src/ZF/Constructible/Separation.thy
 author wenzelm Thu Dec 14 11:24:26 2017 +0100 (21 months ago) changeset 67198 694f29a5433b parent 61798 27f3c10b0b50 child 67443 3abf6a722518 permissions -rw-r--r--
merged
```     1 (*  Title:      ZF/Constructible/Separation.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3 *)
```
```     4
```
```     5 section\<open>Early Instances of Separation and Strong Replacement\<close>
```
```     6
```
```     7 theory Separation imports L_axioms WF_absolute begin
```
```     8
```
```     9 text\<open>This theory proves all instances needed for locale \<open>M_basic\<close>\<close>
```
```    10
```
```    11 text\<open>Helps us solve for de Bruijn indices!\<close>
```
```    12 lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
```
```    13 by simp
```
```    14
```
```    15 lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
```
```    16 lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
```
```    17                    fun_plus_iff_sats
```
```    18
```
```    19 lemma Collect_conj_in_DPow:
```
```    20      "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
```
```    21       ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
```
```    22 by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
```
```    23
```
```    24 lemma Collect_conj_in_DPow_Lset:
```
```    25      "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
```
```    26       ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
```
```    27 apply (frule mem_Lset_imp_subset_Lset)
```
```    28 apply (simp add: Collect_conj_in_DPow Collect_mem_eq
```
```    29                  subset_Int_iff2 elem_subset_in_DPow)
```
```    30 done
```
```    31
```
```    32 lemma separation_CollectI:
```
```    33      "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
```
```    34 apply (unfold separation_def, clarify)
```
```    35 apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
```
```    36 apply simp_all
```
```    37 done
```
```    38
```
```    39 text\<open>Reduces the original comprehension to the reflected one\<close>
```
```    40 lemma reflection_imp_L_separation:
```
```    41       "[| \<forall>x\<in>Lset(j). P(x) \<longleftrightarrow> Q(x);
```
```    42           {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
```
```    43           Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
```
```    44 apply (rule_tac i = "succ(j)" in L_I)
```
```    45  prefer 2 apply simp
```
```    46 apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
```
```    47  prefer 2
```
```    48  apply (blast dest: mem_Lset_imp_subset_Lset)
```
```    49 apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
```
```    50 done
```
```    51
```
```    52 text\<open>Encapsulates the standard proof script for proving instances of
```
```    53       Separation.\<close>
```
```    54 lemma gen_separation:
```
```    55  assumes reflection: "REFLECTS [P,Q]"
```
```    56      and Lu:         "L(u)"
```
```    57      and collI: "!!j. u \<in> Lset(j)
```
```    58                 \<Longrightarrow> Collect(Lset(j), Q(j)) \<in> DPow(Lset(j))"
```
```    59  shows "separation(L,P)"
```
```    60 apply (rule separation_CollectI)
```
```    61 apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu)
```
```    62 apply (rule ReflectsE [OF reflection], assumption)
```
```    63 apply (drule subset_Lset_ltD, assumption)
```
```    64 apply (erule reflection_imp_L_separation)
```
```    65   apply (simp_all add: lt_Ord2, clarify)
```
```    66 apply (rule collI, assumption)
```
```    67 done
```
```    68
```
```    69 text\<open>As above, but typically @{term u} is a finite enumeration such as
```
```    70   @{term "{a,b}"}; thus the new subgoal gets the assumption
```
```    71   @{term "{a,b} \<subseteq> Lset(i)"}, which is logically equivalent to
```
```    72   @{term "a \<in> Lset(i)"} and @{term "b \<in> Lset(i)"}.\<close>
```
```    73 lemma gen_separation_multi:
```
```    74  assumes reflection: "REFLECTS [P,Q]"
```
```    75      and Lu:         "L(u)"
```
```    76      and collI: "!!j. u \<subseteq> Lset(j)
```
```    77                 \<Longrightarrow> Collect(Lset(j), Q(j)) \<in> DPow(Lset(j))"
```
```    78  shows "separation(L,P)"
```
```    79 apply (rule gen_separation [OF reflection Lu])
```
```    80 apply (drule mem_Lset_imp_subset_Lset)
```
```    81 apply (erule collI)
```
```    82 done
```
```    83
```
```    84
```
```    85 subsection\<open>Separation for Intersection\<close>
```
```    86
```
```    87 lemma Inter_Reflects:
```
```    88      "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A \<longrightarrow> x \<in> y,
```
```    89                \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A \<longrightarrow> x \<in> y]"
