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Fh(\024)f Fv(xs\))681 873 y Fx(except)23 b(that)h(w)n(e)f(also)g(allo)n
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Fl(ys)f Fx(has)h(no)g(pre\014x)g(that)681 973 y(satis\014es)j
Fl(P)p Fx(.)h(This)g(de\014nition)g(mak)n(es)f(sense)h(in)g(our)f(con)n
(text)h(where)f(the)i(maximal)e(pre\014x)681 1072 y(should)f(nev)n(er)f
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g(pre\014x)h(that)f(satis\014es)g Fl(P)p Fx(.)805 1282
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(The)h(class)e(of)i(functions)g(w)n(e)f(w)n(an)n(t)g(to)681
1382 y(sp)r(ecify)c(are)e(of)i(t)n(yp)r(e)822 1518 y
Fi(\013)39 b Fv(chopper)80 b(=)f Fi(\013)40 b Fv(list)g
Fh(\))f Fi(\013)g Fv(list)h(list)h Fh(\002)e Fi(\013)g
Fv(list)681 1670 y Fx(The)27 b(sp)r(eci\014cation)h(is)f(a)g(predicate)
822 1813 y Fj(is)p 875 1813 V 27 w(maxchopp)r(er)40 b
Fv(::)g(\()p Fi(\013)f Fv(list)h Fh(\))f Fv(bool\))i
Fh(\))e Fi(\013)g Fv(chopper)i Fh(\))e Fv(bool)681 1965
y Fx(that)33 b(expresses)e(when)j(its)f(second)f(argumen)n(t)g
(correctly)f(c)n(hops)h(up)i(lists)f(according)e(to)681
2064 y(its)d(\014rst)f(argumen)n(t:)822 2207 y Fj(is)p
875 2207 V 27 w(maxchopp)r(er)40 b Fv(P)f(chopper)i Fh(\021)822
2298 y(8)e Fv(xs)g(zs)h(yss.)979 2390 y(\(chopper\(xs\))i(=)d
(\(yss,zs\)\))j(=)979 2481 y(\(xs)e(=)f Fj(concat)g Fv(yss)h(@)g(zs)f
Fh(^)h Fv(\()p Fh(8)f Fv(ys)h Fh(2)f Fj(set)h Fv(yss.)g(ys)f
Fh(6)p Fw(=)g Fv([]\))h Fh(^)979 2572 y Fv(\(case)g(yss)g(of)1097
2664 y([])f Fh(\))g Fj(is)p 1383 2664 V 28 w(maxp)n(ref)h
Fv(P)f([])h(xs)1018 2755 y(|)g(us#uss)g Fh(\))f Fj(is)p
1540 2755 V 28 w(maxp)n(ref)g Fv(P)h(us)g(xs)f Fh(^)1489
2846 y Fv(chopper\()p Fj(concat)r Fv(\(uss\)@zs\))i(=)e(\(uss,zs\)\)\))
681 2998 y Fx(Let's)27 b(recast)g(this)h(in)n(to)f(w)n(ords:)f
Fl(chopper\(xs\))e Fx(returns)i Fl(\(yss,zs\))f Fx(i\013)716
3207 y(1.)41 b(the)28 b(concatenation)e(of)i(the)g(outputs)g(yields)f
(the)h(input,)716 3318 y(2.)41 b(all)27 b(elemen)n(ts)h(of)34
b Fl(yss)26 b Fx(are)h(nonempt)n(y)-7 b(,)27 b(and)716
3428 y(3.)41 b(if)34 b Fl(yss)26 b Fx(is)i(empt)n(y)-7
b(,)27 b(then)h(there)f(is)g(no)g(nonempt)n(y)h(pre\014x)e(of)34
b Fl(xs)27 b Fx(that)g(satis\014es)g Fl(P)p Fx(,)g(and)822
3528 y(if)33 b Fl(yss)43 b(=)g(us#uss)p Fx(,)24 b(then)j
Fl(us)f Fx(is)g(the)h(maximal)f(pre\014x)h(of)32 b Fl(xs)26
b Fx(w.r.t.)h Fl(P)f Fx(and)g(c)n(hopping)822 3627 y(up)i(the)g
(remaining)e(list)i(yields)g Fl(\(uss,zs\))p Fx(.)681
3825 y(Note)38 b(that)h(instead)f(of)h(an)f(unjusti\014ed)h(axiom)f(sp)
r(ecifying)g(a)g(constan)n(t)g Fq(chopp)r(er)p Fx(,)g(the)681
3925 y(predicate)27 b Fq(is)p 1098 3925 25 4 v 29 w(maxchopp)r(er)h
