src/ZF/Constructible/Separation.thy
 author wenzelm Mon Jul 29 18:07:53 2002 +0200 (2002-07-29) changeset 13429 2232810416fc parent 13428 99e52e78eb65 child 13437 01b3fc0cc1b8 permissions -rw-r--r--
tuned;
```     1 header{*Early Instances of Separation and Strong Replacement*}
```
```     2
```
```     3 theory Separation = L_axioms + WF_absolute:
```
```     4
```
```     5 text{*This theory proves all instances needed for locale @{text "M_axioms"}*}
```
```     6
```
```     7 text{*Helps us solve for de Bruijn indices!*}
```
```     8 lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
```
```     9 by simp
```
```    10
```
```    11 lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
```
```    12 lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
```
```    13                    fun_plus_iff_sats
```
```    14
```
```    15 lemma Collect_conj_in_DPow:
```
```    16      "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
```
```    17       ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
```
```    18 by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
```
```    19
```
```    20 lemma Collect_conj_in_DPow_Lset:
```
```    21      "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
```
```    22       ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
```
```    23 apply (frule mem_Lset_imp_subset_Lset)
```
```    24 apply (simp add: Collect_conj_in_DPow Collect_mem_eq
```
```    25                  subset_Int_iff2 elem_subset_in_DPow)
```
```    26 done
```
```    27
```
```    28 lemma separation_CollectI:
```
```    29      "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
```
```    30 apply (unfold separation_def, clarify)
```
```    31 apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
```
```    32 apply simp_all
```
```    33 done
```
```    34
```
```    35 text{*Reduces the original comprehension to the reflected one*}
```
```    36 lemma reflection_imp_L_separation:
```
```    37       "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
```
```    38           {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
```
```    39           Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
```
```    40 apply (rule_tac i = "succ(j)" in L_I)
```
```    41  prefer 2 apply simp
```
```    42 apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
```
```    43  prefer 2
```
```    44  apply (blast dest: mem_Lset_imp_subset_Lset)
```
```    45 apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
```
```    46 done
```
```    47
```
```    48
```
```    49 subsection{*Separation for Intersection*}
```
```    50
```
```    51 lemma Inter_Reflects:
```
```    52      "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
```
```    53                \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]"
```
```    54 by (intro FOL_reflections)
```
```    55
```
```    56 lemma Inter_separation:
```
```    57      "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
```
```    58 apply (rule separation_CollectI)
```
```    59 apply (rule_tac A="{A,z}" in subset_LsetE, blast )
```
```    60 apply (rule ReflectsE [OF Inter_Reflects], assumption)
```
```    61 apply (drule subset_Lset_ltD, assumption)
```
```    62 apply (erule reflection_imp_L_separation)
```
```    63   apply (simp_all add: lt_Ord2, clarify)
```
```    64 apply (rule DPow_LsetI)
```
```    65 apply (rule ball_iff_sats)
```
```    66 apply (rule imp_iff_sats)
```
```    67 apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
```
```    68 apply (rule_tac i=0 and j=2 in mem_iff_sats)
```
```    69 apply (simp_all add: succ_Un_distrib [symmetric])
```
```    70 done
```
```    71
```
```    72 subsection{*Separation for Cartesian Product*}
```
```    73
```
```    74 lemma cartprod_Reflects:
```
```    75      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
```
```    76                 \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
```
```    77                                    pair(**Lset(i),x,y,z))]"
```
```    78 by (intro FOL_reflections function_reflections)
```
```    79
```
```    80 lemma cartprod_separation:
```
```    81      "[| L(A); L(B) |]
```
```    82       ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
```
```    83 apply (rule separation_CollectI)
```
```    84 apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
```
```    85 apply (rule ReflectsE [OF cartprod_Reflects], assumption)
```
```    86 apply (drule subset_Lset_ltD, assumption)
```
```    87 apply (erule reflection_imp_L_separation)
```
```    88   apply (simp_all add: lt_Ord2, clarify)
```
```    89 apply (rule DPow_LsetI)
```
```    90 apply (rename_tac u)
```
```    91 apply (rule bex_iff_sats)
```
```    92 apply (rule conj_iff_sats)
```
```    93 apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
```
```    94 apply (rule sep_rules | simp)+
```
```    95 done
```
```    96
```
```    97 subsection{*Separation for Image*}
```
```    98
```
```    99 lemma image_Reflects:
```
```   100      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
```
```   101            \<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))]"
```
```   102 by (intro FOL_reflections function_reflections)
```
```   103
```
```   104 lemma image_separation:
```
```   105      "[| L(A); L(r) |]
```
```   106       ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
```
```   107 apply (rule separation_CollectI)
```
```   108 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
```
```   109 apply (rule ReflectsE [OF image_Reflects], assumption)
```
```   110 apply (drule subset_Lset_ltD, assumption)
```
```   111 apply (erule reflection_imp_L_separation)
```
```   112   apply (simp_all add: lt_Ord2, clarify)
```
```   113 apply (rule DPow_LsetI)
```
```   114 apply (rule bex_iff_sats)
```
```   115 apply (rule conj_iff_sats)
```
```   116 apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
```
```   117 apply (rule sep_rules | simp)+
```
```   118 done
```
```   119
```
```   120
```
```   121 subsection{*Separation for Converse*}
```
```   122
```
```   123 lemma converse_Reflects:
```
```   124   "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)),
```
```   125      \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
```
```   126                      pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]"
```
```   127 by (intro FOL_reflections function_reflections)
```
```   128
```
```   129 lemma converse_separation:
```
```   130      "L(r) ==> separation(L,
```
```   131          \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
```
```   132 apply (rule separation_CollectI)
```
```   133 apply (rule_tac A="{r,z}" in subset_LsetE, blast )
```
```   134 apply (rule ReflectsE [OF converse_Reflects], assumption)
```
```   135 apply (drule subset_Lset_ltD, assumption)
```
```   136 apply (erule reflection_imp_L_separation)
```
```   137   apply (simp_all add: lt_Ord2, clarify)
```
```   138 apply (rule DPow_LsetI)
```
```   139 apply (rename_tac u)
```
```   140 apply (rule bex_iff_sats)
```
```   141 apply (rule conj_iff_sats)
```
```   142 apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all)
```
```   143 apply (rule sep_rules | simp)+
```
```   144 done
```
```   145
```
```   146
```
```   147 subsection{*Separation for Restriction*}
```
```   148
```
```   149 lemma restrict_Reflects:
```
```   150      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
```
```   151         \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]"
```
```   152 by (intro FOL_reflections function_reflections)
```
```   153
```
```   154 lemma restrict_separation:
```
```   155    "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
```
```   156 apply (rule separation_CollectI)
```
```   157 apply (rule_tac A="{A,z}" in subset_LsetE, blast )
```
```   158 apply (rule ReflectsE [OF restrict_Reflects], assumption)
```
```   159 apply (drule subset_Lset_ltD, assumption)
```
```   160 apply (erule reflection_imp_L_separation)
```
```   161   apply (simp_all add: lt_Ord2, clarify)
```
```   162 apply (rule DPow_LsetI)
```
```   163 apply (rename_tac u)
```
```   164 apply (rule bex_iff_sats)
```
```   165 apply (rule conj_iff_sats)
```
```   166 apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all)
```
```   167 apply (rule sep_rules | simp)+
```
```   168 done
```
```   169
```
```   170
```
```   171 subsection{*Separation for Composition*}
```
```   172
```
```   173 lemma comp_Reflects:
```
```   174      "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   175                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   176                   xy\<in>s & yz\<in>r,
```
```   177         \<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).
```
```   178                   pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
```
```   179                   pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
```
```   180 by (intro FOL_reflections function_reflections)
```
```   181
```
```   182 lemma comp_separation:
```
```   183      "[| L(r); L(s) |]
```
```   184       ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   185                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   186                   xy\<in>s & yz\<in>r)"
```
```   187 apply (rule separation_CollectI)
```
```   188 apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
```
```   189 apply (rule ReflectsE [OF comp_Reflects], assumption)
```
```   190 apply (drule subset_Lset_ltD, assumption)
```
```   191 apply (erule reflection_imp_L_separation)
```
```   192   apply (simp_all add: lt_Ord2, clarify)
```
```   193 apply (rule DPow_LsetI)
```
```   194 apply (rename_tac u)
```
```   195 apply (rule bex_iff_sats)+
```
```   196 apply (rename_tac x y z)
```
```   197 apply (rule conj_iff_sats)
```
```   198 apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
```
```   199 apply (rule sep_rules | simp)+
```
```   200 done
```
```   201
```
```   202 subsection{*Separation for Predecessors in an Order*}
```
```   203
```
```   204 lemma pred_Reflects:
```
```   205      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
```
```   206                     \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]"
```
```   207 by (intro FOL_reflections function_reflections)
```
```   208
```
```   209 lemma pred_separation:
```
```   210      "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
```
```   211 apply (rule separation_CollectI)
```
```   212 apply (rule_tac A="{r,x,z}" in subset_LsetE, blast )
```
```   213 apply (rule ReflectsE [OF pred_Reflects], assumption)
```
```   214 apply (drule subset_Lset_ltD, assumption)
```
```   215 apply (erule reflection_imp_L_separation)
```
```   216   apply (simp_all add: lt_Ord2, clarify)
```
```   217 apply (rule DPow_LsetI)
```
```   218 apply (rename_tac u)
```
```   219 apply (rule bex_iff_sats)
```
```   220 apply (rule conj_iff_sats)
```
```   221 apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats)
```
```   222 apply (rule sep_rules | simp)+
```
```   223 done
```
```   224
```
```   225
```
```   226 subsection{*Separation for the Membership Relation*}
```
```   227
```
```   228 lemma Memrel_Reflects:
```
```   229      "REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
```
```   230             \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]"
```
```   231 by (intro FOL_reflections function_reflections)
```
```   232
```
```   233 lemma Memrel_separation:
```
```   234      "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
```
```   235 apply (rule separation_CollectI)
```
```   236 apply (rule_tac A="{z}" in subset_LsetE, blast )
```
```   237 apply (rule ReflectsE [OF Memrel_Reflects], assumption)
```
```   238 apply (drule subset_Lset_ltD, assumption)
```
```   239 apply (erule reflection_imp_L_separation)
```
```   240   apply (simp_all add: lt_Ord2)
```
```   241 apply (rule DPow_LsetI)
```
```   242 apply (rename_tac u)
```
```   243 apply (rule bex_iff_sats conj_iff_sats)+
```
```   244 apply (rule_tac env = "[y,x,u]" in pair_iff_sats)
```
```   245 apply (rule sep_rules | simp)+
```
```   246 done
```
```   247
```
```   248
```
```   249 subsection{*Replacement for FunSpace*}
```
```   250
```
```   251 lemma funspace_succ_Reflects:
```
```   252  "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
```
```   253             pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
```
```   254             upair(L,cnbf,cnbf,z)),
```
```   255         \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
```
```   256               \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
```
```   257                 pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) &
```
```   258                 is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
```
```   259 by (intro FOL_reflections function_reflections)
```
```   260
```
```   261 lemma funspace_succ_replacement:
```
```   262      "L(n) ==>
```
```   263       strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
```
```   264                 pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
```
```   265                 upair(L,cnbf,cnbf,z))"
```
```   266 apply (rule strong_replacementI)
```
```   267 apply (rule rallI)
```
```   268 apply (rule separation_CollectI)
```
```   269 apply (rule_tac A="{n,A,z}" in subset_LsetE, blast )
```
```   270 apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
```
```   271 apply (drule subset_Lset_ltD, assumption)
```
```   272 apply (erule reflection_imp_L_separation)
```
```   273   apply (simp_all add: lt_Ord2)
```
```   274 apply (rule DPow_LsetI)
```
```   275 apply (rename_tac u)
```
```   276 apply (rule bex_iff_sats)
```
```   277 apply (rule conj_iff_sats)
```
```   278 apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats)
```
```   279 apply (rule sep_rules | simp)+
```
```   280 done
```
```   281
```
```   282
```
```   283 subsection{*Separation for Order-Isomorphisms*}
```
```   284
```
```   285 lemma well_ord_iso_Reflects:
```
```   286   "REFLECTS[\<lambda>x. x\<in>A -->
```
```   287                 (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
```
```   288         \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
```
```   289                 fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
```
```   290 by (intro FOL_reflections function_reflections)
```
```   291
```
```   292 lemma well_ord_iso_separation:
```
```   293      "[| L(A); L(f); L(r) |]
```
```   294       ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
```
```   295                      fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
```
```   296 apply (rule separation_CollectI)
```
```   297 apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast )
```
```   298 apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
```
```   299 apply (drule subset_Lset_ltD, assumption)
```
```   300 apply (erule reflection_imp_L_separation)
```
```   301   apply (simp_all add: lt_Ord2)
```
```   302 apply (rule DPow_LsetI)
```
```   303 apply (rename_tac u)
```
```   304 apply (rule imp_iff_sats)
```
```   305 apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats)
```
```   306 apply (rule sep_rules | simp)+
```
```   307 done
```
```   308
```
```   309
```
```   310 subsection{*Separation for @{term "obase"}*}
```
```   311
```
```   312 lemma obase_reflects:
```
```   313   "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   314              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```   315              order_isomorphism(L,par,r,x,mx,g),
```
```   316         \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
```
```   317              ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
```
```   318              order_isomorphism(**Lset(i),par,r,x,mx,g)]"
```
```   319 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   320
```
```   321 lemma obase_separation:
```
```   322      --{*part of the order type formalization*}
```
```   323      "[| L(A); L(r) |]
```
```   324       ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   325              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```   326              order_isomorphism(L,par,r,x,mx,g))"
```
```   327 apply (rule separation_CollectI)
```
```   328 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
```
```   329 apply (rule ReflectsE [OF obase_reflects], assumption)
```
```   330 apply (drule subset_Lset_ltD, assumption)
```
```   331 apply (erule reflection_imp_L_separation)
```
```   332   apply (simp_all add: lt_Ord2)
```
```   333 apply (rule DPow_LsetI)
```
```   334 apply (rename_tac u)
```
```   335 apply (rule bex_iff_sats)
```
```   336 apply (rule conj_iff_sats)
```
```   337 apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats)
```
```   338 apply (rule sep_rules | simp)+
```
```   339 done
```
```   340
```
```   341
```
```   342 subsection{*Separation for a Theorem about @{term "obase"}*}
```
```   343
```
```   344 lemma obase_equals_reflects:
```
```   345   "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```   346                 ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```   347                 membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```   348                 order_isomorphism(L,pxr,r,y,my,g))),
```
```   349         \<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
```
```   350                 ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
```
```   351                 membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
```
```   352                 order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
```
```   353 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   354
```
```   355
```
```   356 lemma obase_equals_separation:
```
```   357      "[| L(A); L(r) |]
```
```   358       ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```   359                               ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```   360                               membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```   361                               order_isomorphism(L,pxr,r,y,my,g))))"
```
```   362 apply (rule separation_CollectI)
```
```   363 apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
```
```   364 apply (rule ReflectsE [OF obase_equals_reflects], assumption)
```
```   365 apply (drule subset_Lset_ltD, assumption)
```
```   366 apply (erule reflection_imp_L_separation)
```
```   367   apply (simp_all add: lt_Ord2)
```
```   368 apply (rule DPow_LsetI)
```
```   369 apply (rename_tac u)
```
```   370 apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
```
```   371 apply (rule_tac env = "[u,A,r]" in mem_iff_sats)
```
```   372 apply (rule sep_rules | simp)+
```
```   373 done
```
```   374
```
```   375
```
```   376 subsection{*Replacement for @{term "omap"}*}
```
```   377
```
```   378 lemma omap_reflects:
```
```   379  "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   380      ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   381      pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
```
```   382  \<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).
```
```   383         \<exists>par \<in> Lset(i).
```
```   384          ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) &
```
```   385          membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
```
```   386          order_isomorphism(**Lset(i),par,r,x,mx,g))]"
```
```   387 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   388
```
```   389 lemma omap_replacement:
```
```   390      "[| L(A); L(r) |]
```
```   391       ==> strong_replacement(L,
```
```   392              \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   393              ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   394              pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
```
```   395 apply (rule strong_replacementI)
```
```   396 apply (rule rallI)
```
```   397 apply (rename_tac B)
```
```   398 apply (rule separation_CollectI)
```
```   399 apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast )
```
```   400 apply (rule ReflectsE [OF omap_reflects], assumption)
```
```   401 apply (drule subset_Lset_ltD, assumption)
```
```   402 apply (erule reflection_imp_L_separation)
```
```   403   apply (simp_all add: lt_Ord2)
```
```   404 apply (rule DPow_LsetI)
```
```   405 apply (rename_tac u)
```
```   406 apply (rule bex_iff_sats conj_iff_sats)+
```
```   407 apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats)
```
```   408 apply (rule sep_rules | simp)+
```
```   409 done
```
```   410
```
```   411
```
```   412 subsection{*Separation for a Theorem about @{term "obase"}*}
```
```   413
```
```   414 lemma is_recfun_reflects:
```
```   415   "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   416                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   417                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   418                                    fx \<noteq> gx),
```
```   419    \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
```
```   420           pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
```
```   421                 (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
```
```   422                   fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
```
```   423 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   424
```
```   425 lemma is_recfun_separation:
```
```   426      --{*for well-founded recursion*}
```
```   427      "[| L(r); L(f); L(g); L(a); L(b) |]
```
```   428      ==> separation(L,
```
```   429             \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   430                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   431                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   432                                    fx \<noteq> gx))"
```
```   433 apply (rule separation_CollectI)
```
```   434 apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast )
```
```   435 apply (rule ReflectsE [OF is_recfun_reflects], assumption)
```
```   436 apply (drule subset_Lset_ltD, assumption)
```
```   437 apply (erule reflection_imp_L_separation)
```
```   438   apply (simp_all add: lt_Ord2)
```
```   439 apply (rule DPow_LsetI)
```
```   440 apply (rename_tac u)
```
```   441 apply (rule bex_iff_sats conj_iff_sats)+
```
```   442 apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats)
```
```   443 apply (rule sep_rules | simp)+
```
```   444 done
```
```   445
```
```   446
```
```   447 subsection{*Instantiating the locale @{text M_axioms}*}
```
```   448 text{*Separation (and Strong Replacement) for basic set-theoretic constructions
```
```   449 such as intersection, Cartesian Product and image.*}
```
```   450
```
```   451 theorem M_axioms_L: "PROP M_axioms(L)"
```
```   452   apply (rule M_axioms.intro)
```
```   453    apply (rule M_triv_axioms_L)
```
```   454   apply (rule M_axioms_axioms.intro)
```
```   455                apply (assumption | rule
```
```   456                  Inter_separation cartprod_separation image_separation
```
```   457                  converse_separation restrict_separation
```
```   458                  comp_separation pred_separation Memrel_separation
```
```   459                  funspace_succ_replacement well_ord_iso_separation
```
```   460                  obase_separation obase_equals_separation
```
```   461                  omap_replacement is_recfun_separation)+
```
```   462   done
```
```   463
```
```   464 lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L]
```
```   465   and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L]
```
```   466   and sum_closed = M_axioms.sum_closed [OF M_axioms_L]
```
```   467   and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L]
```
```   468   and converse_closed = M_axioms.converse_closed [OF M_axioms_L]
```
```   469   and converse_abs = M_axioms.converse_abs [OF M_axioms_L]
```
```   470   and image_closed = M_axioms.image_closed [OF M_axioms_L]
```
```   471   and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L]
```
```   472   and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L]
```
```   473   and domain_abs = M_axioms.domain_abs [OF M_axioms_L]
```
```   474   and domain_closed = M_axioms.domain_closed [OF M_axioms_L]
```
```   475   and range_abs = M_axioms.range_abs [OF M_axioms_L]
```
```   476   and range_closed = M_axioms.range_closed [OF M_axioms_L]
```
```   477   and field_abs = M_axioms.field_abs [OF M_axioms_L]
```
```   478   and field_closed = M_axioms.field_closed [OF M_axioms_L]
```
```   479   and relation_abs = M_axioms.relation_abs [OF M_axioms_L]
```
```   480   and function_abs = M_axioms.function_abs [OF M_axioms_L]
```
```   481   and apply_closed = M_axioms.apply_closed [OF M_axioms_L]
```
```   482   and apply_abs = M_axioms.apply_abs [OF M_axioms_L]
```
```   483   and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L]
```
```   484   and injection_abs = M_axioms.injection_abs [OF M_axioms_L]
```
```   485   and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L]
```
```   486   and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L]
```
```   487   and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L]
```
```   488   and comp_closed = M_axioms.comp_closed [OF M_axioms_L]
```
```   489   and composition_abs = M_axioms.composition_abs [OF M_axioms_L]
```
```   490   and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L]
```
```   491   and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L]
```
```   492   and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L]
```
```   493   and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L]
```
```   494   and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L]
```
```   495   and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L]
```
```   496   and Int_closed = M_axioms.Int_closed [OF M_axioms_L]
```
```   497   and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L]
```
```   498   and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L]
```
```   499   and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L]
```
```   500   and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L]
```
```   501   and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L]
```
```   502
```
```   503 lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L]
```
```   504   and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L]
```
```   505   and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L]
```
```   506   and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L]
```
```   507   and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L]
```
```   508   and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L]
```
```   509   and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L]
```
```   510   and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L]
```
```   511   and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L]
```
```   512   and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L]
```
```   513   and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L]
```
```   514   and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L]
```
```   515   and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L]
```
```   516   and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L]
```
```   517   and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L]
```
```   518   and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L]
```
```   519   and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L]
```
```   520   and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L]
```
```   521   and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L]
```
```   522   and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L]
```
```   523   and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L]
```
```   524   and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L]
```
```   525   and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L]
```
```   526   and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L]
```
```   527   and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L]
```
```   528   and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L]
```
```   529   and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L]
```
```   530   and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L]
```
```   531   and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L]
```
```   532   and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L]
```
```   533   and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L]
```
```   534   and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L]
```
```   535
```
```   536 lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L]
```
```   537   and membership_abs = M_axioms.membership_abs [OF M_axioms_L]
```
```   538   and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L]
```
```   539   and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L]
```
```   540   and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L]
```
```   541   and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L]
```
```   542   and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L]
```
```   543   and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L]
```
```   544   and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L]
```
```   545   and obase_iff = M_axioms.obase_iff [OF M_axioms_L]
```
```   546   and omap_iff = M_axioms.omap_iff [OF M_axioms_L]
```
```   547   and omap_unique = M_axioms.omap_unique [OF M_axioms_L]
```
```   548   and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L]
```
```   549   and otype_iff = M_axioms.otype_iff [OF M_axioms_L]
```
```   550   and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L]
```
```   551   and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L]
```
```   552   and domain_omap = M_axioms.domain_omap [OF M_axioms_L]
```
```   553   and omap_subset = M_axioms.omap_subset [OF M_axioms_L]
```
```   554   and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L]
```
```   555   and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L]
```
```   556   and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L]
```
```   557   and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L]
```
```   558   and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L]
```
```   559   and obase_equals = M_axioms.obase_equals [OF M_axioms_L]
```
```   560   and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L]
```
```   561   and obase_exists = M_axioms.obase_exists [OF M_axioms_L]
```
```   562   and omap_exists = M_axioms.omap_exists [OF M_axioms_L]
```
```   563   and otype_exists = M_axioms.otype_exists [OF M_axioms_L]
```
```   564   and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L]
```
```   565   and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L]
```
```   566   and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L]
```
```   567   and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L]
```
```   568
```
```   569 declare cartprod_closed [intro, simp]
```
```   570 declare sum_closed [intro, simp]
```
```   571 declare converse_closed [intro, simp]
```
```   572 declare converse_abs [simp]
```
```   573 declare image_closed [intro, simp]
```
```   574 declare vimage_abs [simp]
```
```   575 declare vimage_closed [intro, simp]
```
```   576 declare domain_abs [simp]
```
```   577 declare domain_closed [intro, simp]
```
```   578 declare range_abs [simp]
```
```   579 declare range_closed [intro, simp]
```
```   580 declare field_abs [simp]
```
```   581 declare field_closed [intro, simp]
```
```   582 declare relation_abs [simp]
```
```   583 declare function_abs [simp]
```
```   584 declare apply_closed [intro, simp]
```
```   585 declare typed_function_abs [simp]
```
```   586 declare injection_abs [simp]
```
```   587 declare surjection_abs [simp]
```
```   588 declare bijection_abs [simp]
```
```   589 declare comp_closed [intro, simp]
```
```   590 declare composition_abs [simp]
```
```   591 declare restriction_abs [simp]
```
```   592 declare restrict_closed [intro, simp]
```
```   593 declare Inter_abs [simp]
```
```   594 declare Inter_closed [intro, simp]
```
```   595 declare Int_closed [intro, simp]
```
```   596 declare is_funspace_abs [simp]
```
```   597 declare finite_funspace_closed [intro, simp]
```
```   598
```
```   599 end
```