src/ZF/Constructible/WF_absolute.thy
author paulson
Mon Jul 01 18:16:18 2002 +0200 (2002-07-01)
changeset 13268 240509babf00
parent 13254 5146ccaedf42
child 13269 3ba9be497c33
permissions -rw-r--r--
more use of relativized quantifiers
list_closed
     1 theory WF_absolute = WFrec:
     2 
     3 (*????move to Wellorderings.thy*)
     4 lemma (in M_axioms) wellfounded_on_field_imp_wellfounded:
     5      "wellfounded_on(M, field(r), r) ==> wellfounded(M,r)"
     6 by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)
     7 
     8 lemma (in M_axioms) wellfounded_iff_wellfounded_on_field:
     9      "M(r) ==> wellfounded(M,r) <-> wellfounded_on(M, field(r), r)"
    10 by (blast intro: wellfounded_imp_wellfounded_on
    11                  wellfounded_on_field_imp_wellfounded)
    12 
    13 lemma (in M_axioms) wellfounded_on_subset_A:
    14      "[| wellfounded_on(M,A,r);  B<=A |] ==> wellfounded_on(M,B,r)"
    15 by (simp add: wellfounded_on_def, blast)
    16 
    17 
    18 subsection{*Every well-founded relation is a subset of some inverse image of
    19       an ordinal*}
    20 
    21 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
    22 by (blast intro: wf_rvimage wf_Memrel)
    23 
    24 
    25 constdefs
    26   wfrank :: "[i,i]=>i"
    27     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
    28 
    29 constdefs
    30   wftype :: "i=>i"
    31     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
    32 
    33 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
    34 by (subst wfrank_def [THEN def_wfrec], simp_all)
    35 
    36 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
    37 apply (rule_tac a="a" in wf_induct, assumption)
    38 apply (subst wfrank, assumption)
    39 apply (rule Ord_succ [THEN Ord_UN], blast)
    40 done
    41 
    42 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
    43 apply (rule_tac a1 = "b" in wfrank [THEN ssubst], assumption)
    44 apply (rule UN_I [THEN ltI])
    45 apply (simp add: Ord_wfrank vimage_iff)+
    46 done
    47 
    48 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
    49 by (simp add: wftype_def Ord_wfrank)
    50 
    51 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
    52 apply (simp add: wftype_def)
    53 apply (blast intro: wfrank_lt [THEN ltD])
    54 done
    55 
    56 
    57 lemma wf_imp_subset_rvimage:
    58      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
    59 apply (rule_tac x="wftype(r)" in exI)
    60 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
    61 apply (simp add: Ord_wftype, clarify)
    62 apply (frule subsetD, assumption, clarify)
    63 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
    64 apply (blast intro: wftypeI)
    65 done
    66 
    67 theorem wf_iff_subset_rvimage:
    68   "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
    69 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
    70           intro: wf_rvimage_Ord [THEN wf_subset])
    71 
    72 
    73 subsection{*Transitive closure without fixedpoints*}
    74 
    75 constdefs
    76   rtrancl_alt :: "[i,i]=>i"
    77     "rtrancl_alt(A,r) ==
    78        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
    79                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
    80                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
    81 
    82 lemma alt_rtrancl_lemma1 [rule_format]:
    83     "n \<in> nat
    84      ==> \<forall>f \<in> succ(n) -> field(r).
