src/ZF/Constructible/WF_absolute.thy
 author paulson Thu Jul 11 17:18:28 2002 +0200 (2002-07-11) changeset 13350 626b79677dfa parent 13348 374d05460db4 child 13352 3cd767f8d78b permissions -rw-r--r--
tidied
```     1 header {*Absoluteness for Well-Founded Relations and Well-Founded Recursion*}
```
```     2
```
```     3 theory WF_absolute = WFrec:
```
```     4
```
```     5 subsection{*Every well-founded relation is a subset of some inverse image of
```
```     6       an ordinal*}
```
```     7
```
```     8 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
```
```     9 by (blast intro: wf_rvimage wf_Memrel)
```
```    10
```
```    11
```
```    12 constdefs
```
```    13   wfrank :: "[i,i]=>i"
```
```    14     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
```
```    15
```
```    16 constdefs
```
```    17   wftype :: "i=>i"
```
```    18     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
```
```    19
```
```    20 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
```
```    21 by (subst wfrank_def [THEN def_wfrec], simp_all)
```
```    22
```
```    23 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
```
```    24 apply (rule_tac a=a in wf_induct, assumption)
```
```    25 apply (subst wfrank, assumption)
```
```    26 apply (rule Ord_succ [THEN Ord_UN], blast)
```
```    27 done
```
```    28
```
```    29 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
```
```    30 apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
```
```    31 apply (rule UN_I [THEN ltI])
```
```    32 apply (simp add: Ord_wfrank vimage_iff)+
```
```    33 done
```
```    34
```
```    35 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
```
```    36 by (simp add: wftype_def Ord_wfrank)
```
```    37
```
```    38 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
```
```    39 apply (simp add: wftype_def)
```
```    40 apply (blast intro: wfrank_lt [THEN ltD])
```
```    41 done
```
```    42
```
```    43
```
```    44 lemma wf_imp_subset_rvimage:
```
```    45      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
```
```    46 apply (rule_tac x="wftype(r)" in exI)
```
```    47 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
```
```    48 apply (simp add: Ord_wftype, clarify)
```
```    49 apply (frule subsetD, assumption, clarify)
```
```    50 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
```
```    51 apply (blast intro: wftypeI)
```
```    52 done
```
```    53
```
```    54 theorem wf_iff_subset_rvimage:
```
```    55   "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
```
```    56 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
```
```    57           intro: wf_rvimage_Ord [THEN wf_subset])
```
```    58
```
```    59
```
```    60 subsection{*Transitive closure without fixedpoints*}
```
```    61
```
```    62 constdefs
```
```    63   rtrancl_alt :: "[i,i]=>i"
```
```    64     "rtrancl_alt(A,r) ==
```
```    65        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
```
```    66                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
```
```    67                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
```
```    68
```
```    69 lemma alt_rtrancl_lemma1 [rule_format]:
```
```    70     "n \<in> nat
```
```    71      ==> \<forall>f \<in> succ(n) -> field(r).
```
```    72          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
```
```    73 apply (induct_tac n)
```
```    74 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
```
```    75 apply (rename_tac n f)
```
```    76 apply (rule rtrancl_into_rtrancl)
```
```    77  prefer 2 apply assumption
```
```    78 apply (drule_tac x="restrict(f,succ(n))" in bspec)
```
```    79  apply (blast intro: restrict_type2)
```
```    80 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
```
```    81 done
```
```    82
```
```    83 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
```
```    84 apply (simp add: rtrancl_alt_def)
```
```    85 apply (blast intro: alt_rtrancl_lemma1)
```
```    86 done
```
```    87
```
```    88 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
```
```    89 apply (simp add: rtrancl_alt_def, clarify)
```
```    90 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
```
```    91 apply (erule rtrancl_induct)
```
```    92  txt{*Base case, trivial*}
```
```    93  apply (rule_tac x=0 in bexI)
```
```    94   apply (rule_tac x="lam x:1. xa" in bexI)
```
```    95    apply simp_all
```
```    96 txt{*Inductive step*}
```
```    97 apply clarify
```
```    98 apply (rename_tac n f)
```
```    99 apply (rule_tac x="succ(n)" in bexI)
```
```   100  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
```
