src/ZF/Constructible/Separation.thy
 author paulson Wed Aug 21 15:57:24 2002 +0200 (2002-08-21) changeset 13513 b9e14471629c parent 13505 52a16cb7fefb child 13564 1500a2e48d44 permissions -rw-r--r--
tweaks
```     1 (*  Title:      ZF/Constructible/Separation.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   2002  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header{*Early Instances of Separation and Strong Replacement*}
```
```     8
```
```     9 theory Separation = L_axioms + WF_absolute:
```
```    10
```
```    11 text{*This theory proves all instances needed for locale @{text "M_axioms"}*}
```
```    12
```
```    13 text{*Helps us solve for de Bruijn indices!*}
```
```    14 lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
```
```    15 by simp
```
```    16
```
```    17 lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
```
```    18 lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
```
```    19                    fun_plus_iff_sats
```
```    20
```
```    21 lemma Collect_conj_in_DPow:
```
```    22      "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
```
```    23       ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
```
```    24 by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
```
```    25
```
```    26 lemma Collect_conj_in_DPow_Lset:
```
```    27      "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
```
```    28       ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
```
```    29 apply (frule mem_Lset_imp_subset_Lset)
```
```    30 apply (simp add: Collect_conj_in_DPow Collect_mem_eq
```
```    31                  subset_Int_iff2 elem_subset_in_DPow)
```
```    32 done
```
```    33
```
```    34 lemma separation_CollectI:
```
```    35      "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
```
```    36 apply (unfold separation_def, clarify)
```
```    37 apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
```
```    38 apply simp_all
```
```    39 done
```
```    40
```
```    41 text{*Reduces the original comprehension to the reflected one*}
```
```    42 lemma reflection_imp_L_separation:
```
```    43       "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
```
```    44           {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
```
```    45           Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
```
```    46 apply (rule_tac i = "succ(j)" in L_I)
```
```    47  prefer 2 apply simp
```
```    48 apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
```
```    49  prefer 2
```
```    50  apply (blast dest: mem_Lset_imp_subset_Lset)
```
```    51 apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
```
```    52 done
```
```    53
```
```    54
```
```    55 subsection{*Separation for Intersection*}
```
```    56
```
```    57 lemma Inter_Reflects:
```
```    58      "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
```
```    59                \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]"
```
```    60 by (intro FOL_reflections)
```
```    61
```
```    62 lemma Inter_separation:
```
```    63      "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
```
```    64 apply (rule separation_CollectI)
```
```    65 apply (rule_tac A="{A,z}" in subset_LsetE, blast)
```
```    66 apply (rule ReflectsE [OF Inter_Reflects], assumption)
```
```    67 apply (drule subset_Lset_ltD, assumption)
```
```    68 apply (erule reflection_imp_L_separation)
```
```    69   apply (simp_all add: lt_Ord2, clarify)
```
```    70 apply (rule DPow_LsetI)
```
```    71 apply (rule ball_iff_sats)
```
```    72 apply (rule imp_iff_sats)
```
```    73 apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
```
```    74 apply (rule_tac i=0 and j=2 in mem_iff_sats)
```
```    75 apply (simp_all add: succ_Un_distrib [symmetric])
```
```    76 done
```
```    77
```
```    78 subsection{*Separation for Set Difference*}
```
```    79
```
```    80 lemma Diff_Reflects:
```
```    81      "REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]"
```
```    82 by (intro FOL_reflections)
```
```    83
```
```    84 lemma Diff_separation:
```
```    85      "L(B) ==> separation(L, \<lambda>x. x \<notin> B)"
```
```    86 apply (rule separation_CollectI)
```
```    87 apply (rule_tac A="{B,z}" in subset_LsetE, blast)
```
```    88 apply (rule ReflectsE [OF Diff_Reflects], assumption)
```
```    89 apply (drule subset_Lset_ltD, assumption)
```
```    90 apply (erule reflection_imp_L_separation)
```
```    91   apply (simp_all add: lt_Ord2, clarify)
```
```    92 apply (rule DPow_LsetI)
```
```    93 apply (rule not_iff_sats)
```
```    94 apply (rule_tac env="[x,B]" in mem_iff_sats)
```
```    95 apply (rule sep_rules | simp)+
```
```    96 done
```
```    97
```
```    98 subsection{*Separation for Cartesian Product*}
```
```    99
```
```   100 lemma cartprod_Reflects:
```
```   101      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
```
```   102                 \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
```
```   103                                    pair(**Lset(i),x,y,z))]"
```
```   104 by (intro FOL_reflections function_reflections)
```
```   105
```
```   106 lemma cartprod_separation:
```
```   107      "[| L(A); L(B) |]
```
```   108       ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
```
```   109 apply (rule separation_CollectI)
```
```   110 apply (rule_tac A="{A,B,z}" in subset_LsetE, blast)
```
```   111 apply (rule ReflectsE [OF cartprod_Reflects], assumption)
```
```   112 apply (drule subset_Lset_ltD, assumption)
```
```   113 apply (erule reflection_imp_L_separation)
```
```   114   apply (simp_all add: lt_Ord2, clarify)
```
```   115 apply (rule DPow_LsetI)
```
```   116 apply (rename_tac u)
```
```   117 apply (rule bex_iff_sats)
```
```   118 apply (rule conj_iff_sats)
```
```   119 apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
```
```   120 apply (rule sep_rules | simp)+
```
```   121 done
```
```   122
```
```   123 subsection{*Separation for Image*}
```
```   124
```
```   125 lemma image_Reflects:
```
```   126      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
```
```   127            \<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p))]"
```
```   128 by (intro FOL_reflections function_reflections)
```
```   129
```
```   130 lemma image_separation:
```
```   131      "[| L(A); L(r) |]
```
```   132       ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
```
```   133 apply (rule separation_CollectI)
```
```   134 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
```
```   135 apply (rule ReflectsE [OF image_Reflects], assumption)
```
```   136 apply (drule subset_Lset_ltD, assumption)
```
```   137 apply (erule reflection_imp_L_separation)
```
```   138   apply (simp_all add: lt_Ord2, clarify)
```
```   139 apply (rule DPow_LsetI)
```
```   140 apply (rule bex_iff_sats)
```
```   141 apply (rule conj_iff_sats)
```
```   142 apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
```
```   143 apply (rule sep_rules | simp)+
```
```   144 done
```
```   145
```
```   146
```
```   147 subsection{*Separation for Converse*}
```
```   148
```
```   149 lemma converse_Reflects:
```
```   150   "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
```
```   151      \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
```
```   152                      pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]"
```
```   153 by (intro FOL_reflections function_reflections)
```
```   154
```
```   155 lemma converse_separation:
```
```   156      "L(r) ==> separation(L,
```
```   157          \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
```
```   158 apply (rule separation_CollectI)
```
```   159 apply (rule_tac A="{r,z}" in subset_LsetE, blast)
```
```   160 apply (rule ReflectsE [OF converse_Reflects], assumption)
```
```   161 apply (drule subset_Lset_ltD, assumption)
```
```   162 apply (erule reflection_imp_L_separation)
```
```   163   apply (simp_all add: lt_Ord2, clarify)
```
```   164 apply (rule DPow_LsetI)
```
```   165 apply (rename_tac u)
```
```   166 apply (rule bex_iff_sats)
```
```   167 apply (rule conj_iff_sats)
```
```   168 apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all)
```
```   169 apply (rule sep_rules | simp)+
```
```   170 done
```
```   171
```
```   172
```
```   173 subsection{*Separation for Restriction*}
```
```   174
```
```   175 lemma restrict_Reflects:
```
```   176      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
```
```   177         \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]"
```
```   178 by (intro FOL_reflections function_reflections)
```
```   179
```
```   180 lemma restrict_separation:
```
```   181    "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
```
```   182 apply (rule separation_CollectI)
```
```   183 apply (rule_tac A="{A,z}" in subset_LsetE, blast)
```
```   184 apply (rule ReflectsE [OF restrict_Reflects], assumption)
```
```   185 apply (drule subset_Lset_ltD, assumption)
```
```   186 apply (erule reflection_imp_L_separation)
```
```   187   apply (simp_all add: lt_Ord2, clarify)
```
```   188 apply (rule DPow_LsetI)
```
```   189 apply (rename_tac u)
```
```   190 apply (rule bex_iff_sats)
```
```   191 apply (rule conj_iff_sats)
```
```   192 apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all)
```
```   193 apply (rule sep_rules | simp)+
```
```   194 done
```
```   195
```
```   196
```
```   197 subsection{*Separation for Composition*}
```
```   198
```
```   199 lemma comp_Reflects:
```
