src/ZF/Constructible/AC_in_L.thy
 author skalberg Thu Aug 28 01:56:40 2003 +0200 (2003-08-28) changeset 14171 0cab06e3bbd0 parent 13702 c7cf8fa66534 child 16417 9bc16273c2d4 permissions -rw-r--r--
Extended the notion of letter and digit, such that now one may use greek,
gothic, euler, or calligraphic letters as normal letters.
```     1 (*  Title:      ZF/Constructible/AC_in_L.thy
```
```     2     ID: \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4 *)
```
```     5
```
```     6 header {* The Axiom of Choice Holds in L! *}
```
```     7
```
```     8 theory AC_in_L = Formula:
```
```     9
```
```    10 subsection{*Extending a Wellordering over a List -- Lexicographic Power*}
```
```    11
```
```    12 text{*This could be moved into a library.*}
```
```    13
```
```    14 consts
```
```    15   rlist   :: "[i,i]=>i"
```
```    16
```
```    17 inductive
```
```    18   domains "rlist(A,r)" \<subseteq> "list(A) * list(A)"
```
```    19   intros
```
```    20     shorterI:
```
```    21       "[| length(l') < length(l); l' \<in> list(A); l \<in> list(A) |]
```
```    22        ==> <l', l> \<in> rlist(A,r)"
```
```    23
```
```    24     sameI:
```
```    25       "[| <l',l> \<in> rlist(A,r); a \<in> A |]
```
```    26        ==> <Cons(a,l'), Cons(a,l)> \<in> rlist(A,r)"
```
```    27
```
```    28     diffI:
```
```    29       "[| length(l') = length(l); <a',a> \<in> r;
```
```    30           l' \<in> list(A); l \<in> list(A); a' \<in> A; a \<in> A |]
```
```    31        ==> <Cons(a',l'), Cons(a,l)> \<in> rlist(A,r)"
```
```    32   type_intros list.intros
```
```    33
```
```    34
```
```    35 subsubsection{*Type checking*}
```
```    36
```
```    37 lemmas rlist_type = rlist.dom_subset
```
```    38
```
```    39 lemmas field_rlist = rlist_type [THEN field_rel_subset]
```
```    40
```
```    41 subsubsection{*Linearity*}
```
```    42
```
```    43 lemma rlist_Nil_Cons [intro]:
```
```    44     "[|a \<in> A; l \<in> list(A)|] ==> <[], Cons(a,l)> \<in> rlist(A, r)"
```
```    45 by (simp add: shorterI)
```
```    46
```
```    47 lemma linear_rlist:
```
```    48     "linear(A,r) ==> linear(list(A),rlist(A,r))"
```
```    49 apply (simp (no_asm_simp) add: linear_def)
```
```    50 apply (rule ballI)
```
```    51 apply (induct_tac x)
```
```    52  apply (rule ballI)
```
```    53  apply (induct_tac y)
```
```    54   apply (simp_all add: shorterI)
```
```    55 apply (rule ballI)
```
```    56 apply (erule_tac a=y in list.cases)
```
```    57  apply (rename_tac [2] a2 l2)
```
```    58  apply (rule_tac [2] i = "length(l)" and j = "length(l2)" in Ord_linear_lt)
```
```    59      apply (simp_all add: shorterI)
```
```    60 apply (erule_tac x=a and y=a2 in linearE)
```
```    61     apply (simp_all add: diffI)
```
```    62 apply (blast intro: sameI)
```
```    63 done
```
```    64
```
```    65
```
```    66 subsubsection{*Well-foundedness*}
```
```    67
```
```    68 text{*Nothing preceeds Nil in this ordering.*}
```
```    69 inductive_cases rlist_NilE: " <l,[]> \<in> rlist(A,r)"
```
```    70
```
