src/ZF/Constructible/WF_absolute.thy
author wenzelm
Sat Oct 17 14:43:18 2009 +0200 (2009-10-17)
changeset 32960 69916a850301
parent 21404 eb85850d3eb7
child 46823 57bf0cecb366
permissions -rw-r--r--
eliminated hard tabulators, guessing at each author's individual tab-width;
tuned headers;
     1 (*  Title:      ZF/Constructible/WF_absolute.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 *)
     4 
     5 header {*Absoluteness of Well-Founded Recursion*}
     6 
     7 theory WF_absolute imports WFrec begin
     8 
     9 subsection{*Transitive closure without fixedpoints*}
    10 
    11 definition
    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 
    18 lemma 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) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
    22 apply (induct_tac n)
    23 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
    24 apply (rename_tac n f)
    25 apply (rule rtrancl_into_rtrancl)
    26  prefer 2 apply assumption
    27 apply (drule_tac x="restrict(f,succ(n))" in bspec)
    28  apply (blast intro: restrict_type2)
    29 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
    30 done
    31 
    32 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
    33 apply (simp add: rtrancl_alt_def)
    34 apply (blast intro: alt_rtrancl_lemma1)
    35 done
    36 
    37 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
    38 apply (simp add: rtrancl_alt_def, clarify)
    39 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
    40 apply (erule rtrancl_induct)
    41  txt{*Base case, trivial*}
    42  apply (rule_tac x=0 in bexI)
    43   apply (rule_tac x="lam x:1. xa" in bexI)
    44    apply simp_all
    45 txt{*Inductive step*}
    46 apply clarify
    47 apply (rename_tac n f)
    48 apply (rule_tac x="succ(n)" in bexI)
    49  apply (rule_tac x="lam i: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
    54 done
    55 
    56 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
    57 by (blast del: subsetI
    58           intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
    59 
    60 
    61 definition
    62   rtran_closure_mem :: "[i=>o,i,i,i] => o" where
    63     --{*The property of belonging to @{text "rtran_closure(r)"}*}
    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 --> 
    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 
    75 definition
    76   rtran_closure :: "[i=>o,i,i] => o" where
    77     "rtran_closure(M,r,s) == 
    78         \<forall>A[M]. is_field(M,r,A) -->
    79          (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))"
    80 
    81 definition
    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 
    86 lemma (in M_basic) rtran_closure_mem_iff:
    87      "[|M(A); M(r); M(p)|]
    88       ==> rtran_closure_mem(M,A,r,p) <->
    89           (\<exists>n[M]. n\<in>nat & 
    90            (\<exists>f[M]. f \<in> succ(n) -> A &
    91             (\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
    92                            (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
    93 by (simp add: rtran_closure_mem_def Ord_succ_mem_iff nat_0_le [THEN ltD]) 
    94 
    95 
    96 locale M_trancl = M_basic +
    97   assumes rtrancl_separation:
    98          "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
    99       and wellfounded_trancl_separation:
   100          "[| M(r); M(Z) |] ==> 
   101           separation (M, \<lambda>x. 
   102               \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M]. 
   103                w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
   104 
   105 
   106 lemma (in M_trancl) rtran_closure_rtrancl:
   107      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
   108 apply (simp add: rtran_closure_def rtran_closure_mem_iff 
   109                  rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
   110 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
   111 done
   112 
   113 lemma (in M_trancl) rtrancl_closed [intro,simp]:
   114      "M(r) ==> M(rtrancl(r))"
   115 apply (insert rtrancl_separation [of r "field(r)"])
   116 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
   117                  rtrancl_alt_def rtran_closure_mem_iff)
   118 done
   119 
   120 lemma (in M_trancl) rtrancl_abs [simp]:
   121      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
   122 apply (rule iffI)
   123  txt{*Proving the right-to-left implication*}
   124  prefer 2 apply (blast intro: rtran_closure_rtrancl)
   125 apply (rule M_equalityI)
   126 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
   127                  rtrancl_alt_def rtran_closure_mem_iff)
   128 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype) 
   129 done
   130 
   131 lemma (in M_trancl) trancl_closed [intro,simp]:
   132      "M(r) ==> M(trancl(r))"
   133 by (simp add: trancl_def comp_closed rtrancl_closed)
   134 
   135 lemma (in M_trancl) trancl_abs [simp]:
   136      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
   137 by (simp add: tran_closure_def trancl_def)
   138 
   139 lemma (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^+)"
   141 by (insert wellfounded_trancl_separation [of r Z], simp) 
