2, 1, 0 and z*}
```
```   270 constdefs is_recfun_fm :: "[i, i, i, i]=>i"
```
```   271  "is_recfun_fm(p,r,a,f) ==
```
```   272    Forall(Iff(Member(0,succ(f)),
```
```   273     Exists(Exists(Exists(
```
```   274      And(p,
```
```   275       And(pair_fm(2,0,3),
```
```   276        Exists(Exists(Exists(
```
```   277 	And(pair_fm(5,a#+7,2),
```
```   278 	 And(upair_fm(5,5,1),
```
```   279 	  And(pre_image_fm(r#+7,1,0),
```
```   280 	   And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))"
```
```   281
```
```   282 lemma is_recfun_type [TC]:
```
```   283      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   284       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   285 by (simp add: is_recfun_fm_def)
```
```   286
```
```   287
```
```   288 lemma sats_is_recfun_fm:
```
```   289   assumes MH_iff_sats:
```
```   290       "!!a0 a1 a2 a3.
```
```   291         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|]
```
```   292         ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
```
```   293   shows
```
```   294       "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   295        ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
```
```   296            M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
```
```   297 by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym])
```
```   298
```
```   299 lemma is_recfun_iff_sats:
```
```   300   assumes MH_iff_sats:
```
```   301       "!!a0 a1 a2 a3.
```
```   302         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|]
```
```   303         ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
```
```   304   shows
```
```   305   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   306       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   307    ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
```
```   308 apply (rule iff_sym)
```
```   309 apply (rule iff_trans)
```
```   310 apply (rule sats_is_recfun_fm [of A MH])
```
```   311 apply (rule MH_iff_sats, simp_all)
```
```   312 done
```
```   313 (*FIXME: surely proof can be improved?*)
```
```   314
```
```   315
```
```   316 text{*The additional variable in the premise, namely @{term f'}, is essential.
```
```   317 It lets @{term MH} depend upon @{term x}, which seems often necessary.
```
```   318 The same thing occurs in @{text is_wfrec_reflection}.*}
```
```   319 theorem is_recfun_reflection:
```
```   320   assumes MH_reflection:
```
```   321     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)),
```
```   322                      \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
```
```   323   shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)),
```
```   324              \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
```
```   325 apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
```
```   326 apply (intro FOL_reflections function_reflections
```
```   327              restriction_reflection MH_reflection)
```
```   328 done
```
```   329
```
```   330 subsubsection{*The Operator @{term is_wfrec}*}
```
```   331
```
```   332 text{*The three arguments of @{term p} are always 2, 1, 0*}
```
```   333
```
```   334 (* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o"
```
```   335     "is_wfrec(M,MH,r,a,z) ==
```
```   336       \<exists>f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *)
```
```   337 constdefs is_wfrec_fm :: "[i, i, i, i]=>i"
```
```   338  "is_wfrec_fm(p,r,a,z) ==
```
```   339     Exists(And(is_recfun_fm(p, succ(r), succ(a), 0),
```
```   340            Exists(Exists(Exists(Exists(
```
```   341              And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))"
```
```   342
```
```   343 text{*We call @{term p} with arguments a, f, z by equating them with
```
```   344   the corresponding quantified variables with de Bruijn indices 2, 1, 0.*}
```
```   345
```
```   346 text{*There's an additional existential quantifier to ensure that the
```
```   347       environments in both calls to MH have the same length.*}
```
```   348
```
```   349 lemma is_wfrec_type [TC]:
```
```   350      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   351       ==> is_wfrec_fm(p,x,y,z) \<in> formula"
```
```   352 by (simp add: is_wfrec_fm_def)
```
```   353
```
```   354 lemma sats_is_wfrec_fm:
```
```   355   assumes MH_iff_sats:
```
```   356       "!!a0 a1 a2 a3 a4.
```
```   357         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|]
```
```   358         ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
```
```   359   shows
```
```   360       "[|x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
```
```   361        ==> sats(A, is_wfrec_fm(p,x,y,z), env) <->
```
```   362            is_wfrec(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
```
```   363 apply (frule_tac x=z in lt_length_in_nat, assumption)
```
```   364 apply (frule lt_length_in_nat, assumption)
```
```   365 apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast)
```
```   366 done
```
```   367
```
```   368
```
```   369 lemma is_wfrec_iff_sats:
```
```   370   assumes MH_iff_sats:
```
```   371       "!!a0 a1 a2 a3 a4.
```
```   372         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|]
```
```   373         ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
```
```   374   shows
```
```   375   "[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   376       i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
```
```   377    ==> is_wfrec(**A, MH, x, y, z) <-> sats(A, is_wfrec_fm(p,i,j,k), env)"
```
```   378 by (simp add: sats_is_wfrec_fm [OF MH_iff_sats])
```
```   379
```
```   380 theorem is_wfrec_reflection:
```
```   381   assumes MH_reflection:
```
```   382     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)),
```
```   383                      \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
```
```   384   shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L,x), f(x), g(x), h(x)),
```
```   385                \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
```
```   386 apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
```
```   387 apply (intro FOL_reflections MH_reflection is_recfun_reflection)
```
```   388 done
```
```   389
```
```   390 subsection{*The Locale @{text "M_wfrank"}*}
```
```   391
```
```   392 subsubsection{*Separation for @{term "wfrank"}*}
```
```   393
```
```   394 lemma wfrank_Reflects:
```
```   395  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   396               ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
```
```   397       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   398          ~ (\<exists>f \<in> Lset(i).
```
```   399             M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
```
```   400                         rplus, x, f))]"
```
```   401 by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
```
```   402
```
```   403 lemma wfrank_separation:
```
```   404      "L(r) ==>
```
```   405       separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   406          ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
```
```   407 apply (rule separation_CollectI)
```
```   408 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   409 apply (rule ReflectsE [OF wfrank_Reflects], assumption)
```
```   410 apply (drule subset_Lset_ltD, assumption)
```
```   411 apply (erule reflection_imp_L_separation)
```
```   412   apply (simp_all add: lt_Ord2, clarify)
```
```   413 apply (rule DPow_LsetI)
```
```   414 apply (rename_tac u)
```
```   415 apply (rule ball_iff_sats imp_iff_sats)+
```
```   416 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   417 apply (rule sep_rules | simp)+
```
```   418 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   419 done
```
```   420
```
```   421
```
```   422 subsubsection{*Replacement for @{term "wfrank"}*}
```
```   423
```
```   424 lemma wfrank_replacement_Reflects:
```
```   425  "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
```
```   426         (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   427          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   428                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   429                         is_range(L,f,y))),
```
```   430  \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
```
```   431       (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   432        (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
```
```   433          M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
```
```   434          is_range(**Lset(i),f,y)))]"
```
```   435 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   436              is_recfun_reflection tran_closure_reflection)
```
```   437
```
```   438
```
```   439 lemma wfrank_strong_replacement:
```
```   440      "L(r) ==>
```
```   441       strong_replacement(L, \<lambda>x z.
```
```   442          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   443          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   444                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   445                         is_range(L,f,y)))"
```
```   446 apply (rule strong_replacementI)
```
```   447 apply (rule rallI)
```
```   448 apply (rename_tac B)
```
```   449 apply (rule separation_CollectI)
```
```   450 apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
```
```   451 apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
```
```   452 apply (drule subset_Lset_ltD, assumption)
```
```   453 apply (erule reflection_imp_L_separation)
```
```   454   apply (simp_all add: lt_Ord2)
```
```   455 apply (rule DPow_LsetI)
```
```   456 apply (rename_tac u)
```
```   457 apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
```
```   458 apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
```
```   459 apply (rule sep_rules list.intros app_type tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   460 done
```
```   461
```
```   462
```
```   463 subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
```
```   464
```
```   465 lemma Ord_wfrank_Reflects:
```
```   466  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   467           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   468              is_range(L,f,rangef) -->
```
```   469              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   470              ordinal(L,rangef)),
```
```   471       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   472           ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
```
```   473              is_range(**Lset(i),f,rangef) -->
```
```   474              M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
```
```   475                          rplus, x, f) -->
```
```   476              ordinal(**Lset(i),rangef))]"
```
```   477 by (intro FOL_reflections function_reflections is_recfun_reflection
```
```   478           tran_closure_reflection ordinal_reflection)
```
```   479
```
```   480 lemma  Ord_wfrank_separation:
```
```   481      "L(r) ==>
```
```   482       separation (L, \<lambda>x.
```
```   483          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   484           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   485              is_range(L,f,rangef) -->
```
```   486              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   487              ordinal(L,rangef)))"
```
```   488 apply (rule separation_CollectI)
```
```   489 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   490 apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
```
```   491 apply (drule subset_Lset_ltD, assumption)
```
```   492 apply (erule reflection_imp_L_separation)
```
```   493   apply (simp_all add: lt_Ord2, clarify)
```
```   494 apply (rule DPow_LsetI)
```
```   495 apply (rename_tac u)
```
```   496 apply (rule ball_iff_sats imp_iff_sats)+
```
```   497 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   498 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   499 done
```
```   500
```
```   501
```
```   502 subsubsection{*Instantiating the locale @{text M_wfrank}*}
```
```   503
```
```   504 lemma M_wfrank_axioms_L: "M_wfrank_axioms(L)"
```
```   505   apply (rule M_wfrank_axioms.intro)
```
```   506    apply (assumption | rule
```
```   507      wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
```
```   508   done
```
```   509
```
```   510 theorem M_wfrank_L: "PROP M_wfrank(L)"
```
```   511   apply (rule M_wfrank.intro)
```
```   512      apply (rule M_trancl.axioms [OF M_trancl_L])+
```
```   513   apply (rule M_wfrank_axioms_L)
```
```   514   done
```
```   515
```
```   516 lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L]
```
```   517   and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
```
```   518   and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
```
```   519   and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
```
```   520   and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
```
```   521   and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
```
```   522   and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
```
```   523   and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
```
```   524   and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
```
```   525   and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
```
```   526   and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
```
```   527   and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
```
```   528   and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
```
```   529   and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
```
```   530   and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
```
```   531   and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
```
```   532   and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L]
```
```   533   and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L]
```
```   534   and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L]
```
```   535
```
```   536 declare iterates_closed [intro,simp]
```
```   537 declare Ord_wfrank_range [rule_format]
```
```   538 declare wf_abs [simp]
```
```   539 declare wf_on_abs [simp]
```
```   540
```
```   541
```
```   542 subsection{*For Datatypes*}
```
```   543
```
```   544 subsubsection{*Binary Products, Internalized*}
```
```   545
```
```   546 constdefs cartprod_fm :: "[i,i,i]=>i"
```
```   547 (* "cartprod(M,A,B,z) ==
```
```   548         \<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
```
```   549     "cartprod_fm(A,B,z) ==
```
```   550        Forall(Iff(Member(0,succ(z)),
```
```   551                   Exists(And(Member(0,succ(succ(A))),
```
```   552                          Exists(And(Member(0,succ(succ(succ(B)))),
```
```   553                                     pair_fm(1,0,2)))))))"
```
```   554
```
```   555 lemma cartprod_type [TC]:
```
```   556      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
```
```   557 by (simp add: cartprod_fm_def)
```
```   558
```
```   559 lemma arity_cartprod_fm [simp]:
```
```   560      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   561       ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   562 by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   563
```
```   564 lemma sats_cartprod_fm [simp]:
```
```   565    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   566     ==> sats(A, cartprod_fm(x,y,z), env) <->
```
```   567         cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   568 by (simp add: cartprod_fm_def cartprod_def)
```
```   569
```
```   570 lemma cartprod_iff_sats:
```
```   571       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   572           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   573        ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
```
```   574 by (simp add: sats_cartprod_fm)
```
```   575
```
```   576 theorem cartprod_reflection:
```
```   577      "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
```
```   578                \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
```
```   579 apply (simp only: cartprod_def setclass_simps)
```
```   580 apply (intro FOL_reflections pair_reflection)
```
```   581 done
```
```   582
```
```   583
```
```   584 subsubsection{*Binary Sums, Internalized*}
```
```   585
```
```   586 (* "is_sum(M,A,B,Z) ==
```
```   587        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
```
```   588          3      2       1        0
```
```   589        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
```
```   590        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
```
```   591 constdefs sum_fm :: "[i,i,i]=>i"
```
```   592     "sum_fm(A,B,Z) ==
```
```   593        Exists(Exists(Exists(Exists(
```
```   594         And(number1_fm(2),
```
```   595             And(cartprod_fm(2,A#+4,3),
```
```   596                 And(upair_fm(2,2,1),
```
```   597                     And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
```
```   598
```
```   599 lemma sum_type [TC]:
```
```   600      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
```
```   601 by (simp add: sum_fm_def)
```
```   602
```
```   603 lemma arity_sum_fm [simp]:
```
```   604      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   605       ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   606 by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   607
```
```   608 lemma sats_sum_fm [simp]:
```
```   609    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   610     ==> sats(A, sum_fm(x,y,z), env) <->
```
```   611         is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   612 by (simp add: sum_fm_def is_sum_def)
```
```   613
```
```   614 lemma sum_iff_sats:
```
```   615       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   616           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   617        ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
```
```   618 by simp
```
```   619
```
```   620 theorem sum_reflection:
```
```   621      "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
```
```   622                \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
```
```   623 apply (simp only: is_sum_def setclass_simps)
```
```   624 apply (intro FOL_reflections function_reflections cartprod_reflection)
```
```   625 done
```
```   626
```
```   627
```
```   628 subsubsection{*The Operator @{term quasinat}*}
```
```   629
```
```   630 (* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
```
```   631 constdefs quasinat_fm :: "i=>i"
```
```   632     "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
```
```   633
```
```   634 lemma quasinat_type [TC]:
```
```   635      "x \<in> nat ==> quasinat_fm(x) \<in> formula"
```
```   636 by (simp add: quasinat_fm_def)
```
```   637
```
```   638 lemma arity_quasinat_fm [simp]:
```
```   639      "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
```
```   640 by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   641
```
```   642 lemma sats_quasinat_fm [simp]:
```
```   643    "[| x \<in> nat; env \<in> list(A)|]
```
```   644     ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
```
```   645 by (simp add: quasinat_fm_def is_quasinat_def)
```
```   646
```
```   647 lemma quasinat_iff_sats:
```
```   648       "[| nth(i,env) = x; nth(j,env) = y;
```
```   649           i \<in> nat; env \<in> list(A)|]
```
```   650        ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
```
```   651 by simp
```
```   652
```
```   653 theorem quasinat_reflection:
```
```   654      "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
```
```   655                \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
```
```   656 apply (simp only: is_quasinat_def setclass_simps)
```
```   657 apply (intro FOL_reflections function_reflections)
```
```   658 done
```
```   659
```
```   660
```
```   661 subsubsection{*The Operator @{term is_nat_case}*}
```
```   662 text{*I could not get it to work with the more natural assumption that
```
```   663  @{term is_b} takes two arguments.  Instead it must be a formula where 1 and 0
```
```   664  stand for @{term m} and @{term b}, respectively.*}
```
```   665
```
```   666 (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
```
```   667     "is_nat_case(M, a, is_b, k, z) ==
```
```   668        (empty(M,k) --> z=a) &
```
```   669        (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
```
```   670        (is_quasinat(M,k) | empty(M,z))" *)
```
```   671 text{*The formula @{term is_b} has free variables 1 and 0.*}
```
```   672 constdefs is_nat_case_fm :: "[i, i, i, i]=>i"
```
```   673  "is_nat_case_fm(a,is_b,k,z) ==
```
```   674     And(Implies(empty_fm(k), Equal(z,a)),
```
```   675         And(Forall(Implies(succ_fm(0,succ(k)),
```
```   676                    Forall(Implies(Equal(0,succ(succ(z))), is_b)))),
```
```   677             Or(quasinat_fm(k), empty_fm(z))))"
```
```   678
```
```   679 lemma is_nat_case_type [TC]:
```
```   680      "[| is_b \<in> formula;
```
```   681          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   682       ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
```
```   683 by (simp add: is_nat_case_fm_def)
```
```   684
```
```   685 lemma sats_is_nat_case_fm:
```
```   686   assumes is_b_iff_sats:
```
```   687       "!!a. a \<in> A ==> is_b(a,nth(z, env)) <->
```
```   688                       sats(A, p, Cons(nth(z,env), Cons(a, env)))"
```
```   689   shows
```
```   690       "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   691        ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
```
```   692            is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
```
```   693 apply (frule lt_length_in_nat, assumption)
```
```   694 apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
```
```   695 done
```
```   696
```
```   697 lemma is_nat_case_iff_sats:
```
```   698   "[| (!!a. a \<in> A ==> is_b(a,z) <->
```
```   699                       sats(A, p, Cons(z, Cons(a,env))));
```
```   700       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   701       i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   702    ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
```
```   703 by (simp add: sats_is_nat_case_fm [of A is_b])
```
```   704
```
```   705
```
```   706 text{*The second argument of @{term is_b} gives it direct access to @{term x},
```
```   707   which is essential for handling free variable references.  Without this
```
```   708   argument, we cannot prove reflection for @{term iterates_MH}.*}
```
```   709 theorem is_nat_case_reflection:
```
```   710   assumes is_b_reflection:
```
```   711     "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
```
```   712                      \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
```
```   713   shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
```
```   714                \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
```
```   715 apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
```
```   716 apply (intro FOL_reflections function_reflections
```
```   717              restriction_reflection is_b_reflection quasinat_reflection)
```
```   718 done
```
```   719
```
```   720
```
```   721
```
```   722 subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
```
```   723
```
```   724 (*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   725    "iterates_MH(M,isF,v,n,g,z) ==
```
```   726         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   727                     n, z)" *)
```
```   728 constdefs iterates_MH_fm :: "[i, i, i, i, i]=>i"
```
```   729  "iterates_MH_fm(isF,v,n,g,z) ==
```
```   730     is_nat_case_fm(v,
```
```   731       Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0),
```
```   732                      Forall(Implies(Equal(0,2), isF)))),
```
```   733       n, z)"
```
```   734
```
```   735 lemma iterates_MH_type [TC]:
```
```   736      "[| p \<in> formula;
```
```   737          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   738       ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
```
```   739 by (simp add: iterates_MH_fm_def)
```
```   740
```
```   741 lemma sats_iterates_MH_fm:
```
```   742   assumes is_F_iff_sats:
```
```   743       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   744               ==> is_F(a,b) <->
```
```   745                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
```
```   746   shows
```
```   747       "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   748        ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
```
```   749            iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
```
```   750 apply (frule lt_length_in_nat, assumption)
```
```   751 apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm
```
```   752               is_F_iff_sats [symmetric])
```
```   753 apply (rule is_nat_case_cong)
```
```   754 apply (simp_all add: setclass_def)
```
```   755 done
```
```   756
```
```   757
```
```   758 lemma iterates_MH_iff_sats:
```
```   759   "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   760               ==> is_F(a,b) <->
```
```   761                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env))))));
```
```   762       nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   763       i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   764    ==> iterates_MH(**A, is_F, v, x, y, z) <->
```
```   765        sats(A, iterates_MH_fm(p,i',i,j,k), env)"
```
```   766 apply (rule iff_sym)
```
```   767 apply (rule iff_trans)
```
```   768 apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all)
```
```   769 done
```
```   770 (*FIXME: surely proof can be improved?*)
```
```   771
```
```   772
```
```   773 theorem iterates_MH_reflection:
```
```   774   assumes p_reflection:
```
```   775     "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
```
```   776                      \<lambda>i x. p(**Lset(i), f(x), g(x))]"
```
```   777  shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
```
```   778                \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
```
```   779 apply (simp (no_asm_use) only: iterates_MH_def)
```
```   780 txt{*Must be careful: simplifying with @{text setclass_simps} above would
```
```   781      change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
```
```   782      it would no longer match rule @{text is_nat_case_reflection}. *}
```
```   783 apply (rule is_nat_case_reflection)
```
```   784 apply (simp (no_asm_use) only: setclass_simps)
```
```   785 apply (intro FOL_reflections function_reflections is_nat_case_reflection
```
```   786              restriction_reflection p_reflection)
```
```   787 done
```
```   788
```
```   789
```
```   790
```
```   791 subsection{*@{term L} is Closed Under the Operator @{term list}*}
```
```   792
```
```   793 subsubsection{*The List Functor, Internalized*}
```
```   794
```
```   795 constdefs list_functor_fm :: "[i,i,i]=>i"
```
```   796 (* "is_list_functor(M,A,X,Z) ==
```
```   797         \<exists>n1[M]. \<exists>AX[M].
```
```   798          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
```
```   799     "list_functor_fm(A,X,Z) ==
```
```   800        Exists(Exists(
```
```   801         And(number1_fm(1),
```
```   802             And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
```
```   803
```
```   804 lemma list_functor_type [TC]:
```
```   805      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
```
```   806 by (simp add: list_functor_fm_def)
```
```   807
```
```   808 lemma arity_list_functor_fm [simp]:
```
```   809      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   810       ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   811 by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   812
```
```   813 lemma sats_list_functor_fm [simp]:
```
```   814    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   815     ==> sats(A, list_functor_fm(x,y,z), env) <->
```
```   816         is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   817 by (simp add: list_functor_fm_def is_list_functor_def)
```
```   818
```
```   819 lemma list_functor_iff_sats:
```
```   820   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   821       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   822    ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
```
```   823 by simp
```
```   824
```
```   825 theorem list_functor_reflection:
```
```   826      "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
```
```   827                \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
```
```   828 apply (simp only: is_list_functor_def setclass_simps)
```
```   829 apply (intro FOL_reflections number1_reflection
```
```   830              cartprod_reflection sum_reflection)
```
```   831 done
```
```   832
```
```   833
```
```   834 subsubsection{*Instances of Replacement for Lists*}
```
```   835
```
```   836 lemma list_replacement1_Reflects:
```
```   837  "REFLECTS
```
```   838    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   839          is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
```
```   840     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   841          is_wfrec(**Lset(i),
```
```   842                   iterates_MH(**Lset(i),
```
```   843                           is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
```
```   844 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   845           iterates_MH_reflection list_functor_reflection)
```
```   846
```
```   847
```
```   848 lemma list_replacement1:
```
```   849    "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
```
```   850 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   851 apply (rule strong_replacementI)
```
```   852 apply (rule rallI)
```
```   853 apply (rename_tac B)
```
```   854 apply (rule separation_CollectI)
```
```   855 apply (insert nonempty)
```
```   856 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   857 apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   858 apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
```
```   859 apply (drule subset_Lset_ltD, assumption)
```
```   860 apply (erule reflection_imp_L_separation)
```
```   861   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   862 apply (elim conjE)
```
```   863 apply (rule DPow_LsetI)
```
```   864 apply (rename_tac v)
```
```   865 apply (rule bex_iff_sats conj_iff_sats)+
```
```   866 apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   867 apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
```
```   868             is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   869 done
```
```   870
```
```   871
```
```   872 lemma list_replacement2_Reflects:
```
```   873  "REFLECTS
```
```   874    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   875          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   876            is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
```
```   877                               msn, u, x)),
```
```   878     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   879          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   880           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   881            is_wfrec (**Lset(i),
```
```   882                  iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
```
```   883                      msn, u, x))]"
```
```   884 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   885           iterates_MH_reflection list_functor_reflection)
```
```   886
```
```   887
```
```   888 lemma list_replacement2:
```
```   889    "L(A) ==> strong_replacement(L,
```
```   890          \<lambda>n y. n\<in>nat &
```
```   891                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   892                is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
```
```   893                         msn, n, y)))"
```
```   894 apply (rule strong_replacementI)
```
```   895 apply (rule rallI)
```
```   896 apply (rename_tac B)
```
```   897 apply (rule separation_CollectI)
```
```   898 apply (insert nonempty)
```
```   899 apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
```
```   900 apply (blast intro: L_nat)
```
```   901 apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
```
```   902 apply (drule subset_Lset_ltD, assumption)
```
```   903 apply (erule reflection_imp_L_separation)
```
```   904   apply (simp_all add: lt_Ord2)
```
```   905 apply (rule DPow_LsetI)
```
```   906 apply (rename_tac v)
```
```   907 apply (rule bex_iff_sats conj_iff_sats)+
```
```   908 apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
```
```   909 apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
```
```   910             is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   911 done
```
```   912
```
```   913
```
```   914 subsection{*@{term L} is Closed Under the Operator @{term formula}*}
```
```   915
```
```   916 subsubsection{*The Formula Functor, Internalized*}
```
```   917
```
```   918 constdefs formula_functor_fm :: "[i,i]=>i"
```
```   919 (*     "is_formula_functor(M,X,Z) ==
```
```   920         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
```
```   921            4           3               2       1       0
```
```   922           omega(M,nat') & cartprod(M,nat',nat',natnat) &
```
```   923           is_sum(M,natnat,natnat,natnatsum) &
```
```   924           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
```
```   925           is_sum(M,natnatsum,X3,Z)" *)
```
```   926     "formula_functor_fm(X,Z) ==
```
```   927        Exists(Exists(Exists(Exists(Exists(
```
```   928         And(omega_fm(4),
```
```   929          And(cartprod_fm(4,4,3),
```
```   930           And(sum_fm(3,3,2),
```
```   931            And(cartprod_fm(X#+5,X#+5,1),
```
```   932             And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
```
```   933
```
```   934 lemma formula_functor_type [TC]:
```
```   935      "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
```
```   936 by (simp add: formula_functor_fm_def)
```
```   937
```
```   938 lemma sats_formula_functor_fm [simp]:
```
```   939    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   940     ==> sats(A, formula_functor_fm(x,y), env) <->
```
```   941         is_formula_functor(**A, nth(x,env), nth(y,env))"
```
```   942 by (simp add: formula_functor_fm_def is_formula_functor_def)
```
```   943
```
```   944 lemma formula_functor_iff_sats:
```
```   945   "[| nth(i,env) = x; nth(j,env) = y;
```
```   946       i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   947    ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
```
```   948 by simp
```
```   949
```
```   950 theorem formula_functor_reflection:
```
```   951      "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
```
```   952                \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
```
```   953 apply (simp only: is_formula_functor_def setclass_simps)
```
```   954 apply (intro FOL_reflections omega_reflection
```
```   955              cartprod_reflection sum_reflection)
```
```   956 done
```
```   957
```
```   958 subsubsection{*Instances of Replacement for Formulas*}
```
```   959
```
```   960 lemma formula_replacement1_Reflects:
```
```   961  "REFLECTS
```
```   962    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   963          is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
```
```   964     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   965          is_wfrec(**Lset(i),
```
```   966                   iterates_MH(**Lset(i),
```
```   967                           is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
```
```   968 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   969           iterates_MH_reflection formula_functor_reflection)
```
```   970
```
```   971 lemma formula_replacement1:
```
```   972    "iterates_replacement(L, is_formula_functor(L), 0)"
```
```   973 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   974 apply (rule strong_replacementI)
```
```   975 apply (rule rallI)
```
```   976 apply (rename_tac B)
```
```   977 apply (rule separation_CollectI)
```
```   978 apply (insert nonempty)
```
```   979 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   980 apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   981 apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
```
```   982 apply (drule subset_Lset_ltD, assumption)
```
```   983 apply (erule reflection_imp_L_separation)
```
```   984   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   985 apply (rule DPow_LsetI)
```
```   986 apply (rename_tac v)
```
```   987 apply (rule bex_iff_sats conj_iff_sats)+
```
```   988 apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   989 apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
```
```   990             is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   991 done
```
```   992
```
```   993 lemma formula_replacement2_Reflects:
```
```   994  "REFLECTS
```
```   995    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   996          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   997            is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
```
```   998                               msn, u, x)),
```
```   999     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```  1000          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```  1001           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```  1002            is_wfrec (**Lset(i),
```
```  1003                  iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
```
```  1004                      msn, u, x))]"
```
```  1005 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1006           iterates_MH_reflection formula_functor_reflection)
```
```  1007
```
```  1008
```
```  1009 lemma formula_replacement2:
```
```  1010    "strong_replacement(L,
```
```  1011          \<lambda>n y. n\<in>nat &
```
```  1012                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```  1013                is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
```
```  1014                         msn, n, y)))"
```
```  1015 apply (rule strong_replacementI)
```
```  1016 apply (rule rallI)
```
```  1017 apply (rename_tac B)
```
```  1018 apply (rule separation_CollectI)
```
```  1019 apply (insert nonempty)
```
```  1020 apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
```
```  1021 apply (blast intro: L_nat)
```
```  1022 apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
```
```  1023 apply (drule subset_Lset_ltD, assumption)
```
```  1024 apply (erule reflection_imp_L_separation)
```
```  1025   apply (simp_all add: lt_Ord2)
```
```  1026 apply (rule DPow_LsetI)
```
```  1027 apply (rename_tac v)
```
```  1028 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1029 apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
```
```  1030 apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
```
```  1031             is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```  1032 done
```
```  1033
```
```  1034 text{*NB The proofs for type @{term formula} are virtually identical to those
```
```  1035 for @{term "list(A)"}.  It was a cut-and-paste job! *}
```
```  1036
```
```  1037
```
```  1038 subsection{*Internalized Forms of Data Structuring Operators*}
```
```  1039
```
```  1040 subsubsection{*The Formula @{term is_Inl}, Internalized*}
```
```  1041
```
```  1042 (*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
```
```  1043 constdefs Inl_fm :: "[i,i]=>i"
```
```  1044     "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1045
```
```  1046 lemma Inl_type [TC]:
```
```  1047      "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
```
```  1048 by (simp add: Inl_fm_def)
```
```  1049
```
```  1050 lemma sats_Inl_fm [simp]:
```
```  1051    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1052     ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
```
```  1053 by (simp add: Inl_fm_def is_Inl_def)
```
```  1054
```
```  1055 lemma Inl_iff_sats:
```
```  1056       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1057           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1058        ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
```
```  1059 by simp
```
```  1060
```
```  1061 theorem Inl_reflection:
```
```  1062      "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
```
```  1063                \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
```
```  1064 apply (simp only: is_Inl_def setclass_simps)
```
```  1065 apply (intro FOL_reflections function_reflections)
```
```  1066 done
```
```  1067
```
```  1068
```
```  1069 subsubsection{*The Formula @{term is_Inr}, Internalized*}
```
```  1070
```
```  1071 (*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
```
```  1072 constdefs Inr_fm :: "[i,i]=>i"
```
```  1073     "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1074
```
```  1075 lemma Inr_type [TC]:
```
```  1076      "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
```
```  1077 by (simp add: Inr_fm_def)
```
```  1078
```
```  1079 lemma sats_Inr_fm [simp]:
```
```  1080    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1081     ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
```
```  1082 by (simp add: Inr_fm_def is_Inr_def)
```
```  1083
```
```  1084 lemma Inr_iff_sats:
```
```  1085       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1086           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1087        ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
```
```  1088 by simp
```
```  1089
```
```  1090 theorem Inr_reflection:
```
```  1091      "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
```
```  1092                \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
```
```  1093 apply (simp only: is_Inr_def setclass_simps)
```
```  1094 apply (intro FOL_reflections function_reflections)
```
```  1095 done
```
```  1096
```
```  1097
```
```  1098 subsubsection{*The Formula @{term is_Nil}, Internalized*}
```
```  1099
```
```  1100 (* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
```
```  1101
```
```  1102 constdefs Nil_fm :: "i=>i"
```
```  1103     "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
```
```  1104
```
```  1105 lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
```
```  1106 by (simp add: Nil_fm_def)
```
```  1107
```
```  1108 lemma sats_Nil_fm [simp]:
```
```  1109    "[| x \<in> nat; env \<in> list(A)|]
```
```  1110     ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
```
```  1111 by (simp add: Nil_fm_def is_Nil_def)
```
```  1112
```
```  1113 lemma Nil_iff_sats:
```
```  1114       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1115        ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
```
```  1116 by simp
```
```  1117
```
```  1118 theorem Nil_reflection:
```
```  1119      "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
```
```  1120                \<lambda>i x. is_Nil(**Lset(i),f(x))]"
```
```  1121 apply (simp only: is_Nil_def setclass_simps)
```
```  1122 apply (intro FOL_reflections function_reflections Inl_reflection)
```
```  1123 done
```
```  1124
```
```  1125
```
```  1126 subsubsection{*The Formula @{term is_Cons}, Internalized*}
```
```  1127
```
```  1128
```
```  1129 (*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
```
```  1130 constdefs Cons_fm :: "[i,i,i]=>i"
```
```  1131     "Cons_fm(a,l,Z) ==
```
```  1132        Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
```
```  1133
```
```  1134 lemma Cons_type [TC]:
```
```  1135      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
```
```  1136 by (simp add: Cons_fm_def)
```
```  1137
```
```  1138 lemma sats_Cons_fm [simp]:
```
```  1139    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1140     ==> sats(A, Cons_fm(x,y,z), env) <->
```
```  1141        is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```  1142 by (simp add: Cons_fm_def is_Cons_def)
```
```  1143
```
```  1144 lemma Cons_iff_sats:
```
```  1145       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```  1146           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1147        ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
```
```  1148 by simp
```
```  1149
```
```  1150 theorem Cons_reflection:
```
```  1151      "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
```
```  1152                \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
```
```  1153 apply (simp only: is_Cons_def setclass_simps)
```
```  1154 apply (intro FOL_reflections pair_reflection Inr_reflection)
```
```  1155 done
```
```  1156
```
```  1157 subsubsection{*The Formula @{term is_quasilist}, Internalized*}
```
```  1158
```
```  1159 (* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
```
```  1160
```
```  1161 constdefs quasilist_fm :: "i=>i"
```
```  1162     "quasilist_fm(x) ==
```
```  1163        Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
```
```  1164
```
```  1165 lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
```
```  1166 by (simp add: quasilist_fm_def)
```
```  1167
```
```  1168 lemma sats_quasilist_fm [simp]:
```
```  1169    "[| x \<in> nat; env \<in> list(A)|]
```
```  1170     ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
```
```  1171 by (simp add: quasilist_fm_def is_quasilist_def)
```
```  1172
```
```  1173 lemma quasilist_iff_sats:
```
```  1174       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1175        ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
```
```  1176 by simp
```
```  1177
```
```  1178 theorem quasilist_reflection:
```
```  1179      "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
```
```  1180                \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
```
```  1181 apply (simp only: is_quasilist_def setclass_simps)
```
```  1182 apply (intro FOL_reflections Nil_reflection Cons_reflection)
```
```  1183 done
```
```  1184
```
```  1185
```
```  1186 subsection{*Absoluteness for the Function @{term nth}*}
```
```  1187
```
```  1188
```
```  1189 subsubsection{*The Formula @{term is_hd}, Internalized*}
```
```  1190
```
```  1191 (*   "is_hd(M,xs,H) ==
```
```  1192        (is_Nil(M,xs) --> empty(M,H)) &
```
```  1193        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
```
```  1194        (is_quasilist(M,xs) | empty(M,H))" *)
```
```  1195 constdefs hd_fm :: "[i,i]=>i"
```
```  1196     "hd_fm(xs,H) ==
```
```  1197        And(Implies(Nil_fm(xs), empty_fm(H)),
```
```  1198            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))),
```
```  1199                Or(quasilist_fm(xs), empty_fm(H))))"
```
```  1200
```
```  1201 lemma hd_type [TC]:
```
```  1202      "[| x \<in> nat; y \<in> nat |] ==> hd_fm(x,y) \<in> formula"
```
```  1203 by (simp add: hd_fm_def)
```
```  1204
```
```  1205 lemma sats_hd_fm [simp]:
```
```  1206    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1207     ==> sats(A, hd_fm(x,y), env) <-> is_hd(**A, nth(x,env), nth(y,env))"
```
```  1208 by (simp add: hd_fm_def is_hd_def)
```
```  1209
```
```  1210 lemma hd_iff_sats:
```
```  1211       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1212           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1213        ==> is_hd(**A, x, y) <-> sats(A, hd_fm(i,j), env)"
```
```  1214 by simp
```
```  1215
```
```  1216 theorem hd_reflection:
```
```  1217      "REFLECTS[\<lambda>x. is_hd(L,f(x),g(x)),
```
```  1218                \<lambda>i x. is_hd(**Lset(i),f(x),g(x))]"
```
```  1219 apply (simp only: is_hd_def setclass_simps)
```
```  1220 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1221              quasilist_reflection empty_reflection)
```
```  1222 done
```
```  1223
```
```  1224
```
```  1225 subsubsection{*The Formula @{term is_tl}, Internalized*}
```
```  1226
```
```  1227 (*     "is_tl(M,xs,T) ==
```
```  1228        (is_Nil(M,xs) --> T=xs) &
```
```  1229        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
```
```  1230        (is_quasilist(M,xs) | empty(M,T))" *)
```
```  1231 constdefs tl_fm :: "[i,i]=>i"
```
```  1232     "tl_fm(xs,T) ==
```
```  1233        And(Implies(Nil_fm(xs), Equal(T,xs)),
```
```  1234            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
```
```  1235                Or(quasilist_fm(xs), empty_fm(T))))"
```
```  1236
```
```  1237 lemma tl_type [TC]:
```
```  1238      "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
```
```  1239 by (simp add: tl_fm_def)
```
```  1240
```
```  1241 lemma sats_tl_fm [simp]:
```
```  1242    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1243     ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
```
```  1244 by (simp add: tl_fm_def is_tl_def)
```
```  1245
```
```  1246 lemma tl_iff_sats:
```
```  1247       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1248           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1249        ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
```
```  1250 by simp
```
```  1251
```
```  1252 theorem tl_reflection:
```
```  1253      "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
```
```  1254                \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
```
```  1255 apply (simp only: is_tl_def setclass_simps)
```
```  1256 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1257              quasilist_reflection empty_reflection)
```
```  1258 done
```
```  1259
```
```  1260
```
```  1261 subsubsection{*The Formula @{term is_nth}, Internalized*}
```
```  1262
```
```  1263 (* "is_nth(M,n,l,Z) ==
```
```  1264       \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M].
```
```  1265        2       1       0
```
```  1266        successor(M,n,sn) & membership(M,sn,msn) &
```
```  1267        is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
```
```  1268        is_hd(M,X,Z)" *)
```
```  1269 constdefs nth_fm :: "[i,i,i]=>i"
```
```  1270     "nth_fm(n,l,Z) ==
```
```  1271        Exists(Exists(Exists(
```
```  1272          And(succ_fm(n#+3,1),
```
```  1273           And(Memrel_fm(1,0),
```
```  1274            And(is_wfrec_fm(iterates_MH_fm(tl_fm(1,0),l#+8,2,1,0), 0, n#+3, 2), hd_fm(2,Z#+3)))))))"
```
```  1275
```
```  1276 lemma nth_fm_type [TC]:
```
```  1277  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> nth_fm(x,y,z) \<in> formula"
```
```  1278 by (simp add: nth_fm_def)
```
```  1279
```
```  1280 lemma sats_nth_fm [simp]:
```
```  1281    "[| x < length(env); y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1282     ==> sats(A, nth_fm(x,y,z), env) <->
```
```  1283         is_nth(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```  1284 apply (frule lt_length_in_nat, assumption)
```
```  1285 apply (simp add: nth_fm_def is_nth_def sats_is_wfrec_fm sats_iterates_MH_fm)
```
```  1286 done
```
```  1287
```
```  1288 lemma nth_iff_sats:
```
```  1289       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```  1290           i < length(env); j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1291        ==> is_nth(**A, x, y, z) <-> sats(A, nth_fm(i,j,k), env)"
```
```  1292 by (simp add: sats_nth_fm)
```
```  1293
```
```  1294 theorem nth_reflection:
```
```  1295      "REFLECTS[\<lambda>x. is_nth(L, f(x), g(x), h(x)),
```
```  1296                \<lambda>i x. is_nth(**Lset(i), f(x), g(x), h(x))]"
```
```  1297 apply (simp only: is_nth_def setclass_simps)
```
```  1298 apply (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1299              iterates_MH_reflection hd_reflection tl_reflection)
```
```  1300 done
```
```  1301
```
```  1302 theorem bool_of_o_reflection:
```
```  1303      "REFLECTS [P(L), \<lambda>i. P(**Lset(i))] ==>
```
```  1304       REFLECTS[\<lambda>x. is_bool_of_o(L, P(L,x), f(x)),
```
```  1305                \<lambda>i x. is_bool_of_o(**Lset(i), P(**Lset(i),x), f(x))]"
```
```  1306 apply (simp (no_asm) only: is_bool_of_o_def setclass_simps)
```
```  1307 apply (intro FOL_reflections function_reflections, assumption+)
```
```  1308 done
```
```  1309
```
```  1310
```
```  1311 subsubsection{*An Instance of Replacement for @{term nth}*}
```
```  1312
```
```  1313 lemma nth_replacement_Reflects:
```
```  1314  "REFLECTS
```
```  1315    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1316          is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
```
```  1317     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1318          is_wfrec(**Lset(i),
```
```  1319                   iterates_MH(**Lset(i),
```
```  1320                           is_tl(**Lset(i)), z), memsn, u, y))]"
```
```  1321 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1322           iterates_MH_reflection list_functor_reflection tl_reflection)
```
```  1323
```
```  1324 lemma nth_replacement:
```
```  1325    "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
```
```  1326 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1327 apply (rule strong_replacementI)
```
```  1328 apply (rule rallI)
```
```  1329 apply (rule separation_CollectI)
```
```  1330 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1331 apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1332 apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
```
```  1333 apply (drule subset_Lset_ltD, assumption)
```
```  1334 apply (erule reflection_imp_L_separation)
```
```  1335   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1336 apply (elim conjE)
```
```  1337 apply (rule DPow_LsetI)
```
```  1338 apply (rename_tac v)
```
```  1339 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1340 apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
```
```  1341 apply (rule sep_rules is_nat_case_iff_sats tl_iff_sats
```
```  1342             is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```  1343 done
```
```  1344
```
```  1345
```
```  1346
```
```  1347 subsubsection{*Instantiating the locale @{text M_datatypes}*}
```
```  1348
```
```  1349 lemma M_datatypes_axioms_L: "M_datatypes_axioms(L)"
```
```  1350   apply (rule M_datatypes_axioms.intro)
```
```  1351       apply (assumption | rule
```
```  1352         list_replacement1 list_replacement2
```
```  1353         formula_replacement1 formula_replacement2
```
```  1354         nth_replacement)+
```
```  1355   done
```
```  1356
```
```  1357 theorem M_datatypes_L: "PROP M_datatypes(L)"
```
```  1358   apply (rule M_datatypes.intro)
```
```  1359       apply (rule M_wfrank.axioms [OF M_wfrank_L])+
```
```  1360  apply (rule M_datatypes_axioms_L)
```
```  1361  done
```
```  1362
```
```  1363 lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
```
```  1364   and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
```
```  1365   and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
```
```  1366   and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
```
```  1367   and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
```
```  1368
```
```  1369 declare list_closed [intro,simp]
```
```  1370 declare formula_closed [intro,simp]
```
```  1371 declare list_abs [simp]
```
```  1372 declare formula_abs [simp]
```
```  1373 declare nth_abs [simp]
```
```  1374
```
```  1375
```
```  1376 subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
```
```  1377
```
```  1378 subsubsection{*Instances of Replacement for @{term eclose}*}
```
```  1379
```
```  1380 lemma eclose_replacement1_Reflects:
```
```  1381  "REFLECTS
```
```  1382    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1383          is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
```
```  1384     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1385          is_wfrec(**Lset(i),
```
```  1386                   iterates_MH(**Lset(i), big_union(**Lset(i)), A),
```
```  1387                   memsn, u, y))]"
```
```  1388 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1389           iterates_MH_reflection)
```
```  1390
```
```  1391 lemma eclose_replacement1:
```
```  1392    "L(A) ==> iterates_replacement(L, big_union(L), A)"
```
```  1393 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1394 apply (rule strong_replacementI)
```
```  1395 apply (rule rallI)
```
```  1396 apply (rename_tac B)
```
```  1397 apply (rule separation_CollectI)
```
```  1398 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1399 apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1400 apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
```
```  1401 apply (drule subset_Lset_ltD, assumption)
```
```  1402 apply (erule reflection_imp_L_separation)
```
```  1403   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1404 apply (elim conjE)
```
```  1405 apply (rule DPow_LsetI)
```
```  1406 apply (rename_tac v)
```
```  1407 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1408 apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
```
```  1409 apply (rule sep_rules iterates_MH_iff_sats is_nat_case_iff_sats
```
```  1410              is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1411 done
```
```  1412
```
```  1413
```
```  1414 lemma eclose_replacement2_Reflects:
```
```  1415  "REFLECTS
```
```  1416    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```  1417          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```  1418            is_wfrec (L, iterates_MH (L, big_union(L), A),
```
```  1419                               msn, u, x)),
```
```  1420     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```  1421          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```  1422           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```  1423            is_wfrec (**Lset(i),
```
```  1424                  iterates_MH (**Lset(i), big_union(**Lset(i)), A),
```
```  1425                      msn, u, x))]"
```
```  1426 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1427           iterates_MH_reflection)
```
```  1428
```
```  1429
```
```  1430 lemma eclose_replacement2:
```
```  1431    "L(A) ==> strong_replacement(L,
```
```  1432          \<lambda>n y. n\<in>nat &
```
```  1433                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```  1434                is_wfrec(L, iterates_MH(L,big_union(L), A),
```
```  1435                         msn, n, y)))"
```
```  1436 apply (rule strong_replacementI)
```
```  1437 apply (rule rallI)
```
```  1438 apply (rename_tac B)
```
```  1439 apply (rule separation_CollectI)
```
```  1440 apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
```
```  1441 apply (blast intro: L_nat)
```
```  1442 apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
```
```  1443 apply (drule subset_Lset_ltD, assumption)
```
```  1444 apply (erule reflection_imp_L_separation)
```
```  1445   apply (simp_all add: lt_Ord2)
```
```  1446 apply (rule DPow_LsetI)
```
```  1447 apply (rename_tac v)
```
```  1448 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1449 apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
```
```  1450 apply (rule sep_rules is_nat_case_iff_sats iterates_MH_iff_sats
```
```  1451               is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1452 done
```
```  1453
```
```  1454
```
```  1455 subsubsection{*Instantiating the locale @{text M_eclose}*}
```
```  1456
```
```  1457 lemma M_eclose_axioms_L: "M_eclose_axioms(L)"
```
```  1458   apply (rule M_eclose_axioms.intro)
```
```  1459    apply (assumption | rule eclose_replacement1 eclose_replacement2)+
```
```  1460   done
```
```  1461
```
```  1462 theorem M_eclose_L: "PROP M_eclose(L)"
```
```  1463   apply (rule M_eclose.intro)
```
```  1464        apply (rule M_datatypes.axioms [OF M_datatypes_L])+
```
```  1465   apply (rule M_eclose_axioms_L)
```
```  1466   done
```
```  1467
```
```  1468 lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
```
```  1469   and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
```
```  1470   and transrec_replacementI = M_eclose.transrec_replacementI [OF M_eclose_L]
```
```  1471
```
```  1472 end
```