```    71 inductive_cases rlist_ConsE: " <l', Cons(x,l)> \<in> rlist(A,r)"
```
```    72
```
```    73 lemma not_rlist_Nil [simp]: " <l,[]> \<notin> rlist(A,r)"
```
```    74 by (blast intro: elim: rlist_NilE)
```
```    75
```
```    76 lemma rlist_imp_length_le: "<l',l> \<in> rlist(A,r) ==> length(l') \<le> length(l)"
```
```    77 apply (erule rlist.induct)
```
```    78 apply (simp_all add: leI)
```
```    79 done
```
```    80
```
```    81 lemma wf_on_rlist_n:
```
```    82   "[| n \<in> nat; wf[A](r) |] ==> wf[{l \<in> list(A). length(l) = n}](rlist(A,r))"
```
```    83 apply (induct_tac n)
```
```    84  apply (rule wf_onI2, simp)
```
```    85 apply (rule wf_onI2, clarify)
```
```    86 apply (erule_tac a=y in list.cases, clarify)
```
```    87  apply (simp (no_asm_use))
```
```    88 apply clarify
```
```    89 apply (simp (no_asm_use))
```
```    90 apply (subgoal_tac "\<forall>l2 \<in> list(A). length(l2) = x --> Cons(a,l2) \<in> B", blast)
```
```    91 apply (erule_tac a=a in wf_on_induct, assumption)
```
```    92 apply (rule ballI)
```
```    93 apply (rule impI)
```
```    94 apply (erule_tac a=l2 in wf_on_induct, blast, clarify)
```
```    95 apply (rename_tac a' l2 l')
```
```    96 apply (drule_tac x="Cons(a',l')" in bspec, typecheck)
```
```    97 apply simp
```
```    98 apply (erule mp, clarify)
```
```    99 apply (erule rlist_ConsE, auto)
```
```   100 done
```
```   101
```
```   102 lemma list_eq_UN_length: "list(A) = (\<Union>n\<in>nat. {l \<in> list(A). length(l) = n})"
```
```   103 by (blast intro: length_type)
```
```   104
```
```   105
```
```   106 lemma wf_on_rlist: "wf[A](r) ==> wf[list(A)](rlist(A,r))"
```
```   107 apply (subst list_eq_UN_length)
```
```   108 apply (rule wf_on_Union)
```
```   109   apply (rule wf_imp_wf_on [OF wf_Memrel [of nat]])
```
```   110  apply (simp add: wf_on_rlist_n)
```
```   111 apply (frule rlist_type [THEN subsetD])
```
```   112 apply (simp add: length_type)
```
```   113 apply (drule rlist_imp_length_le)
```
```   114 apply (erule leE)
```
```   115 apply (simp_all add: lt_def)
```
```   116 done
```
```   117
```
```   118
```
```   119 lemma wf_rlist: "wf(r) ==> wf(rlist(field(r),r))"
```
```   120 apply (simp add: wf_iff_wf_on_field)
```
```   121 apply (rule wf_on_subset_A [OF _ field_rlist])
```
```   122 apply (blast intro: wf_on_rlist)
```
```   123 done
```
```   124
```
```   125 lemma well_ord_rlist:
```
```   126      "well_ord(A,r) ==> well_ord(list(A), rlist(A,r))"
```
```   127 apply (rule well_ordI)
```
```   128 apply (simp add: well_ord_def wf_on_rlist)
```
```   129 apply (simp add: well_ord_def tot_ord_def linear_rlist)
```
```   130 done
```
```   131
```
```   132
```
```   133 subsection{*An Injection from Formulas into the Natural Numbers*}
```
```   134
```
```   135 text{*There is a well-known bijection between @{term "nat*nat"} and @{term
```
```   136 nat} given by the expression f(m,n) = triangle(m+n) + m, where triangle(k)
```
```   137 enumerates the triangular numbers and can be defined by triangle(0)=0,
```