Fx(is)f(merely)g(an)h(abbreviation.)805 4035 y(Note)39
b(also)e(that)h(although)g(the)g(sp)r(eci\014cation)g(only)g(sa)n(ys)f
(that)h(the)h(\014rst)f(elemen)n(t)681 4135 y Fl(us)26
b Fx(of)33 b Fl(yss)25 b Fx(m)n(ust)i(b)r(e)g(a)g(maximal)f(pre\014x,)g
(the)h(remaining)f(elemen)n(ts)g(are)g(co)n(v)n(ered)f(b)n(y)h(the)681
4235 y(\\recursiv)n(e")18 b(call)j(of)27 b Fl(chopper)18
b Fx(in)k(the)f(\014nal)g(line.)g(A)g(direct)g(sp)r(eci\014cation)g(of)
g(the)g(maximal)681 4334 y(pre\014x)j(prop)r(ert)n(y)f(for)h(all)h
(elemen)n(ts)f(of)31 b Fl(yss)24 b Fx(is)g(more)g(in)n(v)n(olv)n(ed)f
(b)r(ecause)h(the)h(list)g(of)g(whic)n(h)681 4434 y(they)j(are)e(a)h
(pre\014x)h(is)f(not)h(directly)f(at)g(hand.)805 4544
y(No)n(w)g(that)h(w)n(e)g(ha)n(v)n(e)e(the)i(main)f(sp)r
(eci\014cation,)h(let)g(us)f(lo)r(ok)g(at)h(an)f(implemen)n(tation:)716
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4863 y(2.)41 b(function)28 b Fq(chop)f Fx(iterates)g(the)h(pro)r(cess)e
(of)i(splitting)g(o\013)f(a)h(pre\014x.)2080 5112 y(10)p
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524 y Fi(\013)40 b Fv(splitter)80 b(=)f Fi(\013)39 b
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Fv(list)365 686 y Fx(and)28 b(a)f(separate)f(sp)r(eci\014cation)506
829 y Fj(is)p 559 829 24 4 v 28 w(maxsplitter)38 b Fv(::)i(\()p
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(satis\014es)g Fl(P)g Fx(is)h(found.)490 2164 y(Once)f(y)n(ou)g(come)g
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b Fv(P)g(r)h(ps)f(qs)h(=)f(\(xs,ys\)\))i(=)506 2402 y(\(if)f
Fh(9)f Fv(us.)h(us)g Fh(\024)f Fv(qs)g Fh(^)h Fv(P\(ps@us\))546
2494 y(then)g(xs@ys=ps@qs)i Fh(^)d Fj(is)p 1356 2494
V 27 w(maxp)n(ref)h Fv(P)g(xs)f(\(ps@qs\))i(else)f(\(xs,ys\)=r\))365
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2798 y Fj(is)p 559 2798 V 28 w(maxsplitter)38 b Fv(P)h(\()p
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3058 y(lead)k(to)g(quadratic)f(run)h(times.)h(This)f(problem)f(could)h
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365 3257 y(of)33 b(automata)e(theory)-7 b(,)32 b(this)g(corresp)r(onds)
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3357 y(`error')26 b(state)h(whic)n(h)h(do)r(es)f(not)h(lead)f(to)h(an)n
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3720 y Fj(chop)40 b Fv(::)f Fi(\013)h Fv(splitter)h Fh(\))e
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Fw(=)-13 b Fh(\))585 3903 y Fj(chop)39 b Fv(splitf)i(xs)e(=)h(\(let)g
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4438 y(cursion,)c(whic)n(h)h(is)f(not)h(sho)n(wn.)f(The)g(precondition)
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12 11 bop 681 387 a Fx(is)32 b(necessary)f(to)h(guaran)n(tee)f
