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Publications/ijcar-06.ps
author Christian Urban <christian dot urban at kcl dot ac dot uk>
Wed, 18 May 2016 15:00:08 +0100
changeset 441 a45ca9d73ee4
parent 14 680070975206
permissions -rw-r--r--
typo

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(erm)n(utation)f(action)g(cannot)g(y)n(et)g(b)r(e)h(de\014ned)523
3493 y(directly)-7 b(,)28 b(but)g(needs)f(to)h(b)r(e)g(lifted)g(from)f
(the)h(represen)n(ting)e(t)n(yp)r(e)i(for)f Fp(lam)o
Fy(.\))648 3594 y(The)34 b(nominal)f(datat)n(yp)r(e)h(pac)n(k)-5
b(age)32 b(assumes)h(that)h(ev)n(ery)f(p)r(erm)n(utation)h(action)f
(de-)523 3694 y(\014ned)28 b(for)g(a)f(t)n(yp)r(e)i(satis\014es)e
(three)h(basic)f(prop)r(erties.)g(F)-7 b(or)27 b(this)i(w)n(e)e(use)h
(the)g(terminology)523 3793 y(from)f([11])g(of)h(a)f
Fs(p)l(ermutation)j(typ)l(e)6 b Fy(:)523 3959 y Fd(De\014nition)31
b(2)41 b(\(P)m(erm)m(utation)e(T)m(yp)s(e\).)h Fs(A)35
b(typ)l(e)h Fn(\013)g Fs(wil)t(l)h(b)l(e)f(r)l(eferr)l(e)l(d)g(to)g(as)
42 b Fy(p)r(erm)n(u-)523 4059 y(tation)28 b(t)n(yp)r(e)p
Fs(,)j(written)f Fn(pt)1341 4071 y Fg(\013)1388 4059
y Fs(,)h(pr)l(ovide)l(d)i(the)e(p)l(ermutation)f(action)h(satis\014es)g
(the)f(fol)t(lowing)523 4158 y(thr)l(e)l(e)g(pr)l(op)l(erties:)1398
4319 y Fy(\()p Fn(i)p Fy(\))71 b([])1608 4328 y Ff(\001)1646
4319 y Fn(x)24 b Fy(=)e Fn(x)1369 4442 y Fy(\()p Fn(ii)p
Fy(\))71 b(\()p Fn(\031)1641 4454 y Fr(1)1679 4442 y
Fy(@)p Fn(\031)1791 4454 y Fr(2)1828 4442 y Fy(\))1860
4451 y Ff(\001)1898 4442 y Fn(x)24 b Fy(=)e Fn(\031)2103
4454 y Fr(1)2141 4451 y Ff(\001)2179 4442 y Fy(\()p Fn(\031)2258
4454 y Fr(2)2296 4451 y Ff(\001)2334 4442 y Fn(x)p Fy(\))1340
4565 y(\()p Fn(iii)p Fy(\))71 b Fs(if)30 b Fn(\031)1689
4577 y Fr(1)1750 4565 y Fk(\030)23 b Fn(\031)1885 4577
y Fr(2)1952 4565 y Fs(then)29 b Fn(\031)2183 4577 y Fr(1)2221
4574 y Ff(\001)2259 4565 y Fn(x)24 b Fy(=)e Fn(\031)2464
4577 y Fr(2)2502 4574 y Ff(\001)2540 4565 y Fn(x)523
4725 y Fy(These)38 b(prop)r(erties)g(en)n(tail)h(that)g(the)g(p)r(erm)n
(utations)f(action)g(b)r(eha)n(v)n(es)g(on)g(elemen)n(ts)h(of)523
4825 y(p)r(erm)n(utation)24 b(t)n(yp)r(es)g(as)f(one)h(exp)r(ects,)g
(for)g(example)f(w)n(e)h(ha)n(v)n(e)f Fn(\031)2575 4795
y Fj(\000)p Fr(1)2665 4834 y Ff(\001)2703 4825 y Fy(\()p
Fn(\031)2785 4834 y Ff(\001)2824 4825 y Fn(x)p Fy(\))g(=)g
Fn(x)p Fy(.)i(W)-7 b(e)24 b(note)523 4924 y(that:)p eop
end
%%Page: 6 6
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1284 y Ff(\001)2302 1275 y Fn(x)p Fy(\))918 b(\(7\))523
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(v)n(e)g Fn(\031)2080 1393 y Fj(\000)p Fr(1)2169 1433
y Ff(\001)2207 1424 y Fy(\()p Fn(\031)2289 1433 y Ff(\001)2328
1424 y Fn(x)p Fy(\))d(=)e Fn(x)29 b Fy(for)e Fn(x)h Fy(of)f(t)n(yp)r(e)
h Fn(\013)3130 1436 y Fr(1)3168 1424 y Fy(.)648 1523
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(\014xed)h(as)f(w)n(ell)g([3].)g(The)523 1822 y(de\014nition)h(of)g
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523 1962 y Fd(De\014nition)k(3)41 b(\(Supp)s(ort)32 b(and)g(F)-8
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2262 y Ff(\001)2475 2253 y Fn(x)23 b Fk(6)p Fy(=)g Fn(x)p
Fk(gg)581 2385 y(\017)41 b Fs(A)n(n)27 b(atom)g Fn(a)h
Fs(is)f(said)i(to)e(b)l(e)34 b Fy(fresh)27 b Fs(for)h(an)f
Fn(x)p Fs(,)i(written)e Fn(a)c Fy(#)g Fn(x)p Fs(,)28
b(pr)l(ovide)l(d)h Fn(a)23 b Fk(62)h Fs(supp)5 b Fy(\()p
Fn(x)p Fy(\))p Fs(.)523 2525 y Fy(The)33 b(adv)-5 b(an)n(tage)31
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(y)e(an)h(imp)r(ortan)n(t)g(r^)-42 b(ole)523 2724 y(later)23
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(utation)g(action)g(giv)n(en)f(in)h(\(5\))g(and)523 2824
y(\(6\),)k(one)f(can)g(calculate)g(the)h(supp)r(ort)g(for)f(the)h(t)n
(yp)r(es:)1087 2971 y Fp(name)o Fy(:)226 b Fs(supp)5
b Fy(\()p Fn(a)p Fy(\))24 b(=)f Fk(f)p Fn(a)p Fk(g)1087
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b Fy(\()p Fn(x)2719 3106 y Fr(2)2757 3094 y Fy(\))1087
3218 y Fp(unit)o Fy(:)226 b Fs(supp)5 b Fy(\(\(\)\))25
b(=)d Fi(?)1087 3341 y Fn(\013)15 b Fp(list)n Fy(:)159
b Fs(supp)5 b Fy(\([]\))24 b(=)f Fi(?)1511 3441 y Fs(supp)5
b Fy(\()p Fn(x)24 b Fy(::)f Fn(xs)p Fy(\))h(=)f Fs(supp)5
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b Fy(\()p Fp(Var)13 b Fn(a)p Fy(\))23 b(=)g Fk(f)p Fn(a)p
