# HG changeset patch
# User paulson
# Date 1025109080 -7200
# Node ID 74cb2af8811eef63b1d895630137aef58e66d69a
# Parent  efd5db7dc7ccfcbff372f4bedcb16556f3320e5a
new treatment of wfrec, replacing wf[A](r) by wf(r)

diff -r efd5db7dc7cc -r 74cb2af8811e src/ZF/Constructible/Relative.thy
--- a/src/ZF/Constructible/Relative.thy	Wed Jun 26 12:17:21 2002 +0200
+++ b/src/ZF/Constructible/Relative.thy	Wed Jun 26 18:31:20 2002 +0200
@@ -432,7 +432,7 @@
   and is_recfun_separation:
      --{*for well-founded recursion.  NEEDS RELATIVIZATION*}
      "[| M(A); M(f); M(g); M(a); M(b) |] 
-     ==> separation(M, \<lambda>x. x \<in> A --> \<langle>x,a\<rangle> \<in> r & \<langle>x,b\<rangle> \<in> r & f`x \<noteq> g`x)"
+     ==> separation(M, \<lambda>x. \<langle>x,a\<rangle> \<in> r & \<langle>x,b\<rangle> \<in> r & f`x \<noteq> g`x)"
   and omap_replacement:
      "[| M(A); M(r) |] 
       ==> strong_replacement(M,
diff -r efd5db7dc7cc -r 74cb2af8811e src/ZF/Constructible/WF_absolute.thy
--- a/src/ZF/Constructible/WF_absolute.thy	Wed Jun 26 12:17:21 2002 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy	Wed Jun 26 18:31:20 2002 +0200
@@ -1,10 +1,10 @@
 theory WF_absolute = WFrec:
 
-subsection{*Every well-founded relation is a subset of some inverse image of 
+subsection{*Every well-founded relation is a subset of some inverse image of
       an ordinal*}
 
 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
-by (blast intro: wf_rvimage wf_Memrel )
+by (blast intro: wf_rvimage wf_Memrel)
 
 
 constdefs
@@ -21,7 +21,7 @@
 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
 apply (rule_tac a="a" in wf_induct, assumption)
 apply (subst wfrank, assumption)
-apply (rule Ord_succ [THEN Ord_UN], blast) 
+apply (rule Ord_succ [THEN Ord_UN], blast)
 done
 
 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
@@ -34,19 +34,19 @@
 by (simp add: wftype_def Ord_wfrank)
 
 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
-apply (simp add: wftype_def) 
-apply (blast intro: wfrank_lt [THEN ltD]) 
+apply (simp add: wftype_def)
+apply (blast intro: wfrank_lt [THEN ltD])
 done
 
 
 lemma wf_imp_subset_rvimage:
      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
-apply (rule_tac x="wftype(r)" in exI) 
-apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI) 
-apply (simp add: Ord_wftype, clarify) 
-apply (frule subsetD, assumption, clarify) 
+apply (rule_tac x="wftype(r)" in exI)
+apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
+apply (simp add: Ord_wftype, clarify)
+apply (frule subsetD, assumption, clarify)
 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
-apply (blast intro: wftypeI  ) 
+apply (blast intro: wftypeI)
 done
 
 theorem wf_iff_subset_rvimage:
@@ -59,65 +59,65 @@
 
 constdefs
   rtrancl_alt :: "[i,i]=>i"
-    "rtrancl_alt(A,r) == 
+    "rtrancl_alt(A,r) ==
        {p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
                  (\<exists>x y. p = <x,y> &  f`0 = x & f`n = y) &
                        (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
 
-lemma alt_rtrancl_lemma1 [rule_format]: 
+lemma alt_rtrancl_lemma1 [rule_format]:
     "n \<in> nat
-     ==> \<forall>f \<in> succ(n) -> field(r). 
+     ==> \<forall>f \<in> succ(n) -> field(r).
          (\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) --> \<langle>f`0, f`n\<rangle> \<in> r^*"
-apply (induct_tac n) 
-apply (simp_all add: apply_funtype rtrancl_refl, clarify) 
-apply (rename_tac n f) 
-apply (rule rtrancl_into_rtrancl) 
+apply (induct_tac n)
+apply (simp_all add: apply_funtype rtrancl_refl, clarify)
+apply (rename_tac n f)
+apply (rule rtrancl_into_rtrancl)
  prefer 2 apply assumption
 apply (drule_tac x="restrict(f,succ(n))" in bspec)
- apply (blast intro: restrict_type2) 
-apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI) 
+ apply (blast intro: restrict_type2)
+apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
 done
 
 lemma rtrancl_alt_subset_rtrancl: "rtrancl_alt(field(r),r) <= r^*"
 apply (simp add: rtrancl_alt_def)
-apply (blast intro: alt_rtrancl_lemma1 )  
+apply (blast intro: alt_rtrancl_lemma1)
 done
 
 lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
-apply (simp add: rtrancl_alt_def, clarify) 
-apply (frule rtrancl_type [THEN subsetD], clarify, simp) 
-apply (erule rtrancl_induct) 
+apply (simp add: rtrancl_alt_def, clarify)
+apply (frule rtrancl_type [THEN subsetD], clarify, simp)
+apply (erule rtrancl_induct)
  txt{*Base case, trivial*}
- apply (rule_tac x=0 in bexI) 
-  apply (rule_tac x="lam x:1. xa" in bexI) 
-   apply simp_all 
+ apply (rule_tac x=0 in bexI)
+  apply (rule_tac x="lam x:1. xa" in bexI)
+   apply simp_all
 txt{*Inductive step*}
-apply clarify 
-apply (rename_tac n f) 
-apply (rule_tac x="succ(n)" in bexI) 
+apply clarify
+apply (rename_tac n f)
+apply (rule_tac x="succ(n)" in bexI)
  apply (rule_tac x="lam i:succ(succ(n)). if i=succ(n) then z else f`i" in bexI)
-  apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI) 
-  apply (blast intro: mem_asym)  
- apply typecheck 
- apply auto 
+  apply (simp add: Ord_succ_mem_iff nat_0_le [THEN ltD] leI [THEN ltD] ltI)
+  apply (blast intro: mem_asym)
+ apply typecheck
+ apply auto
 done
 
