src/ZF/Constructible/WF_absolute.thy
author ballarin
Thu Dec 11 18:30:26 2008 +0100 (2008-12-11)
changeset 29223 e09c53289830
parent 21404 eb85850d3eb7
child 32960 69916a850301
permissions -rw-r--r--
Conversion of HOL-Main and ZF to new locales.
     1 (*  Title:      ZF/Constructible/WF_absolute.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4 *)
     5 
     6 header {*Absoluteness of Well-Founded Recursion*}
     7 
     8 theory WF_absolute imports WFrec begin
     9 
    10 subsection{*Transitive closure without fixedpoints*}
    11 
    12 definition
    13   rtrancl_alt :: "[i,i]=>i" where
    14     "rtrancl_alt(A,r) ==
    15        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
    16                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
    17                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
    18 
    19 lemma alt_rtrancl_lemma1 [rule_format]:
    20     "n \<in> nat
    21      ==> \<forall>f \<in> succ(n) -> field(r).
    22          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
    23 apply (induct_tac n)
    24 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
    25 apply (rename_tac n f)
    26 apply (rule rtrancl_into_rtrancl)
    27  prefer 2 apply assumption
    28 apply (drule_tac x="restrict(f,succ(n))" in bspec)
    29  apply (blast intro: restrict_type2)
    30 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
    31 done
    32 
    33 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
    34 apply (simp add: rtrancl_alt_def)
    35 apply (blast intro: alt_rtrancl_lemma1)
    36 done
    37 
    38 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
    39 apply (simp add: rtrancl_alt_def, clarify)
    40 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
    41 apply (erule rtrancl_induct)
    42  txt{*Base case, trivial*}
    43  apply (rule_tac x=0 in bexI)
    44   apply (rule_tac x="lam x:1. xa" in bexI)
    45    apply simp_all
    46 txt{*Inductive step*}
    47 apply clarify
    48 apply (rename_tac n f)
    49 apply (rule_tac x="succ(n)" in bexI)
    50  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
    51   apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
    52   apply (blast intro: mem_asym)
    53  apply typecheck
    54  apply auto
    55 done
    56 
    57 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
    58 by (blast del: subsetI
    59 	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
    60 
    61 
    62 definition
    63   rtran_closure_mem :: "[i=>o,i,i,i] => o" where
    64     --{*The property of belonging to @{text "rtran_closure(r)"}*}
    65     "rtran_closure_mem(M,A,r,p) ==
    66 	      \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M]. 
    67                omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
    68 	       (\<exists>f[M]. typed_function(M,n',A,f) &
    69 		(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
    70 		  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
    71 		  (\<forall>j[M]. j\<in>n --> 
    72 		    (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M]. 
    73 		      fun_apply(M,f,j,fj) & successor(M,j,sj) &
    74 		      fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"
    75 
    76 definition
    77   rtran_closure :: "[i=>o,i,i] => o" where
    78     "rtran_closure(M,r,s) == 
    79         \<forall>A[M]. is_field(M,r,A) -->
    80  	 (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))"
    81 
    82 definition
    83   tran_closure :: "[i=>o,i,i] => o" where
    84     "tran_closure(M,r,t) ==
    85          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
    86 
    87 lemma (in M_basic) rtran_closure_mem_iff:
    88      "[|M(A); M(r); M(p)|]
    89       ==> rtran_closure_mem(M,A,r,p) <->
    90           (\<exists>n[M]. n\<in>nat & 
    91            (\<exists>f[M]. f \<in> succ(n) -> A &
    92             (\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
    93                            (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
    94 by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD]) 
    95 
    96 
    97 locale M_trancl = M_basic +
    98   assumes rtrancl_separation:
    99 	 "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
   100       and wellfounded_trancl_separation:
   101 	 "[| M(r); M(Z) |] ==> 
   102 	  separation (M, \<lambda>x. 
   103 	      \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M]. 