    85          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
    86 apply (induct_tac n)
    87 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
    88 apply (rename_tac n f)
    89 apply (rule rtrancl_into_rtrancl)
    90  prefer 2 apply assumption
    91 apply (drule_tac x="restrict(f,succ(n))" in bspec)
    92  apply (blast intro: restrict_type2)
    93 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
    94 done
    95 
    96 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
    97 apply (simp add: rtrancl_alt_def)
    98 apply (blast intro: alt_rtrancl_lemma1)
    99 done
   100 
   101 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
   102 apply (simp add: rtrancl_alt_def, clarify)
   103 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
   104 apply (erule rtrancl_induct)
   105  txt{*Base case, trivial*}
   106  apply (rule_tac x=0 in bexI)
   107   apply (rule_tac x="lam x:1. xa" in bexI)
   108    apply simp_all
   109 txt{*Inductive step*}
   110 apply clarify
   111 apply (rename_tac n f)
   112 apply (rule_tac x="succ(n)" in bexI)
   113  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
   114   apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
   115   apply (blast intro: mem_asym)
   116  apply typecheck
   117  apply auto
   118 done
   119 
   120 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
   121 by (blast del: subsetI
   122 	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
   123 
   124 
   125 constdefs
   126 
   127   rtran_closure :: "[i=>o,i,i] => o"
   128     "rtran_closure(M,r,s) ==
   129         \<forall>A. M(A) --> is_field(M,r,A) -->
   130  	 (\<forall>p. M(p) -->
   131           (p \<in> s <->
   132            (\<exists>n\<in>nat. M(n) &
   133             (\<exists>n'. M(n') & successor(M,n,n') &
   134              (\<exists>f. M(f) & typed_function(M,n',A,f) &
   135               (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &
   136                    fun_apply(M,f,0,x) & fun_apply(M,f,n,y))) &
   137               (\<forall>i\<in>n. M(i) -->
   138                 (\<forall>i'. M(i') --> successor(M,i,i') -->
   139                  (\<forall>fi. M(fi) --> fun_apply(M,f,i,fi) -->
   140                   (\<forall>fi'. M(fi') --> fun_apply(M,f,i',fi') -->
   141                    (\<forall>q. M(q) --> pair(M,fi,fi',q) --> q \<in> r))))))))))"
   142 
   143   tran_closure :: "[i=>o,i,i] => o"
   144     "tran_closure(M,r,t) ==
   145          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
   146 
   147 
   148 locale M_trancl = M_axioms +
   149 (*THEY NEED RELATIVIZATION*)
   150   assumes rtrancl_separation:
   151      "[| M(r); M(A) |] ==>
   152 	separation
   153 	   (M, \<lambda>p. \<exists>n[M]. n\<in>nat & 
   154                     (\<exists>f[M]. 
   155                      f \<in> succ(n) -> A &
   156                      (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,p) &  
   157                            f`0 = x & f`n = y) &
   158                            (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
   159       and wellfounded_trancl_separation:
   160      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+)"
   161 
   162 
   163 lemma (in M_trancl) rtran_closure_rtrancl:
   164      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
   165 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
   166                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
   167                  Ord_succ_mem_iff M_nat  nat_0_le [THEN ltD], clarify)
   168 apply (rule iffI)
   169  apply clarify
   170  apply simp
   171  apply (rename_tac n f)
   172  apply (rule_tac x=n in bexI)
   173   apply (rule_tac x=f in exI)
   174   apply simp
   175   apply (blast dest: finite_fun_closed dest: transM)
   176  apply assumption
   177 apply clarify
   178 apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)
   179 done
   180 
   181 lemma (in M_trancl) rtrancl_closed [intro,simp]:
   182      "M(r) ==> M(rtrancl(r))"
   183 apply (insert rtrancl_separation [of r "field(r)"])
   184 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
   185                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
   186                  Ord_succ_mem_iff M_nat nat_into_M
   187                  nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
   188 done
   189 
   190 lemma (in M_trancl) rtrancl_abs [simp]:
   191      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
   192 apply (rule iffI)
   193  txt{*Proving the right-to-left implication*}
   194  prefer 2 apply (blast intro: rtran_closure_rtrancl)
   195 apply (rule M_equalityI)
   196 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
   197                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