```   138 triangle(succ(k)) = succ(k + triangle(k)).  Some small amount of effort is
```
```   139 needed to show that f is a bijection.  We already know that such a bijection exists by the theorem @{text well_ord_InfCard_square_eq}:
```
```   140 @{thm[display] well_ord_InfCard_square_eq[no_vars]}
```
```   141
```
```   142 However, this result merely states that there is a bijection between the two
```
```   143 sets.  It provides no means of naming a specific bijection.  Therefore, we
```
```   144 conduct the proofs under the assumption that a bijection exists.  The simplest
```
```   145 way to organize this is to use a locale.*}
```
```   146
```
```   147 text{*Locale for any arbitrary injection between @{term "nat*nat"}
```
```   148       and @{term nat}*}
```
```   149 locale Nat_Times_Nat =
```
```   150   fixes fn
```
```   151   assumes fn_inj: "fn \<in> inj(nat*nat, nat)"
```
```   152
```
```   153
```
```   154 consts   enum :: "[i,i]=>i"
```
```   155 primrec
```
```   156   "enum(f, Member(x,y)) = f ` <0, f ` <x,y>>"
```
```   157   "enum(f, Equal(x,y)) = f ` <1, f ` <x,y>>"
```
```   158   "enum(f, Nand(p,q)) = f ` <2, f ` <enum(f,p), enum(f,q)>>"
```
```   159   "enum(f, Forall(p)) = f ` <succ(2), enum(f,p)>"
```
```   160
```
```   161 lemma (in Nat_Times_Nat) fn_type [TC,simp]:
```
```   162     "[|x \<in> nat; y \<in> nat|] ==> fn`<x,y> \<in> nat"
```
```   163 by (blast intro: inj_is_fun [OF fn_inj] apply_funtype)
```
```   164
```
```   165 lemma (in Nat_Times_Nat) fn_iff:
```
```   166     "[|x \<in> nat; y \<in> nat; u \<in> nat; v \<in> nat|]
```
```   167      ==> (fn`<x,y> = fn`<u,v>) <-> (x=u & y=v)"
```
```   168 by (blast dest: inj_apply_equality [OF fn_inj])
```
```   169
```
```   170 lemma (in Nat_Times_Nat) enum_type [TC,simp]:
```
```   171     "p \<in> formula ==> enum(fn,p) \<in> nat"
```
```   172 by (induct_tac p, simp_all)
```
```   173
```
```   174 lemma (in Nat_Times_Nat) enum_inject [rule_format]:
```
```   175     "p \<in> formula ==> \<forall>q\<in>formula. enum(fn,p) = enum(fn,q) --> p=q"
```
```   176 apply (induct_tac p, simp_all)
```
```   177    apply (rule ballI)
```
```   178    apply (erule formula.cases)
```
```   179    apply (simp_all add: fn_iff)
```
```   180   apply (rule ballI)
```
```   181   apply (erule formula.cases)
```
```   182   apply (simp_all add: fn_iff)
```
```   183  apply (rule ballI)
```
```   184  apply (erule_tac a=qa in formula.cases)
```
```   185  apply (simp_all add: fn_iff)
```
```   186  apply blast
```
```   187 apply (rule ballI)
```
```   188 apply (erule_tac a=q in formula.cases)
```
```   189 apply (simp_all add: fn_iff, blast)
```
```   190 done
```
```   191
```
```   192 lemma (in Nat_Times_Nat) inj_formula_nat:
```
```   193     "(\<lambda>p \<in> formula. enum(fn,p)) \<in> inj(formula, nat)"
```
```   194 apply (simp add: inj_def lam_type)
```
```   195 apply (blast intro: enum_inject)
```
```   196 done
```
```   197
```
```   198 lemma (in Nat_Times_Nat) well_ord_formula:
```
```   199     "well_ord(formula, measure(formula, enum(fn)))"
```
```   200 apply (rule well_ord_measure, simp)
```
```   201 apply (blast intro: enum_inject)
```
```   202 done
```
```   203
```
```   204 lemmas nat_times_nat_lepoll_nat =
```
```   205     InfCard_nat [THEN InfCard_square_eqpoll, THEN eqpoll_imp_lepoll]
```
```   206
```
```   207
```
```   208 text{*Not needed--but interesting?*}
```
```   209 theorem formula_lepoll_nat: "formula \<lesssim> nat"
```
```   210 apply (insert nat_times_nat_lepoll_nat)
```
```   211 apply (unfold lepoll_def)
```
```   212 apply (blast intro: Nat_Times_Nat.inj_formula_nat Nat_Times_Nat.intro)
```
```   213 done
```
```   214
```
```   215
```
```   216 subsection{*Defining the Wellordering on @{term "DPow(A)"}*}
```
```   217
```
```   218 text{*The objective is to build a wellordering on @{term "DPow(A)"} from a
```
```   219 given one on @{term A}.  We first introduce wellorderings for environments,
```
```   220 which are lists built over @{term "A"}.  We combine it with the enumeration of
```
```   221 formulas.  The order type of the resulting wellordering gives us a map from
```
```   222 (environment, formula) pairs into the ordinals.  For each member of @{term
```
```   223 "DPow(A)"}, we take the minimum such ordinal.*}
```
```   224
```
```   225 constdefs
```
```   226   env_form_r :: "[i,i,i]=>i"
```
```   227     --{*wellordering on (environment, formula) pairs*}
```
```   228    "env_form_r(f,r,A) ==
```
```   229       rmult(list(A), rlist(A, r),
```
```   230 	    formula, measure(formula, enum(f)))"
```
```   231
```
```   232   env_form_map :: "[i,i,i,i]=>i"
```
```   233     --{*map from (environment, formula) pairs to ordinals*}
```
```   234    "env_form_map(f,r,A,z)
```
```   235       == ordermap(list(A) * formula, env_form_r(f,r,A)) ` z"
```
```   236
```
```   237   DPow_ord :: "[i,i,i,i,i]=>o"
```
```   238     --{*predicate that holds if @{term k} is a valid index for @{term X}*}
```
```   239    "DPow_ord(f,r,A,X,k) ==
```
```   240            \<exists>env \<in> list(A). \<exists>p \<in> formula.
```
```   241              arity(p) \<le> succ(length(env)) &
```
```   242              X = {x\<in>A. sats(A, p, Cons(x,env))} &
```
```   243              env_form_map(f,r,A,<env,p>) = k"
```
```   244
```
```   245   DPow_least :: "[i,i,i,i]=>i"
```
```   246     --{*function yielding the smallest index for @{term X}*}
```
```   247    "DPow_least(f,r,A,X) == \<mu> k. DPow_ord(f,r,A,X,k)"
```
```   248
```
```   249   DPow_r :: "[i,i,i]=>i"
```
```   250     --{*a wellordering on @{term "DPow(A)"}*}
```
```   251    "DPow_r(f,r,A) == measure(DPow(A), DPow_least(f,r,A))"
```
```   252
```
```   253
```
```   254 lemma (in Nat_Times_Nat) well_ord_env_form_r:
```
```   255     "well_ord(A,r)
```
```   256      ==> well_ord(list(A) * formula, env_form_r(fn,r,A))"
```
```   257 by (simp add: env_form_r_def well_ord_rmult well_ord_rlist well_ord_formula)
```
```   258
```
```   259 lemma (in Nat_Times_Nat) Ord_env_form_map:
```
```   260     "[|well_ord(A,r); z \<in> list(A) * formula|]
```
```   261      ==> Ord(env_form_map(fn,r,A,z))"