```   200      "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   201                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   202                   xy\<in>s & yz\<in>r,
```
```   203         \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
```
```   204                   pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
```
```   205                   pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
```
```   206 by (intro FOL_reflections function_reflections)
```
```   207
```
```   208 lemma comp_separation:
```
```   209      "[| L(r); L(s) |]
```
```   210       ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   211                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   212                   xy\<in>s & yz\<in>r)"
```
```   213 apply (rule separation_CollectI)
```
```   214 apply (rule_tac A="{r,s,z}" in subset_LsetE, blast)
```
```   215 apply (rule ReflectsE [OF comp_Reflects], assumption)
```
```   216 apply (drule subset_Lset_ltD, assumption)
```
```   217 apply (erule reflection_imp_L_separation)
```
```   218   apply (simp_all add: lt_Ord2, clarify)
```
```   219 apply (rule DPow_LsetI)
```
```   220 apply (rename_tac u)
```
```   221 apply (rule bex_iff_sats)+
```
```   222 apply (rename_tac x y z)
```
```   223 apply (rule conj_iff_sats)
```
```   224 apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
```
```   225 apply (rule sep_rules | simp)+
```
```   226 done
```
```   227
```
```   228 subsection{*Separation for Predecessors in an Order*}
```
```   229
```
```   230 lemma pred_Reflects:
```
```   231      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
```
```   232                     \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]"
```
```   233 by (intro FOL_reflections function_reflections)
```
```   234
```
```   235 lemma pred_separation:
```
```   236      "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
```
```   237 apply (rule separation_CollectI)
```
```   238 apply (rule_tac A="{r,x,z}" in subset_LsetE, blast)
```
```   239 apply (rule ReflectsE [OF pred_Reflects], assumption)
```
```   240 apply (drule subset_Lset_ltD, assumption)
```
```   241 apply (erule reflection_imp_L_separation)
```
```   242   apply (simp_all add: lt_Ord2, clarify)
```
```   243 apply (rule DPow_LsetI)
```
```   244 apply (rename_tac u)
```
```   245 apply (rule bex_iff_sats)
```
```   246 apply (rule conj_iff_sats)
```
```   247 apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats)
```
```   248 apply (rule sep_rules | simp)+
```
```   249 done
```
```   250
```
```   251
```
```   252 subsection{*Separation for the Membership Relation*}
```
```   253
```
```   254 lemma Memrel_Reflects:
```
```   255      "REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
```
```   256             \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]"
```
```   257 by (intro FOL_reflections function_reflections)
```
```   258
```
```   259 lemma Memrel_separation:
```
```   260      "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
```
```   261 apply (rule separation_CollectI)
```
```   262 apply (rule_tac A="{z}" in subset_LsetE, blast)
```
```   263 apply (rule ReflectsE [OF Memrel_Reflects], assumption)
```
```   264 apply (drule subset_Lset_ltD, assumption)
```
```   265 apply (erule reflection_imp_L_separation)
```
```   266   apply (simp_all add: lt_Ord2)
```
```   267 apply (rule DPow_LsetI)
```
```   268 apply (rename_tac u)
```
```   269 apply (rule bex_iff_sats conj_iff_sats)+
```
```   270 apply (rule_tac env = "[y,x,u]" in pair_iff_sats)
```
```   271 apply (rule sep_rules | simp)+
```
```   272 done
```
```   273
```
```   274
```
```   275 subsection{*Replacement for FunSpace*}
```
```   276
```
```   277 lemma funspace_succ_Reflects:
```
```   278  "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
```
```   279             pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
```
```   280             upair(L,cnbf,cnbf,z)),
```
```   281         \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
```
```   282               \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
```
```   283                 pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) &
```
```   284                 is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
```
```   285 by (intro FOL_reflections function_reflections)
```
```   286
```
```   287 lemma funspace_succ_replacement:
```
```   288      "L(n) ==>
```
```   289       strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
```
```   290                 pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
```
```   291                 upair(L,cnbf,cnbf,z))"
```
```   292 apply (rule strong_replacementI)
```
```   293 apply (rule rallI)
```
```   294 apply (rule separation_CollectI)
```
```   295 apply (rule_tac A="{n,A,z}" in subset_LsetE, blast)