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3675 y Fr(1)1937 3663 y Fn(t)1967 3675 y Fr(2)2005 3663
y Fy(\))23 b(=)g Fs(supp)5 b Fy(\()p Fn(t)2378 3675 y
Fr(1)2416 3663 y Fy(\))18 b Fk([)h Fs(supp)6 b Fy(\()p
Fn(t)2771 3675 y Fr(2)2808 3663 y Fy(\))1511 3763 y Fs(supp)f
Fy(\()p Fp(Lam)13 b Fn(a)h(t)p Fy(\))23 b(=)g Fs(supp)6
b Fy(\()p Fn(t)p Fy(\))19 b Fk(\000)f(f)p Fn(a)p Fk(g)3299
3368 y Fy(\(8\))523 3908 y(where)35 b(the)g(last)g(clause)g(uses)f(the)
i(fact)f(that)h(alpha-equiv)-5 b(alence)34 b(for)g(the)i(t)n(yp)r(e)f
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4164 y Ff(\001)2778 4155 y Fn(t)2808 4125 y Fj(0)2850
4155 y Fk(^)k Fn(a)23 b Fy(#)h Fn(t)3113 4125 y Fj(0)3136
4155 y Fy(\))3299 4156 y(\(9\))523 4304 y(F)-7 b(or)23
b(p)r(erm)n(utation)g(t)n(yp)r(es)h(the)g(notion)f(of)h(supp)r(ort)g
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Fy(=)f Fn(x)637 b Fy(\(11\))648 4825 y(A)30 b(further)f(restriction)g
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(that)g(con)n(tain)523 4924 y(elemen)n(ts)e(with)g(in\014nite)g(supp)r
(ort:)p eop end
%%Page: 7 7
TeXDict begin 7 6 bop 523 448 a Fd(De\014nition)31 b(4)41
b(\(Finitely)i(Supp)s(orted)g(P)m(erm)m(utation)h(T)m(yp)s(es\).)f
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(ermutation)g(typ)l(es.)523 1408 y(Pr)l(o)l(of.)43 b
Fy(Routine)28 b(pro)r(ofs)f(using)g(the)h(calculations)e(giv)n(en)h(in)
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Fn(x)p Fy(,)i(of)f(a)523 1679 y(p)r(erm)n(utation)g(t)n(yp)r(e)h(has)f
(\014nite)h(supp)r(ort,)f(then)h(there)g(m)n(ust)f(b)r(e)h(a)f(fresh)g
(atom)g(for)g Fn(x)p Fy(,)523 1778 y(since)27 b(there)h(are)e
(in\014nitely)j(man)n(y)e(atoms.)g(Therefore)f(w)n(e)h(ha)n(v)n(e:)523
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g(ther)l(e)g(exists)g(an)523 2049 y(atom)k Fn(a)g Fs(with)g
Fn(a)23 b Fy(#)g Fn(x)p Fs(.)523 2220 y Fy(As)31 b(a)f(result,)h
(whenev)n(er)f(w)n(e)g(need)h(to)g(c)n(ho)r(ose)e(a)i(fresh)f(atom)g
(for)h(an)f Fn(x)i Fy(of)e(p)r(erm)n(utation)523 2320
y(t)n(yp)r(e,)25 b(w)n(e)f(ha)n(v)n(e)f(to)i(mak)n(e)e(sure)h(that)h
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(automat-)523 2419 y(ically)i(p)r(erformed)f(b)n(y)h(Isab)r(elle's)f
(axiomatic)g(t)n(yp)r(e-classes)g([12])g(for)g(most)h(constructions)523
2519 y(o)r(ccurring)h(in)i(informal)f(pro)r(ofs:)f(Isab)r(elle)h(has)g
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2619 y(struction)c(using)f(Lem.)h(2.)g(Unfortunately)-7
b(,)26 b(this)g(is)g(more)f(di\016cult)i(in)g(case)e(of)h(functions,)
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(supp)r(ort,)g(ev)n(en)f(if)h(their)f(domain)g(and)h(co)r(domain)523
2818 y(are)d(\014nitely)i(supp)r(orted)f(p)r(erm)n(utation)g(t)n(yp)r
(es)h(\(see)f([9,)g(Example)g(9]\).)g(Therefore)f(w)n(e)h(ha)n(v)n(e)
523 2917 y(to)k(establish)f(whether)g(a)g(function)i(has)e(\014nite)h
(supp)r(ort)f(on)h(a)f(case-b)n(y-case)e(basis.)h(In)i(or-)523
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(obligations,)f(w)n(e)h(use)h(the)g(auxiliary)e(notion)523
3117 y(of)28 b Fs(supp)l(orts)34 b Fy([3].)523 3272 y
Fd(De\014nition)d(5.)41 b Fs(A)29 b(set)g Fn(S)35 b Fs(of)30
b(atoms)37 b Fy(supp)r(orts)29 b Fs(an)h Fn(x)p Fs(,)h(written)e
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3451 y Fk(8)14 b Fn(a)g(b:)21 b(a)i Fk(62)h Fn(S)37 b
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(appro)n(ximate)d(the)j(supp)r(ort)f(of)h(an)f Fn(x)h
Fy(from)f(\\ab)r(o)n(v)n(e",)e(b)r(ecause)523 3730 y(w)n(e)27
b(can)g(sho)n(w)g(that:)523 3885 y Fd(Lemma)j(3.)40 b
Fs(If)31 b(a)f(set)f Fn(S)34 b Fs(is)d(\014nite)e(and)h
Fn(S)19 b Fs(supp)l(orts)h Fn(x)p Fs(,)31 b(then)f(supp)5
b Fy(\()p Fn(x)p Fy(\))24 b Fk(\022)f Fn(S)5 b Fs(.)523
4056 y(Pr)l(o)l(of.)43 b Fy(By)d(con)n(tradiction)e(w)n(e)h(assume)g
Fs(supp)6 b Fy(\()p Fn(x)p Fy(\))44 b Fk(6\022)f Fn(S)5
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4255 y(\()p Fn(a)14 b(b)p Fy(\))681 4264 y Ff(\001)719
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4264 y Ff(\001)1921 4255 y Fn(x)37 b Fk(6)p Fy(=)e Fn(x)p
Fk(g)h Fy(is)f(a)g(subset)h(of)f Fn(S)5 b Fy(,)35 b(and)h(since)f
Fn(S)40 b Fy(is)523 4355 y(\014nite)28 b(b)n(y)e(assumption,)h(also)f
Fk(f)p Fn(b)c Fk(j)h Fy(\()p Fn(a)14 b(b)p Fy(\))1781
4364 y Ff(\001)1819 4355 y Fn(x)24 b Fk(6)p Fy(=)f Fn(x)p
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4454 y Fn(a)23 b Fk(62)g Fs(supp)6 b Fy(\()p Fn(x)p Fy(\))29
b(whic)n(h)e(giv)n(es)g(the)h(con)n(tradiction.)1287
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(y)g(cases)f(some)h(e\013ectiv)n(e)g(means)g(to)g(decide)h(relativ)n
(ely)e(easily)523 4725 y(whether)40 b(a)f(function)i(has)e(\014nite)i
(supp)r(ort:)e(one)h(only)f(needs)h(to)g(\014nd)g(a)g(\014nite)g(set)g
(of)523 4825 y(atoms)32 b(and)g(then)i(v)n(erify)d(whether)i(this)g
(set)g(supp)r(orts)f(the)h(function.)g(F)-7 b(or)32 b(this)h(w)n(e)f
(use)523 4924 y(the)c(follo)n(wing)f(heuristic:)p eop
end
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1852 y(whic)n(h)28 b(amoun)n(ts)e(to)i(sho)n(wing)e(that)971
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3185 y(ha)n(v)n(e)26 b(\014nite)j(supp)r(ort.)2120 b
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4924 y(directly)-7 b(.)p eop end
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3664 y Fy(and)g Fn(f)1829 3676 y Fr(3)1887 3664 y Fy(ha)n(v)n(e)e
(\014nite)j(supp)r(ort.)f(This)f(condition)h(ensures)523
3764 y(that)28 b(w)n(e)f(can)g(use)h(Prop.)e(1)h(to)h(c)n(hose)e(a)i
(fresh)f(name.)648 3870 y(With)32 b(these)g(t)n(w)n(o)e(conditions)i(w)
n(e)f(can)g(de\014ne)h(a)f(recursion)f(com)n(binator,)g(w)n(e)h(call)h
(it)523 3970 y Fs(rfun)681 3990 y Fg(f)713 3998 y Fc(1)746
3990 y Fg(f)778 3998 y Fc(2)811 3990 y Fg(f)843 3998
y Fc(3)880 3970 y Fy(,)27 b(with)h(the)g(follo)n(wing)f(prop)r(erties:)
523 4177 y Fd(Theorem)j(1)42 b(\(Characteristic)36 b(Equations)g(for)f
(Recursion\).)67 b Fs(If)33 b Fn(f)2957 4189 y Fr(1)2994
4177 y Fs(,)g Fn(f)3093 4189 y Fr(2)3163 4177 y Fs(and)g
Fn(f)3368 4189 y Fr(3)523 4277 y Fs(have)e(\014nite)e(supp)l(ort)h(and)
g Fn(f)1424 4289 y Fr(3)1491 4277 y Fs(satis\014es)g(the)g(F)n(CB,)h
(then:)595 4479 y(rfun)753 4500 y Fg(f)785 4508 y Fc(1)818
4500 y Fg(f)850 4508 y Fc(2)883 4500 y Fg(f)915 4508
y Fc(3)965 4479 y Fy(\()p Fp(Var)13 b Fn(a)p Fy(\))128
b(=)23 b Fn(f)1475 4491 y Fr(1)1526 4479 y Fn(a)595 4581
y Fs(rfun)753 4601 y Fg(f)785 4609 y Fc(1)818 4601 y
Fg(f)850 4609 y Fc(2)883 4601 y Fg(f)915 4609 y Fc(3)965
4581 y Fy(\()p Fp(App)13 b Fn(t)1172 4593 y Fr(1)1223
4581 y Fn(t)1253 4593 y Fr(2)1290 4581 y Fy(\))24 b(=)f
Fn(f)1475 4593 y Fr(2)1526 4581 y Fn(t)1556 4593 y Fr(1)1607
4581 y Fn(t)1637 4593 y Fr(2)1688 4581 y Fy(\()p Fs(rfun)1878
4601 y Fg(f)1910 4609 y Fc(1)1943 4601 y Fg(f)1975 4609
y Fc(2)2008 4601 y Fg(f)2040 4609 y Fc(3)2090 4581 y
Fn(t)2120 4593 y Fr(1)2158 4581 y Fy(\))14 b(\()p Fs(rfun)2394
4601 y Fg(f)2426 4609 y Fc(1)2459 4601 y Fg(f)2491 4609
y Fc(2)2524 4601 y Fg(f)2556 4609 y Fc(3)2606 4581 y
Fn(t)2636 4593 y Fr(2)2674 4581 y Fy(\))595 4682 y Fs(rfun)753
4702 y Fg(f)785 4710 y Fc(1)818 4702 y Fg(f)850 4710
y Fc(2)883 4702 y Fg(f)915 4710 y Fc(3)965 4682 y Fy(\()p
Fp(Lam)f Fn(a)h(t)p Fy(\))84 b(=)23 b Fn(f)1475 4694
y Fr(3)1526 4682 y Fn(a)14 b(t)g Fy(\()p Fs(rfun)1817
4702 y Fg(f)1849 4710 y Fc(1)1882 4702 y Fg(f)1914 4710
y Fc(2)1947 4702 y Fg(f)1979 4710 y Fc(3)2030 4682 y
Fn(t)p Fy(\))382 b Fs(pr)l(ovide)l(d)32 b Fn(a)23 b Fy(#)g(\()p
Fn(f)3034 4694 y Fr(1)3071 4682 y Fn(;)14 b(f)3149 4694
y Fr(2)3186 4682 y Fn(;)g(f)3264 4694 y Fr(3)3301 4682
y Fy(\))p 523 4839 473 4 v 546 4893 a Fo(4)606 4924 y
Fx(W)-6 b(e)25 b(sligh)n(tly)i(adapted)e(the)g(de\014nition)h(of)g
(Pitts)g(to)g(apply)f(to)h(our)g(recursion)g(com)n(binator.)p
eop end
%%Page: 10 10
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(inductiv)n(e)g(relation,)523 548 y(called)27 b Fs(r)l(e)l(c)865
560 y Fg(f)897 568 y Fc(1)930 560 y Fg(f)962 568 y Fc(2)995
560 y Fg(f)1027 568 y Fc(3)1091 548 y Fy(and)g(of)g(t)n(yp)r(e)g(\()p
Fp(lam)16 b Fk(\002)h Fn(\013)p Fy(\))d Fp(set)26 b Fy(where,)h(lik)n
(e)f(ab)r(o)n(v)n(e,)g Fn(\013)h Fy(is)g(assumed)g(to)g(b)r(e)523
648 y(a)g(p)r(erm)n(utation)h(t)n(yp)r(e:)p 1547 760
834 4 v 1547 839 a(\()p Fp(Var)13 b Fn(a;)h(f)1846 851
y Fr(1)1897 839 y Fn(a)p Fy(\))23 b Fk(2)g Fs(r)l(e)l(c)2182
851 y Fg(f)2214 859 y Fc(1)2247 851 y Fg(f)2279 859 y
Fc(2)2312 851 y Fg(f)2344 859 y Fc(3)1272 1013 y Fy(\()p
Fn(t)1334 1025 y Fr(1)1371 1013 y Fn(;)14 b(r)1445 1025
y Fr(1)1483 1013 y Fy(\))23 b Fk(2)g Fs(r)l(e)l(c)1724
1025 y Fg(f)1756 1033 y Fc(1)1789 1025 y Fg(f)1821 1033
y Fc(2)1854 1025 y Fg(f)1886 1033 y Fc(3)2005 1013 y
Fy(\()p Fn(t)2067 1025 y Fr(2)2105 1013 y Fn(;)14 b(r)2179
1025 y Fr(2)2217 1013 y Fy(\))23 b Fk(2)g Fs(r)l(e)l(c)2458
1025 y Fg(f)2490 1033 y Fc(1)2523 1025 y Fg(f)2555 1033
y Fc(2)2588 1025 y Fg(f)2620 1033 y Fc(3)p 1272 1057
1385 4 v 1354 1136 a Fy(\()p Fp(App)13 b Fn(t)1561 1148
y Fr(1)1612 1136 y Fn(t)1642 1148 y Fr(2)1680 1136 y
Fn(;)h(f)1758 1148 y Fr(2)1808 1136 y Fn(t)1838 1148
y Fr(1)1889 1136 y Fn(t)1919 1148 y Fr(2)1970 1136 y
Fn(r)2007 1148 y Fr(1)2059 1136 y Fn(r)2096 1148 y Fr(2)2134
1136 y Fy(\))23 b Fk(2)g Fs(r)l(e)l(c)2375 1148 y Fg(f)2407
1156 y Fc(1)2440 1148 y Fg(f)2472 1156 y Fc(2)2505 1148
y Fg(f)2537 1156 y Fc(3)1367 1311 y Fn(a)g Fy(#)h(\()p
Fn(f)1600 1323 y Fr(1)1637 1311 y Fn(;)14 b(f)1715 1323
y Fr(2)1752 1311 y Fn(;)g(f)1830 1323 y Fr(3)1866 1311
y Fy(\))84 b(\()p Fn(t;)14 b(r)r Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)2362
1323 y Fg(f)2394 1331 y Fc(1)2427 1323 y Fg(f)2459 1331
y Fc(2)2492 1323 y Fg(f)2524 1331 y Fc(3)p 1367 1355
1194 4 v 1477 1433 a Fy(\()p Fp(Lam)13 b Fn(a)h(t;)g(f)1820
1445 y Fr(3)1870 1433 y Fn(a)g(t)g(r)r Fy(\))24 b Fk(2)f
Fs(r)l(e)l(c)2253 1445 y Fg(f)2285 1453 y Fc(1)2317 1445
y Fg(f)2349 1453 y Fc(2)2382 1445 y Fg(f)2414 1453 y
Fc(3)3257 1122 y Fy(\(12\))523 1597 y(With)29 b(this)e(inductiv)n(e)h
(de\014nition)g(comes)f(the)h(follo)n(wing)f(induction)h(principle:)926
1759 y Fk(8)p Fn(a:)14 b(P)24 b Fy(\()p Fp(Var)13 b Fn(a)p
Fy(\))h(\()p Fn(f)1471 1771 y Fr(1)1522 1759 y Fn(a)p
Fy(\))926 1859 y Fk(8)p Fn(t)1003 1871 y Fr(1)1053 1859
y Fn(t)1083 1871 y Fr(2)1134 1859 y Fn(r)1171 1871 y
Fr(1)1223 1859 y Fn(r)1260 1871 y Fr(2)1297 1859 y Fn(:)g(P)26
b(t)1443 1871 y Fr(1)1494 1859 y Fn(r)1531 1871 y Fr(1)1610
1859 y Fk(^)42 b Fn(P)26 b(t)1816 1871 y Fr(2)1867 1859
y Fn(r)1904 1871 y Fr(2)1964 1859 y Fk(\))e Fn(P)h Fy(\()p
Fp(App)13 b Fn(t)2356 1871 y Fr(1)2407 1859 y Fn(t)2437
1871 y Fr(2)2474 1859 y Fy(\))h(\()p Fn(f)2593 1871 y
Fr(2)2645 1859 y Fn(t)2675 1871 y Fr(1)2726 1859 y Fn(t)2756
1871 y Fr(2)2807 1859 y Fn(r)2844 1871 y Fr(1)2895 1859
y Fn(r)2932 1871 y Fr(2)2970 1859 y Fy(\))926 1958 y
Fk(8)p Fn(a)g(t)g(r)n(:)g(a)21 b Fy(#)j(\()p Fn(f)1378
1970 y Fr(1)1415 1958 y Fn(;)14 b(f)1493 1970 y Fr(2)1530
1958 y Fn(;)g(f)1608 1970 y Fr(3)1644 1958 y Fy(\))42
b Fk(^)g Fn(P)26 b(t)14 b(r)25 b Fk(\))e Fn(P)j Fy(\()p
Fp(Lam)13 b Fn(a)h(t)p Fy(\))p 914 2008 2100 4 v 1529
2087 a(\()p Fn(t;)g(r)r Fy(\))24 b Fk(2)g Fs(r)l(e)l(c)1909
2099 y Fg(f)1941 2107 y Fc(1)1974 2099 y Fg(f)2006 2107
y Fc(2)2039 2099 y Fg(f)2071 2107 y Fc(3)2131 2087 y
Fk(\))f Fn(P)i(t)14 b(r)861 b Fy(\(13\))523 2255 y(W)-7
b(e)22 b(shall)g(sho)n(w)f(next)h(that)g(the)g(relation)f
Fs(r)l(e)l(c)1947 2267 y Fg(f)1979 2275 y Fc(1)2012 2267
y Fg(f)2044 2275 y Fc(2)2077 2267 y Fg(f)2109 2275 y
Fc(3)2168 2255 y Fy(de\014nes)g(a)h(function)g(in)g(the)g(sense)g(that)
523 2354 y(for)h(all)g(lam)n(b)r(da-terms)f Fn(t)h Fy(there)g(exists)g
(a)g(unique)g Fn(r)k Fy(so)22 b(that)i(\()p Fn(t;)14
b(r)r Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)2812 2366 y Fg(f)2844
2374 y Fc(1)2877 2366 y Fg(f)2909 2374 y Fc(2)2942 2366
y Fg(f)2974 2374 y Fc(3)3010 2354 y Fy(.)h(F)-7 b(rom)23
b(this)523 2454 y(w)n(e)k(obtain)h(a)f(function)h(from)f
Fp(lam)g Fy(to)g Fn(\013)p Fy(.)648 2553 y(W)-7 b(e)25
b(\014rst)g(sho)n(w)g(that)g(there)h(exists)e(an)h Fn(r)k
Fy(for)24 b(ev)n(ery)g Fn(t)p Fy(.)i(F)-7 b(or)25 b(this)g(w)n(e)g(use)
g(the)h(follo)n(wing)523 2653 y(strong)j(structural)g(induction)i
(principle)f([9{11])e(that)j(the)f(nominal)g(datat)n(yp)r(e)g(pac)n(k)
-5 b(age)523 2753 y(generates)26 b(for)h(the)h(t)n(yp)r(e)g
Fp(lam)o Fy(:)1381 2915 y Fs(\014nite)o Fy(\()p Fn(S)5
b Fy(\))1381 3014 y Fk(8)p Fn(a:)14 b(P)24 b Fy(\()p
Fp(Var)13 b Fn(a)p Fy(\))1381 3114 y Fk(8)p Fn(t)1458
3126 y Fr(1)1494 3114 y Fn(t)1524 3126 y Fr(2)1561 3114
y Fn(:)h(P)26 b(t)1707 3126 y Fr(1)1762 3114 y Fk(^)19
b Fn(P)26 b(t)1945 3126 y Fr(2)2005 3114 y Fk(\))d Fn(P)j
Fy(\()p Fp(App)13 b Fn(t)2397 3126 y Fr(1)2448 3114 y
Fn(t)2478 3126 y Fr(2)2515 3114 y Fy(\))1381 3214 y Fk(8)p
Fn(a)h(t:)g(a)21 b Fk(62)j Fn(S)j Fk(\))d Fn(P)h(t)e
Fk(\))g Fn(P)j Fy(\()p Fp(Lam)13 b Fn(a)h(t)p Fy(\))p
1369 3263 1191 4 v 1910 3337 a Fn(P)25 b(t)1239 b Fy(\(14\))523
3504 y(This)33 b(induction)h(principle)f(is)h(called)e
Fs(str)l(ong)p Fy(,)i(b)r(ecause)e(in)i(the)g(lam)n(b)r(da-case)d(one)i
(do)r(es)523 3604 y(not)27 b(need)g(to)g(establish)f(the)h(prop)r(ert)n
(y)f Fn(P)39 b Fy(for)26 b(all)h(binders)f Fn(a)p Fy(,)h(but)g(only)g
(for)f(binders)h(that)523 3703 y(are)g(not)i(in)f(the)h(\014nite)g(set)
f Fn(S)5 b Fy(.)28 b(With)i(this)e(structural)g(induction)g(principle)h
(the)f(pro)r(of)g(of)523 3803 y(the)g(next)g(lemma)f(is)h(routine.)523
3949 y Fd(Lemma)i(4)41 b(\(T)-8 b(otalit)m(y\).)33 b
Fs(Pr)l(ovide)l(d)g Fn(f)1796 3961 y Fr(1)1833 3949 y
Fs(,)e Fn(f)1930 3961 y Fr(2)1998 3949 y Fs(and)g Fn(f)2201
3961 y Fr(3)2269 3949 y Fs(have)h(\014nite)e(supp)l(ort,)i(then)e(for)i
(al)t(l)523 4048 y Fn(t)e Fs(ther)l(e)g(exists)f(an)h
Fn(r)i Fs(such)e(that)g Fy(\()p Fn(t;)14 b(r)r Fy(\))24
b Fk(2)f Fs(r)l(e)l(c)1944 4060 y Fg(f)1976 4068 y Fc(1)2009
4060 y Fg(f)2041 4068 y Fc(2)2074 4060 y Fg(f)2106 4068
y Fc(3)2143 4048 y Fs(.)523 4208 y(Pr)l(o)l(of.)43 b
Fy(By)24 b(the)g(strong)e(induction)i(principle,)g(where)f(w)n(e)g(tak)
n(e)g Fn(S)29 b Fy(to)23 b(b)r(e)h Fs(supp)6 b Fy(\()p
Fn(f)3083 4220 y Fr(1)3120 4208 y Fn(;)14 b(f)3198 4220
y Fr(2)3235 4208 y Fn(;)g(f)3313 4220 y Fr(3)3350 4208
y Fy(\),)523 4307 y(whic)n(h)25 b(w)n(e)g(kno)n(w)g(b)n(y)g(assumption)
g(is)g(\014nite.)h(Then)g(in)f(the)h(lam)n(b)r(da-case)e(w)n(e)h(can)g
(assume)523 4407 y(that)30 b Fn(a)25 b Fk(62)h Fs(supp)6
b Fy(\()p Fn(f)1097 4419 y Fr(1)1134 4407 y Fn(;)14 b(f)1212
4419 y Fr(2)1249 4407 y Fn(;)g(f)1327 4419 y Fr(3)1364
4407 y Fy(\))29 b(holds,)g(whic)n(h)h(is)f(de\014ned)g(to)h(b)r(e)f
Fn(a)d Fy(#)g(\()p Fn(f)2735 4419 y Fr(1)2772 4407 y
Fn(;)14 b(f)2850 4419 y Fr(2)2887 4407 y Fn(;)g(f)2965
4419 y Fr(3)3002 4407 y Fy(\).)29 b(All)h(cases)523 4506
y(are)d(then)h(routine)f(applying)g(the)h(rules)f(in)h(\(12\).)523
4666 y(Next)g(w)n(e)f(establish)g(that)h(all)g Fn(r)i
Fy(in)e(the)g(relation)f Fs(r)l(e)l(c)2205 4678 y Fg(f)2237
4686 y Fc(1)2270 4678 y Fg(f)2302 4686 y Fc(2)2335 4678
y Fg(f)2367 4686 y Fc(3)2431 4666 y Fy(ha)n(v)n(e)g(\014nite)h(supp)r
(ort.)523 4825 y Fd(Lemma)i(5)41 b(\(Finite)25 b(Supp)s(ort\).)h
Fs(If)g Fn(f)1814 4837 y Fr(1)1851 4825 y Fs(,)f Fn(f)1942
4837 y Fr(2)2004 4825 y Fs(and)h Fn(f)2202 4837 y Fr(3)2264
4825 y Fs(have)h(\014nite)d(supp)l(ort,)i(then)f Fy(\()p
Fn(t;)14 b(r)r Fy(\))25 b Fk(2)523 4924 y Fs(r)l(e)l(c)631
4936 y Fg(f)663 4944 y Fc(1)695 4936 y Fg(f)727 4944
y Fc(2)760 4936 y Fg(f)792 4944 y Fc(3)859 4924 y Fs(implies)31
b(that)f Fn(r)i Fs(has)f(\014nite)e(supp)l(ort.)p eop
end
%%Page: 11 11
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548 y(sho)n(w)37 b(that)g Fn(f)969 560 y Fr(1)1020 548
y Fn(a)g Fy(has)g(\014nite)h(supp)r(ort,)f(whic)n(h)h(w)n(e)f(inferred)
g(in)g(Example)g(1)g(using)g(our)523 648 y(heuristic.)27
b(The)h(application-)f(and)g(lam)n(b)r(da-case)f(are)h(similar.)799
b Fk(u)-55 b(t)523 825 y Fy(In)32 b(order)f(to)h(establish)f(the)h
(\\uniqueness")f(part)h(of)f(Theorem)h(1,)f(w)n(e)h(need)g(the)g(follo)
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Fs(r)l(e)l(c)1896 936 y Fg(f)1928 944 y Fc(1)1961 936
y Fg(f)1993 944 y Fc(2)2026 936 y Fg(f)2058 944 y Fc(3)2130
924 y Fy(is)f Fs(e)l(quivariant)44 b Fy(\(see)35 b([7]\))g(and)g(that)h
(it)523 1024 y(preserv)n(es)26 b(freshness.)523 1202
y Fd(Lemma)k(6)41 b(\(Equiv)-5 b(ariance\).)29 b Fs(If)e
Fy(\()p Fn(t;)14 b(r)r Fy(\))25 b Fk(2)e Fs(r)l(e)l(c)2081
1214 y Fg(f)2113 1222 y Fc(1)2145 1214 y Fg(f)2177 1222
y Fc(2)2210 1214 y Fg(f)2242 1222 y Fc(3)2306 1202 y
Fs(then)k(for)h(al)t(l)g Fn(\031)i Fs(also)e Fy(\()p
Fn(\031)3060 1211 y Ff(\001)3099 1202 y Fn(t;)14 b(\031)3216
1211 y Ff(\001)3254 1202 y Fn(r)r Fy(\))25 b Fk(2)523
1301 y Fn(r)r(ec)637 1320 y Fr(\()p Fg(\031)704 1329
y Ff(\001)742 1320 y Fg(f)774 1328 y Fc(1)807 1320 y
Fr(\)\()p Fg(\031)900 1329 y Ff(\001)938 1320 y Fg(f)970
1328 y Fc(2)1003 1320 y Fr(\)\()p Fg(\031)1096 1329 y
Ff(\001)1134 1320 y Fg(f)1166 1328 y Fc(3)1199 1320 y
Fr(\))1229 1301 y Fs(.)523 1478 y(Pr)l(o)l(of.)43 b Fy(By)24
b(the)h(induction)f(principle)g(giv)n(en)g(in)g(\(13\).)g(All)h(cases)e
(are)g(routine)h(b)n(y)g(pushing)523 1578 y(the)36 b(p)r(erm)n(utation)
f Fn(\031)k Fy(in)n(to)c Fn(t)g Fy(and)g Fn(r)r Fy(,)i(except)e(in)h
(the)f(lam)n(b)r(da-case)f(where)h(w)n(e)g(ha)n(v)n(e)f(to)523
1678 y(apply)27 b(\(10\))h(in)f(order)g(to)g(infer)h(\()p
Fn(\031)1616 1687 y Ff(\001)1655 1678 y Fn(a)p Fy(\))23
b(#)g(\()p Fn(\031)1928 1687 y Ff(\001)1967 1678 y Fy(\()p
Fn(f)2040 1690 y Fr(1)2077 1678 y Fn(;)14 b(f)2155 1690
y Fr(2)2192 1678 y Fn(;)g(f)2270 1690 y Fr(3)2307 1678
y Fy(\)\))28 b(from)f Fn(a)c Fy(#)g(\()p Fn(f)2827 1690
y Fr(1)2864 1678 y Fn(;)14 b(f)2942 1690 y Fr(2)2979
1678 y Fn(;)g(f)3057 1690 y Fr(3)3094 1678 y Fy(\).)201
b Fk(u)-55 b(t)523 1855 y Fd(Lemma)30 b(7)41 b(\(F)-8
b(reshness\).)39 b Fs(If)e Fn(f)1625 1867 y Fr(1)1662
1855 y Fs(,)g Fn(f)1765 1867 y Fr(2)1838 1855 y Fs(and)g
Fn(f)2047 1867 y Fr(3)2121 1855 y Fs(have)g(\014nite)g(supp)l(ort)f
(and)h Fn(f)3049 1867 y Fr(3)3123 1855 y Fs(satis\014es)523
1954 y(the)30 b(F)n(CB,)h(then)f(assuming)f Fy(\()p Fn(t;)14
b(r)r Fy(\))25 b Fk(2)e Fs(r)l(e)l(c)1814 1966 y Fg(f)1846
1974 y Fc(1)1878 1966 y Fg(f)1910 1974 y Fc(2)1943 1966
y Fg(f)1975 1974 y Fc(3)2042 1954 y Fs(and)30 b Fn(a)23
b Fy(#)g(\()p Fn(f)2435 1966 y Fr(1)2472 1954 y Fn(;)14
b(f)2550 1966 y Fr(2)2587 1954 y Fn(;)g(f)2665 1966 y
Fr(3)2702 1954 y Fn(;)g(t)p Fy(\))30 b Fs(implies)h Fn(a)23
b Fy(#)g Fn(r)r Fs(.)523 2131 y(Pr)l(o)l(of.)43 b Fy(By)26
b(the)h(induction)f(principle)g(giv)n(en)g(in)g(\(13\);)g(non-routine)f
(is)h(the)h(lam)n(b)r(da-case.)523 2231 y(In)f(this)g(case,)f(sa)n(y)g
(with)h(the)g(instan)n(tiations)f(\()p Fp(Lam)13 b Fn(a)2191
2201 y Fj(0)2228 2231 y Fn(t)p Fy(\),)26 b(w)n(e)g(ha)n(v)n(e)e(that)i
Fn(a)2871 2201 y Fj(0)2918 2231 y Fy(#)d(\()p Fn(f)3083
2243 y Fr(1)3120 2231 y Fn(;)14 b(f)3198 2243 y Fr(2)3235
2231 y Fn(;)g(f)3313 2243 y Fr(3)3350 2231 y Fy(\).)523
2331 y(W)-7 b(e)24 b(further)f(ha)n(v)n(e)f(that)i Fn(a)f
Fy(#)g(\()p Fn(f)1532 2343 y Fr(1)1569 2331 y Fn(;)14
b(f)1647 2343 y Fr(2)1684 2331 y Fn(;)g(f)1762 2343 y
Fr(3)1799 2331 y Fn(;)g Fp(Lam)f Fn(a)2025 2301 y Fj(0)2062
2331 y Fn(t)p Fy(\))23 b(and)h(ha)n(v)n(e)e(to)h(sho)n(w)g(that)g
Fn(a)g Fy(#)h Fn(f)3167 2343 y Fr(3)3217 2331 y Fn(a)3261
2301 y Fj(0)3298 2331 y Fn(t)14 b(r)r Fy(.)523 2430 y(In)23
b(case)g(that)g Fn(a)g Fy(=)g Fn(a)1168 2400 y Fj(0)1191
2430 y Fy(,)g(w)n(e)g(kno)n(w)f(from)h(the)g(F)n(CB,)g(there)g(exists)g
(an)g Fn(a)2702 2400 y Fj(00)2767 2430 y Fy(suc)n(h)g(that)g
Fn(a)3169 2400 y Fj(00)3235 2430 y Fy(#)g Fn(f)3368 2442
y Fr(3)523 2530 y Fy(and)g Fk(8)14 b Fn(t)g(r)n(:)g(a)901
2500 y Fj(00)966 2530 y Fy(#)23 b Fn(f)1099 2542 y Fr(3)1150
2530 y Fn(a)1194 2500 y Fj(00)1250 2530 y Fn(t)14 b(r)r
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Fn(a)14 b(a)2755 2500 y Fj(00)2797 2530 y Fy(\))24 b(to)g(b)r(oth)g
(sides)f(of)523 2630 y(our)30 b(goal)f(whic)n(h)i(giv)n(es)e
Fn(a)1343 2599 y Fj(00)1413 2630 y Fy(#)f(\(\()p Fn(a)14
b(a)1676 2599 y Fj(00)1719 2630 y Fy(\))1751 2639 y Ff(\001)1790
2630 y Fn(f)1831 2642 y Fr(3)1867 2630 y Fy(\))g Fn(a)1957
2599 y Fj(00)2014 2630 y Fy(\(\()p Fn(a)g(a)2180 2599
y Fj(00)2222 2630 y Fy(\))2254 2639 y Ff(\001)2293 2630
y Fn(t)p Fy(\))g(\(\()p Fn(a)g(a)2535 2599 y Fj(00)2578
2630 y Fy(\))2610 2639 y Ff(\001)2648 2630 y Fn(r)r Fy(\).)32
b(Since)f Fn(a)c Fy(#)i Fn(f)3204 2642 y Fr(3)3271 2630
y Fy(and)523 2729 y Fn(a)567 2699 y Fj(00)643 2729 y
Fy(#)34 b Fn(f)787 2741 y Fr(3)857 2729 y Fy(w)n(e)g(ha)n(v)n(e)f(b)n
(y)g(\(11\))h(that)g(\()p Fn(a)14 b(a)1807 2699 y Fj(00)1849
2729 y Fy(\))1881 2738 y Ff(\001)1920 2729 y Fn(f)1961
2741 y Fr(3)2031 2729 y Fy(=)33 b Fn(f)2170 2741 y Fr(3)2241
2729 y Fy(and)h(hence)g(w)n(e)f(are)g(done.)h(In)g(case)523
2829 y Fn(a)23 b Fk(6)p Fy(=)g Fn(a)722 2799 y Fj(0)768
2829 y Fy(w)n(e)g(can)g(infer)h(from)f Fn(a)g Fy(#)g(\()p
Fn(f)1646 2841 y Fr(1)1683 2829 y Fn(;)14 b(f)1761 2841
y Fr(2)1798 2829 y Fn(;)g(f)1876 2841 y Fr(3)1913 2829
y Fn(;)g Fp(Lam)e Fn(a)2138 2799 y Fj(0)2175 2829 y Fn(t)p
Fy(\))24 b(that)g Fn(a)f Fy(#)g(\()p Fn(f)2669 2841 y
Fr(1)2706 2829 y Fn(;)14 b(f)2784 2841 y Fr(2)2821 2829
y Fn(;)g(f)2899 2841 y Fr(3)2936 2829 y Fn(;)g(t)p Fy(\))23
b(holds)g(and)523 2928 y(th)n(us)28 b(apply)f(the)h(induction)g(h)n(yp)
