(* Title: HOL/WF.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
Well-founded Recursion
*)
WF = Trancl +
consts
wf :: "('a * 'a)set => bool"
cut :: "['a => 'b, ('a * 'a)set, 'a] => 'a => 'b"
wftrec,wfrec :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b] => 'b"
is_recfun :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b, 'a=>'b] => bool"
the_recfun :: "[('a * 'a)set, 'a, ['a,'a=>'b]=>'b] => 'a=>'b"
defs
wf_def "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x.P(x)))"
cut_def "cut f r x == (%y. if (y,x):r then f y else @z.True)"
is_recfun_def "is_recfun r a H f == (f = cut (%x.(H x (cut f r x))) r a)"
the_recfun_def "the_recfun r a H == (@f.is_recfun r a H f)"
wftrec_def "wftrec r a H == H a (the_recfun r a H)"
(*version not requiring transitivity*)
wfrec_def "wfrec r a H == wftrec (trancl r) a (%x f.(H x (cut f r x)))"
end