```
```   296 apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
```
```   297 apply (drule subset_Lset_ltD, assumption)
```
```   298 apply (erule reflection_imp_L_separation)
```
```   299   apply (simp_all add: lt_Ord2)
```
```   300 apply (rule DPow_LsetI)
```
```   301 apply (rename_tac u)
```
```   302 apply (rule bex_iff_sats)
```
```   303 apply (rule conj_iff_sats)
```
```   304 apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats)
```
```   305 apply (rule sep_rules | simp)+
```
```   306 done
```
```   307
```
```   308
```
```   309 subsection{*Separation for Order-Isomorphisms*}
```
```   310
```
```   311 lemma well_ord_iso_Reflects:
```
```   312   "REFLECTS[\<lambda>x. x\<in>A -->
```
```   313                 (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
```
```   314         \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
```
```   315                 fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
```
```   316 by (intro FOL_reflections function_reflections)
```
```   317
```
```   318 lemma well_ord_iso_separation:
```
```   319      "[| L(A); L(f); L(r) |]
```
```   320       ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
```
```   321                      fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
```
```   322 apply (rule separation_CollectI)
```
```   323 apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast)
```
```   324 apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
```
```   325 apply (drule subset_Lset_ltD, assumption)
```
```   326 apply (erule reflection_imp_L_separation)
```
```   327   apply (simp_all add: lt_Ord2)
```
```   328 apply (rule DPow_LsetI)
```
```   329 apply (rename_tac u)
```
```   330 apply (rule imp_iff_sats)
```
```   331 apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats)
```
```   332 apply (rule sep_rules | simp)+
```
```   333 done
```
```   334
```
```   335
```
```   336 subsection{*Separation for @{term "obase"}*}
```
```   337
```
```   338 lemma obase_reflects:
```
```   339   "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   340              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```   341              order_isomorphism(L,par,r,x,mx,g),
```
```   342         \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
```
```   343              ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
```
```   344              order_isomorphism(**Lset(i),par,r,x,mx,g)]"
```
```   345 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   346
```
```   347 lemma obase_separation:
```
```   348      --{*part of the order type formalization*}
```
```   349      "[| L(A); L(r) |]
```
```   350       ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   351              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```   352              order_isomorphism(L,par,r,x,mx,g))"
```
```   353 apply (rule separation_CollectI)
```
```   354 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
```
```   355 apply (rule ReflectsE [OF obase_reflects], assumption)
```
```   356 apply (drule subset_Lset_ltD, assumption)
```
```   357 apply (erule reflection_imp_L_separation)
```
```   358   apply (simp_all add: lt_Ord2)
```
```   359 apply (rule DPow_LsetI)
```
```   360 apply (rename_tac u)
```
```   361 apply (rule bex_iff_sats)
```
```   362 apply (rule conj_iff_sats)
```
```   363 apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats)
```
```   364 apply (rule sep_rules | simp)+
```
```   365 done
```
```   366
```
```   367
```
```   368 subsection{*Separation for a Theorem about @{term "obase"}*}
```
```   369
```
```   370 lemma obase_equals_reflects:
```
```   371   "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```   372                 ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```   373                 membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```   374                 order_isomorphism(L,pxr,r,y,my,g))),
```
```   375         \<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
```
```   376                 ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
```
```   377                 membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
```
```   378                 order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
```
```   379 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   380
```
```   381
```
```   382 lemma obase_equals_separation:
```
```   383      "[| L(A); L(r) |]
```
```   384       ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```   385                               ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```   386                               membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```   387                               order_isomorphism(L,pxr,r,y,my,g))))"
```
```   388 apply (rule separation_CollectI)
```
```   389 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
```
```   390 apply (rule ReflectsE [OF obase_equals_reflects], assumption)
```
```   391 apply (drule subset_Lset_ltD, assumption)
```
```   392 apply (erule reflection_imp_L_separation)
```
```   393   apply (simp_all add: lt_Ord2)
```
```   394 apply (rule DPow_LsetI)
```
```   395 apply (rename_tac u)
```
```   396 apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
```
```   397 apply (rule_tac env = "[u,A,r]" in mem_iff_sats)
```
```   398 apply (rule sep_rules | simp)+
```
```   399 done
```
```   400
```
```   401
```
```   402 subsection{*Replacement for @{term "omap"}*}
```
```   403
```
```   404 lemma omap_reflects:
```
```   405  "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   406      ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   407      pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
```
```   408  \<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i).
```
```   409         \<exists>par \<in> Lset(i).
```
```   410          ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) &
```
```   411          membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
```
```   412          order_isomorphism(**Lset(i),par,r,x,mx,g))]"
```
```   413 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   414
```
```   415 lemma omap_replacement:
```
```   416      "[| L(A); L(r) |]
```
```   417       ==> strong_replacement(L,
```
```   418              \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   419              ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   420              pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
```
```   421 apply (rule strong_replacementI)
```
```   422 apply (rule rallI)
```
```   423 apply (rename_tac B)
```
```   424 apply (rule separation_CollectI)
```
```   425 apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast)
```
```   426 apply (rule ReflectsE [OF omap_reflects], assumption)
```
```   427 apply (drule subset_Lset_ltD, assumption)
```
```   428 apply (erule reflection_imp_L_separation)
```
```   429   apply (simp_all add: lt_Ord2)
```
```   430 apply (rule DPow_LsetI)
```
```   431 apply (rename_tac u)
```
```   432 apply (rule bex_iff_sats conj_iff_sats)+
```
```   433 apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats)
```
```   434 apply (rule sep_rules | simp)+
```
```   435 done
```
```   436
```
```   437
```
```   438 subsection{*Separation for a Theorem about @{term "obase"}*}
```
```   439
```
```   440 lemma is_recfun_reflects:
```
```   441   "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   442                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   443                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   444                                    fx \<noteq> gx),
```
```   445    \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
```
```   446           pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
```
```   447                 (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
```
```   448                   fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
```
```   449 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   450
```
```   451 lemma is_recfun_separation:
```
```   452      --{*for well-founded recursion*}
```
```   453      "[| L(r); L(f); L(g); L(a); L(b) |]
```
```   454      ==> separation(L,
```
```   455             \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   456                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   457                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   458                                    fx \<noteq> gx))"
```
```   459 apply (rule separation_CollectI)
```
```   460 apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast)
```
```   461 apply (rule ReflectsE [OF is_recfun_reflects], assumption)
```
```   462 apply (drule subset_Lset_ltD, assumption)
```
```   463 apply (erule reflection_imp_L_separation)
```
```   464   apply (simp_all add: lt_Ord2)
```
```   465 apply (rule DPow_LsetI)
```
```   466 apply (rename_tac u)
```
```   467 apply (rule bex_iff_sats conj_iff_sats)+
```
```   468 apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats)
```
```   469 apply (rule sep_rules | simp)+
```
```   470 done
```
```   471
```
```   472
```
```   473 subsection{*Instantiating the locale @{text M_axioms}*}
```
```   474 text{*Separation (and Strong Replacement) for basic set-theoretic constructions
```
```   475 such as intersection, Cartesian Product and image.*}
```
```   476
```
```   477 lemma M_axioms_axioms_L: "M_axioms_axioms(L)"
```
```   478   apply (rule M_axioms_axioms.intro)
```
```   479        apply (assumption | rule
```
```   480 	 Inter_separation Diff_separation cartprod_separation image_separation
```
```   481 	 converse_separation restrict_separation
```
```   482 	 comp_separation pred_separation Memrel_separation
```
```   483 	 funspace_succ_replacement well_ord_iso_separation
```
```   484 	 obase_separation obase_equals_separation
```
```   485 	 omap_replacement is_recfun_separation)+
```
```   486   done
```
```   487
```
```   488 theorem M_axioms_L: "PROP M_axioms(L)"
```
```   489 by (rule M_axioms.intro [OF M_triv_axioms_L M_axioms_axioms_L])
```
```   490
```
```   491
```
```   492 lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L]
```
```   493   and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L]
```
```   494   and sum_closed = M_axioms.sum_closed [OF M_axioms_L]
```
```   495   and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L]
```
```   496   and converse_closed = M_axioms.converse_closed [OF M_axioms_L]
```
```   497   and converse_abs = M_axioms.converse_abs [OF M_axioms_L]
```
```   498   and image_closed = M_axioms.image_closed [OF M_axioms_L]
```
```   499   and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L]
```
```   500   and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L]
```
```   501   and domain_abs = M_axioms.domain_abs [OF M_axioms_L]
```
```   502   and domain_closed = M_axioms.domain_closed [OF M_axioms_L]
```
```   503   and range_abs = M_axioms.range_abs [OF M_axioms_L]
```
```   504   and range_closed = M_axioms.range_closed [OF M_axioms_L]
```
```   505   and field_abs = M_axioms.field_abs [OF M_axioms_L]
```
```   506   and field_closed = M_axioms.field_closed [OF M_axioms_L]
```
```   507   and relation_abs = M_axioms.relation_abs [OF M_axioms_L]
```
```   508   and function_abs = M_axioms.function_abs [OF M_axioms_L]
```
```   509   and apply_closed = M_axioms.apply_closed [OF M_axioms_L]
```
```   510   and apply_abs = M_axioms.apply_abs [OF M_axioms_L]
```
```   511   and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L]
```
```   512   and injection_abs = M_axioms.injection_abs [OF M_axioms_L]
```
```   513   and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L]
```
```   514   and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L]
```
```   515   and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L]
```
```   516   and comp_closed = M_axioms.comp_closed [OF M_axioms_L]
```
```   517   and composition_abs = M_axioms.composition_abs [OF M_axioms_L]
```
```   518   and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L]
```
```   519   and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L]
```
```   520   and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L]
```
```   521   and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L]
```
```   522   and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L]
```
```   523   and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L]
```
```   524   and Int_closed = M_axioms.Int_closed [OF M_axioms_L]
```
```   525   and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L]
```
```   526   and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L]
```
```   527   and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L]
```
```   528   and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L]
```
```   529   and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L]
```
```   530
```
```   531 lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L]
```
```   532   and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L]
```
```   533   and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L]
```
```   534   and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L]
```
```   535   and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L]
```
```   536   and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L]
```
```   537   and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L]
```
```   538   and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L]
```
```   539   and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L]
```
```   540   and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L]
```
```   541   and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L]
```
```   542   and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L]
```
```   543   and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L]
```
```   544   and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L]
```
```   545   and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L]
```
```   546   and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L]
```
```   547   and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L]
```
```   548   and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L]
```
```   549   and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L]
```
```   550   and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L]
```
```   551   and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L]
```
```   552   and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L]
```
```   553   and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L]
```
```   554   and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L]
```
```   555   and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L]
```
```   556   and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L]
```
```   557   and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L]
```
```   558   and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L]
```
```   559   and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L]
```
```   560   and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L]
```
```   561   and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L]
```
```   562   and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L]
```
```   563
```
```   564 lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L]
```
```   565   and membership_abs = M_axioms.membership_abs [OF M_axioms_L]
```
```   566   and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L]
```
```   567   and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L]
```
```   568   and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L]
```
```   569   and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L]
```
```   570   and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L]
```
```   571   and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L]
```
```   572   and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L]
```
```   573   and obase_iff = M_axioms.obase_iff [OF M_axioms_L]
```
```   574   and omap_iff = M_axioms.omap_iff [OF M_axioms_L]
```
```   575   and omap_unique = M_axioms.omap_unique [OF M_axioms_L]
```
```   576   and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L]
```
```   577   and otype_iff = M_axioms.otype_iff [OF M_axioms_L]
```
```   578   and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L]
```
```   579   and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L]
```
```   580   and domain_omap = M_axioms.domain_omap [OF M_axioms_L]
```
```   581   and omap_subset = M_axioms.omap_subset [OF M_axioms_L]
```
```   582   and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L]
```
```   583   and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L]
```
```   584   and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L]
```
```   585   and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L]
```
```   586   and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L]
```
```   587   and obase_equals = M_axioms.obase_equals [OF M_axioms_L]
```
```   588   and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L]
```
```   589   and obase_exists = M_axioms.obase_exists [OF M_axioms_L]
```
```   590   and omap_exists = M_axioms.omap_exists [OF M_axioms_L]
```
```   591   and otype_exists = M_axioms.otype_exists [OF M_axioms_L]
```
```   592   and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L]
```
```   593   and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L]
```
```   594   and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L]
```
```   595   and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L]
```
```   596
```
```   597 declare cartprod_closed [intro, simp]
```
```   598 declare sum_closed [intro, simp]
```
```   599 declare converse_closed [intro, simp]
```
```   600 declare converse_abs [simp]
```
```   601 declare image_closed [intro, simp]
```
```   602 declare vimage_abs [simp]
```
```   603 declare vimage_closed [intro, simp]
```
```   604 declare domain_abs [simp]
```
```   605 declare domain_closed [intro, simp]
```
```   606 declare range_abs [simp]
```
```   607 declare range_closed [intro, simp]
```
```   608 declare field_abs [simp]
```
```   609 declare field_closed [intro, simp]
```
```   610 declare relation_abs [simp]
```
```   611 declare function_abs [simp]
```
```   612 declare apply_closed [intro, simp]
```
```   613 declare typed_function_abs [simp]
```
```   614 declare injection_abs [simp]
```
```   615 declare surjection_abs [simp]
```
```   616 declare bijection_abs [simp]
```
```   617 declare comp_closed [intro, simp]
```
```   618 declare composition_abs [simp]
```
```   619 declare restriction_abs [simp]
```
```   620 declare restrict_closed [intro, simp]
```
```   621 declare Inter_abs [simp]
```
```   622 declare Inter_closed [intro, simp]
```
```   623 declare Int_closed [intro, simp]
```
```   624 declare is_funspace_abs [simp]
```
```   625 declare finite_funspace_closed [intro, simp]
```
```   626 declare membership_abs [simp]
```
```   627 declare Memrel_closed  [intro,simp]
```
```   628
```
```   629 end
```