   142 
   143 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
   144       relativized version.  Original version is on theory WF.*}
   145 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
   146 apply (simp add: wf_on_def wf_def)
   147 apply (safe intro!: equalityI)
   148 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
   149 apply (blast elim: tranclE)
   150 done
   151 
   152 lemma (in M_trancl) wellfounded_on_trancl:
   153      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
   154       ==> wellfounded_on(M,A,r^+)"
   155 apply (simp add: wellfounded_on_def)
   156 apply (safe intro!: equalityI)
   157 apply (rename_tac Z x)
   158 apply (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') 
   161 apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
   162 apply (blast dest: transM, simp)
   163 apply (rename_tac y w)
   164 apply (drule_tac x=w in bspec, assumption, clarify)
   165 apply (erule tranclE)
   166   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
   167  apply blast
   168 done
   169 
   170 lemma (in M_trancl) wellfounded_trancl:
   171      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
   172 apply (simp add: wellfounded_iff_wellfounded_on_field)
   173 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
   174    apply blast
   175   apply (simp_all add: trancl_type [THEN field_rel_subset])
   176 done
   177 
   178 
   179 text{*Absoluteness for wfrec-defined functions.*}
   180 
   181 (*first use is_recfun, then M_is_recfun*)
   182 
   183 lemma (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) --> M(H(x,g))|] 
   190    ==> wfrec(r,a,H) = z <-> 
   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})))"
   193 apply (frule wf_trancl) 
   194 apply (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)
   200 done
   201 
   202 
   203 text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
   204       The premise @{term "relation(r)"} is necessary 
   205       before we can replace @{term "r^+"} by @{term r}. *}
   206 theorem (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) --> M(H(x,g))|] 
   210    ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 
   211 apply (frule wfrec_replacement', assumption+) 
   212 apply (simp cong: is_recfun_cong
   213            add: wfrec_relativize trancl_eq_r
   214                 is_recfun_restrict_idem domain_restrict_idem)
   215 done
   216 
   217 theorem (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) --> M(H(x,g))|] 
   221    ==> is_wfrec(M,MH,r,a,z) <-> z=wfrec(r,a,H)" 
   222 by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast) 
   223 
   224 
   225 lemma (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) --> M(H(x,g))|] 
   229    ==> y = <x, wfrec(r, x, H)> <-> 
   230        (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   231 apply safe 
   232  apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x]) 
   233 txt{*converse direction*}
   234 apply (rule sym)
   235 apply (simp add: trans_wfrec_relativize, blast) 
   236 done
   237 
   238 
   239 subsection{*M is closed under well-founded recursion*}
   240 
   241 text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
   242 lemma (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) --> M(H(x,g)) |] 
   246       ==> M(a) --> M(wfrec(r,a,H))"
   247 apply (rule_tac a=a in wf_induct, assumption+)
   248 apply (subst wfrec, assumption, clarify)
   249 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
   250        in rspec [THEN rspec]) 
   251 apply (simp_all add: function_lam) 
   252 apply (blast intro: lam_closed dest: pair_components_in_M) 
   253 done
   254 
   255 text{*Eliminates one instance of replacement.*}
   256 lemma (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))) <->
   259       strong_replacement(M, 
   260            \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   261 apply simp 
   262 apply (rule strong_replacement_cong, blast) 
   263 done
   264 
   265 text{*Useful version for transitive relations*}
   266 theorem (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) --> M(H(x,g)) |] 
   270       ==> M(wfrec(r,a,H))"
   271 apply (frule wfrec_replacement', assumption+) 
   272 apply (frule wfrec_replacement_iff [THEN iffD1]) 
   273 apply (rule wfrec_closed_lemma, assumption+) 
   274 apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 
   275 done
   276 
   277 subsection{*Absoluteness without assuming transitivity*}
   278 lemma (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) --> M(H(x,g))|] 
   285    ==> y = <x, wfrec(r, x, H)> <-> 
   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}))>)"
   288 apply safe  
   289  apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x]) 
   290 txt{*converse direction*}
   291 apply (rule sym)
   292 apply (simp add: wfrec_relativize, blast) 
   293 done
   294 
   295 text{*Full version not assuming transitivity, but maybe not very useful.*}
   296 theorem (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) --> M(H(x,g)) |] 
   301       ==> M(wfrec(r,a,H))"
   302 apply (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) 
   308 done
   309 
   310 end