src/ZF/Constructible/L_axioms.thy
 changeset 13298 b4f370679c65 parent 13291 a73ab154f75c child 13299 3a932abf97e8
```     1.1 --- a/src/ZF/Constructible/L_axioms.thy	Thu Jul 04 16:48:21 2002 +0200
1.2 +++ b/src/ZF/Constructible/L_axioms.thy	Thu Jul 04 16:59:54 2002 +0200
1.3 @@ -288,30 +288,344 @@
1.4  by blast
1.5
1.6
1.7 +subsection{*Internalized formulas for some relativized ones*}
1.8 +
1.9 +subsubsection{*Unordered pairs*}
1.10 +
1.11 +constdefs upair_fm :: "[i,i,i]=>i"
1.12 +    "upair_fm(x,y,z) ==
1.13 +       And(Member(x,z),
1.14 +           And(Member(y,z),
1.15 +               Forall(Implies(Member(0,succ(z)),
1.16 +                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
1.17 +
1.18 +lemma upair_type [TC]:
1.19 +     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
1.21 +
1.22 +lemma arity_upair_fm [simp]:
1.23 +     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1.24 +      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1.25 +by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
1.26 +
1.27 +lemma sats_upair_fm [simp]:
1.28 +   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1.29 +    ==> sats(A, upair_fm(x,y,z), env) <->
1.30 +            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
1.31 +by (simp add: upair_fm_def upair_def)
1.32 +
1.33 +lemma upair_iff_sats:
1.34 +      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1.35 +          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1.36 +       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
1.38 +
1.39 +text{*Useful? At least it refers to "real" unordered pairs*}
1.40 +lemma sats_upair_fm2 [simp]:
1.41 +   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
1.42 +    ==> sats(A, upair_fm(x,y,z), env) <->
1.43 +        nth(z,env) = {nth(x,env), nth(y,env)}"
1.44 +apply (frule lt_length_in_nat, assumption)
1.45 +apply (simp add: upair_fm_def Transset_def, auto)
1.46 +apply (blast intro: nth_type)
1.47 +done
1.48 +
1.49 +subsubsection{*Ordered pairs*}
1.50 +
1.51 +constdefs pair_fm :: "[i,i,i]=>i"
1.52 +    "pair_fm(x,y,z) ==
1.53 +       Exists(And(upair_fm(succ(x),succ(x),0),
1.54 +              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
1.55 +                         upair_fm(1,0,succ(succ(z)))))))"
1.56 +
1.57 +lemma pair_type [TC]:
1.58 +     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
1.60 +
1.61 +lemma arity_pair_fm [simp]:
1.62 +     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
1.63 +      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
1.64 +by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
1.65 +
1.66 +lemma sats_pair_fm [simp]:
1.67 +   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
1.68 +    ==> sats(A, pair_fm(x,y,z), env) <->
1.69 +        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
1.70 +by (simp add: pair_fm_def pair_def)
1.71 +
1.72 +lemma pair_iff_sats:
1.73 +      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
1.74 +          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
1.75 +       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
1.77 +
1.78 +
1.79 +
1.80 +subsection{*Proving instances of Separation using Reflection!*}
1.81 +
1.82 +text{*Helps us solve for de Bruijn indices!*}
1.83 +lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
1.84 +by simp
1.85 +
1.86 +
1.87 +lemma Collect_conj_in_DPow:
1.88 +     "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
1.89 +      ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
1.90 +by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
1.91 +
1.92 +lemma Collect_conj_in_DPow_Lset:
1.93 +     "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
1.94 +      ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
1.95 +apply (frule mem_Lset_imp_subset_Lset)
1.96 +apply (simp add: Collect_conj_in_DPow Collect_mem_eq
1.97 +                 subset_Int_iff2 elem_subset_in_DPow)
1.98 +done
1.99 +
1.100 +lemma separation_CollectI:
1.101 +     "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
1.102 +apply (unfold separation_def, clarify)
1.103 +apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
1.104 +apply simp_all
1.105 +done
1.106 +
1.107 +text{*Reduces the original comprehension to the reflected one*}
1.108 +lemma reflection_imp_L_separation:
1.109 +      "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
1.110 +          {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
1.111 +          Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
1.112 +apply (rule_tac i = "succ(j)" in L_I)
1.113 + prefer 2 apply simp
1.114 +apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
1.115 + prefer 2
1.116 + apply (blast dest: mem_Lset_imp_subset_Lset)
1.117 +apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
1.118 +done
1.119 +
1.120 +
1.121 +subsubsection{*Separation for Intersection*}
1.122 +
1.123 +lemma Inter_Reflects:
1.124 +     "L_Reflects(?Cl, \<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
1.125 +               \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y)"
1.126 +by fast
1.127 +
1.128 +lemma Inter_separation:
1.129 +     "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
1.130 +apply (rule separation_CollectI)
1.131 +apply (rule_tac A="{A,z}" in subset_LsetE, blast )
1.132 +apply (rule ReflectsE [OF Inter_Reflects], assumption)
1.133 +apply (drule subset_Lset_ltD, assumption)
1.134 +apply (erule reflection_imp_L_separation)
1.135 +  apply (simp_all add: lt_Ord2, clarify)
1.136 +apply (rule DPowI2)
1.137 +apply (rule ball_iff_sats)
1.138 +apply (rule imp_iff_sats)
1.139 +apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
1.140 +apply (rule_tac i=0 and j=2 in mem_iff_sats)
1.141 +apply (simp_all add: succ_Un_distrib [symmetric])
1.142 +done
1.143 +
1.144 +subsubsection{*Separation for Cartesian Product*}
1.145 +
1.146 +text{*The @{text simplified} attribute tidies up the reflecting class.*}
1.147 +theorem upair_reflection [simplified,intro]:
1.148 +     "L_Reflects(?Cl, \<lambda>x. upair(L,f(x),g(x),h(x)),
1.149 +                    \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x)))"
1.150 +by (simp add: upair_def, fast)
1.151 +
1.152 +theorem pair_reflection [simplified,intro]:
1.153 +     "L_Reflects(?Cl, \<lambda>x. pair(L,f(x),g(x),h(x)),
1.154 +                    \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x)))"
1.155 +by (simp only: pair_def rex_setclass_is_bex, fast)
1.156 +
1.157 +lemma cartprod_Reflects [simplified]:
1.158 +     "L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
1.159 +                \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
1.160 +                               pair(**Lset(i),x,y,z)))"
1.161 +by fast
1.162 +
1.163 +lemma cartprod_separation:
1.164 +     "[| L(A); L(B) |]
1.165 +      ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
1.166 +apply (rule separation_CollectI)
1.167 +apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
1.168 +apply (rule ReflectsE [OF cartprod_Reflects], assumption)
1.169 +apply (drule subset_Lset_ltD, assumption)
1.170 +apply (erule reflection_imp_L_separation)
1.171 +  apply (simp_all add: lt_Ord2, clarify)
1.172 +apply (rule DPowI2)
1.173 +apply (rename_tac u)
1.174 +apply (rule bex_iff_sats)
1.175 +apply (rule conj_iff_sats)
1.176 +apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
1.177 +apply (rule bex_iff_sats)
1.178 +apply (rule conj_iff_sats)
1.179 +apply (rule mem_iff_sats)
1.180 +apply (blast intro: nth_0 nth_ConsI)
1.181 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.182 +apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
1.183 +apply (simp_all add: succ_Un_distrib [symmetric])
1.184 +done
1.185 +
1.186 +subsubsection{*Separation for Image*}
1.187 +
1.188 +text{*No @{text simplified} here: it simplifies the occurrence of
1.189 +      the predicate @{term pair}!*}
1.190 +lemma image_Reflects:
1.191 +     "L_Reflects(?Cl, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
1.192 +           \<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)))"
1.193 +by fast
1.194 +
1.195 +
1.196 +lemma image_separation:
1.197 +     "[| L(A); L(r) |]
1.198 +      ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
1.199 +apply (rule separation_CollectI)
1.200 +apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
1.201 +apply (rule ReflectsE [OF image_Reflects], assumption)
1.202 +apply (drule subset_Lset_ltD, assumption)
1.203 +apply (erule reflection_imp_L_separation)
1.204 +  apply (simp_all add: lt_Ord2, clarify)
1.205 +apply (rule DPowI2)
1.206 +apply (rule bex_iff_sats)
1.207 +apply (rule conj_iff_sats)
1.208 +apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
1.209 +apply (blast intro: nth_0 nth_ConsI)
1.210 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.211 +apply (rule bex_iff_sats)
1.212 +apply (rule conj_iff_sats)
1.213 +apply (rule mem_iff_sats)
1.214 +apply (blast intro: nth_0 nth_ConsI)
1.215 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.216 +apply (rule pair_iff_sats)
1.217 +apply (blast intro: nth_0 nth_ConsI)
1.218 +apply (blast intro: nth_0 nth_ConsI)
1.219 +apply (blast intro: nth_0 nth_ConsI)
1.220 +apply (simp_all add: succ_Un_distrib [symmetric])
1.221 +done
1.222 +
1.223 +
1.224 +subsubsection{*Separation for Converse*}
1.225 +
1.226 +lemma converse_Reflects:
1.227 +     "L_Reflects(?Cl,
1.228 +        \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
1.229 +     \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
1.230 +                     pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z)))"
1.231 +by fast
1.232 +
1.233 +lemma converse_separation:
1.234 +     "L(r) ==> separation(L,
1.235 +         \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
1.236 +apply (rule separation_CollectI)
1.237 +apply (rule_tac A="{r,z}" in subset_LsetE, blast )
1.238 +apply (rule ReflectsE [OF converse_Reflects], assumption)
1.239 +apply (drule subset_Lset_ltD, assumption)
1.240 +apply (erule reflection_imp_L_separation)
1.241 +  apply (simp_all add: lt_Ord2, clarify)
1.242 +apply (rule DPowI2)
1.243 +apply (rename_tac u)
1.244 +apply (rule bex_iff_sats)
1.245 +apply (rule conj_iff_sats)
1.246 +apply (rule_tac i=0 and j="2" and env="[p,u,r]" in mem_iff_sats, simp_all)
1.247 +apply (rule bex_iff_sats)
1.248 +apply (rule bex_iff_sats)
1.249 +apply (rule conj_iff_sats)
1.250 +apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats, simp_all)
1.251 +apply (rule pair_iff_sats)
1.252 +apply (blast intro: nth_0 nth_ConsI)
1.253 +apply (blast intro: nth_0 nth_ConsI)
1.254 +apply (blast intro: nth_0 nth_ConsI)
1.255 +apply (simp_all add: succ_Un_distrib [symmetric])
1.256 +done
1.257 +
1.258 +
1.259 +subsubsection{*Separation for Restriction*}
1.260 +
1.261 +lemma restrict_Reflects:
1.262 +     "L_Reflects(?Cl, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
1.263 +        \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z)))"
1.264 +by fast
1.265 +
1.266 +lemma restrict_separation:
1.267 +   "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
1.268 +apply (rule separation_CollectI)
1.269 +apply (rule_tac A="{A,z}" in subset_LsetE, blast )
1.270 +apply (rule ReflectsE [OF restrict_Reflects], assumption)
1.271 +apply (drule subset_Lset_ltD, assumption)
1.272 +apply (erule reflection_imp_L_separation)
1.273 +  apply (simp_all add: lt_Ord2, clarify)
1.274 +apply (rule DPowI2)
1.275 +apply (rename_tac u)
1.276 +apply (rule bex_iff_sats)
1.277 +apply (rule conj_iff_sats)
1.278 +apply (rule_tac i=0 and j="2" and env="[x,u,A]" in mem_iff_sats, simp_all)
1.279 +apply (rule bex_iff_sats)
1.280 +apply (rule_tac i=1 and j=0 and k=2 in pair_iff_sats)
1.281 +apply (simp_all add: succ_Un_distrib [symmetric])
1.282 +done
1.283 +
1.284 +
1.285 +subsubsection{*Separation for Composition*}
1.286 +
1.287 +lemma comp_Reflects:
1.288 +     "L_Reflects(?Cl, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
1.289 +		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
1.290 +                  xy\<in>s & yz\<in>r,
1.291 +        \<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).
1.292 +		  pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
1.293 +                  pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r)"
1.294 +by fast
1.295 +
1.296 +lemma comp_separation:
1.297 +     "[| L(r); L(s) |]
1.298 +      ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
1.299 +		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
1.300 +                  xy\<in>s & yz\<in>r)"
1.301 +apply (rule separation_CollectI)
1.302 +apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
1.303 +apply (rule ReflectsE [OF comp_Reflects], assumption)
1.304 +apply (drule subset_Lset_ltD, assumption)
1.305 +apply (erule reflection_imp_L_separation)
1.306 +  apply (simp_all add: lt_Ord2, clarify)
1.307 +apply (rule DPowI2)
1.308 +apply (rename_tac u)
1.309 +apply (rule bex_iff_sats)+
1.310 +apply (rename_tac x y z)
1.311 +apply (rule conj_iff_sats)
1.312 +apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
1.313 +apply (blast intro: nth_0 nth_ConsI)
1.314 +apply (blast intro: nth_0 nth_ConsI)
1.315 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.316 +apply (rule bex_iff_sats)
1.317 +apply (rule conj_iff_sats)
1.318 +apply (rule pair_iff_sats)
1.319 +apply (blast intro: nth_0 nth_ConsI)
1.320 +apply (blast intro: nth_0 nth_ConsI)
1.321 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.322 +apply (rule bex_iff_sats)
1.323 +apply (rule conj_iff_sats)
1.324 +apply (rule pair_iff_sats)
1.325 +apply (blast intro: nth_0 nth_ConsI)
1.326 +apply (blast intro: nth_0 nth_ConsI)
1.327 +apply (blast intro: nth_0 nth_ConsI, simp_all)
1.328 +apply (rule conj_iff_sats)
1.329 +apply (rule mem_iff_sats)
1.330 +apply (blast intro: nth_0 nth_ConsI)
1.331 +apply (blast intro: nth_0 nth_ConsI, simp)
1.332 +apply (rule mem_iff_sats)
1.333 +apply (blast intro: nth_0 nth_ConsI)
1.334 +apply (blast intro: nth_0 nth_ConsI)
1.335 +apply (simp_all add: succ_Un_distrib [symmetric])
1.336 +done
1.337 +
1.338 +
1.339 +
1.340 +
1.341  end
1.342
1.343  (*
1.344
1.345 -  and cartprod_separation:
1.346 -     "[| L(A); L(B) |]
1.347 -      ==> separation(L, \<lambda>z. \<exists>x\<in>A. \<exists>y\<in>B. L(x) & L(y) & pair(L,x,y,z))"
1.348 -  and image_separation:
1.349 -     "[| L(A); L(r) |]
1.350 -      ==> separation(L, \<lambda>y. \<exists>p\<in>r. L(p) & (\<exists>x\<in>A. L(x) & pair(L,x,y,p)))"
1.351 -  and vimage_separation:
1.352 -     "[| L(A); L(r) |]
1.353 -      ==> separation(L, \<lambda>x. \<exists>p\<in>r. L(p) & (\<exists>y\<in>A. L(x) & pair(L,x,y,p)))"
1.354 -  and converse_separation:
1.355 -     "L(r) ==> separation(L, \<lambda>z. \<exists>p\<in>r. L(p) & (\<exists>x y. L(x) & L(y) &
1.356 -				     pair(L,x,y,p) & pair(L,y,x,z)))"
1.357 -  and restrict_separation:
1.358 -     "L(A)
1.359 -      ==> separation(L, \<lambda>z. \<exists>x\<in>A. L(x) & (\<exists>y. L(y) & pair(L,x,y,z)))"
1.360 -  and comp_separation:
1.361 -     "[| L(r); L(s) |]
1.362 -      ==> separation(L, \<lambda>xz. \<exists>x y z. L(x) & L(y) & L(z) &
1.363 -			   (\<exists>xy\<in>s. \<exists>yz\<in>r. L(xy) & L(yz) &
1.364 -		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz)))"
1.365    and pred_separation:
1.366       "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p\<in>r. L(p) & pair(L,y,x,p))"
1.367    and Memrel_separation:
```