src/ZF/WF.thy

simplified ensures_UNIV

(*  Title:      ZF/wf.thy
    ID:         $Id$
    Author:     Tobias Nipkow and Lawrence C Paulson
    Copyright   1994  University of Cambridge

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