   198                  Ord_succ_mem_iff M_nat
   199                  nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
   200  prefer 2 apply assumption
   201  prefer 2 apply blast
   202 apply (rule iffI, clarify)
   203 apply (simp add: nat_0_le [THEN ltD]  apply_funtype, blast, clarify, simp)
   204  apply (rename_tac n f)
   205  apply (rule_tac x=n in bexI)
   206   apply (rule_tac x=f in exI)
   207   apply (blast dest!: finite_fun_closed, assumption)
   208 done
   209 
   210 
   211 lemma (in M_trancl) trancl_closed [intro,simp]:
   212      "M(r) ==> M(trancl(r))"
   213 by (simp add: trancl_def comp_closed rtrancl_closed)
   214 
   215 lemma (in M_trancl) trancl_abs [simp]:
   216      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
   217 by (simp add: tran_closure_def trancl_def)
   218 
   219 
   220 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
   221       relativized version.  Original version is on theory WF.*}
   222 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
   223 apply (simp add: wf_on_def wf_def)
   224 apply (safe intro!: equalityI)
   225 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
   226 apply (blast elim: tranclE)
   227 done
   228 
   229 
   230 lemma (in M_trancl) wellfounded_on_trancl:
   231      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
   232       ==> wellfounded_on(M,A,r^+)"
   233 apply (simp add: wellfounded_on_def)
   234 apply (safe intro!: equalityI)
   235 apply (rename_tac Z x)
   236 apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
   237  prefer 2
   238  apply (blast intro: wellfounded_trancl_separation) 
   239 apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in spec, safe)
   240 apply (blast dest: transM, simp)
   241 apply (rename_tac y w)
   242 apply (drule_tac x=w in bspec, assumption, clarify)
   243 apply (erule tranclE)
   244   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
   245  apply blast
   246 done
   247 
   248 lemma (in M_trancl) wellfounded_trancl:
   249      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
   250 apply (rotate_tac -1)
   251 apply (simp add: wellfounded_iff_wellfounded_on_field)
   252 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
   253    apply blast
   254   apply (simp_all add: trancl_type [THEN field_rel_subset])
   255 done
   256 
   257 text{*Relativized to M: Every well-founded relation is a subset of some
   258 inverse image of an ordinal.  Key step is the construction (in M) of a
   259 rank function.*}
   260 
   261 
   262 (*NEEDS RELATIVIZATION*)
   263 locale M_wfrank = M_trancl +
   264   assumes wfrank_separation':
   265      "M(r) ==>
   266 	separation
   267 	   (M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
   268  and wfrank_strong_replacement':
   269      "M(r) ==>
   270       strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M]. 
   271 		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
   272 		  y = range(f))"
   273  and Ord_wfrank_separation:
   274      "M(r) ==>
   275       separation (M, \<lambda>x. ~ (\<forall>f. M(f) \<longrightarrow>
   276                        is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
   277 
   278 text{*This function, defined using replacement, is a rank function for
   279 well-founded relations within the class M.*}
   280 constdefs
   281  wellfoundedrank :: "[i=>o,i,i] => i"
   282     "wellfoundedrank(M,r,A) ==
   283         {p. x\<in>A, \<exists>y[M]. \<exists>f[M]. 
   284                        p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
   285                        y = range(f)}"
   286 
   287 lemma (in M_wfrank) exists_wfrank:
   288     "[| wellfounded(M,r); M(a); M(r) |]
   289      ==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
   290 apply (rule wellfounded_exists_is_recfun)
   291       apply (blast intro: wellfounded_trancl)
   292      apply (rule trans_trancl)
   293     apply (erule wfrank_separation')
   294    apply (erule wfrank_strong_replacement')
   295 apply (simp_all add: trancl_subset_times)
   296 done
   297 
   298 lemma (in M_wfrank) M_wellfoundedrank:
   299     "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
   300 apply (insert wfrank_strong_replacement' [of r])
   301 apply (simp add: wellfoundedrank_def)
   302 apply (rule strong_replacement_closed)
   303    apply assumption+
   304  apply (rule univalent_is_recfun)
   305    apply (blast intro: wellfounded_trancl)
   306   apply (rule trans_trancl)
   307  apply (simp add: trancl_subset_times, blast)
   308 done
   309 
   310 lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
   311     "[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
   312      ==> \<forall>f. M(f) --> is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
   313 apply (drule wellfounded_trancl, assumption)
   314 apply (rule wellfounded_induct, assumption+)
   315   apply simp
   316  apply (blast intro: Ord_wfrank_separation, clarify)
   317 txt{*The reasoning in both cases is that we get @{term y} such that
   318    @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
   319    @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
   320 apply (rule OrdI [OF _ Ord_is_Transset])
   321  txt{*An ordinal is a transitive set...*}
   322  apply (simp add: Transset_def)
   323  apply clarify
   324  apply (frule apply_recfun2, assumption)
   325  apply (force simp add: restrict_iff)
   326 txt{*...of ordinals.  This second case requires the induction hyp.*}
   327 apply clarify
   328 apply (rename_tac i y)
   329 apply (frule apply_recfun2, assumption)
   330 apply (frule is_recfun_imp_in_r, assumption)
   331 apply (frule is_recfun_restrict)
   332     (*simp_all won't work*)
   333     apply (simp add: trans_trancl trancl_subset_times)+
   334 apply (drule spec [THEN mp], assumption)
   335 apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
   336  apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec)
   337  apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
   338 apply (blast dest: pair_components_in_M)
   339 done
   340 
   341 lemma (in M_wfrank) Ord_range_wellfoundedrank:
   342     "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
   343      ==> Ord (range(wellfoundedrank(M,r,A)))"
   344 apply (frule wellfounded_trancl, assumption)
   345 apply (frule trancl_subset_times)
   346 apply (simp add: wellfoundedrank_def)
   347 apply (rule OrdI [OF _ Ord_is_Transset])
   348  prefer 2
   349  txt{*by our previous result the range consists of ordinals.*}
   350  apply (blast intro: Ord_wfrank_range)
   351 txt{*We still must show that the range is a transitive set.*}
   352 apply (simp add: Transset_def, clarify, simp)
   353 apply (rename_tac x f i u)
   354 apply (frule is_recfun_imp_in_r, assumption)
   355 apply (subgoal_tac "M(u) & M(i) & M(x)")
   356  prefer 2 apply (blast dest: transM, clarify)
   357 apply (rule_tac a=u in rangeI)
   358 apply (rule ReplaceI)
   359   apply (rule_tac x=i in rexI, simp)
   360    apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
   361     apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
   362    apply (simp, simp, blast) 
   363 txt{*Unicity requirement of Replacement*}
   364 apply clarify
   365 apply (frule apply_recfun2, assumption)
   366 apply (simp add: trans_trancl is_recfun_cut)+
   367 done
   368 
   369 lemma (in M_wfrank) function_wellfoundedrank:
   370     "[| wellfounded(M,r); M(r); M(A)|]
   371      ==> function(wellfoundedrank(M,r,A))"
   372 apply (simp add: wellfoundedrank_def function_def, clarify)
   373 txt{*Uniqueness: repeated below!*}
   374 apply (drule is_recfun_functional, assumption)
   375      apply (blast intro: wellfounded_trancl)
   376     apply (simp_all add: trancl_subset_times trans_trancl)
   377 done
   378 
   379 lemma (in M_wfrank) domain_wellfoundedrank:
   380     "[| wellfounded(M,r); M(r); M(A)|]
   381      ==> domain(wellfoundedrank(M,r,A)) = A"
   382 apply (simp add: wellfoundedrank_def function_def)
   383 apply (rule equalityI, auto)
   384 apply (frule transM, assumption)
   385 apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
   386 apply (rule domainI)
   387 apply (rule ReplaceI)
   388   apply (rule_tac x="range(f)" in rexI)
   389   apply simp
   390   apply (rule_tac x=f in rexI, blast, simp_all)
   391 txt{*Uniqueness (for Replacement): repeated above!*}
   392 apply clarify
   393 apply (drule is_recfun_functional, assumption)
   394     apply (blast intro: wellfounded_trancl)
   395     apply (simp_all add: trancl_subset_times trans_trancl)
   396 done
   397 
   398 lemma (in M_wfrank) wellfoundedrank_type:
   399     "[| wellfounded(M,r);  M(r); M(A)|]
   400      ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
   401 apply (frule function_wellfoundedrank [of r A], assumption+)
   402 apply (frule function_imp_Pi)
   403  apply (simp add: wellfoundedrank_def relation_def)
   404  apply blast
   405 apply (simp add: domain_wellfoundedrank)
   406 done
   407 
   408 lemma (in M_wfrank) Ord_wellfoundedrank:
   409     "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
   410      ==> Ord(wellfoundedrank(M,r,A) ` a)"
   411 by (blast intro: apply_funtype [OF wellfoundedrank_type]
   412                  Ord_in_Ord [OF Ord_range_wellfoundedrank])
   413 
   414 lemma (in M_wfrank) wellfoundedrank_eq:
   415      "[| is_recfun(r^+, a, %x. range, f);
   416          wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
   417       ==> wellfoundedrank(M,r,A) ` a = range(f)"
   418 apply (rule apply_equality)
   419  prefer 2 apply (blast intro: wellfoundedrank_type)
   420 apply (simp add: wellfoundedrank_def)
   421 apply (rule ReplaceI)
   422   apply (rule_tac x="range(f)" in rexI) 
   423   apply blast
   424  apply simp_all
   425 txt{*Unicity requirement of Replacement*}
   426 apply clarify
   427 apply (drule is_recfun_functional, assumption)
   428     apply (blast intro: wellfounded_trancl)
   429     apply (simp_all add: trancl_subset_times trans_trancl)
   430 done
   431 
   432 
   433 lemma (in M_wfrank) wellfoundedrank_lt:
   434      "[| <a,b> \<in> r;
   435          wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
   436       ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
   437 apply (frule wellfounded_trancl, assumption)
   438 apply (subgoal_tac "a\<in>A & b\<in>A")
   439  prefer 2 apply blast
   440 apply (simp add: lt_def Ord_wellfoundedrank, clarify)
   441 apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
   442 apply (rename_tac fb)
   443 apply (frule is_recfun_restrict [of concl: "r^+" a])
   444     apply (rule trans_trancl, assumption)
   445    apply (simp_all add: r_into_trancl trancl_subset_times)
   446 txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
   447 apply (simp add: wellfoundedrank_eq)
   448 apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
   449    apply (simp_all add: transM [of a])
   450 txt{*We have used equations for wellfoundedrank and now must use some
   451     for  @{text is_recfun}. *}
   452 apply (rule_tac a=a in rangeI)
   453 apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
   454                  r_into_trancl apply_recfun r_into_trancl)
   455 done
   456 
   457 
   458 lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
   459      "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
   460       ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
   461 apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
   462 apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
   463 apply (simp add: Ord_range_wellfoundedrank, clarify)
   464 apply (frule subsetD, assumption, clarify)
   465 apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
   466 apply (blast intro: apply_rangeI wellfoundedrank_type)
   467 done
   468 
   469 lemma (in M_wfrank) wellfounded_imp_wf:
   470      "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
   471 by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
   472           intro: wf_rvimage_Ord [THEN wf_subset])
   473 
   474 lemma (in M_wfrank) wellfounded_on_imp_wf_on:
   475      "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
   476 apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
   477 apply (rule wellfounded_imp_wf)
   478 apply (simp_all add: relation_def)
   479 done
   480 
   481 
   482 theorem (in M_wfrank) wf_abs [simp]:
   483      "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
   484 by (blast intro: wellfounded_imp_wf wf_imp_relativized)
   485 
   486 theorem (in M_wfrank) wf_on_abs [simp]:
   487      "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
   488 by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
   489 
   490 
   491 text{*absoluteness for wfrec-defined functions.*}
   492 
   493 (*first use is_recfun, then M_is_recfun*)
   494 
   495 lemma (in M_trancl) wfrec_relativize:
   496   "[|wf(r); M(a); M(r);  
   497      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   498           pair(M,x,y,z) & 
   499           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
   500           y = H(x, restrict(g, r -`` {x}))); 
   501      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   502    ==> wfrec(r,a,H) = z <-> 
   503        (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
   504             z = H(a,restrict(f,r-``{a})))"
   505 apply (frule wf_trancl) 
   506 apply (simp add: wftrec_def wfrec_def, safe)
   507  apply (frule wf_exists_is_recfun 
   508               [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"]) 
   509       apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
   510  apply (clarify, rule_tac x=x in rexI) 
   511  apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
   512 done
   513 
   514 
   515 text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
   516       The premise @{term "relation(r)"} is necessary 
   517       before we can replace @{term "r^+"} by @{term r}. *}
   518 theorem (in M_trancl) trans_wfrec_relativize:
   519   "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);
   520      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   521                 pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
   522      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   523    ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))" 