```   101   apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
```
```   102   apply (blast intro: mem_asym)
```
```   103  apply typecheck
```
```   104  apply auto
```
```   105 done
```
```   106
```
```   107 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
```
```   108 by (blast del: subsetI
```
```   109 	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
```
```   110
```
```   111
```
```   112 constdefs
```
```   113
```
```   114   rtran_closure_mem :: "[i=>o,i,i,i] => o"
```
```   115     --{*The property of belonging to @{text "rtran_closure(r)"}*}
```
```   116     "rtran_closure_mem(M,A,r,p) ==
```
```   117 	      \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
```
```   118                omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
```
```   119 	       (\<exists>f[M]. typed_function(M,n',A,f) &
```
```   120 		(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
```
```   121 		  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
```
```   122 		  (\<forall>j[M]. j\<in>n -->
```
```   123 		    (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
```
```   124 		      fun_apply(M,f,j,fj) & successor(M,j,sj) &
```
```   125 		      fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"
```
```   126
```
```   127   rtran_closure :: "[i=>o,i,i] => o"
```
```   128     "rtran_closure(M,r,s) ==
```
```   129         \<forall>A[M]. is_field(M,r,A) -->
```
```   130  	 (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))"
```
```   131
```
```   132   tran_closure :: "[i=>o,i,i] => o"
```
```   133     "tran_closure(M,r,t) ==
```
```   134          \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
```
```   135
```
```   136 lemma (in M_axioms) rtran_closure_mem_iff:
```
```   137      "[|M(A); M(r); M(p)|]
```
```   138       ==> rtran_closure_mem(M,A,r,p) <->
```
```   139           (\<exists>n[M]. n\<in>nat &
```
```   140            (\<exists>f[M]. f \<in> succ(n) -> A &
```
```   141             (\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
```
```   142                            (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)))"
```
```   143 apply (simp add: rtran_closure_mem_def typed_apply_abs
```
```   144                  Ord_succ_mem_iff nat_0_le [THEN ltD], blast)
```
```   145 done
```
```   146
```
```   147 locale M_trancl = M_axioms +
```
```   148   assumes rtrancl_separation:
```
```   149 	 "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
```
```   150       and wellfounded_trancl_separation:
```
```   151 	 "[| M(r); M(Z) |] ==>
```
```   152 	  separation (M, \<lambda>x.
```
```   153 	      \<exists>w[M]. \<exists>wx[M]. \<exists>rp[M].
```
```   154 	       w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
```
```   155
```
```   156
```
```   157 lemma (in M_trancl) rtran_closure_rtrancl:
```
```   158      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
```
```   159 apply (simp add: rtran_closure_def rtran_closure_mem_iff
```
```   160                  rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
```
```   161 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
```
```   162 done
```
```   163
```
```   164 lemma (in M_trancl) rtrancl_closed [intro,simp]:
```
```   165      "M(r) ==> M(rtrancl(r))"
```
```   166 apply (insert rtrancl_separation [of r "field(r)"])
```
```   167 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
```
```   168                  rtrancl_alt_def rtran_closure_mem_iff)
```
```   169 done
```
```   170
```
```   171 lemma (in M_trancl) rtrancl_abs [simp]:
```
```   172      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
```
```   173 apply (rule iffI)
```
```   174  txt{*Proving the right-to-left implication*}
```
```   175  prefer 2 apply (blast intro: rtran_closure_rtrancl)
```
```   176 apply (rule M_equalityI)
```
```   177 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
```
```   178                  rtrancl_alt_def rtran_closure_mem_iff)
```
```   179 apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
```
```   180 done
```
```   181
```
```   182 lemma (in M_trancl) trancl_closed [intro,simp]:
```
```   183      "M(r) ==> M(trancl(r))"
```
```   184 by (simp add: trancl_def comp_closed rtrancl_closed)
```
```   185
```
```   186 lemma (in M_trancl) trancl_abs [simp]:
```
```   187      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
```
```   188 by (simp add: tran_closure_def trancl_def)
```
```   189
```
```   190 lemma (in M_trancl) wellfounded_trancl_separation':
```
```   191      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
```
```   192 by (insert wellfounded_trancl_separation [of r Z], simp)
```
```   193
```
