src/ZF/WF.thy
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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