   524 by (simp cong: is_recfun_cong
   525          add: wfrec_relativize trancl_eq_r
   526                is_recfun_restrict_idem domain_restrict_idem)
   527 
   528 
   529 lemma (in M_trancl) trans_eq_pair_wfrec_iff:
   530   "[|wf(r);  trans(r); relation(r); M(r);  M(y); 
   531      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   532                 pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
   533      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   534    ==> y = <x, wfrec(r, x, H)> <-> 
   535        (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   536 apply safe  
   537  apply (simp add: trans_wfrec_relativize [THEN iff_sym]) 
   538 txt{*converse direction*}
   539 apply (rule sym)
   540 apply (simp add: trans_wfrec_relativize, blast) 
   541 done
   542 
   543 
   544 subsection{*M is closed under well-founded recursion*}
   545 
   546 text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
   547 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
   548      "[|wf(r); M(r); 
   549         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
   550         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   551       ==> M(a) --> M(wfrec(r,a,H))"
   552 apply (rule_tac a=a in wf_induct, assumption+)
   553 apply (subst wfrec, assumption, clarify)
   554 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
   555        in rspec [THEN rspec]) 
   556 apply (simp_all add: function_lam) 
   557 apply (blast intro: dest: pair_components_in_M ) 
   558 done
   559 
   560 text{*Eliminates one instance of replacement.*}
   561 lemma (in M_wfrank) wfrec_replacement_iff:
   562      "strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M]. 
   563                 pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)) <->
   564       strong_replacement(M, 
   565            \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
   566 apply simp 
   567 apply (rule strong_replacement_cong, blast) 
   568 done
   569 
   570 text{*Useful version for transitive relations*}
   571 theorem (in M_wfrank) trans_wfrec_closed:
   572      "[|wf(r); trans(r); relation(r); M(r); M(a);
   573         strong_replacement(M, 
   574              \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   575                     pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
   576         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   577       ==> M(wfrec(r,a,H))"
   578 apply (frule wfrec_replacement_iff [THEN iffD1]) 
   579 apply (rule wfrec_closed_lemma, assumption+) 
   580 apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff) 
   581 done
   582 
   583 section{*Absoluteness without assuming transitivity*}
   584 lemma (in M_trancl) eq_pair_wfrec_iff:
   585   "[|wf(r);  M(r);  M(y); 
   586      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   587           pair(M,x,y,z) & 
   588           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
   589           y = H(x, restrict(g, r -`` {x}))); 
   590      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|] 
   591    ==> y = <x, wfrec(r, x, H)> <-> 
   592        (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) & 
   593             y = <x, H(x,restrict(f,r-``{x}))>)"
   594 apply safe  
   595  apply (simp add: wfrec_relativize [THEN iff_sym]) 
   596 txt{*converse direction*}
   597 apply (rule sym)
   598 apply (simp add: wfrec_relativize, blast) 
   599 done
   600 
   601 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
   602      "[|wf(r); M(r); 
   603         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
   604         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   605       ==> M(a) --> M(wfrec(r,a,H))"
   606 apply (rule_tac a=a in wf_induct, assumption+)
   607 apply (subst wfrec, assumption, clarify)
   608 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)" 
   609        in rspec [THEN rspec]) 
   610 apply (simp_all add: function_lam) 
   611 apply (blast intro: dest: pair_components_in_M ) 
   612 done
   613 
   614 text{*Full version not assuming transitivity, but maybe not very useful.*}
   615 theorem (in M_wfrank) wfrec_closed:
   616      "[|wf(r); M(r); M(a);
   617      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
   618           pair(M,x,y,z) & 
   619           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) & 
   620           y = H(x, restrict(g, r -`` {x}))); 
   621         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |] 
   622       ==> M(wfrec(r,a,H))"
   623 apply (frule wfrec_replacement_iff [THEN iffD1]) 
   624 apply (rule wfrec_closed_lemma, assumption+) 
   625 apply (simp_all add: eq_pair_wfrec_iff) 
   626 done
   627 
   628 end