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userdict/PStoPSxform PStoPSmatrix matrix currentmatrix
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
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1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
1 0 bop 167 -194 2506 3 v 167 106 a FO(PR)l(OOFS)67 b(AND)h(TYPES)p
167 425 V 167 1146 a FN(JEAN-YVES)28 b(GIRARD)167 1315
y FM(T)-5 b(r)l(anslate)l(d)28 b(and)g(with)g(app)l(endic)l(es)g(by)167
1450 y FN(P)-8 b(A)m(UL)29 b(T)-8 b(A)g(YLOR)167 1585
y(YVES)28 b(LAF)m(ONT)167 2889 y FL(CAMBRIDGE)g(UNIVERSITY)f(PRESS)167
3025 y(Cambridge)167 3171 y(New)g(Y)-8 b(ork)230 b(New)27
b(R)-5 b(o)g(chel)5 b(le)167 3306 y(Melb)-5 b(ourne)202
b(Sydney)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
2 1 bop 167 167 a FK(ii)167 404 y(Published)27 b(b)n(y)h(the)g(Press)f
(Syndicate)h(of)g(the)f(Univ)n(ersit)n(y)g(of)h(Cam)n(bridge)167
490 y(The)f(Pitt)g(Building,)f(T)-6 b(rumpington)28 b(Street,)f(Cam)n
(bridge)g(CB2)h(1RP)167 575 y(32)h(East)f(57th)g(Streey)-6
b(,)28 b(New)f(Y)-6 b(ork,)28 b(NY)f(10022,)j(USA)167
660 y(10)f(Stamford)f(Road,)h(Oakleigh,)e(Melb)r(ourne)g(3166,)j
(Australia)187 826 y(c)167 828 y FJ(\015)e FK(Cam)n(bridge)f(Univ)n
(ersit)n(y)g(Press,)g(1989)167 997 y(First)g(Published)g(1989)167
1082 y(Reprin)n(ted)f(with)h(minor)g(corrections)g(1990)167
1167 y(Reprin)n(ted)f(for)i(the)g(W)-6 b(eb)27 b(2003)167
1336 y(Originally)e(prin)n(ted)i(in)g(Great)h(Britain)e(at)i(the)f
(Univ)n(ersit)n(y)g(Press,)g(Cam)n(bridge)167 1504 y
FM(British)i(Libr)l(ary)g(Catalo)l(guing)e(in)h(Public)l(ation)f(Data)h
(available)167 1673 y(Libr)l(ary)h(of)f(Congr)l(ess)g(Catalo)l(guing)g
(in)f(Public)l(ation)g(Data)h(available)167 1842 y FK(ISBN)f(0)h(521)h
(37181)h(3)p eop
PStoPSsaved restore
%%Page: (2,3) 2
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
3 2 bop 2615 167 a FK(iii)167 419 y FI(Preface)167 680
y FK(This)27 b(little)d(b)r(o)r(ok)29 b(comes)e(from)h(a)g(short)g
(graduate)h(course)e(on)i(t)n(yp)r(ed)f FH(\025)p FK(-calculus)e(giv)n
(en)i(at)167 765 y(the)h(Univ)n(ersit)n(\023)-33 b(e)28
b(P)n(aris)h(VI)r(I)g(in)f(the)h(autumn)h(term)e(of)i(1986{7.)43
b(It)29 b(is)g(not)h(in)n(tended)f(to)g(b)r(e)167 850
y(encyclop)r(edic)k(|)h(the)h(Ch)n(urc)n(h-Rosser)g(theorem,)h(for)f
(instance,)h(is)e(not)h(pro)n(v)n(ed)g(|)g(and)167 935
y(the)27 b(selection)f(of)i(topics)f(w)n(as)i(really)d(quite)h
(haphazard.)270 1061 y(Some)d(v)n(ery)h(basic)e(kno)n(wledge)i(of)f
(logic)f(is)g(needed,)i(but)f(w)n(e)g(will)d(nev)n(er)j(go)g(in)n(to)g
(tedious)167 1146 y(details.)65 b(Some)37 b(b)r(o)r(ok)i(in)f(pro)r(of)
g(theory)-6 b(,)42 b(suc)n(h)c(as)g([)p FG(?)q FK(],)i(ma)n(y)f(b)r(e)e
(useful)h(afterw)n(ards)g(to)167 1232 y(complete)26 b(the)h
(information)g(on)i(those)f(p)r(oin)n(ts)g(whic)n(h)f(are)g(lac)n
(king.)270 1358 y(The)41 b(notes)h(w)n(ould)g(nev)n(er)f(ha)n(v)n(e)h
(reac)n(hed)f(the)g(standard)i(of)f(a)g(b)r(o)r(ok)g(without)f(the)167
1443 y(in)n(terest)26 b(tak)n(en)j(in)e(translating)h(\(and)h(in)e(man)
n(y)i(cases)f(rew)n(orking\))f(them)g(b)n(y)i(Yv)n(es)f(Lafon)n(t)167
1528 y(and)c(P)n(aul)g(T)-6 b(a)n(ylor.)34 b(F)-6 b(or)24
b(instance)f(Yv)n(es)g(Lafon)n(t)i(restructured)d(c)n(hapter)h(6)h(and)
g(P)n(aul)g(T)-6 b(a)n(ylor)167 1613 y(c)n(hapter)28
b(8,)g(and)h(some)e(sections)g(ha)n(v)n(e)i(b)r(een)e(dev)n(elop)r(ed)h
(in)n(to)f(detailed)f(app)r(endices.)270 1739 y(The)42
b(translators)h(w)n(ould)g(lik)n(e)d(to)j(thank)g(Luk)n(e)g(Ong,)j
(Christine)41 b(P)n(aulin-Mohring,)167 1824 y(Ramon)34
b(Pino,)h(Mark)f(Ry)n(an,)j(Thomas)d(Streic)n(her,)f(Bill)e(White)h
(and)j(Liz)d(W)-6 b(olf)34 b(for)f(their)167 1909 y(suggestions)f(and)f
(detailed)e(corrections)g(to)i(earlier)d(drafts)j(and)g(also)g(Samson)g
(Abramsky)167 1995 y(for)d(his)f(encouragemen)n(t)h(throughout)i(the)d
(pro)t(ject.)270 2121 y(In)h(the)g FM(r)l(eprinting)33
b FK(an)c(op)r(en)f(problem)f(on)h(page)h(140)g(has)f(b)r(een)g(resolv)
n(ed.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
4 3 bop 167 874 a FF(Con)-5 b(ten)g(ts)1392 4101 y FK(iv)p
eop
PStoPSsaved restore
%%Page: (4,5) 3
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
1 4 bop 167 877 a FF(Chapter)28 b(1)167 1173 y(Sense,)g(Denotation)f
(and)h(Seman)-5 b(tics)167 1496 y FK(Theoretical)28 b(Computing)j(is)e
(not)i(y)n(et)f(a)h(science.)41 b(Man)n(y)32 b(basic)e(concepts)g(ha)n
(v)n(e)h(not)g(b)r(een)167 1581 y(clari\014ed,)26 b(and)j(curren)n(t)d
(w)n(ork)i(in)f(the)g(area)h(ob)r(eys)g(a)g(kind)g(of)f(\\w)n(edding)h
(cak)n(e")g(paradigm:)167 1666 y(for)39 b(instance)g(language)i(design)
e(is)f(reminiscen)n(t)f(of)i(Ptolomeic)e(astronom)n(y)k(|)e(forev)n(er)
167 1751 y(in)33 b(need)g(of)h(further)f(corrections.)51
b(There)33 b(are,)h(ho)n(w)n(ev)n(er,)i(some)d(limited)d(topics)j(suc)n
(h)g(as)167 1836 y(complexit)n(y)c(theory)i(and)g(denotational)g(seman)
n(tics)e(whic)n(h)h(are)g(relativ)n(ely)e(free)i(from)g(this)167
1922 y(criticism.)270 2056 y(In)d(suc)n(h)f(a)h(situation,)f(metho)r
(dological)f(remarks)h(are)g(extremely)e(imp)r(ortan)n(t,)i(since)f(w)n
(e)167 2142 y(ha)n(v)n(e)k(to)f(see)f(metho)r(dology)h(as)g
FM(str)l(ate)l(gy)h FK(and)g(concrete)d(results)h(as)h(of)h(a)f
FM(tactic)l(al)f FK(nature.)270 2276 y(In)36 b(particular)e(what)h(w)n
(e)g(are)g(in)n(terested)f(in)g(is)h(to)g(b)r(e)g(found)i(at)e(the)g
(source)h(of)f(the)167 2361 y(logical)h(whirlp)r(o)r(ol)g(of)i(the)f
(1900's,)42 b(illustrated)35 b(b)n(y)k(the)e(names)g(of)h(F)-6
b(rege,)40 b(L\177)-34 b(ow)n(enheim,)167 2447 y(G\177)g(odel)36
b(and)i(so)g(on.)63 b(The)37 b(reader)f(not)i(acquain)n(ted)f(with)f
(the)g(history)h(of)h(logic)d(should)167 2532 y(consult)28
b([)p FG(?)p FK(].)167 2777 y FE(1.1)114 b(Sense)28 b(and)h(denotation)
h(in)e(logic)167 2936 y FK(Let)i(us)i(start)f(with)f(an)h(example.)44
b(There)30 b(is)h(a)g(standard)h(pro)r(cedure)f(for)g(m)n
(ultiplication,)167 3022 y(whic)n(h)j(yields)f(for)h(the)g(inputs)h(27)
g(and)g(37)g(the)f(result)g(999.)56 b(What)35 b(can)g(w)n(e)e(sa)n(y)j
(ab)r(out)167 3107 y(that?)270 3242 y(A)28 b(\014rst)f(attempt)g(is)g
(to)h(sa)n(y)h(that)f(w)n(e)f(ha)n(v)n(e)i(an)f FM(e)l(quality)1196
3432 y FK(27)16 b FJ(\002)g FK(37)k(=)f(999)167 3622
y(This)30 b(equalit)n(y)g(mak)n(es)g(sense)h(in)f(the)g(mainstream)f
(of)i(mathematics)e(b)n(y)i(sa)n(ying)h(that)e(the)167
3707 y(t)n(w)n(o)k(sides)f FM(denote)h FK(the)f(same)h(in)n(teger)1233
3682 y FD(1)1293 3707 y FK(and)h(that)52 b FJ(\002)f
FK(is)33 b(a)h FM(function)g FK(in)f(the)g(Can)n(torian)167
3793 y(sense)28 b(of)g(a)g(graph.)p 167 3860 1003 3 v
246 3903 a FC(1)273 3925 y FB(By)f FA(inte)m(ger)h FB(w)n(e)g(shall,)f
(throughout,)g(mean)g FA(natur)m(al)i(numb)m(er)p FB(:)34
b(0,)28 b(1,)h(2,...)1402 4101 y FK(1)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
2 5 bop 167 167 a FK(2)610 b Fz(CHAPTER)26 b(1.)63 b(SENSE,)28
b(DENOT)-6 b(A)g(TION)26 b(AND)g(SEMANTICS)270 404 y
FK(This)34 b(is)f(the)h(denotational)h(asp)r(ect,)h(whic)n(h)d(is)g
(undoubtedly)j(correct,)f(but)f(it)f(misses)167 490 y(the)27
b(essen)n(tial)g(p)r(oin)n(t:)270 621 y(There)35 b(is)h(a)g(\014nite)f
FM(c)l(omputation)i FK(pro)r(cess)f(whic)n(h)f(sho)n(ws)i(that)f(the)f
(denotations)i(are)167 706 y(equal.)76 b(It)41 b(is)g(an)h(abuse)g
(\(and)h(this)d(is)h(not)h(c)n(heap)g(philosoph)n(y)g(|)f(it)g(is)f(a)i
(concrete)167 791 y(question\))27 b(to)h(sa)n(y)g(that)f(27)16
b FJ(\002)f FK(37)28 b FM(e)l(quals)f FK(999,)i(since)d(if)g(the)h(t)n
(w)n(o)g(things)h(w)n(e)e(ha)n(v)n(e)i(w)n(ere)e FM(the)167
876 y(same)g FK(then)f(w)n(e)g(w)n(ould)h(nev)n(er)f(feel)f(the)h(need)
g(to)h(state)f(their)g(equalit)n(y)-6 b(.)34 b(Concretely)24
b(w)n(e)h(ask)167 961 y(a)31 b FM(question)p FK(,)h(27)17
b FJ(\002)f FK(37,)33 b(and)f(get)f(an)g FM(answer)p
FK(,)h(999.)46 b(The)31 b(t)n(w)n(o)g(expressions)g(ha)n(v)n(e)h
(di\013eren)n(t)167 1046 y FM(senses)26 b FK(and)i(w)n(e)e(m)n(ust)h
FM(do)g FK(something)g(\(mak)n(e)f(a)h(pro)r(of)h(or)f(a)g
(calculation,)f(or)h(at)g(least)f(lo)r(ok)167 1131 y(in)h(an)h
(encyclop)r(edia\))f(to)h(sho)n(w)g(that)g(these)g(t)n(w)n(o)g
FM(senses)f FK(ha)n(v)n(e)h(the)g(same)f FM(denotation)p
FK(.)270 1263 y(Concerning)j FJ(\002)p FK(,)f(it)f(is)h(incorrect)f(to)
h(sa)n(y)h(that)g(this)f(is)f(a)i(function)f(\(as)h(a)g(graph\))g
(since)167 1348 y(the)g(computer)f(in)g(whic)n(h)h(the)f(program)i(is)e
(loaded)h(has)h(no)g(ro)r(om)f(for)g(an)h(in\014nite)e(graph.)167
1433 y(Hence)34 b(w)n(e)h(ha)n(v)n(e)h(to)f(conclude)g(that)h(w)n(e)e
(are)h(in)g(the)g(presence)f(of)i(a)g FM(\014nitary)f
FK(dynamics)167 1518 y(related)26 b(to)i(this)f(question)h(of)h(sense.)
270 1649 y(Whereas)c(denotation)g(w)n(as)g(mo)r(delled)d(at)j(a)f(v)n
(ery)h(early)f(stage,)h(sense)g(has)g(b)r(een)f(pushed)167
1734 y(to)n(w)n(ards)30 b FM(subje)l(ctivism)p FK(,)g(with)f(the)g
(result)f(that)h(the)g(presen)n(t)g(mathematical)e(treatmen)n(t)i(of)
167 1819 y(sense)38 b(is)g(more)f(or)i(less)e(reduced)h(to)g
FM(syntactic)g FK(manipulation.)67 b(This)38 b(is)f(not)i
FM(a)f(priori)167 1905 y FK(in)f(the)h(essence)f(of)h(the)g(sub)t
(ject,)i(and)f(w)n(e)e(can)h(exp)r(ect)f(in)h(the)f(next)h(decades)g
(to)g(\014nd)167 1990 y(a)c(treatmen)n(t)e(of)h(computation)g(that)h(w)
n(ould)f(com)n(bine)f(the)h(adv)l(an)n(tages)j(of)e(denotational)167
2075 y(seman)n(tics)j(\(mathematical)f(clarit)n(y\))h(with)g(those)i
(of)f(syn)n(tax)i(\(\014nite)e(dynamics\).)67 b(This)167
2160 y(b)r(o)r(ok)33 b(clearly)c(rests)i(on)i(a)e(tradition)g(that)h
(is)e(based)j(on)f(this)f(unfortunate)h(curren)n(t)f(state)167
2245 y(of)26 b(a\013airs:)35 b(in)24 b(the)i(dic)n(hotom)n(y)f(b)r(et)n
(w)n(een)g FM(in\014nite,)g(static)g(denotation)h FK(and)g
FM(\014nite,)f(dynamic)167 2330 y(sense)p FK(,)i(the)h(denotational)g
(side)f(is)g(m)n(uc)n(h)g(more)g(dev)n(elop)r(ed)h(than)g(the)g(other.)
270 2513 y(So,)33 b(one)f(of)g(the)e(most)h(fundamen)n(tal)h
(distinctions)e(in)g(logic)g(is)h(that)g(made)g(b)n(y)h(F)-6
b(rege:)167 2599 y(giv)n(en)28 b(a)g(sen)n(tence)f FH(A)p
FK(,)g(there)g(are)h(t)n(w)n(o)f(w)n(a)n(ys)i(of)f(seeing)f(it:)270
2767 y FJ(\017)34 b FK(as)29 b(a)f(sequence)g(of)g FM(instructions)p
FK(,)f(whic)n(h)g(determine)f(its)h FM(sense)p FK(,)g(for)h(example)f
FH(A)16 b FJ(_)f FH(B)339 2852 y FK(means)28 b(\\)p FH(A)g
FK(or)g FH(B)t FK(",)g FM(etc.)p FK(.)270 3000 y FJ(\017)34
b FK(as)29 b(the)e FM(ide)l(al)h(r)l(esult)34 b FK(found)29
b(b)n(y)g(these)e(op)r(erations:)36 b(this)27 b(is)g(its)g
FM(denotation)p FK(.)339 3161 y(\\Denotation",)34 b(as)e(opp)r(osed)i
(to)d(\\notation",)k(is)c(what)g FM(is)h(denote)l(d)p
FK(,)h(and)g(not)f(what)339 3246 y FM(denotes)p FK(.)51
b(F)-6 b(or)34 b(example)e(the)g(denotation)i(of)g(a)f(logical)e(sen)n
(tence)i(is)f FG(t)g FK(\(true\))h(or)g FG(f)339 3331
y FK(\(false\),)e(and)h(the)e(denotation)h(of)g FH(A)16
b FJ(_)h FH(B)34 b FK(can)d(b)r(e)f(obtained)h(from)g(the)f
(denotations)339 3416 y(of)f FH(A)e FK(and)i FH(B)i FK(b)n(y)d(means)g
(of)g(the)g(truth)f(table)g(for)h(disjunction.)270 3584
y(Tw)n(o)33 b(sen)n(tences)e(whic)n(h)h(ha)n(v)n(e)h(the)f(same)f
(sense)h(ha)n(v)n(e)h(the)f(same)g(denotation,)i(that)f(is)167
3669 y(ob)n(vious;)d(but)f(t)n(w)n(o)f(sen)n(tences)g(with)g(the)g
(same)g(denotation)h(rarely)f(ha)n(v)n(e)h(the)f(same)g(sense.)167
3754 y(F)-6 b(or)41 b(example,)j(tak)n(e)d(a)g(complicated)e
(mathematical)g(equiv)l(alence)h FH(A)19 b FJ(,)g FH(B)t
FK(.)75 b(The)41 b(t)n(w)n(o)167 3840 y(sen)n(tences)30
b(ha)n(v)n(e)i(the)f(same)f(denotation)i(\(they)f(are)f(true)h(at)g
(the)f(same)h(time\))d(but)k(surely)167 3925 y(not)c(the)g(same)f
(sense,)h(otherwise)e(what)i(is)f(the)h(p)r(oin)n(t)f(of)h(sho)n(wing)h
(the)e(equiv)l(alence?)p eop
PStoPSsaved restore
%%Page: (6,7) 4
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
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PStoPSxform concat
3 6 bop 167 167 a Fz(1.1.)64 b(SENSE)28 b(AND)f(DENOT)-6
b(A)g(TION)25 b(IN)i(LOGIC)1035 b FK(3)270 404 y(This)28
b(example)e(allo)n(ws)h(us)h(to)g(in)n(tro)r(duce)f(some)h(asso)r
(ciations)g(of)g(ideas:)270 566 y FJ(\017)34 b FK(sense,)28
b(syn)n(tax,)i(pro)r(ofs;)270 711 y FJ(\017)k FK(denotation,)29
b(truth,)f(seman)n(tics,)f(algebraic)f(op)r(erations.)270
873 y(That)32 b(is)e(the)g(fundamen)n(tal)i(dic)n(hotom)n(y)f(in)f
(logic.)44 b(Ha)n(ving)32 b(said)f(that,)h(the)e(t)n(w)n(o)i(sides)167
958 y(hardly)c(pla)n(y)g(symmetrical)d(r^)-34 b(oles!)167
1162 y Fy(1.1.1)95 b(The)27 b(algebraic)i(tradition)167
1293 y FK(This)23 b(tradition)g(\(b)r(egun)i(b)n(y)f(Bo)r(ole)f(w)n
(ell)f(b)r(efore)h(the)h(time)d(of)k(F)-6 b(rege\))23
b(is)g(based)i(on)f(a)g(radical)167 1378 y(application)33
b(of)g(Oc)n(kham's)h(razor:)47 b(w)n(e)33 b(quite)f(simply)g(discard)h
(the)g(sense,)i(and)g(consider)167 1463 y(only)28 b(the)f(denotation.)
36 b(The)27 b(justi\014cation)g(of)h(this)e(m)n(utilation)g(of)i(logic)
e(is)h(its)f(op)r(erational)167 1549 y(side:)35 b FM(it)27
b(works!)270 1675 y FK(The)20 b(essen)n(tial)f(turning)i(p)r(oin)n(t)f
(whic)n(h)g(established)f(the)h(predominance)g(of)h(this)f(tradition)
167 1760 y(w)n(as)38 b(L\177)-34 b(ow)n(enheim's)38 b(theorem)f(of)h
(1916.)68 b(No)n(w)n(ada)n(ys,)43 b(one)38 b(ma)n(y)g(see)g(Mo)r(del)f
(Theory)h(as)167 1845 y(the)31 b(ric)n(h)g(pa)n(y-o\013)j(from)d(this)g
(epistemological)e(c)n(hoice)i(whic)n(h)g(w)n(as)h(already)g(v)n(ery)g
(old.)47 b(In)167 1931 y(fact,)41 b(considering)d(logic)f(from)h(the)f
(p)r(oin)n(t)i(of)f(view)f(of)i(denotation,)j FM(i.e.)c
FK(the)g FM(r)l(esult)45 b FK(of)167 2016 y(op)r(erations,)33
b(w)n(e)d(disco)n(v)n(er)h(a)g(sligh)n(tly)f(p)r(eculiar)g(kind)h(of)g
(algebra,)i(but)e(one)g(whic)n(h)g(allo)n(ws)167 2101
y(us)e(to)g(in)n(v)n(estigate)e(op)r(erations)i(unfamiliar)e(to)i(more)
f(traditional)f(algebra.)38 b(In)29 b(particular,)167
2186 y(it)23 b(is)h(p)r(ossible)g(to)g(a)n(v)n(oid)i(the)e(limitation)d
(to)k(|)f(shall)f(w)n(e)h(sa)n(y)i(|)e FM(e)l(quational)31
b FK(v)l(arieties,)24 b(and)167 2271 y(consider)33 b(general)g
FM(de\014nable)k FK(structures.)52 b(Th)n(us)34 b(Mo)r(del)g(Theory)f
(rejuv)n(enates)h(the)f(ideas)167 2356 y(and)c(metho)r(ds)e(of)i
(algebra)f(in)f(an)h(often)g(fruitful)e(w)n(a)n(y)-6
b(.)167 2561 y Fy(1.1.2)95 b(The)27 b(syn)m(tactic)i(tradition)167
2692 y FK(On)40 b(the)f(other)h(hand,)k(it)39 b(is)g(imp)r(ossible)e
(to)j(sa)n(y)h(\\forget)f(completely)e(the)h(denotation)167
2777 y(and)i(concen)n(trate)f(on)h(the)f(sense",)k(for)d(the)f(simple)e
(reason)j(that)f(the)g(sense)h(con)n(tains)167 2862 y(the)34
b(denotation,)j(at)e(least)e(implicitly)-6 b(.)51 b(So)35
b(it)e(is)h(not)h(a)g(matter)e(of)i(symmetry)-6 b(.)55
b(In)34 b(fact)167 2947 y(there)d(is)h(hardly)g(an)n(y)i(uni\014ed)e
(syn)n(tactic)f(p)r(oin)n(t)h(of)h(view,)f(b)r(ecause)h(w)n(e)e(ha)n(v)
n(e)i(nev)n(er)f(b)r(een)167 3032 y(able)e(to)g(giv)n(e)g(an)i(op)r
(erational)e(meaning)g(to)h(this)e(m)n(ysterious)h FM(sense)p
FK(.)43 b(The)30 b(only)h(tangible)167 3117 y(realit)n(y)36
b(ab)r(out)i(sense)f(is)f(the)h(w)n(a)n(y)h(it)e(is)g(written,)i(the)f
(formalism;)j(but)d(the)g(formalism)167 3202 y(remains)30
b(an)h(unaccommo)r(dating)h(ob)t(ject)e(of)i(study)-6
b(,)33 b(without)d(true)g(structure,)h(a)g(piece)f(of)167
3287 y FM(soft)e(c)l(amemb)l(ert)p FK(.)270 3414 y(Do)r(es)37
b(this)g(mean)f(that)i(the)e(purely)h(syn)n(tactic)f(approac)n(h)j(has)
f(nothing)g(w)n(orth)n(while)167 3499 y(to)i(sa)n(y?)70
b(Surely)39 b(not,)k(and)d(the)f(famous)i(theorem)d(of)i(Gen)n(tzen)f
(of)g(1934)i(sho)n(ws)g(that)167 3584 y(logic)33 b(p)r(ossesses)h(some)
g(profound)h(symmetries)d(at)i(the)f(syn)n(tactical)g(lev)n(el)f
(\(expressed)h(b)n(y)167 3669 y FM(cut-elimination)p
FK(\).)51 b(Ho)n(w)n(ev)n(er)33 b(these)g(symmetries)e(are)i(blurred)g
(b)n(y)h(the)f(imp)r(erfections)e(of)167 3754 y(syn)n(tax.)36
b(T)-6 b(o)25 b(put)f(it)g(in)f(another)i(w)n(a)n(y)-6
b(,)27 b(they)d(are)g(not)h(symmetries)d(of)j(syn)n(tax,)i(but)d(of)h
(sense.)167 3840 y(F)-6 b(or)31 b(w)n(an)n(t)g(of)f(an)n(ything)i(b)r
(etter,)e(w)n(e)g(m)n(ust)g(express)h(them)e(as)i(prop)r(erties)e(of)i
(syn)n(tax,)i(and)167 3925 y(the)27 b(result)g(is)g(not)h(v)n(ery)g
(prett)n(y)-6 b(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
4 7 bop 167 167 a FK(4)610 b Fz(CHAPTER)26 b(1.)63 b(SENSE,)28
b(DENOT)-6 b(A)g(TION)26 b(AND)g(SEMANTICS)270 404 y
FK(So,)40 b(summing)c(up)h(our)h(opinion)f(ab)r(out)g(this)g
(tradition,)h(it)d(is)h(alw)n(a)n(ys)i(in)e(searc)n(h)h(of)167
490 y(its)j(fundamen)n(tal)h(concepts,)k(whic)n(h)40
b(is)g(to)h(sa)n(y)-6 b(,)46 b(an)c(op)r(erational)f(distinction)e(b)r
(et)n(w)n(een)167 575 y(sense)33 b(and)g(syn)n(tax.)52
b(Or)32 b(to)h(put)g(these)f(things)h(more)f(concretely)-6
b(,)33 b(it)e(aims)h(to)h(\014nd)g(deep)167 660 y(geometrical)25
b FM(invariants)34 b FK(of)28 b(syn)n(tax:)37 b(therein)26
b(is)h(to)h(b)r(e)g(found)h(the)e(sense.)270 879 y(The)g(tradition)g
(called)f(\\syn)n(tactic")h(|)g(for)h(w)n(an)n(t)g(of)g(a)f(nobler)g
(title)e(|)i(nev)n(er)g(reac)n(hed)167 964 y(the)35 b(lev)n(el)e(of)i
(its)g(riv)l(al.)56 b(In)36 b(recen)n(t)d(y)n(ears,)38
b(during)d(whic)n(h)g(the)g(algebraic)f(tradition)g(has)167
1049 y(\015ourished,)29 b(the)e(syn)n(tactic)h(tradition)f(w)n(as)i
(not)f(of)h(note)f(and)h(w)n(ould)f(without)g(doubt)h(ha)n(v)n(e)167
1134 y(disapp)r(eared)k(in)f(one)h(or)g(t)n(w)n(o)f(more)g(decades,)i
(for)f(w)n(an)n(t)g(of)g(an)n(y)h(issue)e(or)h(metho)r(dology)-6
b(.)167 1219 y(The)30 b(disaster)g(w)n(as)g(a)n(v)n(erted)h(b)r(ecause)
f(of)g(computer)g(science)e(|)i(that)h(great)f(manipulator)167
1304 y(of)e(syn)n(tax)h(|)f(whic)n(h)f(p)r(osed)h(it)f(some)g(v)n(ery)h
(imp)r(ortan)n(t)g(theoretical)d(problems.)270 1453 y(Some)37
b(of)f(these)g(problems)g(\(suc)n(h)h(as)g(questions)f(of)h
(algorithmic)d(complexit)n(y\))h(seem)167 1539 y(to)h(require)f(more)g
(the)h(letter)e(than)i(the)g(spirit)e(of)j(logic.)59
b(On)36 b(the)f(other)h(hand)i(all)c(the)167 1624 y(problems)j
(concerning)g(correctness)g(and)i(mo)r(dularit)n(y)e(of)h(programs)g
(app)r(eal)g(in)f(a)h(deep)167 1709 y(w)n(a)n(y)c(to)g(the)f(syn)n
(tactic)g(tradition,)h(to)g FM(pr)l(o)l(of)h(the)l(ory)p
FK(.)53 b(W)-6 b(e)33 b(are)h(led,)g(then,)h(to)e(a)h(revision)167
1794 y(of)c(pro)r(of)h(theory)-6 b(,)31 b(from)f(the)f(fundamen)n(tal)i
(theorem)d(of)j(Herbrand)f(whic)n(h)f(dates)h(bac)n(k)h(to)167
1879 y(1930.)37 b(This)27 b(revision)f(sheds)h(a)h(new)f(ligh)n(t)f(on)
i(those)f(areas)h(whic)n(h)e(one)h(had)h(though)n(t)h(w)n(ere)167
1964 y(\014xed)f(forev)n(er,)g(and)h(where)e(routine)g(had)h(prev)l
(ailed)f(for)h(a)g(long)g(time.)270 2113 y(In)j(the)f(exc)n(hange)i(b)r
(et)n(w)n(een)e(the)g(syn)n(tactic)g(logical)f(tradition)h(and)i
(computer)e(science)167 2198 y(one)37 b(can)g(w)n(ait)e(for)i(new)f
(languages)i(and)g(new)e(mac)n(hines)g(on)h(the)f(computational)h
(side.)167 2283 y(But)32 b(on)h(the)f(logical)f(side)g(\(whic)n(h)g(is)
h(that)g(of)h(the)f(principal)f(author)i(of)g(this)e(b)r(o)r(ok\))j
(one)167 2369 y(can)25 b(at)g(last)f(hop)r(e)h(to)g(dra)n(w)g(on)h(the)
e(conceptual)h(basis)g(whic)n(h)e(has)j(alw)n(a)n(ys)f(b)r(een)g(so)g
(cruelly)167 2454 y(ignored.)167 2716 y FE(1.2)114 b(The)28
b(t)m(w)m(o)g(seman)m(tic)g(traditions)167 2903 y Fy(1.2.1)95
b(T)-8 b(arski)167 3046 y FK(This)38 b(tradition)f(is)h(distinguished)g
(b)n(y)h(an)g(extreme)d(platitude:)56 b(the)38 b(connector)h(\\)p
FJ(_)p FK(")g(is)167 3131 y(translated)34 b(b)n(y)h(\\or",)i(and)e(so)g
(on.)55 b(This)33 b(in)n(terpretation)g(tells)f(us)j(nothing)g
(particularly)167 3216 y(remark)l(able)k(ab)r(out)h(the)g(logical)e
(connectors:)59 b(its)38 b(apparen)n(t)j(lac)n(k)e(of)h(am)n(bition)f
(is)f(the)167 3301 y(underlying)27 b(reason)g(for)g(its)f(op)r
(erationalit)n(y)-6 b(.)35 b(W)-6 b(e)26 b(are)h(only)g(in)n(terested)e
(in)h(the)g(denotation,)167 3386 y FG(t)h FK(or)h FG(f)7
b FK(,)28 b(of)h(a)f(sen)n(tence)f(\(closed)f(expression\))i(of)g(the)g
(syn)n(tax.)251 3609 y(1.)35 b(F)-6 b(or)27 b(atomic)d(sen)n(tences,)i
(w)n(e)g(assume)g(that)g(the)f(denotation)i(is)e(kno)n(wn;)j(for)e
(example:)421 3797 y FJ(\017)35 b FK(3)16 b(+)g(2)k(=)f(5)28
b(has)h(the)e(denotation)h FG(t)p FK(.)421 3919 y FJ(\017)35
b FK(3)16 b(+)g(3)k(=)f(5)28 b(has)h(the)e(denotation)h
FG(f)7 b FK(.)p eop
PStoPSsaved restore
%%Page: (8,9) 5
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
5 8 bop 167 167 a Fz(1.2.)64 b(THE)27 b(TW)n(O)g(SEMANTIC)g(TRADITIONS)
1071 b FK(5)251 404 y(2.)35 b(The)41 b(denotations)h(of)f(the)g
(expressions)g FH(A)19 b FJ(^)g FH(B)t FK(,)45 b FH(A)19
b FJ(_)g FH(B)t FK(,)44 b FH(A)29 b FJ(\))g FH(B)45 b
FK(and)d FJ(:)p FH(A)e FK(are)339 490 y(obtained)29 b(b)n(y)f(means)g
(of)g(a)g(truth)g(table:)948 710 y FH(A)59 b(B)p 1169
736 3 86 v 88 w(A)16 b FJ(^)f FH(B)63 b(A)15 b FJ(_)h
FH(B)63 b(A)19 b FJ(\))h FH(B)62 b FJ(:)p FH(A)p 919
738 1174 3 v 959 814 a FG(t)81 b(t)p 1169 839 3 102 v
175 w(t)213 b(t)230 b(t)185 b(f)959 899 y(f)88 b(t)p
1169 924 3 86 v 175 w(f)220 b(t)230 b(t)185 b(t)959 984
y(t)81 b(f)p 1169 1010 V 182 w(f)220 b(t)229 b(f)959
1069 y(f)88 b(f)p 1169 1095 V 182 w(f)220 b(f)237 b(t)251
1256 y FK(3.)35 b(The)27 b(denotation)g(of)g FJ(8)p FH(\030)s(:)12
b(A)25 b FK(is)h FG(t)g FK(i\013)g(for)g FM(every)32
b FH(a)27 b FK(in)f(the)g(domain)g(of)h(in)n(terpretation)2626
1231 y FD(2)2653 1256 y FK(,)339 1341 y FH(A)p FK([)p
FH(a=\030)s FK(])i(is)e FG(t)p FK(.)35 b(Lik)n(ewise)26
b FJ(9)p FH(\030)s(:)12 b(A)28 b FK(is)f FG(t)g FK(i\013)g
FH(A)p FK([)p FH(a=\030)s FK(])i(is)e FG(t)h FK(for)g
FM(some)33 b FH(a)p FK(.)270 1503 y(Once)h(again,)k(this)c
(de\014nition)g(is)g(ludicrous)g(from)h(the)f(p)r(oin)n(t)h(of)g(view)f
(of)h(logic,)h(but)167 1588 y(en)n(tirely)26 b(adequate)i(for)g(its)f
(purp)r(ose.)37 b(The)27 b(dev)n(elopmen)n(t)g(of)h(Mo)r(del)f(Theory)h
(sho)n(ws)h(this.)167 1793 y Fy(1.2.2)95 b(Heyting)167
1924 y FK(Heyting's)27 b(idea)h(is)f(less)f(w)n(ell)g(kno)n(wn,)j(but)f
(it)e(is)h(di\016cult)g(to)h(imagine)e(a)i(greater)f(disparit)n(y)167
2009 y(b)r(et)n(w)n(een)43 b(the)g(brilliance)d(of)k(the)f(original)g
(idea)g(and)h(the)g(medio)r(crit)n(y)d(of)j(subsequen)n(t)167
2094 y(dev)n(elopmen)n(ts.)35 b(The)27 b(aim)g(is)g(extremely)f(am)n
(bitious:)35 b(to)28 b(mo)r(del)e(not)j(the)e FM(denotation)p
FK(,)h(but)167 2179 y(the)f FM(pr)l(o)l(ofs)p FK(.)270
2305 y(Instead)i(of)g(asking)g(the)f(question)g(\\when)h(is)f(a)g(sen)n
(tence)g FH(A)g FM(true)5 b FK(?",)28 b(w)n(e)g(ask)h(\\what)g(is)167
2391 y(a)g FM(pr)l(o)l(of)44 b FK(of)29 b FH(A)p FK(?".)36
b(By)29 b FM(pr)l(o)l(of)44 b FK(w)n(e)28 b(understand)h(not)g(the)f
(syn)n(tactic)f(formal)h(transcript,)g(but)167 2476 y(the)d(inheren)n
(t)f(ob)t(ject)h(of)h(whic)n(h)e(the)h(written)f(form)h(giv)n(es)g
(only)g(a)h(shado)n(wy)h(re\015ection.)33 b(W)-6 b(e)167
2561 y(tak)n(e)28 b(the)f(view)g(that)h(what)g(w)n(e)f
FM(write)h FK(as)g(a)g(pro)r(of)h(is)e(merely)e(a)k(description)d(of)i
(something)167 2646 y(whic)n(h)38 b(is)g FM(alr)l(e)l(ady)i
FK(a)f(pro)r(cess)g(in)f(itself.)67 b(So)39 b(the)g(reply)f(to)h(our)g
(extremely)e(am)n(bitious)167 2731 y(question)24 b(\(and)g(an)g(imp)r
(ortan)n(t)f(one,)i(if)e(w)n(e)f(read)i(it)e(computationally\))h
(cannot)h(b)r(e)g(a)g FM(formal)167 2816 y(system)p FK(.)251
3066 y(1.)35 b(F)-6 b(or)27 b(atomic)f(sen)n(tences,)g(w)n(e)g(assume)g
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339 3151 y(for)21 b(example,)g(p)r(encil)e(and)i(pap)r(er)g
(calculation)f(serv)n(es)g(as)h(a)g(pro)r(of)g(of)g(\\27)5
b FJ(\002)g FK(37)22 b(=)d(999".)251 3296 y(2.)35 b(A)d(pro)r(of)h(of)f
FH(A)17 b FJ(^)g FH(B)35 b FK(is)c(a)i(pair)e(\()p FH(p;)12
b(q)s FK(\))30 b(consisting)i(of)g(a)h(pro)r(of)g FH(p)e
FK(of)h FH(A)g FK(and)h(a)f(pro)r(of)339 3381 y FH(q)e
FK(of)e FH(B)t FK(.)251 3566 y(3.)35 b(A)28 b(pro)r(of)g(of)h
FH(A)15 b FJ(_)h FH(B)31 b FK(is)c(a)h(pair)f(\()p FH(i;)12
b(p)p FK(\))27 b(with:)421 3728 y FJ(\017)35 b FH(i)19
b FK(=)g(0,)29 b(and)f FH(p)g FK(is)f(a)h(pro)r(of)g(of)g
FH(A)p FK(,)g(or)p 167 3790 1003 3 v 246 3833 a FC(2)273
3854 y Fx(A)p FB([)p Fx(a=\030)s FB(])j(is)g(meta-notation)f(for)i(\\)p
Fx(A)g FB(where)f(all)f(the)h(\(free\))g(o)r(ccurrences)g(of)h
Fx(\030)i FB(ha)n(v)n(e)e(b)r(een)e(replaced)167 3925
y(b)n(y)e Fx(a)p FB(".)34 b(In)28 b(de\014ning)d(this)i(formally)-5
b(,)27 b(w)n(e)i(ha)n(v)n(e)f(to)g(b)r(e)f(careful)g(ab)r(out)f(b)r
(ound)g(v)m(ariables.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
6 9 bop 167 167 a FK(6)610 b Fz(CHAPTER)26 b(1.)63 b(SENSE,)28
b(DENOT)-6 b(A)g(TION)26 b(AND)g(SEMANTICS)421 404 y
FJ(\017)35 b FH(i)19 b FK(=)g(1,)29 b(and)f FH(p)g FK(is)f(a)h(pro)r
(of)g(of)g FH(B)t FK(.)251 595 y(4.)35 b(A)26 b(pro)r(of)i(of)f
FH(A)19 b FJ(\))h FH(B)30 b FK(is)c(a)g(function)h FH(f)8
b FK(,)26 b(whic)n(h)g(maps)g(eac)n(h)h(pro)r(of)g FH(p)f
FK(of)h FH(A)f FK(to)g(a)h(pro)r(of)339 680 y FH(f)8
b FK(\()p FH(p)p FK(\))27 b(of)h FH(B)t FK(.)251 845
y(5.)35 b(In)25 b(general,)h(the)e(negation)i FJ(:)p
FH(A)e FK(is)h(treated)f(as)i FH(A)19 b FJ(\))h(?)j FK(where)i
FJ(?)e FK(is)h(a)i(sen)n(tence)e(with)339 930 y(no)29
b(p)r(ossible)e(pro)r(of.)251 1095 y(6.)35 b(A)28 b(pro)r(of)g(of)g
FJ(8)p FH(\030)s(:)12 b(A)27 b FK(is)f(a)i(function)g
FH(f)8 b FK(,)27 b(whic)n(h)f(maps)i(eac)n(h)g(p)r(oin)n(t)f
FH(a)h FK(of)g(the)f(domain)h(of)339 1180 y(de\014nition)g(to)f(a)i
(pro)r(of)f FH(f)8 b FK(\()p FH(a)p FK(\))27 b(of)h FH(A)p
FK([)p FH(a=\030)s FK(].)251 1345 y(7.)35 b(A)23 b(pro)r(of)i(of)e
FJ(9)p FH(\030)s(:)12 b(A)24 b FK(is)f(a)g(pair)g(\()p
FH(a;)12 b(p)p FK(\))23 b(where)f FH(a)i FK(is)e(a)i(p)r(oin)n(t)g(of)f
(the)g(domain)h(of)g(de\014nition)339 1430 y(and)29 b
FH(p)e FK(is)g(a)h(pro)r(of)h(of)f FH(A)p FK([)p FH(a=\030)s
FK(].)270 1683 y(F)-6 b(or)35 b(example,)g(the)f(sen)n(tence)f
FH(A)24 b FJ(\))g FH(A)34 b FK(is)f(pro)n(v)n(ed)j(b)n(y)e(the)g(iden)n
(tit)n(y)f(function,)j(whic)n(h)167 1768 y(asso)r(ciates)f(to)f(eac)n
(h)h(pro)r(of)g FH(p)f FK(of)h FH(A)p FK(,)h(the)e(same)g(pro)r(of.)57
b(On)34 b(the)h(other)f(hand,)k(ho)n(w)d(can)167 1853
y(w)n(e)f(pro)n(v)n(e)i FH(A)17 b FJ(_)h(:)p FH(A)p FK(?)56
b(W)-6 b(e)34 b(ha)n(v)n(e)i(to)f(b)r(e)f(able)h(to)g(\014nd)g(either)e
(a)i(pro)r(of)h(of)f FH(A)g FK(or)g(a)g(pro)r(of)167
1938 y(of)c FJ(:)p FH(A)p FK(,)g(and)g(this)f(is)g(not)h(p)r(ossible)f
(in)g(general.)43 b(Heyting)30 b(seman)n(tics,)h(then,)g(corresp)r
(onds)167 2024 y(to)d(another)g(logic,)f(the)g FM(intuitionistic)32
b FK(logic)26 b(of)j(Brou)n(w)n(er,)e(whic)n(h)g(w)n(e)g(shall)g(meet)f
(later.)270 2175 y(Undeniably)-6 b(,)48 b(Heyting)43
b(seman)n(tics)f(is)g(v)n(ery)i(original:)66 b(it)42
b(do)r(es)i(not)g(in)n(terpret)d(the)167 2260 y(logical)28
b(op)r(erations)j(b)n(y)g(themselv)n(es,)e(but)h(b)n(y)h(abstract)f
(constructions.)43 b(No)n(w)30 b(w)n(e)f(can)i(see)167
2345 y(that)39 b(these)g(constructions)f(are)h(nothing)h(but)f(t)n(yp)r
(ed)g(\()p FM(i.e.)g FK(mo)r(dular\))f(programs.)70 b(But)167
2430 y(the)33 b(exp)r(erts)f(in)h(the)f(area)i(ha)n(v)n(e)g(seen)e(in)h
(this)f(something)h(v)n(ery)g(di\013eren)n(t:)45 b(a)33
b(functional)167 2515 y(approac)n(h)e(to)f(mathematics.)38
b(In)30 b(other)f(w)n(ords,)h(the)f(seman)n(tics)f(of)i(pro)r(ofs)h(w)n
(ould)e(express)167 2601 y(the)e(v)n(ery)h(essence)f(of)h(mathematics.)
270 2752 y(That)22 b(w)n(as)f(v)n(ery)g(fanciful:)31
b(indeed,)22 b(w)n(e)e(ha)n(v)n(e)i(on)g(the)e(one)h(hand)h(the)f(T)-6
b(arskian)22 b(tradition,)167 2837 y(whic)n(h)h(is)g(commonplace)g(but)
h(honest)h(\(\\)p FJ(_)p FK(")g(means)e(\\or",)j(\\)p
FJ(8)12 b FK(")24 b(means)g(\\for)h(all"\),)e(without)167
2922 y(the)37 b(least)g(pretension.)65 b(Nor)37 b(has)i(it)d
(foundational)j(prosp)r(ects,)i(since)36 b(for)i(foundations,)167
3007 y(one)j(has)h(to)g(giv)n(e)e(an)i(explanation)f(in)g(terms)f(of)h
(something)g(more)f(primitiv)n(e,)i(whic)n(h)167 3092
y(moreo)n(v)n(er)25 b(itself)f(needs)h(its)g(o)n(wn)h(foundation.)36
b(The)25 b(tradition)g(of)h(Heyting)f(is)g(original,)g(but)167
3177 y(fundamen)n(tally)k(has)g(the)g(same)f(problems)g(|)h(G\177)-34
b(odel's)29 b(incompleteness)d(theorem)i(assures)167
3263 y(us,)33 b(b)n(y)g(the)f(w)n(a)n(y)-6 b(,)34 b(that)e(it)e(could)i
(not)g(b)r(e)g(otherwise.)47 b(If)32 b(w)n(e)f(wish)g(to)h(explain)f
FH(A)h FK(b)n(y)g(the)167 3348 y(act)c(of)h(pro)n(ving)g
FH(A)p FK(,)g(w)n(e)e(come)h(up)h(against)g(the)f(fact)g(that)h(the)f
(de\014nition)g(of)h(a)f(pro)r(of)i(uses)167 3433 y(quan)n(ti\014ers)c
(t)n(wice)d(\(for)j FJ(\))g FK(and)g FJ(8)p FK(\).)34
b(Moreo)n(v)n(er)26 b(in)e(the)h FJ(\))h FK(case,)g(one)g(cannot)g(sa)n
(y)h(that)e(the)167 3518 y(domain)j(of)g(de\014nition)f(of)h
FH(f)35 b FK(is)27 b(particularly)f(w)n(ell)g(understo)r(o)r(d!)270
3669 y(Since)38 b(the)h FJ(\))g FK(and)h FJ(8)e FK(cases)h(w)n(ere)f
(problematic)f(\(from)h(this)g(absurd)j(foundational)167
3754 y(p)r(oin)n(t)c(of)h(view\),)h(it)d(has)j(b)r(een)e(prop)r(osed)i
(to)f(add)g(to)g(clauses)f FG(??)h FK(and)g FG(??)g FK(the)g(co)r
(dicil)167 3840 y(\\together)29 b(with)e(a)i(pro)r(of)h(that)e
FH(f)36 b FK(has)29 b(this)f(prop)r(ert)n(y".)39 b(Of)28
b(course)g(that)h(settles)e(nothing,)167 3925 y(and)32
b(the)e(Byzan)n(tine)g(discussions)h(ab)r(out)g(the)g
FM(me)l(aning)36 b FK(whic)n(h)30 b(w)n(ould)h(ha)n(v)n(e)g(to)g(b)r(e)
f(giv)n(en)p eop
PStoPSsaved restore
%%Page: (10,11) 6
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
7 10 bop 167 167 a Fz(1.2.)64 b(THE)27 b(TW)n(O)g(SEMANTIC)g
(TRADITIONS)1071 b FK(7)167 404 y(to)25 b(this)f(co)r(dicil)e(|)j
(discussions)g(without)f(the)g(least)g(mathematical)f(con)n(ten)n(t)i
(|)f(only)h(serv)n(e)167 490 y(to)j(discredit)e(an)i(idea)f(whic)n(h,)g
(w)n(e)g(rep)r(eat,)h(is)f(one)h(of)g(the)f(cornerstones)h(of)g(Logic.)
270 616 y(W)-6 b(e)21 b(shall)e(come)h(across)h(Heyting's)g(idea)f(w)n
(orking)g(in)g(the)h(Curry-Ho)n(w)n(ard)g(isomorphism.)167
701 y(It)32 b(o)r(ccurs)h(in)f(Realisabilit)n(y)e(to)r(o.)51
b(In)33 b(b)r(oth)h(these)e(cases,)j(the)d(foundational)i(pretensions)
167 786 y(ha)n(v)n(e)29 b(b)r(een)f(remo)n(v)n(ed.)37
b(This)27 b(allo)n(ws)g(us)i(to)f(mak)n(e)g(go)r(o)r(d)i(use)e(of)h(an)
g(idea)e(whic)n(h)h(ma)n(y)g(ha)n(v)n(e)167 871 y(sp)r(ectacular)f
(applications)g(in)g(the)h(future.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
8 11 bop 167 874 a FF(Chapter)28 b(2)167 1168 y(Natural)f(Deduction)167
1487 y FK(As)g(w)n(e)f(ha)n(v)n(e)i(said,)f(the)g(syn)n(tactic)f(p)r
(oin)n(t)h(of)h(view)d(sho)n(ws)j(up)g(some)e(profound)j(symmetries)167
1573 y(of)49 b(Logic.)100 b(Gen)n(tzen's)49 b(sequen)n(t)g(calculus)f
(do)r(es)h(this)g(in)f(a)h(particularly)f(satisfying)167
1658 y(manner.)g(Unfortunately)-6 b(,)33 b(the)e(computational)h
(signi\014cance)f(is)g(somewhat)h(obscured)g(b)n(y)167
1743 y(syn)n(tactic)22 b(complications)f(that,)j(although)g(certainly)d
(immaterial,)g(ha)n(v)n(e)i(nev)n(er)g(really)e(b)r(een)167
1828 y(o)n(v)n(ercome.)34 b(That)26 b(is)e(wh)n(y)i(w)n(e)e(presen)n(t)
h(Pra)n(witz')f(natural)i(deduction)f(b)r(efore)g(w)n(e)f(deal)h(with)
167 1913 y(sequen)n(t)j(calculus.)270 2039 y(Natural)47
b(deduction)g(is)g(a)h(sligh)n(tly)d(parado)n(xical)j(system:)75
b(it)46 b(is)g(limited)e(to)j(the)167 2124 y(in)n(tuitionistic)22
b(case)k(\(in)f(the)h(classical)e(case)h(it)g(has)i(no)f(particularly)f
(go)r(o)r(d)i(prop)r(erties\))e(but)167 2209 y(it)30
b(is)h(only)h(satisfactory)g(for)g(the)g(\()p FJ(^)p
FH(;)12 b FJ(\))p FH(;)g FJ(8)p FK(\))30 b(fragmen)n(t)i(of)g(the)f
(language:)45 b(w)n(e)31 b(shall)g(defer)167 2294 y(consideration)e(of)
h FJ(_)f FK(and)h FJ(9)g FK(un)n(til)e(c)n(hapter)h FG(??)q
FK(.)40 b(Y)-6 b(et)29 b(disjunction)g(and)h(existence)e(are)h(the)167
2380 y(t)n(w)n(o)f(most)f FM(typic)l(al)t(ly)35 b FK(in)n(tuitionistic)
24 b(connectors!)270 2553 y(The)35 b(basic)g(idea)g(of)g(natural)h
(deduction)f(is)f(an)i(asymmetry:)50 b(a)36 b(pro)r(of)g(is)f(a)g(v)l
(aguely)167 2638 y(tree-lik)n(e)g(structure)i(\(this)g(view)g(is)f
(more)h(a)i(graphical)e(illusion)f(than)i(a)g(mathematical)167
2723 y(realit)n(y)-6 b(,)27 b(but)i(it)e(is)h(a)g(pleasan)n(t)h
(illusion\))d(with)h(one)i(or)f(more)g(h)n(yp)r(otheses)h(\(p)r
(ossibly)g(none\))167 2808 y(but)38 b(a)h(single)e(conclusion.)66
b(The)38 b(deep)g(symmetry)f(of)h(the)g(calculus)f(is)g(sho)n(wn)j(b)n
(y)e(the)167 2893 y FM(intr)l(o)l(duction)47 b FK(and)42
b FM(elimination)k FK(rules)40 b(whic)n(h)g(matc)n(h)g(eac)n(h)h(other)
g(exactly)-6 b(.)75 b(Observ)n(e,)167 2978 y(inciden)n(tally)-6
b(,)27 b(that)i(with)f(a)h(tree-lik)n(e)d(structure,)j(one)g(can)g(alw)
n(a)n(ys)h(decide)d(uniquely)i(what)167 3063 y(w)n(as)g(the)f
FM(last)35 b FK(rule)27 b(used,)i(whic)n(h)f(is)g(something)g(w)n(e)g
(could)g(not)h(sa)n(y)g(if)f(there)f(w)n(ere)h(sev)n(eral)167
3149 y(conclusions.)1402 4101 y(8)p eop
PStoPSsaved restore
%%Page: (12,13) 7
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
9 12 bop 167 167 a Fz(2.1.)64 b(THE)27 b(CALCULUS)1745
b FK(9)167 404 y FE(2.1)114 b(The)28 b(calculus)167 564
y FK(W)-6 b(e)27 b(shall)g(use)h(the)f(notation)1396
718 y FJ(\001)1396 756 y(\001)1396 794 y(\001)1380 851
y FH(A)167 1013 y FK(to)e(designate)f(a)h FM(de)l(duction)30
b FK(of)c FH(A)p FK(,)e(that)h(is,)g(ending)g(at)g FH(A)p
FK(.)34 b(The)24 b(deduction)h(will)c(b)r(e)k(written)167
1098 y(as)30 b(a)f(\014nite)g(tree,)f(and)i(in)f(particular,)g(the)f
(tree)g(will)f(ha)n(v)n(e)j(lea)n(v)n(es)f(lab)r(elled)e(b)n(y)i(sen)n
(tences.)167 1183 y(F)-6 b(or)28 b(these)f(sen)n(tences,)h(there)f(are)
g(t)n(w)n(o)h(p)r(ossible)f(states,)h FM(de)l(ad)36 b
FK(or)27 b FM(alive)p FK(.)270 1319 y(In)h(the)f(usual)h(state,)g(a)g
(sen)n(tence)f(is)g(aliv)n(e,)g(that)h(is)e(to)i(sa)n(y)h(it)d(tak)n
(es)j(an)f(activ)n(e)f(part)h(in)167 1404 y(the)33 b(pro)r(of:)47
b(w)n(e)32 b(sa)n(y)i(it)e(is)g(a)h FM(hyp)l(othesis)p
FK(.)53 b(The)33 b(t)n(ypical)f(case)h(is)f(illustrated)f(b)n(y)i(the)g
(\014rst)167 1489 y(rule)h(of)i(natural)g(deduction,)i(whic)n(h)d(allo)
n(ws)g(us)h(to)f(form)h(a)g(deduction)f(consisting)h(of)g(a)167
1574 y(single)27 b(sen)n(tence:)1380 1767 y FH(A)167
1959 y FK(Here)i FH(A)h FK(is)g(b)r(oth)i(the)e(leaf)g(and)h(the)g(ro)r
(ot;)h(logically)-6 b(,)30 b(w)n(e)g(deduce)g FH(A)p
FK(,)h(but)g(that)g(w)n(as)g(easy)167 2044 y(b)r(ecause)d
FH(A)f FK(w)n(as)h(assumed!)270 2180 y(No)n(w)c(a)g(sen)n(tence)f(at)g
(a)h(leaf)f(can)h(b)r(e)f(dead,)i(when)f(it)e(no)j(longer)e(pla)n(ys)h
(an)g(activ)n(e)f(part)h(in)167 2265 y(the)29 b(pro)r(of.)40
b(Dead)28 b(sen)n(tences)h(are)f(obtained)i(b)n(y)f(killing)e(liv)n(e)g
(ones.)40 b(The)28 b(t)n(ypical)g(example)167 2350 y(is)f(the)g
FJ(\))p FK(-in)n(tro)r(duction)h(rule:)1293 2516 y([)p
FH(A)p FK(])1329 2574 y FJ(\001)1329 2612 y(\001)1329
2650 y(\001)1310 2707 y FH(B)p 1230 2752 218 4 v 1468
2768 a FJ(\)I)1230 2829 y FH(A)19 b FJ(\))h FH(B)167
2990 y FK(It)28 b(m)n(ust)h(b)r(e)f(understo)r(o)r(d)i(th)n(us:)38
b(starting)29 b(from)g(a)g(deduction)f(of)h FH(B)t FK(,)g(in)f(whic)n
(h)g(w)n(e)g(c)n(ho)r(ose)167 3076 y(a)j(certain)f(n)n(um)n(b)r(er)h
(of)g(o)r(ccurrences)f(of)h FH(A)g FK(as)g FM(hyp)l(otheses)h
FK(\(the)f(n)n(um)n(b)r(er)f(is)h(arbitrary:)41 b(0,)167
3161 y(1,)30 b(250,)i(.)11 b(.)g(.)g(\),)31 b(w)n(e)e(form)h(a)f(new)g
(deduction)h(of)g(whic)n(h)e(the)h(conclusion)h(is)e
FH(A)21 b FJ(\))g FH(B)t FK(,)30 b(but)g(in)167 3246
y(whic)n(h)f(all)f(these)i(o)r(ccurrences)e(of)i FH(A)g
FK(ha)n(v)n(e)g(b)r(een)g FM(dischar)l(ge)l(d)p FK(,)h
FM(i.e.)f FK(killed.)39 b(There)29 b(ma)n(y)h(b)r(e)167
3331 y(other)e(o)r(ccurrences)e(of)j FH(A)e FK(whic)n(h)g(w)n(e)g(ha)n
(v)n(e)i(c)n(hosen)f(not)g(to)g(disc)n(harge.)270 3467
y(This)43 b(rule)g(illustrates)e(v)n(ery)i(w)n(ell)f(the)h(illusion)e
(of)j(the)f(tree-lik)n(e)e(notation:)67 b(it)43 b(is)167
3552 y(of)36 b(critical)d(imp)r(ortance)i(to)h(kno)n(w)h
FM(when)j FK(a)c(h)n(yp)r(othesis)h(w)n(as)f(disc)n(harged,)j(and)e(so)
f(it)f(is)167 3637 y(essen)n(tial)30 b(to)i(record)f(this.)46
b(But)31 b(if)g(w)n(e)g(do)h(this)f(in)g(the)g(example)f(ab)r(o)n(v)n
(e,)k(this)d(means)h(w)n(e)167 3722 y(ha)n(v)n(e)j(to)f(link)e(the)i
(crossed)g FH(A)f FK(with)g(the)g(line)f(of)i(the)g FJ(\)I)39
b FK(rule;)d(but)e(it)f(is)g(no)h(longer)g(a)167 3807
y(gen)n(uine)28 b(tree)e(w)n(e)h(are)h(considering!)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
10 13 bop 167 167 a FK(10)1139 b Fz(CHAPTER)27 b(2.)63
b(NA)-6 b(TURAL)26 b(DEDUCTION)167 404 y Fy(2.1.1)95
b(The)27 b(rules)270 537 y FJ(\017)34 b FM(Hyp)l(othesis)6
b FK(:)116 b FH(A)270 684 y FJ(\017)34 b FM(Intr)l(o)l(ductions)6
b FK(:)688 893 y FJ(\001)688 931 y(\001)688 969 y(\001)672
1026 y FH(A)889 893 y FJ(\001)889 931 y(\001)889 969
y(\001)870 1026 y FH(B)p 672 1070 255 4 v 947 1086 a
FJ(^I)706 1147 y FH(A)15 b FJ(^)h FH(B)1433 869 y FK([)p
FH(A)p FK(])1469 928 y FJ(\001)1469 966 y(\001)1469 1004
y(\001)1450 1061 y FH(B)p 1370 1106 218 4 v 1608 1122
a FJ(\)I)1370 1182 y FH(A)j FJ(\))h FH(B)2122 886 y FJ(\001)2122
924 y(\001)2122 962 y(\001)2106 1019 y FH(A)p 2054 1064
156 4 v 2230 1080 a FJ(8I)2054 1140 y(8)p FH(\030)s(:)12
b(A)270 1394 y FJ(\017)34 b FM(Eliminations)6 b FK(:)581
1576 y FJ(\001)581 1614 y(\001)581 1652 y(\001)498 1709
y FH(A)16 b FJ(^)f FH(B)p 498 1754 186 4 v 705 1770 a
FJ(^)p FK(1)p FJ(E)565 1831 y FH(A)1072 1576 y FJ(\001)1072
1614 y(\001)1072 1652 y(\001)988 1709 y FH(A)h FJ(^)f
FH(B)p 988 1754 V 1195 1770 a FJ(^)p FK(2)p FJ(E)1053
1831 y FH(B)1495 1576 y FJ(\001)1495 1614 y(\001)1495
1652 y(\001)1479 1709 y FH(A)1776 1576 y FJ(\001)1776
1614 y(\001)1776 1652 y(\001)1678 1709 y FH(A)k FJ(\))h
FH(B)p 1479 1754 417 4 v 1916 1770 a FJ(\)E)1659 1831
y FH(B)2276 1567 y FJ(\001)2276 1605 y(\001)2276 1643
y(\001)2208 1701 y(8)p FH(\030)s(:)12 b(A)p 2188 1746
195 4 v 2403 1762 a FJ(8E)2188 1822 y FH(A)p FK([)p FH(a=\030)s
FK(])339 2021 y(The)28 b(rule)e FJ(\))q(E)34 b FK(is)27
b(traditionally)f(called)g FM(mo)l(dus)j(p)l(onens)p
FK(.)167 2232 y(Some)f(remarks:)270 2318 y(All)40 b(the)g(rules,)k
(except)d FJ(\)I)5 b FK(,)45 b(preserv)n(e)c(the)g(sto)r(c)n(k)g(of)h
(h)n(yp)r(otheses:)63 b(for)42 b(example,)167 2403 y(the)32
b(h)n(yp)r(otheses)i(in)d(the)h(deduction)h(ab)r(o)n(v)n(e)g(whic)n(h)f
(ends)h(in)f FJ(\)E)6 b FK(,)34 b(are)e(those)h(of)g(the)f(t)n(w)n(o)
167 2489 y(immediate)25 b(sub-deductions.)270 2575 y(F)-6
b(or)28 b(w)n(ell-kno)n(wn)d(logical)h(reasons,)i(it)d(is)h(necessary)h
(to)g(restrict)e FJ(8I)31 b FK(to)c(the)f(case)h(where)167
2660 y(the)j(v)l(ariable)529 2635 y FD(1)587 2660 y FH(\030)k
FK(is)c(not)h(free)f(in)g(an)n(y)h(h)n(yp)r(othesis)h(\(it)d(ma)n(y)-6
b(,)32 b(on)f(the)f(other)h(hand,)i(b)r(e)d(free)167
2745 y(in)d(a)h(dead)g(leaf)5 b(\).)270 2832 y(The)47
b(fundamen)n(tal)g(symmetry)e(of)j(the)e(system)g(is)g(the)g
FM(intr)l(o)l(duction/elimination)167 2917 y FK(symmetry)-6
b(,)44 b(whic)n(h)c(replaces)g(the)g FM(hyp)l(othesis/c)l(onclusion)48
b FK(symmetry)40 b(that)h(cannot)h(b)r(e)167 3002 y(implemen)n(ted)25
b(in)i(this)g(con)n(text.)167 3242 y FE(2.2)114 b(Computational)29
b(signi\014cance)167 3399 y FK(W)-6 b(e)20 b(shall)f(re-examine)f(the)h
(natural)i(deduction)e(system)h(in)f(the)h(ligh)n(t)e(of)j(Heyting)e
(seman)n(tics;)167 3484 y(w)n(e)32 b(shall)g(supp)r(ose)i(\014xed)f
(the)f(in)n(terpretation)g(of)h(atomic)e(form)n(ulae)i(and)g(also)g
(the)f(range)167 3570 y(of)d(the)f(quan)n(ti\014ers.)38
b(A)28 b(form)n(ula)g FH(A)g FK(will)e(b)r(e)i(seen)g(as)h(the)f(set)g
(of)h(its)e(p)r(ossible)h(deductions;)167 3655 y(instead)g(of)g(sa)n
(ying)g(\\)p FH(\016)j FK(pro)n(v)n(es)d FH(A)p FK(",)g(w)n(e)f(shall)g
(sa)n(y)i(\\)p FH(\016)22 b FJ(2)e FH(A)p FK(".)p 167
3719 1003 3 v 246 3763 a FC(1)273 3784 y FB(The)28 b(v)m(ariable)e
Fx(\030)31 b FB(b)r(elongs)c(to)h(the)f FA(obje)m(ct)i(language)f
FB(\(it)f(ma)n(y)h(stand)g(for)g(a)g(n)n(um)n(b)r(er,)g(a)g
(data-record,)167 3854 y(an)i(ev)n(en)n(t\).)43 b(W)-5
b(e)31 b(reserv)n(e)g Fx(x)p FB(,)g Fx(y)r FB(,)g Fx(z)i
FB(for)e Fx(\025)p FB(-calculus)e(v)m(ariables,)h(whic)n(h)f(w)n(e)j
(shall)d(in)n(tro)r(duce)g(in)g(the)h(next)167 3925 y(section.)p
eop
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PStoPSxform concat
11 14 bop 167 167 a Fz(2.2.)64 b(COMPUT)-6 b(A)g(TIONAL)26
b(SIGNIFICANCE)1093 b FK(11)270 404 y(The)33 b(rules)e(of)j(natural)f
(deduction)f(then)h(app)r(ear)g(as)g(a)h(sp)r(ecial)d(w)n(a)n(y)i(of)g
(constructing)167 490 y(functions:)g(a)22 b(deduction)f(of)i
FH(A)e FK(on)h(the)g(h)n(yp)r(otheses)h FH(B)1630 501
y FD(1)1658 490 y FH(;)12 b(:)g(:)g(:)g(;)g(B)1865 501
y Fw(n)1920 490 y FK(can)22 b(b)r(e)f(seen)h(as)g(a)g(function)167
575 y FH(t)p FK([)p FH(x)250 586 y FD(1)278 575 y FH(;)12
b(:)g(:)g(:)g(;)g(x)472 586 y Fw(n)505 575 y FK(])25
b(whic)n(h)f(asso)r(ciates)h(to)h(elemen)n(ts)c FH(b)1466
586 y Fw(i)1506 575 y FJ(2)e FH(B)1625 586 y Fw(i)1670
575 y FK(a)25 b(result)f FH(t)p FK([)p FH(b)1996 586
y FD(1)2024 575 y FH(;)12 b(:)g(:)g(:)g(;)g(b)2208 586
y Fw(n)2241 575 y FK(])20 b FJ(2)g FH(A)p FK(.)34 b(In)25
b(fact,)167 660 y(for)31 b(this)g(corresp)r(ondence)g(to)h(b)r(e)f
(exact,)h(one)g(has)g(to)f(w)n(ork)h(with)e FM(p)l(ar)l(c)l(els)i(of)g
(hyp)l(otheses)6 b FK(:)167 745 y(the)30 b(same)g(form)n(ula)g
FH(B)k FK(ma)n(y)d(in)f(general)g(app)r(ear)h(sev)n(eral)f(times)e
(among)k(the)e(h)n(yp)r(otheses,)167 830 y(and)c(t)n(w)n(o)f(o)r
(ccurrences)f(of)i FH(B)i FK(in)d(the)f(same)h(parcel)f(will)e(corresp)
r(ond)k(to)f(the)g(same)f(v)l(ariable.)270 959 y(This)43
b(is)g(a)h(little)c(m)n(ysterious,)47 b(but)d(it)e(will)f(quic)n(kly)i
(b)r(ecome)f(clearer)g(with)g(some)167 1044 y(examples.)167
1251 y Fy(2.2.1)95 b(In)m(terpretation)28 b(of)f(the)h(rules)251
1382 y FK(1.)35 b(A)46 b(deduction)g(consisting)f(of)i(a)f(single)f(h)n
(yp)r(othesis)i FH(A)e FK(is)h(represen)n(ted)f(b)n(y)h(the)339
1468 y(expression)32 b FH(x)p FK(,)g(where)f FH(x)g FK(is)f(a)i(v)l
(ariable)f(for)g(an)h(elemen)n(t)d(of)j FH(A)p FK(.)46
b(Later,)32 b(if)e(w)n(e)h(ha)n(v)n(e)339 1553 y(other)37
b(o)r(ccurrences)e(of)i FH(A)p FK(,)h(w)n(e)e(shall)g(c)n(ho)r(ose)h
(the)f(same)g FH(x)p FK(,)j(or)d(another)h(v)l(ariable,)339
1638 y(dep)r(ending)h(up)r(on)h(whether)e(or)h(not)h(those)f(other)f(o)
r(ccurrences)g(are)h(in)f(the)g(same)339 1723 y(parcel.)251
1869 y(2.)e(If)52 b(a)g(deduction)f(has)i(b)r(een)e(obtained)h(b)n(y)h
(means)e(of)i FJ(^I)k FK(from)51 b(t)n(w)n(o)h(others)339
1954 y(corresp)r(onding)39 b(to)f FH(u)p FK([)p FH(x)993
1965 y FD(1)1021 1954 y FH(;)12 b(:)g(:)g(:)g(;)g(x)1215
1965 y Fw(n)1248 1954 y FK(])38 b(and)h FH(v)s FK([)p
FH(x)1548 1965 y FD(1)1576 1954 y FH(;)12 b(:)g(:)g(:)f(;)h(x)1769
1965 y Fw(n)1803 1954 y FK(],)40 b(then)e(w)n(e)f(asso)r(ciate)h(to)g
(our)339 2039 y(deduction)23 b(the)f(pair)f FJ(h)p FH(u)p
FK([)p FH(x)1039 2050 y FD(1)1068 2039 y FH(;)12 b(:)g(:)g(:)g(;)g(x)
1262 2050 y Fw(n)1295 2039 y FK(])p FH(;)g(v)s FK([)p
FH(x)1439 2050 y FD(1)1467 2039 y FH(;)g(:)g(:)g(:)g(;)g(x)1661
2050 y Fw(n)1694 2039 y FK(])p FJ(i)p FK(,)24 b(since)d(a)i(pro)r(of)g
(of)g(a)g(conjunction)339 2124 y(is)34 b(a)h FM(p)l(air)p
FK(.)57 b(W)-6 b(e)34 b(ha)n(v)n(e)h(made)f FH(u)h FK(and)g
FH(v)j FK(dep)r(end)c(on)i(the)e(same)g(v)l(ariables;)k(indeed,)339
2209 y(the)f(c)n(hoice)g(of)g(v)l(ariables)g(of)h FH(u)f
FK(and)i FH(v)h FK(is)c(correlated,)j(b)r(ecause)e(some)g(parcels)g(of)
339 2294 y(h)n(yp)r(otheses)29 b(will)c(b)r(e)j(iden)n(ti\014ed.)251
2440 y(3.)35 b(If)46 b(a)g(deduction)g(ends)g(in)g FJ(^)p
FK(1)p FJ(E)6 b FK(,)51 b(and)c FH(t)p FK([)p FH(x)1570
2451 y FD(1)1598 2440 y FH(;)12 b(:)g(:)g(:)g(;)g(x)1792
2451 y Fw(n)1825 2440 y FK(])46 b(w)n(as)g(asso)r(ciated)g(with)f(the)
339 2525 y(immediate)f(sub-deduction,)51 b(then)46 b(w)n(e)f(shall)h
(asso)r(ciate)g FH(\031)2035 2500 y FD(1)2062 2525 y
FH(t)p FK([)p FH(x)2145 2536 y FD(1)2173 2525 y FH(;)12
b(:)g(:)g(:)g(;)g(x)2367 2536 y Fw(n)2400 2525 y FK(])46
b(to)h(our)339 2610 y(pro)r(of.)c(That)30 b(is)f(the)h
FM(\014rst)f(pr)l(oje)l(ction)p FK(,)j(since)c FH(t)p
FK(,)j(as)f(a)g(pro)r(of)h(of)f(a)g(conjunction,)h(has)339
2695 y(to)d(b)r(e)g(a)g(pair.)35 b(Lik)n(ewise,)26 b(the)h
FJ(^)p FK(2)p FJ(E)35 b FK(rule)26 b(in)n(v)n(olv)n(es)i(the)f
FM(se)l(c)l(ond)i(pr)l(oje)l(ction)k FH(\031)2459 2670
y FD(2)2487 2695 y FK(.)339 2853 y(Although)46 b(this)e(is)h(not)g(v)n
(ery)g(formal,)k(it)44 b(will)e(b)r(e)j(necessary)h(to)f(consider)f
(the)339 2939 y(fundamen)n(tal)29 b(equations:)637 3146
y FH(\031)679 3120 y FD(1)707 3146 y FJ(h)p FH(u;)12
b(v)s FJ(i)20 b FK(=)f FH(u)299 b(\031)1340 3120 y FD(2)1367
3146 y FJ(h)p FH(u;)12 b(v)s FJ(i)20 b FK(=)g FH(v)300
b FJ(h)p FH(\031)2023 3120 y FD(1)2051 3146 y FH(t;)12
b(\031)2149 3120 y FD(2)2177 3146 y FH(t)p FJ(i)20 b
FK(=)f FH(t)339 3353 y FK(These)25 b(equations)h(\(and)f(the)g(similar)
d(ones)k(w)n(e)e(shall)g(ha)n(v)n(e)i(o)r(ccasion)f(to)g(write)e(do)n
(wn\))339 3438 y(are)28 b(the)f(essence)g(of)h(the)g(corresp)r(ondence)
g(b)r(et)n(w)n(een)f(logic)f(and)j(computer)e(science.)251
3584 y(4.)35 b(If)26 b(a)g(deduction)g(ends)h(in)e FJ(\)I)5
b FK(,)27 b(let)e FH(v)j FK(b)r(e)e(the)g(term)e(asso)r(ciated)i(with)f
(the)h(immediate)339 3669 y(sub-deduction;)39 b(this)33
b(immediate)f(sub-deduction)j(is)e(unam)n(biguously)j(determined)339
3754 y(at)d(the)g(lev)n(el)e(of)i(parcels)f(of)h(h)n(yp)r(otheses,)j(b)
n(y)d(sa)n(ying)h(that)f(a)h(whole)e FH(A)p FK(-parcel)f(has)339
3840 y(b)r(een)h(disc)n(harged.)48 b(If)32 b FH(x)g FK(is)f(a)h(v)l
(ariable)g(asso)r(ciated)g(to)g(this)f(parcel,)h(then)g(w)n(e)f(ha)n(v)
n(e)339 3925 y(a)e(function)g FH(v)s FK([)p FH(x;)12
b(x)842 3936 y FD(1)870 3925 y FH(;)g(:)g(:)g(:)g(;)g(x)1064
3936 y Fw(n)1097 3925 y FK(].)38 b(W)-6 b(e)28 b(shall)g(asso)r(ciate)h
(to)f(our)h(deduction)g(the)f(function)p eop
PStoPSsaved restore
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PStoPSmatrix setmatrix
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PStoPSxform concat
12 15 bop 167 167 a FK(12)1139 b Fz(CHAPTER)27 b(2.)63
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415 y Fw(n)677 404 y FK(])42 b(whic)n(h)f(maps)g(eac)n(h)h(argumen)n(t)
g FH(a)f FK(of)h FH(A)f FK(to)h FH(v)s FK([)p FH(a;)12
b(x)2184 415 y FD(1)2212 404 y FH(;)g(:)g(:)g(:)f(;)h(x)2405
415 y Fw(n)2439 404 y FK(].)76 b(The)339 490 y(notation)29
b(is)e FH(\025x:)12 b(v)s FK([)p FH(x;)g(x)968 501 y
FD(1)996 490 y FH(;)g(:)g(:)g(:)g(;)g(x)1190 501 y Fw(n)1223
490 y FK(])28 b(in)f(whic)n(h)g FH(x)h FK(is)f(b)r(ound.)339
604 y(Observ)n(e)h(that)g FM(binding)f FK(corresp)r(onds)i(to)e
FM(dischar)l(ge)p FK(.)251 748 y(5.)35 b(The)30 b(case)f(of)h(a)g
(deduction)g(ending)g(with)f FJ(\)E)36 b FK(is)29 b(treated)g(b)n(y)h
(considering)f(the)h(t)n(w)n(o)339 833 y(functions)40
b FH(t)p FK([)p FH(x)736 844 y FD(1)764 833 y FH(;)12
b(:)g(:)g(:)g(;)g(x)958 844 y Fw(n)991 833 y FK(])39
b(and)h FH(u)p FK([)p FH(x)1296 844 y FD(1)1325 833 y
FH(;)12 b(:)g(:)g(:)f(;)h(x)1518 844 y Fw(n)1552 833
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918 y(sub-deductions.)i(F)-6 b(or)30 b(\014xed)g(v)l(alues)f(of)h
FH(x)1485 929 y FD(1)1513 918 y FH(;)12 b(:)g(:)g(:)f(;)h(x)1706
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FH(A)g FK(to)g FH(B)t FK(,)339 1003 y(and)g FH(u)f FK(is)f(an)h(elemen)
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FK([)p FH(x)1205 1217 y FD(1)1233 1206 y FH(;)12 b(:)g(:)g(:)f(;)h(x)
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1217 y FD(1)1616 1206 y FH(;)g(:)g(:)g(:)g(;)g(x)1810
1217 y Fw(n)1843 1206 y FK(])339 1408 y(represen)n(ts)28
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1760 y(\()p FH(\025x:)12 b(v)s FK(\))g FH(u)58 b FK(=)h
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FH(t)p FK(\))270 2090 y(The)38 b(rules)f(for)h FJ(8)f
FK(ec)n(ho)h(those)h(for)f FJ(\))p FK(:)56 b(they)38
b(do)h(not)f(add)h(m)n(uc)n(h,)i(so)d(w)n(e)g(shall)f(in)167
2175 y(future)30 b(omit)e(them)h(from)g(our)h(discussion.)42
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2260 y(the)41 b(b)r(oring)i(\014rst-order)e(quan)n(ti\014er)h(b)n(y)h
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PStoPSxform concat
13 16 bop 167 167 a Fz(2.2.)64 b(COMPUT)-6 b(A)g(TIONAL)26
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e(lead)h(to)g(equations)167 620 y(b)r(et)n(w)n(een)e(deductions.)36
b(F)-6 b(or)28 b(example:)1039 810 y FJ(\001)1039 848
y(\001)1039 886 y(\001)1023 943 y FH(A)1240 810 y FJ(\001)1240
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255 4 v 1298 1003 a FJ(^I)1057 1064 y FH(A)15 b FJ(^)h
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969 y(\001)1934 1007 y(\001)1918 1064 y FH(A)1039 1397
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a FJ(^I)1057 1651 y FH(A)f FJ(^)h FH(B)p 1057 1696 186
4 v 1264 1712 a FJ(^)p FK(2)p FJ(E)1122 1772 y FH(B)1526
1513 y FK(\\equals")1936 1518 y FJ(\001)1936 1556 y(\001)1936
1594 y(\001)1918 1651 y FH(B)854 2175 y FJ(\001)854 2213
y(\001)854 2251 y(\001)838 2308 y FH(A)1100 1995 y FK([)p
FH(A)p FK(])1136 2054 y FJ(\001)1136 2092 y(\001)1136
2130 y(\001)1117 2187 y FH(B)p 1037 2232 218 4 v 1275
2248 a FJ(\)I)1037 2308 y FH(A)j FJ(\))h FH(B)p 838 2353
417 4 v 1275 2369 a FJ(\)E)1018 2430 y FH(B)1526 2135
y FK(\\equals")1936 2053 y FJ(\001)1936 2091 y(\001)1936
2129 y(\001)1920 2186 y FH(A)1936 2227 y FJ(\001)1936
2265 y(\001)1936 2303 y(\001)1918 2360 y FH(B)167 2613
y FK(What)30 b(w)n(e)f(ha)n(v)n(e)h(written)e(is)h(clear,)f(pro)n
(vided)i(that)g(w)n(e)f(observ)n(e)h(carefully)e(what)i(happ)r(ens)167
2699 y(in)35 b(the)g(last)g(case:)51 b FM(al)t(l)42 b
FK(the)35 b(disc)n(harged)h(h)n(yp)r(otheses)h(are)e(replaced)g(b)n(y)h
(\(copies)f(of)5 b(\))35 b(the)167 2784 y(deduction)28
b(ending)f(in)h FH(A)p FK(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
14 17 bop 167 874 a FF(Chapter)28 b(3)167 1168 y(The)g(Curry-Ho)-5
b(w)g(ard)28 b(Isomorphism)167 1487 y FK(W)-6 b(e)27
b(ha)n(v)n(e)i(seen)f(that)g(Heyting's)g(ideas)f(p)r(erform)g(v)n(ery)h
(w)n(ell)e(in)h(the)h(framew)n(ork)f(of)i(natural)167
1573 y(deduction.)36 b(W)-6 b(e)27 b(shall)g(exploit)g(this)g(remark)h
(b)n(y)g(establishing)f(a)i FM(formal)f FK(system)f(of)i(t)n(yp)r(ed)
167 1658 y(terms)39 b(for)i(discussing)g(the)f(functional)g(ob)t(jects)
g(whic)n(h)g(lie)f(b)r(ehind)h(the)g(pro)r(ofs.)75 b(The)167
1743 y(signi\014cance)30 b(of)h(the)f(system)g(will)d(b)r(e)j(giv)n(en)
h(b)n(y)g(means)f(of)h(the)f(functional)g(equations)i(w)n(e)167
1828 y(ha)n(v)n(e)h(written)d(do)n(wn.)50 b(In)32 b(fact,)h(these)f
(equations)h(ma)n(y)f(b)r(e)g(read)g(in)g(t)n(w)n(o)g(di\013eren)n(t)f
(w)n(a)n(ys,)167 1913 y(whic)n(h)c(re-iterate)e(the)j(dic)n(hotom)n(y)g
(b)r(et)n(w)n(een)f(sense)g(and)i(denotation:)270 2074
y FJ(\017)34 b FK(as)41 b(the)f FM(e)l(quations)g FK(whic)n(h)f
(de\014ne)h(the)g(equalit)n(y)g(of)g(terms,)i(in)e(other)g(w)n(ords)g
(the)339 2159 y(equalit)n(y)27 b(of)i(denotations)f(\(the)f
FM(static)h FK(viewp)r(oin)n(t\).)270 2303 y FJ(\017)34
b FK(as)27 b FM(r)l(ewrite)f FK(rules)g(whic)n(h)f(allo)n(ws)h(us)h(to)
f(calculate)f(terms)g(b)n(y)i(reduction)f(to)h(a)f(normal)339
2388 y(form.)64 b(That)37 b(is)f(an)i(op)r(erational,)i
FM(dynamic)d FK(viewp)r(oin)n(t,)i(the)e(only)g(truly)f(fruitful)339
2474 y(view)27 b(for)h(this)f(asp)r(ect)h(of)g(logic.)270
2635 y(Of)37 b(course)g(the)f(second)i(viewp)r(oin)n(t)e(is)g
(under-dev)n(elop)r(ed)h(b)n(y)g(comparison)g(with)f(the)167
2720 y(\014rst)44 b(one,)49 b(as)44 b(w)n(as)h(the)f(case)g(in)f
(Logic!)84 b(F)-6 b(or)45 b(example)e FM(denotational)h
FK(seman)n(tics)f(of)167 2805 y(programs)c(\(Scott's)e(seman)n(tics,)j
(for)d(example\))g(ab)r(ound:)57 b(for)38 b(this)f(kind)h(of)g(seman)n
(tics,)167 2890 y(nothing)h(c)n(hanges)g(throughout)i(the)d(execution)g
(of)g(a)h(program.)68 b(On)38 b(the)g(other)h(hand,)167
2975 y(there)d(is)h(hardly)h(an)n(y)g(civilised)c FM(op)l(er)l(ational)
39 b FK(seman)n(tics)d(of)i(programs)g(\(w)n(e)e(exclude)h
FM(ad)167 3060 y(ho)l(c)h FK(seman)n(tics)e(whic)n(h)h(crudely)g
(paraphrase)i(the)e(steps)h(to)n(w)n(ard)g(normalisation\).)64
b(The)167 3146 y(establishmen)n(t)28 b(of)j(a)f(truly)f(op)r(erational)
h(seman)n(tics)f(of)h(algorithms)g(is)f(p)r(erhaps)i(the)e(most)167
3231 y(imp)r(ortan)n(t)e(problem)g(in)g(computer)g(science.)270
3357 y(The)h(corresp)r(ondence)g(b)r(et)n(w)n(een)f(t)n(yp)r(es)h(and)g
(prop)r(ositions)g(w)n(as)h(set)e(out)h(in)f([)p FG(?)q
FK(].)1385 4101 y(14)p eop
PStoPSsaved restore
%%Page: (18,19) 10
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
15 18 bop 167 167 a Fz(3.1.)64 b(LAMBD)n(A)27 b(CALCULUS)1549
b FK(15)167 404 y FE(3.1)114 b(Lam)m(b)s(da)28 b(Calculus)167
601 y Fy(3.1.1)95 b(T)m(yp)s(es)167 753 y FK(When)48
b(w)n(e)f(think)g(of)h(pro)r(ofs)h(in)e(the)g(spirit)f(of)i(Heyting,)k
(form)n(ulae)47 b(b)r(ecome)g FM(typ)l(es)p FK(.)167
839 y(Sp)r(eci\014cally:)251 1057 y(1.)35 b(A)n(tomic)26
b(t)n(yp)r(es)i FH(T)812 1068 y FD(1)841 1057 y FH(;)12
b(:)g(:)g(:)f(;)h(T)1035 1068 y Fw(n)1096 1057 y FK(are)28
b(t)n(yp)r(es.)251 1236 y(2.)35 b(If)28 b FH(U)35 b FK(and)28
b FH(V)44 b FK(are)27 b(t)n(yp)r(es,)h(then)g FH(U)7
b FJ(\002)p FH(V)43 b FK(and)29 b FH(U)7 b FJ(!)p FH(V)43
b FK(are)28 b(t)n(yp)r(es.)251 1415 y(3.)35 b(The)j(only)h(t)n(yp)r(es)
f(are)h(\(for)f(the)g(time)e(b)r(eing\))i(those)h(obtained)f(b)n(y)h
(means)g(of)f FG(??)339 1500 y FK(and)29 b FG(??)q FK(.)270
1718 y(This)40 b(corresp)r(onds)g(to)g(the)g(\()p FJ(^)p
FH(;)12 b FJ(\))p FK(\))39 b(fragmen)n(t)h(of)g(prop)r(ositional)g
(calculus:)58 b(atomic)167 1803 y(prop)r(ositions)43
b(are)g(written)e FH(T)1012 1814 y Fw(i)1032 1803 y FK(,)47
b(\\)p FJ(^)p FK(")c(b)r(ecomes)f(\\)p FJ(\002)q FK(")h(\(Cartesian)f
(pro)r(duct\))h(and)h(\\)p FJ(\))p FK(")167 1888 y(b)r(ecomes)27
b(\\)p FJ(!)q FK(".)167 2143 y Fy(3.1.2)95 b(T)-8 b(erms)167
2295 y FK(Pro)r(ofs)27 b(b)r(ecome)f FM(terms)6 b FK(;)26
b(more)g(precisely)-6 b(,)25 b(a)i(pro)r(of)g(of)g FH(A)f
FK(\(as)h(a)g(form)n(ula\))f(b)r(ecomes)f(a)i FM(term)167
2380 y(of)h(typ)l(e)g FH(A)g FK(\(as)g(a)g(t)n(yp)r(e\).)35
b(Sp)r(eci\014cally:)251 2598 y(1.)g(The)28 b(v)l(ariables)f
FH(x)815 2573 y Fw(T)815 2616 y FD(0)854 2598 y FH(;)12
b(:)g(:)g(:)g(;)g(x)1048 2573 y Fw(T)1048 2616 y(n)1087
2598 y FH(;)g(:)g(:)g(:)27 b FK(are)h(terms)e(of)j(t)n(yp)r(e)e
FH(T)10 b FK(.)251 2778 y(2.)35 b(If)27 b FH(u)h FK(and)g
FH(v)i FK(are)c(terms)g(of)i(t)n(yp)r(es)f(resp)r(ectiv)n(ely)e
FH(U)34 b FK(and)28 b FH(V)16 b FK(,)27 b(then)g FJ(h)p
FH(u;)12 b(v)s FJ(i)28 b FK(is)e(a)i(term)e(of)339 2863
y(t)n(yp)r(e)i FH(U)7 b FJ(\002)p FH(V)16 b FK(.)251
3042 y(3.)35 b(If)27 b FH(t)f FK(is)g(a)h(term)e(of)i(t)n(yp)r(e)g
FH(U)7 b FJ(\002)p FH(V)42 b FK(then)27 b FH(\031)1400
3016 y FD(1)1427 3042 y FH(t)g FK(and)g FH(\031)1658
3016 y FD(2)1686 3042 y FH(t)f FK(are)h(terms)e(of)i(t)n(yp)r(es)g
(resp)r(ectiv)n(ely)339 3127 y FH(U)35 b FK(and)29 b
FH(V)15 b FK(.)251 3306 y(4.)35 b(If)25 b FH(v)i FK(is)d(a)h(term)e(of)
i(t)n(yp)r(e)g FH(V)40 b FK(and)26 b FH(x)1258 3281 y
Fw(U)1258 3324 y(n)1324 3306 y FK(is)e(a)h(v)l(ariable)f(of)h(t)n(yp)r
(e)g FH(U)32 b FK(then)24 b FH(\025x)2267 3281 y Fw(U)2267
3324 y(n)2309 3306 y FH(:)12 b(v)28 b FK(is)23 b(a)i(term)339
3391 y(of)h(t)n(yp)r(e)g FH(U)7 b FJ(!)q FH(V)15 b FK(.)35
b(In)26 b(general)f(w)n(e)g(shall)f(supp)r(ose)j(that)f(w)n(e)f(ha)n(v)
n(e)i(settled)d(questions)i(of)339 3476 y(the)j(c)n(hoice)g(of)h(b)r
(ound)h(v)l(ariables)e(and)h(of)g(substitution,)g(b)n(y)g(some)f(means)
g(or)h(other,)339 3562 y(whic)n(h)h(allo)n(ws)g(us)i(to)e(disregard)h
(the)g(names)g(of)g(b)r(ound)h(v)l(ariables,)g(the)e(idea)g(b)r(eing)
339 3647 y(that)d(a)h(b)r(ound)g(v)l(ariable)e(has)i(no)f(individualit)
n(y)-6 b(.)251 3826 y(5.)35 b(If)c FH(t)f FK(and)h FH(u)g
FK(are)f(terms)f(of)i(t)n(yp)r(es)g(resp)r(ectiv)n(ely)d
FH(U)7 b FJ(!)q FH(V)46 b FK(and)31 b FH(U)7 b FK(,)31
b(then)g FH(t)12 b(u)30 b FK(is)g(a)g(term)339 3911 y(of)f(t)n(yp)r(e)e
FH(V)16 b FK(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
16 19 bop 167 167 a FK(16)639 b Fz(CHAPTER)26 b(3.)63
b(THE)27 b(CURR)-6 b(Y-HO)n(W)e(ARD)26 b(ISOMORPHISM)167
404 y FE(3.2)114 b(Denotational)30 b(signi\014cance)167
567 y FK(T)n(yp)r(es)k(represen)n(t)f(the)h(kind)g(of)g(ob)t(ject)g
(under)g(discussion.)54 b(F)-6 b(or)34 b(example)f(an)i(ob)t(ject)f(of)
167 652 y(t)n(yp)r(e)28 b FH(U)7 b FJ(!)p FH(V)43 b FK(is)27
b(a)h(function)g(from)f FH(U)35 b FK(to)28 b FH(V)15
b FK(,)28 b(and)h(an)f(ob)t(ject)f(of)h(t)n(yp)r(e)g
FH(U)7 b FJ(\002)p FH(V)43 b FK(is)27 b(an)h(ordered)167
737 y(pair)34 b(consisting)g(of)h(an)g(ob)t(ject)f(of)h
FH(U)42 b FK(and)35 b(an)h(ob)t(ject)e(of)h FH(V)15 b
FK(.)56 b(The)34 b(meaning)g(of)h(atomic)167 822 y(t)n(yp)r(es)28
b(is)f(not)h(imp)r(ortan)n(t)f(|)h(it)f(dep)r(ends)h(on)g(the)g(con)n
(text.)270 1024 y(The)42 b(terms)f(follo)n(w)h(v)n(ery)g(precisely)e
(the)i(\014v)n(e)h(sc)n(hemes)e(whic)n(h)g(w)n(e)h(ha)n(v)n(e)h(used)g
(for)167 1109 y(Heyting)27 b(seman)n(tics)g(and)h(natural)g(deduction.)
