%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: n8kurz.dvi %%CreationDate: Wed Oct 28 13:08:59 1998 %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips n8kurz.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.10.28:1304 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 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1411 y FC(\()p Fv(k)s FC(+)r(1\)\),)9 b(where)j Fv(x)903 1396 y Fs(1)903 1423 y Ft(k)948 1411 y FC(is)e(the)h(constan)o(t)g(family)c Fv(x)1365 1417 y Ft(k)1385 1411 y FC(.)j(The)g(de\014nition)g(of)262 1461 y Fu(")282 1471 y Ft(\032)p Fs(!)p Ft(\033)369 1461 y FC(has)k(to)g(b)q(e)h(mo)q(di\014ed)d(accordingly)m(.)h(This)h(idea)f (corresp)q(onds)j(to)e(a)g(represen)o(tation)262 1511 y(of)h(terms)h(in)g(the)h(st)o(yle)g(of)i Fw(de)f(Br)o(uijn)e FC([9].)f(An)h(adv)n(an)o(tage)g(of)g(this)g(approac)o(h)g(is)g(that) 262 1561 y(w)o(e)c(get)h(the)h(same)d(normal)g(form)g(ev)o(en)i(if)f (the)h(terms)g(are)g(only)e(equal)i(up)f(to)h(renaming)e(of)262 1611 y(b)q(ound)j(v)n(ariables.)324 1662 y(The)e(pro)q(of)f(of)h (correctness)i(is)e(easy:)g(Since)g(for)g(the)g(t)o(yp)q(ed)g(lam)o(b)q (da)e(calculus)i(without)262 1712 y(constan)o(ts)18 b(w)o(e)h(ha)o(v)o (e)e(preserv)n(ation)i(of)e(v)n(alues,)h(i.e.)f([)-7 b([)p Fv(M)5 b FC(])-7 b(])1194 1718 y Ft(\030)1228 1712 y FC(=)19 b([)-7 b([)p Fp(lnf)r FC(\()p Fv(M)5 b FC(\)])-7 b(])1436 1718 y Ft(\030)1471 1712 y FC(for)18 b(all)f(terms)262 1762 y Fv(M)j FC(and)c(en)o(vironmen)o(ts)f Fv(\030)r FC(,)h(w)o(e)g(only)f(ha)o(v)o(e)h(to)g(v)o(erify)f Fu(#)p FC(\([)-7 b([)p Fv(N)5 b FC(])-7 b(])1243 1768 y Fs(")1260 1762 y FC(\))15 b(=)g Fv(N)21 b FC(for)16 b(normal)e(terms)262 1811 y Fv(N)5 b FC(,)15 b(whic)o(h)i(is)f(straigh)o(tforw)o(ard.)g(The) h(situation)e(is)i(di\013eren)o(t)g(when)g(w)o(e)g(add)f(constan)o(ts) 262 1861 y(together)g(with)g(rewrite)h(rules,)f(since)g(then)h(preserv) n(ation)f(of)f(v)n(alues)h(\(in)f(our)h(mo)q(del\))f(is)262 1911 y(false)e(in)g(general.)h(Ho)o(w)o(ev)o(er,)f(correctness)k(of)c (normalization)e(b)o(y)i(ev)n(aluation)f(still)h(holds,)262 1961 y(but)h(needs)h(to)f(b)q(e)g(pro)o(v)o(en)g(b)o(y)g(a)f (di\013eren)o(t)i(metho)q(d.)324 2012 y(The)g(structure)j(of)c(the)i (pap)q(er)g(is)f(as)g(follo)o(ws.)e(In)j(section)g(2)e(w)o(e)i(presen)o (t)h(the)e(simply)262 2062 y(t)o(yp)q(ed)h Fv(\025)p FC(-calculus)g(with)f(constan)o(ts)i(and)e(pairing)g(and)g(giv)o(e)h (some)e(examples)h(of)g(higher)262 2112 y(order)k(rewrite)g(systems.)g (Then)g(w)o(e)g(inductiv)o(ely)f(de\014ne)h(a)f(relation)h Fv(M)24 b Fu(\000)-7 b(!)19 b Fv(Q)p FC(,)f(with)262 2162 y(the)g(in)o(tended)h(meaning)d(that)i Fv(M)23 b FC(is)18 b(normalizable)e(with)i(long)f(normal)f(form)g Fv(Q)p FC(,)i(and)262 2212 y(pro)o(v)o(e)g(in)h(section)g(4)g(the)g (correctness)j(of)c(normalization)e(b)o(y)i(ev)n(aluation)g(b)o(y)g (sho)o(wing)262 2261 y(that)h Fv(M)26 b Fu(\000)-7 b(!)20 b Fv(Q)g FC(implies)d Fu(#)p FC(\([)-7 b([)p Fv(M)5 b FC(])-7 b(])825 2267 y Fs(")842 2261 y FC(\))21 b(=)g Fv(Q)p FC(.)e(Hence)i(the)f(mapping)d Fv(M)26 b Fu(7!)20 b(#)p FC(\([)-7 b([)p Fv(M)5 b FC(])-7 b(])1622 2267 y Fs(")1639 2261 y FC(\))20 b(is)262 2311 y(a)e(normalization)e (function.)h(In)i(order)g(to)g(de\014ne)g(the)h(seman)o(tics)e([)-7 b([)p Fv(M)5 b FC(])-7 b(])16 b(of)i(a)h(term)f Fv(M)262 2361 y FC(prop)q(erly)c(w)o(e)g(use)h(domain)c(theory)m(.)j(This)f(is)h (describ)q(ed)i(brie\015y)e(in)f(section)i(3.)324 2412 y(In)i(subsection)g(4.2)f(w)o(e)h(sho)o(w)g(ho)o(w)f(to)h(in)o(terpret) h(a)e(constan)o(t)h Fv(c)g FC(more)f(e\016cien)o(tly)g(if)262 2462 y(all)e(its)i(rules)h(are)f(of)f(some)g(sp)q(ecial)h(forms.)e(In)i (fact,)g(most)e(of)i(the)g(rewrite)h(rules)f(in)g(the)p eop %%Page: 3 3 3 2 bop 262 224 a Fw(Minlog)14 b FC(system)g(ha)o(v)o(e)g(one)h(of)e (these)j(forms.)c(Again,)h(normalization)f(b)o(y)i(ev)n(aluation)f(is) 262 274 y(sho)o(wn)g(to)h(remain)f(correct.)324 324 y(The)20 b(\014nal)g(section)h(5)f(con)o(tains)g(a)g(review)g(of)g(the)h(relev)n (an)o(t)f(literature)g([1,)6 b(5,)g(7,)h(8],)262 374 y(and)19 b(also)h(a)g(comparison)e(of)i(the)g(run-times)g(of)f(our)h (algorithm)e(with)i(those)g(coming)262 424 y(from)15 b(the)j(other)h(setups.)f(The)g(results)h(clearly)e(supp)q(ort)h(our)g (claim)d(that)j(the)g(presen)o(t)262 473 y(approac)o(h)c(\(with)h (function)f(spaces)i(as)f(higher)g(t)o(yp)q(e)g(domains\))e(leads)h(to) h(a)f(m)o(uc)o(h)g(faster)262 523 y(implem)o(en)o(tation.)262 586 y Fn(A)n(cknow)r(le)n(dgements)p FC(.)20 b(The)h(presen)o(t)h(w)o (ork)e(has)h(b)q(ene\014tted)h(considerably)f(from)d(ideas)262 635 y(of)c(F)m(elix)f(Joac)o(himski)g(and)i(Ralph)f(Matthes)h (concerning)h(strategies)g(for)e(normalization)262 685 y(pro)q(ofs,)f(including)f Fv(\021)q FC(-expansion)i(and)f(primitiv)o (e)e(recursion.)k(In)e(particular,)g(the)h(idea)g(to)262 735 y(emplo)o(y)h(the)j(inductiv)o(e)g(de\014nition)f(of)g(the)i (relation)e Fv(M)22 b Fu(\000)-6 b(!)17 b Fv(Q)g FC(is)h(essen)o (tially)f(due)i(to)262 785 y(them.)12 b(W)m(e)i(also)f(w)o(an)o(t)g(to) h(thank)g(Holger)g(Benl)g(for)f(illuminating)d(discussions.)262 918 y Fx(2)56 b(A)19 b(simply)d(t)n(yp)r(ed)i Fm(\025)p Fx(-calculus)g(with)h(constan)n(ts)262 1018 y Fl(2.1)47 b(T)o(yp)q(es)16 b(and)f(terms;)g(rewrite)f(rules)262 1102 y FC(W)m(e)e(start)i(from)d(a)i(giv)o(en)g(set)h(of)e Fn(gr)n(ound)j(typ)n(es)p FC(.)d Fn(T)m(yp)n(es)17 b FC(are)c(inductiv)o(ely)g(generated)h(from)262 1152 y(ground)f(t)o(yp)q (es)i Fv(\034)k FC(b)o(y)13 b Fv(\032)f Fu(!)f Fv(\033)16 b FC(and)d Fv(\032)d Fu(\002)f Fv(\033)q FC(.)14 b Fn(T)m(erms)i FC(are)615 1233 y Fv(x)639 1217 y Ft(\032)1063 1233 y FC(t)o(yp)q(ed)f(v)n(ariables,)615 1282 y Fv(c)633 1267 y Ft(\032)1063 1282 y FC(constan)o(ts,)615 1332 y(\()p Fv(\025x)679 1317 y Ft(\032)698 1332 y Fv(M)743 1317 y Ft(\033)765 1332 y FC(\))781 1317 y Ft(\032)p Fs(!)p Ft(\033)1063 1332 y FC(abstractions,)615 1382 y(\()p Fv(M)676 1367 y Ft(\032)p Fs(!)p Ft(\033)748 1382 y Fv(N)786 1367 y Ft(\032)805 1382 y FC(\))821 1367 y Ft(\033)1063 1382 y FC(applications,)615 1432 y Fu(h)p Fv(M)676 1412 y Ft(\032)671 1443 y Fr(0)695 1432 y Fv(;)7 b(M)759 1417 y Ft(\033)754 1442 y Fr(1)780 1432 y Fu(i)796 1417 y Ft(\032)p Fs(\002)p Ft(\033)1063 1432 y FC(pairing,)615 1482 y Fv(\031)639 1488 y Fr(0)657 1482 y FC(\()p Fv(M)718 1467 y Ft(\032)p Fs(\002)p Ft(\033)783 1482 y FC(\))799 1467 y Ft(\032)819 1482 y FC(,)13 b Fv(\031)868 1488 y Fr(1)886 1482 y FC(\()p Fv(M)947 1467 y Ft(\032)p Fs(\002)p Ft(\033)1013 1482 y FC(\))1029 1467 y Ft(\033)1063 1482 y FC(pro)r(jections.)262 1565 y(Ground)h(t)o(yp)q(es)h(will)e(alw)o(a)o (ys)g(b)q(e)i(denoted)h(b)o(y)e Fv(\034)5 b FC(.)14 b(W)m(e)g (sometimes)e(write)j Fv(M)5 b FC(0)14 b(for)g Fv(\031)1607 1571 y Fr(0)1625 1565 y FC(\()p Fv(M)5 b FC(\))262 1615 y(and)13 b Fv(M)5 b FC(1)13 b(for)g Fv(\031)508 1621 y Fr(1)526 1615 y FC(\()p Fv(M)5 b FC(\).)13 b(Tw)o(o)g(terms)g Fv(M)19 b FC(and)13 b Fv(N)18 b FC(are)c(called)f Fv(\013)p Fn(-e)n(qual)18 b FC({)13 b(written)h Fv(M)i FC(=)1629 1621 y Ft(\013)1665 1615 y Fv(N)262 1665 y FC({)h(if)g(they)i(are)f (equal)g(up)g(to)g(renaming)f(of)g(b)q(ound)h(v)n(ariables.)f Fv(\003)1333 1671 y Ft(\032)1370 1665 y FC(denotes)i(the)g(set)g(of)262 1715 y(all)12 b(terms)h(of)g(t)o(yp)q(e)i Fv(\032)p FC(.)e Fv(M)5 b Fq(N)19 b FC(denotes)c(\()p Fv(:)7 b(:)g(:)e FC(\()p Fv(M)g(N)1039 1721 y Fr(1)1058 1715 y FC(\))p Fv(N)1107 1721 y Fr(2)1133 1715 y Fv(:)i(:)g(:)e FC(\))p Fv(N)1237 1721 y Ft(n)1260 1715 y FC(,)14 b(where)g(some)f(of)g(the)h Fv(N)1660 1721 y Ft(i)1675 1715 y FC('s)262 1765 y(ma)o(y)g(b)q(e)j(0)e (or)h(1.)g(By)g Fp(FV)p FC(\()p Fv(M)5 b FC(\))16 b(w)o(e)h(denote)g (the)g(set)g(of)e(v)n(ariables)h(o)q(ccurring)g(free)h(in)f Fv(M)5 b FC(.)262 1814 y(By)20 b Fv(M)373 1820 y Ft(x)395 1814 y FC([)p Fv(N)5 b FC(])19 b(w)o(e)i(mean)e(substitution)i(of)f(ev) o(ery)i(free)f(o)q(ccurrence)j(of)c Fv(x)g FC(in)g Fv(M)26 b FC(b)o(y)20 b Fv(N)5 b FC(,)262 1864 y(renaming)15 b(b)q(ound)h(v)n(ariables)g(if)g(necessary)m(.)i(Similarly)13 b Fv(M)1207 1870 y Fk(x)1231 1864 y FC([)p Fq(N)5 b FC(])16 b(denotes)i(sim)o(ultaneous)262 1914 y(substitution.)d(Finally)g Fq(\032)g Fu(!)g Fv(\033)i FC(stands)g(for)e Fv(\032)1006 1920 y Fr(1)1041 1914 y Fu(!)f FC(\()p Fv(\032)1134 1920 y Fr(2)1169 1914 y Fu(!)g Fv(:)7 b(:)g(:)e FC(\()p Fv(\032)1317 1920 y Ft(n)1356 1914 y Fu(!)14 b Fv(\033)q FC(\))7 b Fv(:)g(:)g(:)f FC(\))16 b(and)g Fv(\025)p Fq(x)q Fv(r)262 1964 y FC(abbreviates)e Fv(\025x)530 1970 y Fr(1)556 1964 y Fv(:)7 b(:)g(:)e(\025x)659 1970 y Ft(n)681 1964 y Fv(r)q FC(.)324 2014 y(F)m(or)10 b(the)h(constan)o(ts)g Fv(c)661 1999 y Ft(\032)690 2014 y FC(w)o(e)g(assume)f(that)g(some)f (rewrite)j(rules)e(of)g(the)h(form)d Fv(c)p Fq(K)15 b Fu(7\000)-6 b(!)11 b Fv(N)262 2064 y FC(are)e(giv)o(en,)g(where)h Fp(FV)p FC(\()p Fv(N)5 b FC(\))12 b Fu(\022)g Fp(FV)p FC(\()p Fq(K)s FC(\))e(and)f Fv(c)p Fq(K)s FC(,)g Fv(N)14 b FC(ha)o(v)o(e)9 b(the)h(same)f(t)o(yp)q(e)h(\(not)f(necessarily)262 2113 y(a)14 b(ground)g(t)o(yp)q(e\).)g(Moreo)o(v)o(er,)h(for)f(an)o(y)g (t)o(w)o(o)g Fv(c)p FC(-rules)g Fv(c)p Fq(K)i Fu(7\000)-7 b(!)12 b Fv(N)19 b FC(and)c Fv(c)p Fq(K)1455 2095 y Fs(0)1479 2113 y Fu(7\000)-7 b(!)12 b Fv(N)1596 2098 y Fs(0)1622 2113 y FC(with)262 2163 y(equal)g(pro)r(jection)i(mark)o(ers)e(0)p Fv(;)7 b FC(1)12 b(w)o(e)i(require)g(that)f Fq(K)j FC(and)d Fq(K)1264 2145 y Fs(0)1289 2163 y FC(are)g(of)g(the)h(same)e(length.) 262 2213 y(F)m(or)g(example,)g(if)g Fv(c)h FC(is)h(of)e(t)o(yp)q(e)i (\()p Fv(\034)j Fu(!)11 b Fv(\034)16 b Fu(!)11 b Fv(\034)5 b FC(\))j Fu(\002)g FC(\()p Fv(\034)17 b Fu(!)11 b Fv(\034)5 b FC(\),)12 b(then)i(the)g(rules)g Fv(c)p FC(0)p Fv(x)1530 2219 y Fr(1)1548 2213 y Fv(x)1572 2219 y Fr(1)1602 2213 y Fu(7\000)-7 b(!)12 b Fv(a)262 2263 y FC(and)h Fv(c)p FC(1)p Fv(x)e Fu(7\000)-6 b(!)11 b Fv(b)i FC(are)i(admitted.)324 2313 y(Since)g(w)o(e)g(allo)o(w)o(ed)f(almost)f(arbitrary)h(rewrite)i (rules,)f(it)f(ma)o(y)f(happ)q(en)j(that)f(a)f(term)262 2363 y(can)f(b)q(e)h(rewritten)g(b)o(y)f(di\013eren)o(t)i(rules.)e(In)g (order)h(to)f(obtain)g(a)g(deterministic)g(pro)q(cedure)262 2412 y(w)o(e)g(assume)h(that)f(for)g(ev)o(ery)i(constan)o(t)f Fv(c)913 2397 y Fk(\032)p Fs(!)p Ft(\033)1002 2412 y FC(w)o(e)g(are)g(giv)o(en)f(a)g(function)g Fp(sel)1481 2418 y Ft(c)1511 2412 y FC(computing)262 2462 y(for)j(ev)o(ery)i(tuple) f(of)f(terms)h Fq(M)771 2444 y Fk(\032)809 2462 y FC(either)h(a)f(rule) g Fv(c)p Fq(K)i Fu(7\000)-6 b(!)16 b Fv(N)5 b FC(,)16 b(in)g(whic)o(h)h(case)h Fq(M)k FC(is)17 b(an)p eop %%Page: 4 4 4 3 bop 262 224 a FC(instance)15 b(of)f Fq(K)s FC(,)g(i.e.)g Fq(M)j FC(=)c Fq(K)761 230 y Fk(x)785 224 y FC([)p Fq(L)p FC(])o(,)h(or)h(else)g(the)g(message)g(\\)p Fp(no)p FC(-)o Fp(match)p FC(",)f(in)g(whic)o(h)g(case)262 274 y Fq(M)21 b FC(do)q(esn't)16 b(matc)o(h)f(an)o(y)g(rule,)h(i.e.)f(there)i(is)f (no)g(rule)g Fv(c)p Fq(K)i Fu(7\000)-6 b(!)14 b Fv(N)21 b FC(suc)o(h)c(that)f Fq(M)21 b FC(is)16 b(an)262 324 y(instance)e(of)f Fq(K)t FC(.)g(Clearly)g Fp(sel)728 330 y Ft(c)758 324 y FC(should)h(b)q(e)g(compatible)f(with)g Fv(\013)p FC(-equalit)o(y)m(.)324 374 y(F)m(or)i(readabilit)o(y)f(a)h (rule)h Fv(c)p Fq(K)h Fu(7\000)-6 b(!)13 b Fv(\025)p Fq(y)q Fv(N)21 b FC(with)15 b Fq(y)h FC(distinct)g(v)n(ariables)f(not)g (free)h(in)f Fq(K)262 423 y 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Fv(\025y)q(\025z)r FC(\()p Fv(z)r(x)p FC(\()p Fv(Rxy)q(z)r FC(\)\).)j(A)e(simpli\014ed)f(sc)o(heme)h(of)262 1011 y(the)e(same)f(form)f(giv)o(es)i(a)f(cases)j(construct:)782 1096 y Fp(if)10 b FC(0)p Fv(y)q(z)k Fu(7\000)-7 b(!)11 b Fv(y)q(;)665 1158 y Fp(if)e FC(\()p Fw(Suc)f Fv(x)p FC(\))p Fv(y)q(z)14 b Fu(7\000)-7 b(!)11 b Fv(z)r(:)262 1255 y FC(W)m(e)i(ma)o(y)f(also)h(add)h(a)g(rewrite)g(rule)h(\(due)f (to)g Fw(McCar)m(thy)h FC([11]\))669 1353 y Fp(if)10 b FC(\()p Fp(if)f Fv(xy)q(z)r FC(\))p Fv(uv)14 b Fu(7\000)-7 b(!)11 b Fp(if)f Fv(x)p FC(\()p Fp(if)f Fv(y)q(uv)q FC(\)\()p Fp(if)i Fv(z)r(uv)q FC(\))p Fv(:)262 1438 y FC(Moreo)o(v)o(er)h(w)o(e)h (can)f(write)h(do)o(wn)f(rules)h(according)f(to)g(the)h(usual)f (recursiv)o(e)i(de\014nitions)e(of)262 1488 y(addition)g(and)i(m)o (ultiplicatio)o(n,)d(and)j(then)g(also)f(the)i(rewrite)g(rule)613 1573 y Fp(mult)p FC(\()p Fp(add)8 b Fv(xy)q FC(\))p Fv(z)14 b Fu(7\000)-7 b(!)11 b Fp(add)q FC(\()p Fp(mult)c Fv(xz)r FC(\)\()p Fp(mult)f Fv(y)q(z)r FC(\))p Fv(:)262 1658 y FC(Sim)o(ultaneous)12 b(recursion)j(ma)o(y)d(b)q(e)i(treated)h(as)f (w)o(ell,)f(e.g.)581 1743 y Fp(o)q(dd)8 b FC(0)j Fu(7\000)-6 b(!)11 b Fw(Suc)c FC(0)324 b Fp(even)9 b FC(0)i Fu(7\000)-7 b(!)11 b FC(0)p Fv(;)471 1805 y Fp(o)q(dd)p FC(\()p Fw(Suc)d Fv(x)p FC(\))j Fu(7\000)-6 b(!)11 b Fp(even)d Fv(x)209 b Fp(even)q FC(\()p Fw(Suc)8 b Fv(x)p FC(\))j Fu(7\000)-7 b(!)11 b Fp(o)q(dd)d Fv(x:)262 1890 y FC(\(b\))13 b(W)m(e)h(can)g(also) f(deal)g(with)g(in\014nitely)g(branc)o(hing)g(trees)j(lik)o(e)d(the)h Fw(Br)o(ouwer)g FC(ordinals)262 1940 y(of)9 b(t)o(yp)q(e)i Fu(O)q FC(.)f(W)m(e)f(ha)o(v)o(e)h(constructors)j(0)864 1925 y Fs(O)903 1940 y FC(and)d Fw(Sup)1054 1925 y Fr(\()p Ft(\023)p Fs(!O)q Fr(\))p Fs(!)p Ft(\023)1200 1940 y FC(,)f(and)h(for)g(recursion)h(constan)o(ts)262 1995 y Fw(Rec)342 1974 y Fs(O)q(!)p Ft(\032)p Fs(!)p Fr(\(\()p Ft(\023)p Fs(!O)q Fr(\))p Fs(!)p Fr(\()p Ft(\023)p Fs(!)p Ft(\032)p Fr(\))p Fs(!)p Ft(\032)p Fr(\))p Fs(!)p Ft(\032)342 2000 y(\032)802 1995 y FC(.)i(The)i(rewrite)f(rules)h(for)f Fw(Rec)f FC(are:)749 2080 y Fw(Rec)p FC(0)p Fv(y)q(z)h Fu(7\000)-6 b(!)11 b Fv(y)q(;)633 2143 y Fw(Rec)p FC(\()p Fw(Sup)d Fv(x)p FC(\))p Fv(y)q(z)14 b Fu(7\000)-6 b(!)11 b Fv(z)r(x)p FC(\()p Fv(\025u)c Fw(Rec)p FC(\()p Fv(xu)p FC(\))p Fv(y)q(z)r FC(\))p Fv(:)262 2228 y FC(\(c\))12 b(It)f(is)h(w)o(ell)e(kno)o(wn)i(that)f(b)o(y)g(the)h Fw(Curr)m(y)q FC(-)p Fw(Ho)o(w)l(ard)h FC(corresp)q(ondence)i(natural)c (deduc-)262 2278 y(tion)h(pro)q(ofs)h(can)g(b)q(e)h(written)f(as)g Fv(\025)p FC(-terms)g(with)g(form)o(ulas)d(as)k(t)o(yp)q(es.)f(T)m(o)f (use)i(normaliza-)262 2327 y(tion)g(b)o(y)g(ev)n(aluation)g(for)g (normalizing)e(pro)q(ofs)j(w)o(e)g(ma)o(y)d(also)i(in)o(tro)q(duce)i(a) e(ground)h(t)o(yp)q(e)262 2377 y Fp(ex)f FC(with)f(constructors)j(and)e (destructors)523 2462 y(\()p Fu(9)562 2445 y Fr(+)562 2472 y Ft(\032)579 2476 y Fj(0)595 2472 y Ft(;\032)622 2476 y Fj(1)641 2462 y FC(\))657 2445 y Ft(\032)674 2449 y Fj(0)690 2445 y 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Fv(Q)896 2234 y FC(if)i Fp(sel)978 2240 y Ft(c)995 2234 y FC(\()p Fq(M)1063 2217 y Fh(0)1077 2234 y FC(\))f(=)f Fv(c)p Fq(K)k Fu(7\000)-7 b(!)11 b Fv(N)19 b FC(and)14 b Fq(M)1484 2217 y Fh(0)1502 2234 y FC(=)e Fq(K)1589 2240 y Fk(x)1613 2234 y FC([)p Fq(L)o FC(])324 2345 y Fw(P)l(assApp)p FC(.)542 2425 y Fq(M)17 b Fu(\000)-7 b(!)11 b Fq(M)736 2407 y Fs(0)789 2425 y Fq(P)19 b Fu(\000)-7 b(!)11 b Fq(P)953 2407 y Fs(0)p 542 2444 423 2 v 590 2483 a Fv(c)p Fq(M)5 b(P)18 b Fu(\000)-7 b(!)11 b Fv(c)p Fq(M)857 2465 y Fs(0)868 2483 y Fq(P)905 2465 y Fs(0)1011 2453 y FC(if)i Fp(sel)1093 2459 y Ft(c)1110 2453 y FC(\()p Fq(M)1178 2436 y Fh(0)1192 2453 y FC(\))e(=)h Fp(no)p FC(-)p Fp(match)p eop %%Page: 6 6 6 5 bop 324 224 a FC(F)m(or)15 b(readabilit)o(y)f(w)o(e)i(will)e(write) h Fw(Red)h FC(in)f(the)h(follo)o(wing)d(form)g(and)j(alw)o(a)o(ys)e (assume)262 274 y(that)f Fv(c)p Fq(K)i Fu(7\000)-6 b(!)11 b Fv(N)18 b FC(is)c(the)h(selected)g(rule.)539 365 y Fq(M)i Fu(\000)-7 b(!)11 b Fq(K)725 371 y Fk(x)748 365 y FC([)p Fq(L)p FC(])82 b Fv(N)918 371 y Fk(x)942 365 y FC([)p Fq(L)p FC(])p Fq(P)18 b Fu(\000)-7 b(!)11 b Fv(Q)p 539 384 617 2 v 733 422 a(c)p Fq(M)5 b(P)18 b Fu(\000)-6 b(!)11 b Fv(Q)1203 393 y FC(if)i Fv(c)p Fq(K)h Fu(7\000)-6 b(!)11 b Fv(N)262 495 y FC(The)k(set)h Fw(Lnf)g FC(of)f(terms)f(in)h(long)f(normal)f(form)g(is)i(de\014ned)h(as)g (follo)o(ws:)d Fv(\025xM)5 b FC(,)14 b Fu(h)p Fv(M)r(;)7 b(N)e Fu(i)p FC(,)262 544 y Fv(x)p Fq(M)23 b FC(and)18 b Fv(c)p Fq(M)5 b(N)23 b FC(are)c(in)f Fw(Lnf)h FC(if)f Fv(M)r(;)7 b(N)r(;)g Fq(M)e Fv(;)i Fq(N)23 b FC(are)c(and)f Fp(sel)1288 550 y Ft(c)1305 544 y FC(\()p Fv(c)p Fq(M)5 b FC(\))19 b(=)h Fp(no)p FC(-)p Fp(match)p FC(.)d(It)262 594 y(is)e(ob)o(vious)g(that)h Fv(M)k Fu(\000)-7 b(!)15 b Fv(Q)g FC(implies)f(that)i Fv(Q)f Fu(2)f Fw(Lnf)q FC(.)i(F)m (urthermore)f(it)h(can)g(b)q(e)g(sho)o(wn)262 644 y(that)h(in)h(this)g (case)g Fv(M)23 b FC(normalizes)16 b(to)i Fv(Q)g FC(in)f(the)h(usual)g (sense)h(w.r.t.)e Fv(\014)r FC(-reduction,)i Fv(\021)q FC(-)262 694 y(expansion)c(and)h(the)g(rewrite)h(rules)f(for)f(the)i (constan)o(ts.)f(Con)o(v)o(ersely)m(,)f(if)g Fv(M)20 b FC(is)c(strongly)262 744 y(normalizable)e(w.r.t.)h(these)j (reductions)g(\(i.e.)d(ev)o(ery)j(reduction)f(sequence)i(terminates\)) 262 793 y(then)13 b Fv(M)k Fu(\000)-7 b(!)11 b Fv(Q)i FC(for)g(some)f Fv(Q)p FC(.)h(W)m(e)f(omit)g(the)h(pro)q(ofs)g(of)g (these)i(facts)e(since)h(they)g(will)e(not)262 843 y(b)q(e)i(needed)h (in)f(the)g(sequel.)g(Although)g Fv(M)i Fu(\000)-6 b(!)11 b Fv(Q)i FC(implies)f(that)i Fv(Q)g FC(is)f(a)h(normal)e(form)g(of)262 893 y Fv(M)5 b FC(,)11 b(the)i(con)o(v)o(erse)h(is)e(not)h(true)g(in)f (general.)g(T)m(o)g(see)h(this,)f(consider)h(the)g(non-terminating)262 943 y(rewrite)i(rules)g Fp(mult)p Fv(x)607 928 y Ft(\023)621 943 y FC(0)d Fu(7\000)-6 b(!)12 b FC(0)i(and)h Fu(?)884 928 y Ft(\023)910 943 y Fu(7\000)-7 b(!)13 b(?)p FC(.)g(Then)i(0)f(is)h (a)f(normal)f(form)g(of)h Fp(mult)o Fu(?)p FC(0,)262 993 y(but)g Fp(mult)o Fu(?)p FC(0)d Fu(\000)-7 b(!)12 b FC(0)h(do)q(es)i(not)f(hold.)f(Moreo)o(v)o(er,)h(the)h(relation)f Fv(M)i Fu(\000)-6 b(!)11 b Fv(Q)j FC(clearly)g(is)g(not)262 1043 y(closed)g(under)h(substitution.)e(Ho)o(w)o(ev)o(er,)h(it)g(is)f (closed)i(under)g(substitution)f(of)f(v)n(ariables.)262 1137 y Fl(Lemma)i(1.)21 b Fn(If)c Fv(M)k Fu(\000)-7 b(!)16 b Fv(Q)p Fn(,)h(then)h Fv(M)877 1143 y Ft(x)898 1137 y FC([)p Fv(z)r FC(])e Fu(\000)-7 b(!)16 b Fv(Q)1075 1143 y Ft(x)1096 1137 y FC([)p Fv(z)r FC(])g Fn(with)h(a)h(derivation)f (of)h(the)f(same)262 1187 y(height.)262 1280 y(Pr)n(o)n(of.)j FC(W)m(e)14 b(use)i(induction)f(on)f(the)i(heigh)o(t)f(of)f(the)h (deriv)n(ation)g(of)f Fv(M)k Fu(\000)-7 b(!)13 b Fv(Q)p FC(,)h(and)h(only)262 1330 y(treat)f(the)h(rule)f Fw(Et)m(a)734 1367 y Fv(M)5 b(y)13 b Fu(\000)-6 b(!)11 b Fv(Q)p 722 1386 214 2 v 722 1424 a(M)16 b Fu(\000)-6 b(!)11 b Fv(\025y)q(Q)982 1395 y FC(for)j Fv(y)k(=)-26 b Fu(2)11 b Fp(FV)p FC(\()p Fv(M)5 b FC(\))262 1494 y(with)12 b Fv(M)17 b FC(of)12 b(t)o(yp)q(e)h Fv(\032)f Fu(!)f Fv(\033)q FC(.)h(In)g(case)i Fv(x)d FC(=)h Fv(y)i FC(there)g(is)e(nothing)g(to)g(sho)o(w,)g(since)h (then)h Fv(x)e FC(do)q(es)262 1544 y(not)h(o)q(ccur)i(free)g(in)e(the)i (conclusion)f Fv(M)i Fu(\000)-7 b(!)11 b Fv(\025y)q(Q)p FC(.)k(So)e(assume)h Fv(x)d Fu(6)p FC(=)h Fv(y)q FC(.)324 1596 y Fn(Sub)n(c)n(ase)19 b Fv(y)c Fu(6)p FC(=)f Fv(z)r FC(.)h(W)m(e)g(m)o(ust)f(sho)o(w)h Fv(M)929 1602 y Ft(x)950 1596 y FC([)p Fv(z)r FC(])e Fu(\000)-6 b(!)13 b Fv(\025y)q(Q)1167 1602 y Ft(x)1189 1596 y FC([)p Fv(z)r FC(])o(.)i(By)g(induction)g(h)o (yp)q(othesis)262 1645 y Fv(M)302 1651 y Ft(x)323 1645 y FC([)p Fv(z)r FC(])o Fv(y)21 b Fu(\000)-6 b(!)19 b Fv(Q)528 1651 y Ft(x)548 1645 y FC([)p Fv(z)r FC(],)f(since)h Fv(x)h Fu(6)p FC(=)f Fv(y)q FC(.)g(Therefore)h(b)o(y)f Fw(Et)m(a)g Fv(M)1269 1651 y Ft(x)1290 1645 y FC([)p Fv(z)r FC(])g Fu(\000)-7 b(!)19 b Fv(\025y)q(Q)1518 1651 y Ft(x)1540 1645 y FC([)p Fv(z)r FC(])o(,)g(since)262 1695 y Fv(y)f(=)-26 b Fu(2)11 b Fp(FV)p FC(\()p Fv(M)442 1701 y Ft(x)463 1695 y FC([)p Fv(z)r FC(])o(\))j(\(b)q(ecause)i(of)d Fv(y)h Fu(6)p FC(=)e Fv(z)r FC(\).)324 1746 y Fn(Sub)n(c)n(ase)21 b Fv(y)d FC(=)f Fv(z)r FC(.)f(W)m(e)g(m)o(ust)g(sho)o(w)h Fv(M)943 1752 y Ft(x)964 1746 y FC([)p Fv(y)q FC(])f Fu(\000)-6 b(!)16 b FC(\()p Fv(\025y)q(Q)p FC(\))1219 1752 y Ft(x)1241 1746 y FC([)p Fv(y)q FC(])g(=)h Fv(\025uQ)1432 1752 y Ft(x;y)1480 1746 y FC([)p Fv(y)q(;)7 b(u)p FC(])16 b(with)h(a)262 1796 y(new)i(v)n(ariable)f Fv(u)p FC(.)g(By)h(induction) g(h)o(yp)q(othesis)g Fv(M)5 b(u)20 b Fu(\000)-7 b(!)19 b Fv(Q)1244 1802 y Ft(y)1264 1796 y FC([)p Fv(u)p FC(])f(with)g(a)h (deriv)n(ation)f(of)262 1846 y(the)g(same)g(heigh)o(t)g(as)g(that)g (for)g Fv(M)5 b(y)20 b Fu(\000)-6 b(!)18 b Fv(Q)p FC(,)f(hence)j(again) d(b)o(y)h(induction)g(h)o(yp)q(othesis)262 1896 y Fv(M)302 1902 y Ft(x)323 1896 y FC([)p Fv(y)q FC(])p Fv(u)11 b Fu(\000)-7 b(!)11 b Fv(Q)514 1902 y Ft(x;y)563 1896 y FC([)p Fv(y)q(;)c(u)p FC(])o(.)13 b(Therefore)i(b)o(y)g Fw(Et)m(a)f Fv(M)1054 1902 y Ft(x)1075 1896 y FC([)p Fv(y)q FC(])d Fu(\000)-7 b(!)12 b Fv(\025uQ)1291 1902 y Ft(x;y)1339 1896 y FC([)p Fv(y)q(;)7 b(u)p FC(])o(.)262 2028 y Fl(2.4)47 b(T)l(erm)16 b(families)262 2119 y FC(Since)e (normalization)d(b)o(y)j(ev)n(aluation)f(needs)i(to)f(create)i(b)q (ound)e(v)n(ariables)f(when)i(\\reify-)262 2169 y(ing")10 b(abstract)i(ob)r(jects)h(of)e(higher)g(t)o(yp)q(e,)h(it)f(is)g(useful) g(to)h(follo)o(w)e Fw(de)j(Br)o(uijn)p FC('s)e([9])f(st)o(yle)i(of)262 2218 y(represen)o(ting)h(b)q(ound)e(v)n(ariables)g(in)g(terms.)f(This)i (is)f(done)h(here)g({)f(as)h(in)f([4])f({)h(b)o(y)g(means)g(of)262 2268 y Fn(term)j(families)p FC(.)f(A)h(term)g(family)d(is)j(a)g (parametrized)g(v)o(ersion)g(of)g(a)g(giv)o(en)g(term)f Fv(M)5 b FC(.)14 b(The)262 2318 y(idea)d(is)h(that)g(the)g(term)g (family)c(of)k Fv(M)k FC(at)c(index)g Fv(k)h FC(repro)q(duces)h Fv(M)j FC(with)11 b(b)q(ound)h(v)n(ariables)262 2368 y(renamed)h(starting)h(at)g Fv(k)q FC(.)f(F)m(or)g(example,)f(for)706 2462 y Fv(M)k FC(:=)c Fv(\025u\025v)q(:c)p FC(\()p Fv(\025x:v)q(x)p FC(\)\()p Fv(\025y)q(\025z)r(:z)r(u)p FC(\))p eop %%Page: 7 7 7 6 bop 262 224 a FC(the)14 b(asso)q(ciated)h(term)e(family)e Fv(M)802 209 y Fs(1)851 224 y FC(at)i(index)h(3)g(yields)567 310 y Fv(M)612 292 y Fs(1)647 310 y FC(\(3\))d(:=)h Fv(\025x)815 316 y Fr(3)833 310 y Fv(\025x)881 316 y Fr(4)900 310 y Fv(:c)p FC(\()p Fv(\025x)994 316 y Fr(5)1012 310 y Fv(:x)1048 316 y Fr(4)1066 310 y Fv(x)1090 316 y Fr(5)1108 310 y FC(\)\()p Fv(\025x)1188 316 y Fr(5)1207 310 y Fv(\025x)1255 316 y Fr(6)1273 310 y Fv(:x)1309 316 y Fr(6)1327 310 y Fv(x)1351 316 y Fr(3)1370 310 y FC(\))p Fv(:)262 395 y FC(W)m(e)h(denote)i(terms)e(b)o(y)h Fv(M)r(;)7 b(N)r(;)g(K)q(;)g(:)g (:)g(:)e FC(,)13 b(and)h(term)f(families)e(b)o(y)j Fv(r)o(;)7 b(s;)g(t;)g(:)g(:)g(:)t FC(.)324 445 y(T)m(o)13 b(ev)o(ery)i(term)e Fv(M)639 430 y Ft(\032)672 445 y FC(w)o(e)h(assign)g(a)f(term)g(family) e Fv(M)1161 430 y Fs(1)1201 445 y FC(:)i Fg(N)d Fu(!)h Fv(\003)1350 451 y Ft(\032)1383 445 y FC(b)o(y)387 530 y Fv(x)411 513 y Fs(1)446 530 y FC(\()p Fv(k)q FC(\))h(:=)f Fv(x;)288 592 y FC(\()p Fv(\025y)q(M)5 b FC(\))410 575 y Fs(1)446 592 y FC(\()p Fv(k)q FC(\))12 b(:=)f Fv(\025x)616 598 y Ft(k)636 592 y FC(\()p Fv(M)692 598 y Ft(y)713 592 y FC([)p Fv(x)749 598 y Ft(k)768 592 y FC(])780 572 y Fs(1)815 592 y FC(\()p Fv(k)f FC(+)g(1\)\))p Fv(;)41 b Fu(h)p Fv(M)1067 598 y Fr(0)1086 592 y Fv(;)7 b(M)1145 598 y Fr(1)1163 592 y Fu(i)1179 575 y Fs(1)1215 592 y FC(\()p Fv(k)q FC(\))k(:=)h Fu(h)p Fv(M)1398 575 y Fs(1)1393 603 y Fr(0)1433 592 y FC(\()p Fv(k)q FC(\))p Fv(;)7 b(M)1552 575 y Fs(1)1547 603 y Fr(1)1587 592 y FC(\()p Fv(k)q FC(\))p Fu(i)p Fv(;)296 655 y FC(\()p Fv(M)e(N)g FC(\))411 638 y Fs(1)446 655 y FC(\()p Fv(k)q FC(\))12 b(:=)f Fv(M)613 638 y Fs(1)648 655 y FC(\()p Fv(k)q FC(\))p Fv(N)741 638 y Fs(1)776 655 y FC(\()p Fv(k)q FC(\))p Fv(;)222 b(\031)1089 661 y Ft(i)1102 655 y FC(\()p Fv(M)5 b FC(\))1179 638 y Fs(1)1215 655 y FC(\()p Fv(k)q FC(\))11 b(:=)h Fv(\031)1361 661 y Ft(i)1374 655 y FC(\()p Fv(M)1435 638 y Fs(1)1470 655 y FC(\()p Fv(k)q FC(\)\))p Fv(:)262 740 y FC(Application)18 b(of)h(a)g(term)g(family)d Fv(r)6 b FC(:)15 b Fg(N)k Fu(!)h Fv(\003)1016 746 y Ft(\032)p Fs(!)p Ft(\033)1108 740 y FC(to)g(a)f(term)g(family)d Fv(s)5 b FC(:)16 b Fg(N)j Fu(!)h Fv(\003)1636 746 y Ft(\032)1675 740 y FC(is)262 790 y(the)d(family)c Fv(r)q(s)5 b FC(:)15 b Fg(N)f Fu(!)i Fv(\003)669 796 y Ft(\033)707 790 y FC(de\014ned)i(b)o (y)f(\()p Fv(r)q(s)p FC(\)\()p Fv(k)q FC(\))g(:=)f Fv(r)q FC(\()p Fv(k)q FC(\))p Fv(s)p FC(\()p Fv(k)q FC(\),)h(and)f(similarly)e (for)i(pair-)262 840 y(ing)e Fu(h)p Fv(r)367 846 y Fr(0)385 840 y Fv(;)7 b(r)423 846 y Fr(1)441 840 y Fu(i)p FC(\()p Fv(k)q FC(\))14 b(:=)f Fu(h)p Fv(r)618 846 y Fr(0)636 840 y FC(\()p Fv(k)q FC(\))p Fv(;)7 b(r)729 846 y Fr(1)747 840 y FC(\()p Fv(k)q FC(\))p Fu(i)15 b FC(and)g(pro)r(jections)h Fv(\031)1154 846 y Ft(i)1167 840 y FC(\()p Fv(r)q FC(\)\()p Fv(k)q FC(\))e(:=)f Fv(\031)1369 846 y Ft(i)1382 840 y FC(\()p Fv(r)q FC(\()p Fv(k)q FC(\)\).)i(Hence)h(e.g.)262 890 y(\()p Fv(M)5 b(N)g FC(\))377 875 y Fs(1)423 890 y FC(=)12 b Fv(M)512 875 y Fs(1)547 890 y Fv(N)585 875 y Fs(1)620 890 y FC(.)324 939 y(Note)17 b(that)f(in)g(con)o(trast)h(to) f(our)g(con)o(v)o(en)o(tion)g(to)h(consider)g(terms)f(up)g(to)g(b)q (ound)h(re-)262 989 y(naming,)f(the)j(de\014nition)g(of)f(\()p Fv(\025y)q(M)5 b FC(\))866 974 y Fs(1)921 989 y FC(refers)21 b(to)e(a)f(particular)h(c)o(hoice)g(of)g(the)g(b)q(ound)262 1039 y(v)n(ariable)11 b Fv(y)q FC(.)j(P)o(art)f(a)g(of)f(the)i(follo)o (wing)c(lemma)g(sho)o(ws)j(that)g(nev)o(ertheless)j(this)d (de\014nition)262 1089 y(is)k(compatible)f(with)i(our)f(con)o(v)o(en)o (tion.)g(In)h(the)g(rest)h(of)e(this)h(subsection)h(w)o(e)f(drop)g(the) 262 1139 y(con)o(v)o(en)o(tion)c(to)h(iden)o(tify)f Fv(\013)p FC(-equal)g(terms.)g(Hence)j Fv(M)h FC(=)13 b Fv(N)20 b FC(means)14 b(that)h Fv(M)20 b FC(and)14 b Fv(N)20 b FC(are)262 1188 y(literally)15 b(iden)o(tical)h(as)h(opp)q(osed)g(to) g Fv(M)k FC(=)952 1194 y Ft(\013)993 1188 y Fv(N)5 b FC(,)16 b(whic)o(h)h(means)f(equalit)o(y)g(up)h(to)f(b)q(ound)262 1238 y(renaming.)324 1288 y(W)m(e)e(let)h Fv(k)f(>)f Fp(FV)p FC(\()p Fv(M)5 b FC(\))15 b(mean)e(that)i Fv(k)h FC(is)e(greater)i(than)f(all)e Fv(i)j FC(suc)o(h)f(that)g Fv(x)1501 1268 y Ft(\032)1501 1300 y(i)1533 1288 y Fu(2)d Fp(FV)p FC(\()p Fv(M)5 b FC(\))262 1338 y(for)13 b(some)g(t)o(yp)q(e)h Fv(\032)p FC(.)262 1419 y Fl(Lemma)h(2.)37 b Fn(a.)20 b(If)15 b Fv(M)h FC(=)691 1425 y Ft(\013)727 1419 y Fv(N)5 b Fn(,)14 b(then)h Fv(M)929 1404 y Fs(1)976 1419 y FC(=)d Fv(N)1058 1404 y Fs(1)1093 1419 y Fn(.)280 1468 y(b.)20 b(If)15 b Fv(k)d(>)g Fp(FV)p FC(\()p Fv(M)5 b FC(\))p Fn(,)14 b(then)i Fv(M)748 1453 y Fs(1)783 1468 y FC(\()p Fv(k)q FC(\))11 b(=)881 1474 y Ft(\013)917 1468 y Fv(M)5 b Fn(.)262 1549 y(Pr)n(o)n(of.)20 b FC(a.)e(Induction)i(on)f(the)h (heigh)o(t)f(of)f Fv(M)5 b FC(.)19 b(Only)g(the)h(case)g(where)h Fv(M)j FC(and)19 b Fv(N)24 b FC(are)262 1599 y(abstractions)c(is)g (critical.)g(So)g(assume)g Fv(\025y)965 1584 y Ft(\032)985 1599 y Fv(M)27 b FC(=)1084 1605 y Ft(\013)1130 1599 y Fv(\025z)1175 1584 y Ft(\032)1194 1599 y Fv(N)5 b FC(.)20 b(Then)h Fv(M)1419 1605 y Ft(y)1439 1599 y FC([)p Fv(P)6 b FC(])20 b(=)1548 1605 y Ft(\013)1595 1599 y Fv(N)1628 1605 y Ft(z)1647 1599 y FC([)p Fv(P)6 b FC(])262 1649 y(for)17 b(all)f(terms)i Fv(P)543 1634 y Ft(\032)562 1649 y FC(.)f(In)h(particular)f Fv(M)882 1655 y Ft(y)902 1649 y FC([)p Fv(x)938 1655 y Ft(k)958 1649 y FC(])h(=)1020 1655 y Ft(\013)1062 1649 y Fv(N)1095 1655 y Ft(z)1114 1649 y FC([)p Fv(x)1150 1655 y Ft(k)1170 1649 y FC(])f(for)g(arbitrary) h Fv(k)h Fu(2)f Fg(N)p FC(.)d(Hence)262 1699 y Fv(M)302 1705 y Ft(y)322 1699 y FC([)p Fv(x)358 1705 y Ft(k)377 1699 y FC(])389 1678 y Fs(1)424 1699 y FC(\()p Fv(k)10 b FC(+)g(1\))h(=)h Fv(N)639 1705 y Ft(z)659 1699 y FC([)p Fv(x)695 1705 y Ft(k)714 1699 y FC(])726 1678 y Fs(1)761 1699 y FC(\()p Fv(k)e FC(+)g(1\),)j(b)o(y)h(induction)f(h)o(yp)q (othesis.)h(Therefore)297 1784 y(\()p Fv(\025y)q(M)5 b FC(\))419 1767 y Fs(1)456 1784 y FC(\()p Fv(k)q FC(\))11 b(=)h Fv(\025x)614 1790 y Ft(k)635 1784 y FC(\()p Fv(M)691 1790 y Ft(y)711 1784 y FC([)p Fv(x)747 1790 y Ft(k)767 1784 y FC(])778 1763 y Fs(1)813 1784 y FC(\()p Fv(k)f FC(+)e(1\)\))j(=)g Fv(\025x)1060 1790 y Ft(k)1080 1784 y FC(\()p Fv(N)1129 1790 y Ft(z)1149 1784 y FC([)p Fv(x)1185 1790 y Ft(k)1204 1784 y FC(])1216 1763 y Fs(1)1251 1784 y FC(\()p Fv(k)e FC(+)g(1\)\))h(=)h(\()p Fv(\025z)r(N)5 b FC(\))1564 1767 y Fs(1)1600 1784 y FC(\()p Fv(k)q FC(\))p Fv(:)262 1869 y FC(b.)13 b(Induction)h(on)f(the)i(heigh)o(t)e(of)h Fv(M)5 b FC(.)13 b(W)m(e)g(only)g(consider)i(the)f(case)h Fv(\025y)q(M)5 b FC(.)14 b(The)g(assump-)262 1919 y(tion)i Fv(k)j(>)f Fp(FV)p FC(\()p Fv(\025y)q(M)5 b FC(\))18 b(implies)e Fv(x)802 1925 y Ft(k)844 1919 y Fv(=)-25 b Fu(2)17 b Fp(FV)p FC(\()p Fv(\025y)q(M)5 b FC(\))19 b(and)e(hence)i Fv(\025y)q(M)k FC(=)1421 1925 y Ft(\013)1463 1919 y Fv(\025x)1511 1925 y Ft(k)1531 1919 y FC(\()p Fv(M)1587 1925 y Ft(y)1608 1919 y FC([)p Fv(x)1644 1925 y Ft(k)1663 1919 y FC(]\).)262 1969 y(F)m(urthermore)17 b Fv(k)c FC(+)e(1)18 b Fv(>)g Fp(FV)p FC(\()p Fv(M)781 1975 y Ft(y)801 1969 y FC([)p Fv(x)837 1975 y Ft(k)857 1969 y FC(])o(\),)f(and)h(hence)h Fv(M)1157 1975 y Ft(y)1177 1969 y FC([)p Fv(x)1213 1975 y Ft(k)1233 1969 y FC(])1244 1948 y Fs(1)1279 1969 y FC(\()p Fv(k)13 b FC(+)f(1\))18 b(=)1461 1975 y Ft(\013)1502 1969 y Fv(M)1542 1975 y Ft(y)1562 1969 y FC([)p Fv(x)1598 1975 y Ft(k)1618 1969 y FC(],)f(b)o(y)262 2019 y(induction)c(h)o(yp)q(othesis.)h(Therefore) 410 2104 y(\()p Fv(\025y)q(M)5 b FC(\))532 2087 y Fs(1)568 2104 y FC(\()p Fv(k)q FC(\))12 b(=)f Fv(\025x)726 2110 y Ft(k)747 2104 y FC(\()p Fv(M)803 2110 y Ft(y)823 2104 y FC([)p Fv(x)859 2110 y Ft(k)879 2104 y FC(])890 2083 y Fs(1)925 2104 y FC(\()p Fv(k)g FC(+)e(1\)\))j(=)1112 2110 y Ft(\013)1147 2104 y Fv(\025x)1195 2110 y Ft(k)1216 2104 y FC(\()p Fv(M)1272 2110 y Ft(y)1292 2104 y FC([)p Fv(x)1328 2110 y Ft(k)1348 2104 y FC(])o(\))g(=)1419 2110 y Ft(\013)1454 2104 y Fv(\025y)q(M)r(:)122 b Fu(u)-28 b(t)324 2190 y FC(Let)15 b Fp(ext)q FC(\()p Fv(r)q FC(\))f(:=)f Fv(r)q FC(\()p Fv(k)q FC(\),)i(where)h Fv(k)g FC(is)f(the)h(least)f(n)o (um)o(b)q(er)g(greater)h(than)f(all)f Fv(i)h FC(suc)o(h)h(that)262 2239 y(some)d(v)n(ariable)f(of)i(the)g(form)e Fv(x)762 2220 y Ft(\032)762 2251 y(i)795 2239 y FC(o)q(ccurs)j(\(free)g(or)f(b)q (ound\))g(in)g Fv(r)q FC(\(0\).)262 2321 y Fl(Lemma)h(3.)21 b Fp(ext)541 2287 y Fo(\000)560 2321 y Fv(M)605 2306 y Fs(1)640 2287 y Fo(\001)671 2321 y FC(=)703 2327 y Ft(\013)738 2321 y Fv(M)5 b Fn(.)262 2402 y(Pr)n(o)n(of.)20 b FC(Use)c(part)f(b)g(of)g(the)h(lemma)c(ab)q(o)o(v)o(e,)i(and)i(the)f (fact)h(that)f Fp(ext)q FC(\()p Fv(M)1433 2387 y Fs(1)1468 2402 y FC(\))f(=)1530 2408 y Ft(\013)1567 2402 y Fv(M)1612 2387 y Fs(1)1647 2402 y FC(\()p Fv(k)q FC(\))262 2452 y(where)h Fv(k)d(>)g Fp(FV)p FC(\()p Fv(M)5 b FC(\).)p eop %%Page: 8 8 8 7 bop 262 224 a Fx(3)56 b(Domain)17 b(theoretic)g(seman)n(tics)g(of)i (simply)d(t)n(yp)r(ed)i Fm(\025)p Fx(-calculi)262 330 y FC(In)c(this)h(section)g(w)o(e)g(shall)f(discuss)i(the)f(domain)d (theoretic)k(seman)o(tics)e(of)g(simply)f(t)o(yp)q(ed)262 379 y(lam)o(b)q(da)g(calculi)j(in)f(general.)h(Although)g(the)h (constructions)g(b)q(elo)o(w)f(are)g(standard)h(\(see)262 429 y(e.g.)c(the)h(b)q(o)q(oks)h(of)h Fw(Lambek/Scott)f FC([10])e(or)h Fw(Cr)o(ole)g FC([6]\),)f(w)o(e)h(discuss)h(them)f(in)f (some)262 479 y(detail)d(in)g(order)h(to)g(mak)o(e)e(the)i(pap)q(er)h (accessible)g(also)e(for)g(readers)i(not)f(famili)o(ar)d(with)i(this) 262 529 y(sub)r(ject.)i(Most)g(constructions)h(mak)o(e)d(sense)j(in)e (an)g(arbitrary)h(cartesian)g(closed)g(category)262 579 y(\(ccc\).)j(Ho)o(w)o(ev)o(er)g(w)o(e)g(will)f(con\014ne)h(ourselv)o (es)h(to)e(the)i(domain)c(seman)o(tics)i(and)h(will)e(only)262 629 y(o)q(ccasionally)f(commen)o(t)g(on)i(the)g(categorical)g(asp)q (ects.)324 680 y(It)c(is)h(w)o(ell-kno)o(wn)e(that)h Fw(Scott)p FC(-domains)e(with)i(con)o(tin)o(uous)g(functions)h(form)d (a)j(carte-)262 729 y(sian)h(closed)i(category)g Fw(Dom)p FC(.)f(The)g(pro)q(duct)h Fv(D)9 b Fu(\002)f Fv(E)15 b FC(is)d(the)i(set-theoretic)h(pro)q(duct)f(with)262 779 y(the)g(comp)q(onen)o(t-wise)g(ordering.)f(The)i(exp)q(onen)o(tial) e([)p Fv(D)g Fu(!)e Fv(E)r FC(])j(is)g(the)g(con)o(tin)o(uous)g(func-) 262 829 y(tion)k(space)i(with)e(the)h(p)q(oin)o(t)o(wise)g(ordering.)f (The)h(terminal)e(ob)r(ject)j(is)f(the)g(one)g(p)q(oin)o(t)262 879 y(space)13 b Fl(1)f FC(:=)f Fu(f?g)g FC(\(there)k(is)d(no)h (initial)d(ob)r(ject)k(and)e(there)j(are)e(no)f(copro)q(ducts\).)i(In)f (order)262 929 y(to)k(cop)q(e)i(with)e(the)h(categorical)g(in)o (terpretation)g(w)o(e)g(will)e(iden)o(tify)h(an)g(elemen)o(t)g Fv(x)h FC(of)f(a)262 978 y(domain)11 b Fv(D)k FC(with)f(the)h(mapping)c (from)h Fl(1)i FC(to)g Fv(D)h FC(with)f(v)n(alue)f Fv(x)p FC(.)324 1029 y(Besides)f(the)f(cartesian)g(closedness)h(w)o(e)f(also)e (use)j(the)f(fact)f(that)g Fw(Dom)h FC(is)g(closed)g(under)262 1079 y(in\014nite)k(pro)q(ducts)i(and)f(that)g(there)h(is)f(a)f (\014xed)h(p)q(oin)o(t)g(op)q(erator)g Fw(Fix)t FC(:)e(\()p Fv(D)j Fu(!)d Fv(D)q FC(\))i Fu(!)e Fv(D)262 1129 y FC(assigning)f(to)g (ev)o(ery)i(con)o(tin)o(uous)f(function)f Fv(f)d FC(:)j Fv(D)g Fu(!)e Fv(D)16 b FC(its)e(least)g(\014xed)g(p)q(oin)o(t)g Fw(Fix)o FC(\()p Fv(f)t FC(\))f Fu(2)262 1179 y Fv(D)q FC(.)k(F)m(urthermore)g(w)o(e)g(will)f(use)i(that)f(partial)f(families) f(of)h(terms)h(form)f(a)h(domain)e(and)262 1229 y(some)10 b(basic)h(op)q(erations)h(on)f(terms)g(and)g(term)g(famili)o(es)e(are)j (con)o(tin)o(uous)f(and)g(hence)i(exist)262 1278 y(as)h(morphisms)e(in) i(the)h(category)m(.)e(An)o(y)i(other)g(ccc)g(with)f(these)i(prop)q (erties)g(w)o(ould)d(do)h(as)262 1328 y(w)o(ell.)324 1379 y(Elemen)o(ts)19 b(of)h(a)g(pro)q(duct)h(domain)c Fv(D)953 1385 y 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y(to)i(ev)o(ery)h(ground)f(t)o(yp)q(e)h Fv(\034)k FC(a)15 b(domain)e Fu(I)s FC(\()p Fv(\034)5 b FC(\).)14 b(Giv)o(en)h(suc)o(h)h(an)f(in)o(terpretation)h(w)o(e)f (de\014ne)262 1630 y(domains)c([)-7 b([)p Fv(\032)p FC(])g(])480 1614 y Fs(I)516 1630 y FC(for)13 b(ev)o(ery)i(t)o(yp)q(e)f Fv(\032)h FC(b)o(y)351 1723 y([)-7 b([)p Fv(\034)5 b FC(])-7 b(])408 1706 y Fs(I)441 1723 y FC(:=)11 b Fu(I)s FC(\()p Fv(\034)5 b FC(\))p Fv(;)48 b FC([)-7 b([)p Fv(\032)11 b Fu(!)g Fv(\033)q FC(])-7 b(])781 1706 y Fs(I)815 1723 y FC(:=)11 b([[)-7 b([)p Fv(\032)p FC(])g(])937 1706 y Fs(I)970 1723 y Fu(!)11 b FC([)-7 b([)p Fv(\033)q FC(])g(])1082 1706 y Fs(I)1103 1723 y FC(])p Fv(;)47 b FC([)-7 b([)p Fv(\032)9 b Fu(\002)h Fv(\033)q FC(])-7 b(])1305 1706 y Fs(I)1338 1723 y FC(:=)12 b([)-7 b([)p Fv(\032)p FC(])g(])1449 1706 y Fs(I)1479 1723 y Fu(\002)10 b FC([)-7 b([)p Fv(\033)q FC(])g(])1580 1706 y Fs(I)1601 1723 y Fv(:)262 1817 y FC(W)m(e)16 b(write)h([)-7 b([)p Fv(\032)483 1823 y Fr(1)501 1817 y Fv(;)7 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FC(])g(])1056 1129 y Fs(I)1089 1146 y Fu(2)11 b Fw(Dom)q Fv(:)262 1279 y FC(F)m(or)i(ev)o(ery)i(term)e Fv(M)j Fu(2)11 b Fv(\003)669 1285 y Ft(\032)689 1279 y FC(\()p Fv(x)729 1285 y Fr(1)747 1279 y Fv(;)c(:)g(:)g(:)e(;)i(x)864 1285 y Ft(n)886 1279 y FC(\))14 b(w)o(e)g(de\014ne)h(a)e(con)o(tin)o (uous)h(function)466 1370 y([)-7 b([)p Fv(M)5 b FC(])-7 b(])545 1353 y Fs(I)570 1370 y FC(:)14 b Fw(Env)e Fu(!)f FC([)-7 b([)p Fv(\032)p FC(])g(])797 1353 y Fs(I)818 1370 y Fv(;)48 b FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])957 1353 y Fs(I)957 1380 y Ft(\030)990 1370 y FC(:=)11 b([)-7 b([)p Fv(M)5 b FC(])-7 b(])1124 1353 y Fs(I)1124 1380 y Fk(x)1146 1370 y FC(\([)p Fv(\030)r FC(\()p Fv(x)1234 1376 y Fr(1)1252 1370 y FC(\))p Fv(;)7 b(:)g(:)g(:)e(;)i(\030)r FC(\()p Fv(x)1421 1376 y Ft(n)1443 1370 y FC(\)]\))p Fv(:)262 1461 y FC(F)m(ormally)14 b(this)j(de\014nition)g(dep)q(ends)h (on)f(a)g(particular)g(c)o(hoice)h(of)e(the)i(list)f(of)f(v)n(ariables) 262 1511 y Fv(x)286 1517 y Fr(1)304 1511 y Fv(;)7 b(:)g(:)g(:)e(;)i(x) 421 1517 y Ft(n)442 1511 y FC(.)16 b(Ho)o(w)o(ev)o(er,)g(b)q(ecause)i (of)d(the)i(w)o(ell-kno)o(wn)d(coincidence)j(prop)q(ert)o(y)g(in)f (fact)g(it)262 1561 y(do)q(es)e(not.)262 1648 y Fl(Lemma)h(4.)21 b Fn(If)14 b Fv(M)j Fu(2)11 b Fv(\003)657 1654 y Ft(\032)676 1648 y FC(\()p Fv(y)712 1654 y Fr(1)731 1648 y Fv(;)c(:)g(:)g(:)e(;)i (y)844 1654 y Ft(m)875 1648 y FC(\))15 b Fn(then)689 1739 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])768 1722 y Fs(I)768 1749 y Ft(\030)800 1739 y FC(=)12 b([)-7 b([)p Fv(M)5 b FC(])-7 b(])923 1722 y Fs(I)923 1749 y Fk(y)944 1739 y FC(\()p Fv(\030)r FC(\()p Fv(y)1016 1745 y Fr(1)1035 1739 y FC(\))p Fv(;)7 b(:)g(:)g(:)e(;)i(\030)r FC(\()p Fv(y)1200 1745 y Ft(m)1232 1739 y FC(\)\))p Fv(:)324 1830 y FC(F)m(rom)12 b(this)i(w)o(e)g(easily)f(get)h(the)h(familia)o(r) c(equations)856 1921 y([)-7 b([)p Fv(c)p FC(])g(])908 1904 y Fs(I)908 1931 y Ft(\030)940 1921 y FC(=)12 b Fu(I)s FC(\()p Fv(c)p FC(\))p Fv(;)850 1991 y FC([)-7 b([)p Fv(x)p FC(])g(])908 1974 y Fs(I)908 2001 y Ft(\030)940 1991 y FC(=)12 b Fv(\030)r FC(\()p Fv(x)p FC(\))p Fv(;)727 2060 y FC([)-7 b([)p Fv(\025xM)5 b FC(])-7 b(])854 2043 y Fs(I)854 2070 y Ft(\030)875 2060 y FC(\()p Fv(a)p FC(\))11 b(=)h([)-7 b([)p Fv(M)5 b FC(])-7 b(])1063 2043 y Fs(I)1063 2072 y Ft(\030)q Fr([)p Ft(x)p Fs(7!)p Ft(a)p Fr(])1168 2060 y Fv(;)791 2132 y FC([)g([)p Fv(M)5 b(N)g FC(])-7 b(])908 2115 y Fs(I)908 2143 y Ft(\030)940 2132 y FC(=)12 b([)-7 b([)p Fv(M)5 b FC(])-7 b(])1063 2115 y Fs(I)1063 2143 y Ft(\030)1084 2099 y Fo(\000)1103 2132 y FC([)g([)p Fv(N)5 b FC(])-7 b(])1175 2115 y Fs(I)1175 2143 y Ft(\030)1196 2099 y Fo(\001)1215 2132 y Fv(;)743 2202 y FC([)g([)p Fu(h)p Fv(M)r(;)7 b(N)e Fu(i)p FC(])-7 b(])908 2185 y Fs(I)908 2212 y Ft(\030)940 2202 y FC(=)12 b([[)-7 b([)p Fv(M)5 b FC(])-7 b(])1075 2185 y Fs(I)1075 2212 y Ft(\030)1096 2202 y Fv(;)7 b FC([)-7 b([)p Fv(N)5 b FC(])-7 b(])1187 2185 y Fs(I)1187 2212 y Ft(\030)1207 2202 y FC(])p Fv(;)759 2271 y FC([)g([)p Fv(\031)800 2277 y Ft(i)813 2271 y FC(\()p Fv(M)5 b FC(\)])-7 b(])907 2254 y Fs(I)907 2282 y Ft(\030)940 2271 y FC(=)12 b Fv(\031)1008 2277 y Ft(i)1022 2271 y FC(\([)-7 b([)p Fv(M)5 b FC(])-7 b(])1117 2254 y Fs(I)1117 2282 y Ft(\030)1138 2271 y FC(\))p Fv(:)262 2363 y FC(In)15 b(man)o(y)e(cases)k(the)f(in)o(terpretation)g Fu(I)i FC(of)d(the)h(constan)o(ts)g(will)e(ha)o(v)o(e)h(to)g(b)q(e)h (de\014ned)h(re-)262 2412 y(cursiv)o(ely)m(,)10 b(b)o(y)i(e.g.)f (referring)h(to)f([)-7 b([)p Fv(M)5 b FC(])-7 b(])861 2397 y Fs(I)894 2412 y FC(for)11 b(sev)o(eral)h(terms)f Fv(M)5 b FC(.)11 b(This)h(causes)h(no)e(problem,)262 2462 y(since)k(the)g(functionals)g([)-7 b([)p Fv(M)711 2447 y Ft(\032)729 2462 y FC(])g(])746 2447 y Fs(I)772 2462 y FC(:)14 b Fw(Env)g Fu(!)e FC([)-7 b([)p Fv(\032)p FC(])g(])14 b(dep)q(end)i(con)o(tin)o(uously)e(on)g Fu(I)s FC(,)g(where)i Fu(I)i FC(is)p eop %%Page: 10 10 10 9 bop 262 224 a FC(to)10 b(b)q(e)h(considered)h(as)f(an)f(elemen)o (t)g(of)g(the)i(in\014nite)e(pro)q(duct)h Fv(\005)1251 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FC(Induction)14 b(on)f Fv(M)5 b FC(.)262 1804 y Fl(Lemma)15 b(7.)21 b Fn(If)14 b Fv(M)615 1781 y Fj(1)589 1804 y Fu(\000)-7 b(!)655 1810 y Ft(\014)689 1804 y Fv(N)20 b Fn(or)15 b Fv(M)878 1781 y Fj(1)852 1804 y Fu(\000)-7 b(!)919 1810 y Ft(\021)951 1804 y Fv(N)5 b Fn(,)14 b(then)h FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])1187 1810 y Ft(\030)1215 1804 y FC(=)12 b([)-7 b([)p Fv(N)5 b FC(])-7 b(])1331 1810 y Ft(\030)1347 1804 y Fn(.)262 1951 y Fx(4)56 b(Normalization)16 b(b)n(y)j(ev)m(aluation)262 2064 y FC(W)m(e)e(no)o(w)h(consider)i(a)e (sp)q(ecial)g(mo)q(del,)f(whose)i(ground)f(t)o(yp)q(e)h(ob)r(jects)g (are)g(made)e(from)262 2114 y(syn)o(tactic)f(material.)d(W)m(e)j(let)g Fg(N)810 2090 y Fd([)796 2114 y Fu(\000)-25 b(!)14 b Fv(\003)888 2120 y Ft(\032)923 2114 y FC(denote)i(the)h(set)g(of)e (partial)g(term)g(famili)o(es,)e(i.e.)262 2169 y(partial)c(functions)h (from)f(the)i(in)o(tegers)g(to)f(the)h(set)h(of)d(terms)i(of)e(t)o(yp)q (e)i Fv(\032)p FC(.)g Fg(N)1445 2145 y Fd([)1431 2169 y 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Fu(#)375 391 y Ft(\032)398 381 y FC(:)g([)-7 b([)p Fv(\032)p FC(])g(])10 b Fu(!)h FC(\()p Fg(N)614 357 y Fd([)600 381 y Fu(\000)-26 b(!)11 b Fv(\003)688 387 y Ft(\032)707 381 y FC(\))42 b(\(\\reify"\))83 b Fu(")1024 391 y Ft(\032)1047 381 y FC(:)14 b(\()p Fg(N)1145 357 y Fd([)1130 381 y Fu(\000)-25 b(!)11 b Fv(\003)1219 387 y Ft(\032)1238 381 y FC(\))h Fu(!)f FC([)-7 b([)p Fv(\032)p FC(])g(])40 b(\(\\re\015ect"\))r Fv(:)262 463 y FC(These)17 b(functions)f(will)e(b)q(e)j(used)g(when)f(de\014ning)g(the)g(in)o (terpretation)h(of)e(the)h(constan)o(ts)262 512 y(as)f(w)o(ell)g(as)g (in)g(our)h(\014nal)f(goal,)f(normalization)e(b)o(y)j(ev)n(aluation.)f Fu(#)1330 523 y Ft(\032)1365 512 y FC(and)h Fu(")1468 523 y Ft(\032)1502 512 y FC(are)h(de\014ned)262 562 y(sim)o (ultaneously)11 b(b)o(y)j(induction)f(on)h(the)h(t)o(yp)q(e)f Fv(\032)p FC(.)382 644 y Fu(#)403 654 y Ft(\034)424 644 y FC(\()p Fv(r)q FC(\))d(:=)h Fv(r)o(;)568 b Fu(")1162 654 y Ft(\034)1182 644 y FC(\()p Fv(r)q FC(\))12 b(:=)f Fv(r)o(;)273 711 y Fu(#)294 721 y Ft(\032)p Fs(!)p Ft(\033)366 711 y FC(\()p Fv(a)p FC(\)\()p Fv(k)q FC(\))h(:=)g Fv(\025x)591 691 y Ft(\032)591 723 y(k)611 711 y Fu(#)632 721 y Ft(\033)654 677 y Fo(\000)673 711 y Fv(a)p FC(\()p Fu(")732 721 y Ft(\032)751 711 y FC(\()p Fv(x)791 694 y Fs(1)791 721 y Ft(k)826 711 y FC(\)\))858 677 y Fo(\001)878 711 y FC(\()p Fv(k)q FC(+1\))p Fv(;)41 b Fu(")1060 721 y Ft(\032)p Fs(!)p Ft(\033)1132 711 y FC(\()p Fv(r)q FC(\)\()p Fv(b)p FC(\))12 b(:=)f Fu(")1322 721 y Ft(\033)1345 711 y FC(\()p Fv(r)c Fu(#)1408 721 y Ft(\032)1428 711 y FC(\()p Fv(b)p FC(\)\))p Fv(;)276 778 y Fu(#)297 788 y Ft(\032)p Fs(\002)p Ft(\033)362 778 y FC(\([)p Fv(a;)g(b)p FC(]\))j(:=)i Fu(h#)579 788 y Ft(\032)599 778 y FC(\()p Fv(a)p FC(\))p Fv(;)7 b Fu(#)692 788 y Ft(\033)715 778 y FC(\()p Fv(b)p FC(\))p Fu(i)p Fv(;)303 b Fu(")1117 788 y Ft(\032)p Fs(\002)p Ft(\033)1182 778 y FC(\()p Fv(r)q FC(\))12 b(:=)f([)p Fu(")1334 788 y Ft(\032)1353 778 y FC(\()p Fv(\031)1393 784 y Fr(0)1411 778 y FC(\()p Fv(r)q FC(\)\))p Fv(;)c Fu(")1519 788 y Ft(\033)1541 778 y FC(\()p Fv(\031)1581 784 y Fr(1)1600 778 y FC(\()p Fv(r)q FC(\)\)])p Fv(:)262 860 y FC(Note)15 b(that)f(for)h Fv(a)540 866 y Ft(i)566 860 y Fu(2)e FC([)-7 b([)p Fv(\032)645 866 y Ft(i)658 860 y FC(])g(])14 b(w)o(e)h(ha)o(v)o(e)f Fu(")868 870 y Fk(\032)p Fs(!)p Ft(\033)943 860 y FC(\()p Fv(r)q FC(\)\()p Fv(a)1033 866 y Fr(1)1052 860 y Fv(;)7 b(:)g(:)g(:)e(a)1148 866 y Ft(n)1171 860 y FC(\))13 b(=)g Fu(")1265 870 y Ft(\033)1288 860 y FC(\()7 b Fu(#)1332 870 y Ft(\032)1349 874 y Fj(1)1367 860 y FC(\()p Fv(a)1405 866 y Fr(1)1424 860 y FC(\))g Fv(:)g(:)g(:)e Fu(#)1523 870 y Ft(\032)1540 874 y Ff(n)1563 860 y FC(\()p Fv(a)1601 866 y Ft(n)1623 860 y FC(\),)15 b(to)262 909 y(whic)o(h)e(w)o(e)h(refer)h(b)o(y)803 991 y Fu(")p FC(\()p Fv(r)q FC(\)\()p Fq(a)q FC(\))c(=)h Fu(")p FC(\()p Fv(r)c Fu(#)p FC(\()p Fq(a)p FC(\)\))p Fv(:)488 b FC(\(1\))324 1073 y(W)m(e)13 b(no)o(w)h(de\014ne)h(the)g(v)n (alues)f(of)f(the)i(constan)o(ts)g Fv(c)f FC(in)f(our)h(sp)q(ecial)h (mo)q(del.)d(Note)i(that)262 1122 y(w)o(e)j(can)g(view)f Fu(")h FC(as)g(an)f(en)o(vironmen)o(t:)f(it)i(assigns)g(to)f(ev)o(ery)i (v)n(ariable)e Fv(x)g FC(of)g(t)o(yp)q(e)i Fv(\032)f FC(the)262 1172 y(v)n(alue)c Fu(")p FC(\()p Fv(x)430 1157 y Fs(1)465 1172 y FC(\))e Fu(2)g FC([)-7 b([)p Fv(\032)p FC(])g(].)267 1325 y Fu(I)s FC(\()p Fv(c)p FC(\)\()p Fq(a)p FC(\))12 b(:=)468 1227 y Fo(8)468 1264 y(>)468 1277 y(<)468 1352 y(>)468 1364 y(:)505 1260 y FC([)-7 b([)p Fv(N)5 b FC(])-7 b(])577 1245 y Fs(I)577 1276 y Fr([)p Fk(x)o Fs(7!)p Fr([)h([)p Fk(L)q Fr(])g(])689 1266 y Fd(I)689 1286 y(")710 1276 y Fr(])763 1260 y FC(if)13 b Fp(sel)846 1266 y Ft(c)862 1260 y FC(\()p Fp(ext)q FC(\()p Fu(#)p FC(\()p Fq(a)p FC(\)\)\))f(=)g Fv(c)p Fq(K)s Fu(7\000)-6 b(!)o Fv(N)19 b FC(and)14 b Fp(ext)p FC(\()p Fu(#)p FC(\()p Fq(a)q FC(\)\)=)p Fq(K)1614 1266 y Fk(x)1638 1260 y FC([)p Fq(L)p FC(])505 1332 y Fu(")p FC(\()p Fv(c)560 1317 y Fs(1)595 1332 y Fu(#)p FC(\()p Fq(a)p FC(\)\))73 b(if)13 b Fp(sel)846 1338 y Ft(c)862 1332 y FC(\()p Fp(ext)q FC(\()p Fu(#)p FC(\()p Fq(a)p FC(\)\)\))f(=)g Fp(no)p FC(-)p Fp(match)505 1392 y Fu(?)225 b FC(otherwise,)15 b(i.e.)d Fp(ext)q FC(\()p Fu(#)p FC(\()p Fq(a)p FC(\)\))i(is)g(unde\014ned)q Fv(:)324 1475 y FC(Note)h(that)h (this)f(in)f(general)i(is)f(a)g(recursiv)o(e)h(de\014nition,)f(since)h (the)f(terms)g Fv(N)20 b FC(and)15 b Fq(L)262 1524 y FC(ma)o(y)c(con)o(tain)j Fv(c)p FC(.)f(W)m(e)h(no)o(w)f(obtain)g(the)i (correctness)i(of)c(normalization)e(b)o(y)j(ev)n(aluation:)262 1602 y Fl(Theorem)g(1.)21 b Fn(If)15 b Fv(M)h Fu(\000)-7 b(!)11 b Fv(Q)p Fn(,)k(then)g Fu(#)874 1568 y Fo(\000)893 1602 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])972 1608 y Fs(")989 1568 y Fo(\001)1020 1602 y FC(=)11 b Fv(Q)1096 1587 y Fs(1)1131 1602 y Fn(.)262 1679 y(Pr)n(o)n(of.)20 b FC(By)d(induction)g(on)g(the)h(heigh)o(t)f(of)g(the)h(deriv)n(ation)f (of)f Fv(M)23 b Fu(\000)-7 b(!)17 b Fv(Q)p FC(.)f(F)m(or)h(brevit)o(y) 262 1729 y(w)o(e)d(lea)o(v)o(e)f(out)h(the)h(rules)f(concerning)h(pro)q (duct)g(t)o(yp)q(es,)f(since)h(their)f(treatmen)o(t)g(do)q(es)h(not)262 1779 y(bring)e(up)h(an)o(y)f(new)i(issues.)324 1829 y Fn(Case)i Fw(Et)m(a)p FC(,)c(i.e.)734 1869 y Fv(M)5 b(y)13 b Fu(\000)-6 b(!)11 b Fv(Q)p 722 1888 214 2 v 722 1926 a(M)16 b Fu(\000)-6 b(!)11 b Fv(\025y)q(Q)982 1897 y FC(for)j Fv(y)k(=)-26 b Fu(2)11 b Fp(FV)p FC(\()p Fv(M)5 b FC(\))262 1988 y(with)18 b Fv(M)24 b FC(of)18 b(t)o(yp)q(e)h Fv(\032)h Fu(!)g Fv(\033)q FC(.)e(By)h(lemma)d(1)j(w)o(e)g(then)g(ha)o (v)o(e)g Fv(M)5 b(x)1317 1994 y Ft(k)1357 1988 y Fu(\000)-7 b(!)19 b Fv(Q)1476 1994 y Ft(y)1496 1988 y FC([)p Fv(x)1532 1994 y Ft(k)1552 1988 y FC(])f(with)h(a)262 2038 y(deriv)n(ation)10 b(of)h(the)i(same)d(heigh)o(t)i(as)g(that)f(of)g(the)i(giv)o(en)e (deriv)n(ation)f(of)h Fv(M)5 b(y)14 b Fu(\000)-7 b(!)11 b Fv(Q)p FC(,)g(hence)262 2088 y(b)o(y)i(induction)h(h)o(yp)q(othesis)g Fu(#)729 2054 y Fo(\000)748 2088 y FC([)-7 b([)p Fv(M)5 b(x)834 2094 y Ft(k)853 2088 y FC(])-7 b(])870 2094 y Fs(")888 2054 y Fo(\001)919 2088 y FC(=)12 b Fv(Q)996 2094 y Ft(y)1015 2088 y FC([)p Fv(x)1051 2094 y Ft(k)1071 2088 y FC(])1083 2067 y Fs(1)1118 2088 y FC(.)h(W)m(e)h(obtain)505 2186 y Fu(#)525 2153 y Fo(\000)544 2186 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])623 2192 y Fs(")641 2153 y Fo(\001)660 2186 y FC(\()p Fv(k)q FC(\))11 b(=)h Fv(\025x)818 2192 y Ft(k)839 2140 y Fo(\020)863 2186 y Fu(#)884 2153 y Fo(\000)903 2186 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])982 2192 y Fs(")999 2186 y FC(\()p Fu(")p FC(\()p Fv(x)1076 2169 y Fs(1)1076 2196 y Ft(k)1111 2186 y FC(\)\))1143 2153 y Fo(\001)1163 2186 y FC(\()p Fv(k)10 b FC(+)f(1\))1289 2140 y Fo(\021)726 2277 y FC(=)j Fv(\025x)818 2283 y Ft(k)839 2231 y Fo(\020)863 2277 y Fu(#)884 2244 y Fo(\000)903 2277 y FC([)-7 b([)p Fv(M)5 b(x)989 2283 y Ft(k)1008 2277 y FC(])-7 b(])1025 2283 y Fs(")1043 2244 y Fo(\001)1062 2277 y FC(\()p Fv(k)11 b FC(+)e(1\))1189 2231 y Fo(\021)726 2369 y FC(=)j Fv(\025x)818 2375 y Ft(k)839 2323 y Fo(\020)863 2369 y Fv(Q)896 2375 y Ft(y)916 2369 y FC([)p Fv(x)952 2375 y Ft(k)972 2369 y FC(])983 2348 y Fs(1)1018 2369 y FC(\()p Fv(k)f FC(+)e(1\))1145 2323 y Fo(\021)1356 2369 y FC(b)o(y)k(IH)726 2447 y(=)f(\()p Fv(\025y)q(Q)p FC(\))880 2430 y Fs(1)917 2447 y FC(\()p Fv(k)q FC(\))p Fv(:)p eop %%Page: 12 12 12 11 bop 262 224 a FC(In)13 b(the)i(follo)o(wing)c(cases)16 b(the)e(rules)h(are)f(of)g(ground)f(t)o(yp)q(e)i Fv(\034)5 b FC(,)13 b(so)h(w)o(e)g(use)h(that)f Fu(#)p FC(\([)-7 b([)p Fv(M)5 b FC(])-7 b(])1625 230 y Fs(")1642 224 y FC(\))12 b(=)262 274 y([)-7 b([)p Fv(M)5 b FC(])-7 b(])341 280 y Fs(")372 274 y FC(for)13 b Fv(M)j Fu(2)c Fv(\003)560 280 y Ft(\034)594 274 y FC(.)324 326 y Fn(Case)17 b Fw(Bet)m(a)p FC(,)c(i.e.)841 376 y Fv(M)881 382 y Ft(x)902 376 y FC([)p Fv(N)5 b FC(])o Fq(P)19 b Fu(\000)-7 b(!)11 b Fv(Q)p 821 395 323 2 v 821 433 a FC(\()p Fv(\025xM)5 b FC(\))p Fv(N)g Fq(P)18 b Fu(\000)-7 b(!)11 b Fv(Q)597 560 y FC([)-7 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(ailable,)262 1123 y(as)k(a)h(\014xed)g(constan)o(t)h(with)e(a)h (\014xed)g(in)o(terpretation.)262 1245 y Fq(f)5 b Fl(-rules.)21 b FC(There)c(is)e(a)h(\014nal)g(t)o(yp)q(e)g(of)f(rules)i({)f(to)g(b)q (e)g(called)g Fv(f)t FC(-rules)h({)f(whic)o(h)g(w)o(e)g(w)o(an)o(t)262 1294 y(to)e(consider.)h(Their)g(usefulness)h(comes)e(up)g(when)h(w)o(e) g(w)o(ork)g(with)f(concrete)i(data)f(t)o(yp)q(es)262 1344 y(lik)o(e)c(the)i(t)o(yp)q(e)g Fv(\023)f FC(of)g(natural)f(n)o(um) o(b)q(ers)h(and)h(w)o(an)o(t)f(to)g(emplo)o(y)e(e.g.)i(the)h(usual)f(p) q(olynomial)262 1394 y(normal)7 b(form)i(of)g(terms.)h(F)m(or)f (instance,)i(the)f(term)g(\()p Fv(n)r FC(+)r(3\)\()p Fv(n)1218 1379 y Fr(2)1237 1394 y FC(+)r(2)p Fv(n)r FC(+)r(5\))f (should)h(normalize)262 1444 y(to)f Fv(n)333 1429 y Fr(3)353 1444 y FC(+)q(5)p Fv(n)432 1429 y Fr(2)452 1444 y FC(+)q(11)p Fv(n)q FC(+)q(15.)g(Abstractly)m(,)h(what)g(w)o(e)g(ha)o(v)o(e)g(here)h (is)e(a)h(function)g Fp(no)o(rm)s FC(:)k Fv(\003)1568 1450 y Ft(\034)1600 1444 y Fu(!)d Fv(\003)1682 1450 y Ft(\034)262 1494 y FC(for)h(a)g(ground)g(t)o(yp)q(e)h Fv(\034)5 b FC(,)12 b(and)g(it)g(will)f(turn)i(out)f(that)h(all)e(w)o (e)i(need)g(to)f(pro)o(v)o(e)h(theorem)f(1)g(are)262 1544 y(the)i(follo)o(wing)d(prop)q(erties:)543 1631 y Fp(no)o(rm)631 1614 y Fr(2)661 1631 y FC(=)h Fp(no)o(rm)855 b FC(\(5\))543 1693 y Fp(no)o(rm)o FC(\()p Fv(M)692 1676 y Fs(1)727 1693 y FC(\()p Fv(k)q FC(\)\))12 b(=)g Fp(no)o(rm)o FC(\()p Fv(M)5 b FC(\))1019 1676 y Fs(1)1054 1693 y FC(\()p Fv(k)q FC(\))540 b(\(6\))543 1755 y(if)13 b Fv(M)18 b FC(is)c Fv(cde)p FC(-normal,)d(then)k Fp(no)o(rm)o FC(\()p Fv(M)5 b FC(\))14 b(is)g Fv(cde)p FC(-normal)225 b(\(7\))262 1842 y(where)13 b Fv(cde)p FC(-normal)d(refers)k(to)e(the)h Fv(c)p FC(-rules,)g(the)g Fv(d)p FC(-rules)f(and)h(the)g Fv(e)p FC(-rules)g(\(and)f(of)g(course)262 1892 y(the)i Fv(\014)r FC(-rule\).)g(Giv)o(en)g(suc)o(h)g(a)g(function)f Fp(no)o(rm)p FC(,)g(w)o(e)h(add)g(all)e(rules)j(of)e(the)i(form)529 1979 y Fv(f)t Fq(M)j Fu(7\000)-7 b(!)11 b Fp(no)o(rm)o FC(\()p Fv(f)t Fq(M)6 b FC(\))42 b(pro)o(vided)14 b(b)q(oth)g(are)g (di\013eren)o(t)q Fv(:)214 b FC(\(8\))262 2067 y(with)10 b Fq(M)16 b FC(and)11 b Fv(f)t Fq(M)16 b FC(of)11 b(ground)f(t)o(yp)q (es;)i(these)g(should)f(b)q(e)g(the)h(only)e(rules)h(for)g Fv(f)t FC(.)g(W)m(e)f(de\014ne)262 2116 y(the)k Fp(sel)377 2122 y Ft(f)413 2116 y FC(function)f(b)o(y)325 2242 y Fp(sel)370 2248 y Ft(f)391 2242 y FC(\()p Fq(M)6 b FC(\))11 b(:=)543 2171 y Fo(\()576 2214 y Fv(f)t Fq(M)17 b Fu(7\000)-6 b(!)11 b Fp(no)o(rm)o FC(\()p Fv(f)t Fq(M)6 b FC(\))42 b(if)13 b Fv(f)t Fq(M)20 b FC(and)13 b Fp(no)o(rm)p FC(\()p Fv(f)t Fq(M)6 b FC(\))14 b(are)g(di\013eren)o(t,)576 2274 y Fp(no)p FC(-)p Fp(match)242 b FC(otherwise.)262 2365 y(The)14 b(in)o(terpretation)g(is)g(de\014ned)h(b)o(y)734 2452 y Fu(I)s FC(\()p Fv(f)t FC(\)\()p Fq(r)r FC(\)\()p Fv(k)q FC(\))d(:=)g Fp(no)o(rm)o FC(\()p Fv(f)t Fq(r)r FC(\()p Fv(k)q FC(\)\))p Fv(:)p eop %%Page: 15 15 15 14 bop 262 224 a Fl(Extension)12 b(of)j(theorem)e(1.)20 b FC(W)m(e)12 b(will)g(sho)o(w)h(that)g(theorem)f(1)h(\(i.e.)f Fv(M)k Fu(\000)-6 b(!)11 b Fv(Q)h FC(implies)262 274 y Fu(#)282 240 y Fo(\000)301 274 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])380 280 y Fs(")397 240 y Fo(\001)430 274 y FC(=)13 b Fv(Q)508 259 y Fs(1)543 274 y FC(\))i(con)o(tin)o(ues)h(to) f(hold)f(for)h(suc)o(h)g(constan)o(ts)h Fv(d)p FC(,)e Fv(e)h FC(and)g Fv(f)20 b FC(with)15 b(in)o(terpre-)262 324 y(tations)e(as)h(giv)o(en)f(ab)q(o)o(v)o(e.)262 435 y Fl(Lemma)i(8.)21 b Fn(F)m(or)14 b(normal)h(terms)f Fv(M)20 b Fn(we)14 b(have)831 536 y Fu(#)852 503 y Fo(\000)871 536 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])950 542 y Fs(")967 503 y Fo(\001)998 536 y FC(=)11 b Fv(M)1086 519 y Fs(1)1121 536 y Fv(:)262 642 y Fn(Pr)n(o)n(of.)20 b FC(Induction)14 b(on)f(the)i(heigh)o(t)e(of)h Fv(M)5 b FC(.)324 697 y Fn(Case)20 b Fv(\025y)476 682 y Ft(\032)496 697 y Fv(M)541 682 y Ft(\033)563 697 y FC(.)d(This)g(is)g(similar)d(to,)i(but)i(a)e (little)h(simpler)e(than)i(the)h(case)g Fw(Et)m(a)f FC(in)262 747 y(theorem)c(1.)459 848 y Fu(#)479 814 y Fo(\000)498 848 y FC([)-7 b([)p Fv(\025y)q(M)5 b FC(])-7 b(])622 854 y Fs(")641 814 y Fo(\001)660 848 y FC(\()p Fv(k)q FC(\))11 b(=)h Fv(\025x)818 854 y Ft(k)839 802 y Fo(\020)863 848 y Fu(#)884 814 y Fo(\000)903 848 y FC([)-7 b([)p Fv(\025y)q(M)5 b FC(])-7 b(])1027 854 y Fs(")1045 848 y FC(\()p Fu(")p FC(\()p Fv(x)1122 831 y Fs(1)1122 858 y Ft(k)1157 848 y FC(\)\))1189 814 y Fo(\001)1209 848 y FC(\()p Fv(k)10 b FC(+)f(1\))1335 802 y Fo(\021)726 939 y FC(=)j Fv(\025x)818 945 y Ft(k)839 893 y Fo(\020)863 939 y Fu(#)884 905 y Fo(\000)903 939 y FC([)-7 b([)p Fv(M)960 945 y Ft(y)979 939 y FC([)p Fv(x)1015 945 y Ft(k)1035 939 y FC(]])g(])1064 945 y Fs(")1082 905 y Fo(\001)1101 939 y FC(\()p Fv(k)10 b FC(+)g(1\))1228 893 y Fo(\021)726 1018 y FC(=)i Fv(\025x)818 1024 y Ft(k)839 984 y Fo(\000)858 1018 y Fv(M)898 1024 y Ft(y)918 1018 y FC([)p Fv(x)954 1024 y Ft(k)973 1018 y FC(])985 997 y Fs(1)1020 1018 y FC(\()p Fv(k)e FC(+)g(1\))1147 984 y Fo(\001)1402 1018 y FC(b)o(y)j(IH)726 1080 y(=)f(\()p Fv(\025y)q(M)5 b FC(\))892 1063 y Fs(1)929 1080 y FC(\()p Fv(k)q FC(\))p Fv(:)324 1186 y Fn(Case)17 b Fu(h)p Fv(M)r(;)7 b(N)e Fu(i)p FC(.)13 b(Easy)m(.)324 1241 y Fn(Case)k FC(\()p Fv(x)p Fq(M)5 b FC(\))536 1226 y Ft(\034)557 1241 y FC(.)690 1342 y([)-7 b([)p Fv(x)p Fq(M)t FC(])g(])799 1348 y Fs(")829 1342 y FC(=)12 b Fu(")p FC(\()p Fv(x)934 1325 y Fs(1)969 1342 y FC(\)\([)-7 b([)p Fq(M)t FC(])g(])1086 1348 y Fs(")1105 1342 y FC(\))829 1404 y(=)12 b Fv(x)897 1387 y Fs(1)932 1404 y Fu(#)p FC(\([)-7 b([)p Fq(M)t FC(])g(])1054 1410 y Fs(")1072 1404 y FC(\))74 b(b)o(y)15 b(\(1\))829 1466 y(=)d Fv(x)897 1449 y Fs(1)932 1466 y Fq(M)984 1448 y Fs(1)1162 1466 y FC(b)o(y)i(IH)829 1535 y(=)873 1502 y Fo(\000)892 1535 y Fv(x)p Fq(M)968 1502 y Fo(\001)987 1510 y Fs(1)1022 1535 y Fv(:)324 1641 y Fn(Case)24 b FC(\()p Fv(c)p Fq(M)5 b(P)i FC(\))574 1626 y Ft(\034)595 1641 y FC(.)20 b(Since)h Fv(c)p Fq(M)6 b(P)27 b FC(is)21 b(normal,)d(w)o(e)k(ha)o(v)o(e)e Fp(sel)1294 1647 y Ft(c)1311 1641 y FC(\()p Fq(M)5 b FC(\))23 b(=)h Fp(no)p FC(-)p Fp(match)d FC(b)o(y)262 1691 y(our)f(general)g (requiremen)o(ts)h(on)f Fp(sel)853 1697 y Ft(c)870 1691 y FC(,)g(and)g(b)o(y)g(induction)g(h)o(yp)q(othesis)h(and)f(lemma)d(3) 262 1741 y Fp(sel)306 1747 y Ft(c)323 1741 y FC(\()p Fp(ext)p FC(\()p Fu(#)p FC(\([)-7 b([)p Fv(M)5 b FC(])-7 b(])523 1747 y Fs(")541 1741 y FC(\)\)\))12 b(=)f Fp(sel)689 1747 y Ft(c)706 1741 y FC(\()p Fp(ext)p FC(\()p Fq(M)843 1723 y Fs(1)878 1741 y FC(\)\))h(=)f Fp(sel)1010 1747 y Ft(c)1027 1741 y FC(\()p Fq(M)5 b FC(\))12 b(=)f Fp(no)p FC(-)p Fp(match)p FC(,)j(hence)621 1842 y([)-7 b([)p Fv(c)p Fq(M)t(P)7 b FC(])-7 b(])761 1848 y Fs(")790 1842 y FC(=)12 b Fu(I)s FC(\()p Fv(c)p FC(\)\([)-7 b([)p Fq(M)t FC(])g(])1011 1848 y Fs(")1030 1842 y FC(\)\([)g([)p Fq(P)6 b FC(])-7 b(])1132 1848 y Fs(")1150 1842 y FC(\))790 1909 y(=)12 b Fu(")855 1875 y Fo(\000)874 1909 y Fv(c)892 1892 y Fs(1)927 1909 y Fu(#)p FC(\([)-7 b([)p Fq(M)t FC(])g(])1049 1915 y Fs(")1067 1909 y FC(\))1083 1875 y Fo(\001)1103 1909 y FC([)g([)p Fq(P)5 b FC(])-7 b(])1172 1915 y Fs(")790 1971 y FC(=)12 b Fv(c)852 1954 y Fs(1)887 1971 y Fu(#)p FC(\([)-7 b([)p Fq(M)t FC(])g(])1009 1977 y Fs(")1028 1971 y FC(\))p Fu(#)p FC(\([)g([)p Fq(P)5 b FC(])-7 b(])1150 1977 y Fs(")1169 1971 y FC(\))47 b(b)o(y)14 b(\(1\))790 2033 y(=)e Fv(c)852 2016 y Fs(1)887 2033 y Fq(M)939 2015 y Fs(1)975 2033 y Fq(P)1011 2015 y Fs(1)1232 2033 y FC(b)o(y)i(IH)790 2102 y(=)834 2069 y Fo(\000)853 2102 y Fv(c)p Fq(M)5 b(P)960 2069 y Fo(\001)979 2077 y Fs(1)1014 2102 y Fv(:)324 2208 y Fn(Case)16 b FC(\()p Fv(d)p Fq(M)5 b(P)h FC(\))569 2193 y Ft(\034)590 2208 y FC(.)12 b(W)m(e)h(ha)o(v)o(e)f(only)g(used)i(that)f(the)g(in)o (terpretation)g(of)f(the)i(constan)o(ts)f Fv(c)262 2258 y FC(satis\014es)j Fu(I)s FC(\()p Fv(c)p FC(\)\()p Fq(a)p FC(\))e(=)g Fu(")p FC(\()p Fv(c)665 2243 y Fs(1)700 2258 y Fu(#)p FC(\()p Fq(a)p FC(\)\))i(if)e Fp(sel)895 2264 y Ft(c)912 2258 y FC(\()p Fp(ext)p FC(\()p Fu(#)p FC(\()p Fq(a)q FC(\)\)\))g(=)g Fp(no)p FC(-)p Fp(match)p FC(,)h(and)g(this)g (also)g(holds)262 2308 y(for)e(the)i(constan)o(ts)f Fv(d)p FC(.)324 2363 y Fn(Case)k FC(\()p Fv(e)p Fq(M)6 b(P)h FC(\))570 2347 y Ft(\034)590 2363 y FC(.)15 b(Since)h Fv(e)p Fq(M)6 b(P)22 b FC(is)15 b(normal,)e(w)o(e)j(again)e(ha)o(v)o(e) h Fp(sel)1364 2369 y Ft(e)1382 2363 y FC(\()p Fq(M)5 b FC(\))15 b(=)f Fp(no)p FC(-)p Fp(match)p FC(.)262 2412 y(By)k(induction)f(h)o(yp)q(othesis)i(and)e(\(4\))h Fp(sel)927 2418 y Ft(e)945 2412 y FC(\([)-7 b([)p Fq(M)5 b FC(])-7 b(])1047 2418 y Fs(")1065 2412 y FC(\(0\)\))18 b(=)g Fp(sel)1247 2418 y Ft(e)1264 2412 y FC(\()p Fq(M)1333 2394 y Fs(1)1368 2412 y FC(\(0\)\))g(=)h Fp(sel)1550 2418 y Ft(e)1568 2412 y FC(\()p Fq(M)5 b FC(\))18 b(=)262 2462 y Fp(no)p FC(-)p Fp(match)o FC(.)c(The)g(pro)q(of)g(then)g(pro)q (ceeds)i(as)e(b)q(efore.)p eop %%Page: 16 16 16 15 bop 324 224 a Fn(Case)17 b FC(\()p 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Fv(:)599 b Fu(u)-28 b(t)262 736 y Fl(Theorem)14 b(2.)21 b Fn(If)15 b Fv(M)h Fu(\000)-7 b(!)11 b Fv(Q)k Fn(w.r.t.)e(al)r(l)h(typ)n(es)h(of)g(rules)f(ab)n(ove,) i(then)f Fu(#)1412 702 y Fo(\000)1431 736 y FC([)-7 b([)p Fv(M)5 b FC(])-7 b(])1510 742 y Fs(")1527 702 y Fo(\001)1557 736 y FC(=)12 b Fv(Q)1634 721 y Fs(1)1669 736 y Fn(.)262 833 y(Pr)n(o)n(of.)20 b FC(As)14 b(for)g(theorem)f(1;)g(w)o(e)i(only)e (ha)o(v)o(e)h(to)f(add)h(an)g(argumen)o(t)f(for)g(the)i(new)f(rules)h (in)262 883 y(case)f Fw(Red)p FC(.)324 936 y Fn(Case)j Fv(d)p Fq(N)494 942 y Fk(z)515 936 y FC([)p Fq(K)s FC(])11 b Fu(7\000)-7 b(!)11 b Fv(P)698 942 y Fk(z)720 936 y FC([)p Fq(K)r FC(])j(for)f Fw(Red)p FC(,)h(i.e.)589 1028 y Fq(M)i Fu(\000)-6 b(!)11 b Fq(N)775 1034 y Fk(z)796 1028 y FC([)p Fq(K)851 1034 y Fk(x)875 1028 y FC([)p Fq(L)p FC(])o(])83 b Fv(P)1051 1034 y Fk(z)1072 1028 y FC([)p Fq(K)1127 1034 y Fk(x)1150 1028 y FC([)p Fq(L)p FC(]])o Fq(P)18 b Fu(\000)-6 b(!)11 b Fv(Q)p 589 1046 787 2 v 865 1084 a(d)p Fq(M)5 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Fk(x)701 2274 y FC([)p Fq(L)p FC(])o(])g(])772 2280 y Fs(")802 2274 y FC(=)12 b Fu(#)867 2240 y Fo(\000)886 2274 y FC([)-7 b([)p Fq(K)945 2280 y Fk(x)969 2274 y FC([)p Fq(L)o FC(]])g(])1040 2280 y Fs(")1058 2240 y Fo(\001)1089 2274 y FC(=)12 b Fq(K)1176 2280 y Fk(x)1199 2274 y FC([)p Fq(L)p FC(])1254 2253 y Fs(1)1303 2274 y FC(b)o(y)h(lemma)e(8.)324 2326 y Fn(Case)17 b Fv(e)p Fq(K)e Fu(7\000)-7 b(!)11 b Fv(P)607 2332 y Fk(z)628 2326 y FC([)p Fq(K)s FC(])j(for)f Fw(Red)p FC(,)h(i.e.)633 2418 y Fq(M)j Fu(\000)-7 b(!)11 b Fq(K)818 2424 y Fk(x)842 2418 y FC([)p Fq(L)p FC(])82 b Fv(P)1006 2424 y Fk(z)1027 2418 y FC([)p Fq(K)1082 2424 y Fk(x)1106 2418 y FC([)p Fq(L)p FC(])o(])p Fq(P)18 b Fu(\000)-7 b(!)11 b Fv(Q)p 633 2437 699 2 v 867 2475 a(e)p Fq(M)5 b(P)18 b Fu(\000)-6 b(!)11 b Fv(Q)p eop %%Page: 17 17 17 16 bop 262 224 a FC(where)21 b Fv(e)p Fq(K)k Fu(7\000)-7 b(!)22 b Fv(P)588 230 y Fk(z)609 224 y FC([)p Fq(K)s FC(])d(is)h(the)h(selected)h(rule,)e(i.e.)f Fp(sel)1200 230 y Ft(e)1218 224 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1356 y Fs(")941 1350 y FC(\)\()p Fv(k)q FC(\))695 1413 y(=)12 b Fp(no)o(rm)o FC(\()p Fv(f)t FC([)-7 b([)p Fq(M)5 b FC(])-7 b(])953 1419 y Fs(")972 1413 y FC(\()p Fv(k)q FC(\)\))49 b(b)o(y)13 b(de\014nition)h(of)f Fu(I)s FC(\()p Fv(f)t FC(\))695 1475 y(=)f Fp(no)o(rm)o FC(\()p Fv(f)t Fq(K)911 1457 y Fs(1)947 1475 y FC(\()p Fv(k)q FC(\)\))74 b(b)o(y)13 b(IH)695 1537 y(=)f Fp(no)o(rm)o FC(\(\()p Fv(f)t Fq(K)5 b FC(\))944 1520 y Fs(1)979 1537 y FC(\()p Fv(k)q FC(\)\))695 1599 y(=)12 b Fp(no)o(rm)o FC(\()p Fv(f)t Fq(K)t FC(\))927 1582 y Fs(1)963 1599 y FC(\()p Fv(k)q FC(\))74 b(b)o(y)14 b(\(6\))695 1662 y(=)e Fv(Q)772 1644 y Fs(1)807 1662 y FC(\()p Fv(k)q FC(\))p Fv(:)801 b Fu(u)-28 b(t)262 1785 y Fx(5)56 b(Comparison)262 1875 y FC(The)14 b(di\013erence)j(of)d(the)h(presen)o(t)h(approac)o(h)e(to)g (those)i(of)d(other)i(authors)g(is)g(mainly)c(that)262 1925 y Fu(!)k FC(is)g(mo)q(delled)g(simply)f(b)o(y)h(the)i(function)e (space.)i(This)e(means)g(that)h(w)o(e)g(only)f(use)i(the)262 1975 y(prop)q(erties)h(of)e(application)f(and)h(abstraction)h(giv)o(en) f(in)g(the)h(de\014nition)f(of)g(a)h(cartesian)262 2025 y(closed)12 b(category)m(.)g(Due)g(to)g(this)g(fact)g(w)o(e)g(can)h (use)g(an)f(in)o(ternal)f(ev)n(aluation)g(of)g(a)h(program-)262 2074 y(ming)i(language)i(lik)o(e)g Fw(Scheme)p FC(.)h(In)g(other)h (approac)o(hes)f([1,)6 b(5,)g(7])17 b(ev)n(aluation)e(has)i(to)g(b)q(e) 262 2124 y(de\014ned)12 b(b)o(y)g(hand.)f(A)h(comparison)e(b)q(et)o(w)o (een)j(an)e(in)o(ternal)h(and)f(a)h(hand)f(made)g(ev)n(aluation)262 2174 y(sho)o(ws)18 b(that)f(the)i(former)d(is)i(m)o(uc)o(h)e(more)h (e\016cien)o(t.)h(W)m(e)f(tested)i(this)f(in)f Fw(Scheme)i FC(and)262 2224 y(presen)o(t)c(a)f(table)f(with)h(the)g(run-times)g (for)f(some)g(appropriate)h(examples)f(b)q(elo)o(w.)262 2339 y Fl(5.1)47 b(Related)14 b(w)o(ork)262 2412 y FC(In)d([1])f Fw(Al)m(tenkir)o(ch)p FC(,)h Fw(Hofmann)i FC(and)f Fw(Streicher)f FC(giv)o(e)f(a)i(categorical)f(explanation)f(of)262 2462 y(normalization)h(b)o(y)j(ev)n(aluation.)f(Therefore)j(they)f(describ)q (e)h(terms)f(as)f(morphisms)e(from)p eop %%Page: 18 18 18 17 bop 262 224 a FC(a)16 b(con)o(text)i(to)f(their)h(t)o(yp)q(es.)g (The)f(mo)q(del)f(is)h(not)g(an)g(arbitrary)g(cartesian)h(closed)g (cate-)262 274 y(gory)c(but)h(a)f(presheaf)i(o)o(v)o(er)f(a)g(category) g Fv(W)21 b FC(of)14 b(w)o(eak)o(enings)g(with)h(con)o(texts)h(as)f(ob) r(jects.)262 324 y(Roughly)i(sp)q(eaking)j(w)o(eak)o(enings)f(are)g (pro)r(jections)h(on)f(\014nite)h(sequences)i(of)c(v)n(ariables)262 374 y(and)d(a)f(presheaf)j(can)e(b)q(e)h(seen)g(as)g(a)f(v)n(ariable)f (set)i(or)f(a)g(pro)q(of)g(relev)n(an)o(t)g Fw(Kripke)g FC(struc-)262 423 y(ture.)j(In)f(the)h(in)o(terpretation)g(of)f Fv(\003)843 429 y Ft(\032)p Fs(!)p Ft(\033)933 423 y FC(ob)r(jects)i(dep)q(end)g(on)e([)-7 b([)p Fv(\003)1334 429 y Ft(\032)1353 423 y FC(])g(])16 b(as)i(w)o(ell)f(as)g(on)h(the)262 473 y(morphisms)9 b(of)j Fv(W)18 b FC(\(cf.)11 b(the)i(implication)c (in)j(a)f Fw(Kripke)i FC(structure\).)h(In)e(their)g(pro)q(of)g(they) 262 523 y(mak)o(e)d(use)j(of)f(a)g(category)h(that)f(has)h(b)q(oth,)f (a)g(seman)o(tical)e(and)i(a)g(syn)o(tactical)h(comp)q(onen)o(t,)262 573 y(called)f(glueing)g(mo)q(del.)e(They)j(also)f(describ)q(e)j(a)d (corresp)q(ondence)k(b)q(et)o(w)o(een)e(in)o(tuitionistic)262 623 y(completeness)h(pro)q(ofs)g(and)g(normalization.)324 675 y Fw(Cubric)p FC(,)9 b Fw(D)o(ybjer)i FC(and)f Fw(Scott)g FC([7])f(ha)o(v)o(e)h(a)g(similar)e(aim,)g(namely)g(to)j(giv)o(e)f(a)g (normal-)262 725 y(ization)15 b(pro)q(of)h(with)f(as)i(little)e(syn)o (tactic)i(prop)q(erties)g(as)g(p)q(ossible.)f(They)g(also)g(describ)q (e)262 775 y(terms)e(as)h(morphisms)e(and)i(use)h(a)e(presheaf)i(mo)q (del)e(o)o(v)o(er)h(these)h(terms,)e(but)i(here)g(cate-)262 825 y(gories)c(are)h(equipp)q(ed)g(with)g(a)f(partial)f(equiv)n(alence) i(relation.)f(So)g(the)h(function)f Fu(#)h FC(\(called)262 874 y Fp(quote)h FC(in)g([1,)6 b(5,)g(7]\))13 b(app)q(ears)i(as)f (natural)f(isomorphism)e(b)q(et)o(w)o(een)k(the)g Fw(Yoned)o(a)g FC(em)o(b)q(ed-)262 924 y(ding)j(and)i(an)f(in)o(terpretation,)h(that)f (in)o(terprets)i(atoms)e(b)o(y)g Fw(Yoned)o(a)p FC(.)i(The)f(function) 262 974 y Fp(quote)e FC(and)f(its)g(in)o(v)o(erse)h(are)g(used)g(when)g (the)g(univ)o(ersal)f(prop)q(ert)o(y)i(of)d(the)i(category)g(of)262 1024 y(terms)c(is)h(sho)o(wn.)f(In)h(the)h(app)q(endix)f(they)g(in)o(v) o(estigate)g(some)f Fv(\025)p FC(-theories,)i(but)f(it)f(seems)262 1074 y(that)f(these)j(are)e(to)q(o)g(restricted)i(for)d(practical)h (purp)q(oses.)324 1126 y Fw(Coquand)j FC(and)e Fw(D)o(ybjer)g FC([5])f(also)h(in)o(terpret)h(terms)g(in)e(a)h(mo)q(del)f(and)h (reconstruct)262 1176 y(the)g(normal)e(forms)g(via)h(the)h(function)g Fp(quote)q FC(.)f(The)i Fv(\025)p FC(-calculus)f(is)f(giv)o(en)h(b)o(y) f Fv(S)r FC(-)i(and)e Fv(K)s FC(-)262 1226 y(com)o(binators,)d(so)i (they)h(get)f(sligh)o(tly)f(di\013eren)o(t)i(normal)d(forms.)g(The)j (mo)q(del)d(is)i(a)g(glueing)262 1276 y(mo)q(del,)i(therefore)j(it)f (is)g(not)g(extensional)h(and)f(application)f(is)h(application)e(of)i (the)h(se-)262 1325 y(man)o(tic)13 b(part.)h(They)i(emphasize)e(that)h (they)h(use)f(an)g(in)o(tuitionistic)f(metalanguage)f(and)262 1375 y(ha)o(v)o(e)g(implemen)o(ted)f(the)i(algorithm)e(in)h Fw(Alf)p FC(.)324 1428 y Fw(D)o(anvy)18 b FC([8])c(successfully)j(uses) f(the)g Fu(#)f FC(and)g Fu(")h FC(functions)f(for)g(partial)f(ev)n (aluation,)g(in)262 1478 y(particular)f(for)h(compiler)e(construction.) j(His)e(setting)i(is)e(the)i(t)o(w)o(o-lev)o(el)e Fv(\025)p FC(-calculus.)262 1616 y Fl(5.2)47 b(Comparison)14 b(of)i(algorithms) 262 1712 y FC(W)m(e)11 b(tested)i(di\013eren)o(t)g(w)o(a)o(ys)e(to)h (normalize)e(simply)g(t)o(yp)q(ed)i Fv(\025)p FC(-terms.)f(The)h (\014rst)h(is)f(normal-)262 1762 y(ization)i(b)o(y)g(ev)n(aluation)g (using)h(an)f(in)o(ternal)h(ev)n(al)f(function)h(of)f(the)i(giv)o(en)e (programming)262 1812 y(language.)i(The)i(second)i(is)d(normalization)e (b)o(y)j(ev)n(aluation)f(with)g(a)h(user)h(de\014ned)g(ev)n(al)262 1862 y(function.)11 b(The)i(third)g(is)f(normalization)e(b)o(y)i(means) g(of)g(a)g(user)i(de\014ned)g Fv(\014)r FC(-con)o(v)o(ersion.)e(In)262 1912 y(situations)j(where)h(application)e(is)i(not)f(the)h(usual)g (one,)f(e.g.)f(in)i(presheaf)g(mo)q(dels)e(or)i(in)262 1961 y(glueing)d(mo)q(dels,)f(and)i(for)g(t)o(yp)q(ed)h(programmi)o(ng) c(language)j(lik)o(e)f(Standard)h(ML,)g(whose)262 2011 y(ev)n(al)f(is)g(not)h(accessible)h(to)f(the)h(user,)f(the)h(\014rst)f (w)o(a)o(y)g(is)f(excluded.)324 2064 y(T)m(o)e(test)i(the)f(di\013eren) o(t)h(normalization)c(algorithms)g(w)o(e)j(used)h(iterated)f (functions,)f(i.e.)262 2113 y Fv(M)302 2119 y Ft(nm)369 2113 y FC(:=)j(\()p Fp(it)468 2096 y Fr(3)468 2124 y Ft(n)491 2113 y Fp(it)516 2096 y Fr(2)516 2124 y Ft(m)547 2113 y FC(\)\()p Fv(\025x)627 2098 y Fr(0)646 2113 y Fv(x)p FC(\))686 2098 y Fr(1)705 2113 y FC(,)h(where)i Fp(it)879 2096 y Ft(i)p Fr(+2)879 2124 y Ft(n)950 2113 y FC(:=)d Fv(\025f)1056 2098 y Ft(i)p Fr(+1)1113 2113 y Fv(\025x)1161 2098 y Ft(i)1175 2113 y FC(\()p Fv(f)e(:)7 b(:)g(:)e FC(\()p Fv(f)t(x)p FC(\))i Fv(:)g(:)g(:)f FC(\))16 b(with)g Fv(n)f FC(o)q(ccur-)262 2163 y(rences)h(of)e Fv(f)t FC(.)h(Here)h(the)g(t)o(yp)q(e)f(0)f(is)h(an)o(y)f(ground)h(t)o (yp)q(e)g(and)g Fv(i)10 b FC(+)g(1)j(:=)f Fv(i)i Fu(!)e Fv(i)p FC(.)j(So)f Fp(it)1589 2145 y Ft(i)1589 2174 y(n)1626 2163 y FC(is)h(of)262 2213 y(t)o(yp)q(e)g Fv(i)h FC(and)f Fv(M)508 2219 y Ft(nm)575 2213 y FC(is)g(of)g(t)o(yp)q(e)h(1)f(with)g Fv(\025x)942 2198 y Fr(0)960 2213 y Fv(x)g FC(as)g(its)h(normal)d (form.)g(The)i(p)q(oin)o(t)g(in)g(these)262 2263 y(examples)e(is)h (that)g(the)h(result)g(is)f(small)d(\(alw)o(a)o(ys)j Fv(\025x)1109 2248 y Fr(0)1127 2263 y Fv(x)p FC(\),)g(but)g(due)h(to)f (iterated)h(function)262 2313 y(applications)c(it)h(tak)o(es)g(man)o(y) f(steps)i(to)f(reac)o(h)h(the)g(normal)d(form.)g(The)j(table)f(sho)o (ws)g(that)262 2363 y(normalization)h(b)o(y)j(ev)n(aluation)f(with)g (an)h(in)o(ternal)g(ev)n(aluation)f(is)h(ab)q(out)g(t)o(w)o(en)o(t)o(y) g(times)262 2412 y(faster)f(than)h(the)f(v)o(ersion)h(with)f(self-made) f(ev)n(aluation,)f(and)i(this)g(again)f(is)i(faster)f(than)262 2462 y(a)j(recursiv)o(ely)h(de\014ned)h(normalization.)15 b(The)k(run-times)f(are)h(giv)o(en)g(in)f(seconds)i(resp.)p eop %%Page: 19 19 19 18 bop 262 224 a FC(min)o(utes)12 b(and)i(seconds:)p 542 265 881 2 v 541 315 2 50 v 628 315 V 638 300 a(normalization)p 895 315 V 11 w(normalization)p 1158 315 V 56 w(recursiv)o(e)p 1421 315 V 541 364 V 628 364 V 635 350 a(b)o(y)f(an)h(in)o(ternal)p 895 364 V 25 w(b)o(y)g(a)g(de\014ned)p 1156 364 V 29 w(normalization)p 1421 364 V 541 414 V 628 414 V 669 399 a(ev)n(aluation)p 895 414 V 77 w(ev)n(aluation)p 1156 414 V 1421 414 V 542 416 881 2 v 541 466 2 50 v 548 451 a Fv(M)588 457 y Fr(45)p 628 466 V 752 451 a FC(0)p 895 466 V 243 w(1)p 1156 466 V 242 w(3)p 1421 466 V 541 516 V 548 501 a Fv(M)588 507 y Fr(55)p 628 516 V 752 501 a FC(0)p 895 516 V 243 w(2)p 1156 516 V 231 w(14)p 1421 516 V 541 565 V 548 550 a Fv(M)588 556 y Fr(56)p 628 565 V 752 550 a FC(0)p 895 565 V 243 w(6)p 1156 565 V 125 w(stac)o(k)h(o)o(v)o(er\015o)o(w)p 1422 565 V 541 615 V 548 600 a Fv(M)588 606 y Fr(66)p 628 615 V 752 600 a FC(2)p 895 615 V 232 w(36)p 1156 615 V 1421 615 V 541 665 V 548 650 a Fv(M)588 656 y Fr(67)p 628 665 V 752 650 a FC(4)p 895 665 V 216 w(1:27)p 1156 665 V 1421 665 V 541 715 V 548 700 a Fv(M)588 706 y Fr(76)p 628 715 V 742 700 a FC(10)p 895 715 V 205 w(3:34)p 1156 715 V 1421 715 V 541 765 V 548 750 a Fv(M)588 756 y Fr(77)p 628 765 V 742 750 a FC(31)p 895 765 V 195 w(10:13)p 1156 765 V 1421 765 V 541 814 V 548 800 a Fv(M)588 806 y Fr(78)p 628 814 V 725 800 a FC(1:16)p 895 814 V 179 w(25:28)p 1156 814 V 1421 814 V 541 864 V 548 849 a Fv(M)588 855 y Fr(88)p 628 864 V 715 849 a FC(10:12)p 895 864 V 157 w(207:59)p 1156 864 V 1421 864 V 542 866 881 2 v 262 934 a(No)o(w)e(w)o(e)g(giv)o(e)g(the)h(main)e(de\014nitions)h(of)g(an)g (implemen)o(tation)d(in)j Fw(Scheme)p FC(.)h(W)m(e)f(restrict)262 984 y(ourselv)o(es)h(to)g(the)h(case)f(of)g(closed)g(terms)g(since)g(b) o(y)g Fv(\025)p FC(-abstraction)g(w)o(e)g(can)g(bind)g(all)f(free)262 1033 y(v)n(ariables.)f(First)i(our)g Fu(#)g FC(and)g Fu(")o FC(-functions:)262 1161 y Fb(\(define)20 b(\(reify)g(type\))305 1211 y(\(if)h(\(ground-type?)e(type\))392 1261 y(\(lambda)i(\(x\))g (x\))392 1310 y(\(let)g(\(\(reflect-rho)e(\(reflect)h(\(arg-type)g (type\)\)\))523 1360 y(\(reify-sigma)f(\(reify)i(\(val-type)f (type\)\)\)\))436 1410 y(\(lambda)g(\(a\))479 1460 y(\(lambda)h(\(k\)) 523 1510 y(\(let)g(\(\(xk)g(\(mvar)g(k\)\)\))567 1560 y(\(abst)f(xk)i(\(\(reify-sigma)894 1609 y(\(a)f(\(reflect-rho)e (\(lambda)i(\(l\))g(xk\)\)\)\))785 1659 y(\(+)g(k)h (1\)\)\)\)\)\)\)\)\))262 1759 y(\(define)e(\(reflect)g(type\))305 1809 y(\(if)h(\(ground-type?)e(type\))392 1858 y(\(lambda)i(\(x\))g (x\))392 1908 y(\(let)g(\(\(reify-rho)f(\(reify)g(\(arg-type)g (type\)\)\))523 1958 y(\(reflect-sigma)f(\(reflect)h(\(val-type)g (type\)\)\)\))436 2008 y(\(lambda)g(\(r\))479 2058 y(\(lambda)h(\(a\)) 523 2107 y(\(reflect-sigma)545 2157 y(\(lambda)f(\(k\))588 2207 y(\(app)h(\(r)h(k\))f(\(\(reify-rho)f(a\))h(k\)\)\)\)\)\)\)\)\)) 262 2335 y FC(Normalization)11 b(with)i(in)o(ternal)h(ev)n(aluation:) 262 2462 y Fb(\(define)20 b(\(ev1)h(x\))g(\(eval)g(x)h (\(the-environmen)o(t\)\)\))p eop %%Page: 20 20 20 19 bop 262 224 a Fb(\(define)20 b(\(norm1)g(r)i(rho\))f(\(\(\(reify) f(rho\))h(\(ev1)g(r\)\))g(0\)\))262 358 y FC(Normalization)11 b(with)i(de\014ned)i(ev)n(aluation:)262 493 y Fb(\(define)20 b(\(ev2)h(M\))305 543 y(\(lambda)f(\(env\))349 593 y(\(cond)g (\(\(variable?)g(M\))h(\(cadr)g(\(assq)g(M)g(env\)\)\))479 643 y(\(\(application?)e(M\))501 692 y(\(\(\(ev2)i(\(operator)f(M\)\))h (env\))g(\(\(ev2)g(\(argument)f(M\)\))h(env\)\)\))479 742 y(\(\(abstraction?)e(M\))501 792 y(\(lambda)i(\(a\))g(\(\(ev2)g (\(kernel)f(M\)\))785 842 y(\(cons)g(\(list)h(\(abstvar)f(M\))i(a\))f (env\)\)\)\))479 892 y(\(else)g(#f\)\)\)\))262 991 y(\(define)f (\(norm2)g(r)i(rho\))f(\(\(\(reify)f(rho\))h(\(\(ev2)g(r\))g(\(\)\)\))g (0\)\))262 1126 y FC(Finally)12 b(a)h(recursiv)o(e)j(normalization:)262 1260 y Fb(\(define)k(\(norm3)g(r\))305 1310 y(\(cond)h(\(\(variable?)e (r\))j(r\))436 1360 y(\(\(application?)d(r\))458 1410 y(\(let)i(\(\(op)g(\(norm3)f(\(operator)g(r\)\)\))588 1460 y(\(arg)h(\(norm3)g(\(argument)f(r\)\)\)\))501 1509 y(\(if)h(\(abstraction?)f(op\))588 1559 y(\(let)h(\(\(x)h(\(abstvar)e (op\)\))719 1609 y(\(s)i(\(kernel)e(op\)\)\))632 1659 y(\(norm3)h(\(substitute)e(s)j(x)f(arg\)\)\))588 1709 y(\(app)g(op)h(arg\)\)\)\))436 1759 y(\(\(abstraction?)d(r\))458 1808 y(\(abst)h(\(abstvar)h(r\))g(\(norm3)f(\(kernel)h(r\)\)\)\)\)\)) 262 1943 y FC(The)d(auxiliary)e(de\014nitions)i(are)g(not)g(men)o (tioned)f(and)g(hop)q(efully)g(ob)o(vious.)g(So)h(for)f(in-)262 1992 y(stance)11 b Fb(mvar)e FC(pro)q(duces)j(v)n(ariables)d Fv(x)845 1998 y Ft(k)876 1992 y FC(from)f Fv(k)q FC(,)i Fb(abst)f FC(constructs)j(a)e Fv(\025)p FC(-abstraction)g Fv(\025x)1637 1998 y Ft(k)1658 1992 y Fv(M)262 2042 y FC(from)e Fv(x)380 2048 y Ft(k)411 2042 y FC(and)i Fv(M)5 b FC(,)10 b Fb(app)g FC(constructs)j(an)d(application)f Fv(M)c(N)15 b FC(from)9 b Fv(M)15 b FC(and)c Fv(N)5 b FC(,)10 b(and)g Fb(abstvar)262 2092 y FC(resp.)k Fb(kernel)f FC(gets)h(out)g Fv(x)689 2098 y Ft(k)723 2092 y FC(resp.)g Fv(M)19 b FC(of)13 b Fv(\025x)976 2098 y Ft(k)997 2092 y Fv(M)5 b FC(.)262 2227 y Fx(References)281 2325 y FB(1.)20 b(Thorsten)14 b(Altenkirc)o(h,)g(Martin)g(Hofmann,)e(and)i(Thomas)f (Streic)o(her.)18 b(Categorical)c(recon-)331 2371 y(struction)f(of)e(a) g(reduction)i(free)d(normalizatio)q(n)k(pro)q(of.)g(In)d Fa(CTCS'95,)h(Cambridge)p FB(,)d(v)o(olume)331 2417 y(953)j(of)f Fa(L)n(e)n(ctur)n(e)g(Notes)f(in)i(Computer)f(Scienc)n(e)p FB(,)e(pages)j(182{199.)g(Springer)h(V)m(erlag,)f(Berlin,)331 2462 y(Heidelb)q(erg,)j(New)e(Y)m(ork,)f(1995.)p eop %%Page: 21 21 21 20 bop 281 224 a FB(2.)20 b(Holger)h(Benl,)f(Ulric)o(h)h(Berger,)e (Helm)o(ut)h(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg,)h(Monik)n(a)g(Seisen)o (b)q(erger,)h(and)331 270 y(W)m(olfgang)16 b(Zub)q(er.)21 b(Pro)q(of)14 b(theory)g(at)g(w)o(ork:)g(Program)g(dev)o(elopmen)o(t)i (in)f(the)f(Minlog)i(sys-)331 315 y(tem.)j(T)m(o)14 b(app)q(ear:)g (Automated)g(Deduction)i({)e(A)f(Basis)i(for)f(Applications)j(\(eds)d (W.)f(Bib)q(el)331 361 y(and)h(P)m(.)e(Sc)o(hmitt\),)i(V)m(olume)g(I:)e (F)m(oundations,)j(Klu)o(w)o(er)e(Academic)h(Publishers,)i(1998.)281 407 y(3.)k(Ulric)o(h)28 b(Berger.)57 b(Con)o(tin)o(uous)28 b(functionals)g(of)e(dep)q(enden)o(t)i(and)f(trans\014nite)h(t)o(yp)q (es.)331 452 y(Habilitatio)q(nssc)o(h)q(rift,)15 b(Mathematisc)o(hes)g (Institut)f(der)g(Univ)o(ersit\177)-19 b(at)14 b(M)q(\177)-20 b(unc)o(hen,)14 b(1997.)281 498 y(4.)20 b(Ulric)o(h)13 b(Berger)f(and)g(Helm)o(ut)g(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg.)k(An)c (in)o(v)o(erse)g(of)f(the)h(ev)n(aluation)i(functional)331 544 y(for)f(t)o(yp)q(ed)h Fz(\025)p FB({calculus.)19 b(In)13 b(R.)g(V)m(em)o(uri,)g(editor,)h Fa(Pr)n(o)n(c)n(e)n(e)n(dings) d(of)i(the)f(Sixth)g(A)o(nnual)g(IEEE)331 589 y(Symp)n(osium)d(on)i(L)n (o)n(gic)e(in)i(Computer)f(Scienc)n(e)p FB(,)d(pages)k(203{211.)f(IEEE) g(Computer)h(So)q(ciet)o(y)331 635 y(Press,)i(Los)g(Alamitos,)i(1991.) 281 681 y(5.)20 b(Thierry)d(Co)q(quand)f(and)g(P)o(eter)f(Dyb)r(jer.)25 b(In)o(tuitionistic)18 b(mo)q(del)f(constructions)g(and)f(nor-)331 726 y(malization)g(pro)q(ofs.)h Fa(Mathematic)n(al)12 b(Structur)n(es)f(in)i(Computer)g(Scienc)n(e)p FB(,)d(7:73{94,)j(1997.) 281 772 y(6.)20 b(Ro)o(y)14 b(L.)e(Crole.)18 b Fa(Cate)n(gories)11 b(for)j(T)m(yp)n(es)p FB(.)h(Cam)o(bridge)g(Univ)o(ersit)o(y)f(Press,)f (1993.)281 818 y(7.)20 b(Djordje)d(Cubric,)f(P)o(eter)g(Dyb)r(jer,)g (and)h(Philip)h(Scott.)26 b(Normalization)19 b(and)e(the)f(Y)m(oneda) 331 863 y(em)o(b)q(edding.)j(T)m(o)13 b(app)q(ear:)h(Mathematical)h (Structures)f(in)g(Computer)f(Science,)h(1998.)281 909 y(8.)20 b(Olivier)i(Dan)o(vy)m(.)39 b(Pragmatics)21 b(of)f(t)o(yp)q (e-directed)i(partial)f(ev)n(aluation.)40 b(In)20 b(O.)g(Dan)o(vy)m(,) 331 955 y(R.)c(Gl)q(\177)-20 b(uc)o(k,)17 b(and)g(P)m(.)f(Thiemann,)i (editors,)f Fa(Partial)e(Evaluation)p FB(,)e(v)o(olume)k(1110)g(of)f Fa(LNCS)p FB(,)331 1000 y(pages)e(73{94.)g(Springer)g(V)m(erlag,)g (Berlin,)g(Heidelb)q(erg,)h(New)d(Y)m(ork,)h(1996.)281 1046 y(9.)20 b(N.G.)e(de)g(Bruijn.)33 b(Lam)o(b)q(da)19 b(calculus)h(notation)g(with)f(nameless)g(dummies,)g(a)g(to)q(ol)g(for) 331 1092 y(automatic)14 b(form)o(ula)f(manipulatio)q(n,)i(with)d (application)k(to)c(the)g(Ch)o(urc)o(h{Rosser)i(theorem.)331 1137 y Fa(Indagatione)o(s)d(Math.)p FB(,)g(34:381{392,)j(1972.)262 1183 y(10.)20 b(J.)11 b(Lam)o(b)q(ek)i(and)f(P)m(.)f(Scott.)k Fa(Intr)n(o)n(duction)9 b(to)j(higher)f(or)n(der)h(c)n(ate)n(goric)n (al)d(lo)n(gic)p FB(,)h(v)o(olume)j(7)f(of)331 1229 y Fa(Cambridge)h(Studies)e(in)i(A)n(dvanc)n(e)n(d)e(Mathematics)p FB(.)j(Cam)o(bridge)h(Univ)o(ersit)o(y)f(Press,)f(1986.)262 1274 y(11.)20 b(John)10 b(McCarth)o(y)m(.)h(Recursiv)o(e)f(functions)h (of)e(sym)o(b)q(olic)i(expressions)h(and)d(their)h(computation)331 1320 y(b)o(y)k(mac)o(hine.)k Fa(Communic)n(ations)11 b(of)i(the)g(A)o(CM)p FB(,)g(3\(4\):184{195,)g(1960.)262 1366 y(12.)20 b(Helm)o(ut)11 b(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg)h(and)f (Karl)g(Stro)q(etmann.)j(F)m(rom)c(higher)i(order)e(terms)h(to)f (circuits.)331 1411 y(In)15 b(M.L.)e(Dalla)k(Chiara,)e(K.)e(Do)q(ets,)i (D.)f(Mundici,)i(and)f(J.)f(v)n(an)h(Ben)o(them,)f(editors,)i Fa(L)n(o)n(gic)331 1457 y(and)e(Scienti\014c)e(Metho)n(ds.)i(Pr)n(o)n (c)n(e)n(e)n(dings)e(\(V)m(ol.)i(1\))g(of)h(the)f(T)m(enth)h (Internation)o(al)d(Congr)n(ess)331 1503 y(of)j(L)n(o)n(gic,)f(Metho)n (dolo)n(gy)e(and)i(Philosophy)f(of)h(Scienc)n(e,)f(Flor)n(enc)n(e,)g(A) o(ugust)g(1995)p FB(,)g(v)o(olume)331 1548 y(259)g(of)e Fa(Synthese)f(Libr)n(ary)p FB(,)g(pages)j(209{220,)f(Dordrec)o(h)o(t,)h (Boston,)f(London,)h(1997.)f(Klu)o(w)o(er)331 1594 y(Academic)i (Publishers.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF