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b(rom)33b Fn(S)19 b Fs(supp)l(orts)h Fn(x)35 b Fy(follo)n(ws)e(that)h(for)f(all)h Fn(b)f Fk(62)h Fn(S)39 b Fy(w)n(e)33 b(ha)n(v)n(e)5234255 y(\()p Fn(a)14 b(b)p Fy(\))681 4264 y Ff(\001)7194255 y Fn(x)37 b Fy(=)e Fn(x)p Fy(.)i(Hence)e(the)h(set)fFk(f)p Fn(b)h Fk(j)g Fy(\()p Fn(a)14 b(b)p Fy(\))18834264 y Ff(\001)1921 4255 y Fn(x)37 b Fk(6)p Fy(=)e Fn(x)pFk(g)h Fy(is)f(a)g(subset)h(of)f Fn(S)5 b Fy(,)35 b(and)h(since)fFn(S)40 b Fy(is)523 4355 y(\014nite)28 b(b)n(y)e(assumption,)h(also)fFk(f)p Fn(b)c Fk(j)h Fy(\()p Fn(a)14 b(b)p Fy(\))17814364 y Ff(\001)1819 4355 y Fn(x)24 b Fk(6)p Fy(=)f Fn(x)pFk(g)k Fy(m)n(ust)g(b)r(e)g(\014nite.)h(But)f(this)g(implies)h(that)5234454 y Fn(a)23 b Fk(62)g Fs(supp)6 b Fy(\()p Fn(x)p Fy(\))29b(whic)n(h)e(giv)n(es)g(the)h(con)n(tradiction.)1287b Fk(u)-55 b(t)523 4625 y Fy(Lem.)32 b(3)g(giv)n(es)g(us)g(in)g(man)n(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 eopend%%Page: 8 8TeXDict begin 8 7 bop 523 448 a Fd(Heuristic)31 b(1)41b Fs(Assume)28 b(an)g(HOL-function,)f(say)i(fn)7 b(,)28b(is)h(given)g(as)f(a)h(lamb)l(da-term.)g(The)523 548y(supp)l(ort)i(of)g(the)g(tuple)g(c)l(onsisting)f(of)i(the)f(fr)l(e)l(e)g(variables)h(of)g(fn)37 b(supp)l(orts)30 b(this)h(function,)523648 y(mor)l(e)h(formal)t(ly)i(we)e(have)h(supp)6 b Fy(\()pFn(F)12 b(V)19 b Fy(\()p Fs(fn)6 b Fy(\)\))28 b Fs(supp)l(orts)33b(fn)7 b(,)32 b(wher)l(e)g(we)g(assume)g Fn(F)12 b(V)51b Fs(is)32 b(de-)523 747 y(\014ne)l(d)d(as)h(usual,)f(exc)l(ept)h(that)f(we)h(gr)l(oup)f(the)h(fr)l(e)l(e)g(variables)h(in)f(tuples,)f(inste)l(ad)h(of)g(\014nite)523 847 y(sets.)523 995 y Fy(This)f(is)f(a)g(heuristic,)g(b)r(ecause)h(it)g(can)f(v)n(ery)f(lik)n(ely)h(not)h(b)r(e)f(established)h(as)f(a)g(lemma)g(in-)523 1094 y(side)d(Isab)r(elle/HOL,)f(since)h(it)g(is)g(a)g(prop)r(ert)n(y)f(ab)r(out)h(HOL-functions.)g(Nev)n(ertheless)f(the)523 1194 y(heuristic)h(is)g(extremely)g(helpful)h(for)e(deciding)h(whether)g(a)g(function)h(has)e(\014nite)i(supp)r(ort.)523 1294 y(Consider)h(the)h(follo)n(wing)e(t)n(w)n(o)h(examples:)523 1454 y Fs(Example)k(1.)42 b Fy(Giv)n(en)23b(a)g(function)g Fs(fn)1692 1407 y Fr(def)1703 1454 yFy(=)34 b Fn(f)1843 1466 y Fr(1)1893 1454 y Fn(c)23 bFy(where)g Fn(f)2229 1466 y Fr(1)2289 1454 y Fy(is)g(a)f(function)i(of)f(t)n(yp)r(e)g Fp(name)e Fk(\))i Fn(\013)p Fy(.)523 1553y(W)-7 b(e)26 b(also)e(assume)h(that)g Fn(f)1331 1565y Fr(1)1393 1553 y Fy(has)g(\014nite)h(supp)r(ort.)f(The)g(question)g(is)h(whether)f Fs(fn)32 b Fy(has)24 b(\014nite)523 1653y(supp)r(ort?)35 b(The)g(free)f(v)-5 b(ariables)34 b(of)gFs(fn)42 b Fy(are)34 b Fn(f)1975 1665 y Fr(1)2046 1653y Fy(and)h Fn(c)p Fy(,)g(that)g(means)f Fn(F)12 b(V)19b Fy(\()p Fs(fn)7 b Fy(\))35 b(=)g(\()p Fn(f)3240 1665y Fr(1)3277 1653 y Fn(;)14 b(c)p Fy(\).)523 1753 y(According)40b(to)g(our)g(heuristic)h(w)n(e)f(ha)n(v)n(e)g(to)g(v)n(erify)g(whether)h Fs(supp)5 b Fy(\()p Fn(f)2828 1765 y Fr(1)2866 1753y Fn(;)14 b(c)p Fy(\))g Fs(supp)l(orts)20 b(fn)7 b Fy(,)5231852 y(whic)n(h)28 b(amoun)n(ts)e(to)i(sho)n(wing)e(that)9712012 y Fk(8)p Fn(a)14 b(b:)22 b(a)h Fk(62)g Fs(supp)6b Fy(\()p Fn(f)1544 2024 y Fr(1)1581 2012 y Fn(;)14 b(c)pFy(\))32 b Fk(^)h Fn(b)23 b Fk(62)g Fs(supp)6 b Fy(\()pFn(f)2185 2024 y Fr(1)2222 2012 y Fn(;)14 b(c)p Fy(\))37b Fk(\))46 b Fy(\()p Fn(a)14 b(b)p Fy(\))2651 2021 yFf(\001)2689 2012 y Fs(fn)30 b Fy(=)22 b Fs(fn)523 2172y Fy(T)-7 b(o)30 b(do)g(so)f(w)n(e)h(can)g(assume)f(b)n(y)h(the)h(de\014nition)f(of)h(freshness)e(\(Def.)i(3\))f(that)hFn(a)c Fy(#)h(\()p Fn(f)3263 2184 y Fr(1)3300 2172 yFn(;)14 b(c)p Fy(\))523 2272 y(and)37 b Fn(b)h Fy(#)g(\()pFn(f)948 2284 y 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Fy(there)g(exists)g(a)g(unique)g Fn(r)k Fy(so)22 b(that)i(\()p Fn(t;)14b(r)r Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)2812 2366 y Fg(f)28442374 y Fc(1)2877 2366 y Fg(f)2909 2374 y Fc(2)2942 2366y Fg(f)2974 2374 y Fc(3)3010 2354 y Fy(.)h(F)-7 b(rom)23b(this)523 2454 y(w)n(e)k(obtain)h(a)f(function)h(from)fFp(lam)g Fy(to)g Fn(\013)p Fy(.)648 2553 y(W)-7 b(e)25b(\014rst)g(sho)n(w)g(that)g(there)h(exists)e(an)h Fn(r)kFy(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)gFp(lam)o Fy(:)1381 2915 y Fs(\014nite)o Fy(\()p Fn(S)5b Fy(\))1381 3014 y Fk(8)p Fn(a:)14 b(P)24 b Fy(\()pFp(Var)13 b Fn(a)p Fy(\))1381 3114 y Fk(8)p Fn(t)14583126 y Fr(1)1494 3114 y Fn(t)1524 3126 y Fr(2)1561 3114y Fn(:)h(P)26 b(t)1707 3126 y Fr(1)1762 3114 y Fk(^)19b Fn(P)26 b(t)1945 3126 y Fr(2)2005 3114 y Fk(\))d Fn(P)jFy(\()p Fp(App)13 b Fn(t)2397 3126 y Fr(1)2448 3114 yFn(t)2478 3126 y Fr(2)2515 3114 y Fy(\))1381 3214 y Fk(8)pFn(a)h(t:)g(a)21 b Fk(62)j Fn(S)j Fk(\))d Fn(P)h(t)eFk(\))g Fn(P)j Fy(\()p Fp(Lam)13 b Fn(a)h(t)p Fy(\))p1369 3263 1191 4 v 1910 3337 a Fn(P)25 b(t)1239 b Fy(\(14\))5233504 y(This)33 b(induction)h(principle)f(is)h(called)eFs(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.)5233949 y Fd(Lemma)i(4)41 b(\(T)-8 b(otalit)m(y\).)33 bFs(Pr)l(ovide)l(d)g Fn(f)1796 3961 y Fr(1)1833 3949 yFs(,)e Fn(f)1930 3961 y Fr(2)1998 3949 y Fs(and)g Fn(f)22013961 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)hFn(r)i Fs(such)e(that)g Fy(\()p Fn(t;)14 b(r)r Fy(\))24b Fk(2)f Fs(r)l(e)l(c)1944 4060 y Fg(f)1976 4068 y Fc(1)20094060 y Fg(f)2041 4068 y Fc(2)2074 4060 y Fg(f)2106 4068y Fc(3)2143 4048 y Fs(.)523 4208 y(Pr)l(o)l(of.)43 bFy(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(\()pFn(f)3083 4220 y Fr(1)3120 4208 y Fn(;)14 b(f)3198 4220y Fr(2)3235 4208 y Fn(;)g(f)3313 4220 y Fr(3)3350 4208y Fy(\),)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)6b Fy(\()p Fn(f)1097 4419 y Fr(1)1134 4407 y Fn(;)14 b(f)12124419 y Fr(2)1249 4407 y Fn(;)g(f)1327 4419 y Fr(3)13644407 y Fy(\))29 b(holds,)g(whic)n(h)h(is)f(de\014ned)g(to)h(b)r(e)fFn(a)d Fy(#)g(\()p Fn(f)2735 4419 y Fr(1)2772 4407 yFn(;)14 b(f)2850 4419 y Fr(2)2887 4407 y Fn(;)g(f)29654419 y Fr(3)3002 4407 y Fy(\).)29 b(All)h(cases)523 4506y(are)d(then)h(routine)f(applying)g(the)h(rules)f(in)h(\(12\).)5234666 y(Next)g(w)n(e)f(establish)g(that)h(all)g Fn(r)iFy(in)e(the)g(relation)f Fs(r)l(e)l(c)2205 4678 y Fg(f)22374686 y Fc(1)2270 4678 y Fg(f)2302 4686 y Fc(2)2335 4678y 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(\(Finite)25 b(Supp)s(ort\).)hFs(If)g Fn(f)1814 4837 y Fr(1)1851 4825 y Fs(,)f Fn(f)19424837 y Fr(2)2004 4825 y Fs(and)h Fn(f)2202 4837 y Fr(3)22644825 y Fs(have)h(\014nite)d(supp)l(ort,)i(then)f Fy(\()pFn(t;)14 b(r)r Fy(\))25 b Fk(2)523 4924 y Fs(r)l(e)l(c)6314936 y Fg(f)663 4944 y Fc(1)695 4936 y Fg(f)727 4944y Fc(2)760 4936 y Fg(f)792 4944 y Fc(3)859 4924 y Fs(implies)31b(that)f Fn(r)i Fs(has)f(\014nite)e(supp)l(ort.)p eopend%%Page: 11 11TeXDict begin 11 10 bop 523 448 a Fs(Pr)l(o)l(of.)43b 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)523548 y(sho)n(w)37 b(that)g Fn(f)969 560 y Fr(1)1020 548y 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.)27b(The)h(application-)f(and)g(lam)n(b)r(da-case)f(are)h(similar.)799b 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)iFs(r)l(e)l(c)1896 936 y Fg(f)1928 944 y Fc(1)1961 936y Fg(f)1993 944 y Fc(2)2026 936 y Fg(f)2058 944 y Fc(3)2130924 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 1202y Fd(Lemma)k(6)41 b(\(Equiv)-5 b(ariance\).)29 b Fs(If)eFy(\()p Fn(t;)14 b(r)r Fy(\))25 b Fk(2)e Fs(r)l(e)l(c)20811214 y Fg(f)2113 1222 y Fc(1)2145 1214 y Fg(f)2177 1222y Fc(2)2210 1214 y Fg(f)2242 1222 y Fc(3)2306 1202 yFs(then)k(for)h(al)t(l)g Fn(\031)i Fs(also)e Fy(\()pFn(\031)3060 1211 y Ff(\001)3099 1202 y Fn(t;)14 b(\031)32161211 y Ff(\001)3254 1202 y Fn(r)r Fy(\))25 b Fk(2)5231301 y Fn(r)r(ec)637 1320 y Fr(\()p Fg(\031)704 1329y Ff(\001)742 1320 y Fg(f)774 1328 y Fc(1)807 1320 yFr(\)\()p Fg(\031)900 1329 y Ff(\001)938 1320 y Fg(f)9701328 y Fc(2)1003 1320 y Fr(\)\()p Fg(\031)1096 1329 yFf(\001)1134 1320 y Fg(f)1166 1328 y Fc(3)1199 1320 yFr(\))1229 1301 y Fs(.)523 1478 y(Pr)l(o)l(of.)43 b Fy(By)24b(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)5231678 y(apply)27 b(\(10\))h(in)f(order)g(to)g(infer)h(\()pFn(\031)1616 1687 y Ff(\001)1655 1678 y Fn(a)p Fy(\))23b(#)g(\()p Fn(\031)1928 1687 y Ff(\001)1967 1678 y Fy(\()pFn(f)2040 1690 y Fr(1)2077 1678 y Fn(;)14 b(f)2155 1690y Fr(2)2192 1678 y Fn(;)g(f)2270 1690 y Fr(3)2307 1678y Fy(\)\))28 b(from)f Fn(a)c Fy(#)g(\()p Fn(f)2827 1690y Fr(1)2864 1678 y Fn(;)14 b(f)2942 1690 y Fr(2)29791678 y Fn(;)g(f)3057 1690 y Fr(3)3094 1678 y Fy(\).)201b Fk(u)-55 b(t)523 1855 y Fd(Lemma)30 b(7)41 b(\(F)-8b(reshness\).)39 b Fs(If)e Fn(f)1625 1867 y Fr(1)16621855 y Fs(,)g Fn(f)1765 1867 y Fr(2)1838 1855 y Fs(and)gFn(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)5231954 y(the)30 b(F)n(CB,)h(then)f(assuming)f Fy(\()p Fn(t;)14b(r)r Fy(\))25 b Fk(2)e Fs(r)l(e)l(c)1814 1966 y Fg(f)18461974 y Fc(1)1878 1966 y Fg(f)1910 1974 y Fc(2)1943 1966y Fg(f)1975 1974 y Fc(3)2042 1954 y Fs(and)30 b Fn(a)23b Fy(#)g(\()p Fn(f)2435 1966 y Fr(1)2472 1954 y Fn(;)14b(f)2550 1966 y Fr(2)2587 1954 y Fn(;)g(f)2665 1966 yFr(3)2702 1954 y Fn(;)g(t)p Fy(\))30 b Fs(implies)h Fn(a)23b Fy(#)g Fn(r)r Fs(.)523 2131 y(Pr)l(o)l(of.)43 b Fy(By)26b(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(\()p Fp(Lam)13 b Fn(a)21912201 y Fj(0)2228 2231 y Fn(t)p Fy(\),)26 b(w)n(e)g(ha)n(v)n(e)e(that)iFn(a)2871 2201 y Fj(0)2918 2231 y Fy(#)d(\()p Fn(f)30832243 y Fr(1)3120 2231 y Fn(;)14 b(f)3198 2243 y Fr(2)32352231 y Fn(;)g(f)3313 2243 y Fr(3)3350 2231 y Fy(\).)5232331 y(W)-7 b(e)24 b(further)f(ha)n(v)n(e)f(that)i Fn(a)fFy(#)g(\()p Fn(f)1532 2343 y Fr(1)1569 2331 y Fn(;)14b(f)1647 2343 y Fr(2)1684 2331 y Fn(;)g(f)1762 2343 yFr(3)1799 2331 y Fn(;)g Fp(Lam)f Fn(a)2025 2301 y Fj(0)20622331 y Fn(t)p Fy(\))23 b(and)h(ha)n(v)n(e)e(to)h(sho)n(w)g(that)gFn(a)g Fy(#)h Fn(f)3167 2343 y Fr(3)3217 2331 y Fn(a)32612301 y Fj(0)3298 2331 y Fn(t)14 b(r)r Fy(.)523 2430 y(In)23b(case)g(that)g Fn(a)g Fy(=)g Fn(a)1168 2400 y Fj(0)11912430 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)gFn(a)3169 2400 y Fj(00)3235 2430 y Fy(#)g Fn(f)3368 2442y Fr(3)523 2530 y Fy(and)g Fk(8)14 b Fn(t)g(r)n(:)g(a)9012500 y Fj(00)966 2530 y Fy(#)23 b Fn(f)1099 2542 y Fr(3)11502530 y Fn(a)1194 2500 y Fj(00)1250 2530 y Fn(t)14 b(r)rFy(.)24 b(Using)g(\(10\))f(w)n(e)g(apply)g(the)h(sw)n(apping)f(\()pFn(a)14 b(a)2755 2500 y Fj(00)2797 2530 y Fy(\))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)eFn(a)1343 2599 y Fj(00)1413 2630 y Fy(#)f(\(\()p Fn(a)14b(a)1676 2599 y Fj(00)1719 2630 y Fy(\))1751 2639 y Ff(\001)17902630 y Fn(f)1831 2642 y Fr(3)1867 2630 y Fy(\))g Fn(a)19572599 y Fj(00)2014 2630 y Fy(\(\()p Fn(a)g(a)2180 2599y Fj(00)2222 2630 y Fy(\))2254 2639 y Ff(\001)2293 2630y Fn(t)p Fy(\))g(\(\()p Fn(a)g(a)2535 2599 y Fj(00)25782630 y Fy(\))2610 2639 y Ff(\001)2648 2630 y Fn(r)r Fy(\).)32b(Since)f Fn(a)c Fy(#)i Fn(f)3204 2642 y Fr(3)3271 2630y Fy(and)523 2729 y Fn(a)567 2699 y Fj(00)643 2729 yFy(#)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(\()p Fn(a)14 b(a)1807 2699 y Fj(00)18492729 y Fy(\))1881 2738 y Ff(\001)1920 2729 y Fn(f)19612741 y Fr(3)2031 2729 y Fy(=)33 b Fn(f)2170 2741 y Fr(3)22412729 y Fy(and)h(hence)g(w)n(e)f(are)g(done.)h(In)g(case)5232829 y Fn(a)23 b Fk(6)p Fy(=)g Fn(a)722 2799 y Fj(0)7682829 y Fy(w)n(e)g(can)g(infer)h(from)f Fn(a)g Fy(#)g(\()pFn(f)1646 2841 y Fr(1)1683 2829 y Fn(;)14 b(f)1761 2841y Fr(2)1798 2829 y Fn(;)g(f)1876 2841 y Fr(3)1913 2829y Fn(;)g Fp(Lam)e Fn(a)2138 2799 y Fj(0)2175 2829 y Fn(t)pFy(\))24 b(that)g Fn(a)f Fy(#)g(\()p Fn(f)2669 2841 yFr(1)2706 2829 y Fn(;)14 b(f)2784 2841 y Fr(2)2821 2829y Fn(;)g(f)2899 2841 y Fr(3)2936 2829 y Fn(;)g(t)p Fy(\))23b(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)27b(w)n(e)g(can)h(sho)n(w)e(the)i(crucial)f(lemma)h(ab)r(out)fFs(r)l(e)l(c)2216 3118 y Fg(f)2248 3126 y Fc(1)2280 3118y Fg(f)2312 3126 y Fc(2)2345 3118 y Fg(f)2377 3126 yFc(3)2442 3106 y Fy(b)r(eing)h(a)f(\\function".)523 3283y Fd(Lemma)j(8)41 b(\(Uniqueness\).)31 b Fs(If)f Fn(f)16843295 y Fr(1)1721 3283 y Fs(,)h Fn(f)1818 3295 y Fr(2)18843283 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 yFs(satis\014es)523 3383 y(the)f(F)n(CB,)h(then)f Fy(\()pFn(t;)14 b(r)r Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)1451 3395y Fg(f)1483 3403 y Fc(1)1516 3395 y Fg(f)1548 3403 yFc(2)1580 3395 y Fg(f)1612 3403 y Fc(3)1679 3383 y Fs(and)30b Fy(\()p Fn(t;)14 b(r)1978 3353 y Fj(0)2003 3383 y Fy(\))23b Fk(2)g Fs(r)l(e)l(c)2244 3395 y Fg(f)2276 3403 y Fc(1)23093395 y Fg(f)2341 3403 y Fc(2)2373 3395 y Fg(f)2405 3403y Fc(3)2472 3383 y Fs(implies)31 b(that)f Fn(r)c Fy(=)dFn(r)3116 3353 y Fj(0)3140 3383 y Fs(.)523 3560 y(Pr)l(o)l(of.)43b 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)36b(the)h(lam)n(b)r(da-case.)e(By)h(assumption)f(w)n(e)h(kno)n(w)g(that)h(\()p Fp(Lam)12 b Fn(a)i(t;)g(f)2744 3672 y Fr(3)27953660 y Fn(a)g(t)g(r)r Fy(\))38 b Fk(2)g Fs(r)l(e)l(c)32063672 y Fg(f)3238 3680 y Fc(1)3271 3672 y Fg(f)3303 3680y Fc(2)3336 3672 y Fg(f)3368 3680 y Fc(3)523 3759 y Fy(from)24b(whic)n(h)g(w)n(e)g(can)g(infer)h(that)g Fn(a)e Fy(#)g(\()pFn(f)1816 3771 y Fr(1)1853 3759 y Fn(;)14 b(f)1931 3771y Fr(2)1968 3759 y Fn(;)g(f)2046 3771 y Fr(3)2083 3759y Fy(\))24 b(and)h(\()p Fn(t;)14 b(r)r Fy(\))24 b Fk(2)fFs(r)l(e)l(c)2678 3771 y Fg(f)2710 3779 y Fc(1)2743 3771y Fg(f)2775 3779 y Fc(2)2807 3771 y Fg(f)2839 3779 yFc(3)2876 3759 y Fy(;)i(the)g(induction)523 3859 y(h)n(yp)r(othesis)19b(states)g(that)g(for)g(all)g Fn(r)1588 3829 y Fj(0)16123859 y Fy(,)g(\()p Fn(t;)14 b(r)1792 3829 y Fj(0)18163859 y Fy(\))24 b Fk(2)f Fs(r)l(e)l(c)2058 3871 y Fg(f)20903879 y Fc(1)2123 3871 y Fg(f)2155 3879 y Fc(2)2187 3871y Fg(f)2219 3879 y Fc(3)2276 3859 y Fy(implies)c Fn(r)26b Fy(=)d Fn(r)2739 3829 y Fj(0)2763 3859 y Fy(.)c(Using)g(the)h(second)523 3959 y(assumption)g(\()p Fp(Lam)13 b Fn(b)h(t)12123928 y Fj(0)1235 3959 y Fn(;)28 b(r)1325 3928 y Fj(0)13493959 y Fy(\))23 b Fk(2)h Fs(r)l(e)l(c)1590 3971 y Fg(f)16223979 y Fc(1)1655 3971 y Fg(f)1687 3979 y Fc(2)1720 3971y 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)25b Fy(=)e Fn(f)2964 3971 y Fr(3)3015 3959 y Fn(b)14 b(t)30953928 y Fj(0)3131 3959 y Fn(r)3170 3928 y Fj(0)3215 3959y Fy(holds)523 4058 y(for)26 b(all)h Fp(Lam)13 b Fn(b)h(t)9894028 y Fj(0)1038 4058 y Fy(suc)n(h)27 b(that)g Fn(b)cFy(#)g(\()p Fn(f)1628 4070 y Fr(1)1665 4058 y Fn(;)14b(f)1743 4070 y Fr(2)1780 4058 y Fn(;)g(f)1858 4070 yFr(3)1895 4058 y Fy(\))27 b(and)g Fp(Lam)12 b Fn(a)i(t)23b Fy(=)g Fp(Lam)12 b Fn(b)i(t)2682 4028 y Fj(0)2705 4058y Fy(.)27 b(The)g(latter)g(implies)523 4158 y(b)n(y)g(\(9\))h(that)g(either)1066 4342 y(\()p Fn(a)c Fy(=)e Fn(b)c Fk(^)hFn(t)k Fy(=)g Fn(t)1552 4308 y Fj(0)1575 4342 y Fy(\))46b(or)g(\()p Fn(a)23 b Fk(6)p Fy(=)g Fn(b)18 b Fk(^)gFn(t)23 b Fy(=)g(\()p Fn(a)14 b(b)p Fy(\))2387 4351 yFf(\001)2425 4342 y Fn(t)2455 4308 y Fj(0)2497 4342 yFk(^)19 b Fn(a)k Fy(#)g Fn(t)2760 4308 y Fj(0)2783 4342y 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 4595y 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)19744625 y Fn(t;)g(r)2080 4595 y Fj(0)2104 4625 y Fy(\))23b Fk(2)g Fs(r)l(e)l(c)2345 4637 y Fg(f)2377 4645 y Fc(1)24104637 y Fg(f)2442 4645 y Fc(2)2475 4637 y Fg(f)2507 4645y Fc(3)2565 4625 y Fy(and)d(b)n(y)h(Lem.)g(6)g(also)f(that)5234725 y(\()p Fn(t;)14 b Fy(\()p Fn(a)g(b)p Fy(\))780 4734y Ff(\001)818 4725 y Fn(r)857 4695 y Fj(0)881 4725 yFy(\))24 b Fk(2)f Fs(r)l(e)l(c)1123 4737 y Fg(f)11554745 y Fc(1)1187 4737 y Fg(f)1219 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Fn(t)g(r)n(:)g(b)1953916 y Fj(0)2001 947 y Fy(#)27 b Fn(f)2138 959 y Fr(3)2189947 y Fn(b)2225 916 y Fj(0)2262 947 y Fn(t)14 b(r)32b Fy(holds.)e(If)g Fn(b)d Fy(=)f Fn(b)2893 916 y Fj(0)2946947 y Fy(w)n(e)j(are)g(done;)523 1046 y(otherwise)19b(w)n(e)g(use)h(\(10\))f(and)h(apply)f(the)h(sw)n(apping)f(\()pFn(b)14 b(b)2284 1016 y Fj(0)2306 1046 y Fy(\))20 b(to)g(b)r(oth)g(sides)f(of)h Fn(b)j Fy(#)g(\()p Fn(f)3142 1058 y Fr(3)31931046 y Fn(b)14 b(t)3273 1016 y Fj(0)3310 1046 y Fn(r)33491016 y Fj(0)3373 1046 y Fy(\))523 1146 y(whic)n(h)33b(giv)n(es)e Fn(b)1010 1116 y Fj(0)1064 1146 y Fy(#)h(\(\()pFn(b)14 b(b)1315 1116 y Fj(0)1338 1146 y Fy(\))1370 1155y Ff(\001)1408 1146 y Fn(f)1449 1158 y Fr(3)1486 1146y Fy(\))g Fn(b)1568 1116 y Fj(0)1605 1146 y Fy(\(\()pFn(b)g(b)1755 1116 y Fj(0)1778 1146 y Fy(\))1810 1155y Ff(\001)1849 1146 y Fn(t)1879 1116 y Fj(0)1902 1146y Fy(\))g(\(\()p Fn(b)g(b)2098 1116 y Fj(0)2121 1146y Fy(\))2153 1155 y Ff(\001)2191 1146 y Fn(r)2230 1116y Fj(0)2254 1146 y Fy(\).)33 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Fj(0)1535 1445 y Fn(r)1574 1415 y Fj(0)15981445 y Fy(.)38 b(The)g(left-hand)h(side)f(of)g(this)g(equation)g(is)g(equal)f(to)523 1545 y Fn(f)564 1557 y Fr(3)615 1545y Fn(a)14 b Fy(\(\()p Fn(a)g(b)p Fy(\))863 1554 y Ff(\001)9011545 y Fn(t)931 1514 y Fj(0)954 1545 y Fy(\))g(\(\()pFn(a)g(b)p Fy(\))1190 1554 y Ff(\001)1229 1545 y Fn(r)12681514 y Fj(0)1292 1545 y Fy(\))28 b(whic)n(h)f(is)h(what)f(w)n(e)g(had)h(to)f(sho)n(w.)883 b Fk(u)-55 b(t)523 1719 y Fy(T)-7b(o)25 b(pro)n(v)n(e)e(our)h(theorem)g(ab)r(out)h(structural)f(recursion)g(w)n(e)g(de\014ne)h Fs(rfun)2813 1740 y Fg(f)28451748 y Fc(1)2878 1740 y Fg(f)2910 1748 y Fc(2)2943 1740y Fg(f)2975 1748 y Fc(3)3026 1719 y Fn(t)g Fy(to)f(b)r(e)i(the)5231819 y(unique)e Fn(r)j Fy(so)d(that)g(\()p Fn(t;)14 b(r)rFy(\))25 b Fk(2)e Fs(r)l(e)l(c)1509 1831 y Fg(f)15411839 y Fc(1)1573 1831 y Fg(f)1605 1839 y Fc(2)1638 1831y Fg(f)1670 1839 y Fc(3)1707 1819 y Fy(.)h(This)g(is)g(a)g(standard)f(construction)h(in)g(HOL-based)523 1919 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