251 1290 y(1.)35 b(A)28 b(v)l(ariable)g FH(x)722 1265
y Fw(T)790 1290 y FK(of)h(t)n(yp)r(e)f FH(T)38 b FK(represen)n(ts)28
b(an)n(y)i(term)d FH(t)h FK(of)h(t)n(yp)r(e)f FH(T)39
b FK(\(pro)n(vided)28 b(that)h FH(x)2633 1265 y Fw(T)339
1375 y FK(is)e(replaced)g(b)n(y)h FH(t)p FK(\).)251 1531
y(2.)35 b FJ(h)p FH(u;)12 b(v)s FJ(i)29 b FK(is)e(the)g(ordered)h(pair)
f(of)h FH(u)g FK(and)h FH(v)s FK(.)251 1687 y(3.)35 b
FH(\031)381 1661 y FD(1)409 1687 y FH(t)27 b FK(and)i
FH(\031)642 1661 y FD(2)670 1687 y FH(t)e FK(are)h(resp)r(ectiv)n(ely)d
(the)j(\014rst)f(and)i(second)f(pro)t(jection)f(of)i
FH(t)p FK(.)251 1843 y(4.)35 b FH(\025x)418 1817 y Fw(U)461
1843 y FH(:)12 b(v)30 b FK(is)e(the)g(function)g(whic)n(h)g(to)g(an)n
(y)h FH(u)g FK(of)g(t)n(yp)r(e)f FH(U)35 b FK(asso)r(ciates)29
b FH(v)s FK([)p FH(u=x)p FK(],)g(that)g(is)e FH(v)339
1928 y FK(in)g(whic)n(h)g FH(x)665 1902 y Fw(U)735 1928
y FK(is)g(regarded)h(as)g(an)h(abbreviation)f(for)g FH(u)p
FK(.)251 2084 y(5.)35 b FH(t)12 b(u)28 b FK(is)f(the)g(result)g(of)h
(applying)g(the)g(function)f FH(t)h FK(to)g(the)f(argumen)n(t)h
FH(u)p FK(.)270 2315 y(Denotationally)-6 b(,)28 b(w)n(e)f(ha)n(v)n(e)i
(the)e(follo)n(wing)g(\()p FM(primary)p FK(\))h(equations)466
2518 y FH(\031)508 2492 y FD(1)535 2518 y FJ(h)p FH(u;)12
b(v)s FJ(i)21 b FK(=)e FH(u)299 b(\031)1169 2492 y FD(2)1197
2518 y FJ(h)p FH(u;)12 b(v)s FJ(i)20 b FK(=)f FH(v)302
b FK(\()p FH(\025x)1891 2492 y Fw(U)1933 2518 y FH(:)12
b(v)s FK(\))p FH(u)19 b FK(=)h FH(v)s FK([)p FH(u=x)p
FK(])167 2720 y(together)28 b(with)e(the)i FM(se)l(c)l(ondary)h
FK(equations)556 2922 y FJ(h)p FH(\031)625 2896 y FD(1)653
2922 y FH(t;)12 b(\031)751 2896 y FD(2)778 2922 y FH(t)p
FJ(i)20 b FK(=)f FH(t)389 b(\025x)1416 2896 y Fw(U)1458
2922 y FH(:)12 b(t)g(x)20 b FK(=)f FH(t)78 b FK(\()p
FH(x)28 b FK(not)g(free)f(in)g FH(t)p FK(\))167 3124
y(whic)n(h)g(ha)n(v)n(e)i(nev)n(er)e(b)r(een)g(giv)n(en)h(adequate)h
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(these)e(equations)h(is)f(consisten)n(t)h(and)g(decidable.)270
3529 y(By)33 b FM(c)l(onsistent)p FK(,)h(w)n(e)e(mean)g(that)h(the)g
(equalit)n(y)f FH(x)23 b FK(=)f FH(y)s FK(,)34 b(where)d
FH(x)i FK(and)h FH(y)g FK(are)f(distinct)167 3614 y(v)l(ariables,)28
b(cannot)g(b)r(e)g(pro)n(v)n(ed.)270 3754 y(Although)45
b(this)f(result)e(holds)j(for)f(the)g(whole)f(set)h(of)h(equations,)k
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(three.)46 b(It)32 b(is)f(a)h(consequence)g(of)g(the)f
FM(Chur)l(ch-R)l(osser)i(pr)l(op)l(erty)40 b FK(and)167
3925 y(the)27 b FM(normalisation)i(the)l(or)l(em)f FK(\(c)n(hapter)g
FG(??)q FK(\).)p eop
PStoPSsaved restore
%%Page: (20,21) 11
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
17 20 bop 167 167 a Fz(3.3.)64 b(OPERA)-6 b(TIONAL)26
b(SIGNIFICANCE)1204 b FK(17)167 404 y FE(3.3)114 b(Op)s(erational)29
b(signi\014cance)167 560 y FK(In)22 b(general,)h FM(terms)g
FK(will)c(represen)n(t)i FM(pr)l(o)l(gr)l(ams)p FK(.)36
b(The)22 b(purp)r(ose)i(of)e(a)h(program)g(is)e(to)i(calculate)167
645 y(\(or)32 b(at)g(least)g(put)g(in)f(a)i(con)n(v)n(enien)n(t)f
(form\))f(its)g(denotation.)50 b(The)31 b FM(typ)l(e)i
FK(of)g(a)f(program)h(is)167 730 y(seen)e(as)h(a)g FM(sp)l(e)l(ci\014c)
l(ation)p FK(,)h FM(i.e.)e FK(what)h(the)f(program)h(\(abstractly\))f
(do)r(es.)47 b FM(A)32 b(priori)g FK(it)e(is)h(a)167
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(the)i(sum)f(of)i(t)n(w)n(o)e(in)n(tegers".)270 942 y(What)j(is)e(the)h
(relev)l(an)n(t)f(part)h(of)h(this)e(commen)n(tary?)38
b(In)29 b(other)g(w)n(ords,)h(when)f(w)n(e)f(giv)n(e)167
1028 y(this)k(kind)g(of)h(information,)g(are)f(w)n(e)g(b)r(eing)g
FM(su\016ciently)f FK(precise)f(|)j(for)f(example,)h(ough)n(t)167
1113 y(one)27 b(to)f(sa)n(y)h(in)f(what)g(w)n(a)n(y)h(this)f
(calculation)e(is)i(done?)35 b(Or)25 b FM(to)l(o)i FK(precise)e(|)h(is)
f(it)g(enough)j(to)167 1198 y(sa)n(y)h(that)f(the)f(program)i(tak)n(es)
f(t)n(w)n(o)g(in)n(tegers)f(as)h(argumen)n(ts)g(and)h(returns)e(an)i
(in)n(teger?)270 1325 y(In)38 b(terms)d(of)j(syn)n(tax,)j(the)c(answ)n
(er)g(is)f(not)h(clear:)53 b(for)37 b(example)f(the)h(t)n(yp)r(e)g
(systems)167 1410 y(en)n(visaged)52 b(in)f(this)g(b)r(o)r(ok)i(concern)
e(themselv)n(es)g(only)g(with)g(the)g(most)h(elemen)n(tary)167
1495 y(information)36 b(\(sending)h(in)n(tegers)e(to)i(in)n(tegers\),)h
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1580 y(of)31 b([)p FG(?)q FK(],)g(giv)n(e)f(information)g(ab)r(out)i
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(a)167 1665 y(denotational)d(kind.)270 1793 y(A)n(t)i(a)g(more)g
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b(should)g(see)f(a)h(t)n(yp)r(e)g(as)g(an)g(instruction)f(for)h
FM(plugging)e FK(things)h(together.)72 b(Let)39 b(us)167
1963 y(imagine)g(that)i(w)n(e)f(program)h(with)f FM(mo)l(dules)p
FK(,)k FM(i.e.)d FK(closed)f(units,)j(whic)n(h)d(w)n(e)g(can)h(plug)167
2048 y(together.)52 b(A)33 b(mo)r(dule)g(is)f(absolutely)h(closed,)i(w)
n(e)d(ha)n(v)n(e)i(no)g(righ)n(t)f(to)h(op)r(en)g(it.)51
b(W)-6 b(e)33 b(just)167 2133 y(ha)n(v)n(e)26 b(the)f(abilit)n(y)e(to)i
(use)g(it)f(or)h(not,)i(and)f(to)f(c)n(ho)r(ose)g(the)g(manner)g(of)g
(use)g(\(plugging\).)35 b(The)167 2218 y(t)n(yp)r(e)28
b(of)g(a)h(mo)r(dule)e(is)g(of)h(course)g(completely)d(determined)h(b)n
(y)j(all)d(the)i(p)r(ossible)f FM(pluggings)167 2303
y FK(it)36 b(allo)n(ws)h(without)g(crashing.)64 b(In)38
b(particular,)g(one)g(can)g(alw)n(a)n(ys)f(substitute)g(a)h(mo)r(dule)
167 2388 y(with)30 b(another)i(of)f(the)g(same)g(t)n(yp)r(e,)h(in)e
(the)h(ev)n(en)n(t)g(of)h(a)f(breakdo)n(wn,)j(or)d(for)g(the)g(purp)r
(ose)167 2474 y(of)d(optimisation.)270 2601 y(This)38
b(idea)f(of)i FM(arbitr)l(ary)g(pluggings)e FK(seems)g
FM(mathematisable)p FK(,)k(but)d(to)g(attempt)g(this)167
2686 y(w)n(ould)28 b(lead)f(us)h(to)r(o)g(far)g(astra)n(y)-6
b(.)270 2861 y(A)35 b(term)e(of)i(t)n(yp)r(e)f FH(T)10
b FK(,)36 b(sa)n(y)g FH(t)p FK(,)g(whic)n(h)e(dep)r(ends)h(on)g(v)l
(ariables)f FH(x)2035 2872 y FD(1)2064 2861 y FH(;)12
b(x)2134 2872 y FD(2)2162 2861 y FH(;)g(:)g(:)g(:)f(;)h(x)2355
2872 y Fw(n)2423 2861 y FK(of)35 b(t)n(yp)r(es)167 2946
y(resp)r(ectiv)n(ely)i FH(U)606 2957 y FD(1)634 2946
y FH(;)12 b(:)g(:)g(:)f(;)h(U)835 2957 y Fw(n)868 2946
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(of)h(substituting)167 3031 y(for)f FH(x)326 3042 y Fw(i)383
3031 y FK(the)f(terms)f FH(u)761 3042 y Fw(i)819 3031
y FK(of)i(t)n(yp)r(es)f FH(U)1155 3042 y Fw(i)1175 3031
y FK(,)j(but)e(as)g(a)f FM(plugging)f FK(instruction.)64
b(The)37 b(term)f(has)167 3117 y(places)31 b(\(sym)n(b)r(olised,)g
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(ariables\))f(in)g(whic)n(h)167 3202 y(w)n(e)k(can)h(plug)f
FM(inputs)h FK(of)g(appropriate)g(t)n(yp)r(e:)51 b(for)36
b(example,)h(to)e(eac)n(h)h(o)r(ccurrence)e(of)i FH(x)2652
3213 y Fw(i)167 3287 y FK(corresp)r(onds)g(the)e(p)r(ossibilit)n(y)e
(of)j(plugging)g(in)f(a)h(term)e FH(u)1783 3298 y Fw(i)1838
3287 y FK(of)i(t)n(yp)r(e)f FH(U)2141 3298 y Fw(i)2161
3287 y FK(,)i(the)e(same)g(term)167 3372 y(b)r(eing)26
b(sim)n(ultaneously)g(plugged)h(in)f(eac)n(h)h(instance.)34
b(But)27 b(also,)g FH(t)f FK(itself,)f(b)r(eing)h(of)h(t)n(yp)r(e)g
FH(T)10 b FK(,)167 3457 y(is)31 b(a)h(plugging)g(instruction,)g(so)g
(that)g(it)f(can)h(b)r(e)f(plugged)h(in)g(an)n(y)g(v)l(ariable)g
FH(y)h FK(of)f(t)n(yp)r(e)g FH(T)167 3542 y FK(app)r(earing)d(in)e
(another)h(term.)270 3669 y(This)i(w)n(a)n(y)h(of)g(seeing)e(v)l
(ariables)h(and)h(v)l(alues)g(as)f(dual)h(asp)r(ects)f(of)h(the)f(same)
f(plugging)167 3754 y(phenomenon,)36 b(allo)n(ws)c(us)i(to)f(view)f
(the)h(execution)g(of)g(an)h(algorithm)f(as)g(a)h(symmetrical)167
3840 y(input/output)e(pro)r(cess.)46 b(The)31 b(true)g(op)r(erational)g
(in)n(terpretation)f(of)h(the)g(sc)n(hemes)f(is)h(still)167
3925 y(in)c(an)h(em)n(bry)n(onic)f(state)h(\(see)f(app)r(endix)h
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PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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userdict/PStoPSmatrix matrix currentmatrix put
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18 21 bop 167 167 a FK(18)639 b Fz(CHAPTER)26 b(3.)63
b(THE)27 b(CURR)-6 b(Y-HO)n(W)e(ARD)26 b(ISOMORPHISM)270
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b(,)45 b(w)n(e)167 490 y(ha)n(v)n(e)33 b(an)f FM(ad)g(ho)l(c)g
FK(notion,)i(whic)n(h)d(is)g(not)h(so)g(bad:)45 b(w)n(e)31
b(shall)g(mak)n(e)g(the)h(equations)g(of)g FG(??)167
575 y FK(asymmetric)k(and)j(turn)f(them)f(in)n(to)g(rewrite)f(rules.)64
b(This)38 b FM(r)l(ewriting)43 b FK(ma)n(y)38 b(b)r(e)g(seen)f(as)167
660 y(an)e(em)n(bry)n(onic)e(program)i(calculating)e(the)h(terms)f(in)g
(question.)56 b(That)34 b(is)f(not)i(to)r(o)g(bad,)167
745 y(b)r(ecause)23 b(the)f(op)r(erational)g(seman)n(tics)g(whic)n(h)f
(w)n(e)h(lac)n(k)g(is)g(surely)g(v)n(ery)h(close)e(to)i(this)f(pro)r
(cess)167 830 y(of)28 b(calculation,)f(itself)e(based)k(on)f(the)g
(fundamen)n(tal)g(symmetries)d(of)j(logic.)270 956 y(So)j(one)e(could)g
(hop)r(e)h(to)g(mak)n(e)f(progress)i(at)e(the)g(op)r(erational)g(lev)n
(el)f(b)n(y)i(a)g(close)e(study)167 1041 y(of)g(normalisation.)167
1276 y FE(3.4)114 b(Con)m(v)m(ersion)167 1431 y FK(A)27
b(term)g(is)g FM(normal)35 b FK(if)26 b(none)j(of)f(its)f(subterms)g
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FD(1)668 1603 y FJ(h)p FH(u;)12 b(v)s FJ(i)432 b FH(\031)1302
1578 y FD(2)1329 1603 y FJ(h)p FH(u;)12 b(v)s FJ(i)432
b FK(\()p FH(\025x)2027 1578 y Fw(U)2069 1603 y FH(:)12
b(v)s FK(\))g FH(u)167 1776 y FK(A)27 b(term)g FH(t)g
FM(c)l(onverts)g FK(to)h(a)g(term)f FH(t)1087 1750 y
Fv(0)1131 1776 y FK(when)g(one)h(of)h(the)e(follo)n(wing)g(three)g
(cases)g(holds:)471 1965 y FH(t)58 b FK(=)h FH(\031)709
1939 y FD(1)736 1965 y FJ(h)p FH(u;)12 b(v)s FJ(i)293
b FH(t)59 b FK(=)f FH(\031)1427 1939 y FD(2)1455 1965
y FJ(h)p FH(u;)12 b(v)s FJ(i)293 b FH(t)59 b FK(=)f(\()p
FH(\025x)2210 1939 y Fw(U)2252 1965 y FH(:)12 b(v)s FK(\))p
FH(u)454 2050 y(t)479 2025 y Fv(0)554 2050 y FK(=)59
b FH(u)467 b(t)1198 2025 y Fv(0)1273 2050 y FK(=)58 b
FH(v)474 b(t)1917 2025 y Fv(0)1992 2050 y FK(=)58 b FH(v)s
FK([)p FH(u=x)p FK(])167 2243 y FH(t)27 b FK(is)g(called)f(the)i
FM(r)l(e)l(dex)g FK(and)g FH(t)960 2218 y Fv(0)1004 2243
y FK(the)g FM(c)l(ontr)l(actum)p FK(;)g(they)g(are)g(alw)n(a)n(ys)g(of)
g(the)f(same)h(t)n(yp)r(e.)270 2416 y(A)f(term)e FH(u)i
FM(r)l(e)l(duc)l(es)796 2390 y FD(1)851 2416 y FK(to)g(a)g(term)f
FH(v)j FK(when)e(there)e(is)h(a)h(sequence)g(of)g(con)n(v)n(ersions)g
(from)g FH(u)167 2501 y FK(to)e FH(v)s FK(,)h(that)f(is)f(a)h(sequence)
g FH(u)20 b FK(=)f FH(t)1060 2512 y FD(0)1088 2501 y
FH(;)k(t)1155 2512 y FD(1)1183 2501 y FH(;)12 b(:)g(:)g(:)f(;)h(t)1362
2512 y Fw(n)p Fv(\000)p FD(1)1459 2501 y FH(;)24 b(t)1527
2512 y Fw(n)1579 2501 y FK(=)19 b FH(v)28 b FK(suc)n(h)e(that)f(for)g
FH(i)19 b FK(=)g(0)p FH(;)12 b FK(1)p FH(;)g(:)g(:)g(:)h(;)f(n)f
FJ(\000)g FK(1,)167 2586 y FH(t)192 2597 y Fw(i)p FD(+1)305
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2586 y FK(b)n(y)h(replacing)e(a)i(redex)f(b)n(y)h(its)e(con)n(tractum.)
40 b(W)-6 b(e)29 b(write)e FH(u)21 b Fu( )g FH(v)167
2671 y FK(for)28 b(\\)p FH(u)g FK(reduces)f(to)h FH(v)s
FK(":)36 b(\\)p Fu( )p FK(")29 b(is)e(re\015exiv)n(e)g(and)i(transitiv)
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FH(u)h FK(suc)n(h)g(that)h FH(t)22 b Fu( )h FH(u)33 b
FK(and)g(whic)n(h)e(is)g(normal.)48 b(W)-6 b(e)167 2928
y(shall)27 b(see)g(in)g(the)g(follo)n(wing)g(c)n(hapter)h(that)g
(normal)f(forms)h(exist)f(and)h(are)g(unique.)270 3101
y(W)-6 b(e)38 b(shall)f(w)n(an)n(t)h(to)g(discuss)g(normal)g(forms)f
(in)g(detail,)i(and)g(for)f(this)g(purp)r(ose)g(the)167
3186 y(follo)n(wing)24 b(de\014nition,)g(whic)n(h)g(is)g(essen)n(tial)f
(to)i(the)f(study)i(of)f FM(untyp)l(e)l(d)g FH(\025)p
FK(-calculus,)f(is)g(useful:)167 3312 y FG(Lemma)h FK(A)i(term)g
FH(t)g FK(is)g(normal)g(i\013)g(it)g(is)g(in)g FM(he)l(ad)h(normal)g
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3495 y FD(1)1602 3484 y FH(u)1641 3495 y FD(2)1681 3484
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b(If)28 b(it)f(is)g(an)i(application,)e FH(t)20 b FK(=)g
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userdict/PStoPSsaved save put
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b(cannot)h(sa)n(y)h(it)d(is)167 648 y(an)d(isomorphism:)42
b(this)32 b(requires)e(that)j(structures)e(of)h(the)g(same)f(kind)h
(already)g(exist)f(on)167 733 y(either)26 b(side.)270
866 y(In)20 b(fact)f(the)h(tradition)e(of)i(normalisation)f(exists)g
(indep)r(enden)n(tly)f(for)i(natural)g(deduction:)167
951 y(a)30 b(pro)r(of)i(is)d(normal)h(when)g(it)f(do)r(es)h(not)h(con)n
(tain)f(an)n(y)h(sequence)f(of)h(an)f(in)n(tro)r(duction)g(and)167
1037 y(an)e(elimination)d(rule:)482 1221 y FJ(\001)482
1259 y(\001)482 1297 y(\001)466 1354 y FH(A)683 1221
y FJ(\001)683 1259 y(\001)683 1297 y(\001)665 1354 y
FH(B)p 466 1399 255 4 v 741 1414 a FJ(^I)500 1475 y FH(A)15
b FJ(^)h FH(B)p 500 1520 186 4 v 707 1536 a FJ(^)p FK(1)p
FJ(E)567 1596 y FH(A)1146 1221 y FJ(\001)1146 1259 y(\001)1146
1297 y(\001)1130 1354 y FH(A)1347 1221 y FJ(\001)1347
1259 y(\001)1347 1297 y(\001)1329 1354 y FH(B)p 1130
1399 255 4 v 1406 1414 a FJ(^I)1165 1475 y FH(A)f FJ(^)h
FH(B)p 1165 1520 186 4 v 1371 1536 a FJ(^)p FK(2)p FJ(E)1230
1596 y FH(B)1811 1377 y FJ(\001)1811 1415 y(\001)1811
1453 y(\001)1795 1510 y FH(A)2058 1197 y FK([)p FH(A)p
FK(])2093 1256 y FJ(\001)2093 1294 y(\001)2093 1332 y(\001)2075
1389 y FH(B)p 1994 1434 218 4 v 2232 1450 a FJ(\)I)1994
1510 y FH(A)j FJ(\))h FH(B)p 1795 1555 417 4 v 2232 1571
a FJ(\)E)1975 1632 y FH(B)270 1789 y FK(F)-6 b(or)27
b(eac)n(h)f(of)g(these)f(con\014gurations,)j(it)d(is)g(p)r(ossible)g
(to)h(de\014ne)g(a)g(notion)g(of)g FM(c)l(onversion)p
FK(.)167 1874 y(In)34 b(c)n(hapter)h FG(??)p FK(,)h(w)n(e)e
FM(identi\014e)l(d)g FK(deductions)h(b)n(y)g(the)f(w)n(ord)g
(\\equals";)39 b(w)n(e)34 b(no)n(w)h(consider)167 1959
y(these)29 b(iden)n(ti\014cations)e(as)j FM(r)l(ewriting)p
FK(,)f(the)g(left)f(mem)n(b)r(er)f(of)i(the)g(equalit)n(y)f(b)r(eing)h
(rewritten)167 2044 y(to)f(the)f(righ)n(t)h(one.)270
2177 y(That)f(w)n(e)e(ha)n(v)n(e)h(an)h(isomorphism)e(follo)n(ws)g
(from)g(the)h(fact)g(that,)g(mo)r(dulo)g(the)f(bijection)167
2262 y(w)n(e)34 b(ha)n(v)n(e)h(already)g(in)n(tro)r(duced,)h(the)e
(notions)h(of)f FM(c)l(onversion)p FK(,)i FM(normality)f
FK(and)h FM(r)l(e)l(duction)167 2347 y FK(in)n(tro)r(duced)g(in)g(the)g
(t)n(w)n(o)h(cases)f(\(and)i(indep)r(enden)n(tly)-6 b(,)38
b(from)e(the)h(historical)d(viewp)r(oin)n(t\))167 2433
y(corresp)r(ond)c(p)r(erfectly)-6 b(.)37 b(In)29 b(particular)f(the)g
FM(normal)h(form)h(the)l(or)l(em)f FK(w)n(e)g(announced)h(in)e
FG(??)167 2518 y FK(has)j(an)g(exact)f(coun)n(terpart)h(in)f(natural)g
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167 2603 y FM(he)l(ad)c(normal)g(forms)h FK(in)e(section)g
FG(??)p FK(.)270 2790 y(Ha)n(ving)34 b(said)e(this,)i(the)e(in)n
(terest)f(in)h(an)h(isomorphism)f(lies)f(in)h(a)h(di\013erence)e(b)r
(et)n(w)n(een)167 2875 y(the)24 b(t)n(w)n(o)g(participan)n(ts,)h
(otherwise)e(what)i(is)f(the)g(p)r(oin)n(t)g(of)h(it?)32
b(In)25 b(the)f(case)g(whic)n(h)f(in)n(terests)167 2960
y(us,)43 b(the)c(functional)g(side)f(p)r(ossesses)j(an)f(op)r
(erational)f(asp)r(ect)g(alien)f(to)i(formal)f(pro)r(ofs.)167
3045 y(The)29 b(pro)r(of)h(side)e(is)g(distinguished)g(b)n(y)i(its)e
(logical)f(asp)r(ect,)j FM(a)f(priori)g FK(alien)f(to)h(algorithmic)167
3130 y(considerations.)270 3264 y(The)e(comparison)h(of)g(the)f(t)n(w)n
(o)g(alien)f(viewp)r(oin)n(ts)h(has)h(some)f(deep)h(consequences)f
(from)167 3349 y(a)h(metho)r(dological)e(p)r(oin)n(t)h(of)g(view)f
(\(tec)n(hnically)f(none,)j(seen)f(at)g(the)g(w)n(eak)h(tec)n(hnical)d
(lev)n(el)167 3434 y(of)j(the)g(t)n(w)n(o)f(traditions\):)270
3605 y FJ(\017)34 b FK(All)26 b(go)r(o)r(d)j(\(constructiv)n(e\))e
(logic)f(m)n(ust)i(ha)n(v)n(e)g(an)h(op)r(erational)f(side.)270
3754 y FJ(\017)34 b FK(Con)n(v)n(ersely)-6 b(,)23 b(one)f(cannot)g(w)n
(ork)f(with)g(t)n(yp)r(ed)g(calculi)e(without)h(regard)i(to)f(the)g
(implicit)339 3840 y(symmetries,)35 b(whic)n(h)g(are)g(those)h(of)f
(Logic.)58 b(In)36 b(general,)g(the)f(\\impro)n(v)n(emen)n(ts")h(of)339
3925 y(t)n(yping)29 b(based)f(on)h(logical)d(atro)r(cities)f(do)k(not)f
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21 24 bop 167 167 a Fz(3.6.)64 b(RELEV)-8 b(ANCE)27 b(OF)g(THE)g
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22 25 bop 167 874 a FF(Chapter)28 b(4)167 1168 y(The)g(Normalisation)e
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(o)g(results)g(whic)n(h)f(ensure)h(that)h(the)e(t)n(yp)r(ed)i
FH(\025)p FK(-calculus)167 1573 y(b)r(eha)n(v)n(es)48
b(w)n(ell)c(computationally)-6 b(.)92 b(The)46 b FM(Normalisation)i
(The)l(or)l(em)g FK(pro)n(vides)e(for)h(the)167 1658
y(existence)39 b(of)i(a)g(normal)g(form,)i(whilst)c(the)i
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1743 y(uniqueness.)e(In)29 b(fact)g(w)n(e)g(shall)f(simply)g(state)h
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1828 y(really)d(a)i(matter)f(of)h(t)n(yp)r(e)g(theory)g(and)h(is)e(w)n
(ell)e(co)n(v)n(ered)j(in)f(the)g(literature,)f FM(e.g.)h
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n(w)n(o)g(forms:)270 2069 y FJ(\017)34 b FK(a)26 b FM(we)l(ak)33
b FK(one)26 b(\(there)f(is)g FM(some)h FK(terminating)e(strategy)i(for)
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(form.)36 b(That)28 b(is)f(wh)n(y)h(one)g(sp)r(eaks)h(of)f
FM(we)l(ak)f(normalisation)p FK(.)270 2032 y(Ha)n(ving)g(said)e(that,)i
(it)d(is)h(p)r(ossible)g(to)h(\014nd)h(the)e(normal)g(form)h(b)n(y)g
(en)n(umerating)f(all)f(the)167 2117 y(reductions)29
b(in)h(one)g(step,)g(all)f(the)g(reductions)h(in)f(t)n(w)n(o)h(steps,)h
(and)g(so)f(on)h(un)n(til)d(a)i(normal)167 2203 y(form)25
b(is)f(found.)36 b(This)25 b(inelegan)n(t)f(pro)r(cedure)h(is)f
(justi\014ed)h(b)n(y)h(the)f(fact)g(that)g(there)g(are)g(only)167
2288 y(\014nitely)h(man)n(y)j(reductions)e(of)h(length)g
FH(n)f FK(starting)h(from)f(a)i(\014xed)f(term)e FH(t)p
FK(.)270 2420 y(The)31 b(strong)h(normalisation)e(theorem)f(will)f
(simplify)h(the)h(situation)h(b)n(y)g(guaran)n(teeing)167
2505 y(that)26 b(all)d(normalisation)i(strategies)f(are)h(go)r(o)r(d,)j
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b(if)f FH(T)756 3255 y Fw(i)804 3244 y FK(is)g(atomic.)270
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b(=)f FH(@)t FK(\()p FH(\031)865 3750 y FD(2)893 3776
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FJ(h)p FH(u;)12 b(v)s FJ(i)p FK(.)270 3925 y FJ(\017)34
b FH(@)t FK(\(\()p FH(\025x:)12 b(v)s FK(\))g FH(u)p
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PStoPSxform concat
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y(The)27 b FM(de)l(gr)l(e)l(e)h FH(d)p FK(\()p FH(t)p
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h(the)f(redexes)g(it)g(con)n(tains.)36 b(By)167 490 y(con)n(v)n(en)n
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b FK(A)24 b(redex)h FH(r)h FK(has)g(t)n(w)n(o)f(degrees:)33
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FK(Inside)f FH(t)p FK([)p FH(u=x)p FK(],)i(one)f(\014nds:)270
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b(new)f(redexes,)53 b(in)48 b(the)g(case)h(where)e FH(x)i
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FH(r)r FK(\))19 b FH(>)h(@)t FK(\()p FH(T)10 b FK(\))27
b(\(ob)n(vious\).)167 2522 y FG(Lemma)e FK(If)i FH(t)20
b Fu( )g FH(u)28 b FK(then)f FH(d)p FK(\()p FH(u)p FK(\))20
b FJ(\024)f FH(d)p FK(\()p FH(t)p FK(\).)167 2728 y FG(Pro)r(of)33
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3907 y(the)28 b(t)n(yp)r(e)g(of)g FH(r)r FK(,)f(and)i
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PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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PStoPSxform concat
26 29 bop 167 167 a FK(26)776 b Fz(CHAPTER)26 b(4.)63
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Fy(4.3.3)95 b(Con)m(v)m(ersion)27 b(of)g(maximal)g(degree)167
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(that)e(all)e(the)167 634 y(redexes)26 b(strictly)f(con)n(tained)h(in)g
FH(r)j FK(ha)n(v)n(e)e(degree)f(less)g(than)h FH(n)p
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b(example)g(if)f(one)i(replaces)e(\()p FH(\025x:)12 b
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PStoPSxform concat
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1643 y(is)e(strongly)h(normalisable:)270 1804 y FJ(\017)34
b FM(internalisation)5 b FK(:)48 b(this)34 b(consists)g(of)g(a)g
(tortuous)h(translation)f(of)h(the)e(calculus)g(in)n(to)339
1889 y(itself)k(in)g(suc)n(h)i(a)g(w)n(a)n(y)g(as)g(to)f(pro)n(v)n(e)i
(strong)f(normalisation)f(b)n(y)h(means)f(of)h(w)n(eak)339
1975 y(normalisation.)c(Gandy)29 b(w)n(as)g(the)e(\014rst)h(to)g(use)f
(this)g(tec)n(hnique)g([)p FG(?)q FK(].)270 2118 y FJ(\017)34
b FM(r)l(e)l(ducibility)6 b FK(:)65 b(w)n(e)41 b(in)n(tro)r(duce)g(a)h
(prop)r(ert)n(y)g(of)g(\\hereditary)g(calculabilit)n(y")d(whic)n(h)339
2204 y(allo)n(ws)g(us)g(to)g(manipulate)g(complex)f(com)n(binatorial)g
(information.)69 b(This)38 b(is)g(the)339 2289 y(metho)r(d)c(w)n(e)g
(shall)f(follo)n(w,)i(since)e(it)g(is)h(the)f(only)i(one)f(whic)n(h)g
(generalises)e(to)j(v)n(ery)339 2374 y(complicated)26
b(situations.)36 b(This)27 b(metho)r(d)g(will)e(b)r(e)j(the)f(sub)t
(ject)g(of)h(c)n(hapter)g FG(??)q FK(.)p 167 3790 1003
3 v 246 3833 a FC(1)273 3854 y FB(A)d(\014nitely)e(branc)n(hing)i(tree)
g(with)g(no)g(in\014nite)e(branc)n(h)i(is)g(\014nite.)32
b(Unless)25 b(the)g(branc)n(hes)g(are)h(lab)r(elled)167
3925 y(\(as)i(they)f(usually)f(are\),)i(this)f(requires)f(the)h(axiom)h
(of)g(Choice.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
28 31 bop 167 874 a FF(Chapter)28 b(5)167 1168 y(Sequen)-5
b(t)28 b(Calculus)167 1487 y FK(The)22 b FM(se)l(quent)f(c)l(alculus)p
FK(,)h(due)h(to)f(Gen)n(tzen,)g(is)g(the)f(prettiest)f(illustration)g
(of)j(the)e(symmetries)167 1573 y(of)33 b(Logic.)51 b(It)33
b(presen)n(ts)g(n)n(umerous)g(analogies)g(with)f(natural)h(deduction,)i
(without)d(b)r(eing)167 1658 y(limited)24 b(to)k(the)g(in)n
(tuitionistic)c(case.)270 1831 y(This)37 b(calculus)f(is)h(generally)f
(ignored)h(b)n(y)h(computer)f(scien)n(tists)2071 1805
y FD(1)2097 1831 y FK(.)64 b(Y)-6 b(et)36 b(it)h(underlies)167
1916 y(essen)n(tial)22 b(ideas:)33 b(for)24 b(example,)f
Fr(PROLOG)i FK(is)d(an)i(implemen)n(tation)d(of)j(a)f(fragmen)n(t)h(of)
g(sequen)n(t)167 2001 y(calculus,)h(and)i(the)f(\\tableaux")h(used)f
(in)f(automatic)h(theorem-pro)n(ving)g(are)f(just)h(a)h(sp)r(ecial)167
2086 y(case)i(of)h(this)e(calculus.)39 b(In)29 b(other)h(w)n(ords,)g
(it)e(is)g(used)i(un)n(wittingly)e(b)n(y)i(man)n(y)g(p)r(eople,)f(but)
167 2171 y(mixed)k(with)f FM(c)l(ontr)l(ol)41 b FK(features,)36
b FM(i.e.)d FK(programming)h(devices.)52 b(What)34 b(mak)n(es)g(ev)n
(erything)167 2256 y(w)n(ork)c(is)g(the)f(sequen)n(t)i(calculus)d(with)
h(its)g(deep)h(symmetries,)f(and)h(not)h(particular)e(tric)n(ks.)167
2341 y(So)24 b(it)e(is)g(di\016cult)g(to)i(consider,)f(sa)n(y)-6
b(,)26 b(the)d(theory)g(of)h Fr(PROLOG)h FK(without)d(kno)n(wing)j
(thoroughly)167 2427 y(the)i(subtleties)f(of)i(sequen)n(t)g(calculus.)
270 2600 y(F)-6 b(rom)40 b(an)f(algorithmic)e(viewp)r(oin)n(t,)k(the)e
(sequen)n(t)h(calculus)e(has)i(no)f FM(Curry-Howar)l(d)167
2685 y(isomorphism)p FK(,)d(b)r(ecause)c(of)g(the)g(m)n(ultitude)e(of)j
(w)n(a)n(ys)g(of)g(writing)d(the)i(same)g(pro)r(of.)50
b(This)167 2770 y(prev)n(en)n(ts)31 b(us)g(from)g(using)g(it)f(as)i(a)f
(t)n(yp)r(ed)g FH(\025)p FK(-calculus,)g(although)h(w)n(e)e(glimpse)g
(some)g(deep)167 2855 y(structure)k(of)h(this)f(kind,)j(probably)f
(link)n(ed)e(with)g(parallelism.)53 b(But)35 b(it)f(requires)f(a)i(new)
167 2940 y(approac)n(h)30 b(to)d(the)h(syn)n(tax,)h(for)f(example)f
(natural)h(deductions)g(with)e(sev)n(eral)h(conclusions.)p
167 3860 1003 3 v 246 3903 a FC(1)273 3925 y FB(An)g(exception)f(is)h
([)p Ft(?)q FB(].)1385 4101 y FK(28)p eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
29 32 bop 167 167 a Fz(5.1.)64 b(THE)27 b(CALCULUS)1710
b FK(29)167 404 y FE(5.1)114 b(The)28 b(calculus)167
580 y Fy(5.1.1)95 b(Sequen)m(ts)167 710 y FK(A)26 b FM(se)l(quent)g
FK(is)g(an)h(expression)998 711 y FH(A)p 1001 731 39
3 v 1069 710 a FJ(`)1132 711 y FH(B)p 1135 731 43 3 v
1214 710 a FK(where)1417 711 y FH(A)p 1420 731 39 3 v
1495 710 a FK(and)1632 711 y FH(B)p 1636 731 43 3 v 1715
710 a FK(are)f(\014nite)g(sequences)h(of)g(form)n(ulae)167
796 y FH(A)219 807 y FD(1)247 796 y FH(;)12 b(:)g(:)g(:)f(;)h(A)453
807 y Fw(n)514 796 y FK(and)28 b FH(B)704 807 y FD(1)733
796 y FH(;)12 b(:)g(:)g(:)f(;)h(B)939 807 y Fw(m)986
796 y FK(.)270 918 y(The)44 b(na)-8 b(\177)-27 b(\020v)n(e)45
b(\(denotational\))f(in)n(terpretation)f(is)g(that)h(the)g(conjunction)
g(of)h(the)f FH(A)2653 929 y Fw(i)167 1003 y FK(implies)25
b(the)i(disjunction)g(of)i(the)e FH(B)1147 1014 y Fw(j)1173
1003 y FK(.)36 b(In)27 b(particular,)270 1148 y FJ(\017)34
b FK(if)407 1149 y FH(A)p 411 1168 39 3 v 487 1148 a
FK(is)26 b(empt)n(y)-6 b(,)28 b(the)g(sequen)n(t)g(asserts)g(the)f
(disjunction)g(of)i(the)e FH(B)2136 1159 y Fw(j)2162
1148 y FK(;)270 1287 y FJ(\017)34 b FK(if)407 1288 y
FH(A)p 411 1308 V 487 1287 a FK(is)26 b(empt)n(y)i(and)915
1288 y FH(B)p 918 1308 43 3 v 998 1287 a FK(is)f(just)h
FH(B)1265 1298 y FD(1)1293 1287 y FK(,)g(it)f(asserts)h
FH(B)1694 1298 y FD(1)1722 1287 y FK(;)270 1427 y FJ(\017)34
b FK(if)407 1428 y FH(B)p 411 1448 V 491 1427 a FK(is)27
b(empt)n(y)-6 b(,)27 b(it)g(asserts)h(the)f(negation)i(of)f(the)f
(conjunction)h(of)h(the)e FH(A)2346 1438 y Fw(i)2366
1427 y FK(;)270 1567 y FJ(\017)34 b FK(if)407 1568 y
FH(A)p 411 1587 39 3 v 487 1567 a FK(and)625 1568 y FH(B)p
629 1587 43 3 v 709 1567 a FK(are)27 b(empt)n(y)-6 b(,)28
b(it)f(asserts)h(con)n(tradiction.)167 1769 y Fy(5.1.2)95
b(Structural)27 b(rules)167 1899 y FK(These)38 b(rules,)j(whic)n(h)e
(seem)e(not)j(to)f(sa)n(y)h(an)n(ything)g(at)f(all,)i(imp)r(ose)d(a)h
(certain)f(w)n(a)n(y)i(of)167 1984 y(managing)29 b(the)e(\\slots")i(in)
e(whic)n(h)g(one)h(writes)e(form)n(ulae.)35 b(They)27
b(are:)251 2129 y(1.)35 b(The)28 b FM(exchange)e FK(rules)719
2312 y FH(A)p 723 2332 39 3 v -1 x(;)12 b(C)q(;)g(D)r(;)973
2312 y(A)p 977 2332 V -30 x Fv(0)1061 2311 y FJ(`)1124
2312 y FH(B)p 1127 2332 43 3 v 719 2356 461 4 v 1200
2372 a FJ(L)p Fq(X)719 2434 y FH(A)p 723 2453 39 3 v
-1 x(;)g(D)r(;)g(C)q(;)973 2434 y(A)p 977 2453 V -31
x Fv(0)1061 2433 y FJ(`)1124 2434 y FH(B)p 1127 2453
43 3 v 1674 2311 a(A)19 b FJ(`)1808 2312 y FH(B)p 1812
2332 V 1864 2311 a(;)12 b(C)q(;)g(D)r(;)2066 2312 y(B)p
2070 2332 V 2122 2282 a Fv(0)p 1674 2356 465 4 v 2160
2372 a FJ(R)p Fq(X)1674 2433 y FH(A)19 b FJ(`)1808 2434
y FH(B)p 1812 2453 43 3 v 1864 2433 a(;)12 b(D)r(;)g(C)q(;)2066
2434 y(B)p 2070 2453 V 2122 2403 a Fv(0)339 2615 y FK(These)38
b(rules)f(express)h(in)g(some)f(w)n(a)n(y)i(the)f FM(c)l(ommutativity)h
FK(of)g(logic,)g(b)n(y)g(allo)n(wing)339 2700 y(p)r(erm)n(utation)28
b(of)g(form)n(ulae)f(on)h(either)e(side)h(of)h(the)g(sym)n(b)r(ol)f(\\)
p FJ(`)p FK(".)251 2840 y(2.)35 b(The)28 b FM(we)l(akening)e
FK(rules)875 3016 y FH(A)p 878 3036 39 3 v 946 3015 a
FJ(`)1009 3016 y FH(B)p 1012 3036 43 3 v 832 3060 276
4 v 1128 3076 a FJ(L)p Fq(W)832 3138 y FH(A)p 836 3157
39 3 v -1 x(;)12 b(C)25 b FJ(`)1051 3138 y FH(B)p 1055
3157 43 3 v 1775 3016 a(A)p 1779 3036 39 3 v 1847 3015
a FJ(`)1909 3016 y FH(B)p 1913 3036 43 3 v 1735 3060
272 4 v 2027 3076 a FJ(R)p Fq(W)1735 3138 y FH(A)p 1738
3157 39 3 v 1806 3137 a FJ(`)19 b FH(C)q(;)1949 3138
y(B)p 1953 3157 43 3 v 339 3319 a FK(as)29 b(their)d(name)i(suggests,)h
(allo)n(w)d(replacemen)n(t)g(of)i(a)g(sequen)n(t)g(b)n(y)h(a)f(w)n(eak)
n(er)f(one.)251 3459 y(3.)35 b(The)28 b FM(c)l(ontr)l(action)g
FK(rules)792 3622 y FH(A)p 796 3642 39 3 v -1 x(;)12
b(C)q(;)g(C)25 b FJ(`)1093 3622 y FH(B)p 1096 3642 43
3 v 792 3666 357 4 v 1169 3681 a FJ(L)p Fq(C)833 3743
y FH(A)p 836 3763 39 3 v -1 x(;)12 b(C)24 b FJ(`)1052
3743 y FH(B)p 1056 3763 43 3 v 1715 3622 a(A)p 1718 3642
39 3 v 1786 3621 a FJ(`)c FH(C)q(;)12 b(C)q(;)2011 3622
y(B)p 2015 3642 43 3 v 1715 3666 353 4 v 2088 3681 a
FJ(R)p Fq(C)1755 3743 y FH(A)p 1759 3763 39 3 v 1827
3742 a FJ(`)19 b FH(C)q(;)1970 3743 y(B)p 1974 3763 43
3 v 339 3925 a FK(express)28 b(the)g(idemp)r(otence)e(of)i(conjunction)
g(and)h(disjunction.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
30 33 bop 167 167 a FK(30)1200 b Fz(CHAPTER)26 b(5.)63
b(SEQUENT)27 b(CALCULUS)270 404 y FK(In)41 b(fact,)j(con)n(trary)d(to)g
(p)r(opular)h(b)r(elief,)f(these)g(rules)e(are)i(the)f(most)g(imp)r
(ortan)n(t)h(of)167 490 y(the)32 b(whole)f(calculus,)h(for,)h(without)f
(ha)n(ving)h(written)d(a)i(single)f(logical)g(sym)n(b)r(ol,)i(w)n(e)e
(ha)n(v)n(e)167 575 y(practically)g(determined)g(the)h(future)h(b)r
(eha)n(viour)h(of)f(the)g(logical)e(op)r(erations.)51
b(Y)-6 b(et)32 b(these)167 660 y(rules,)25 b(if)e(they)i(are)g(ob)n
(vious)h(from)f(the)f(denotational)h(p)r(oin)n(t)g(of)g(view,)g(should)
g(b)r(e)g(examined)167 745 y(closely)h(from)i(the)f(op)r(erational)h(p)
r(oin)n(t)f(of)i(view,)d(esp)r(ecially)g(the)h FM(c)l(ontr)l(action)p
FK(.)270 867 y(It)f(is)g(p)r(ossible)f(to)h(en)n(visage)h(v)l(arian)n
(ts)g(on)g(the)e(sequen)n(t)i(calculus,)e(in)h(whic)n(h)f(these)h
(rules)167 953 y(are)34 b(ab)r(olished)g(or)g(extremely)e(restricted.)
52 b(That)34 b(seems)f(to)h(ha)n(v)n(e)h(some)e(v)n(ery)i(b)r
(ene\014cial)167 1038 y(e\013ects,)27 b(leading)h(to)g(linear)f(logic)g
([)p FG(?)p FK(].)37 b(But)28 b(without)f(going)i(that)f(far,)h
(certain)e(w)n(ell-kno)n(wn)167 1123 y(restrictions)20
b(on)j(the)f(sequen)n(t)h(calculus)e(seem)g(to)h(ha)n(v)n(e)h(no)g
(purp)r(ose)g(apart)g(from)f(con)n(trolling)167 1208
y(the)27 b(structural)g(rules,)g(as)i(w)n(e)e(shall)g(see)g(in)g(the)g
(follo)n(wing)g(sections.)167 1410 y Fy(5.1.3)95 b(The)27
b(in)m(tuitionistic)i(case)167 1541 y FK(Essen)n(tially)-6
b(,)30 b(the)g(in)n(tuitionistic)d(sequen)n(t)k(calculus)e(is)h
(obtained)h(b)n(y)g(restricting)d(the)i(form)167 1626
y(of)g(sequen)n(ts:)38 b(an)30 b FM(intuitionistic)f(se)l(quent)f
FK(is)h(a)g(sequen)n(t)1739 1627 y FH(A)p 1743 1647 39
3 v 1811 1626 a FJ(`)1875 1627 y FH(B)p 1878 1647 43
3 v 1980 1626 a FK(where)2186 1627 y FH(B)p 2189 1647
V 2270 1626 a FK(is)g(a)g(sequence)167 1711 y(formed)f(from)h
FM(at)f(most)i(one)e FK(form)n(ula.)38 b(In)29 b(the)f(in)n
(tuitionistic)d(sequen)n(t)30 b(calculus,)d(the)i(only)167
1796 y(structural)e(rule)g(on)i(the)f(righ)n(t)g(is)f
FJ(R)p Fq(W)i FK(since)e FJ(R)p Fq(X)g FK(and)i FJ(R)p
Fq(C)f FK(assume)g(sev)n(eral)g(form)n(ulae)f(on)167
1881 y(the)g(righ)n(t.)270 2004 y(The)43 b(in)n(tuitionistic)d
(restriction)h(is)i(in)f(fact)h(a)h(mo)r(di\014cation)f(to)g(the)g
(managemen)n(t)167 2089 y(of)i(the)f(form)n(ulae)h(|)f(the)g
(particular)g(place)g(distinguished)g(b)n(y)h(the)g(sym)n(b)r(ol)f
FJ(`)h FK(is)f(a)167 2174 y(place)g(where)f(con)n(traction)i(is)f
(forbidden)g(|)h(and)g(from)g(that,)k(n)n(umerous)c(prop)r(erties)167
2259 y(follo)n(w.)50 b(On)33 b(the)f(other)h(hand,)j(this)c(c)n(hoice)f
(is)i(made)f(at)h(the)g(exp)r(ense)f(of)i(the)e(left/righ)n(t)167
2344 y(symmetry.)50 b(A)33 b(b)r(etter)f(result)g(is)g(without)h(doubt)
h(obtained)f(b)n(y)h(forbidding)f(con)n(traction)167
2429 y(\(and)c(w)n(eak)n(ening\))e(altogether,)h(whic)n(h)f(allo)n(ws)g
(the)g(symmetry)g(to)h(reapp)r(ear.)270 2552 y(Otherwise,)36
b(the)f(in)n(tuitionistic)d(sequen)n(t)j(calculus)g(will)d(b)r(e)j
(obtained)h(b)n(y)g(restricting)167 2637 y(to)29 b(the)g(in)n
(tuitionistic)d(sequen)n(ts,)31 b(and)f(preserving)f(|)g(apart)h(from)f
(one)h(exception)e(|)h(the)167 2722 y(classical)d(rules)g(of)j(the)e
(calculus.)167 2924 y Fy(5.1.4)95 b(The)27 b(\\iden)m(tit)m(y")j(group)
251 3055 y FK(1.)35 b(F)-6 b(or)30 b(ev)n(ery)g(form)n(ula)f
FH(C)34 b FK(there)29 b(is)g(the)g FM(identity)g(axiom)51
b FH(C)26 b FJ(`)21 b FH(C)26 b FK(.)41 b(In)29 b(fact)g(one)h(could)
339 3140 y(limit)25 b(it)h(to)i(the)g(case)f(of)h(atomic)f
FH(C)5 b FK(,)28 b(but)g(this)f(is)g(rarely)g(done.)251
3280 y(2.)35 b(The)28 b FM(cut)f(rule)1101 3450 y FH(A)p
1104 3470 39 3 v 1172 3449 a FJ(`)20 b FH(C)q(;)1316
3450 y(B)p 1320 3470 43 3 v 82 w(A)p 1454 3470 39 3 v
-30 x Fv(0)1518 3449 y FH(;)12 b(C)25 b FJ(`)1686 3450
y FH(B)p 1689 3470 43 3 v 1742 3420 a Fv(0)p 1101 3494
658 4 v 1779 3510 a Fq(Cut)1233 3572 y FH(A)p 1237 3591
39 3 v -1 x(;)1316 3572 y(A)p 1319 3591 V -30 x Fv(0)1404
3571 y FJ(`)1466 3572 y FH(B)p 1470 3591 43 3 v 1522
3571 a(;)1553 3572 y(B)p 1557 3591 V 1609 3542 a Fv(0)339
3754 y FK(is)32 b(another)g(w)n(a)n(y)h(of)f(expressing)g(the)g(iden)n
(tit)n(y)-6 b(.)47 b(The)31 b(iden)n(tit)n(y)g(axiom)g(sa)n(ys)j(that)e
FH(C)339 3840 y FK(\(on)d(the)f(left\))f(is)h(stronger)h(than)g
FH(C)34 b FK(\(on)29 b(the)f(righ)n(t\);)g(this)g(rule)f(states)i(the)f
(con)n(v)n(erse)339 3925 y(truth,)g FM(i.e.)g FH(C)33
b FK(\(on)28 b(the)g(righ)n(t\))f(is)g(stronger)h(than)g
FH(C)33 b FK(\(on)28 b(the)g(left\).)p eop
PStoPSsaved restore
%%Page: (34,35) 18
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
31 34 bop 167 167 a Fz(5.1.)64 b(THE)27 b(CALCULUS)1710
b FK(31)270 404 y(The)34 b(iden)n(tit)n(y)f(axiom)h(is)f(absolutely)h
(necessary)g(to)h(an)n(y)g(pro)r(of,)i(to)d(start)g(things)g(o\013.)167
490 y(That)41 b(is)e(undoubtedly)j(wh)n(y)f(the)f(cut)g(rule,)j(whic)n
(h)d(represen)n(ts)f(the)i(dual,)i(symmetric)167 575
y(asp)r(ect)29 b(can)f(b)r(e)h(eliminated,)d(b)n(y)j(means)g(of)g(a)g
(di\016cult)e(theorem)h(\(pro)n(v)n(ed)h(in)f(c)n(hapter)g
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(w)f(us)g(to)h(pass)h(from)e(the)g(righ)n(t)g(hand)i(side)339
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FK(2.)35 b FM(Conjunction)p FK(:)h(on)28 b(the)f(left,)g(t)n(w)n(o)h
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y FJ(\017)34 b FK(The)28 b(immediate)d(subform)n(ulae)j(of)g
FH(A)15 b FJ(^)h FH(B)t FK(,)27 b FH(A)16 b FJ(_)f FH(B)32
b FK(and)d FH(A)19 b FJ(\))h FH(B)31 b FK(are)d FH(A)f
FK(and)i FH(B)t FK(.)270 3688 y FJ(\017)34 b FK(The)28
b(only)g(immediate)d(subform)n(ula)j(of)g FJ(:)p FH(A)f
FK(is)g FH(A)p FK(.)270 3840 y FJ(\017)34 b FK(The)42
b(immediate)d(subform)n(ulae)i(of)i FJ(8)p FH(\030)s(:)12
b(A)40 b FK(and)j FJ(9)p FH(\030)s(:)12 b(A)42 b FK(are)f(the)h(form)n
(ulae)f FH(A)p FK([)p FH(a=\030)s FK(])339 3925 y(where)27
b FH(a)h FK(is)f(an)n(y)i(term.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
34 37 bop 167 167 a FK(34)1200 b Fz(CHAPTER)26 b(5.)63
b(SEQUENT)27 b(CALCULUS)270 404 y FK(No)n(w)39 b(it)f(is)g(clear)g
(that)h(all)f(the)h(rules)f(|)h(except)f(the)h(cut)f(|)h(ha)n(v)n(e)h
(the)e(prop)r(ert)n(y)167 490 y(that)c(the)f(premises)e(are)i(made)h
(up)f(of)h(subform)n(ulae)g(of)g(the)f(conclusion.)52
b(In)33 b(particular,)167 575 y(a)38 b(cut-free)e(pro)r(of)i(of)g(a)g
(sequen)n(t)f(uses)h(only)f(subform)n(ulae)h(of)g(its)e(form)n(ulae.)64
b(W)-6 b(e)37 b(shall)167 660 y(pro)n(v)n(e)32 b(the)g(corresp)r
(onding)g(result)f(for)h(natural)f(deduction)h(in)f(section)g
FG(??)q FK(.)47 b(This)31 b(is)g(v)n(ery)167 745 y(in)n(teresting)37
b(for)h FM(automate)l(d)i(de)l(duction)p FK(.)68 b(Of)38
b(course,)j(it)c(is)h(not)h(enough)g(to)g(mak)n(e)f(the)167
830 y(predicate)i(calculus)h FM(de)l(cidable)p FK(,)j(since)d(w)n(e)f
(ha)n(v)n(e)j(an)f(in\014nit)n(y)f(of)h(subform)n(ulae)g(for)f(the)167
915 y(sen)n(tences)27 b(with)g(quan)n(ti\014ers.)167
1230 y Fy(5.2.3)95 b(Asymmetrical)27 b(in)m(terpretation)167
1365 y FK(W)-6 b(e)43 b(ha)n(v)n(e)g(describ)r(ed)f(the)h(iden)n(tit)n
(y)f(axiom)g(and)i(the)f(cut)g(rule)e(as)j(the)f(t)n(w)n(o)g(faces)g
(of)167 1450 y(\\)p FH(A)34 b FK(is)e FH(A)p FK(".)53
b(No)n(w,)35 b(in)e(the)h(absence)f(of)h(cut,)h(the)e(situation)g(is)g
(suddenly)h(v)n(ery)g(di\013eren)n(t:)167 1535 y(w)n(e)c(can)h(no)g
(longer)f(express)g(that)h FH(A)f FK(\(on)h(the)f(righ)n(t\))g(is)f
(stronger)i(than)h FH(A)d FK(\(on)i(the)f(left\).)167
1620 y(Then)k(there)f(arises)g(the)g(p)r(ossibilit)n(y)f(of)j
(splitting)d FH(A)h FK(in)n(to)h(t)n(w)n(o)g(in)n(terpretations)e
FH(A)2490 1594 y Fv(L)2561 1620 y FK(and)167 1705 y FH(A)219
1679 y Fv(R)264 1705 y FK(,)c(whic)n(h)f(need)g(not)i(necessarily)d
(coincide.)34 b(Let)27 b(us)h(b)r(e)f(more)g(precise.)270
1839 y(In)41 b(a)g(sen)n(tence,)i(w)n(e)d(can)h(de\014ne)f(the)g
FM(signatur)l(e)h FK(of)g(an)g(o)r(ccurrence)e(of)i(an)g(atomic)167
1924 y(predicate,)27 b(+1)h(or)g FJ(\000)p FK(1:)36 b(the)27
b(signature)h(is)f(the)h(parit)n(y)g(of)g(the)f(n)n(um)n(b)r(er)h(of)g
(times)e(that)j(this)167 2009 y(sym)n(b)r(ol)f(has)h(b)r(een)f
(negated.)37 b(Concretely)-6 b(,)28 b FH(P)38 b FK(retains)27
b(the)h(signature)g(whic)n(h)g(it)f(had)i(in)e FH(A)p
FK(,)167 2094 y(when)k(it)f(is)g(considered)g(in)h FH(A)16
b FJ(^)h FH(B)t FK(,)31 b FH(B)21 b FJ(^)16 b FH(A)p
FK(,)32 b FH(A)16 b FJ(_)h FH(B)t FK(,)31 b FH(B)21 b
FJ(_)16 b FH(A)p FK(,)32 b FH(B)26 b FJ(\))c FH(A)p FK(,)32
b FJ(8)p FH(\030)s(:)12 b(A)30 b FK(and)i FJ(9)p FH(\030)s(:)12
b(A)p FK(,)167 2180 y(and)29 b(rev)n(erses)e(it)f(in)h
FJ(:)p FH(A)h FK(and)g FH(A)20 b FJ(\))g FH(B)t FK(.)270
2314 y(In)29 b(a)g(sequen)n(t)g(to)r(o,)g(w)n(e)f(can)h(de\014ne)f(the)
g(signature)h(of)g(an)g(o)r(ccurrence)e(of)i(a)g(predicate:)167
2399 y(if)f FH(P)39 b FK(o)r(ccurs)29 b(in)g FH(A)f FK(on)i(the)f(left)
f(of)h(\\)p FJ(`)p FK(",)i(the)e(signature)g(is)f(rev)n(ersed,)h(if)f
FH(P)40 b FK(o)r(ccurs)29 b(on)g(the)167 2484 y(righ)n(t,)e(it)g(is)g
(conserv)n(ed.)270 2618 y(The)h(rules)f(of)h(the)f(sequen)n(t)h
(calculus)e(\(apart)j(from)e(the)h(iden)n(tit)n(y)e(axiom)h(and)i(the)f
(cut\))167 2703 y(preserv)n(e)37 b(the)g(signature:)56
b(in)37 b(other)g(w)n(ords,)k(they)d(relate)e(o)r(ccurrences)g(with)h
(the)g(same)167 2788 y(signature.)44 b(The)30 b(iden)n(tit)n(y)f(axiom)
h(sa)n(ys)i(that)f(the)f(negativ)n(e)g(o)r(ccurrences)g(\(signature)g
FJ(\000)p FK(1\))167 2873 y(are)f(stronger)g(than)h(the)f(p)r(ositiv)n
(e)f(ones;)i(the)e(cut)h(sa)n(ys)h(the)f(opp)r(osite.)39
b(So)30 b(in)e(the)h(absence)167 2958 y(of)38 b(cut,)j(there)c(is)g
(the)h(p)r(ossibilit)n(y)e(of)i(giving)g(asymmetric)e(in)n
(terpretations)h(to)h(sequen)n(t)167 3044 y(calculus:)c
FH(A)26 b FK(do)r(es)h(not)g(ha)n(v)n(e)g(the)f(same)h(meaning)f(when)g
(it)g(is)f(on)j(the)e(righ)n(t)g(as)h(when)f(it)g(is)167
3129 y(on)i(the)g(left)e(of)i(\\)p FJ(`)p FK(".)270 3347
y FJ(\017)34 b FH(A)391 3321 y Fv(R)466 3347 y FK(is)29
b(obtained)h(b)n(y)h(replacing)d(the)h(p)r(ositiv)n(e)g(o)r(ccurrences)
g(of)h(the)f(predicate)g FH(P)39 b FK(b)n(y)339 3432
y FH(P)393 3407 y Fv(R)467 3432 y FK(and)28 b(the)g(negativ)n(e)f(ones)
i(b)n(y)f FH(P)1318 3407 y Fv(L)1355 3432 y FK(.)270
3583 y FJ(\017)34 b FH(A)391 3557 y Fv(L)459 3583 y FK(is)29
b(obtained)i(b)n(y)g(replacing)e(the)h(p)r(ositiv)n(e)g(o)r(ccurrences)
f(of)i(the)f(predicate)f FH(P)40 b FK(b)n(y)339 3668
y FH(P)393 3642 y Fv(L)458 3668 y FK(and)29 b(the)e(negativ)n(e)h(ones)
g(b)n(y)h FH(P)1310 3642 y Fv(R)1355 3668 y FK(.)270
3840 y(The)38 b(atomic)f(sym)n(b)r(ols)i FH(P)1001 3814
y Fv(R)1084 3840 y FK(and)g FH(P)1287 3814 y Fv(L)1363
3840 y FK(are)f(tied)f(together)h(b)n(y)h(a)f(condition,)j(namely)167
3925 y FH(P)221 3899 y Fv(L)278 3925 y FJ(\))20 b FH(P)422
3899 y Fv(R)467 3925 y FK(.)p eop
PStoPSsaved restore
%%Page: (38,39) 20
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
35 38 bop 167 167 a Fz(5.3.)64 b(SEQUENT)27 b(CALCULUS)g(AND)f(NA)-6
b(TURAL)27 b(DEDUCTION)500 b FK(35)270 404 y(It)37 b(is)f(easy)h(to)g
(see)f(that)h(this)f(kind)h(of)g(asymmetrical)d(in)n(terpretation)h(is)
h(consisten)n(t)167 490 y(with)27 b(the)g(system)g(without)h(cut,)f(in)
n(terpreting)1491 491 y FH(A)p 1494 510 39 3 v 1562 490
a FJ(`)1625 491 y FH(B)p 1628 510 43 3 v 1728 490 a FK(b)n(y)1848
491 y FH(A)p 1851 510 39 3 v 1899 461 a Fv(L)1956 490
y FJ(`)2019 491 y FH(B)p 2022 510 43 3 v 2075 461 a Fv(R)2140
490 y FK(.)270 627 y(The)18 b(sequen)n(t)g(calculus)e(seems)h(to)h
(lend)f(itself)f(to)h(some)h(m)n(uc)n(h)f(more)g(subtle)g(asymmetrical)
167 712 y(in)n(terpretations,)27 b(esp)r(ecially)e(in)i(linear)f
(logic.)167 960 y FE(5.3)114 b(Sequen)m(t)29 b(Calculus)g(and)f
(Natural)h(Deduction)167 1121 y FK(W)-6 b(e)28 b(shall)g(consider)g
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(the)f(fragmen)n(t)167 1206 y(without)33 b FJ(_)p FK(,)i
FJ(9)f FK(or)g FJ(:)p FK(.)53 b(W)-6 b(e)33 b(restrict)e(ourselv)n(es)j
(to)f(sequen)n(ts)i(of)f(the)f(form)2299 1207 y FH(A)p
2302 1226 39 3 v 2374 1206 a FJ(`)24 b FH(B)k FK(;)36
b(the)167 1291 y(corresp)r(ondence)28 b(with)f(natural)h(deduction)f
(is)g(giv)n(en)h(as)g(follo)n(ws:)270 1467 y FJ(\017)34
b FK(T)-6 b(o)31 b(a)f(pro)r(of)h(of)806 1468 y FH(A)p
810 1488 V 879 1467 a FJ(`)21 b FH(B)56 b FK(corresp)r(onds)31
b(a)f(deduction)g(of)h FH(B)i FK(under)e(the)f(h)n(yp)r(otheses,)339
1552 y(or)e(rather)g(parcels)f(of)h(h)n(yp)r(otheses,)1333
1553 y FH(A)p 1336 1573 V -1 x FK(.)270 1705 y FJ(\017)34
b FK(Con)n(v)n(ersely)-6 b(,)36 b(a)f(deduction)f(of)g
FH(B)k FK(under)c(the)g(\(parcels)f(of)5 b(\))34 b(h)n(yp)r(otheses)
2378 1706 y FH(A)p 2382 1726 V 2464 1705 a FK(can)g(b)r(e)339
1790 y(represen)n(ted)27 b(in)g(the)h(sequen)n(t)g(calculus,)e(but)j
(unfortunately)f(not)g(uniquely)-6 b(.)270 1985 y(F)g(rom)29
b(a)g(pro)r(of)h(of)810 1986 y FH(A)p 813 2006 V 882
1985 a FJ(`)20 b FH(B)k FK(,)29 b(w)n(e)f(build)g(a)h(deduction)g(of)g
FH(B)t FK(,)g(of)g(whic)n(h)e(the)i(h)n(yp)r(otheses)167
2070 y(are)f(parcels,)f(eac)n(h)g(parcel)g(corresp)r(onding)i(in)e(a)h
(precise)e(w)n(a)n(y)i(to)g(a)g(form)n(ula)g(of)2370
2071 y FH(A)p 2373 2091 V 2421 2070 a FK(.)251 2246 y(1.)35
b(The)28 b(axiom)47 b FH(A)19 b FJ(`)h FH(A)47 b FK(b)r(ecomes)27
b(the)g(deduction)106 b FH(A)78 b FK(.)251 2399 y(2.)35
b(If)28 b(the)f(last)h(rule)e(is)h(a)h(cut)1150 2601
y FH(A)p 1153 2621 V 1221 2600 a FJ(`)20 b FH(B)1418
2601 y(A)p 1421 2621 V 1469 2571 a Fv(0)1486 2600 y FH(;)12
b(B)23 b FJ(`)d FH(C)p 1150 2645 560 4 v 1730 2661 a
Fq(Cut)1286 2723 y FH(A)p 1289 2742 39 3 v 1337 2722
a(;)1368 2723 y(A)p 1372 2742 V -30 x Fv(0)1456 2722
y FJ(`)g FH(C)339 3031 y FK(and)37 b(the)e(deductions)h
FH(\016)i FK(of)1238 2941 y FH(A)p 1241 2960 V 1254 3001
a FJ(\001)1254 3039 y(\001)1254 3077 y(\001)1236 3134
y FH(B)1428 3031 y FK(and)e FH(\016)1608 3005 y Fv(0)1659
3031 y FK(of)1851 2945 y FH(A)p 1855 2964 V -30 x Fv(0)1919
2944 y FH(;)12 b(B)1919 3005 y FJ(\001)1919 3043 y(\001)1919
3081 y(\001)1902 3138 y FH(C)2142 3031 y FK(are)36 b(asso)r(ciated)g
(to)339 3193 y(the)c(sub-pro)r(ofs)i(ab)r(o)n(v)n(e)f(the)e(t)n(w)n(o)h
(premises,)f(then)h(w)n(e)f(asso)r(ciate)h(to)f(our)i(pro)r(of)f(the)
339 3278 y(deduction)g FH(\016)700 3253 y Fv(0)747 3278
y FK(where)f(all)g(the)g(o)r(ccurrences)g(of)h FH(B)k
FK(in)31 b(the)h(parcel)e(it)h(represen)n(ts)g(are)339
3363 y(replaced)c(b)n(y)h FH(\016)s FK(:)1397 3766 y
FH(A)p 1401 3786 V -30 x Fv(0)1465 3765 y FH(;)1533 3572
y(A)p 1536 3591 V 1549 3632 a FJ(\001)1549 3670 y(\001)1549
3708 y(\001)1531 3765 y FH(B)1482 3827 y FJ(\001)1482
3865 y(\001)1482 3903 y(\001)1465 3960 y FH(C)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
36 39 bop 167 167 a FK(36)1200 b Fz(CHAPTER)26 b(5.)63
b(SEQUENT)27 b(CALCULUS)339 404 y FK(In)d(general)f(the)g(h)n(yp)r
(otheses)h(in)f(the)g(parcel)f(in)1618 405 y FH(A)p 1622
425 39 3 v 1693 404 a FK(are)h(proliferated,)g(but)h(the)f(n)n(um)n(b)r
(er)339 490 y(is)28 b(preserv)n(ed)g(b)n(y)h(putting)g(in)f(the)g(same)
g(parcel)f(afterw)n(ards)i(the)f(h)n(yp)r(otheses)i(whic)n(h)339
575 y(came)c(from)h(the)f(same)h(parcel)f(b)r(efore)g(and)i(ha)n(v)n(e)
g(b)r(een)e(duplicated.)34 b(No)27 b(regrouping)339 660
y(o)r(ccurs)h(b)r(et)n(w)n(een)828 661 y FH(A)p 831 680
V 907 660 a FK(and)1046 661 y FH(A)p 1049 680 V -30 x
Fv(0)1114 660 y FK(.)251 796 y(3.)35 b(The)28 b(rule)e
FJ(L)p Fq(X)1204 958 y FH(A)p 1208 978 V -1 x(;)12 b(C)q(;)g(D)r(;)1458
958 y(A)p 1462 978 V -30 x Fv(0)1546 957 y FJ(`)20 b
FH(B)p 1204 1002 461 4 v 1685 1018 a FJ(L)p Fq(X)1204
1079 y FH(A)p 1208 1099 39 3 v -1 x(;)12 b(D)r(;)g(C)q(;)1458
1079 y(A)p 1462 1099 V -30 x Fv(0)1546 1078 y FJ(`)20
b FH(B)339 1253 y FK(is)27 b(in)n(terpreted)f(as)j(the)e(iden)n(tit)n
(y:)34 b(the)28 b(same)f(deduction)h(b)r(efore)f(and)i(after)e(the)h
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1545 y FH(A)p 1333 1565 V 1401 1544 a FJ(`)19 b FH(B)p
1287 1589 276 4 v 1583 1605 a FJ(L)p Fq(W)1287 1666 y
FH(A)p 1290 1686 39 3 v -1 x(;)12 b(C)25 b FJ(`)19 b
FH(B)339 1840 y FK(is)25 b(in)n(terpreted)g(as)h(the)g(creation)f(of)h
(a)g(mo)r(c)n(k)g(parcel)f(formed)g(from)h(zero)f(o)r(ccurrences)339
1925 y(of)k FH(C)5 b FK(.)35 b(W)-6 b(eak)n(ening)29
b(is)e(then)g(the)h(p)r(ossibilit)n(y)e(of)i(forming)f(empt)n(y)g
(parcels.)251 2062 y(5.)35 b(The)28 b(rule)e FJ(L)p Fq(C)1257
2217 y FH(A)p 1261 2237 V -1 x(;)12 b(C)q(;)g(C)25 b
FJ(`)20 b FH(B)p 1257 2261 357 4 v 1634 2277 a FJ(L)p
Fq(C)1298 2338 y FH(A)p 1301 2358 39 3 v 1349 2337 a(;)12
b(C)25 b FJ(`)20 b FH(B)339 2512 y FK(is)33 b(in)n(terpreted)f(as)i
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b(in)n(to)g(one.)54 b(Con)n(traction)34 b(is)339 2597
y(then)28 b(the)f(p)r(ossibilit)n(y)f(of)j(forming)e(big)h(parcels.)251
2734 y(6.)35 b(The)28 b(rule)e FJ(R^)1192 2895 y FH(A)p
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3036 39 3 v 1269 3016 a(;)1300 3017 y(A)p 1304 3036 V
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3191 y FK(will)23 b(b)r(e)j(in)n(terpreted)e(b)n(y)i
FJ(^I)5 b FK(:)35 b(supp)r(ose)27 b(that)f(deductions)g(ending)g(in)f
FH(B)k FK(and)e FH(C)k FK(ha)n(v)n(e)339 3276 y(b)r(een)g(constructed)g
(to)h(represen)n(t)e(the)h(pro)r(ofs)h(ab)r(o)n(v)n(e)g(the)f(t)n(w)n
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3635 y(\001)1322 3673 y(\001)1304 3730 y FH(B)1507 3536
y(A)p 1511 3556 V -30 x Fv(0)1532 3597 y FJ(\001)1532
3635 y(\001)1532 3673 y(\001)1514 3730 y FH(C)p 1304
3774 265 4 v 1589 3790 a FJ(^I)1342 3851 y FH(B)20 b
FJ(^)15 b FH(C)p eop
PStoPSsaved restore
%%Page: (40,41) 21
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
37 40 bop 167 167 a Fz(5.3.)64 b(SEQUENT)27 b(CALCULUS)g(AND)f(NA)-6
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b(B)24 b FJ(`)19 b FH(C)p 638 800 354 4 v 1013 816 a
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FH(\031)j FK(to)c(b)r(e)h(cut-free)e(in)i(the)f(case)h(where)e
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PStoPSxform concat
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FG(F)p FK(.)270 1896 y(F)-6 b(or)29 b(simple)c(t)n(yp)r(ed)k
FH(\025)p FK(-calculus,)e(there)g(is)g(pro)r(of)i(theoretic)d(tec)n
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b(As)35 b(a)167 2236 y(result,)f(strong)h(normalisation)e(implies)d
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FG(P)-23 b(A)27 b FK(\(Second)h(Incompleteness)e(Theorem\).)270
2461 y(Accordingly)44 b(w)n(e)g(ha)n(v)n(e)h(to)g(use)f(a)h(strong)h
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2546 y(in)n(tro)r(duce)27 b(an)h(abstract)g(notion)g(called)e
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2716 y(normalisation,)c(w)n(e)e(iden)n(tify)h(the)g(three)f(imp)r
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b(e)26 b(de\014ne)g(a)h(set)f Fq(RED)798 3226 y Fw(T)863
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FK(terms)f(of)i(t)n(yp)r(e)f FH(T)10 b FK("\))26 b(b)n(y)h(induction)f
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(where)e(com)n(binatorial)h(in)n(tuition)e(fails)i(is)167
490 y(its)d(logical)f(complexit)n(y)-6 b(.)34 b(Indeed,)28
b(w)n(e)f(ha)n(v)n(e:)451 642 y FH(t)20 b FJ(2)f Fq(RED)697
653 y Fw(U)5 b Fv(!)p Fw(V)1113 642 y FK(i\013)284 b
FJ(8)p FH(u)19 b FK(\()p FH(u)g FJ(2)h Fq(RED)1842 653
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805 y Fw(U)1544 794 y FK(has)d(b)r(een)g(negated,)k(and)d(a)f(univ)n
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FM(neutr)l(al)38 b FK(if)30 b(it)g(is)g(not)i(of)167
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b FK(or)g FH(\025x:)12 b(v)s FK(.)36 b(In)27 b(other)h(w)n(ords,)h
(neutral)e(terms)f(are)i(those)g(of)g(the)f(form:)602
1671 y FH(x)435 b(\031)1118 1645 y FD(1)1145 1671 y FH(t)g(\031)1647
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b(conditions)f(that)h(in)n(terest)e(us)j(are)e(the)g(follo)n(wing:)201
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2106 y FJ(2)20 b Fq(RED)1638 2117 y Fw(T)1677 2106 y
FK(.)201 2239 y(\()p FG(CR)30 b(3)p FK(\))e(If)i FH(t)f
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2324 y FH(t)364 2298 y Fv(0)400 2324 y FJ(2)20 b Fq(RED)602
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PStoPSxform concat
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3218 y FH(t)364 3192 y Fv(0)393 3218 y FH(u)f FK(is)g(reducible.)34
b(So)28 b FH(t)985 3192 y Fv(0)1029 3218 y FK(is)f(reducible.)201
3360 y(\()p FG(CR)21 b(3)p FK(\))f(Let)g FH(t)g FK(b)r(e)g(neutral)g
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3360 y FK(one)h(step)f(from)g FH(t)h FK(are)f(reducible.)31
b(Let)20 b FH(u)339 3445 y FK(b)r(e)32 b(a)g(reducible)e(term)h(of)h(t)
n(yp)r(e)g FH(U)7 b FK(;)34 b(w)n(e)d(w)n(an)n(t)h(to)h(sho)n(w)f(that)
g FH(t)12 b(u)32 b FK(is)f(reducible.)46 b(By)339 3530
y(induction)23 b(h)n(yp)r(othesis)h(\()p FG(CR)f(1)p
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FH(u)g FK(is)g(strongly)g(normalisable;)339 3615 y(so)29
b(w)n(e)e(can)h(reason)g(b)n(y)h(induction)e(on)h FH(\027)t
FK(\()p FH(u)p FK(\).)339 3767 y(In)g(one)g(step,)g FH(t)12
b(u)27 b FK(con)n(v)n(erts)i(to)421 3925 y FJ(\017)35
b FH(t)516 3899 y Fv(0)544 3925 y FH(u)28 b FK(with)f
FH(t)798 3899 y Fv(0)842 3925 y FK(one)h(step)f(from)h
FH(t)p FK(;)f(but)h FH(t)1518 3899 y Fv(0)1562 3925 y
FK(is)f(reducible,)f(so)i FH(t)2069 3899 y Fv(0)2098
3925 y FH(u)f FK(is.)p eop
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PStoPSmatrix setmatrix
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PStoPSxform concat
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b(STR)n(ONG)27 b(NORMALISA)-6 b(TION)26 b(THEOREM)421
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FH(u)p FK(.)50 b FH(u)1502 379 y Fv(0)1551 404 y FK(is)32
b(reducible)e(b)n(y)j(induction)f(h)n(yp)r(othesis)491
490 y(\()p FG(CR)25 b(2)p FK(\))f(for)g FH(U)7 b FK(,)26
b(and)f FH(\027)t FK(\()p FH(u)1194 464 y Fv(0)1211 490
y FK(\))20 b FH(<)f(\027)t FK(\()p FH(u)p FK(\);)26 b(so)f(the)f
(induction)g(h)n(yp)r(othesis)h(for)g FH(u)2508 464 y
Fv(0)2549 490 y FK(tells)491 575 y(us)j(that)g FH(t)12
b(u)814 549 y Fv(0)858 575 y FK(is)27 b(reducible.)421
678 y FJ(\017)35 b FK(There)c(is)f(no)i(other)g(p)r(ossibilit)n(y)-6
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f(\()p FH(t)g FK(is)g(not)491 763 y(of)d(the)g(form)f
FH(\025x:)12 b(v)s FK(\).)339 901 y(In)26 b(ev)n(ery)g(case,)h(w)n(e)e
(ha)n(v)n(e)i(seen)e(that)i(the)e(neutral)h(term)e FH(t)12
b(u)26 b FK(con)n(v)n(erts)g(in)n(to)g(reducible)339
986 y(terms)k(only)-6 b(.)46 b(The)31 b(induction)g(h)n(yp)r(othesis)g
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b(us)h(to)h(conclude)339 1071 y(that)c FH(t)12 b(u)28
b FK(is)f(reducible,)f(and)i(so)h FH(t)e FK(is)g(reducible.)980
b Fu(\003)167 1301 y FE(6.3)114 b(Reducibilit)m(y)28
b(theorem)167 1476 y Fy(6.3.1)95 b(P)m(airing)167 1607
y FG(Lemma)25 b FK(If)i FH(u)h FK(and)h FH(v)h FK(are)e(reducible,)e
(then)i(so)g(is)f FJ(h)p FH(u;)12 b(v)s FJ(i)p FK(.)167
1757 y FG(Pro)r(of)33 b FK(Because)g(of)g(\()p FG(CR)g(1)p
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1816 y FD(1)391 1842 y FJ(h)p FH(u;)12 b(v)s FJ(i)28
b FK(is)f(reducible.)33 b(This)27 b(term)g(con)n(v)n(erts)h(to:)270
1957 y FJ(\017)34 b FH(u)p FK(,)28 b(whic)n(h)f(is)g(reducible.)270
2089 y FJ(\017)34 b FH(\031)381 2064 y FD(1)409 2089
y FJ(h)p FH(u)475 2064 y Fv(0)492 2089 y FH(;)12 b(v)s
FJ(i)p FK(,)30 b(with)e FH(u)837 2064 y Fv(0)883 2089
y FK(one)i(step)f(from)h FH(u)p FK(.)40 b FH(u)1475 2064
y Fv(0)1521 2089 y FK(is)29 b(reducible)e(b)n(y)j(\()p
FG(CR)g(2)p FK(\),)f(and)h(w)n(e)f(ha)n(v)n(e)339 2175
y FH(\027)t FK(\()p FH(u)443 2149 y Fv(0)460 2175 y FK(\))20
b FH(<)f(\027)t FK(\()p FH(u)p FK(\);)27 b(so)f(the)e(induction)h(h)n
(yp)r(othesis)h(tells)d(us)j(that)g(this)e(term)g(is)h(reducible.)270
2307 y FJ(\017)34 b FH(\031)381 2281 y FD(1)409 2307
y FJ(h)p FH(u;)12 b(v)542 2281 y Fv(0)559 2307 y FJ(i)p
FK(,)28 b(with)e FH(v)830 2281 y Fv(0)874 2307 y FK(one)i(step)f(from)g
FH(v)s FK(:)35 b(this)27 b(term)f(is)h(reducible)e(for)j(similar)d
(reasons.)167 2422 y(In)30 b(ev)n(ery)f(case,)h(the)f(neutral)g(term)f
FH(\031)1202 2397 y FD(1)1230 2422 y FJ(h)p FH(u;)12
b(v)s FJ(i)30 b FK(con)n(v)n(erts)g(to)f(reducible)f(terms)g(only)-6
b(,)31 b(and)f(b)n(y)167 2507 y(\()p FG(CR)e(3)p FK(\))f(it)f(is)h
(reducible.)34 b(Lik)n(ewise)26 b FH(\031)1230 2482 y
FD(2)1257 2507 y FJ(h)p FH(u;)12 b(v)s FJ(i)p FK(,)29
b(and)f(so)h FJ(h)p FH(u;)12 b(v)s FJ(i)28 b FK(is)f(reducible.)369
b Fu(\003)167 2706 y Fy(6.3.2)95 b(Abstraction)167 2836
y FG(Lemma)25 b FK(If)i(for)h(all)f(reducible)e FH(u)j
FK(of)g(t)n(yp)r(e)g FH(U)7 b FK(,)28 b FH(v)s FK([)p
FH(u=x)p FK(])h(is)e(reducible,)f(then)h(so)i(is)e FH(\025x:)12
b(v)s FK(.)167 2986 y FG(Pro)r(of)30 b FK(W)-6 b(e)29
b(w)n(an)n(t)h(to)g(sho)n(w)g(that)g(\()p FH(\025x:)12
b(v)s FK(\))g FH(u)29 b FK(is)g(reducible)f(for)i(all)e(reducible)g
FH(u)p FK(.)41 b(Again)30 b(w)n(e)167 3071 y(reason)f(b)n(y)f
(induction)f(on)h FH(\027)t FK(\()p FH(v)s FK(\))16 b(+)f
FH(\027)t FK(\()p FH(u)p FK(\).)270 3174 y(The)28 b(term)e(\()p
FH(\025x:)12 b(v)s FK(\))g FH(u)28 b FK(con)n(v)n(erts)g(to)270
3289 y FJ(\017)34 b FH(v)s FK([)p FH(u=x)p FK(],)29 b(whic)n(h)e(is)g
(reducible)f(b)n(y)i(h)n(yp)r(othesis.)270 3422 y FJ(\017)34
b FK(\()p FH(\025x:)12 b(v)512 3396 y Fv(0)529 3422 y
FK(\))g FH(u)31 b FK(with)f FH(v)839 3396 y Fv(0)887
3422 y FK(one)i(step)f(from)g FH(v)s FK(;)i(so)f FH(v)1564
3396 y Fv(0)1611 3422 y FK(is)f(reducible,)f FH(\027)t
FK(\()p FH(v)2113 3396 y Fv(0)2130 3422 y FK(\))22 b
FH(<)g(\027)t FK(\()p FH(v)s FK(\),)32 b(and)g(the)339
3507 y(induction)c(h)n(yp)r(othesis)g(tells)e(us)i(that)g(this)f(term)f
(is)h(reducible.)270 3639 y FJ(\017)34 b FK(\()p FH(\025x:)12
b(v)s FK(\))g FH(u)590 3614 y Fv(0)644 3639 y FK(with)35
b FH(u)853 3614 y Fv(0)907 3639 y FK(one)i(step)g(from)g
FH(u)p FK(:)54 b FH(u)1535 3614 y Fv(0)1589 3639 y FK(is)36
b(reducible,)h FH(\027)t FK(\()p FH(u)2106 3614 y Fv(0)2123
3639 y FK(\))26 b FH(<)g(\027)t FK(\()p FH(u)p FK(\),)40
b(and)e(w)n(e)339 3724 y(conclude)28 b(similarly)-6 b(.)167
3840 y(In)30 b(ev)n(ery)f(case)g(the)g(neutral)g(term)f(\()p
FH(\025x:)12 b(v)s FK(\))g FH(u)29 b FK(con)n(v)n(erts)h(to)g
(reducible)d(terms)h(only)-6 b(,)31 b(and)f(b)n(y)167
3925 y(\()p FG(CR)e(3)p FK(\))f(it)f(is)h(reducible.)34
b(So)28 b FH(\025x:)12 b(v)31 b FK(is)c(reducible.)1070
b Fu(\003)p eop
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535 y FK(No)n(w)h(w)n(e)f(can)h(pro)n(v)n(e)g(the)167
661 y FG(Theorem)f FK(All)f(terms)g(are)i(reducible.)167
787 y(Hence,)f(b)n(y)h(\()p FG(CR)g(1)p FK(\),)f(w)n(e)g(ha)n(v)n(e)i
(the)167 914 y FG(Corollary)c FK(All)h(terms)g(are)i(strongly)g
(normalisable.)270 1087 y(In)d(the)e(pro)r(of)i(of)g(the)f(theorem,)g
(w)n(e)f(need)h(a)h(stronger)f(induction)g(h)n(yp)r(othesis)h(to)f
(handle)167 1172 y(the)35 b(case)g(of)g(abstraction.)58
b(This)35 b(is)f(the)h(purp)r(ose)h(of)f(the)g(follo)n(wing)f(prop)r
(osition,)j(from)167 1257 y(whic)n(h)27 b(the)g(theorem)g(follo)n(ws)g
(b)n(y)i(putting)e FH(u)1385 1268 y Fw(i)1425 1257 y
FK(=)19 b FH(x)1537 1268 y Fw(i)1557 1257 y FK(.)167
1430 y FG(Prop)r(osition)28 b FK(Let)i FH(t)h FK(b)r(e)f
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b(to)g(b)r(e)g(reducible\),)e(and)i(supp)r(ose)h(all)167
1515 y(the)d(free)g(v)l(ariables)h(of)g FH(t)f FK(are)h(among)h
FH(x)1253 1526 y FD(1)1281 1515 y FH(;)12 b(:)g(:)g(:)f(;)h(x)1474
1526 y Fw(n)1537 1515 y FK(of)30 b(t)n(yp)r(es)g FH(U)1858
1526 y FD(1)1886 1515 y FH(;)12 b(:)g(:)g(:)g(;)g(U)2088
1526 y Fw(n)2121 1515 y FK(.)41 b(If)30 b FH(u)2296 1526
y FD(1)2324 1515 y FH(;)12 b(:)g(:)g(:)f(;)h(u)2517 1526
y Fw(n)2580 1515 y FK(are)167 1600 y(reducible)26 b(terms)g(of)i(t)n
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FH(u)1515 1611 y FD(1)1543 1600 y FH(=x)1616 1611 y FD(1)1645
1600 y FH(;)12 b(:)g(:)g(:)f(;)h(u)1838 1611 y Fw(n)1872
1600 y FH(=x)1945 1611 y Fw(n)1978 1600 y FK(])28 b(is)f(reducible.)167
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1794 V -1 x FK(])i(for)g FH(t)p FK([)p FH(u)1703 1785
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y FH(=x)2133 1785 y Fw(n)2167 1774 y FK(].)251 1935 y(1.)35
b FH(t)43 b FK(is)f FH(x)535 1946 y Fw(i)555 1935 y FK(:)65
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FH(u)1859 1946 y Fw(i)1921 1935 y FK(is)g(reducible,)j(then)d
FH(u)2563 1946 y Fw(i)2626 1935 y FK(is)339 2020 y(reducible";)27
b(details)f(are)i(left)e(to)i(the)f(reader.)251 2164
y(2.)35 b FH(t)30 b FK(is)g FH(\031)513 2138 y FD(1)540
2164 y FH(w)r FK(:)40 b(b)n(y)31 b(induction)f(h)n(yp)r(othesis,)h(for)
g(ev)n(ery)f(sequence)2027 2165 y FH(u)p 2031 2185 V
2097 2164 a FK(of)g(reducible)e(terms,)339 2249 y FH(w)r
FK([)421 2250 y FH(u)p 425 2270 V 461 2249 a(=)495 2250
y(x)p 499 2270 V -1 x FK(])e(is)e(reducible.)33 b(That)25
b(means)g(that)h FH(\031)1557 2223 y FD(1)1584 2249 y
FK(\()p FH(w)r FK([)1693 2250 y FH(u)p 1697 2270 V -1
x(=)1766 2250 y(x)p 1770 2270 V 1806 2249 a FK(]\))f(is)g(reducible,)e
(but)j(this)e(term)339 2334 y(is)j(nothing)i(other)f(than)g
FH(\031)1063 2309 y FD(1)1091 2334 y FH(w)r FK([)1173
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y FH(u)p 1457 2355 V -1 x(=)1527 2335 y(x)p 1531 2355
V -1 x FK(].)251 2478 y(3.)35 b FH(t)28 b FK(is)f FH(\031)508
2452 y FD(2)535 2478 y FH(w)r FK(:)36 b(as)28 b FG(??)q
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FK([)2095 2623 y FH(u)p 2099 2642 V 2135 2622 a(=)2169
2623 y(x)p 2173 2642 V -1 x FK(])f(are)g(reducible.)339
2707 y(Lemma)c FG(??)i FK(sa)n(ys)g(that)f FH(t)p FK([)1054
2708 y FH(u)p 1057 2728 V -1 x(=)1127 2708 y(x)p 1131
2728 V -1 x FK(])20 b(=)f FJ(h)p FH(v)s FK([)1372 2708
y FH(u)p 1376 2728 V 1412 2707 a(=)1446 2708 y(x)p 1450
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1621 2728 V 1657 2707 a(=)1691 2708 y(x)p 1695 2728 V
-1 x FK(])p FJ(i)29 b FK(is)e(reducible.)251 2851 y(5.)35
b FH(t)k FK(is)g FH(w)14 b(v)s FK(:)58 b(b)n(y)40 b(induction)f(h)n(yp)
r(othesis)g FH(w)r FK([)1537 2852 y FH(u)p 1541 2871
V 1577 2851 a(=)1611 2852 y(x)p 1615 2871 V -1 x FK(])h(and)g
FH(v)s FK([)1926 2852 y FH(u)p 1929 2871 V -1 x(=)1999
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2936 y(so)c(\(b)n(y)g(de\014nition\))e(is)h FH(w)r FK([)1080
2937 y FH(u)p 1083 2956 V -1 x(=)1153 2937 y(x)p 1157
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399 3042 V -1 x(=)468 3022 y(x)p 472 3042 V 508 3021
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FH(w)r FK([)2025 3166 y FH(u)p 2029 3185 V 2065 3165
a(=)2099 3166 y(x)p 2103 3185 V -1 x(;)12 b(v)s(=y)s
FK(])29 b(is)g(reducible)339 3250 y(for)22 b(all)f FH(v)j
FK(of)f(t)n(yp)r(e)f FH(V)15 b FK(.)33 b(Lemma)21 b FG(??)i
FK(sa)n(ys)g(that)f FH(t)p FK([)1625 3251 y FH(u)p 1628
3270 V -1 x(=)1698 3251 y(x)p 1702 3270 V 1738 3250 a
FK(])d(=)h FH(\025y)s(:)12 b FK(\()p FH(w)r FK([)2067
3251 y FH(u)p 2069 3270 V 2105 3250 a(=)2139 3251 y(x)p
2143 3270 V 2179 3250 a FK(]\))21 b(is)g(reducible.)10
b Fu(\003)p eop
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PStoPSmatrix setmatrix
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userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
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1743 y FM(no)p FK(.)270 1869 y(T)-6 b(o)24 b(obtain)f(more)f
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b(new)34 b(t)n(yp)r(es,)j(or)e(new)f(terms,)h(often)g(b)r(oth)g
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b Fq(D)12 b FH(u)592 2140 y Fv(0)620 2165 y FH(v)656
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2251 y(w)n(e)j(ha)n(v)n(e)i FH(\027)t FK(\()p FH(u)889
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FJ(\017)35 b FH(u)28 b FK(\(if)f FH(t)19 b FK(=)g Fq(O)p
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Fp(Int)1286 2684 y FH(:)h Fq(S)g FH(z)s FK(\))g FH(y)s
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2837 y FH(:)12 b(\025z)1569 2812 y Fv(0)f Fp(Int)1654
2837 y FH(:)h Fq(S)g FH(z)s FK(\))g(\()p FH(t)p FK([)p
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FK(.)270 3106 y(Among)52 b(easy)f(exercises)f(in)g(this)g(st)n(yle,)57
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FK(\).)p eop
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FK(\()p Fq(Int)o FJ(!)h Fq(Int)p FK(\),)27 b(the)h(function)f
Fq(it)p FK(,)g(is:)1200 1278 y FH(\025x)1279 1252 y Fp(Int)p
Fv(!)p Fp(Int)1447 1278 y FH(:)12 b Fq(it)o FK(\()p FH(x)p
FK(\))270 1546 y(It)26 b(is)f(easy)h(to)g(see)g(that)g(b)n(y)g
(\014nite)f(iteration)f(of)j(some)e(reasonable)i(function)e
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2128 y(for)k FH(y)22 b FK(=)d Fq(O)27 b FK(to)h FH(f)607
2139 y FD(0)635 2128 y FK(,)g(and)h(for)p 931 2090 42
3 v 28 w FH(n)e FK(to)h Fq(it)1129 2099 y Fw(n)1174 2128
y FH(f)1208 2139 y FD(0)1236 2128 y FK(.)270 2363 y(T)-6
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n(t)f(of)g FH(v)j FK(in)c Fq(R)12 b FH(u)g(v)j(t)28 b
FK(is)g(frequen)n(tly)167 2448 y(un)n(used.)43 b(One)29
b(w)n(ould)h(prefer)f(an)i(iterator)d Fq(It)i FK(instead)g(of)g(the)f
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FH(v)14 b FK(\()p Fq(It)e FH(u)g(v)i(t)p FK(\))270 3002
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FK(\()p Fq(S)12 b FH(t)p FK(\))22 b Fu( )i FH(t)32 b
FK(only)g(when)h FH(t)f FK(is)g(of)g(the)g(form)p 1465
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FK(.)48 b(This)32 b(is)g(a)167 3427 y(little)c(anno)n(ying)33
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1570 1177 42 3 v 27 w FH(n)p FK(.)270 1353 y FJ(\017)34
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FK(or)g Fq(F)p FK(.)270 1489 y FJ(\017)34 b FK(If)28
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b(that)e(can)g(again)h(b)r(e)e(done)i(in)e FG(P)-23 b(A)p
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(signi\014can)n(t)f(in)n(terest)f(\(G\177)-34 b(odel)27
b(n)n(um)n(b)r(ering\).)270 1848 y(Summing)38 b(up,)j(the)d
(termination)f(is)g(pro)n(v)l(able)i(in)e(arithmetic:)54
b(w)n(e)37 b(sa)n(y)i(that)g FJ(j)p FH(t)p FJ(j)e FK(is)167
1933 y FM(pr)l(ovably)28 b(total)35 b FK(in)27 b FG(P)-23
b(A)p FK(.)270 2107 y(The)27 b(con)n(v)n(erse)h(is)e(true:)35
b(let)26 b FH(f)34 b FK(b)r(e)27 b(a)h(recursiv)n(e)e(function,)h(pro)n
(v)l(ably)i(total)e(in)f FG(P)-23 b(A)p FK(,)27 b(then)167
2192 y(one)32 b(can)g(\014nd)g(a)g(term)e(of)i(t)n(yp)r(e)f
Fq(Int)p FJ(!)12 b Fq(Int)32 b FK(in)e FG(T)p FK(,)i(suc)n(h)g(that)g
FH(f)8 b FK(\()p FH(n)p FK(\))21 b(=)h FJ(j)p FH(t)p
FJ(j)p FK(\()p FH(n)p FK(\))30 b(for)i(all)e FH(n)p FK(.)47
b(In)167 2277 y(other)30 b(w)n(ords,)i(the)d(expressiv)n(e)h(p)r(o)n(w)
n(er)g(of)h(the)f(system)f FG(T)i FK(is)e(enormous,)j(and)f(m)n(uc)n(h)
f(more)167 2362 y(than)f(what)g(is)f(feasible)808 2336
y FD(1)863 2362 y FK(on)h(a)g(computer!)37 b(The)28 b(further)g
(generalisations)g(are)g(not)h(aiming)167 2447 y(to)e(increase)e(the)h
(class)g(of)h(represen)n(table)f(functions,)h(whic)n(h)f(is)f(already)i
(to)r(o)g(big,)g(but)g(only)167 2532 y(to)36 b(enlarge)f(the)h(class)f
(of)i(particular)d(algorithms)i(calculating)e(simple)g(giv)n(en)i
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(v)n(ed.)270 2744 y(W)-6 b(e)33 b(do)g(not)g(w)n(an)n(t)g(to)f(giv)n(e)
g(a)h(pro)r(of)h(of)f(this)e(con)n(v)n(erse)i(here,)g(since)f(w)n(e)f
(consider)h(the)167 2829 y(\(more)27 b(delicate\))e(case)j(of)g(system)
f FG(F)h FK(in)f FG(??)q FK(.)p 167 3790 1003 3 v 246
3833 a FC(1)273 3854 y FB(In)34 b(the)g(sense)h(of)g
FA(c)m(omplexity)p FB(.)57 b(Th)n(us)35 b(for)g(instance)e
FA(hyp)m(er)m(exp)m(onential)j FB(algorithms,)g(suc)n(h)e(as)i(the)167
3925 y(pro)r(of)27 b(of)h(cut)f(elimination,)e(are)j(not)g(feasible.)p
eop
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PStoPSxform concat
53 56 bop 167 881 a FF(Chapter)28 b(8)167 1182 y(Coherence)g(Spaces)167
1508 y FK(The)40 b(earliest)e(w)n(ork)i(in)g(denotational)g(seman)n
(tics)f(w)n(as)i(done)f(b)n(y)h([)p FG(?)q FK(])f(for)g(the)g(un)n(t)n
(yp)r(ed)167 1594 y FH(\025)p FK(-calculus,)22 b(and)h(m)n(uc)n(h)e
(has)i(b)r(een)e(written)f(since)g(then.)34 b(His)20
b(approac)n(h)k(is)d(c)n(haracterised)f(b)n(y)167 1679
y FM(c)l(ontinuity)p FK(,)29 b FM(i.e.)f FK(the)h(preserv)l(ation)f(of)
h(directed)e(joins.)38 b(In)28 b(this)g(c)n(hapter,)h(a)g(no)n(v)n(el)f
(kind)h(of)167 1764 y(domain)c(theory)g(is)g(in)n(tro)r(duced,)g(in)f
(whic)n(h)g(w)n(e)h(also)g(ha)n(v)n(e)h(\(and)g(preserv)n(e\))e(meets)f
(b)r(ounded)167 1849 y(ab)r(o)n(v)n(e)30 b(\()p FM(pul)t(lb)l(acks)p
FK(\).)38 b(This)29 b(prop)r(ert)n(y)-6 b(,)30 b(called)d
FM(stability)p FK(,)j(w)n(as)f(originally)f(in)n(tro)r(duced)h(b)n(y)g
([)p FG(?)q FK(])167 1934 y(in)34 b(an)h(attempt)e(to)i(giv)n(e)f(a)h
(seman)n(tic)e(c)n(haracterisation)g(of)i FM(se)l(quential)f
FK(algorithms.)54 b(W)-6 b(e)167 2019 y(shall)31 b(\014nd)h(that)g
(this)f(seman)n(tics)g(is)g(w)n(ell)e(adapted)k(to)f(system)g
FG(F)f FK(and)i(leads)e(us)h(to)n(w)n(ards)167 2104 y(linear)26
b(logic.)167 2365 y FE(8.1)114 b(General)27 b(ideas)167
2530 y FK(The)18 b(fundamen)n(tal)g(idea)g(of)g(denotational)h(seman)n
(tics)e(is)g(to)h(in)n(terpret)f(reduction)g(\(a)h(dynamic)167
2615 y(notion\))35 b(b)n(y)g(equalit)n(y)g(\(a)g(static)e(notion\).)57
b(T)-6 b(o)35 b(put)h(it)d(in)h(another)i(w)n(a)n(y)-6
b(,)38 b(w)n(e)c(mo)r(del)f(the)167 2700 y(in)n(v)l(arian)n(ts)j(of)f
(the)g(calculi.)56 b(This)35 b(said,)i(there)d(are)i(mo)r(dels)e(and)i
(mo)r(dels:)50 b(it)34 b(has)i(b)r(een)167 2786 y(kno)n(wn)46
b(since)f(G\177)-34 b(odel)44 b(\(1930\))j(ho)n(w)f(to)f(construct)g
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y(extensions.)36 b(This)27 b(is)g(certainly)f(not)i(what)g(w)n(e)f
(mean,)h(b)r(ecause)f(it)g(giv)n(es)h(no)g FM(information)p
FK(.)270 3018 y(W)-6 b(e)32 b(ha)n(v)n(e)i(in)d(mind)h(rather)g(to)h
(tak)n(e)f(literally)d(the)j(na)-8 b(\177)-27 b(\020v)n(e)34
b(in)n(terpretation)c(|)j(that)f(an)167 3103 y(ob)t(ject)j(of)h(t)n(yp)
r(e)f FH(U)25 b FJ(!)18 b FH(V)51 b FK(is)34 b(a)i(function)f(from)g
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(things,)167 3273 y(w)n(e)k(try)g(to)g(a)n(v)n(oid)h(b)r(eing)f
(obsessed)h(b)n(y)g(completeness,)i(but)d(instead)g(lo)r(ok)h(for)f
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(idea)f(whic)n(h)g(comes)g(to)h(mind)e(is)h(the)h(follo)n(wing:)270
3763 y FJ(\017)34 b FK(t)n(yp)r(e)28 b(=)f(set.)270 3925
y FJ(\017)34 b FH(U)23 b FJ(!)16 b FH(V)43 b FK(is)27
b(the)h(set)f(of)h(all)f(functions)h(\(in)e(the)i(set-theoretic)d
(sense\))j(from)f FH(U)35 b FK(to)28 b FH(V)15 b FK(.)1385
4101 y(53)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
54 57 bop 167 167 a FK(54)1205 b Fz(CHAPTER)27 b(8.)63
b(COHERENCE)26 b(SP)-6 b(A)n(CES)270 404 y FK(This)37
b(in)n(terpretation)e(is)h(all)f(v)n(ery)i(w)n(ell,)g(but)g(it)f(do)r
(es)h(not)g(explain)f(an)n(ything.)64 b(The)167 490 y(computationally)
41 b(in)n(teresting)f(ob)t(jects)i(just)f(get)h(dro)n(wned)g(in)f(a)h
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b(function)h(spaces)g(also)g(quic)n(kly)f(b)r(ecome)g(enormous.)270
783 y(Kreisel)e(had)k(the)e(follo)n(wing)g(idea)g(\(hereditarily)f
(e\013ectiv)n(e)f(op)r(erations\):)270 1020 y FJ(\017)34
b FK(t)n(yp)r(e)28 b(=)f(partial)g(equiv)l(alence)g(relation)f(on)j(I)
-12 b(N.)270 1178 y FJ(\017)34 b FH(U)25 b FJ(!)19 b
FH(V)50 b FK(is)35 b(the)g(set)g(of)g(\(co)r(des)h(of)5
b(\))35 b(partial)f(recursiv)n(e)g(functions)i FH(f)43
b FK(suc)n(h)35 b(that,)j(if)339 1263 y FH(x)12 b(U)19
b(y)s FK(,)27 b(then)h FH(f)8 b FK(\()p FH(x)p FK(\))k
FH(V)26 b(f)8 b FK(\()p FH(y)s FK(\),)26 b(sub)t(ject)i(to)g(the)f
(equiv)l(alence)g(relation:)604 1502 y FH(f)20 b FK(\()p
FH(U)i FJ(!)16 b FH(V)g FK(\))c FH(g)267 b FK(i\013)e
FJ(8)p FH(x;)12 b(y)21 b FK(\()p FH(x)12 b(U)18 b(y)k
FJ(\))e FH(f)8 b FK(\()p FH(x)p FK(\))k FH(V)26 b(g)s
FK(\()p FH(y)s FK(\)\))270 1804 y(This)40 b(stic)n(ks)g(more)f(closely)
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b(,)26 b(it)e(seems,)i(for)g(in)f(fact)h(it)f(hardly)h(do)r(es)h(more)e
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r(dulo)i(some)f(unexciting)g(co)r(ding.)270 2182 y(Scott's)i(idea)e(is)
g(m)n(uc)n(h)g(b)r(etter:)270 2367 y FJ(\017)34 b FK(t)n(yp)r(e)28
b(=)f(top)r(ological)h(space.)270 2525 y FJ(\017)34 b
FH(U)23 b FJ(!)16 b FH(V)43 b FK(=)28 b(con)n(tin)n(uous)h(functions)e
(from)h FH(U)35 b FK(to)28 b FH(V)15 b FK(.)270 2710
y(No)n(w)24 b(it)e(is)h(w)n(ell)e(kno)n(wn)k(that)f(top)r(ology)h(do)r
(es)f(not)g(lend)f(itself)e(w)n(ell)g(to)j(the)f(construction)167
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2880 y(p)r(oin)n(t)n(wise,)h(or)h(uniformly)e(in)h(some)h(w)n(a)n(y)
1282 2854 y FD(1)1311 2880 y FK(?)270 3024 y(T)-6 b(o)25
b(resolv)n(e)f(these)g(problems,)h(Scott)f(w)n(as)h(led)f(to)g(imp)r
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(spaces)h(whic)n(h)e(are)h(far)h(remo)n(v)n(ed)f(from)g(the)g
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3168 y FD(2)455 3194 y FK(.)47 b(In)32 b(fact)g(his)f(spaces)h(are)f
(really)f(only)i(partially)e(ordered)i(sets)f(with)g(directed)167
3279 y(joins:)j(the)25 b(top)r(ology)h(is)e(an)h(inciden)n(tal)e
(feature.)34 b(So)25 b(it)f(is)g(natural)h(to)g(ask)h(oneself)e
(whether)167 3364 y(p)r(erhaps)29 b(the)e(top)r(ological)g(in)n
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b(whic)n(h)c(has)i(b)r(een)f(set)g(out)g(in)g(a)h(computer)167
3925 y(science)27 b(con)n(text)h(b)n(y)f([)p Ft(?)p FB(,)h
Ft(?)p FB(,)g Ft(?)p FB(,)g Ft(?)p FB(].)p eop
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userdict/PStoPSsaved save put
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862 y(ii\))41 b(Binary)h(completeness:)62 b(if)42 b FH(M)37
b FJ(\032)29 b(A)42 b FK(and)h(if)f FJ(8)p FH(a)1710
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862 y FJ(2)29 b FH(M)37 b FK(\()p FH(a)2105 873 y FD(1)2153
862 y FJ([)20 b FH(a)2256 873 y FD(2)2313 862 y FJ(2)30
b(A)p FK(\),)46 b(then)339 894 y Fo(S)410 947 y FH(M)27
b FJ(2)19 b(A)p FK(.)270 1106 y(In)28 b(particular,)f(w)n(e)g(ha)n(v)n
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b FJ(2)h(A)p FK(.)270 1232 y(The)31 b(reader)g(ma)n(y)g(consider)g(a)h
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g(its)f(\014nite)167 1402 y(subsets\))28 b(and)h(satis\014es)e(the)h
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Fn(?)617 1585 y FJ(f)p FG(t)p FJ(g)233 b(f)p FG(f)7 b
FJ(g)743 1734 y Fs(@)701 1692 y(@)868 1734 y(\000)910
1692 y(\000)1643 1785 y Fn(?)1618 1585 y FJ(f)p FK(0)p
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2509 y Fn(?)1117 2342 y FJ(f)p FK(0)p FJ(g)146 b(f)p
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FH(;)12 b FK(2)q FJ(g)80 b(f)p FK(0)p FH(;)12 b FK(2)q
FJ(g)81 b(f)p FK(0)p FH(;)12 b FK(1)q FJ(g)1311 2458
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2458 a(\021)1528 2419 y(\021)1561 2397 y(\021)1311 2275
y(@)1252 2216 y(@)1219 2183 y(@)1470 2275 y(\000)1528
2216 y(\000)1561 2183 y(\000)1219 2275 y(\000)1278 2216
y(\000)1311 2183 y(\000)1236 2291 y(\010)1295 2262 y(\010)1353
2233 y(\010)1412 2203 y(\010)1471 2174 y(\010)1528 2145
y(\010)1561 2275 y(@)1503 2216 y(@)1470 2183 y(@)1545
2291 y(H)1486 2262 y(H)1427 2233 y(H)1369 2203 y(H)1310
2174 y(H)1253 2145 y(H)167 2651 y FK(is)44 b(not.)87
b(Ho)n(w)n(ev)n(er)45 b(w)n(e)f(shall)g(see)g(that)h(coherence)f
(spaces)h(are)g(b)r(etter)e(regarded)i(as)167 2736 y(undirected)27
b(graphs.)167 2940 y Fy(8.2.1)95 b(The)27 b(w)m(eb)h(of)g(a)f
(coherence)i(space)167 3087 y FK(Consider)c FJ(jAj)572
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y FJ(A)20 b FK(=)f FJ(f)p FH(\013)h FK(:)39 b FJ(f)p
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FJ(jAj)f FK(are)g(called)f FM(tokens)p FK(,)h(and)167
3172 y(the)i FM(c)l(oher)l(enc)l(e)h(r)l(elation)g(mo)l(dulo)g
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3319 y Fv(0)813 3344 y FK(\(mo)r(d)24 b FJ(A)p FK(\))386
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3344 y FJ(g)20 b(2)f(A)167 3516 y FK(whic)n(h)36 b(is)h(a)g(re\015exiv)
n(e)g(symmetric)d(relation,)39 b(so)e FJ(jAj)g FK(equipp)r(ed)g(with)
2193 3494 y FH(_)2193 3537 y(^)2326 3516 y FK(is)f(a)i(graph,)167
3601 y(called)26 b(the)h FM(web)g FK(of)h FJ(A)p FK(.)p
167 3649 1003 3 v 246 3692 a FC(3)273 3713 y FB(The)d(term)g
FA(esp)m(ac)m(e)g(c)m(oh)n(\023)-29 b(er)m(ent)26 b FB(is)f(used)f(in)g
(the)h(F)-5 b(renc)n(h)25 b(text,)h(and)f(indeed)e(Plotkin)h(has)h
(also)g(used)g(the)167 3784 y(w)n(ord)30 b FA(c)m(oher)m(ent)f
FB(to)h(refer)f(to)g(this)f(binary)g(condition.)38 b(Ho)n(w)n(ev)n(er)
31 b FA(c)m(oher)m(ent)e(sp)m(ac)m(e)g FB(is)g(established,)f(alb)r
(eit)167 3854 y(p)r(eculiar,)33 b(usage)g(for)h(a)f(space)h(with)e(a)h
(basis)g(of)g(compact)h(op)r(en)e(sets,)j(also)e(called)f(a)h
FA(sp)m(e)m(ctr)m(al)h(sp)m(ac)m(e)p FB(.)167 3925 y(Consequen)n(tly)-5
b(,)27 b(the)g(term)h(w)n(as)h(mo)r(di\014ed)c(in)i(translation.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
56 59 bop 167 167 a FK(56)1205 b Fz(CHAPTER)27 b(8.)63
b(COHERENCE)26 b(SP)-6 b(A)n(CES)270 404 y FK(F)g(or)44
b(example,)h(the)e(w)n(eb)f(of)i FJ(B)-10 b FM(o)l(ol)51
b FK(consists)43 b(of)g(the)f(tok)n(ens)i FG(t)e FK(and)i
FG(f)7 b FK(,)47 b(whic)n(h)42 b(are)167 490 y(incoheren)n(t;)j
(similarly)36 b(the)j(w)n(eb)g(of)h FJ(I)-7 b FM(nt)46
b FK(is)39 b(a)h FM(discr)l(ete)f(gr)l(aph)46 b FK(whose)39
b(p)r(oin)n(ts)h(are)f(the)167 575 y(in)n(tegers.)c(Suc)n(h)28
b(domains)g(w)n(e)f(call)f FM(\015at)p FK(.)270 792 y(The)43
b(construction)g(of)h(the)f(w)n(eb)g(of)h(a)f(coherence)g(space)g(is)f
(a)i(bijection)e(b)r(et)n(w)n(een)167 877 y(coherence)28
b(spaces)i(and)g(\(re\015exiv)n(e-symmetric\))c(graphs.)41
b(F)-6 b(rom)29 b(the)g(w)n(eb)g(w)n(e)g(reco)n(v)n(er)f(the)167
962 y(coherence)f(space)h(b)n(y:)627 1179 y FH(a)20 b
FJ(2)f(A)d(,)g FH(a)k FJ(\032)g(jAj)15 b(^)g(8)p FH(\013)1295
1190 y FD(1)1322 1179 y FH(;)d(\013)1397 1190 y FD(2)1444
1179 y FJ(2)20 b FH(a)g FK(\()p FH(\013)1638 1190 y FD(1)1685
1157 y FH(_)1685 1200 y(^)1773 1179 y(\013)1817 1190
y FD(2)1892 1179 y FK(\(mo)r(d)j FJ(A)p FK(\)\))270 1397
y(So)39 b(in)e(the)h(terminology)f(of)h(Graph)h(Theory)-6
b(,)41 b(a)d(p)r(oin)n(t)g(is)f(exactly)g(a)i FM(clique)p
FK(,)g FM(i.e.)f FK(a)167 1482 y(complete)26 b(subgraph.)167
1712 y Fy(8.2.2)95 b(In)m(terpretation)167 1854 y FK(The)30
b(aim)g(is)f(to)i(in)n(terpret)e(a)i(t)n(yp)r(e)f(b)n(y)h(a)g
(coherence)f(space)g FJ(A)p FK(,)i(and)f(a)g(term)e(of)i(this)f(t)n(yp)
r(e)167 1939 y(b)n(y)f(a)f(p)r(oin)n(t)h(of)f FJ(A)g
FK(\(coheren)n(t)g(subset)h(of)f FJ(jAj)p FK(,)g(in\014nite)f(in)g
(general:)36 b(w)n(e)28 b(write)e FJ(A)2374 1950 y Fm(\014n)2466
1939 y FK(for)i(the)167 2024 y(set)f(of)h FM(\014nite)f
FK(p)r(oin)n(ts\).)270 2241 y(T)-6 b(o)30 b(w)n(ork)g(in)f(an)h
(e\013ectiv)n(e)d(manner)i(with)g(p)r(oin)n(ts)g(of)h
FJ(A)p FK(,)g(it)e(is)h(necessary)g(to)h(in)n(tro)r(duce)167
2326 y(a)f(notion)g(of)g FM(\014nite)f(appr)l(oximation)p
FK(.)40 b(An)29 b(appro)n(ximan)n(t)h(of)f FH(a)20 b
FJ(2)h(A)28 b FK(is)g(an)n(y)i(subset)f FH(a)2487 2301
y Fv(0)2533 2326 y FK(of)g FH(a)p FK(.)167 2412 y(Condition)23
b(\(i\))f(for)i(coherence)f(spaces)h(ensures)f(that)h(appro)n(ximan)n
(ts)h(are)e(still)e(in)i FJ(A)p FK(.)34 b(Ab)r(o)n(v)n(e)167
2497 y(all,)27 b(there)f(are)i(enough)h FM(\014nite)e
FK(appro)n(ximan)n(ts)i(to)f FH(a)p FK(:)270 2688 y FJ(\017)34
b FH(a)28 b FK(is)f(the)h(union)g(of)g(its)f(set)g(of)h(\014nite)f
(appro)n(ximan)n(ts.)270 2850 y FJ(\017)34 b FK(The)28
b(set)f FH(I)34 b FK(of)28 b(\014nite)f(appro)n(ximan)n(ts)i(is)e
FM(dir)l(e)l(cte)l(d)p FK(.)36 b(In)28 b(other)f(w)n(ords,)374
3037 y(i\))f FH(I)34 b FK(is)27 b(nonempt)n(y)h(\()p
Fn(?)20 b FJ(2)f FH(I)6 b FK(\).)374 3159 y(ii\))26 b(If)h
FH(a)576 3133 y Fv(0)593 3159 y FH(;)12 b(a)660 3133
y Fv(00)710 3159 y FJ(2)19 b FH(I)6 b FK(,)28 b(one)g(can)g(\014nd)g
FH(a)20 b FJ(2)g FH(I)33 b FK(suc)n(h)28 b(that)g FH(a)1799
3133 y Fv(0)1816 3159 y FH(;)12 b(a)1883 3133 y Fv(00)1933
3159 y FJ(\032)19 b FH(a)28 b FK(\(tak)n(e)g FH(a)19
b FK(=)h FH(a)2417 3133 y Fv(0)2449 3159 y FJ([)c FH(a)2548
3133 y Fv(00)2578 3159 y FK(\).)270 3402 y(This)28 b(comes)f(from)g
(the)g(follo)n(wing)g(idea:)270 3592 y FJ(\017)34 b FK(On)h(the)f(one)h
(hand)g(w)n(e)f(ha)n(v)n(e)h(the)g FM(true)f FK(\(or)g
FM(total)p FK(\))g(ob)t(jects)g(of)h FJ(A)p FK(.)56 b(F)-6
b(or)35 b(example,)339 3677 y(in)27 b FJ(B)-10 b FM(o)l(ol)9
b FK(,)28 b(the)f(singletons)g FJ(f)p FG(t)p FJ(g)h FK(and)h
FJ(f)p FG(f)7 b FJ(g)p FK(,)29 b(in)e FJ(I)-7 b FM(nt)7
b FK(,)28 b FJ(f)p FK(0)p FJ(g)p FK(,)g FJ(f)p FK(1)p
FJ(g)p FK(,)h FJ(f)p FK(2)p FJ(g)p FK(,)g FM(etc.)270
3840 y FJ(\017)34 b FK(On)c(the)f(other)g(hand,)i(the)e(appro)n(ximan)n
(ts,)i(of)f(whic)n(h,)f(in)g(the)g(t)n(w)n(o)g(simplistic)d(cases)339
3925 y(considered,)i Fn(?)f FK(is)g(the)g(only)h(example.)35
b(They)28 b(are)f FM(p)l(artial)i FK(ob)t(jects.)p eop
PStoPSsaved restore
%%Page: (60,61) 31
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
57 60 bop 167 167 a Fz(8.3.)64 b(ST)-6 b(ABLE)28 b(FUNCTIONS)1547
b FK(57)270 404 y(The)27 b(addition)h(of)f(partial)g(ob)t(jects)g(has)h
(m)n(uc)n(h)g(the)f(same)g(signi\014cance)f(as)i(in)f(recursion)167
490 y(theory)-6 b(,)27 b(where)d(w)n(e)h(shift)g(from)g(total)f(to)i
(partial)e(functions:)35 b(for)26 b(example,)e(to)i(the)f(in)n(tegers)
167 575 y(\(represen)n(ted)i(b)n(y)h(singletons\))f(w)n(e)g(add)i(the)f
(\\unde\014ned")h Fn(?)p FK(.)270 714 y(One)i(should)i(not,)g(ho)n(w)n
(ev)n(er,)f(attac)n(h)h(to)r(o)f(m)n(uc)n(h)f(imp)r(ortance)f(to)i
(this)f(\014rst)g(in)n(tuition.)167 799 y(F)-6 b(or)32
b(example,)h(it)d(is)i(misguided)f(to)h(seek)g(to)g(iden)n(tify)f(the)g
(total)h(p)r(oin)n(ts)g(of)g(an)h(arbitrary)167 884 y(coherence)d
(space)g FJ(A)p FK(.)44 b(One)30 b(migh)n(t)f(p)r(erhaps)j(think)e
(that)h(the)f(total)g(p)r(oin)n(ts)h(of)g FJ(A)f FK(are)g(the)167
969 y(maximal)c(p)r(oin)n(ts,)i FM(i.e.)g FK(suc)n(h)g(that:)715
1169 y FJ(8)p FH(\013)19 b FJ(2)g(jAj)g FK(\()p FJ(8)p
FH(\013)1110 1143 y Fv(0)1146 1169 y FJ(2)g FH(a)h(\013)1332
1146 y(_)1332 1189 y(^)1420 1169 y(\013)1464 1143 y Fv(0)1528
1169 y FK(\(mo)r(d)k FJ(A)p FK(\)\))19 b FJ(\))h FH(\013)g
FJ(2)f FH(a)167 1368 y FK(whic)n(h)30 b(indeed)g(they)g(are)h(|)f(in)g
(the)g(simple)f(cases)h(\(in)n(tegers,)h(b)r(o)r(oleans,)h
FM(etc.)p FK(\).)43 b(Ho)n(w)n(ev)n(er)167 1453 y(w)n(e)28
b(w)n(ould)h(lik)n(e)e(to)i(de\014ne)g(totalit)n(y)e(in)i(coherence)e
(spaces)i(whic)n(h)f(are)h(the)f(in)n(terpretations)167
1538 y(of)j(complex)e(t)n(yp)r(es,)j(using)e(form)n(ulae)g(analogous)j
(to)d(the)g(ones)h(for)g(reducibilit)n(y)c(\(see)i FG(??)q
FK(\).)167 1623 y(These)23 b(are)h(of)h(greater)e(and)i(greater)f
(logical)e(complexit)n(y)1711 1598 y FD(4)1738 1623 y
FK(,)i(and)h(altogether)f(unpredictable,)167 1708 y(whilst)30
b(the)g(notion)i(of)g(maximalit)n(y)d(remains)h(desp)r(erately)h(\005)
1875 1683 y FD(0)1875 1726 y(2)1902 1708 y FK(,)h(so)g(one)g(cannot)g
(hop)r(e)g(for)167 1793 y(a)g(coincidence.)47 b(In)32
b(fact,)h(for)f(an)n(y)h(giv)n(en)f(coherence)f(space)h(there)f(are)h
(man)n(y)h(notions)g(of)167 1879 y(totalit)n(y)-6 b(,)31
b(just)g(as)g(there)e(are)i(man)n(y)g FM(r)l(e)l(ducibility)g(c)l
(andidates)g FK(\(c)n(hapter)f FG(??)q FK(\))g(for)h(the)f(same)167
1964 y(t)n(yp)r(e.)38 b(In)29 b(fact)g(the)f(seman)n(tics)g
(partialises)f(ev)n(erything,)i(and)g(the)g(total)f(ob)t(jects)g(get)h
(a)g(bit)167 2049 y(lost)e(inside)g(it.)270 2248 y(The)40
b(functions)g(from)g FJ(A)g FK(to)g FJ(B)i FK(will)37
b(b)r(e)j(seen)g(as)g(functions)g(de\014ned)h(uniquely)e(b)n(y)167
2333 y(their)33 b(appro)n(ximan)n(ts,)38 b(and)d(in)f(this)f(w)n(a)n(y)
i(\\con)n(tin)n(uous".)58 b(Here)33 b(it)g(is)h(p)r(ossible)g(to)g(use)
h(a)167 2418 y(top)r(ological)28 b(language)i(where)d(the)h(subsets)h
FJ(f)p FH(a)21 b FK(:)40 b FH(a)1575 2429 y Fv(\016)1623
2418 y FJ(\032)20 b FH(a)p FJ(g)29 b FK(of)f FJ(A)p FK(,)h(for)g
FH(a)2132 2429 y Fv(\016)2188 2418 y FK(\014nite,)f(are)g(op)r(en.)167
2503 y(Ho)n(w)n(ev)n(er)e(whereas)h(in)f(Scott-st)n(yle)g(domain)h
(theory)g(the)f(functions)h(b)r(et)n(w)n(een)f(domains)h(are)167
2589 y(exactly)35 b(those)i(whic)n(h)e(are)h(con)n(tin)n(uous)h(for)f
(this)g(top)r(ology)-6 b(,)40 b(this)35 b(will)e(no)k(longer)f(b)r(e)f
(so)167 2674 y(here.)167 2925 y FE(8.3)114 b(Stable)29
b(functions)167 3086 y FK(Giv)n(en)e(t)n(w)n(o)h(coherence)f(spaces)h
FJ(A)g FK(and)h FJ(B)r FK(,)f(a)g(function)g FH(F)36
b FK(from)28 b FJ(A)g FK(to)g FJ(B)h FK(is)e FM(stable)g
FK(if:)201 3265 y(i\))g FH(a)311 3239 y Fv(0)347 3265
y FJ(\032)20 b FH(a)f FJ(2)h(A)g(\))g FH(F)10 b FK(\()p
FH(a)828 3239 y Fv(0)843 3265 y FK(\))20 b FJ(\032)f
FH(F)10 b FK(\()p FH(a)p FK(\))201 3420 y(ii\))26 b FH(F)10
b FK(\()376 3367 y Fo(S)434 3385 y Fv(")434 3440 y Fw(i)p
Fv(2)p Fw(I)524 3420 y FH(a)560 3431 y Fw(i)580 3420
y FK(\))20 b(=)700 3367 y Fo(S)759 3385 y Fv(")759 3440
y Fw(i)p Fv(2)p Fw(I)849 3420 y FH(F)10 b FK(\()p FH(a)967
3431 y Fw(i)986 3420 y FK(\))106 b(\(directed)25 b(union\))201
3574 y(iii\))g FH(a)349 3585 y FD(1)393 3574 y FJ([)16
b FH(a)492 3585 y FD(2)539 3574 y FJ(2)k(A)g(\))g FH(F)10
b FK(\()p FH(a)890 3585 y FD(1)932 3574 y FJ(\\)16 b
FH(a)1031 3585 y FD(2)1059 3574 y FK(\))k(=)f FH(F)10
b FK(\()p FH(a)1297 3585 y FD(1)1324 3574 y FK(\))15
b FJ(\\)h FH(F)10 b FK(\()p FH(a)1547 3585 y FD(2)1574
3574 y FK(\))106 b(\()p FG(St)p FK(\))p 167 3649 1003
3 v 246 3692 a FC(4)273 3713 y FB(The)23 b(logical)g(complexit)n(y)g
(of)g(a)h(form)n(ula)g(is)f(essen)n(tially)f(determined)g(b)n(y)h(the)h
(n)n(um)n(b)r(er)e(of)i(alternations)167 3784 y(of)33
b(quan)n(ti\014ers.)49 b(In)32 b(particular,)h(w)n(e)h(sa)n(y)f(that)f
(a)i(form)n(ula)e Fl(8)p Fx(x:)10 b Fl(9)p Fx(x)1775
3763 y Fk(0)1789 3784 y Fx(:)g Fl(8)p Fx(x)1883 3763
y Fk(00)1911 3784 y Fx(:)20 b(:)10 b(:)g(:)g(P)e FB(\()p
Fx(x;)i(x)2188 3763 y Fk(0)2204 3784 y Fx(;)g(x)2264
3763 y Fk(00)2293 3784 y Fx(;)g(:)g(:)g(:)q FB(\))32
b(where)h Fx(P)167 3854 y FB(is)j(a)i(primitiv)n(e)c(recursiv)n(e)j
(predicate,)i(is)d(of)h(logical)f(complexit)n(y)g(\005)1852
3833 y FC(0)1852 3869 y Fj(n)1884 3854 y FB(,)k(where)d
Fx(n)g FB(is)f(the)h(n)n(um)n(b)r(er)f(of)167 3925 y(quan)n(ti\014ers.)
d(Similarly)-5 b(,)25 b Fl(9)p Fx(x:)10 b Fl(8)p Fx(x)915
3903 y Fk(0)929 3925 y Fx(:)g Fl(9)p Fx(x)1022 3903 y
Fk(00)1051 3925 y Fx(:)20 b(:)10 b(:)g(:)h(P)d FB(\()p
Fx(x;)i(x)1329 3903 y Fk(0)1344 3925 y Fx(;)g(x)1404
3903 y Fk(00)1434 3925 y Fx(;)g(:)g(:)g(:)p FB(\))28
b(is)f(of)g(logical)g(complexit)n(y)f(\006)2259 3903
y FC(0)2259 3939 y Fj(n)2292 3925 y FB(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
58 61 bop 167 167 a FK(58)1205 b Fz(CHAPTER)27 b(8.)63
b(COHERENCE)26 b(SP)-6 b(A)n(CES)270 404 y FK(The)32
b(\014rst)g(condition)f(sa)n(ys)i(that)g FH(F)40 b FK(preserv)n(es)32
b(appro)n(ximation:)45 b(if)30 b(w)n(e)i(pro)n(vide)g(more)167
490 y(information)25 b(to)g(start)g(o\013)h(with)f(\()p
FH(a)g FK(rather)g(than)h FH(a)1552 464 y Fv(0)1569 490
y FK(\))f(then)g(w)n(e)g(get)g(more)f(bac)n(k)i(at)g(the)f(end.)167
575 y(Alternativ)n(ely)-6 b(,)26 b FH(F)36 b FK(only)28
b(uses)g FM(p)l(ositive)h FK(information)e(ab)r(out)i(its)d(argumen)n
(ts.)270 734 y(The)i(second)g(states)g(con)n(tin)n(uit)n(y:)839
973 y FH(F)10 b FK(\()p FH(a)p FK(\))19 b(=)1076 920
y Fo(S)1135 939 y Fv(")1163 973 y FJ(f)p FH(F)10 b FK(\()p
FH(a)1316 984 y Fv(\016)1343 973 y FK(\))19 b(:)40 b
FH(a)1484 984 y Fv(\016)1531 973 y FJ(\032)20 b FH(a;)31
b(a)1728 984 y Fv(\016)1784 973 y FK(\014nite)o FJ(g)167
1212 y FK(This)c(sp)r(ecial)f(case)i(of)g(\(ii\))d(is)i(in)g(fact)h
(equiv)l(alen)n(t)f(to)h(it.)270 1371 y(Considering)i(a)h(coherence)e
(space)h(as)h(a)g(category)f(in)g(whic)n(h)f(the)h(morphisms)g(from)f
FH(a)2655 1345 y Fv(0)167 1456 y FK(to)34 b FH(a)h FK(are)f(inclusions)
f FH(a)821 1430 y Fv(0)862 1456 y FJ(\032)24 b FH(a)p
FK(,)36 b(the)e(\014rst)g(condition)g(states)g(that)h(a)f(stable)g
(function)g(is)g(a)167 1541 y FM(functor)29 b FK(and)h(the)f(second)h
(that)g(this)e(preserv)n(es)h FM(\014lter)l(e)l(d)g(c)l(olimits)p
FK(.)41 b(These)29 b(t)n(w)n(o)g(conditions)167 1626
y(are)37 b(en)n(tirely)d(familiar)h(from)h(the)h(top)r(ological)f
(setting;)41 b(this)36 b(is)g(no)h(longer)g(true)f(of)h(the)167
1711 y(last)28 b(condition)f(|)h(the)g(stabilit)n(y)e(prop)r(ert)n(y)j
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1796 y(signi\014cance.)k(It)26 b(lo)r(oks)g(a)h(bit)e(p)r(eculiar)f(at)
i(\014rst)g(sigh)n(t,)h(but)f(in)f(terms)g(of)i(categories)e(it)g(just)
167 1882 y(sa)n(ys)k(that)f(the)f(pullbac)n(k)1316 2055
y FH(a)1352 2066 y FD(1)1396 2055 y FJ([)16 b FH(a)1495
2066 y FD(2)1137 2301 y FH(a)1173 2312 y FD(1)1638 2301
y FH(a)1674 2312 y FD(2)1316 2556 y FH(a)1352 2567 y
FD(1)1396 2556 y FJ(\\)g FH(a)1495 2567 y FD(2)1219 2241
y Fs(\000)1278 2182 y(\000)1311 2149 y(\000)-59 b(\022)1561
2241 y(@)1503 2182 y(@)1470 2149 y(@)g(I)1311 2492 y(@)1252
2433 y(@)1219 2400 y(@)g(I)1470 2492 y(\000)1528 2433
y(\000)1561 2400 y(\000)g(\022)167 2755 y FK(m)n(ust)28
b(b)r(e)f(preserv)n(ed.)37 b(The)27 b(in)n(ten)n(tion)g(is)g(that)i
(this)e(should)i(hold)f(for)g(an)n(y)h(set)f FJ(f)p FH(a)2402
2766 y FD(1)2430 2755 y FH(;)12 b(a)2497 2766 y FD(2)2525
2755 y FH(;)g(:)g(:)g(:)p FJ(g)167 2840 y FK(whic)n(h)27
b(is)g(b)r(ounded)i(ab)r(o)n(v)n(e,)g(not)g(just)f(\014nite)f(ones,)h
(but)g(in)f(the)h(con)n(text)f(of)i(strongly)f(\014nite)167
2926 y(appro)n(ximation)35 b(\()p FM(i.e.)g FK(the)f(fact)h(that)g(the)
g(appro)n(ximating)g(elemen)n(ts)e(ha)n(v)n(e)i(only)g(\014nitely)167
3011 y(man)n(y)h(elemen)n(ts)e(b)r(elo)n(w)g(them,)j(whic)n(h)e(is)f
(not)i(in)f(general)g(true)g(in)g(Scott's)h(theory\))g(w)n(e)167
3096 y(don't)29 b(need)e(to)h(sa)n(y)h(this.)270 3255
y(Let)h(us)h(giv)n(e)f(an)h(example)f(to)h(sho)n(w)g(that)g(the)f(h)n
(yp)r(othesis)h(of)g(coherence)e(b)r(et)n(w)n(een)h FH(a)2644
3266 y FD(1)167 3340 y FK(and)e FH(a)341 3351 y FD(2)396
3340 y FK(cannot)g(b)r(e)e(lifted.)34 b(W)-6 b(e)26 b(w)n(an)n(t)i(to)f
(b)r(e)f(able)h(to)g(represen)n(t)f(all)f(functions)j(from)e(I)-12
b(N)27 b(to)167 3425 y(I)-12 b(N)24 b(as)g(stable)g(functions)g(from)g
FJ(I)-7 b FM(nt)31 b FK(to)24 b FJ(I)-7 b FM(nt)8 b FK(,)24
b(in)g(particular)f FH(f)8 b FK(\(0\))19 b(=)g FH(f)8
b FK(\(1\))19 b(=)h(0,)25 b FH(f)8 b FK(\()p FH(n)i FK(+)g(2\))19
b(=)g(1.)167 3510 y(This)41 b(forces)h FH(F)10 b FK(\()p
FJ(f)p FK(0)p FJ(g)p FK(\))29 b(=)g FH(F)10 b FK(\()p
FJ(f)p FK(1)p FJ(g)p FK(\))30 b(=)f FJ(f)p FK(0)p FJ(g)p
FK(,)46 b FH(F)10 b FK(\()p FJ(f)p FH(n)15 b FK(+)g(2)q
FJ(g)p FK(\))29 b(=)h FJ(f)p FK(1)p FJ(g)p FK(,)46 b(and)d(b)n(y)f
(monotonicit)n(y)-6 b(,)167 3595 y FH(F)10 b FK(\()p
Fn(?)p FK(\))20 b(=)g Fn(?)p FK(.)41 b(No)n(w,)30 b FH(F)10
b FK(\()p FJ(f)p FK(0)p FJ(g)16 b(\\)g(f)p FK(1)p FJ(g)p
FK(\))22 b(=)e FH(F)10 b FK(\()p Fn(?)p FK(\))20 b(=)g
Fn(?)h FJ(6)p FK(=)f FH(F)10 b FK(\()p FJ(f)p FK(0)p
FJ(g)p FK(\))16 b FJ(\\)g FH(F)10 b FK(\()p FJ(f)p FK(1)p
FJ(g)p FK(\);)30 b(w)n(e)f(are)h(sa)n(v)n(ed)g(b)n(y)167
3680 y(the)d(incoherence)g(of)h(0)g(and)h(1,)f(whic)n(h)f(mak)n(es)h
FJ(f)p FK(0)p FJ(g)16 b([)g(f)p FK(1)p FJ(g)28 b FH(=)-42
b FJ(2)20 b(I)-7 b FM(nt)7 b FK(.)270 3840 y(W)-6 b(e)30
b(shall)f(see)g(that)h(this)f(prop)r(ert)n(y)h(forces)g(the)f
(existence)f(of)i(a)g FM(le)l(ast)g FK(appro)n(ximan)n(t)h(in)167
3925 y(certain)26 b(cases,)i(simply)f(b)n(y)h(taking)g(the)g(in)n
(tersection)d(of)k(a)f(set)f(whic)n(h)g(is)g(b)r(ounded)i(ab)r(o)n(v)n
(e.)p eop
PStoPSsaved restore
%%Page: (62,63) 32
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
59 62 bop 167 167 a Fz(8.3.)64 b(ST)-6 b(ABLE)28 b(FUNCTIONS)1547
b FK(59)167 404 y Fy(8.3.1)95 b(Stable)28 b(functions)f(on)g(a)g
(\015at)h(space)167 536 y FK(Let)f(us)h(lo)r(ok)g(at)g(the)g(stable)f
(functions)h FH(F)37 b FK(from)27 b FJ(I)-7 b FM(nt)35
b FK(to)28 b FJ(I)-7 b FM(nt)7 b FK(:)270 700 y FJ(\017)34
b FK(If)28 b FH(F)10 b FK(\()p Fn(?)p FK(\))18 b(=)h
FJ(f)p FH(n)p FJ(g)p FK(,)29 b(then)e FH(F)10 b FK(\()p
FH(a)p FK(\))19 b(=)g FJ(f)p FH(n)p FJ(g)28 b FK(for)g(all)f
FH(a)20 b FJ(2)f(I)-7 b FM(nt)7 b FK(.)270 845 y FJ(\017)34
b FK(Otherwise,)23 b FH(F)10 b FK(\()p Fn(?)p FK(\))19
b(=)g Fn(?)p FK(:)33 b(w)n(e)24 b(consider)g(the)g(partial)f(function)h
FH(f)8 b FK(,)25 b(de\014ned)g(exactly)e(on)339 930 y(the)f(in)n
(tegers)f FH(n)i FK(suc)n(h)f(that)h FH(F)10 b FK(\()p
FJ(f)p FH(n)p FJ(g)p FK(\))19 b FJ(6)p FK(=)g Fn(?)p
FK(,)k(in)f(whic)n(h)f(case)h(w)n(e)f(put)i FJ(f)p FH(f)8
b FK(\()p FH(n)p FK(\))p FJ(g)19 b FK(=)g FH(F)10 b FK(\()p
FJ(f)p FH(n)p FJ(g)p FK(\),)339 1023 y(and)29 b(w)n(e)e(write)f
FH(F)j FK(=)925 1004 y Fo(e)912 1023 y FH(f)8 b FK(.)270
1186 y(So)29 b(w)n(e)e(ha)n(v)n(e)i(found:)270 1350 y
FJ(\017)34 b FK(the)28 b(constan)n(ts)h(\\b)n(y)g(v)n(o)r(cation")40
b(_)-30 b FH(n)p FK(:)46 b(_)-30 b FH(n)p FK(\()p FH(a)p
FK(\))20 b(=)f FJ(f)p FH(n)p FJ(g)p FK(;)270 1503 y FJ(\017)34
b FK(the)19 b(functions)759 1485 y Fo(e)746 1503 y FH(f)8
b FK(,)20 b(amongst)g(whic)n(h)e(are)g(the)g(\\constan)n(ts")1898
1485 y Fo(e)1886 1503 y FH(f)7 b FK(\()p Fn(?)p FK(\))19
b(=)h Fn(?)p FK(,)2235 1485 y Fo(e)2223 1503 y FH(f)7
b FK(\()p FJ(f)p FH(m)p FJ(g)p FK(\))20 b(=)f FJ(f)p
FH(n)p FJ(g)p FK(,)339 1588 y(whic)n(h)27 b(only)h(di\013er)f(from)g
(the)h(\014rst)g(b)n(y)g(the)f(v)l(alue)h(at)g Fn(?)p
FK(.)167 1794 y Fy(8.3.2)95 b(P)m(arallel)29 b(Or)167
1926 y FK(Let)f(us)h(lo)r(ok)g(for)g(all)f(the)g(stable)g(functions)h
(of)g(t)n(w)n(o)g(argumen)n(ts)h(from)e FJ(B)-10 b FM(o)l(ol)9
b FK(,)29 b FJ(B)-10 b FM(o)l(ol)37 b FK(to)29 b FJ(B)-10
b FM(o)l(ol)167 2011 y FK(whic)n(h)25 b(represen)n(t)g(the)h
(disjunction)g(in)f(the)h(sense)f(that)i FH(F)10 b FK(\()p
FJ(f)p FH(\013)p FJ(g)p FH(;)i FJ(f)p FH(\014)t FJ(g)p
FK(\))19 b(=)g FJ(f)p FH(\013)d FJ(_)g FH(\014)t FJ(g)26
b FK(for)g(ev)n(ery)167 2096 y(substitution)i(of)g FG(t)f
FK(and)i FG(f)35 b FK(for)28 b FH(\013)g FK(and)g FH(\014)t
FK(.)270 2273 y(W)-6 b(e)42 b(m)n(ust)f(ha)n(v)n(e)i
FH(F)10 b FK(\()p FH(a)893 2247 y Fv(0)909 2273 y FH(;)i(b)969
2247 y Fv(0)985 2273 y FK(\))30 b FJ(\032)f FH(F)10 b
FK(\()p FH(a;)i(b)p FK(\))41 b(when)h FH(a)1607 2247
y Fv(0)1653 2273 y FJ(\032)29 b FH(a)42 b FK(and)h FH(b)1997
2247 y Fv(0)2043 2273 y FJ(\032)29 b FH(b)p FK(.)78 b(In)42
b(particular,)167 2358 y(if)53 b FH(F)10 b FK(\()p Fn(?)p
FH(;)i Fn(?)p FK(\))36 b(=)i FJ(f)p FG(t)p FJ(g)54 b
FK(\(or)f FJ(f)p FG(f)7 b FJ(g)p FK(\),)61 b(then)54
b FH(F)63 b FK(tak)n(es)54 b(constan)n(tly)g(the)g(v)l(alue)g
FG(t)f FK(\(or)h FG(f)7 b FK(\),)167 2443 y(whic)n(h)44
b(is)g(imp)r(ossible.)85 b(Similarly)42 b(w)n(e)i(ha)n(v)n(e)i
FH(F)10 b FK(\()p FJ(f)p FG(f)d FJ(g)p FH(;)12 b Fn(?)p
FK(\))31 b(=)h FH(F)10 b FK(\()p Fn(?)p FH(;)i FJ(f)p
FG(f)7 b FJ(g)p FK(\))31 b(=)g Fn(?)45 b FK(b)r(ecause)167
2528 y FH(F)10 b FK(\()p FJ(f)p FG(f)d FJ(g)p FH(;)12
b Fn(?)p FK(\))19 b FJ(\032)g FH(F)10 b FK(\()p FJ(f)p
FG(f)d FJ(g)p FH(;)12 b FJ(f)p FG(t)p FJ(g)p FK(\))20
b(=)f FJ(f)p FG(t)p FJ(g)28 b FK(and)h FH(F)10 b FK(\()p
FJ(f)p FG(f)d FJ(g)p FH(;)12 b Fn(?)p FK(\))18 b FJ(\032)i
FH(F)10 b FK(\()p FJ(f)p FG(f)d FJ(g)p FH(;)12 b FJ(f)p
FG(f)7 b FJ(g)p FK(\))20 b(=)f FJ(f)p FG(f)7 b FJ(g)p
FK(.)270 2656 y FH(F)j FK(\()p FJ(f)p FG(t)p FJ(g)p FH(;)i
Fn(?)p FK(\))21 b(=)h FJ(f)p FG(t)p FJ(g)31 b FK(is)f(p)r(ossible,)h
(but)g(then)g FH(F)10 b FK(\()p Fn(?)p FH(;)i FJ(f)p
FG(t)p FJ(g)p FK(\))21 b(=)g Fn(?)p FK(:)42 b(indeed,)31
b(if)f(w)n(e)g(write)f(the)167 2741 y(third)e(condition)g(for)h(t)n(w)n
(o)g(argumen)n(ts:)283 2917 y FH(a)319 2928 y FD(1)363
2917 y FJ([)16 b FH(a)462 2928 y FD(2)509 2917 y FJ(2)k(B)-10
b FM(o)l(ol)24 b FJ(^)15 b FH(b)811 2928 y FD(1)855 2917
y FJ([)h FH(b)947 2928 y FD(2)995 2917 y FJ(2)j(B)-10
b FM(o)l(ol)28 b FJ(\))20 b FH(F)10 b FK(\()p FH(a)1417
2928 y FD(1)1460 2917 y FJ(\\)16 b FH(a)1559 2928 y FD(2)1587
2917 y FH(;)c(b)1647 2928 y FD(1)1691 2917 y FJ(\\)j
FH(b)1782 2928 y FD(2)1810 2917 y FK(\))20 b(=)f FH(F)10
b FK(\()p FH(a)2048 2928 y FD(1)2075 2917 y FH(;)i(b)2135
2928 y FD(1)2163 2917 y FK(\))k FJ(\\)f FH(F)10 b FK(\()p
FH(a)2386 2928 y FD(2)2413 2917 y FH(;)i(b)2473 2928
y FD(2)2502 2917 y FK(\))167 3094 y(and)30 b(apply)f(it)f(for)h
FH(a)724 3105 y FD(1)772 3094 y FK(=)20 b FJ(f)p FG(t)p
FJ(g)p FK(,)30 b FH(a)1032 3105 y FD(2)1080 3094 y FK(=)20
b Fn(?)p FK(,)29 b FH(b)1286 3105 y FD(1)1334 3094 y
FK(=)20 b Fn(?)p FK(,)29 b FH(b)1540 3105 y FD(2)1589
3094 y FK(=)20 b FJ(f)p FG(t)p FJ(g)p FK(,)29 b(then)g
FH(F)10 b FK(\()p Fn(?)p FH(;)i FJ(f)p FG(t)p FJ(g)p
FK(\))19 b(=)h FJ(f)p FG(t)p FJ(g)29 b FK(w)n(ould)167
3179 y(giv)n(e)e(us)h FH(F)10 b FK(\()p Fn(?)p FH(;)i
Fn(?)p FK(\))18 b(=)i FJ(f)p FG(t)p FJ(g)p FK(.)270 3307
y(By)28 b(symmetry)-6 b(,)28 b(w)n(e)f(ha)n(v)n(e)h(obtained)g(t)n(w)n
(o)g(functions:)270 3471 y FJ(\017)34 b FH(F)384 3482
y FD(1)412 3471 y FK(\()p FJ(f)p FG(t)p FJ(g)p FH(;)12
b Fn(?)p FK(\))19 b(=)g FH(F)790 3482 y FD(1)818 3471
y FK(\()p FJ(f)p FG(t)p FJ(g)p FH(;)12 b FJ(f)p FG(t)p
FJ(g)p FK(\))19 b(=)h FH(F)1243 3482 y FD(1)1270 3471
y FK(\()p FJ(f)p FG(t)p FJ(g)p FH(;)12 b FJ(f)p FG(f)7
b FJ(g)p FK(\))20 b(=)g FH(F)1696 3482 y FD(1)1723 3471
y FK(\()p FJ(f)p FG(f)7 b FJ(g)p FH(;)12 b FJ(f)p FG(t)p
FJ(g)p FK(\))20 b(=)g FJ(f)p FG(t)p FJ(g)270 3616 y(\017)34
b FH(F)384 3627 y FD(1)412 3616 y FK(\()p FJ(f)p FG(f)7
b FJ(g)p FH(;)12 b FJ(f)p FG(f)7 b FJ(g)p FK(\))21 b(=)e
FJ(f)p FG(f)7 b FJ(g)270 3761 y(\017)34 b FH(F)384 3772
y FD(1)412 3761 y FK(\()p Fn(?)p FH(;)12 b Fn(?)p FK(\))19
b(=)g FH(F)744 3772 y FD(1)771 3761 y FK(\()p FJ(f)p
FG(f)7 b FJ(g)p FH(;)12 b Fn(?)p FK(\))20 b(=)f FH(F)1150
3772 y FD(1)1178 3761 y FK(\()p Fn(?)p FH(;)12 b FJ(f)p
FG(t)p FJ(g)p FK(\))19 b(=)g FH(F)1556 3772 y FD(1)1584
3761 y FK(\()p Fn(?)p FH(;)12 b FJ(f)p FG(f)7 b FJ(g)p
FK(\))19 b(=)h Fn(?)167 3925 y FK(and)29 b FH(F)351 3936
y FD(2)378 3925 y FK(\()p FH(a;)12 b(b)p FK(\))20 b(=)f
FH(F)666 3936 y FD(1)694 3925 y FK(\()p FH(b;)12 b(a)p
FK(\).)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
60 63 bop 167 167 a FK(60)1205 b Fz(CHAPTER)27 b(8.)63
b(COHERENCE)26 b(SP)-6 b(A)n(CES)270 404 y FK(There)27
b(remains)g(another)h(solution:)270 521 y FJ(\017)34
b FH(F)384 532 y FD(3)412 521 y FK(\()p FJ(f)p FG(t)p
FJ(g)p FH(;)12 b FJ(f)p FG(t)p FJ(g)p FK(\))20 b(=)f
FH(F)837 532 y FD(3)864 521 y FK(\()p FJ(f)p FG(f)7 b
FJ(g)p FH(;)12 b FJ(f)p FG(t)p FJ(g)p FK(\))21 b(=)e
FH(F)1290 532 y FD(3)1317 521 y FK(\()p FJ(f)p FG(t)p
FJ(g)p FH(;)12 b FJ(f)p FG(f)7 b FJ(g)p FK(\))20 b(=)g
FJ(f)p FG(t)p FJ(g)270 654 y(\017)34 b FH(F)384 665 y
FD(3)412 654 y FK(\()p FJ(f)p FG(f)7 b FJ(g)p FH(;)12
b FJ(f)p FG(f)7 b FJ(g)p FK(\))21 b(=)e FJ(f)p FG(f)7
b FJ(g)270 786 y(\017)34 b Fn(?)28 b FK(otherwise.)270
933 y(The)g(stabilit)n(y)e(condition)h(w)n(as)h(used)h(to)e(eliminate)e
(the)i(case)h(of:)270 1049 y FJ(\017)34 b FH(F)384 1060
y FD(0)412 1049 y FK(\()p FJ(f)p FG(t)p FJ(g)p FH(;)12
b Fn(?)p FK(\))19 b(=)g FH(F)790 1060 y FD(0)818 1049
y FK(\()p Fn(?)p FH(;)12 b FJ(f)p FG(t)p FJ(g)p FK(\))19
b(=)g FJ(f)p FG(t)p FJ(g)270 1166 y FK(What)31 b(ha)n(v)n(e)f(w)n(e)g
(got)g(against)h(this)e(example?)41 b(It)29 b(violates)g(a)i(principle)
c(of)j FM(le)l(ast)g(data)5 b FK(:)167 1251 y(w)n(e)33
b(ha)n(v)n(e)i FH(F)494 1262 y FD(0)522 1251 y FK(\()p
FJ(f)p FG(t)p FJ(g)p FH(;)12 b FJ(f)p FG(t)p FJ(g)p FK(\))24
b(=)g FJ(f)p FG(t)p FJ(g)p FK(;)37 b(w)n(e)d(seek)f(to)h(\014nd)h(a)g
(least)e(appro)n(ximan)n(t)i(to)f(the)g(pair)g(of)167
1336 y(argumen)n(ts)e FJ(f)p FG(t)p FJ(g)p FH(;)12 b
FJ(f)p FG(t)p FJ(g)31 b FK(whic)n(h)g(already)g(giv)n(es)g
FJ(f)p FG(t)p FJ(g)p FK(;)i(no)n(w)f(w)n(e)f(ha)n(v)n(e)h(at)f(our)h
(disp)r(osal)f Fn(?)p FH(;)12 b FJ(f)p FG(t)p FJ(g)167
1421 y FK(and)29 b FJ(f)p FG(t)p FJ(g)p FH(;)12 b Fn(?)27
b FK(whic)n(h)g(are)h(minimal)c(\()p Fn(?)p FH(;)12 b
Fn(?)27 b FK(do)r(es)i(not)f(w)n(ork\))g(and)g(distinct.)270
1536 y(Of)c(course,)i(kno)n(wing)f(that)g(w)n(e)f(alw)n(a)n(ys)g(ha)n
(v)n(e)i(a)f(distinguished)e(\()p FM(le)l(ast)p FK(\))h(solution)g
(\(rather)167 1621 y(than)k(man)n(y)g FM(minimal)g FK(solutions\))f
(for)g(a)h(problem)e(of)i(this)e(kind)i(radically)d(simpli\014es)g(a)j
(lot)167 1707 y(of)g(calculations.)167 1936 y FE(8.4)114
b(Direct)27 b(pro)s(duct)i(of)f(t)m(w)m(o)g(coherence)f(spaces)167
2091 y FK(A)g(function)h FH(F)37 b FK(of)28 b(t)n(w)n(o)g(argumen)n
(ts,)g(mapping)g FJ(A)p FH(;)12 b FJ(B)31 b FK(to)c FJ(C)32
b FK(is)27 b(stable)g(when:)201 2208 y(i\))g FH(a)311
2182 y Fv(0)347 2208 y FJ(\032)20 b FH(a)f FJ(2)h(A)c(^)f
FH(b)707 2182 y Fv(0)744 2208 y FJ(\032)k FH(b)h FJ(2)g(B)i(\))e
FH(F)10 b FK(\()p FH(a)1210 2182 y Fv(0)1225 2208 y FH(;)i(b)1285
2182 y Fv(0)1302 2208 y FK(\))19 b FJ(\032)h FH(F)10
b FK(\()p FH(a;)i(b)p FK(\))201 2340 y(ii\))26 b FH(F)10
b FK(\()376 2287 y Fo(S)434 2306 y Fv(")434 2361 y Fw(i)p
Fv(2)p Fw(I)524 2340 y FH(a)560 2351 y Fw(i)580 2340
y FH(;)611 2287 y Fo(S)670 2306 y Fv(")670 2361 y Fw(j)s
Fv(2)p Fw(J)772 2340 y FH(b)801 2351 y Fw(j)828 2340
y FK(\))19 b(=)947 2287 y Fo(S)1006 2306 y Fv(")1006
2362 y FD(\()p Fw(i;j)s FD(\))p Fv(2)p Fw(I)t Fv(\002)p
Fw(J)1242 2340 y FH(F)10 b FK(\()p FH(a)1360 2351 y Fw(i)1379
2340 y FH(;)i(b)1439 2351 y Fw(j)1465 2340 y FK(\))106
b(\(directed)26 b(union\))201 2479 y(iii\))f FH(a)349
2490 y FD(1)393 2479 y FJ([)16 b FH(a)492 2490 y FD(2)539
2479 y FJ(2)k(A)c(^)f FH(b)769 2490 y FD(1)813 2479 y
FJ([)h FH(b)905 2490 y FD(2)953 2479 y FJ(2)j(B)j(\))e
FH(F)10 b FK(\()p FH(a)1295 2490 y FD(1)1338 2479 y FJ(\\)16
b FH(a)1437 2490 y FD(2)1465 2479 y FH(;)c(b)j FJ(\\)h
FH(b)1632 2490 y FD(2)1660 2479 y FK(\))k(=)f FH(F)10
b FK(\()p FH(a)1898 2490 y FD(1)1925 2479 y FH(;)i(b)1985
2490 y FD(1)2013 2479 y FK(\))k FJ(\\)f FH(F)10 b FK(\()p
FH(a)2236 2490 y FD(2)2263 2479 y FH(;)i(b)2323 2490
y FD(2)2351 2479 y FK(\))270 2596 y(Lik)n(ewise)43 b(w)n(e)g(de\014ne)h
(stabilit)n(y)f(in)g(an)n(y)i(n)n(um)n(b)r(er)f(of)h(argumen)n(ts.)85
b(Observ)n(e)44 b(that,)167 2681 y(whereas)24 b(separate)g
FM(c)l(ontinuity)31 b FK(su\016ces)25 b(for)f(join)n(t)g(con)n(tin)n
(uit)n(y)-6 b(,)25 b(stabilit)n(y)e(in)g(t)n(w)n(o)h(argumen)n(ts)167
2766 y(is)h(equiv)l(alen)n(t)f(to)i(stabilit)n(y)e(in)g(eac)n(h)i
(separately)-6 b(,)26 b(together)f(with)g(the)g(additional)f(condition)
167 2851 y(that)k(the)f(pullbac)n(k)1345 2895 y(\()p
FH(a;)12 b(b)p FK(\))1019 3212 y(\()p FH(a;)g(b)1142
3187 y Fv(0)1159 3212 y FK(\))468 b(\()p FH(a)1717 3187
y Fv(0)1733 3212 y FH(;)12 b(b)p FK(\))1328 3530 y(\()p
FH(a)1391 3504 y Fv(0)1408 3530 y FH(;)g(b)1468 3504
y Fv(0)1484 3530 y FK(\))1186 3111 y Fs(\000)1244 3053
y(\000)1277 3020 y(\000)-59 b(\022)1595 3111 y(@)1536
3053 y(@)1503 3020 y(@)g(I)1277 3429 y(@)1219 3370 y(@)1186
3337 y(@)g(I)1503 3429 y(\000)1562 3370 y(\000)1595 3337
y(\000)g(\022)167 3604 y FK(\(where)27 b FH(a)434 3578
y Fv(0)470 3604 y FJ(\032)19 b FH(a)h FJ(2)g(A)27 b FK(and)i
FH(b)918 3578 y Fv(0)954 3604 y FJ(\032)20 b FH(b)g FJ(2)f(B)r
FK(\))28 b(b)r(e)f(preserv)n(ed.)270 3754 y(W)-6 b(e)34
b(w)n(ould)g(lik)n(e)e(to)i(a)n(v)n(oid)g(studying)h(stable)e
(functions)h(of)h(t)n(w)n(o)f(\(or)f(more\))g(v)l(ariables)167
3840 y(and)h(so)g(reduce)f(them)f(to)i(the)f(unary)h(case.)52
b(F)-6 b(or)34 b(this)f(w)n(e)g(shall)f(in)n(tro)r(duce)h(the)g(\()p
FM(dir)l(e)l(ct)p FK(\))167 3925 y FM(pr)l(o)l(duct)c
FJ(A)16 b FK(&)f FJ(B)30 b FK(of)e(t)n(w)n(o)g(coherence)f(spaces.)36
b(The)27 b(notation)i(comes)e(from)g(linear)f(logic.)p
eop
PStoPSsaved restore
%%Page: (64,65) 33
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
61 64 bop 167 167 a Fz(8.5.)64 b(THE)27 b(FUNCTION-SP)-6
b(A)n(CE)1461 b FK(61)270 404 y(If)28 b FJ(A)g FK(and)h
FJ(B)h FK(are)d(t)n(w)n(o)h(coherence)f(spaces,)h(w)n(e)f(de\014ne)h
FJ(A)16 b FK(&)f FJ(B)30 b FK(b)n(y:)743 607 y FJ(jA)16
b FK(&)f FJ(B)r(j)k FK(=)h FJ(jAj)15 b FK(+)g FJ(jB)r(j)k
FK(=)g FJ(f)p FK(1)p FJ(g)e(\002)e(jAj)g([)h(f)p FK(2)p
FJ(g)g(\002)g(jB)r(j)518 826 y FK(\(1)p FH(;)c(\013)p
FK(\))701 804 y FH(_)701 847 y(^)790 826 y FK(\(1)p FH(;)g(\013)926
801 y Fv(0)943 826 y FK(\))47 b(\(mo)r(d)23 b FJ(A)16
b FK(&)f FJ(B)s FK(\))86 b(i\013)27 b FH(\013)1653 804
y(_)1653 847 y(^)1741 826 y(\013)1785 801 y Fv(0)1849
826 y FK(\(mo)r(d)c FJ(A)p FK(\))520 997 y(\(2)p FH(;)12
b(\014)t FK(\))701 975 y FH(_)701 1017 y(^)790 997 y
FK(\(2)p FH(;)g(\014)925 971 y Fv(0)941 997 y FK(\))47
b(\(mo)r(d)24 b FJ(A)16 b FK(&)f FJ(B)r FK(\))88 b(i\013)27
b FH(\014)1651 975 y(_)1651 1017 y(^)1740 997 y(\014)1783
971 y Fv(0)1846 997 y FK(\(mo)r(d)c FJ(B)r FK(\))527
1167 y(\(1)p FH(;)12 b(\013)p FK(\))710 1145 y FH(_)710
1188 y(^)799 1167 y FK(\(2)p FH(;)g(\014)t FK(\))47 b(\(mo)r(d)23
b FJ(A)16 b FK(&)f FJ(B)s FK(\))95 b(for)28 b(all)e FH(\013)20
b FJ(2)g(jAj)27 b FK(and)h FH(\014)c FJ(2)19 b(jB)r(j)167
1388 y FK(In)33 b(particular,)h(the)f(p)r(oin)n(ts)g(of)h
FJ(A)17 b FK(&)g FJ(B)36 b FK(\(coheren)n(t)c(subsets)i(of)g
FJ(jA)15 b FK(&)g FJ(B)s(j)p FK(\))32 b(can)i(b)r(e)f(written)167
1473 y(uniquely)e(as)h FJ(f)p FK(1)p FJ(g)18 b(\002)e
FH(a)h FJ([)g(f)p FK(2)p FJ(g)g(\002)g FH(b)31 b FK(with)g
FH(a)22 b FJ(2)g(A)p FK(,)33 b FH(b)23 b FJ(2)f(B)r FK(.)47
b(The)31 b(reader)g(is)f(in)n(vited)h(to)g(sho)n(w)167
1558 y(that)j(this)f(is)g(the)g(pro)r(duct)h(in)f(the)h(categorical)e
(sense)i(\(w)n(e)e(shall)h(return)h(to)f(this)g(in)g(the)167
1643 y(next)28 b(c)n(hapter)g(when)f(w)n(e)g(de\014ne)h(the)g(in)n
(terpretation\).)270 1808 y(Giv)n(en)f(a)g(stable)f(function)h
FH(F)35 b FK(from)27 b FJ(A)p FH(;)12 b FJ(B)29 b FK(to)e
FJ(C)t FK(,)g(w)n(e)f(de\014ne)h(a)g(function)f FH(G)h
FK(from)g FJ(A)14 b FK(&)g FJ(B)167 1893 y FK(to)28 b
FJ(C)k FK(b)n(y:)940 2059 y FH(G)p FK(\()p FJ(f)p FK(1)p
FJ(g)17 b(\002)f FH(a)f FJ([)h(f)p FK(2)p FJ(g)h(\002)e
FH(b)p FK(\))20 b(=)f FH(F)10 b FK(\()p FH(a;)i(b)p FK(\))167
2224 y(It)37 b(is)f(immediate)f(that)i FH(G)h FK(is)f(stable;)k(con)n
(v)n(ersely)c(the)g(same)g(form)n(ula)g(de\014nes,)k(from)c(a)167
2309 y(stable)26 b(unary)i(function)f FH(G)p FK(,)g(a)h(stable)e
(binary)h(function)g FH(F)10 b FK(,)26 b(and)i(the)e(t)n(w)n(o)h
(transformations)167 2394 y(are)h(in)n(v)n(erse.)167
2627 y FE(8.5)114 b(The)28 b(F)-10 b(unction-Space)167
2782 y FK(W)k(e)33 b(started)g(with)f(the)h(idea)g(that)g(\\t)n(yp)r(e)
h(=)f(coherence)f(space".)53 b(The)32 b(previous)i(section)167
2867 y(de\014nes)29 b(a)h(pro)r(duct)f(of)g(coherence)f(spaces)i
(corresp)r(onding)f(to)h(the)e(pro)r(duct)i(of)f(t)n(yp)r(es,)h(but)167
2952 y(what)35 b(do)g(w)n(e)g(do)g(with)f(the)g(arro)n(w?)57
b(W)-6 b(e)34 b(w)n(ould)h(lik)n(e)e(to)i(de\014ne)g
FJ(A)17 b(!)i(B)37 b FK(as)e(the)f(set)h(of)167 3038
y(stable)h(functions)h(from)g FJ(A)f FK(to)h FJ(B)r FK(,)j(but)d(this)f
(is)g(not)h(presen)n(ted)g(as)g(a)g(coherence)f(space.)167
3123 y(So)30 b(w)n(e)e(shall)h(giv)n(e)g(a)g(particular)f(represen)n
(tation)h(of)h(the)f(set)f(of)i(stable)f(functions)g(in)g(suc)n(h)167
3208 y(a)f(w)n(a)n(y)h(as)f(to)g(mak)n(e)f(it)g(a)h(coherence)f(space.)
167 3410 y Fy(8.5.1)95 b(The)27 b(trace)i(of)e(a)h(stable)g(function)
167 3541 y FG(Lemma)d FK(Let)i FH(F)37 b FK(b)r(e)27
b(a)h(stable)g(function)f(from)h FJ(A)g FK(to)g FJ(B)r
FK(,)g(and)g(let)f FH(a)19 b FJ(2)h(A)p FK(,)28 b FH(\014)23
b FJ(2)d FH(F)10 b FK(\()p FH(a)p FK(\);)27 b(then)201
3686 y(i\))g(it)g(is)g(p)r(ossible)g(to)g(\014nd)i FH(a)954
3697 y Fv(\016)1002 3686 y FJ(\032)19 b FH(a)28 b FK(\014nite)f(suc)n
(h)h(that)g FH(\014)23 b FJ(2)d FH(F)10 b FK(\()p FH(a)1882
3697 y Fv(\016)1909 3686 y FK(\).)201 3826 y(ii\))29
b(if)h FH(a)404 3837 y Fv(\016)462 3826 y FK(is)g(c)n(hosen)h(minimal)d
(for)j(the)f(inclusion)f(among)j(the)e(solutions)h(to)f(\(i\),)g(then)h
FH(a)2644 3837 y Fv(\016)339 3911 y FK(is)c FM(le)l(ast)p
FK(,)h(and)h(is)e(in)g(particular)g FM(unique)p FK(.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
62 65 bop 167 167 a FK(62)1205 b Fz(CHAPTER)27 b(8.)63
b(COHERENCE)26 b(SP)-6 b(A)n(CES)167 404 y FG(Pro)r(of)201
586 y FK(i\))53 b(W)-6 b(rite)53 b FH(a)38 b FK(=)690
534 y Fo(S)749 552 y Fv(")749 607 y Fw(i)p Fv(2)p Fw(I)839
586 y FH(a)875 597 y Fw(i)895 586 y FK(,)61 b(where)53
b(the)g FH(a)1390 597 y Fw(i)1464 586 y FK(are)h(the)g(\014nite)f
(subsets)h(of)h FH(a)p FK(.)114 b(Then)339 683 y FH(F)10
b FK(\()p FH(a)p FK(\))19 b(=)576 630 y Fo(S)635 648
y Fv(")635 703 y Fw(i)p Fv(2)p Fw(I)725 683 y FH(F)10
b FK(\()p FH(a)843 694 y Fw(i)862 683 y FK(\),)28 b(and)h(if)d
FH(\014)e FJ(2)19 b FH(F)10 b FK(\()p FH(a)p FK(\),)27
b FH(\014)d FJ(2)19 b FH(F)10 b FK(\()p FH(a)1709 694
y Fw(i)1726 701 y Fi(0)1753 683 y FK(\))27 b(for)h(some)g
FH(i)2119 694 y FD(0)2165 683 y FJ(2)20 b FH(I)6 b FK(.)201
839 y(ii\))50 b(Supp)r(ose)i FH(a)652 850 y Fv(\016)731
839 y FK(is)f(minimal,)j(and)e(let)e FH(a)1476 814 y
Fv(0)1528 839 y FJ(\032)36 b FH(a)52 b FK(suc)n(h)f(that)h
FH(\014)40 b FJ(2)c FH(F)10 b FK(\()p FH(a)2348 814 y
Fv(0)2364 839 y FK(\).)105 b(Then)339 924 y FH(a)375
935 y Fv(\016)422 924 y FJ([)18 b FH(a)523 899 y Fv(0)566
924 y FJ(\032)26 b FH(a)h FJ(2)f(A)p FK(,)40 b(so)d FH(a)1032
935 y Fv(\016)1079 924 y FJ([)18 b FH(a)1180 899 y Fv(0)1223
924 y FJ(2)26 b(A)37 b FK(and)i FH(\014)30 b FJ(2)c FH(F)10
b FK(\()p FH(a)1798 935 y Fv(\016)1825 924 y FK(\))18
b FJ(\\)h FH(F)10 b FK(\()p FH(a)2054 899 y Fv(0)2069
924 y FK(\))26 b(=)g FH(F)10 b FK(\()p FH(a)2320 935
y Fv(\016)2366 924 y FJ(\\)18 b FH(a)2467 899 y Fv(0)2484
924 y FK(\).)63 b(As)339 1009 y FH(a)375 1020 y Fv(\016)438
1009 y FK(is)34 b(minimal,)g(this)g(forces)g FH(a)1207
1020 y Fv(\016)1260 1009 y FJ(\032)24 b FH(a)1375 1020
y Fv(\016)1421 1009 y FJ([)17 b FH(a)1521 984 y Fv(0)1538
1009 y FK(,)36 b(so)g FH(a)1726 1020 y Fv(\016)1778 1009
y FJ(\032)25 b FH(a)1894 984 y Fv(0)1910 1009 y FK(,)37
b(and)e FH(a)2147 1020 y Fv(\016)2210 1009 y FK(is)f(indeed)g
FM(le)l(ast)p FK(.)339 1095 y(T)-6 b(o)29 b(put)f(this)f(another)h(w)n
(a)n(y)-6 b(,)29 b(w)n(e)f(ha)n(v)n(e)g(said)g(that)g(w)n(e)f(in)n
(tend)h(stabilit)n(y)e(to)i(mean)f(the)339 1180 y(in)n(tersection)f(of)
i(an)g FM(arbitr)l(ary)g FK(family)e(whic)n(h)h(is)f(b)r(ounded)j(ab)r
(o)n(v)n(e,)g(and)g(here)d(w)n(e)h(are)339 1265 y(just)h(taking)h(the)e
(in)n(tersection)f(of)i(the)f(\014nite)g FH(a)1619 1239
y Fv(0)1655 1265 y FJ(\032)20 b FH(a)28 b FK(suc)n(h)g(that)g
FH(\014)23 b FJ(2)d FH(F)10 b FK(\()p FH(a)2355 1239
y Fv(0)2370 1265 y FK(\).)201 b Fu(\003)270 1498 y FK(The)28
b FM(tr)l(ac)l(e)g FJ(T)-6 b FH(r)r FK(\()p FH(F)10 b
FK(\))27 b(is)g(the)g(set)h(of)g(pairs)f(\()p FH(a)1418
1509 y Fv(\016)1446 1498 y FH(;)12 b(\014)t FK(\))28
b(suc)n(h)g(that:)201 1680 y(i\))f FH(a)311 1691 y Fv(\016)367
1680 y FK(is)g(a)h(\014nite)f(p)r(oin)n(t)g(of)h FJ(A)g
FK(and)h FH(\014)23 b FJ(2)d(jB)r(j)201 1836 y FK(ii\))26
b FH(\014)e FJ(2)19 b FH(F)10 b FK(\()p FH(a)541 1847
y Fv(\016)568 1836 y FK(\))201 1993 y(iii\))25 b(if)i
FH(a)417 1968 y Fv(0)453 1993 y FJ(\032)20 b FH(a)564
2004 y Fv(\016)619 1993 y FK(and)29 b FH(\014)24 b FJ(2)19
b FH(F)10 b FK(\()p FH(a)1005 1968 y Fv(0)1021 1993 y
FK(\))27 b(then)h FH(a)1273 1968 y Fv(0)1309 1993 y FK(=)19
b FH(a)1418 2004 y Fv(\016)1446 1993 y FK(.)270 2226
y FJ(T)-6 b FH(r)r FK(\()p FH(F)10 b FK(\))28 b(determines)d
FH(F)37 b FK(uniquely)27 b(b)n(y)i(the)e(form)n(ula)700
2430 y(\()p FG(App)p FK(\))78 b FH(F)10 b FK(\()p FH(a)p
FK(\))18 b(=)h FJ(f)p FH(\014)24 b FK(:)39 b FJ(9)p FH(a)1447
2441 y Fv(\016)1495 2430 y FJ(\032)20 b FH(a)f FK(\()p
FH(a)1688 2441 y Fv(\016)1716 2430 y FH(;)12 b(\014)t
FK(\))19 b FJ(2)h(T)-6 b FH(r)r FK(\()p FH(F)10 b FK(\))p
FJ(g)167 2634 y FK(whic)n(h)26 b(results)f(immediately)e(from)k(the)f
(lemma.)33 b(In)27 b(particular)e(the)h(function)h FH(F)h
FJ(7!)20 b(T)-6 b FH(r)r FK(\()p FH(F)10 b FK(\))167
2719 y(is)27 b(1{1.)270 2923 y(Consider)22 b(for)g(example)f(the)h
(stable)f(function)h FH(F)1560 2934 y FD(1)1609 2923
y FK(from)g FJ(B)-10 b FM(o)l(ol)15 b FK(&)7 b FJ(B)-10
b FM(o)l(ol)30 b FK(to)22 b FJ(B)-10 b FM(o)l(ol)30 b
FK(in)n(tro)r(duced)167 3008 y(in)55 b FG(??)p FK(.)36
b(The)27 b(elemen)n(ts)f(of)i(its)f(trace)g FJ(T)-6 b
FH(r)r FK(\()p FH(F)1340 3019 y FD(1)1368 3008 y FK(\))27
b(are:)423 3212 y(\()p FJ(f)p FK(\(1)p FH(;)12 b FG(t)p
FK(\))p FJ(g)p FH(;)32 b FG(t)p FK(\))255 b(\()p FJ(f)p
FK(\(1)p FH(;)12 b FG(f)7 b FK(\))p FH(;)25 b FK(\(2)p
FH(;)12 b FG(t)p FK(\))o FJ(g)p FH(;)32 b FG(t)p FK(\))256
b(\()p FJ(f)p FK(\(1)p FH(;)12 b FG(f)7 b FK(\))p FH(;)24
b FK(\(2)p FH(;)12 b FG(f)7 b FK(\))p FJ(g)p FH(;)32
b FG(f)7 b FK(\))167 3416 y(W)-6 b(e)27 b(can)h(read)g(this)f(as)i(the)
e(sp)r(eci\014cation:)270 3598 y FJ(\017)34 b FK(if)27
b(the)h(\014rst)f(argumen)n(t)h(is)f FM(true)p FK(,)h(the)f(result)g
(is)g FM(true)p FK(;)270 3754 y FJ(\017)34 b FK(if)27
b(the)h(\014rst)f(argumen)n(t)h(is)f FM(false)h FK(and)g(the)g(second)g
FM(true)p FK(,)f(the)h(result)e(is)h FM(true)p FK(;)270
3911 y FJ(\017)34 b FK(if)27 b(the)h(\014rst)f(argumen)n(t)h(is)f
FM(false)h FK(and)g(the)g(second)g FM(false)p FK(,)g(the)f(result)g(is)
g FM(false)p FK(.)p eop
PStoPSsaved restore
%%Page: (66,67) 34
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
63 66 bop 167 167 a Fz(8.5.)64 b(THE)27 b(FUNCTION-SP)-6
b(A)n(CE)1461 b FK(63)167 404 y Fy(8.5.2)95 b(Represen)m(tation)28
b(of)f(the)h(function)f(space)167 535 y FG(Prop)r(osition)d
FK(As)i FH(F)35 b FK(v)l(aries)26 b(o)n(v)n(er)h(the)f(stable)g
(functions)g(from)g FJ(A)g FK(to)h FJ(B)r FK(,)g(their)e(traces)g(giv)n
(e)167 620 y(the)i(p)r(oin)n(ts)h(of)g(a)g(coherence)f(space,)h
(written)e FJ(A)16 b(!)g(B)r FK(.)167 773 y FG(Pro)r(of)29
b FK(Let)g(us)h(de\014ne)g(the)f(coherence)f(space)i
FJ(C)j FK(b)n(y)d FJ(jC)t(j)20 b FK(=)h FJ(A)1859 784
y Fm(\014n)1938 773 y FJ(\002)16 b(jB)r(j)29 b FK(\()p
FJ(A)2209 784 y Fm(\014n)2302 773 y FK(is)g(the)g(set)g(of)167
858 y(\014nite)e(p)r(oin)n(ts)h(of)g FJ(A)p FK(\))g(where)e(\()p
FH(a)1020 869 y FD(1)1048 858 y FH(;)12 b(\014)1118 869
y FD(1)1146 858 y FK(\))1192 836 y FH(_)1192 879 y(^)1281
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
65 68 bop 167 167 a Fz(8.5.)64 b(THE)27 b(FUNCTION-SP)-6
b(A)n(CE)1461 b FK(65)167 404 y Fy(8.5.4)95 b(P)m(artial)29
b(functions)167 535 y FK(Let)43 b(us)g(see)g(ho)n(w)h(this)e
(construction)i(w)n(orks)g(b)n(y)f(calculating)f FJ(I)-7
b FM(nt)28 b FJ(!)20 b(I)-7 b FM(nt)7 b FK(.)82 b(W)-6
b(e)43 b(ha)n(v)n(e)167 620 y FJ(I)-7 b FM(nt)267 631
y Fm(\014n)350 620 y FJ(')20 b FK(I)-12 b(N)15 b FJ([)h(f)p
Fn(?)p FJ(g)27 b FK(and)i FJ(jI)-7 b FM(nt)7 b FJ(j)19
b FK(=)g(I)-12 b(N,)28 b(so)g FJ(jI)-7 b FM(nt)22 b FJ(!)17
b(I)-7 b FM(nt)7 b FJ(j)19 b(')g FK(\(I)-12 b(N)15 b
FJ([)h(f)p Fn(?)p FJ(g)p FK(\))g FJ(\002)f FK(I)-12 b(N)28
b(where)201 782 y(i\))f(\()p FH(n;)12 b(m)p FK(\))481
759 y FH(_)481 802 y(^)569 782 y FK(\()p FH(n)637 756
y Fv(0)654 782 y FH(;)g(m)745 756 y Fv(0)761 782 y FK(\))28
b(if)f FH(n)19 b FK(=)h FH(n)1059 756 y Fv(0)1095 782
y FJ(\))g FH(m)g FK(=)f FH(m)1398 756 y Fv(0)201 925
y FK(ii\))26 b(\()p Fn(?)p FH(;)12 b(m)p FK(\))513 903
y FH(_)513 946 y(^)602 925 y FK(\()p Fn(?)p FH(;)g(m)p
FK(\))167 1087 y(with)44 b(incoherence)g(otherwise.)86
b(This)44 b(is)g(the)h FM(dir)l(e)l(ct)h(sum)f FK(\(see)f(section)g
FG(??)q FK(\))h(of)g(the)167 1172 y(coherence)23 b(space)h(whic)n(h)f
(represen)n(ts)g(partial)g(functions)h(with)e(the)i(space)g(whic)n(h)f
(represen)n(ts)167 1257 y(the)j(constan)n(ts)h(\\b)n(y)h(v)n(o)r
(cation".)36 b(Let)26 b(us)g(ignore)g(the)g(latter)f(part)h(and)i
(concen)n(trate)d(on)i(the)167 1342 y(space)h FJ(P)-6
b(F)34 b FK(de\014ned)28 b(on)h(the)e(w)n(eb)h(I)-12
b(N)15 b FJ(\002)g FK(I)-12 b(N)28 b(b)n(y)g(condition)g(\(i\).)270
1468 y(What)39 b(is)f(the)g(order)g(relation)g(on)h FJ(P)-6
b(F)6 b FK(?)67 b(W)-6 b(ell)37 b FH(f)d FJ(2)28 b(P)-6
b(F)45 b FK(is)37 b(a)i(set)f(of)h(pairs)f(\()p FH(n;)12
b(m)p FK(\))167 1553 y(suc)n(h)38 b(that)h(if)e(\()p
FH(n;)12 b(m)p FK(\))p FH(;)g FK(\()p FH(n;)g(m)956 1528
y Fv(0)972 1553 y FK(\))27 b FJ(2)g FH(f)45 b FK(then)38
b FH(m)27 b FK(=)f FH(m)1578 1528 y Fv(0)1595 1553 y
FK(,)41 b(whic)n(h)c(is)g(just)h(the)f(usual)i(\\graph")167
1639 y(represen)n(tation)29 b(of)g(a)g(partial)g(function.)39
b(Since)28 b(the)h(Berry)f(order)h(corresp)r(onds)h(simply)e(to)167
1724 y(con)n(tainmen)n(t,)f(it)g(is)g(the)g(usual)h(extension)g(order)g
(on)g(partial)f(functions.)270 1850 y(In)40 b(the)e(Berry)h(order,)i
(the)e(partial)f(functions)1632 1831 y Fo(e)1619 1850
y FH(f)47 b FK(and)40 b(the)e(constan)n(ts)j(b)n(y)e(v)n(o)r(cation)178
1935 y(_)-30 b FH(n)34 b FK(are)g(incomparable.)53 b(Ho)n(w)n(ev)n(er)
33 b FM(p)l(ointwise)i FK(w)n(e)e(ha)n(v)n(e)1736 1916
y Fo(e)1724 1935 y FH(f)e(<)1874 1920 y FK(_)1867 1935
y(0)j(for)g(an)n(y)h(partial)e(function)167 2020 y(whic)n(h)h(tak)n(es)
h(no)g(other)f(v)l(alue)h(than)g(zero,)h(of)f(whic)n(h)f(there)g(are)g
(in\014nitely)f(man)n(y)-6 b(.)57 b(One)167 2105 y(adv)l(an)n(tage)30
b(of)d(our)h(seman)n(tics)e(is)g(that)i(it)e(a)n(v)n(oids)i(this)e
(phenomenon)i(of)g FM(c)l(omp)l(act)2403 2080 y FD(5)2459
2105 y FK(ob)t(jects)167 2190 y(with)f(in\014nitely)f(man)n(y)i(ob)t
(jects)g(b)r(elo)n(w)f(them.)270 2275 y(Another)37 b(consequence)f(of)h
(the)e(Berry)h(order)g(arises)g(at)g(an)h(ev)n(en)f(simpler)e(t)n(yp)r
(e:)53 b(in)167 2361 y(the)40 b(function-space)h FJ(S)-6
b FM(gl)25 b FJ(!)20 b(S)-6 b FM(gl)6 b FK(,)44 b(where)c
FJ(S)-6 b FM(gl)46 b FK(is)40 b(the)g(coherence)g(space)h(with)f(just)h
(one)167 2446 y(tok)n(en)30 b(\(section)f FG(??)p FK(\).)41
b(In)30 b(the)f(p)r(oin)n(t)n(wise)f(\(Scott\))h(order,)h(the)f(iden)n
(tit)n(y)f(function)i(is)f(b)r(elo)n(w)167 2531 y(the)e(constan)n(t)h
(\\b)n(y)h(v)n(o)r(cation")f FJ(f\017)q(g)p FK(,)g(whilst)d(in)i(the)g
(Berry)g(order)g(they)g(are)h(incomparable.)167 2616
y(This)e(means)h(that)g(in)f(the)g(stable)g(seman)n(tics,)g(unlik)n(e)g
(the)h(Scott)f(seman)n(tics,)g(it)g(is)g(p)r(ossible)167
2701 y(for)k(a)g(test)g(program)g(to)g(succeed)f(on)i(the)e(iden)n(tit)
n(y)g(\(whic)n(h)g(reads)h(its)f(input\))h(but)g(fail)e(on)167
2786 y(the)f(constan)n(t)i(\(whic)n(h)e(do)r(es)h(not\).)p
167 3711 1003 3 v 246 3763 a FC(5)273 3784 y FB(The)35
b(notion)f(of)h(compactness)h(in)e(top)r(ology)g(is)h(purely)f
(order-theoretic:)48 b(if)34 b Fx(a)21 b Fl(\024)2288
3740 y Fh(S)2337 3754 y Fk(")2374 3784 y Fx(I)39 b FB(for)d(some)167
3854 y FA(dir)m(e)m(cte)m(d)c FB(set)g Fx(I)37 b FB(then)31
b Fx(a)19 b Fl(\024)g Fx(b)32 b FB(for)g(some)h Fx(b)19
b Fl(2)g Fx(I)5 b FB(.)47 b(Besides)31 b(Scott's)h(domain)g(theory)-5
b(,)33 b(this)e(also)h(arises)g(in)167 3925 y(ring)27
b(theory)g(as)h(No)r(etherianness)f(and)g(in)f(univ)n(ersal)h(algebra)g
(as)i(\014nite)d(presen)n(tabilit)n(y)-5 b(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
66 69 bop 167 875 a FF(Chapter)28 b(9)167 1169 y(Denotational)f(Seman)
-5 b(tics)28 b(of)g(T)167 1489 y FK(The)40 b(constructions)g(of)h(c)n
(hapter)f FG(??)h FK(pro)n(vide)f(a)g(nice)f(denotational)i(seman)n
(tics)e(of)h(the)167 1574 y(systems)28 b(w)n(e)f(ha)n(v)n(e)h(already)g
(considered.)167 1811 y FE(9.1)114 b(Simple)27 b(t)m(yp)s(ed)i
(calculus)167 1967 y FK(W)-6 b(e)27 b(prop)r(ose)i(here)e(to)h(in)n
(terpret)f(the)g(simple)f(t)n(yp)r(ed)i(calculus,)f(based)h(on)h
FJ(!)f FK(and)h FJ(\002)p FK(.)35 b(The)167 2052 y(essen)n(tial)26
b(idea)i(is)e(that:)270 2216 y FJ(\017)34 b FH(\025)p
FK(-abstraction)29 b(turns)f(a)g(function)g(\()p FH(x)19
b FJ(7!)h FH(t)p FK([)p FH(x)p FK(]\))28 b(in)n(to)f(an)i(ob)t(ject;)
270 2361 y FJ(\017)34 b FK(application)27 b(asso)r(ciates)h(to)g(an)g
(ob)t(ject)g FH(t)f FK(of)i(t)n(yp)r(e)e FH(U)7 b FJ(!)q
FH(V)43 b FK(a)28 b(function)g FH(u)20 b FJ(7!)g FH(t)12
b(u)p FK(.)270 2525 y(In)52 b(other)g(w)n(ords,)58 b(application)51
b(and)i FH(\025)p FK(-abstraction)f(are)f(t)n(w)n(o)h(m)n(utually)f(in)
n(v)n(erse)167 2610 y(op)r(erations)28 b(whic)n(h)f(iden)n(tify)g(ob)t
(jects)g(of)h(t)n(yp)r(e)g FH(U)7 b FJ(!)q FH(V)43 b
FK(and)29 b(functions)f(from)f FH(U)35 b FK(to)28 b FH(V)15
b FK(.)270 2738 y(So)29 b(w)n(e)e(shall)g(in)n(terpret)f(them)h(as)h
(follo)n(ws:)270 2901 y FJ(\017)34 b FH(\025)p FK(-abstraction)f(b)n(y)
g(the)e(op)r(eration)i(whic)n(h)e(maps)h(a)g(stable)g(function)g(from)g
FJ(A)g FK(to)g FJ(B)339 2987 y FK(to)c(its)f(trace,)g(a)h(p)r(oin)n(t)g
(of)g FJ(A)16 b(!)g(B)r FK(;)270 3132 y FJ(\017)34 b
FK(application)29 b(b)n(y)i(the)e(op)r(eration)h(whic)n(h)f(maps)h(a)g
(p)r(oin)n(t)g(of)g FJ(A)16 b(!)h(B)32 b FK(to)e(the)f(function)339
3217 y(of)g(whic)n(h)d(it)h(is)g(the)g(trace.)167 3423
y Fy(9.1.1)95 b(T)m(yp)s(es)167 3555 y FK(Supp)r(ose)34
b(w)n(e)d(ha)n(v)n(e)j(\014xed)f(for)f(eac)n(h)g(atomic)g(t)n(yp)r(e)g
FH(S)1622 3566 y Fw(i)1675 3555 y FK(a)g(coherence)g(space)g([)-12
b([)q FH(S)2327 3566 y Fw(i)2347 3555 y FK(])g(])q(;)34
b(then)f(w)n(e)167 3640 y(de\014ne)28 b([)-12 b([)p FH(T)10
b FK(])-12 b(])29 b(for)f(eac)n(h)f(t)n(yp)r(e)h FH(T)38
b FK(b)n(y:)574 3816 y([)-12 b([)q FH(U)7 b FJ(\002)p
FH(V)16 b FK(])-12 b(])20 b(=)f([)-12 b([)p FH(U)7 b
FK(])-12 b(])17 b(&)e([)-12 b([)q FH(V)15 b FK(])-12
b(])408 b([)-12 b([)q FH(U)7 b FJ(!)p FH(V)16 b FK(])-12
b(])20 b(=)f([)-12 b([)q FH(U)7 b FK(])-12 b(])16 b FJ(!)g
FK([)-12 b([)q FH(V)16 b FK(])-12 b(])1385 4101 y(66)p
eop
PStoPSsaved restore
%%Page: (70,71) 36
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
67 70 bop 167 167 a Fz(9.1.)64 b(SIMPLE)28 b(TYPED)f(CALCULUS)1319
b FK(67)167 404 y Fy(9.1.2)95 b(T)-8 b(erms)167 538 y
FK(If)25 b FH(t)p FK([)p FH(x)321 549 y FD(1)349 538
y FH(;)12 b(:)g(:)g(:)g(;)g(x)543 549 y Fw(n)576 538
y FK(])25 b(is)g(a)h(term)e(of)h(t)n(yp)r(e)h FH(T)35
b FK(dep)r(ending)25 b(on)h(free)f(v)l(ariables)g FH(x)2125
549 y Fw(i)2170 538 y FK(of)h(t)n(yp)r(e)f FH(S)2450
549 y Fw(i)2495 538 y FK(\(some)167 623 y(of)30 b(the)e
FH(x)415 634 y Fw(i)464 623 y FK(ma)n(y)i(not)f(actually)g(o)r(ccur)g
(in)f FH(t)p FK(\),)h(w)n(e)g(asso)r(ciate)g(to)g(it)f(a)i(stable)e
(function)i([)-12 b([)p FH(t)p FK(])g(])30 b(of)167 708
y FH(n)e FK(argumen)n(ts)g(from)f([)-12 b([)q FH(S)812
719 y FD(1)840 708 y FK(])g(])q FH(;)12 b(:)g(:)g(:)f(;)h
FK([)-12 b([)q FH(S)1090 719 y Fw(n)1124 708 y FK(])g(])28
b(to)g([)-12 b([)p FH(T)10 b FK(])-12 b(])q(:)251 876
y(1.)35 b FH(t)p FK([)p FH(x)422 887 y FD(1)451 876 y
FH(;)12 b(:)g(:)g(:)f(;)h(x)644 887 y Fw(n)677 876 y
FK(])23 b(=)f FH(x)834 887 y Fw(i)855 876 y FK(:)43 b(then)32
b([)-12 b([)q FH(t)p FK(])g(]\()p FH(a)1224 887 y FD(1)1252
876 y FH(;)12 b(:)g(:)g(:)g(;)g(a)1443 887 y Fw(n)1476
876 y FK(\))23 b(=)f FH(a)1638 887 y Fw(i)1658 876 y
FK(;)34 b(the)e(stabilit)n(y)e(of)i(this)g(function)g(is)339
962 y(immediate.)251 1110 y(2.)j FH(t)20 b FK(=)f FJ(h)p
FH(u;)12 b(v)s FJ(i)p FK(;)29 b(w)n(e)f(ha)n(v)n(e)g(at)g(our)h(disp)r
(osal)f(functions)g([)-12 b([)p FH(u)p FK(])g(])29 b(and)g([)-12
b([)q FH(v)s FK(])g(])28 b(from)f([)-12 b([)q FH(S)2334
1121 y FD(1)2362 1110 y FK(])g(])q FH(;)12 b(:)g(:)g(:)f(;)h
FK([)-12 b([)q FH(S)2612 1121 y Fw(n)2646 1110 y FK(])g(])339
1195 y(to)29 b([)-12 b([)p FH(U)7 b FK(])-12 b(])29 b(and)g([)-12
b([)q FH(V)15 b FK(])-12 b(])29 b(resp)r(ectiv)n(ely)-6
b(.)35 b(Consider)28 b(the)f(stable)h(binary)h(function)f
FJ(P)-18 b FM(air)8 b FK(,)28 b(from)339 1280 y([)-12
b([)q FH(U)7 b FK(])-12 b(])q FH(;)12 b FK([)-12 b([)p
FH(V)16 b FK(])-12 b(])28 b(to)g([)-12 b([)p FH(U)7 b
FK(])-12 b(])17 b(&)e([)-12 b([)q FH(V)15 b FK(])-12
b(])q(,)28 b(de\014ned)g(b)n(y:)1046 1493 y FJ(P)-18
b FM(air)8 b FK(\()p FH(a;)k(b)p FK(\))19 b(=)h FJ(f)p
FK(1)p FJ(g)c(\002)g FH(a)g FJ([)f(f)p FK(2)p FJ(g)i(\002)e
FH(b)339 1707 y FK(W)-6 b(e)28 b(put)f([)-12 b([)q FH(t)p
FK(])g(])q(\()p FH(a)735 1718 y FD(1)762 1707 y FH(;)12
b(:)g(:)g(:)g(;)g(a)953 1718 y Fw(n)986 1707 y FK(\))20
b(=)f FJ(P)-18 b FM(air)8 b FK(\([)-12 b([)q FH(u)p FK(])g(]\()p
FH(a)1412 1718 y FD(1)1440 1707 y FH(;)12 b(:)g(:)g(:)g(;)g(a)1631
1718 y Fw(n)1664 1707 y FK(\))p FH(;)g FK([)-12 b([)q
FH(v)s FK(])g(]\()p FH(b)1867 1718 y FD(1)1895 1707 y
FH(;)12 b(:)g(:)g(:)g(;)g(b)2079 1718 y Fw(n)2112 1707
y FK(\)\);)27 b(this)g(function)h(is)339 1792 y(still)e(stable.)251
1940 y(3.)35 b FH(t)21 b FK(=)g FH(\031)502 1915 y FD(1)530
1940 y FH(w)32 b FK(or)e FH(t)21 b FK(=)g FH(\031)865
1915 y FD(2)892 1940 y FH(w)r FK(.)43 b(Here)28 b(again)j(w)n(e)f(comp)
r(ose)g(with)e(one)j(of)f(the)g(follo)n(wing)f(t)n(w)n(o)339
2025 y(stable)f(functions:)624 2239 y(\005)676 2213 y
FD(1)703 2239 y FK(\()p FH(c)p FK(\))19 b(=)h FJ(f)p
FH(\013)g FK(:)39 b(\(1)p FH(;)12 b(\013)p FK(\))20 b
FJ(2)f FH(c)p FJ(g)285 b FK(\005)1688 2213 y FD(2)1715
2239 y FK(\()p FH(c)p FK(\))19 b(=)g FJ(f)p FH(\014)24
b FK(:)39 b(\(2)p FH(;)12 b(\014)t FK(\))20 b FJ(2)f
FH(c)p FJ(g)251 2484 y FK(4.)35 b FH(t)24 b FK(=)f FH(\025x:)12
b(v)s FK(;)36 b(b)n(y)d(h)n(yp)r(othesis)h(w)n(e)f(already)g(ha)n(v)n
(e)h(a)f(\()p FH(n)18 b FK(+)e(1\)-ary)34 b(stable)f(function)g([)-12
b([)q FH(v)s FK(])g(])339 2569 y(from)29 b([)-12 b([)534
2570 y FH(S)p 538 2590 34 3 v 581 2569 a FK(])g(])q FH(;)12
b FK([)-12 b([)p FH(U)7 b FK(])-12 b(])29 b(to)g([)-12
b([)q FH(V)15 b FK(])-12 b(])q(;)29 b(in)f(particular,)g(for)1561
2570 y FH(a)p 1565 2590 23 3 v 1626 2569 a FK(\014xed,)i(the)e
(function)h FH(b)20 b FJ(7!)h FK([)-12 b([)q FH(v)s FK(])g(]\()2474
2570 y FH(a)p 2478 2590 V -1 x(;)12 b(b)p FK(\))29 b(is)339
2654 y(stable)f(from)f([)-12 b([)q FH(U)7 b FK(])-12
b(])28 b(to)g([)-12 b([)p FH(V)16 b FK(])-12 b(])28 b(and)h(so)f(one)g
(can)g(de\014ne)g([)-12 b([)q FH(t)p FK(])g(]\()1870
2655 y FH(a)p 1874 2675 V -1 x FK(\))20 b(=)f FJ(T)-6
b FH(r)r FK(\()p FH(b)20 b FJ(7!)g FK([)-12 b([)q FH(v)s
FK(])g(]\()2373 2655 y FH(a)p 2377 2675 V -1 x(;)12 b(b)p
FK(\)\).)339 2815 y(Chec)n(king)38 b(that)f([)-12 b([)q
FH(t)p FK(])g(])38 b(is)e(stable)h(is)g(a)h(b)r(oring)g(but)f(straigh)n
(tforw)n(ard)i(exercise.)63 b(F)-6 b(or)339 2900 y(example,)35
b(in)f(the)g(case)g(where)f FH(n)25 b FK(=)f(1,)36 b(w)n(e)e(ha)n(v)n
(e)h(to)f(sho)n(w)h(that)g(if)e FH(F)43 b FK(is)34 b(a)g(stable)339
2986 y(function)28 b(from)f FJ(A)16 b FK(&)g FJ(B)29
b FK(to)f FJ(C)t FK(,)g(it)f(induces)g(a)h(stable)f(function)h
FH(G)g FK(from)f FJ(A)h FK(to)g FJ(B)18 b(!)e(C)t FK(,)339
3071 y(b)n(y)1053 3284 y FH(G)p FK(\()p FH(a)p FK(\))k(=)f
FJ(T)-6 b FH(r)r FK(\()p FH(b)21 b FJ(7!)f FH(F)10 b
FK(\()p FJ(P)-18 b FM(air)7 b FK(\()p FH(a;)12 b(b)p
FK(\)\)\))339 3498 y(Then)28 b FH(G)g FK(itself)e(has)i(a)g(trace,)g
(for)g(whic)n(h)e(w)n(e)i(shall)f(just)h(giv)n(e)f(the)g(form)n(ula:)
818 3711 y FJ(T)-6 b FH(r)r FK(\()p FH(G)p FK(\))21 b(=)e
FJ(f)p FK(\()p FH(a;)12 b FK(\()p FH(b;)g(\015)t FK(\)\))19
b(:)39 b(\()p FJ(P)-18 b FM(air)8 b FK(\()p FH(a;)k(b)p
FK(\))p FH(;)g(\015)t FK(\))19 b FJ(2)h(T)-6 b FH(r)r
FK(\()p FH(F)10 b FK(\))p FJ(g)339 3925 y FK(It)28 b(is)f(not)h(a)g
(pro)r(of,)h(but)f(it)e(should)j(b)r(e)e(enough)i(to)f(con)n(vince)g
(us!)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
68 71 bop 167 167 a FK(68)730 b Fz(CHAPTER)26 b(9.)63
b(DENOT)-6 b(A)g(TIONAL)26 b(SEMANTICS)h(OF)g(T)251 404
y FK(5.)35 b FH(t)20 b FK(=)f FH(w)14 b(u)25 b FK(with)f
FH(w)j FK(of)e(t)n(yp)r(e)g FH(U)7 b FJ(!)p FH(V)16 b
FK(,)25 b FH(u)g FK(of)g(t)n(yp)r(e)g FH(U)7 b FK(;)26
b(w)n(e)e(de\014ne)h(the)g(function)g FJ(A)p FM(pp)k
FK(from)339 490 y([)-12 b([)q FH(U)7 b FJ(!)q FH(V)15
b FK(])-12 b(])q FH(;)12 b FK([)-12 b([)p FH(U)7 b FK(])-12
b(])29 b(to)e([)-12 b([)q FH(V)16 b FK(])-12 b(])28 b(b)n(y:)919
669 y FJ(A)p FM(pp)5 b FK(\()p FH(f)t(;)12 b(a)p FK(\))19
b(=)h FJ(f)p FH(\014)j FK(:)39 b FJ(9)p FH(a)1532 680
y Fv(\016)1580 669 y FJ(\032)20 b FH(a)f FK(\()p FH(a)1773
680 y Fv(\016)1801 669 y FH(;)12 b(\014)t FK(\))19 b
FJ(2)h FH(f)8 b FJ(g)339 848 y FK(It)26 b(is)g(immediate)d(that)k
FJ(A)p FM(pp)k FK(is)26 b(stable;)g(so)h(w)n(e)f(de\014ne)h([)-12
b([)p FH(t)p FK(])g(])q(\()1936 849 y FH(s)p 1940 869
19 3 v -1 x FK(\))19 b(=)h FJ(A)p FM(pp)5 b FK(\([)-12
b([)p FH(w)r FK(])g(])q(\()2377 849 y FH(s)p 2381 869
V -1 x FK(\))p FH(;)12 b FK([)-12 b([)p FH(u)p FK(])g(])q(\()2586
849 y FH(s)p 2590 869 V -1 x FK(\)\))270 973 y(As)37
b(an)g(exercise,)g(one)g(can)g(calculate)e(the)i(traces)f(of)h
FJ(P)-18 b FM(air)8 b FK(,)39 b(\005)2038 947 y FD(1)2066
973 y FK(,)g(\005)2176 947 y FD(2)2203 973 y FK(,)g FJ(A)p
FM(pp)j FK(and)37 b(the)167 1058 y(function)28 b(in)f
FG(??)h FK(whic)n(h)f(tak)n(es)h FH(F)37 b FK(to)28 b
FH(G)p FK(.)167 1288 y FE(9.2)114 b(Prop)s(erties)27
b(of)h(the)g(in)m(terpretation)167 1443 y FK(Essen)n(tially)-6
b(,)32 b(as)h(w)n(e)f(ha)n(v)n(e)h(said,)g(con)n(v)n(ersion)g(b)r
(ecomes)e(denotational)i(equalit)n(y:)44 b(if)31 b FH(t)23
b Fu( )g FH(u)167 1528 y FK(then)28 b([)-12 b([)p FH(t)p
FK(])g(])20 b(=)g([)-12 b([)p FH(u)p FK(])g(])q(.)35
b(T)-6 b(o)28 b(sho)n(w)h(this,)e(w)n(e)g(use:)354 1683
y(\005)406 1657 y FD(1)434 1683 y FK(\()p FJ(P)-18 b
FM(air)8 b FK(\()p FH(a;)k(b)p FK(\)\))19 b(=)h FH(a)187
b FK(\005)1130 1657 y FD(2)1158 1683 y FK(\()p FJ(P)-18
b FM(air)8 b FK(\()p FH(a;)k(b)p FK(\)\))19 b(=)g FH(b)188
b FJ(A)p FM(pp)5 b FK(\()p FJ(T)-6 b FH(r)r FK(\()p FH(F)10
b FK(\))p FH(;)i(a)p FK(\))19 b(=)g FH(F)10 b FK(\()p
FH(a)p FK(\))167 1837 y(The)43 b(last)g(form)n(ula)g(is)g(to)h(b)r(e)f
(used)h(in)f(conjunction)h(with)e(a)i(substitution)f(prop)r(ert)n(y:)
167 1922 y(consider)27 b FH(v)s FK([)495 1923 y FH(x)p
499 1943 26 3 v -1 x(;)12 b(u)p FK([)623 1923 y FH(x)p
627 1943 V 663 1922 a FK(])p FH(=y)s FK(];)28 b(one)g(can)g(asso)r
(ciate)f(to)h(this)f(t)n(w)n(o)h(stable)f(functions:)270
2047 y FJ(\017)34 b FK(b)n(y)29 b(calculating)d(the)h(in)n
(terpretation)g(of)h(this)f(term;)270 2181 y FJ(\017)34
b FK(b)n(y)c(forming)g(the)f(\()p FH(n)16 b FK(+)f(1\)-ary)31
b(function)e([)-12 b([)q FH(v)s FK(])g(]\()1585 2182
y FH(a)p 1589 2202 23 3 v -1 x(;)12 b(b)p FK(\),)30 b(the)f
FH(n)p FK(-ary)h(function)g([)-12 b([)p FH(u)p FK(])g(])q(\()2469
2182 y FH(a)p 2473 2202 V -1 x FK(\))29 b(and)339 2266
y(then)f([)-12 b([)q FH(v)s FK(])g(]\()617 2267 y FH(a)p
621 2287 V -1 x(;)12 b FK([)-12 b([)q FH(u)p FK(])g(]\()803
2267 y FH(a)p 807 2287 V -1 x FK(\)\).)167 2391 y(The)28
b(t)n(w)n(o)g(functions)g(so)g(obtained)h(are)f(equal,)g(as)g(can)g(b)r
(e)g(sho)n(wn)h(without)f(di\016cult)n(y)f(\(but)167
2476 y(what)h(a)g(b)r(ore!\))36 b(b)n(y)28 b(induction)f(on)h
FH(v)s FK(.)270 2593 y(This)g(prop)r(ert)n(y)g(is)f(used)h(th)n(us)g
(\(omitting)e(the)i(auxiliary)f(v)l(ariables\):)501 2748
y([)-12 b([)q(\()p FH(\025x:)12 b(v)s FK(\))g FH(u)p
FK(])-12 b(])20 b(=)f FJ(A)p FM(pp)5 b FK(\()p FJ(T)-6
b FH(r)r FK(\()p FH(a)20 b FJ(7!)g FK([)-12 b([)p FH(v)s
FK(])g(])q(\()p FH(a)p FK(\)\))p FH(;)12 b FK([)-12 b([)p
FH(u)p FK(])g(])q(\))19 b(=)g([)-12 b([)q FH(v)s FK(])g(]\([)g([)q
FH(u)p FK(])g(])q(\))19 b(=)g([)-12 b([)q FH(v)s FK([)p
FH(u=x)p FK(]])g(])270 2934 y(In)32 b(fact,)h(the)e(secondary)i
(equations,)g(whic)n(h)e(w)n(e)g(k)n(eep)h(meeting)e(but)i(ha)n(v)n(e)g
(not)g(tak)n(en)167 3019 y(seriously)-6 b(,)27 b(are)h(also)g
(satis\014ed:)544 3174 y FJ(P)-18 b FM(air)8 b FK(\(\005)747
3148 y FD(1)775 3174 y FK(\()p FH(c)p FK(\))p FH(;)k
FK(\005)942 3148 y FD(2)969 3174 y FK(\()p FH(c)p FK(\)\))18
b(=)i FH(c)376 b FJ(T)-6 b FH(r)r FK(\()p FH(a)21 b FJ(7!)f(A)p
FM(pp)5 b FK(\()p FH(f)t(;)12 b(a)p FK(\)\))19 b(=)g
FH(f)270 3360 y FK(Categorically)-6 b(,)40 b(what)f(w)n(e)f(ha)n(v)n(e)
h(sho)n(wn)g(is)f(that)g(&)g(and)h FJ(!)g FK(are)f(the)g(pro)r(duct)h
(and)167 3445 y(exp)r(onen)n(tial)d(for)h(a)g FM(Cartesian)g(close)l(d)
g(c)l(ate)l(gory)g FK(whose)g(ob)t(jects)f(are)h(coherence)f(spaces)167
3530 y(and)f(whose)g(morphisms)f(are)g(stable)g(maps.)56
b(Ho)n(w)n(ev)n(er,)36 b(w)n(e)e(ha)n(v)n(e)h(forgotten)g(one)g(thing:)
167 3615 y(comp)r(osition!)g(But)27 b(it)g(is)g(easy)h(to)g(sho)n(w)h
(that)f(the)f(trace)g(of)h FH(G)16 b FJ(\016)g FH(F)37
b FK(is)310 3770 y FJ(f)p FK(\()p FH(a)408 3781 y FD(1)452
3770 y FJ([)15 b FH(:::)i FJ([)e FH(a)686 3781 y Fw(k)717
3770 y FH(;)d(\015)t FK(\))18 b(:)40 b(\()p FJ(f)p FH(\014)993
3781 y FD(1)1021 3770 y FH(;)12 b(:::;)g(\014)1179 3781
y Fw(k)1209 3770 y FJ(g)p FH(;)g(\015)t FK(\))19 b FJ(2)h(T)-6
b FH(r)r FK(\()p FH(F)10 b FK(\))p FH(;)31 b FK(\()p
FH(a)1715 3781 y FD(1)1743 3770 y FH(;)12 b(\014)1813
3781 y FD(1)1840 3770 y FK(\))p FH(;)g(:::;)g FK(\()p
FH(a)2049 3781 y Fw(k)2080 3770 y FH(;)g(\014)2150 3781
y Fw(k)2180 3770 y FK(\))19 b FJ(2)g(T)-6 b FH(r)r FK(\()p
FH(G)p FK(\))p FJ(g)167 3925 y FK(where)27 b FH(F)37
b FK(and)28 b FH(G)g FK(are)g(stable)f(functions)h(from)f
FJ(A)h FK(to)g FJ(B)i FK(and)f(from)e FJ(B)j FK(to)e
FJ(C)k FK(resp)r(ectiv)n(ely)-6 b(.)p eop
PStoPSsaved restore
%%Page: (72,73) 37
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
69 72 bop 167 167 a Fz(9.3.)64 b(G)404 149 y(\177)391
167 y(ODEL'S)28 b(SYSTEM)1642 b FK(69)167 404 y FE(9.3)114
b(G\177)-57 b(odel's)27 b(system)167 580 y Fy(9.3.1)95
b(Bo)s(oleans)167 710 y FK(W)-6 b(e)27 b(shall)g(in)n(terpret)f(the)i
(t)n(yp)r(e)f Fq(Bo)r(ol)h FK(b)n(y)g FJ(B)-10 b FM(o)l(ol)9
b FK(:)689 892 y([)-12 b([)q Fq(T)p FK(])g(])20 b(=)f
FJ(T)958 852 y FD(def)966 892 y FK(=)26 b FJ(f)p FG(t)p
FJ(g)523 b FK([)-12 b([)p Fq(F)p FK(])g(])21 b(=)e FJ(F)1932
852 y FD(def)1940 892 y FK(=)27 b FJ(f)p FG(f)7 b FJ(g)270
1059 y Fq(D)12 b FH(u)g(v)i(t)28 b FK(is)f(in)n(terpreted)g(using)h(a)h
(ternary)f(stable)f(function)h FJ(D)j FK(from)c FJ(A)p
FK(,)i FJ(A)p FK(,)f FJ(B)-10 b FM(o)l(ol)37 b FK(to)28
b FJ(A)p FK(,)167 1144 y(de\014ned)g(b)n(y)444 1312 y
FJ(D)r FK(\()p FH(a;)12 b(b;)g Fn(?)p FK(\))20 b(=)f
Fn(?)277 b FJ(D)r FK(\()p FH(a;)12 b(b;)g FJ(f)p FG(t)p
FJ(g)p FK(\))20 b(=)g FH(a)277 b FJ(D)r FK(\()p FH(a;)12
b(b;)g FJ(f)p FG(f)7 b FJ(g)p FK(\))21 b(=)e FH(b)167
1479 y FK(and)29 b(so)f(w)n(e)f(put)h([)-12 b([)q Fq(D)12
b FH(u)g(v)i(t)p FK(])-12 b(])20 b(=)f FJ(D)r FK(\([)-12
b([)q FH(u)p FK(])g(])p FH(;)12 b FK([)-12 b([)q FH(v)s
FK(])g(])p FH(;)12 b FK([)-12 b([)q FH(t)p FK(])g(]\).)270
1647 y(In)39 b(particular,)g(the)f(fact)g(that)g(terms)f(of)h(G\177)-34
b(odel's)38 b(system)g(can)g(b)r(e)g(in)n(terpreted)e(b)n(y)167
1732 y(stable)27 b(functions)g(mak)n(es)g(it)f(imp)r(ossible)f(to)j
(de\014ne)f FM(p)l(ar)l(al)t(lel)f(or)p FK(.)36 b(Indeed,)27
b(if)g(the)f(equations)516 1900 y FH(t)12 b FJ(h)p Fq(T)p
FH(;)g(x)p FJ(i)21 b Fu( )f Fq(T)349 b FH(t)12 b FJ(h)p
FH(x;)g Fq(T)p FJ(i)20 b Fu( )g Fq(T)350 b FH(t)12 b
FJ(h)p Fq(F)p FH(;)g Fq(F)p FJ(i)20 b Fu( )g Fq(F)167
2067 y FK(had)29 b(a)f(solution)f(in)g FG(T)p FK(,)h(w)n(e)f(w)n(ould)h
(ha)n(v)n(e)468 2235 y([)-12 b([)q FH(t)p FK(])g(]\()p
FJ(T)18 b FH(;)12 b Fn(?)p FK(\))19 b(=)g FJ(T)320 b
FK([)-12 b([)p FH(t)p FK(])g(])q(\()p Fn(?)p FH(;)12
b FJ(T)17 b FK(\))j(=)f FJ(T)319 b FK([)-12 b([)q FH(t)p
FK(])g(]\()p FJ(F)7 b FH(;)12 b FJ(F)7 b FK(\))19 b(=)g
FJ(F)167 2402 y FK(whic)n(h)27 b(corresp)r(onds)i(to)f(the)f
(non-stable)h(function)g(called)e FH(F)1822 2413 y FD(0)1877
2402 y FK(in)h FG(??)q FK(.)167 2605 y Fy(9.3.2)95 b(In)m(tegers)167
2736 y FK(The)27 b(ob)n(vious)h(idea)f(for)g(in)n(terpreting)f
Fq(Int)h FK(is)f(the)h(coherence)f(space)i FJ(I)-7 b
FM(nt)34 b FK(in)n(tro)r(duced)27 b(in)f(the)167 2821
y(previous)i(c)n(hapter:)343 3002 y([)-12 b([)q Fq(O)p
FK(])g(])20 b(=)f FJ(O)618 2962 y FD(def)625 3002 y FK(=)27
b FJ(f)p FK(0)p FJ(g)177 b FK([)-12 b([)q Fq(S)12 b FH(t)p
FK(])-12 b(])20 b(=)f FJ(S)5 b FK(\([)-12 b([)p FH(t)p
FK(])g(])q(\))27 b(with)g FJ(S)5 b FK(\()p Fn(?)p FK(\))19
b(=)g Fn(?)p FH(;)31 b FJ(S)5 b FK(\()p FJ(f)p FH(n)p
FJ(g)p FK(\))20 b(=)f FJ(f)p FH(n)d FK(+)g(1)p FJ(g)270
3169 y FK(This)34 b(in)n(terpretation)f(w)n(orks)j(only)e
FM(by)h(values)p FK(;)i(indeed,)f(it)d(is)h(easy)h(to)g(\014nd)g
FH(u)g FK(and)g FH(v)167 3255 y FK(suc)n(h)28 b(that)723
3422 y Fq(R)12 b FH(u)g(v)j Fq(O)k Fu( )h Fq(T)557 b(R)12
b FH(u)g(v)j FK(\()p Fq(S)d FH(x)p FK(\))19 b Fu( )h
Fq(F)167 3590 y FK(If)28 b FH(F)36 b FK(is)27 b(the)h(function)f(whic)n
(h)g(in)n(terprets)g FH(x)19 b FJ(7!)h Fq(R)12 b FH(u)g(v)j(x)p
FK(,)28 b(this)f(forces)720 3757 y FH(F)10 b FK(\()p
FJ(O)r FK(\))18 b(=)i FJ(f)p FG(t)p FJ(g)553 b FH(F)10
b FK(\()p FJ(S)5 b FK(\()p Fn(?)p FK(\)\))18 b(=)h FJ(f)p
FG(f)7 b FJ(g)167 3925 y FK(but)28 b FJ(S)5 b FK(\()p
Fn(?)p FK(\))19 b(=)g Fn(?)h FJ(\032)f(O)r FK(,)28 b(con)n(tradiction.)
p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
70 73 bop 167 167 a FK(70)730 b Fz(CHAPTER)26 b(9.)63
b(DENOT)-6 b(A)g(TIONAL)26 b(SEMANTICS)h(OF)g(T)270 404
y FK(What)i(is)e(wrong)i(with)e FJ(I)-7 b FM(nt)7 b FK(?)35
b(If)28 b(w)n(e)f(apply)i FJ(S)k FK(to)28 b Fn(?)f FK(\(empt)n(y)h
(information\),)f(w)n(e)h(obtain)167 490 y Fn(?)h FK(again,)j(whereas)d
(w)n(e)g(kno)n(w)i(something)f(more,)f(namely)h(that)g(w)n(e)f(ha)n(v)n
(e)i(a)f FM(suc)l(c)l(essor)g FK(|)167 575 y(a)e(piece)e(of)i
(information)f(whic)n(h)g(ma)n(y)h(w)n(ell)e(b)r(e)h(su\016cien)n(t)h
(for)g(a)g(recursion)f(step.)270 722 y(Therefore,)37
b(w)n(e)d(m)n(ust)h(revise)f(our)i(in)n(terpretation,)f(adding)h(0)1996
697 y FD(+)2073 722 y FK(for)g(the)f(information)167
807 y(\\b)r(eing)h(a)h(successor",)i FM(i.e.)d FK(something)g
FH(>)25 b FK(0,)38 b(and)g(more)d(generally)-6 b(,)38
b FH(p)2170 782 y FD(+)2247 807 y FK(for)e(something)167
892 y(greater)27 b(than)i FH(p)p FK(.)35 b(Let)27 b(us)h(de\014ne)1057
867 y FD(1)1113 892 y FJ(I)-7 b FM(nt)1213 867 y FD(+)1282
892 y FK(b)n(y)29 b FJ(jI)-7 b FM(nt)1502 867 y FD(+)1544
892 y FJ(j)19 b FK(=)h FJ(f)p FK(0)p FH(;)12 b FK(0)1791
867 y FD(+)1833 892 y FH(;)g FK(1)p FH(;)g FK(1)1963
867 y FD(+)2006 892 y FH(;)g(:)g(:)g(:)p FJ(g)27 b FK(with:)402
1108 y FH(p)456 1086 y(_)456 1129 y(^)544 1108 y(q)j
FK(i\013)d FH(p)19 b FK(=)g FH(q)237 b(p)1122 1083 y
FD(+)1183 1086 y FH(_)1183 1129 y(^)1271 1108 y(q)30
b FK(i\013)d FH(p)19 b(<)g(q)237 b(p)1849 1083 y FD(+)1910
1086 y FH(_)1910 1129 y(^)1999 1108 y(q)2033 1083 y FD(+)2101
1108 y FK(for)28 b(all)e FH(p;)12 b(q)270 1379 y FK(T)-6
b(o)28 b(see)e(ho)n(w)i(it)e(all)f(w)n(orks)j(out,)g(let)d(us)j(lo)r
(ok)f(for)g(the)g(maximal)e(p)r(oin)n(ts.)36 b(If)27
b FH(a)19 b FJ(2)h(I)-7 b FM(nt)2557 1354 y FD(+)2626
1379 y FK(is)167 1464 y(maximal,)26 b(either:)270 1654
y FJ(\017)34 b FK(some)g FH(p)24 b FJ(2)g FH(a)p FK(;)38
b(then)c FH(a)h FK(con)n(tains)f(no)h(other)g FH(q)s
FK(,)g(nor)f(do)r(es)h(it)e(con)n(tain)i(an)n(y)g FH(q)2463
1629 y FD(+)2538 1654 y FK(with)339 1753 y FH(p)20 b
FJ(\024)f FH(q)s FK(.)34 b(So)27 b FH(a)19 b FJ(\032)k
Fo(e)-42 b FH(p)838 1713 y FD(def)846 1753 y FK(=)26
b FJ(f)p FK(0)995 1727 y FD(+)1038 1753 y FH(;)12 b(:)g(:)g(:)f(;)h
FK(\()p FH(p)j FJ(\000)h FK(1\))1401 1727 y FD(+)1443
1753 y FH(;)c(p)o FJ(g)p FK(;)27 b(but)g(this)e(set)h(is)f(coheren)n
(t,)h(and)h(as)g FH(a)f FK(is)339 1838 y(maximal)h(it)f(m)n(ust)i(b)r
(e)f(equal)h(to)j Fo(e)-42 b FH(p)o FK(.)270 2013 y FJ(\017)34
b FH(a)f FK(con)n(tains)f(no)h FH(p)p FK(;)h(then)e FH(a)23
b FJ(\032)g Fo(f)-70 b FJ(1)1276 1974 y FD(def)1283 2013
y FK(=)30 b FJ(f)p FK(0)1436 1988 y FD(+)1479 2013 y
FH(;)12 b FK(1)1544 1988 y FD(+)1586 2013 y FH(;)g FK(2)1651
1988 y FD(+)1693 2013 y FH(;)g(:)g(:)g(:)p FJ(g)32 b
FK(whic)n(h)f(is)h(coheren)n(t,)h(so)g FH(a)f FK(is)339
2098 y(equal)c(to)g(this)f(in\014nite)f(set.)270 2288
y(The)i(in)n(terpretation)e(is)h(as)h(follo)n(ws:)393
2504 y FJ(O)21 b FK(=)e FJ(f)p FK(0)p FJ(g)227 b(S)5
b FK(\()p FH(a)p FK(\))19 b(=)g FJ(f)p FK(0)1173 2478
y FD(+)1216 2504 y FJ(g)d([)f(f)p FH(i)h FK(+)f(1)20
b(:)39 b FH(i)20 b FJ(2)f FH(a)p FJ(g)d([)g(f)p FK(\()p
FH(i)f FK(+)g(1\))2076 2478 y FD(+)2138 2504 y FK(:)39
b FH(i)2220 2478 y FD(+)2281 2504 y FJ(2)20 b FH(a)p
FJ(g)167 2720 y FK(In)28 b(particular)e(the)i(n)n(umeral)p
975 2681 35 3 v 27 w FH(p)19 b FK(=)g Fq(S)1140 2694
y Fw(p)1180 2720 y Fq(O)28 b FK(will)c(b)r(e)k(in)n(terpreted)e(b)n(y)
32 b Fo(e)-43 b FH(p)p FK(.)270 2867 y(It)33 b(remains)f(to)h(in)n
(terpret)e(recursion:)45 b(giv)n(en)33 b(a)g(coherence)f(space)h
FJ(A)p FK(,)h(a)g(p)r(oin)n(t)e FH(o)24 b FJ(2)f(A)167
2952 y FK(and)31 b(a)g(stable)e(function)h FH(F)40 b
FK(from)30 b FJ(A)p FH(;)12 b FJ(I)-7 b FM(nt)1298 2927
y FD(+)1370 2952 y FK(to)30 b FJ(A)p FK(,)h(w)n(e)e(shall)h(construct)g
(a)g(stable)g(function)167 3038 y FH(G)e FK(from)f FJ(I)-7
b FM(nt)517 3012 y FD(+)586 3038 y FK(to)28 b FJ(A)g
FK(whic)n(h)f(satis\014es:)386 3253 y FH(G)p FK(\()p
FJ(O)r FK(\))20 b(=)f FH(o)219 b(G)p FK(\()p FJ(S)5 b
FK(\()p FH(a)p FK(\)\))20 b(=)f FH(F)10 b FK(\()p FH(G)p
FK(\()p FH(a)p FK(\))p FH(;)i(a)p FK(\))218 b FH(G)p
FK(\()p FH(a)p FK(\))20 b(=)f Fn(?)28 b FK(if)e(0)p FH(;)12
b FK(0)2260 3228 y FD(+)2331 3253 y FH(=)-42 b FJ(2)19
b FH(a)270 3469 y(G)35 b FK(is)e(actually)g(w)n(ell-de\014ned)g(on)h
(the)g(\014nite)f(p)r(oin)n(ts)h(of)h FJ(I)-7 b FM(nt)1932
3444 y FD(+)1974 3469 y FK(;)37 b(it)c(is)g(easily)g(sho)n(wn)i(to)167
3554 y(b)r(e)c(monotone)h(and)g(hence)e(extends)h(to)g(a)h(con)n(tin)n
(uous,)h(and)f(indeed)e(stable,)h(function)g(on)167 3639
y(in\014nite)26 b(p)r(oin)n(ts.)36 b(In)28 b(particular,)f
FH(G)p FK(\()p Fo(f)-70 b FJ(1)p FK(\))20 b(=)1348 3586
y Fo(S)1407 3605 y Fv(")1434 3639 y FJ(f)p FH(G)p FK(\()p
FJ(S)1598 3614 y Fw(n)1632 3639 y FK(\()p Fn(?)p FK(\)\))f(:)39
b FH(n)20 b FJ(2)f FK(I)-12 b(N)p FJ(g)p FK(.)p 167 3719
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(space)g(but)f(a)h(dI-domain)f(\(section)g Ft(??)p FB(\).)48
b(The)32 b(di\013erence)f(is)h(that)f(w)n(e)i(admit)167
3925 y(the)27 b(tok)n(en)h(1)473 3903 y FC(+)540 3925
y FB(in)e(the)h(absence)h(of)g(0)1053 3903 y FC(+)1092
3925 y FB(,)g(although)f(it)f(is)h(di\016cult)f(to)h(see)h(what)g(this)
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PStoPSmatrix setmatrix
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PStoPSxform concat
71 74 bop 167 167 a Fz(9.3.)64 b(G)404 149 y(\177)391
167 y(ODEL'S)28 b(SYSTEM)1642 b FK(71)270 404 y(In)28
b(fact,)g(if)f FH(a)625 379 y Fv(0)661 404 y FJ(\032)19
b FH(a)28 b FK(is)f(the)h(largest)f(subset)h(of)g(the)f(form)270
566 y FJ(\017)38 b Fo(e)-43 b FH(p)20 b FK(=)f FJ(f)p
FK(0)536 540 y FD(+)578 566 y FH(;)12 b(:)g(:)g(:)g(;)g
FK(\()p FH(p)j FJ(\000)g FK(1\))941 540 y FD(+)983 566
y FH(;)d(p)p FJ(g)20 b FK(=)f FJ(S)1225 540 y Fw(p)1253
566 y FJ(O)r FK(,)28 b(or)270 727 y FJ(\017)k FK(\027)-50
b FH(p)394 688 y FD(def)401 727 y FK(=)27 b FJ(f)p FK(0)551
702 y FD(+)593 727 y FH(;)12 b FK(1)658 702 y FD(+)701
727 y FH(;)g(:::;)g FK(\()p FH(p)j FJ(\000)g FK(1\))1028
702 y FD(+)1070 727 y FJ(g)20 b FK(=)f FJ(S)1246 702
y Fw(p)1274 727 y Fn(?)167 889 y FK(then)30 b FH(G)p
FK(\()p FH(a)448 863 y Fv(0)465 889 y FK(\))20 b(=)h
FH(G)p FK(\()p FH(a)p FK(\))30 b(\(assuming)g FH(F)38
b FK(has)31 b(this)e(prop)r(ert)n(y\),)i(so)f(\(b)n(y)g(induction\))f
(no)h(term)f(of)167 974 y FG(T)f FK(in)n(v)n(olv)n(es)f
FH(p)h FK(or)f FH(p)699 948 y FD(+)768 974 y FK(in)g(its)g(seman)n
(tics)g(without)g FJ(f)p FK(0)1602 948 y FD(+)1645 974
y FH(;)12 b(:)g(:)g(:)f FK(\()p FH(p)k FJ(\000)h FK(1\))1977
948 y FD(+)2019 974 y FJ(g)28 b FK(as)g(w)n(ell.)270
1147 y(As)k(an)g(exercise,)e(one)i(can)g(try)f(to)h(calculate)e
(directly)f(a)j(stable)f(function)g(from)g FJ(I)-7 b
FM(nt)2630 1121 y FD(+)167 1232 y FK(to)28 b FJ(I)-7
b FM(nt)356 1206 y FD(+)425 1232 y FK(whic)n(h)27 b(represen)n(ts)g
(the)h(predecessor.)167 1436 y Fy(9.3.3)95 b(In\014nit)m(y)28
b(and)f(\014xed)h(p)s(oin)m(t)167 1567 y FK(What)d(is)g(the)g(r^)-34
b(ole)24 b(of)h(the)g(ob)t(ject)g Fo(f)-70 b FJ(1)p FK(?)34
b(W)-6 b(e)25 b(see)f(that)i(it)e(is)g(a)i(\014xed)f(p)r(oin)n(t)g(of)h
(the)f(successor:)167 1652 y FJ(S)5 b FK(\()p Fo(f)-70
b FJ(1)p FK(\))20 b(=)h Fo(f)-70 b FJ(1)p FK(.)39 b(One)29
b(could)f(try)h(to)g(add)h(it)e(to)h(the)g(syn)n(tax)h(of)g
FG(T)p FK(,)f(with)f(the)g FM(nonc)l(onver)l(gent)167
1737 y FK(rewriting)e(rule)g FJ(1)20 b Fu( )g Fq(S)12
b FJ(1)p FK(.)36 b(W)-6 b(e)27 b(see,)h(b)n(y)g(using)g(the)f
(iterator,)g(that)1076 1910 y Fq(It)11 b FH(u)h(v)j FJ(1)20
b Fu( )g FH(v)15 b FK(\()p Fq(It)c FH(u)h(v)i FJ(1)p
FK(\))167 2083 y(and)29 b(so)f FJ(1)p FK(,)g(com)n(bined)f(with)g
(recursion,)g(giv)n(es)h(us)g(access)f(to)h(the)f FM(\014xe)l(d)h(p)l
(oint)p FK(,)g Fq(Y)r FK(.)270 2210 y(In)i(the)f(denotational)g(seman)n
(tics,)g(the)g(tok)n(en)h FH(\013)f FK(o)r(ccurs)g(in)g(the)g(in)n
(terpretation)f(of)h Fq(Y)r FH(f)167 2295 y FK(whenev)n(er)35
b FJ(h)p FH(a;)12 b(\013)p FJ(i)37 b FK(o)r(ccurs)f(in)f(the)g(trace)g
(of)h(\(the)f(in)n(terpretation)g(of)5 b(\))36 b FH(f)43
b FK(and)36 b(the)g(clique)167 2380 y FH(a)k FK(o)r(ccurs)h(in)e(the)h
(in)n(terpretation)f(of)i Fq(Y)r FH(f)8 b FK(.)72 b(Hardly)40
b(surprisingly)-6 b(,)43 b(this)c(is)h(a)g(recursiv)n(e)167
2465 y(de\014nition,)32 b(and)g(it)f(is)f(obtained)i(b)n(y)g(rep)r
(eatedly)f(applying)h FH(f)38 b FK(to)32 b Fn(?)p FK(.)46
b(The)31 b(tok)n(ens)h(of)g(the)167 2550 y(in)n(terpretation)26
b(of)i Fq(Y)h FK(itself)d(can)i(therefore)f(b)r(e)g(describ)r(ed)g(in)g
(terms)g(of)h(\014nite)f(trees.)270 2723 y(It)39 b(is)f(not)h(our)g
(purp)r(ose)h(here)e(to)g(discuss)h(the)f(programming)h(applications)f
(of)h(the)167 2808 y(\014xed)e(p)r(oin)n(t)f(\(general)g(recursion\),)i
(an)f(idea)f(whic)n(h)f(is)h(curren)n(tly)f(rather)h(alien)f(to)i(t)n
(yp)r(e)167 2893 y(systems,)j(although)e(the)f(denotational)h(seman)n
(tics)e(accommo)r(dates)h(it)g(v)n(ery)g(w)n(ell.)62
b(But)167 2978 y(fundamen)n(tally)-6 b(,)28 b(what)g(do)r(es)g(this)f
(mean?)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
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1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
72 75 bop 167 874 a FF(Chapter)28 b(10)167 1168 y(Sums)g(in)f(Natural)g
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(description)g(of)i(those)f(parts)h(of)f(natural)h(deduction)f(whose)
167 1573 y(b)r(eha)n(viour)i(is)f(not)h(so)g(prett)n(y)-6
b(,)35 b(although)g(they)e(sho)n(w)i(precisely)c(the)i(features)g(whic)
n(h)g(are)167 1658 y(most)26 b(t)n(ypical)e(of)j(in)n(tuitionism.)32
b(F)-6 b(or)26 b(this)f(fragmen)n(t,)i(our)f(syn)n(tactic)f(metho)r(ds)
h(are)f(frankly)167 1743 y(inadequate,)i(and)g(only)g(a)f(complete)e
(recasting)i(of)g(the)g(ideas)g(could)g(allo)n(w)f(us)i(to)f(progress.)
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(connectors)i(to)f(put)h(bac)n(k:)40 b FJ(:)12 b FK(,)29
b FJ(_)h FK(and)g FJ(9)p FK(.)40 b(F)-6 b(or)30 b FJ(:)12
b FK(,)167 1913 y(it)27 b(is)g(common)g(to)h(add)h(a)f(sym)n(b)r(ol)f
FJ(?)g FK(\(absurdit)n(y\))h(and)g(in)n(terp)r(et)f FJ(:)p
FH(A)g FK(as)h FH(A)19 b FJ(\))h(?)p FK(.)270 2036 y(The)28
b(rules)f(are:)427 2237 y FJ(\001)427 2274 y(\001)427
2312 y(\001)411 2370 y FH(A)p 344 2414 186 4 v 551 2430
a FJ(_)p FK(1)p FJ(I)344 2491 y FH(A)15 b FJ(_)h FH(B)936
2237 y FJ(\001)936 2274 y(\001)936 2312 y(\001)917 2370
y FH(B)p 853 2414 V 1059 2430 a FJ(_)p FK(2)p FJ(I)853
2491 y FH(A)f FJ(_)g FH(B)1444 2272 y FJ(\001)1444 2310
y(\001)1444 2348 y(\001)1361 2405 y FH(A)g FJ(_)h FH(B)1694
2213 y FK([)p FH(A)p FK(])1729 2272 y FJ(\001)1729 2310
y(\001)1729 2348 y(\001)1712 2405 y FH(C)1931 2213 y
FK([)p FH(B)t FK(])1969 2272 y FJ(\001)1969 2310 y(\001)1969
2348 y(\001)1951 2405 y FH(C)p 1361 2450 645 4 v 2026
2465 a FJ(_E)1656 2526 y FH(C)2311 2236 y FJ(\001)2311
2274 y(\001)2311 2312 y(\001)2294 2370 y(?)p 2294 2415
55 4 v 2369 2431 a(?E)2294 2491 y FH(C)822 2761 y FJ(\001)822
2799 y(\001)822 2837 y(\001)734 2899 y FH(A)p FK([)p
FH(a=\030)s FK(])p 734 2944 195 4 v 949 2960 a FJ(9I)754
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2747 y FK([)p FH(A)p FK(])1937 2806 y FJ(\001)1937 2844
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3281 y(use)d(of)g(the)f(rule)g FJ(9E)6 b FK(.)36 b(There)27
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3514 y FE(10.1)114 b(Defects)28 b(of)g(the)g(system)167
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FJ(9)p FK(\))f(are)f(excellen)n(t!)167 3754 y(Moreo)n(v)n(er,)24
b(if)e(y)n(ou)i(men)n(tally)e(turn)h(them)e(upside-do)n(wn,)k(y)n(ou)f
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b FJ(^)p FK(1)p FJ(E)6 b FK(,)34 b FJ(^)o FK(2)p FJ(E)6
b FK(,)34 b FJ(8E)j FK(\(in)31 b(linear)f(logic,)h(there)g(is)g(only)g
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3925 y FM(ar)l(e)e FK(actually)f(turned)h(o)n(v)n(er\).)1385
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PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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userdict/PStoPSmatrix matrix currentmatrix put
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PStoPSxform concat
73 76 bop 167 167 a Fz(10.2.)65 b(ST)-6 b(AND)n(ARD)26
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b FH(C)g FK(pla)n(ys)29 b(the)f(r^)-34 b(ole)27 b(of)i(a)g(con)n(text,)
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(concession)g(to)g(sequen)n(t)g(calculus.)270 909 y(In)j(fact,)g(the)g
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(tradicts)i(the)g(idea)f(that)i(natural)f(deductions)g(are)g(the)g
(\\real)g(ob)t(jects")167 1079 y(b)r(ehind)42 b(the)h(pro)r(ofs.)81
b(Indeed,)47 b(w)n(e)42 b(cannot)h(decen)n(tly)f(w)n(ork)h(with)e(the)i
(full)e(fragmen)n(t)167 1164 y(without)27 b(iden)n(tifying)g
FM(a)h(priori)35 b FK(di\013eren)n(t)27 b(deductions,)h(for)g(example:)
456 1459 y FJ(\001)456 1497 y(\001)456 1535 y(\001)373
1592 y FH(A)15 b FJ(_)h FH(B)706 1400 y FK([)p FH(A)p
FK(])741 1459 y FJ(\001)741 1497 y(\001)741 1535 y(\001)724
1592 y FH(C)943 1400 y FK([)p FH(B)t FK(])980 1459 y
FJ(\001)980 1497 y(\001)980 1535 y(\001)963 1592 y FH(C)p
373 1637 645 4 v 1038 1653 a FJ(_E)668 1713 y FH(C)p
665 1758 60 4 v 745 1774 a Fq(r)665 1835 y FH(D)1334
1540 y FK(and)1734 1580 y FJ(\001)1734 1618 y(\001)1734
1656 y(\001)1651 1713 y FH(A)f FJ(_)h FH(B)1984 1400
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1535 y(\001)2002 1592 y FH(C)p 2000 1637 V 2079 1653
a Fq(r)2000 1713 y FH(D)2251 1400 y FK([)p FH(B)t FK(])2288
1459 y FJ(\001)2288 1497 y(\001)2288 1535 y(\001)2271
1592 y FH(C)p 2269 1637 V 2348 1653 a Fq(r)2269 1713
y FH(D)p 1651 1758 677 4 v 2348 1774 a FJ(_E)1960 1835
y FH(D)270 2053 y FK(F)-6 b(ortunately)g(,)27 b(this)e(kind)h(of)g
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(satisfying)f(Ch)n(urc)n(h-Rosser)g(and)h(strong)g(normalisation.)167
2224 y(Nev)n(ertheless,)59 b(ev)n(en)54 b(though)h(the)f(damage)h(is)e
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3751 y FE(10.2)114 b(Standard)30 b(con)m(v)m(ersions)167
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PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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 closepath}put initclip
PStoPSxform concat
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542 y FH(A)2082 583 y FJ(\001)2082 621 y(\001)2082 659
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PStoPSxform concat
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1046 y FK(ends)d(in)f(a)h(\\go)r(o)r(d")i(elimination.)270
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3899 a FJ(_E)1224 3960 y FH(E)2111 3665 y FK(con)n(v)n(erts)28
b(to)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
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1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
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PStoPSxform concat
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FJ(\001)818 483 y(\001)818 521 y(\001)732 578 y FH(C)21
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eop
PStoPSsaved restore
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PStoPSmatrix setmatrix
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PStoPSxform concat
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y FH(t)h FK(is)f(of)i(t)n(yp)r(e)f FH(U)7 b FK(.)167
2565 y(The)27 b(comm)n(utation)h(for)f FH(")879 2576
y Fw(U)949 2565 y FK(is)g(set)g(out)h(in)g(\014v)n(e)g(cases:)1213
2894 y FH(\031)1255 2868 y FD(1)1282 2894 y FK(\()p FH(")1341
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b Fu( )h FH(")1745 2990 y Fw(V)1800 2979 y FH(t)1220
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PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
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FH(R)q FK(,)h FH(S)t FK(,)g(and)g FH(u)p FK(,)f FH(v)s
FK(,)h FH(t)e FK(are)h(of)g(resp)r(ectiv)n(e)339 1013
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(natural.)p eop
PStoPSsaved restore
%%Page: (84,85) 43
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
81 84 bop 167 874 a FF(Chapter)28 b(11)167 1168 y(System)f(F)167
1488 y FK(System)36 b FG(F)h FK([)p FG(?)p FK(])g(arises)f(as)h(an)g
(extension)g(of)g(the)f(simple)f(t)n(yp)r(ed)i(calculus,)g(obtained)g
(b)n(y)167 1573 y(adding)26 b(an)f(op)r(eration)g(of)h(abstraction)f
(on)g(t)n(yp)r(es.)35 b(This)24 b(op)r(eration)i(is)e(extremely)e(p)r
(o)n(w)n(erful)167 1659 y(and)29 b(in)e(particular)g(all)g(the)g(usual)
h(data-t)n(yp)r(es)i(\(in)n(tegers,)d(lists,)f FM(etc.)p
FK(\))35 b(are)28 b(de\014nable.)36 b(The)167 1744 y(system)23
b(w)n(as)i(in)n(tro)r(duced)e(in)h(the)f(con)n(text)h(of)g(pro)r(of)h
(theory)f([)p FG(?)q FK(],)h(but)f(it)e(w)n(as)j(indep)r(enden)n(tly)
167 1829 y(disco)n(v)n(ered)i(in)g(computer)h(science)d([)p
FG(?)q FK(].)270 1956 y(The)41 b(most)f(primitiv)n(e)d(v)n(ersion)k(of)
g(the)f(system)g(is)g(set)g(out)h(here:)61 b(it)39 b(is)h(based)h(on)
167 2041 y(implication)25 b(and)k(univ)n(ersal)e(quan)n(ti\014cation.)
37 b(W)-6 b(e)28 b(shall)f(con)n(ten)n(t)h(ourselv)n(es)g(with)f
(de\014ning)167 2126 y(the)g(system)h(and)g(giving)g(some)f
(illustrations)f(of)i(its)f(expressiv)n(e)g(p)r(o)n(w)n(er.)167
2363 y FE(11.1)114 b(The)28 b(calculus)167 2518 y FM(T)-5
b(yp)l(es)49 b FK(are)f(de\014ned)g(starting)g(from)f
FM(typ)l(e)i(variables)e FH(X)r(;)12 b(Y)t(;)g(Z)q(;)g(:)g(:)g(:)48
b FK(b)n(y)h(means)e(of)i(t)n(w)n(o)167 2603 y(op)r(erations:)251
2766 y(1.)35 b(if)27 b FH(U)35 b FK(and)29 b FH(V)43
b FK(are)27 b(t)n(yp)r(es,)i(then)e FH(U)7 b FJ(!)q FH(V)43
b FK(is)27 b(a)h(t)n(yp)r(e.)251 2910 y(2.)35 b(if)27
b FH(V)43 b FK(is)27 b(a)h(t)n(yp)r(e,)g(and)h FH(X)k
FK(a)28 b(t)n(yp)r(e)g(v)l(ariable,)g(then)f(\005)p FH(X)r(:)12
b(V)43 b FK(is)27 b(a)h(t)n(yp)r(e.)167 3073 y(There)f(are)g(\014v)n(e)
h(sc)n(hemes)f(for)h(forming)g FM(terms)6 b FK(:)251
3236 y(1.)35 b FM(variables:)k FH(x)700 3210 y Fw(T)739
3236 y FH(;)12 b(y)807 3210 y Fw(T)845 3236 y FH(;)g(z)911
3210 y Fw(T)950 3236 y FH(;)g(:)g(:)g(:)28 b FK(of)g(t)n(yp)r(e)g
FH(T)10 b FK(,)251 3380 y(2.)35 b FM(applic)l(ation:)41
b FH(tu)27 b FK(of)h(t)n(yp)r(e)g FH(V)16 b FK(,)27 b(where)g
FH(t)h FK(is)f(of)h(t)n(yp)r(e)g FH(U)7 b FJ(!)p FH(V)43
b FK(and)29 b FH(u)f FK(is)f(of)h(t)n(yp)r(e)g FH(U)7
b FK(,)251 3525 y(3.)35 b FH(\025)p FM(-abstr)l(action:)k
FH(\025x)871 3499 y Fw(U)914 3525 y FH(:)12 b(v)29 b
FK(of)d(t)n(yp)r(e)h FH(U)7 b FJ(!)q FH(V)15 b FK(,)27
b(where)e FH(x)1714 3499 y Fw(U)1783 3525 y FK(is)g(a)i(v)l(ariable)f
(of)h(t)n(yp)r(e)f FH(U)34 b FK(and)27 b FH(v)339 3610
y FK(is)g(of)i(t)n(yp)r(e)e FH(V)16 b FK(,)251 3754 y(4.)35
b FM(universal)g(abstr)l(action:)57 b FK(if)34 b FH(v)39
b FK(is)c(a)h(term)f(of)h(t)n(yp)r(e)g FH(V)16 b FK(,)38
b(then)d(w)n(e)h(can)g(form)g(\003)p FH(X)r(:)12 b(v)339
3840 y FK(of)33 b(t)n(yp)r(e)f(\005)p FH(X)r(:)12 b(V)j
FK(,)33 b(so)g(long)g(as)f(the)g(v)l(ariable)g FH(X)37
b FK(is)32 b(not)g(free)g(in)f(the)h(t)n(yp)r(e)g(of)g(a)h(free)339
3925 y(v)l(ariable)28 b(of)g FH(v)s FK(.)1385 4101 y(81)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
82 85 bop 167 167 a FK(82)1550 b Fz(CHAPTER)26 b(11.)64
b(SYSTEM)28 b(F)251 404 y FK(5.)35 b FM(universal)g(applic)l(ation)41
b FK(\(sometimes)33 b(called)h FM(extr)l(action)p FK(\):)50
b(if)35 b FH(t)g FK(is)f(a)i(term)e(of)i(t)n(yp)r(e)339
490 y(\005)p FH(X)r(:)12 b(V)43 b FK(and)29 b FH(U)35
b FK(is)27 b(a)h(t)n(yp)r(e,)g(then)f FH(t)12 b(U)35
b FK(is)27 b(a)h(term)e(of)i(t)n(yp)r(e)g FH(V)15 b FK([)p
FH(U)t(=X)6 b FK(].)270 663 y(As)27 b(w)n(ell)d(as)j(the)f(usual)h
FM(c)l(onversions)f FK(for)h(application/)p FH(\025)p
FK(-abstraction,)g(there)e(is)h(one)h(for)167 748 y(the)g(other)h(pair)
f(of)i(sc)n(hemes:)1094 921 y(\(\003)p FH(X)r(:)12 b(v)s
FK(\))g FH(U)25 b Fu( )20 b FH(v)s FK([)p FH(U)t(=X)6
b FK(])167 1135 y FG(Con)n(v)n(en)n(tion)53 b FK(W)-6
b(e)27 b(shall)g(write)f FH(U)1132 1146 y FD(1)1160 1135
y FJ(!)p FH(U)1277 1146 y FD(2)1305 1135 y FJ(!)12 b
FH(:)g(:)g(:)g(U)1527 1146 y Fw(n)1560 1135 y FJ(!)p
FH(V)k FK(,)28 b(without)f(paren)n(theses,)h(for)1005
1308 y FH(U)1052 1319 y FD(1)1080 1308 y FJ(!)q FK(\()p
FH(U)1225 1319 y FD(2)1252 1308 y FJ(!)12 b FH(:)g(:)g(:)g
FK(\()p FH(U)1501 1319 y Fw(n)1534 1308 y FJ(!)p FH(V)k
FK(\))c FH(:)g(:)g(:)o FK(\))167 1482 y(and)29 b(similarly)-6
b(,)25 b FH(f)19 b(u)698 1493 y FD(1)738 1482 y FH(u)777
1493 y FD(2)817 1482 y FH(:)12 b(:)g(:)f(u)948 1493 y
Fw(n)1009 1482 y FK(for)28 b(\()p FH(:)12 b(:)g(:)g FK(\(\()p
FH(f)19 b(u)1385 1493 y FD(1)1413 1482 y FK(\))12 b FH(u)1491
1493 y FD(2)1519 1482 y FK(\))g FH(:)g(:)g(:)o FK(\))g
FH(u)1716 1493 y Fw(n)1749 1482 y FK(.)167 1717 y FE(11.2)114
b(Commen)m(ts)167 1872 y FK(First)32 b(let)g(us)h(illustrate)e(the)i
(restriction)e(on)i(v)l(ariables)g(in)g(univ)n(ersal)f(abstraction:)47
b(if)33 b(w)n(e)167 1957 y(could)g(form)g(\003)p FH(X)r(:)12
b(x)711 1931 y Fw(X)758 1957 y FK(,)35 b(what)e(w)n(ould)g(then)g(b)r
(e)g(the)g(t)n(yp)r(e)g(of)h(the)e(free)g(v)l(ariable)h
FH(x)g FK(in)g(this)167 2042 y(expression?)55 b(On)34
b(the)f(other)i(hand,)i(w)n(e)d FM(c)l(an)39 b FK(form)34
b(\003)p FH(X)r(:)12 b(\025x)1860 2016 y Fw(X)1908 2042
y FH(:)g(x)1978 2016 y Fw(X)2060 2042 y FK(of)34 b(t)n(yp)r(e)g(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)g FK(,)167 2127 y(whic)n(h)27
b(is)g(the)g(iden)n(tit)n(y)g(of)h(an)n(y)h(t)n(yp)r(e.)270
2300 y(The)e(na)-8 b(\177)-27 b(\020v)n(e)29 b(in)n(terpretation)d(of)i
(the)f(\\\005")h(t)n(yp)r(e)f(is)g(that)g(an)h(ob)t(ject)f(of)h(t)n(yp)
r(e)g(\005)p FH(X)r(:)12 b(V)42 b FK(is)27 b(a)167 2385
y(function)h(whic)n(h,)f(to)h(ev)n(ery)f(t)n(yp)r(e)h
FH(U)7 b FK(,)28 b(asso)r(ciates)g(an)g(ob)t(ject)g(of)g(t)n(yp)r(e)g
FH(V)15 b FK([)p FH(U)t(=X)6 b FK(].)270 2511 y(This)28
b(in)n(terpretation)f(runs)i(up)g(against)g(a)g(problem)e(of)i
FM(size)5 b FK(:)37 b(in)28 b(order)g(to)h(understand)167
2597 y(\005)p FH(X)r(:)12 b(V)j FK(,)42 b(it)37 b(is)h(necessary)h(to)g
(kno)n(w)h FM(al)t(l)45 b FK(the)38 b FH(V)16 b FK([)p
FH(U)t(=X)6 b FK(].)68 b(But)39 b(among)h(all)d(the)i
FH(V)15 b FK([)p FH(U)t(=X)6 b FK(])167 2682 y(there)38
b(are)h(some)f(whic)n(h)g(are)h(\(in)f(general\))g(more)g(complex)g
(than)h(the)g(t)n(yp)r(e)g(whic)n(h)f(w)n(e)167 2767
y(seek)33 b(to)h(mo)r(del,)g(for)g(example)f FH(V)16
b FK([\005)p FH(X)r(:)c(V)j(=X)6 b FK(].)54 b(So)34 b(there)f(is)g(a)h
(circularit)n(y)d(in)i(the)g(na)-8 b(\177)-27 b(\020v)n(e)167
2852 y(in)n(terpretation,)28 b(and)h(one)h(can)f(exp)r(ect)f(the)g(w)n
(orst)h(to)g(happ)r(en.)40 b(In)28 b(fact)h(it)f(all)f(w)n(orks)i(out,)
167 2937 y(but)i(the)f(system)g(is)g(extremely)e(sensitiv)n(e)i(to)g
(mo)r(di\014cations)g(whic)n(h)g(are)h(not)g(of)g(a)g(logical)167
3022 y(nature.)270 3148 y(W)-6 b(e)30 b(can)h(nev)n(ertheless)d(mak)n
(e)i(\(a)h(bit\))e(more)g(precise)f(the)i(idea)g(of)g(a)h(function)f
(de\014ned)167 3233 y(on)39 b FM(al)t(l)44 b FK(t)n(yp)r(es:)56
b(in)37 b(some)h(sense,)i(a)e(function)g(of)h(univ)n(ersal)e(t)n(yp)r
(e)h(m)n(ust)g(b)r(e)f(\\uniform",)167 3319 y FM(i.e.)32
b FK(do)h(the)e(same)h(thing)g(on)h(all)d(t)n(yp)r(es.)49
b FH(\025)p FK(-abstraction)32 b(accommo)r(dates)g(a)h(certain)d(dose)
167 3404 y(of)37 b(non-uniformit)n(y)-6 b(,)39 b(for)e(example)e(w)n(e)
h(can)h(de\014ne)g(a)g(function)f(b)n(y)h(cases)g(\(if)e(.)11
b(.)g(.)75 b(then)167 3489 y(.)11 b(.)g(.)76 b(else\).)62
b(Suc)n(h)38 b(a)f(kind)h(of)f(de\014nition)g(is)f(inconceiv)l(able)g
(for)h(univ)n(ersal)f(abstraction:)167 3574 y(the)f(v)l(alues)h(tak)n
(en)h(b)n(y)f(an)g(ob)t(ject)g(of)g(univ)n(ersal)f(t)n(yp)r(e)h(on)g
(di\013eren)n(ts)f(t)n(yp)r(es)h(ha)n(v)n(e)g(to)g(b)r(e)167
3659 y(essen)n(tially)h(\\the)h(same")h(\(see)f FG(??)q
FK(\).)68 b(It)38 b(still)e(remains)h(to)i(mak)n(e)g(this)f(v)l(ague)h
(in)n(tuition)167 3744 y(precise)26 b(b)n(y)i(appropriate)h(seman)n
(tic)d(considerations.)p eop
PStoPSsaved restore
%%Page: (86,87) 44
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
83 86 bop 167 167 a Fz(11.3.)65 b(REPRESENT)-6 b(A)g(TION)25
b(OF)j(SIMPLE)g(TYPES)903 b FK(83)167 404 y FE(11.3)114
b(Represen)m(tation)29 b(of)f(simple)e(t)m(yp)s(es)167
564 y FK(A)k(large)g(part)h(of)g(the)f(in)n(terest)f(in)h
FG(F)h FK(is)f(in)g(the)g(p)r(ossibilit)n(y)f(of)i(de\014ning)g
(commonly)f(used)167 649 y(t)n(yp)r(es)e(in)f(it;)g(w)n(e)g(shall)g
(dev)n(ote)h(the)f(rest)g(of)h(the)g(c)n(hapter)g(to)f(this.)167
864 y Fy(11.3.1)94 b(Bo)s(oleans)167 999 y FK(W)-6 b(e)27
b(de\014ne)h Fq(Bo)r(ol)g FK(\(not)g(the)f(one)h(of)g(system)g
FG(T)p FK(\))f(as)h(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p
FH(X)g FJ(!)p FH(X)34 b FK(with)567 1208 y Fq(T)633 1169
y FD(def)641 1208 y FK(=)27 b(\003)p FH(X)r(:)12 b(\025x)939
1183 y Fw(X)986 1208 y FH(:)g(\025y)1094 1183 y Fw(X)1141
1208 y FH(:)g(x)400 b Fq(F)1670 1169 y FD(def)1677 1208
y FK(=)27 b(\003)p FH(X)r(:)12 b(\025x)1975 1183 y Fw(X)2023
1208 y FH(:)g(\025y)2131 1183 y Fw(X)2178 1208 y FH(:)g(y)167
1400 y FK(and)29 b(if)e FH(u;)12 b(v)s(;)g(t)27 b FK(are)g(of)i(resp)r
(ectiv)n(e)d(t)n(yp)r(es)i FH(U;)12 b(U;)g Fq(Bo)r(ol)26
b FK(w)n(e)h(de\014ne)h Fq(D)12 b FH(u)g(v)i(t)27 b FK(of)h(t)n(yp)r(e)
g FH(U)35 b FK(b)n(y)1165 1605 y Fq(D)12 b FH(u)g(v)i(t)1369
1566 y FD(def)1377 1605 y FK(=)26 b FH(t)12 b(U)19 b(u)12
b(v)270 1844 y FK(Let)28 b(us)g(calculate)e Fq(D)12 b
FH(u)g(v)i Fq(T)27 b FK(and)i Fq(D)12 b FH(u)g(v)i Fq(F)p
FK(:)854 2063 y Fq(D)e FH(u)g(v)i Fq(T)67 b FK(=)g(\(\003)p
FH(X)r(:)12 b(\025x)1493 2037 y Fw(X)1541 2063 y FH(:)g(\025y)1649
2037 y Fw(X)1696 2063 y FH(:)g(x)p FK(\))g FH(U)18 b(u)12
b(v)1120 2148 y Fu( )59 b FK(\()p FH(\025x)1355 2122
y Fw(U)1397 2148 y FH(:)12 b(\025y)1505 2122 y Fw(U)1546
2148 y FH(:)g(x)p FK(\))g FH(u)g(v)1120 2233 y Fu( )59
b FK(\()p FH(\025y)1353 2207 y Fw(U)1394 2233 y FH(:)12
b(u)p FK(\))g FH(v)1120 2318 y Fu( )59 b FH(u)862 2488
y Fq(D)12 b FH(u)g(v)i Fq(F)67 b FK(=)g(\(\003)p FH(X)r(:)12
b(\025x)1493 2463 y Fw(X)1541 2488 y FH(:)g(\025y)1649
2463 y Fw(X)1696 2488 y FH(:)g(y)s FK(\))g FH(U)17 b(u)12
b(v)1120 2573 y Fu( )59 b FK(\()p FH(\025x)1355 2548
y Fw(U)1397 2573 y FH(:)12 b(\025y)1505 2548 y Fw(U)1546
2573 y FH(:)g(y)s FK(\))g FH(u)g(v)1120 2659 y Fu( )59
b FK(\()p FH(\025y)1353 2633 y Fw(U)1394 2659 y FH(:)12
b(y)1462 2633 y Fw(U)1503 2659 y FK(\))g FH(v)1120 2744
y Fu( )59 b FH(v)167 3047 y Fy(11.3.2)94 b(Pro)s(duct)27
b(of)g(t)m(yp)s(es)167 3198 y FK(W)-6 b(e)27 b(de\014ne)h
FH(U)7 b FJ(\002)p FH(V)678 3159 y FD(def)686 3198 y
FK(=)27 b(\005)p FH(X)r(:)12 b FK(\()p FH(U)7 b FJ(!)p
FH(V)15 b FJ(!)q FH(X)6 b FK(\))p FJ(!)p FH(X)33 b FK(with)965
3407 y FJ(h)p FH(u;)12 b(v)s FJ(i)1146 3368 y FD(def)1153
3407 y FK(=)27 b(\003)p FH(X)r(:)12 b(\025x)1451 3382
y Fw(U)5 b Fv(!)p Fw(V)11 b Fv(!)p Fw(X)1678 3407 y FH(:)h(x)g(u)g(v)
270 3599 y FK(The)28 b(pro)t(jections)f(are)h(de\014ned)g(as)g(follo)n
(ws:)527 3804 y FH(\031)569 3778 y FD(1)596 3804 y FH(t)641
3764 y FD(def)648 3804 y FK(=)f FH(t)12 b(U)19 b FK(\()p
FH(\025x)938 3778 y Fw(U)980 3804 y FH(:)12 b(\025y)1088
3778 y Fw(V)1130 3804 y FH(:)g(x)p FK(\))359 b FH(\031)1628
3778 y FD(2)1656 3804 y FH(t)1700 3764 y FD(def)1708
3804 y FK(=)27 b FH(t)12 b(V)27 b FK(\()p FH(\025x)1999
3778 y Fw(U)2041 3804 y FH(:)12 b(\025y)2149 3778 y Fw(V)2191
3804 y FH(:)g(y)s FK(\))p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
84 87 bop 167 167 a FK(84)1550 b Fz(CHAPTER)26 b(11.)64
b(SYSTEM)28 b(F)270 404 y FK(Let)g(us)g(calculate)e FH(\031)826
379 y FD(1)853 404 y FJ(h)p FH(u;)12 b(v)s FJ(i)29 b
FK(and)f FH(\031)1222 379 y FD(2)1250 404 y FJ(h)p FH(u;)12
b(v)s FJ(i)p FK(:)607 567 y FH(\031)649 542 y FD(1)676
567 y FJ(h)p FH(u;)g(v)s FJ(i)68 b FK(=)f(\(\003)p FH(X)r(:)12
b(\025x)1269 542 y Fw(U)5 b Fv(!)o Fw(V)12 b Fv(!)o Fw(X)1495
567 y FH(:)g(x)g(u)g(v)s FK(\))g FH(U)18 b FK(\()p FH(\025x)1874
542 y Fw(U)1916 567 y FH(:)12 b(\025y)2024 542 y Fw(V)2067
567 y FH(:)g(x)p FK(\))896 653 y Fu( )59 b FK(\()p FH(\025x)1131
627 y Fw(U)5 b Fv(!)p Fw(V)11 b Fv(!)p Fw(U)1351 653
y FH(:)h(x)g(u)g(v)s FK(\))g(\()p FH(\025x)1665 627 y
Fw(U)1707 653 y FH(:)g(\025y)1815 627 y Fw(V)1857 653
y FH(:)g(x)p FK(\))896 738 y Fu( )59 b FK(\()p FH(\025x)1131
712 y Fw(U)1173 738 y FH(:)12 b(\025y)1281 712 y Fw(V)1323
738 y FH(:)g(x)p FK(\))g FH(u)g(v)896 823 y Fu( )59 b
FK(\()p FH(\025y)1129 797 y Fw(V)1171 823 y FH(:)12 b(u)p
FK(\))g FH(v)896 908 y Fu( )59 b FH(u)607 1078 y(\031)649
1053 y FD(2)676 1078 y FJ(h)p FH(u;)12 b(v)s FJ(i)68
b FK(=)f(\(\003)p FH(X)r(:)12 b(\025x)1269 1053 y Fw(U)5
b Fv(!)o Fw(V)12 b Fv(!)o Fw(X)1495 1078 y FH(:)g(x)g(u)g(v)s
FK(\))g FH(V)27 b FK(\()p FH(\025x)1876 1053 y Fw(U)1918
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(ariable)f FH(X)33 b FK(\(w)n(e)27 b(shall)g(con)n(tin)n(ue)g(to)h
(write)e FH(S)2652 3463 y Fw(i)167 3537 y FK(for)i FH(S)319
3548 y Fw(i)339 3537 y FK([)p FH(X=)p FK(\002]\))g(and)h(w)n(e)e(in)n
(tro)r(duce:)966 3731 y FH(T)j FK(=)19 b(\005)p FH(X)r(:)12
b(S)1293 3742 y FD(1)1321 3731 y FJ(!)p FH(S)1433 3742
y FD(2)1462 3731 y FJ(!)g FH(:)g(:)g(:)f(S)1678 3742
y Fw(n)1712 3731 y FJ(!)q FH(X)167 3925 y FK(W)-6 b(e)27
b(shall)g(see)g(that)h FH(T)38 b FK(has)29 b(a)f(go)r(o)r(d)h(claim)c
(to)j(represen)n(t)f(\002.)p eop
PStoPSsaved restore
%%Page: (90,91) 46
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
87 90 bop 167 167 a Fz(11.4.)65 b(REPRESENT)-6 b(A)g(TION)25
b(OF)j(A)f(FREE)g(STR)n(UCTURE)693 b FK(87)167 404 y
Fy(11.4.2)94 b(Represen)m(tation)28 b(of)f(the)h(constructors)167
553 y FK(W)-6 b(e)37 b(ha)n(v)n(e)h(to)g(\014nd)g(an)g(ob)t(ject)f
FH(f)1091 564 y Fw(i)1148 553 y FK(for)h(eac)n(h)f(t)n(yp)r(e)h
FH(S)1649 564 y Fw(i)1669 553 y FK([)p FH(T)8 b(=X)e
FK(].)65 b(In)38 b(other)f(w)n(ords,)j(w)n(e)d(are)167
639 y(lo)r(oking)29 b(for)h(a)f(function)h FH(f)900 650
y Fw(i)949 639 y FK(whic)n(h)e(tak)n(es)i FH(k)1371 650
y Fw(i)1420 639 y FK(argumen)n(ts)f(of)h(t)n(yp)r(es)f
FH(T)2084 613 y Fw(i)2074 656 y(j)2104 639 y FK([)p FH(T)8
b(=X)e FK(])30 b(and)g(returns)167 724 y(a)e(v)l(alue)g(of)g(t)n(yp)r
(e)g FH(T)10 b FK(.)270 886 y(Let)38 b FH(x)448 897 y
FD(1)476 886 y FH(;)12 b(:)g(:)g(:)g(;)g(x)670 897 y
Fw(k)696 904 y Ff(i)756 886 y FK(b)r(e)38 b(the)g(argumen)n(ts)h(of)g
FH(f)1478 897 y Fw(i)1498 886 y FK(.)68 b(As)38 b FH(X)44
b FK(o)r(ccurs)38 b FM(p)l(ositively)45 b FK(in)38 b
FH(T)2496 861 y Fw(i)2486 904 y(j)2516 886 y FK(,)j(the)167
971 y(canonical)27 b(function)h FH(h)789 982 y Fw(i)836
971 y FK(of)g(t)n(yp)r(e)g FH(T)10 b FJ(!)q FH(X)33 b
FK(de\014ned)28 b(b)n(y)414 1218 y FH(h)454 1229 y Fw(i)485
1218 y FH(x)20 b FK(=)f FH(x)12 b(X)17 b(y)779 1187 y
Fw(S)809 1194 y Fi(1)776 1235 y FD(1)848 1218 y FH(:)12
b(:)g(:)g(y)978 1192 y Fw(S)1008 1198 y Ff(n)975 1235
y Fw(n)1287 1218 y FK(\(where)27 b FH(X)r(;)12 b(y)1642
1229 y FD(1)1669 1218 y FH(;)g(:)g(:)g(:)f(;)h(y)1857
1229 y Fw(n)1918 1218 y FK(are)27 b(parameters\))167
1464 y(induces)34 b(a)h(function)g FH(T)825 1438 y Fw(i)815
1481 y(j)845 1464 y FK([)p FH(h)904 1475 y Fw(i)923 1464
y FK(])g(from)f FH(T)1201 1438 y Fw(i)1191 1481 y(j)1221
1464 y FK([)p FH(T)8 b(=X)e FK(])35 b(to)g FH(T)1585
1438 y Fw(i)1575 1481 y(j)1640 1464 y FK(dep)r(ending)g(on)g
FH(X)r(;)12 b(y)2214 1475 y FD(1)2241 1464 y FH(;)g(:)g(:)g(:)g(;)g(y)
2430 1475 y Fw(n)2463 1464 y FK(.)56 b(This)167 1549
y(function)25 b(could)g(b)r(e)g(de\014ned)g(formally)-6
b(,)25 b(but)g(w)n(e)g(shall)f(see)g(it)g(m)n(uc)n(h)h(b)r(etter)f
(with)g(examples.)270 1711 y(Finally)j(w)n(e)g(put)h
FH(t)772 1722 y Fw(j)817 1711 y FK(=)19 b FH(T)940 1686
y Fw(i)930 1729 y(j)960 1711 y FK([)p FH(h)1019 1722
y Fw(i)1039 1711 y FK(])12 b FH(x)1109 1722 y Fw(j)1163
1711 y FK(for)28 b FH(j)23 b FK(=)c(1)p FH(;)12 b(:)g(:)g(:)g(;)g(k)
1623 1722 y Fw(i)1670 1711 y FK(and)29 b(w)n(e)e(de\014ne)761
1968 y FH(f)795 1979 y Fw(i)827 1968 y FH(x)866 1979
y FD(1)906 1968 y FH(:)12 b(:)g(:)f(x)1037 1979 y Fw(k)1063
1986 y Ff(i)1105 1968 y FK(=)19 b(\003)p FH(X)r(:)12
b(\025y)1393 1937 y Fw(S)1423 1944 y Fi(1)1390 1985 y
FD(1)1450 1968 y FH(:)23 b(:)12 b(:)g(:)g(\025y)1662
1942 y Fw(S)1692 1948 y Ff(n)1659 1985 y Fw(n)1725 1968
y FH(:)g(y)1790 1979 y Fw(i)1821 1968 y FH(t)1846 1979
y FD(1)1885 1968 y FH(:)g(:)g(:)g(t)2003 1979 y Fw(k)2029
1986 y Ff(i)167 2364 y Fy(11.4.3)94 b(Induction)167 2513
y FK(The)41 b(question)g(of)h(kno)n(wing)g(whether)e(the)h(only)g(ob)t
(jects)g(of)h(t)n(yp)r(e)f FH(T)51 b FK(whic)n(h)40 b(one)i(can)167
2598 y(construct)f(are)f(indeed)h(those)g(generated)g(from)f(the)h
FH(f)1738 2609 y Fw(i)1798 2598 y FK(is)g(hard;)48 b(the)40
b(answ)n(er)i(is)e FM(yes)p FK(,)167 2683 y(almost!)35
b(W)-6 b(e)27 b(shall)g(come)g(bac)n(k)i(to)f(this)f(in)g
FG(??)q FK(.)270 2846 y(A)34 b(preliminary)d(indication)h(of)i(this)f
(fact)h(is)f(the)g(p)r(ossibilit)n(y)f(of)i(de\014ning)g(a)g(function)
167 2931 y(b)n(y)29 b(induction)e(on)i(the)f(construction)f(of)i(\002.)
36 b(W)-6 b(e)28 b(start)g(o\013)g(with)f(a)i(t)n(yp)r(e)f
FH(U)35 b FK(and)29 b(functions)167 3016 y FH(g)200 3027
y FD(1)228 3016 y FH(;)12 b(:)g(:)g(:)g(;)g(g)416 3027
y Fw(n)475 3016 y FK(of)27 b(t)n(yp)r(es)f FH(S)784 3027
y Fw(i)805 3016 y FK([)p FH(U)t(=X)6 b FK(])26 b(\()p
FH(i)19 b FK(=)g(1)p FH(;)12 b(:)g(:)g(:)g(;)g(n)p FK(\).)35
b(W)-6 b(e)26 b(w)n(ould)g(lik)n(e)f(to)h(de\014ne)h(a)f(function)h
FH(h)f FK(of)167 3101 y(t)n(yp)r(e)i FH(T)10 b FJ(!)p
FH(U)35 b FK(satisfying:)347 3347 y FH(h)12 b FK(\()p
FH(f)460 3358 y Fw(i)491 3347 y FH(x)530 3358 y FD(1)569
3347 y FH(:)g(:)g(:)g(x)701 3358 y Fw(k)727 3365 y Ff(i)749
3347 y FK(\))19 b(=)g FH(g)901 3358 y Fw(i)933 3347 y
FH(u)972 3358 y FD(1)1012 3347 y FH(:)12 b(:)g(:)g(u)1144
3358 y Fw(k)1170 3365 y Ff(i)1371 3347 y FK(where)27
b FH(u)1614 3358 y Fw(j)1659 3347 y FK(=)20 b FH(T)1783
3322 y Fw(i)1773 3365 y(j)1803 3347 y FK([)p FH(h)p FK(])12
b FH(x)1932 3358 y Fw(j)1985 3347 y FK(for)28 b FH(j)23
b FK(=)c(1)p FH(;)12 b(:)g(:)g(:)g(;)g(k)2445 3358 y
Fw(i)167 3594 y FK(F)-6 b(or)28 b(this)f(w)n(e)g(put)h
FH(h)12 b(x)20 b FK(=)f FH(x)12 b(U)19 b(g)1005 3605
y FD(1)1044 3594 y FH(:)12 b(:)g(:)g(g)1170 3605 y Fw(n)1231
3594 y FK(and)29 b(the)e(previous)h(equation)g(is)f(clearly)f
(satis\014ed.)270 3840 y(This)33 b(represen)n(tation)g(of)h(inductiv)n
(e)e(t)n(yp)r(es)h(w)n(as)h(inspired)e(b)n(y)i(a)g(1970)h(man)n
(uscript)e(of)167 3925 y(Martin-L\177)-34 b(of.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
88 91 bop 167 167 a FK(88)1550 b Fz(CHAPTER)26 b(11.)64
b(SYSTEM)28 b(F)167 404 y FE(11.5)114 b(Represen)m(tation)29
b(of)f(inductiv)m(e)g(t)m(yp)s(es)167 559 y FK(All)d(the)h
(de\014nitions)g(giv)n(en)h(in)f FG(??)h FK(\(except)f(the)g(existen)n
(tial)e(t)n(yp)r(e\))j(are)f(particular)g(cases)h(of)167
644 y(what)h(w)n(e)f(describ)r(e)f(in)h FG(??)q FK(:)36
b(they)28 b(do)g(not)g(come)f(out)h(of)g(a)g(hat.)251
791 y(1.)35 b(The)28 b FM(b)l(o)l(ole)l(an)35 b FK(t)n(yp)r(e)28
b(has)i(t)n(w)n(o)e(constan)n(ts,)i(whic)n(h)e(will)d(then)j(giv)n(e)h
FH(f)2171 802 y FD(1)2227 791 y FK(and)g FH(f)2400 802
y FD(2)2456 791 y FK(of)g(t)n(yp)r(e)339 876 y(b)r(o)r(olean:)49
b(so)35 b FH(S)778 887 y FD(1)830 876 y FK(=)24 b FH(S)950
887 y FD(2)1002 876 y FK(=)g FH(X)40 b FK(and)35 b Fq(Bo)r(ol)24
b FK(=)f(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)g FJ(!)p
FH(X)g FK(.)54 b(It)34 b(is)f(easy)i(to)f(sho)n(w)339
962 y(that)k Fq(T)f FK(and)h Fq(F)f FK(are)f(indeed)h(the)f(0-ary)i
(functions)g(de\014ned)f(in)f FG(??)i FK(and)g(that)f(the)339
1047 y(induction)28 b(op)r(eration)g(is)f(nothing)h(other)g(than)g
Fq(D)p FK(.)251 1187 y(2.)35 b(The)e FM(pr)l(o)l(duct)41
b FK(t)n(yp)r(e)33 b(has)h(a)f(function)g FH(f)1429 1198
y FD(1)1490 1187 y FK(of)g(t)n(w)n(o)g(argumen)n(ts,)i(one)f(of)f(t)n
(yp)r(e)g FH(U)40 b FK(and)339 1272 y(one)29 b(of)g(t)n(yp)r(e)g
FH(V)16 b FK(.)38 b(So)29 b(w)n(e)f(ha)n(v)n(e)i FH(S)1244
1283 y FD(1)1292 1272 y FK(=)20 b FH(U)7 b FJ(!)q FH(V)15
b FJ(!)q FH(X)6 b FK(,)29 b(whic)n(h)e(explains)i(the)f(translation.)
339 1357 y(The)33 b(pairing)g(function)h(\014ts)f(in)g(w)n(ell)e(with)h
(the)h(general)g(case)g(of)h FG(??)q FK(,)g(but)g(the)f(t)n(w)n(o)339
1442 y(pro)t(jections)c(go)h(outside)f(this)f(treatmen)n(t:)37
b(they)29 b(are)g(in)g(fact)g(more)f(easy)i(to)f(handle)339
1527 y(than)g(the)e(indirect)f(sc)n(heme)g(resulting)h(from)g(a)h(mec)n
(hanical)e(application)h(of)i FG(??)p FK(.)251 1667 y(3.)35
b(The)d FM(sum)38 b FK(t)n(yp)r(e)32 b(has)h(t)n(w)n(o)f(functions)h
(\(the)e(canonical)h(injections\),)g(so)g FH(S)2357 1678
y FD(1)2408 1667 y FK(=)23 b FH(U)7 b FJ(!)p FH(X)339
1753 y FK(and)37 b FH(S)528 1764 y FD(2)582 1753 y FK(=)25
b FH(V)16 b FJ(!)p FH(X)6 b FK(.)60 b(The)35 b(in)n(terpretation)g(of)h
FG(??)h FK(matc)n(hes)e(faithfully)g(the)h(general)339
1838 y(sc)n(heme.)251 1978 y(4.)f(The)41 b FM(empty)47
b FK(t)n(yp)r(e)41 b(has)h(nothing,)j(so)c FH(n)29 b
FK(=)f(0.)75 b(The)41 b(function)f FH(")2193 1989 y Fw(U)2276
1978 y FK(is)g(indeed)g(its)339 2063 y(induction)28 b(op)r(erator.)270
2210 y(Let)g(us)g(no)n(w)g(turn)g(to)g(some)f(more)g(complex)g
(examples.)167 2412 y Fy(11.5.1)94 b(In)m(tegers)167
2543 y FK(The)36 b(in)n(teger)f(t)n(yp)r(e)h(has)h(t)n(w)n(o)g
(functions:)53 b Fq(O)36 b FK(of)g(t)n(yp)r(e)h(in)n(teger)e(and)i
Fq(S)g FK(from)f(in)n(tegers)f(to)167 2628 y(in)n(tegers,)27
b(whic)n(h)g(giv)n(es)g FH(S)863 2639 y FD(1)911 2628
y FK(=)20 b FH(X)33 b FK(and)c FH(S)1256 2639 y FD(2)1303
2628 y FK(=)20 b FH(X)6 b FJ(!)p FH(X)g FK(,)27 b(so)983
2807 y Fq(Int)1082 2768 y FD(def)1090 2807 y FK(=)f(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)p FK(\()p FH(X)g FJ(!)p FH(X)g
FK(\))p FJ(!)p FH(X)270 2973 y FK(In)28 b(the)g(t)n(yp)r(e)f
Fq(Int)p FK(,)h(the)g(in)n(teger)e FH(n)i FK(will)d(b)r(e)i(represen)n
(ted)g(b)n(y)p 763 3100 42 3 v 763 3139 a FH(n)20 b FK(=)f(\003)p
FH(X)r(:)12 b(\025x)1114 3114 y Fw(X)1162 3139 y FH(:)g(\025y)1270
3114 y Fw(X)t Fv(!)o Fw(X)1411 3139 y FH(:)27 b(y)14
b FK(\()p FH(y)f FK(\()p FH(y)h(:)e(:)g(:)g FK(\()p FH(y)1457
3183 y Fo(|)p 1489 3183 114 9 v 114 w({z)p 1667 3183
V 114 w(})1454 3243 y Fx(n)28 b FB(o)r(ccurrences)1836
3139 y FH(x)p FK(\))12 b FH(:)g(:)g(:)p FK(\)\))270 3423
y(By)28 b(in)n(terc)n(hanging)g FH(S)859 3434 y FD(1)915
3423 y FK(and)h FH(S)1096 3434 y FD(2)1124 3423 y FK(,)f(one)g(could)f
(represen)n(t)g Fq(Int)h FK(b)n(y)g(the)g(v)l(arian)n(t)1050
3589 y(\005)p FH(X)r(:)12 b FK(\()p FH(X)6 b FJ(!)p FH(X)g
FK(\))p FJ(!)p FK(\()p FH(X)g FJ(!)p FH(X)g FK(\))167
3754 y(whic)n(h)35 b(giv)n(es)g(essen)n(tially)f(the)h(same)g(thing.)59
b(In)35 b(this)g(case,)i(the)e(in)n(terpretation)f(of)i
FH(n)g FK(is)167 3840 y(immediate:)f(it)28 b(is)g(the)h(function)f
(whic)n(h)h(to)g(an)n(y)h(t)n(yp)r(e)f FH(U)36 b FK(and)30
b(function)f FH(f)36 b FK(of)29 b(t)n(yp)r(e)g FH(U)7
b FJ(!)q FH(U)167 3925 y FK(asso)r(ciates)28 b(the)f(function)h
FH(f)929 3899 y Fw(n)962 3925 y FK(,)f FM(i.e.)h FH(f)35
b FK(iterated)27 b FH(n)g FK(times.)p eop
PStoPSsaved restore
%%Page: (92,93) 47
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
89 92 bop 167 167 a Fz(11.5.)65 b(REPRESENT)-6 b(A)g(TION)25
b(OF)j(INDUCTIVE)d(TYPES)763 b FK(89)270 404 y(Let)28
b(us)g(write)e(the)h(basic)g(functions:)402 583 y Fq(O)473
543 y FD(def)481 583 y FK(=)f(\003)p FH(X)r(:)12 b(\025x)778
557 y Fw(X)826 583 y FH(:)g(\025y)934 557 y Fw(X)t Fv(!)o
Fw(X)1076 583 y FH(:)g(x)235 b Fq(S)12 b FH(t)1476 543
y FD(def)1483 583 y FK(=)27 b(\003)p FH(X)r(:)12 b(\025x)1781
557 y Fw(X)1829 583 y FH(:)g(\025y)1937 557 y Fw(X)t
Fv(!)o Fw(X)2078 583 y FH(:)g(y)i FK(\()p FH(t)e(X)17
b(x)12 b(y)s FK(\))167 761 y(Of)27 b(course,)h(w)n(e)f(ha)n(v)n(e)i
Fq(O)19 b FK(=)p 918 707 35 3 v 20 w(0)28 b(and)g Fq(S)p
1168 722 42 3 v 12 w FH(n)20 b Fu( )p 1319 707 130 3
v 20 w FH(n)p FK(+1.)270 889 y(As)25 b(to)g(the)f(induction)g(op)r
(erator,)i(it)e(is)f(in)h(fact)h(the)f FM(iter)l(ator)33
b Fq(It)p FK(,)25 b(whic)n(h)f(tak)n(es)h(an)g(ob)t(ject)167
974 y(of)j(t)n(yp)r(e)g FH(U)7 b FK(,)28 b(a)g(function)g(of)g(t)n(yp)r
(e)g FH(U)7 b FJ(!)p FH(U)35 b FK(and)29 b(returns)e(a)h(result)f(of)h
(t)n(yp)r(e)g FH(U)7 b FK(:)737 1189 y Fq(It)k FH(u)h(f)20
b(t)67 b FK(=)f FH(t)12 b(U)19 b(u)12 b(f)711 1360 y
Fq(It)f FH(u)h(f)20 b Fq(O)67 b FK(=)f(\(\003)p FH(X)r(:)12
b(\025x)1353 1334 y Fw(X)1401 1360 y FH(:)g(\025y)1509
1334 y Fw(X)t Fv(!)o Fw(X)1651 1360 y FH(:)g(x)p FK(\))g
FH(U)18 b(u)12 b(f)980 1445 y Fu( )59 b FK(\()p FH(\025x)1215
1419 y Fw(U)1257 1445 y FH(:)12 b(\025y)1365 1419 y Fw(U)5
b Fv(!)o Fw(U)1495 1445 y FH(:)12 b(x)p FK(\))g FH(u)g(f)980
1530 y Fu( )59 b FK(\()p FH(\025y)1213 1504 y Fw(U)5
b Fv(!)o Fw(U)1343 1530 y FH(:)12 b(u)p FK(\))g FH(f)980
1615 y Fu( )59 b FH(u)633 1785 y Fq(It)12 b FH(u)g(f)19
b FK(\()p Fq(S)12 b FH(t)p FK(\))67 b(=)f(\(\003)p FH(X)r(:)12
b(\025x)1353 1760 y Fw(X)1401 1785 y FH(:)g(\025y)1509
1760 y Fw(X)t Fv(!)o Fw(X)1651 1785 y FH(:)g(y)h FK(\()p
FH(t)f(X)17 b(x)12 b(y)s FK(\)\))g FH(U)18 b(u)12 b(f)980
1870 y Fu( )59 b FK(\()p FH(\025x)1215 1845 y Fw(U)1257
1870 y FH(:)12 b(\025y)1365 1845 y Fw(U)5 b Fv(!)o Fw(U)1495
1870 y FH(:)12 b(y)i FK(\()p FH(t)e(U)18 b(x)12 b(y)s
FK(\)\))g FH(u)g(f)980 1955 y Fu( )59 b FK(\()p FH(\025y)1213
1930 y Fw(U)5 b Fv(!)o Fw(U)1343 1955 y FH(:)12 b(y)i
FK(\()p FH(t)e(U)18 b(u)12 b(y)s FK(\)\))g FH(f)980 2040
y Fu( )59 b FH(f)20 b FK(\()p FH(t)12 b(U)18 b(u)12 b(f)c
FK(\))989 2126 y(=)66 b FH(f)20 b FK(\()p Fq(It)11 b
FH(u)h(f)19 b(t)p FK(\))270 2352 y(It)28 b(is)f(not)h(true)f(that)h
Fq(It)12 b FH(u)g(f)p 1015 2298 V 19 w(n)p FK(+1)20 b
Fu( )g FH(f)f FK(\()p Fq(It)11 b FH(u)h(f)p 1493 2313
42 3 v 19 w(n)q FK(\),)27 b(but)h(b)r(oth)h(terms)d(reduce)h(to)1101
2530 y FH(f)19 b FK(\()p FH(f)g FK(\()p FH(f)g(:)12 b(:)g(:)g
FK(\()p FH(f)1101 2574 y Fo(|)p 1133 2574 124 9 v 124
w({z)p 1321 2574 V 124 w(})1071 2641 y Fx(n)p FB(+1)28
b(o)r(ccurrences)1528 2530 y FH(u)p FK(\))12 b FH(:)g(:)g(:)p
FK(\)\))167 2811 y(so)28 b(at)g(least)f Fq(It)11 b FH(u)h(f)p
670 2758 130 3 v 19 w(n)p FK(+1)20 b FJ(\030)g FH(f)f
FK(\()p Fq(It)11 b FH(u)h(f)p 1133 2773 42 3 v 19 w(n)p
FK(\))28 b(,)f(where)g(\\)p FJ(\030)p FK(")h(is)f(the)g(equiv)l(alence)
g(closure)g(of)h(\\)p Fu( )p FK(".)167 2897 y(In)23 b(fact,)g(\\)p
Fu( )p FK(")h(satis\014es)f(the)f(Ch)n(urc)n(h-Rosser)h(prop)r(ert)n(y)
-6 b(,)25 b(so)e(that)g(t)n(w)n(o)g(terms)e(are)i(equiv)l(alen)n(t)167
2982 y(i\013)k(they)h(reduce)f(to)h(a)g(common)f(one.)270
3110 y(While)35 b(w)n(e)i(are)f(on)i(the)e(sub)t(ject,)j(let)d(us)h
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(in)167 3195 y(terms)28 b(of)i FM(iter)l(ation)p FK(.)41
b(Let)29 b FH(u)h FK(b)r(e)f(of)h(t)n(yp)r(e)f FH(U)7
b FK(,)30 b FH(f)37 b FK(of)30 b(t)n(yp)r(e)f FH(U)7
b FJ(!)q Fq(Int)p FJ(!)p FH(U)g FK(.)41 b(W)-6 b(e)29
b(construct)h FH(g)i FK(of)167 3281 y(t)n(yp)r(e)c FH(U)7
b FJ(\002)p Fq(Int)p FJ(!)p FH(U)g FJ(\002)p Fq(Int)28
b FK(b)n(y)900 3459 y FH(g)22 b FK(=)e FH(\025x)1108
3433 y Fw(U)5 b Fv(\002)p Fp(Int)1246 3459 y FH(:)12
b FJ(h)p FH(f)20 b FK(\()p FH(\031)1427 3433 y FD(1)1454
3459 y FH(x)p FK(\))12 b(\()p FH(\031)1601 3433 y FD(2)1628
3459 y FH(x)p FK(\))p FH(;)g Fq(S)g FH(\031)1817 3433
y FD(2)1845 3459 y FH(x)p FJ(i)167 3637 y FK(In)28 b(particular,)f
FH(g)14 b FJ(h)p FH(u;)p 744 3598 V 12 w(n)q FJ(i)20
b Fu( )g FJ(h)p FH(f)f(u)p 1054 3598 V 12 w(n;)p 1126
3583 130 3 v 12 w(n)p FK(+1)q FJ(i)p FK(.)35 b(So)29
b(if)e Fq(It)11 b FJ(h)p FH(u;)p 1658 3583 35 3 v 12
w FK(0)q FJ(i)h FH(g)p 1779 3598 42 3 v 14 w(n)20 b FJ(\030)f(h)p
FH(t)1966 3648 y Fw(n)2000 3637 y FH(;)p 2031 3598 V
12 w(n)p FJ(i)28 b FK(then:)550 3815 y Fq(It)11 b FJ(h)p
FH(u;)p 702 3761 35 3 v 12 w FK(0)q FJ(i)h FH(g)p 823
3761 130 3 v 14 w(n)p FK(+1)20 b FJ(\030)g FH(g)14 b
FK(\()p Fq(It)d FJ(h)p FH(u;)p 1273 3761 35 3 v 12 w
FK(0)q FJ(i)h FH(g)p 1395 3776 42 3 v 15 w(n)p FK(\))19
b FJ(\030)h FH(g)14 b FJ(h)p FH(t)1656 3826 y Fw(n)1689
3815 y FH(;)p 1720 3776 V 12 w(n)p FJ(i)20 b(\030)g(h)p
FH(f)f(t)1988 3826 y Fw(n)p 2033 3776 V 2033 3815 a FH(n;)p
2105 3761 130 3 v 12 w(n)p FK(+1)q FJ(i)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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 closepath}put initclip
PStoPSxform concat
90 93 bop 167 167 a FK(90)1550 b Fz(CHAPTER)26 b(11.)64
b(SYSTEM)28 b(F)270 413 y FK(Finally)-6 b(,)27 b(consider)g
Fq(R)12 b FH(u)g(f)20 b(t)1001 374 y FD(def)1009 413
y FK(=)27 b FH(\031)1132 388 y FD(1)1159 413 y FK(\()p
Fq(It)11 b FJ(h)p FH(u;)p 1338 359 35 3 v 12 w FK(0)q
FJ(i)h FH(g)j(t)p FK(\).)35 b(W)-6 b(e)27 b(ha)n(v)n(e:)635
586 y Fq(R)12 b FH(u)g(f)p 796 532 V 20 w FK(0)20 b FJ(\030)f
FH(u)468 b Fq(R)12 b FH(u)g(f)p 1592 532 130 3 v 20 w(n)p
FK(+1)20 b FJ(\030)f FH(f)g FK(\()p Fq(R)12 b FH(u)g(f)p
2055 547 42 3 v 19 w(n)q FK(\))p 2136 547 V 12 w FH(n)270
759 y FK(The)33 b(second)h(equation)g(for)g(recursion)e(is)h
(satis\014ed)g(b)n(y)i(v)l(alues)e(only)-6 b(,)36 b FM(i.e.)d
FK(for)h(eac)n(h)f FH(n)167 844 y FK(separately)-6 b(.)67
b(W)-6 b(e)37 b(mak)n(e)h(no)h(secret)e(of)h(the)g(fact)g(that)g(this)f
(is)h(a)g(defect)f(of)i(system)e FG(F)p FK(.)167 929
y(Indeed,)28 b(if)f(w)n(e)g(program)h(the)g(predecessor)f(function)747
1102 y Fq(p)n(red)12 b(O)20 b FK(=)f Fq(O)579 b(p)n(red)13
b FK(\()p Fq(S)f FH(x)p FK(\))20 b(=)f FH(x)167 1274
y FK(the)40 b(second)h(equation)g(will)c(only)k(b)r(e)f(satis\014ed)h
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FK(,)i(whic)n(h)d(means)167 1359 y(that)32 b(the)g(program)h(decomp)r
(oses)f(the)g(argumen)n(t)g FH(x)h FK(completely)c(in)n(to)j
Fq(S)12 b(S)g(S)g FH(:)g(:)g(:)g Fq(S)g(O)p FK(,)33 b(then)167
1445 y(reconstructs)43 b(it)f(lea)n(ving)i(out)g(the)f(last)g(sym)n(b)r
(ol)g Fq(S)p FK(.)83 b(Of)43 b(course)h(it)e(w)n(ould)h(b)r(e)h(more)
167 1530 y(economical)26 b(to)i(remo)n(v)n(e)f(the)h(\014rst)f
(instead!)167 1734 y Fy(11.5.2)94 b(Lists)167 1864 y
FH(U)29 b FK(b)r(eing)22 b(a)g(t)n(yp)r(e,)h(w)n(e)f(w)n(an)n(t)g(to)h
(form)e(the)h(t)n(yp)r(e)g Fq(List)12 b FH(U)7 b FK(,)23
b(whose)f(ob)t(jects)g(are)g(\014nite)f(sequences)167
1949 y(\()p FH(u)233 1960 y FD(1)261 1949 y FH(;)12 b(:)g(:)g(:)g(;)g
(u)455 1960 y Fw(n)488 1949 y FK(\))27 b(of)h(t)n(yp)r(e)g
FH(U)7 b FK(.)36 b(W)-6 b(e)27 b(ha)n(v)n(e)i(t)n(w)n(o)f(functions:)
270 2110 y FJ(\017)34 b FK(the)28 b(sequence)f(\(\))g(of)i(t)n(yp)r(e)e
Fq(List)12 b FH(U)7 b FK(,)28 b(and)h(hence)e FH(S)1672
2121 y FD(1)1720 2110 y FK(=)19 b FH(X)6 b FK(;)270 2254
y FJ(\017)34 b FK(the)28 b(function)g(whic)n(h)f(maps)i(an)f(ob)t(ject)
g FH(u)g FK(of)g(t)n(yp)r(e)g FH(U)35 b FK(and)29 b(a)g(sequence)e(\()p
FH(u)2390 2265 y FD(1)2418 2254 y FH(;)12 b(:)g(:)g(:)g(;)g(u)2612
2265 y Fw(n)2645 2254 y FK(\))339 2339 y(to)28 b(\()p
FH(u;)12 b(u)564 2350 y FD(1)592 2339 y FH(;)g(:)g(:)g(:)g(;)g(u)786
2350 y Fw(n)819 2339 y FK(\).)36 b(So)28 b FH(S)1043
2350 y FD(2)1091 2339 y FK(=)19 b FH(U)7 b FJ(!)q FH(X)f
FJ(!)p FH(X)g FK(.)270 2499 y(Mec)n(hanically)27 b(applying)h(the)f
(general)g(sc)n(heme,)g(w)n(e)g(get)757 2727 y Fq(List)11
b FH(U)986 2687 y FD(def)994 2727 y FK(=)66 b(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)p FK(\()p FH(U)h FJ(!)p FH(X)f
FJ(!)p FH(X)g FK(\))p FJ(!)p FH(X)860 2916 y Fq(nil)986
2877 y FD(def)994 2916 y FK(=)66 b(\003)p FH(X)r(:)12
b(\025x)1331 2891 y Fw(X)1379 2916 y FH(:)g(\025y)1487
2891 y Fw(U)5 b Fv(!)o Fw(X)t Fv(!)o Fw(X)1717 2916 y
FH(:)12 b(x)713 3021 y Fq(cons)h FH(u)f(t)986 2982 y
FD(def)994 3021 y FK(=)66 b(\003)p FH(X)r(:)12 b(\025x)1331
2995 y Fw(X)1379 3021 y FH(:)g(\025y)1487 2995 y Fw(U)5
b Fv(!)o Fw(X)t Fv(!)o Fw(X)1717 3021 y FH(:)12 b(y)i(u)e
FK(\()p FH(t)g(X)17 b(x)12 b(y)s FK(\))167 3234 y(So)28
b(the)g(sequence)f(\()p FH(u)747 3245 y FD(1)775 3234
y FH(;)12 b(:)g(:)g(:)g(;)g(u)969 3245 y Fw(n)1002 3234
y FK(\))28 b(is)e(represen)n(ted)h(b)n(y)730 3406 y(\003)p
FH(X)r(:)12 b(\025x)947 3381 y Fw(X)995 3406 y FH(:)g(\025y)1103
3381 y Fw(U)5 b Fv(!)o Fw(X)t Fv(!)o Fw(X)1333 3406 y
FH(:)12 b(y)i(u)1451 3417 y FD(1)1491 3406 y FK(\()p
FH(y)f(u)1604 3417 y FD(2)1644 3406 y FH(:)f(:)g(:)g
FK(\()p FH(y)i(u)1851 3417 y Fw(n)1896 3406 y FH(x)p
FK(\))e FH(:)g(:)g(:)p FK(\))167 3579 y(whic)n(h)27 b(w)n(e)g
(recognise,)g(replacing)g FH(y)j FK(b)n(y)e Fq(cons)h
FK(and)f FH(x)g FK(b)n(y)g Fq(nil)o FK(,)g(as)897 3752
y Fq(cons)12 b FH(u)1074 3763 y FD(1)1114 3752 y FK(\()p
Fq(cons)h FH(u)1319 3763 y FD(2)1358 3752 y FH(:)f(:)g(:)g
FK(\()p Fq(cons)h FH(u)1656 3763 y Fw(n)1701 3752 y Fq(nil)o
FK(\))f FH(:)g(:)g(:)o FK(\))167 3925 y(This)27 b(last)g(term)g(could)g
(b)r(e)h(obtained)g(b)n(y)g(reducing)f(\()p FH(u)1659
3936 y FD(1)1687 3925 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1881
3936 y Fw(n)1914 3925 y FK(\))g(\()p Fq(List)f FH(U)c
FK(\))12 b Fq(nil)f(cons)q FK(.)p eop
PStoPSsaved restore
%%Page: (94,95) 48
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
91 94 bop 167 167 a Fz(11.5.)65 b(REPRESENT)-6 b(A)g(TION)25
b(OF)j(INDUCTIVE)d(TYPES)763 b FK(91)270 404 y(The)25
b(b)r(eha)n(viour)g(of)g(lists)f(is)g(v)n(ery)g(similar)e(to)j(that)g
(of)h(in)n(tegers.)33 b(W)-6 b(e)24 b(ha)n(v)n(e)i(in)e(particular)167
490 y(an)i(iteration)f(on)h(lists:)33 b(if)25 b FH(W)35
b FK(is)25 b(a)h(t)n(yp)r(e,)g FH(w)i FK(is)d(of)h(t)n(yp)r(e)g
FH(W)10 b FK(,)25 b FH(f)33 b FK(is)25 b(of)h(t)n(yp)r(e)g
FH(U)7 b FJ(!)q FH(W)j FJ(!)p FH(W)g FK(,)26 b(one)167
575 y(can)i(de\014ne)g(for)g FH(t)f FK(of)h(t)n(yp)r(e)g
Fq(List)12 b FH(U)35 b FK(the)27 b(term)f Fq(It)12 b
FH(w)i(f)19 b(t)27 b FK(of)i(t)n(yp)r(e)e FH(W)38 b FK(b)n(y)1140
753 y Fq(It)11 b FH(w)j(f)19 b(t)1356 714 y FD(def)1363
753 y FK(=)27 b FH(t)12 b(W)21 b(w)14 b(f)167 918 y FK(whic)n(h)27
b(satis\014es)558 1083 y Fq(It)12 b FH(w)i(f)19 b Fq(nil)f
Fu( )i FH(w)394 b Fq(It)11 b FH(w)j(f)19 b FK(\()p Fq(cons)13
b FH(u)f(t)p FK(\))19 b Fu( )h FH(f)f(u)12 b FK(\()p
Fq(It)f FH(w)j(f)20 b(t)p FK(\))167 1285 y FG(Examples)270
1430 y FJ(\017)34 b Fq(It)12 b(nil)e(cons)j FH(t)20 b
Fu( )g FH(t)27 b FK(for)h(all)e FH(t)i FK(of)g(the)f(form)h(\()p
FH(u)1503 1441 y FD(1)1531 1430 y FH(;)12 b(:)g(:)g(:)f(;)h(u)1724
1441 y Fw(n)1758 1430 y FK(\).)270 1570 y FJ(\017)34
b FK(If)i FH(W)g FK(=)25 b Fq(List)12 b FH(V)51 b FK(where)36
b FH(V)51 b FK(is)36 b(another)h(t)n(yp)r(e,)h(and)f
FH(f)c FK(=)26 b FH(\025x)2024 1545 y Fw(U)2066 1570
y FH(:)12 b(\025y)2174 1545 y Fp(List)d Fw(W)2315 1570
y FH(:)j Fq(cons)g FK(\()p FH(g)i(x)p FK(\))e FH(y)339
1655 y FK(where)27 b FH(g)j FK(is)d(of)h(t)n(yp)r(e)g
FH(U)7 b FJ(!)q FH(V)15 b FK(,)28 b(it)f(is)g(easy)h(to)g(see)f(that)
948 1848 y Fq(It)11 b(nil)g FH(f)19 b FK(\()p FH(u)1202
1859 y FD(1)1230 1848 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1424
1859 y Fw(n)1457 1848 y FK(\))20 b Fu( )g FK(\()p FH(g)14
b(u)1707 1859 y FD(1)1735 1848 y FH(;)e(:)g(:)g(:)f(;)h(g)j(u)1976
1859 y Fw(n)2009 1848 y FK(\))270 2110 y(Using)28 b(a)g(pro)r(duct)g(t)
n(yp)r(e,)g(w)n(e)f(can)h(obtain)g(a)h(recursion)e(op)r(erator)h(\(b)n
(y)g(v)l(alues\):)822 2273 y Fq(R)12 b FH(v)j(f)k Fq(nil)58
b FJ(\030)g FH(v)542 2358 y Fq(R)12 b FH(v)j(f)k FK(\()p
FH(u)765 2369 y FD(1)793 2358 y FH(;)12 b(:)g(:)g(:)g(;)g(u)987
2369 y Fw(n)1020 2358 y FK(\))59 b FJ(\030)f FH(f)20
b(u)1312 2369 y FD(1)1351 2358 y FK(\()p FH(u)1417 2369
y FD(2)1445 2358 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1639 2369
y Fw(n)1672 2358 y FK(\))g(\()p Fq(R)g FH(v)i(f)20 b
FK(\()p FH(u)1961 2369 y FD(2)1989 2358 y FH(;)12 b(:)g(:)g(:)f(;)h(u)
2182 2369 y Fw(n)2216 2358 y FK(\)\))167 2518 y(with)27
b FH(v)k FK(of)d(t)n(yp)r(e)g FH(V)44 b FK(and)29 b FH(f)35
b FK(of)29 b(t)n(yp)r(e)f FH(U)7 b FJ(!)q Fq(List)k FH(U)c
FJ(!)q FH(V)15 b FJ(!)q FH(V)g FK(.)37 b(This)27 b(enables)h(us)g(to)g
(de\014ne,)h(for)167 2603 y(example,)35 b(the)e(truncation)h(of)g(a)h
(list)d(b)n(y)i(remo)n(v)l(al)g(of)g(its)f(\014rst)h(elemen)n(t)e(\(if)
h(an)n(y\),)j(in)d(an)167 2688 y(analogous)d(w)n(a)n(y)e(to)g(the)g
(predecessor:)724 2853 y Fq(tail)10 b(nil)18 b FK(=)i
Fq(nil)555 b(tail)o FK(\()p Fq(cons)12 b FH(u)g(t)p FK(\))20
b(=)f FH(t)167 3018 y FK(where)27 b(the)g(second)h(equation)h(is)e
(only)h(satis\014ed)f(for)h FH(t)g FK(of)g(the)f(form)h(\()p
FH(u)2110 3029 y FD(1)2138 3018 y FH(;)12 b(:)g(:)g(:)f(;)h(u)2331
3029 y Fw(n)2365 3018 y FK(\).)270 3140 y(As)28 b(an)g(exercise,)e
(de\014ne)i(b)n(y)g(iteration:)270 3285 y FJ(\017)34
b FM(c)l(onc)l(atenation)5 b FK(:)36 b(\()p FH(u)877
3296 y FD(1)905 3285 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1099
3296 y Fw(n)1132 3285 y FK(\))19 b(@)h(\()p FH(v)1312
3296 y FD(1)1340 3285 y FH(;)12 b(:)g(:)g(:)f(;)h(v)1527
3296 y Fw(m)1575 3285 y FK(\))19 b(=)g(\()p FH(u)1760
3296 y FD(1)1788 3285 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1982
3296 y Fw(n)2015 3285 y FH(;)g(v)2079 3296 y FD(1)2107
3285 y FH(;)g(:)g(:)g(:)g(;)g(v)2295 3296 y Fw(m)2342
3285 y FK(\))270 3425 y FJ(\017)34 b FM(r)l(eversal)7
b FK(:)36 b Fq(reverse)11 b FK(\()p FH(u)906 3436 y FD(1)934
3425 y FH(;)h(:)g(:)g(:)g(;)g(u)1128 3436 y Fw(n)1161
3425 y FK(\))20 b(=)f(\()p FH(u)1347 3436 y Fw(n)1380
3425 y FH(;)12 b(:)g(:)g(:)g(;)g(u)1574 3436 y FD(1)1602
3425 y FK(\))270 3607 y Fq(List)g FH(U)35 b FK(dep)r(ends)28
b(on)g FH(U)7 b FK(,)28 b(but)g(the)f(de\014nition)g(w)n(e)g(ha)n(v)n
(e)h(giv)n(en)g(is)f(in)g(fact)g(uniform)g(in)g(it,)167
3692 y(so)h(w)n(e)f(can)h(de\014ne)638 3814 y Fq(Nil)57
b FK(=)i(\003)p FH(X)r(:)12 b Fq(nil)n FK([)p FH(X)6
b FK(])118 b(of)28 b(t)n(yp)r(e)g(\005)p FH(X)r(:)12
b Fq(List)f FH(X)578 3899 y Fq(Cons)60 b FK(=)f(\003)p
FH(X)r(:)12 b Fq(cons)p FK([)p FH(X)6 b FK(])59 b(of)28
b(t)n(yp)r(e)g(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p Fq(List)11
b FH(X)6 b FJ(!)q Fq(List)11 b FH(X)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
92 95 bop 167 167 a FK(92)1550 b Fz(CHAPTER)26 b(11.)64
b(SYSTEM)28 b(F)167 404 y Fy(11.5.3)94 b(Binary)28 b(trees)167
535 y FK(W)-6 b(e)27 b(are)h(in)n(terested)e(in)h(\014nite)g(binary)h
(trees.)35 b(F)-6 b(or)28 b(this,)f(w)n(e)g(ha)n(v)n(e)i(t)n(w)n(o)f
(functions:)270 695 y FJ(\017)34 b FK(the)28 b(tree)e(consisting)i
(only)g(of)g(its)f(ro)r(ot,)h(so)g FH(S)1571 706 y FD(1)1619
695 y FK(=)19 b FH(X)6 b FK(;)270 838 y FJ(\017)34 b
FK(the)28 b(construction)f(of)i(a)f(tree)e(from)i(t)n(w)n(o)f(trees,)g
(so)i FH(S)1773 849 y FD(2)1821 838 y FK(=)19 b FH(X)6
b FJ(!)p FH(X)g FJ(!)p FH(X)g FK(.)632 1098 y Fq(Bintree)898
1059 y FD(def)905 1098 y FK(=)66 b(\005)p FH(X)r(:)12
b(X)6 b FJ(!)p FK(\()p FH(X)g FJ(!)p FH(X)g FJ(!)p FH(X)g
FK(\))p FJ(!)p FH(X)771 1288 y Fq(nil)898 1249 y FD(def)905
1288 y FK(=)66 b(\003)p FH(X)r(:)12 b(\025x)1242 1262
y Fw(X)1290 1288 y FH(:)g(\025y)1398 1262 y Fw(X)t Fv(!)o
Fw(X)t Fv(!)p Fw(X)1634 1288 y FH(:)g(x)558 1393 y Fq(couple)f
FH(u)h(v)898 1353 y FD(def)905 1393 y FK(=)66 b(\003)p
FH(X)r(:)12 b(\025x)1242 1367 y Fw(X)1290 1393 y FH(:)g(\025y)1398
1367 y Fw(X)t Fv(!)o Fw(X)t Fv(!)p Fw(X)1634 1393 y FH(:)g(y)i
FK(\()p FH(u)e(X)17 b(x)12 b(y)s FK(\))g(\()p FH(v)h(X)18
b(x)12 b(y)s FK(\))270 1631 y(Iteration)25 b(on)h(trees)f(is)f(then)h
(de\014ned)h(b)n(y)g Fq(It)12 b FH(w)i(f)19 b(t)1608
1592 y FD(def)1616 1631 y FK(=)27 b FH(t)12 b(W)21 b(w)14
b(f)33 b FK(when)25 b FH(W)35 b FK(is)24 b(a)i(t)n(yp)r(e,)g
FH(w)i FK(of)167 1716 y(t)n(yp)r(e)g FH(W)10 b FK(,)27
b FH(f)35 b FK(of)28 b(t)n(yp)r(e)g FH(W)10 b FJ(!)p
FH(W)g FJ(!)p FH(W)38 b FK(and)28 b FH(t)g FK(of)g(t)n(yp)r(e)g
Fq(Bintree)o FK(.)35 b(It)28 b(satis\014es:)457 1888
y Fq(It)12 b FH(w)i(f)19 b Fq(nil)f Fu( )i FH(w)293 b
Fq(It)11 b FH(w)j(f)19 b FK(\()p Fq(couple)12 b FH(u)g(v)s
FK(\))19 b Fu( )h FH(f)f FK(\()p Fq(It)11 b FH(w)j(f)19
b(u)p FK(\))12 b(\()p Fq(It)f FH(w)j(f)20 b(v)s FK(\))167
2179 y Fy(11.5.4)94 b(T)-8 b(rees)28 b(of)f(branc)m(hing)h(t)m(yp)s(e)g
Fe(U)167 2310 y FK(There)f(are)g(t)n(w)n(o)h(functions:)270
2470 y FJ(\017)34 b FK(the)28 b(tree)e(consisting)i(only)g(of)g(its)f
(ro)r(ot,)h(so)g FH(S)1571 2481 y FD(1)1619 2470 y FK(=)19
b FH(X)6 b FK(;)270 2613 y FJ(\017)34 b FK(the)21 b(construction)f(of)i
(a)f(tree)e(from)i(a)g(family)e(\()p FH(t)1591 2624 y
Fw(u)1623 2613 y FK(\))1650 2624 y Fw(u)p Fv(2)p Fw(U)1774
2613 y FK(of)i(trees,)g(so)h FH(S)2158 2624 y FD(2)2205
2613 y FK(=)e(\()p FH(U)7 b FJ(!)p FH(X)f FK(\))p FJ(!)p
FH(X)g FK(.)647 2877 y Fq(T)-6 b(ree)12 b FH(U)898 2838
y FD(def)906 2877 y FK(=)65 b(\005)p FH(X)r(:)12 b(X)6
b FJ(!)p FK(\(\()p FH(U)h FJ(!)p FH(X)f FK(\))p FJ(!)q
FH(X)g FK(\))p FJ(!)p FH(X)771 3067 y Fq(nil)898 3028
y FD(def)906 3067 y FK(=)65 b(\003)p FH(X)r(:)12 b(\025x)1242
3041 y Fw(X)1290 3067 y FH(:)g(\025y)1398 3041 y FD(\()p
Fw(U)5 b Fv(!)p Fw(X)t FD(\))p Fv(!)p Fw(X)1667 3067
y FH(:)12 b(x)602 3172 y Fq(collect)e FH(f)898 3132 y
FD(def)906 3172 y FK(=)65 b(\003)p FH(X)r(:)12 b(\025x)1242
3146 y Fw(X)1290 3172 y FH(:)g(\025y)1398 3146 y FD(\()p
Fw(U)5 b Fv(!)p Fw(X)t FD(\))p Fv(!)p Fw(X)1667 3172
y FH(:)12 b(y)i FK(\()p FH(\025z)1848 3146 y Fw(U)1890
3172 y FH(:)e(f)19 b(z)c(X)i(x)12 b(y)s FK(\))270 3369
y(The)31 b(\(trans\014nite\))f(iteration)f(is)h(de\014ned)h(b)n(y)g
Fq(It)12 b FH(w)i(h)e(t)1739 3330 y FD(def)1747 3369
y FK(=)29 b FH(t)12 b(W)21 b(w)14 b(h)30 b FK(when)h
FH(W)40 b FK(is)30 b(a)h(t)n(yp)r(e,)167 3454 y FH(w)f
FK(of)e(t)n(yp)r(e)g FH(W)10 b FK(,)27 b FH(f)35 b FK(of)28
b(t)n(yp)r(e)g(\()p FH(U)7 b FJ(!)q FH(W)j FK(\))p FJ(!)p
FH(W)37 b FK(and)29 b FH(t)e FK(of)h(t)n(yp)r(e)g Fq(Bintree)o
FK(.)35 b(It)28 b(satis\014es:)479 3627 y Fq(It)11 b
FH(w)j(h)e Fq(nil)18 b Fu( )i FH(w)314 b Fq(It)12 b FH(w)i(h)e
FK(\()p Fq(collect)d FH(f)f FK(\))19 b Fu( )h FH(h)12
b FK(\()p FH(\025x)1918 3601 y Fw(U)1959 3627 y FH(:)g
Fq(It)f FH(w)j(h)e FK(\()p FH(f)19 b(x)p FK(\)\))270
3840 y(Notice)h(that)i Fq(Bintree)e FK(could)h(b)r(e)g(treated)g(as)h
(the)f(t)n(yp)r(e)h(of)f(trees)g(with)f(b)r(o)r(olean)i(branc)n(hing)
167 3925 y(t)n(yp)r(e,)28 b(without)f(substan)n(tial)h(alteration.)p
eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
93 96 bop 167 167 a Fz(11.6.)65 b(THE)27 b(CURR)-6 b(Y-HO)n(W)e(ARD)25
b(ISOMORPHISM)924 b FK(93)270 404 y(Just)30 b(as)f(w)n(e)f(can)h
(abstract)h(on)f FH(U)36 b FK(in)28 b Fq(List)11 b FH(U)c
FK(,)30 b(the)e(same)g(thing)h(is)f(p)r(ossible)g(with)g(trees.)167
490 y(This)21 b(p)r(oten)n(tial)f(for)h(abstraction)h(sho)n(ws)g(up)g
(the)f(mo)r(dularit)n(y)f(of)i FG(F)f FK(v)n(ery)g(w)n(ell:)30
b(for)22 b(example,)167 575 y(one)37 b(can)f(de\014ne)h(the)f(mo)r
(dule)f Fq(Collect)24 b FK(=)h(\003)p FH(X)r(:)12 b Fq(collect)n
FK([)p FH(X)6 b FK(],)38 b(whic)n(h)e(can)g(subsequen)n(tly)h(b)r(e)167
660 y(used)i(b)n(y)h(sp)r(ecifying)e(the)h(t)n(yp)r(e)g
FH(X)6 b FK(.)69 b(Of)39 b(course,)j(w)n(e)c(see)h(the)f(v)l(alue)h(of)
h(this)e(in)h(more)167 745 y(complicated)21 b(cases:)34
b(w)n(e)22 b(only)i(write)d(the)i(program)h(once,)g(but)g(it)e(can)h(b)
r(e)g(applied)g(\(plugged)167 830 y(in)n(to)k(other)h(mo)r(dules\))f
(in)g(a)h(great)g(v)l(ariet)n(y)g(of)g(situations.)167
1065 y FE(11.6)114 b(The)28 b(Curry-Ho)m(w)m(ard)h(Isomorphism)167
1220 y FK(The)24 b(t)n(yp)r(es)g(in)f FG(F)h FK(are)g(nothing)h(other)f
(than)g(prop)r(ositions)h(quan)n(ti\014ed)g(at)f(the)g
FM(se)l(c)l(ond)g(or)l(der)p FK(,)167 1305 y(and)31 b(the)f
(isomorphism)f(w)n(e)h(ha)n(v)n(e)h(already)f(established)g(for)g(the)g
(arro)n(w)h(extends)f(to)g(these)167 1390 y(quan)n(ti\014ers:)845
1548 y FJ(\001)845 1586 y(\001)845 1624 y(\001)829 1681
y FH(A)p 764 1726 181 4 v 965 1742 a FJ(8)1005 1716 y
FD(2)1032 1742 y FJ(I)764 1802 y(8)p FH(X)r(:)12 b(A)1785
1539 y FJ(\001)1785 1577 y(\001)1785 1615 y(\001)1704
1673 y(8)p FH(X)r(:)g(A)p 1673 1718 244 4 v 1937 1733
a FJ(8)1977 1708 y FD(2)2004 1733 y FJ(E)1673 1794 y
FH(A)p FK([)p FH(B)t(=X)6 b FK(])167 1954 y(whic)n(h)27
b(corresp)r(ond)h(exactly)g(to)g(univ)n(ersal)f(abstraction)h(and)h
(application.)270 2081 y(If)38 b FH(t)f FK(of)g(t)n(yp)r(e)h
FH(A)e FK(represen)n(ts)h(the)g(part)h(of)f(the)g(deduction)g(ab)r(o)n
(v)n(e)i FJ(8)2209 2055 y FD(2)2236 2081 y FJ(I)5 b FK(,)40
b(then)e(\003)p FH(X)r(:)12 b(t)167 2166 y FK(represen)n(ts)39
b(the)g(whole)g(deduction.)71 b(The)40 b(usual)g(restriction)d(on)j(v)l
(ariables)g(in)f(natural)167 2251 y(deduction)26 b(\()p
FH(X)32 b FK(not)27 b(free)f(in)f(the)i(h)n(yp)r(otheses\))g(corresp)r
(onds)h(exactly)-6 b(,)26 b(as)h(w)n(e)f(can)h(see)f(here,)167
2336 y(to)i(the)f(restriction)f(on)i(the)g(formation)f(of)h(univ)n
(ersal)g(abstraction.)270 2462 y(Lik)n(ewise,)40 b FJ(8)625
2437 y FD(2)652 2462 y FJ(E)45 b FK(corresp)r(onds)39
b(to)g(an)g(application)f(to)g(t)n(yp)r(e)h FH(B)t FK(.)67
b(T)-6 b(o)39 b(b)r(e)f(completely)167 2547 y(precise,)32
b(in)h(the)f(case)h(where)e FH(X)39 b FK(do)r(es)33 b(not)g(app)r(ear)h
(in)e FH(A)p FK(,)i(one)f(should)g(sp)r(ecify)f(what)h
FH(B)167 2632 y FK(has)c(b)r(een)e(substituted.)270 2806
y(The)e(con)n(v)n(ersion)h(rule)e(\(\003)p FH(X)r(:)12
b(v)s FK(\))g FH(U)26 b Fu( )20 b FH(v)s FK([)p FH(U)t(=X)6
b FK(])25 b(corresp)r(onds)h(exactly)f(to)g(what)h(w)n(e)f(w)n(an)n(t)
167 2891 y(for)j(natural)g(deduction:)660 3026 y FJ(\001)660
3063 y(\001)660 3101 y(\001)644 3159 y FH(A)p 579 3203
181 4 v 781 3219 a FJ(8)821 3194 y FD(2)848 3219 y FJ(I)579
3280 y(8)p FH(X)r(:)12 b(A)p 548 3325 244 4 v 812 3341
a FJ(8)852 3315 y FD(2)879 3341 y FJ(E)548 3401 y FH(A)p
FK([)p FH(B)t(=X)6 b FK(])1303 3150 y(con)n(v)n(erts)28
b(to)2132 3145 y FJ(\001)2132 3182 y(\001)2132 3220 y(\001)2020
3282 y FH(A)p FK([)p FH(B)t(=X)6 b FK(])p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
94 97 bop 167 874 a FF(Chapter)28 b(12)167 1168 y(Coherence)g(Seman)-5
b(tics)28 b(of)g(the)f(Sum)167 1487 y FK(Here)i(w)n(e)h(consider)g(the)
h(denotational)g(seman)n(tics)e(of)i Fq(Emp)g FK(and)g(+)f(\(corresp)r
(onding)i(to)e FJ(?)167 1573 y FK(and)f FJ(_)p FK(\))e(in)n(tro)r
(duced)h(in)f(c)n(hapter)g FG(??)q FK(.)270 1746 y Fq(Emp)i
FK(is)g(naturally)g(in)n(terpreted)e(as)j(the)f(coherence)f(space)h
FJ(E)-17 b FM(mp)34 b FK(whose)c(w)n(eb)f(is)f(empt)n(y)-6
b(,)167 1831 y(and)29 b(the)e(in)n(terpretation)f(of)j
FH(")988 1842 y Fw(U)1069 1831 y FK(follo)n(ws)e(immediately)1673
1805 y FD(1)1697 1831 y FK(.)270 2004 y(The)k(sum,)i(on)f(the)e(other)i
(hand,)h(p)r(oses)f(some)f(delicate)e(problems.)46 b(When)31
b FJ(A)g FK(and)i FJ(B)167 2089 y FK(are)i(t)n(w)n(o)h(coherence)e
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(namely)c(the)167 2174 y FM(dir)l(e)l(ct)25 b(sum)31
b FK(in)n(tro)r(duced)24 b(b)r(elo)n(w.)34 b(Unfortunately)-6
b(,)26 b(the)e FH(\016)j FK(sc)n(heme)d(is)g(not)h(in)n(terpreted.)33
b(This)167 2259 y(ob)t(jection)27 b(also)h(holds)g(for)g(other)g(kinds)
g(of)g(seman)n(tics,)f(for)h(example)f(Scott)g(domains.)270
2385 y(After)39 b(examining)f(and)i(rejecting)e(a)i(certain)e(n)n(um)n
(b)r(er)h(of)g(fudged)h(alternativ)n(es,)i(w)n(e)167
2471 y(are)c(led)g(bac)n(k)h(to)g(the)f(original)f(solution,)k(whic)n
(h)d(w)n(ould)h(w)n(ork)f(with)g FM(line)l(ar)46 b FK(functions)167
2556 y(\()p FM(i.e.)28 b FK(preserving)f(unions\),)h(and)h(w)n(e)e
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(as:)1291 2729 y(!)p FJ(A)16 b(\010)g FK(!)p FJ(B)167
2902 y FK(It)23 b(is)f(this)g(decomp)r(osition)h(whic)n(h)f(is)h(the)f
(origin)h(of)g(linear)f(logic:)32 b(the)23 b(op)r(erations)g
FJ(\010)g FK(\(direct)167 2987 y(sum\))k(and)i(!)16 b
(\(linearisation\))26 b(are)h(in)g(fact)h(logical)e(op)r(erations)i(in)
f(their)g(o)n(wn)h(righ)n(t.)p 167 3578 1003 3 v 246
3622 a FC(1)273 3643 y FB(The)g(reader)g(familiar)e(with)h(category)i
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b FB(an)28 b(initial)d(ob)s(ject.)167 3713 y(This)h(is)g(to)h(b)r(e)e
(exp)r(ected)h(in)f(an)n(y)i(reasonable)g(category)g(of)g(domains,)f(b)
r(ecause)g(there)g(can)h(b)r(e)f(no)g(initial)167 3784
y(ob)s(ject)33 b(in)f(a)h(non-degenerate)f(Cartesian)g(closed)h
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167 3854 y(will)f(b)r(e)h(if)g(there)h(are)g(\014xp)r(oin)n(ts\).)52
b(With)33 b(linear)f(logic,)j(the)f(problem)e(v)m(anishes)h(b)r(ecause)
h(w)n(e)h(do)e(not)167 3925 y(require)27 b(a)h FA(Cartesian)f
FB(closed)h(category)-5 b(.)1385 4101 y FK(94)p eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
95 98 bop 167 167 a Fz(12.1.)65 b(DIRECT)26 b(SUM)1776
b FK(95)167 404 y FE(12.1)114 b(Direct)27 b(sum)167 560
y FK(The)50 b(problem)f(with)h(sum)g(t)n(yp)r(es)h(arises)e(from)h(the)
h(imp)r(ossibilit)n(y)c(of)k(de\014ning)f(the)167 645
y(in)n(terpretation)26 b(b)n(y)j(means)e(of)i(the)e(direct)f(sum:)743
863 y FJ(jA)15 b(\010)g(B)s(j)k FK(=)g FJ(jAj)c FK(+)h
FJ(jB)r(j)j FK(=)g FJ(f)p FK(1)p FJ(g)d(\002)g(jAj)f([)g(f)p
FK(2)p FJ(g)i(\002)e(jB)r(j)631 1109 y FK(\(1)p FH(;)d(\013)p
FK(\))813 1087 y FH(_)813 1129 y(^)902 1109 y FK(\(1)p
FH(;)g(\013)1038 1083 y Fv(0)1055 1109 y FK(\))47 b(\(mo)r(d)23
b FJ(A)16 b(\010)g(B)r FK(\))86 b(if)27 b FH(\013)1747
1087 y(_)1747 1129 y(^)1835 1109 y(\013)1879 1083 y Fv(0)1943
1109 y FK(\(mo)r(d)d FJ(A)p FK(\))632 1279 y(\(2)p FH(;)12
b(\014)t FK(\))813 1257 y FH(_)813 1300 y(^)902 1279
y FK(\(2)p FH(;)g(\014)1037 1253 y Fv(0)1053 1279 y FK(\))47
b(\(mo)r(d)24 b FJ(A)16 b(\010)f(B)r FK(\))88 b(if)27
b FH(\014)1745 1257 y(_)1745 1300 y(^)1834 1279 y(\014)1877
1253 y Fv(0)1940 1279 y FK(\(mo)r(d)d FJ(B)r FK(\))167
1431 y(with)j(incoherence)f(otherwise.)270 1559 y(Domain-theoretically)
-6 b(,)37 b(this)e(amoun)n(ts)j(to)f(taking)g(the)f(disjoin)n(t)g
(union)h(with)e(the)h Fn(?)167 1644 y FK(elemen)n(t)25
b(iden)n(ti\014ed,)i(so)h(it)f(is)g(sometimes)e(called)h(an)j
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(\(stable\))g(functions)h FJ(I)-7 b FM(nj)1446 1793 y
FD(1)1503 1820 y FK(from)28 b FJ(A)h FK(to)g FJ(A)17
b(\010)e(B)32 b FK(and)e FJ(I)-7 b FM(nj)2308 1793 y
FD(2)2365 1820 y FK(from)29 b FJ(B)i FK(to)167 1905 y
FJ(A)16 b(\010)f(B)30 b FK(b)n(y)638 2081 y FJ(I)-7 b
FM(nj)740 2054 y FD(1)767 2081 y FK(\()p FH(a)p FK(\))20
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b(=)g FJ(f)p FK(2)p FJ(g)e(\002)e FH(b)167 2257 y FK(ev)n(ery)27
b(ob)t(ject)f(of)h(the)g(coherence)e(space)i FJ(A)15
b(\010)f(B)29 b FK(can)e(b)r(e)f(written)f FJ(I)-7 b
FM(nj)2064 2230 y FD(1)2092 2257 y FK(\()p FH(a)p FK(\))26
b(for)h(some)g FH(a)19 b FJ(2)h(A)167 2342 y FK(or)33
b FJ(I)-7 b FM(nj)363 2315 y FD(2)391 2342 y FK(\()p
FH(b)p FK(\))32 b(for)h(some)g FH(b)24 b FJ(2)f(B)r FK(.)51
b(This)32 b(expression)h(is)f(unique,)i(except)f(in)f(the)g(case)h(of)g
(the)167 2427 y(empt)n(y)26 b(set:)34 b Fn(?)19 b FK(=)h
FJ(I)-7 b FM(nj)768 2400 y FD(1)796 2427 y Fn(?)20 b
FK(=)f FJ(I)-7 b FM(nj)1046 2400 y FD(2)1073 2427 y Fn(?)p
FK(.)35 b(This)26 b(non-uniqueness)h(of)g(the)e(decomp)r(osition)h(mak)
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(casewise)616 2689 y FH(H)5 b FK(\()p FJ(I)-7 b FM(nj)807
2662 y FD(1)835 2689 y FK(\()p FH(a)p FK(\)\))19 b(=)g
FH(F)10 b FK(\()p FH(a)p FK(\))448 b FH(H)5 b FK(\()p
FJ(I)-7 b FM(nj)1828 2662 y FD(2)1856 2689 y FK(\()p
FH(b)p FK(\)\))19 b(=)h FH(G)p FK(\()p FH(b)p FK(\))167
2865 y(from)31 b(t)n(w)n(o)h(stable)e(functions)i FH(F)40
b FK(from)31 b FJ(A)h FK(to)f FJ(C)36 b FK(and)c FH(G)g
FK(from)f FJ(B)i FK(to)f FJ(C)t FK(.)46 b(Indeed)31 b(this)g(fails)167
2950 y(for)d(the)f(argumen)n(t)h Fn(?)p FK(,)g(since)e
FH(F)10 b FK(\()p Fn(?)p FK(\))27 b(has)h(no)h(reason)f(to)g(b)r(e)g
(equal)f(to)h FH(G)p FK(\()p Fn(?)p FK(\).)167 3187 y
FE(12.2)114 b(Lifted)28 b(sum)167 3343 y FK(A)k(\014rst)f(solution)h
(is)f(giv)n(en)h(b)n(y)h(adding)g(t)n(w)n(o)f FM(tags)g
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FH(;)12 b(\014)t FK(\))167 3513 y(but)28 b(not)g(with)f(the)g(\(1)p
FH(;)12 b(\013)p FK(\).)270 3640 y(W)-6 b(e)28 b(can)g(then)g
(de\014ne:)553 3817 y FJ(q)606 3787 y FD(1)634 3817 y
FK(\()p FH(a)p FK(\))20 b(=)f FJ(f)p FK(1)p FJ(g)d([)g(I)-7
b FM(nj)1102 3789 y FD(1)1130 3817 y FK(\()p FH(a)p FK(\))386
b FJ(q)1659 3787 y FD(2)1687 3817 y FK(\()p FH(b)p FK(\))19
b(=)g FJ(f)p FK(2)p FJ(g)e([)e(I)-7 b FM(nj)2147 3789
y FD(2)2175 3817 y FK(\()p FH(b)p FK(\))p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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 closepath}put initclip
PStoPSxform concat
96 99 bop 167 167 a FK(96)537 b Fz(CHAPTER)27 b(12.)64
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e(de\014nition)h(is)g(p)r(ossible:)648 597 y FH(H)5 b
FK(\()p FJ(q)791 567 y FD(1)818 597 y FK(\()p FH(a)p
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FH(H)5 b FK(\()p FJ(q)1795 567 y FD(2)1823 597 y FK(\()p
FH(b)p FK(\)\))20 b(=)f FH(G)p FK(\()p FH(b)p FK(\))1025
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FK(\))25 b(and)j(ask)f(whether)e FH(\015)e FJ(2)d FH(G)p
FK(\()p FH(a)p FK(\).)270 1202 y(This)28 b(solution)f(in)n(terprets)f
(the)i(standard)h(con)n(v)n(ersion)f(sc)n(hemes:)504
1394 y FH(\016)22 b(x:)12 b(u)20 b(y)s(:)12 b(v)21 b
FK(\()p FH(\023)859 1368 y FD(1)887 1394 y FH(r)r FK(\))f
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1368 y FD(2)1961 1394 y FH(s)p FK(\))f Fu( )g FH(v)s
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b FK(\()p FH(\023)1656 1560 y FD(1)1684 1586 y FH(x)p
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FJ(q)1063 1749 y FD(1)1091 1778 y FK(\()p FH(a)p FK(\))464
b FH(H)5 b FK(\()p FJ(q)1787 1749 y FD(2)1815 1778 y
FK(\()p FH(b)p FK(\)\))19 b(=)g FJ(q)2070 1749 y FD(2)2098
1778 y FK(\()p FH(b)p FK(\))1025 1970 y FH(H)5 b FK(\()p
FH(c)p FK(\))19 b(=)h Fn(?)78 b FK(if)26 b FH(c)16 b
FJ(\\)f(f)p FK(1)p FH(;)d FK(2)q FJ(g)20 b FK(=)f Fn(?)167
2163 y FK(do)r(es)32 b(not)g(alw)n(a)n(ys)h(satisfy)f
FH(H)5 b FK(\()p FH(c)p FK(\))22 b(=)g FH(c)p FK(.)48
b(In)31 b(fact)h(this)f(equation)h(is)f(satis\014ed)h(only)g(for)g
FH(c)g FK(of)167 2248 y(the)27 b(form)h FJ(q)511 2218
y FD(1)539 2248 y FK(\()p FH(a)p FK(\),)f FJ(q)728 2218
y FD(2)756 2248 y FK(\()p FH(b)p FK(\))g(or)h Fn(?)p
FK(.)270 2383 y(On)36 b(the)g(other)g(hand,)k(the)c(comm)n(uting)f(con)
n(v)n(ersions)i(do)g(hold:)53 b(let)34 b FH(t)26 b FJ(7!)g
Fq(E)12 b FH(t)35 b FK(b)r(e)h(an)167 2468 y(elimination)31
b(of)j(the)g(form)f FH(\031)972 2443 y FD(1)1000 2468
y FH(t)p FK(,)i(or)f FH(\031)1216 2443 y FD(2)1244 2468
y FH(t)p FK(,)h(or)f FH(t)12 b(w)r FK(,)36 b(or)e FH(")1688
2479 y Fw(U)1742 2468 y FH(t)p FK(,)h(or)f FH(\016)26
b(x)2012 2443 y Fv(0)2029 2468 y FH(:)12 b(u)2099 2443
y Fv(0)2139 2468 y FH(y)2176 2443 y Fv(0)2192 2468 y
FH(:)g(v)2259 2443 y Fv(0)2299 2468 y FH(t)p FK(.)54
b(W)-6 b(e)34 b(w)n(an)n(t)167 2554 y(to)28 b(c)n(hec)n(k)f(that)i
Fq(E)12 b FK(\()p FH(\016)21 b(x:)12 b(u)20 b(y)s(:)12
b(v)22 b(t)p FK(\))27 b(and)i FH(\016)22 b(x:)12 b FK(\()p
Fq(E)g FH(u)p FK(\))19 b FH(y)s(:)12 b FK(\()p Fq(E)g
FH(v)s FK(\))18 b FH(t)28 b FK(ha)n(v)n(e)g(the)g(same)f(in)n
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b([)q Fq(E)12 b FH(u)p FK(])-12 b(])q(\()p FH(a)p FK(\).)49
b(In)167 2724 y(the)37 b(case)f(where)g FH(c)18 b FJ(\\)h(f)p
FK(1)p FH(;)12 b FK(2)q FJ(g)26 b FK(=)g Fn(?)p FK(,)39
b(w)n(e)d(get)h(on)h(the)e(one)h(hand)i FH(E)t FK(\()p
Fn(?)p FK(\))d(where)g FH(E)41 b FK(is)36 b(the)167 2809
y(stable)26 b(function)h(corresp)r(onding)h(to)f Fq(E)p
FK(,)g(and)h(on)g(the)e(other)h Fn(?)p FK(;)g(but)g(it)f(is)g(easy)h
(to)g(see)g(that)167 2894 y FH(E)t FK(\()p Fn(?)p FK(\))19
b(=)g Fn(?)28 b FK(\()p FH(E)j FK(is)c FM(strict)p FK(\))h(in)f(all)f
(the)h(cases)h(in)f(question.)270 3030 y(Ha)n(ving)35
b(said)g(this,)g(the)f(presence)f(of)i(an)g(equation)g(\(ho)n(w)n(ev)n
(er)f(minor\))f(whic)n(h)h(is)g(not)167 3115 y(in)n(terpreted)d(means)i
(w)n(e)g(m)n(ust)f(reject)g(the)g(seman)n(tics.)51 b(Ev)n(en)33
b(if)f(w)n(e)g(are)h(unsure)g(ho)n(w)h(to)167 3200 y(use)28
b(it,)e(the)i(equation)1076 3392 y FH(\016)22 b(x:)12
b FK(\()p FH(\023)1250 3367 y FD(1)1278 3392 y FH(x)p
FK(\))20 b FH(y)s(:)12 b FK(\()p FH(\023)1483 3367 y
FD(2)1510 3392 y FH(y)s FK(\))19 b FH(t)g FK(=)g FH(t)167
3584 y FK(pla)n(ys)26 b(a)f(part)h(in)f(the)g(implicit)c(symmetries)i
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3669 y(lo)r(oking)33 b(for)h(a)f(mo)r(del)g(at)g(an)n(y)h(price,)g(but)
f(for)g(a)h(con)n(vincing)f(one.)52 b(F)-6 b(or)34 b(that,)h(ev)n(en)e
(the)167 3754 y(secondary)24 b(connectors)f(\(suc)n(h)g(as)g
FJ(_)p FK(\))f(and)i(the)e(marginal)g(equations)h(are)g(precious,)g(b)r
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b(By)27 b(trying)g(to)167 3925 y(analyse)h(this)f(discord,)h(one)g(can)
g(hop)r(e)g(to)g(\014nd)h(some)e(prop)r(erties)g(hidden)h(in)f(the)g
(syn)n(tax.)p eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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 closepath}put initclip
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PStoPSxform concat
97 100 bop 167 167 a Fz(12.2.)65 b(LIFTED)27 b(SUM)1788
b FK(97)167 404 y Fy(12.2.1)94 b(dI-domains)167 536 y
FK(There)29 b(is)g(a)h(simple)d(solution,)j(but)g(it)f(requires)g(the)g
(abandonmen)n(t)j(of)e(coherence)e(spaces:)167 621 y(let)39
b(us)h(simply)e(sa)n(y)k(that)e(in)f FJ(A)20 b(q)f(B)r
FK(,)43 b(w)n(e)c(only)i(consider)e(suc)n(h)i(ob)t(jects)f(as)g
FJ(q)2436 592 y FD(1)2464 621 y FH(a)p FK(,)j FJ(q)2615
592 y FD(2)2643 621 y FH(b)167 706 y FK(and)32 b Fn(?)p
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b(but)167 792 y(the)28 b(structure)f(so)i(obtained)f(is)f(no)i(longer)f
(a)g(coherence)g(space:)36 b(indeed,)27 b(if)h FH(\013)20
b FJ(2)f(jAj)p FK(,)28 b(then)167 877 y FJ(q)220 847
y FD(1)248 877 y FH(\013)20 b FK(=)f FJ(f)p FK(1)p FH(;)12
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FJ(A)16 b(q)g(B)r FK(,)28 b(but)g(not)g(its)f(subset)h
FJ(f)p FK(\(1)p FH(;)12 b(\013)p FK(\))q FJ(g)p FK(.)270
1005 y(In)33 b(fact,)g(w)n(e)e(see)h(that)g(it)f(is)g(necessary)h(to)h
(add)g(to)f(the)g(idea)f(of)i FM(c)l(oher)l(enc)l(e)k
FK(a)32 b FM(p)l(artial)167 1090 y(or)l(der)g(r)l(elation)5
b FK(,)33 b(here)e(1)23 b FH(<)f FK(\(1)p FH(;)12 b(\013)p
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b(W)-6 b(e)31 b(are)g(in)n(terested)f(in)h(coheren)n(t)g(subsets)167
1175 y(of)h(the)f(space)g(whic)n(h)g(are)g FM(downwar)l(ds-close)l(d)7
b FK(:)43 b(if)31 b FH(\013)1612 1150 y Fv(0)1650 1175
y FH(<)22 b(\013)h FJ(2)f FH(a)p FK(,)32 b(then)f FH(\013)2158
1150 y Fv(0)2197 1175 y FJ(2)22 b FH(a)p FK(.)47 b(According)167
1260 y(to)28 b([)p FG(?)p FK(],)g(the)f(tok)n(ens)i(should)f(b)r(e)f
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f(when)167 1346 y(t)n(w)n(o)k(ev)n(en)n(ts)h FM(may)37
b FK(co-exist)30 b(and)h(the)f(partial)f(order)h FH(\013)1668
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b(is)f(called)f(an)j FM(event)d(structur)l(e)p FK(;)167
1516 y([)p FG(?)q FK(])h(c)n(haracterises)g(the)g(resulting)g(spaces,)h
(whic)n(h)f(are)h(exactly)f([)p FG(?)p FK(]'s)h(original)f
FM(dI-domains)p FK(.)270 1644 y(As)35 b(an)g(example,)g(one)f(can)h
(re-de\014ne)f(the)g FM(lazy)h(natur)l(al)f(numb)l(ers)p
FK(,)i FJ(I)-7 b FM(nt)2288 1619 y FD(+)2330 1644 y FK(,)36
b(whic)n(h)e(w)n(e)167 1729 y(met)27 b(in)h(section)f
FG(??)q FK(.)38 b(Clearly)26 b(w)n(e)i(w)n(an)n(t)h FH(p)1323
1704 y FD(+)1384 1729 y FH(<)20 b(q)30 b FK(and)g FH(p)1694
1704 y FD(+)1755 1729 y FH(<)20 b(q)1863 1704 y FD(+)1932
1729 y FK(for)29 b FH(p)19 b(<)h(q)s FK(;)28 b(one)g(ma)n(y)h(then)167
1814 y(sho)n(w)34 b(that)f(the)g(p)r(oin)n(ts)h(of)f(the)g(corresp)r
(onding)h(dI-domain)g FJ(I)-7 b FM(nt)1966 1789 y Fw(<)2041
1814 y FK(are)33 b(just)g(the)j Fo(e)-42 b FH(p)p FK(,)31
b(\027)-49 b FH(p)p FK(,)34 b Fn(?)167 1900 y FK(and)29
b Fo(f)-70 b FJ(1)p FK(.)36 b(The)27 b(three)g(spaces)h(satisfy)g(the)f
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b FJ(')19 b(S)-6 b FM(gl)21 b FJ(\010)16 b(I)-7 b FM(nt)163
b FJ(I)-7 b FM(nt)1060 2052 y FD(+)1121 2077 y FJ(')19
b(S)-6 b FM(gl)22 b FJ(\010)15 b FK(\()p FJ(S)-6 b FM(gl)21
b FK(&)16 b FJ(I)-7 b FM(nt)1679 2052 y FD(+)1721 2077
y FK(\))156 b FJ(I)-7 b FM(nt)2004 2052 y Fw(<)2065 2077
y FJ(')20 b(E)-17 b FM(mp)20 b FJ(q)c(I)-7 b FM(nt)2440
2052 y Fw(<)167 2255 y FK(where)30 b FJ(S)-6 b FM(gl)36
b FK(is)30 b(the)g(coherence)f(space)i(with)f(just)g(one)h(tok)n(en)g
(\(section)f FG(??)q FK(\).)44 b(This)29 b(ma)n(y)i(b)r(e)167
2340 y(used)d(as)g(an)h(alternativ)n(e)d(w)n(a)n(y)j(of)f(de\014ning)g
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(b)n(y)h(this)e(in)n(terpretation)g(is)g(limited,)f(b)r(ecause)i(one)g
(can)g(require)167 2554 y(that)21 b(for)h(all)e FH(\013)f
FJ(2)h(jAj)p FK(,)i(the)f(set)f(of)i FH(\013)1122 2528
y Fv(0)1158 2554 y FH(<)d(\013)j FK(b)r(e)e(\014nite,)i(whic)n(h)f
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b(a)h(\014nite)e(set)g(is)g(alw)n(a)n(ys)i(\014nite,)e(and)i(so)f(w)n
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2724 y(Scott)f(domains.)270 2902 y(Seman)n(tically)-6
b(,)34 b(there)f(is)g(nothing)i(else)e(to)h(quarrel)f(with)g(ab)r(out)i
(this)e(in)n(terpretation,)167 2987 y(whic)n(h)28 b(accoun)n(ts)i(for)g
(all)d(reasonable)j(constructions.)40 b(But)29 b(on)h(the)e(other)h
(hand,)i(it)d(forces)167 3072 y(us)44 b(to)g(lea)n(v)n(e)g(the)f(class)
h(of)g(coherence)f(spaces,)49 b(and)44 b(uses)h(an)f(order)g(relation)f
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d(of)29 b(the)e(system.)270 3286 y(This)j(leads)g(us)h(to)g(lo)r(ok)f
(for)h(something)f(else,)g(whic)n(h)g(do)r(es)h(preserv)n(e)e(this)h
(class.)43 b(The)167 3371 y(price)c(will)f(b)r(e)i(a)h(more)f
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f(are)167 3456 y(basically)26 b(only)h(in)n(terested)e(in)h(the)g(sum)h
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35 b FM(line)l(arity)p FK(.)270 3669 y(The)29 b(in)n(terpretation)e(w)n
(e)g(shall)h(giv)n(e)g(is)g(manifestly)f(not)i(asso)r(ciativ)n(e.)37
b(It)28 b(is)g(in)n(teresting)167 3754 y(to)44 b(remark)g(that)g(Winsk)
n(el's)f(in)n(terpretation)g(is)g(not)i(either:)66 b(indeed,)48
b(if)43 b FJ(A)p FH(;)12 b FJ(B)r FH(;)g FJ(C)48 b FK(are)167
3840 y(coherence)23 b(spaces)i(considered)f(as)g(ev)n(en)n(t)h
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3925 y(\()p FJ(A)16 b(q)f(B)r FK(\))h FJ(q)g(C)31 b FK(and)e
FJ(A)16 b(q)f FK(\()p FJ(B)j(q)e(C)t FK(\))27 b(are)h(not)g(the)f
(same:)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
98 101 bop 167 167 a FK(98)537 b Fz(CHAPTER)27 b(12.)64
b(COHERENCE)25 b(SEMANTICS)j(OF)f(THE)g(SUM)276 435 y
FK(\(1)p FH(;)12 b FK(\(1)p FH(;)g(\013)p FK(\)\))53
b(\(1)p FH(;)12 b FK(\(2)p FH(;)g(\014)t FK(\)\))341
686 y(\(1)p FH(;)g FK(1\))181 b(\(1)p FH(;)12 b FK(2\))178
b(\(2)p FH(;)12 b(\015)t FK(\))567 941 y(1)467 b(2)p
416 585 3 84 v 750 585 V 1084 835 V 492 835 a Fs(@)468
810 y(@)635 835 y(\000)659 810 y(\000)651 1103 y FK(\()p
FJ(A)16 b(q)f(B)r FK(\))h FJ(q)g(C)2283 435 y FK(\(2)p
FH(;)c FK(\(2)p FH(;)g(\015)t FK(\)\))-614 b(\(2)p FH(;)12
b FK(\(1)p FH(;)g(\014)t FK(\)\))2345 686 y(\(2)p FH(;)g
FK(2\))-487 b(\(2)p FH(;)12 b FK(1\))-492 b(\(1)p FH(;)12
b(\013)p FK(\))2237 941 y(2)-535 b(1)p 2420 585 V 2086
585 V 1752 835 V 2288 835 a Fs(\000)2313 810 y(\000)2146
835 y(@)2121 810 y(@)1820 1103 y FJ(A)16 b(q)g FK(\()p
FJ(B)i(q)d(C)t FK(\))167 1412 y FE(12.3)114 b(Linearit)m(y)167
1571 y FK(W)-6 b(e)32 b(ha)n(v)n(e)h(already)f(remark)n(ed)g(that)g
(the)g(op)r(eration)g FH(t)23 b FJ(7!)g FH(t)12 b(u)31
b FK(is)h(strict,)f FM(i.e.)h FK(preserv)n(es)g Fn(?)p
FK(.)167 1656 y(Better)c(than)i(this)f(it)g(is)g FM(line)l(ar)p
FK(.)40 b(Let)29 b(us)h(lo)r(ok)g(no)n(w)g(at)g(what)g(that)g(can)g
(mean.)41 b(Let)29 b FH(E)k FK(b)r(e)167 1741 y(the)27
b(function)h(from)g FJ(A)15 b(!)i(B)30 b FK(to)d FJ(B)j
FK(de\014ned)f(b)n(y)699 1927 y FH(E)t FK(\()p FH(f)8
b FK(\))19 b(=)g FH(f)8 b FK(\()p FH(a)p FK(\))78 b(where)27
b FH(a)h FK(is)f(a)h(giv)n(en)f(ob)t(ject)h(of)g FJ(A)p
FK(.)270 2159 y(Let)35 b(us)h(w)n(ork)g(out)f FJ(T)-6
b FH(r)r FK(\()p FH(E)t FK(\):)51 b(w)n(e)35 b(ha)n(v)n(e)h(to)g
(\014nd)g(all)e(the)h FH(\014)28 b FJ(2)d FH(E)t FK(\()p
FH(f)8 b FK(\))35 b(with)f FH(f)43 b FK(minimal.)167
2244 y(No)n(w)31 b FH(\014)26 b FJ(2)c FH(E)t FK(\()p
FH(f)8 b FK(\))22 b(=)f FH(f)8 b FK(\()p FH(a)p FK(\))31
b(i\013)f(there)h(exists)f(some)h FH(a)1567 2255 y Fv(\016)1618
2244 y FJ(\032)21 b FH(a)32 b FK(suc)n(h)g(that)f(\()p
FH(a)2146 2255 y Fv(\016)2174 2244 y FH(;)12 b(\014)t
FK(\))22 b FJ(2)g FH(f)8 b FK(.)45 b(So)32 b(the)167
2330 y(minimal)c FH(f)39 b FK(are)31 b(the)h(singletons)f
FJ(f)p FK(\()p FH(a)1192 2341 y Fv(\016)1220 2330 y FH(;)12
b(\014)t FK(\))o FJ(g)32 b FK(with)e FH(a)1588 2341 y
Fv(\016)1638 2330 y FJ(\032)22 b FH(a)p FK(,)33 b FH(a)1839
2341 y Fv(\016)1898 2330 y FK(\014nite,)f(and)g(the)f(ob)t(jects)h(of)
167 2415 y FJ(T)-6 b FH(r)r FK(\()p FH(E)t FK(\))28 b(are)g(of)g(the)f
(form)701 2601 y(\()p FJ(f)p FK(\()p FH(a)826 2612 y
Fv(\016)854 2601 y FH(;)12 b(\014)t FK(\))p FJ(g)p FH(;)24
b(\014)t FK(\))77 b(with)27 b FH(\014)d FJ(2)19 b(jB)r(j)p
FK(,)27 b FH(a)1641 2612 y Fv(\016)1689 2601 y FJ(\032)19
b FH(a)p FK(,)28 b FH(a)1882 2612 y Fv(\016)1938 2601
y FK(\014nite.)270 2833 y(A)34 b(stable)f(function)h
FH(F)43 b FK(from)34 b FJ(A)g FK(to)g FJ(B)j FK(is)c
FM(line)l(ar)g FK(precisely)f(when)i FJ(T)-6 b FH(r)r
FK(\()p FH(F)10 b FK(\))34 b(consists)g(of)167 2918 y(pairs)28
b(\()p FJ(f)p FH(\013)p FJ(g)p FH(;)12 b(\014)t FK(\))27
b(with)g FH(\013)20 b FJ(2)f(jAj)27 b FK(and)i FH(\014)23
b FJ(2)d(jB)r(j)p FK(.)167 3131 y Fy(12.3.1)94 b(Characterisation)28
b(in)g(terms)e(of)h(preserv)-5 b(ation)167 3265 y FK(Let)27
b(us)h(lo)r(ok)g(at)g(some)g(of)g(the)f(prop)r(erties)g(of)h(linear)e
(functions.)201 3435 y(i\))51 b FH(F)10 b FK(\()p Fn(?)p
FK(\))35 b(=)h Fn(?)p FK(.)105 b(Indeed,)58 b(to)51 b(ha)n(v)n(e)h
FH(\014)41 b FJ(2)36 b FH(F)10 b FK(\()p Fn(?)p FK(\),)56
b(w)n(e)50 b(need)i FH(a)2101 3446 y Fv(\016)2165 3435
y FJ(\032)36 b Fn(?)51 b FK(suc)n(h)h(that)339 3520 y(\()p
FH(a)402 3531 y Fv(\016)430 3520 y FH(;)12 b(\014)t FK(\))19
b FJ(2)h(T)-6 b FH(r)r FK(\()p FH(F)10 b FK(\);)27 b(but)h
FH(a)1004 3531 y Fv(\016)1052 3520 y FK(=)19 b Fn(?)28
b FK(and)g(so)h(cannot)f(b)r(e)g(a)g(singleton.)201 3669
y(ii\))81 b(If)i FH(a)514 3680 y FD(1)573 3669 y FJ([)31
b FH(a)687 3680 y FD(2)774 3669 y FJ(2)58 b(A)p FK(,)97
b(then)82 b FH(F)10 b FK(\()p FH(a)1385 3680 y FD(1)1444
3669 y FJ([)31 b FH(a)1558 3680 y FD(2)1586 3669 y FK(\))58
b(=)g FH(F)10 b FK(\()p FH(a)1901 3680 y FD(1)1928 3669
y FK(\))31 b FJ([)h FH(F)10 b FK(\()p FH(a)2183 3680
y FD(2)2210 3669 y FK(\).)200 b(Clearly)339 3754 y FH(F)10
b FK(\()p FH(a)457 3765 y FD(1)485 3754 y FK(\))19 b
FJ([)h FH(F)10 b FK(\()p FH(a)716 3765 y FD(2)743 3754
y FK(\))30 b FJ(\032)g FH(F)10 b FK(\()p FH(a)1003 3765
y FD(1)1050 3754 y FJ([)20 b FH(a)1153 3765 y FD(2)1181
3754 y FK(\).)81 b(Con)n(v)n(ersely)-6 b(,)47 b(if)41
b FH(\014)34 b FJ(2)d FH(F)10 b FK(\()p FH(a)2045 3765
y FD(1)2092 3754 y FJ([)20 b FH(a)2195 3765 y FD(2)2223
3754 y FK(\),)46 b(that)d(means)339 3840 y(there)27 b(is)g(some)g
FH(a)807 3814 y Fv(0)843 3840 y FJ(\032)19 b FH(a)953
3851 y FD(1)996 3840 y FJ([)c FH(a)1094 3851 y FD(2)1150
3840 y FK(suc)n(h)28 b(that)f(\()p FH(a)1526 3814 y Fv(0)1543
3840 y FH(;)12 b(\014)t FK(\))19 b FJ(2)g(T)-6 b FH(r)r
FK(\()p FH(F)10 b FK(\);)27 b(but)h FH(a)2116 3814 y
Fv(0)2160 3840 y FK(is)f(a)g(singleton,)g(so)339 3925
y FH(a)375 3899 y Fv(0)412 3925 y FJ(\032)19 b FH(a)522
3936 y FD(1)550 3925 y FK(,)28 b(in)f(whic)n(h)g(case)g
FH(\014)d FJ(2)19 b FH(F)10 b FK(\()p FH(a)1281 3936
y FD(1)1308 3925 y FK(\),)28 b(or)g FH(a)1507 3899 y
Fv(0)1543 3925 y FJ(\032)20 b FH(a)1654 3936 y FD(2)1682
3925 y FK(,)27 b(in)g(whic)n(h)g(case)h FH(\014)23 b
FJ(2)d FH(F)10 b FK(\()p FH(a)2413 3936 y FD(2)2440 3925
y FK(\).)p eop
PStoPSsaved restore
%%Page: (102,103) 52
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
99 102 bop 167 167 a Fz(12.3.)65 b(LINEARITY)1834 b FK(99)270
404 y(These)39 b(prop)r(erties)f(c)n(haracterise)f(the)i(stable)f
(functions)h(whic)n(h)f(are)h(linear;)k(indeed,)167 490
y(if)27 b FH(\014)c FJ(2)d FH(F)10 b FK(\()p FH(a)p FK(\))26
b(with)h FH(a)h FK(minimal,)d FH(a)j FK(m)n(ust)f(b)r(e)h(a)g
(singleton:)201 652 y(i\))f FH(F)10 b FK(\()p Fn(?)p
FK(\))18 b(=)h Fn(?)p FK(,)28 b(so)g FH(a)20 b FJ(6)p
FK(=)f Fn(?)p FK(.)201 797 y(ii\))27 b(if)h FH(a)21 b
FK(=)f FH(a)531 771 y Fv(0)564 797 y FJ([)15 b FH(a)662
771 y Fv(00)693 797 y FK(,)29 b(then)g FH(F)10 b FK(\()p
FH(a)p FK(\))19 b(=)h FH(F)10 b FK(\()p FH(a)1260 771
y Fv(0)1276 797 y FK(\))16 b FJ([)g FH(F)10 b FK(\()p
FH(a)1500 771 y Fv(00)1529 797 y FK(\),)29 b(so)g FH(\014)24
b FJ(2)d FH(F)10 b FK(\()p FH(a)1943 771 y Fv(0)1958
797 y FK(\))29 b(or)g FH(\014)24 b FJ(2)d FH(F)10 b FK(\()p
FH(a)2353 771 y Fv(00)2382 797 y FK(\);)29 b(so,)h(if)e
FH(a)339 882 y FK(is)d(not)h(a)f(singleton,)h(w)n(e)e(can)i(\014nd)g(a)
g(decomp)r(osition)e FH(a)c FK(=)f FH(a)1962 856 y Fv(0)1991
882 y FJ([)12 b FH(a)2086 856 y Fv(00)2141 882 y FK(whic)n(h)24
b(con)n(tradicts)339 967 y(the)k(minimalit)n(y)c(of)k
FH(a)p FK(.)270 1171 y(Prop)r(erties)f(\(i\))f(and)j(\(ii\))c(com)n
(bine)i(with)g(preserv)l(ation)h(of)g(\014ltered)e(unions)j(\()p
FG(Lin)p FK(\):)731 1334 y(if)e FH(A)19 b FJ(\032)h(A)p
FK(,)27 b(and)i(for)f(all)f FH(a)1433 1345 y FD(1)1461
1334 y FH(;)12 b(a)1528 1345 y FD(2)1575 1334 y FJ(2)20
b FH(A)p FK(,)27 b FH(a)1776 1345 y FD(1)1820 1334 y
FJ([)15 b FH(a)1918 1345 y FD(2)1966 1334 y FJ(2)20 b(A)p
FK(,)915 1419 y(then)27 b FH(F)10 b FK(\()1158 1367 y
Fo(S)1228 1419 y FH(A)p FK(\))19 b(=)1399 1367 y Fo(S)1458
1419 y FJ(f)p FH(F)10 b FK(\()p FH(a)p FK(\))19 b(:)39
b FH(a)20 b FJ(2)f FH(A)p FJ(g)167 1582 y FK(Observ)n(e)34
b(that)g(this)f(condition)h(is)f(in)g(the)h(spirit)e(of)i(coherence)f
(spaces,)j(whic)n(h)e(m)n(ust)f(b)r(e)167 1667 y(closed)h(under)h
(pairwise-b)r(ounded)g(unions.)57 b(So)35 b(w)n(e)f(can)h(de\014ne)g
FM(line)l(ar)g(stable)e(functions)167 1752 y FK(from)27
b FJ(A)h FK(to)g FJ(B)i FK(b)n(y)f(\()p FG(Lin)p FK(\))c(and)k(\()p
FG(St)p FK(\):)702 1915 y(if)e FH(a)806 1926 y FD(1)850
1915 y FJ([)15 b FH(a)948 1926 y FD(2)996 1915 y FJ(2)k(A)28
b FK(then)g FH(F)10 b FK(\()p FH(a)1426 1926 y FD(1)1469
1915 y FJ(\\)15 b FH(a)1567 1926 y FD(2)1595 1915 y FK(\))20
b(=)f FH(F)10 b FK(\()p FH(a)1833 1926 y FD(1)1860 1915
y FK(\))16 b FJ(\\)f FH(F)10 b FK(\()p FH(a)2083 1926
y FD(2)2110 1915 y FK(\))167 2078 y(the)27 b(monotonicit)n(y)h(of)g
FH(F)37 b FK(b)r(eing)27 b(a)h(consequence)g(of)g(\()p
FG(Lin)p FK(\).)167 2283 y Fy(12.3.2)94 b(Linear)29 b(implication)167
2414 y FK(W)-6 b(e)37 b(stra)n(y)n(ed)h(from)f(the)h(trace)e(to)i(giv)n
(e)f(a)h(c)n(haracterisation)f(in)f(terms)h(of)h(preserv)l(ation.)167
2499 y(Returning)25 b(to)h(it,)f(if)g(w)n(e)g(kno)n(w)h(that)g
FH(F)35 b FK(is)25 b(linear,)f(w)n(e)h(can)h(discard)g(the)f(singleton)
g(sym)n(b)r(ols)167 2585 y(in)i FJ(T)-6 b FH(r)r FK(\()p
FH(F)10 b FK(\):)933 2760 y FJ(T)-6 b FM(rlin)5 b FK(\()p
FH(F)10 b FK(\))19 b(=)g FJ(f)p FK(\()p FH(\013;)12 b(\014)t
FK(\))19 b(:)40 b FH(\014)23 b FJ(2)d FH(F)10 b FK(\()p
FH(\013)p FK(\))p FJ(g)270 2935 y FK(The)29 b(set)g(of)g(all)f(the)h
FJ(T)-6 b FM(rlin)5 b FK(\()p FH(F)10 b FK(\))28 b(as)h
FH(F)39 b FK(v)l(aries)28 b(o)n(v)n(er)i(stable)e(linear)g(functions)h
(from)g FJ(A)g FK(to)167 3020 y FJ(B)i FK(forms)e(a)h(coherence)e
(space)h FJ(A)16 b Fu(\()g FJ(B)32 b FK(\()p FM(line)l(ar)c(implic)l
(ation)p FK(\),)i(with)e FJ(jA)16 b Fu(\()g FJ(B)r(j)k
FK(=)g FJ(jAj)c(\002)g(jB)r(j)167 3105 y FK(and)29 b(\()p
FH(\013;)12 b(\014)t FK(\))497 3083 y FH(_)497 3125 y(^)586
3105 y FK(\()p FH(\013)657 3079 y Fv(0)673 3105 y FH(;)g(\014)747
3079 y Fv(0)763 3105 y FK(\))47 b(\(mo)r(d)24 b FJ(A)15
b Fu(\()h FJ(B)s FK(\))27 b(if)201 3267 y(i\))g FH(\013)339
3245 y(_)339 3288 y(^)427 3267 y(\013)471 3242 y Fv(0)535
3267 y FK(\(mo)r(d)d FJ(A)p FK(\))19 b FJ(\))h FH(\014)973
3245 y(_)973 3288 y(^)1061 3267 y(\014)1104 3242 y Fv(0)1168
3267 y FK(\(mo)r(d)j FJ(B)r FK(\))201 3412 y(ii\))j FH(\014)357
3390 y(^)357 3433 y(_)445 3412 y(\014)488 3386 y Fv(0)551
3412 y FK(\(mo)r(d)e FJ(B)r FK(\))c FJ(\))g FH(\013)983
3390 y(^)983 3433 y(_)1071 3412 y(\013)1115 3386 y Fv(0)1179
3412 y FK(\(mo)r(d)j FJ(A)p FK(\))167 3575 y(in)k(whic)n(h)g(w)n(e)g
(in)n(tro)r(duce)g(the)h(abbreviation:)760 3750 y FH(\013)824
3728 y(^)824 3770 y(_)912 3750 y(\013)956 3724 y Fv(0)1020
3750 y FK(\(mo)r(d)c FJ(A)p FK(\))j(for)h FJ(:)p FK(\()p
FH(\013)1561 3728 y(_)1561 3770 y(^)1650 3750 y(\013)1694
3724 y Fv(0)1710 3750 y FK(\))g(or)g FH(\013)19 b FK(=)h
FH(\013)2035 3724 y Fv(0)167 3925 y FK(for)28 b FM(inc)l(oher)l(enc)l
(e)p FK(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
100 103 bop 167 167 a FK(100)503 b Fz(CHAPTER)27 b(12.)64
b(COHERENCE)25 b(SEMANTICS)j(OF)f(THE)g(SUM)270 404 y
FK(Immediately)61 b(w)n(e)i(can)h(see)e(the)h(essen)n(tial)f(prop)r
(ert)n(y)i(of)g(linear)e(implication:)167 490 y FM(antisymmetry)p
FK(.)83 b(If)42 b(w)n(e)h(de\014ne,)k(for)c(a)g(coherence)g(space)g
FJ(A)p FK(,)k(the)42 b(space)i FJ(A)2391 464 y Fv(?)2475
490 y FK(\()p FM(line)l(ar)167 575 y(ne)l(gation)p FK(\))27
b(b)n(y)1243 775 y FJ(jA)1319 750 y Fv(?)1361 775 y FJ(j)19
b FK(=)g FJ(jAj)803 976 y FH(\013)867 954 y(_)867 997
y(^)955 976 y(\013)999 950 y Fv(0)1063 976 y FK(\(mo)r(d)24
b FJ(A)1302 950 y Fv(?)1344 976 y FK(\))78 b(i\013)f
FH(\013)1650 954 y(^)1650 997 y(_)1739 976 y(\013)1783
950 y Fv(0)1846 976 y FK(\(mo)r(d)24 b FJ(A)p FK(\))167
1177 y(then)e(the)g(map)g(\()p FH(\013;)12 b(\014)t FK(\))19
b FJ(7!)h FK(\()p FH(\014)t(;)12 b(\013)p FK(\))22 b(is)f(an)i
(isomorphism)e(from)h FJ(A)7 b Fu(\()g FJ(B)25 b FK(to)d
FJ(B)2145 1151 y Fv(?)2194 1177 y Fu(\()7 b FJ(A)2335
1151 y Fv(?)2377 1177 y FK(.)34 b(In)22 b(other)167 1262
y(w)n(ords,)28 b(\()p FH(\013;)12 b(\014)t FK(\))580
1240 y FH(_)580 1282 y(^)668 1262 y FK(\()p FH(\013)739
1236 y Fv(0)756 1262 y FH(;)g(\014)830 1236 y Fv(0)846
1262 y FK(\))47 b(\(mo)r(d)23 b FJ(A)16 b Fu(\()g FJ(B)s
FK(\))27 b(i\013)g(\()p FH(\014)t(;)12 b(\013)p FK(\))1649
1240 y FH(_)1649 1282 y(^)1738 1262 y FK(\()p FH(\014)1808
1236 y Fv(0)1824 1262 y FH(;)g(\013)1899 1236 y Fv(0)1916
1262 y FK(\))46 b(\(mo)r(d)24 b FJ(B)2220 1236 y Fv(?)2278
1262 y Fu(\()16 b FJ(A)2428 1236 y Fv(?)2469 1262 y FK(\).)270
1402 y(What)38 b(is)f(the)g(meaning)h(of)f(this?)64 b(A)37
b(stable)g(function)h(tak)n(es)g(an)g(input)f(of)h FJ(A)g
FK(and)167 1487 y(returns)c(an)i(output)f(of)g FJ(B)r
FK(.)57 b(When)35 b(the)f(function)g(is)g(linear,)h(this)f(pro)r(cess)h
(can)g(b)r(e)f(seen)167 1572 y(dually)f(as)h(returning)f(an)i(input)e
(of)h FJ(A)g FK(\()p FM(i.e.)f FK(an)h(output)h(of)f
FJ(A)1917 1546 y Fv(?)1959 1572 y FK(\))f(from)g(an)h(output)h(of)f
FJ(B)167 1657 y FK(\()p FM(i.e.)28 b FK(an)h(input)f(of)h
FJ(B)739 1631 y Fv(?)781 1657 y FK(\).)38 b(So)29 b(the)f(linear)f
(implication)e(in)n(tro)r(duces)j(a)h(symmetrical)c(form)k(of)167
1742 y(functional)h(dep)r(endence,)h(the)f(dualit)n(y)f(of)i(r^)-34
b(oles)30 b(of)g(the)g(argumen)n(t)h(and)g(the)f(result)f(b)r(eing)167
1827 y(expressed)c(b)n(y)h(the)e FM(line)l(ar)h(ne)l(gation)30
b FJ(A)20 b(7!)g(A)1395 1802 y Fv(?)1437 1827 y FK(.)34
b(This)25 b(is)f(analogous)j(to)e(transp)r(osition)h(\(not)167
1912 y(in)n(v)n(ersion\))h(in)g(Linear)g(Algebra.)270
2052 y(T)-6 b(o)35 b(mak)n(e)f(this)f(relev)l(an)n(t,)i(w)n(e)e(ha)n(v)
n(e)i(to)f(sho)n(w)h(that)f(linearit)n(y)e(is)h(not)h(an)h(exceptional)
167 2137 y(phenomenon,)29 b(and)g(w)n(e)e(shall)g(b)r(e)g(able)g(to)h
(symmetrise)e(the)h(functional)h(situations.)167 2389
y FE(12.4)114 b(Linearisation)167 2551 y FK(Let)27 b
FJ(A)h FK(b)r(e)f(a)i(coherence)d(space.)36 b(W)-6 b(e)27
b(can)h(de\014ne)g(the)g(space)g(!)p FJ(A)g FK(\(\\of)g(course)g
FJ(A)p FK("\))g(b)n(y)924 2751 y FJ(j)p FK(!)p FJ(Aj)19
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PStoPSsaved restore
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PStoPSxform concat
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(duct)p FK(.)36 b(The)24 b(dual)i(\(linear)d(negation\))i(of)h(the)e
(tensor)h(pro)r(duct)167 2212 y(is)i(called)f(the)h FM(p)l(ar)36
b FK(or)28 b FM(tensor)g(sum)p FK(:)1079 2385 y(\()p
FJ(A)16 b(\012)f(B)r FK(\))1324 2351 y Fv(?)1385 2385
y FK(=)k FJ(A)1514 2360 y Fv(?)1572 2385 y Fc(O)c FJ(B)1690
2360 y Fv(?)167 2558 y FK(Comparing)28 b(this)f(with)g(the)g
FM(line)l(ar)h(implic)l(ation)g FK(w)n(e)f(ha)n(v)n(e)926
2738 y FJ(A)16 b Fu(\()g FJ(B)22 b FK(=)d FJ(A)1289 2713
y Fv(?)1346 2738 y Fc(O)d FJ(B)22 b FK(=)d(\()p FJ(A)d(\012)f(B)1775
2713 y Fv(?)1817 2738 y FK(\))1844 2699 y Fv(?)270 2999
y FK(Finally)-6 b(,)35 b(eac)n(h)f(of)g(the)g(four)g(asso)r(ciativ)n(e)
g(binary)g(op)r(ertions)g FJ(\010)p FK(,)i(&,)f FJ(\012)e
FK(and)i Fc(O)f FK(has)h(a)167 3085 y(unit,)d(resp)r(ectiv)n(ely)e
(called)g FG(0)p FK(,)i FJ(>)p FK(,)f FG(1)g FK(and)i
FJ(?)d FK(\(see)h(section)g FG(??)q FK(\).)47 b(Ho)n(w)n(ev)n(er)31
b(for)h(coherence)167 3170 y(spaces)c(they)g(coincide)e(in)h(pairs:)270
3331 y FJ(\017)34 b FG(0)20 b FK(=)f FJ(>)f FK(=)h FJ(E)-17
b FM(mp)5 b FK(,)28 b(where)f FJ(jE)-17 b FM(mp)t FJ(j)19
b FK(=)h Fn(?)270 3475 y FJ(\017)34 b FG(1)20 b FK(=)f
FJ(?)f FK(=)h FJ(S)-6 b FM(gl)6 b FK(,)28 b(where)f FJ(jS)-6
b FM(gl)5 b FJ(j)19 b FK(=)g FJ(f\017)q(g)p FK(.)167
3636 y(Whic)n(h)40 b(of)h(these)g(is)f(the)g FM(terminal)h(obje)l(ct)47
b FK(for)41 b(coherence)f(spaces)h(and)h(stable)e(maps?)167
3721 y(F)-6 b(or)28 b(linear)d(maps?)35 b(Ho)n(w)27 b(do)h(these)f(t)n
(yp)r(es)g(relate)f(to)h FM(absur)l(dity)34 b FK(and)28
b FM(tautolo)l(gy)34 b FK(in)27 b(natural)167 3806 y(deduction?)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
104 107 bop 167 878 a FF(Chapter)28 b(13)167 1175 y(Cut)f(Elimination)g
(\(Hauptsatz\))167 1505 y FK(Gen)n(tzen's)32 b(theorem,)h(one)g(of)f
(the)g(most)h(imp)r(ortan)n(t)e(in)h(logic,)h(is)e(not)i(v)n(ery)g(far)
f(remo)n(v)n(ed)167 1590 y(from)26 b(normalisation)f(in)h(natural)g
(deduction,)h(whic)n(h)e(is)h(to)g(a)h(large)e(exten)n(t)h(inspired)f
(b)n(y)i(it.)167 1675 y(In)e(a)g(sligh)n(tly)f(mo)r(di\014ed)h(form,)g
(it)f(is)g(at)h(the)g(ro)r(ot)g(of)h(languages)g(suc)n(h)g(as)f
Fr(PROLOG)p FK(.)i(In)e(other)167 1760 y(w)n(ords,)j(it)f(is)g(a)g
(result)g(whic)n(h)g(ev)n(ery)n(one)h(should)g(see)f(pro)n(v)n(ed)h(at)
g(least)f(once.)35 b(Ho)n(w)n(ev)n(er)27 b(the)167 1845
y(pro)r(of)i(is)e(v)n(ery)i(delicate)c(and)k(\014ddly)-6
b(.)37 b(So)29 b(w)n(e)f(shall)f(b)r(egin)h(b)n(y)g(p)r(oin)n(ting)g
(out)h(the)e(k)n(ey)i(cases)167 1930 y(whic)n(h)35 b(it)f(is)g(imp)r
(ortan)n(t)h(to)h(understand.)59 b(Afterw)n(ards)35 b(w)n(e)g(shall)f
(dev)n(elop)i(the)f(detailed)167 2015 y(pro)r(of,)29
b(whose)f(in)n(tricacies)d(are)i(less)g(in)n(teresting.)167
2264 y FE(13.1)114 b(The)28 b(k)m(ey)g(cases)167 2425
y FK(The)f(aim)g(is)g(to)h(eliminate)c(cuts)k(of)g(the)f(sp)r(ecial)g
(form)1014 2611 y FH(A)p 1018 2631 39 3 v 1086 2610 a
FJ(`)19 b FH(C)q(;)1229 2611 y(B)p 1233 2631 43 3 v 83
w(A)p 1368 2631 39 3 v -30 x Fv(0)1432 2610 y FH(;)12
b(C)25 b FJ(`)1600 2611 y FH(B)p 1603 2631 43 3 v 1656
2581 a Fv(0)p 1014 2655 658 4 v 1693 2671 a Fq(Cut)1147
2732 y FH(A)p 1151 2752 39 3 v -1 x(;)1230 2732 y(A)p
1233 2752 V 1281 2702 a Fv(0)1317 2731 y FJ(`)1380 2732
y FH(B)p 1384 2752 43 3 v 1436 2731 a(;)1467 2732 y(B)p
1470 2752 V 1523 2702 a Fv(0)167 2917 y FK(where)31 b(the)g(left)f
(premise)f(is)i(a)h(righ)n(t)f(logical)e(rule)i(and)h(the)f(righ)n(t)g
(premise)f(a)i(left)e(logical)167 3002 y(rule,)36 b(so)f(that)g(b)r
(oth)h(in)n(tro)r(duce)e(the)h(main)f(sym)n(b)r(ol)h(of)g
FH(C)5 b FK(.)57 b(These)35 b(cases)g(enligh)n(ten)e(the)167
3088 y(deep)28 b(symmetries)d(of)j(logical)e(rules,)h(whic)n(h)g(matc)n
(h)h(eac)n(h)f(other)h(exactly)-6 b(.)251 3264 y(1.)35
b FJ(R^)28 b FK(and)g FJ(L)p FK(1)p FJ(^)712 3467 y FH(A)p
715 3486 39 3 v 783 3466 a FJ(`)20 b FH(C)q(;)927 3467
y(B)p 931 3486 43 3 v 82 w(A)p 1065 3486 39 3 v -30 x
Fv(0)1149 3466 y FJ(`)g FH(D)r(;)1302 3467 y(B)p 1305
3486 43 3 v 1358 3437 a Fv(0)p 712 3511 663 4 v 1395
3526 a FJ(R^)736 3588 y FH(A)p 739 3608 39 3 v 787 3587
a(;)818 3588 y(A)p 822 3608 V -30 x Fv(0)906 3587 y FJ(`)g
FH(C)h FJ(^)15 b FH(D)r(;)1191 3588 y(B)p 1195 3608 43
3 v 1247 3587 a(;)1278 3588 y(B)p 1282 3608 V 1334 3558
a Fv(0)1718 3467 y FH(A)p 1721 3486 39 3 v 1769 3437
a Fv(00)1799 3466 y FH(;)d(C)25 b FJ(`)1967 3467 y FH(B)p
1970 3486 43 3 v 2023 3437 a Fv(00)p 1649 3511 473 4
v 2142 3526 a FJ(L)p FK(1)p FJ(^)1649 3588 y FH(A)p 1652
3608 39 3 v -30 x Fv(00)1731 3587 y FH(;)12 b(C)20 b
FJ(^)c FH(D)22 b FJ(`)2035 3588 y FH(B)p 2039 3608 43
3 v 2091 3558 a Fv(00)p 736 3632 1386 4 v 2142 3648 a
Fq(Cut)1118 3709 y FH(A)p 1121 3729 39 3 v 1169 3708
a(;)1200 3709 y(A)p 1204 3729 V -30 x Fv(0)1268 3708
y FH(;)1299 3709 y(A)p 1303 3729 V -30 x Fv(00)1401 3708
y FJ(`)1463 3709 y FH(B)p 1467 3729 43 3 v 1519 3708
a(;)1550 3709 y(B)p 1554 3729 V 1606 3679 a Fv(0)1622
3708 y FH(;)1653 3709 y(B)p 1657 3729 V 1709 3679 a Fv(00)339
3925 y FK(is)27 b(replaced)g(b)n(y)1368 4101 y(104)p
eop
PStoPSsaved restore
%%Page: (108,109) 55
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
105 108 bop 167 167 a Fz(13.1.)65 b(THE)27 b(KEY)f(CASES)1617
b FK(105)1087 390 y FH(A)p 1091 410 39 3 v 1158 389 a
FJ(`)20 b FH(C)q(;)1302 390 y(B)p 1306 410 43 3 v 83
w(A)p 1440 410 39 3 v 1488 360 a Fv(00)1518 389 y FH(;)12
b(C)25 b FJ(`)1686 390 y FH(B)p 1689 410 43 3 v 1742
360 a Fv(00)p 1087 434 685 4 v 1792 450 a Fq(Cut)1220
511 y FH(A)p 1223 531 39 3 v -1 x(;)1303 511 y(A)p 1306
531 V 1354 481 a Fv(00)1404 510 y FJ(`)1466 511 y FH(B)p
1470 531 43 3 v 1522 510 a(;)1553 511 y(B)p 1557 531
V 1609 481 a Fv(00)1119 571 y FK(=)-17 b(=)h(=)g(=)g(=)f(=)h(=)g(=)g(=)
g(=)g(=)g(=)g(=)g(=)g(=)g(=)1119 633 y FH(A)p 1122 652
39 3 v 1170 632 a(;)1201 633 y(A)p 1205 652 V -30 x Fv(0)1269
632 y FH(;)1300 633 y(A)p 1304 652 V -30 x Fv(00)1402
632 y FJ(`)1464 633 y FH(B)p 1468 652 43 3 v 1520 632
a(;)1551 633 y(B)p 1555 652 V 1607 603 a Fv(0)1623 632
y FH(;)1654 633 y(B)p 1658 652 V 1710 603 a Fv(00)339
841 y FK(where)32 b(the)f(double)i(bar)f(denotes)h(a)f(certain)f(n)n
(um)n(b)r(er)h(of)g(structural)g(rules,)g(in)g(this)339
926 y(case)c(w)n(eak)n(ening)g(and)h(exc)n(hange.)251
1177 y(2.)35 b FJ(R^)28 b FK(and)g FJ(L)p FK(2)p FJ(^)712
1373 y FH(A)p 715 1393 39 3 v 783 1372 a FJ(`)20 b FH(C)q(;)927
1373 y(B)p 931 1393 43 3 v 82 w(A)p 1065 1393 39 3 v
-30 x Fv(0)1149 1372 y FJ(`)g FH(D)r(;)1302 1373 y(B)p
1305 1393 43 3 v 1358 1343 a Fv(0)p 712 1417 663 4 v
1395 1433 a FJ(R^)736 1495 y FH(A)p 739 1514 39 3 v 787
1494 a(;)818 1495 y(A)p 822 1514 V -31 x Fv(0)906 1494
y FJ(`)g FH(C)h FJ(^)15 b FH(D)r(;)1191 1495 y(B)p 1195
1514 43 3 v 1247 1494 a(;)1278 1495 y(B)p 1282 1514 V
1334 1464 a Fv(0)1715 1373 y FH(A)p 1719 1393 39 3 v
-30 x Fv(00)1797 1372 y FH(;)d(D)21 b FJ(`)1969 1373
y FH(B)p 1973 1393 43 3 v 2025 1343 a Fv(00)p 1649 1417
473 4 v 2142 1433 a FJ(L)p FK(2)p FJ(^)1649 1495 y FH(A)p
1652 1514 39 3 v -31 x Fv(00)1731 1494 y FH(;)12 b(C)20
b FJ(^)c FH(D)22 b FJ(`)2035 1495 y FH(B)p 2039 1514
43 3 v 2091 1464 a Fv(00)p 736 1538 1386 4 v 2142 1554
a Fq(Cut)1118 1616 y FH(A)p 1121 1636 39 3 v 1169 1615
a(;)1200 1616 y(A)p 1204 1636 V -30 x Fv(0)1268 1615
y FH(;)1299 1616 y(A)p 1303 1636 V -30 x Fv(00)1401 1615
y FJ(`)1463 1616 y FH(B)p 1467 1636 43 3 v 1519 1615
a(;)1550 1616 y(B)p 1554 1636 V 1606 1586 a Fv(0)1622
1615 y FH(;)1653 1616 y(B)p 1657 1636 V 1709 1586 a Fv(00)339
1824 y FK(is)27 b(replaced)g(similarly)e(b)n(y)1064 2033
y FH(A)p 1067 2053 39 3 v -30 x Fv(0)1152 2032 y FJ(`)19
b FH(D)r(;)1304 2033 y(B)p 1308 2053 43 3 v 1360 2003
a Fv(0)1455 2033 y FH(A)p 1459 2053 39 3 v -30 x Fv(00)1537
2032 y FH(;)12 b(D)21 b FJ(`)1709 2033 y FH(B)p 1713
2053 43 3 v 1765 2003 a Fv(00)p 1064 2077 731 4 v 1816
2093 a Fq(Cut)1203 2155 y FH(A)p 1207 2174 39 3 v -30
x Fv(0)1272 2154 y FH(;)1303 2155 y(A)p 1306 2174 V 1354
2125 a Fv(00)1404 2154 y FJ(`)1466 2155 y FH(B)p 1470
2174 43 3 v 1522 2125 a Fv(0)1539 2154 y FH(;)1570 2155
y(B)p 1573 2174 V 1626 2125 a Fv(00)1119 2214 y FK(=)-17
b(=)h(=)g(=)g(=)f(=)h(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)1119
2276 y FH(A)p 1122 2296 39 3 v 1170 2275 a(;)1201 2276
y(A)p 1205 2296 V -30 x Fv(0)1269 2275 y FH(;)1300 2276
y(A)p 1304 2296 V -30 x Fv(00)1402 2275 y FJ(`)1464 2276
y FH(B)p 1468 2296 43 3 v 1520 2275 a(;)1551 2276 y(B)p
1555 2296 V 1607 2246 a Fv(0)1623 2275 y FH(;)1654 2276
y(B)p 1658 2296 V 1710 2246 a Fv(00)251 2517 y FK(3.)35
b FJ(R)p FK(1)p FJ(_)28 b FK(and)h FJ(L_)781 2713 y FH(A)p
784 2732 39 3 v 852 2712 a FJ(`)19 b FH(C)q(;)995 2713
y(B)p 999 2732 43 3 v 710 2756 413 4 v 1143 2772 a FJ(R)p
FK(1)p FJ(_)710 2834 y FH(A)p 713 2854 39 3 v 781 2833
a FJ(`)h FH(C)h FJ(_)15 b FH(D)r(;)1066 2834 y(B)p 1070
2854 43 3 v 1432 2713 a(A)p 1435 2732 39 3 v 1483 2683
a Fv(0)1500 2712 y FH(;)d(C)25 b FJ(`)1667 2713 y FH(B)p
1671 2732 43 3 v 1723 2683 a Fv(0)1818 2713 y FH(A)p
1821 2732 39 3 v -30 x Fv(00)1900 2712 y FH(;)12 b(D)21
b FJ(`)2072 2713 y FH(B)p 2075 2732 43 3 v 2128 2683
a Fv(00)p 1432 2756 727 4 v 2178 2772 a FJ(L_)1457 2834
y FH(A)p 1461 2854 39 3 v -30 x Fv(0)1525 2833 y FH(;)1556
2834 y(A)p 1560 2854 V -30 x Fv(00)1638 2833 y FH(;)12
b(C)21 b FJ(_)16 b FH(D)21 b FJ(`)1943 2834 y FH(B)p
1946 2854 43 3 v 1999 2804 a Fv(0)2015 2833 y FH(;)2046
2834 y(B)p 2050 2854 V 2102 2804 a Fv(00)p 710 2878 1423
4 v 2153 2894 a Fq(Cut)1110 2955 y FH(A)p 1114 2975 39
3 v -1 x(;)1193 2955 y(A)p 1196 2975 V -30 x Fv(0)1261
2954 y FH(;)1292 2955 y(A)p 1295 2975 V -30 x Fv(00)1393
2954 y FJ(`)1456 2955 y FH(B)p 1459 2975 43 3 v 1512
2954 a(;)1543 2955 y(B)p 1546 2975 V 1599 2925 a Fv(0)1615
2954 y FH(;)1646 2955 y(B)p 1649 2975 V 1702 2925 a Fv(00)339
3163 y FK(is)27 b(replaced)g(b)n(y)1101 3373 y FH(A)p
1104 3393 39 3 v 1172 3372 a FJ(`)20 b FH(C)q(;)1316
3373 y(B)p 1320 3393 43 3 v 82 w(A)p 1454 3393 39 3 v
-30 x Fv(0)1518 3372 y FH(;)12 b(C)25 b FJ(`)1686 3373
y FH(B)p 1689 3393 43 3 v 1742 3343 a Fv(0)p 1101 3417
658 4 v 1779 3432 a Fq(Cut)1233 3494 y FH(A)p 1237 3514
39 3 v -1 x(;)1316 3494 y(A)p 1319 3514 V -30 x Fv(0)1404
3493 y FJ(`)1466 3494 y FH(B)p 1470 3514 43 3 v 1522
3493 a(;)1553 3494 y(B)p 1557 3514 V 1609 3464 a Fv(0)1119
3554 y FK(=)-17 b(=)h(=)g(=)g(=)f(=)h(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g
(=)g(=)1119 3615 y FH(A)p 1122 3635 39 3 v 1170 3614
a(;)1201 3615 y(A)p 1205 3635 V -30 x Fv(0)1269 3614
y FH(;)1300 3615 y(A)p 1304 3635 V -30 x Fv(00)1402 3614
y FJ(`)1464 3615 y FH(B)p 1468 3635 43 3 v 1520 3614
a(;)1551 3615 y(B)p 1555 3635 V 1607 3585 a Fv(0)1623
3614 y FH(;)1654 3615 y(B)p 1658 3635 V 1710 3585 a Fv(00)339
3824 y FK(This)28 b(is)e(the)i(dual)g(of)g(case)f FG(??)q
FK(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
106 109 bop 167 167 a FK(106)710 b Fz(CHAPTER)26 b(13.)64
b(CUT)27 b(ELIMINA)-6 b(TION)26 b(\(HA)n(UPTSA)-6 b(TZ\))251
404 y FK(4.)35 b FJ(R)p FK(2)p FJ(_)28 b FK(and)h FJ(L_)776
603 y FH(A)p 780 623 39 3 v 848 602 a FJ(`)19 b FH(D)r(;)1000
603 y(B)p 1004 623 43 3 v 710 647 413 4 v 1143 663 a
FJ(R)p FK(2)p FJ(_)710 724 y FH(A)p 713 744 39 3 v 781
723 a FJ(`)h FH(C)h FJ(_)15 b FH(D)r(;)1066 724 y(B)p
1070 744 43 3 v 1432 603 a(A)p 1435 623 39 3 v 1483 573
a Fv(0)1500 602 y FH(;)d(C)25 b FJ(`)1667 603 y FH(B)p
1671 623 43 3 v 1723 573 a Fv(0)1818 603 y FH(A)p 1821
623 39 3 v -30 x Fv(00)1900 602 y FH(;)12 b(D)21 b FJ(`)2072
603 y FH(B)p 2075 623 43 3 v 2128 573 a Fv(00)p 1432
647 727 4 v 2178 663 a FJ(L_)1457 724 y FH(A)p 1461 744
39 3 v -30 x Fv(0)1525 723 y FH(;)1556 724 y(A)p 1560
744 V -30 x Fv(00)1638 723 y FH(;)12 b(C)21 b FJ(_)16
b FH(D)21 b FJ(`)1943 724 y FH(B)p 1946 744 43 3 v 1999
694 a Fv(0)2015 723 y FH(;)2046 724 y(B)p 2050 744 V
2102 694 a Fv(00)p 710 768 1423 4 v 2153 784 a Fq(Cut)1110
846 y FH(A)p 1114 865 39 3 v -1 x(;)1193 846 y(A)p 1196
865 V -30 x Fv(0)1261 845 y FH(;)1292 846 y(A)p 1295
865 V -30 x Fv(00)1393 845 y FJ(`)1456 846 y FH(B)p 1459
865 43 3 v 1512 845 a(;)1543 846 y(B)p 1546 865 V 1599
816 a Fv(0)1615 845 y FH(;)1646 846 y(B)p 1649 865 V
1702 816 a Fv(00)339 1057 y FK(is)27 b(replaced)g(b)n(y)1080
1269 y FH(A)p 1084 1289 39 3 v 1152 1268 a FJ(`)19 b
FH(D)r(;)1304 1269 y(B)p 1308 1289 43 3 v 83 w(A)p 1442
1289 39 3 v 1490 1239 a Fv(00)1520 1268 y FH(;)12 b(D)22
b FJ(`)1693 1269 y FH(B)p 1696 1289 43 3 v 1748 1239
a Fv(00)p 1080 1313 699 4 v 1799 1329 a Fq(Cut)1220 1390
y FH(A)p 1223 1410 39 3 v -1 x(;)1303 1390 y(A)p 1306
1410 V 1354 1360 a Fv(00)1404 1389 y FJ(`)1466 1390 y
FH(B)p 1470 1410 43 3 v 1522 1389 a(;)1553 1390 y(B)p
1557 1410 V 1609 1360 a Fv(00)1119 1450 y FK(=)-17 b(=)h(=)g(=)g(=)f(=)
h(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)1119 1512 y FH(A)p
1122 1531 39 3 v 1170 1511 a(;)1201 1512 y(A)p 1205 1531
V -31 x Fv(0)1269 1511 y FH(;)1300 1512 y(A)p 1304 1531
V -31 x Fv(00)1402 1511 y FJ(`)1464 1512 y FH(B)p 1468
1531 43 3 v 1520 1511 a(;)1551 1512 y(B)p 1555 1531 V
1607 1481 a Fv(0)1623 1511 y FH(;)1654 1512 y(B)p 1658
1531 V 1710 1481 a Fv(00)339 1723 y FK(This)28 b(is)e(the)i(dual)g(of)g
(case)f FG(??)q FK(.)251 2019 y(5.)35 b FJ(R:)28 b FK(and)g
FJ(L:)1175 1898 y FH(A)p 1179 1918 39 3 v -1 x(;)12 b(C)25
b FJ(`)1394 1898 y FH(B)p 1398 1918 43 3 v 1154 1942
319 4 v 1493 1958 a FJ(R:)1154 2019 y FH(A)p 1157 2039
39 3 v 1225 2018 a FJ(`)19 b(:)p FH(C)q(;)1415 2019 y(B)p
1419 2039 43 3 v 1772 1898 a(A)p 1775 1918 39 3 v 1823
1868 a Fv(0)1860 1897 y FJ(`)g FH(C)q(;)2003 1898 y(B)p
