1(* Title: ZF/Constructible/WF_absolute.thy 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3*) 4 5section \<open>Absoluteness of Well-Founded Recursion\<close> 6 7theory WF_absolute imports WFrec begin 8 9subsection\<open>Transitive closure without fixedpoints\<close> 10 11definition 12 rtrancl_alt :: "[i,i]=>i" where 13 "rtrancl_alt(A,r) == 14 {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A. 15 (\<exists>x y. p = <x,y> & f`0 = x & f`n = y) & 16 (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}" 17 18lemma alt_rtrancl_lemma1 [rule_format]: 19 "n \<in> nat 20 ==> \<forall>f \<in> succ(n) -> field(r). 21 (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) \<longrightarrow> \<langle>f`0, f`n\<rangle> \<in> r^*" 22apply (induct_tac n) 23apply (simp_all add: apply_funtype rtrancl_refl, clarify) 24apply (rename_tac n f) 25apply (rule rtrancl_into_rtrancl) 26 prefer 2 apply assumption 27apply (drule_tac x="restrict(f,succ(n))" in bspec) 28 apply (blast intro: restrict_type2) 29apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI) 30done 31 32lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) \<subseteq> r^*" 33apply (simp add: rtrancl_alt_def) 34apply (blast intro: alt_rtrancl_lemma1) 35done 36 37lemma rtrancl_subset_rtrancl_alt: "r^* \<subseteq> rtrancl_alt(field(r),r)" 38apply (simp add: rtrancl_alt_def, clarify) 39apply (frule rtrancl_type [THEN subsetD], clarify, simp) 40apply (erule rtrancl_induct) 41 txt\<open>Base case, trivial\<close> 42 apply (rule_tac x=0 in bexI) 43 apply (rule_tac x="\<lambda>x\<in>1. xa" in bexI) 44 apply simp_all 45txt\<open>Inductive step\<close> 46apply clarify 47apply (rename_tac n f) 48apply (rule_tac x="succ(n)" in bexI) 49 apply (rule_tac x="\<lambda>i\<in>succ(succ(n)). if i=succ(n) then z else f`i" in bexI) 50 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI) 51 apply (blast intro: mem_asym) 52 apply typecheck 53 apply auto 54done 55 56lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*" 57by (blast del: subsetI 58 intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt) 59 60 61definition 62 rtran_closure_mem :: "[i=>o,i,i,i] => o" where 63 \<comment> \<open>The property of belonging to \<open>rtran_closure(r)\<close>\<close> 64 "rtran_closure_mem(M,A,r,p) == 65 \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M]. 66 omega(M,nnat) & n\<in>nnat & successor(M,n,n') & 67 (\<exists>f[M]. typed_function(M,n',A,f) & 68 (\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) & 69 fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) & 70 (\<forall>j[M]. j\<in>n \<longrightarrow> 71 (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M]. 72 fun_apply(M,f,j,fj) & successor(M,j,sj) & 73 fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))" 74 75definition 76 rtran_closure :: "[i=>o,i,i] => o" where 77 "rtran_closure(M,r,s) == 78 \<forall>A[M]. is_field(M,r,A) \<longrightarrow> 79 (\<forall>p[M]. p \<in> s \<longleftrightarrow> rtran_closure_mem(M,A,r,p))" 80 81definition 82 tran_closure :: "[i=>o,i,i] => o" where 83 "tran_closure(M,r,t) == 84 \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" 85 86locale M_trancl = M_basic + 87 assumes rtrancl_separation: 88 "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))" 89 and wellfounded_trancl_separation: 90 "[| M(r); M(Z) |] ==> 91 separation (M, \<lambda>x. 92 \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M]. 93 w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)" 94 and M_nat [iff] : "M(nat)" 95 96lemma (in M_trancl) rtran_closure_mem_iff: 97 "[|M(A); M(r); M(p)|] 98 ==> rtran_closure_mem(M,A,r,p) \<longleftrightarrow> 99 (\<exists>n[M]. n\<in>nat & 100 (\<exists>f[M]. f \<in> succ(n) -> A & 101 (\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) & 102 (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))" 103 apply (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD]) 104done 105 106lemma (in M_trancl) rtran_closure_rtrancl: 107 "M(r) ==> rtran_closure(M,r,rtrancl(r))" 108apply (simp add: rtran_closure_def rtran_closure_mem_iff 109 rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def) 110apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 111done 112 113lemma (in M_trancl) rtrancl_closed [intro,simp]: 114 "M(r) ==> M(rtrancl(r))" 115apply (insert rtrancl_separation [of r "field(r)"]) 116apply (simp add: rtrancl_alt_eq_rtrancl [symmetric] 117 rtrancl_alt_def rtran_closure_mem_iff) 118done 119 120lemma (in M_trancl) rtrancl_abs [simp]: 121 "[| M(r); M(z) |] ==> rtran_closure(M,r,z) \<longleftrightarrow> z = rtrancl(r)" 122apply (rule iffI) 123 txt\<open>Proving the right-to-left implication\<close> 124 prefer 2 apply (blast intro: rtran_closure_rtrancl) 125apply (rule M_equalityI) 126apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric] 127 rtrancl_alt_def rtran_closure_mem_iff) 128apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 129done 130 131lemma (in M_trancl) trancl_closed [intro,simp]: 132 "M(r) ==> M(trancl(r))" 133by (simp add: trancl_def) 134 135lemma (in M_trancl) trancl_abs [simp]: 136 "[| M(r); M(z) |] ==> tran_closure(M,r,z) \<longleftrightarrow> z = trancl(r)" 137by (simp add: tran_closure_def trancl_def) 138 139lemma (in M_trancl) wellfounded_trancl_separation': 140 "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)" 141by (insert wellfounded_trancl_separation [of r Z], simp) 142 143text\<open>Alternative proof of \<open>wf_on_trancl\<close>; inspiration for the 144 relativized version. Original version is on theory WF.\<close> 145lemma "[| wf[A](r); r-``A \<subseteq> A |] ==> wf[A](r^+)" 146apply (simp add: wf_on_def wf_def) 147apply (safe) 148apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec) 149apply (blast elim: tranclE) 150done 151 152lemma (in M_trancl) wellfounded_on_trancl: 153 "[| wellfounded_on(M,A,r); r-``A \<subseteq> A; M(r); M(A) |] 154 ==> wellfounded_on(M,A,r^+)" 155apply (simp add: wellfounded_on_def) 156apply (safe intro!: equalityI) 157apply (rename_tac Z x) 158apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})") 159 prefer 2 160 apply (blast intro: wellfounded_trancl_separation') 161apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe) 162apply (blast dest: transM, simp) 163apply (rename_tac y w) 164apply (drule_tac x=w in bspec, assumption, clarify) 165apply (erule tranclE) 166 apply (blast dest: transM) (*transM is needed to prove M(xa)*) 167 apply blast 168done 169 170lemma (in M_trancl) wellfounded_trancl: 171 "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)" 172apply (simp add: wellfounded_iff_wellfounded_on_field) 173apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl) 174 apply blast 175 apply (simp_all add: trancl_type [THEN field_rel_subset]) 176done 177 178 179text\<open>Absoluteness for wfrec-defined functions.\<close> 180 181(*first use is_recfun, then M_is_recfun*) 182 183lemma (in M_trancl) wfrec_relativize: 184 "[|wf(r); M(a); M(r); 185 strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 186 pair(M,x,y,z) & 187 is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 188 y = H(x, restrict(g, r -`` {x}))); 189 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] 190 ==> wfrec(r,a,H) = z \<longleftrightarrow> 191 (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 192 z = H(a,restrict(f,r-``{a})))" 193apply (frule wf_trancl) 194apply (simp add: wftrec_def wfrec_def, safe) 195 apply (frule wf_exists_is_recfun 196 [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"]) 197 apply (simp_all add: trans_trancl function_restrictI trancl_subset_times) 198 apply (clarify, rule_tac x=x in rexI) 199 apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times) 200done 201 202 203text\<open>Assuming \<^term>\<open>r\<close> is transitive simplifies the occurrences of \<open>H\<close>. 204 The premise \<^term>\<open>relation(r)\<close> is necessary 205 before we can replace \<^term>\<open>r^+\<close> by \<^term>\<open>r\<close>.\<close> 206theorem (in M_trancl) trans_wfrec_relativize: 207 "[|wf(r); trans(r); relation(r); M(r); M(a); 208 wfrec_replacement(M,MH,r); relation2(M,MH,H); 209 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] 210 ==> wfrec(r,a,H) = z \<longleftrightarrow> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 211apply (frule wfrec_replacement', assumption+) 212apply (simp cong: is_recfun_cong 213 add: wfrec_relativize trancl_eq_r 214 is_recfun_restrict_idem domain_restrict_idem) 215done 216 217theorem (in M_trancl) trans_wfrec_abs: 218 "[|wf(r); trans(r); relation(r); M(r); M(a); M(z); 219 wfrec_replacement(M,MH,r); relation2(M,MH,H); 220 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] 221 ==> is_wfrec(M,MH,r,a,z) \<longleftrightarrow> z=wfrec(r,a,H)" 222by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast) 223 224 225lemma (in M_trancl) trans_eq_pair_wfrec_iff: 226 "[|wf(r); trans(r); relation(r); M(r); M(y); 227 wfrec_replacement(M,MH,r); relation2(M,MH,H); 228 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] 229 ==> y = <x, wfrec(r, x, H)> \<longleftrightarrow> 230 (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)" 231apply safe 232 apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x]) 233txt\<open>converse direction\<close> 234apply (rule sym) 235apply (simp add: trans_wfrec_relativize, blast) 236done 237 238 239subsection\<open>M is closed under well-founded recursion\<close> 240 241text\<open>Lemma with the awkward premise mentioning \<open>wfrec\<close>.\<close> 242lemma (in M_trancl) wfrec_closed_lemma [rule_format]: 243 "[|wf(r); M(r); 244 strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>); 245 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |] 246 ==> M(a) \<longrightarrow> M(wfrec(r,a,H))" 247apply (rule_tac a=a in wf_induct, assumption+) 248apply (subst wfrec, assumption, clarify) 249apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 250 in rspec [THEN rspec]) 251apply (simp_all add: function_lam) 252apply (blast intro: lam_closed dest: pair_components_in_M) 253done 254 255text\<open>Eliminates one instance of replacement.\<close> 256lemma (in M_trancl) wfrec_replacement_iff: 257 "strong_replacement(M, \<lambda>x z. 258 \<exists>y[M]. pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))) \<longleftrightarrow> 259 strong_replacement(M, 260 \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)" 261apply simp 262apply (rule strong_replacement_cong, blast) 263done 264 265text\<open>Useful version for transitive relations\<close> 266theorem (in M_trancl) trans_wfrec_closed: 267 "[|wf(r); trans(r); relation(r); M(r); M(a); 268 wfrec_replacement(M,MH,r); relation2(M,MH,H); 269 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |] 270 ==> M(wfrec(r,a,H))" 271apply (frule wfrec_replacement', assumption+) 272apply (frule wfrec_replacement_iff [THEN iffD1]) 273apply (rule wfrec_closed_lemma, assumption+) 274apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 275done 276 277subsection\<open>Absoluteness without assuming transitivity\<close> 278lemma (in M_trancl) eq_pair_wfrec_iff: 279 "[|wf(r); M(r); M(y); 280 strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 281 pair(M,x,y,z) & 282 is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 283 y = H(x, restrict(g, r -`` {x}))); 284 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|] 285 ==> y = <x, wfrec(r, x, H)> \<longleftrightarrow> 286 (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 287 y = <x, H(x,restrict(f,r-``{x}))>)" 288apply safe 289 apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x]) 290txt\<open>converse direction\<close> 291apply (rule sym) 292apply (simp add: wfrec_relativize, blast) 293done 294 295text\<open>Full version not assuming transitivity, but maybe not very useful.\<close> 296theorem (in M_trancl) wfrec_closed: 297 "[|wf(r); M(r); M(a); 298 wfrec_replacement(M,MH,r^+); 299 relation2(M,MH, \<lambda>x f. H(x, restrict(f, r -`` {x}))); 300 \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |] 301 ==> M(wfrec(r,a,H))" 302apply (frule wfrec_replacement' 303 [of MH "r^+" "\<lambda>x f. H(x, restrict(f, r -`` {x}))"]) 304 prefer 4 305 apply (frule wfrec_replacement_iff [THEN iffD1]) 306 apply (rule wfrec_closed_lemma, assumption+) 307 apply (simp_all add: eq_pair_wfrec_iff func.function_restrictI) 308done 309 310end 311