```
```   262 by (simp add: env_form_map_def Ord_ordermap well_ord_env_form_r)
```
```   263
```
```   264 lemma DPow_imp_ex_DPow_ord:
```
```   265     "X \<in> DPow(A) ==> \<exists>k. DPow_ord(fn,r,A,X,k)"
```
```   266 apply (simp add: DPow_ord_def)
```
```   267 apply (blast dest!: DPowD)
```
```   268 done
```
```   269
```
```   270 lemma (in Nat_Times_Nat) DPow_ord_imp_Ord:
```
```   271      "[|DPow_ord(fn,r,A,X,k); well_ord(A,r)|] ==> Ord(k)"
```
```   272 apply (simp add: DPow_ord_def, clarify)
```
```   273 apply (simp add: Ord_env_form_map)
```
```   274 done
```
```   275
```
```   276 lemma (in Nat_Times_Nat) DPow_imp_DPow_least:
```
```   277     "[|X \<in> DPow(A); well_ord(A,r)|]
```
```   278      ==> DPow_ord(fn, r, A, X, DPow_least(fn,r,A,X))"
```
```   279 apply (simp add: DPow_least_def)
```
```   280 apply (blast dest: DPow_imp_ex_DPow_ord intro: DPow_ord_imp_Ord LeastI)
```
```   281 done
```
```   282
```
```   283 lemma (in Nat_Times_Nat) env_form_map_inject:
```
```   284     "[|env_form_map(fn,r,A,u) = env_form_map(fn,r,A,v); well_ord(A,r);
```
```   285        u \<in> list(A) * formula;  v \<in> list(A) * formula|]
```
```   286      ==> u=v"
```
```   287 apply (simp add: env_form_map_def)
```
```   288 apply (rule inj_apply_equality [OF bij_is_inj, OF ordermap_bij,
```
```   289                                 OF well_ord_env_form_r], assumption+)
```
```   290 done
```
```   291
```
```   292 lemma (in Nat_Times_Nat) DPow_ord_unique:
```
```   293     "[|DPow_ord(fn,r,A,X,k); DPow_ord(fn,r,A,Y,k); well_ord(A,r)|]
```
```   294      ==> X=Y"
```
```   295 apply (simp add: DPow_ord_def, clarify)
```
```   296 apply (drule env_form_map_inject, auto)
```
```   297 done
```
```   298
```
```   299 lemma (in Nat_Times_Nat) well_ord_DPow_r:
```
```   300     "well_ord(A,r) ==> well_ord(DPow(A), DPow_r(fn,r,A))"
```
```   301 apply (simp add: DPow_r_def)
```
```   302 apply (rule well_ord_measure)
```
```   303  apply (simp add: DPow_least_def Ord_Least)
```
```   304 apply (drule DPow_imp_DPow_least, assumption)+
```
```   305 apply simp
```
```   306 apply (blast intro: DPow_ord_unique)
```
```   307 done
```
```   308
```
```   309 lemma (in Nat_Times_Nat) DPow_r_type:
```
```   310     "DPow_r(fn,r,A) \<subseteq> DPow(A) * DPow(A)"
```
```   311 by (simp add: DPow_r_def measure_def, blast)
```
```   312
```
```   313
```
```   314 subsection{*Limit Construction for Well-Orderings*}
```
```   315
```
```   316 text{*Now we work towards the transfinite definition of wellorderings for
```
```   317 @{term "Lset(i)"}.  We assume as an inductive hypothesis that there is a family
```
```   318 of wellorderings for smaller ordinals.*}
```
```   319
```
```   320 constdefs
```
```   321   rlimit :: "[i,i=>i]=>i"
```
```   322   --{*Expresses the wellordering at limit ordinals.  The conditional
```
```   323       lets us remove the premise @{term "Limit(i)"} from some theorems.*}
```
```   324     "rlimit(i,r) ==
```
```   325        if Limit(i) then
```
```   326 	 {z: Lset(i) * Lset(i).