2007 1918 43 3 v 2060 1868 a Fv(0)p 1746 1942 356 4 v
2122 1958 a FJ(L:)1746 2019 y FH(A)p 1750 2039 39 3 v
-30 x Fv(0)1815 2018 y FH(;)12 b FJ(:)p FH(C)24 b FJ(`)2029
2019 y FH(B)p 2033 2039 43 3 v 2085 1989 a Fv(0)p 1154
2063 948 4 v 2122 2079 a Fq(Cut)1431 2141 y FH(A)p 1435
2160 39 3 v -1 x(;)1514 2141 y(A)p 1518 2160 V -30 x
Fv(0)1602 2140 y FJ(`)1664 2141 y FH(B)p 1668 2160 43
3 v 1720 2140 a(;)1751 2141 y(B)p 1755 2160 V 1807 2111
a Fv(0)339 2455 y FK(is)j(replaced)g(b)n(y)1101 2667
y FH(A)p 1104 2687 39 3 v 1152 2637 a Fv(0)1188 2666
y FJ(`)20 b FH(C)q(;)1332 2667 y(B)p 1336 2687 43 3 v
1388 2637 a Fv(0)1483 2667 y FH(A)p 1487 2687 39 3 v
-1 x(;)12 b(C)25 b FJ(`)1702 2667 y FH(B)p 1706 2687
43 3 v 1101 2711 658 4 v 1779 2727 a Fq(Cut)1233 2789
y FH(A)p 1237 2808 39 3 v -30 x Fv(0)1302 2788 y FH(;)1333
2789 y(A)p 1336 2808 V 1404 2788 a FJ(`)1466 2789 y FH(B)p
1470 2808 43 3 v 1522 2759 a Fv(0)1539 2788 y FH(;)1570
2789 y(B)p 1573 2808 V 1233 2848 a FK(=)-17 b(=)g(=)h(=)g(=)g(=)g(=)g
(=)g(=)f(=)1233 2910 y FH(A)p 1237 2930 39 3 v -1 x(;)1316
2910 y(A)p 1319 2930 V -30 x Fv(0)1404 2909 y FJ(`)1466
2910 y FH(B)p 1470 2930 43 3 v 1522 2909 a(;)1553 2910
y(B)p 1557 2930 V 1609 2880 a Fv(0)339 3121 y FK(Note)28
b(the)f(switc)n(h.)251 3272 y(6.)35 b FJ(R\))28 b FK(and)h
FJ(L\))729 3471 y FH(A)p 733 3491 39 3 v -1 x(;)12 b(C)25
b FJ(`)19 b FH(D)r(;)1038 3471 y(B)p 1042 3491 43 3 v
690 3515 444 4 v 1155 3531 a FJ(R\))690 3592 y FH(A)p
693 3612 39 3 v 761 3591 a FJ(`)h FH(C)25 b FJ(\))20
b FH(D)r(;)1078 3592 y(B)p 1081 3612 43 3 v 1432 3471
a(A)p 1435 3491 39 3 v -30 x Fv(0)1520 3470 y FJ(`)f
FH(C)q(;)1663 3471 y(B)p 1667 3491 43 3 v 1720 3441 a
Fv(0)1814 3471 y FH(A)p 1818 3491 39 3 v -30 x Fv(00)1896
3470 y FH(;)12 b(D)22 b FJ(`)2068 3471 y FH(B)p 2072
3491 43 3 v 2124 3441 a Fv(00)p 1432 3515 723 4 v 2175
3531 a FJ(L\))1440 3592 y FH(A)p 1443 3612 39 3 v -30
x Fv(0)1508 3591 y FH(;)1539 3592 y(A)p 1543 3612 V -30
x Fv(00)1621 3591 y FH(;)12 b(C)25 b FJ(\))20 b FH(D)h
FJ(`)1957 3592 y FH(B)p 1960 3612 43 3 v 2013 3562 a
Fv(0)2029 3591 y FH(;)2060 3592 y(B)p 2064 3612 V 2116
3562 a Fv(00)p 690 3636 1457 4 v 2167 3652 a Fq(Cut)1107
3714 y FH(A)p 1111 3733 39 3 v -1 x(;)1190 3714 y(A)p
1193 3733 V -30 x Fv(0)1258 3713 y FH(;)1289 3714 y(A)p
1292 3733 V -30 x Fv(00)1390 3713 y FJ(`)1453 3714 y
FH(B)p 1456 3733 43 3 v 1509 3713 a(;)1540 3714 y(B)p
1543 3733 V 1596 3684 a Fv(0)1612 3713 y FH(;)1643 3714
y(B)p 1646 3733 V 1699 3684 a Fv(00)339 3925 y FK(is)27
b(replaced)g(b)n(y)p eop
PStoPSsaved restore
%%Page: (110,111) 56
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
107 110 bop 167 167 a Fz(13.1.)65 b(THE)27 b(KEY)f(CASES)1617
b FK(107)767 390 y FH(A)p 770 410 39 3 v 818 360 a Fv(0)855
389 y FJ(`)19 b FH(C)q(;)998 390 y(B)p 1002 410 43 3
v 1055 360 a Fv(0)1149 390 y FH(A)p 1153 410 39 3 v -1
x(;)12 b(C)25 b FJ(`)19 b FH(D)r(;)1458 390 y(B)p 1462
410 43 3 v 767 434 748 4 v 1535 450 a Fq(Cut)899 511
y FH(A)p 903 531 39 3 v -30 x Fv(0)968 510 y FH(;)999
511 y(A)p 1002 531 V 1070 510 a FJ(`)1132 511 y FH(B)p
1136 531 43 3 v 1188 481 a Fv(0)1205 510 y FH(;)12 b(D)r(;)1326
511 y(B)p 1329 531 V 899 571 a FK(=)-29 b(=)-16 b(=)g(=)g(=)g(=)g(=)g
(=)g(=)g(=)g(=)g(=)-30 b(=)899 633 y FH(A)p 903 652 39
3 v -1 x(;)982 633 y(A)p 986 652 V -30 x Fv(0)1070 632
y FJ(`)19 b FH(D)r(;)1222 633 y(B)p 1226 652 43 3 v 1278
632 a(;)1309 633 y(B)p 1313 652 V 1365 603 a Fv(0)1752
633 y FH(A)p 1756 652 39 3 v -30 x Fv(00)1834 632 y FH(;)12
b(D)22 b FJ(`)2006 633 y FH(B)p 2010 652 43 3 v 2062
603 a Fv(00)p 899 677 1193 4 v 2113 692 a Fq(Cut)1185
754 y FH(A)p 1189 774 39 3 v -1 x(;)1268 754 y(A)p 1271
774 V 1319 724 a Fv(0)1336 753 y FH(;)1367 754 y(A)p
1370 774 V 1418 724 a Fv(00)1468 753 y FJ(`)1531 754
y FH(B)p 1534 774 43 3 v 1586 753 a(;)1617 754 y(B)p
1621 774 V 1673 724 a Fv(0)1690 753 y FH(;)1721 754 y(B)p
1724 774 V 1777 724 a Fv(00)339 980 y FK(So)29 b(the)e(problem)g(is)g
(solv)n(ed)h(b)n(y)g FM(two)k FK(cuts.)251 1253 y(7.)j
FJ(R8)27 b FK(and)i FJ(L8)943 1467 y FH(A)p 947 1486
39 3 v 1015 1466 a FJ(`)19 b FH(C)q(;)1158 1467 y(B)p
1162 1486 43 3 v 892 1510 375 4 v 1287 1526 a FJ(R8)892
1588 y FH(A)p 895 1608 39 3 v 963 1587 a FJ(`)h(8)p FH(\030)s(:)12
b(C)q(;)1211 1588 y(B)p 1214 1608 43 3 v 1533 1467 a(A)p
1537 1486 39 3 v -30 x Fv(0)1601 1466 y FH(;)g(C)5 b
FK([)p FH(a=\030)s FK(])22 b FJ(`)1911 1467 y FH(B)p
1915 1486 43 3 v 1967 1437 a Fv(0)p 1533 1510 451 4 v
2005 1526 a FJ(L8)1553 1588 y FH(A)p 1556 1608 39 3 v
1604 1558 a Fv(0)1621 1587 y FH(;)12 b FJ(8)p FH(\030)s(:)g(C)24
b FJ(`)1892 1588 y FH(B)p 1895 1608 43 3 v 1948 1558
a Fv(0)p 892 1632 1073 4 v 1985 1648 a Fq(Cut)1232 1709
y FH(A)p 1235 1729 39 3 v -1 x(;)1315 1709 y(A)p 1318
1729 V 1366 1679 a Fv(0)1402 1708 y FJ(`)1465 1709 y
FH(B)p 1468 1729 43 3 v 1521 1708 a(;)1552 1709 y(B)p
1555 1729 V 1608 1679 a Fv(0)339 1935 y FK(is)j(replaced)g(b)n(y)956
2163 y FH(A)p 960 2182 39 3 v 1027 2162 a FJ(`)20 b FH(C)5
b FK([)p FH(a=\030)s FK(])p FH(;)1316 2163 y(B)p 1321
2182 43 3 v 84 w(A)p 1456 2182 39 3 v -30 x Fv(0)1520
2162 y FH(;)12 b(C)5 b FK([)p FH(a=\030)s FK(])22 b FJ(`)1830
2163 y FH(B)p 1834 2182 43 3 v 1886 2133 a Fv(0)p 956
2207 947 4 v 1923 2222 a Fq(Cut)1233 2284 y FH(A)p 1237
2304 39 3 v -1 x(;)1316 2284 y(A)p 1319 2304 V -30 x
Fv(0)1404 2283 y FJ(`)1466 2284 y FH(B)p 1470 2304 43
3 v 1522 2283 a(;)1553 2284 y(B)p 1557 2304 V 1609 2254
a Fv(0)339 2510 y FK(where)27 b FH(a)h FK(is)f(substituted)h(for)f
FH(\030)32 b FK(throughout)d(the)f(left-hand)f(sub-pro)r(of.)251
2668 y(8.)35 b FJ(R9)28 b FK(and)h FJ(L9)881 2881 y FH(A)p
885 2901 39 3 v 953 2880 a FJ(`)19 b FH(C)5 b FK([)p
FH(a=\030)s FK(])p FH(;)1241 2881 y(B)p 1247 2901 43
3 v 881 2925 418 4 v 1320 2941 a FJ(R9)903 3003 y FH(A)p
906 3022 39 3 v 974 3002 a FJ(`)20 b(9)p FH(\030)s(:)12
b(C)q(;)1221 3003 y(B)p 1225 3022 43 3 v 1618 2881 a(A)p
1621 2901 39 3 v 1669 2851 a Fv(0)1686 2880 y FH(;)g(C)25
b FJ(`)1853 2881 y FH(B)p 1857 2901 43 3 v 1909 2851
a Fv(0)p 1566 2925 412 4 v 1998 2941 a FJ(L9)1566 3003
y FH(A)p 1569 3022 39 3 v -31 x Fv(0)1634 3002 y FH(;)12
b FJ(9)p FH(\030)s(:)g(C)25 b FJ(`)1905 3003 y FH(B)p
1909 3022 43 3 v 1961 2972 a Fv(0)p 903 3046 1075 4 v
1998 3062 a Fq(Cut)1244 3124 y FH(A)p 1248 3144 39 3
v -1 x(;)1327 3124 y(A)p 1330 3144 V 1378 3094 a Fv(0)1414
3123 y FJ(`)1477 3124 y FH(B)p 1481 3144 43 3 v 1533
3123 a(;)1564 3124 y(B)p 1567 3144 V 1620 3094 a Fv(0)339
3350 y FK(is)i(replaced)g(b)n(y)956 3577 y FH(A)p 960
3597 39 3 v 1027 3576 a FJ(`)20 b FH(C)5 b FK([)p FH(a=\030)s
FK(])p FH(;)1316 3577 y(B)p 1321 3597 43 3 v 84 w(A)p
1456 3597 39 3 v -30 x Fv(0)1520 3576 y FH(;)12 b(C)5
b FK([)p FH(a=\030)s FK(])22 b FJ(`)1830 3577 y FH(B)p
1834 3597 43 3 v 1886 3547 a Fv(0)p 956 3621 947 4 v
1923 3637 a Fq(Cut)1233 3699 y FH(A)p 1237 3718 39 3
v -1 x(;)1316 3699 y(A)p 1319 3718 V -30 x Fv(0)1404
3698 y FJ(`)1466 3699 y FH(B)p 1470 3718 43 3 v 1522
3698 a(;)1553 3699 y(B)p 1557 3718 V 1609 3669 a Fv(0)339
3925 y FK(This)28 b(is)e(the)i(dual)g(of)g(case)f FG(??)q
FK(.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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 closepath}put initclip
PStoPSxform concat
108 111 bop 167 167 a FK(108)710 b Fz(CHAPTER)26 b(13.)64
b(CUT)27 b(ELIMINA)-6 b(TION)26 b(\(HA)n(UPTSA)-6 b(TZ\))167
404 y FE(13.2)114 b(The)28 b(principal)g(lemma)167 571
y FK(The)f FM(de)l(gr)l(e)l(e)34 b FH(@)t FK(\()p FH(A)p
FK(\))27 b(of)h(a)g FM(formula)34 b FK(is)26 b(de\014ned)j(b)n(y:)270
764 y FJ(\017)34 b FH(@)t FK(\()p FH(A)p FK(\))20 b(=)f(1)28
b(for)g FH(A)f FK(atomic)270 927 y FJ(\017)34 b FH(@)t
FK(\()p FH(A)16 b FJ(^)f FH(B)t FK(\))20 b(=)f FH(@)t
FK(\()p FH(A)c FJ(_)h FH(B)t FK(\))j(=)h FH(@)t FK(\()p
FH(A)f FJ(\))h FH(B)t FK(\))f(=)g(max)q(\()p FH(@)t FK(\()p
FH(A)p FK(\))p FH(;)12 b(@)t FK(\()p FH(B)t FK(\)\))i(+)i(1)270
1090 y FJ(\017)34 b FH(@)t FK(\()p FJ(:)p FH(A)p FK(\))20
b(=)f FH(@)t FK(\()p FJ(8)p FH(\030)s(:)12 b(A)p FK(\))18
b(=)i FH(@)t FK(\()p FJ(9)p FH(\030)s(:)12 b(A)p FK(\))19
b(=)h FH(@)t FK(\()p FH(A)p FK(\))15 b(+)g(1)270 1283
y(so)29 b(that)f FH(@)t FK(\()p FH(A)p FK([)p FH(a=\030)s
FK(]\))20 b(=)g FH(@)t FK(\()p FH(A)p FK(\).)270 1433
y(The)32 b FM(de)l(gr)l(e)l(e)38 b FK(of)33 b(a)f FM(cut)g(rule)37
b FK(is)31 b(de\014ned)i(to)f(b)r(e)g(the)g(degree)f(of)i(the)f(form)n
(ula)g(whic)n(h)f(it)167 1518 y(eliminates.)45 b(The)32
b(k)n(ey)g(cases)g(considered)f(ab)r(o)n(v)n(e)i(replace)e(a)h(cut)g(b)
n(y)g(one)h(or)f(t)n(w)n(o)g(cuts)g(of)167 1603 y(lo)n(w)n(er)27
b(degree.)270 1752 y(The)c FM(de)l(gr)l(e)l(e)29 b FH(d)p
FK(\()p FH(\031)s FK(\))22 b(of)h(a)h FM(pr)l(o)l(of)39
b FK(is)23 b(the)f(sup)i(of)f(the)g(degrees)g(of)g(its)f(cut)h(rules,)g
(so)h FH(d)p FK(\()p FH(\031)s FK(\))18 b(=)i(0)167 1838
y(i\013)27 b FH(\031)j FK(is)d(cut-free.)270 1987 y(The)32
b FM(height)38 b FH(h)p FK(\()p FH(\031)s FK(\))30 b(of)i(a)g
FM(pr)l(o)l(of)48 b FK(is)31 b(that)h(of)f(its)g(asso)r(ciated)h(tree:)
42 b(if)30 b FH(\031)35 b FK(ends)c(in)g(a)h(rule)167
2072 y(whose)23 b(premises)f(are)g(pro)n(v)n(ed)i(b)n(y)g
FH(\031)1124 2083 y FD(1)1152 2072 y FH(;)12 b(:)g(:)g(:)g(;)g(\031)
1346 2083 y Fw(n)1402 2072 y FK(\()p FH(n)19 b FK(=)g(0)p
FH(;)12 b FK(1)24 b(or)f(2\))g(then)g FH(h)p FK(\()p
FH(\031)s FK(\))c(=)g(sup)q(\()p FH(h)p FK(\()p FH(\031)2475
2083 y Fw(i)2494 2072 y FK(\)\))8 b(+)g(1.)270 2292 y(The)22
b(principal)f(lemma)f(sa)n(ys)k(that)e(the)g(\014nal)h(cut)f(rule)f
(can)h(b)r(e)g(eliminated.)31 b(Its)22 b(complex)167
2377 y(form)n(ulation)27 b(tak)n(es)h(accoun)n(t)h(of)f(the)f
(structural)g(rules)g(whic)n(h)g(in)n(terfere)f(with)g(cuts.)167
2597 y FG(Notation)79 b FK(If)642 2598 y FH(A)p 646 2618
39 3 v 735 2597 a FK(is)40 b(a)h(sequence)f(of)h(form)n(ulae,)j(then)
1795 2598 y FH(A)p 1798 2618 V 1854 2597 a FJ(\000)19
b FH(C)46 b FK(denotes)2290 2598 y FH(A)p 2294 2618 V
2382 2597 a FK(where)40 b(an)167 2682 y FM(arbitr)l(ary)35
b FK(n)n(um)n(b)r(er)28 b(of)g(o)r(ccurrences)e(of)j(the)e(form)n(ula)g
FH(C)33 b FK(ha)n(v)n(e)c(b)r(een)e(deleted.)167 2902
y FG(Lemma)22 b FK(Let)j FH(C)30 b FK(b)r(e)25 b(a)g(form)n(ula)g(of)h
(degree)e FH(d)p FK(,)i(and)g FH(\031)s(;)12 b(\031)1690
2877 y Fv(0)1731 2902 y FK(pro)r(ofs)26 b(of)2040 2903
y FH(A)p 2043 2923 V 2111 2902 a FJ(`)2174 2903 y FH(B)p
2177 2923 43 3 v 2274 2902 a FK(and)2430 2903 y FH(A)p
2433 2923 39 3 v -29 x Fv(0)2518 2902 y FJ(`)2580 2903
y FH(B)p 2584 2923 43 3 v 2636 2874 a Fv(0)167 2988 y
FK(of)h(degrees)e(less)h(than)g FH(d)p FK(.)35 b(Then)26
b(w)n(e)g(can)g(mak)n(e)g(a)h(pro)r(of)1694 2962 y FD(1)1749
2988 y FH($)h FK(of)1935 2989 y FH(A)p 1938 3008 39 3
v 1986 2988 a(;)2017 2989 y(A)p 2021 3008 V -30 x Fv(0)2087
2988 y FJ(\000)13 b FH(C)25 b FJ(`)2292 2989 y FH(B)p
2295 3008 43 3 v 2349 2988 a FJ(\000)13 b FH(C)q(;)2498
2989 y(B)p 2502 3008 V 2555 2959 a Fv(0)2617 2988 y FK(of)167
3073 y(degree)27 b(less)g(than)h FH(d)p FK(.)167 3293
y FG(Pro)r(of)42 b FH($)h FK(is)e(constructed)g(b)n(y)i(induction)e(on)
h FH(h)p FK(\()p FH(\031)s FK(\))18 b(+)i FH(h)p FK(\()p
FH(\031)1870 3267 y Fv(0)1885 3293 y FK(\),)45 b(but)d(unfortunately)h
(not)167 3378 y(symmetrically)25 b(in)j FH(\031)j FK(and)f
FH(\031)955 3352 y Fv(0)971 3378 y FK(:)37 b(at)28 b(some)g(stages)i
(preference)d(is)g(giv)n(en)i(to)f FH(\031)s FK(,)h(or)g(to)f
FH(\031)2497 3352 y Fv(0)2513 3378 y FK(,)h(and)167 3463
y FH($)g FK(is)e(irrev)n(ersibly)f(a\013ected)h(b)n(y)h(this)g(c)n
(hoice.)270 3612 y(T)-6 b(o)41 b(simplify)c(matters,)42
b(w)n(e)d(shall)g(supp)r(ose)i(that)f(in)1793 3613 y
FH(A)p 1797 3633 39 3 v -29 x Fv(0)1868 3612 y FJ(\000)19
b FH(C)45 b FK(and)2187 3613 y FH(B)p 2190 3633 43 3
v 2250 3612 a FJ(\000)19 b FH(C)45 b FK(w)n(e)39 b(ha)n(v)n(e)167
3698 y(remo)n(v)n(ed)24 b FM(al)t(l)31 b FK(the)24 b(o)r(ccurrences)g
(of)h FH(C)5 b FK(.)34 b(This)24 b(allo)n(ws)g(us)h(to)g(a)n(v)n(oid)g
(length)n(y)g(circumlo)r(cutions)167 3783 y(without)i(making)h(an)n(y)h
(essen)n(tial)d(di\013erence)g(to)i(the)g(pro)r(of.)p
167 3860 1003 3 v 246 3903 a FC(1)273 3925 y Fx($)g FB(is)g(a)f(v)m
(arian)n(t)h(of)f Fx(\031)r FB(,)i(not)e(of)h Fx(!)r
FB(.)p eop
PStoPSsaved restore
%%Page: (112,113) 57
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
109 112 bop 167 167 a Fz(13.2.)65 b(THE)27 b(PRINCIP)-6
b(AL)26 b(LEMMA)1341 b FK(109)270 404 y(The)30 b(last)g(rule)f
Fq(r)j FK(of)e FH(\031)j FK(has)e(premises)1353 405 y
FH(A)p 1357 425 39 3 v 23 x Fw(i)1446 404 y FJ(`)1511
405 y FH(B)p 1514 425 43 3 v 1567 428 a Fw(i)1638 404
y FK(pro)n(v)n(ed)g(b)n(y)g FH(\031)2011 415 y Fw(i)2031
404 y FK(,)g(and)g(the)f(last)g(rule)f Fq(r)2655 379
y Fv(0)167 490 y FK(of)f FH(\031)292 464 y Fv(0)336 490
y FK(has)g(premises)768 491 y FH(A)p 772 510 39 3 v -30
x Fv(0)820 513 y Fw(j)865 490 y FJ(`)928 491 y FH(B)p
931 510 43 3 v 984 461 a Fv(0)984 513 y Fw(j)1057 490
y FK(pro)n(v)n(ed)g(b)n(y)h FH(\031)1428 464 y Fv(0)1425
507 y Fw(j)1451 490 y FK(.)35 b(There)27 b(are)h(sev)n(eral)f(cases)g
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b(There)27 b(are)h(t)n(w)n(o)g(sub)r(cases:)421 773 y
FJ(\017)35 b FH(\031)c FK(pro)n(v)n(es)49 b FH(C)25 b
FJ(`)20 b FH(C)25 b FK(.)36 b(Then)28 b(a)h(pro)r(of)g
FH($)h FK(of)48 b FH(C)q(;)1769 774 y(A)p 1773 794 39
3 v -30 x Fv(0)1842 773 y FJ(\000)15 b FH(C)25 b FJ(`)2049
774 y FH(B)p 2053 794 43 3 v 2105 744 a Fv(0)2169 773
y FK(is)j(obtained)g(from)491 858 y FH(\031)533 833 y
Fv(0)577 858 y FK(b)n(y)g(means)g(of)g(structural)f(rules.)421
961 y FJ(\017)35 b FH(\031)f FK(pro)n(v)n(es)54 b FH(D)25
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FK(If)i(the)h(atomic)e(sequen)n(t)1167 2821 y FH(A)p
1171 2841 V 1240 2820 a FJ(`)1305 2821 y FH(B)p 1309
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b FH(B)1081 2880 y Fv(0)1155 2906 y FK(pro)n(v)l(able)35
b(without)e(w)n(eak)n(ening)h(or)g(con)n(traction,)i(suc)n(h)167
2991 y(that:)270 3145 y FJ(\017)339 3146 y FH(A)p 343
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3926 y FH(A)p 2126 3945 V -30 x Fv(0)2174 3948 y FD(1)2202
3925 y FK(,)h FH(B)2305 3899 y Fv(0)2340 3925 y FK(=)20
b FH(B)2470 3899 y Fv(0)2466 3942 y FD(1)2494 3925 y
FK(.)p eop
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b(CUT)27 b(ELIMINA)-6 b(TION)26 b(\(HA)n(UPTSA)-6 b(TZ\))251
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b FH(C)q(;)1304 578 y(B)p 1308 597 43 3 v 1361 600 a
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698 y FJ(`)1466 699 y FH(B)p 1470 719 43 3 v 1522 722
a FD(1)1550 698 y FH(;)1581 699 y(B)p 1585 719 V 1637
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y FJ(`)g FH(B)2334 874 y Fv(0)2330 917 y FD(2)2418 900
y FK(and)37 b(t)n(w)n(o)339 985 y(cases)28 b(arise:)421
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1164 1147 y FH(A)p 1168 1167 V -30 x Fv(0)1252 1146 y
FK(=)1325 1147 y FH(A)p 1329 1167 V -30 x Fv(0)1377 1170
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1347 y FH(A)p 1963 1367 V -30 x Fv(0)2047 1346 y FK(=)2120
1347 y FH(A)p 2124 1367 V -30 x Fv(0)2172 1369 y FD(1)2200
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1448 y FD(2)875 1431 y FK(.)1723 b Fu(\003)270 1633 y
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1634 y FH(A)p 2389 1654 V -29 x Fv(0)2484 1633 y FK(empt)n(y)167
1719 y(and)44 b FH(B)377 1693 y Fv(0)424 1719 y FK(=)30
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PStoPSxform concat
113 116 bop 167 880 a FF(Chapter)28 b(14)167 1179 y(Strong)g
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2198 y FM(se)l(c)l(ond)28 b(or)l(der)g(arithmetic)k FG(P)-23
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b(the)f(Axiom)g(\(sc)n(heme\))f(of)i(comprehension)1059
3061 y FJ(9)p FH(X)r(:)12 b FJ(8)p FH(\030)s(:)g FK(\()p
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3440 y(the)32 b(\014rst)g(order)h(v)l(ariables)f(range)g(o)n(v)n(er)h
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3925 y(try)28 b(to)g(extend)f(it)g(to)h(system)f FG(F)p
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PStoPSxform concat
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3711 y(to)f(substitution:)167 3925 y FG(Lemma)d Fq(RED)584
3936 y Fw(T)7 b FD([)p Fw(V)f(=)s(Y)k FD(])753 3925 y
FK([)772 3926 y FJ(R)p 776 3945 47 3 v -1 x FH(=)866
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3936 y Fw(T)1215 3925 y FK([)1234 3926 y FJ(R)p 1238
3945 47 3 v -1 x FH(=)1328 3926 y(X)p 1332 3945 50 3
v 1391 3925 a(;)12 b Fq(RED)1557 3936 y Fw(V)1600 3925
y FK([)1619 3926 y FJ(R)p 1623 3945 47 3 v -1 x FH(=)1713
3926 y(X)p 1717 3945 50 3 v 1776 3925 a FK(])p FH(=)t(Y)k
FK(])p eop
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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PStoPSxform concat
117 120 bop 167 167 a Fz(14.3.)65 b(REDUCIBILITY)25 b(THEOREM)1286
b FK(117)270 404 y(Here)29 b(w)n(e)g(mak)n(e)h(hidden)g(use)f(of)i(the)
e(comprehension)h(sc)n(heme,)f(since,)g(in)h(order)f(to)h(b)r(e)167
490 y(able)24 b(to)h(use)g(the)g FM(pr)l(e)l(dic)l(ate)31
b Fq(RED)1070 501 y Fw(V)1113 490 y FK([)1132 491 y FJ(R)p
1136 510 47 3 v -1 x FH(=)1226 491 y(X)p 1230 510 50
3 v 1289 490 a FK(])25 b(as)g(a)g(parameter,)g(it)f(is)g(necessary)h
(to)g(kno)n(w)h(that)167 575 y(it)h(is)g(a)h FM(set)p
FK(.)270 740 y(This)d(lemma)f(is)h(pro)n(v)n(ed)h(b)n(y)h(a)e(straigh)n
(tforw)n(ard)i(induction)e(on)h FH(T)10 b FK(.)35 b(The)25
b(only)h(di\016cult)n(y)167 825 y(w)n(as)i(to)g(form)n(ulate)f(it)g
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b FH(w)r FK([)p FH(V)8 b(=)t(Y)16 b FK(])k FJ(2)f Fq(RED)2067
1305 y Fw(W)2123 1294 y FK([)2142 1295 y FJ(R)p 2146
1315 47 3 v -1 x FH(=)2236 1295 y(X)p 2240 1315 50 3
v 2300 1294 a(;)12 b FJ(S)5 b FH(=)t(Y)15 b FK(],)28
b(then)167 1380 y(\003)p FH(Y)t(:)12 b(w)21 b FJ(2)f
Fq(RED)562 1391 y FD(\005)p Fw(Y)s(:)8 b(W)710 1380 y
FK([)729 1381 y FJ(R)p 733 1400 47 3 v -1 x FH(=)823
1381 y(X)p 827 1400 50 3 v 886 1380 a FK(].)167 1631
y FG(Pro)r(of)37 b FK(W)-6 b(e)36 b(ha)n(v)n(e)h(to)g(sho)n(w)g(that)g
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b Fq(RED)1673 1642 y Fw(W)1730 1631 y FK([)1749 1632
y FJ(R)p 1753 1652 47 3 v -1 x FH(=)1843 1632 y(X)p 1847
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FK(])37 b(for)g(ev)n(ery)f(t)n(yp)r(e)h FH(V)167 1717
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FH(\027)t FK(\()p FH(w)r FK(\).)37 b(Con)n(v)n(erting)29
b(a)f(redex)167 1802 y(of)g(\(\003)p FH(Y)t(:)12 b(w)r
FK(\))g FH(V)42 b FK(giv)n(es:)270 2016 y FJ(\017)34
b FK(\(\003)p FH(Y)t(:)12 b(w)540 1990 y Fv(0)557 2016
y FK(\))g FH(V)35 b FK(with)19 b FH(\027)t FK(\()p FH(w)941
1990 y Fv(0)958 2016 y FK(\))g FH(<)g(\027)t FK(\()p
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y Fw(W)1792 2016 y FK([)1811 2017 y FJ(R)p 1815 2036
47 3 v -1 x FH(=)1905 2017 y(X)p 1909 2036 50 3 v 1968
2016 a(;)12 b FJ(S)5 b FH(=)t(Y)16 b FK(])k(b)n(y)h(the)f(induction)339
2101 y(h)n(yp)r(othesis.)270 2277 y FJ(\017)34 b FH(w)r
FK([)p FH(V)8 b(=)t(Y)16 b FK(])28 b(whic)n(h)f(is)g(in)g
Fq(RED)1094 2288 y Fw(W)1151 2277 y FK([)1170 2278 y
FJ(R)p 1174 2298 47 3 v -1 x FH(=)1264 2278 y(X)p 1268
2298 50 3 v 1327 2277 a(;)12 b FJ(S)5 b FH(=)t(Y)16 b
FK(])27 b(b)n(y)i(assumption.)270 2491 y(So)g(the)e(result)g(follo)n
(ws)g(from)g(\()p FG(CR)h(3)p FK(\).)1271 b Fu(\003)167
2743 y Fy(14.2.3)94 b(Univ)m(ersal)29 b(application)167
2960 y FG(Lemma)c FK(If)i FH(t)20 b FJ(2)f Fq(RED)769
2971 y FD(\005)p Fw(Y)s(:)8 b(W)917 2960 y FK([)936 2961
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b FJ(2)20 b Fq(RED)1634 2971 y Fw(W)7 b FD([)p Fw(V)f(=)s(Y)k
FD(])1821 2960 y FK([)1840 2961 y FJ(R)p 1844 2981 47
3 v -1 x FH(=)1934 2961 y(X)p 1938 2981 50 3 v 1997 2960
a FK(])28 b(for)g(ev)n(ery)f(t)n(yp)r(e)h FH(V)15 b FK(.)167
3212 y FG(Pro)r(of)36 b FK(By)h(h)n(yp)r(othesis)g FH(t)12
b(V)41 b FJ(2)26 b Fq(RED)1196 3223 y Fw(W)1253 3212
y FK([)1272 3213 y FJ(R)p 1276 3233 47 3 v -1 x FH(=)1366
3213 y(X)p 1370 3233 50 3 v 1429 3212 a(;)12 b FJ(S)5
b FH(=)t(Y)15 b FK(])37 b(for)f(ev)n(ery)h(candidate)f
FJ(S)5 b FK(.)62 b(W)-6 b(e)36 b(just)167 3297 y(tak)n(e)28
b FJ(S)d FK(=)19 b Fq(RED)597 3308 y Fw(V)640 3297 y
FK([)659 3298 y FJ(R)p 663 3318 47 3 v -1 x FH(=)753
3298 y(X)p 757 3318 50 3 v 816 3297 a FK(])28 b(and)g(the)g(result)e
(follo)n(ws)h(from)h(lemma)e FG(??)q FK(.)576 b Fu(\003)167
3580 y FE(14.3)114 b(Reducibilit)m(y)28 b(theorem)167
3754 y FK(A)j(term)g FH(t)g FK(of)h(t)n(yp)r(e)g FH(T)42
b FK(is)31 b(said)h FM(r)l(e)l(ducible)k FK(if)31 b(it)g(is)g(in)g
Fq(RED)1790 3765 y Fw(T)1829 3754 y FK([)1848 3755 y
FJ(S)-6 b(N)p 1852 3775 91 3 v 1952 3754 a FH(=)1986
3755 y(X)p 1990 3775 50 3 v 2049 3754 a FK(],)33 b(where)e
FH(X)2385 3765 y FD(1)2413 3754 y FH(;)12 b(:)g(:)g(:)g(;)g(X)2625
3765 y Fw(m)167 3840 y FK(are)25 b(the)g(free)g(t)n(yp)r(e)g(v)l
(ariables)g(of)h FH(T)10 b FK(,)25 b(and)i FJ(S)-6 b(N)1401
3851 y Fw(i)1446 3840 y FK(is)25 b(the)g(set)g(of)g(strongly)h
(normalisable)e(terms)167 3925 y(of)k(t)n(yp)r(e)g FH(X)467
3936 y Fw(i)487 3925 y FK(.)p eop
PStoPSsaved restore
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PStoPSmatrix setmatrix
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PStoPSxform concat
118 121 bop 167 167 a FK(118)702 b Fz(CHAPTER)26 b(14.)64
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(the)167 531 y FG(Theorem)e FK(All)f(terms)g(of)i FG(F)g
FK(are)f(reducible.)167 657 y(and)i(hence,)e(b)n(y)h(\()p
FG(CR)g(1)p FK(\),)f(the)167 783 y FG(Corollary)e FK(All)h(terms)g(of)i
FG(F)g FK(are)f(strongly)i(normalisable.)270 956 y(W)-6
b(e)29 b(need)h(a)f(more)g(general)g(result,)g(whic)n(h)f(uses)i
(substitution)f FM(twic)l(e)g FK(\(once)g(for)h(t)n(yp)r(es,)167
1041 y(and)j(again)h(for)e(terms\))f(and)j(from)e(whic)n(h)f(the)h
(theorem)g(follo)n(ws)f(b)n(y)i(putting)g FJ(R)2429 1052
y Fw(i)2471 1041 y FK(=)23 b FJ(S)-6 b(N)2652 1052 y
Fw(i)167 1126 y FK(and)29 b FH(u)345 1137 y Fw(j)390
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1253 y FG(Prop)r(osition)40 b FK(Let)i FH(t)g FK(b)r(e)g(a)h(term)e(of)
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(v)l(ariables)i(of)167 1338 y FH(t)c FK(are)g(among)i
FH(x)640 1349 y FD(1)668 1338 y FH(;)12 b(:)g(:)g(:)f(;)h(x)861
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y FD(1)319 1423 y FH(;)g(:)g(:)g(:)f(;)h(U)520 1434 y
Fw(n)582 1423 y FK(are)28 b(among)h FH(X)986 1434 y FD(1)1014
1423 y FH(;)12 b(:)g(:)g(:)g(;)g(X)1226 1434 y Fw(m)1273
1423 y FK(.)37 b(If)29 b FJ(R)1464 1434 y FD(1)1491 1423
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1423 y FK(are)28 b(reducibilit)n(y)d(candidates)k(of)167
1508 y(t)n(yp)r(es)k FH(V)399 1519 y FD(1)427 1508 y
FH(;)12 b(:)g(:)g(:)f(;)h(V)621 1519 y Fw(m)700 1508
y FK(and)34 b FH(u)883 1519 y FD(1)911 1508 y FH(;)12
b(:)g(:)g(:)f(;)h(u)1104 1519 y Fw(n)1170 1508 y FK(are)32
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FK(])32 b(whic)n(h)167 1593 y(are)c(in)f Fq(RED)507 1604
y Fw(U)541 1611 y Fi(1)568 1593 y FK([)587 1594 y FJ(R)p
591 1614 47 3 v -1 x FH(=)681 1594 y(X)p 685 1614 50
3 v 744 1593 a FK(])p FH(;)12 b(:)g(:)g(:)g(;)g Fq(RED)1053
1604 y Fw(U)1087 1610 y Ff(n)1119 1593 y FK([)1138 1594
y FJ(R)p 1142 1614 47 3 v -1 x FH(=)1232 1594 y(X)p 1236
1614 50 3 v 1296 1593 a FK(])27 b(then)h FH(t)p FK([)1548
1594 y FH(V)p 1552 1614 43 3 v 1604 1593 a(=)1638 1594
y(X)p 1642 1614 50 3 v 1701 1593 a FK(][)1739 1594 y
FH(u)p 1743 1614 26 3 v -1 x(=)1812 1594 y(x)p 1816 1614
V 1852 1593 a FK(])20 b FJ(2)f Fq(RED)2092 1604 y Fw(T)2131
1593 y FK([)2150 1594 y FJ(R)p 2154 1614 47 3 v -1 x
FH(=)2244 1594 y(X)p 2248 1614 50 3 v 2307 1593 a FK(].)270
1766 y(The)28 b(pro)r(of)h(is)d(similar)f(to)j FG(??)q
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PStoPSxform concat
119 122 bop 167 874 a FF(Chapter)28 b(15)167 1168 y(Represen)-5
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(aim)f(to)h(study)g(the)g(\\strength")h(of)g(system)e
FG(F)h FK(with)f(a)h(view)f(to)167 1573 y(iden)n(tifying)26
b(the)i(class)f(of)h(algorithms)f(whic)n(h)g(are)g(represen)n(table.)34
b(F)-6 b(or)28 b(example,)f(if)g FH(f)35 b FK(is)27 b(a)167
1658 y(closed)f(term)g(of)h(t)n(yp)r(e)g Fq(Int)p FJ(!)q
Fq(Int)p FK(,)g(it)f(giv)n(es)h(rise)e(to)i(a)h(function)f(\(in)f(the)g
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b FK(from)i(I)-12 b(N)27 b(to)h(I)-12 b(N)27 b(b)n(y)1195
1916 y FH(f)8 b FK(\()p 1264 1877 42 3 v FH(n)p FK(\))19
b Fu( )p 1441 1855 176 3 v 20 w FJ(j)p FH(f)8 b FJ(j)p
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FH(f)8 b FJ(j)33 b FK(is)h(recursiv)n(e,)g(indeed)g(w)n(e)f(ha)n(v)n(e)
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y(namely:)270 2336 y FJ(\017)h FK(write)26 b(the)i(term)e
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y FJ(\017)34 b FK(normalise)44 b(it:)69 b(an)n(y)46 b(normalisation)e
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2564 y(normalisation)d(theorem)g(sa)n(ys)h(that)g(all)e(reduction)h
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FK(metho)r(d)29 b(due)h(to)f(Martin-L\177)-34 b(of,)31
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3691 y(that)f(if)f(a)h(function)g(is)f(pro)n(v)l(ably)i(total)e(then)h
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PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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PStoPSxform concat
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b(Numerals)167 710 y FG(Prop)r(osition)29 b FK(An)n(y)k(closed)e
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FJ(!)p FH(X)g FK(\))p FJ(!)p FH(X)38 b FK(is)31 b(a)167
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y FH(:)g(\025y)1488 1194 y Fw(X)t Fv(!)o Fw(X)1630 1219
y FH(:)g(v)167 1388 y FK(where)31 b FH(v)j FK(is)c(of)i(t)n(yp)r(e)g
FH(X)6 b FK(,)32 b(and)g(so)g(cannot)h(b)r(e)e(an)h(abstraction.)47
b(W)-6 b(e)31 b(pro)n(v)n(e)h(b)n(y)h(induction)167 1473
y(that)28 b FH(v)i FK(is)d(of)i(the)e(form)1111 1643
y FH(y)14 b FK(\()p FH(y)g FK(\()p FH(y)g(:)e(:)g(:)f
FK(\()p FH(y)1111 1686 y Fo(|)p 1143 1686 114 9 v 114
w({z)p 1321 1686 V 114 w(})1109 1746 y Fx(n)27 b FB(o)r(ccurrences)1491
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y(where)27 b FH(n)h FK(is)f(an)h(in)n(teger.)270 2016
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FH(w)510 2076 y Fv(0)538 2101 y FH(u)577 2076 y Fv(0)622
2101 y FK(or)g FH(w)762 2076 y Fv(0)791 2101 y FH(U)845
2076 y Fv(0)861 2101 y FK(.)37 b(But)28 b(the)g(t)n(yp)r(es)g(of)h
FH(x)f FK(and)h FH(y)i FK(are)d(simpler)d(than)k(that)g(of)f
FH(w)2546 2076 y Fv(0)2563 2101 y FK(,)g(so)167 2186
y FH(w)218 2161 y Fv(0)261 2186 y FK(is)f(an)g(abstraction)h(and)g
FH(w)h FK(is)d(a)h(redex:)35 b(con)n(tradiction.)g(So)27
b FH(v)j FK(is)c FH(x)p FK(,)h(in)g(whic)n(h)f(case)h(our)167
2272 y(result)i(holds)h(with)e FH(n)21 b FK(=)g(0,)31
b(or)f FH(v)i FK(is)d FH(y)14 b(v)1253 2246 y Fv(0)1299
2272 y FK(and)31 b(w)n(e)e(apply)h(the)f(induction)g(h)n(yp)r(othesis)i
(to)f FH(v)2656 2246 y Fv(0)167 2357 y FK(of)e(t)n(yp)r(e)g
FH(X)6 b FK(.)2125 b Fu(\003)167 2526 y FG(Remark)75
b FK(If)39 b(w)n(e)e(had)j(tak)n(en)f(the)f(v)l(arian)n(t)h(\005)p
FH(X)r(:)12 b FK(\()p FH(X)6 b FJ(!)o FH(X)g FK(\))p
FJ(!)p FK(\()p FH(X)g FJ(!)p FH(X)g FK(\))38 b(w)n(e)g(w)n(ould)g(ha)n
(v)n(e)167 2611 y(obtained)28 b(almost)f(the)h(same)f(result,)g(but)h
(in)f(addition)h(there)e(is)h(a)h(v)l(arian)n(t)h(for)f(1:)1194
2780 y(\003)p FH(X)r(:)12 b(\025y)1409 2755 y Fw(X)t
Fv(!)o Fw(X)1550 2780 y FH(:)g(y)270 2949 y FK(This)39
b(phenomenon)h(is)f(one)g(of)h(the)e(little)e(imp)r(erfections)h(of)i
(the)g(syn)n(tax.)72 b(Similar)167 3034 y(features)32
b(arise)g(with)f(inductiv)n(e)h(data)h(t)n(yp)r(es,)h
FM(i.e.)e FK(the)h(closed)e(normal)h(forms)h(of)g(t)n(yp)r(e)f
FH(T)167 3120 y FK(are)j(\\almost")g(the)g(terms)e(obtained)j(b)n(y)f
(com)n(bining)g(the)f(functions)h FH(f)2151 3131 y Fw(i)2171
3120 y FK(,)i(but)e(in)g(general)167 3205 y(only)28 b(\\almost".)270
3329 y(Ha)n(ving)d(said)g(this,)f(the)g(recursion)g(sc)n(heme)f(for)i
(inductiv)n(e)e(t)n(yp)r(es,)i(de\014ned)g(\(morally\))e(in)167
3414 y(terms)j(of)h(the)f FH(f)601 3425 y Fw(i)621 3414
y FK(,)h(sho)n(ws)h(that)f(\(in)f(a)h(sense)g(to)g(b)r(e)f(made)h
(precise\))e(the)h(terms)g(constructed)167 3499 y(from)36
b(the)g FH(f)509 3510 y Fw(i)565 3499 y FK(are)g(\\dense")i(among)f
(the)f(others.)62 b(T)-6 b(o)37 b(return)f(to)g(our)h(p)r(et)f(sub)t
(ject,)i(the)167 3584 y(syn)n(tax)29 b(seems)e(to)g(b)r(e)h(to)r(o)g
(rigid)e(and)j(m)n(uc)n(h)e(to)r(o)h(arti\014cial)e(to)i(allo)n(w)e(a)i
(satisfactory)g(study)167 3669 y(of)43 b(suc)n(h)f(di\016culties.)77
b(Undoubtedly)43 b(they)f(cannot)h(b)r(e)f(resolv)n(ed)g(otherwise)f
(than)i(b)n(y)167 3754 y(means)24 b(of)g(an)g(op)r(erational)g(seman)n
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(distinguish)167 3840 y(b)r(et)n(w)n(een\))k(algorithms)h(b)r(ey)n(ond)
i(what)e(can)h(b)r(e)f(done)h(with)e(normalisation,)i(whic)n(h)e(is)h
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eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
121 124 bop 167 167 a Fz(15.1.)65 b(REPRESENT)-6 b(ABLE)26
b(FUNCTIONS)1138 b FK(121)167 404 y Fy(15.1.2)94 b(T)-8
b(otal)28 b(recursiv)m(e)h(functions)167 535 y FK(Let)35
b(us)h(return)f(to)h(the)f(original)g(question,)j(whic)n(h)c(w)n(as)j
(to)e(c)n(haracterise)g(the)g(functions)167 620 y(whic)n(h)i(are)g
(represen)n(table)f(in)h FG(F)p FK(.)g(W)-6 b(e)37 b(ha)n(v)n(e)i(seen)
e(that)g(suc)n(h)h(functions)g(are)f(recursiv)n(e,)167
705 y FM(i.e.)28 b FK(calculable.)167 868 y FG(Prop)r(osition)d
FK(There)i(is)g(a)h(total)g(recursiv)n(e)e(function)i(whic)n(h)e(is)h
(not)i(represen)n(table)d(in)h FG(F)p FK(.)167 1030 y
FG(Pro)r(of)40 b FK(The)h(function)f(whic)n(h)g(w)n(e)g(shall)f(tak)n
(e)i(is)f(the)g(normalisation)f(op)r(eration.)74 b(W)-6
b(e)167 1115 y(represen)n(t)33 b(terms)g(in)h(a)g(formal)g(language)h
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167 1285 y(b)n(y)38 b(an)g(in)n(teger\))e(and)i(yields)e(another.)66
b(This)36 b(function)i(is)e(univ)n(ersal)h(\(in)f(the)h(sense)h(of)167
1370 y(T)-6 b(uring\))30 b(with)e(resp)r(ect)h(to)h(the)f(functions)h
(represen)n(table)e(in)i FG(F)p FK(,)f(and)i(so)f(cannot)h(itself)c(b)r
(e)167 1456 y(represen)n(ted)g(in)g FG(F)p FK(.)270 1576
y(More)h(precisely:)270 1716 y FJ(\017)34 b FH(N)7 b
FK(\()p FH(n)p FK(\))20 b(=)g FH(m)28 b FK(if)e FH(n)i
FK(co)r(des)g(the)f(term)g FH(t)p FK(,)g FH(m)h FK(co)r(des)g
FH(u)g FK(and)h FH(u)e FK(is)g(the)h(normal)f(form)g(of)i
FH(t)p FK(.)270 1855 y FJ(\017)34 b FH(N)7 b FK(\()p
FH(n)p FK(\))20 b(=)g(0)28 b(if)e FH(n)i FK(do)r(es)g(not)h(co)r(de)e
(an)n(y)i(term)d(of)j FG(F)p FK(.)270 1994 y(On)f(the)f(other)h(hand)h
(w)n(e)e(ha)n(v)n(e)i(the)e(functions:)270 2134 y FJ(\017)34
b FH(A)p FK(\()p FH(m;)12 b(n)p FK(\))32 b(=)f FH(p)44
b FK(if)g FH(m;)12 b(n;)g(p)45 b FK(are)g(the)f(co)r(des)h(of)g
FH(t;)12 b(u;)g(v)48 b FK(suc)n(h)d(that)g FH(v)35 b
FK(=)c FH(t)12 b(u)p FK(,)49 b(with)339 2219 y FH(A)p
FK(\()p FH(m;)12 b(n)p FK(\))20 b(=)f(0)28 b(otherwise.)270
2358 y FJ(\017)34 b FH(])p FK(\()p FH(n)p FK(\))20 b(=)f
FH(m)28 b FK(if)f FH(m)g FK(co)r(des)p 988 2319 42 3
v 28 w FH(n)p FK(.)270 2496 y FJ(\017)34 b FH([)p FK(\()p
FH(m)p FK(\))20 b(=)f FH(n)28 b FK(if)f FH(m)g FK(is)g(the)h(co)r(de)f
(of)i(the)e(n)n(umeral)p 1636 2457 V 27 w FH(n)p FK(,)h(with)e
FH([)p FK(\()p FH(m)p FK(\))20 b(=)f(0)28 b(otherwise.)270
2636 y(No)n(w)g(consider:)975 2798 y FH(D)r FK(\()p FH(n)p
FK(\))19 b(=)h FH([)p FK(\()p FH(N)7 b FK(\()p FH(A)p
FK(\()p FH(n;)12 b(])p FK(\()p FH(n)p FK(\)\)\)\))i(+)h(1)270
2961 y(This)44 b(is)f(certainly)f(a)i(total)f(recursiv)n(e)g(function,)
48 b(but)c(it)f(cannot)h(b)r(e)g(represen)n(ted)167 3046
y(in)39 b FG(F)p FK(.)h(Indeed,)j(supp)r(ose)e(that)f
FH(t)g FK(of)g(t)n(yp)r(e)g Fq(Int)p FJ(!)p Fq(Int)g
FK(represen)n(ts)f FH(D)j FK(and)f(let)d FH(n)i FK(b)r(e)f(the)167
3131 y(co)r(de)h(of)h FH(t)p FK(.)74 b(Then)40 b FH(A)p
FK(\()p FH(n;)12 b(])p FK(\()p FH(n)p FK(\)\))40 b(is)f(the)h(co)r(de)h
(of)g FH(t)p 1626 3092 V 12 w(n)p FK(,)i(and)f FH(N)7
b FK(\()p FH(A)p FK(\()p FH(n;)12 b(])p FK(\()p FH(n)p
FK(\)\)\))39 b(that)i(of)g(its)167 3216 y(normal)28 b(form.)36
b(But)28 b(b)n(y)h(de\014nition)e(of)i FH(t)p FK(,)f
FH(t)p 1348 3177 V 12 w(n)20 b Fu( )p 1499 3155 154 3
v 20 w FH(D)r FK(\()p FH(n)p FK(\),)28 b(so)h FH(N)7
b FK(\()p FH(A)p FK(\()p FH(n;)12 b(])p FK(\()p FH(n)p
FK(\)\)\))19 b(=)h FH(])p FK(\()p FH(D)r FK(\()p FH(n)p
FK(\)\))27 b(and)167 3301 y FH([)p FK(\()p FH(N)7 b FK(\()p
FH(A)p FK(\()p FH(n;)12 b(])p FK(\()p FH(n)p FK(\)\)\)\))18
b(=)i FH(D)r FK(\()p FH(n)p FK(\))27 b(whence)g FH(D)r
FK(\()p FH(n)p FK(\))20 b(=)f FH(D)r FK(\()p FH(n)p FK(\))c(+)h(1:)36
b(con)n(tradiction.)270 3422 y(F)-6 b(or)45 b(an)n(y)f(reasonable)g(co)
r(ding,)k FH(A)p FK(,)g FH(])43 b FK(and)i FH([)e FK(are)g(ob)n
(viously)i(represen)n(table)d(in)h FG(F)p FK(,)167 3507
y(so)28 b FH(N)35 b FK(itself)26 b(is)h FM(not)34 b FK(represen)n
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(v)n(ery)h(famous)g(result)e(in)g(Recursion)h(Theory)167
3754 y(\(due)44 b(to)g(T)-6 b(uring\),)48 b(namely)43
b(that)h(the)g(set)f(of)h(total)g(recursiv)n(e)e(functions)i(cannot)h
(b)r(e)167 3840 y(en)n(umerated)36 b(b)n(y)h(a)g(single)e(total)h
(recursiv)n(e)f(function.)61 b(In)37 b(particular)e(it)g(applies)h(to)g
(all)167 3925 y(sorts)28 b(of)g(calculi,)e(t)n(yp)r(ed)i(or)g(un)n(t)n
(yp)r(ed,)h(whic)n(h)d(satisfy)i(the)g(normalisation)f(theorem.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
122 125 bop 167 167 a FK(122)837 b Fz(CHAPTER)26 b(15.)64
b(REPRESENT)-6 b(A)g(TION)26 b(THEOREM)167 404 y Fy(15.1.3)94
b(Pro)m(v)-5 b(ably)28 b(total)g(functions)167 536 y
FK(A)k(recursiv)n(e)e(function)i FH(f)39 b FK(whic)n(h)32
b(is)f(total)g(from)h(I)-12 b(N)32 b(to)g(I)-12 b(N)31
b(is)h(called)e FM(pr)l(ovably)i(total)39 b FK(in)32
b(a)167 621 y(system)h(of)g(arithmetic)e FG(A)h FK(if)g
FG(A)h FK(pro)n(v)n(es)h(the)f(form)n(ula)g(whic)n(h)f(expresses)h
(\\for)h(all)e FH(n)p FK(,)i(the)167 706 y(program)d
FH(e)p FK(,)g(with)f(input)g FH(n)p FK(,)i(terminates)c(and)k(returns)e
(an)h(in)n(teger")f(for)h(some)f(algorithm)167 791 y
FH(e)36 b FK(represen)n(ting)g FH(f)8 b FK(.)62 b(The)36
b(precise)f(form)n(ulation)h(dep)r(ends)h(on)g(ho)n(w)g(w)n(e)f(write)f
(programs)167 876 y(formally)27 b(in)g FG(A)p FK(.)35
b(F)-6 b(or)28 b(example,)f(with)g(the)g(Kleene)f(notation:)971
1051 y FG(A)h FK(pro)n(v)n(es)i FJ(8)p FH(n:)12 b FJ(9)p
FH(m:)g Fq(T)1565 1062 y FD(1)1592 1051 y FK(\()p FH(e;)g(n;)g(m)p
FK(\))167 1227 y(where)32 b Fq(T)423 1238 y FD(1)451
1227 y FK(\()p FH(e;)12 b(n;)g(m)p FK(\))33 b(means)g(that)h(the)f
(program)h FH(e)f FK(terminates)e(with)h(output)i FH(m)g
FK(if)e(giv)n(en)167 1312 y(input)d FH(n)p FK(.)38 b(This)29
b(ma)n(y)g(itself)d(b)r(e)j(expressed)g(as)g FJ(9)p FH(m)1550
1286 y Fv(0)1567 1312 y FH(:)12 b(P)e FK(\()p FH(n;)i(m;)g(m)1902
1286 y Fv(0)1919 1312 y FK(\))28 b(where)g FH(P)39 b
FK(is)28 b(a)h(primitiv)n(e)167 1397 y(recursiv)n(e)40
b(predicate)h(and)i FH(m)1004 1371 y Fv(0)1062 1397 y
FK(is)e(the)h(\\transcript")g(of)g(the)f(computation.)78
b(The)42 b(t)n(w)n(o)167 1482 y(quan)n(ti\014ers)34 b
FJ(9)p FH(m:)12 b FJ(9)p FH(m)744 1457 y Fv(0)761 1482
y FH(:)45 b FK(can)34 b(b)r(e)f(replaced)g(b)n(y)h(a)f(single)g(one)g
FJ(9)p FH(p:)45 b FK(using)34 b(some)f(\(primitiv)n(e)167
1567 y(recursiv)n(e\))g(co)r(ding)h(of)h(pairs.)55 b(W)-6
b(e)34 b(prefer)f(to)i(b)r(e)f(no)h(more)f(sp)r(eci\014c)f(ab)r(out)i
(this)f(precise)167 1652 y(form)n(ulation,)27 b(but)h(w)n(e)g(notice)e
(that)i(termination)e(is)h(expressed)h(b)n(y)h(a)f(\005)2144
1627 y FD(0)2144 1670 y(2)2199 1652 y FK(form)n(ula)2429
1627 y FD(1)2457 1652 y FK(.)270 1828 y(In)39 b FG(??)q
FK(,)i(w)n(e)d(sa)n(w)h(that)f(the)g(functions)h(represen)n(table)e(in)
h FG(T)g FK(are)h(pro)n(v)l(ably)h(total)e(in)167 1913
y(P)n(eano)29 b(arithmetic)24 b FG(P)-23 b(A)p FK(,)28
b(and)g(the)g(con)n(v)n(erse)g(is)f(also)h(true.)35 b(Here)26
b(w)n(e)h(ha)n(v)n(e:)167 2088 y FG(Prop)r(osition)f
FK(The)h(functions)h(represen)n(table)f(in)g FG(F)h FK(are)g(pro)n(v)l
(ably)h(total)f(in)f FM(se)l(c)l(ond)h(or)l(der)167 2173
y FK(P)n(eano)h(arithmetic)24 b FG(P)-23 b(A)806 2184
y Fb(2)838 2173 y FK(.)167 2349 y FG(Pro)r(of)34 b FK(An)h(ob)t(ject)f
FH(f)41 b FK(of)35 b(t)n(yp)r(e)g Fq(Int)p FJ(!)p Fq(Int)f
FK(giv)n(es)h(rise)e(to)h(an)h(algorithm)f(whic)n(h,)h(giv)n(en)f(an)
167 2434 y(in)n(teger)25 b FH(n)p FK(,)i(returns)f FJ(j)p
FH(f)8 b FJ(j)p FK(\()p FH(n)p FK(\);)25 b(w)n(e)h(ha)n(v)n(e)h
(describ)r(ed)e(ho)n(w)i(to)g(do)f(this)g(already)-6
b(.)36 b(No)n(w)26 b(w)n(e)g(w)n(an)n(t)167 2519 y(to)k(sho)n(w)h(that)
f(this)f(program)i(terminates.)40 b(W)-6 b(e)29 b(mak)n(e)h(use)g(of)g
(the)g(strong)h(normalisation)167 2604 y(theorem,)26
b(and)g(b)n(y)h(examining)e(the)h(mathematical)e(principles)f(emplo)n
(y)n(ed)j(in)f(the)h(pro)r(of)h(w)n(e)167 2689 y(obtain)h(the)g
(result.)270 2816 y(What)39 b(matters)f(is)f(essen)n(tially)g(the)h
(reducibilit)n(y)d(of)j FH(f)46 b FK(alone)38 b(\(together)g(with)g
(that)167 2901 y(of)c(the)f(n)n(umerals,)h(whic)n(h)f(is)g
(immediate\).)49 b(W)-6 b(e)33 b(only)h(use)f(\014nitely)f(man)n(y)i
(reducibilities,)167 2987 y(whic)n(h)28 b(sa)n(v)n(es)i(us)f(from)f
(the)h(fact)f(that)h(\(as)g(in)f FG(T)p FK(\))h(reducibilit)n(y)c(is)j
(not)i(globally)e(de\014nable.)167 3072 y(The)i(reducibilit)n(y)d
(predicates)j(are)g(de\014nable)g(b)n(y)h(second)g(order)f(quan)n
(ti\014cation)h(o)n(v)n(er)g(sets)167 3157 y(of)d(\(terms)f(co)r(ded)g
(as\))i(in)n(tegers.)34 b(The)28 b(mathematical)d(principles)g(w)n(e)j
(ha)n(v)n(e)g(used)g(are:)270 3320 y FJ(\017)34 b FK(induction)28
b(on)g(the)f(reducibilit)n(y)e(predicates)i(for)h(the)f(t)n(yp)r(es)h
(in)n(v)n(olv)n(ed)g(in)f FH(f)8 b FK(,)270 3464 y FJ(\017)34
b FK(the)e(comprehension)f(sc)n(heme)f(and)j(second)f(order)f(quan)n
(ti\014cation,)i(whic)n(h)e(allo)n(w)g(us)339 3550 y(to)d(de\014ne)g(a)
g(reducibilit)n(y)d(candidate)j(from)f(a)h(parametrised)f(reducibilit)n
(y)-6 b(.)167 3712 y(But)28 b FG(P)-23 b(A)400 3723 y
Fb(2)458 3712 y FK(is)27 b(precisely)f(the)h(system)h(of)g(arithmetic)d
(with)h(induction,)i(comprehension)f(and)167 3798 y(second)h(order)g
(quan)n(ti\014cation.)1606 b Fu(\003)p 167 3860 1003
3 v 246 3903 a FC(1)273 3925 y FB(See)27 b(fo)r(otnote)h(page)g
Ft(??)o FB(.)p eop
PStoPSsaved restore
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userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
123 126 bop 167 167 a Fz(15.2.)65 b(PR)n(OOFS)27 b(INTO)f(PR)n(OGRAMS)
1268 b FK(123)167 404 y FG(Remark)48 b FK(Let)25 b(us)g(p)r(oin)n(t)g
(out)g(brie\015y)f(the)h(status)g(of)g(functions)g(whic)n(h)f(are)h
(pro)n(v)l(ably)h(total)167 490 y(in)h(a)h(system)f(of)i(arithmetic)c
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PStoPSxform concat
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
129 132 bop 167 167 a Fz(15.2.)65 b(PR)n(OOFS)27 b(INTO)f(PR)n(OGRAMS)
1268 b FK(129)270 404 y(If)32 b FH(T)41 b FK(is)30 b(a)i(t)n(yp)r(e)f
(with)f(free)h(v)l(ariables)g FH(X)1392 415 y FD(1)1420
404 y FH(;)12 b(:)g(:)g(:)f(;)h(X)1631 415 y Fw(p)1691
404 y FK(w)n(e)30 b(de\014ne)i(inductiv)n(ely)d(a)j(term)e
FH(\023)2633 415 y Fw(T)167 490 y FK(of)e(t)n(yp)r(e)g
FJ(h)-12 b(h)p FH(T)10 b FJ(i)-12 b(i)30 b FK(with)c(free)h(\014rst)h
(order)g(v)l(ariables)f FH(x)1533 501 y FD(1)1561 490
y FH(;)12 b(:)g(:)g(:)g(;)g(x)1755 501 y Fw(p)1810 490
y FK(of)28 b(t)n(yp)r(es)g FH(X)2137 501 y FD(1)2166
490 y FH(;)12 b(:)g(:)g(:)f(;)h(X)2377 501 y Fw(p)2405
490 y FK(:)270 658 y FJ(\017)34 b FH(\023)363 669 y Fw(X)431
658 y FK(=)19 b FH(x)543 633 y Fw(X)270 807 y FJ(\017)34
b FH(\023)363 818 y Fw(U)5 b Fv(!)p Fw(V)515 807 y FK(=)19
b FH(\025y)665 781 y Fv(h)-8 b(h)p Fw(U)5 b Fv(i)-8 b(i)767
807 y FH(:)12 b(\023)822 818 y Fw(V)893 807 y FK(\(note)28
b(that)g FH(y)i FK(do)r(es)e(not)g(o)r(ccur)g(in)f FH(\023)1883
818 y Fw(V)1926 807 y FK(\))270 955 y FJ(\017)34 b FH(\023)363
966 y FD(\005)p Fw(X)q(:)9 b(V)528 955 y FK(=)19 b(\003)p
FH(X)r(:)12 b(\025x)818 930 y Fw(X)866 955 y FH(:)g(\023)921
966 y Fw(V)992 955 y FK(\(where)26 b FH(x)i FM(may)35
b FK(o)r(ccur)27 b(in)h FH(\023)1748 966 y Fw(V)1791
955 y FK(\))167 1124 y(In)g(particular,)f(if)f FH(T)38
b FK(is)27 b(closed,)g FJ(h)-12 b(h)p FH(T)10 b FJ(i)-12
b(i)30 b FK(is)d(inhabited)g(b)n(y)h(the)g(closed)f(term)f
FH(\023)2220 1135 y Fw(T)2259 1124 y FK(,)i(for)g(instance)611
1308 y FJ(h)-12 b(h)p FK(\005)p FH(X)r(:)12 b(X)6 b FJ(i)-12
b(i)21 b FK(=)e(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)84
b FK(and)79 b FH(\023)1623 1319 y FD(\005)p Fw(X)q(:)9
b(X)1792 1308 y FK(=)20 b(\003)p FH(X)r(:)12 b(\025x)2083
1282 y Fw(X)2130 1308 y FH(:)g(x)270 1589 y FK(If)30
b FH(t)g FK(is)f(term)f(of)i(t)n(yp)r(e)g FH(T)40 b FK(with)29
b(free)g(t)n(yp)r(e)g(v)l(ariables)h FH(X)1789 1600 y
FD(1)1817 1589 y FH(;)12 b(:)g(:)g(:)g(;)g(X)2029 1600
y Fw(p)2087 1589 y FK(and)30 b(free)f(\014rst)h(order)167
1674 y(v)l(ariables)e FH(y)492 1685 y FD(1)520 1674 y
FH(;)12 b(:)g(:)g(:)f(;)h(y)708 1685 y Fw(q)763 1674
y FK(of)30 b(t)n(yp)r(es)e FH(U)1082 1685 y FD(1)1110
1674 y FH(;)12 b(:)g(:)g(:)g(;)g(U)1312 1685 y Fw(q)1367
1674 y FK(w)n(e)28 b(de\014ne)h(inductiv)n(ely)e(a)i(term)e
FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)31 b FK(\(without)167
1760 y(junk\))24 b(of)f(t)n(yp)r(e)g FJ(h)-12 b(h)p FH(T)10
b FJ(i)-12 b(i)25 b FK(with)d(free)g(t)n(yp)r(e)i(v)l(ariables)e
FH(X)1530 1771 y FD(1)1558 1760 y FH(;)12 b(:)g(:)g(:)g(;)g(X)1770
1771 y Fw(p)1821 1760 y FK(and)24 b(free)e(\014rst)h(order)g(v)l
(ariables)167 1845 y FH(x)206 1856 y FD(1)234 1845 y
FH(;)12 b(:)g(:)g(:)g(;)g(x)428 1856 y Fw(p)456 1845
y FH(;)g(y)521 1856 y FD(1)548 1845 y FH(;)g(:)g(:)g(:)g(;)g(y)737
1856 y Fw(q)791 1845 y FK(of)28 b(t)n(yp)r(es)g FH(X)1118
1856 y FD(1)1146 1845 y FH(;)12 b(:)g(:)g(:)g(;)g(X)1358
1856 y Fw(p)1386 1845 y FH(;)g FJ(h)-12 b(h)p FH(U)1506
1856 y FD(1)1534 1845 y FJ(i)g(i)p FH(;)12 b(:)g(:)g(:)h(;)f
FJ(h)-12 b(h)p FH(U)1821 1856 y Fw(q)1849 1845 y FJ(i)g(i)p
FK(:)270 2013 y FJ(\017)34 b(h)-12 b(h)p FH(y)418 1988
y Fw(T)458 2013 y FJ(i)g(i)20 b FK(=)f FH(y)630 1988
y Fv(h)-8 b(h)p Fw(T)7 b Fv(i)-8 b(i)270 2162 y FJ(\017)34
b(h)-12 b(h)p FH(\025y)458 2136 y Fw(U)501 2162 y FH(:)12
b(v)s FJ(i)-12 b(i)20 b FK(=)f FH(\025y)780 2136 y Fv(h)-8
b(h)p Fw(U)5 b Fv(i)-8 b(i)882 2162 y FH(:)12 b FJ(h)-12
b(h)p FH(v)s FJ(i)g(i)270 2310 y(\017)34 b(h)-12 b(h)p
FH(t)12 b(u)p FJ(i)-12 b(i)22 b FK(=)d FJ(h)-12 b(h)p
FH(t)p FJ(i)g(i)12 b(h)-12 b(h)p FH(u)p FJ(i)g(i)270
2459 y(\017)34 b(h)-12 b(h)p FK(\003)p FH(X)r(:)12 b(v)s
FJ(i)-12 b(i)22 b FK(=)d(\003)p FH(X)r(:)12 b(\025x)909
2433 y Fw(X)957 2459 y FH(:)g FJ(h)-12 b(h)p FH(v)s FJ(i)g(i)29
b FK(\(note)f(that)g FH(x)g FK(ma)n(y)g(o)r(ccur)f(in)g
FJ(h)-12 b(h)p FH(v)s FJ(i)g(i)p FK(\))270 2607 y FJ(\017)34
b(h)-12 b(h)p FH(t)12 b(U)7 b FJ(i)-12 b(i)22 b FK(=)d
FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)12 b(h)-12 b(h)p FH(U)7
b FJ(i)-12 b(i)12 b FH(\023)904 2618 y Fw(U)270 2755
y FJ(\017)34 b(h)-12 b(h)p FK(\012)p FJ(i)g(i)22 b FK(=)d
FH(\023)592 2766 y Fp(Emp)710 2755 y FK(=)g(\003)p FH(X)r(:)12
b(\025x)1000 2730 y Fw(X)1048 2755 y FH(:)g(x)167 2924
y FK(Again)28 b(the)f(reader)g(can)h(c)n(hec)n(k)g(the)f(follo)n(wing)g
(prop)r(erties)836 3104 y FJ(h)-12 b(h)p FH(t)p FK([)p
FH(u=y)1032 3078 y Fw(U)1075 3104 y FK(])p FJ(i)g(i)60
b FK(=)e FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)p FK([)p FJ(h)g(h)p
FH(u)p FJ(i)g(i)t FH(=y)1634 3078 y Fv(h)k(h)p Fw(U)5
b Fv(i)-8 b(i)1737 3104 y FK(])940 3275 y FH(\023)964
3286 y Fw(T)7 b FD([)p Fw(U)r(=X)t FD(])1196 3275 y FK(=)58
b FH(\023)1332 3286 y Fw(T)1371 3275 y FK([)p FJ(h)-12
b(h)p FH(U)7 b FJ(i)-12 b(i)s FH(=X)6 b FK(][)p FH(\023)1690
3286 y Fw(U)1732 3275 y FH(=x)1805 3249 y Fv(h)-8 b(h)p
Fw(U)5 b Fv(i)-8 b(i)1909 3275 y FK(])840 3446 y FJ(h)c(h)p
FH(t)p FK([)p FH(U)t(=X)6 b FK(])p FJ(i)-12 b(i)61 b
FK(=)d FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)p FK([)p FJ(h)g(h)p
FH(U)7 b FJ(i)-12 b(i)t FH(=X)6 b FK(][)p FH(\023)1737
3457 y Fw(U)1780 3446 y FH(=x)1853 3421 y Fv(h)-8 b(h)p
Fw(U)5 b Fv(i)-8 b(i)1956 3446 y FK(])167 3626 y(whic)n(h)27
b(are)g(needed)h(for)g(the)f(preserv)l(ation)h(of)g(con)n(v)n(ersions:)
1017 3810 y(if)f FH(t)20 b Fu( )g FH(u)28 b FK(then)f
FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)22 b Fu( )e FJ(h)-12 b(h)p
FH(u)p FJ(i)g(i)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
130 133 bop 167 167 a FK(130)837 b Fz(CHAPTER)26 b(15.)64
b(REPRESENT)-6 b(A)g(TION)26 b(THEOREM)270 404 y FK(No)n(w)i(w)n(e)f
(see)g(that)830 558 y FJ(h)-12 b(h)p Fq(Int)q FJ(i)g(i)60
b FK(=)e(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)g FJ(!)p
FK(\()p FH(X)g FJ(!)p FH(X)g FK(\))p FJ(!)p FH(X)868
728 y FJ(h)-12 b(h)p 910 689 42 3 v FH(n)q FJ(i)g(i)60
b FK(=)e(\003)p FH(X)r(:)12 b(\025x)1383 703 y Fw(X)1431
728 y FH(:)g(\025y)1539 703 y Fw(X)1586 728 y FH(:)g(\025z)1692
703 y Fw(X)t Fv(!)p Fw(X)1834 728 y FH(:)g(z)1900 703
y Fw(n)1945 728 y FH(y)503 988 y Fq(w)n(eak)n(en)p 721
950 V 12 w FH(n)20 b Fu( )g FJ(h)-12 b(h)p 914 950 V
FH(n)q FJ(i)g(i)337 b FK(and)g Fq(contract)12 b FJ(h)-12
b(h)p 2073 950 V FH(n)q FJ(i)g(i)21 b Fu( )p 2268 950
V 20 w FH(n)167 1162 y FK(Finally)-6 b(,)27 b(a)h(term)e
FH(t)i FK(of)g(t)n(yp)r(e)g Fq(Int)p FJ(!)p Fq(Int)g
FK(with)e(junk)j(can)f(b)r(e)g(replaced)e(b)n(y)901 1335
y FH(t)926 1309 y Fv(0)962 1335 y FK(=)19 b FH(\025z)1110
1309 y Fp(Int)1171 1335 y FH(:)12 b Fq(contract)p FK(\()p
FJ(h)-12 b(h)p FH(t)p FJ(i)g(i)p FK(\()p Fq(w)n(eak)n(en)14
b FH(z)s FK(\)\))167 1508 y(without)27 b(junk.)2037 b
Fu(\003)p eop
PStoPSsaved restore
%%Page: (134,135) 68
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
131 134 bop 167 875 a FF(App)5 b(endix)28 b(A)167 1169
y(Seman)-5 b(tics)28 b(of)g(System)f(F)167 1377 y FI(b)m(y)h(P)m(aul)f
(T)-10 b(a)m(ylor)167 1698 y FK(In)35 b(this)f(app)r(endix)h(w)n(e)f
(shall)g(giv)n(e)g(a)h(seman)n(tics)f(for)h(system)f
FG(F)g FK(in)g(terms)g(of)h(coherence)167 1783 y(spaces.)g(In)23
b(particular)g(w)n(e)g(shall)f(in)n(terpret)g(univ)n(ersal)h
(abstraction)h(b)n(y)g(means)g(of)g(a)g(kind)g(of)167
1868 y(\\trace",)f(sho)n(wing)e(that)g(the)g(primary)f(and)i(secondary)
g(equations)f(hold.)34 b(W)-6 b(e)20 b(shall)g(examine)167
1954 y(the)26 b(w)n(a)n(y)h(in)f(whic)n(h)f(its)h(terms)f(are)h
(\\uniform")h(o)n(v)n(er)g(all)e(t)n(yp)r(es.)35 b(Finally)25
b(w)n(e)h(shall)g(attempt)167 2039 y(to)42 b(calculate)f(some)h(univ)n
(ersal)g(t)n(yp)r(es)g(suc)n(h)h(as)g Fq(Emp)30 b FK(=)f(\005)p
FH(X)r(:)12 b(X)6 b FK(,)45 b Fq(Sgl)30 b FK(=)g(\005)p
FH(X)r(:)12 b(X)25 b FJ(!)20 b FH(X)6 b FK(,)167 2124
y Fq(Bo)r(ol)19 b FK(=)h(\005)p FH(X)r(:)12 b(X)21 b
FJ(!)16 b FH(X)21 b FJ(!)16 b FH(X)34 b FK(and)28 b Fq(Int)20
b FK(=)f(\005)p FH(X)r(:)12 b(X)21 b FJ(!)16 b FK(\()p
FH(X)21 b FJ(!)c FH(X)6 b FK(\))15 b FJ(!)h FH(X)6 b
FK(.)167 2363 y FE(A.1)114 b(T)-10 b(erms)27 b(of)h(univ)m(ersal)g(t)m
(yp)s(e)167 2539 y Fy(A.1.1)95 b(Finite)28 b(appro)m(ximation)167
2671 y FK(W)-6 b(e)34 b(ha)n(v)n(e)i(already)e(said)h(in)f(section)f
FG(??)i FK(that)g(a)g(term)e(\003)p FH(X)r(:)12 b(t)34
b FK(of)h(univ)n(ersal)f(t)n(yp)r(e)h(\005)p FH(X)r(:)12
b(T)167 2757 y FK(is)29 b(in)n(tended)g(to)h(b)r(e)g(a)g(function)g
(whic)n(h)f(assigns)h(to)g(an)n(y)h(t)n(yp)r(e)f FH(U)37
b FK(a)30 b(term)e FH(t)p FK([)p FH(U)t(=X)6 b FK(])30
b(of)g(t)n(yp)r(e)167 2842 y FH(T)10 b FK([)p FH(U)t(=X)c
FK(].)35 b(In)26 b(particular,)g(the)g(in)n(terpretation)f(of)h(\003)p
FH(X)r(:)12 b(\025x:)g(x)27 b FK(is)e(to)i(b)r(e)f(the)g(function)g
(whic)n(h)167 2927 y(assigns)j(to)f(an)n(y)g(coherence)f(space)h
FJ(A)g FK(\(the)f(trace)g(of)5 b(\))28 b(the)f(iden)n(tit)n(y)g
(function,)h FM(i.e.)995 3106 y FJ(I)-7 b FH(d)1062 3080
y Fv(A)1125 3106 y FK(=)20 b FJ(f)p FK(\()p FJ(f)p FH(\013)p
FJ(g)p FH(;)12 b(\013)p FK(\))20 b(:)39 b FH(\013)20
b FJ(2)g(jAjg)167 3285 y FK(But)34 b(w)n(e)g(ha)n(v)n(e)h(a)f(problem)g
(of)g FM(size)p FK(:)49 b(there)33 b(is)g(a)i(prop)r(er)f(class)g(of)h
(coherence)e(spaces,)j(so)167 3370 y(ho)n(w)28 b(can)g(this)f(b)r(e)h
(a)g(legitimate)c(function?)270 3499 y(W)-6 b(e)42 b(can)h(solv)n(e)f
(this)g(problem)g(in)f(the)h(same)g(w)n(a)n(y)i(as)e(w)n(e)g(did)g(for)
h(functions,)j(b)n(y)167 3584 y(requiring)30 b(that)i(ev)n(ery)f
(domain)h(b)r(e)f(expressible)f(as)i(a)g(\\limit")d(of)j(\014nite)e
(domains.)47 b(Then)167 3669 y(b)n(y)29 b(con)n(tin)n(uit)n(y)e(w)n(e)h
(can)g(deriv)n(e)f(the)g(v)l(alue)h(of)h(a)f(univ)n(ersal)g(term)e(at)i
(an)h(arbitrary)f(domain)167 3754 y(from)38 b(its)f(v)l(alues)h(at)g
(\014nite)f(domains.)67 b(Since)37 b(there)g(are)h(only)g(coun)n(tably)
h(man)n(y)g(\014nite)167 3840 y(domains)c(up)h(to)f(isomorphism,)i(the)
d(function)i(is)e(de\014ned)i(b)n(y)f(a)h FM(set)f FK(|)g(so)g(long)h
(as)g(w)n(e)167 3925 y(ensure)28 b(that)g(its)e(v)l(alues)i(at)g
(isomorphic)f(domains)h(are)f(equal)h(\(along)g(the)g(isomorphisms\).)
1368 4101 y(131)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
132 135 bop 167 167 a FK(132)947 b Fz(APPENDIX)26 b(A.)55
b(SEMANTICS)27 b(OF)g(SYSTEM)h(F)167 404 y Fy(A.1.2)95
b(Saturated)28 b(domains)167 545 y FK(There)22 b(is)g(a)i(common)e(but)
i(misleading)d(alternativ)n(e)h(solution.)33 b(W)-6 b(e)23
b(c)n(ho)r(ose)h(a)f(\\big")h(domain)167 630 y(\012)29
b(whic)n(h)g(is)g(saturated)h(under)g(all)e(the)h(relev)l(an)n(t)g(op)r
(erations)h(on)g(t)n(yp)r(es,)h(and)f(restrict)e(our)167
715 y(notion)37 b(of)h(domain)f FJ(A)g FK(to)g(\\sub)r(domains")j(of)d
(\012.)63 b(Th)n(us)38 b(for)f(instance)g(if)f FJ(A)h
FK(is)f(suc)n(h)i(a)167 801 y(sub)r(domain)d(then)f(w)n(e)f(require)f
FJ(A)18 b(!)g(A)34 b FK(to)g(b)r(e)f(one)i(also;)i(in)c(particular)g
(\012)17 b FJ(!)h FK(\012)33 b(is)h(one.)167 886 y(Then)26
b(the)g(iden)n(tit)n(y)-6 b(,)25 b(b)r(eing)h(an)g(elemen)n(t)e(of)i
(\012)13 b FJ(!)h FK(\012,)26 b(whic)n(h)f(is)g(iden)n(ti\014ed)g(with)
g(a)h(subspace)167 971 y(of)32 b(\012,)h(is)e(an)i(elemen)n(t)c(of)k
(\012.)47 b(Scott's)33 b FJ(P)-6 b FH(!)34 b FK(mo)r(del)d([)p
FG(?)p FK(])h(is)f(a)h(w)n(ell-kno)n(wn)f(example)g(of)i(this)167
1056 y(approac)n(h,)38 b(and)d([)p FG(?)q FK(])f(examined)g(this)g(in)f
(detail)g(as)i(a)f(notion)h(of)g(mo)r(del)e(of)i(the)f(un)n(t)n(yp)r
(ed)167 1141 y(lam)n(b)r(da)28 b(calculus)656 1115 y
FD(1)683 1141 y FK(.)270 1287 y(Ho)n(w)n(ev)n(er,)h(b)r(esides)g(the)f
(fact)h(that)g(not)h(all)d(domains)j(are)e(represen)n(ted,)h(this)f
(approac)n(h)167 1373 y(has)h(sev)n(eral)e(pitfalls.)270
1537 y FJ(\017)34 b FK(Whereas)f(in)g(set)f(theory)i(the)e(notions)i
(of)g(elemen)n(t)c(and)k(t)n(yp)r(e)f(are)g(confused,)j(here)339
1622 y(w)n(e)c(ha)n(v)n(e)i(to)f(distinguish)f(b)r(et)n(w)n(een)f(\012)
i(as)g(the)f(\\univ)n(erse)h(of)g(elemen)n(ts")e(and)i(some)339
1707 y(domain)i FJ(V)40 b FK(whose)34 b(elemen)n(ts)e(ma)n(y)i(serv)n
(e)g(as)h(names)f(of)h(t)n(yp)r(es)f(|)g(a)g(\\univ)n(erse)h(of)339
1792 y(t)n(yp)r(es".)270 1952 y FJ(\017)f FK(It)e(is)f(not)h(go)r(o)r
(d)h(enough)h(to)e(construct)f(suc)n(h)i(a)f FJ(V)37
b FK(with)31 b(the)h(prop)r(ert)n(y)g(that)g(ev)n(ery)339
2038 y(domain)25 b(b)r(e)g(named)g(b)n(y)h(a)f(p)r(oin)n(t)g(of)g
FJ(V)6 b FK(:)34 b(this)24 b(is)g(lik)n(e)f(the)i(\\b)n(y)h(v)l(alues")
f(in)n(terpretation)339 2123 y(of)31 b(recursiv)n(e)d(functions.)42
b(W)-6 b(e)29 b(need)h(that)g(ev)n(ery)f FM(variable)g
FK(domain)h(b)r(e)f(named)h(b)n(y)h(a)339 2208 y(term)i(\(with)g(the)g
(same)h(free)f(v)l(ariables\))h(of)g(t)n(yp)r(e)g FJ(V)6
b FK(.)54 b(The)34 b(ob)n(vious)h(c)n(hoice)e(is)g(the)339
2293 y FM(c)l(ate)l(gory)28 b FK(of)g(domains)g(and)g(em)n(b)r
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b(It)339 2378 y(is,)e(ho)n(w)n(ev)n(er,)g(p)r(ossible)f(to)g(\\co)n(v)n
(er")h(it)e(with)g(a)h(domain,)i(although)f(the)f(tec)n(hniques)339
2463 y(required)25 b(for)h(this,)g(whic)n(h)f(are)h(set)f(out)h(in)g([)
p FG(?)p FK(],)h FJ(x)p FK(5.6,)g(are)f(m)n(uc)n(h)f(more)h(di\016cult)
e(than)339 2548 y(the)k(construction)f(of)i(\012.)270
2709 y FJ(\017)34 b FK(Isomorphic)24 b(t)n(yp)r(es)g(ma)n(y)g(b)r(e)g
(represen)n(ted)f(b)n(y)h(di\013eren)n(t)f(elemen)n(ts)f(of)i
FJ(V)6 b FK(,)25 b(and)g(there)e(is)339 2794 y(nothing)29
b(to)g(force)e(the)h(v)l(alues)g(of)h(univ)n(ersal)e(terms)g(at)h(suc)n
(h)h(elemen)n(ts)d(to)i(b)r(e)g(equal.)339 2879 y(This)i(means)h(that)g
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FH(X)r(:)12 b(\025x:)g(x)339 3134 y FK(is)27 b(v)n(ery)h(uneconomical.)