 lemma rtrancl_alt_eq_rtrancl: "rtrancl_alt(field(r),r) = r^*"
 by (blast del: subsetI
-	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt) 
+	  intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
 
 
 constdefs
 
   rtran_closure :: "[i=>o,i,i] => o"
-    "rtran_closure(M,r,s) == 
+    "rtran_closure(M,r,s) ==
         \<forall>A. M(A) --> is_field(M,r,A) -->
- 	 (\<forall>p. M(p) --> 
-          (p \<in> s <-> 
-           (\<exists>n\<in>nat. M(n) & 
+ 	 (\<forall>p. M(p) -->
+          (p \<in> s <->
+           (\<exists>n\<in>nat. M(n) &
             (\<exists>n'. M(n') & successor(M,n,n') &
              (\<exists>f. M(f) & typed_function(M,n',A,f) &
-              (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &  
+              (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &
                    fun_apply(M,f,0,x) & fun_apply(M,f,n,y))) &
               (\<forall>i\<in>n. M(i) -->
                 (\<forall>i'. M(i') --> successor(M,i,i') -->
@@ -126,7 +126,7 @@
                    (\<forall>q. M(q) --> pair(M,fi,fi',q) --> q \<in> r))))))))))"
 
   tran_closure :: "[i=>o,i,i] => o"
-    "tran_closure(M,r,t) == 
+    "tran_closure(M,r,t) ==
          \<exists>s. M(s) & rtran_closure(M,r,s) & composition(M,r,s,t)"
 
 
@@ -142,335 +142,350 @@
      "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z)"
 
 
-lemma (in M_trancl) rtran_closure_rtrancl: 
+lemma (in M_trancl) rtran_closure_rtrancl:
      "M(r) ==> rtran_closure(M,r,rtrancl(r))"
-apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric] 
+apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
-                 Ord_succ_mem_iff M_nat  nat_0_le [THEN ltD], clarify) 
-apply (rule iffI) 
- apply clarify 
- apply simp 
- apply (rename_tac n f) 
- apply (rule_tac x=n in bexI) 
-  apply (rule_tac x=f in exI) 
+                 Ord_succ_mem_iff M_nat  nat_0_le [THEN ltD], clarify)
+apply (rule iffI)
+ apply clarify
+ apply simp
+ apply (rename_tac n f)
+ apply (rule_tac x=n in bexI)
+  apply (rule_tac x=f in exI)
   apply simp
   apply (blast dest: finite_fun_closed dest: transM)
  apply assumption
 apply clarify
-apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)  
+apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)
 done
 
-lemma (in M_trancl) rtrancl_closed [intro,simp]: 
+lemma (in M_trancl) rtrancl_closed [intro,simp]:
      "M(r) ==> M(rtrancl(r))"
-apply (insert rtrancl_separation [of r "field(r)"]) 
-apply (simp add: rtrancl_alt_eq_rtrancl [symmetric] 
+apply (insert rtrancl_separation [of r "field(r)"])
+apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
                  Ord_succ_mem_iff M_nat
                  nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
 done
 
-lemma (in M_trancl) rtrancl_abs [simp]: 
+lemma (in M_trancl) rtrancl_abs [simp]:
      "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
 apply (rule iffI)
  txt{*Proving the right-to-left implication*}
- prefer 2 apply (blast intro: rtran_closure_rtrancl) 
+ prefer 2 apply (blast intro: rtran_closure_rtrancl)
 apply (rule M_equalityI)
-apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric] 
+apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
                  rtrancl_alt_def field_closed typed_apply_abs apply_closed
                  Ord_succ_mem_iff M_nat
-                 nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype) 
+                 nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
  prefer 2 apply assumption
  prefer 2 apply blast
-apply (rule iffI, clarify) 
-apply (simp add: nat_0_le [THEN ltD]  apply_funtype, blast, clarify, simp) 
- apply (rename_tac n f) 
- apply (rule_tac x=n in bexI) 
+apply (rule iffI, clarify)
+apply (simp add: nat_0_le [THEN ltD]  apply_funtype, blast, clarify, simp)
+ apply (rename_tac n f)
+ apply (rule_tac x=n in bexI)
   apply (rule_tac x=f in exI)
   apply (blast dest!: finite_fun_closed, assumption)
 done
 
 
-lemma (in M_trancl) trancl_closed [intro,simp]: 
+lemma (in M_trancl) trancl_closed [intro,simp]:
      "M(r) ==> M(trancl(r))"
-by (simp add: trancl_def comp_closed rtrancl_closed) 
+by (simp add: trancl_def comp_closed rtrancl_closed)
 
-lemma (in M_trancl) trancl_abs [simp]: 
+lemma (in M_trancl) trancl_abs [simp]:
      "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
-by (simp add: tran_closure_def trancl_def) 
+by (simp add: tran_closure_def trancl_def)
 
 
-text{*Alternative proof of @{text wf_on_trancl}; inspiration for the 
+text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
       relativized version.  Original version is on theory WF.*}
 lemma "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)"
-apply (simp add: wf_on_def wf_def) 
+apply (simp add: wf_on_def wf_def)
 apply (safe intro!: equalityI)
-apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec) 
-apply (blast elim: tranclE) 
+apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
+apply (blast elim: tranclE)
 done
 
 
 lemma (in M_trancl) wellfounded_on_trancl:
      "[| wellfounded_on(M,A,r);  r-``A <= A; M(r); M(A) |]
-      ==> wellfounded_on(M,A,r^+)" 
-apply (simp add: wellfounded_on_def) 
+      ==> wellfounded_on(M,A,r^+)"
+apply (simp add: wellfounded_on_def)
 apply (safe intro!: equalityI)
 apply (rename_tac Z x)
-apply (subgoal_tac "M({x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z})") 
- prefer 2 
- apply (simp add: wellfounded_trancl_separation) 
-apply (drule_tac x = "{x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec) 
+apply (subgoal_tac "M({x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z})")
+ prefer 2
+ apply (simp add: wellfounded_trancl_separation)
+apply (drule_tac x = "{x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
 apply safe
-apply (blast dest: transM, simp) 
-apply (rename_tac y w) 
+apply (blast dest: transM, simp)
+apply (rename_tac y w)
 apply (drule_tac x=w in bspec, assumption, clarify)
 apply (erule tranclE)
   apply (blast dest: transM)   (*transM is needed to prove M(xa)*)
- apply blast 
+ apply blast
 done
 