```
```   327 	  \<exists>x' x. z = <x',x> &
```
```   328 		 (lrank(x') < lrank(x) |
```
```   329 		  (lrank(x') = lrank(x) & <x',x> \<in> r(succ(lrank(x)))))}
```
```   330        else 0"
```
```   331
```
```   332   Lset_new :: "i=>i"
```
```   333   --{*This constant denotes the set of elements introduced at level
```
```   334       @{term "succ(i)"}*}
```
```   335     "Lset_new(i) == {x \<in> Lset(succ(i)). lrank(x) = i}"
```
```   336
```
```   337 lemma Limit_Lset_eq2:
```
```   338     "Limit(i) ==> Lset(i) = (\<Union>j\<in>i. Lset_new(j))"
```
```   339 apply (simp add: Limit_Lset_eq)
```
```   340 apply (rule equalityI)
```
```   341  apply safe
```
```   342  apply (subgoal_tac "Ord(y)")
```
```   343   prefer 2 apply (blast intro: Ord_in_Ord Limit_is_Ord)
```
```   344  apply (simp_all add: Limit_is_Ord Lset_iff_lrank_lt Lset_new_def
```
```   345                       Ord_mem_iff_lt)
```
```   346  apply (blast intro: lt_trans)
```
```   347 apply (rule_tac x = "succ(lrank(x))" in bexI)
```
```   348  apply (simp add: Lset_succ_lrank_iff)
```
```   349 apply (blast intro: Limit_has_succ ltD)
```
```   350 done
```
```   351
```
```   352 lemma wf_on_Lset:
```
```   353     "wf[Lset(succ(j))](r(succ(j))) ==> wf[Lset_new(j)](rlimit(i,r))"
```
```   354 apply (simp add: wf_on_def Lset_new_def)
```
```   355 apply (erule wf_subset)
```
```   356 apply (simp add: rlimit_def, force)
```
```   357 done
```
```   358
```
```   359 lemma wf_on_rlimit:
```
```   360     "(\<forall>j<i. wf[Lset(j)](r(j))) ==> wf[Lset(i)](rlimit(i,r))"
```
```   361 apply (case_tac "Limit(i)")
```
```   362  prefer 2
```
```   363  apply (simp add: rlimit_def wf_on_any_0)
```
```   364 apply (simp add: Limit_Lset_eq2)
```
```   365 apply (rule wf_on_Union)
```
```   366   apply (rule wf_imp_wf_on [OF wf_Memrel [of i]])
```
```   367  apply (blast intro: wf_on_Lset Limit_has_succ Limit_is_Ord ltI)
```
```   368 apply (force simp add: rlimit_def Limit_is_Ord Lset_iff_lrank_lt Lset_new_def
```
```   369                        Ord_mem_iff_lt)
```
```   370 done
```
```   371
```
```   372 lemma linear_rlimit:
```
```   373     "[|Limit(i); \<forall>j<i. linear(Lset(j), r(j)) |]
```
```   374      ==> linear(Lset(i), rlimit(i,r))"
```
```   375 apply (frule Limit_is_Ord)
```
```   376 apply (simp add: Limit_Lset_eq2 Lset_new_def)
```
```   377 apply (simp add: linear_def rlimit_def Ball_def lt_Ord Lset_iff_lrank_lt)
```
```   378 apply (simp add: ltI, clarify)
```
```   379 apply (rename_tac u v)
```
```   380 apply (rule_tac i="lrank(u)" and j="lrank(v)" in Ord_linear_lt, simp_all)
```
```   381 apply (drule_tac x="succ(lrank(u) Un lrank(v))" in ospec)
```
```   382  apply (simp add: ltI)
```
```   383 apply (drule_tac x=u in spec, simp)
```
```   384 apply (drule_tac x=v in spec, simp)
```
```   385 done
```
```   386
```
```   387 lemma well_ord_rlimit:
```
```   388     "[|Limit(i); \<forall>j<i. well_ord(Lset(j), r(j)) |]
```
```   389      ==> well_ord(Lset(i), rlimit(i,r))"
```
```   390 by (blast intro: well_ordI wf_on_rlimit well_ord_is_wf
```
```   391                            linear_rlimit well_ord_is_linear)
```
```   392
```
```   393 lemma rlimit_cong:
```
```   394      "(!!j. j<i ==> r'(j) = r(j)) ==> rlimit(i,r) = rlimit(i,r')"
```
```   395 apply (simp add: rlimit_def, clarify)
```
```   396 apply (rule refl iff_refl Collect_cong ex_cong conj_cong)+
```
```   397 apply (simp add: Limit_is_Ord Lset_lrank_lt)
```
```   398 done
```
```   399
```
```   400
```
```   401 subsection{*Transfinite Definition of the Wellordering on @{term "L"}*}
```
```   402
```
```   403 constdefs
```
```   404  L_r :: "[i, i] => i"
```
```   405   "L_r(f) == %i.
```
```   406       transrec3(i, 0, \<lambda>x r. DPow_r(f, r, Lset(x)),
```
```   407                 \<lambda>x r. rlimit(x, \<lambda>y. r`y))"
```
```   408
```
```   409 subsubsection{*The Corresponding Recursion Equations*}
```
```   410 lemma [simp]: "L_r(f,0) = 0"
```
```   411 by (simp add: L_r_def)
```
```   412
```
```   413 lemma [simp]: "L_r(f, succ(i)) = DPow_r(f, L_r(f,i), Lset(i))"
```
```   414 by (simp add: L_r_def)
```
```   415
```
```   416 text{*The limit case is non-trivial because of the distinction between
```
```   417 object-level and meta-level abstraction.*}
```
```   418 lemma [simp]: "Limit(i) ==> L_r(f,i) = rlimit(i, L_r(f))"
```
```   419 by (simp cong: rlimit_cong add: transrec3_Limit L_r_def ltD)
```
```   420
```
```   421 lemma (in Nat_Times_Nat) L_r_type:
```
```   422     "Ord(i) ==> L_r(fn,i) \<subseteq> Lset(i) * Lset(i)"
```
```   423 apply (induct i rule: trans_induct3_rule)
```
```   424   apply (simp_all add: Lset_succ DPow_r_type well_ord_DPow_r rlimit_def
```
```   425                        Transset_subset_DPow [OF Transset_Lset], blast)
```
```   426 done
```
```   427
```
```   428 lemma (in Nat_Times_Nat) well_ord_L_r:
```
```   429     "Ord(i) ==> well_ord(Lset(i), L_r(fn,i))"
```
```   430 apply (induct i rule: trans_induct3_rule)
```
```   431 apply (simp_all add: well_ord0 Lset_succ L_r_type well_ord_DPow_r
```
```   432                      well_ord_rlimit ltD)
```
```   433 done
```
```   434
```
```   435 lemma well_ord_L_r:
```
```   436     "Ord(i) ==> \<exists>r. well_ord(Lset(i), r)"
```
```   437 apply (insert nat_times_nat_lepoll_nat)
```
```   438 apply (unfold lepoll_def)
```
```   439 apply (blast intro: Nat_Times_Nat.well_ord_L_r Nat_Times_Nat.intro)
```
```   440 done
```
```   441
```
```   442
```
```   443 text{*Locale for proving results under the assumption @{text "V=L"}*}
```
```   444 locale V_equals_L =
```
```   445   assumes VL: "L(x)"
```
```   446
```
```   447 text{*The Axiom of Choice holds in @{term L}!  Or, to be precise, the
```
```   448 Wellordering Theorem.*}
```
```   449 theorem (in V_equals_L) AC: "\<exists>r. well_ord(x,r)"
```
```   450 apply (insert Transset_Lset VL [of x])
```
```   451 apply (simp add: Transset_def L_def)
```
```   452 apply (blast dest!: well_ord_L_r intro: well_ord_subset)
```
```   453 done
```
```   454
```
```   455 end
```