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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 +TeXDict begin 11 10 bop 523 448 a Fs(Pr)l(o)l(of.)43 +b Fy(By)32 b(the)g(induction)g(principle)g(giv)n(e)f(in)h(\(13\).)f(In) +h(the)h(v)-5 b(ariable-case)29 b(w)n(e)j(ha)n(v)n(e)e(to)523 +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) +n(w-)523 924 y(ing)j(t)n(w)n(o)f(lemmas)h(establishing)f(that)i +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 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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 +Fy(.)24 b(Using)g(\(10\))f(w)n(e)g(apply)g(the)h(sw)n(apping)f(\()p +Fn(a)14 b(a)2755 2500 y Fj(00)2797 2530 y Fy(\))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 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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 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