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