r(othesis.)1499 b Fk(u)-55 b(t)523 3106 y Fy(No)n(w)27
b(w)n(e)g(can)h(sho)n(w)e(the)i(crucial)f(lemma)h(ab)r(out)f
Fs(r)l(e)l(c)2216 3118 y Fg(f)2248 3126 y Fc(1)2280 3118
y Fg(f)2312 3126 y Fc(2)2345 3118 y Fg(f)2377 3126 y
Fc(3)2442 3106 y Fy(b)r(eing)h(a)f(\\function".)523 3283
y Fd(Lemma)j(8)41 b(\(Uniqueness\).)31 b Fs(If)f Fn(f)1684
3295 y Fr(1)1721 3283 y Fs(,)h Fn(f)1818 3295 y Fr(2)1884
3283 y Fs(and)g Fn(f)2087 3295 y Fr(3)2154 3283 y Fs(have)g(\014nite)e
(supp)l(ort)h(and)h Fn(f)3056 3295 y Fr(3)3123 3283 y
Fs(satis\014es)523 3383 y(the)f(F)n(CB,)h(then)f Fy(\()p
Fn(t;)14 b(r)r Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)1451 3395
y Fg(f)1483 3403 y Fc(1)1516 3395 y Fg(f)1548 3403 y
Fc(2)1580 3395 y Fg(f)1612 3403 y Fc(3)1679 3383 y Fs(and)30
b Fy(\()p Fn(t;)14 b(r)1978 3353 y Fj(0)2003 3383 y Fy(\))23
b Fk(2)g Fs(r)l(e)l(c)2244 3395 y Fg(f)2276 3403 y Fc(1)2309
3395 y Fg(f)2341 3403 y Fc(2)2373 3395 y Fg(f)2405 3403
y Fc(3)2472 3383 y Fs(implies)31 b(that)f Fn(r)c Fy(=)d
Fn(r)3116 3353 y Fj(0)3140 3383 y Fs(.)523 3560 y(Pr)l(o)l(of.)43
b Fy(By)26 b(the)g(induction)h(principle)f(giv)n(en)f(in)i(\(13\);)f
(again)f(the)h(only)g(non-routine)f(case)523 3660 y(is)36
b(the)h(lam)n(b)r(da-case.)e(By)h(assumption)f(w)n(e)h(kno)n(w)g(that)h
(\()p Fp(Lam)12 b Fn(a)i(t;)g(f)2744 3672 y Fr(3)2795
3660 y Fn(a)g(t)g(r)r Fy(\))38 b Fk(2)g Fs(r)l(e)l(c)3206
3672 y Fg(f)3238 3680 y Fc(1)3271 3672 y Fg(f)3303 3680
y Fc(2)3336 3672 y Fg(f)3368 3680 y Fc(3)523 3759 y Fy(from)24
b(whic)n(h)g(w)n(e)g(can)g(infer)h(that)g Fn(a)e Fy(#)g(\()p
Fn(f)1816 3771 y Fr(1)1853 3759 y Fn(;)14 b(f)1931 3771
y Fr(2)1968 3759 y Fn(;)g(f)2046 3771 y Fr(3)2083 3759
y Fy(\))24 b(and)h(\()p Fn(t;)14 b(r)r Fy(\))24 b Fk(2)f
Fs(r)l(e)l(c)2678 3771 y Fg(f)2710 3779 y Fc(1)2743 3771
y Fg(f)2775 3779 y Fc(2)2807 3771 y Fg(f)2839 3779 y
Fc(3)2876 3759 y Fy(;)i(the)g(induction)523 3859 y(h)n(yp)r(othesis)19
b(states)g(that)g(for)g(all)g Fn(r)1588 3829 y Fj(0)1612
3859 y Fy(,)g(\()p Fn(t;)14 b(r)1792 3829 y Fj(0)1816
3859 y Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)2058 3871 y Fg(f)2090
3879 y Fc(1)2123 3871 y Fg(f)2155 3879 y Fc(2)2187 3871
y Fg(f)2219 3879 y Fc(3)2276 3859 y Fy(implies)c Fn(r)26
b Fy(=)d Fn(r)2739 3829 y Fj(0)2763 3859 y Fy(.)c(Using)g(the)h(second)
523 3959 y(assumption)g(\()p Fp(Lam)13 b Fn(b)h(t)1212
3928 y Fj(0)1235 3959 y Fn(;)28 b(r)1325 3928 y Fj(0)1349
3959 y Fy(\))23 b Fk(2)h Fs(r)l(e)l(c)1590 3971 y Fg(f)1622
3979 y Fc(1)1655 3971 y Fg(f)1687 3979 y Fc(2)1720 3971
y Fg(f)1752 3979 y Fc(3)1810 3959 y Fy(w)n(e)c(need)h(to)g(sho)n(w)f
(that)h Fn(f)2620 3971 y Fr(3)2671 3959 y Fn(a)14 b(t)g(r)25
b Fy(=)e Fn(f)2964 3971 y Fr(3)3015 3959 y Fn(b)14 b(t)3095
3928 y Fj(0)3131 3959 y Fn(r)3170 3928 y Fj(0)3215 3959
y Fy(holds)523 4058 y(for)26 b(all)h Fp(Lam)13 b Fn(b)h(t)989
4028 y Fj(0)1038 4058 y Fy(suc)n(h)27 b(that)g Fn(b)c
Fy(#)g(\()p Fn(f)1628 4070 y Fr(1)1665 4058 y Fn(;)14
b(f)1743 4070 y Fr(2)1780 4058 y Fn(;)g(f)1858 4070 y
Fr(3)1895 4058 y Fy(\))27 b(and)g Fp(Lam)12 b Fn(a)i(t)23
b Fy(=)g Fp(Lam)12 b Fn(b)i(t)2682 4028 y Fj(0)2705 4058
y Fy(.)27 b(The)g(latter)g(implies)523 4158 y(b)n(y)g(\(9\))h(that)g
(either)1066 4342 y(\()p Fn(a)c Fy(=)e Fn(b)c Fk(^)h
Fn(t)k Fy(=)g Fn(t)1552 4308 y Fj(0)1575 4342 y Fy(\))46
b(or)g(\()p Fn(a)23 b Fk(6)p Fy(=)g Fn(b)18 b Fk(^)g
Fn(t)23 b Fy(=)g(\()p Fn(a)14 b(b)p Fy(\))2387 4351 y
Ff(\001)2425 4342 y Fn(t)2455 4308 y Fj(0)2497 4342 y
Fk(^)19 b Fn(a)k Fy(#)g Fn(t)2760 4308 y Fj(0)2783 4342
y Fy(\))h Fn(:)523 4526 y Fy(The)30 b(\014rst)g(case)f(is)h(routine)g
(b)r(ecause)g(b)n(y)g(the)g(induction)g(h)n(yp)r(othesis)g(w)n(e)g(can)
g(infer)g(that)523 4625 y Fn(r)c Fy(=)c Fn(r)712 4595
y Fj(0)737 4625 y Fy(.)f(In)g(the)g(second)g(case)f(w)n(e)g(ha)n(v)n(e)
g(\(\()p Fn(a)14 b(b)p Fy(\))1935 4634 y Ff(\001)1974
4625 y Fn(t;)g(r)2080 4595 y Fj(0)2104 4625 y Fy(\))23
b Fk(2)g Fs(r)l(e)l(c)2345 4637 y Fg(f)2377 4645 y Fc(1)2410
4637 y Fg(f)2442 4645 y Fc(2)2475 4637 y Fg(f)2507 4645
y Fc(3)2565 4625 y Fy(and)d(b)n(y)h(Lem.)g(6)g(also)f(that)523
4725 y(\()p Fn(t;)14 b Fy(\()p Fn(a)g(b)p Fy(\))780 4734
y Ff(\001)818 4725 y Fn(r)857 4695 y Fj(0)881 4725 y
Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)1123 4737 y Fg(f)1155
4745 y Fc(1)1187 4737 y Fg(f)1219 4745 y Fc(2)1252 4737
y Fg(f)1284 4745 y Fc(3)1347 4725 y Fy(\(where)j(w)n(e)f(also)g(use)h
(\(11\))f(and)h(the)g(facts)g Fn(a)d Fy(#)g(\()p Fn(f)2946
4737 y Fr(1)2983 4725 y Fn(;)14 b(f)3061 4737 y Fr(2)3098
4725 y Fn(;)g(f)3176 4737 y Fr(3)3213 4725 y Fy(\))26
b(and)523 4825 y Fn(b)d Fy(#)g(\()p Fn(f)747 4837 y Fr(1)784
4825 y Fn(;)14 b(f)862 4837 y Fr(2)899 4825 y Fn(;)g(f)977
4837 y Fr(3)1014 4825 y Fy(\)\).)21 b(By)g(induction)g(h)n(yp)r
(othesis)g(w)n(e)f(can)h(therefore)f(infer)h(that)g Fn(r)26
b Fy(=)d(\()p Fn(a)14 b(b)p Fy(\))3281 4834 y Ff(\001)3319
4825 y Fn(r)3358 4795 y Fj(0)3382 4825 y Fy(.)523 4924
y(Hence)28 b(w)n(e)f(ha)n(v)n(e)f(to)i(sho)n(w)f(that)h
Fn(f)1612 4936 y Fr(3)1662 4924 y Fn(a)14 b Fy(\(\()p
Fn(a)g(b)p Fy(\))1910 4933 y Ff(\001)1948 4924 y Fn(t)1978
4894 y Fj(0)2002 4924 y Fy(\))g(\(\()p Fn(a)g(b)p Fy(\))2238
4933 y Ff(\001)2276 4924 y Fn(r)2315 4894 y Fj(0)2339
4924 y Fy(\))24 b(=)e Fn(f)2523 4936 y Fr(3)2574 4924
y Fn(b)14 b(t)2654 4894 y Fj(0)2690 4924 y Fn(r)2729
4894 y Fj(0)2781 4924 y Fy(holds.)p eop end
%%Page: 12 12
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1016 y Fj(0)3373 1046 y Fy(\))523 1146 y(whic)n(h)33
b(giv)n(es)e Fn(b)1010 1116 y Fj(0)1064 1146 y Fy(#)h(\(\()p
Fn(b)14 b(b)1315 1116 y Fj(0)1338 1146 y Fy(\))1370 1155
y Ff(\001)1408 1146 y Fn(f)1449 1158 y Fr(3)1486 1146
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Fn(r)2867 946 y Fr(2)523 1122 y Fy(where)26 b(in)g(the)h(second)f
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(for)f(all)h(terms)g(not)h(in)523 1222 y(its)h(scop)r(e.)648
1323 y(Alb)r(eit)39 b(not)g(y)n(et)f(supp)r(orted)h(b)n(y)f(the)h
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1422 y Fy(\()p Fp(lam)20 b Fk(\002)523 1511 y Fq(h)-10
b(h)562 1522 y Fp(name)737 1511 y Fq(i)g(i)776 1522 y
Fp(lam)o Fy(\))p Fp(")24 b Fy(where)f(w)n(e)g(ha)n(v)n(e)f(t)n(w)n(o)h
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Fp(Letrec)523 1622 y Fy(is)617 1770 y Fs(rfun)775 1791
y Fg(f)807 1799 y Fc(1)851 1791 y Fg(f)883 1799 y Fc(2)927
1791 y Fg(f)959 1799 y Fc(3)1003 1791 y Fg(f)1035 1799
y Fc(4)1079 1791 y Fg(f)1111 1799 y Fc(5)1162 1770 y
Fy(\()p Fp(Letrec)12 b Fn(a)i(t)1558 1782 y Fr(1)1608
1770 y Fn(b)g(t)1688 1782 y Fr(2)1725 1770 y Fy(\))24
b(=)e Fn(f)1909 1782 y Fr(5)1960 1770 y Fn(a)14 b(t)2048
1782 y Fr(1)2099 1770 y Fn(b)g(t)2179 1782 y Fr(2)2229
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1791 y Fg(::)d(f)2579 1799 y Fc(5)2629 1770 y Fn(t)2659
1782 y Fr(1)2697 1770 y Fy(\))j(\()p Fs(rfun)2933 1791
y Fg(f)2965 1799 y Fc(1)3009 1791 y Fg(::)d(f)3092 1799
y Fc(5)3143 1770 y Fn(t)3173 1782 y Fr(2)3210 1770 y
Fy(\))2223 1894 y Fs(pr)l(ovide)l(d)32 b Fn(a)22 b Fy(#)i(\()p
Fn(f)2783 1906 y Fr(1)2820 1894 y Fn(;)14 b(f)2898 1906
y Fr(2)2935 1894 y Fn(;)g(f)3013 1906 y Fr(3)3049 1894
y Fn(;)g(f)3127 1906 y Fr(4)3164 1894 y Fn(;)g(f)3242
1906 y Fr(5)3279 1894 y Fy(\))2223 2017 y Fs(and)30 b
Fn(b)23 b Fy(#)g(\()p Fn(f)2608 2029 y Fr(1)2645 2017
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2029 y Fr(3)2875 2017 y Fn(;)g(f)2953 2029 y Fr(4)2989
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2029 y Fr(1)3208 2017 y Fy(\))2223 2140 y Fs(and)30 b
Fn(a)23 b Fk(6)p Fy(=)g Fn(b)523 2329 y Fy(where)39 b(w)n(e)h(need)g
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2329 y Fy(since)c Fn(t)1979 2341 y Fr(1)2056 2329 y Fy(is)g(not)g(in)g
(the)h(scop)r(e)e(of)h(the)g(binder)g Fn(b)p Fy(.)523
2429 y(Ho)n(w)n(ev)n(er,)29 b(in)h(case)g(w)n(e)g(ha)n(v)n(e)g(more)f
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2529 y(further)j(need)g(to)g(add)g(constrain)n(ts)f(that)h(mak)n(e)g
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2628 y(these)k(generalisations)d(the)j(pro)r(ofs)f(w)n(e)g(ha)n(v)n(e)g
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2728 y(datat)n(yp)r(es.)523 2999 y Ft(6)112 b(Conclusion)523
3204 y Fy(W)-7 b(e)27 b(presen)n(ted)f(a)h(structural)e(recursion)h
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3303 y(tails)g(w)n(ere)f(giv)n(en)g(for)g(the)h(nominal)g(datat)n(yp)r
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523 3403 y(case|further)c(details)h(are)g(giv)n(en)f(in)i([9].)f(F)-7
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