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1347 y(applied)d(to)h(longer)e(and)i(longer)e(pre\014xes,)h(this)h
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y Fh(\))f Fi(\013)h Fv(splitter)822 1872 y Fj(auto)p
970 1872 V 27 w(split)e Fv(A)h(q)h(r)f(ps)h([])197 b(=)39
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3231 2196 25 4 v 29 w(split)f Fx(and)681 2295 y(need)23
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2395 y(on)k Fl(xs)p Fx(\))g(to)h(sho)n(w)822 2542 y Fj(auto)p
970 2542 24 4 v 27 w(split)38 b Fv(A)h(\()p Fj(delta)g
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b(1979.)404 3676 y(7.)42 b(C.)26 b(Kreitz.)32 b(Constructiv)n(e)21
b(automata)g(theory)f(implemen)n(ted)e(with)j(the)f(Nuprl)g(pro)r(of)i
(dev)n(el-)505 3768 y(opmen)n(t)g(system.)33 b(T)-6 b(ec)n(hnical)25
b(Rep)r(ort)e(TR)h(86-779,)i(Dept.)e(of)g(Computer)f(Science,)i
(Cornell)505 3859 y(Univ)n(ersit)n(y)-6 b(,)25 b(1986.)404
3950 y(8.)42 b(T.)26 b(Nipk)n(o)n(w.)34 b(Winsk)n(el)25
b(is)h(\(almost\))g(righ)n(t:)g(T)-6 b(o)n(w)n(ards)26
b(a)g(mec)n(hanized)f(seman)n(tics)g(textb)r(o)r(ok.)505
4041 y(In)37 b(V.)25 b(Chandru)37 b(and)g(V.)25 b(Vina)n(y)-6
b(,)37 b(editors,)h Fa(F)-6 b(oundations)41 b(of)d(Softwar)l(e)i(T)-6
b(e)l(chnolo)l(gy)40 b(and)505 4133 y(The)l(or)l(etic)l(al)35
b(Computer)f(Scienc)l(e)p Fw(,)f(v)n(olume)d(1180)k(of)e
Fa(L)l(e)l(ct.)i(Notes)h(in)d(Comp.)h(Sci.)p Fw(,)e(pages)505
4224 y(180{192.)e(Springer-V)-6 b(erlag,)26 b(1996.)404
4315 y(9.)42 b(L.)25 b(C.)36 b(P)n(aulson.)62 b(Generic)36
b(automatic)e(pro)r(of)i(to)r(ols.)63 b(In)34 b(R.)25
b(V)-6 b(ero\013,)35 b(editor,)g Fa(A)n(utomate)l(d)505
4407 y(R)l(e)l(asoning)j(and)f(its)g(Applic)l(ations)p
Fw(.)g(MIT)f(Press,)h(1997.)65 b(Also)37 b(Rep)r(ort)e(396,)i(Computer)
505 4498 y(Lab)r(oratory)-6 b(,)26 b(Univ)n(ersit)n(y)f(of)h(Cam)n
(bridge.)365 4589 y(10.)43 b(L.)25 b(C.)19 b(P)n(aulson.)33
b(A)17 b(generic)i(tableau)g(pro)n(v)n(er)f(and)f(its)i(in)n(tegration)
g(with)g(Isab)r(elle.)32 b(T)-6 b(ec)n(hnical)505 4681
y(Rep)r(ort)25 b(441,)i(Univ)n(ersit)n(y)e(of)i(Cam)n(bridge,)e
(Computer)g(Lab)r(oratory)-6 b(,)27 b(1998.)365 4772
y(11.)43 b(S.)25 b(Thompson.)63 b(Regular)35 b(expressions)h(and)f
(automata)h(using)f(Miranda.)64 b(Av)l(ailable)36 b(at)505
4863 y Fv(http://www.cs.ukc.ac.uk/pubs/)q(1995/)q(212)p
Fw(,)c(1995.)1765 5112 y Fx(15)p eop
%%Page: 16 16
16 15 bop 681 387 a Fw(This)26 b(article)h(w)n(as)g(t)n(yp)r(eset)e
(using)h(the)f(L)1859 370 y Fg(A)1892 387 y Fw(T)1934
403 y(E)1978 387 y(X)g(macro)g(pac)n(k)l(age)h(with)g(the)g(LLNCS2E)g
(class.)2080 5112 y Fx(16)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF