1(* Title: ZF/Constructible/Separation.thy 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3*) 4 5section\<open>Early Instances of Separation and Strong Replacement\<close> 6 7theory Separation imports L_axioms WF_absolute begin 8 9text\<open>This theory proves all instances needed for locale \<open>M_basic\<close>\<close> 10 11text\<open>Helps us solve for de Bruijn indices!\<close> 12lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x" 13by simp 14 15lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI 16lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats 17 fun_plus_iff_sats 18 19lemma 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)" 22by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) 23 24lemma 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))" 27apply (frule mem_Lset_imp_subset_Lset) 28apply (simp add: Collect_conj_in_DPow Collect_mem_eq 29 subset_Int_iff2 elem_subset_in_DPow) 30done 31 32lemma separation_CollectI: 33 "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))" 34apply (unfold separation_def, clarify) 35apply (rule_tac x="{x\<in>z. P(x)}" in rexI) 36apply simp_all 37done 38 39text\<open>Reduces the original comprehension to the reflected one\<close> 40lemma 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)})" 44apply (rule_tac i = "succ(j)" in L_I) 45 prefer 2 apply simp 46apply (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) 49apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) 50done 51 52text\<open>Encapsulates the standard proof script for proving instances of 53 Separation.\<close> 54lemma 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)" 60apply (rule separation_CollectI) 61apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu) 62apply (rule ReflectsE [OF reflection], assumption) 63apply (drule subset_Lset_ltD, assumption) 64apply (erule reflection_imp_L_separation) 65 apply (simp_all add: lt_Ord2, clarify) 66apply (rule collI, assumption) 67done 68 69text\<open>As above, but typically \<^term>\<open>u\<close> is a finite enumeration such as 70 \<^term>\<open>{a,b}\<close>; thus the new subgoal gets the assumption 71 \<^term>\<open>{a,b} \<subseteq> Lset(i)\<close>, which is logically equivalent to 72 \<^term>\<open>a \<in> Lset(i)\<close> and \<^term>\<open>b \<in> Lset(i)\<close>.\<close> 73lemma 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)" 79apply (rule gen_separation [OF reflection Lu]) 80apply (drule mem_Lset_imp_subset_Lset) 81apply (erule collI) 82done 83 84 85subsection\<open>Separation for Intersection\<close> 86 87lemma 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]" 90by (intro FOL_reflections) 91 92lemma Inter_separation: 93 "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A \<longrightarrow> x\<in>y)" 94apply (rule gen_separation [OF Inter_Reflects], simp) 95apply (rule DPow_LsetI) 96 txt\<open>I leave this one example of a manual proof. The tedium of manually 97 instantiating \<^term>\<open>i\<close>, \<^term>\<open>j\<close> and \<^term>\<open>env\<close> is obvious.\<close> 98apply (rule ball_iff_sats) 99apply (rule imp_iff_sats) 100apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) 101apply (rule_tac i=0 and j=2 in mem_iff_sats) 102apply (simp_all add: succ_Un_distrib [symmetric]) 103done 104 105subsection\<open>Separation for Set Difference\<close> 106 107lemma Diff_Reflects: 108 "REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]" 109by (intro FOL_reflections) 110 111lemma Diff_separation: 112 "L(B) ==> separation(L, \<lambda>x. x \<notin> B)" 113apply (rule gen_separation [OF Diff_Reflects], simp) 114apply (rule_tac env="[B]" in DPow_LsetI) 115apply (rule sep_rules | simp)+ 116done 117 118subsection\<open>Separation for Cartesian Product\<close> 119 120lemma 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))]" 124by (intro FOL_reflections function_reflections) 125 126lemma 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)))" 129apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto) 130apply (rule_tac env="[A,B]" in DPow_LsetI) 131apply (rule sep_rules | simp)+ 132done 133 134subsection\<open>Separation for Image\<close> 135 136lemma 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))]" 139by (intro FOL_reflections function_reflections) 140 141lemma 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)))" 144apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto) 145apply (rule_tac env="[A,r]" in DPow_LsetI) 146apply (rule sep_rules | simp)+ 147done 148 149 150subsection\<open>Separation for Converse\<close> 151 152lemma 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))]" 156by (intro FOL_reflections function_reflections) 157 158lemma 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)))" 161apply (rule gen_separation [OF converse_Reflects], simp) 162apply (rule_tac env="[r]" in DPow_LsetI) 163apply (rule sep_rules | simp)+ 164done 165 166 167subsection\<open>Separation for Restriction\<close> 168 169lemma 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))]" 172by (intro FOL_reflections function_reflections) 173 174lemma restrict_separation: 175 "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))" 176apply (rule gen_separation [OF restrict_Reflects], simp) 177apply (rule_tac env="[A]" in DPow_LsetI) 178apply (rule sep_rules | simp)+ 179done 180 181 182subsection\<open>Separation for Composition\<close> 183 184lemma 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]" 191by (intro FOL_reflections function_reflections) 192 193lemma 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)" 198apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto) 199txt\<open>Subgoals after applying general ``separation'' rule: 200 @{subgoals[display,indent=0,margin=65]}\<close> 201apply (rule_tac env="[r,s]" in DPow_LsetI) 202txt\<open>Subgoals ready for automatic synthesis of a formula: 203 @{subgoals[display,indent=0,margin=65]}\<close> 204apply (rule sep_rules | simp)+ 205done 206 207 208subsection\<open>Separation for Predecessors in an Order\<close> 209 210lemma 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)]" 213by (intro FOL_reflections function_reflections) 214 215lemma pred_separation: 216 "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))" 217apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto) 218apply (rule_tac env="[r,x]" in DPow_LsetI) 219apply (rule sep_rules | simp)+ 220done 221 222 223subsection\<open>Separation for the Membership Relation\<close> 224 225lemma 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]" 228by (intro FOL_reflections function_reflections) 229 230lemma Memrel_separation: 231 "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)" 232apply (rule gen_separation [OF Memrel_Reflects nonempty]) 233apply (rule_tac env="[]" in DPow_LsetI) 234apply (rule sep_rules | simp)+ 235done 236 237 238subsection\<open>Replacement for FunSpace\<close> 239 240lemma 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))]" 248by (intro FOL_reflections function_reflections) 249 250lemma 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))" 255apply (rule strong_replacementI) 256apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects], 257 auto) 258apply (rule_tac env="[n,B]" in DPow_LsetI) 259apply (rule sep_rules | simp)+ 260done 261 262 263subsection\<open>Separation for a Theorem about \<^term>\<open>is_recfun\<close>\<close> 264 265lemma 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)]" 274by (intro FOL_reflections function_reflections fun_plus_reflections) 275 276lemma 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))" 284apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"], 285 auto) 286apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI) 287apply (rule sep_rules | simp)+ 288done 289 290 291subsection\<open>Instantiating the locale \<open>M_basic\<close>\<close> 292text\<open>Separation (and Strong Replacement) for basic set-theoretic constructions 293such as intersection, Cartesian Product and image.\<close> 294 295lemma 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 power_ax)+ 302 done 303 304theorem M_basic_L: " M_basic(L)" 305by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L]) 306 307interpretation L: M_basic L by (rule M_basic_L) 308 309 310end 311