```
```    90 by (intro FOL_reflections)
```
```    91
```
```    92 lemma Inter_separation:
```
```    93      "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A \<longrightarrow> x\<in>y)"
```
```    94 apply (rule gen_separation [OF Inter_Reflects], simp)
```
```    95 apply (rule DPow_LsetI)
```
```    96  txt\<open>I leave this one example of a manual proof.  The tedium of manually
```
```    97       instantiating @{term i}, @{term j} and @{term env} is obvious.\<close>
```
```    98 apply (rule ball_iff_sats)
```
```    99 apply (rule imp_iff_sats)
```
```   100 apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
```
```   101 apply (rule_tac i=0 and j=2 in mem_iff_sats)
```
```   102 apply (simp_all add: succ_Un_distrib [symmetric])
```
```   103 done
```
```   104
```
```   105 subsection\<open>Separation for Set Difference\<close>
```
```   106
```
```   107 lemma Diff_Reflects:
```
```   108      "REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]"
```
```   109 by (intro FOL_reflections)
```
```   110
```
```   111 lemma Diff_separation:
```
```   112      "L(B) ==> separation(L, \<lambda>x. x \<notin> B)"
```
```   113 apply (rule gen_separation [OF Diff_Reflects], simp)
```
```   114 apply (rule_tac env="[B]" in DPow_LsetI)
```
```   115 apply (rule sep_rules | simp)+
```
```   116 done
```
```   117
```
```   118 subsection\<open>Separation for Cartesian Product\<close>
```
```   119
```
```   120 lemma cartprod_Reflects:
```
```   121      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
```
```   122                 \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
```
```   123                                    pair(##Lset(i),x,y,z))]"
```
```   124 by (intro FOL_reflections function_reflections)
```
```   125
```
```   126 lemma cartprod_separation:
```
```   127      "[| L(A); L(B) |]
```
```   128       ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
```
```   129 apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto)
```
```   130 apply (rule_tac env="[A,B]" in DPow_LsetI)
```
```   131 apply (rule sep_rules | simp)+
```
```   132 done
```
```   133
```
```   134 subsection\<open>Separation for Image\<close>
```
```   135
```
```   136 lemma image_Reflects:
```
```   137      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
```
```   138            \<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))]"
```
```   139 by (intro FOL_reflections function_reflections)
```
```   140
```
```   141 lemma image_separation:
```
```   142      "[| L(A); L(r) |]
```
```   143       ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
```
```   144 apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto)
```
```   145 apply (rule_tac env="[A,r]" in DPow_LsetI)
```
```   146 apply (rule sep_rules | simp)+
```
```   147 done
```
```   148
```
```   149
```
```   150 subsection\<open>Separation for Converse\<close>
```
```   151
```
```   152 lemma converse_Reflects:
```
```   153   "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)),
```
```   154      \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
```
```   155                      pair(##Lset(i),x,y,p) & pair(##Lset(i),y,x,z))]"
```
```   156 by (intro FOL_reflections function_reflections)
```
```   157
```
```   158 lemma converse_separation:
```
```   159      "L(r) ==> separation(L,
```
```   160          \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
```
```   161 apply (rule gen_separation [OF converse_Reflects], simp)
```
```   162 apply (rule_tac env="[r]" in DPow_LsetI)
```
```   163 apply (rule sep_rules | simp)+
```
```   164 done
```
```   165
```
```   166
```
```   167 subsection\<open>Separation for Restriction\<close>
```
```   168
```
```   169 lemma restrict_Reflects:
```
```   170      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
```
```   171         \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(##Lset(i),x,y,z))]"
```
```   172 by (intro FOL_reflections function_reflections)
```
```   173
```
```   174 lemma restrict_separation:
```
```   175    "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
```
```   176 apply (rule gen_separation [OF restrict_Reflects], simp)
```
```   177 apply (rule_tac env="[A]" in DPow_LsetI)
```
```   178 apply (rule sep_rules | simp)+
```
```   179 done
```
```   180
```
```   181
```
```   182 subsection\<open>Separation for Composition\<close>
```
```   183
```
```   184 lemma comp_Reflects:
```
```   185      "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   186                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   187                   xy\<in>s & yz\<in>r,
```
```   188         \<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).
```
```   189                   pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) &
```
```   190                   pair(##Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
```
```   191 by (intro FOL_reflections function_reflections)
```
```   192
```
```   193 lemma comp_separation:
```
```   194      "[| L(r); L(s) |]
```
```   195       ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
```
```   196                   pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
```
```   197                   xy\<in>s & yz\<in>r)"
```
```   198 apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto)
```
```   199 txt\<open>Subgoals after applying general ``separation'' rule:
```
```   200      @{subgoals[display,indent=0,margin=65]}\<close>
```
```   201 apply (rule_tac env="[r,s]" in DPow_LsetI)
```
```   202 txt\<open>Subgoals ready for automatic synthesis of a formula:
```
```   203      @{subgoals[display,indent=0,margin=65]}\<close>
```
```   204 apply (rule sep_rules | simp)+
```
```   205 done
```
```   206
```
```   207
```
```   208 subsection\<open>Separation for Predecessors in an Order\<close>
```
```   209
```
```   210 lemma pred_Reflects:
```
```   211      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
```
```   212                     \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(##Lset(i),y,x,p)]"
```
```   213 by (intro FOL_reflections function_reflections)
```
```   214
```
```   215 lemma pred_separation:
```
```   216      "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
```
```   217 apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto)
```
```   218 apply (rule_tac env="[r,x]" in DPow_LsetI)
```
```   219 apply (rule sep_rules | simp)+
```
```   220 done
```
```   221
```
```   222
```
```   223 subsection\<open>Separation for the Membership Relation\<close>
```
```   224
```
```   225 lemma Memrel_Reflects:
```
```   226      "REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
```
```   227             \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(##Lset(i),x,y,z) & x \<in> y]"
```
```   228 by (intro FOL_reflections function_reflections)
```
```   229
```
```   230 lemma Memrel_separation:
```
```   231      "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
```
```   232 apply (rule gen_separation [OF Memrel_Reflects nonempty])
```
```   233 apply (rule_tac env="[]" in DPow_LsetI)
```
```   234 apply (rule sep_rules | simp)+
```
```   235 done
```
```   236
```
```   237
```
```   238 subsection\<open>Replacement for FunSpace\<close>
```
```   239
```
```   240 lemma funspace_succ_Reflects:
```
```   241  "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
```
```   242             pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
```
```   243             upair(L,cnbf,cnbf,z)),
```
```   244         \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
```
```   245               \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
```
```   246                 pair(##Lset(i),f,b,p) & pair(##Lset(i),n,b,nb) &
```
```   247                 is_cons(##Lset(i),nb,f,cnbf) & upair(##Lset(i),cnbf,cnbf,z))]"
```
```   248 by (intro FOL_reflections function_reflections)
```
```   249
```
```   250 lemma funspace_succ_replacement:
```
```   251      "L(n) ==>
```
```   252       strong_replacement(L, \<lambda>p z. \<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 apply (rule strong_replacementI)
```
```   256 apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects],
```
```   257        auto)
```
```   258 apply (rule_tac env="[n,B]" in DPow_LsetI)
```
```   259 apply (rule sep_rules | simp)+
```
```   260 done
```
```   261
```
```   262
```
```   263 subsection\<open>Separation for a Theorem about @{term "is_recfun"}\<close>
```
```   264
```
```   265 lemma is_recfun_reflects:
```
```   266   "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   267                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   268                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   269                                    fx \<noteq> gx),
```
```   270    \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
```
```   271           pair(##Lset(i),x,a,xa) & xa \<in> r & pair(##Lset(i),x,b,xb) & xb \<in> r &
```
```   272                 (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(##Lset(i),f,x,fx) &
```
```   273                   fun_apply(##Lset(i),g,x,gx) & fx \<noteq> gx)]"
```
```   274 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```   275
```
```   276 lemma is_recfun_separation:
```
```   277      \<comment>\<open>for well-founded recursion\<close>
```
```   278      "[| L(r); L(f); L(g); L(a); L(b) |]
```
```   279      ==> separation(L,
```
```   280             \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
```
```   281                 pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
```
```   282                 (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
```
```   283                                    fx \<noteq> gx))"
```
```   284 apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"],
```
```   285             auto)
```
```   286 apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI)
```
```   287 apply (rule sep_rules | simp)+
```
```   288 done
```
```   289
```
```   290
```
```   291 subsection\<open>Instantiating the locale \<open>M_basic\<close>\<close>
```
```   292 text\<open>Separation (and Strong Replacement) for basic set-theoretic constructions
```
```   293 such as intersection, Cartesian Product and image.\<close>
```
```   294
```
```   295 lemma M_basic_axioms_L: "M_basic_axioms(L)"
```
```   296   apply (rule M_basic_axioms.intro)
```
```   297        apply (assumption | rule
```
```   298          Inter_separation Diff_separation cartprod_separation image_separation
```
```   299          converse_separation restrict_separation
```
```   300          comp_separation pred_separation Memrel_separation
```
```   301          funspace_succ_replacement is_recfun_separation)+
```
```   302   done
```
```   303
```
```   304 theorem M_basic_L: "PROP M_basic(L)"
```
```   305 by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])
```
```   306
```
```   307 interpretation L?: M_basic L by (rule M_basic_L)
```
```   308
```
```   309
```
```   310 end
```