   104 	       w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
   105 
   106 
   107 lemma (in M_trancl) rtran_closure_rtrancl:
   108      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
   109 apply (simp add: rtran_closure_def rtran_closure_mem_iff 
   110                  rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
   111 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
   112 done
   113 
   114 lemma (in M_trancl) rtrancl_closed [intro,simp]:
   115      "M(r) ==> M(rtrancl(r))"
   116 apply (insert rtrancl_separation [of r "field(r)"])
   117 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
   118                  rtrancl_alt_def rtran_closure_mem_iff)
   119 done
   120 
   121 lemma (in M_trancl) rtrancl_abs [simp]:
   122      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
   123 apply (rule iffI)
   124  txt{*Proving the right-to-left implication*}
   125  prefer 2 apply (blast intro: rtran_closure_rtrancl)
   126 apply (rule M_equalityI)
   127 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
   128                  rtrancl_alt_def rtran_closure_mem_iff)
   129 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
   130 done
   131 
   132 lemma (in M_trancl) trancl_closed [intro,simp]:
   133      "M(r) ==> M(trancl(r))"
   134 by (simp add: trancl_def comp_closed rtrancl_closed)
   135 
   136 lemma (in M_trancl) trancl_abs [simp]:
   137      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
   138 by (simp add: tran_closure_def trancl_def)
   139 
   140 lemma (in M_trancl) wellfounded_trancl_separation':
   141      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
   142 by (insert wellfounded_trancl_separation [of r Z], simp) 
   143 
   144 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
   145       relativized version.  Original version is on theory WF.*}
   146 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
   147 apply (simp add: wf_on_def wf_def)
   148 apply (safe intro!: equalityI)
   149 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
   150 apply (blast elim: tranclE)
   151 done
   152 
   153 lemma (in M_trancl) wellfounded_on_trancl:
   154      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
   155       ==> wellfounded_on(M,A,r^+)"
   156 apply (simp add: wellfounded_on_def)
   157 apply (safe intro!: equalityI)
   158 apply (rename_tac Z x)
   159 apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
   160  prefer 2
   161  apply (blast intro: wellfounded_trancl_separation') 
   162 apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
   163 apply (blast dest: transM, simp)
   164 apply (rename_tac y w)
   165 apply (drule_tac x=w in bspec, assumption, clarify)
   166 apply (erule tranclE)
   167   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
   168  apply blast
   169 done
   170 
   171 lemma (in M_trancl) wellfounded_trancl:
   172      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
   173 apply (simp add: wellfounded_iff_wellfounded_on_field)
   174 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
   175    apply blast
   176   apply (simp_all add: trancl_type [THEN field_rel_subset])
   177 done
   178 
   179 
   180 text{*Absoluteness for wfrec-defined functions.*}
   181 
   182 (*first use is_recfun, then M_is_recfun*)
   183 
   184 lemma (in M_trancl) wfrec_relativize:
   185   "[|wf(r); M(a); M(r);  
   186      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   187           pair(M,x,y,z) & 
   188           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
   189           y = H(x, restrict(g, r -`` {x}))); 
   190      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   191    ==> wfrec(r,a,H) = z <-> 
   192        (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
   193             z = H(a,restrict(f,r-``{a})))"
   194 apply (frule wf_trancl) 
   195 apply (simp add: wftrec_def wfrec_def, safe)
   196  apply (frule wf_exists_is_recfun 
   197               [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"]) 
   198       apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
   199  apply (clarify, rule_tac x=x in rexI) 
   200  apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
   201 done
   202 
   203 
   204 text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
   205       The premise @{term "relation(r)"} is necessary 
   206       before we can replace @{term "r^+"} by @{term r}. *}
   207 theorem (in M_trancl) trans_wfrec_relativize:
   208   "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);
   209      wfrec_replacement(M,MH,r);  relation2(M,MH,H);
   210      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   211    ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 
   212 apply (frule wfrec_replacement', assumption+) 
   213 apply (simp cong: is_recfun_cong
   214            add: wfrec_relativize trancl_eq_r
   215                 is_recfun_restrict_idem domain_restrict_idem)
   216 done
   217 
   218 theorem (in M_trancl) trans_wfrec_abs:
   219   "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);  M(z);
   220      wfrec_replacement(M,MH,r);  relation2(M,MH,H);
   221      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   222    ==> is_wfrec(M,MH,r,a,z) <-> z=wfrec(r,a,H)" 
   223 by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast) 
   224 
   225 
   226 lemma (in M_trancl) trans_eq_pair_wfrec_iff:
   227   "[|wf(r);  trans(r); relation(r); M(r);  M(y); 
   228      wfrec_replacement(M,MH,r);  relation2(M,MH,H);
   229      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   230    ==> y = <x, wfrec(r, x, H)> <-> 
   231        (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   232 apply safe 
   233  apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x]) 
   234 txt{*converse direction*}
   235 apply (rule sym)
   236 apply (simp add: trans_wfrec_relativize, blast) 
   237 done
   238 
   239 
   240 subsection{*M is closed under well-founded recursion*}
   241 
   242 text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
   243 lemma (in M_trancl) wfrec_closed_lemma [rule_format]:
   244      "[|wf(r); M(r); 
   245         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
   246         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   247       ==> M(a) --> M(wfrec(r,a,H))"
   248 apply (rule_tac a=a in wf_induct, assumption+)
   249 apply (subst wfrec, assumption, clarify)
   250 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
   251        in rspec [THEN rspec]) 
   252 apply (simp_all add: function_lam) 
   253 apply (blast intro: lam_closed dest: pair_components_in_M) 
   254 done
   255 
   256 text{*Eliminates one instance of replacement.*}
   257 lemma (in M_trancl) wfrec_replacement_iff:
   258      "strong_replacement(M, \<lambda>x z. 
   259           \<exists>y[M]. pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g))) <->
   260       strong_replacement(M, 
   261            \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   262 apply simp 
   263 apply (rule strong_replacement_cong, blast) 
   264 done
   265 
   266 text{*Useful version for transitive relations*}
   267 theorem (in M_trancl) trans_wfrec_closed:
   268      "[|wf(r); trans(r); relation(r); M(r); M(a);
   269        wfrec_replacement(M,MH,r);  relation2(M,MH,H);
   270         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   271       ==> M(wfrec(r,a,H))"
   272 apply (frule wfrec_replacement', assumption+) 
   273 apply (frule wfrec_replacement_iff [THEN iffD1]) 
   274 apply (rule wfrec_closed_lemma, assumption+) 
   275 apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 
   276 done
   277 
   278 subsection{*Absoluteness without assuming transitivity*}
   279 lemma (in M_trancl) eq_pair_wfrec_iff:
   280   "[|wf(r);  M(r);  M(y); 
   281      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   282           pair(M,x,y,z) & 
   283           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
   284           y = H(x, restrict(g, r -`` {x}))); 
   285      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   286    ==> y = <x, wfrec(r, x, H)> <-> 
   287        (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
   288             y = <x, H(x,restrict(f,r-``{x}))>)"
   289 apply safe  
   290  apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x]) 
   291 txt{*converse direction*}
   292 apply (rule sym)
   293 apply (simp add: wfrec_relativize, blast) 
   294 done
   295 
   296 text{*Full version not assuming transitivity, but maybe not very useful.*}
   297 theorem (in M_trancl) wfrec_closed:
   298      "[|wf(r); M(r); M(a);
   299         wfrec_replacement(M,MH,r^+);  
   300         relation2(M,MH, \<lambda>x f. H(x, restrict(f, r -`` {x})));
   301         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   302       ==> M(wfrec(r,a,H))"
   303 apply (frule wfrec_replacement' 
   304                [of MH "r^+" "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
   305    prefer 4
   306    apply (frule wfrec_replacement_iff [THEN iffD1]) 
   307    apply (rule wfrec_closed_lemma, assumption+) 
   308      apply (simp_all add: eq_pair_wfrec_iff func.function_restrictI) 
   309 done
   310 
   311 end