270 3295 y FJ(\017)34 b FK(It)49 b(is)e(p)r(ossible)h(to)h(mo)r(del)f
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3380 y(Constructions,)28 b(using)g(the)f(category)h(of)g(em)n(b)r
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(can)h(b)r(e)f(rigidly)e(em)n(b)r(edded)i(\()p Ft(??)o
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PStoPSxform concat
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b(our)h(case)f(w)n(e)f(w)n(an)n(t)i(it)e(to)h(b)r(e)g
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FM(cf.)167 1433 y FK(section)33 b FG(??)q FK(\),)j(and)f(in)f(the)g
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(to)i(what)167 1518 y(exten)n(t)27 b(this)g(is)g(true.)270
1737 y(W)-6 b(e)42 b(obtain)g(p)r(o)n(w)n(er)g(from)f(this)g(condition)
g(b)n(y)h(man)n(ufacturing)h(automorphisms)f(to)167 1822
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1907 y(itself)e(\(either)h(lifted)f(or)j(amalgamated)f(on)h(some)f(sub)
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y(\\left-righ)n(t")28 b(symmetry.)39 b(\(W)-6 b(e)29
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e(the)h(next)167 2077 y(section.\))68 b(Giv)n(en)38 b(a)i(subspace)f
(inclusion)f FJ(A)28 b(\032)f(B)r FK(,)42 b(a)d(\\uniform")g(elemen)n
(t)e(of)i FJ(B)21 b FK(+)2562 2088 y Fv(A)2624 2077 y
FJ(B)167 2162 y FK(cannot)32 b(b)r(e)e(in)g(either)f(the)h(left)f(or)i
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b(This)23 b(is)h(the)f(con)n(undrum)i(of)g(the)e(donk)n(ey)j(whic)n(h)d
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y(its)g(left)f(and)j(righ)n(t.)270 2636 y(There)42 b(is)f(a)h(similar)e
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b(for)e(\014elds)g(whic)n(h)f(underlies)g(Galois)167
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2806 y(the)33 b(automorphisms)g(of)h FH(M)40 b FK(\014xing)33
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FH(K)5 b FK(.)51 b(F)-6 b(or)34 b(\014elds,)g FH(M)40
b FK(is)32 b(the)167 2892 y FM(normal)c(closur)l(e)f
FK(|)h(a)g(more)f(complex)g(construction)g(than)i(our)f
FJ(B)18 b FK(+)2062 2903 y Fv(A)2120 2892 y FJ(B)r FK(.)270
3110 y(Uniformit)n(y)k(with)g(resp)r(ect)g(to)i(automorphisms)g(is)e(a)
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3195 y(including)31 b(Scott's.)51 b(Ho)n(w)n(ev)n(er)32
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b(to)h(sub)r(domains:)56 b(the)38 b(v)l(alue)g(of)g(a)g(univ)n(ersal)f
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3366 y FK(than)29 b(that)g(at)f FJ(B)i FK(\(where)e FJ(A)20
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y(space)c(mo)r(del)e(is)h FM(uniform)p FK(.)270 3925
y(W)-6 b(e)28 b(mak)n(e)f(this)h(v)l(ague)h(argumen)n(t)f(precise)e(in)
h(A.4.1.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
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PStoPSxform concat
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g(functions)g(to)g(de\014ne)h(when)f(an)h(elemen)n(t)c
FH(a)k FK(of)f FJ(A)h FK(is)e(\\less)h(than")167 1180
y(an)35 b(elemen)n(t)c FH(b)j FK(of)g FJ(B)j FK(\(but)d(not)g
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1657 y(and)i(re\015ects)e(atoms)h(and)h(the)e(coherence)g(relation.)37
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FJ(i)22 b FK(:)39 b FH(\013)20 b FJ(2)g(jAjg)937 2324
y(T)-6 b FH(r)r FK(\()p FH(p)p FK(\))20 b(=)58 b FJ(fhf)p
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b FH(\013)20 b FJ(2)g(jAjg)167 2520 y FK(W)-6 b(e)25
b(shall)g(often)h(write)e FH(e)c FK(:)f FJ(A)h(!)g(B)28
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2998 y(of)33 b(the)f(set)g(of)h(represen)n(ted)f(v)l(alues.)50
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3199 a FC(2)273 3220 y FB(There)35 b(are)g(reasons)h(for)f(w)n(eak)n
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(remain)f(v)m(alid.)48 b(Ho)n(w)n(ev)n(er)35 b(for)e
FA(rigid)167 3643 y FB(adjunctions)26 b(1)519 3652 y
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g(order.)246 3692 y FC(3)273 3713 y FB(In)e(fact)h Fl(\024)f
FB(is)g(not)h(a)g(partial)e(order)h(but)g(a)h(category)-5
b(,)27 b(b)r(ecause)f(it)e(dep)r(ends)g(on)i Fx(e)p FB(.)34
b(Applying)23 b(this)i(to)167 3784 y(a)k(functor)f Fl(T)14
b FB(,)29 b(w)n(e)h(obtain)d(a)i(category)g(with)f(ob)s(jects)h(the)f
(pairs)f(\()p Fl(A)p Fx(;)10 b(b)p FB(\))29 b(for)f Fx(b)17
b Fl(2)g(T)e FB(\()p Fl(A)p FB(\))28 b(and)g(morphisms)167
3854 y(giv)n(en)g(in)f(this)g(w)n(a)n(y)i(b)n(y)f(em)n(b)r(eddings;)f
(this)g(is)g(called)g(the)h FA(total)h(c)m(ate)m(gory)g
FB(or)f FA(Gr)m(othendie)m(ck)h(\014br)m(ation)f FB(of)167
3925 y Fl(T)42 b FB(and)28 b(is)f(written)f(\006)647
3920 y Fl(\016)689 3925 y Fx(X)q(:)10 b Fl(T)16 b FB(.)p
eop
PStoPSsaved restore
%%Page: (138,139) 70
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
135 138 bop 167 167 a Fz(A.2.)63 b(RIGID)27 b(EMBEDDINGS)1499
b FK(135)270 404 y(Observ)n(e)34 b(then)h(that)f(for)h(inclusions)e
(the)h(em)n(b)r(edding)f(is)h(just)g(the)g(iden)n(tit)n(y)f(and)i(the)
167 490 y(pro)t(jection)27 b(is)g(the)h(restriction:)995
735 y FH(e)p FK(\()p FH(a)p FK(\))20 b(=)f FH(a)157 b(p)p
FK(\()p FH(b)p FK(\))19 b(=)g FH(b)d FJ(\\)g(jAj)167
1131 y Fy(A.2.1)95 b(F)-8 b(unctorialit)m(y)29 b(of)e(arro)m(w)167
1280 y FK(The)k(reason)i(for)f(using)g(pairs)f(of)h(maps)g(for)g(appro)
n(ximations)h(is)e(that)h(w)n(e)f(need)g(to)h(mak)n(e)167
1365 y(the)g(function-space)g(functorial)g(\(p)r(ositiv)n(e\))f(in)g
(its)g(\014rst)i(argumen)n(t:)44 b(if)32 b FJ(A)2221
1340 y Fv(0)2269 1365 y FK(appro)n(ximates)167 1450 y
FJ(A)c FK(then)g(w)n(e)f(need)g FJ(A)740 1425 y Fv(0)773
1450 y FJ(!)16 b(B)30 b FK(to)e(appro)n(ximate)g FJ(A)15
b(!)i(B)30 b FK(and)e(not)g FM(vic)l(e)g(versa)p FK(.)270
1613 y(Indeed)23 b(if)e FH(e)f FK(:)g FJ(A)703 1587 y
Fv(0)739 1613 y Fu(\032)g FJ(A)i FK(and)i FH(f)j FK(:)19
b FJ(B)1196 1587 y Fv(0)1233 1613 y Fu(\032)h FJ(B)k
FK(then)f(w)n(e)e(ha)n(v)n(e)j FH(e)8 b FJ(!)f FH(f)27
b FK(:)20 b(\()p FJ(A)2117 1587 y Fv(0)2141 1613 y FJ(!)8
b(B)2267 1587 y Fv(0)2284 1613 y FK(\))19 b Fu(\032)h
FK(\()p FJ(A)8 b(!)g(B)r FK(\))167 1698 y(b)n(y)927 1937
y(\()p FH(e)16 b FJ(!)g FH(f)8 b FK(\))1157 1912 y FD(+)1198
1937 y FK(\()p FH(t)1250 1912 y Fv(0)1266 1937 y FK(\)\()p
FH(a)p FK(\))59 b(=)46 b FH(f)1584 1912 y FD(+)1626 1937
y FK(\()p FH(t)1678 1912 y Fv(0)1694 1937 y FK(\()p FH(e)1753
1912 y Fv(\000)1794 1937 y FH(a)p FK(\)\))927 2023 y(\()p
FH(e)16 b FJ(!)g FH(f)8 b FK(\))1157 1997 y Fv(\000)1198
2023 y FK(\()p FH(t)p FK(\)\()p FH(a)1340 1997 y Fv(0)1356
2023 y FK(\))59 b(=)46 b FH(f)1584 1997 y Fv(\000)1626
2023 y FK(\()p FH(t)p FK(\()p FH(e)1737 1997 y FD(+)1778
2023 y FH(a)1814 1997 y Fv(0)1831 2023 y FK(\)\))167
2267 y(for)29 b FH(a)21 b FJ(2)f(A)p FK(,)29 b FH(a)542
2242 y Fv(0)579 2267 y FJ(2)21 b(A)703 2242 y Fv(0)719
2267 y FK(,)29 b FH(t)21 b FK(:)f FJ(A)c(!)h(B)31 b FK(and)f
FH(t)1253 2242 y Fv(0)1289 2267 y FK(:)21 b FJ(A)1385
2242 y Fv(0)1418 2267 y FJ(!)16 b(B)1552 2242 y Fv(0)1569
2267 y FK(.)38 b(\(W)-6 b(e)29 b(lea)n(v)n(e)f(the)g(reader)h(to)g(c)n
(hec)n(k)f(the)167 2352 y(inequalities.\))270 2515 y(Recall)33
b(that)i(the)g(tok)n(ens)h(of)f FJ(A)18 b(!)g(B)37 b
FK(are)e(of)h(the)e(form)h(\()p FH(a;)12 b(\014)t FK(\))34
b(where)g FH(a)h FK(is)f(a)i(clique)167 2600 y(\(\014nite)g(coheren)n
(t)h(subset\))g(of)h FJ(jAj)e FK(and)i FH(\014)j FK(is)36
b(a)h(tok)n(en)h(of)g FJ(jB)r(j)p FK(.)63 b(If)37 b FH(e)26
b FK(:)g FJ(jA)2262 2574 y Fv(0)2278 2600 y FJ(j)g Fu(\032)h
FJ(jAj)36 b FK(and)167 2685 y FH(f)27 b FK(:)20 b FJ(jB)335
2659 y Fv(0)351 2685 y FJ(j)f Fu(\032)h FJ(jB)r(j)25
b FK(are)h(rigid)e(em)n(b)r(eddings)i(then)f(the)h(e\013ect)e(on)i(the)
g(tok)n(en)g(\()p FH(a)2199 2659 y Fv(0)2215 2685 y FH(;)12
b(\014)2289 2659 y Fv(0)2306 2685 y FK(\))25 b(of)h FJ(A)2495
2659 y Fv(0)2524 2685 y FJ(!)13 b(B)2655 2659 y Fv(0)167
2770 y FK(is)27 b(simply)f(the)i(corresp)r(onding)g(renaming)f
(throughout,)j FM(i.e.)e FK(\()p FH(e)1910 2745 y FD(+)1952
2770 y FH(a)1988 2745 y Fv(0)2004 2770 y FH(;)12 b(f)c(\014)2120
2745 y Fv(0)2136 2770 y FK(\).)270 2933 y(In)39 b(particular)e(the)h
(tok)n(en)h(\()p FJ(f)p FH(\013)1148 2907 y Fv(0)1165
2933 y FJ(g)p FH(;)12 b(\013)1275 2907 y Fv(0)1291 2933
y FK(\))38 b(of)h FJ(I)-7 b FH(d)1517 2907 y Fv(A)1557
2890 y Fa(0)1615 2933 y FK(is)37 b(mapp)r(ed)i(to)f(\()p
FJ(f)p FH(e\013)2214 2907 y Fv(0)2231 2933 y FJ(g)p FH(;)12
b(e\013)2373 2907 y Fv(0)2390 2933 y FK(\),)41 b(so)d(the)167
3018 y(iden)n(tit)n(y)26 b(is)h(uniform)h(in)f(the)g(sense)h(that)1059
3263 y FJ(I)-7 b FH(d)1126 3238 y Fv(A)1166 3221 y Fa(0)1205
3263 y FK(=)19 b FJ(I)-7 b FH(d)1345 3238 y Fv(A)1404
3263 y FJ(\\)16 b(jA)1543 3238 y Fv(0)1574 3263 y FJ(!)h(A)1717
3238 y Fv(0)1733 3263 y FJ(j)167 3509 y FK(where)27 b
FJ(A)427 3483 y Fv(0)463 3509 y Fu(\032)20 b FJ(A)28
b FK(is)f(a)h(subspace.)270 3754 y(Coherence)53 b(spaces)g(and)h(rigid)
d(em)n(b)r(eddings)i(|)g(or)g(equiv)l(alen)n(tly)f FM(G)6
b FK(raphs)54 b(and)167 3840 y FM(em)5 b FK(b)r(eddings)29
b(|)g(form)f(a)h(category)g FG(Gem)o FK(,)g(and)g(w)n(e)f(ha)n(v)n(e)h
(sho)n(wn)h(that)f FJ(!)f FK(is)g(a)h FM(c)l(ovariant)167
3925 y FK(functor)f(of)g(t)n(w)n(o)g(argumen)n(ts)g(from)g
FG(Gem)o FH(;)12 b FG(Gem)27 b FK(to)h FG(Gem)p FK(.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
136 139 bop 167 167 a FK(136)947 b Fz(APPENDIX)26 b(A.)55
b(SEMANTICS)27 b(OF)g(SYSTEM)h(F)167 404 y FE(A.3)114
b(In)m(terpretation)29 b(of)e(T)m(yp)s(es)167 582 y FK(W)-6
b(e)28 b(can)h(use)f(this)g(to)g(express)g(an)n(y)i(t)n(yp)r(e)e
FH(T)38 b FK(of)29 b FG(F)f FK(with)f FH(n)i FK(free)e(t)n(yp)r(e)h(v)l
(ariables)g FH(X)2434 593 y FD(1)2462 582 y FH(;)12 b(:::;)g(X)2638
593 y Fw(n)167 667 y FK(as)28 b(a)g(functor)g([)-12 b([)q
FH(T)10 b FK(])-12 b(])20 b(:)g FG(Gem)888 637 y Fw(n)941
667 y FJ(!)g FG(Gem)27 b FK(as)h(follo)n(ws:)251 887
y(1.)35 b(If)28 b FH(T)38 b FK(is)27 b(a)h(constan)n(t)g(t)n(yp)r(e)g
(then)g(w)n(e)f(assign)h(to)g(it)f(a)h(coherence)f(space)h
FJ(T)45 b FK(and)1193 1182 y([)-12 b([)p FH(T)10 b FK(])-12
b(])q(\()p FJ(A)1379 1193 y FD(1)1407 1182 y FH(;)12
b(:::;)g FJ(A)1582 1193 y Fw(n)1615 1182 y FK(\))20 b(=)f
FJ(T)339 1477 y FK(An)n(y)29 b(morphism)d(is)h(mapp)r(ed)h(to)g(the)g
(iden)n(tit)n(y)e(on)i FJ(T)19 b FK(.)251 1657 y(2.)35
b(If)28 b FH(T)38 b FK(is)27 b(the)g(v)l(ariable)g FH(X)1008
1668 y Fw(i)1056 1657 y FK(then)h(the)f(functor)h(is)f(the)g
FH(i)p FK(th)h(pro)t(jection)1169 1952 y([)-12 b([)q
FH(X)1253 1963 y Fw(i)1273 1952 y FK(])g(]\()p FJ(A)1382
1963 y FD(1)1410 1952 y FH(;)12 b(:::;)g FJ(A)1585 1963
y Fw(n)1619 1952 y FK(\))19 b(=)h FJ(A)1795 1963 y Fw(i)339
2247 y FK(and)29 b(similarly)c(on)j(morphisms.)251 2428
y(3.)35 b(If)27 b FH(T)36 b FK(is)25 b FH(U)c FJ(!)14
b FH(V)i FK(,)27 b(and)g FH(U)33 b FK(and)28 b FH(V)41
b FK(ha)n(v)n(e)28 b(b)r(een)e(in)n(terpreted)e(b)n(y)j(the)f(functors)
h([)-12 b([)q FH(U)7 b FK(])-12 b(])27 b(and)339 2513
y([)-12 b([)q FH(V)16 b FK(])-12 b(])28 b(then)635 2807
y([)-12 b([)p FH(U)23 b FJ(!)16 b FH(V)g FK(])-12 b(]\()p
FJ(A)982 2818 y FD(1)1010 2807 y FH(;)12 b(:::;)g FJ(A)1185
2818 y Fw(n)1219 2807 y FK(\))19 b(=)h([)-12 b([)p FH(U)7
b FK(])-12 b(])q(\()p FJ(A)1529 2818 y FD(1)1557 2807
y FH(;)12 b(:::;)g FJ(A)1732 2818 y Fw(n)1766 2807 y
FK(\))j FJ(!)h FK([)-12 b([)q FH(V)15 b FK(])-12 b(])q(\()p
FJ(A)2086 2818 y FD(1)2114 2807 y FH(;)12 b(:::;)g FJ(A)2289
2818 y Fw(n)2322 2807 y FK(\))339 3102 y(Its)28 b(v)l(alue)g(on)g
(morphisms)f(is)g(as)h(giv)n(en)g(at)g(the)f(end)h(of)g(the)g(previous)
g(section.)167 3323 y(This)c(de\014nition)f(resp)r(ects)h(substitution)
g(of)h(t)n(yp)r(es)f FH(U)1593 3334 y FD(1)1621 3323
y FH(;)12 b(:::;)g(U)1787 3334 y Fw(n)1845 3323 y FK(for)24
b(the)g(v)l(ariables)g FH(X)2415 3334 y FD(1)2443 3323
y FH(;)12 b(:::;)g(X)2619 3334 y Fw(n)2653 3323 y FK(:)167
3408 y([)-12 b([)q FH(T)10 b FK([)p FH(U)310 3419 y Fw(i)330
3408 y FH(=X)421 3419 y Fw(i)441 3408 y FK(]])-12 b(])20
b(=)g([)-12 b([)p FH(T)10 b FK(])-12 b(])q(\([)g([)p
FH(U)783 3419 y FD(1)811 3408 y FK(])g(])q FH(;)12 b(:::;)g
FK([)-12 b([)p FH(U)1030 3419 y Fw(n)1063 3408 y FK(])g(])q(\).)270
3669 y(Because)60 b(of)h(functorialit)n(y)-6 b(,)67 b(w)n(e)59
b(immediately)e(kno)n(w)k(that)f(if)f FJ(A)2265 3644
y Fv(0)2325 3669 y FJ(')42 b(A)60 b FK(then)167 3754
y([)-12 b([)q FH(T)10 b FK(])-12 b(]\()p FJ(A)353 3729
y Fv(0)370 3754 y FK(\))32 b FJ(')g FK([)-12 b([)p FH(T)10
b FK(])-12 b(])q(\()p FJ(A)p FK(\).)89 b(It)45 b(is)g(con)n(v)n(enien)n
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3840 y(if)29 b FJ(A)293 3814 y Fv(0)331 3840 y FJ(\032)21
b(A)30 b FK(is)f(a)h FM(subsp)l(ac)l(e)h FK(then)f(the)f(induced)h(em)n
(b)r(edding)f([)-12 b([)q FH(T)10 b FK(])-12 b(])q(\()p
FJ(A)2010 3814 y Fv(0)2026 3840 y FK(\))21 b Fu(\032)g
FK([)-12 b([)q FH(T)10 b FK(])-12 b(])q(\()p FJ(A)p FK(\))29
b(is)h(also)g(a)167 3925 y FM(subsp)l(ac)l(e)e FK(inclusion.)p
eop
PStoPSsaved restore
%%Page: (140,141) 71
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
137 140 bop 167 167 a Fz(A.3.)63 b(INTERPRET)-6 b(A)g(TION)26
b(OF)h(TYPES)1188 b FK(137)167 404 y Fy(A.3.1)95 b(T)-8
b(ok)m(ens)27 b(for)g(univ)m(ersal)i(t)m(yp)s(es)167
535 y FK(The)j(in)n(terpretation)f(is)h FM(c)l(ontinuous)p
FK(:)45 b(if)32 b FH(\014)26 b FJ(2)d(j)p FK([)-12 b([)p
FH(T)10 b FK(])-12 b(])q(\()p FJ(A)p FK(\))p FJ(j)32
b FK(then)g(there)g(is)g(a)g(\014nite)g(subspace)167
620 y FJ(A)223 595 y Fv(0)263 620 y Fu(\032)23 b FJ(A)32
b FK(suc)n(h)h(that)g FH(\014)27 b FJ(2)c(j)p FK([)-12
b([)p FH(T)10 b FK(])-12 b(]\()p FJ(A)1117 595 y Fv(0)1134
620 y FK(\))p FJ(j)p FK(.)49 b(\(Categorically)-6 b(,)32
b(w)n(e)g(w)n(ould)g(sa)n(y)i(that)f(the)f(functor)167
706 y(preserv)n(es)j FM(\014lter)l(e)l(d)g(c)l(olimits)p
FK(.\))58 b(This)35 b(means)g(that,)i(as)f(in)f(section)f
FG(??)q FK(,)j(w)n(e)e(ma)n(y)g(restrict)167 791 y(atten)n(tion)28
b(to)f(\014nite)g(coherence)g(spaces.)36 b(F)-6 b(or)28
b(an)h(arbitrary)e(coherence)g(space)h FJ(A)p FK(,)768
965 y FJ(j)p FK([)-12 b([)p FH(T)10 b FK(])-12 b(])q(\()p
FJ(A)p FK(\))p FJ(j)19 b FK(=)1113 912 y Fo(S)1172 931
y Fv(")1199 965 y FJ(fj)p FK([)-12 b([)q FH(T)10 b FK(])-12
b(]\()p FJ(A)1440 939 y Fv(0)1457 965 y FK(\))p FJ(j)19
b FK(:)39 b FJ(A)1637 939 y Fv(0)1673 965 y Fu(\032)20
b FJ(A)28 b FK(\014nite)o FJ(g)270 1181 y FK(But)g(more)f(than)i(this,)
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1330 y Fv(0)528 1355 y FH(;)12 b FJ(A)615 1330 y Fv(00)665
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FK([)-12 b([)p FH(T)10 b FK(])-12 b(]\()p FJ(A)1296 1330
y Fv(0)1313 1355 y FK(\))p FJ(j)p FH(;)12 b FJ(j)p FK([)-12
b([)o FH(T)10 b FK(])-12 b(])q(\()p FJ(A)1596 1330 y
Fv(00)1626 1355 y FK(\))p FJ(j)27 b FK(then)h FH(\014)23
b FJ(2)d(j)p FK([)-12 b([)p FH(T)10 b FK(])-12 b(]\()p
FJ(A)2196 1330 y Fv(0)2229 1355 y FJ(\\)15 b(A)2347 1330
y Fv(00)2377 1355 y FK(\))p FJ(j)167 1529 y FM(i.e.)30
b FK(the)g(functor)h(preserv)n(es)f FM(pul)t(lb)l(acks)1235
1504 y FD(4)1262 1529 y FK(.)43 b(F)-6 b(or)30 b(a)h(stable)f
(function,)h(if)e(w)n(e)h(kno)n(w)h FH(\014)25 b FJ(2)d
FH(f)8 b FK(\()p FH(a)p FK(\),)167 1615 y(then)25 b(there)f(is)h(a)g
(least)f FH(a)834 1589 y Fv(0)871 1615 y FJ(\032)19 b
FH(a)25 b FK(suc)n(h)h(that)f FH(\014)f FJ(2)19 b FH(f)8
b FK(\()p FH(a)1549 1589 y Fv(0)1565 1615 y FK(\).)34
b(W)-6 b(e)25 b(ha)n(v)n(e)h(a)g(similar)2193 1589 y
FD(5)2243 1615 y FK(prop)r(ert)n(y)g(here:)167 1700 y(if)h
FH(\014)c FJ(2)d(j)p FK([)-12 b([)p FH(T)10 b FK(])-12
b(]\()p FJ(A)p FK(\))p FJ(j)28 b FK(then)f(there)g(is)g(a)h(least)f
(subspace)i FJ(A)1637 1674 y Fv(0)1673 1700 y Fu(\032)20
b FJ(A)28 b FK(with)e FH(\014)e FJ(2)19 b(j)p FK([)-12
b([)p FH(T)10 b FK(])-12 b(])q(\()p FJ(A)2351 1674 y
Fv(0)2367 1700 y FK(\))p FJ(j)p FK(.)270 1826 y(The)28
b(tok)n(en)g FH(\014)j FK(of)d([)-12 b([)q FH(T)10 b
FK(])-12 b(]\()p FJ(A)p FK(\))27 b(therefore)g(in)n(trinsically)d
(carries)j(with)f(it)h(a)g(particular)g(\014nite)167
1912 y(subspace)36 b FJ(A)523 1886 y Fv(0)564 1912 y
FJ(\032)24 b(A)p FK(,)37 b(namely)d(the)g(least)g(subspace)i(on)f(whic)
n(h)f(it)g(can)h(b)r(e)f(de\014ned.)57 b(It)35 b(is)167
1997 y(not)d(di\016cult)f(to)h(see)f(that,)i(in)e(terms)f(of)i(the)g(w)
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FH(\013)167 2082 y FK(whic)n(h)d(o)r(ccur)g(in)g(the)h(expression)f
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(tok)n(en)h(o)r(ccurring)167 2167 y(in)h FH(\014)28 b
FK(=)23 b(\()p FJ(f)p FH(\013)p FJ(g)p FH(;)12 b(\013)p
FK(\))34 b(is)f FH(\013)p FK(,)j(and)e(the)f(corresp)r(onding)i
(\014nite)e(space)h(is)e FJ(S)-6 b FM(gl)6 b FK(,)35
b(whose)f(w)n(eb)g(is)f(a)167 2252 y(singleton,)27 b
FJ(f\017)q(g)p FK(.)270 2379 y(W)-6 b(e)35 b(shall)f(see)h(later)e
(that)j(the)e(pairs)h FJ(hA)p FH(;)12 b(\014)t FJ(i)p
FK(,)38 b(where)c FH(\014)28 b FJ(2)d(j)p FK([)-12 b([)p
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3558 y(_)2481 3601 y(^)2569 3580 y(e)2601 3591 y FD(2)2629
3580 y FH(\014)167 3666 y FK(in)f FJ(B)r FK(.)2298 b
Fu(\003)p 167 3719 1003 3 v 246 3763 a FC(7)273 3784
y FB(It)27 b(is)g(the)g(total)h(category)g(\006)916 3779
y Fl(\016)958 3784 y Fx(X)q(:)10 b Fl(T)16 b FB(\()p
Fx(X)5 b FB(\))26 b(whic)n(h)h(w)n(e)i(met)e(in)g(section)g
Ft(??)o FB(.)246 3833 y FC(8)273 3854 y FB(Note)f(that)f(this)f
(equalit)n(y)h(only)g(holds)f(for)i FA(typ)m(e)h FB(v)m(ariables)d(and)
h(not)h(for)g(dep)r(endency)d(o)n(v)n(er)k(ordinary)167
3925 y(domains.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
142 145 bop 167 167 a FK(142)947 b Fz(APPENDIX)26 b(A.)55
b(SEMANTICS)27 b(OF)g(SYSTEM)h(F)270 404 y FK(The)g(con)n(v)n(erse)g
(holds:)167 538 y FG(Lemma)36 b FK(Let)i FH(\014)31 b
FJ(2)d(jT)17 b FK(\()p FJ(A)p FK(\))p FJ(j)38 b FK(b)r(e)h(suc)n(h)g
(that)g(\(i\))f FJ(A)g FK(is)g(minimal)e(for)j FH(\014)j
FK(and)e(\(ii\))d FH(\014)42 b FK(has)167 623 y(coheren)n(t)31
b(images)f(under)h(an)n(y)h(pair)e(of)i(em)n(b)r(eddings)e(of)i
FJ(A)f FK(in)n(to)f(another)i(domain.)45 b(Then)167 708
y(there)27 b(is)g(an)h(ob)t(ject)g FH(\034)766 719 y
Fv(hA)p Fw(;\014)r Fv(i)919 708 y FK(of)g(t)n(yp)r(e)g
FJ(T)46 b FK(whose)28 b(v)l(alue)g(at)g FJ(T)18 b FK(\()p
FJ(B)r FK(\))27 b(is)1056 895 y FJ(fT)19 b FK(\()p FH(e)p
FK(\)\()p FH(\014)t FK(\))f(:)40 b FH(e)19 b FK(:)h FJ(A)g
Fu(\032)g FJ(B)r(g)167 1082 y FK(and)29 b(moreo)n(v)n(er)e(this)g(is)g
FM(atomic)p FK(,)i FM(i.e.)e FK(has)i(no)f(non)n(trivial)f(sub)r(ob)t
(ject.)568 b Fu(\003)270 1215 y FK(T)-6 b(o)29 b(test)e(this)g
(condition)h(w)n(e)f(only)h(need)g(to)g(consider)g(graphs)h(up)f(to)h
(t)n(wice)d(the)h(size)g(of)167 1300 y FJ(jAj)p FK(,)h(and)h(so)g(it)e
(is)g(a)h(\014nite)903 1274 y FD(9)959 1300 y FK(calculation)e(to)j
(determine)d(whether)h FJ(hA)p FH(;)12 b(\014)t FJ(i)29
b FK(satis\014es)f(it.)36 b(F)-6 b(or)167 1385 y(an)n(y)30
b(giv)n(en)e(t)n(yp)r(e)h(these)f(tok)n(ens)i(are)e(recursiv)n(ely)f
(en)n(umerable.)37 b(Because)29 b FH(\034)2221 1396 y
Fv(hA)p Fw(;\014)r Fv(i)2375 1385 y FK(is)f(atomic,)167
1470 y(w)n(e)i(m)n(ust)h(ha)n(v)n(e)g(just)g FM(one)f
FK(tok)n(en)i(for)f(\005)p FH(X)r(:)12 b FJ(T)17 b FK(\()p
FH(X)6 b FK(\),)31 b(so)h FJ(hA)p FH(;)12 b(\014)t FJ(i)31
b FK(and)h FJ(hA)2107 1444 y Fv(0)2124 1470 y FH(;)12
b(\014)2198 1444 y Fv(0)2214 1470 y FJ(i)31 b FK(are)f(iden)n(ti\014ed)
167 1555 y(for)e(an)n(y)h FH(e)19 b FK(:)h FJ(A)g(')f(A)708
1530 y Fv(0)753 1555 y FK(with)26 b FH(e\014)e FK(=)19
b FH(\014)1125 1530 y Fv(0)1141 1555 y FK(.)270 1742
y(W)-6 b(e)28 b(still)d(ha)n(v)n(e)k(to)e(sa)n(y)i(when)f(these)f(tok)n
(ens)i(are)e(coheren)n(t.)167 1875 y FG(Lemma)i FK(Let)h
FH(\014)623 1886 y FD(1)673 1875 y FJ(2)22 b(jT)c FK(\()p
FJ(A)901 1886 y FD(1)929 1875 y FK(\))p FJ(j)31 b FK(and)h
FH(\014)1188 1886 y FD(2)1238 1875 y FJ(2)22 b(jT)c FK(\()p
FJ(A)1466 1886 y FD(2)1494 1875 y FK(\))o FJ(j)31 b FK(eac)n(h)h
(satisfy)f(these)g(conditions.)46 b(Then)167 1960 y FH(\034)197
1971 y Fv(hA)256 1978 y Fi(1)281 1971 y Fw(;\014)323
1978 y Fi(1)347 1971 y Fv(i)369 1960 y FK(\()p FJ(B)r
FK(\))491 1938 y FH(_)491 1981 y(^)579 1960 y(\034)609
1971 y Fv(hA)668 1978 y Fi(2)693 1971 y Fw(;\014)735
1978 y Fi(2)760 1971 y Fv(i)782 1960 y FK(\()p FJ(B)r
FK(\))22 b(at)g(ev)n(ery)f(coherence)g(space)h FJ(B)i
FK(i\013)e(for)g(ev)n(ery)f(pair)h(of)g(em)n(b)r(eddings)167
2045 y FH(e)199 2056 y FD(1)246 2045 y FK(:)e FJ(A)341
2056 y FD(1)389 2045 y Fu(\032)g FJ(C)t FK(,)28 b FH(e)607
2056 y FD(2)654 2045 y FK(:)20 b FJ(A)749 2056 y FD(2)796
2045 y Fu(\032)g FJ(C)t FK(,)28 b(w)n(e)f(ha)n(v)n(e)i
FJ(T)18 b FK(\()p FH(e)1367 2056 y FD(1)1395 2045 y FK(\)\()p
FH(\014)t FK(\))1538 2023 y FH(_)1538 2066 y(^)1626 2045
y FJ(T)h FK(\()p FH(e)1742 2056 y FD(2)1769 2045 y FK(\)\()p
FH(\014)t FK(\).)705 b Fu(\003)270 2178 y FK(Finally)33
b(this)g(enables)g(us)h(to)g(calculate)d(the)j(univ)n(ersal)f
(abstraction)h(of)g(an)n(y)g(v)l(ariable)167 2263 y(coherence)27
b(space.)167 2397 y FG(Prop)r(osition)38 b FK(Let)i FJ(T)47
b FK(:)29 b FG(Gem)f FJ(!)h FG(Gem)40 b FK(b)r(e)g(a)h(stable)f
(functor.)74 b(Then)41 b(its)f(univ)n(ersal)167 2482
y(abstraction,)24 b(\005)p FH(X)r(:)12 b FJ(T)17 b FK(\()p
FH(X)6 b FK(\),)23 b(is)e(the)g(coherence)g(space)h(whose)g(tok)n(ens)g
(are)g(equiv)l(alence)f(classes)167 2567 y(of)28 b(pairs)g
FJ(hA)p FH(;)12 b(\014)t FJ(i)28 b FK(suc)n(h)g(that)270
2737 y FJ(\017)34 b FH(\014)24 b FJ(2)19 b(jT)f FK(\()p
FJ(A)p FK(\))p FJ(j)270 2887 y(\017)34 b(A)29 b FK(is)f(minimal)e(for)j
(this,)g FM(i.e.)f FK(if)g FJ(A)1292 2861 y Fv(0)1329
2887 y FJ(\032)20 b(A)29 b FK(and)h FH(\014)24 b FJ(2)c(jT)e
FK(\()p FJ(A)1918 2861 y Fv(0)1934 2887 y FK(\))p FJ(j)28
b FK(then)h FJ(A)2228 2861 y Fv(0)2265 2887 y FK(=)20
b FJ(A)29 b FK(\(so)g FJ(A)g FK(is)339 2972 y(\014nite\).)270
3121 y FJ(\017)34 b FK(for)28 b(an)n(y)h(t)n(w)n(o)f(rigid)f(em)n(b)r
(eddings)g FH(e)1297 3132 y FD(1)1325 3121 y FH(;)12
b(e)1388 3132 y FD(2)1435 3121 y FK(:)20 b FJ(A)g Fu(\032)g
FJ(B)r FK(,)28 b(w)n(e)f(ha)n(v)n(e)1172 3338 y FJ(T)18
b FK(\()p FH(e)1287 3349 y FD(1)1315 3338 y FK(\)\()p
FH(\014)t FK(\))1457 3316 y FH(_)1457 3359 y(^)1546 3338
y FJ(T)g FK(\()p FH(e)1661 3349 y FD(2)1689 3338 y FK(\)\()p
FH(\014)t FK(\))339 3555 y(in)27 b FJ(T)19 b FK(\()p
FJ(B)r FK(\).)270 3705 y FJ(\017)34 b(hA)p FH(;)12 b(\014)t
FJ(i)27 b FK(is)f(iden)n(ti\014ed)f(with)g FJ(hA)1170
3679 y Fv(0)1187 3705 y FH(;)12 b(\014)1261 3679 y Fv(0)1277
3705 y FJ(i)27 b FK(i\013)e FH(e)20 b FK(:)g FJ(A)f(')h(A)1713
3679 y Fv(0)1756 3705 y FK(and)27 b FJ(T)18 b FK(\()p
FH(e)p FK(\)\()p FH(\014)t FK(\))h(=)g FH(\014)2267 3679
y Fv(0)2310 3705 y FK(\(so)26 b FJ(jAj)g FK(ma)n(y)339
3790 y(b)r(e)i(tak)n(en)g(to)g(b)r(e)g(a)g(subset)g(of)g(I)-12
b(N\).)p 167 3860 1003 3 v 246 3903 a FC(9)273 3925 y
FB(Though)27 b(it)g(w)n(ould)g(app)r(ear)g(to)h(b)r(e)f(exp)r(onen)n
(tial)f(in)g Fl(jAj)1593 3903 y FC(2)1620 3925 y FB(.)p
eop
PStoPSsaved restore
%%Page: (146,147) 74
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
143 146 bop 167 167 a Fz(A.4.)63 b(INTERPRET)-6 b(A)g(TION)26
b(OF)h(TERMS)1173 b FK(143)270 404 y FJ(\017)34 b(hA)p
FH(;)12 b(\014)t FJ(i)32 b FK(is)f(coheren)n(t)g(with)g
FJ(hA)1168 379 y Fv(0)1185 404 y FH(;)12 b(\014)1259
379 y Fv(0)1275 404 y FJ(i)32 b FK(i\013)e(for)i(ev)n(ery)f(pair)g(of)h
(em)n(b)r(eddings)f FH(e)22 b FK(:)h FJ(A)f Fu(\032)h
FJ(B)339 490 y FK(and)29 b FH(e)510 464 y Fv(0)546 490
y FK(:)20 b FJ(A)641 464 y Fv(0)677 490 y Fu(\032)g FJ(B)30
b FK(w)n(e)d(ha)n(v)n(e)i FJ(T)18 b FK(\()p FH(e)p FK(\)\()p
FH(\014)t FK(\))1379 467 y FH(_)1379 510 y(^)1467 490
y FJ(T)g FK(\()p FH(e)1582 464 y Fv(0)1599 490 y FK(\)\()p
FH(\014)1696 464 y Fv(0)1712 490 y FK(\).)167 669 y FG(Pro)r(of)36
b FK(\005)p FH(X)r(:)12 b FJ(T)17 b FK(\()p FH(X)6 b
FK(\))35 b(is)g(a)h(coherence)f(space)h(b)r(ecause)g(if)e(an)n(y)j
FJ(hA)p FH(;)12 b(\014)t FJ(i)36 b FK(o)r(ccurs)g(in)f(a)h(p)r(oin)n(t)
167 754 y(then)30 b(so)h(do)r(es)g(the)e(whole)h(of)g
FH(\034)1029 765 y Fv(hA)p Fw(;\014)r Fv(i)1155 754 y
FK(,)h(and)g(an)n(y)h(coheren)n(t)d(union)i(of)g(these)e(giv)n(es)h
(rise)f(to)i(a)167 840 y(uniform)c(elemen)n(t.)1934 b
Fu(\003)270 961 y FK(One)39 b(ough)n(t)h(to)e(pro)n(v)n(e)i(that)f(if)e
FJ(T)d FK(:)28 b FG(Gem)18 b FJ(\002)g FG(Gem)27 b FJ(!)h
FG(Gem)38 b FK(is)g(stable)g(then)h(so)g(is)167 1046
y(\005)p FH(X)r(:)12 b FJ(T)42 b FK(:)25 b FG(Gem)f FJ(!)h
FG(Gem)p FK(,)37 b(and)e(also)g(c)n(hec)n(k)g(that)g(the)g(p)r(ositiv)n
(e)f(and)h(negativ)n(e)g(criterion)167 1131 y(remains)27
b(v)l(alid.)167 1333 y Fy(A.4.3)95 b(In)m(terpretation)28
b(of)f(F)167 1464 y FK(Let)e(us)h(sum)f(up)h(b)n(y)h(setting)e(out)h
(in)f(full)f(the)h(coherence)g(space)g(seman)n(tics)g(of)h
FG(F)p FK(.)f(The)g FM(typ)l(e)167 1549 y FH(U)34 b FK(in)27
b FH(n)g FK(free)f(v)l(ariables)827 1550 y FH(X)p 831
1570 50 3 v 917 1549 a FK(is)g(in)n(terpreted)g(as)h(a)h(stable)f
(functor)g([)-12 b([)q FH(U)7 b FK(])-12 b(])20 b(:)g
FG(Gem)2276 1519 y Fw(n)2329 1549 y FJ(!)g FG(Gem)26
b FK(as)167 1634 y(in)h FJ(x)p FK(A.3,)h(with)f(the)h(additional)f
(clause)251 1777 y(4.)35 b(If)21 b FH(U)26 b FK(=)20
b(\005)p FH(X)r(:)12 b(T)30 b FK(then)20 b(the)g(w)n(eb)g(of)h([)-12
b([)q FH(U)7 b FK(])-12 b(])q(\()1383 1778 y FJ(A)p 1387
1798 43 3 v -1 x FK(\))20 b(is)f(giv)n(en)i(as)g(in)f(the)g(preceding)f
(prop)r(osition,)339 1862 y(where)31 b FJ(T)18 b FK(\()p
FH(X)6 b FK(\))21 b(=)h([)-12 b([)q FH(T)10 b FK(])-12
b(]\()947 1863 y FJ(A)p 951 1883 V -1 x FH(;)12 b(X)6
b FK(\).)46 b(The)30 b(em)n(b)r(edding)h(induced)g(b)n(y)2064
1863 y FH(e)p 2067 1883 19 3 v 2118 1862 a FK(:)2159
1863 y FJ(A)2215 1838 y Fv(0)p 2162 1883 60 3 v 2254
1862 a Fu(\032)2354 1863 y FJ(A)p 2357 1883 43 3 v 2441
1862 a FK(is)f(tak)n(es)339 1948 y(tok)n(ens)f(of)f([)-12
b([)q FH(U)7 b FK(])-12 b(]\()776 1949 y FJ(A)832 1923
y Fv(0)p 780 1968 60 3 v 849 1948 a FK(\))27 b(to)h(the)g(corresp)r
(onding)g(tok)n(ens)h(with)d FH(\013)1988 1922 y Fv(0)1988
1965 y Fw(i)2036 1948 y FK(replaced)g(b)n(y)j FH(e)2445
1959 y Fw(i)2465 1948 y FH(\013)2509 1922 y Fv(0)2509
1965 y Fw(i)2529 1948 y FK(.)270 2170 y(The)40 b FM(term)g
FH(t)g FK(of)h(t)n(yp)r(e)f FH(T)50 b FK(with)40 b FH(m)g
FK(free)f(v)l(ariables)1759 2171 y FH(x)p 1763 2190 26
3 v 1839 2170 a FK(of)h(t)n(yp)r(es)2134 2171 y FH(U)p
2137 2190 41 3 v 2228 2170 a FK(\(the)g(free)f(t)n(yp)r(e)167
2255 y(v)l(ariables)29 b(of)h FH(T)6 b(;)621 2256 y(U)p
625 2276 V 705 2255 a FK(b)r(eing)897 2256 y FH(X)p 901
2276 50 3 v 960 2255 a FK(\))29 b(is)g(in)n(terpreted)f(as)i(an)h
(assignmen)n(t)e(to)h(eac)n(h)2261 2256 y FJ(A)p 2264
2276 43 3 v 2346 2255 a FK(of)h(a)e(stable)167 2340 y(function)764
2504 y([)-12 b([)q FH(t)p FK(])g(]\()869 2505 y FJ(A)p
873 2525 V -1 x FK(\))20 b(:)f([)-12 b([)q FH(U)1084
2515 y FD(1)1112 2504 y FK(])g(]\()1165 2505 y FJ(A)p
1169 2525 V -1 x FK(\))16 b(&)f FH(:::)h FK(&)g([)-12
b([)p FH(U)1549 2515 y Fw(m)1596 2504 y FK(])g(])q(\()1650
2505 y FJ(A)p 1654 2525 V -1 x FK(\))15 b FJ(!)h FK([)-12
b([)q FH(T)10 b FK(])-12 b(]\()1964 2505 y FJ(A)p 1968
2525 V -1 x FK(\))167 2668 y(suc)n(h)28 b(that)g(for)591
2669 y FH(e)p 595 2689 19 3 v 643 2668 a FK(:)681 2669
y FJ(A)737 2643 y Fv(0)p 685 2689 60 3 v 774 2668 a Fu(\032)872
2669 y FJ(A)p 875 2689 43 3 v 955 2668 a FK(and)h FH(b)1123
2679 y Fw(j)1169 2668 y FJ(2)19 b FK([)-12 b([)q FH(U)1309
2679 y Fw(j)1335 2668 y FK(])g(]\()1388 2669 y FJ(A)p
1392 2689 V -1 x FK(\))28 b(the)f FM(uniformity)i(e)l(quation)e
FK(holds:)794 2832 y([)-12 b([)q FH(T)10 b FK(])-12 b(]\()924
2833 y FH(e)p 928 2853 19 3 v -1 x FK(\))983 2807 y Fv(\000)1025
2832 y FK(\([)g([)p FH(t)p FK(])g(]\()1156 2833 y FJ(A)p
1160 2853 43 3 v -1 x FK(\)\()1266 2833 y FH(b)p 1270
2853 16 3 v -1 x FK(\)\))20 b(=)f([)-12 b([)p FH(t)p
FK(])g(])q(\()1547 2833 y FJ(A)1603 2807 y Fv(0)p 1551
2853 60 3 v 1619 2832 a FK(\)\([)g([)1700 2833 y FH(U)p
1703 2853 41 3 v 1754 2832 a FK(])g(]\()1807 2833 y FH(e)p
1811 2853 19 3 v -1 x FK(\))1866 2807 y Fv(\000)1908
2832 y FK(\()1935 2833 y FH(b)p 1939 2853 16 3 v -1 x
FK(\)\))167 2996 y(In)28 b(detail,)251 3139 y(1.)35 b(The)28
b FM(variable)f FH(x)786 3150 y Fw(j)839 3139 y FK(is)g(in)n(terpreted)
f(b)n(y)j(the)e FH(j)t FK(th)g(pro)r(duct)h(pro)t(jection.)1263
3331 y([)-12 b([)p FH(x)1328 3342 y Fw(j)1354 3331 y
FK(])g(])q(\()1408 3332 y FJ(A)p 1412 3351 43 3 v -1
x FK(\)\()1518 3332 y FH(b)p 1522 3351 16 3 v -1 x FK(\))19
b(=)g FH(b)1695 3342 y Fw(j)251 3549 y FK(2.)35 b(The)e(in)n
(terpretation)e(of)h FH(\025)p FM(-abstr)l(action)h FH(\025x:)12
b(u)33 b FK(is)f(giv)n(en)g(in)g(terms)f(of)i(that)g(of)g
FH(u)f FK(b)n(y)339 3634 y(the)c(trace)606 3825 y([)-12
b([)p FH(\025x:)12 b(u)p FK(])-12 b(])q(\()835 3826 y
FJ(A)p 839 3846 43 3 v -1 x FK(\)\()945 3826 y FH(b)p
949 3846 16 3 v -1 x FK(\))20 b(=)f FJ(f)p 1129 3754
149 3 v(h)p FH(c;)p 1217 3768 34 3 v 12 w(\016)r FJ(i)h
FK(:)39 b FH(\016)22 b FJ(2)e FK([)-12 b([)p FH(u)p FK(])g(])q(\()1594
3826 y FJ(A)p 1598 3846 43 3 v -1 x FK(\)\()1704 3826
y FH(b)p 1708 3846 16 3 v -1 x(;)12 b(c)p FK(\))p FH(;)39
b FK(with)27 b FH(c)g FK(minimal)m FJ(g)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
144 147 bop 167 167 a FK(144)947 b Fz(APPENDIX)26 b(A.)55
b(SEMANTICS)27 b(OF)g(SYSTEM)h(F)251 404 y FK(3.)35 b(The)28
b FM(applic)l(ation)g FH(uv)j FK(is)c(in)n(terpreted)f(using)i(the)f
(form)n(ula)h(\()p FG(App)p FK(\))f(of)h(section)f FG(??)q
FK(:)697 640 y([)-12 b([)p FH(uv)s FK(])g(])q(\()852
641 y FJ(A)p 856 661 43 3 v -1 x FK(\)\()962 641 y FH(b)p
966 661 16 3 v -1 x FK(\))19 b(=)g FJ(f)p FH(\016)j FK(:)40
b FJ(9)p FH(c)19 b FJ(\032)g FK([)-12 b([)q FH(v)s FK(])g(])q(\()1536
641 y FJ(A)p 1540 661 43 3 v -1 x FK(\)\()1646 641 y
FH(b)p 1650 661 16 3 v -1 x FK(\))p FH(:)p 1733 569 149
3 v 12 w FJ(h)p FH(c;)p 1821 583 34 3 v 12 w(\016)q FJ(i)20
b(2)g FK([)-12 b([)p FH(u)p FK(])g(])q(\()2086 641 y
FJ(A)p 2090 661 43 3 v -1 x FK(\)\()2196 641 y FH(b)p
2200 661 16 3 v -1 x FK(\))p FJ(g)251 893 y FK(4.)35
b(The)28 b FM(universal)f(abstr)l(action)p FK(,)h(\003)p
FH(X)r(:)12 b(v)s FK(,)27 b(is)g(also)h(giv)n(en)g(b)n(y)g(a)g
(\\trace":)559 1113 y([)-12 b([\003)p FH(X)r(:)12 b(v)s
FK(])-12 b(])q(\()813 1114 y FJ(A)p 817 1134 43 3 v -1
x FK(\)\()923 1114 y FH(b)p 927 1134 16 3 v -1 x FK(\))19
b(=)g FJ(f)p FK([)p FJ(hC)t FH(;)12 b(\016)s FJ(i)p FK(])20
b(:)40 b FH(\016)21 b FJ(2)f FK([)-12 b([)p FH(v)s FK(])g(])q(\()1618
1114 y FJ(A)p 1622 1134 43 3 v -1 x FH(;)12 b FJ(C)t
FK(\)\()1800 1114 y FH(b)p 1804 1134 16 3 v -1 x FK(\))p
FH(;)39 b FK(with)27 b FJ(C)32 b FK(minimal)m FJ(g)339
1333 y FK(where)f([)p FJ(hC)t FH(;)12 b(\016)s FJ(i)p
FK(])32 b(denotes)g(the)f(equiv)l(alence)f(class:)43
b FJ(hC)t FH(;)12 b(\016)s FJ(i)32 b FK(is)e(iden)n(ti\014ed)h(with)f
FJ(hC)2547 1308 y Fv(0)2564 1333 y FH(;)12 b(\016)2629
1308 y Fv(0)2645 1333 y FJ(i)339 1419 y FK(whenev)n(er)28
b FH(e)19 b FK(:)h FJ(C)k(')19 b(C)913 1393 y Fv(0)957
1419 y FK(and)29 b([)-12 b([)q FH(v)s FK(])g(]\()1212
1420 y FJ(A)p 1216 1439 43 3 v -1 x FH(;)12 b(e)p FK(\)\()1385
1420 y FH(b)p 1389 1439 16 3 v -1 x FK(\)\()p FH(\016)s
FK(\))18 b(=)h FH(\016)1654 1393 y Fv(0)1670 1419 y FK(.)251
1569 y(5.)35 b(The)28 b FM(universal)f(applic)l(ation)p
FK(,)h FH(tU)7 b FK(,)28 b(is)f(giv)n(en)h(b)n(y)g(an)g(application)f
(form)n(ula)647 1790 y([)-12 b([)p FH(tU)7 b FK(])-12
b(])q(\()806 1791 y FJ(A)p 810 1810 43 3 v -1 x FK(\)\()916
1791 y FH(b)p 920 1810 16 3 v -1 x FK(\))19 b(=)g FJ(f)p
FH(\016)j FK(:)40 b FJ(9)p FH(e)19 b FK(:)h FJ(C)k Fu(\032)c
FK([)-12 b([)p FH(U)7 b FK(])-12 b(])q(\()1633 1791 y
FJ(A)p 1637 1810 43 3 v -1 x FK(\))p FH(:)12 b FK([)p
FJ(hC)t FH(;)g(\016)s FJ(i)p FK(])20 b FJ(2)f FK([)-12
b([)q FH(t)p FK(])g(]\()2136 1791 y FJ(A)p 2140 1810
V -1 x FK(\)\()2246 1791 y FH(b)p 2250 1810 16 3 v -1
x FK(\))p FJ(g)270 2167 y FK(The)22 b(con)n(v)n(ersion)h(rules)e(are)h
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(b)r(et)n(w)n(een)167 2252 y(ob)t(jects)30 b(of)g(\005)p
FH(X)r(:)12 b FJ(T)18 b FK(\()p FH(X)6 b FK(\))29 b(and)i(v)l(ariable)f
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(section)g FG(??)q FK(\).)167 2582 y FE(A.5)114 b(Examples)167
2762 y Fy(A.5.1)95 b(Of)28 b(course)167 2897 y FK(W)-6
b(e)41 b(aim)f(to)i(calculate)d(the)i(coherence)g(space)g(denotations)h
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b(using)i(system)f FG(F)g FK(in)g(section)g FG(??)q FK(,)g(whic)n(h)g
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FM(existential)167 3067 y FK(t)n(yp)r(es.)61 b(These)35
b(are)h(all)e(essen)n(tially)h(deriv)n(ed)1417 3041 y
FD(10)1505 3067 y FK(from)g(\005)p FH(X)r(:)12 b FK(\()p
FJ(U)25 b(!)18 b FH(X)6 b FK(\))18 b FJ(!)g FH(X)6 b
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FJ(U)7 b FK(,)167 3372 y(and)29 b(so)f(a)g(tok)n(en)g(for)g(this)f(t)n
(yp)r(e)h(is)f(of)h(the)g(form)938 3577 y FJ(hS)-6 b
FM(gl)6 b FH(;)p 1089 3502 758 3 v 12 w FJ(hf)p 1151
3516 181 3 v(h)p FH(u)1217 3588 y Fw(i)1238 3577 y FH(;)p
1269 3538 36 3 v 12 w FJ(\017i)20 b FK(:)39 b FH(i)19
b FK(=)h(1)p FH(;)12 b(:::;)g(k)r FJ(g)p FH(;)p 1783
3538 V 12 w FJ(\017)q(ii)p 167 3649 1003 3 v 223 3692
a FC(10)273 3713 y FB([)-10 b([)p Fd(Bo)r(ol)q FB(])g(])30
b(is)f(also)g(a)h(sp)r(ecial)e(case)i(if)e(w)n(e)j(admit)d(the)h(t)n(w)
n(o-elemen)n(t)g(discrete)g(p)r(oset)g(\(not)f(a)i(coherence)167
3784 y(space\))k(for)g(the)f(domain)g Fl(U)6 b FB(,)35
b(in)e(a)h(category)h(with)e(copro)r(ducts.)52 b(The)34
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b Fl(')18 b FB(\()p Fl(U)6 b(\002)o(V)f FB(\))14 b Fl(!)g
Fx(X)5 b FB(,)167 3925 y(\()p Fl(A)13 b(!)f Fx(X)5 b
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b FB(\()p Fx(X)g FB(\))11 b Fl(!)i Fx(Y)g FB(\))j Fl(')g
FB(\(\006)1779 3920 y Fl(\016)1821 3925 y Fx(X)q(:)10
b Fl(V)5 b FB(\()p Fx(X)g FB(\)\))13 b Fl(!)f Fx(Y)h
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(denotationally)-6 b(,)806 891 y(\005)p FH(X)r(:)12 b
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b FJ(')20 b FK(\(!\(\(!)o FJ(U)7 b FK(\))1672 857 y Fv(?)1714
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FJ(U)167 1066 y FK(whic)n(h)27 b(\(b)n(y)h(a)g(sligh)n(t)f(abuse\))h(w)
n(e)g(shall)e(call)g FJ(:)-12 b(:U)7 b FK(.)270 1193
y(The)28 b(e\013ect)e(of)j(the)e(program)1017 1383 y
FJ(hS)-6 b FM(gl)7 b FH(;)p 1169 1307 599 3 v 12 w FJ(hf)p
1231 1321 189 3 v(h)p FH(u)1297 1394 y FD(1)1326 1383
y FH(;)p 1357 1343 36 3 v 12 w FJ(\017i)p FH(;)p 1450
1321 189 3 v 12 w FJ(h)p FH(u)1516 1394 y FD(2)1545 1383
y FH(;)p 1576 1343 36 3 v 12 w FJ(\017ig)p FH(;)p 1704
1343 V 12 w FJ(\017i)q(i)167 1557 y FK(at)32 b(the)f(t)n(yp)r(e)h
FJ(A)g FK(and)h(giv)n(en)f(the)f(stable)h(function)f
FH(f)f FK(:)23 b FJ(U)30 b(!)22 b(A)32 b FK(is)f(to)h(examine)f(the)h
(trace)167 1642 y FJ(T)-6 b FH(r)r FK(\()p FH(f)8 b FK(\))28
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FH(;)p 1792 1603 45 3 v 12 w(\013)p FJ(i)g FM(and)g FJ(h)p
FH(u)2094 1653 y FD(2)2123 1642 y FH(;)p 2154 1603 V
12 w(\013)p FJ(i)h FK(lie)d(in)h(it.)36 b(This)167 1727
y(generalises)26 b(the)i(in)n(tersection)d(w)n(e)j(found)h(in)e([)-12
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FG(F)g FK(that)h Fn(?)f FK(b)r(e)167 1981 y(added)29
b(to)f FJ(U)7 b FK(,)27 b(since)g(a)h(program)g(of)h(t)n(yp)r(e)e
FJ(:)-12 b(:U)35 b FK(is)27 b(under)h(no)h(obligation)e(to)h
(terminate.)270 2108 y(What)c(seems)f(sligh)n(tly)f(p)r(eculiar)g(is)h
(that)h(w)n(e)f(ma)n(y)h(ha)n(v)n(e)g FH(u)1836 2119
y FD(1)1884 2108 y FJ(\024)19 b FH(u)1997 2119 y FD(2)2025
2108 y FK(,)25 b(t)n(w)n(o)f(\014nite)e(p)r(oin)n(ts)i(\(or)167
2193 y(cliques\))e(of)h FJ(U)7 b FK(,)25 b(whic)n(h)d(giv)n(e)i(rise)e
(to)h FM(atomic)h FK(tok)n(ens)g(of)g(t)n(yp)r(e)g FJ(:)-12
b(:U)30 b FK(\(on)24 b(some)f(functions)h(one)167 2278
y(will)d(output)k FH(\013)f FK(and)h(the)f(other)g(not,)h(and)g(on)g
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2363 y(of)31 b(the)f FM(stable)g FK(in)n(terpretation)f(and)i(the)f
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(sum)g(and)h(existen)n(tial)c(t)n(yp)r(es.)633 2967 y(\005)p
FH(X)r(:)12 b FK(\()p FJ(U)22 b(!)16 b(V)22 b(!)16 b
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FH(X)r(:)g FK(\()p FJ(V)21 b(!)16 b FH(Y)f FK(\)\))g
FJ(!)i FH(Y)34 b FJ(')20 b(:)-12 b(:)p FK(\(\006)1770
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3840 y(where)39 b(for)g(a)h(v)l(ariable)f(t)n(yp)r(e)g
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y(whic)n(h)27 b(w)n(e)g(met)g(in)g(section)g FG(??)p
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PStoPSxform concat
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1084 624 56 3 v 12 w(\015)1120 674 y Fw(i)1140 663 y
FJ(i)g FK(induces)f(a)h(\\transition)g(relation")f(on)h
FJ(A)p FK(.)270 789 y(The)g FM(line)l(ar)f(numer)l(al)p
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b(so)g(there)f(are)h(still)e(quite)h(a)i(lot)e(of)i(them!)167
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2581 y(\(pro)r(of)29 b(nets\).)270 2725 y FJ(\017)34
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4 v 1782 1380 a Fq(Cut)1155 1441 y FH(A)p 1159 1461 39
3 v -1 x(;)1238 1441 y(D)p 1241 1461 46 3 v 1317 1440
a FJ(`)1379 1441 y FH(B)p 1383 1461 43 3 v 1435 1440
a(;)1466 1441 y(E)p 1470 1461 42 3 v 167 1597 a FK(reduces)i(to)739
1726 y FJ(\001)739 1764 y(\001)739 1802 y(\001)654 1861
y FH(A)p 657 1880 39 3 v 725 1860 a FJ(`)788 1861 y FH(B)p
791 1880 43 3 v 566 1920 a FK(=)-31 b(=)-16 b(=)g(=)g(=)g(=)g(=)g(=)g
(=)-31 b(=)566 1982 y FH(A)p 569 2002 39 3 v -1 x(;)649
1982 y(D)p 652 2002 46 3 v 727 1981 a FJ(`)790 1982 y
FH(B)p 793 2002 43 3 v 846 1981 a(;)877 1982 y(E)p 880
2002 42 3 v 1331 1792 a FK(or)28 b(to)2053 1726 y FJ(\001)2053
1764 y(\001)2053 1802 y(\001)1964 1861 y FH(D)p 1968
1880 46 3 v 2043 1860 a FJ(`)2106 1861 y FH(E)p 2109
1880 42 3 v 1880 1920 a FK(=)-31 b(=)-16 b(=)g(=)g(=)g(=)g(=)g(=)g(=)
-31 b(=)1880 1982 y FH(A)p 1883 2002 39 3 v 1931 1981
a(;)1962 1982 y(D)p 1966 2002 46 3 v 2041 1981 a FJ(`)2104
1982 y FH(B)p 2107 2002 43 3 v 2160 1981 a(;)2191 1982
y(E)p 2194 2002 42 3 v 167 2142 a FK(\(where)25 b(the)h(double)g(bar)h
(is)f(a)g(succession)g(of)h(w)n(eak)n(enings)f(and)i(exc)n(hanges\))f
(dep)r(ending)f(on)167 2227 y(whether)h(w)n(e)g(lo)r(ok)h(at)g(the)f
(left)g(or)h(at)f(the)h(righ)n(t)f(side)g(\014rst.)270
2350 y(In)g(particular,)f(if)f(w)n(e)h(ha)n(v)n(e)i(t)n(w)n(o)e(pro)r
(ofs)i FH(\031)h FK(and)f FH(\031)1627 2324 y Fv(0)1669
2350 y FK(of)f(the)f(same)g(form)n(ula)h FH(B)t FK(,)f(and)i
FH(C)k FK(is)167 2435 y(an)n(y)d(form)n(ula,)f(the)f(pro)r(of)1075
2591 y FH(\031)1086 2632 y FJ(\001)1086 2670 y(\001)1086
2708 y(\001)1036 2766 y(`)20 b FH(B)p 995 2811 201 4
v 1216 2827 a FJ(R)p Fq(W)995 2887 y FJ(`)g FH(C)q(;)12
b(B)1556 2591 y(\031)1598 2562 y Fv(0)1575 2632 y FJ(\001)1575
2670 y(\001)1575 2708 y(\001)1526 2766 y(`)19 b FH(B)p
1489 2811 193 4 v 1702 2827 a FJ(L)p Fq(W)1489 2887 y
FH(C)25 b FJ(`)19 b FH(B)p 995 2932 653 4 v 1669 2948
a Fq(Cut)1157 3003 y FJ(`)h FH(B)t(;)12 b(B)p 1157 3048
206 4 v 1383 3064 a FJ(R)p Fq(C)1200 3124 y FJ(`)20 b
FH(B)167 3261 y FK(reduces)27 b(to)728 3391 y FH(\031)739
3432 y FJ(\001)739 3470 y(\001)739 3508 y(\001)690 3566
y(`)19 b FH(B)690 3626 y FK(=)-21 b(=)e(=)690 3687 y
FJ(`)19 b FH(B)1331 3472 y FK(or)28 b(to)2034 3403 y
FH(\031)2076 3374 y Fv(0)2053 3444 y FJ(\001)2053 3482
y(\001)2053 3520 y(\001)2003 3578 y(`)20 b FH(B)2003
3639 y FK(=)-21 b(=)f(=)2003 3699 y FJ(`)20 b FH(B)167
3840 y FK(where)26 b(the)g(double)h(bar)h(is)e(a)h(w)n(eak)n(ening)g
(\(with)f(an)h(exc)n(hange)h(in)e(the)g(\014rst)h(case\))g(follo)n(w)n
(ed)167 3925 y(b)n(y)h(a)g(con)n(traction.)p eop
PStoPSsaved restore
%%Page: (154,155) 78
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
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userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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PStoPSxform concat
151 154 bop 167 167 a Fz(B.2.)63 b(LINEAR)27 b(SEQUENT)g(CALCULUS)1174
b FK(151)270 404 y(But)28 b(y)n(ou)h(will)c(certainly)h(admit)h(that)h
(in)f(b)r(oth)h(cases,)1346 565 y FJ(`)20 b FH(B)1346
626 y FK(=)-21 b(=)f(=)1346 687 y FJ(`)20 b FH(B)167
837 y FK(is)28 b(essen)n(tially)e(nothing.)38 b(So)29
b FH(\031)i FK(and)e FH(\031)1215 811 y Fv(0)1260 837
y FK(are)f(obtained)h(b)n(y)g(reducing)e(the)h(same)g(pro)r(of,)i(and)
167 922 y(they)e(m)n(ust)f(b)r(e)h(denotationally)f(equal.)270
1099 y(More)38 b(generally)-6 b(,)38 b(all)e(the)h(pro)r(ofs)h(of)f(a)h
(giv)n(en)f(sequen)n(t)1870 1100 y FH(A)p 1874 1120 39
3 v 1948 1099 a FJ(`)2017 1100 y FH(B)p 2021 1120 43
3 v 2110 1099 a FK(are)g(iden)n(ti\014ed.)62 b(So)167
1185 y(classical)23 b(logic)h(is)g FM(inc)l(onsistent)p
FK(,)h(not)h(from)e(a)i FM(lo)l(gic)l(al)e FK(viewp)r(oin)n(t)g(\()p
FJ(?)g FK(is)g(not)h(pro)n(v)l(able\),)i(but)167 1270
y(from)34 b(an)h FM(algorithmic)g FK(one.)56 b(This)34
b(is)g(also)g(expressed)h(b)n(y)g(the)f(fact)g(\(noticed)g(b)n(y)h(Jo)n
(y)n(al\))167 1355 y(that)29 b FM(any)g(Cartesian)g(close)l(d)g(c)l
(ate)l(gory)g(with)g(an)f(initial)h(obje)l(ct)f FK(0)h
FM(such)g(that)g FK(0)2320 1329 y FD(0)2345 1313 y Ff(A)2404
1355 y FJ(')20 b FH(A)28 b FM(is)h(a)167 1440 y(p)l(oset)f
FK(\(see)f([)p FG(?)q FK(])h(page)g(67\).)270 1617 y(Of)35
b(course,)i(our)e(example)f(sho)n(ws)i(that)f(cut)g(elimination)c(in)k
(sequen)n(t)g(calculus)f(do)r(es)167 1703 y(not)28 b(satisfy)h(the)e
(Ch)n(urc)n(h-Rosser)i(prop)r(ert)n(y:)36 b(it)27 b(ev)n(en)h(div)n
(erges)g(in)f(the)h(w)n(orst)g(w)n(a)n(y!)37 b(There)167
1788 y(are)28 b(t)n(w)n(o)f(options)i(to)f(eliminate)c(this)j
(pathology:)270 1952 y FJ(\017)34 b FK(making)28 b(the)g(calculus)e
(asymmetric:)34 b(this)27 b(leads)g(to)h FM(intuitionistic)g(lo)l(gic)p
FK(;)270 2098 y FJ(\017)34 b FK(forbidding)25 b(structural)e(rules,)h
(except)f(the)h FM(exchange)e FK(whic)n(h)h(is)g(harmless:)33
b(this)23 b(leads)339 2183 y(to)28 b FM(line)l(ar)g(lo)l(gic)p
FK(.)167 2421 y FE(B.2)114 b(Linear)27 b(Sequen)m(t)j(Calculus)167
2577 y FK(W)-6 b(e)27 b(simply)f(discard)h FM(we)l(akening)f
FK(and)i FM(c)l(ontr)l(action)p FK(.)36 b FM(Exchange)p
FK(,)26 b FM(identity)h FK(and)i FM(cut)d FK(are)h(left)167
2662 y(unc)n(hanged,)j(but)e(logical)e(rules)h(need)h(some)g(adjustmen)
n(ts:)36 b(for)28 b(example,)g(the)f(rules)g(for)h FJ(^)167
2747 y FK(are)f(no)n(w)h(inadequate)g(\(since)e(cut)h(elimination)e(in)
h FG(??)j FK(requires)d(w)n(eak)n(enings\).)36 b(In)27
b(fact,)h(w)n(e)167 2832 y(need)g FM(two)f FK(conjunctions:)36
b(a)28 b FM(tensor)g(pr)l(o)l(duct)g FK(\(or)g FM(cumulative)f(c)l
(onjunction)p FK(\))575 3001 y FH(A)p 579 3021 39 3 v
-1 x(;)12 b(C)q(;)g(D)22 b FJ(`)880 3001 y FH(B)p 884
3021 43 3 v 546 3045 421 4 v 987 3061 a FJ(L\012)546
3122 y FH(A)p 549 3142 39 3 v 597 3121 a(;)12 b(C)21
b FJ(\012)16 b FH(D)21 b FJ(`)910 3122 y FH(B)p 913 3142
43 3 v 1468 3004 a(A)p 1472 3024 39 3 v 1540 3003 a FJ(`)e
FH(C)q(;)1683 3004 y(B)p 1687 3024 43 3 v 83 w(A)p 1822
3024 39 3 v -30 x Fv(0)1906 3003 y FJ(`)g FH(D)r(;)2058
3004 y(B)p 2062 3024 43 3 v 2114 2974 a Fv(0)p 1468 3048
663 4 v 2152 3064 a FJ(R\012)1488 3126 y FH(A)p 1492
3145 39 3 v -1 x(;)1571 3126 y(A)p 1574 3145 V -31 x
Fv(0)1659 3125 y FJ(`)g FH(C)i FJ(\012)16 b FH(D)r(;)1952
3126 y(B)p 1955 3145 43 3 v 2008 3125 a(;)2039 3126 y(B)p
2042 3145 V 2095 3095 a Fv(0)167 3296 y FK(and)29 b(a)f
FM(dir)l(e)l(ct)g(pr)l(o)l(duct)h FK(\(or)f FM(alternative)e(c)l
(onjunction)p FK(\):)378 3462 y FH(A)p 382 3482 39 3
v -1 x(;)12 b(C)25 b FJ(`)598 3462 y FH(B)p 601 3482
43 3 v 306 3506 421 4 v 747 3522 a FJ(L)p FK(1)p Fc(N)306
3584 y FH(A)p 309 3603 39 3 v 357 3583 a(;)12 b(C)21
b Fc(N)16 b FH(D)22 b FJ(`)670 3584 y FH(B)p 674 3603
43 3 v 1094 3462 a(A)p 1097 3482 39 3 v 1145 3461 a(;)12
b(D)22 b FJ(`)1318 3462 y FH(B)p 1321 3482 43 3 v 1023
3506 421 4 v 1465 3522 a FJ(L)p FK(2)p Fc(N)1023 3584
y FH(A)p 1027 3603 39 3 v -1 x(;)12 b(C)21 b Fc(N)16
b FH(D)21 b FJ(`)1388 3584 y FH(B)p 1391 3603 43 3 v
1741 3462 a(A)p 1745 3482 39 3 v 1812 3461 a FJ(`)f FH(C)q(;)1956
3462 y(B)p 1960 3482 43 3 v 83 w(A)p 2094 3482 39 3 v
2162 3461 a FJ(`)g FH(D)r(;)2315 3462 y(B)p 2318 3482
43 3 v 1741 3506 630 4 v 2391 3522 a FJ(R)p Fc(N)1846
3584 y FH(A)p 1849 3603 39 3 v 1917 3583 a FJ(`)g FH(C)h
Fc(N)15 b FH(D)r(;)2210 3584 y(B)p 2214 3603 43 3 v 270
3754 a FK(Dually)-6 b(,)32 b(w)n(e)f(shall)g(ha)n(v)n(e)h(a)g
FM(tensor)f(sum)g Fc(O)g FK(\(dual)h(of)f FJ(\012)p FK(\))g(and)i(a)e
FM(dir)l(e)l(ct)h(sum)g FJ(\010)f FK(\(dual)167 3840
y(of)j Fc(N)p FK(\),)g(with)f(symmetrical)d(rules:)46
b(left)32 b(b)r(ecoming)h(righ)n(t)g(and)h FM(vic)l(e)f(versa)p
FK(.)52 b(There)33 b(is)f(an)167 3925 y(easy)c(w)n(a)n(y)h(to)f(a)n(v)n
(oid)g(this)f(b)r(oring)h(rep)r(etition,)e(b)n(y)j(using)f
(asymmetrical)d(sequen)n(ts.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
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PStoPSxform concat
152 155 bop 167 167 a FK(152)987 b Fz(APPENDIX)27 b(B.)55
b(WHA)-6 b(T)27 b(IS)g(LINEAR)g(LOGIC?)270 404 y FK(F)-6
b(or)29 b(this,)e(w)n(e)g(in)n(tro)r(duce)g(the)g FM(line)l(ar)h(ne)l
(gation)p FK(:)270 556 y FJ(\017)34 b FK(Eac)n(h)28 b(atomic)e(form)n
(ula)h(is)g(giv)n(en)g(in)f(t)n(w)n(o)i(forms:)35 b(p)r(ositiv)n(e)26
b(\()p FH(A)p FK(\))h(and)h(negativ)n(e)f(\()p FH(A)2585
530 y Fv(?)2626 556 y FK(\).)339 641 y(By)h(de\014nition,)g(the)f
(linear)f(negation)i(of)h FH(A)e FK(is)g FH(A)1675 615
y Fv(?)1716 641 y FK(,)h(and)h FM(vic)l(e)e(versa)p FK(.)270
782 y FJ(\017)34 b FK(Linear)28 b(negation)g(is)f(extended)g(to)h(comp)
r(osed)g(form)n(ulae)g(b)n(y)g FM(de)f(Mor)l(gan)h FK(la)n(ws:)601
950 y(\()p FH(A)16 b FJ(\012)f FH(B)t FK(\))849 925 y
Fv(?)949 950 y FK(=)59 b FH(A)1114 925 y Fv(?)1171 950
y Fc(O)15 b FH(B)1297 925 y Fv(?)1673 950 y FK(\()p FH(A)g
Fc(N)g FH(B)t FK(\))1920 925 y Fv(?)2021 950 y FK(=)58
b FH(A)2185 925 y Fv(?)2242 950 y FJ(\010)15 b FH(B)2368
925 y Fv(?)601 1036 y FK(\()p FH(A)h Fc(O)f FH(B)t FK(\))849
1010 y Fv(?)949 1036 y FK(=)59 b FH(A)1114 1010 y Fv(?)1171
1036 y FJ(\012)15 b FH(B)1297 1010 y Fv(?)1673 1036 y
FK(\()p FH(A)g FJ(\010)g FH(B)t FK(\))1920 1010 y Fv(?)2021
1036 y FK(=)58 b FH(A)2185 1010 y Fv(?)2242 1036 y Fc(N)15
b FH(B)2368 1010 y Fv(?)270 1205 y FK(Linear)45 b(negation)h(is)f(not)h
(itself)d(a)j(connector:)71 b(for)45 b(example,)k(if)c
FH(A)g FK(and)h FH(B)j FK(are)167 1290 y(atomic)34 b(form)n(ulae,)j(\()
p FH(A)17 b FJ(\012)h FH(B)947 1265 y Fv(?)989 1290 y
FK(\))1016 1265 y Fv(?)1092 1290 y FK(is)35 b(just)g(a)h(meta-notation)
f(for)h FH(A)2022 1265 y Fv(?)2081 1290 y Fc(O)18 b FH(B)t
FK(,)37 b(whic)n(h)d(is)h(also)167 1376 y(con)n(v)n(en)n(tionally)e
(written)f(as)i FH(A)17 b Fu(\()g FH(B)38 b FK(\()p FM(line)l(ar)33
b(implic)l(ation)p FK(\).)53 b(Note)33 b(that)g FH(A)2281
1350 y Fv(??)2395 1376 y FK(is)f(alw)n(a)n(ys)167 1461
y FM(e)l(qual)27 b FK(to)h FH(A)p FK(.)270 1584 y(A)g(t)n(w)n(o-sided)f
(sequen)n(t)1038 1752 y FH(A)1090 1763 y FD(1)1118 1752
y FH(;)12 b(:)g(:)g(:)f(;)h(A)1324 1763 y Fw(n)1377 1752
y FJ(`)19 b FH(B)1491 1763 y FD(1)1520 1752 y FH(;)12
b(:)g(:)g(:)f(;)h(B)1726 1763 y Fw(m)167 1920 y FK(is)27
b(replaced)g(b)n(y:)1021 2088 y FJ(`)20 b FH(A)1136 2063
y Fv(?)1136 2106 y FD(1)1177 2088 y FH(;)12 b(:)g(:)g(:)g(;)g(A)1384
2063 y Fv(?)1384 2106 y Fw(n)1425 2088 y FH(;)g(B)1508
2099 y FD(1)1536 2088 y FH(;)g(:)g(:)g(:)g(;)g(B)1743
2099 y Fw(m)167 2256 y FK(In)28 b(particular,)f(the)g(iden)n(tit)n(y)f
(axiom)i(b)r(ecomes)46 b FJ(`)20 b FH(A)1605 2231 y Fv(?)1646
2256 y FH(;)12 b(A)47 b FK(and)29 b(the)e(cut:)1074 2420
y FJ(`)19 b FH(C)q(;)1217 2421 y(A)p 1221 2441 39 3 v
1367 2420 a FJ(`)h FH(C)1484 2391 y Fv(?)1526 2420 y
FH(;)1557 2421 y(B)p 1560 2441 43 3 v 1074 2465 540 4
v 1634 2481 a Fq(Cut)1243 2542 y FJ(`)1305 2543 y FH(A)p
1309 2562 39 3 v -1 x(;)1388 2543 y(B)p 1391 2562 43
3 v 167 2700 a FK(Of)27 b(course,)h(the)g(only)g(structural)e(rule)h
(is)1186 2851 y FJ(`)1249 2852 y FH(A)p 1252 2872 39
3 v -1 x(;)12 b(C)q(;)g(D)r(;)1503 2852 y(B)p 1507 2872
43 3 v 1186 2896 373 4 v 1580 2912 a Fq(X)1186 2973 y
FJ(`)1249 2974 y FH(A)p 1252 2993 39 3 v -1 x(;)g(D)r(;)g(C)q(;)1503
2974 y(B)p 1507 2993 43 3 v 167 3131 a FK(and)29 b(the)e(logical)f
(rules)h(are)h(no)n(w)g(expressed)g(b)n(y:)660 3282 y
FJ(`)20 b FH(C)q(;)804 3283 y(A)p 808 3303 39 3 v 954
3282 a FJ(`)f FH(D)r(;)1106 3283 y(B)p 1110 3303 43 3
v 660 3327 503 4 v 1183 3343 a FJ(\012)695 3404 y(`)h
FH(C)h FJ(\012)16 b FH(D)r(;)989 3405 y(A)p 992 3424
39 3 v 1040 3404 a(;)1071 3405 y(B)p 1075 3424 43 3 v
1761 3282 a FJ(`)j FH(C)q(;)12 b(D)r(;)1994 3283 y(A)p
1998 3303 39 3 v 1731 3327 345 4 v 2097 3343 a Fc(O)1731
3404 y FJ(`)20 b FH(C)h Fc(O)15 b FH(D)r(;)2024 3405
y(A)p 2028 3424 39 3 v 416 3645 a FJ(`)k FH(C)q(;)559
3646 y(A)p 563 3666 V 709 3645 a FJ(`)h FH(D)r(;)862
3646 y(A)p 865 3666 V 416 3690 498 4 v 934 3706 a Fc(N)492
3766 y FJ(`)g FH(C)h Fc(N)15 b FH(D)r(;)785 3767 y(A)p
789 3787 39 3 v 1312 3645 a FJ(`)20 b FH(C)q(;)1456 3646
y(A)p 1460 3666 V 1238 3690 345 4 v 1603 3706 a FK(1)p
FJ(\010)1238 3766 y(`)f FH(C)i FJ(\010)16 b FH(D)r(;)1531
3767 y(A)p 1534 3787 39 3 v 2011 3645 a FJ(`)k FH(D)r(;)2164
3646 y(A)p 2168 3666 V 1941 3690 345 4 v 2307 3706 a
FK(2)p FJ(\010)1941 3766 y(`)g FH(C)h FJ(\010)15 b FH(D)r(;)2234
3767 y(A)p 2238 3787 39 3 v 167 3925 a FK(There)27 b(is)g(nothing)h
(deep)g(in)f(this)g(con)n(v)n(en)n(tion:)36 b(it)27 b(is)g(just)h(a)g
(matter)e(of)j(econom)n(y!)p eop
PStoPSsaved restore
%%Page: (156,157) 79
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
153 156 bop 167 167 a Fz(B.2.)63 b(LINEAR)27 b(SEQUENT)g(CALCULUS)1174
b FK(153)270 404 y(Units)27 b(\()p FG(1)g FK(for)h FJ(\012)p
FK(,)g FJ(?)e FK(for)i Fc(O)p FK(,)g FJ(>)e FK(for)i
Fc(N)f FK(and)i FG(0)e FK(for)h FJ(\010)p FK(\))f(are)h(also)g(in)n
(tro)r(duced:)479 650 y FG(1)519 625 y Fv(?)580 650 y
FK(=)20 b FJ(?)311 b(?)1077 625 y Fv(?)1137 650 y FK(=)19
b FG(1)312 b FJ(>)1618 625 y Fv(?)1678 650 y FK(=)19
b FG(0)312 b(0)2143 625 y Fv(?)2204 650 y FK(=)19 b FJ(>)p
431 1053 103 4 v 554 1069 a FG(1)431 1130 y FJ(`)g FG(1)900
1023 y FJ(`)962 1024 y FH(A)p 966 1043 39 3 v 857 1067
200 4 v 1077 1083 a FJ(?)857 1144 y(`)g(?)p FH(;)1006
1145 y(A)p 1009 1165 39 3 v 1396 1043 200 4 v 1616 1059
a FJ(>)1396 1119 y(`)g(>)p FH(;)1545 1120 y(A)p 1548
1140 39 3 v 1935 1018 a FK(\(no)28 b(rule)f(for)h FG(0)p
FK(\))270 1445 y(Finally)-6 b(,)38 b(the)e(lost)g(structural)f(rules)h
(come)f(bac)n(k)j(with)d(a)i(logical)d(dressing,)39 b
FM(via)d FK(the)167 1530 y(mo)r(dalities)h(!)12 b FH(A)27
b FK(\()p FM(of)h(c)l(ourse)g FH(A)p FK(\))f(and)41 b(?)12
b FH(A)26 b FK(\()p FM(why)i(not)f FH(A)p FK(\):)718
1776 y(\()12 b(!)g FH(A)p FK(\))867 1750 y Fv(?)927 1776
y FK(=)g(?)g FH(A)1090 1750 y Fv(?)1681 1776 y FK(\()g(?)g
FH(A)p FK(\))1844 1750 y Fv(?)1904 1776 y FK(=)g(!)g
FH(A)2053 1750 y Fv(?)375 2157 y FJ(`)20 b FH(B)t(;)j
FK(?)581 2158 y FH(A)p 584 2178 V 364 2202 281 4 v 664
2218 a FK(!)364 2279 y FJ(`)12 b FK(!)g FH(B)t(;)23 b
FK(?)593 2280 y FH(A)p 596 2299 39 3 v 942 2157 a FJ(`)1004
2158 y FH(A)p 1008 2178 V 880 2202 238 4 v 1138 2218
a Fq(W)q FK(?)880 2279 y FJ(`)12 b FK(?)g FH(B)t(;)1067
2280 y(A)p 1070 2299 39 3 v 1433 2157 a FJ(`)g FK(?)g
FH(B)t(;)23 b FK(?)12 b FH(B)t(;)1763 2158 y(A)p 1766
2178 V 1433 2202 381 4 v 1834 2218 a Fq(C)p FK(?)1505
2279 y FJ(`)g FK(?)g FH(B)t(;)1692 2280 y(A)p 1694 2299
39 3 v 2126 2157 a FJ(`)19 b FH(B)t(;)2275 2158 y(A)p
2279 2178 V 2108 2202 238 4 v 2366 2218 a Fq(D)p FK(?)2108
2279 y FJ(`)12 b FK(?)g FH(B)t(;)2295 2280 y(A)p 2297
2299 39 3 v 167 2515 a FK(The)35 b(last)g(is)f(called)g
FM(der)l(eliction)5 b FK(:)50 b(it)34 b(is)h(equiv)l(alen)n(t)g(to)g
(the)g(axiom)g FH(B)t Fu(\()12 b FK(?)g FH(B)t FK(,)37
b(or)e(dually)167 2600 y(!)12 b FH(B)20 b Fu(\()15 b
FH(B)t FK(.)270 2762 y(This)42 b(allo)n(ws)g(us)h(to)f(represen)n(t)g
(in)n(tuitionistic)d(form)n(ulae)j(in)g(linear)f(logic,)k
FM(via)i FK(the)167 2847 y(follo)n(wing)27 b(de\014nitions)274
3093 y FH(A)15 b FJ(^)g FH(B)24 b FK(=)19 b FH(A)d Fc(N)f
FH(B)110 b(A)16 b FJ(_)f FH(B)24 b FK(=)12 b(!)g FH(A)p
FJ(\010)g FK(!)g FH(B)109 b(A)19 b FJ(\))h FH(B)k FK(=)12
b(!)g FH(A)i Fu(\()i FH(B)111 b FJ(:)p FH(A)19 b FK(=)12
b(!)g FH(A)i Fu(\()i FG(0)167 3339 y FK(in)28 b(suc)n(h)i(a)f(w)n(a)n
(y)h(that)f(an)h(in)n(tuitionistic)c(form)n(ula)j(is)f(v)l(alid)h
(i\013)f(its)g(translation)h(is)f(pro)n(v)l(able)167
3424 y(in)j(Linear)g(Sequen)n(t)h(Calculus)e(\(so,)j(for)f(example,)f
(dereliction)e(expresses)j(that)f FH(B)c FJ(\))22 b FH(B)t
FK(\).)167 3509 y(This)40 b(translation)g(is)g(in)g(fact)h(used)g(for)g
(the)f(coherence)f(seman)n(tics)h(of)h(t)n(yp)r(ed)g(lam)n(b)r(da)167
3594 y(calculus)27 b(\(c)n(hapters)g FG(??)q FK(,)h FG(??)q
FK(,)f FG(??)i FK(and)g(app)r(endix)f FG(??)q FK(\).)270
3840 y(It)35 b(is)f(also)i(p)r(ossible)e(to)h(add)h(\(\014rst)f(and)i
(second)e(order\))g(quan)n(ti\014ers,)j(but)d(the)g(main)167
3925 y(features)28 b(of)g(linear)e(logic)g(are)i(already)g(con)n
(tained)g(in)f(the)g(prop)r(ositional)h(fragmen)n(t.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
154 157 bop 167 167 a FK(154)987 b Fz(APPENDIX)27 b(B.)55
b(WHA)-6 b(T)27 b(IS)g(LINEAR)g(LOGIC?)167 404 y FE(B.3)114
b(Pro)s(of)27 b(nets)167 560 y FK(Here,)g(w)n(e)g(shall)f(concen)n
(trate)i(on)g(the)f(so-called)g FM(multiplic)l(ative)32
b FK(fragmen)n(t)c(of)g(linear)e(logic,)167 645 y FM(i.e.)40
b FK(the)g(connectors)g FJ(\012)p FK(,)j FG(1)p FK(,)g
Fc(O)d FK(and)h FJ(?)p FK(.)71 b(In)41 b(this)e(fragmen)n(t,)44
b(rules)39 b(are)h FM(c)l(onservative)167 730 y FK(o)n(v)n(er)31
b(con)n(texts:)41 b(the)30 b(con)n(text)g(in)g(the)f(conclusion)h(is)g
(the)g(disjoin)n(t)g(union)g(of)h(those)g(of)f(the)167
815 y(premises.)47 b(The)32 b(rules)f(for)h Fc(N)g FK(and)h
FJ(>)d FK(are)i(not,)i(and)f(if)e(w)n(e)h(renounce)g(these)g
(connectors,)167 900 y(w)n(e)27 b(m)n(ust)h(renounce)f(their)g(duals)h
FJ(\010)f FK(and)i FG(0)p FK(.)270 1027 y(F)-6 b(rom)36
b(an)g(algorithmic)d(viewp)r(oin)n(t,)k(this)d(fragmen)n(t)i(is)e(v)n
(ery)i FM(unexpr)l(essive)p FK(,)h(but)f(this)167 1112
y(restriction)27 b(is)h(necessary)i(if)e(w)n(e)g(w)n(an)n(t)i(to)f(tac)
n(kle)f(problems)h(progressiv)n(ely)-6 b(.)40 b(F)-6
b(urthermore,)167 1197 y(m)n(ultiplicativ)n(e)50 b(connectors)k(and)g
(rules)f(can)h(b)r(e)g(generalised)e(to)i(mak)n(e)g(a)g(gen)n(uine)167
1282 y(programming)28 b(language)860 1256 y FD(1)889
1282 y FK(.)270 1409 y(Sequen)n(t)h(pro)r(ofs)f(con)n(tain)g(a)g(lot)g
(of)g(redundancy:)37 b(in)27 b(a)h(rule)e(suc)n(h)i(as)1225
1566 y FJ(`)20 b FH(C)q(;)12 b(D)r(;)1459 1567 y(A)p
1463 1587 39 3 v 1196 1611 345 4 v 1561 1627 a Fc(O)1196
1688 y FJ(`)19 b FH(C)i Fc(O)16 b FH(D)r(;)1489 1689
y(A)p 1492 1708 39 3 v 167 1852 a FK(the)32 b(con)n(text)550
1853 y FH(A)p 553 1873 V 601 1852 a FK(,)i(whic)n(h)e(pla)n(ys)h(a)f
(passiv)n(e)h(r^)-34 b(ole,)33 b(is)f(rewritten)e(without)i(an)n(y)h(c)
n(hange.)51 b(By)167 1937 y(exp)r(elling)37 b(all)h(those)i(b)r(oring)f
(con)n(texts,)k(w)n(e)38 b(obtain)i(the)f FM(substanti\014que)f(mo)l
(el)t(le)g FK(of)h(the)167 2022 y(pro)r(of,)29 b(called)d(the)h
FM(pr)l(o)l(of)j(net)p FK(.)270 2149 y(F)-6 b(or)29 b(example,)d(the)i
(pro)r(of)852 2319 y FJ(`)20 b FH(A;)12 b(A)1050 2290
y Fv(?)1189 2319 y FJ(`)19 b FH(B)t(;)12 b(B)1394 2290
y Fv(?)p 852 2364 584 4 v 1457 2380 a FJ(\012)889 2441
y(`)20 b FH(A)15 b FJ(\012)h FH(B)t(;)c(A)1229 2412 y
Fv(?)1270 2441 y FH(;)g(B)1357 2412 y Fv(?)1644 2441
y FJ(`)19 b FH(C)q(;)12 b(C)1841 2412 y Fv(?)p 889 2486
995 4 v 1905 2501 a FJ(\012)973 2562 y(`)20 b FK(\()p
FH(A)15 b FJ(\012)h FH(B)t FK(\))f FJ(\012)g FH(C)q(;)d(A)1502
2533 y Fv(?)1544 2562 y FH(;)g(B)1631 2533 y Fv(?)1673
2562 y FH(;)g(C)1758 2533 y Fv(?)973 2623 y FK(=)-28
b(=)-16 b(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g
(=)g(=)g(=)g(=)-29 b(=)973 2683 y FJ(`)20 b FH(A)1088
2654 y Fv(?)1129 2683 y FH(;)12 b(B)1216 2654 y Fv(?)1258
2683 y FH(;)g FK(\()p FH(A)j FJ(\012)g FH(B)t FK(\))h
FJ(\012)f FH(C)q(;)d(C)1757 2654 y Fv(?)p 946 2728 882
4 v 1848 2744 a Fc(O)946 2805 y FJ(`)19 b FH(A)1060 2776
y Fv(?)1117 2805 y Fc(O)d FH(B)1244 2776 y Fv(?)1286
2805 y FH(;)c FK(\()p FH(A)j FJ(\012)g FH(B)t FK(\))g
FJ(\012)h FH(C)q(;)c(C)1785 2776 y Fv(?)167 2966 y FK(b)r(ecomes)575
3290 y FH(A)78 b(B)p 571 3335 194 4 v 571 3411 a(A)15
b FJ(\012)h FH(B)878 3412 y(C)p 558 3456 388 4 v 558
3533 a FK(\()p FH(A)f FJ(\012)g FH(B)t FK(\))h FJ(\012)f
FH(C)1371 3550 y(C)1425 3525 y Fv(?)1953 3423 y FH(A)2005
3394 y Fv(?)2125 3423 y FH(B)2181 3394 y Fv(?)p 1949
3468 278 4 v 1949 3544 a FH(A)2001 3515 y Fv(?)2058 3544
y Fc(O)h FH(B)2185 3515 y Fv(?)p 618 3134 1370 3 v 617
3216 3 84 v 1986 3350 3 218 v 735 3050 1420 3 v 734 3216
3 168 v 2153 3350 3 301 v 902 3267 502 3 v 901 3333 3
67 v 1402 3483 3 218 v 167 3649 1003 3 v 246 3692 a FC(1)273
3713 y FB(The)25 b(idea)g(is)g(to)h(use,)g(not)f(a)h(\014xed)f(logic,)g
(but)g(an)g(extensible)f(one.)34 b(The)25 b(program)h(declares)g(its)e
(o)n(wn)167 3784 y(connectors)h(\()p FA(i.e.)g FB(p)r(olymorphic)d(t)n
(yp)r(es\))j(and)f(rules)g(\()p FA(i.e.)h FB(constructors)g(and)g
(destructors\),)g(and)f(describ)r(es)167 3854 y(the)29
b(con)n(v)n(ersions)g(\()p FA(i.e.)g FB(the)g(program\).)40
b(Cut)29 b(elimination)d(is)j(in)f(fact)h FA(p)m(ar)m(al)s(lel)g(c)m
(ommunic)m(ation)i(b)m(etwe)m(en)167 3925 y(pr)m(o)m(c)m(esses)p
FB(.)j(In)28 b(this)e(language,)i(logic)f(do)r(es)g(not)h(ensure)f
FA(termination)p FB(,)h(but)f FA(absenc)m(e)h(of)f(de)m(ad)s(lo)m(ck)p
FB(.)p eop
PStoPSsaved restore
%%Page: (158,159) 80
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
155 158 bop 167 167 a Fz(B.3.)63 b(PR)n(OOF)27 b(NETS)1755
b FK(155)167 404 y(whic)n(h)27 b(could)g(also)h(come)f(from:)852
610 y FJ(`)20 b FH(A;)12 b(A)1050 581 y Fv(?)1189 610
y FJ(`)19 b FH(B)t(;)12 b(B)1394 581 y Fv(?)p 852 655
584 4 v 1457 671 a FJ(\012)889 731 y(`)20 b FH(A)15 b
FJ(\012)h FH(B)t(;)c(A)1229 702 y Fv(?)1270 731 y FH(;)g(B)1357
702 y Fv(?)889 792 y FK(=)-16 b(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g
(=)g(=)889 853 y FJ(`)20 b FH(A)1004 824 y Fv(?)1045
853 y FH(;)12 b(B)1132 824 y Fv(?)1174 853 y FH(;)g(A)j
FJ(\012)h FH(B)p 862 897 565 4 v 1447 913 a Fc(O)862
974 y FJ(`)j FH(A)976 945 y Fv(?)1034 974 y Fc(O)c FH(B)1160
945 y Fv(?)1202 974 y FH(;)d(A)j FJ(\012)g FH(B)862 1035
y FK(=)-26 b(=)-17 b(=)h(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)g(=)-26
b(=)862 1095 y FJ(`)19 b FH(A)d FJ(\012)f FH(B)t(;)d(A)1201
1066 y Fv(?)1258 1095 y Fc(O)k FH(B)1385 1066 y Fv(?)1644
1095 y FJ(`)j FH(C)q(;)12 b(C)1841 1066 y Fv(?)p 862
1140 1023 4 v 1905 1156 a FJ(\012)932 1217 y(`)20 b FK(\()p
FH(A)15 b FJ(\012)g FH(B)t FK(\))h FJ(\012)f FH(C)q(;)d(A)1461
1187 y Fv(?)1519 1217 y Fc(O)j FH(B)1645 1187 y Fv(?)1687
1217 y FH(;)d(C)1772 1187 y Fv(?)270 1431 y FK(Essen)n(tially)-6
b(,)27 b(w)n(e)g(lose)g(the)h(\(inessen)n(tial\))d(application)i(order)
h(of)g(rules.)270 1654 y(A)n(t)g(this)f(p)r(oin)n(t,)i(precise)d
(de\014nitions)i(are)g(needed.)36 b(A)28 b FM(pr)l(o)l(of)i(structur)l
(e)e FK(is)f(just)h(a)h(graph)167 1739 y(built)d(from)i(the)f(follo)n
(wing)g(comp)r(onen)n(ts:)270 1934 y FJ(\017)34 b FM(link)8
b FK(:)1396 2248 y FH(A)95 b(A)1595 2222 y Fv(?)p 1422
2082 168 3 v 1421 2164 3 84 v 1588 2164 V 270 2474 a
FJ(\017)34 b FM(cut)7 b FK(:)1380 2711 y FH(A)78 b(A)1562
2682 y Fv(?)p 1380 2756 224 4 v 270 3096 a FJ(\017)34
b FM(lo)l(gic)l(al)28 b(rules)6 b FK(:)692 3333 y FH(A)78
b(B)p 688 3378 194 4 v 688 3455 a(A)16 b FJ(\012)f FH(B)1235
3333 y(A)78 b(B)p 1231 3378 V 1231 3455 a(A)16 b Fc(O)f
FH(B)p 1774 3357 79 4 v 1793 3433 a FG(1)p 2202 3357
94 4 v 388 w FJ(?)270 3754 y FK(Eac)n(h)31 b(form)n(ula)e(m)n(ust)h(b)r
(e)g(the)f(conclusion)h(of)g(exactly)f(one)i(rule)d(and)j(a)f(premise)e
(of)j(at)167 3840 y(most)24 b(one)g(rule.)33 b(F)-6 b(orm)n(ulae)24
b(whic)n(h)f(are)h(not)g(premises)f(are)g(called)g FM(c)l(onclusions)g
(of)i(the)e(pr)l(o)l(of)167 3925 y(structur)l(e)5 b FK(:)36
b(these)27 b(conclusions)g(are)h(not)g(ordered.)36 b(Links)27
b(and)i(cuts)f(are)f(symmetrical.)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
156 159 bop 167 167 a FK(156)987 b Fz(APPENDIX)27 b(B.)55
b(WHA)-6 b(T)27 b(IS)g(LINEAR)g(LOGIC?)270 404 y FM(Pr)l(o)l(of)h(nets)
e FK(are)g(pro)r(of)h(structures)f(whic)n(h)f(are)h(constructed)g
(according)h(to)f(the)g(rules)g(of)167 490 y(Linear)h(Sequen)n(t)h
(Calculus:)270 649 y FJ(\017)34 b FK(Links)28 b(are)g(pro)r(of)g(nets.)
270 792 y FJ(\017)34 b FK(If)24 b FH(A)f FK(is)g(a)h(conclusion)g(of)g
(a)g(pro)r(of)h(net)e FH(\027)28 b FK(and)d FH(A)1633
767 y Fv(?)1698 792 y FK(is)e(a)h(conclusion)g(of)g(a)g(pro)r(of)h(net)
e FH(\027)2636 767 y Fv(0)2653 792 y FK(,)1352 997 y
FH(\027)1362 1038 y FJ(\001)1362 1075 y(\001)1362 1113
y(\001)1346 1171 y FH(A)1564 983 y(\027)1602 954 y Fv(0)1582
1024 y FJ(\001)1582 1062 y(\001)1582 1100 y(\001)1545
1171 y FH(A)1597 1142 y Fv(?)p 1346 1215 293 4 v 339
1463 a FK(is)k(a)h(pro)r(of)h(net.)270 1606 y FJ(\017)34
b FK(If)26 b FH(A)f FK(is)g(a)h(conclusion)g(of)g(a)g(pro)r(of)h(net)e
FH(\027)30 b FK(and)d FH(B)i FK(is)c(a)h(conclusion)g(of)g(a)g(pro)r
(of)h(net)e FH(\027)2636 1581 y Fv(0)2653 1606 y FK(,)1371
1797 y FH(\027)1381 1838 y FJ(\001)1381 1876 y(\001)1381
1914 y(\001)1365 1971 y FH(A)1564 1797 y(\027)1602 1768
y Fv(0)1582 1838 y FJ(\001)1582 1876 y(\001)1582 1914
y(\001)1563 1971 y FH(B)p 1365 2016 255 4 v 1395 2092
a(A)15 b FJ(\012)h FH(B)339 2269 y FK(is)27 b(a)h(pro)r(of)h(net.)270
2412 y FJ(\017)34 b FK(If)28 b FH(A)f FK(and)i FH(B)i
FK(are)d(conclusions)g(of)g(the)f(same)g(pro)r(of)i(net)f
FH(\027)t FK(,)1473 2578 y FH(\027)1381 2637 y FJ(\001)1381
2675 y(\001)1381 2713 y(\001)1365 2770 y FH(A)1582 2637
y FJ(\001)1582 2675 y(\001)1582 2713 y(\001)1563 2770
y FH(B)p 1330 2815 324 4 v 1395 2891 a(A)15 b Fc(O)h
FH(B)339 3068 y FK(is)27 b(a)h(pro)r(of)h(net.)270 3224
y FJ(\017)p 359 3190 40 4 v 359 3267 a FG(1)446 3224
y FK(is)e(a)h(pro)r(of)g(net.)270 3380 y FJ(\017)34 b
FK(If)28 b FH(\027)k FK(is)27 b(a)h(pro)r(of)h(net,)1473
3546 y FH(\027)p 1465 3677 55 4 v 1465 3754 a FJ(?)339
3925 y FK(is)e(a)h(pro)r(of)h(net.)p eop
PStoPSsaved restore
%%Page: (160,161) 81
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
157 160 bop 167 167 a Fz(B.4.)63 b(CUT)27 b(ELIMINA)-6
b(TION)1557 b FK(157)270 404 y(There)40 b(is)f(a)i(funn)n(y)h
(correctness)d(criterion)f(\(the)i FM(long)g(trip)g FK(condition,)j
(see)d([)p FG(?)q FK(]\))g(to)167 490 y(c)n(haracterise)28
b(pro)r(of)j(nets)e(among)i(pro)r(of)g(structures.)41
b(F)-6 b(or)30 b(example,)f(the)g(follo)n(wing)g(pro)r(of)167
575 y(structure)1327 916 y FH(A)78 b(B)p 1323 960 194
4 v 1323 1037 a(A)15 b Fc(O)g FH(B)1039 1026 y(A)1091
1000 y Fv(?)1705 1026 y FH(B)1761 1000 y Fv(?)p 1086
777 268 3 v 1084 943 3 168 v 1352 842 3 67 v 1486 777
268 3 v 1752 943 3 168 v 1485 842 3 67 v 167 1216 a FK(is)50
b(not)h(a)g(pro)r(of)h(net,)57 b(and)51 b(indeed,)56
b(do)r(es)c(not)f(satisfy)g(the)f(long)h(trip)f(condition.)167
1301 y(Unfortunately)-6 b(,)39 b(this)d(criterion)f(w)n(orks)i(only)f
(for)h(the)f(\()p FJ(\012)p FH(;)12 b Fc(O)p FH(;)g FG(1)p
FK(\))35 b(fragmen)n(t)i(of)g(the)f(logic)167 1386 y(\(not)28
b FJ(?)p FK(\).)167 1646 y FE(B.4)114 b(Cut)29 b(elimination)167
1812 y FK(Pro)r(ofs)g(nets)e(pro)n(vide)h(a)g(v)n(ery)g(nice)e(framew)n
(ork)i(for)g(describing)f(cut)g(elimination.)270 1959
y(Con)n(v)n(ersions)i(are)f(purely)f(lo)r(cal:)676 2257
y FH(A)806 2261 y(A)858 2232 y Fv(?)1029 2128 y FJ(\001)1029
2166 y(\001)1029 2204 y(\001)1012 2261 y FH(A)p 806 2306
258 4 v 690 2303 a FJ(\001)690 2341 y(\001)690 2378 y(\001)p
701 2091 134 3 v 700 2190 3 101 v 834 2190 V 1384 2257
a Fu( )1794 2124 y FJ(\001)1794 2162 y(\001)1794 2200
y(\001)1778 2257 y FH(A)1792 2303 y FJ(\001)1792 2341
y(\001)1792 2378 y(\001)360 2503 y(\001)360 2541 y(\001)360
2579 y(\001)344 2636 y FH(A)561 2503 y FJ(\001)561 2541
y(\001)561 2579 y(\001)543 2636 y FH(B)p 344 2681 255
4 v 375 2757 a(A)15 b FJ(\012)h FH(B)783 2489 y FJ(\001)783
2527 y(\001)783 2565 y(\001)746 2636 y FH(A)798 2607
y Fv(?)1026 2489 y FJ(\001)1026 2527 y(\001)1026 2565
y(\001)987 2636 y FH(B)1043 2607 y Fv(?)p 746 2681 339
4 v 777 2757 a FH(A)829 2728 y Fv(?)886 2757 y Fc(O)g
FH(B)1013 2728 y Fv(?)p 375 2802 680 4 v 1419 2681 a
Fu( )1839 2564 y FJ(\001)1839 2601 y(\001)1839 2639 y(\001)1823
2697 y FH(A)2059 2550 y FJ(\001)2059 2588 y(\001)2059
2626 y(\001)2022 2697 y FH(A)2074 2667 y Fv(?)p 1823
2741 293 4 v 2212 2564 a FJ(\001)2212 2601 y(\001)2212
2639 y(\001)2194 2697 y FH(B)2436 2550 y FJ(\001)2436
2588 y(\001)2436 2626 y(\001)2397 2697 y FH(B)2453 2667
y Fv(?)p 2194 2741 301 4 v 843 3008 40 4 v 843 3085 a
FG(1)p 1030 3008 55 4 v 147 w FJ(?)p 843 3130 242 4 v
1419 3103 a Fu( )334 b FK(\(nothing\))167 3404 y FG(Prop)r(osition)25
b FK(The)j(con)n(v)n(ersions)g(preserv)n(e)f(the)h(prop)r(ert)n(y)g(of)
g(b)r(eing)f(a)i(pro)r(of)f(net.)270 3551 y(T)-6 b(o)34
b(pro)n(v)n(e)f(this,)h(y)n(ou)g(sho)n(w)f(that)g(con)n(v)n(ersions)h
(of)f(pro)r(of)h(nets)e(re\015ect)g(con)n(v)n(ersions)i(of)167
3637 y(sequen)n(t)28 b(pro)r(ofs,)h(or)f(alternativ)n(ely)-6
b(,)26 b(y)n(ou)j(mak)n(e)f(use)g(of)g(the)f(long)h(trip)f(condition.)
216 b Fu(\003)167 3852 y FG(Prop)r(osition)25 b FK(An)n(y)j(pro)r(of)h
(net)e(reduces)h(to)g(a)g(\(unique\))f(cut)g(free)g(one.)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
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PStoPSxform concat
158 161 bop 167 167 a FK(158)987 b Fz(APPENDIX)27 b(B.)55
b(WHA)-6 b(T)27 b(IS)g(LINEAR)g(LOGIC?)270 404 y FK(F)-6
b(or)29 b(example,)d(the)i(pro)r(of)h(net)333 736 y FH(A)78
b(B)p 329 781 194 4 v 329 858 a(A)15 b FJ(\012)h FH(B)116
b(C)p 315 902 388 4 v 315 979 a FK(\()p FH(A)16 b FJ(\012)f
FH(B)t FK(\))g FJ(\012)h FH(C)979 997 y(C)1033 971 y
Fv(?)1337 992 y FH(A)1389 963 y Fv(?)1446 992 y Fc(O)g
FH(B)1573 963 y Fv(?)1693 992 y FH(A)f FJ(\012)g FH(B)p
1337 1037 550 4 v 2131 997 a(B)2187 971 y Fv(?)2467 997
y FH(A)2519 971 y Fv(?)p 359 580 1003 3 v 358 663 3 84
v 1360 930 3 351 v 493 497 1053 3 v 491 663 3 168 v 1544
930 3 435 v 660 714 351 3 v 658 779 3 67 v 1009 930 3
218 v 1729 747 769 3 v 1727 930 3 184 v 2496 930 V 1862
831 301 3 v 1861 930 3 101 v 2162 930 V 270 1153 a FK(reduces)28
b(\(in)f(three)f(steps\))i(to)575 1489 y FH(A)78 b(B)p
571 1534 194 4 v 571 1610 a(A)15 b FJ(\012)h FH(B)878
1611 y(C)p 558 1655 388 4 v 558 1732 a FK(\()p FH(A)f
FJ(\012)g FH(B)t FK(\))h FJ(\012)f FH(C)1371 1749 y(C)1425
1724 y Fv(?)1872 1749 y FH(B)1928 1724 y Fv(?)2208 1749
y FH(A)2260 1724 y Fv(?)p 735 1333 1170 3 v 734 1415
3 84 v 1903 1682 3 351 v 618 1249 1621 3 v 617 1415 3
168 v 2237 1682 3 435 v 902 1466 502 3 v 901 1549 3 84
v 1402 1682 3 218 v 270 1946 a FK(T)-6 b(o)26 b(pro)n(v)n(e)g(the)f
(prop)r(osition,)i(it)d(is)g(enough)j(to)f(see)f(that)g
Fu( )h FK(de\014nes)g(a)f FM(terminating)g FK(and)167
2031 y FM(c)l(on\015uent)j FK(relation,)e(and)j(a)f(normal)f(form)h(is)
f(necessarily)f(cut)h(free,)g(unless)h(it)e(con)n(tains)1308
2326 y FH(A)1360 2297 y Fv(?)1480 2326 y FH(A)p 1308
2371 224 4 v 1336 2165 168 3 v 1335 2264 3 101 v 1502
2264 V 167 2529 a FK(whic)n(h)j(is)f(imp)r(ossible)g(in)g(a)i(pro)r(of)
h(net.)40 b FM(T)-5 b(ermination)30 b FK(is)e(ob)n(vious)j(\(the)e
(size)f(decreases)h(at)167 2614 y(eac)n(h)i(step\))f(and)i
FM(c)l(on\015uenc)l(e)e FK(comes)g(from)g(the)h(fact)f(that)h(con)n(v)n
(ersions)g(are)g(purely)f(lo)r(cal,)167 2699 y(the)d(only)h(p)r
(ossible)f(con\015icts)h(b)r(eing:)475 2957 y FH(A)640
2953 y(A)692 2924 y Fv(?)811 2953 y FH(A)p 640 2998 224
4 v 955 2957 a(A)1007 2932 y Fv(?)p 501 2791 168 3 v
500 2890 3 101 v 667 2890 V 835 2791 168 3 v 834 2890
3 101 v 1001 2890 V 1364 2957 a FK(and)1829 2806 y FJ(\001)1829
2844 y(\001)1829 2882 y(\001)1792 2953 y FH(A)1844 2924
y Fv(?)1998 2953 y FH(A)p 1792 2998 258 4 v 76 w(A)2178
2924 y Fv(?)2348 2820 y FJ(\001)2348 2858 y(\001)2348
2896 y(\001)2332 2953 y FH(A)p 2126 2998 V 2021 2791
151 3 v 2020 2890 3 101 v 2170 2890 V 167 3114 a FK(The)f(reader)h(can)
g(easily)e(c)n(hec)n(k)i(the)f(con\015uence)h(in)f(b)r(oth)i(cases.)689
b Fu(\003)270 3285 y FK(It)40 b(is)g(imp)r(ortan)n(t)g(to)h(notice)e
(that)i(cuts)f(are)h(eliminated)c(in)j(arbitrary)g(order:)61
b(cut)167 3370 y(elimination)25 b(is)i(a)h FM(p)l(ar)l(al)t(lel)34
b FK(pro)r(cess.)270 3540 y(A)28 b(link)1156 3762 y FH(A)15
b FJ(\012)g FH(B)103 b(A)1500 3737 y Fv(?)1557 3762 y
Fc(O)16 b FH(B)1684 3737 y Fv(?)p 1253 3619 335 3 v 1251
3701 3 84 v 1585 3701 V 167 3925 a FK(can)28 b(alw)n(a)n(ys)g(b)r(e)g
(replaced)e(b)n(y)p eop
PStoPSsaved restore
%%Page: (162,163) 82
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
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/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
159 162 bop 167 167 a Fz(B.4.)63 b(CUT)27 b(ELIMINA)-6
b(TION)1557 b FK(159)1160 566 y FH(A)78 b(B)p 1156 611
194 4 v 1156 687 a(A)15 b FJ(\012)g FH(B)1452 574 y(A)1504
545 y Fv(?)1624 574 y FH(B)1680 545 y Fv(?)p 1448 619
278 4 v 1448 696 a FH(A)1500 667 y Fv(?)1557 696 y Fc(O)h
FH(B)1684 667 y Fv(?)p 1186 419 301 3 v 1185 501 3 84
v 1485 501 V 1319 335 335 3 v 1318 501 3 168 v 1652 501
V 167 818 a FK(and)29 b(similarly)24 b(for)k FG(1)f FK(and)i
FJ(?)p FK(.)34 b(So)29 b(w)n(e)e(can)h(also)g(restrict)d(links)i(to)h
FM(atomic)k FK(form)n(ulae.)270 978 y(Consider)23 b(no)n(w)g(a)g(cut)f
(free)f(pro)r(of)j(net)e(with)g(\014xed)h(conclusions.)33
b(Since)22 b(the)g(logical)f(rules)167 1063 y(follo)n(w)35
b(faithfully)g(the)h(structure)f(of)i(these)e(conclusions,)k(our)d(pro)
r(of)h(net)f(is)f(completely)167 1148 y(determined)26
b(b)n(y)i(its)f(\(atomic\))f(links.)35 b(So)28 b(our)g(\014rst)g
(example)f(comes)g(to)558 1553 y(\()p FH(A)15 b FJ(\012)g
FH(B)t FK(\))h FJ(\012)f FH(C)1371 1554 y(C)1425 1528
y Fv(?)1949 1548 y FH(A)2001 1522 y Fv(?)2058 1548 y
Fc(O)h FH(B)2185 1522 y Fv(?)p 618 1304 1370 3 v 617
1487 3 184 v 1986 1487 V 751 1221 1403 3 v 750 1487 3
268 v 2153 1487 V 919 1388 485 3 v 917 1487 3 101 v 1402
1487 V 167 1687 a FK(whic)n(h)30 b(is)g(just)h(an)g(in)n(v)n(olutiv)n
(e)f(p)r(erm)n(utation,)h(sending)g(an)g(\(o)r(ccurrence)e(of)5
b(\))31 b(atom)g(to)g(\(an)167 1773 y(o)r(ccurrence)c(of)5
b(\))27 b(its)g(negation.)270 1932 y(The)38 b(cut)g(itself)e(has)j(a)f
(natural)g(in)n(terpretation)f(in)g(terms)g(of)i(those)f(p)r(erm)n
(utations.)167 2017 y(Instead)28 b(of)g(eliminating)d(it)i(in)333
2326 y FH(A)78 b(B)p 329 2371 194 4 v 329 2447 a(A)15
b FJ(\012)h FH(B)116 b(C)p 315 2492 388 4 v 315 2568
a FK(\()p FH(A)16 b FJ(\012)f FH(B)t FK(\))g FJ(\012)h
FH(C)979 2586 y(C)1033 2560 y Fv(?)1337 2582 y FH(A)1389
2553 y Fv(?)1446 2582 y Fc(O)g FH(B)1573 2553 y Fv(?)1693
2582 y FH(A)f FJ(\012)g FH(B)p 1337 2626 550 4 v 2131
2586 a(B)2187 2560 y Fv(?)2467 2586 y FH(A)2519 2560
y Fv(?)p 359 2170 1003 3 v 358 2252 3 84 v 1360 2519
3 351 v 493 2086 1053 3 v 491 2252 3 168 v 1544 2519
3 435 v 660 2303 351 3 v 658 2386 3 84 v 1009 2519 3
218 v 1729 2337 769 3 v 1727 2519 3 184 v 2496 2519 V
1862 2420 301 3 v 1861 2519 3 101 v 2162 2519 V 167 2720
a FK(y)n(ou)29 b(connect)e(the)h(p)r(erm)n(utations)315
3121 y(\()p FH(A)16 b FJ(\012)f FH(B)t FK(\))g FJ(\012)h
FH(C)979 3122 y(C)1033 3096 y Fv(?)1323 3116 y FH(A)1375
3090 y Fv(?)1432 3116 y Fc(O)f FH(B)1558 3090 y Fv(?)1698
3116 y FH(A)h FJ(\012)f FH(B)2131 3122 y(B)2187 3096
y Fv(?)2467 3122 y FH(A)2519 3096 y Fv(?)p 376 2872 986
3 v 375 3055 3 184 v 1360 3055 V 509 2789 1036 3 v 508
3055 3 268 v 1544 3055 V 676 2956 335 3 v 675 3055 3
101 v 1009 3055 V 1729 2872 769 3 v 1727 3055 3 184 v
2496 3055 V 1862 2956 301 3 v 1861 3055 3 101 v 2162
3055 V 1361 3206 368 3 v 1360 3205 3 51 v 1727 3205 V
1545 3256 318 3 v 1544 3255 3 101 v 1861 3255 V 167 3389
a FK(to)28 b(obtain)g(the)f(normal)h(form)f(b)n(y)i(iteration:)315
3790 y(\()p FH(A)16 b FJ(\012)f FH(B)t FK(\))g FJ(\012)h
FH(C)979 3791 y(C)1033 3766 y Fv(?)2131 3791 y FH(B)2187
3766 y Fv(?)2467 3791 y FH(A)2519 3766 y Fv(?)p 376 3542
986 3 v 375 3724 3 184 v 1360 3875 3 335 v 509 3458 1036
3 v 508 3724 3 268 v 1544 3925 3 468 v 676 3625 335 3
v 675 3724 3 101 v 1009 3724 V 1729 3542 769 3 v 1727
3875 3 335 v 2496 3724 3 184 v 1862 3625 301 3 v 1861
3925 3 301 v 2162 3724 3 101 v 1361 3876 368 3 v 1545
3926 318 3 v eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
160 163 bop 167 167 a FK(160)987 b Fz(APPENDIX)27 b(B.)55
b(WHA)-6 b(T)27 b(IS)g(LINEAR)g(LOGIC?)270 404 y FK(This)k
FM(turb)l(o)h(cut)f(elimination)g FK(mec)n(hanism)f(is)h(the)g(basic)g
(idea)g(for)h(generalising)e(pro)r(of)167 490 y(nets)e(to)f(non-m)n
(ultiplicativ)n(e)e(connectiv)n(es)i(\()p FM(ge)l(ometry)h(of)g(inter)l
(action)p FK(\).)167 720 y FE(B.5)114 b(Pro)s(of)27 b(nets)i(and)g
(natural)g(deduction)167 875 y FK(It)24 b(is)g(fair)g(to)h(sa)n(y)h
(that)f(pro)r(of)h(nets)e(are)h(the)f(natural)h(deductions)g(of)g
(linear)f(logic,)g(but)h(with)167 960 y(t)n(w)n(o)j(notable)g
(di\013erences:)270 1082 y FJ(\017)34 b FK(Thanks)c(to)d(linearit)n(y)
-6 b(,)26 b(there)h(is)g(no)i(need)e(for)h FM(p)l(ar)l(c)l(els)h(of)f
(hyp)l(otheses)p FK(.)270 1217 y FJ(\017)34 b FK(Thanks)d(to)f(linear)d
(negation,)k(there)d(is)h(no)h(need)f(for)g FM(dischar)l(ge)h
FK(or)g(for)f FM(elimination)339 1302 y(rules)p FK(.)270
1424 y(F)-6 b(or)47 b(example,)j(if)45 b(w)n(e)h(follo)n(w)f(the)h(ob)n
(vious)h(analogy)h(b)r(et)n(w)n(een)e(the)g(in)n(tuitionistic)167
1509 y(implication)25 b FH(A)19 b FJ(\))h FH(B)32 b FK(and)c(the)g
(linear)e(one)i FH(A)15 b Fu(\()h FH(B)24 b FK(=)19 b
FH(A)1728 1484 y Fv(?)1785 1509 y Fc(O)c FH(B)t FK(,)28
b(the)f(in)n(tro)r(duction)1293 1651 y([)p FH(A)p FK(])1329
1709 y FJ(\001)1329 1747 y(\001)1329 1785 y(\001)1310
1842 y FH(B)p 1230 1887 218 4 v 1468 1903 a FJ(\)I)1230
1963 y FH(A)19 b FJ(\))h FH(B)167 2087 y FK(corresp)r(onds)29
b(to)1288 2468 y FH(A)1340 2439 y Fv(?)1497 2294 y FH(A)1513
2335 y FJ(\001)1513 2373 y(\001)1513 2411 y(\001)1495
2468 y FH(B)p 1288 2513 263 4 v 1302 2589 a(A)1354 2560
y Fv(?)1411 2589 y Fc(O)15 b FH(B)p 1336 2145 184 3 v
1335 2395 3 251 v 1519 2228 3 84 v 167 2718 a FK(and)29
b(the)e(elimination)e(\()p FM(mo)l(dus)k(p)l(onens)p
FK(\))1229 2851 y FJ(\001)1229 2889 y(\001)1229 2927
y(\001)1130 2984 y FH(A)20 b FJ(\))g FH(B)1511 2851 y
FJ(\001)1511 2889 y(\001)1511 2927 y(\001)1495 2984 y
FH(A)p 1130 3029 417 4 v 1567 3044 a FJ(\))q(E)1310 3105
y FH(B)167 3223 y FK(to)1122 3449 y FJ(\001)1122 3487
y(\001)1122 3525 y(\001)1014 3596 y FH(A)1066 3567 y
Fv(?)1123 3596 y Fc(O)15 b FH(B)1412 3342 y FJ(\001)1412
3380 y(\001)1412 3418 y(\001)1396 3475 y FH(A)113 b(B)1617
3446 y Fv(?)p 1396 3519 263 4 v 1410 3596 a FH(A)15 b
FJ(\012)g FH(B)1603 3567 y Fv(?)p 1014 3641 632 4 v 1759
3600 a FH(B)p 1587 3268 201 3 v 1585 3400 3 134 v 1786
3534 3 268 v 167 3723 a FK(whic)n(h)27 b(sho)n(ws)i(that)f
FM(mo)l(dus)h(p)l(onens)f FK(is)e(written)g(upside)i(do)n(wn!)270
3840 y(So)j(linear)c(logic)i(is)f(not)i(just)g(another)g(exotic)e
(logic:)38 b(it)28 b(giv)n(es)h(new)g(insigh)n(t)g(in)n(to)g(basic)167
3925 y(notions)f(whic)n(h)f(had)i(seemed)e(to)g(b)r(e)h(\014xed)g
(forev)n(er.)p eop
PStoPSsaved restore
%%Page: (164,165) 83
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
161 164 bop 167 877 a FF(Bibliograph)-5 b(y)167 1200
y FK([Abr87])36 b(S.)26 b(Abramsky)-6 b(,)28 b FM(Domain)f(the)l(ory)g
(and)g(the)f(lo)l(gic)g(of)h(observable)e(pr)l(op)l(erties)p
FK(,)j(Ph.D.)409 1285 y(thesis)f(\(Queen)g(Mary)h(College,)e(Univ)n
(ersit)n(y)h(of)h(London,)i(1987\).)167 1409 y([Abr88])36
b(S.)26 b(Abramsky)-6 b(,)28 b(Domain)e(theory)h(in)f(logical)f(form,)i
FM(A)n(nnals)f(of)h(Pur)l(e)f(and)g(Applie)l(d)409 1494
y(L)l(o)l(gic)32 b FG(51)27 b FK(\(1991\))i(1{77.)167
1618 y([AbrVic)n(k])34 b(S.)42 b(Abramsky)h(and)g(S.J.)g(Vic)n(k)n
(ers,)h(Quan)n(tales,)i(Observ)l(ational)c(Logic)g(and)409
1703 y(Pro)r(cess)51 b(Seman)n(tics,)57 b FM(Mathematic)l(al)52
b(Structur)l(es)f(in)g(Computer)h(Scienc)l(e)p FK(,)57
b FG(3)409 1788 y FK(\(1993\))29 b(161{227.)167 1912
y([Barendregt])35 b(H.)50 b(Barendregt,)55 b FM(The)50
b(lamb)l(da-c)l(alculus:)79 b(its)50 b(syntax)g(and)g(semantics)p
FK(,)409 1997 y(North-Holland)27 b(\(1980\).)167 2121
y([Barwise])34 b(J.)25 b(Barwise)f(\(ed.\),)i FM(Handb)l(o)l(ok)g(of)f
(mathematic)l(al)h(lo)l(gic)p FK(,)g(North-Holland)e(\(1977\).)167
2245 y([Berry])34 b(G.)e(Berry)-6 b(,)33 b(Stable)f(Mo)r(dels)g(of)g(T)
n(yp)r(ed)h(lam)n(b)r(da-calculi,)d(in:)44 b FM(Pr)l(o)l(c)l(e)l(e)l
(dings)33 b(of)f(the)409 2330 y(\014fth)27 b(ICALP)h(Confer)l(enc)l(e,)
g FK(Springer-V)-6 b(erlag)27 b(LNCS)g FG(62)g FK(\(Udine,)g(1978\))i
(72{89.)167 2454 y([BTM])42 b(V.)k(Breazu-T)-6 b(annen)48
b(and)f(A.)g(Mey)n(er,)k(P)n(olymorphism)46 b(is)g(conserv)l(ativ)n(e)h
(o)n(v)n(er)409 2539 y(simple)26 b(t)n(yp)r(es,)i(in)f(the)h(pro)r
(ceedings)f(of)i(the)e(second)i(IEEE)e(symp)r(osium)h(on)g
FM(L)l(o)l(gic)409 2624 y(in)f(Computer)i(Scienc)l(e)i
FK(\(Cornell,)26 b(1987\).)167 2748 y([BruLon])36 b(K.)e(Bruce)f(and)j
(G.)f(Longo,)j(A)c(mo)r(dest)g(mo)r(del)f(of)i(records,)h(inheritance)d
(and)409 2833 y(b)r(ounded)e(quan)n(ti\014cation,)f(in)f(the)g(pro)r
(ceedings)g(of)h(the)f(third)f(IEEE)h(symp)r(osium)409
2918 y(on)f FM(L)l(o)l(gic)h(in)e(Computer)i(Scienc)l(e)i
FK(\(Edin)n(burgh,)e(1988\).)167 3042 y([CAML])35 b Fr(CAML)p
FK(,)28 b FM(the)g(r)l(efer)l(enc)l(e)f(manual)p FK(,)h(Pro)t(jet)f(F)
-6 b(ormel,)27 b(INRIA-ENS)f(\(P)n(aris,)i(1987\))167
3166 y([Co)r(quand])37 b(T.)32 b(Co)r(quand,)k FM(Une)c(th)n(\023)-33
b(eorie)33 b(des)f(c)l(onstructions)p FK(,)i(Th)n(\022)-33
b(ese)33 b(de)g(troisi)n(\022)-33 b(eme)31 b(cycle)409
3251 y(\(Univ)n(ersit)n(\023)-33 b(e)26 b(P)n(aris)i(VI)r(I,)f(1985\).)
