src/ZF/Constructible/Rec_Separation.thy
 author paulson Wed Jul 31 18:30:25 2002 +0200 (2002-07-31) changeset 13440 cdde97e1db1c parent 13437 01b3fc0cc1b8 child 13441 d6d620639243 permissions -rw-r--r--
some progress towards "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 subsection{*The Locale @{text "M_trancl"}*}
```
```    20
```
```    21 subsubsection{*Separation for Reflexive/Transitive Closure*}
```
```    22
```
```    23 text{*First, The Defining Formula*}
```
```    24
```
```    25 (* "rtran_closure_mem(M,A,r,p) ==
```
```    26       \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
```
```    27        omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
```
```    28        (\<exists>f[M]. typed_function(M,n',A,f) &
```
```    29         (\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
```
```    30           fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
```
```    31         (\<forall>j[M]. j\<in>n -->
```
```    32           (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
```
```    33             fun_apply(M,f,j,fj) & successor(M,j,sj) &
```
```    34             fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
```
```    35 constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
```
```    36  "rtran_closure_mem_fm(A,r,p) ==
```
```    37    Exists(Exists(Exists(
```
```    38     And(omega_fm(2),
```
```    39      And(Member(1,2),
```
```    40       And(succ_fm(1,0),
```
```    41        Exists(And(typed_function_fm(1, A#+4, 0),
```
```    42         And(Exists(Exists(Exists(
```
```    43               And(pair_fm(2,1,p#+7),
```
```    44                And(empty_fm(0),
```
```    45                 And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
```
```    46             Forall(Implies(Member(0,3),
```
```    47              Exists(Exists(Exists(Exists(
```
```    48               And(fun_apply_fm(5,4,3),
```
```    49                And(succ_fm(4,2),
```
```    50                 And(fun_apply_fm(5,2,1),
```
```    51                  And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
```
```    52
```
```    53
```
```    54 lemma rtran_closure_mem_type [TC]:
```
```    55  "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
```
```    56 by (simp add: rtran_closure_mem_fm_def)
```
```    57
```
```    58 lemma arity_rtran_closure_mem_fm [simp]:
```
```    59      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```    60       ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```    61 by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```    62
```
```    63 lemma sats_rtran_closure_mem_fm [simp]:
```
```    64    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```    65     ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
```
```    66         rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```    67 by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
```
```    68
```
```    69 lemma rtran_closure_mem_iff_sats:
```
```    70       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```    71           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```    72        ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
```
```    73 by (simp add: sats_rtran_closure_mem_fm)
```
```    74
```
```    75 theorem rtran_closure_mem_reflection:
```
```    76      "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
```
```    77                \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
```
```    78 apply (simp only: rtran_closure_mem_def setclass_simps)
```
```    79 apply (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    80 done
```
```    81
```
```    82 text{*Separation for @{term "rtrancl(r)"}.*}
```
```    83 lemma rtrancl_separation:
```
```    84      "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
```
```    85 apply (rule separation_CollectI)
```
```    86 apply (rule_tac A="{r,A,z}" in subset_LsetE, blast )
```
```    87 apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
```
```    88 apply (drule subset_Lset_ltD, assumption)
```
```    89 apply (erule reflection_imp_L_separation)
```
```    90   apply (simp_all add: lt_Ord2)
```
```    91 apply (rule DPow_LsetI)
```
```    92 apply (rename_tac u)
```
```    93 apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
```
```    94 apply (rule sep_rules | simp)+
```
```    95 done
```
```    96
```
```    97
```
```    98 subsubsection{*Reflexive/Transitive Closure, Internalized*}
```
```    99
```
```   100 (*  "rtran_closure(M,r,s) ==
```
```   101         \<forall>A[M]. is_field(M,r,A) -->
```
```   102          (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
```
```   103 constdefs rtran_closure_fm :: "[i,i]=>i"
```
```   104  "rtran_closure_fm(r,s) ==
```
```   105    Forall(Implies(field_fm(succ(r),0),
```
```   106                   Forall(Iff(Member(0,succ(succ(s))),
```
```   107                              rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
```
```   108
```
```   109 lemma rtran_closure_type [TC]:
```
```   110      "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
```
```   111 by (simp add: rtran_closure_fm_def)
```
```   112
```
```   113 lemma arity_rtran_closure_fm [simp]:
```
```   114      "[| x \<in> nat; y \<in> nat |]
```
```   115       ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   116 by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   117
```
```   118 lemma sats_rtran_closure_fm [simp]:
```
```   119    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   120     ==> sats(A, rtran_closure_fm(x,y), env) <->
```
```   121         rtran_closure(**A, nth(x,env), nth(y,env))"
```
```   122 by (simp add: rtran_closure_fm_def rtran_closure_def)
```
```   123
```
```   124 lemma rtran_closure_iff_sats:
```
```   125       "[| nth(i,env) = x; nth(j,env) = y;
```
```   126           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   127        ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
```
```   128 by simp
```
```   129
```
```   130 theorem rtran_closure_reflection:
```
```   131      "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
```
```   132                \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
```
```   133 apply (simp only: rtran_closure_def setclass_simps)
```
```   134 apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
```
```   135 done
```
```   136
```
```   137
```
```   138 subsubsection{*Transitive Closure of a Relation, Internalized*}
```
```   139
```
```   140 (*  "tran_closure(M,r,t) ==
```
```   141          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
```
```   142 constdefs tran_closure_fm :: "[i,i]=>i"
```
```   143  "tran_closure_fm(r,s) ==
```
```   144    Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
```
```   145
```
```   146 lemma tran_closure_type [TC]:
```
```   147      "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
```
```   148 by (simp add: tran_closure_fm_def)
```
```   149
```
```   150 lemma arity_tran_closure_fm [simp]:
```
```   151      "[| x \<in> nat; y \<in> nat |]
```
```   152       ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   153 by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   154
```
```   155 lemma sats_tran_closure_fm [simp]:
```
```   156    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   157     ==> sats(A, tran_closure_fm(x,y), env) <->
```
```   158         tran_closure(**A, nth(x,env), nth(y,env))"
```
```   159 by (simp add: tran_closure_fm_def tran_closure_def)
```
```   160
```
```   161 lemma tran_closure_iff_sats:
```
```   162       "[| nth(i,env) = x; nth(j,env) = y;
```
```   163           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   164        ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
```
```   165 by simp
```
```   166
```
```   167 theorem tran_closure_reflection:
```
```   168      "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
```
```   169                \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
```
```   170 apply (simp only: tran_closure_def setclass_simps)
```
```   171 apply (intro FOL_reflections function_reflections
```
```   172              rtran_closure_reflection composition_reflection)
```
```   173 done
```
```   174
```
```   175
```
```   176 subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
```
```   177
```
```   178 lemma wellfounded_trancl_reflects:
```
```   179   "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   180                  w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
```
```   181    \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
```
```   182        w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
```
```   183        wx \<in> rp]"
```
```   184 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   185           tran_closure_reflection)
```
```   186
```
```   187
```
```   188 lemma wellfounded_trancl_separation:
```
```   189          "[| L(r); L(Z) |] ==>
```
```   190           separation (L, \<lambda>x.
```
```   191               \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
```
```   192                w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
```
```   193 apply (rule separation_CollectI)
```
```   194 apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast )
```
```   195 apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
```
```   196 apply (drule subset_Lset_ltD, assumption)
```
```   197 apply (erule reflection_imp_L_separation)
```
```   198   apply (simp_all add: lt_Ord2)
```
```   199 apply (rule DPow_LsetI)
```
```   200 apply (rename_tac u)
```
```   201 apply (rule bex_iff_sats conj_iff_sats)+
```
```   202 apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats)
```
```   203 apply (rule sep_rules tran_closure_iff_sats | simp)+
```
```   204 done
```
```   205
```
```   206
```
```   207 subsubsection{*Instantiating the locale @{text M_trancl}*}
```
```   208
```
```   209 lemma M_trancl_axioms_L: "M_trancl_axioms(L)"
```
```   210   apply (rule M_trancl_axioms.intro)
```
```   211    apply (assumption | rule rtrancl_separation wellfounded_trancl_separation)+
```
```   212   done
```
```   213
```
```   214 theorem M_trancl_L: "PROP M_trancl(L)"
```
```   215 by (rule M_trancl.intro
```
```   216          [OF M_triv_axioms_L M_axioms_axioms_L M_trancl_axioms_L])
```
```   217
```
```   218 lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
```
```   219   and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
```
```   220   and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L]
```
```   221   and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L]
```
```   222   and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L]
```
```   223   and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L]
```
```   224   and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L]
```
```   225   and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L]
```
```   226   and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L]
```
```   227   and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L]
```
```   228   and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L]
```
```   229   and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L]
```
```   230   and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L]
```
```   231
```
```   232 declare rtrancl_closed [intro,simp]
```
```   233 declare rtrancl_abs [simp]
```
```   234 declare trancl_closed [intro,simp]
```
```   235 declare trancl_abs [simp]
```
```   236
```
```   237
```
```   238 subsection{*Well-Founded Recursion!*}
```
```   239
```
```   240 (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
```
```   241    "M_is_recfun(M,MH,r,a,f) ==
```
```   242      \<forall>z[M]. z \<in> f <->
```
```   243             5      4       3       2       1           0
```
```   244             (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
```
```   245                pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
```
```   246                pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
```
```   247                xa \<in> r & MH(x, f_r_sx, y))"
```
```   248 *)
```
```   249
```
```   250 text{*The three arguments of @{term p} are always 5, 0, 4.*}
```
```   251 constdefs is_recfun_fm :: "[i, i, i, i]=>i"
```
```   252  "is_recfun_fm(p,r,a,f) ==
```
```   253    Forall(Iff(Member(0,succ(f)),
```
```   254     Exists(Exists(Exists(Exists(Exists(Exists(
```
```   255      And(pair_fm(5,4,6),
```
```   256       And(pair_fm(5,a#+7,3),
```
```   257        And(upair_fm(5,5,2),
```
```   258         And(pre_image_fm(r#+7,2,1),
```
```   259          And(restriction_fm(f#+7,1,0),
```
```   260           And(Member(3,r#+7), p))))))))))))))"
```
```   261
```
```   262
```
```   263 lemma is_recfun_type [TC]:
```
```   264      "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   265       ==> is_recfun_fm(p,x,y,z) \<in> formula"
```
```   266 by (simp add: is_recfun_fm_def)
```
```   267
```
```   268 lemma sats_is_recfun_fm:
```
```   269   assumes MH_iff_sats:
```
```   270       "!!a0 a1 a2 a3 a4 a5 a6.
```
```   271         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A|] ==>
```
```   272         MH(a5, a0, a4) <->
```
```   273        sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,Cons(a5,Cons(a6,env))))))))"
```
```   274   shows
```
```   275       "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   276        ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
```
```   277            M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
```
```   278 by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
```
```   279 (*
```
```   280 apply (rule ball_cong bex_cong iff_cong conj_cong refl iff_refl) +
```
```   281  sats(A, p,
```
```   282    Cons(xf, Cons(xe, Cons(xd, Cons(xc, Cons(xb, Cons(xaa, Cons(xa, env))))))))
```
```   283 \<longleftrightarrow> MH(xaa, xf, xb)
```
```   284
```
```   285 MH(nth(5,env), nth(0,env), nth(4,env)) <-> sats(A, p, env);
```
```   286 *)
```
```   287
```
```   288 (*      "!!x y z. [|x\<in>A; y\<in>A; z\<in>A|] ==> MH(x,y,z) <-> sats(A, p, env)"
```
```   289 *)
```
```   290
```
```   291 lemma is_recfun_iff_sats:
```
```   292   assumes MH_iff_sats:
```
```   293       "!!a0 a1 a2 a3 a4 a5 a6.
```
```   294         [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A|] ==>
```
```   295         MH(a5, a0, a4) <->
```
```   296        sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,Cons(a5,Cons(a6,env))))))))"
```
```   297   shows
```
```   298   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   299       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   300    ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
```
```   301 apply (rule iff_sym)
```
```   302 apply (rule iff_trans)
```
```   303 apply (rule sats_is_recfun_fm [of A MH])
```
```   304 apply (rule MH_iff_sats, simp_all)
```
```   305 done
```
```   306 (*FIXME: surely proof can be improved?*)
```
```   307
```
```   308
```
```   309 text{*The additional variable in the premise, namely @{term f'}, is essential.
```
```   310 It lets @{term MH} depend upon @{term x}, which seems often necessary.
```
```   311 The same thing occurs in @{text is_wfrec_reflection}.*}
```
```   312 theorem is_recfun_reflection:
```
```   313   assumes MH_reflection:
```
```   314     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)),
```
```   315                      \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
```
```   316   shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)),
```
```   317              \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
```
```   318 apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
```
```   319 apply (intro FOL_reflections function_reflections
```
```   320              restriction_reflection MH_reflection)
```
```   321 done
```
```   322
```
```   323 text{*Currently, @{text sats}-theorems for higher-order operators don't seem
```
```   324 useful.  Reflection theorems do work, though.  This one avoids the repetition
```
```   325 of the @{text MH}-term. *}
```
```   326 theorem is_wfrec_reflection:
```
```   327   assumes MH_reflection:
```
```   328     "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)),
```
```   329                      \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
```
```   330   shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L,x), f(x), g(x), h(x)),
```
```   331                \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
```
```   332 apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
```
```   333 apply (intro FOL_reflections MH_reflection is_recfun_reflection)
```
```   334 done
```
```   335
```
```   336 subsection{*The Locale @{text "M_wfrank"}*}
```
```   337
```
```   338 subsubsection{*Separation for @{term "wfrank"}*}
```
```   339
```
```   340 lemma wfrank_Reflects:
```
```   341  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   342               ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
```
```   343       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   344          ~ (\<exists>f \<in> Lset(i).
```
```   345             M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
```
```   346                         rplus, x, f))]"
```
```   347 by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
```
```   348
```
```   349 lemma wfrank_separation:
```
```   350      "L(r) ==>
```
```   351       separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   352          ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
```
```   353 apply (rule separation_CollectI)
```
```   354 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   355 apply (rule ReflectsE [OF wfrank_Reflects], assumption)
```
```   356 apply (drule subset_Lset_ltD, assumption)
```
```   357 apply (erule reflection_imp_L_separation)
```
```   358   apply (simp_all add: lt_Ord2, clarify)
```
```   359 apply (rule DPow_LsetI)
```
```   360 apply (rename_tac u)
```
```   361 apply (rule ball_iff_sats imp_iff_sats)+
```
```   362 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   363 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   364 done
```
```   365
```
```   366
```
```   367 subsubsection{*Replacement for @{term "wfrank"}*}
```
```   368
```
```   369 lemma wfrank_replacement_Reflects:
```
```   370  "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
```
```   371         (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   372          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   373                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   374                         is_range(L,f,y))),
```
```   375  \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
```
```   376       (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   377        (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
```
```   378          M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
```
```   379          is_range(**Lset(i),f,y)))]"
```
```   380 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   381              is_recfun_reflection tran_closure_reflection)
```
```   382
```
```   383
```
```   384 lemma wfrank_strong_replacement:
```
```   385      "L(r) ==>
```
```   386       strong_replacement(L, \<lambda>x z.
```
```   387          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   388          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   389                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   390                         is_range(L,f,y)))"
```
```   391 apply (rule strong_replacementI)
```
```   392 apply (rule rallI)
```
```   393 apply (rename_tac B)
```
```   394 apply (rule separation_CollectI)
```
```   395 apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
```
```   396 apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
```
```   397 apply (drule subset_Lset_ltD, assumption)
```
```   398 apply (erule reflection_imp_L_separation)
```
```   399   apply (simp_all add: lt_Ord2)
```
```   400 apply (rule DPow_LsetI)
```
```   401 apply (rename_tac u)
```
```   402 apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
```
```   403 apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
```
```   404 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   405 done
```
```   406
```
```   407
```
```   408 subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
```
```   409
```
```   410 lemma Ord_wfrank_Reflects:
```
```   411  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   412           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   413              is_range(L,f,rangef) -->
```
```   414              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   415              ordinal(L,rangef)),
```
```   416       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
```
```   417           ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
```
```   418              is_range(**Lset(i),f,rangef) -->
```
```   419              M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
```
```   420                          rplus, x, f) -->
```
```   421              ordinal(**Lset(i),rangef))]"
```
```   422 by (intro FOL_reflections function_reflections is_recfun_reflection
```
```   423           tran_closure_reflection ordinal_reflection)
```
```   424
```
```   425 lemma  Ord_wfrank_separation:
```
```   426      "L(r) ==>
```
```   427       separation (L, \<lambda>x.
```
```   428          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   429           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   430              is_range(L,f,rangef) -->
```
```   431              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   432              ordinal(L,rangef)))"
```
```   433 apply (rule separation_CollectI)
```
```   434 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   435 apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
```
```   436 apply (drule subset_Lset_ltD, assumption)
```
```   437 apply (erule reflection_imp_L_separation)
```
```   438   apply (simp_all add: lt_Ord2, clarify)
```
```   439 apply (rule DPow_LsetI)
```
```   440 apply (rename_tac u)
```
```   441 apply (rule ball_iff_sats imp_iff_sats)+
```
```   442 apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
```
```   443 apply (rule sep_rules is_recfun_iff_sats | simp)+
```
```   444 done
```
```   445
```
```   446
```
```   447 subsubsection{*Instantiating the locale @{text M_wfrank}*}
```
```   448
```
```   449 lemma M_wfrank_axioms_L: "M_wfrank_axioms(L)"
```
```   450   apply (rule M_wfrank_axioms.intro)
```
```   451    apply (assumption | rule
```
```   452      wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
```
```   453   done
```
```   454
```
```   455 theorem M_wfrank_L: "PROP M_wfrank(L)"
```
```   456   apply (rule M_wfrank.intro)
```
```   457      apply (rule M_trancl.axioms [OF M_trancl_L])+
```
```   458   apply (rule M_wfrank_axioms_L)
```
```   459   done
```
```   460
```
```   461 lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L]
```
```   462   and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
```
```   463   and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
```
```   464   and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
```
```   465   and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
```
```   466   and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
```
```   467   and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
```
```   468   and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
```
```   469   and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
```
```   470   and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
```
```   471   and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
```
```   472   and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
```
```   473   and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
```
```   474   and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
```
```   475   and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
```
```   476   and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
```
```   477   and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L]
```
```   478   and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L]
```
```   479   and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L]
```
```   480
```
```   481 declare iterates_closed [intro,simp]
```
```   482 declare Ord_wfrank_range [rule_format]
```
```   483 declare wf_abs [simp]
```
```   484 declare wf_on_abs [simp]
```
```   485
```
```   486
```
```   487 subsection{*For Datatypes*}
```
```   488
```
```   489 subsubsection{*Binary Products, Internalized*}
```
```   490
```
```   491 constdefs cartprod_fm :: "[i,i,i]=>i"
```
```   492 (* "cartprod(M,A,B,z) ==
```
```   493         \<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
```
```   494     "cartprod_fm(A,B,z) ==
```
```   495        Forall(Iff(Member(0,succ(z)),
```
```   496                   Exists(And(Member(0,succ(succ(A))),
```
```   497                          Exists(And(Member(0,succ(succ(succ(B)))),
```
```   498                                     pair_fm(1,0,2)))))))"
```
```   499
```
```   500 lemma cartprod_type [TC]:
```
```   501      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
```
```   502 by (simp add: cartprod_fm_def)
```
```   503
```
```   504 lemma arity_cartprod_fm [simp]:
```
```   505      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   506       ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   507 by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   508
```
```   509 lemma sats_cartprod_fm [simp]:
```
```   510    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   511     ==> sats(A, cartprod_fm(x,y,z), env) <->
```
```   512         cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   513 by (simp add: cartprod_fm_def cartprod_def)
```
```   514
```
```   515 lemma cartprod_iff_sats:
```
```   516       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   517           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   518        ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
```
```   519 by (simp add: sats_cartprod_fm)
```
```   520
```
```   521 theorem cartprod_reflection:
```
```   522      "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
```
```   523                \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
```
```   524 apply (simp only: cartprod_def setclass_simps)
```
```   525 apply (intro FOL_reflections pair_reflection)
```
```   526 done
```
```   527
```
```   528
```
```   529 subsubsection{*Binary Sums, Internalized*}
```
```   530
```
```   531 (* "is_sum(M,A,B,Z) ==
```
```   532        \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
```
```   533          3      2       1        0
```
```   534        number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
```
```   535        cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
```
```   536 constdefs sum_fm :: "[i,i,i]=>i"
```
```   537     "sum_fm(A,B,Z) ==
```
```   538        Exists(Exists(Exists(Exists(
```
```   539         And(number1_fm(2),
```
```   540             And(cartprod_fm(2,A#+4,3),
```
```   541                 And(upair_fm(2,2,1),
```
```   542                     And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
```
```   543
```
```   544 lemma sum_type [TC]:
```
```   545      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
```
```   546 by (simp add: sum_fm_def)
```
```   547
```
```   548 lemma arity_sum_fm [simp]:
```
```   549      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   550       ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   551 by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   552
```
```   553 lemma sats_sum_fm [simp]:
```
```   554    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   555     ==> sats(A, sum_fm(x,y,z), env) <->
```
```   556         is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   557 by (simp add: sum_fm_def is_sum_def)
```
```   558
```
```   559 lemma sum_iff_sats:
```
```   560       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   561           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   562        ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
```
```   563 by simp
```
```   564
```
```   565 theorem sum_reflection:
```
```   566      "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
```
```   567                \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
```
```   568 apply (simp only: is_sum_def setclass_simps)
```
```   569 apply (intro FOL_reflections function_reflections cartprod_reflection)
```
```   570 done
```
```   571
```
```   572
```
```   573 subsubsection{*The Operator @{term quasinat}*}
```
```   574
```
```   575 (* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
```
```   576 constdefs quasinat_fm :: "i=>i"
```
```   577     "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
```
```   578
```
```   579 lemma quasinat_type [TC]:
```
```   580      "x \<in> nat ==> quasinat_fm(x) \<in> formula"
```
```   581 by (simp add: quasinat_fm_def)
```
```   582
```
```   583 lemma arity_quasinat_fm [simp]:
```
```   584      "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
```
```   585 by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   586
```
```   587 lemma sats_quasinat_fm [simp]:
```
```   588    "[| x \<in> nat; env \<in> list(A)|]
```
```   589     ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
```
```   590 by (simp add: quasinat_fm_def is_quasinat_def)
```
```   591
```
```   592 lemma quasinat_iff_sats:
```
```   593       "[| nth(i,env) = x; nth(j,env) = y;
```
```   594           i \<in> nat; env \<in> list(A)|]
```
```   595        ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
```
```   596 by simp
```
```   597
```
```   598 theorem quasinat_reflection:
```
```   599      "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
```
```   600                \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
```
```   601 apply (simp only: is_quasinat_def setclass_simps)
```
```   602 apply (intro FOL_reflections function_reflections)
```
```   603 done
```
```   604
```
```   605
```
```   606 subsubsection{*The Operator @{term is_nat_case}*}
```
```   607 text{*I could not get it to work with the more natural assumption that
```
```   608  @{term is_b} takes two arguments.  Instead it must be a formula where 1 and 0
```
```   609  stand for @{term m} and @{term b}, respectively.*}
```
```   610
```
```   611 (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
```
```   612     "is_nat_case(M, a, is_b, k, z) ==
```
```   613        (empty(M,k) --> z=a) &
```
```   614        (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
```
```   615        (is_quasinat(M,k) | empty(M,z))" *)
```
```   616 text{*The formula @{term is_b} has free variables 1 and 0.*}
```
```   617 constdefs is_nat_case_fm :: "[i, i, i, i]=>i"
```
```   618  "is_nat_case_fm(a,is_b,k,z) ==
```
```   619     And(Implies(empty_fm(k), Equal(z,a)),
```
```   620         And(Forall(Implies(succ_fm(0,succ(k)),
```
```   621                    Forall(Implies(Equal(0,succ(succ(z))), is_b)))),
```
```   622             Or(quasinat_fm(k), empty_fm(z))))"
```
```   623
```
```   624 lemma is_nat_case_type [TC]:
```
```   625      "[| is_b \<in> formula;
```
```   626          x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   627       ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
```
```   628 by (simp add: is_nat_case_fm_def)
```
```   629
```
```   630 lemma sats_is_nat_case_fm:
```
```   631   assumes is_b_iff_sats:
```
```   632       "!!a. a \<in> A ==> is_b(a,nth(z, env)) <->
```
```   633                       sats(A, p, Cons(nth(z,env), Cons(a, env)))"
```
```   634   shows
```
```   635       "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   636        ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
```
```   637            is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
```
```   638 apply (frule lt_length_in_nat, assumption)
```
```   639 apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
```
```   640 done
```
```   641
```
```   642 lemma is_nat_case_iff_sats:
```
```   643   "[| (!!a. a \<in> A ==> is_b(a,z) <->
```
```   644                       sats(A, p, Cons(z, Cons(a,env))));
```
```   645       nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   646       i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   647    ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
```
```   648 by (simp add: sats_is_nat_case_fm [of A is_b])
```
```   649
```
```   650
```
```   651 text{*The second argument of @{term is_b} gives it direct access to @{term x},
```
```   652   which is essential for handling free variable references.  Without this
```
```   653   argument, we cannot prove reflection for @{term iterates_MH}.*}
```
```   654 theorem is_nat_case_reflection:
```
```   655   assumes is_b_reflection:
```
```   656     "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
```
```   657                      \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
```
```   658   shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
```
```   659                \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
```
```   660 apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
```
```   661 apply (intro FOL_reflections function_reflections
```
```   662              restriction_reflection is_b_reflection quasinat_reflection)
```
```   663 done
```
```   664
```
```   665
```
```   666
```
```   667 subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
```
```   668
```
```   669 (*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   670    "iterates_MH(M,isF,v,n,g,z) ==
```
```   671         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   672                     n, z)" *)
```
```   673 constdefs iterates_MH_fm :: "[i, i, i, i, i]=>i"
```
```   674  "iterates_MH_fm(isF,v,n,g,z) ==
```
```   675     is_nat_case_fm(v,
```
```   676       Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0),
```
```   677                      Forall(Implies(Equal(0,2), isF)))),
```
```   678       n, z)"
```
```   679
```
```   680 lemma iterates_MH_type [TC]:
```
```   681      "[| p \<in> formula;
```
```   682          v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   683       ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
```
```   684 by (simp add: iterates_MH_fm_def)
```
```   685
```
```   686 lemma sats_iterates_MH_fm:
```
```   687   assumes is_F_iff_sats:
```
```   688       "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   689               ==> is_F(a,b) <->
```
```   690                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
```
```   691   shows
```
```   692       "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
```
```   693        ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
```
```   694            iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
```
```   695 apply (frule lt_length_in_nat, assumption)
```
```   696 apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm
```
```   697               is_F_iff_sats [symmetric])
```
```   698 apply (rule is_nat_case_cong)
```
```   699 apply (simp_all add: setclass_def)
```
```   700 done
```
```   701
```
```   702
```
```   703 lemma iterates_MH_iff_sats:
```
```   704   "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
```
```   705               ==> is_F(a,b) <->
```
```   706                   sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env))))));
```
```   707       nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   708       i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
```
```   709    ==> iterates_MH(**A, is_F, v, x, y, z) <->
```
```   710        sats(A, iterates_MH_fm(p,i',i,j,k), env)"
```
```   711 apply (rule iff_sym)
```
```   712 apply (rule iff_trans)
```
```   713 apply (rule sats_iterates_MH_fm [of A is_F], blast)
```
```   714 apply simp_all
```
```   715 done
```
```   716 (*FIXME: surely proof can be improved?*)
```
```   717
```
```   718
```
```   719 theorem iterates_MH_reflection:
```
```   720   assumes p_reflection:
```
```   721     "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
```
```   722                      \<lambda>i x. p(**Lset(i), f(x), g(x))]"
```
```   723  shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
```
```   724                \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
```
```   725 apply (simp (no_asm_use) only: iterates_MH_def)
```
```   726 txt{*Must be careful: simplifying with @{text setclass_simps} above would
```
```   727      change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
```
```   728      it would no longer match rule @{text is_nat_case_reflection}. *}
```
```   729 apply (rule is_nat_case_reflection)
```
```   730 apply (simp (no_asm_use) only: setclass_simps)
```
```   731 apply (intro FOL_reflections function_reflections is_nat_case_reflection
```
```   732              restriction_reflection p_reflection)
```
```   733 done
```
```   734
```
```   735
```
```   736
```
```   737 subsection{*@{term L} is Closed Under the Operator @{term list}*}
```
```   738
```
```   739 subsubsection{*The List Functor, Internalized*}
```
```   740
```
```   741 constdefs list_functor_fm :: "[i,i,i]=>i"
```
```   742 (* "is_list_functor(M,A,X,Z) ==
```
```   743         \<exists>n1[M]. \<exists>AX[M].
```
```   744          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
```
```   745     "list_functor_fm(A,X,Z) ==
```
```   746        Exists(Exists(
```
```   747         And(number1_fm(1),
```
```   748             And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
```
```   749
```
```   750 lemma list_functor_type [TC]:
```
```   751      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
```
```   752 by (simp add: list_functor_fm_def)
```
```   753
```
```   754 lemma arity_list_functor_fm [simp]:
```
```   755      "[| x \<in> nat; y \<in> nat; z \<in> nat |]
```
```   756       ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
```
```   757 by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
```
```   758
```
```   759 lemma sats_list_functor_fm [simp]:
```
```   760    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```   761     ==> sats(A, list_functor_fm(x,y,z), env) <->
```
```   762         is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```   763 by (simp add: list_functor_fm_def is_list_functor_def)
```
```   764
```
```   765 lemma list_functor_iff_sats:
```
```   766   "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```   767       i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```   768    ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
```
```   769 by simp
```
```   770
```
```   771 theorem list_functor_reflection:
```
```   772      "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
```
```   773                \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
```
```   774 apply (simp only: is_list_functor_def setclass_simps)
```
```   775 apply (intro FOL_reflections number1_reflection
```
```   776              cartprod_reflection sum_reflection)
```
```   777 done
```
```   778
```
```   779
```
```   780 subsubsection{*Instances of Replacement for Lists*}
```
```   781
```
```   782 lemma list_replacement1_Reflects:
```
```   783  "REFLECTS
```
```   784    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   785          is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
```
```   786     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   787          is_wfrec(**Lset(i),
```
```   788                   iterates_MH(**Lset(i),
```
```   789                           is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
```
```   790 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   791           iterates_MH_reflection list_functor_reflection)
```
```   792
```
```   793 lemma list_replacement1:
```
```   794    "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
```
```   795 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   796 apply (rule strong_replacementI)
```
```   797 apply (rule rallI)
```
```   798 apply (rename_tac B)
```
```   799 apply (rule separation_CollectI)
```
```   800 apply (insert nonempty)
```
```   801 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   802 apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   803 apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
```
```   804 apply (drule subset_Lset_ltD, assumption)
```
```   805 apply (erule reflection_imp_L_separation)
```
```   806   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   807 apply (elim conjE)
```
```   808 apply (rule DPow_LsetI)
```
```   809 apply (rename_tac v)
```
```   810 apply (rule bex_iff_sats conj_iff_sats)+
```
```   811 apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   812 apply (rule sep_rules | simp)+
```
```   813 apply (simp add: is_wfrec_def)
```
```   814 apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
```
```   815             is_recfun_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   816 done
```
```   817
```
```   818 lemma list_replacement2_Reflects:
```
```   819  "REFLECTS
```
```   820    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   821          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   822            is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
```
```   823                               msn, u, x)),
```
```   824     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   825          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   826           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   827            is_wfrec (**Lset(i),
```
```   828                  iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
```
```   829                      msn, u, x))]"
```
```   830 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   831           iterates_MH_reflection list_functor_reflection)
```
```   832
```
```   833
```
```   834 lemma list_replacement2:
```
```   835    "L(A) ==> strong_replacement(L,
```
```   836          \<lambda>n y. n\<in>nat &
```
```   837                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   838                is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
```
```   839                         msn, n, y)))"
```
```   840 apply (rule strong_replacementI)
```
```   841 apply (rule rallI)
```
```   842 apply (rename_tac B)
```
```   843 apply (rule separation_CollectI)
```
```   844 apply (insert nonempty)
```
```   845 apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
```
```   846 apply (blast intro: L_nat)
```
```   847 apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
```
```   848 apply (drule subset_Lset_ltD, assumption)
```
```   849 apply (erule reflection_imp_L_separation)
```
```   850   apply (simp_all add: lt_Ord2)
```
```   851 apply (rule DPow_LsetI)
```
```   852 apply (rename_tac v)
```
```   853 apply (rule bex_iff_sats conj_iff_sats)+
```
```   854 apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
```
```   855 apply (rule sep_rules | simp)+
```
```   856 apply (simp add: is_wfrec_def)
```
```   857 apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
```
```   858             is_recfun_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   859 done
```
```   860
```
```   861
```
```   862 subsection{*@{term L} is Closed Under the Operator @{term formula}*}
```
```   863
```
```   864 subsubsection{*The Formula Functor, Internalized*}
```
```   865
```
```   866 constdefs formula_functor_fm :: "[i,i]=>i"
```
```   867 (*     "is_formula_functor(M,X,Z) ==
```
```   868         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
```
```   869            4           3               2       1       0
```
```   870           omega(M,nat') & cartprod(M,nat',nat',natnat) &
```
```   871           is_sum(M,natnat,natnat,natnatsum) &
```
```   872           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
```
```   873           is_sum(M,natnatsum,X3,Z)" *)
```
```   874     "formula_functor_fm(X,Z) ==
```
```   875        Exists(Exists(Exists(Exists(Exists(
```
```   876         And(omega_fm(4),
```
```   877          And(cartprod_fm(4,4,3),
```
```   878           And(sum_fm(3,3,2),
```
```   879            And(cartprod_fm(X#+5,X#+5,1),
```
```   880             And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
```
```   881
```
```   882 lemma formula_functor_type [TC]:
```
```   883      "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
```
```   884 by (simp add: formula_functor_fm_def)
```
```   885
```
```   886 lemma sats_formula_functor_fm [simp]:
```
```   887    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```   888     ==> sats(A, formula_functor_fm(x,y), env) <->
```
```   889         is_formula_functor(**A, nth(x,env), nth(y,env))"
```
```   890 by (simp add: formula_functor_fm_def is_formula_functor_def)
```
```   891
```
```   892 lemma formula_functor_iff_sats:
```
```   893   "[| nth(i,env) = x; nth(j,env) = y;
```
```   894       i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```   895    ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
```
```   896 by simp
```
```   897
```
```   898 theorem formula_functor_reflection:
```
```   899      "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
```
```   900                \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
```
```   901 apply (simp only: is_formula_functor_def setclass_simps)
```
```   902 apply (intro FOL_reflections omega_reflection
```
```   903              cartprod_reflection sum_reflection)
```
```   904 done
```
```   905
```
```   906 subsubsection{*Instances of Replacement for Formulas*}
```
```   907
```
```   908 lemma formula_replacement1_Reflects:
```
```   909  "REFLECTS
```
```   910    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```   911          is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
```
```   912     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```   913          is_wfrec(**Lset(i),
```
```   914                   iterates_MH(**Lset(i),
```
```   915                           is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
```
```   916 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   917           iterates_MH_reflection formula_functor_reflection)
```
```   918
```
```   919 lemma formula_replacement1:
```
```   920    "iterates_replacement(L, is_formula_functor(L), 0)"
```
```   921 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```   922 apply (rule strong_replacementI)
```
```   923 apply (rule rallI)
```
```   924 apply (rename_tac B)
```
```   925 apply (rule separation_CollectI)
```
```   926 apply (insert nonempty)
```
```   927 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```   928 apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
```
```   929 apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
```
```   930 apply (drule subset_Lset_ltD, assumption)
```
```   931 apply (erule reflection_imp_L_separation)
```
```   932   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```   933 apply (rule DPow_LsetI)
```
```   934 apply (rename_tac v)
```
```   935 apply (rule bex_iff_sats conj_iff_sats)+
```
```   936 apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
```
```   937 apply (rule sep_rules | simp)+
```
```   938 apply (simp add: is_wfrec_def)
```
```   939 apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
```
```   940             is_recfun_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   941 done
```
```   942
```
```   943 lemma formula_replacement2_Reflects:
```
```   944  "REFLECTS
```
```   945    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```   946          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```   947            is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
```
```   948                               msn, u, x)),
```
```   949     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```   950          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```   951           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```   952            is_wfrec (**Lset(i),
```
```   953                  iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
```
```   954                      msn, u, x))]"
```
```   955 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```   956           iterates_MH_reflection formula_functor_reflection)
```
```   957
```
```   958
```
```   959 lemma formula_replacement2:
```
```   960    "strong_replacement(L,
```
```   961          \<lambda>n y. n\<in>nat &
```
```   962                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```   963                is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
```
```   964                         msn, n, y)))"
```
```   965 apply (rule strong_replacementI)
```
```   966 apply (rule rallI)
```
```   967 apply (rename_tac B)
```
```   968 apply (rule separation_CollectI)
```
```   969 apply (insert nonempty)
```
```   970 apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
```
```   971 apply (blast intro: L_nat)
```
```   972 apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
```
```   973 apply (drule subset_Lset_ltD, assumption)
```
```   974 apply (erule reflection_imp_L_separation)
```
```   975   apply (simp_all add: lt_Ord2)
```
```   976 apply (rule DPow_LsetI)
```
```   977 apply (rename_tac v)
```
```   978 apply (rule bex_iff_sats conj_iff_sats)+
```
```   979 apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
```
```   980 apply (rule sep_rules | simp)+
```
```   981 apply (simp add: is_wfrec_def)
```
```   982 apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
```
```   983             is_recfun_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```   984 done
```
```   985
```
```   986 text{*NB The proofs for type @{term formula} are virtually identical to those
```
```   987 for @{term "list(A)"}.  It was a cut-and-paste job! *}
```
```   988
```
```   989
```
```   990 subsection{*Internalized Forms of Data Structuring Operators*}
```
```   991
```
```   992 subsubsection{*The Formula @{term is_Inl}, Internalized*}
```
```   993
```
```   994 (*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
```
```   995 constdefs Inl_fm :: "[i,i]=>i"
```
```   996     "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```   997
```
```   998 lemma Inl_type [TC]:
```
```   999      "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
```
```  1000 by (simp add: Inl_fm_def)
```
```  1001
```
```  1002 lemma sats_Inl_fm [simp]:
```
```  1003    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1004     ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
```
```  1005 by (simp add: Inl_fm_def is_Inl_def)
```
```  1006
```
```  1007 lemma Inl_iff_sats:
```
```  1008       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1009           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1010        ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
```
```  1011 by simp
```
```  1012
```
```  1013 theorem Inl_reflection:
```
```  1014      "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
```
```  1015                \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
```
```  1016 apply (simp only: is_Inl_def setclass_simps)
```
```  1017 apply (intro FOL_reflections function_reflections)
```
```  1018 done
```
```  1019
```
```  1020
```
```  1021 subsubsection{*The Formula @{term is_Inr}, Internalized*}
```
```  1022
```
```  1023 (*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
```
```  1024 constdefs Inr_fm :: "[i,i]=>i"
```
```  1025     "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
```
```  1026
```
```  1027 lemma Inr_type [TC]:
```
```  1028      "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
```
```  1029 by (simp add: Inr_fm_def)
```
```  1030
```
```  1031 lemma sats_Inr_fm [simp]:
```
```  1032    "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1033     ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
```
```  1034 by (simp add: Inr_fm_def is_Inr_def)
```
```  1035
```
```  1036 lemma Inr_iff_sats:
```
```  1037       "[| nth(i,env) = x; nth(k,env) = z;
```
```  1038           i \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1039        ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
```
```  1040 by simp
```
```  1041
```
```  1042 theorem Inr_reflection:
```
```  1043      "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
```
```  1044                \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
```
```  1045 apply (simp only: is_Inr_def setclass_simps)
```
```  1046 apply (intro FOL_reflections function_reflections)
```
```  1047 done
```
```  1048
```
```  1049
```
```  1050 subsubsection{*The Formula @{term is_Nil}, Internalized*}
```
```  1051
```
```  1052 (* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
```
```  1053
```
```  1054 constdefs Nil_fm :: "i=>i"
```
```  1055     "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
```
```  1056
```
```  1057 lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
```
```  1058 by (simp add: Nil_fm_def)
```
```  1059
```
```  1060 lemma sats_Nil_fm [simp]:
```
```  1061    "[| x \<in> nat; env \<in> list(A)|]
```
```  1062     ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
```
```  1063 by (simp add: Nil_fm_def is_Nil_def)
```
```  1064
```
```  1065 lemma Nil_iff_sats:
```
```  1066       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1067        ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
```
```  1068 by simp
```
```  1069
```
```  1070 theorem Nil_reflection:
```
```  1071      "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
```
```  1072                \<lambda>i x. is_Nil(**Lset(i),f(x))]"
```
```  1073 apply (simp only: is_Nil_def setclass_simps)
```
```  1074 apply (intro FOL_reflections function_reflections Inl_reflection)
```
```  1075 done
```
```  1076
```
```  1077
```
```  1078 subsubsection{*The Formula @{term is_Cons}, Internalized*}
```
```  1079
```
```  1080
```
```  1081 (*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
```
```  1082 constdefs Cons_fm :: "[i,i,i]=>i"
```
```  1083     "Cons_fm(a,l,Z) ==
```
```  1084        Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
```
```  1085
```
```  1086 lemma Cons_type [TC]:
```
```  1087      "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
```
```  1088 by (simp add: Cons_fm_def)
```
```  1089
```
```  1090 lemma sats_Cons_fm [simp]:
```
```  1091    "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
```
```  1092     ==> sats(A, Cons_fm(x,y,z), env) <->
```
```  1093        is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
```
```  1094 by (simp add: Cons_fm_def is_Cons_def)
```
```  1095
```
```  1096 lemma Cons_iff_sats:
```
```  1097       "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
```
```  1098           i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
```
```  1099        ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
```
```  1100 by simp
```
```  1101
```
```  1102 theorem Cons_reflection:
```
```  1103      "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
```
```  1104                \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
```
```  1105 apply (simp only: is_Cons_def setclass_simps)
```
```  1106 apply (intro FOL_reflections pair_reflection Inr_reflection)
```
```  1107 done
```
```  1108
```
```  1109 subsubsection{*The Formula @{term is_quasilist}, Internalized*}
```
```  1110
```
```  1111 (* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
```
```  1112
```
```  1113 constdefs quasilist_fm :: "i=>i"
```
```  1114     "quasilist_fm(x) ==
```
```  1115        Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
```
```  1116
```
```  1117 lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
```
```  1118 by (simp add: quasilist_fm_def)
```
```  1119
```
```  1120 lemma sats_quasilist_fm [simp]:
```
```  1121    "[| x \<in> nat; env \<in> list(A)|]
```
```  1122     ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
```
```  1123 by (simp add: quasilist_fm_def is_quasilist_def)
```
```  1124
```
```  1125 lemma quasilist_iff_sats:
```
```  1126       "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
```
```  1127        ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
```
```  1128 by simp
```
```  1129
```
```  1130 theorem quasilist_reflection:
```
```  1131      "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
```
```  1132                \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
```
```  1133 apply (simp only: is_quasilist_def setclass_simps)
```
```  1134 apply (intro FOL_reflections Nil_reflection Cons_reflection)
```
```  1135 done
```
```  1136
```
```  1137
```
```  1138 subsection{*Absoluteness for the Function @{term nth}*}
```
```  1139
```
```  1140
```
```  1141 subsubsection{*The Formula @{term is_hd}, Internalized*}
```
```  1142
```
```  1143 (*   "is_hd(M,xs,H) ==
```
```  1144        (is_Nil(M,xs) --> empty(M,H)) &
```
```  1145        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
```
```  1146        (is_quasilist(M,xs) | empty(M,H))" *)
```
```  1147 constdefs hd_fm :: "[i,i]=>i"
```
```  1148     "hd_fm(xs,H) ==
```
```  1149        And(Implies(Nil_fm(xs), empty_fm(H)),
```
```  1150            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))),
```
```  1151                Or(quasilist_fm(xs), empty_fm(H))))"
```
```  1152
```
```  1153 lemma hd_type [TC]:
```
```  1154      "[| x \<in> nat; y \<in> nat |] ==> hd_fm(x,y) \<in> formula"
```
```  1155 by (simp add: hd_fm_def)
```
```  1156
```
```  1157 lemma sats_hd_fm [simp]:
```
```  1158    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1159     ==> sats(A, hd_fm(x,y), env) <-> is_hd(**A, nth(x,env), nth(y,env))"
```
```  1160 by (simp add: hd_fm_def is_hd_def)
```
```  1161
```
```  1162 lemma hd_iff_sats:
```
```  1163       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1164           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1165        ==> is_hd(**A, x, y) <-> sats(A, hd_fm(i,j), env)"
```
```  1166 by simp
```
```  1167
```
```  1168 theorem hd_reflection:
```
```  1169      "REFLECTS[\<lambda>x. is_hd(L,f(x),g(x)),
```
```  1170                \<lambda>i x. is_hd(**Lset(i),f(x),g(x))]"
```
```  1171 apply (simp only: is_hd_def setclass_simps)
```
```  1172 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1173              quasilist_reflection empty_reflection)
```
```  1174 done
```
```  1175
```
```  1176
```
```  1177 subsubsection{*The Formula @{term is_tl}, Internalized*}
```
```  1178
```
```  1179 (*     "is_tl(M,xs,T) ==
```
```  1180        (is_Nil(M,xs) --> T=xs) &
```
```  1181        (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
```
```  1182        (is_quasilist(M,xs) | empty(M,T))" *)
```
```  1183 constdefs tl_fm :: "[i,i]=>i"
```
```  1184     "tl_fm(xs,T) ==
```
```  1185        And(Implies(Nil_fm(xs), Equal(T,xs)),
```
```  1186            And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
```
```  1187                Or(quasilist_fm(xs), empty_fm(T))))"
```
```  1188
```
```  1189 lemma tl_type [TC]:
```
```  1190      "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
```
```  1191 by (simp add: tl_fm_def)
```
```  1192
```
```  1193 lemma sats_tl_fm [simp]:
```
```  1194    "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
```
```  1195     ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
```
```  1196 by (simp add: tl_fm_def is_tl_def)
```
```  1197
```
```  1198 lemma tl_iff_sats:
```
```  1199       "[| nth(i,env) = x; nth(j,env) = y;
```
```  1200           i \<in> nat; j \<in> nat; env \<in> list(A)|]
```
```  1201        ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
```
```  1202 by simp
```
```  1203
```
```  1204 theorem tl_reflection:
```
```  1205      "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
```
```  1206                \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
```
```  1207 apply (simp only: is_tl_def setclass_simps)
```
```  1208 apply (intro FOL_reflections Nil_reflection Cons_reflection
```
```  1209              quasilist_reflection empty_reflection)
```
```  1210 done
```
```  1211
```
```  1212
```
```  1213 subsubsection{*The Formula @{term is_nth}, Internalized*}
```
```  1214
```
```  1215 (* "is_nth(M,n,l,Z) ==
```
```  1216       \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M].
```
```  1217        2       1       0
```
```  1218        successor(M,n,sn) & membership(M,sn,msn) &
```
```  1219        is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
```
```  1220        is_hd(M,X,Z)"
```
```  1221 constdefs nth_fm :: "[i,i,i]=>i"
```
```  1222     "nth_fm(n,l,Z) ==
```
```  1223        Exists(Exists(Exists(
```
```  1224          And(successor_fm(n#+3,1),
```
```  1225           And(membership_fm(1,0),
```
```  1226            And(
```
```  1227  *)
```
```  1228
```
```  1229 theorem nth_reflection:
```
```  1230      "REFLECTS[\<lambda>x. is_nth(L, f(x), g(x), h(x)),
```
```  1231                \<lambda>i x. is_nth(**Lset(i), f(x), g(x), h(x))]"
```
```  1232 apply (simp only: is_nth_def setclass_simps)
```
```  1233 apply (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1234              iterates_MH_reflection hd_reflection tl_reflection)
```
```  1235 done
```
```  1236
```
```  1237 theorem bool_of_o_reflection:
```
```  1238      "REFLECTS [P(L), \<lambda>i. P(**Lset(i))] ==>
```
```  1239       REFLECTS[\<lambda>x. is_bool_of_o(L, P(L,x), f(x)),
```
```  1240                \<lambda>i x. is_bool_of_o(**Lset(i), P(**Lset(i),x), f(x))]"
```
```  1241 apply (simp (no_asm) only: is_bool_of_o_def setclass_simps)
```
```  1242 apply (intro FOL_reflections function_reflections)
```
```  1243 apply assumption+
```
```  1244 done
```
```  1245
```
```  1246
```
```  1247 subsubsection{*An Instance of Replacement for @{term nth}*}
```
```  1248
```
```  1249 lemma nth_replacement_Reflects:
```
```  1250  "REFLECTS
```
```  1251    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1252          is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
```
```  1253     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1254          is_wfrec(**Lset(i),
```
```  1255                   iterates_MH(**Lset(i),
```
```  1256                           is_tl(**Lset(i)), z), memsn, u, y))]"
```
```  1257 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1258           iterates_MH_reflection list_functor_reflection tl_reflection)
```
```  1259
```
```  1260 lemma nth_replacement:
```
```  1261    "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
```
```  1262 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1263 apply (rule strong_replacementI)
```
```  1264 apply (rule rallI)
```
```  1265 apply (rule separation_CollectI)
```
```  1266 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1267 apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1268 apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
```
```  1269 apply (drule subset_Lset_ltD, assumption)
```
```  1270 apply (erule reflection_imp_L_separation)
```
```  1271   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1272 apply (elim conjE)
```
```  1273 apply (rule DPow_LsetI)
```
```  1274 apply (rename_tac v)
```
```  1275 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1276 apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
```
```  1277 apply (rule sep_rules | simp)+
```
```  1278 apply (simp add: is_wfrec_def)
```
```  1279 apply (rule sep_rules is_nat_case_iff_sats tl_iff_sats
```
```  1280             is_recfun_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
```
```  1281 done
```
```  1282
```
```  1283
```
```  1284
```
```  1285 subsubsection{*Instantiating the locale @{text M_datatypes}*}
```
```  1286
```
```  1287 lemma M_datatypes_axioms_L: "M_datatypes_axioms(L)"
```
```  1288   apply (rule M_datatypes_axioms.intro)
```
```  1289       apply (assumption | rule
```
```  1290         list_replacement1 list_replacement2
```
```  1291         formula_replacement1 formula_replacement2
```
```  1292         nth_replacement)+
```
```  1293   done
```
```  1294
```
```  1295 theorem M_datatypes_L: "PROP M_datatypes(L)"
```
```  1296   apply (rule M_datatypes.intro)
```
```  1297       apply (rule M_wfrank.axioms [OF M_wfrank_L])+
```
```  1298  apply (rule M_datatypes_axioms_L);
```
```  1299  done
```
```  1300
```
```  1301 lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
```
```  1302   and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
```
```  1303   and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
```
```  1304   and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
```
```  1305   and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
```
```  1306
```
```  1307 declare list_closed [intro,simp]
```
```  1308 declare formula_closed [intro,simp]
```
```  1309 declare list_abs [simp]
```
```  1310 declare formula_abs [simp]
```
```  1311 declare nth_abs [simp]
```
```  1312
```
```  1313
```
```  1314 subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
```
```  1315
```
```  1316 subsubsection{*Instances of Replacement for @{term eclose}*}
```
```  1317
```
```  1318 lemma eclose_replacement1_Reflects:
```
```  1319  "REFLECTS
```
```  1320    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
```
```  1321          is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
```
```  1322     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
```
```  1323          is_wfrec(**Lset(i),
```
```  1324                   iterates_MH(**Lset(i), big_union(**Lset(i)), A),
```
```  1325                   memsn, u, y))]"
```
```  1326 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1327           iterates_MH_reflection)
```
```  1328
```
```  1329 lemma eclose_replacement1:
```
```  1330    "L(A) ==> iterates_replacement(L, big_union(L), A)"
```
```  1331 apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
```
```  1332 apply (rule strong_replacementI)
```
```  1333 apply (rule rallI)
```
```  1334 apply (rename_tac B)
```
```  1335 apply (rule separation_CollectI)
```
```  1336 apply (subgoal_tac "L(Memrel(succ(n)))")
```
```  1337 apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
```
```  1338 apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
```
```  1339 apply (drule subset_Lset_ltD, assumption)
```
```  1340 apply (erule reflection_imp_L_separation)
```
```  1341   apply (simp_all add: lt_Ord2 Memrel_closed)
```
```  1342 apply (elim conjE)
```
```  1343 apply (rule DPow_LsetI)
```
```  1344 apply (rename_tac v)
```
```  1345 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1346 apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
```
```  1347 apply (rule sep_rules | simp)+
```
```  1348 apply (simp add: is_wfrec_def)
```
```  1349 apply (rule sep_rules iterates_MH_iff_sats is_nat_case_iff_sats
```
```  1350              is_recfun_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1351 done
```
```  1352
```
```  1353
```
```  1354 lemma eclose_replacement2_Reflects:
```
```  1355  "REFLECTS
```
```  1356    [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
```
```  1357          (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
```
```  1358            is_wfrec (L, iterates_MH (L, big_union(L), A),
```
```  1359                               msn, u, x)),
```
```  1360     \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
```
```  1361          (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
```
```  1362           successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
```
```  1363            is_wfrec (**Lset(i),
```
```  1364                  iterates_MH (**Lset(i), big_union(**Lset(i)), A),
```
```  1365                      msn, u, x))]"
```
```  1366 by (intro FOL_reflections function_reflections is_wfrec_reflection
```
```  1367           iterates_MH_reflection)
```
```  1368
```
```  1369
```
```  1370 lemma eclose_replacement2:
```
```  1371    "L(A) ==> strong_replacement(L,
```
```  1372          \<lambda>n y. n\<in>nat &
```
```  1373                (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
```
```  1374                is_wfrec(L, iterates_MH(L,big_union(L), A),
```
```  1375                         msn, n, y)))"
```
```  1376 apply (rule strong_replacementI)
```
```  1377 apply (rule rallI)
```
```  1378 apply (rename_tac B)
```
```  1379 apply (rule separation_CollectI)
```
```  1380 apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
```
```  1381 apply (blast intro: L_nat)
```
```  1382 apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
```
```  1383 apply (drule subset_Lset_ltD, assumption)
```
```  1384 apply (erule reflection_imp_L_separation)
```
```  1385   apply (simp_all add: lt_Ord2)
```
```  1386 apply (rule DPow_LsetI)
```
```  1387 apply (rename_tac v)
```
```  1388 apply (rule bex_iff_sats conj_iff_sats)+
```
```  1389 apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
```
```  1390 apply (rule sep_rules | simp)+
```
```  1391 apply (simp add: is_wfrec_def)
```
```  1392 apply (rule sep_rules is_nat_case_iff_sats iterates_MH_iff_sats
```
```  1393               is_recfun_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
```
```  1394 done
```
```  1395
```
```  1396
```
```  1397 subsubsection{*Instantiating the locale @{text M_eclose}*}
```
```  1398
```
```  1399 lemma M_eclose_axioms_L: "M_eclose_axioms(L)"
```
```  1400   apply (rule M_eclose_axioms.intro)
```
```  1401    apply (assumption | rule eclose_replacement1 eclose_replacement2)+
```
```  1402   done
```
```  1403
```
```  1404 theorem M_eclose_L: "PROP M_eclose(L)"
```
```  1405   apply (rule M_eclose.intro)
```
```  1406        apply (rule M_datatypes.axioms [OF M_datatypes_L])+
```
```  1407   apply (rule M_eclose_axioms_L)
```
```  1408   done
```
```  1409
```
```  1410 lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
```
```  1411   and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
```
```  1412   and transrec_replacementI = M_eclose.transrec_replacementI [OF M_eclose_L]
```
```  1413
```
```  1414 end
```