--- a/src/ZF/Constructible/Relative.thy Tue Jul 09 10:44:41 2002 +0200
+++ b/src/ZF/Constructible/Relative.thy Tue Jul 09 10:44:53 2002 +0200
@@ -764,9 +764,13 @@
ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) &
pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))"
and is_recfun_separation:
- --{*for well-founded recursion. NEEDS RELATIVIZATION*}
- "[| M(A); M(f); M(g); M(a); M(b) |]
- ==> separation(M, \<lambda>x. \<langle>x,a\<rangle> \<in> r & \<langle>x,b\<rangle> \<in> r & f`x \<noteq> g`x)"
+ --{*for well-founded recursion*}
+ "[| M(r); M(f); M(g); M(a); M(b) |]
+ ==> separation(M,
+ \<lambda>x. \<exists>xa[M]. \<exists>xb[M].
+ pair(M,x,a,xa) & xa \<in> r & pair(M,x,b,xb) & xb \<in> r &
+ (\<exists>fx[M]. \<exists>gx[M]. fun_apply(M,f,x,fx) & fun_apply(M,g,x,gx) &
+ fx \<noteq> gx))"
lemma (in M_axioms) cartprod_iff_lemma:
"[| M(C); \<forall>u[M]. u \<in> C <-> (\<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}});