```   194 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
```
```   195       relativized version.  Original version is on theory WF.*}
```
```   196 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
```
```   197 apply (simp add: wf_on_def wf_def)
```
```   198 apply (safe intro!: equalityI)
```
```   199 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
```
```   200 apply (blast elim: tranclE)
```
```   201 done
```
```   202
```
```   203 lemma (in M_trancl) wellfounded_on_trancl:
```
```   204      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
```
```   205       ==> wellfounded_on(M,A,r^+)"
```
```   206 apply (simp add: wellfounded_on_def)
```
```   207 apply (safe intro!: equalityI)
```
```   208 apply (rename_tac Z x)
```
```   209 apply (subgoal_tac "M({x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+})")
```
```   210  prefer 2
```
```   211  apply (blast intro: wellfounded_trancl_separation')
```
```   212 apply (drule_tac x = "{x\<in>A. \<exists>w[M]. w \<in> Z & \<langle>w,x\<rangle> \<in> r^+}" in rspec, safe)
```
```   213 apply (blast dest: transM, simp)
```
```   214 apply (rename_tac y w)
```
```   215 apply (drule_tac x=w in bspec, assumption, clarify)
```
```   216 apply (erule tranclE)
```
```   217   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
```
```   218  apply blast
```
```   219 done
```
```   220
```
```   221 lemma (in M_trancl) wellfounded_trancl:
```
```   222      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
```
```   223 apply (rotate_tac -1)
```
```   224 apply (simp add: wellfounded_iff_wellfounded_on_field)
```
```   225 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
```
```   226    apply blast
```
```   227   apply (simp_all add: trancl_type [THEN field_rel_subset])
```
```   228 done
```
```   229
```
```   230 text{*Relativized to M: Every well-founded relation is a subset of some
```
```   231 inverse image of an ordinal.  Key step is the construction (in M) of a
```
```   232 rank function.*}
```
```   233
```
```   234
```
```   235 locale M_wfrank = M_trancl +
```
```   236   assumes wfrank_separation:
```
```   237      "M(r) ==>
```
```   238       separation (M, \<lambda>x.
```
```   239          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
```
```   240          ~ (\<exists>f[M]. M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f)))"
```
```   241  and wfrank_strong_replacement:
```
```   242      "M(r) ==>
```
```   243       strong_replacement(M, \<lambda>x z.
```
```   244          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
```
```   245          (\<exists>y[M]. \<exists>f[M]. pair(M,x,y,z)  &
```
```   246                         M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f) &
```
```   247                         is_range(M,f,y)))"
```
```   248  and Ord_wfrank_separation:
```
```   249      "M(r) ==>
```
```   250       separation (M, \<lambda>x.
```
```   251          \<forall>rplus[M]. tran_closure(M,r,rplus) -->
```
```   252           ~ (\<forall>f[M]. \<forall>rangef[M].
```
```   253              is_range(M,f,rangef) -->
```
```   254              M_is_recfun(M, rplus, x, \<lambda>x f y. is_range(M,f,y), f) -->
```
```   255              ordinal(M,rangef)))"
```
```   256
```
```   257 text{*Proving that the relativized instances of Separation or Replacement
```
```   258 agree with the "real" ones.*}
```
```   259
```
```   260 lemma (in M_wfrank) wfrank_separation':
```
```   261      "M(r) ==>
```
```   262       separation
```
```   263 	   (M, \<lambda>x. ~ (\<exists>f[M]. is_recfun(r^+, x, %x f. range(f), f)))"
```
```   264 apply (insert wfrank_separation [of r])
```
```   265 apply (simp add: is_recfun_abs [of "%x. range"])
```
```   266 done
```
```   267
```
```   268 lemma (in M_wfrank) 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 apply (insert wfrank_strong_replacement [of r])
```
```   274 apply (simp add: is_recfun_abs [of "%x. range"])
```
```   275 done
```
```   276
```
```   277 lemma (in M_wfrank) Ord_wfrank_separation':
```
```   278      "M(r) ==>
```
```   279       separation (M, \<lambda>x.
```
```   280          ~ (\<forall>f[M]. is_recfun(r^+, x, \<lambda>x. range, f) --> Ord(range(f))))"
```
```   281 apply (insert Ord_wfrank_separation [of r])
```
```   282 apply (simp add: is_recfun_abs [of "%x. range"])
```
```   283 done
```
```   284
```
```   285 text{*This function, defined using replacement, is a rank function for
```
```   286 well-founded relations within the class M.*}
```
```   287 constdefs
```
```   288  wellfoundedrank :: "[i=>o,i,i] => i"
```
```   289     "wellfoundedrank(M,r,A) ==
```
```   290         {p. x\<in>A, \<exists>y[M]. \<exists>f[M].
