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(* Title: ZF/wf.thy
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ID: $Id$
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Author: Tobias Nipkow and Lawrence C Paulson
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Copyright 1994 University of Cambridge
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Well-founded Recursion
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*)
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WF = Trancl + mono + equalities +
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consts
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wf :: i=>o
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wf_on :: [i,i]=>o ("wf[_]'(_')")
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wftrec,wfrec :: [i, i, [i,i]=>i] =>i
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wfrec_on :: [i, i, i, [i,i]=>i] =>i ("wfrec[_]'(_,_,_')")
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is_recfun :: [i, i, [i,i]=>i, i] =>o
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the_recfun :: [i, i, [i,i]=>i] =>i
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defs
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(*r is a well-founded relation*)
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wf_def "wf(r) == ALL Z. Z=0 | (EX x:Z. ALL y. <y,x>:r --> ~ y:Z)"
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(*r is well-founded relation over A*)
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wf_on_def "wf_on(A,r) == wf(r Int A*A)"
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is_recfun_def "is_recfun(r,a,H,f) ==
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(f = (lam x: r-``{a}. H(x, restrict(f, r-``{x}))))"
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the_recfun_def "the_recfun(r,a,H) == (THE 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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(*public version. Does not require r to be transitive*)
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wfrec_def "wfrec(r,a,H) == wftrec(r^+, a, %x f. H(x, restrict(f,r-``{x})))"
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wfrec_on_def "wfrec[A](r,a,H) == wfrec(r Int A*A, a, H)"
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end
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