src/ZF/WF.thy
 author clasohm Tue, 06 Feb 1996 12:27:17 +0100 changeset 1478 2b8c2a7547ab parent 1401 0c439768f45c child 2469 b50b8c0eec01 permissions -rw-r--r--
expanded tabs
```
(*  Title:      ZF/wf.thy
ID:         \$Id\$
Author:     Tobias Nipkow and Lawrence C Paulson

Well-founded Recursion
*)

WF = Trancl + "mono" + "equalities" +
consts
wf           :: i=>o
wf_on        :: [i,i]=>o                      ("wf[_]'(_')")

wftrec,wfrec :: [i, i, [i,i]=>i] =>i
wfrec_on     :: [i, i, i, [i,i]=>i] =>i       ("wfrec[_]'(_,_,_')")
is_recfun    :: [i, i, [i,i]=>i, i] =>o
the_recfun   :: [i, i, [i,i]=>i] =>i

defs
(*r is a well-founded relation*)
wf_def         "wf(r) == ALL Z. Z=0 | (EX x:Z. ALL y. <y,x>:r --> ~ y:Z)"

(*r is well-founded relation over A*)
wf_on_def      "wf_on(A,r) == wf(r Int A*A)"

is_recfun_def  "is_recfun(r,a,H,f) ==
(f = (lam x: r-``{a}. H(x, restrict(f, r-``{x}))))"

the_recfun_def "the_recfun(r,a,H) == (THE f.is_recfun(r,a,H,f))"

wftrec_def     "wftrec(r,a,H) == H(a, the_recfun(r,a,H))"

(*public version.  Does not require r to be transitive*)
wfrec_def "wfrec(r,a,H) == wftrec(r^+, a, %x f. H(x, restrict(f,r-``{x})))"

wfrec_on_def   "wfrec[A](r,a,H) == wfrec(r Int A*A, a, H)"

end
```