```
```   291                        p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
```
```   292                        y = range(f)}"
```
```   293
```
```   294 lemma (in M_wfrank) exists_wfrank:
```
```   295     "[| wellfounded(M,r); M(a); M(r) |]
```
```   296      ==> \<exists>f[M]. is_recfun(r^+, a, %x f. range(f), f)"
```
```   297 apply (rule wellfounded_exists_is_recfun)
```
```   298       apply (blast intro: wellfounded_trancl)
```
```   299      apply (rule trans_trancl)
```
```   300     apply (erule wfrank_separation')
```
```   301    apply (erule wfrank_strong_replacement')
```
```   302 apply (simp_all add: trancl_subset_times)
```
```   303 done
```
```   304
```
```   305 lemma (in M_wfrank) M_wellfoundedrank:
```
```   306     "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
```
```   307 apply (insert wfrank_strong_replacement' [of r])
```
```   308 apply (simp add: wellfoundedrank_def)
```
```   309 apply (rule strong_replacement_closed)
```
```   310    apply assumption+
```
```   311  apply (rule univalent_is_recfun)
```
```   312    apply (blast intro: wellfounded_trancl)
```
```   313   apply (rule trans_trancl)
```
```   314  apply (simp add: trancl_subset_times, blast)
```
```   315 done
```
```   316
```
```   317 lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
```
```   318     "[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
```
```   319      ==> \<forall>f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
```
```   320 apply (drule wellfounded_trancl, assumption)
```
```   321 apply (rule wellfounded_induct, assumption+)
```
```   322   apply simp
```
```   323  apply (blast intro: Ord_wfrank_separation', clarify)
```
```   324 txt{*The reasoning in both cases is that we get @{term y} such that
```
```   325    @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
```
```   326    @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
```
```   327 apply (rule OrdI [OF _ Ord_is_Transset])
```
```   328  txt{*An ordinal is a transitive set...*}
```
```   329  apply (simp add: Transset_def)
```
```   330  apply clarify
```
```   331  apply (frule apply_recfun2, assumption)
```
```   332  apply (force simp add: restrict_iff)
```
```   333 txt{*...of ordinals.  This second case requires the induction hyp.*}
```
```   334 apply clarify
```
```   335 apply (rename_tac i y)
```
```   336 apply (frule apply_recfun2, assumption)
```
```   337 apply (frule is_recfun_imp_in_r, assumption)
```
```   338 apply (frule is_recfun_restrict)
```
```   339     (*simp_all won't work*)
```
```   340     apply (simp add: trans_trancl trancl_subset_times)+
```
```   341 apply (drule spec [THEN mp], assumption)
```
```   342 apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
```
```   343  apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
```
```   344 apply assumption
```
```   345  apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
```
```   346 apply (blast dest: pair_components_in_M)
```
```   347 done
```
```   348
```
```   349 lemma (in M_wfrank) Ord_range_wellfoundedrank:
```
```   350     "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
```
```   351      ==> Ord (range(wellfoundedrank(M,r,A)))"
```
```   352 apply (frule wellfounded_trancl, assumption)
```
```   353 apply (frule trancl_subset_times)
```
```   354 apply (simp add: wellfoundedrank_def)
```
```   355 apply (rule OrdI [OF _ Ord_is_Transset])
```
```   356  prefer 2
```
```   357  txt{*by our previous result the range consists of ordinals.*}
```
```   358  apply (blast intro: Ord_wfrank_range)
```
```   359 txt{*We still must show that the range is a transitive set.*}
```
```   360 apply (simp add: Transset_def, clarify, simp)
```
```   361 apply (rename_tac x i f u)
```
```   362 apply (frule is_recfun_imp_in_r, assumption)
```
```   363 apply (subgoal_tac "M(u) & M(i) & M(x)")
```
```   364  prefer 2 apply (blast dest: transM, clarify)
```
```   365 apply (rule_tac a=u in rangeI)
```
```   366 apply (rule_tac x=u in ReplaceI)
```
```   367   apply simp
```
```   368   apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI)
```
```   369    apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
```
```   370   apply simp
```
```   371 apply blast
```
```   372 txt{*Unicity requirement of Replacement*}
```
```   373 apply clarify
```
```   374 apply (frule apply_recfun2, assumption)
```
```   375 apply (simp add: trans_trancl is_recfun_cut)
```
```   376 done
```
```   377
```
```   378 lemma (in M_wfrank) function_wellfoundedrank:
```
```   379     "[| wellfounded(M,r); M(r); M(A)|]
```
```   380      ==> function(wellfoundedrank(M,r,A))"
```
```   381 apply (simp add: wellfoundedrank_def function_def, clarify)
```
```   382 txt{*Uniqueness: repeated below!*}
```
```   383 apply (drule is_recfun_functional, assumption)
```
```   384      apply (blast intro: wellfounded_trancl)
```
```   385     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   386 done
```
```   387
```
```   388 lemma (in M_wfrank) domain_wellfoundedrank:
```
```   389     "[| wellfounded(M,r); M(r); M(A)|]
```
```   390      ==> domain(wellfoundedrank(M,r,A)) = A"
```
```   391 apply (simp add: wellfoundedrank_def function_def)
```
```   392 apply (rule equalityI, auto)
```
```   393 apply (frule transM, assumption)
```
```   394 apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
```
```   395 apply (rule_tac b="range(f)" in domainI)
```
```   396 apply (rule_tac x=x in ReplaceI)
```
```   397   apply simp
```
```   398   apply (rule_tac x=f in rexI, blast, simp_all)
```
```   399 txt{*Uniqueness (for Replacement): repeated above!*}
```
```   400 apply clarify
```
```   401 apply (drule is_recfun_functional, assumption)
```
```   402     apply (blast intro: wellfounded_trancl)
```
```   403     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   404 done
```
```   405
```
```   406 lemma (in M_wfrank) wellfoundedrank_type:
```
```   407     "[| wellfounded(M,r);  M(r); M(A)|]
```
```   408      ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
```
```   409 apply (frule function_wellfoundedrank [of r A], assumption+)
```
```   410 apply (frule function_imp_Pi)
```
```   411  apply (simp add: wellfoundedrank_def relation_def)
```
```   412  apply blast
```
```   413 apply (simp add: domain_wellfoundedrank)
```
```   414 done
```
```   415
```
```   416 lemma (in M_wfrank) Ord_wellfoundedrank:
```
```   417     "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
```
```   418      ==> Ord(wellfoundedrank(M,r,A) ` a)"
```
```   419 by (blast intro: apply_funtype [OF wellfoundedrank_type]
```
```   420                  Ord_in_Ord [OF Ord_range_wellfoundedrank])
```
```   421
```
```   422 lemma (in M_wfrank) wellfoundedrank_eq:
```
```   423      "[| is_recfun(r^+, a, %x. range, f);
```
```   424          wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
```
```   425       ==> wellfoundedrank(M,r,A) ` a = range(f)"
```
```   426 apply (rule apply_equality)
```
```   427  prefer 2 apply (blast intro: wellfoundedrank_type)
```
```   428 apply (simp add: wellfoundedrank_def)
```
```   429 apply (rule ReplaceI)
```
```   430   apply (rule_tac x="range(f)" in rexI)
```
```   431   apply blast
```
```   432  apply simp_all
```
```   433 txt{*Unicity requirement of Replacement*}
```
```   434 apply clarify
```
```   435 apply (drule is_recfun_functional, assumption)
```
```   436     apply (blast intro: wellfounded_trancl)
```
```   437     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   438 done
```
```   439
```
```   440
```
```   441 lemma (in M_wfrank) wellfoundedrank_lt:
```
```   442      "[| <a,b> \<in> r;
```
```   443          wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
```
```   444       ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
```
```   445 apply (frule wellfounded_trancl, assumption)
```
```   446 apply (subgoal_tac "a\<in>A & b\<in>A")
```
```   447  prefer 2 apply blast
```
```   448 apply (simp add: lt_def Ord_wellfoundedrank, clarify)
```
```   449 apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
```
```   450 apply (rename_tac fb)
```
```   451 apply (frule is_recfun_restrict [of concl: "r^+" a])
```
```   452     apply (rule trans_trancl, assumption)
```
```   453    apply (simp_all add: r_into_trancl trancl_subset_times)
```
```   454 txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
```
```   455 apply (simp add: wellfoundedrank_eq)
```
```   456 apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
```
```   457    apply (simp_all add: transM [of a])
```
```   458 txt{*We have used equations for wellfoundedrank and now must use some
```
```   459     for  @{text is_recfun}. *}
```
```   460 apply (rule_tac a=a in rangeI)
```
```   461 apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
```
```   462                  r_into_trancl apply_recfun r_into_trancl)
```
```   463 done
```
```   464
```
```   465
```
```   466 lemma (in M_wfrank) wellfounded_imp_subset_rvimage:
```
```   467      "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
```
```   468       ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
```
```   469 apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
```
```   470 apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
```
```   471 apply (simp add: Ord_range_wellfoundedrank, clarify)
```
```   472 apply (frule subsetD, assumption, clarify)
```
```   473 apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
```
```   474 apply (blast intro: apply_rangeI wellfoundedrank_type)
```
```   475 done
```
```   476
```
```   477 lemma (in M_wfrank) wellfounded_imp_wf:
```
```   478      "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
```
```   479 by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
```
```   480           intro: wf_rvimage_Ord [THEN wf_subset])
```
```   481
```
```   482 lemma (in M_wfrank) wellfounded_on_imp_wf_on:
```
```   483      "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
```
```   484 apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
```
```   485 apply (rule wellfounded_imp_wf)
```
```   486 apply (simp_all add: relation_def)
```
```   487 done
```
```   488
```
```   489
```
```   490 theorem (in M_wfrank) wf_abs [simp]:
```
```   491      "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
```
```   492 by (blast intro: wellfounded_imp_wf wf_imp_relativized)
```
```   493
```
```   494 theorem (in M_wfrank) wf_on_abs [simp]:
```
```   495      "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
```
```   496 by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
```
```   497
```
```   498
```
```   499 text{*absoluteness for wfrec-defined functions.*}
```
```   500
```
```   501 (*first use is_recfun, then M_is_recfun*)
```
```   502
```
```   503 lemma (in M_trancl) wfrec_relativize:
```
```   504   "[|wf(r); M(a); M(r);
```
```   505      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
```
```   506           pair(M,x,y,z) &
```
```   507           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
```
```   508           y = H(x, restrict(g, r -`` {x})));
```
```   509      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   510    ==> wfrec(r,a,H) = z <->
```
```   511        (\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
```
```   512             z = H(a,restrict(f,r-``{a})))"
```
```   513 apply (frule wf_trancl)
```
```   514 apply (simp add: wftrec_def wfrec_def, safe)
```
```   515  apply (frule wf_exists_is_recfun
```
```   516               [of concl: "r^+" a "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
```
```   517       apply (simp_all add: trans_trancl function_restrictI trancl_subset_times)
```
```   518  apply (clarify, rule_tac x=x in rexI)
```
```   519  apply (simp_all add: the_recfun_eq trans_trancl trancl_subset_times)
```
```   520 done
```
```   521
```
```   522
```
```   523 text{*Assuming @{term r} is transitive simplifies the occurrences of @{text H}.
```
```   524       The premise @{term "relation(r)"} is necessary
```
```   525       before we can replace @{term "r^+"} by @{term r}. *}
```
```   526 theorem (in M_trancl) trans_wfrec_relativize:
```
```   527   "[|wf(r);  trans(r);  relation(r);  M(r);  M(a);
```
```   528      strong_replacement(M, \<lambda>x z. \<exists>y[M].
```
```   529                 pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
```
```   530      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   531    ==> wfrec(r,a,H) = z <-> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
```
```   532 by (simp cong: is_recfun_cong
```
```   533          add: wfrec_relativize trancl_eq_r
```
```   534                is_recfun_restrict_idem domain_restrict_idem)
```
```   535
```
```   536
```
```   537 lemma (in M_trancl) trans_eq_pair_wfrec_iff:
```
```   538   "[|wf(r);  trans(r); relation(r); M(r);  M(y);
```
```   539      strong_replacement(M, \<lambda>x z. \<exists>y[M].
```
```   540                 pair(M,x,y,z) & (\<exists>g[M]. is_recfun(r,x,H,g) & y = H(x,g)));
```
```   541      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   542    ==> y = <x, wfrec(r, x, H)> <->
```
```   543        (\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
```
```   544 apply safe
```
```   545  apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x])
```
```   546 txt{*converse direction*}
```
```   547 apply (rule sym)
```
```   548 apply (simp add: trans_wfrec_relativize, blast)
```
```   549 done
```
```   550
```
```   551
```
```   552 subsection{*M is closed under well-founded recursion*}
```
```   553
```
```   554 text{*Lemma with the awkward premise mentioning @{text wfrec}.*}
```
```   555 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
```
```   556      "[|wf(r); M(r);
```
```   557         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
```
```   558         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
```
```   559       ==> M(a) --> M(wfrec(r,a,H))"
```
```   560 apply (rule_tac a=a in wf_induct, assumption+)
```
```   561 apply (subst wfrec, assumption, clarify)
```
```   562 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
```
```   563        in rspec [THEN rspec])
```
```   564 apply (simp_all add: function_lam)
```
```   565 apply (blast intro: dest: pair_components_in_M )
```
```   566 done
```
```   567
```
```   568 text{*Eliminates one instance of replacement.*}
```
```   569 lemma (in M_wfrank) wfrec_replacement_iff:
```
```   570      "strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
```
```   571                 pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)) <->
```
```   572       strong_replacement(M,
```
```   573            \<lambda>x y. \<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
```
```   574 apply simp
```
```   575 apply (rule strong_replacement_cong, blast)
```
```   576 done
```
```   577
```
```   578 text{*Useful version for transitive relations*}
```
```   579 theorem (in M_wfrank) trans_wfrec_closed:
```
```   580      "[|wf(r); trans(r); relation(r); M(r); M(a);
```
```   581         strong_replacement(M,
```
```   582              \<lambda>x z. \<exists>y[M]. \<exists>g[M].
```
```   583                     pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g));
```
```   584         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
```
```   585       ==> M(wfrec(r,a,H))"
```
```   586 apply (frule wfrec_replacement_iff [THEN iffD1])
```
```   587 apply (rule wfrec_closed_lemma, assumption+)
```
```   588 apply (simp_all add: wfrec_replacement_iff trans_eq_pair_wfrec_iff)
```
```   589 done
```
```   590
```
```   591 section{*Absoluteness without assuming transitivity*}
```
```   592 lemma (in M_trancl) eq_pair_wfrec_iff:
```
```   593   "[|wf(r);  M(r);  M(y);
```
```   594      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
```
```   595           pair(M,x,y,z) &
```
```   596           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
```
```   597           y = H(x, restrict(g, r -`` {x})));
```
```   598      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   599    ==> y = <x, wfrec(r, x, H)> <->
```
```   600        (\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
```
```   601             y = <x, H(x,restrict(f,r-``{x}))>)"
```
```   602 apply safe
```
```   603  apply (simp add: wfrec_relativize [THEN iff_sym, of concl: _ x])
```
```   604 txt{*converse direction*}
```
```   605 apply (rule sym)
```
```   606 apply (simp add: wfrec_relativize, blast)
```
```   607 done
```
```   608
```
```   609 lemma (in M_wfrank) wfrec_closed_lemma [rule_format]:
```
```   610      "[|wf(r); M(r);
```
```   611         strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
```
```   612         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
```
```   613       ==> M(a) --> M(wfrec(r,a,H))"
```
```   614 apply (rule_tac a=a in wf_induct, assumption+)
```
```   615 apply (subst wfrec, assumption, clarify)
```
```   616 apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
```
```   617        in rspec [THEN rspec])
```
```   618 apply (simp_all add: function_lam)
```
```   619 apply (blast intro: dest: pair_components_in_M )
```
```   620 done
```
```   621
```
```   622 text{*Full version not assuming transitivity, but maybe not very useful.*}
```
```   623 theorem (in M_wfrank) wfrec_closed:
```
```   624      "[|wf(r); M(r); M(a);
```
```   625      strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
```
```   626           pair(M,x,y,z) &
```
```   627           is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
```
```   628           y = H(x, restrict(g, r -`` {x})));
```
```   629         \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g)) |]
```
```   630       ==> M(wfrec(r,a,H))"
```
```   631 apply (frule wfrec_replacement_iff [THEN iffD1])
```
```   632 apply (rule wfrec_closed_lemma, assumption+)
```
```   633 apply (simp_all add: eq_pair_wfrec_iff)
```
```   634 done
```
```   635
```
```   636 end
```