+(*????move to Wellorderings.thy*)
+lemma (in M_axioms) wellfounded_on_field_imp_wellfounded:
+     "wellfounded_on(M, field(r), r) ==> wellfounded(M,r)"
+by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast)
+
+lemma (in M_axioms) wellfounded_iff_wellfounded_on_field:
+     "M(r) ==> wellfounded(M,r) <-> wellfounded_on(M, field(r), r)"
+by (blast intro: wellfounded_imp_wellfounded_on
+                 wellfounded_on_field_imp_wellfounded)
+
+lemma (in M_axioms) wellfounded_on_subset_A:
+     "[| wellfounded_on(M,A,r);  B<=A |] ==> wellfounded_on(M,B,r)"
+by (simp add: wellfounded_on_def, blast)
+
+
+
+lemma (in M_trancl) wellfounded_trancl:
+     "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
+apply (rotate_tac -1)
+apply (simp add: wellfounded_iff_wellfounded_on_field)
+apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
+   apply blast
+  apply (simp_all add: trancl_type [THEN field_rel_subset])
+done
 
 text{*Relativized to M: Every well-founded relation is a subset of some
-inverse image of an ordinal.  Key step is the construction (in M) of a 
+inverse image of an ordinal.  Key step is the construction (in M) of a
 rank function.*}
 
 
 (*NEEDS RELATIVIZATION*)
 locale M_recursion = M_trancl +
   assumes wfrank_separation':
-     "[| M(r); M(A) |] ==>
+     "M(r) ==>
 	separation
-	   (M, \<lambda>x. x \<in> A --> 
-		~(\<exists>f. M(f) \<and> is_recfun(r^+, x, %x f. range(f), f)))"
+	   (M, \<lambda>x. ~ (\<exists>f. M(f) & is_recfun(r^+, x, %x f. range(f), f)))"
  and wfrank_strong_replacement':
      "M(r) ==>
       strong_replacement(M, \<lambda>x z. \<exists>y f. M(y) & M(f) &
-		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) & 
+		  pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
 		  y = range(f))"
  and Ord_wfrank_separation:
-     "[| M(r); M(A) |] ==>
-      separation (M, \<lambda>x. x \<in> A \<longrightarrow>
-                \<not> (\<forall>f. M(f) \<longrightarrow>
+     "M(r) ==>
+      separation (M, \<lambda>x. ~ (\<forall>f. M(f) \<longrightarrow>
                        is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
 
-constdefs 
+text{*This function, defined using replacement, is a rank function for
+well-founded relations within the class M.*}
+constdefs
  wellfoundedrank :: "[i=>o,i,i] => i"
-    "wellfoundedrank(M,r,A) == 
-        {p. x\<in>A, \<exists>y f. M(y) & M(f) & 
-                       p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) & 
+    "wellfoundedrank(M,r,A) ==
+        {p. x\<in>A, \<exists>y f. M(y) & M(f) &
+                       p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
                        y = range(f)}"
 
 lemma (in M_recursion) exists_wfrank:
-    "[| wellfounded(M,r); r \<subseteq> A*A; a\<in>A; M(r); M(A) |]
+    "[| wellfounded(M,r); M(a); M(r) |]
      ==> \<exists>f. M(f) & is_recfun(r^+, a, %x f. range(f), f)"
-apply (rule wellfounded_exists_is_recfun [of A]) 
-apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-apply (rule trans_trancl [THEN trans_imp_trans_on], assumption+)
-apply (simp_all add: trancl_subset_times) 
+apply (rule wellfounded_exists_is_recfun)
+      apply (blast intro: wellfounded_trancl)
+     apply (rule trans_trancl)
+    apply (erule wfrank_separation')
+   apply (erule wfrank_strong_replacement')
+apply (simp_all add: trancl_subset_times)
 done
 
 lemma (in M_recursion) M_wellfoundedrank:
-    "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A) |] 
-     ==> M(wellfoundedrank(M,r,A))"
-apply (insert wfrank_strong_replacement' [of r]) 
-apply (simp add: wellfoundedrank_def) 
-apply (rule strong_replacement_closed) 
+    "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))"
+apply (insert wfrank_strong_replacement' [of r])
+apply (simp add: wellfoundedrank_def)
+apply (rule strong_replacement_closed)
    apply assumption+
- apply (rule univalent_is_recfun) 
-     apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-    apply (rule trans_on_trancl) 
-   apply (simp add: trancl_subset_times) 
-  apply blast+
+ apply (rule univalent_is_recfun)
+   apply (blast intro: wellfounded_trancl)
+  apply (rule trans_trancl)
+ apply (simp add: trancl_subset_times)
+apply blast
 done
 
 lemma (in M_recursion) Ord_wfrank_range [rule_format]:
-    "[| wellfounded(M,r); r \<subseteq> A*A; a\<in>A; M(r); M(A) |]
+    "[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
      ==> \<forall>f. M(f) --> is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
-apply (subgoal_tac "wellfounded_on(M, A, r^+)") 
- prefer 2
- apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-apply (erule wellfounded_on_induct2, assumption+)
-apply (simp add: trancl_subset_times) 
-apply (blast intro: Ord_wfrank_separation, clarify)
+apply (drule wellfounded_trancl, assumption)
+apply (rule wellfounded_induct, assumption+)
+  apply (simp add:);
+ apply (blast intro: Ord_wfrank_separation);
+apply (clarify)
 txt{*The reasoning in both cases is that we get @{term y} such that
-   @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that 
+   @{term "\<langle>y, x\<rangle> \<in> r^+"}.  We find that
    @{term "f`y = restrict(f, r^+ -`` {y})"}. *}
 apply (rule OrdI [OF _ Ord_is_Transset])
  txt{*An ordinal is a transitive set...*}
- apply (simp add: Transset_def) 
+ apply (simp add: Transset_def)
  apply clarify
- apply (frule apply_recfun2, assumption) 
+ apply (frule apply_recfun2, assumption)
  apply (force simp add: restrict_iff)
-txt{*...of ordinals.  This second case requires the induction hyp.*} 
-apply clarify 
+txt{*...of ordinals.  This second case requires the induction hyp.*}
+apply clarify
 apply (rename_tac i y)
-apply (frule apply_recfun2, assumption) 
-apply (frule is_recfun_imp_in_r, assumption) 
-apply (frule is_recfun_restrict) 
+apply (frule apply_recfun2, assumption)
+apply (frule is_recfun_imp_in_r, assumption)
+apply (frule is_recfun_restrict)
     (*simp_all won't work*)
-    apply (simp add: trans_on_trancl trancl_subset_times)+  
+    apply (simp add: trans_trancl trancl_subset_times)+
 apply (drule spec [THEN mp], assumption)
 apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
- apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec) 
+ apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec)
  apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
 apply (blast dest: pair_components_in_M)
 done
 
 lemma (in M_recursion) Ord_range_wellfoundedrank:
-    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |] 
+    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A) |]
      ==> Ord (range(wellfoundedrank(M,r,A)))"
-apply (subgoal_tac "wellfounded_on(M, A, r^+)") 
- prefer 2
- apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-apply (frule trancl_subset_times) 
+apply (frule wellfounded_trancl, assumption)
+apply (frule trancl_subset_times)
 apply (simp add: wellfoundedrank_def)
 apply (rule OrdI [OF _ Ord_is_Transset])
  prefer 2
- txt{*by our previous result the range consists of ordinals.*} 
- apply (blast intro: Ord_wfrank_range) 
+ txt{*by our previous result the range consists of ordinals.*}
+ apply (blast intro: Ord_wfrank_range)
 txt{*We still must show that the range is a transitive set.*}
 apply (simp add: Transset_def, clarify, simp)
-apply (rename_tac x i f u)   
-apply (frule is_recfun_imp_in_r, assumption) 
-apply (subgoal_tac "M(u) & M(i) & M(x)") 
- prefer 2 apply (blast dest: transM, clarify) 
-apply (rule_tac a=u in rangeI) 
-apply (rule ReplaceI) 
-  apply (rule_tac x=i in exI, simp) 
+apply (rename_tac x i f u)
+apply (frule is_recfun_imp_in_r, assumption)
+apply (subgoal_tac "M(u) & M(i) & M(x)")
+ prefer 2 apply (blast dest: transM, clarify)
+apply (rule_tac a=u in rangeI)
+apply (rule ReplaceI)
+  apply (rule_tac x=i in exI, simp)
   apply (rule_tac x="restrict(f, r^+ -`` {u})" in exI)
-  apply (blast intro: is_recfun_restrict trans_on_trancl dest: apply_recfun2)
+  apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2)
  apply blast
-txt{*Unicity requirement of Replacement*} 
+txt{*Unicity requirement of Replacement*}
 apply clarify
-apply (frule apply_recfun2, assumption) 
-apply (simp add: trans_on_trancl is_recfun_cut)+
+apply (frule apply_recfun2, assumption)
+apply (simp add: trans_trancl is_recfun_cut)+
 done
 
 lemma (in M_recursion) function_wellfoundedrank:
-    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
+    "[| wellfounded(M,r); M(r); M(A)|]
      ==> function(wellfoundedrank(M,r,A))"
-apply (simp add: wellfoundedrank_def function_def, clarify) 
+apply (simp add: wellfoundedrank_def function_def, clarify)
 txt{*Uniqueness: repeated below!*}
 apply (drule is_recfun_functional, assumption)
-    apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-    apply (simp_all add: trancl_subset_times 
-                         trans_trancl [THEN trans_imp_trans_on]) 
-apply (blast dest: transM) 
+     apply (blast intro: wellfounded_trancl)
+    apply (simp_all add: trancl_subset_times trans_trancl)
 done
 
 lemma (in M_recursion) domain_wellfoundedrank:
-    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
+    "[| wellfounded(M,r); M(r); M(A)|]
      ==> domain(wellfoundedrank(M,r,A)) = A"
-apply (simp add: wellfoundedrank_def function_def) 
+apply (simp add: wellfoundedrank_def function_def)
 apply (rule equalityI, auto)
-apply (frule transM, assumption)  
-apply (frule exists_wfrank, assumption+, clarify) 
-apply (rule domainI) 
+apply (frule transM, assumption)
+apply (frule_tac a=x in exists_wfrank, assumption+, clarify)
+apply (rule domainI)
 apply (rule ReplaceI)
-apply (rule_tac x="range(f)" in exI)
-apply simp  
-apply (rule_tac x=f in exI, blast, assumption)
+  apply (rule_tac x="range(f)" in exI)
+  apply simp
+  apply (rule_tac x=f in exI, blast, assumption)
 txt{*Uniqueness (for Replacement): repeated above!*}
 apply clarify
 apply (drule is_recfun_functional, assumption)
-    apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-    apply (simp_all add: trancl_subset_times 
-                         trans_trancl [THEN trans_imp_trans_on]) 
+    apply (blast intro: wellfounded_trancl)
+    apply (simp_all add: trancl_subset_times trans_trancl)
 done
 
 lemma (in M_recursion) wellfoundedrank_type:
-    "[| wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
+    "[| wellfounded(M,r);  M(r); M(A)|]
      ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
-apply (frule function_wellfoundedrank, assumption+) 
-apply (frule function_imp_Pi) 
- apply (simp add: wellfoundedrank_def relation_def) 
- apply blast  
+apply (frule function_wellfoundedrank [of r A], assumption+)
+apply (frule function_imp_Pi)
+ apply (simp add: wellfoundedrank_def relation_def)
+ apply blast
 apply (simp add: domain_wellfoundedrank)
 done
 
 lemma (in M_recursion) Ord_wellfoundedrank:
-    "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |] 
+    "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A;  M(r); M(A) |]
      ==> Ord(wellfoundedrank(M,r,A) ` a)"
 by (blast intro: apply_funtype [OF wellfoundedrank_type]
                  Ord_in_Ord [OF Ord_range_wellfoundedrank])
 
 lemma (in M_recursion) wellfoundedrank_eq:
      "[| is_recfun(r^+, a, %x. range, f);
-         wellfounded(M,r);  a \<in> A; r \<subseteq> A*A;  M(f); M(r); M(A)|] 
+         wellfounded(M,r);  a \<in> A; M(f); M(r); M(A)|]
       ==> wellfoundedrank(M,r,A) ` a = range(f)"
-apply (rule apply_equality) 
- prefer 2 apply (blast intro: wellfoundedrank_type ) 
+apply (rule apply_equality)
+ prefer 2 apply (blast intro: wellfoundedrank_type)
 apply (simp add: wellfoundedrank_def)
 apply (rule ReplaceI)
-  apply (rule_tac x="range(f)" in exI) 
-  apply blast 
+  apply (rule_tac x="range(f)" in exI)
+  apply blast
  apply assumption
-txt{*Unicity requirement of Replacement*} 
+txt{*Unicity requirement of Replacement*}
 apply clarify
 apply (drule is_recfun_functional, assumption)
-    apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
-    apply (simp_all add: trancl_subset_times 
-                         trans_trancl [THEN trans_imp_trans_on])
-apply (blast dest: transM) 
+    apply (blast intro: wellfounded_trancl)
+    apply (simp_all add: trancl_subset_times trans_trancl)
 done
 
 
 lemma (in M_recursion) wellfoundedrank_lt:
      "[| <a,b> \<in> r;
-         wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|] 
+         wellfounded(M,r); r \<subseteq> A*A;  M(r); M(A)|]
       ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b"
-apply (subgoal_tac "wellfounded_on(M, A, r^+)") 
- prefer 2
- apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+apply (frule wellfounded_trancl, assumption)
 apply (subgoal_tac "a\<in>A & b\<in>A")
  prefer 2 apply blast
-apply (simp add: lt_def Ord_wellfoundedrank, clarify)   
-apply (frule exists_wfrank [of concl: _ b], assumption+, clarify) 
+apply (simp add: lt_def Ord_wellfoundedrank, clarify)
+apply (frule exists_wfrank [of concl: _ b], assumption+, clarify)
 apply (rename_tac fb)
-apply (frule is_recfun_restrict [of concl: _ a])
-    apply (rule trans_on_trancl, assumption)
-   apply (simp_all add: r_into_trancl trancl_subset_times) 
+apply (frule is_recfun_restrict [of concl: "r^+" a])
+    apply (rule trans_trancl, assumption)
+   apply (simp_all add: r_into_trancl trancl_subset_times)
 txt{*Still the same goal, but with new @{text is_recfun} assumptions.*}
-apply (simp add: wellfoundedrank_eq) 
+apply (simp add: wellfoundedrank_eq)
 apply (frule_tac a=a in wellfoundedrank_eq, assumption+)
    apply (simp_all add: transM [of a])
 txt{*We have used equations for wellfoundedrank and now must use some
     for  @{text is_recfun}. *}
-apply (rule_tac a=a in rangeI) 
-apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff 
-                 r_into_trancl apply_recfun r_into_trancl)  
+apply (rule_tac a=a in rangeI)
+apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff
+                 r_into_trancl apply_recfun r_into_trancl)
 done
 
 
 lemma (in M_recursion) wellfounded_imp_subset_rvimage:
-     "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|] 
+     "[|wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
       ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
 apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI)
 apply (rule_tac x="wellfoundedrank(M,r,A)" in exI)
-apply (simp add: Ord_range_wellfoundedrank, clarify) 
-apply (frule subsetD, assumption, clarify) 
+apply (simp add: Ord_range_wellfoundedrank, clarify)
+apply (frule subsetD, assumption, clarify)
 apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD])
-apply (blast intro: apply_rangeI wellfoundedrank_type) 
+apply (blast intro: apply_rangeI wellfoundedrank_type)
 done
 
-lemma (in M_recursion) wellfounded_imp_wf: 
-     "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)" 
+lemma (in M_recursion) wellfounded_imp_wf:
+     "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)"
 by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage
           intro: wf_rvimage_Ord [THEN wf_subset])
 
-lemma (in M_recursion) wellfounded_on_imp_wf_on: 
-     "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)" 
-apply (simp add: wellfounded_on_iff_wellfounded wf_on_def) 
+lemma (in M_recursion) wellfounded_on_imp_wf_on:
+     "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)"
+apply (simp add: wellfounded_on_iff_wellfounded wf_on_def)
 apply (rule wellfounded_imp_wf)
-apply (simp_all add: relation_def)  
+apply (simp_all add: relation_def)
 done
 
 
-theorem (in M_recursion) wf_abs [simp]: 
+theorem (in M_recursion) wf_abs [simp]:
      "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)"
-by (blast intro: wellfounded_imp_wf wf_imp_relativized) 
+by (blast intro: wellfounded_imp_wf wf_imp_relativized)
 
-theorem (in M_recursion) wf_on_abs [simp]: 
+theorem (in M_recursion) wf_on_abs [simp]:
      "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)"
-by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized) 
+by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized)
 
 end
diff -r efd5db7dc7cc -r 74cb2af8811e src/ZF/Constructible/WFrec.thy
--- a/src/ZF/Constructible/WFrec.thy	Wed Jun 26 12:17:21 2002 +0200
+++ b/src/ZF/Constructible/WFrec.thy	Wed Jun 26 18:31:20 2002 +0200
@@ -37,71 +37,75 @@
 apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
 done
 
-lemma trans_on_Int_eq [simp]:
-      "[| trans[A](r); <y,x> \<in> r;  r \<subseteq> A*A |] 
+(*????GET RID OF [simp]*)
+lemma trans_Int_eq [simp]:
+      "[| trans(r); <y,x> \<in> r |] 
        ==> r -`` {y} \<inter> r -`` {x} = r -`` {y}"
-by (blast intro: trans_onD) 
+by (blast intro: transD) 
 
-lemma trans_on_Int_eq2 [simp]:
-      "[| trans[A](r); <y,x> \<in> r;  r \<subseteq> A*A |] 
+lemma trans_Int_eq2 [simp]:
+      "[| trans(r); <y,x> \<in> r |] 
        ==> r -`` {x} \<inter> r -`` {y} = r -`` {y}"
-by (blast intro: trans_onD) 
+by (blast intro: transD) 
 
 
-text{*Stated using @{term "trans[A](r)"} rather than
+text{*Stated using @{term "trans(r)"} rather than
       @{term "transitive_rel(M,A,r)"} because the latter rewrites to
       the former anyway, by @{text transitive_rel_abs}.
-      As always, theorems should be expressed in simplified form.*}
+      As always, theorems should be expressed in simplified form.
+      The last three M-premises are redundant because of @{term "M(r)"}, 
+      but without them we'd have to undertake
+      more work to set up the induction formula.*}
 lemma (in M_axioms) is_recfun_equal [rule_format]: 
     "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
-       wellfounded_on(M,A,r);  trans[A](r); 
-       M(A); M(f); M(g); M(a); M(b); 
-       r \<subseteq> A*A;  x\<in>A |] 
+       wellfounded(M,r);  trans(r);
+       M(f); M(g); M(r); M(x); M(a); M(b) |] 
      ==> <x,a> \<in> r --> <x,b> \<in> r --> f`x=g`x"
 apply (frule_tac f="f" in is_recfun_type) 
 apply (frule_tac f="g" in is_recfun_type) 
 apply (simp add: is_recfun_def)
-apply (erule wellfounded_on_induct2, assumption+) 
-apply (force intro: is_recfun_separation, clarify)
+apply (erule_tac a=x in wellfounded_induct) 
+apply assumption+
+txt{*Separation to justify the induction*}
+ apply (force intro: is_recfun_separation)
+txt{*Now the inductive argument itself*}
+apply (clarify ); 
 apply (erule ssubst)+
 apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
 apply (rename_tac x1)
 apply (rule_tac t="%z. H(x1,z)" in subst_context) 
 apply (subgoal_tac "ALL y : r-``{x1}. ALL z. <y,z>:f <-> <y,z>:g")
- apply (blast intro: trans_onD) 
+ apply (blast intro: transD) 
 apply (simp add: apply_iff) 
-apply (blast intro: trans_onD sym) 
+apply (blast intro: transD sym) 
 done
 
 lemma (in M_axioms) is_recfun_cut: 
     "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
-       wellfounded_on(M,A,r); trans[A](r); 
-       M(A); M(f); M(g); M(a); M(b); 
-       r \<subseteq> A*A;  <b,a>\<in>r |]   
+       wellfounded(M,r); trans(r); 
+       M(f); M(g); M(r); <b,a> \<in> r |]   
       ==> restrict(f, r-``{b}) = g"
 apply (frule_tac f="f" in is_recfun_type) 
 apply (rule fun_extension) 
-apply (blast intro: trans_onD restrict_type2) 
+apply (blast intro: transD restrict_type2) 
 apply (erule is_recfun_type, simp) 
-apply (blast intro: is_recfun_equal trans_onD) 
+apply (blast intro: is_recfun_equal transD dest: transM) 
 done
 
 lemma (in M_axioms) is_recfun_functional:
      "[|is_recfun(r,a,H,f);  is_recfun(r,a,H,g);  
-       wellfounded_on(M,A,r); trans[A](r); 
-       M(A); M(f); M(g); M(a); 
-       r \<subseteq> A*A |]   
+       wellfounded(M,r); trans(r); 
+       M(f); M(g); M(r) |]   
       ==> f=g"
 apply (rule fun_extension)
 apply (erule is_recfun_type)+
-apply (blast intro!: is_recfun_equal) 
+apply (blast intro!: is_recfun_equal dest: transM) 
 done
 
 text{*Tells us that is_recfun can (in principle) be relativized.*}
 lemma (in M_axioms) is_recfun_relativize:
-     "[| M(r); M(a); M(f); 
-       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |] ==>
-       is_recfun(r,a,H,f) <->
+  "[| M(r); M(f); \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |] 
+   ==> is_recfun(r,a,H,f) <->
        (\<forall>z. z \<in> f <-> (\<exists>x y. M(x) & M(y) & z=<x,y> & <x,a> \<in> r & 
                               y = H(x, restrict(f, r-``{x}))))";
 apply (simp add: is_recfun_def vimage_closed restrict_closed lam_def)
@@ -118,7 +122,7 @@
  prefer 2
  apply (simp add: function_def) 
 apply (frule pair_components_in_M, assumption) 
-  apply (simp add: is_recfun_imp_function function_restrictI restrict_closed vimage_closed) 
+  apply (simp add: is_recfun_imp_function function_restrictI) 
 done
 
 (* ideas for further weaking the H-closure premise:
@@ -136,23 +140,23 @@
 *)
 
 lemma (in M_axioms) is_recfun_restrict:
-     "[| wellfounded_on(M,A,r); trans[A](r); is_recfun(r,x,H,f); \<langle>y,x\<rangle> \<in> r; 
-       M(A); M(r); M(f); 
-       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)); r \<subseteq> A * A |]
+     "[| wellfounded(M,r); trans(r); is_recfun(r,x,H,f); \<langle>y,x\<rangle> \<in> r; 
+       M(r); M(f); 
+       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |]
        ==> is_recfun(r, y, H, restrict(f, r -`` {y}))"
 apply (frule pair_components_in_M, assumption, clarify) 
 apply (simp (no_asm_simp) add: is_recfun_relativize restrict_iff)
 apply safe
   apply (simp_all add: vimage_singleton_iff is_recfun_type [THEN apply_iff]) 
   apply (frule_tac x=xa in pair_components_in_M, assumption)
-  apply (frule_tac x=xa in apply_recfun, blast intro: trans_onD)  
+  apply (frule_tac x=xa in apply_recfun, blast intro: transD)  
   apply (simp add: is_recfun_type [THEN apply_iff] 
-                   is_recfun_imp_function function_restrictI) 
-apply (blast intro: apply_recfun dest: trans_onD)+
+                   is_recfun_imp_function function_restrictI)
+apply (blast intro: apply_recfun dest: transD)
 done
  
 lemma (in M_axioms) restrict_Y_lemma:
-     "[| wellfounded_on(M,A,r); trans[A](r); M(A); M(r); r \<subseteq> A \<times> A;
+   "[| wellfounded(M,r); trans(r); M(r);
        \<forall>x g. M(x) \<and> M(g) & function(g) --> M(H(x,g));  M(Y);
        \<forall>b. M(b) -->
 	   b \<in> Y <->
@@ -161,10 +165,10 @@
 		   (\<exists>g. M(g) \<and> b = \<langle>x,y\<rangle> \<and> is_recfun(r,x,H,g) \<and> y = H(x,g)));
           \<langle>x,a1\<rangle> \<in> r; M(f); is_recfun(r,x,H,f) |]
        ==> restrict(Y, r -`` {x}) = f"
-apply (subgoal_tac "ALL y : r-``{x}. ALL z. <y,z>:Y <-> <y,z>:f") 
-apply (simp (no_asm_simp) add: restrict_def) 
-apply (thin_tac "All(?P)")+  --{*essential for efficiency*}
-apply (frule is_recfun_type [THEN fun_is_rel], blast)
+apply (subgoal_tac "\<forall>y \<in> r-``{x}. \<forall>z. <y,z>:Y <-> <y,z>:f") 
+ apply (simp (no_asm_simp) add: restrict_def) 
+ apply (thin_tac "All(?P)")+  --{*essential for efficiency*}
+ apply (frule is_recfun_type [THEN fun_is_rel], blast)
 apply (frule pair_components_in_M, assumption, clarify) 
 apply (rule iffI)
  apply (frule_tac y="<y,z>" in transM, assumption )
@@ -174,17 +178,16 @@
 txt{*Opposite inclusion: something in f, show in Y*}
 apply (frule_tac y="<y,z>" in transM, assumption, simp) 
 apply (rule_tac x=y in bexI)
-prefer 2 apply (blast dest: trans_onD)
+prefer 2 apply (blast dest: transD)
 apply (rule_tac x=z in exI, simp) 
 apply (rule_tac x="restrict(f, r -`` {y})" in exI) 
 apply (simp add: vimage_closed restrict_closed is_recfun_restrict
                  apply_recfun is_recfun_type [THEN apply_iff]) 
 done
 
-(*FIXME: use this lemma just below*)
 text{*For typical applications of Replacement for recursive definitions*}
 lemma (in M_axioms) univalent_is_recfun:
-     "[|wellfounded_on(M,A,r); trans[A](r); r \<subseteq> A*A; M(r); M(A)|]
+     "[|wellfounded(M,r); trans(r); M(r)|]
       ==> univalent (M, A, \<lambda>x p. \<exists>y. M(y) &
                     (\<exists>f. M(f) & p = \<langle>x, y\<rangle> & is_recfun(r,x,H,f) & y = H(x,f)))"
 apply (simp add: univalent_def) 
@@ -194,69 +197,67 @@
 text{*Proof of the inductive step for @{text exists_is_recfun}, since
       we must prove two versions.*}
 lemma (in M_axioms) exists_is_recfun_indstep:
-    "[|a1 \<in> A;  \<forall>y. \<langle>y, a1\<rangle> \<in> r --> (\<exists>f. M(f) & is_recfun(r, y, H, f)); 
-       wellfounded_on(M,A,r); trans[A](r); 
+    "[|\<forall>y. \<langle>y, a1\<rangle> \<in> r --> (\<exists>f. M(f) & is_recfun(r, y, H, f)); 
+       wellfounded(M,r); trans(r); M(r); M(a1);
        strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                    pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
-       M(A); M(r); r \<subseteq> A * A;
        \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g))|]   
       ==> \<exists>f. M(f) & is_recfun(r,a1,H,f)"
-apply (frule_tac y=a1 in transM, assumption)
 apply (drule_tac A="r-``{a1}" in strong_replacementD)
-  apply blast
+  apply blast 
  txt{*Discharge the "univalent" obligation of Replacement*}
- apply (clarsimp simp add: univalent_def)
- apply (blast dest!: is_recfun_functional)
+ apply (simp add: univalent_is_recfun) 
 txt{*Show that the constructed object satisfies @{text is_recfun}*} 
 apply clarify 
 apply (rule_tac x=Y in exI)  
-apply (simp (no_asm_simp) add: is_recfun_relativize vimage_closed restrict_closed) 
+apply (simp (no_asm_simp) add: is_recfun_relativize) 
 (*Tried using is_recfun_iff2 here.  Much more simplification takes place
   because an assumption can kick in.  Not sure how to relate the new 
   proof state to the current one.*)
 apply safe
-txt{*Show that elements of @{term Y} are in the right relationship.*}
-apply (frule_tac x=z and P="%b. M(b) --> ?Q(b)" in spec)
-apply (erule impE, blast intro: transM)
-txt{*We have an element of  @{term Y}, so we have x, y, z*} 
-apply (frule_tac y=z in transM, assumption, clarify)
-apply (simp add: vimage_closed restrict_closed restrict_Y_lemma [of A r H]) 
+ txt{*Show that elements of @{term Y} are in the right relationship.*}
+ apply (frule_tac x=z and P="%b. M(b) --> ?Q(b)" in spec)
+ apply (erule impE, blast intro: transM)
+ txt{*We have an element of  @{term Y}, so we have x, y, z*} 
+ apply (frule_tac y=z in transM, assumption, clarify)
+ apply (simp add: restrict_Y_lemma [of r H]) 
 txt{*one more case*}
-apply (simp add: vimage_closed restrict_closed )
+apply (simp)
 apply (rule_tac x=x in bexI) 
-prefer 2 apply blast 
+ prefer 2 apply blast 
 apply (rule_tac x="H(x, restrict(Y, r -`` {x}))" in exI) 
-apply (simp add: vimage_closed restrict_closed )
+apply (simp)
 apply (drule_tac x1=x in spec [THEN mp], assumption, clarify) 
 apply (rule_tac x=f in exI) 
-apply (simp add: restrict_Y_lemma [of A r H]) 
+apply (simp add: restrict_Y_lemma [of r H]) 
 done
 
-
 text{*Relativized version, when we have the (currently weaker) premise
-      @{term "wellfounded_on(M,A,r)"}*}
+      @{term "wellfounded(M,r)"}*}
 lemma (in M_axioms) wellfounded_exists_is_recfun:
-    "[|wellfounded_on(M,A,r);  trans[A](r);  a\<in>A; 
-       separation(M, \<lambda>x. x \<in> A --> ~ (\<exists>f. M(f) \<and> is_recfun(r, x, H, f)));
+    "[|wellfounded(M,r);  trans(r);  
+       separation(M, \<lambda>x. ~ (\<exists>f. M(f) \<and> is_recfun(r, x, H, f)));
        strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                    pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
-       M(A);  M(r);  r \<subseteq> A*A;  
+       M(r);  M(a);  
        \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |]   
       ==> \<exists>f. M(f) & is_recfun(r,a,H,f)"
-apply (rule wellfounded_on_induct2, assumption+, clarify)
+apply (rule wellfounded_induct, assumption+, clarify)
 apply (rule exists_is_recfun_indstep, assumption+)
 done
 
-lemma (in M_axioms) wf_exists_is_recfun:
-    "[|wf[A](r);  trans[A](r);  a\<in>A; 
+lemma (in M_axioms) wf_exists_is_recfun [rule_format]:
+    "[|wf(r);  trans(r);  
        strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                    pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
-       M(A);  M(r);  r \<subseteq> A*A;  
+        M(r);  
        \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |]   
-      ==> \<exists>f. M(f) & is_recfun(r,a,H,f)"        
-apply (rule wf_on_induct2, assumption+)
-apply (frule wf_on_imp_relativized)  
-apply (rule exists_is_recfun_indstep, assumption+)
+      ==> M(a) --> (\<exists>f. M(f) & is_recfun(r,a,H,f))"
+apply (rule wf_induct, assumption+)
+apply (frule wf_imp_relativized)
+apply (intro impI)
+apply (rule exists_is_recfun_indstep)
+      apply (blast dest: pair_components_in_M)+
 done
 
 constdefs
@@ -377,12 +378,10 @@
 lemma (in M_recursion) exists_oadd:
     "[| Ord(j);  M(i);  M(j) |]
      ==> \<exists>f. M(f) & is_recfun(Memrel(succ(j)), j, %x g. i Un g``x, f)"
-apply (rule wf_exists_is_recfun) 
-apply (rule wf_Memrel [THEN wf_imp_wf_on]) 
-apply (rule trans_Memrel [THEN trans_imp_trans_on], simp)  
-apply (rule succI1) 
-apply (blast intro: oadd_strong_replacement') 
-apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
+apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
+    apply (simp add: ); 
+   apply (blast intro: oadd_strong_replacement') 
+  apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
 done
 
 lemma (in M_recursion) exists_oadd_fun:
@@ -491,12 +490,10 @@
 lemma (in M_recursion) exists_omult:
     "[| Ord(j);  M(i);  M(j) |]
      ==> \<exists>f. M(f) & is_recfun(Memrel(succ(j)), j, %x g. THE z. omult_eqns(i,x,g,z), f)"
-apply (rule wf_exists_is_recfun) 
-apply (rule wf_Memrel [THEN wf_imp_wf_on]) 
-apply (rule trans_Memrel [THEN trans_imp_trans_on], simp)  
-apply (rule succI1) 
-apply (blast intro: omult_strong_replacement') 
-apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
+apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel])
+    apply (simp add: );
+   apply (blast intro: omult_strong_replacement') 
+  apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
 apply (blast intro: the_omult_eqns_closed) 
 done
 
diff -r efd5db7dc7cc -r 74cb2af8811e src/ZF/Constructible/Wellorderings.thy
--- a/src/ZF/Constructible/Wellorderings.thy	Wed Jun 26 12:17:21 2002 +0200
+++ b/src/ZF/Constructible/Wellorderings.thy	Wed Jun 26 18:31:20 2002 +0200
@@ -85,6 +85,16 @@
      "[|wellfounded_on(M,A,r); r \<subseteq> A*A|] ==> wellfounded(M,r)"
 by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff)
 
+(*Consider the least z in domain(r) such that P(z) does not hold...*)
+lemma (in M_axioms) wellfounded_induct: 
+     "[| wellfounded(M,r); M(a); M(r); separation(M, \<lambda>x. ~P(x));  
+         \<forall>x. M(x) & (\<forall>y. <y,x> \<in> r --> P(y)) --> P(x) |]
+      ==> P(a)";
+apply (simp (no_asm_use) add: wellfounded_def)
+apply (drule_tac x="{z \<in> domain(r). ~P(z)}" in spec)
+apply (blast dest: transM)
+done
+
 lemma (in M_axioms) wellfounded_on_induct: 
      "[| a\<in>A;  wellfounded_on(M,A,r);  M(A);  
        separation(M, \<lambda>x. x\<in>A --> ~P(x));