src/ZF/Constructible/WF_absolute.thy
 author paulson Wed Jun 26 18:31:20 2002 +0200 (2002-06-26) changeset 13251 74cb2af8811e parent 13247 e3c289f0724b child 13254 5146ccaedf42 permissions -rw-r--r--
new treatment of wfrec, replacing wf[A](r) by wf(r)
```     1 theory WF_absolute = WFrec:
```
```     2
```
```     3 subsection{*Every well-founded relation is a subset of some inverse image of
```
```     4       an ordinal*}
```
```     5
```
```     6 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
```
```     7 by (blast intro: wf_rvimage wf_Memrel)
```
```     8
```
```     9
```
```    10 constdefs
```
```    11   wfrank :: "[i,i]=>i"
```
```    12     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
```
```    13
```
```    14 constdefs
```
```    15   wftype :: "i=>i"
```
```    16     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
```
```    17
```
```    18 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
```
```    19 by (subst wfrank_def [THEN def_wfrec], simp_all)
```
```    20
```
```    21 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
```
```    22 apply (rule_tac a="a" in wf_induct, assumption)
```
```    23 apply (subst wfrank, assumption)
```
```    24 apply (rule Ord_succ [THEN Ord_UN], blast)
```
```    25 done
```
```    26
```
```    27 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
```
```    28 apply (rule_tac a1 = "b" in wfrank [THEN ssubst], assumption)
```
```    29 apply (rule UN_I [THEN ltI])
```
```    30 apply (simp add: Ord_wfrank vimage_iff)+
```
```    31 done
```
```    32
```
```    33 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
```
```    34 by (simp add: wftype_def Ord_wfrank)
```
```    35
```
```    36 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
```
```    37 apply (simp add: wftype_def)
```
```    38 apply (blast intro: wfrank_lt [THEN ltD])
```
```    39 done
```
```    40
```
```    41
```
```    42 lemma wf_imp_subset_rvimage:
```
```    43      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
```
```    44 apply (rule_tac x="wftype(r)" in exI)
```
```    45 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
```
```    46 apply (simp add: Ord_wftype, clarify)
```
```    47 apply (frule subsetD, assumption, clarify)
```
```    48 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
```
```    49 apply (blast intro: wftypeI)
```
```    50 done
```
```    51
```
```    52 theorem wf_iff_subset_rvimage:
```
```    53   "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
```
```    54 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
```
```    55           intro: wf_rvimage_Ord [THEN wf_subset])
```
```    56
```
```    57
```
```    58 subsection{*Transitive closure without fixedpoints*}
```
```    59
```
```    60 constdefs
```
```    61   rtrancl_alt :: "[i,i]=>i"
```
```    62     "rtrancl_alt(A,r) ==
```
```    63        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
```
```    64                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
```
```    65                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
```
```    66
```
```    67 lemma alt_rtrancl_lemma1 [rule_format]:
```
```    68     "n \<in> nat
```
```    69      ==> \<forall>f \<in> succ(n) -> field(r).
```
```    70          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
```
```    71 apply (induct_tac n)
```
```    72 apply (simp_all add: apply_funtype rtrancl_refl, clarify)
```
```    73 apply (rename_tac n f)
```
```    74 apply (rule rtrancl_into_rtrancl)
```
```    75  prefer 2 apply assumption
```
```    76 apply (drule_tac x="restrict(f,succ(n))" in bspec)
```
```    77  apply (blast intro: restrict_type2)
```
```    78 apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
```
```    79 done
```
```    80
```
```    81 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
```
```    82 apply (simp add: rtrancl_alt_def)
```
```    83 apply (blast intro: alt_rtrancl_lemma1)
```
```    84 done
```
```    85
```
```    86 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
```
```    87 apply (simp add: rtrancl_alt_def, clarify)
```
```    88 apply (frule rtrancl_type [THEN subsetD], clarify, simp)
```
```    89 apply (erule rtrancl_induct)
```
```    90  txt{*Base case, trivial*}
```
```    91  apply (rule_tac x=0 in bexI)
```
```    92   apply (rule_tac x="lam x:1. xa" in bexI)
```
```    93    apply simp_all
```
```    94 txt{*Inductive step*}
```
```    95 apply clarify
```
```    96 apply (rename_tac n f)
```
```    97 apply (rule_tac x="succ(n)" in bexI)
```
```    98  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
```
```    99   apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
```
```   100   apply (blast intro: mem_asym)
```
```   101  apply typecheck
```
```   102  apply auto
```
```   103 done
```
```   104
```
```   105 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
```
```   106 by (blast del: subsetI
```
```   107 	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
```
```   108
```
```   109
```
```   110 constdefs
```
```   111
```
```   112   rtran_closure :: "[i=>o,i,i] => o"
```
```   113     "rtran_closure(M,r,s) ==
```
```   114         \<forall>A. M(A) --> is_field(M,r,A) -->
```
```   115  	 (\<forall>p. M(p) -->
```
```   116           (p \<in> s <->
```
```   117            (\<exists>n\<in>nat. M(n) &
```
```   118             (\<exists>n'. M(n') & successor(M,n,n') &
```
```   119              (\<exists>f. M(f) & typed_function(M,n',A,f) &
```
```   120               (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &
```
```   121                    fun_apply(M,f,0,x) & fun_apply(M,f,n,y))) &
```
```   122               (\<forall>i\<in>n. M(i) -->
```
```   123                 (\<forall>i'. M(i') --> successor(M,i,i') -->
```
```   124                  (\<forall>fi. M(fi) --> fun_apply(M,f,i,fi) -->
```
```   125                   (\<forall>fi'. M(fi') --> fun_apply(M,f,i',fi') -->
```
```   126                    (\<forall>q. M(q) --> pair(M,fi,fi',q) --> q \<in> r))))))))))"
```
```   127
```
```   128   tran_closure :: "[i=>o,i,i] => o"
```
```   129     "tran_closure(M,r,t) ==
```
```   130          \<exists>s. M(s) & rtran_closure(M,r,s) & composition(M,r,s,t)"
```
```   131
```
```   132
```
```   133 locale M_trancl = M_axioms +
```
```   134 (*THEY NEED RELATIVIZATION*)
```
```   135   assumes rtrancl_separation:
```
```   136      "[| M(r); M(A) |] ==>
```
```   137 	separation
```
```   138 	   (M, \<lambda>p. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
```
```   139                     (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
```
```   140                           (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r))"
```
```   141       and wellfounded_trancl_separation:
```
```   142      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z)"
```
```   143
```
```   144
```
```   145 lemma (in M_trancl) rtran_closure_rtrancl:
```
```   146      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
```
```   147 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
```
```   148                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
```
```   149                  Ord_succ_mem_iff M_nat  nat_0_le [THEN ltD], clarify)
```
```   150 apply (rule iffI)
```
```   151  apply clarify
```
```   152  apply simp
```
```   153  apply (rename_tac n f)
```
```   154  apply (rule_tac x=n in bexI)
```
```   155   apply (rule_tac x=f in exI)
```
```   156   apply simp
```
```   157   apply (blast dest: finite_fun_closed dest: transM)
```
```   158  apply assumption
```
```   159 apply clarify
```
```   160 apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)
```
```   161 done
```
```   162
```
```   163 lemma (in M_trancl) rtrancl_closed [intro,simp]:
```
```   164      "M(r) ==> M(rtrancl(r))"
```
```   165 apply (insert rtrancl_separation [of r "field(r)"])
```
```   166 apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
```
```   167                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
```
```   168                  Ord_succ_mem_iff M_nat
```
```   169                  nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
```
```   170 done
```
```   171
```
```   172 lemma (in M_trancl) rtrancl_abs [simp]:
```
```   173      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
```
```   174 apply (rule iffI)
```
```   175  txt{*Proving the right-to-left implication*}
```
```   176  prefer 2 apply (blast intro: rtran_closure_rtrancl)
```
```   177 apply (rule M_equalityI)
```
```   178 apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
```
```   179                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
```
```   180                  Ord_succ_mem_iff M_nat
```
```   181                  nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
```
```   182  prefer 2 apply assumption
```
```   183  prefer 2 apply blast
```
```   184 apply (rule iffI, clarify)
```
```   185 apply (simp add: nat_0_le [THEN ltD]  apply_funtype, blast, clarify, simp)
```
```   186  apply (rename_tac n f)
```
```   187  apply (rule_tac x=n in bexI)
```
```   188   apply (rule_tac x=f in exI)
```
```   189   apply (blast dest!: finite_fun_closed, assumption)
```
```   190 done
```
```   191
```
```   192
```
```   193 lemma (in M_trancl) trancl_closed [intro,simp]:
```
```   194      "M(r) ==> M(trancl(r))"
```
```   195 by (simp add: trancl_def comp_closed rtrancl_closed)
```
```   196
```
```   197 lemma (in M_trancl) trancl_abs [simp]:
```
```   198      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
```
```   199 by (simp add: tran_closure_def trancl_def)
```
```   200
```
```   201
```
```   202 text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
```
```   203       relativized version.  Original version is on theory WF.*}
```
```   204 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
```
```   205 apply (simp add: wf_on_def wf_def)
```
```   206 apply (safe intro!: equalityI)
```
```   207 apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
```
```   208 apply (blast elim: tranclE)
```
```   209 done
```
```   210
```
```   211
```
```   212 lemma (in M_trancl) wellfounded_on_trancl:
```
```   213      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
```
```   214       ==> wellfounded_on(M,A,r^+)"
```
```   215 apply (simp add: wellfounded_on_def)
```
```   216 apply (safe intro!: equalityI)
```
```   217 apply (rename_tac Z x)
```
```   218 apply (subgoal_tac "M({x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z})")
```
```   219  prefer 2
```
```   220  apply (simp add: wellfounded_trancl_separation)
```
```   221 apply (drule_tac x = "{x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
```
```   222 apply safe
```
```   223 apply (blast dest: transM, simp)
```
```   224 apply (rename_tac y w)
```
```   225 apply (drule_tac x=w in bspec, assumption, clarify)
```
```   226 apply (erule tranclE)
```
```   227   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
```
```   228  apply blast
```
```   229 done
```
```   230
```
```   231 (*????move to Wellorderings.thy*)
```
```   232 lemma (in M_axioms) wellfounded_on_field_imp_wellfounded:
```
```   233      "wellfounded_on(M, field(r), r) ==> wellfounded(M,r)"
```
```   234 by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)
```
```   235
```
```   236 lemma (in M_axioms) wellfounded_iff_wellfounded_on_field:
```
```   237      "M(r) ==> wellfounded(M,r) <-> wellfounded_on(M, field(r), r)"
```
```   238 by (blast intro: wellfounded_imp_wellfounded_on
```
```   239                  wellfounded_on_field_imp_wellfounded)
```
```   240
```
```   241 lemma (in M_axioms) wellfounded_on_subset_A:
```
```   242      "[| wellfounded_on(M,A,r);  B<=A |] ==> wellfounded_on(M,B,r)"
```
```   243 by (simp add: wellfounded_on_def, blast)
```
```   244
```
```   245
```
```   246
```
```   247 lemma (in M_trancl) wellfounded_trancl:
```
```   248      "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
```
```   249 apply (rotate_tac -1)
```
```   250 apply (simp add: wellfounded_iff_wellfounded_on_field)
```
```   251 apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
```
```   252    apply blast
```
```   253   apply (simp_all add: trancl_type [THEN field_rel_subset])
```
```   254 done
```
```   255
```
```   256 text{*Relativized to M: Every well-founded relation is a subset of some
```
```   257 inverse image of an ordinal.  Key step is the construction (in M) of a
```
```   258 rank function.*}
```
```   259
```
```   260
```
```   261 (*NEEDS RELATIVIZATION*)
```
```   262 locale M_recursion = M_trancl +
```
```   263   assumes wfrank_separation':
```
```   264      "M(r) ==>
```
```   265 	separation
```
```   266 	   (M, \<lambda>x. ~ (\<exists>f. M(f) & is_recfun(r^+, x, %x f. range(f), f)))"
```
```   267  and wfrank_strong_replacement':
```
```   268      "M(r) ==>
```
```   269       strong_replacement(M, \<lambda>x z. \<exists>y f. M(y) & M(f) &
```
```   270 		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
```
```   271 		  y = range(f))"
```
```   272  and Ord_wfrank_separation:
```
```   273      "M(r) ==>
```
```   274       separation (M, \<lambda>x. ~ (\<forall>f. M(f) \<longrightarrow>
```
```   275                        is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
```
```   276
```
```   277 text{*This function, defined using replacement, is a rank function for
```
```   278 well-founded relations within the class M.*}
```
```   279 constdefs
```
```   280  wellfoundedrank :: "[i=>o,i,i] => i"
```
```   281     "wellfoundedrank(M,r,A) ==
```
```   282         {p. x\<in>A, \<exists>y f. M(y) & M(f) &
```
```   283                        p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
```
```   284                        y = range(f)}"
```
```   285
```
```   286 lemma (in M_recursion) exists_wfrank:
```
```   287     "[| wellfounded(M,r); M(a); M(r) |]
```
```   288      ==> \<exists>f. M(f) & is_recfun(r^+, a, %x f. range(f), f)"
```
```   289 apply (rule wellfounded_exists_is_recfun)
```
```   290       apply (blast intro: wellfounded_trancl)
```
```   291      apply (rule trans_trancl)
```
```   292     apply (erule wfrank_separation')
```
```   293    apply (erule wfrank_strong_replacement')
```
```   294 apply (simp_all add: trancl_subset_times)
```
```   295 done
```
```   296
```
```   297 lemma (in M_recursion) M_wellfoundedrank:
```
```   298     "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
```
```   299 apply (insert wfrank_strong_replacement' [of r])
```
```   300 apply (simp add: wellfoundedrank_def)
```
```   301 apply (rule strong_replacement_closed)
```
```   302    apply assumption+
```
```   303  apply (rule univalent_is_recfun)
```
```   304    apply (blast intro: wellfounded_trancl)
```
```   305   apply (rule trans_trancl)
```
```   306  apply (simp add: trancl_subset_times)
```
```   307 apply blast
```
```   308 done
```
```   309
```
```   310 lemma (in M_recursion) 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 add:);
```
```   316  apply (blast intro: Ord_wfrank_separation);
```
```   317 apply (clarify)
```
```   318 txt{*The reasoning in both cases is that we get @{term y} such that
```
```   319    @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
```
```   320    @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
```
```   321 apply (rule OrdI [OF _ Ord_is_Transset])
```
```   322  txt{*An ordinal is a transitive set...*}
```
```   323  apply (simp add: Transset_def)
```
```   324  apply clarify
```
```   325  apply (frule apply_recfun2, assumption)
```
```   326  apply (force simp add: restrict_iff)
```
```   327 txt{*...of ordinals.  This second case requires the induction hyp.*}
```
```   328 apply clarify
```
```   329 apply (rename_tac i y)
```
```   330 apply (frule apply_recfun2, assumption)
```
```   331 apply (frule is_recfun_imp_in_r, assumption)
```
```   332 apply (frule is_recfun_restrict)
```
```   333     (*simp_all won't work*)
```
```   334     apply (simp add: trans_trancl trancl_subset_times)+
```
```   335 apply (drule spec [THEN mp], assumption)
```
```   336 apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
```
```   337  apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec)
```
```   338  apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
```
```   339 apply (blast dest: pair_components_in_M)
```
```   340 done
```
```   341
```
```   342 lemma (in M_recursion) Ord_range_wellfoundedrank:
```
```   343     "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
```
```   344      ==> Ord (range(wellfoundedrank(M,r,A)))"
```
```   345 apply (frule wellfounded_trancl, assumption)
```
```   346 apply (frule trancl_subset_times)
```
```   347 apply (simp add: wellfoundedrank_def)
```
```   348 apply (rule OrdI [OF _ Ord_is_Transset])
```
```   349  prefer 2
```
```   350  txt{*by our previous result the range consists of ordinals.*}
```
```   351  apply (blast intro: Ord_wfrank_range)
```
```   352 txt{*We still must show that the range is a transitive set.*}
```
```   353 apply (simp add: Transset_def, clarify, simp)
```
```   354 apply (rename_tac x i f u)
```
```   355 apply (frule is_recfun_imp_in_r, assumption)
```
```   356 apply (subgoal_tac "M(u) & M(i) & M(x)")
```
```   357  prefer 2 apply (blast dest: transM, clarify)
```
```   358 apply (rule_tac a=u in rangeI)
```
```   359 apply (rule ReplaceI)
```
```   360   apply (rule_tac x=i in exI, simp)
```
```   361   apply (rule_tac x="restrict(f, r^+ -`` {u})" in exI)
```
```   362   apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
```
```   363  apply blast
```
```   364 txt{*Unicity requirement of Replacement*}
```
```   365 apply clarify
```
```   366 apply (frule apply_recfun2, assumption)
```
```   367 apply (simp add: trans_trancl is_recfun_cut)+
```
```   368 done
```
```   369
```
```   370 lemma (in M_recursion) function_wellfoundedrank:
```
```   371     "[| wellfounded(M,r); M(r); M(A)|]
```
```   372      ==> function(wellfoundedrank(M,r,A))"
```
```   373 apply (simp add: wellfoundedrank_def function_def, clarify)
```
```   374 txt{*Uniqueness: repeated below!*}
```
```   375 apply (drule is_recfun_functional, assumption)
```
```   376      apply (blast intro: wellfounded_trancl)
```
```   377     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   378 done
```
```   379
```
```   380 lemma (in M_recursion) domain_wellfoundedrank:
```
```   381     "[| wellfounded(M,r); M(r); M(A)|]
```
```   382      ==> domain(wellfoundedrank(M,r,A)) = A"
```
```   383 apply (simp add: wellfoundedrank_def function_def)
```
```   384 apply (rule equalityI, auto)
```
```   385 apply (frule transM, assumption)
```
```   386 apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
```
```   387 apply (rule domainI)
```
```   388 apply (rule ReplaceI)
```
```   389   apply (rule_tac x="range(f)" in exI)
```
```   390   apply simp
```
```   391   apply (rule_tac x=f in exI, blast, assumption)
```
```   392 txt{*Uniqueness (for Replacement): repeated above!*}
```
```   393 apply clarify
```
```   394 apply (drule is_recfun_functional, assumption)
```
```   395     apply (blast intro: wellfounded_trancl)
```
```   396     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   397 done
```
```   398
```
```   399 lemma (in M_recursion) wellfoundedrank_type:
```
```   400     "[| wellfounded(M,r);  M(r); M(A)|]
```
```   401      ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
```
```   402 apply (frule function_wellfoundedrank [of r A], assumption+)
```
```   403 apply (frule function_imp_Pi)
```
```   404  apply (simp add: wellfoundedrank_def relation_def)
```
```   405  apply blast
```
```   406 apply (simp add: domain_wellfoundedrank)
```
```   407 done
```
```   408
```
```   409 lemma (in M_recursion) Ord_wellfoundedrank:
```
```   410     "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
```
```   411      ==> Ord(wellfoundedrank(M,r,A) ` a)"
```
```   412 by (blast intro: apply_funtype [OF wellfoundedrank_type]
```
```   413                  Ord_in_Ord [OF Ord_range_wellfoundedrank])
```
```   414
```
```   415 lemma (in M_recursion) wellfoundedrank_eq:
```
```   416      "[| is_recfun(r^+, a, %x. range, f);
```
```   417          wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
```
```   418       ==> wellfoundedrank(M,r,A) ` a = range(f)"
```
```   419 apply (rule apply_equality)
```
```   420  prefer 2 apply (blast intro: wellfoundedrank_type)
```
```   421 apply (simp add: wellfoundedrank_def)
```
```   422 apply (rule ReplaceI)
```
```   423   apply (rule_tac x="range(f)" in exI)
```
```   424   apply blast
```
```   425  apply assumption
```
```   426 txt{*Unicity requirement of Replacement*}
```
```   427 apply clarify
```
```   428 apply (drule is_recfun_functional, assumption)
```
```   429     apply (blast intro: wellfounded_trancl)
```
```   430     apply (simp_all add: trancl_subset_times trans_trancl)
```
```   431 done
```
```   432
```
```   433
```
```   434 lemma (in M_recursion) wellfoundedrank_lt:
```
```   435      "[| <a,b> \<in> r;
```
```   436          wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
```
```   437       ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
```
```   438 apply (frule wellfounded_trancl, assumption)
```
```   439 apply (subgoal_tac "a\<in>A & b\<in>A")
```
```   440  prefer 2 apply blast
```
```   441 apply (simp add: lt_def Ord_wellfoundedrank, clarify)
```
```   442 apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
```
```   443 apply (rename_tac fb)
```
```   444 apply (frule is_recfun_restrict [of concl: "r^+" a])
```
```   445     apply (rule trans_trancl, assumption)
```
```   446    apply (simp_all add: r_into_trancl trancl_subset_times)
```
```   447 txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
```
```   448 apply (simp add: wellfoundedrank_eq)
```
```   449 apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
```
```   450    apply (simp_all add: transM [of a])
```
```   451 txt{*We have used equations for wellfoundedrank and now must use some
```
```   452     for  @{text is_recfun}. *}
```
```   453 apply (rule_tac a=a in rangeI)
```
```   454 apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
```
```   455                  r_into_trancl apply_recfun r_into_trancl)
```
```   456 done
```
```   457
```
```   458
```
```   459 lemma (in M_recursion) wellfounded_imp_subset_rvimage:
```
```   460      "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
```
```   461       ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
```
```   462 apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
```
```   463 apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
```
```   464 apply (simp add: Ord_range_wellfoundedrank, clarify)
```
```   465 apply (frule subsetD, assumption, clarify)
```
```   466 apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
```
```   467 apply (blast intro: apply_rangeI wellfoundedrank_type)
```
```   468 done
```
```   469
```
```   470 lemma (in M_recursion) wellfounded_imp_wf:
```
```   471      "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
```
```   472 by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
```
```   473           intro: wf_rvimage_Ord [THEN wf_subset])
```
```   474
```
```   475 lemma (in M_recursion) wellfounded_on_imp_wf_on:
```
```   476      "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
```
```   477 apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
```
```   478 apply (rule wellfounded_imp_wf)
```
```   479 apply (simp_all add: relation_def)
```
```   480 done
```
```   481
```
```   482
```
```   483 theorem (in M_recursion) wf_abs [simp]:
```
```   484      "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
```
```   485 by (blast intro: wellfounded_imp_wf wf_imp_relativized)
```
```   486
```
```   487 theorem (in M_recursion) wf_on_abs [simp]:
```
```   488      "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
```
```   489 by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
```
```   490
```
```   491 end
```