167 3375 y([CGW86])36 b(Th.)h(Co)r(quand,)43 b(C.A.)37
b(Gun)n(ter)g(and)i(G.)f(Winsk)n(el,)h(dI-domains)f(as)g(a)g(mo)r(del)
409 3460 y(of)h(p)r(olymorphism,)i(in)c(Main,)42 b(Melton,)f(Mislo)n(v)
n(e)c(and)j(Sc)n(hmidt)e(\(eds.\),)j FM(Thir)l(d)409
3545 y(Workshop)c(on)g(the)f(Mathematic)l(al)h(F)-5 b(oundations)37
b(of)f(Pr)l(o)l(gr)l(amming)j(L)l(anguag)d(e)409 3631
y(Semantics)p FK(,)22 b(Springer{V)-6 b(erlag)22 b(Lecture)e(Notes)i
(in)f(Computer)g(Science)f FG(298)h FK(\(1987\))409 3716
y(344{363.)167 3840 y([CGW87])36 b(Th.)c(Co)r(quand,)k(C.A.)c(Gun)n
(ter)g(and)i(G.)e(Winsk)n(el,)h(Domain-theoretic)d(mo)r(dels)409
3925 y(of)e(p)r(olymorphism,)f FM(Information)h(and)g(Computation)34
b FG(81)27 b FK(\(1989\))i(123{167.)1368 4101 y(161)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
162 165 bop 167 167 a FK(162)1852 b Fz(BIBLIOGRAPHY)167
404 y FK([CurryF)-6 b(eys])35 b(H.B.)26 b(Curry)g(and)h(R.)f(F)-6
b(eys,)27 b FM(Combinatory)g(L)l(o)l(gic)f(I)p FK(,)g(North-Holland)f
(\(1958\).)167 531 y([Gallier])32 b(J.)d(Gallier,)c FM(L)l(o)l(gic)j
(for)g(Computer)h(Scienc)l(e)p FK(,)d(Harp)r(er)i(and)g(Ro)n(w)g
(\(1986\).)167 658 y([Gandy])36 b(R.O.)28 b(Gandy)-6
b(,)29 b FM(Pr)l(o)l(of)g(of)f(str)l(ong)g(normalisation)p
FK(,)g(in)g([)p FG(?)p FK(].)167 785 y([Gir71])35 b(J.Y.)j(Girard,)h
(Une)e(extension)g(de)h(l'in)n(terpr)n(\023)-33 b(etation)36
b(de)h(G\177)-34 b(odel)37 b(\022)-34 b(a)38 b(l'analyse,)i(et)409
870 y(son)35 b(application)f(\022)-34 b(a)35 b(l')n(\023)-33
b(elimination)32 b(des)i(coupures)h(dans)h(l'analyse)e(et)g(la)g(th)n
(\023)-33 b(eorie)409 956 y(des)34 b(t)n(yp)r(es,)h(in:)47
b(J.E.)34 b(F)-6 b(enstad)35 b(\(ed.\),)g FM(Pr)l(o)l(c)l(e)l(e)l
(dings)g(of)e(the)h(Sc)l(andinavian)f(L)l(o)l(gic)409
1041 y(Symp)l(osium)p FK(,)d(North-Holland)d(\(1971\))i(63{92.)167
1168 y([Gir72])35 b(J.Y.)29 b(Girard,)f FM(Interpr)n(\023)-33
b(etation)28 b(fonctionnel)t(le)e(et)g(\023)-33 b(elimination)28
b(des)g(c)l(oupur)l(es)i(dans)409 1253 y(l'arithm)n(\023)-33
b(etique)22 b(d'or)l(dr)l(e)i(sup)n(\023)-33 b(erieur)p
FK(,)23 b(Th)n(\022)-33 b(ese)23 b(de)f(do)r(ctorat)h(d')n(\023)-33
b(etat)23 b(\(Univ)n(ersit)n(\023)-33 b(e)21 b(P)n(aris)409
1338 y(VI)r(I,)27 b(1972\).)167 1465 y([Gir85])35 b(J.Y.)d(Girard,)g
(Normal)e(F)-6 b(unctors,)33 b(p)r(o)n(w)n(er)e(series)f(and)j(lam)n(b)
r(da-calculus,)e FM(A)n(nnals)409 1550 y(of)d(Pur)l(e)f(and)h(Applie)l
(d)g(L)l(o)l(gic)33 b FG(37)27 b FK(\(1988\))i(129{177.)167
1677 y([Gir86])35 b(J.Y.)23 b(Girard,)g(The)f(system)g
FG(F)g FK(of)h(v)l(ariable)f(t)n(yp)r(es,)i(\014fteen)e(y)n(ears)h
(later,)f FM(The)l(or)l(etic)l(al)409 1762 y(Computer)28
b(Scienc)l(e)k FG(45)27 b FK(\(1986\))i(159{192.)167
1889 y([Gir87])35 b(J.Y.)28 b(Girard,)f(Linear)g(logic,)g
FM(The)l(or)l(etic)l(al)i(Computer)f(Scienc)l(e)k FG(50)27
b FK(\(1987\))i(1{102.)167 2016 y([Gir])103 b(J.Y.)22
b(Girard,)h FM(Pr)l(o)l(of)h(the)l(ory)f(and)f(lo)l(gic)l(al)g(c)l
(omplexity)p FK(,)i(Bibliop)r(olis)19 b(\(Nap)r(oli,)j(1987\).)167
2143 y([Gir87a])35 b(J.Y.)28 b(Girard,)g FM(T)-5 b(owar)l(ds)29
b(a)e(ge)l(ometry)i(of)e(inter)l(action)p FK(,)h(in)f([)p
FG(?)q FK(],)h(69{108.)167 2270 y([Gir88])35 b(J.Y.)i(Girard,)h
(Geometry)d(of)i(in)n(teraction)e(I:)h(in)n(terpretation)f(of)i(System)
f FG(F)p FK(,)g(in:)409 2355 y FM(Pr)l(o)l(c)l(e)l(e)l(dings)29
b(of)f(the)f(ASL)h(me)l(eting)33 b FK(\(P)n(ado)n(v)l(a,)d(1988\),)g
(221{260.)167 2482 y([GirLaf)5 b(])34 b(J.Y.)47 b(Girard)f(and)h(Y.)e
(Lafon)n(t,)52 b(Linear)46 b(logic)f(and)i(lazy)e(computation,)51
b(in:)409 2567 y FM(T)-5 b(APSOFT)27 b('87)p FK(,)i(v)n(ol.)e
FG(2)p FK(,)h(Springer-V)-6 b(erlag)27 b(LNCS)g FG(250)g
FK(\(Pisa,)g(1987\).)167 2694 y([GLR])56 b(J.-Y.)26 b(Girard,)g(Y.)f
(Lafon)n(t,)i(L.)f(Regnier)f(\(eds.\),)h FM(A)l(dvanc)l(es)g(in)f(Line)
l(ar)h(L)l(o)l(gic)p FK(,)h(Lon-)409 2779 y(don)h(Mathematical)e(So)r
(ciet)n(y)h(Lecture)f(Note)h(Series)f(222,)j(Cam)n(bridge)d(Univ)n
(ersit)n(y)409 2864 y(Press)h(\(1995\))167 2991 y([GOS])58
b(J.-Y.)37 b(Girard,)h(M.)f(Ok)l(ada,)k(A.)36 b(Scedro)n(v)h(\(eds.\),)
i(Linear)d(Logic,)j(to)e(app)r(ear)h(in)409 3076 y FM(The)l(or)l(etic)l
(al)29 b(Computer)f(Scienc)l(e)k FK(\(2003\).)167 3203
y([GrSc])54 b(J.W.)40 b(Gra)n(y)g(and)h(A.)e(Scedro)n(v)h(\(eds.\),)i
FM(Cate)l(gories)e(in)f(c)l(omputer)h(scienc)l(e)f(and)409
3288 y(lo)l(gic)p FK(,)27 b(American)f(Mathematical)h(So)r(ciet)n(y)g
(\(Boulder,)g(1987\).)167 3415 y([HinSel])33 b(J.R.)k(Hindley)f(and)i
(J.P)-6 b(.)38 b(Seldin,)g FM(T)-5 b(o)37 b(H.B.)g(Curry:)54
b(Essays)38 b(on)f(c)l(ombinatory)409 3501 y(lo)l(gic,)27
b(L)l(amb)l(da)j(Calculus)d(and)h(F)-5 b(ormalism)p FK(,)28
b(Academic)e(Press)h(\(1980\).)167 3627 y([Ho)n(w)n(ard])36
b(W.A.)27 b(Ho)n(w)n(ard,)i FM(The)e(formulae-as-typ)l(es)i(notion)e
(of)h(c)l(onstruction)p FK(,)g(in)f([)p FG(?)q FK(].)167
3754 y([Hyland])35 b(J.M.E.)55 b(Hyland,)60 b(The)53
b(e\013ectiv)n(e)f(top)r(os,)61 b(in)53 b FM(L.E.J.)g(Br)l(ouwer)h(c)l
(entenary)409 3840 y(symp)l(osium)p FK(,)e(A.S.)46 b(T)-6
b(ro)r(elstra)44 b(and)i(D.S.)g(v)l(an)g(Dalen)f(\(eds.\),)k
(North-Holland)409 3925 y(\(1982\))29 b(165{216.)p eop
PStoPSsaved restore
%%Page: (166,167) 84
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
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PStoPSxform concat
163 166 bop 167 167 a Fz(BIBLIOGRAPHY)1849 b FK(163)167
404 y([HylPit])33 b(J.M.E.)f(Hyland)g(and)g(A.M.)g(Pitts,)f
FM(The)g(the)l(ory)i(of)e(c)l(onstructions:)43 b(c)l(ate)l(goric)l(al)
409 490 y(semantics)28 b(and)f(top)l(os-the)l(or)l(etic)i(mo)l(dels)p
FK(,)g(in)e([)p FG(?)p FK(],)h(137{199.)167 609 y([Jung])59
b(A.)24 b(Jung,)j FM(Cartesian)e(close)l(d)f(c)l(ate)l(gories)h(of)g
(domains)p FK(,)h(Ph.)f(D.)f(thesis)g(\(T)-6 b(ec)n(hnisc)n(he)409
694 y(Ho)r(c)n(hsc)n(h)n(ule)27 b(Darmstadt,)h(1988\).)167
813 y([Ko)n(w)n(alski])34 b(R.)24 b(Ko)n(w)n(alski,)h
FM(L)l(o)l(gic)g(for)g(pr)l(oblem)f(solving)30 b FK([)p
Fr(PROLOG)p FK(],)c(North-Holland)e(\(1979\).)167 932
y([Ko)n(ymans])36 b(C.P)-6 b(.J.)33 b(Ko)n(ymans,)j FM(Mo)l(dels)d(of)g
(the)f FH(\025)p FM(-c)l(alculus)p FK(,)i(Cen)n(truum)e(v)n(o)r(or)i
(Wiskunde)409 1017 y(en)27 b(Informatica,)h FG(9)f FK(\(1984\).)167
1136 y([KrLev])34 b(G.)23 b(Kreisel)d(and)j(A.)f(L)n(\023)-33
b(evy)-6 b(,)26 b(Re\015ection)21 b(principles)f(and)j(their)f(use)g
(for)h(establishing)409 1221 y(the)35 b(complexit)n(y)e(of)i(axiomatic)
f(systems,)j FM(Z.)e(Math.)h(L)l(o)l(gik)f(Grund)t(lagen)f(Math.)409
1306 y FG(33)27 b FK(\(1968\).)167 1425 y([KriP)n(ar])34
b(J.L.)26 b(Krivine)d(and)j(M.)f(P)n(arigot,)h(Programming)f(with)f
(pro)r(ofs,)j FM(Sixth)e(symp)l(osium)409 1510 y(on)i(somputation)i
(the)l(ory)35 b FK(\(W)-6 b(endisc)n(h-Rietz,)25 b(1987\).)167
1629 y([Laf87])38 b(Y.)24 b(Lafon)n(t,)j FM(L)l(o)l(giques,)f(c)l(at)n
(\023)-33 b(egories)25 b(et)f(machines)p FK(,)i(Th)n(\022)-33
b(ese)25 b(de)g(do)r(ctorat)g(\(Univ)n(ersit)n(\023)-33
b(e)409 1714 y(P)n(aris)27 b(VI)r(I,)g(1988\).)167 1833
y([Laf88])38 b(Y.)27 b(Lafon)n(t,)i(The)e(linear)e(abstract)j(mac)n
(hine,)e FM(The)l(or)l(etic)l(al)j(Computer)f(Scienc)l(e)j
FG(59)409 1918 y FK(\(1988\))e(157{180.)167 2037 y([LamSco])35
b(J.)f(Lam)n(b)r(ek)g(and)h(P)-6 b(.J.)34 b(Scott,)h
FM(A)n(n)e(intr)l(o)l(duction)i(to)e(higher)g(or)l(der)i(c)l(ate)l
(goric)l(al)409 2122 y(lo)l(gic)p FK(,)27 b(Cam)n(bridge)h(Univ)n
(ersit)n(y)e(Press)h(\(1986\).)167 2241 y([Lei83])42
b(D.)31 b(Leiv)l(an)n(t,)h(Reasoning)g(ab)r(out)h(functional)e
(programs)h(and)h(complexit)n(y)d(classes)409 2326 y(asso)r(ciated)23
b(with)e(t)n(yp)r(e)i(disciplines,)d FM(Twenty)j(fourth)f(annual)g
(symp)l(osium)j(on)d(foun-)409 2411 y(dations)30 b(of)g(c)l(omputer)g
(scienc)l(e,)f FK(IEEE)g(Computer)g(So)r(ciet)n(y)g(Press,)h(\(W)-6
b(ashington)409 2496 y(DC,)27 b(1983\).)167 2615 y([Lei90])42
b(D.)23 b(Leiv)l(an)n(t,)h(Con)n(tracting)f(pro)r(ofs)h(to)f(programs,)
i(in:)33 b(P)n(ergiorgio)22 b(Odifreddi)f(\(ed.\),)409
2700 y FM(L)l(o)l(gic)28 b(in)g(Computer)g(Scienc)l(e)p
FK(,)f(Academic)e(Press)j(\(1990\).)167 2819 y([ML70])36
b(P)-6 b(.)32 b(Martin-L\177)-34 b(of,)33 b FM(A)f(c)l(onstruction)g
(of)g(the)f(pr)l(ovable)h(wel)t(l-or)l(dering)e(of)i(the)f(the)l(ory)
409 2904 y(of)d(sp)l(e)l(cies)34 b FK(\(unpublished\).)167
3023 y([ML84])i(P)-6 b(.)28 b(Martin-L\177)-34 b(of,)28
b FM(Intuitionistic)f(typ)l(e)h(the)l(ories)p FK(,)h(Bibliop)r(olis)c
(\(Nap)r(oli,)h(1984\))167 3142 y([Pra)n(witz])34 b(D.)j(Pra)n(witz,)j
(Ideas)e(and)h(results)e(in)h(pro)r(of-theory)-6 b(,)42
b(in:)56 b FM(Pr)l(o)l(c)l(e)l(e)l(dings)39 b(of)f(the)409
3227 y(se)l(c)l(ond)28 b(Sc)l(andinavian)g(lo)l(gic)f(symp)l(osium)36
b FK(,)27 b(North-Holland)h(\(1971\))h(237{309.)167 3346
y([Reynolds])35 b(J.C.)30 b(Reynolds,)g(T)-6 b(o)n(w)n(ards)30
b(theory)g(of)g(t)n(yp)r(e)f(structure,)g FM(Paris)h(c)l(ol)t(lo)l
(quium)f(on)409 3431 y(pr)l(o)l(gr)l(amming)p FK(,)h(Springer-V)-6
b(erlag)27 b(LNCS)g FG(19)g FK(\(1974\).)167 3550 y([ReyPlo])34
b(J.C.)24 b(Reynolds)f(and)h(G.)f(Plotkin,)g FM(On)f(functors)h(expr)l
(essible)f(in)h(the)f(p)l(olymorphic)409 3636 y(lamb)l(da)28
b(c)l(alculus)p FK(.)167 3754 y([ERobinson])36 b(E.)42
b(Robinson,)47 b(Logical)42 b(asp)r(ects)h(of)g(denotational)g(seman)n
(tics)e(in:)64 b(D.H.)409 3840 y(Pitt,)25 b(A.)g(P)n(oign)n(\023)-33
b(e)27 b(and)f(D.E.)g(Rydeheard)g(\(eds.\),)g FM(Cate)l(gory)g(the)l
(ory)h(and)f(c)l(omputer)409 3925 y(scienc)l(e)32 b FK(LNCS)c
FG(283)p FK(,)e(Springer-V)-6 b(erlag)27 b(\(Edin)n(burgh,)i(1987\).)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
164 167 bop 167 167 a FK(164)1852 b Fz(BIBLIOGRAPHY)167
404 y FK([JARobinson])36 b(J.A.)g(Robinson,)k(A)35 b(mac)n(hine)g
(orien)n(ted)g(logic)g(based)i(on)g(the)f(resolution)409
490 y(principle,)30 b FM(Journal)h(of)h(the)g(Asso)l(ciation)g(of)g
(Computing)g(Machinery)38 b FG(12)31 b FK(\(1965\))409
575 y(23{41.)167 695 y([Scott69])36 b(D.)42 b(Scott,)47
b(Outline)41 b(of)i(a)g(mathematical)e(theory)i(of)g(computation,)k(in)
42 b FM(4th)409 780 y(A)n(nnual)52 b(Princ)l(eton)h(Confer)l(enc)l(e)g
(on)f(Information)h(Scienc)l(es)f(and)h(Systems)p FK(,)409
865 y(Princeton)27 b(Univ)n(ersit)n(y)f(\(1970\))j(169{176.)167
986 y([Scott76])36 b(D.)29 b(Scott,)i(Data)f(t)n(yp)r(es)f(as)i
(lattices,)d FM(SIAM)h(Journal)g(of)h(Computing)36 b
FG(5)30 b FK(\(1976\))409 1071 y(522{587.)167 1191 y([Scott82])36
b(D.)f(Scott,)j(Domains)d(for)h(denotational)g(seman)n(tics,)g(in:)51
b FM(ICALP)35 b('82)p FK(,)j(LNCS)409 1276 y FG(140)p
FK(,)26 b(Springer-V)-6 b(erlag)27 b(\(Aarh)n(us,)i(1982\).)167
1396 y([ScoGun])36 b(D.S.)28 b(Scott)f(and)h(C.A.)g(Gun)n(ter,)f(Seman)
n(tic)f(domains,)i FM(Handb)l(o)l(ok)h(of)e(Computer)409
1482 y(Scienc)l(e)p FK(,)f(North-Holland)i(\(1988\).)167
1602 y([Seely])48 b(R.A.G.)33 b(Seely)-6 b(,)33 b(Linear)f(logic,)h
(*-autonomous)j(categories)c(and)h(cofree)f(algebras,)409
1687 y(in)27 b([)p FG(?)q FK(].)167 1807 y([Sm)n(yth])35
b(M.)28 b(Sm)n(yth,)g(P)n(o)n(w)n(erdomains)f(and)h(predicate)e
(transformers:)36 b(a)27 b(top)r(ological)g(view)409
1892 y(in:)34 b(J.)26 b(Diaz)e(\(ed.\),)i FM(A)n(utomata,)h(L)l
(anguages)e(and)h(Pr)l(o)l(gr)l(amming)p FK(,)i(Springer-V)-6
b(erlag)409 1978 y(LNCS)27 b FG(154)g FK(\(1983\))i(662{675.)167
2098 y([T)-6 b(ait])79 b(W.W.)47 b(T)-6 b(ait,)51 b(In)n(tensional)46
b(in)n(terpretation)g(of)h(functionals)g(of)g(\014nite)f(t)n(yp)r(e)h
(I,)409 2183 y FM(Journal)27 b(of)h(Symb)l(olic)g(L)l(o)l(gic)33
b FG(32)27 b FK(\(1967\))i(198{212.)167 2303 y([T)-6
b(a)n(y86])37 b(P)-6 b(.)29 b(T)-6 b(a)n(ylor,)29 b FM(R)l(e)l(cursive)
g(domains,)h(indexe)l(d)e(c)l(ate)l(gory)h(the)l(ory)h(and)e(p)l
(olymorphism)p FK(,)409 2388 y(Ph.D.)g(thesis)f(\(Univ)n(ersit)n(y)f
(of)i(Cam)n(bridge,)f(1986\).)167 2509 y([T)-6 b(a)n(y88])37
b(P)-6 b(.)28 b(T)-6 b(a)n(ylor,)28 b(An)f(algebraic)g(approac)n(h)j
(to)e(stable)f(domains,)h FM(Journal)g(of)f(Pur)l(e)h(and)409
2594 y(Applie)l(d)g(A)n(lgebr)l(a)33 b FG(64)27 b FK(\(1990\))i
(171{203.)167 2714 y([T)-6 b(a)n(y89])37 b(P)-6 b(.)30
b(T)-6 b(a)n(ylor,)30 b FM(The)g(tr)l(ac)l(e)g(factorisation)g(of)g
(stable)e(functors)p FK(,)i(1989,)i(a)n(v)l(ailable)d(from)409
2799 y Fr(www.di.unito.it/)p FJ(\030)p Fr(pt/stable)167
2920 y FK([T)-6 b(a)n(y89a])37 b(P)-6 b(.)49 b(T)-6 b(a)n(ylor,)55
b(Quan)n(titativ)n(e)47 b(Domains,)54 b(Group)r(oids)c(and)g(Linear)e
(Logic,)53 b(in:)409 3005 y(D.)41 b(Pitt)f(\(ed.\),)45
b FM(Cate)l(gory)e(The)l(ory)f(and)g(Computer)g(Scienc)l(e)k
FK(Springer-V)-6 b(erlag)409 3090 y(\(Manc)n(hester,)28
b(1989\))h(155{181.)167 3210 y([T)-6 b(a)n(y99])37 b(P)-6
b(.)36 b(T)-6 b(a)n(ylor,)38 b FM(Pr)l(actic)l(al)d(F)-5
b(oundations)36 b(of)g(Mathematics)p FK(,)h(Cam)n(bridge)e(Univ)n
(ersit)n(y)409 3295 y(Press)27 b(\(1999\).)167 3416 y([v)l(anHeijeno)r
(ort])35 b(J.)54 b(v)l(an)g(Heijeno)r(ort,)59 b FM(F)-5
b(r)l(om)53 b(F)-5 b(r)l(e)l(ge)53 b(to)g(G\177)-35 b(odel,)60
b(a)53 b(sour)l(c)l(e)h(b)l(o)l(ok)f(in)409 3501 y(mathematic)l(al)28
b(lo)l(gic,)g(1879{1931)p FK(,)i(Harv)l(ard)e(Univ)n(ersit)n(y)f(Press)
g(\(1967\).)167 3621 y([Vic)n(k)n(ers])34 b(S.)28 b(Vic)n(k)n(ers,)e
FM(T)-5 b(op)l(olo)l(gy)29 b(via)f(lo)l(gic)p FK(,)g(Cam)n(bridge)f
(Univ)n(ersit)n(y)f(Press)h(\(1989\).)167 3741 y([Winsk)n(el])34
b(G.)46 b(Winsk)n(el,)51 b(Ev)n(en)n(t)46 b(structures,)51
b(in:)73 b FM(A)l(dvanc)l(e)l(d)47 b(c)l(ourse)g(on)g(Petri)e(nets)p
FK(,)409 3826 y(Springer-V)-6 b(erlag)27 b(LNCS)g FG(255)g
FK(\(1987\).)p eop
PStoPSsaved restore
%%Page: (168,169) 85
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
165 168 bop 167 729 a FF(Index)167 1034 y FG(Notation)284
1122 y FK(v)l(ariables,)23 b FH(x)p FK(,)h FH(y)s FK(,)e
FH(z)343 1209 y FK(ob)t(ject)h(language,)h FH(\030)s
FK(,)g FH(\021)343 1297 y FK(second)g(order,)f FH(X)6
b FK(,)23 b FH(Y)15 b FK(,)23 b FH(Z)284 1385 y FK(terms,)g
FH(t)p FK(,)g FH(u)p FK(,)g FH(v)s FK(,)g FH(w)343 1472
y FK(ob)t(ject)g(language,)h FH(a)p FK(,)g FH(b)284 1560
y FK(t)n(yp)r(es,)g FH(S)t FK(,)g FH(T)10 b FK(,)23 b
FH(U)7 b FK(,)23 b FH(V)16 b FK(,)23 b FH(W)284 1647
y FK(prop)r(ositions,)h FH(A)p FK(,)f FH(B)t FK(,)g FH(C)5
b FK(,)24 b FH(D)284 1735 y FK(coherence)f(spaces,)g
FJ(A)p FK(,)h FJ(B)r FK(,)f FJ(C)343 1822 y FK(p)r(oin)n(ts,)h
FH(a)p FK(,)f FH(b)p FK(,)g FH(c)343 1910 y FK(tok)n(ens,)h
FH(\013)p FK(,)g FH(\014)t FK(,)f FH(\015)284 1998 y
FK(n)n(um)n(b)r(ers,)h FH(m)p FK(,)f FH(n)p FK(,)h FH(p)p
FK(,)e FH(q)167 2085 y FK(Brac)n(k)n(ets)284 2173 y(denotation,)i([)-12
b([)q FH(T)10 b FK(])-12 b(])q(,)23 b([)-12 b([)p FH(t)p
FK(])g(])284 2260 y(pairing,)23 b(\()p FH(a;)12 b(b)p
FK(\))23 b(and)h FJ(h)p FH(a;)12 b(b)p FJ(i)284 2348
y FK(set,)23 b FJ(f)p FH(n)d FK(:)39 b FH(P)10 b FK([)p
FH(n)p FK(])p FJ(g)284 2436 y FK(substitution,)23 b FH(t)p
FK([)p FH(u=x)p FK(])284 2523 y(w)n(eb,)g FJ(jAj)167
2611 y FK(Connectiv)n(es)g(on)g(t)n(yp)r(es)284 2698
y(conjunction,)h FJ(^)284 2786 y FK(direct)e(pro)r(duct,)i(&)284
2874 y(direct)e(sum,)h FJ(\010)284 2961 y FK(disjunction,)g
FJ(_)284 3049 y FK(function-space,)h FJ(!)284 3136 y
FK(implication,)d FJ(\))284 3224 y FK(linear)h(implication,)e
Fu(\()284 3312 y FK(pro)r(duct,)k FJ(\002)284 3399 y
FK(sum,)g(+,)545 3346 y Fo(`)284 3487 y FK(tensor)g(pro)r(duct,)f
FJ(\012)284 3574 y FK(tensor)h(sum)f(or)g(par,)h Fc(O)167
3662 y FK(Quan)n(ti\014ers)284 3750 y(existen)n(tial,)d
FJ(9)284 3837 y FK(existen)n(tial)g(t)n(yp)r(e,)j(\006,)f(\006)884
3831 y FJ(\016)935 3837 y FK(,)f FJ(r)284 3925 y FK(univ)n(ersal,)h
FJ(8)1640 1034 y FK(univ)n(ersal)f(t)n(yp)r(e,)i(\005)1522
1119 y(Relations)1640 1204 y(coherence,)1993 1182 y FH(_)1993
1225 y(^)1640 1310 y FK(de\014nitional)e(equalit)n(y)-6
b(,)2276 1271 y FD(def)2283 1310 y FK(=)1640 1395 y(em)n(b)r(edding)22
b(and)i(pro)t(jection,)f Fu(\032)p FK(,)h FJ(\000)-15
b FH(.)1640 1481 y FK(if)22 b(and)i(only)f(if)g(\(i\013)5
b(\),)41 b FJ(\()-12 b(\))1640 1566 y FK(incoherence,)2050
1544 y FH(^)2050 1586 y(_)1640 1651 y FK(in)n(tercon)n(v)n(ertible)20
b(with,)i FJ(\030)1640 1736 y FK(isomorphism,)g FJ(')1640
1821 y FK(reduces)g(\(con)n(v)n(erts\))h(to,)h Fu( )1640
1906 y FK(result)e(of)h(function,)g FJ(7!)1640 1992 y
FK(sequen)n(t,)g FJ(`)1522 2077 y FK(Miscellaneous)1640
2162 y(comp)r(osition,)f FJ(\016)1640 2247 y FK(directed)f(union)j(and)
g(join,)2385 2194 y Fo(S)2444 2213 y Fv(")2471 2247 y
FK(,)2514 2194 y Fo(W)2572 2213 y Fv(")1640 2332 y FK(negation,)g
FJ(:)e FK(\(linear,)2246 2307 y Fv(?)2288 2332 y FK(\))1640
2417 y(of)h(course)g(and)h(wh)n(y)g(not,)g(!,)f(?)1640
2503 y(sequence,)1945 2504 y FH(A)p 1948 2523 39 3 v
1522 2647 a FG(A)p FK(bramsky)-6 b(,)25 b(i,)d(55)1522
2732 y(abstraction)i(\()p FH(\025)p FK(\))1640 2818 y(con)n(v)n
(ersion,)f(13)1640 2903 y(in)n(tro)r(duction,)f(12,)j(20)1640
2988 y(realisabilit)n(y)-6 b(,)20 b(127)1640 3073 y(reducibilit)n(y)-6
b(,)20 b(45)1640 3158 y(seman)n(tics,)i(68,)i(144)1640
3243 y(syn)n(tax,)h(15,)f(82)1522 3329 y(absurdit)n(y)g(\()p
FJ(?)p FK(\),)e(6,)h(95)1640 3414 y(comm)n(uting)f(con)n(v)n(ersion,)i
(78)1640 3499 y(denotation)g(\()p FG(f)7 b FK(\),)23
b FM(se)l(e)28 b FK(b)r(o)r(oleans)c FM(and)1757 3584
y FK(unde\014ned)g(ob)t(ject)f(\()p Fn(?)p FK(\))1640
3669 y(empt)n(y)f(t)n(yp)r(e)i(\()p Fq(Emp)e FK(and)j
FH(")2353 3680 y Fw(U)2394 3669 y FK(\),)e(80)1640 3754
y(linear)e(logic)h(\()p FJ(?)g FK(and)i FG(0)p FK(\),)e(154)1640
3840 y(natural)h(deduction)g(\()p FJ(?E)6 b FK(\),)22
b(73)1640 3925 y(realisabilit)n(y)-6 b(,)20 b(129)1368
4101 y(165)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
166 169 bop 167 167 a FK(166)2175 b Fz(INDEX)284 404
y FK(sequen)n(t)24 b(calculus)e(\()p Fn(?)d FJ(`)h Fn(?)p
FK(\),)i(29)167 491 y(Ac)n(k)n(ermann's)h(function,)h(51)167
577 y(algebraic)e(domain,)i(56)167 663 y(aliv)n(e)e(h)n(yp)r(othesis,)i
(9)167 749 y(all)e(\()p FJ(8)p FK(\),)g FM(se)l(e)28
b FK(univ)n(ersal)23 b(quan)n(ti\014er)167 835 y(alternations)g(of)g
(quan)n(ti\014ers,)h(58,)g(124)167 922 y(alternativ)n(e)e(conjunction)h
(\(&\),)g FM(se)l(e)402 1007 y FK(direct)e(pro)r(duct)167
1093 y(amalgamated)i(sum,)g(96,)i(134)167 1179 y(analysis)e(\(second)h
(order)f(arithmetic\),)402 1264 y(114)167 1350 y(and,)h
FM(se)l(e)29 b FK(conjunction)167 1437 y(application)284
1523 y(con)n(v)n(ersion,)24 b(13)284 1609 y(elimination,)d(12,)j(20)284
1695 y(realisabilit)n(y)-6 b(,)21 b(127)284 1781 y(reducibilit)n(y)-6
b(,)21 b(43)284 1868 y(seman)n(tics)h(\()p FJ(A)p FM(pp)5
b FK(\),)23 b(69)284 1954 y(stabilit)n(y=Berry)e(order,)i(65)284
2040 y(syn)n(tax,)i(15,)f(82)284 2126 y(trace)f(form)n(ula)g(\()p
FG(App)p FK(\),)f(63,)i(64,)h(144)167 2212 y(appro)n(ximation)f(of)f(p)
r(oin)n(ts)h(and)g(domains,)402 2297 y(57,)g(134)167
2384 y(arro)n(w)f(t)n(yp)r(e,)h FM(se)l(e)k FK(implication)20
b FM(and)402 2469 y FK(function-space)167 2555 y(asso)r(ciativit)n(y)i
(of)i(sum)f(t)n(yp)r(e,)g(98)167 2641 y(asymmetrical)e(in)n
(terpretation,)h(34)167 2727 y(atomic)g(form)n(ulae,)h(4,)g(5,)h(30,)g
(112,)h(160)167 2814 y(atomic)d(p)r(oin)n(ts,)i FM(se)l(e)k
FK(tok)n(ens)167 2900 y(atomic)22 b(sequen)n(ts,)i(112)167
2986 y(atomic)e(t)n(yp)r(es,)i(15,)g(48)167 3072 y(automated)g
(deduction,)f(28,)h(34)167 3158 y(automorphisms,)g(134)167
3245 y(axiom)284 3331 y(comprehension,)f(114,)i(118,)g(123)284
3417 y(excluded)e(middle,)e(6,)j(156)284 3503 y(h)n(yp)r(othesis,)g(10)
284 3589 y(iden)n(tit)n(y)-6 b(,)23 b(30)284 3676 y(link,)g(156)284
3762 y(prop)r(er,)h(112)167 3925 y FG(B)p FK(ad)g(elimination,)c(77)
1522 404 y(Barendregt,)k(22)1522 491 y(Berry)-6 b(,)23
b(54)1522 577 y(Berry)g(order)g(\()p FJ(\024)1977 588
y FD(B)2015 577 y FK(\),)g(65,)h(66,)g(135,)h(146)1522
663 y(b)r(eta)e(\()p FH(\014)t FK(\))g(rule,)f FM(se)l(e)28
b FK(con)n(v)n(ersion)1522 749 y(binary)c(completeness,)d(56)1522
835 y(binary)j(trees)e(\()p Fq(Bintree)o FK(\),)h(93)1522
921 y(binding)g(v)l(ariables,)g(5,)h(12,)g(15,)g(83,)g(161)1522
1008 y(Bo)r(ole,)f(3)1522 1094 y(b)r(o)r(oleans,)h(4)1640
1180 y(coherence)e(space)1698 1266 y FJ(B)-10 b FM(o)l(ol)9
b FK(,)23 b(56,)h(60,)g(70)1698 1352 y([)-12 b([)q(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)g FJ(!)p FH(X)g FK(])-12
b(],)23 b(140)1640 1439 y(comm)n(uting)f(con)n(v)n(ersion,)i(86)1640
1525 y(con)n(v)n(ersion,)f(48)1640 1611 y(denotation)h(\()p
FG(t)e FK(and)i FG(f)7 b FK(\),)24 b(4)1640 1697 y(in)e
FG(F)h FK(\(\005)p FH(X)r(:)12 b(X)6 b FJ(!)p FH(X)g
FJ(!)p FH(X)g FK(\),)22 b(84,)j(140)1640 1783 y(in)d
FG(T)h FK(\()p Fq(Bo)r(ol)p FK(,)g Fq(T)p FK(,)g Fq(F)p
FK(\),)g(48,)i(50,)f(70)1640 1869 y(in)e(system)h FG(F)p
FK(,)g(84)1640 1956 y(totalit)n(y)-6 b(,)22 b(149)1522
2042 y(b)r(ounded)j(meet,)d FM(se)l(e)28 b FK(pullbac)n(k)1522
2128 y(b)r(oundedly)c(complete)e(domains,)h(140)1522
2214 y(Brou)n(w)n(er,)g(6)1522 2300 y(b)n(y)h(v)l(alues,)g(51,)g(70,)g
(91,)g(133)1522 2463 y Fq(C)p FK(,)f Fq(C)p FK(?,)g FM(se)l(e)29
b FK(con)n(traction)1522 2549 y(camem)n(b)r(ert,)22 b(3)1522
2635 y Fr(CAML)p FK(,)i(81)1522 2721 y(candidate)1640
2808 y(reducibilit)n(y)-6 b(,)20 b(43,)k(115,)h(116)1640
2894 y(totalit)n(y)-6 b(,)22 b(58,)i(149)1522 2980 y(Can)n(tor,)g(1)
1522 3066 y(Cartesian)f(closed)f(category)-6 b(,)25 b(54,)f(62,)g(67,)
1757 3151 y(69,)g(95,)g(152)1522 3237 y(Cartesian)f(natural)g
(transformation,)h FM(se)l(e)1757 3323 y FK(Berry)f(order)1522
3409 y(Cartesian)g(pro)r(duct,)h FM(se)l(e)k FK(pro)r(duct)1522
3495 y(casewise)22 b(de\014nition)h(\()p Fq(D)p FK(\),)f(48,)i(83,)g
(97)1522 3581 y(category)-6 b(,)24 b(59,)h(95,)f(133,)g(135)1522
3667 y(c)n(haracteristic)d(subgroup,)k(134)1522 3753
y(Ch)n(urc)n(h-Rosser)f(prop)r(ert)n(y)-6 b(,)24 b(16,)g(22,)g(49,)1757
3839 y(74,)g(79,)g(90,)g(114,)h(152,)g(159)1522 3925
y(clique,)d(57,)i(62,)g(101,)h(138)p eop
PStoPSsaved restore
%%Page: (170,171) 86
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
167 170 bop 167 167 a Fz(INDEX)2173 b FK(167)167 404
y(closed)22 b(normal)h(form,)g(19,)h(52,)g(121)167 490
y(co)r(closure,)e(135)167 576 y(coherence)g(space,)i(56)284
662 y(b)r(o)r(oleans)343 748 y FJ(B)-10 b FM(o)l(ol)9
b FK(,)23 b(56,)h(60,)g(70)343 834 y([)-12 b([)q(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)o FH(X)g FJ(!)q FH(X)g FK(])-12
b(],)23 b(140)284 920 y(coherence)g(relation)e(\()898
898 y FH(_)898 941 y(^)986 920 y FK(\),)i(56)284 1006
y(direct)f(pro)r(duct)h(\(&\),)g(62)284 1092 y(direct)f(sum)h(\()p
FJ(\010)p FK(\),)f(96,)i(103)284 1178 y(empt)n(y)f(t)n(yp)r(e)g(\()p
FJ(E)-17 b FM(mp)5 b FK(\),)23 b(104,)i(139)284 1264
y(function)f(space)f(\()p FJ(!)p FK(\),)g(64,)h(102,)h(138)284
1350 y(in)n(tegers)343 1435 y(\015at)f(\()p FJ(I)-7 b
FM(nt)7 b FK(\),)23 b(56,)h(60,)g(66,)g(70)343 1521 y(lazy)f(\()p
FJ(I)-7 b FM(nt)614 1496 y FD(+)655 1521 y FK(\),)23
b(71,)i(98)343 1607 y([)-12 b([)q(\005)p FH(X)r(:)12
b(X)6 b FJ(!)o FK(\()p FH(X)g FJ(!)p FH(X)g FK(\))p FJ(!)p
FH(X)g FK(\)])-12 b(])q(,)23 b(147)284 1693 y(linear)f(implication)e
(\()p Fu(\()p FK(\),)j(100,)i(104,)402 1778 y(138)284
1864 y(linear)d(negation)i(\()p FJ(A)839 1839 y Fv(?)880
1864 y FK(\),)f(100,)i(138)284 1950 y(of)f(course)f(\(!\),)g(101,)i
(138,)f(145)284 2036 y FJ(P)-18 b FM(air)9 b FK(,)23
b(\005)503 2011 y FD(1)530 2036 y FK(,)h(\005)625 2011
y FD(2)652 2036 y FK(,)f(68)284 2122 y(partial)g(functions)g(\()p
FJ(P)-6 b(F)6 b FK(\),)23 b(66)284 2208 y(\005)g(t)n(yp)r(es,)h(143)284
2294 y(seman)n(tics,)f(67,)h(132)284 2380 y(singleton)f(\()p
FJ(S)-6 b FM(gl)6 b FK(\),)23 b(104,)h(139)284 2466 y(tensor)g(pro)r
(duct)f(\()p FJ(\012)p FK(\),)g(104,)h(138)284 2552 y(tensor)g(sum)f
(or)g(par)h(\()p Fc(O)o FK(\),)f(104)284 2638 y(tok)n(ens)h(and)g(w)n
(eb,)f(56)167 2723 y(coheren)n(t)g(or)g(sp)r(ectral)f(space,)i(56)167
2809 y Fq(collect)d FK(\(forming)h(trees\),)h(94)167
2895 y(comm)n(unicating)f(pro)r(cesses,)h(155)167 2981
y(comm)n(utativit)n(y)f(of)h(logic,)f(29)167 3067 y(comm)n(uting)g(con)
n(v)n(ersion)i(of)f(sum,)h FM(se)l(e)402 3152 y FK(con)n(v)n(ersion)167
3238 y(compact,)f(59,)h(66)167 3324 y(compact-op)r(en)g(top)r(ology)-6
b(,)24 b(55)167 3410 y(complemen)n(tary)e(graph,)i FM(se)l(e)29
b FK(linear)402 3495 y(negation)167 3581 y(complete)21
b(subgraph,)26 b FM(se)l(e)i FK(clique)167 3667 y(complexit)n(y)284
3753 y(algorithmic,)22 b(53,)i(111,)g(143)284 3839 y(logical,)e(42,)i
(58,)g(114,)h(122,)g(124,)f(140)167 3925 y(comp)r(onen)n(ts)g(\()p
FH(\031)616 3899 y FD(1)666 3925 y FK(and)g FH(\031)842
3899 y FD(2)870 3925 y FK(\))1640 404 y(elimination,)c(19)1640
490 y(reducibilit)n(y)-6 b(,)20 b(43)1522 576 y(comp)r(osition)1640
662 y(stable)i(functions,)i(69)1522 748 y(comprehension)f(sc)n(heme,)f
(114,)j(118,)f(123,)1757 833 y(126)1522 919 y(computational)f
(signi\014cance,)g(1,)g(11,)i(17,)1757 1004 y(112,)g(120)1522
1090 y(con\015uen)n(t)f(relation,)e FM(se)l(e)29 b FK(Ch)n(urc)n
(h-Rosser)1757 1175 y(prop)r(ert)n(y)1522 1261 y(conjunction,)24
b(5)1640 1347 y(and)g(pro)r(duct,)g(11,)g(15,)g(19)1640
1433 y Fq(conj)f FK(in)g Fq(Bo)r(ol)p FK(,)g(50)1640
1519 y(con)n(v)n(ersion,)g(13)1640 1605 y(cut)f(elimination,)f(105)1640
1691 y(in)h FG(F)p FK(:)35 b(\005)p FH(X)r(:)12 b FK(\()p
FH(U)7 b FJ(!)p FH(V)15 b FJ(!)q FH(X)6 b FK(\))p FJ(!)p
FH(X)g FK(,)23 b(84)1640 1777 y(linear)e(logic,)h(152)1640
1863 y(natural)h(deduction)1698 1949 y FJ(^I)5 b FK(,)24
b FJ(^)p FK(1)p FJ(E)30 b FK(and)24 b FJ(^)p FK(2)p FJ(E)6
b FK(,)24 b(10)1640 2034 y(realisabilit)n(y)-6 b(,)20
b(126)1640 2120 y(sequen)n(t)j(calculus)1698 2206 y FJ(L)p
FK(1)p FJ(^)p FK(,)h FJ(L)p FK(2)p FJ(^)f FK(and)h FJ(R^)o
FK(,)f(31)1522 2292 y Fq(cons)h FK(\(add)g(to)f(list\),)f(91)1522
2378 y(consistency)1640 2464 y(equational,)h(16,)h(23,)g(152)1640
2550 y(logical)d(\()p Fq(consis)p FK(\),)j(42,)g(114,)g(124)1522
2636 y(constan)n(ts)h(b)n(y)e(v)n(o)r(cation,)h(60,)g(66)1522
2722 y(constructors,)g FM(se)l(e)k FK(data)c(t)n(yp)r(es)g(in)e
FG(F)1522 2808 y FK(con)n(tin)n(uit)n(y)-6 b(,)23 b(54,)h(59,)h(137)
1522 2894 y(con)n(traction)1640 2980 y FJ(L)p Fq(C)d
FK(and)i FJ(R)p Fq(C)p FK(,)f(29)1640 3065 y(linear)e(logic)h(\()p
Fq(C)p FK(?\),)h(154)1522 3151 y(con)n(tractum,)g(18,)h
FM(se)l(e)29 b FK(redex)1522 3237 y(con)n(trol)23 b(features)g(of)h
Fr(PROLOG)p FK(,)g(28)1522 3323 y(con)n(v)n(ersion)g(\()p
Fu( )p FK(\),)f(18)1640 3409 y(b)r(ogus)h(example,)f(75)1640
3495 y(b)r(o)r(oleans)h(\()p Fq(D)p FK(\),)e(48)1640
3581 y(comm)n(uting,)g(74,)i(78,)g(85,)g(97,)h(103)1640
3667 y(conjunction)e(\()p FJ(^)p FK(\),)g(13)1640 3753
y(degree,)f(25)1640 3839 y(denotational)h(equalit)n(y)-6
b(,)23 b(69,)h(132)1640 3925 y(disjunction)e(\()p FJ(_)p
FK(\),)h(75,)h(97)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
168 171 bop 167 167 a FK(168)2175 b Fz(INDEX)284 404
y FK(existen)n(tial)21 b(quan)n(ti\014er)j(\()p FJ(9)p
FK(\),)f(75)167 491 y(con)n(v)n(ersion)h(\()p Fu( )p
FK(\))284 577 y(implication)d(\()p FJ(\))p FK(\),)i(13)284
663 y(in)g FG(F)p FK(,)g(83,)h(94)284 749 y(in\014nit)n(y)f(\()p
FJ(1)p FK(\),)g(72)284 835 y(in)n(tegers)g(\()p Fq(R)p
FK(,)g Fq(It)p FK(\),)g(48,)h(51)284 921 y FH(\025)p
FK(-calculus,)f(11,)h(16,)g(18,)g(69)284 1007 y(linear)e(logic)g(\(pro)
r(of)i(nets\),)f(158)284 1094 y(natural)h(deduction,)f(13,)h(20)284
1180 y(reducibilit)n(y)-6 b(,)21 b(43,)j(116)284 1266
y(rewrite)d(rules,)h(14)284 1352 y(second)i(order,)f(94)167
1438 y(Co)r(quand,)i(116,)g(133)167 1524 y(correctness)d(criterion)284
1610 y(for)i(pro)r(of)g(nets,)f(158)284 1696 y(for)h(tok)n(ens)g(of)f
(\005)g(t)n(yp)r(es,)g(139,)i(142)167 1783 y Fq(couple)e
FK(\(forming)f(trees\),)g(93)167 1869 y(\()p FG(CR)h(1)p
FJ(\000)p FG(3)o FK(\),)g FM(se)l(e)28 b FK(reducibilit)n(y)167
1955 y(cum)n(ulativ)n(e)22 b(conjunction,)h FM(se)l(e)29
b FK(tensor)402 2040 y(pro)r(duct)167 2126 y(Curry-Ho)n(w)n(ard)24
b(isomorphism,)e(5,)i(150)284 2212 y(conjunction)g(and)g(pro)r(duct,)g
(14)284 2298 y(disjunction)f(and)h(sum,)f(80)284 2385
y(implication)e(and)j(functions,)f(14)284 2471 y(none)h(in)f(sequen)n
(t)g(calculus,)f(28)284 2557 y(second)i(order,)f(94)167
2643 y(cut)g(rule)284 2729 y Fq(Cut)q FK(,)g(30)284 2815
y(elimination)e(of,)i(3,)h(105,)g(151,)h(158)284 2901
y(linear)d(logic,)g(153,)j(156,)f(158)284 2988 y(natural)g(deduction,)f
(35,)h(40)284 3074 y(not)g(Ch)n(urc)n(h-Rosser,)g(150)284
3160 y(pro)r(ofs)h(without,)e(33,)h(159)284 3246 y(restriction)d(of,)j
(112)167 3408 y Fq(D)p FK(,)f FJ(D)r FK(,)g FM(se)l(e)28
b FK(casewise)22 b(de\014nition)167 3494 y(data)i(t)n(yp)r(es)f(in)g
FG(F)p FK(,)g(87,)h(89)167 3580 y(dead)g(h)n(yp)r(othesis,)g(9)167
3666 y(deadlo)r(c)n(k,)g(155)167 3752 y(deduction)f(\(natural\),)g(9)
167 3839 y(degree,)g(24,)h(109)284 3925 y FH(@)t FK(\(\),)f(of)h(form)n
(ula)f(or)g(t)n(yp)r(e)1640 404 y FH(d)p FK(\(\),)f(of)i(cut)f(or)g
(redex)1522 491 y FJ(D)-10 b FM(elin)5 b FK(,)23 b FM(se)l(e)29
b FK(linearisation)1522 577 y(denotation,)24 b(1)1522
663 y(denotational)g(seman)n(tics,)e(14,)i(54,)g(67,)1757
748 y(95,)g(132)1522 834 y(dereliction)c(\()p Fq(D)p
FK(?\),)i(154)1522 920 y(dI-domains,)i(71,)g(98)1522
1006 y(direct)e(pro)r(duct)h(\(&\))1640 1093 y(coherence)f(space,)h
(61,)h(67)1640 1179 y(linear)d(logic,)h(152)1522 1265
y(direct)g(sum)h(\()p FJ(\010)p FK(\))1640 1351 y(coherence)f(space,)h
(96,)h(103,)h(146)1698 1437 y(example,)e(66)1640 1523
y(linear)e(logic,)h(152)1522 1609 y(directed)g(joins,)h(57,)h(59,)g(66)
1522 1695 y(disc)n(harge,)g(9,)f(12,)h(37,)g(73,)h(161)1522
1782 y(discrete)d(graph,)i FM(se)l(e)k FK(\015at)c(domain)1522
1868 y(disjunction,)f(5,)h(6,)f(95)1640 1954 y(and)h(sum,)f(81)1640
2040 y(comm)n(uting)f(con)n(v)n(ersion,)i(78)1640 2126
y(con)n(v)n(ersion,)f(75)1640 2212 y(cut)f(elimination,)f(106)1640
2298 y Fq(disj)i FK(in)f Fq(Bo)r(ol)p FK(,)h(50)1640
2385 y(in)n(tuitionistic)c(prop)r(ert)n(y)-6 b(,)24 b(8,)g(33)1640
2471 y(linear)d(logic)h(\()p FJ(\010)h FK(and)h Fc(O)p
FK(\),)f(152)1640 2557 y(natural)g(deduction)1698 2643
y FJ(_)p FK(1)p FJ(I)5 b FK(,)25 b FJ(_)o FK(2)p FJ(I)k
FK(and)c FJ(_E)6 b FK(,)23 b(73)1640 2729 y(sequen)n(t)g(calculus)1698
2815 y FJ(L_)p FK(,)g FJ(R)p FK(1)p FJ(_)g FK(and)h FJ(R)p
FK(2)p FJ(_)p FK(,)g(31)1698 2901 y(in)n(tuitionistic)c
FJ(L_)p FK(,)j(32)1522 2988 y(domain)h(theory)-6 b(,)23
b(56,)i(132)1640 3074 y(dI-domains,)e(71,)i(98)1640 3160
y(domain)e(equations,)h(98)1640 3246 y(Girard)e FM(versus)h
FK(Scott,)g(54,)h(66,)h(98)1640 3332 y(L-domains,)e(140)1640
3418 y(lifted)e(sum,)i(96)1522 3504 y(donk)n(ey)-6 b(,)25
b(134)1522 3590 y(dynamic,)e(2,)h(14,)g(54)1522 3752
y FJ(9)p FK(,)g FM(se)l(e)k FK(existen)n(tial)21 b(quan)n(ti\014er)1522
3839 y(elimination,)g(8,)i(48)1640 3925 y FJ(^)p FK(1)p
FJ(E)6 b FK(,)24 b FJ(^)p FK(2)p FJ(E)6 b FK(,)24 b FJ(\)E)30
b FK(and)24 b FJ(8E)6 b FK(,)23 b(10)p eop
PStoPSsaved restore
%%Page: (172,173) 87
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
169 172 bop 167 167 a Fz(INDEX)2173 b FK(169)284 404
y Fq(R)24 b FK(and)g Fq(D)e FK(in)h FG(T)p FK(,)g(48)167
493 y(elimination)284 581 y FJ(_E)6 b FK(,)24 b FJ(?E)k
FK(and)c FJ(9E)6 b FK(,)24 b(73)284 669 y FJ(8)324 644
y FD(2)351 669 y FJ(E)6 b FK(,)24 b(94,)g(125)284 757
y(application)f(and)h(comp)r(onen)n(ts,)g(19)284 846
y(go)r(o)r(d)h(and)f(bad,)g(77)284 934 y(left)e(logical)g(rules,)g(37,)
i(40)284 1022 y(linear)e(logic,)g(161)284 1110 y(linearit)n(y)f(of,)j
(99,)g(103)167 1199 y(em)n(b)r(edding-pro)t(jection)e(pair,)h(133,)i
(134)167 1287 y(empt)n(y)e(t)n(yp)r(e)284 1375 y(and)i(absurdit)n(y)-6
b(,)24 b(80)284 1463 y(coherence)f(space)g(\()p FJ(E)-17
b FM(mp)5 b FK(\),)23 b(95,)h(104,)402 1548 y(139)284
1637 y Fq(Emp)f FK(and)h FH(")604 1648 y Fw(U)646 1637
y FK(,)g(80)284 1725 y(in)f FG(F)p FK(:)34 b Fq(Emp)20
b FK(=)f(\005)p FH(X)r(:)12 b(X)6 b FK(,)23 b(85,)h(139)284
1813 y(linear)e(logic)g(\()p FJ(?)f FK(and)k FG(0)p FK(\),)d(154)284
1901 y(realisabilit)n(y)-6 b(,)21 b(129)167 1990 y(equalisers,)h(137)
167 2078 y(equations)i(b)r(et)n(w)n(een)e(terms)g(and)i(pro)r(ofs,)402
2163 y FM(se)l(e)k FK(con)n(v)n(ersion)167 2251 y(espace)23
b(coh)n(\023)-33 b(eren)n(t,)24 b(56)167 2339 y(eta)f(rule,)f
FM(se)l(e)28 b FK(secondary)d(equations)167 2428 y(ev)l(aluation,)f
FM(se)l(e)k FK(application)167 2516 y(ev)n(en)n(t)23
b(structures,)g(98)167 2604 y(exc)n(hange)284 2692 y
FJ(L)p Fq(X)g FK(and)h FJ(R)p Fq(X)p FK(,)e(29)284 2781
y(linear)g(logic)g(\()p Fq(X)p FK(\),)g(153)167 2869
y(existen)n(tial)f(quan)n(ti\014er,)i(5,)h(6)284 2957
y(comm)n(uting)f(con)n(v)n(ersion,)g(78)284 3045 y(con)n(v)n(ersion,)h
(75)284 3134 y(cut)f(elimination,)d(108)284 3222 y(in)n(tuitionistic)g
(prop)r(ert)n(y)-6 b(,)24 b(8,)g(33)284 3310 y(natural)g(deduction)343
3398 y FJ(9I)29 b FK(and)24 b FJ(9E)6 b FK(,)24 b(73)284
3487 y(sequen)n(t)g(calculus)343 3575 y FJ(L9)f FK(and)h
FJ(R9)p FK(,)f(32)167 3663 y(existen)n(tial)e(t)n(yp)r(e)i(in)g
FG(F)f FK(\(\006,)h FJ(r)p FK(\),)g(86,)h(145)167 3751
y(exp)r(onen)n(tial)284 3840 y(ob)t(ject,)f FM(se)l(e)29
b FK(implication)20 b FM(and)402 3925 y FK(function-space)1640
404 y(pro)r(cess,)j FM(se)l(e)28 b FK(complexit)n(y)-6
b(,)1757 490 y(algorithmic)1522 576 y(expressiv)n(e)23
b(p)r(o)n(w)n(er,)g(50,)h(89,)g(155)1522 662 y(extraction,)f
FM(se)l(e)28 b FK(univ)n(ersal)23 b(application)1522
824 y FG(F)g FK(\(Girard-Reynolds)g(system\))1640 910
y(represen)n(table)f(functions,)h(120)1640 996 y(seman)n(tics,)f(132)
1640 1082 y(strong)i(normalisation,)e(42,)i(114)1640
1168 y(syn)n(tax,)h(82)1522 1254 y FH(F)1567 1265 y FD(0)1618
1254 y FK(\(parallel)c(or\),)i(61,)h(70)1522 1341 y(false)1640
1427 y(denotation)g(\()p FG(f)7 b FK(,)23 b FJ(F)7 b
FK(,)23 b Fq(F)p FK(\),)g FM(se)l(e)28 b FK(b)r(o)r(oleans)1640
1513 y(prop)r(osition)23 b(\()p FJ(?)p FK(\),)f FM(se)l(e)28
b FK(absurdit)n(y)1522 1599 y(feasible,)22 b FM(se)l(e)28
b FK(complexit)n(y)-6 b(,)22 b(algorithmic)1522 1685
y(\014elds)h(of)g(n)n(um)n(b)r(ers,)h(134)1522 1771 y(\014ltered)e
(colimits,)e(59,)k(137)1522 1857 y(\014nite)1640 1944
y(appro)n(ximation,)g(57,)g(66,)g(132,)g(134)1640 2030
y(branc)n(hing)g(tree,)e(27)1640 2116 y(normalisation,)g(24)1640
2202 y(p)r(oin)n(ts)h(\()p FJ(A)1930 2213 y Fm(\014n)1994
2202 y FK(\),)f(57)1640 2288 y(presen)n(tabilit)n(y)-6
b(,)22 b(66)1640 2374 y(sense)h(and)h(denotation,)g(2)1640
2460 y(v)n(ery)-6 b(,)23 b(59,)i(66)1522 2547 y(\014xed)f(p)r(oin)n(t,)
f(72,)h(95)1522 2633 y(\015at)g(domain,)f(57,)h(60,)g(66,)h(70,)f(140)
1522 2719 y(for)g(all)d(\()p FJ(8)p FK(\),)h FM(se)l(e)29
b FK(univ)n(ersal)22 b(quan)n(ti\014er)1522 2805 y(for)i(some)e(\()p
FJ(9)p FK(\),)h FM(se)l(e)29 b FK(existen)n(tial)21 b(quan)n(ti\014er)
1522 2891 y(F)-6 b(rege,)23 b(1,)h(2)1522 2977 y(function,)g(1,)f(17)
1640 3063 y(Berry)f(order)h(\()p FJ(\024)2094 3074 y
FD(B)2132 3063 y FK(\),)g(65)1640 3149 y(comp)r(osition,)f(69)1640
3236 y(con)n(tin)n(uous,)i(55,)g(58)1640 3322 y(\014xed)f(p)r(oin)n(t,)
h(72)1640 3408 y(graph,)g(1,)g(66)1640 3494 y(linear,)d(99,)k(148)1640
3580 y(not)e(represen)n(table,)f(122)1640 3666 y(on)h(pro)r(ofs,)i(6,)f
(11)1640 3752 y(on)f(t)n(yp)r(es,)h(83,)g(132,)h(136)1640
3839 y(partial,)d(60,)i(66)1640 3925 y(partial)e(recursiv)n(e,)g(55)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
170 173 bop 167 167 a FK(170)2175 b Fz(INDEX)167 404
y FK(function)284 492 y(p)r(oin)n(t)n(wise)22 b(order,)i(66)284
580 y(p)r(olynomial)f(resolution,)f(147)284 668 y(pro)n(v)l(ably)j
(total,)e(52,)h(123)284 755 y(recursion,)f(50,)h(90,)g(120)284
843 y(represen)n(table,)e(52,)j(121)284 931 y(sequen)n(tial,)e(54)284
1018 y(stable,)g(58,)h(62,)g(68)284 1106 y(total)f(recursiv)n(e,)f(122)
284 1194 y(trace)h(\()p FJ(T)-6 b FH(r)r FK(\),)23 b(62)284
1282 y(t)n(w)n(o)h(argumen)n(ts,)f(61)167 1369 y(function-space)284
1457 y(and)i(implication,)20 b(12,)k(15,)g(20)284 1545
y(in)f FG(F)p FK(,)g(82)284 1632 y FH(\025)p FK(-calculus,)g(12,)h(15)
284 1720 y(linear)e(decomp)r(osition,)h(101)284 1808
y(seman)n(tics,)g(54,)h(62,)g(64,)g(67,)g(136)167 1896
y(functor,)g(59,)g(134,)g(136,)h(141)167 2085 y FG(G)p
FK(allier,)c(28)167 2173 y(Galois)i(Theory)-6 b(,)23
b(134)167 2261 y(Gandy)-6 b(,)25 b(27)167 2348 y(garbage)g(collection,)
20 b(150)167 2436 y FG(Gem)o FK(,)k(136)167 2524 y(general)f
(recursion,)f(72)167 2612 y(Gen)n(tzen,)h(3,)g(28,)i(105)167
2699 y(geometry)e(of)g(in)n(teraction,)f(4,)i(160)167
2787 y(Girard,)f(30,)h(42,)g(80,)g(82,)g(114,)h(124,)g(150)167
2875 y(goals)f(in)e Fr(PROLOG)p FK(,)j(112)167 2962 y(G\177)-34
b(odel,)23 b(1,)h(6,)f(47,)h(54)284 3050 y(incompleteness)d(theorem,)i
(42,)h(114)284 3138 y(n)n(um)n(b)r(ering,)f(53)284 3226
y FJ(::)p FK(-translation,)h(124)167 3313 y(go)r(o)r(d)h(elimination,)
20 b(77)167 3401 y(graph)284 3489 y(em)n(b)r(edding,)j(133,)i(134)284
3576 y(function,)f(1,)f(66)284 3664 y(pro)r(duct,)h(104,)h(138)284
3752 y(w)n(eb,)e(56)167 3840 y(Grothendiec)n(k)g(\014bration)g(\(\006)
902 3834 y FJ(\016)953 3840 y FK(\),)g(135,)i(137,)402
3925 y(141)1522 404 y FG(H)p FK(auptsatz)f(\(cut)f(elimination\),)d(3,)
k(105,)1757 490 y(151,)h(158)1522 576 y(head)f(normal)f(form,)g(19,)h
(52,)g(76,)g(121)1522 662 y(heigh)n(t)f(of)h(a)f(pro)r(of)h(\()p
FH(h)p FK(\),)f(109)1522 748 y(Herbrand,)h(4)1522 834
y(hereditarily)d(e\013ectiv)n(e)g(op)r(erations,)j(55)1522
920 y(Heyting,)f(5,)h(15,)g(80,)g(120)1640 1007 y(arithmetic)c(\()p
FG(H)-12 b(A)2110 1018 y Fb(2)2142 1007 y FK(\),)23 b(124)1522
1093 y(Horn)h(clause,)e FM(se)l(e)29 b FK(in)n(tuitionistic)19
b(sequen)n(t)1522 1179 y(Ho)n(w)n(ard,)24 b FM(se)l(e)29
b FK(Curry-Ho)n(w)n(ard)1757 1264 y(isomorphism)1522
1350 y(Hyland,)24 b(133)1522 1436 y(h)n(yp)r(erexp)r(onen)n(tial)g
(function,)f(111)1522 1523 y(h)n(yp)r(otheses,)i(9)1640
1609 y(disc)n(harge,)e(9,)h(161)1640 1695 y(parcels)e(of,)i(11,)g(36,)g
(40,)g(161)1640 1781 y(subform)n(ula)f(prop)r(ert)n(y)-6
b(,)24 b(76)1640 1867 y(v)l(ariables,)f(11)1522 2030
y FJ(I)5 b FK(,)24 b FM(se)l(e)k FK(in)n(tro)r(duction)1522
2116 y(idemp)r(otence)22 b(of)h(logic,)f(29)1522 2202
y(iden)n(ti\014cation)g(of)i(terms)e(and)i(pro)r(ofs,)g
FM(se)l(e)1757 2287 y FK(con)n(v)n(ersion)1522 2374 y(iden)n(tit)n(y)
1640 2460 y(axiom,)f(30,)h(112,)h(156)1640 2546 y(h)n(yp)r(othesis,)f
(10)1640 2632 y(maximal)d(in)i(Berry)f(order,)i(65,)g(135)1640
2718 y(p)r(olymorphic,)e(83,)i(132,)h(136,)g(138)1640
2804 y(pro)r(of)f(of)f FH(A)d FJ(\))g FH(A)p FK(,)i(6)1522
2891 y(if,)h FM(se)l(e)28 b FK(casewise)22 b(de\014nition)1522
2977 y(implication,)f(5)1640 3063 y(and)j(function-space,)f(12,)h(15,)h
(20)1640 3149 y(con)n(v)n(ersion,)e(13)1640 3235 y(cut)f(elimination,)f
(107)1640 3321 y(linear)g(\()p Fu(\()p FK(\),)i(100,)i(153)1640
3408 y(natural)e(deduction)1698 3494 y FJ(\))q(I)28 b
FK(and)c FJ(\))q(E)6 b FK(,)24 b(10)1640 3580 y(realisabilit)n(y)-6
b(,)20 b(127)1640 3666 y(seman)n(tics,)i(54)1640 3752
y(sequen)n(t)h(calculus)1698 3839 y FJ(L\))g FK(and)h
FJ(R\))q FK(,)f(32)1522 3925 y(inclusion)f(order,)h(56)p
eop
PStoPSsaved restore
%%Page: (174,175) 88
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
171 174 bop 167 167 a Fz(INDEX)2173 b FK(171)167 404
y(incoherence)22 b(\()585 382 y FH(^)585 425 y(_)674
404 y FK(\),)h(100)167 490 y(incompleteness)e(theorem,)h(6,)i(42,)g
(114,)402 575 y(124)167 661 y(inductiv)n(e)e(data)i(t)n(yp)r(es,)f(87,)
i(121)167 747 y(inductiv)n(e)d(de\014nition)g(of)i(+,)f
FJ(\002)p FK(,)g FM(etc.)p FK(,)f(50)167 833 y(in\014nite)g(static)g
(denotation,)i(2)167 919 y(in\014nit)n(y)f(\()p Fo(f)-70
b FJ(1)23 b FK(and)h FJ(1)p FK(\),)f(71)167 1005 y(initial)d(ob)t
(ject,)j(95,)h(152)167 1091 y(input,)f(1,)h(17)167 1177
y(in)n(tegers,)e(1)284 1263 y(coherence)h(space)343 1349
y(\015at)h(\()p FJ(I)-7 b FM(nt)7 b FK(\),)23 b(56,)h(60,)g(66,)g(70)
343 1435 y(lazy)f(\()p FJ(I)-7 b FM(nt)614 1409 y FD(+)655
1435 y FK(\),)23 b(71,)i(98)343 1521 y([)-12 b([)q(\005)p
FH(X)r(:)12 b(X)6 b FJ(!)o FK(\()p FH(X)g FJ(!)p FH(X)g
FK(\))p FJ(!)p FH(X)g FK(\)])-12 b(])q(,)23 b(147)284
1607 y(con)n(v)n(ersion,)h(48)284 1692 y(dI-domain)g(\()p
FJ(I)-7 b FM(nt)743 1667 y Fw(<)785 1692 y FK(\),)22
b(98)284 1778 y(in)h FG(F)p FK(,)g(89,)h(121,)h(147)284
1864 y(in)e FG(H)-12 b(A)475 1875 y Fb(2)507 1864 y FK(,)23
b(125)284 1950 y(in)g FG(T)g FK(\()p Fq(Int)p FK(,)g
Fq(O)p FK(,)g Fq(S)p FK(,)g Fq(R)p FK(\),)h(48,)g(70)284
2036 y(iteration)e(\()p Fq(It)p FK(\),)h(51,)h(70,)g(90)284
2122 y(linear)e(t)n(yp)r(e)h(\()p Fq(LInt)p FK(\),)g(148)284
2208 y(normal)g(form,)g(52,)h(121)284 2294 y(realisabilit)n(y)c(\()p
Fq(Nat)q FK(\),)i(126,)j(127)284 2380 y(recursor)e(\()p
Fq(R)p FK(\),)g(48,)h(91)284 2466 y(totalit)n(y)-6 b(,)23
b(149)167 2552 y(in)n(ternalisation,)f(27)167 2638 y(in)n(tersection,)f
FM(se)l(e)29 b FK(conjunction)284 2723 y(b)r(ounded)c(ab)r(o)n(v)n(e,)g
FM(se)l(e)j FK(pullbac)n(k)284 2809 y(in)23 b([)-12 b([)p
Fq(Bo)r(ol)p FK(])g(])q(,)23 b(140)167 2895 y(in)n(tro)r(duction,)g(8,)
g(48)284 2981 y FJ(^I)5 b FK(,)24 b FJ(\)I)29 b FK(and)24
b FJ(8I)5 b FK(,)23 b(10)284 3067 y Fq(O)p FK(,)g Fq(S)p
FK(,)h Fq(T)f FK(and)h Fq(F)f FK(in)g FG(T)p FK(,)g(48)284
3153 y FJ(_)p FK(1)p FJ(I)5 b FK(,)25 b FJ(_)o FK(2)p
FJ(I)30 b FK(and)24 b FJ(9I)5 b FK(,)23 b(73)284 3239
y FJ(8)324 3213 y FD(2)351 3239 y FJ(I)5 b FK(,)24 b(94,)g(125)284
3325 y(linear)e(logic,)g(161)284 3411 y(pairing)h(and)h
FH(\025)p FK(-abstraction,)g(11,)g(12,)402 3496 y(19)284
3582 y(righ)n(t)f(logical)f(rules,)g(37,)i(40)284 3668
y(sums,)g(81)167 3754 y(in)n(tuitionism,)d(6,)i(150)167
3840 y(in)n(tuitionistic)d(sequen)n(t,)j(8,)h(30,)g(32,)g(33,)402
3925 y(112,)g(152)1522 404 y(in)n(v)n(ersion)f(in)f(linear)g(algebra,)i
(101)1522 490 y(isomorphisms,)f(132{134)1522 575 y(iteration)f(\()p
Fq(It)o FK(\),)h(51,)i(70,)f(90)1522 724 y FG(J)p FK(oin,)f
FM(se)l(e)29 b FK(disjunction)1640 809 y(preserv)l(ation)23
b(\(linearit)n(y\),)e(99)1522 895 y(join)n(t)i(con)n(tin)n(uit)n(y)g
(and)h(stabilit)n(y)-6 b(,)22 b(61)1522 980 y(Jung,)j(133,)g(140)1522
1129 y FG(K)p FK(leene,)d(123)1522 1214 y(K\177)-34 b(onig's)24
b(lemma,)d(27)1522 1300 y(Ko)n(ymans,)j(133)1522 1385
y(Kreisel,)d(55)1522 1508 y Fv(L)1560 1534 y FK(,)i FM(se)l(e)28
b FK(asymmetrical)21 b(in)n(terpretation)1522 1619 y
FJ(L)i FK(\(left)e(logical)h(rules\),)g FM(se)l(e)28
b FK(sequen)n(t)1757 1704 y(calculus)1522 1790 y FH(`)p
FK(\()p FH(t)p FK(\),)23 b(length)g(of)g(normal)g(form,)g(49)1522
1875 y(Lafon)n(t,)i(150)1522 1961 y FH(\025)p FK(,)f
FM(se)l(e)k FK(abstraction)1522 2046 y(\003,)23 b FM(se)l(e)29
b FK(univ)n(ersal)22 b(abstraction)1522 2131 y FH(\025)p
FK(-calculus,)h(12,)h(15)1640 2217 y(Ch)n(urc)n(h-Rosser)f(prop)r(ert)n
(y)-6 b(,)24 b(22)1640 2302 y(con)n(v)n(ersion,)f(18)1640
2388 y(head)g(normal)g(form,)g(19)1640 2473 y(natural)g(deduction,)g
(11,)h(19)1640 2558 y(normalisation,)e(24,)i(42)1640
2644 y(second)f(order)g(p)r(olymorphic,)g(82)1640 2729
y(seman)n(tics,)f(54,)i(67)1640 2815 y(un)n(t)n(yp)r(ed,)g(19,)g(22,)g
(133)1522 2900 y(last)f(rule)f(of)h(a)h(pro)r(of,)g(8,)g(33)1522
2985 y(lazy)f(ev)l(aluation,)h(150)1522 3071 y(lazy)f(natural)g(n)n(um)
n(b)r(ers,)h(71,)g(98)1522 3156 y(L-domains,)g(140)1522
3242 y(least)f(appro)n(ximan)n(t,)h(59,)g(137)1522 3327
y(left)e(logical)g(rules,)g FM(se)l(e)28 b FK(sequen)n(t)c(calculus)
1522 3412 y(lifted)d(sum,)j(96)1522 3498 y(limit-colimit)17
b(coincidence,)k(137)1522 3583 y(linear)h(algebra,)h(101)1522
3669 y(linear)f(logic,)g(30,)i(35,)g(74,)g(161)1640 3754
y(Ch)n(urc)n(h-Rosser)f(prop)r(ert)n(y)-6 b(,)24 b(159)1640
3839 y(cut)e(rule,)h(153,)h(156,)h(158)1522 3925 y(linear)d(logic)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
172 175 bop 167 167 a FK(172)2175 b Fz(INDEX)284 404
y FK(direct)22 b(pro)r(duct)h(\(&\),)g(61,)h(104,)h(152)284
491 y(direct)d(sum)h(\()p FJ(\010)p FK(\),)f(96,)i(103,)h(146,)g(152)
284 577 y(implication)c(\()p Fu(\()p FK(\),)i(100,)h(104,)h(153)284
663 y(in)n(tegers)e(\()p Fq(LInt)p FK(\),)g(148)284 749
y(in)n(tuitionistic)d(logic,)i(154)284 835 y(linear)g(maps,)i(99,)g
(101)284 921 y(link)e(rule,)h(156)284 1008 y(natural)h(deduction,)f
(161)284 1094 y(negation)h(\()591 1068 y Fv(?)633 1094
y FK(\),)f(100,)h(138,)h(153)284 1180 y(notation)f(for)g(tok)n(ens,)g
(138)284 1266 y(of)g(course)f(\(!\),)g(101,)i(145,)f(154)284
1352 y(p)r(olynomial)f(resolution,)f(147)284 1438 y(pro)r(of)i(nets,)g
(155)284 1524 y(reducibilit)n(y)-6 b(,)21 b(115)284 1611
y(seman)n(tics,)i(95)284 1697 y(sequen)n(t)h(calculus,)e(152)284
1783 y(sum)h(decomp)r(osition,)g(95,)h(103,)h(146)284
1869 y(syn)n(tax,)g(150)284 1955 y(tensor)f(pro)r(duct)f(\()p
FJ(\012)p FK(\),)g(104,)h(146,)h(152,)402 2040 y(156)284
2127 y(tensor)f(sum)f(or)g(par)h(\()p Fc(O)o FK(\),)f(104,)i(152,)402
2212 y(156)284 2298 y(trace)e(\()p FJ(T)-6 b FM(rlin)5
b FK(\),)23 b(100)284 2384 y(units)g(\()p FG(1)p FK(,)g
FJ(?)p FK(,)f FJ(>)f FK(and)k FG(0)p FK(\),)d(104,)j(154)284
2470 y(wh)n(y)f(not)g(\(?)o(\),)f(102,)i(154)167 2556
y(link)d(axiom,)h(156)167 2642 y(lists,)f(47,)i(91)167
2729 y(lo)r(cally)d(compact)i(space,)h(55)167 2815 y(logical)e(rules,)g
(31)284 2901 y(cut)h(elimination,)d(105,)25 b(112)284
2987 y(linear)d(logic,)g(153,)j(156)284 3073 y(natural)f(deduction,)f
(37)167 3159 y(logical)f(view)g(of)h(top)r(ology)-6 b(,)25
b(55)167 3245 y(long)e(trip)g(condition,)f(158)167 3332
y(L\177)-34 b(ow)n(enheim,)22 b(1,)i(3)167 3494 y FG(M)p
FK(artin-L\177)-34 b(of,)24 b(88)167 3580 y Fq(match)e
FK(\(patterns\),)i(81)167 3666 y(maximally)d(consisten)n(t)i
(extensions,)g(54)167 3752 y(meet,)f FM(se)l(e)28 b FK(conjunction)284
3839 y(b)r(ounded)d(ab)r(o)n(v)n(e,)g FM(se)l(e)j FK(pullbac)n(k)167
3925 y(memory)22 b(allo)r(cation,)h(150)1522 404 y(mo)r(d,)h(coherence)
e(relation,)g(56)1522 490 y(mo)r(dalities,)f(101,)k(154)1522
576 y(mo)r(del)d(theory)-6 b(,)24 b(3)1522 662 y(mo)r(dules,)f(17,)h
(47)1522 748 y(mo)r(dus)g(p)r(onens)g(\()p FJ(\)E)6 b
FK(\),)24 b FM(se)l(e)k FK(elimination)1522 834 y(Moggi,)c(141)1522
992 y(I)-12 b(N)20 b(=)f FJ(f)p FK(0)p FH(;)12 b FK(1)p
FH(;)g FK(2)p FH(;)g(:::)r FJ(g)p FK(,)23 b(set)g(of)g(in)n(tegers)1522
1077 y Fq(Nat)h FK(predicate)d(in)i FG(H)-12 b(A)2141
1088 y Fb(2)2173 1077 y FK(,)23 b(126)1522 1163 y(natural)h(deduction,)
f(8)1640 1249 y FJ(^)p FK(,)g FJ(\))g FK(and)h FJ(8)p
FK(,)f(10)1640 1335 y FJ(_)p FK(,)g FJ(?)f FK(and)i FJ(9)p
FK(,)f(73)1640 1421 y(con)n(v)n(ersion,)g(13,)i(20,)f(75,)g(78)1640
1507 y FH(\025)p FK(-calculus,)e(11,)i(19)1640 1593 y(linear)d(logic,)h
(161)1640 1679 y(normalisation,)g(24,)i(42)1640 1765
y(second)f(order,)h(94,)g(125)1640 1850 y(sequen)n(t)f(calculus,)f(35)
1640 1936 y(subform)n(ula)h(prop)r(ert)n(y)-6 b(,)24
b(76)1522 2022 y(natural)g(n)n(um)n(b)r(ers,)f FM(se)l(e)28
b FK(in)n(tegers)1522 2108 y(negation,)c(5)1640 2194
y FH(A)15 b FJ(_)h(:)p FH(A)p FK(,)22 b(6)1640 2294 y
FJ(:)p FH(A)1758 2254 y FD(def)1766 2294 y FK(=)k FH(A)20
b FJ(\))g(?)p FK(,)i(6,)h(73)1640 2379 y(cut)f(elimination,)f(107)1640
2465 y(linear)g(\()1858 2440 y Fv(?)1900 2465 y FK(\),)i(100,)h(138,)h
(153)1640 2551 y Fq(neg)e FK(in)g Fq(Bo)r(ol)p FK(,)g(50)1640
2637 y(sequen)n(t)g(calculus)f(\()p FJ(L:)p FK(,)h FJ(R:)p
FK(\),)f(31)1522 2723 y(neutralit)n(y)-6 b(,)23 b(43,)h(49,)g(116)1522
2809 y Fq(nil)e FK(\(empt)n(y)h(tree)f(or)h(list\),)e(91)1522
2895 y FG(N)-12 b(J)23 b FK(\(Pra)n(witz')f(system\),)h
FM(se)l(e)28 b FK(natural)1757 2980 y(deduction)1522
3066 y(No)r(etherian,)23 b(66)1522 3152 y(noncon)n(v)n(ergen)n(t)i
(rewriting,)c(72)1522 3238 y(nondeterminism,)h(150)1522
3323 y(normal)h(closure)f(of)i(a)f(\014eld,)g(134)1522
3409 y(normal)g(form,)g(18,)h(76)1640 3495 y(cut-free,)e(41)1640
3581 y(existence,)f(24)1640 3667 y(head)i(normal)g(form,)g(19)1640
3753 y(in)n(tegers,)f(52,)i(121)1640 3839 y(linear)d(logic,)h(159)1522
3925 y(normal)h(form)p eop
PStoPSsaved restore
%%Page: (176,177) 89
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
173 176 bop 167 167 a Fz(INDEX)2173 b FK(173)284 404
y(uniqueness,)24 b(22)167 490 y(normalisation)e(theorem)284
575 y(for)i FG(F)p FK(,)f(114)284 660 y(for)h FG(T)p
FK(,)f(49)284 745 y(length)g(of)h(pro)r(cess)f(\()p FH(\027)t
FK(\),)g(27,)h(43,)g(49)284 830 y(linear)e(logic,)g(155)284
915 y(strong,)j(42)284 1000 y(w)n(eak,)f(22,)g(24)167
1085 y(not)g(\()p FJ(:)p FK(\),)e FM(se)l(e)29 b FK(negation)167
1228 y Fq(O)p FK(,)23 b FJ(O)i FK(\(zero\),)d FM(se)l(e)29
b FK(in)n(tegers)167 1313 y(ob)t(ject)23 b(language,)h(10)167
1399 y(Oc)n(kham's)f(razor,)h(3)167 1484 y(of)g(course)f(\(!\),)g(101,)
h(138,)h(145,)f(154)167 1569 y(op)r(erational)f(seman)n(tics,)f(14,)i
(121)167 1654 y(or,)f FM(se)l(e)29 b FK(disjunction)23
b FM(and)g FK(parallel)e(or)167 1739 y(orthogonal)k(\()539
1713 y Fv(?)580 1739 y FK(\),)e FM(se)l(e)29 b FK(linear)21
b(negation)167 1824 y(output,)j(17)167 1967 y FG(P)-23
b(A)o FK(,)24 b FG(P)-23 b(A)391 1978 y Fb(2)422 1967
y FK(,)23 b FM(se)l(e)28 b FK(P)n(eano)c(arithmetic)167
2052 y(pairing)284 2137 y(conjunction,)g(11)284 2222
y(in)f FG(F)p FK(,)g(84)284 2308 y(in)n(tro)r(duction,)g(19)284
2393 y(seman)n(tics)f(\()p FJ(P)-18 b FM(air)9 b FK(\),)23
b(61,)h(68)167 2478 y(par)g(\()p Fc(O)o FK(\),)f FM(se)l(e)28
b FK(tensor)c(sum)167 2563 y(parallel)d(or,)j(50,)g(60,)g(70,)g(81)167
2648 y(parallel)d(pro)r(cess,)j(150,)g(159)167 2733 y(parcels)e(of)i(h)
n(yp)r(otheses,)g(11,)g(36,)h(40,)f(161)167 2818 y(partial)e(equiv)l
(alence)g(relation,)g(55)167 2903 y(partial)g(function,)i(60)284
2988 y(coherence)f(space)g(\()p FJ(P)-6 b(F)6 b FK(\),)23
b(66)284 3074 y(recursiv)n(e,)f(55)167 3159 y(partial)g(ob)t(jects,)h
(57)167 3244 y Fr(PASCAL)p FK(,)i(47)167 3329 y(pattern)e(matc)n(hing,)
g(81)167 3414 y(P)n(eano)h(arithmetic)c(\()p FG(P)-23
b(A)p FK(\),)23 b(42,)h(53)284 3499 y(second)g(order)f(\()p
FG(P)-23 b(A)805 3510 y Fb(2)836 3499 y FK(\),)23 b(114,)i(123)167
3584 y(p)r(eculiar,)d(134)167 3669 y(p)r(erm)n(utations,)h(134)167
3754 y(\005,)g FM(se)l(e)28 b FK(system)23 b FG(F)167
3840 y FH(\031)209 3814 y FD(1)237 3840 y FK(,)g FH(\031)321
3814 y FD(2)348 3840 y FK(,)g(\005)442 3814 y FD(1)470
3840 y FK(,)g(\005)564 3814 y FD(2)592 3840 y FK(,)g
FM(se)l(e)28 b FK(comp)r(onen)n(ts)167 3925 y(Pitts,)22
b(133)1522 404 y(Plotkin,)h(56)1522 490 y(plugging,)h(17)1522
575 y(p)r(olynomial,)f(148)1522 660 y FJ(P)-6 b FH(!)s
FK(,)23 b FM(se)l(e)28 b FK(Scott)1522 745 y(p)r(ositiv)n(e)22
b(negativ)n(e,)i(34,)g(87,)g(139,)g(142)1522 830 y(p)r(oten)n(tial)e
(tok)n(ens,)j(138)1522 915 y(Pra)n(witz,)d(8,)i(80)1522
1000 y(predecessor)f(\()p Fq(p)n(red)p FK(\),)h(51,)g(72,)g(91)1522
1085 y(preserv)l(ation)g(of)f(joins)g(\(linearit)n(y\),)e(99)1522
1170 y(primary)i(equations,)h FM(se)l(e)k FK(con)n(v)n(ersion)1522
1256 y(principal)22 b(branc)n(h)i(or)f(premise,)e(75,)j(76)1522
1341 y(pro)r(duct)1640 1426 y(and)g(conjunction,)g(11,)g(15,)g(19)1640
1511 y(coherence)e(space)h(\(&\),)g(61,)h(68)1640 1596
y(in)e FG(F)p FK(,)h(84,)h(145)1640 1681 y(linear)d(logic)h(\()p
FJ(\012)h FK(and)h(&\),)e(152)1640 1766 y(pro)t(jection,)g(11,)j(61,)f
(68,)g(84)1522 1851 y(programs,)h(17,)f(53,)g(72,)g(84,)g(124)1522
1936 y(pro)t(jection)f(\(em)n(b)r(edding\),)f(135,)j(142)1522
2022 y Fr(PROLOG)p FK(,)g(28,)f(105,)h(112)1522 2107
y(pro)r(of,)g(5)1522 2192 y(pro)r(of)f(net,)f(155)1522
2277 y(pro)r(of)h(of)g(program,)g(53)1522 2362 y(pro)r(of)g(structure,)
f(156)1522 2447 y(pro)n(v)l(ably)i(total,)e(53,)h(124)1522
2532 y(Ptolomeic)d(astronom)n(y)-6 b(,)25 b(1)1522 2617
y(pullbac)n(k,)e(54,)h(59,)h(61,)f(65,)g(137,)g(141)1522
2760 y FG(Q)p FK(uan)n(ti\014ers,)g FM(se)l(e)k FK(univ)n(ersal,)23
b(existen)n(tial)1757 2845 y FM(and)g FK(second)h(order)1522
2988 y FJ(R)f FK(\(righ)n(t)f(logical)g(rules\),)g FM(se)l(e)28
b FK(sequen)n(t)1757 3074 y(calculus)1522 3133 y Fv(R)1568
3159 y FK(,)23 b FM(se)l(e)28 b FK(asymmetrical)21 b(in)n(terpretation)
1522 3244 y(real)h(n)n(um)n(b)r(ers,)i(114)1522 3329
y(realisabilit)n(y)-6 b(,)21 b(7)1522 3414 y(recursion)1640
3499 y(and)j(iteration,)e(51,)i(70,)g(90)1640 3584 y(recurrence)d
(relation,)h(50)1640 3669 y(recursor)g(\()p Fq(R)p FK(\),)i(48)1640
3754 y(seman)n(tics,)e(70)1522 3840 y(redex,)h(18,)h(24,)g(48)1522
3925 y(reducibilit)n(y)-6 b(,)21 b(27,)j(58,)g(123)p
eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
174 177 bop 167 167 a FK(174)2175 b Fz(INDEX)284 404
y FK(in)23 b FG(F)p FK(,)g(115)284 491 y(in)g FG(T)p
FK(,)g(49)284 577 y FH(\025)p FK(-calculus,)g(42)167
663 y(reduction,)g(18)167 749 y(re\015exiv)n(e)f(symmetric)f(relation,)
h(57)167 835 y(represen)n(table)g(functions,)h(52,)i(120)167
921 y(resolution)d(metho)r(d,)i(112)167 1008 y(rewrite)d(rules,)h
FM(se)l(e)28 b FK(con)n(v)n(ersion)167 1094 y(Reynolds,)23
b(82)167 1180 y(righ)n(t)g(logical)e(rules,)i FM(se)l(e)28
b FK(sequen)n(t)402 1265 y(calculus)167 1351 y(rigid)22
b(em)n(b)r(eddings,)h(134)167 1437 y(ring)g(theory)-6
b(,)24 b(66)167 1523 y(Robinson,)g(112)167 1610 y(Rosser,)f
FM(se)l(e)29 b FK(Ch)n(urc)n(h-Rosser)23 b(prop)r(ert)n(y)167
1772 y Fq(S)p FK(,)g FJ(S)28 b FK(\(successor\),)23 b
FM(se)l(e)29 b FK(in)n(tegers)167 1858 y(saturated)24
b(domains,)f(133)167 1944 y(Scott,)g(54,)h(55,)g(64,)h(66,)f(133)167
2030 y(second)g(incompleteness)d(theorem,)h(42)167 2116
y(second)i(order)f(logic,)f(94,)i(114,)g(123)167 2203
y(secondary)g(equations,)g(16,)g(69,)h(81,)f(85,)402
2288 y(97,)g(132)167 2374 y(seman)n(tic)e(de\014nabilit)n(y)-6
b(,)22 b(140)167 2460 y(sense,)h(1)167 2546 y(separabilit)n(y)-6
b(,)22 b(134)167 2632 y(sequen)n(t)h(calculus,)g(28)284
2719 y Fq(Cut)h FK(rule,)e(30)284 2805 y(linear)g(logic,)g(150)284
2891 y(logical)g(rules,)g(31)284 2977 y(natural)i(deduction,)f(35)284
3063 y Fr(PROLOG)p FK(,)i(112)284 3149 y(structural)e(rules,)f(29)167
3235 y(sequen)n(tial)g(algorithm,)h(54)167 3322 y(side)f(e\013ects,)h
(150)167 3408 y(\006)g(\(existen)n(tial)e(t)n(yp)r(e\))i(in)f
FG(F)p FK(,)h(86,)h(145)167 3494 y(\006)170 3488 y FJ(\016)244
3494 y FK(\(total)e(category\),)i(135)167 3580 y(signature,)f(34,)i
(87,)f(139)167 3666 y(simple)d(t)n(yp)r(es,)i(84,)i(139,)f(145)167
3752 y(singleton)f(t)n(yp)r(e)g(\()p FJ(S)-6 b FM(gl)5
b FK(\),)23 b(104,)i(139)167 3839 y(size)d(problem,)g(83,)i(112,)h(132)
167 3925 y(Sm)n(yth,)f(55)1522 404 y FJ(S)-6 b(N)33 b
FK(\(strongly)23 b(normalisable)f(terms\),)1757 490 y(119)1522
576 y(sp)r(eci\014cation,)h(17)1522 662 y(sp)r(ectral)f(space,)i(56)
1522 748 y(stabilit)n(y)-6 b(,)22 b(54,)i(134,)h(137)1640
834 y(de\014nition)d(\()p FG(St)p FK(\),)h(58,)h(100)1522
920 y(static,)f(2,)g(14,)h(54)1522 1006 y(strict)e(\(preserv)n(e)g
Fn(?)p FK(\),)h(97)1522 1093 y(strong)h(\014niteness,)f(59,)h(66)1522
1179 y(strong)g(normalisation,)f(26,)h(42,)g(114)1522
1265 y(structural)f(rules)1640 1351 y(cut)f(elimination,)f(106,)j(112)
1640 1437 y(linear)d(logic,)h(152,)j(153)1640 1523 y(natural)e
(deduction,)g(36)1640 1610 y(sequen)n(t)g(calculus,)f(29)1522
1696 y(sub)r(domain,)i(134)1522 1782 y(subform)n(ula)g(prop)r(ert)n(y)
1640 1868 y FH(\025)p FK(-calculus,)e(19)1640 1954 y(natural)h
(deduction,)g(76)1640 2040 y(sequen)n(t)g(calculus,)f(33)1522
2127 y(substan)n(ti\014que)j(mo)r(elle,)20 b(155)1522
2213 y(substitution,)j(5,)h(25,)g(69,)g(112,)h(118)1522
2299 y(subuniformit)n(y)-6 b(,)24 b(134)1522 2385 y(successor)f(\()p
Fq(S)h FK(and)g FJ(S)5 b FK(\),)23 b FM(se)l(e)28 b FK(in)n(tegers)1522
2471 y(sum)23 b(t)n(yp)r(e,)h(95)1640 2557 y(+,)e FH(\023)1759
2532 y FD(1)1788 2557 y FK(,)h FH(\023)1854 2532 y FD(2)1905
2557 y FK(and)h FH(\016)s FK(,)f(81)1640 2643 y(and)h(disjunction,)f
(81)1640 2730 y(coherence)f(space,)h(103,)i(146)1640
2816 y(linear)c(decomp)r(osition,)i(96,)h(103)1640 2902
y(linear)d(logic)h(\()p FJ(\010)h FK(and)h Fc(O)p FK(\),)f(152)1522
2988 y(symmetry)-6 b(,)23 b(10,)h(18,)g(28,)g(30,)g(31,)g(97,)h(105,)
1757 3073 y(134)1522 3235 y Fq(T)p FK(,)e FG(t)p FK(,)g
FJ(T)c FK(,)k FM(se)l(e)28 b FK(true)1522 3322 y FG(T)23
b FK(\(G\177)-34 b(odel's)23 b(system\),)g(47,)h(67,)g(70,)g(123)1522
3408 y(tableaux,)g(28)1522 3494 y(T)-6 b(ait,)23 b(42,)h(49,)g(114)1522
3580 y(T)-6 b(ak)n(euti,)23 b(125)1522 3666 y(T)-6 b(arski,)24
b(4)1522 3752 y(tautology)1640 3839 y(linear)d(logic)h(\()p
FG(1)h FK(and)h FJ(>)p FK(\),)d(154)1522 3925 y(T)-6
b(a)n(ylor,)24 b(133)p eop
PStoPSsaved restore
%%Page: (178,179) 90
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 -15.874016 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
/showpage{}def/copypage{}def/erasepage{}def
PStoPSxform concat
175 178 bop 167 167 a Fz(INDEX)2173 b FK(175)167 404
y(tensor)23 b(pro)r(duct)h(\()p FJ(\012)p FK(\))284 491
y(coherence)f(space,)g(104,)i(138,)f(146)284 577 y(linear)e(logic,)g
(152)167 663 y(tensor)h(sum)g(or)h(par)f(\()p Fc(O)p
FK(\))284 749 y(coherence)g(space,)g(104)284 835 y(linear)f(logic,)g
(152)167 922 y(terminal)f(ob)t(ject,)i(104)167 1008 y(terminating)e
(relation,)i FM(se)l(e)402 1093 y FK(normalisation)167
1179 y(terms)284 1265 y(in)g FG(F)p FK(,)g(82)284 1352
y(in)g FG(H)-12 b(A)475 1363 y Fb(2)507 1352 y FK(,)23
b(125)284 1438 y(in)g FG(T)p FK(,)g(68)284 1524 y FH(\025)p
FK(-calculus,)g(15)284 1610 y(ob)t(ject)g(language,)i(5)167
1696 y(theory)e(of)h(constructions,)f(116,)i(133)167
1782 y(tok)n(en,)f(56,)g(64,)g(137)167 1869 y(top)r(ological)f(space,)g
(55)167 1955 y(total)g(category)h(\(\006)640 1949 y FJ(\016)691
1955 y FK(\),)e(135,)j(137,)f(141,)h(146)167 2041 y(total)e(ob)t
(jects,)g(57,)h(149)167 2127 y(total)f(recursiv)n(e)f(function,)h(53,)h
(122)167 2213 y(trace)e(\()p FJ(T)-6 b FH(r)r FK(\),)24
b(62,)g(67,)g(144)284 2300 y(linear)e(\()p FJ(T)-6 b
FM(rlin)5 b FK(\),)23 b(100)167 2386 y(transp)r(osition)g(in)g(linear)f
(algebra,)h(101)167 2472 y(trees,)f(8,)i(47,)g(93)167
2558 y(true)284 2644 y(denotation)g(\()p FG(t)p FK(,)f
FJ(T)18 b FK(,)23 b Fq(T)p FK(\),)g FM(se)l(e)402 2730
y FK(b)r(o)r(oleans)284 2816 y(prop)r(osition)h(\()p
FJ(>)p FK(\),)d FM(se)l(e)29 b FK(tautology)167 2902
y(turb)r(o)24 b(cut)e(elimination,)f(160)167 2988 y(T)-6
b(uring,)23 b(122)167 3074 y(t)n(yp)r(e)g(v)l(ariables,)g(82)167
3161 y(t)n(yp)r(es,)h(15,)g(54,)g(67)167 3323 y FG(U)p
FK(nde\014ned)g(ob)t(ject)f(\()p Fn(?)p FK(\),)f(56,)i(96,)g(129,)402
3409 y(139,)g(146,)h(149)167 3495 y(uni\014cation,)e(113)167
3581 y(uniform)g(con)n(tin)n(uit)n(y)-6 b(,)23 b(55)167
3667 y(uniformit)n(y)f(of)i(\005)e(t)n(yp)r(es,)i(83,)g(132,)h(134,)402
3752 y(143)167 3838 y(units)e(\()p FG(0)p FK(,)f FJ(>)p
FK(,)g FG(1)h FK(and)h FJ(?)p FK(\),)e(104)167 3925 y(univ)n(ersal)h
(algebra,)g(66)1522 404 y(univ)n(ersal)g(domain,)g(133)1522
490 y(univ)n(ersal)g(program)h(\(T)-6 b(uring\),)23 b(122)1522
575 y(univ)n(ersal)g(quan)n(ti\014er,)g(5,)h(6)1640 660
y(cut)e(elimination,)f(108)1640 745 y(natural)i(deduction)1698
830 y FJ(8I)28 b FK(and)c FJ(8E)6 b FK(,)23 b(10)1640
915 y(sequen)n(t)g(calculus)1698 1000 y FJ(L8)f FK(and)i
FJ(R8)p FK(,)e(32)1522 1085 y(univ)n(ersal)h(quan)n(ti\014er)g
(\(second)h(order\),)1757 1170 y(82,)g(126)1640 1256
y FJ(8)1680 1230 y FD(2)1707 1256 y FJ(I)k FK(and)c FJ(8)1947
1230 y FD(2)1974 1256 y FJ(E)6 b FK(,)24 b(94,)g(125)1640
1341 y(reducibilit)n(y)-6 b(,)20 b(118)1640 1426 y(seman)n(tics,)i
(132,)i(143,)h(149)1522 1570 y FG(V)p FK(ariable)d(coherence)g(spaces,)
i(141)1522 1655 y(v)l(ariables)1640 1740 y(h)n(yp)r(otheses,)g(11,)g
(15,)g(19)1640 1825 y(ob)t(ject)e(language,)j(10,)f(125)1640
1910 y(t)n(yp)r(e,)f(82,)h(125)1522 1995 y(v)n(ery)g(\014nite,)e(59,)i
(66)1522 2080 y(Vic)n(k)n(ers,)e(55)1522 2224 y FG(W)p
FK(eak)i(normalisation,)e(24)1522 2309 y(w)n(eak)n(ening)1640
2394 y FJ(L)p Fq(W)h FK(and)h FJ(R)p Fq(W)q FK(,)f(29)1640
2479 y(linear)e(logic)h(\()p Fq(W)q FK(?\),)g(154)1522
2564 y(w)n(eb,)h(56,)h(135)1522 2650 y(wh)n(y)g(not)f(\(?\),)g(102,)h
(154)1522 2735 y(Winsk)n(el,)e(98,)i(133)1522 2879 y
Fq(X)p FK(,)f FJ(L)p Fq(X)p FK(,)f FJ(R)p Fq(X)h FK(\(exc)n(hange\),)g
(29,)h(153)1522 3022 y Fq(Y)f FK(\(\014xed)g(p)r(oin)n(t\),)g(72)1522
3166 y FG(Z)p FK(ero)1640 3251 y FG(0)p FK(,)f(unit)h(of)h
FJ(\010)p FK(,)f(154)1640 3336 y Fq(O)g FK(and)h FJ(O)r
FK(,)f FM(se)l(e)28 b FK(in)n(tegers)p eop
PStoPSsaved restore
userdict/PStoPSsaved save put
PStoPSmatrix setmatrix
819.779528 400.818898 translate
90 rotate
1.000000 dup scale
userdict/PStoPSmatrix matrix currentmatrix put
userdict/PStoPSclip{0 0 moveto
 595.000000 0 rlineto 0 842.000000 rlineto -595.000000 0 rlineto
 closepath}put initclip
PStoPSxform concat
showpage
PStoPSsaved restore
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF