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y(cut)3308 2021 y FA(\(1\))523 2183 y(The)32 b(cut)g(in)g(this)h(proof)
e(can)h(be)g(eliminated)f(by)h(reducing)e(it)j(to)f(one)g(of)g(the)g
(follo)n(wing)e(tw)o(o)523 2283 y(normalforms)1169 2460
y Fs(A)p 1250 2448 10 38 v 1259 2432 42 4 v 88 w(A)84
b(A)p 1545 2448 10 38 v 1555 2432 42 4 v 88 w(A)p 1169
2480 509 4 v 1209 2554 a(A;)14 b(A)p 1388 2542 10 38
v 1398 2525 42 4 v 88 w(A)p Fr(^)q Fs(A)1719 2497 y Fr(^)1774
2510 y Fq(R)p 1209 2590 430 4 v 1258 2663 a Fs(A)p 1339
2651 10 38 v 1348 2635 42 4 v 88 w(A)p Fr(^)q Fs(A)1679
2609 y(contr)1872 2621 y Fq(L)2005 2460 y Fs(A)p 2086
2448 10 38 v 2096 2432 42 4 v 89 w(A)83 b(A)p 2382 2448
10 38 v 2391 2432 42 4 v 88 w(A)p 2005 2480 509 4 v 2045
2554 a(A;)14 b(A)p 2225 2542 10 38 v 2234 2525 42 4 v
88 w(A)p Fr(^)q Fs(A)2555 2497 y Fr(^)2610 2510 y Fq(R)p
2045 2590 430 4 v 2094 2663 a Fs(A)p 2175 2651 10 38
v 2185 2635 42 4 v 89 w(A)p Fr(^)p Fs(A)2516 2609 y(contr)2709
2621 y Fq(L)p 1258 2683 1167 4 v 1509 2756 a Fs(A)p Fr(_)q
Fs(A)p 1707 2744 10 38 v 1717 2728 42 4 v 88 w(A)p Fr(^)q
Fs(A;)g(A)p Fr(^)p Fs(A)2466 2700 y Fr(_)2521 2712 y
Fq(L)p 1509 2792 665 4 v 1617 2866 a Fs(A)p Fr(_)q Fs(A)p
1816 2854 10 38 v 1825 2837 42 4 v 88 w(A)p Fr(^)q Fs(A)2215
2812 y(contr)2408 2824 y Fq(R)3308 2866 y FA(\(2\))1146
3103 y Fs(A)p 1227 3091 10 38 v 1236 3074 42 4 v 88 w(A)83
b(A)p 1522 3091 10 38 v 1532 3074 42 4 v 89 w(A)p 1146
3123 509 4 v 1167 3196 a(A)19 b Fr(_)f Fs(A)p 1402 3184
10 38 v 1412 3167 42 4 v 89 w(A;)c(A)1695 3140 y Fr(_)1751
3152 y Fq(L)p 1167 3232 467 4 v 1216 3306 a Fs(A)19 b
Fr(_)g Fs(A)p 1452 3294 10 38 v 1461 3277 42 4 v 88 w(A)1675
3251 y(contr)1868 3263 y Fq(R)2005 3103 y Fs(A)p 2086
3091 10 38 v 2096 3074 42 4 v 89 w(A)83 b(A)p 2382 3091
10 38 v 2391 3074 42 4 v 88 w(A)p 2005 3123 509 4 v 2026
3196 a(A)19 b Fr(_)g Fs(A)p 2262 3184 10 38 v 2271 3167
42 4 v 88 w(A;)14 b(A)2555 3140 y Fr(_)2610 3152 y Fq(L)p
2026 3232 467 4 v 2076 3306 a Fs(A)19 b Fr(_)g Fs(A)p
2311 3294 10 38 v 2321 3277 42 4 v 88 w(A)2534 3251 y(contr)2727
3263 y Fq(R)p 1216 3325 1227 4 v 1497 3399 a Fs(A)p Fr(_)q
Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1913 3387 10 38 v 1922 3370
42 4 v 88 w(A)p Fr(^)q Fs(A)2485 3342 y Fr(^)2540 3355
y Fq(R)p 1497 3435 665 4 v 1606 3508 a Fs(A)p Fr(_)p
Fs(A)p 1804 3496 10 38 v 1814 3480 42 4 v 89 w(A)p Fr(^)p
Fs(A)2203 3454 y(contr)2396 3466 y Fq(L)3308 3508 y FA(\(3\))523
3671 y(which)i(are)h(obtained,)e(respecti)n(v)o(ely)-5
b(,)15 b(by)h(either)g(permuting)f(the)i(cut)f(to)h(the)g(left)g(o)o(v)
o(er)f(the)g Fs(contr)3322 3683 y Fq(R)3377 3671 y FA(-)523
3770 y(rule)f(or)h(to)f(the)h(right)f(o)o(v)o(er)f(the)i
Fs(contr)1597 3782 y Fq(L)1647 3770 y FA(-rule.)e(Another)h(e)o(xample)
f(sho)n(wing)g(that)i(in)f(classical)i(logic)523 3870
y(one)j(can,)f(in)i(general,)e(reach)g(more)g(than)h(one)g(normalform)d
(is)k(Lafont')-5 b(s)20 b(proof)e([10,)i(P)o(age)f(151].)648
3973 y(In)26 b(light)h(of)f(the)h(absence)f(of)g(the)h(Church-Rosser)e
(property)g(for)h(cut-elimination)e(in)j(clas-)523 4072
y(sical)h(logic)e(and)g(in)h(light)g(of)f(the)h(w)o(ork)f(on)g
(double-ne)o(gation)d(translations,)i(there)i(seem)g(to)g(be)523
4172 y(ob)o(vious)k(questions:)g(What)i(is)g(the)g(correspondence)28
b(between)k(cut-elimination)e(in)j(classical)523 4272
y(logic)15 b(and)g(the)h(embeddings)d(of)i(classical)h(proofs)e(into)i
(intuitionistic)e(logic)i(via)f(double-ne)o(gation)523
4371 y(translations?)23 b(Since)h(cut-elimination)e(in)i
(intuitionistic)g(logic)f(is)i(Church-Rosser)m(,)3030
4341 y Fp(1)3085 4371 y FA(which)f(re-)523 4471 y(striction)e(is)h
(tacitly)f(enforced)f(by)g(a)i(double-ne)o(gation)18
b(translation)j(so)i(that)f(eliminating)f(cuts)h(in)523
4570 y(the)i(double-ne)o(gated)c(v)o(ersion)j(of)h(a)g(classical)h
(proof)e(leads)h(to)g(only)g(a)g(single)g(normalform?)d(Or)p
523 4654 473 4 v 558 4710 a Fo(1)606 4742 y Fx(As)j(mentioned)i
(earlier)f(we)f(ignore)h(what)g(we)g(belie)n(v)o(e)g(to)f(be)h
(super\002cial)g(v)n(ariations)g(between)g(dif)n(fer)o(-)606
4833 y(ent)c(normalforms)g(reachable)h(from)f(an)g(intuitionistic)f
(sequent-proof.)i(Ho)n(we)n(v)o(er)f(see)g([18])g(for)g(a)f(more)606
4924 y(thorough)h(analysis)e(of)g(this)g(aspect.)p eop
end
%%Page: 3 3
TeXDict begin 3 2 bop 523 448 a FA(more)24 b(concisely)f(ask)o(ed,)i
(do)f(double-ne)o(gation)c(translations)k(correspond)e(to)i(particular)
f(strate-)523 548 y(gies)e(of)g(ho)n(w)f(to)h(eliminate)g(cuts?)g(Does)
g(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(lead)g(to)h(the)g
(same)523 648 y(normalform)i(\(by)j Ft(same)g FA(we)h(mean)e
(corresponding)e(to)j(one)g(particular)f(normalform)e(obtained)523
747 y(by)j(cut-elimination)e(in)j(classical)g(logic\)?)f(If)g(not,)g
(then)g(can)g(one)g(\002nd)g(for)g(e)n(v)o(ery)f(normalform)523
847 y(of)h(a)g(classical)h(proof)d(a)j(corresponding)22
b(double-ne)o(gation)g(translation)j(that)h(will)h(produce)c(the)523
946 y(double-ne)o(gated)g(v)o(ersion)j(of)i(this)g(normalform\227that)c
(is,)k(can)f(e)n(v)o(ery)f(reduction)g(sequence)g(in)523
1046 y(classical)16 b(logic)f(be)h(simulated)f(by)g(a)h(\(probably)d
(carefully)h(chosen\))g(double)g(ne)o(gation)f(translation)523
1146 y(and)26 b(performing)e(cut-elimination)h(in)i(intuitionistic)f
(logic?)h(Are)g(there)f(an)o(y)g(double-ne)o(gation)523
1245 y(translations)20 b(that)g(lead)g(to)g(normalforms)e(that)i(ha)n
(v)o(e)g(no)f(equi)n(v)n(alent)g(amongst)g(the)h(normalforms)523
1345 y(reachable)i(by)h(cut-elimination)f(in)h(classical)h(logic?)f
(Can)h(one)f(characterise)f(someho)n(w)-5 b(,)22 b(which)523
1445 y(normalforms)16 b(can)i(be)g(reached)f(by)h(double-ne)o(gation)c
(translations)k(and)g(which)g(can)g(not?)g(In)g(this)523
1544 y(paper)h(we)i(conjecture)d(answers)i(for)g(all)h(these)f
(questions.)648 1654 y(Although)k(some)i(special)h(cases)g(seem)g(to)f
(be)h(answered)e(by)h(e)o(xisting)g(w)o(ork,)f(for)h(e)o(xample)523
1753 y([5,)13 b(6,)g(14],)21 b(we)i(are)g(una)o(w)o(are)e(of)h(an)o(y)g
(w)o(ork)g(that)g(treat)h(these)g(questions)e(in)i(full)f(generality)-5
b(.)21 b(The)523 1853 y(answers)27 b(we)g(shall)g(gi)n(v)o(e)f(to)h
(these)f(questions)g(are)h(a)g(lot)g(inspired)f(by)g(the)h(comments)e
(made)h(in)523 1952 y([6,)d(Sec.)g(7].)g(Ho)n(we)n(v)o(er)m(,)e(there)i
(only)g(one)g(half)g(of)g(the)g(correspondence)d(is)k(considered,)e
(namely)523 2052 y(ho)n(w)31 b(their)h(v)o(ersion)f(of)h(classical)g
(logic)g(and)f(cut-elimination)f(can)i(be)g(embedded)e(via)i(some)523
2152 y(speci\002c)21 b(double-ne)o(gation)c(translations)j(into)g
(intuitionistic)h(logic.)f(W)-7 b(e)22 b(conjecture)d(also)j(a)f(cor)n
(-)523 2251 y(respondence)i(in)i(the)g(other)f(direction,)g(namely)g
(that)h(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(and)523
2351 y(corresponding)12 b(reduction)i(sequences)h(can)g(be)g(simulated)
g(by)g(their)h(cut-elimination)d(procedure.)523 2451
y(Since)19 b(we)g(shall)h(use)f(as)h(\223point)e(of)g(reference\224)f
(a)j(more)e(general)g(cut-elimination)f(procedure)f(for)523
2550 y(cut-elimination)h(in)j(classical)g(logic)f(than)g(the)g(one)g
(described)f(in)h([6],)g(we)h(are)f(also)h(able)f(to)g(dra)o(w)523
2650 y(the)26 b(conclusion)e(that)h(gi)n(v)o(en)g(our)g(conjecture)f
(is)i(true,)f(then)g(double-ne)o(gation)d(translations)j(are)523
2749 y(not)20 b(enough)e(to)i(describe)g(the)g Ft(full)h
FA(computational)c(meaning)i(of)h(a)h(classical)g(proof.)648
2859 y(As)h(can)f(be)h(seen)g(to)f(answer)h(the)f(correspondence)d
(questions,)j(we)h(\002rst)g(ha)n(v)o(e)f(to)h(mak)o(e)f(pre-)523
2959 y(cise)28 b(what)e(we)i(mean)e(by)g(cut-elimination)f(in)i
(classical)h(logic.)e(Most)h(cut-elimination)e(proce-)523
3058 y(dures,)d(including)f(Gentzen')-5 b(s)23 b(original)f(one,)g
(only)g(terminate)g(if)h(a)h(particular)d(strate)o(gy)h(for)h(cut-)523
3158 y(elimination)e(is)i(emplo)o(yed.)c(Common)i(e)o(xamples)g(being)g
(an)h(innermost)e(reduction)g(strate)o(gy)-5 b(,)21 b(or)523
3257 y(the)d(elimination)g(of)g(the)g(cut)h(with)f(the)h(highest)f
(rank.)f(Using)h(those)g(cut-elimination)f(procedures)523
3357 y(we)29 b(cannot)f(characterise)f(what)i(the)f(set)i(of)e
Ft(all)h FA(normalforms)d(of)i(a)h(classical)h(proof)d(is\227the)o(y)
523 3457 y(w)o(ould)c(produce)e(only)i(one)g(or)h(a)g(limited)f(number)
f(of)h(normalforms.)e(W)-7 b(e)25 b(shall)f(therefore)d(base)523
3556 y(our)f(ar)o(guments)f(on)i(the)g(cut-elimination)e(procedure)f
(de)n(v)o(eloped)h(by)h(Urban)h(and)f(Bierman)h([20,)523
3656 y(21],)h(which)g(is)i(lik)o(e)g(Gentzen')-5 b(s)22
b(procedure)f(e)o(xcept)g(it)j(imposes)f(one)f(slight)h(restriction)f
(on)h(ho)n(w)523 3756 y(commuting)g(cuts)j(need)f(to)h(be)g(analysed.)e
(Since)i(this)g(cut-elimination)d(procedure)g(is)k(strongly)523
3855 y(normalising,)g(we)i(can)g(calculate)f(all)h(cut-free)f
(normalforms)e(of)j(a)g(classical)g(proof.)e(Because)523
3955 y(this)19 b(procedure)d(is)k(not)f(Church-Rosser)m(,)d(the)j
(collection)f(of)g(normalforms)e(for)i(a)i(classical)f(proof)523
4054 y(contains)f(in)h(general)e(more)h(than)g(one)g(element\227as)g
(can)h(be)f(seen)h(for)f(e)o(xample)f(with)i(the)g(proofs)523
4154 y(\(2\))24 b(and)g(\(3\).)f(As)j(this)f(cut-elimination)d
(procedure)g(puts)j(only)e(v)o(ery)h(slight)g(restrictions)g(on)g(the)
523 4254 y(process)i(of)h(cut-elimination)d(we)j(belie)n(v)o(e)f(a)h
(good)e(case)i(can)g(be)g(made)f(that)g(the)h(collection)f(of)523
4353 y(normalforms)18 b(calculated)i(by)h(this)g(procedure)d(includes)i
(all)i(\223essential\224)f(normalforms.)3167 4323 y Fp(2)3218
4353 y FA(Ho)n(w-)523 4453 y(e)n(v)o(er)e(this)i(is)g(a)g(point)e(we)i
(shall)f(not)g(be)g(concerned)e(with)j(in)f(this)h(paper)-5
b(.)p 523 4563 473 4 v 558 4619 a Fo(2)606 4650 y Fx(Making)31
b(such)e(a)h(case)f(is)g(hopeless)h(for)f(other)h(strongly-normalising)
h(cut-elimination)e(procedures,)606 4742 y(lik)o(e)f(the)g(one)h(by)f
(Dragalin)g([7],)g(because)h(although)g(the)o(y)f(are)g
(strongly-normalising,)i(the)o(y)e(enforce)606 4833 y(quite)21
b(strong)g(restrictions)g(on)g(ho)n(w)h(cuts)f(can)g(be)g(eliminated.)g
(F)o(or)f(e)o(xample)i(Dragalin)e(does)i(not)f(allo)n(w)606
4924 y(\(multi\)cuts)e(to)f(permute)i(o)o(v)o(er)f(other)g
(\(multi\)cuts,)g(see)g([19].)p eop end
%%Page: 4 4
TeXDict begin 4 3 bop 648 448 a FA(The)30 b(cut-elimination)f
(procedure)f(of)j(Urban)f(and)g(Bierman)g(will)i(be)f(described)e(in)i
(more)523 548 y(detail)24 b(in)g(Sec.)h(3,)f(together)e(with)i(a)h(v)n
(ariant\227the)e(colour)f(protocol\227de)n(v)o(eloped)e(by)j(Danos)h
(et)523 648 y(al.)d([6,)12 b(12].)20 b(Beforehand,)e(ho)n(we)n(v)o(er)m
(,)f(we)k(present)f(some)g(preliminaries)f(about)g(double-ne)o(gation)
523 747 y(translations)28 b(in)h(Sec.)f(2.)h(W)-7 b(e)29
b(will)h(state)f(the)f(conjecture)f(in)i(Sec.)f(4,)h(gi)n(v)o(e)f(some)
g(e)n(vidence)f(on)523 847 y(why)e(this)h(conjecture)e(is)j(plausible)e
(and)g(present)g(some)g(ideas)h(on)f(ho)n(w)g(to)h(pro)o(v)o(e)e(it.)i
(In)g(Sec.)f(5)523 946 y(we)32 b(shall)g(dra)o(w)f(some)h(conclusions)e
(with)i(respect)g(to)g(the)g(computational)d(interpretation)h(of)523
1046 y(classical)21 b(proofs.)523 1300 y Fu(2)99 b(Pr)n(eliminaries)25
b(on)g(Double-Negation)h(T)-7 b(ranslations)523 1487
y FA(W)g(e)21 b(assume)e(the)h(reader)e(has)i(acquaintance)d(with)j
(sequent-calculus)d(formulations)g(of)j(classical)523
1586 y(and)e(intuitionistic)f(logic.)h(Because)g(there)g(e)o(xist)g
(sequents)g(that)h(are)f(pro)o(v)n(able)e(in)i(classical)h(logic,)523
1686 y(b)n(ut)25 b(unpro)o(v)n(able)d(in)j(intuitionistic)g(logic,)g
(the)g(interesting)f(point)g(of)h(double-ne)o(gation)c(transla-)523
1786 y(tions)g(is)i(that)e(one)g(can)g(embed)f(classical)j(logic)e
(into)g(intuitionistic)f(logic)h(so)h(that)g(pro)o(v)n(ability)d(is)523
1885 y(preserv)o(ed.)f(F)o(or)i(e)o(xample)f(the)h(follo)n(wing)e
(translation)i(de\002ned)f(o)o(v)o(er)g(formulae)1515
2075 y Fs(A)1577 2044 y Fn(\003)1639 2028 y Fp(def)1643
2075 y Fm(=)28 b Fr(::)p Fs(A)22 b FA(with)e Fs(A)h FA(being)e(atomic)
1390 2199 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1576 2169 y Fn(\003)1639
2152 y Fp(def)1643 2199 y Fm(=)28 b Fr(:)p Fm(\()p Fs(B)1890
2169 y Fn(\003)1929 2199 y Fm(\))1325 2324 y(\()p Fs(B)t
Fr(^)q Fs(C)6 b Fm(\))1577 2294 y Fn(\003)1639 2277 y
Fp(def)1643 2324 y Fm(=)28 b Fs(B)1803 2294 y Fn(\003)1841
2324 y Fr(^)q Fs(C)1962 2294 y Fn(\003)1316 2449 y Fm(\()p
Fs(B)t Fr(\033)p Fs(C)6 b Fm(\))1577 2419 y Fn(\003)1639
2402 y Fp(def)1643 2449 y Fm(=)28 b Fs(B)1803 2419 y
Fn(\003)1841 2449 y Fr(\033)p Fs(C)1971 2419 y Fn(\003)1325
2574 y Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))1577 2544
y Fn(\003)1639 2527 y Fp(def)1643 2574 y Fm(=)28 b Fr(:)p
Fm(\()p Fr(:)p Fm(\()p Fs(B)1977 2544 y Fn(\003)2017
2574 y Fm(\))p Fr(^:)p Fm(\()p Fs(C)2256 2544 y Fn(\003)2295
2574 y Fm(\)\))3308 2313 y FA(\(4\))523 2737 y(can)20
b(be)g(used)g(to)g(sho)n(w)g(that)h(e)n(v)o(ery)d(classical)k(proof)c
(with)i(the)h(end-sequent)1854 2904 y Fs(\000)p 1935
2892 10 38 v 1945 2876 42 4 v 100 w(\001)523 3072 y FA(can)f(be)g
(translated)g(to)g(an)g(intuitionistic)g(proof)e(with)j(the)f
(end-sequent)1747 3240 y Fs(\000)1810 3209 y Fn(\003)1848
3240 y Fs(;)14 b Fr(:)p Fs(\001)2009 3209 y Fn(\003)p
2066 3228 10 38 v 2075 3211 42 4 v 2158 3240 a Fs(:)1127
b FA(\(5\))523 3407 y(W)-7 b(e)28 b(use)e(the)h(con)m(v)o(ention)c
(that)j(if)h Fs(\000)39 b FA(is)27 b(the)f(sequent-conte)o(xt)e
Fr(f)p Fs(B)2502 3419 y Fl(1)2539 3407 y Fs(;)14 b(:)g(:)g(:)f(;)h(B)
2786 3419 y Fq(n)2831 3407 y Fr(g)27 b FA(then)f Fs(\000)3133
3377 y Fn(\003)3197 3407 y FA(stands)523 3507 y(for)f(the)g
(sequent-conte)o(xt)d Fr(f)p Fs(B)1432 3477 y Fn(\003)1428
3527 y Fl(1)1470 3507 y Fs(;)14 b(:)g(:)g(:)f(;)h(B)1721
3477 y Fn(\003)1717 3527 y Fq(n)1763 3507 y Fr(g)p FA(.)25
b(Similarly)f(for)h Fr(:)p Fs(\001)2432 3477 y Fn(\003)2471
3507 y FA(.)g(W)-7 b(e)27 b(shall)e(also)h(use)f(the)g(con-)523
3606 y(v)o(ention)17 b(that)h Fs(A)h FA(stands)f(for)f(an)h(atomic)g
(formula)f(and)g Fs(B)t(;)d(:)g(:)g(:)19 b FA(for)e(arbitrary)g
(formulae.)f(A)i(similar)523 3706 y(embedding)g(can)i(be)g(obtained)f
(with)h(the)g(translation:)1729 3895 y Fs(A)1791 3865
y Fn(\016)1852 3848 y Fp(def)1857 3895 y Fm(=)28 b Fr(::)p
Fs(A)1604 4020 y Fm(\()p Fr(:)p Fs(B)t Fm(\))1790 3990
y Fn(\016)1852 3973 y Fp(def)1857 4020 y Fm(=)g Fr(:)p
Fm(\()p Fs(B)2104 3990 y Fn(\016)2143 4020 y Fm(\))1539
4145 y(\()p Fs(B)t Fr(^)p Fs(C)6 b Fm(\))1790 4115 y
Fn(\016)1852 4098 y Fp(def)1857 4145 y Fm(=)28 b Fr(::)p
Fm(\()p Fs(B)2159 4115 y Fn(\016)2198 4145 y Fr(^)q Fs(C)2319
4115 y Fn(\016)2357 4145 y Fm(\))1529 4270 y(\()p Fs(B)t
Fr(\033)p Fs(C)6 b Fm(\))1790 4240 y Fn(\016)1852 4223
y Fp(def)1857 4270 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159
4240 y Fn(\016)2198 4270 y Fr(\033)p Fs(C)2328 4240 y
Fn(\016)2366 4270 y Fm(\))1539 4395 y(\()p Fs(B)t Fr(_)p
Fs(C)6 b Fm(\))1790 4365 y Fn(\016)1852 4348 y Fp(def)1857
4395 y Fm(=)28 b Fr(::)p Fm(\()p Fs(B)2159 4365 y Fn(\016)2198
4395 y Fr(_)q Fs(C)2319 4365 y Fn(\016)2357 4395 y Fm(\))3308
4133 y FA(\(6\))648 4558 y(The)i(usual)h(proof)e(\(see)i(for)g(e)o
(xample)e([4]\))h(for)h(establishing)f(that)h(e)n(v)o(ery)e(classical)j
(proof)523 4657 y(can)21 b(be)g(translated)f(to)i(an)f(intuitionistic)f
(proof)g(proceeds)f(inducti)n(v)o(ely)g(by)i(translating)f(stepwise)523
4757 y(e)n(v)o(ery)f(inference)f(rule)i(in)h(a)f(proof.)f(F)o(or)g
(instance)h(an)g(axiom)g(of)g(the)g(form)1858 4924 y
Fs(A)p 1938 4912 10 38 v 1948 4896 42 4 v 88 w(A)p eop
end
%%Page: 5 5
TeXDict begin 5 4 bop 523 448 a FA(is)21 b(translated)f(by)f
Fm(\()p Fr(\000)p Fm(\))1175 418 y Fn(\003)1235 448 y
FA(to)h(the)g(proof)1674 603 y Fr(::)p Fs(A)p 1865 591
10 38 v 1875 574 42 4 v 89 w Fr(::)p Fs(A)p 1628 623
527 4 v 1628 696 a Fr(::)p Fs(A;)14 b Fr(:::)p Fs(A)p
2084 684 10 38 v 2094 668 42 4 v 2195 635 a Fr(:)2250
647 y Fq(L)523 865 y FA(which)i(conforms)g(with)h(the)g(desired)f
(property)f(stated)i(in)g(\(5\).)f(When)h(translating)f(a)i(proof)d
(ending)523 964 y(with)20 b(an)h Fr(_)846 976 y Fq(L)896
964 y FA(-rule)1648 1021 y Fm(:)1466 1094 y Fs(B)t(;)14
b(\000)1621 1106 y Fl(1)p 1676 1082 10 38 v 1686 1066
42 4 v 1746 1094 a Fs(\001)1815 1106 y Fl(1)2114 1021
y Fm(:)1935 1094 y Fs(C)q(;)g(\000)2083 1106 y Fl(2)p
2140 1082 10 38 v 2149 1066 42 4 v 2209 1094 a Fs(\001)2278
1106 y Fl(2)p 1466 1130 851 4 v 1505 1204 a Fs(B)t Fr(_)q
Fs(C)q(;)g(\000)1776 1216 y Fl(1)1813 1204 y Fs(;)g(\000)1901
1216 y Fl(2)p 1957 1192 10 38 v 1967 1175 42 4 v 2027
1204 a Fs(\001)2096 1216 y Fl(1)2133 1204 y Fs(;)g(\001)2239
1216 y Fl(2)2357 1147 y Fr(_)2413 1160 y Fq(L)3308 1204
y FA(\(7\))523 1345 y(then)20 b(by)g(induction)e(hypothesis)g(we)j(ha)n
(v)o(e)f(tw)o(o)g(intuitionistic)g(proofs)f(ending)f(with)1522
1476 y Fm(:)1268 1549 y Fs(B)1335 1519 y Fn(\003)1374
1549 y Fs(;)c(\000)1474 1519 y Fn(\003)1462 1570 y Fl(1)1511
1549 y Fs(;)g Fr(:)p Fs(\001)1672 1519 y Fn(\003)1672
1570 y Fl(1)p 1729 1537 10 38 v 1739 1521 42 4 v 1882
1549 a FA(and)2338 1476 y Fm(:)2085 1549 y Fs(C)2150
1519 y Fn(\003)2188 1549 y Fs(;)g(\000)2288 1519 y Fn(\003)2276
1570 y Fl(2)2326 1549 y Fs(;)g Fr(:)p Fs(\001)2487 1519
y Fn(\003)2487 1570 y Fl(2)p 2544 1537 10 38 v 2554 1521
42 4 v 2637 1549 a Fs(:)523 1718 y FA(W)-7 b(e)21 b(can)f(then)g(form)f
(the)h(intuitionistic)g(proof)1486 1849 y Fm(:)1232 1923
y Fs(B)1299 1892 y Fn(\003)1338 1923 y Fs(;)14 b(\000)1438
1892 y Fn(\003)1426 1943 y Fl(1)1475 1923 y Fs(;)g Fr(:)p
Fs(\001)1636 1892 y Fn(\003)1636 1943 y Fl(1)p 1693 1911
10 38 v 1703 1894 42 4 v 1223 1963 549 4 v 1223 2037
a Fs(\000)1286 2007 y Fn(\003)1274 2058 y Fl(1)1324 2037
y Fs(;)g Fr(:)p Fs(\001)1485 2007 y Fn(\003)1485 2058
y Fl(1)p 1542 2025 10 38 v 1551 2008 42 4 v 1611 2037
a Fr(:)p Fs(B)1733 2007 y Fn(\003)1814 1975 y Fr(:)1869
1987 y Fq(R)2268 1849 y Fm(:)2016 1923 y Fs(C)2081 1892
y Fn(\003)2119 1923 y Fs(;)g(\000)2219 1892 y Fn(\003)2207
1943 y Fl(2)2257 1923 y Fs(;)g Fr(:)p Fs(\001)2418 1892
y Fn(\003)2418 1943 y Fl(2)p 2475 1911 10 38 v 2484 1894
42 4 v 2006 1963 548 4 v 2006 2037 a Fs(\000)2069 2007
y Fn(\003)2057 2058 y Fl(2)2107 2037 y Fs(;)g Fr(:)p
Fs(\001)2268 2007 y Fn(\003)2268 2058 y Fl(2)p 2325 2025
10 38 v 2335 2008 42 4 v 2395 2037 a Fr(:)p Fs(C)2515
2007 y Fn(\003)2595 1975 y Fr(:)2650 1987 y Fq(R)p 1223
2077 1331 4 v 1338 2151 a Fs(\000)1401 2121 y Fn(\003)1389
2172 y Fl(1)1439 2151 y Fs(;)g(\000)1539 2121 y Fn(\003)1527
2172 y Fl(2)1577 2151 y Fs(;)g Fr(:)p Fs(\001)1738 2121
y Fn(\003)1738 2172 y Fl(1)1776 2151 y Fs(;)g Fr(:)p
Fs(\001)1937 2121 y Fn(\003)1937 2172 y Fl(2)p 1994 2139
10 38 v 2004 2123 42 4 v 2064 2151 a Fr(:)p Fs(B)2186
2121 y Fn(\003)2224 2151 y Fr(^)q(:)p Fs(C)2400 2121
y Fn(\003)2595 2094 y Fr(^)2650 2107 y Fq(R)p 1260 2192
1258 4 v 1260 2271 a Fr(:)p Fm(\()p Fr(:)p Fs(B)1469
2240 y Fn(\003)1508 2271 y Fr(^:)p Fs(C)1683 2240 y Fn(\003)1722
2271 y Fm(\))p Fs(;)g(\000)1854 2240 y Fn(\003)1842 2291
y Fl(1)1892 2271 y Fs(;)g(\000)1992 2240 y Fn(\003)1980
2291 y Fl(2)2030 2271 y Fs(;)g Fr(:)p Fs(\001)2191 2240
y Fn(\003)2191 2291 y Fl(1)2229 2271 y Fs(;)g Fr(:)p
Fs(\001)2390 2240 y Fn(\003)2390 2291 y Fl(2)p 2447 2259
10 38 v 2457 2242 42 4 v 2558 2203 a Fr(:)2613 2215 y
Fq(L)3308 2271 y FA(\(8\))523 2439 y(as)32 b(the)f(translation)g(of)g
(\(7\).)f(F)o(or)h(the)g(sak)o(e)g(of)g(more)g(clarity)g(we)g(will)h
(omit)f(in)h(what)f(follo)n(ws)523 2539 y(the)22 b(sequent-conte)o(xts)
d(whene)n(v)o(er)h(the)o(y)h(are)h(unimportant.)d(Thus)j(we)g(shall)g
(gi)n(v)o(e)f(for)h(the)f(proof-)523 2638 y(fragment)e(sho)n(wn)g(in)h
(\(8\))g(only)f(the)h(follo)n(wing)f(simpli\002ed)h(inference)e(rules:)
1636 2769 y Fm(:)1551 2843 y Fs(B)1618 2813 y Fn(\003)p
1675 2831 10 38 v 1684 2814 42 4 v 1523 2863 249 4 v
1542 2925 10 38 v 1551 2908 42 4 v 1611 2937 a Fr(:)p
Fs(B)1733 2907 y Fn(\003)1814 2875 y Fr(:)1869 2887 y
Fq(R)2118 2769 y Fm(:)2034 2843 y Fs(C)2099 2813 y Fn(\003)p
2156 2831 10 38 v 2166 2814 42 4 v 2006 2863 247 4 v
2025 2925 10 38 v 2034 2908 42 4 v 2094 2937 a Fr(:)p
Fs(C)2214 2907 y Fn(\003)2295 2875 y Fr(:)2350 2887 y
Fq(R)p 1523 2957 730 4 v 1675 3018 10 38 v 1685 3002
42 4 v 1745 3030 a Fr(:)p Fs(B)1867 3000 y Fn(\003)1906
3030 y Fr(^:)p Fs(C)2081 3000 y Fn(\003)2295 2974 y Fr(^)2350
2986 y Fq(R)p 1597 3050 583 4 v 1597 3129 a Fr(:)p Fm(\()p
Fr(:)p Fs(B)1806 3099 y Fn(\003)1845 3129 y Fr(^)q(:)p
Fs(C)2021 3099 y Fn(\003)2059 3129 y Fm(\))p 2110 3117
10 38 v 2120 3101 42 4 v 2221 3062 a Fr(:)2276 3074 y
Fq(L)523 3298 y FA(When)i(translating)f(a)i(classical)g(proof)e(ending)
f(with)j(an)f Fr(^)2243 3310 y Fq(R)2298 3298 y FA(-rule)1759
3429 y Fm(:)p 1711 3490 10 38 v 1721 3473 42 4 v 1781
3502 a Fs(B)1996 3429 y Fm(:)p 1949 3490 10 38 v 1959
3473 42 4 v 73 x Fs(C)p 1693 3522 392 4 v 1769 3583 10
38 v 1778 3567 42 4 v 1838 3595 a(B)t Fr(^)q Fs(C)2126
3539 y Fr(^)2181 3551 y Fq(R)3308 3595 y FA(\(9\))523
3764 y(we)h(ha)n(v)o(e)e(by)h(induction)e(hypothesis)h(tw)o(o)h
(intuitionistic)g(proofs)f(ending)g(in)1663 3895 y Fm(:)1550
3969 y Fr(:)p Fs(B)1672 3938 y Fn(\003)p 1729 3957 10
38 v 1739 3940 42 4 v 1882 3969 a FA(and)2197 3895 y
Fm(:)2085 3969 y Fr(:)p Fs(C)2205 3938 y Fn(\003)p 2262
3957 10 38 v 2272 3940 42 4 v 2355 3969 a Fs(:)523 4137
y FA(In)g(order)f(to)i(form)e(an)h(intuitionistic)g(proof)f(ending)g
(with)h(the)g(sequent)g Fr(:)p Fm(\()p Fs(B)2773 4107
y Fn(\003)2812 4137 y Fr(^)p Fs(C)2932 4107 y Fn(\003)2971
4137 y Fm(\))p 3021 4125 10 38 v 3031 4109 42 4 v 88
w FA(,)g(we)h(need)523 4237 y(to)i(e)o(xploit)f(the)h(property)e(of)i
Fm(\()p Fr(\000)p Fm(\))1511 4207 y Fn(\003)1572 4237
y FA(that)g(one)g(can)g(al)o(w)o(ays)g(pro)o(v)o(e)e
(intuitionistically)h(the)h(sequent)523 4336 y Fr(::)p
Fm(\()p Fr(\000)p Fm(\))762 4306 y Fn(\003)p 819 4324
10 38 v 829 4308 42 4 v 889 4336 a Fm(\()p Fr(\000)p
Fm(\))1018 4306 y Fn(\003)1056 4336 y FA(.)g(F)o(or)f(e)o(xample)f(in)h
(the)g(atomic)g(case)h(one)e(has)i(for)e Fr(::)p Fs(A)2723
4306 y Fn(\003)p 2781 4324 10 38 v 2790 4308 42 4 v 2850
4336 a Fs(A)2912 4306 y Fn(\003)2973 4336 y FA(the)h(intuition-)523
4436 y(istic)g(proof:)1727 4519 y Fr(:)p Fs(A)p 1863
4507 10 38 v 1872 4490 42 4 v 88 w Fr(:)p Fs(A)p 1681
4539 416 4 v 1681 4612 a Fr(:)p Fs(A;)14 b Fr(::)p Fs(A)p
2027 4600 10 38 v 2036 4584 42 4 v 2138 4551 a Fr(:)2193
4563 y Fq(L)p 1671 4648 434 4 v 1671 4722 a Fr(:)p Fs(A)p
1807 4710 10 38 v 1817 4693 42 4 v 89 w Fr(:::)p Fs(A)2147
4660 y Fr(:)2202 4672 y Fq(R)p 1625 4742 527 4 v 1625
4815 a Fr(::::)p Fs(A;)g Fr(:)p Fs(A)p 2082 4803 10 38
v 2091 4786 42 4 v 2193 4753 a Fr(:)2248 4765 y Fq(L)p
1616 4851 545 4 v 1616 4924 a Fr(::::)p Fs(A)p 1918 4912
10 38 v 1928 4896 42 4 v 90 w Fr(::)p Fs(A)2202 4863
y Fr(:)2257 4875 y Fq(R)p eop end
%%Page: 6 6
TeXDict begin 6 5 bop 523 448 a FA(Using)26 b(proofs)e(for)i
Fr(::)p Fs(B)1287 418 y Fn(\003)p 1344 436 10 38 v 1354
420 42 4 v 1414 448 a Fs(B)1481 418 y Fn(\003)1546 448
y FA(and)f Fr(::)p Fs(C)1867 418 y Fn(\003)p 1925 436
10 38 v 1934 420 42 4 v 1994 448 a Fs(C)2059 418 y Fn(\003)2097
448 y FA(,)i(we)f(can)g(construct)f(the)h(follo)n(wing)e(trans-)523
548 y(lated)c(proof)f(for)g(\(9\):)963 694 y Fm(:)850
768 y Fr(:)p Fs(B)972 738 y Fn(\003)p 1030 756 10 38
v 1039 739 42 4 v 823 788 304 4 v 841 850 10 38 v 851
833 42 4 v 911 862 a Fr(::)p Fs(B)1088 831 y Fn(\003)1168
799 y Fr(:)1223 811 y Fq(R)1554 788 y Fm(:)1361 862 y
Fr(::)p Fs(B)1538 831 y Fn(\003)p 1596 850 10 38 v 1605
833 42 4 v 1665 862 a Fs(B)1732 831 y Fn(\003)p 823 881
948 4 v 1218 943 10 38 v 1228 927 42 4 v 1288 955 a Fs(B)1355
925 y Fn(\003)1812 907 y Fs(cut)2148 694 y Fm(:)2036
768 y Fr(:)p Fs(C)2156 738 y Fn(\003)p 2213 756 10 38
v 2223 739 42 4 v 2008 788 303 4 v 2027 850 10 38 v 2036
833 42 4 v 2096 862 a Fr(::)p Fs(C)2271 831 y Fn(\003)2352
799 y Fr(:)2407 811 y Fq(R)2736 788 y Fm(:)2545 862 y
Fr(::)p Fs(C)2720 831 y Fn(\003)p 2777 850 10 38 v 2787
833 42 4 v 2847 862 a Fs(C)2912 831 y Fn(\003)p 2008
881 942 4 v 2402 943 10 38 v 2412 927 42 4 v 2472 955
a Fs(C)2537 925 y Fn(\003)2992 907 y Fs(cut)p 1200 975
1376 4 v 1730 1037 10 38 v 1739 1020 42 4 v 1799 1049
a(B)1866 1019 y Fn(\003)1905 1049 y Fr(^)p Fs(C)2025
1019 y Fn(\003)2616 992 y Fr(^)2672 1004 y Fq(R)p 1651
1069 472 4 v 1651 1148 a Fr(:)p Fm(\()p Fs(B)1805 1118
y Fn(\003)1844 1148 y Fr(^)q Fs(C)1965 1118 y Fn(\003)2003
1148 y Fm(\))p 2054 1136 10 38 v 2063 1119 42 4 v 2165
1081 a Fr(:)2220 1093 y Fq(L)523 1332 y FA(W)-7 b(e)31
b(shall)f(refer)f(to)h(the)f(cuts)h(introduced)e(by)h(the)h(double-ne)o
(gation)25 b(translation)k(as)h Ft(auxiliary)523 1432
y(cuts)p FA(.)25 b(F)o(or)f(a)h(number)e(of)h(reasons)g(\(one)g(of)g
(them)g(being)g(to)g(minimise)g(the)h(amount)e(of)h(writing\))523
1531 y(we)d(shall)f(use)h(a)f(ne)n(w)g(inference)f(rule,)g(namely)p
1746 1686 10 38 v 1756 1670 42 4 v 1816 1698 a Fr(::)p
Fs(B)p 1728 1718 266 4 v 1802 1780 10 38 v 1811 1763
42 4 v 1871 1792 a(B)2035 1730 y Fr(::)2145 1742 y Fq(R)523
1976 y FA(to)h(stand)g(for)g(auxiliary)f(cuts,)h(which)g(ha)n(v)o(e)f
(al)o(w)o(ays)i(the)f(form:)1644 2122 y Fm(:)1491 2195
y Fs(\000)p 1572 2183 10 38 v 1582 2167 42 4 v 100 w
Fr(::)p Fs(B)2058 2122 y Fm(:)1903 2195 y Fr(::)p Fs(B)p
2099 2183 10 38 v 2109 2167 42 4 v 92 w(B)p 1491 2215
745 4 v 1755 2289 a(\000)p 1836 2277 10 38 v 1845 2260
42 4 v 99 w(B)2277 2241 y(cut)j(:)523 2473 y FA(Clearly)-5
b(,)28 b(this)h(ne)n(w)g(rule)g(does)f(not)g(af)n(fect)h(the)g(pro)o(v)
n(ability)d(of)i(sequents.)g(Issues)i(whether)d(the)523
2573 y Fr(::)633 2585 y Fq(R)688 2573 y FA(-rule)22 b(\(or)g(an)g
(auxiliary)f(cut\))i(af)n(fects)f(the)g(beha)n(viour)f(under)g
(cut-elimination)f(are)j(delayed)523 2672 y(until)f(Sec.)g(4.)g(W)m
(ith)g(this)g(ne)n(w)g(inference)e(rule)i(we)g(can)g(gi)n(v)o(e)f(the)g
(translation)g(of)h(\(9\))f(more)g(com-)523 2772 y(pactly)f(as:)1553
2836 y Fm(:)1440 2909 y Fr(:)p Fs(B)1562 2879 y Fn(\003)p
1620 2897 10 38 v 1629 2881 42 4 v 1413 2929 304 4 v
1431 2991 10 38 v 1441 2974 42 4 v 1501 3003 a Fr(::)p
Fs(B)1678 2973 y Fn(\003)1758 2941 y Fr(:)1813 2953 y
Fq(R)p 1413 3023 304 4 v 1487 3085 10 38 v 1496 3068
42 4 v 1556 3097 a Fs(B)1623 3067 y Fn(\003)1758 3035
y Fr(::)1868 3047 y Fq(R)2146 2836 y Fm(:)2034 2909 y
Fr(:)p Fs(C)2154 2879 y Fn(\003)p 2211 2897 10 38 v 2221
2881 42 4 v 2006 2929 303 4 v 2025 2991 10 38 v 2034
2974 42 4 v 2094 3003 a Fr(::)p Fs(C)2269 2973 y Fn(\003)2350
2941 y Fr(:)2405 2953 y Fq(R)p 2006 3023 303 4 v 2080
3085 10 38 v 2090 3068 42 4 v 2150 3097 a Fs(C)2215 3067
y Fn(\003)2350 3035 y Fr(::)2460 3047 y Fq(R)p 1468 3117
786 4 v 1703 3178 10 38 v 1713 3162 42 4 v 1773 3190
a Fs(B)1840 3160 y Fn(\003)1878 3190 y Fr(^)p Fs(C)1998
3160 y Fn(\003)2295 3134 y Fr(^)2350 3146 y Fq(R)p 1625
3210 472 4 v 1625 3289 a Fr(:)p Fm(\()p Fs(B)1779 3259
y Fn(\003)1818 3289 y Fr(^)p Fs(C)1938 3259 y Fn(\003)1976
3289 y Fm(\))p 2027 3277 10 38 v 2037 3261 42 4 v 2138
3222 a Fr(:)2193 3234 y Fq(L)523 3441 y FA(The)h(translations)f(for)g
(the)h(rules)f Fs(contr)1694 3453 y Fq(L)1745 3441 y
FA(,)h Fs(contr)1980 3453 y Fq(R)2035 3441 y FA(,)g Fs(w)r(eak)2264
3453 y Fq(L)2314 3441 y FA(,)g Fs(w)r(eak)2543 3453 y
Fq(R)2598 3441 y FA(,)g Fr(:)2695 3453 y Fq(L)2745 3441
y FA(,)g Fr(:)2842 3453 y Fq(R)2897 3441 y FA(,)g Fr(_)2995
3453 y Fq(R)3045 3461 y Fk(i)3075 3441 y FA(,)g Fr(^)3173
3453 y Fq(L)3219 3461 y Fk(i)3249 3441 y FA(,)g Fr(\033)3355
3453 y Fq(L)523 3541 y FA(and)f Fr(\033)728 3553 y Fq(R)803
3541 y FA(are)h(left)f(as)h(e)o(x)o(ercises)e(to)i(the)f(reader)-5
b(.)648 3641 y(W)e(e)21 b(can)f(no)n(w)g(gi)n(v)o(e)f(the)h(double-ne)o
(gation)c(translations)j(of)h(the)g(tw)o(o)h(subproofs)1033
3808 y Fs(A)p 1114 3796 10 38 v 1123 3779 42 4 v 88 w(A)84
b(A)p 1409 3796 10 38 v 1419 3779 42 4 v 88 w(A)p 1033
3828 509 4 v 1054 3901 a(A)19 b Fr(_)f Fs(A)p 1289 3889
10 38 v 1299 3872 42 4 v 89 w(A;)c(A)1583 3845 y Fr(_)1638
3857 y Fq(L)p 1054 3937 467 4 v 1104 4011 a Fs(A)k Fr(_)h
Fs(A)p 1339 3999 10 38 v 1348 3982 42 4 v 88 w(A)1562
3956 y(contr)1755 3968 y Fq(R)2142 3808 y Fs(A)p 2222
3796 10 38 v 2232 3779 42 4 v 88 w(A)83 b(A)p 2518 3796
10 38 v 2527 3779 42 4 v 88 w(A)p 2142 3828 509 4 v 2181
3901 a(A;)14 b(A)p 2361 3889 10 38 v 2370 3872 42 4 v
88 w(A)p Fr(^)q Fs(A)2691 3845 y Fr(^)2746 3857 y Fq(R)p
2181 3937 430 4 v 2231 4011 a Fs(A)p 2311 3999 10 38
v 2321 3982 42 4 v 88 w(A)p Fr(^)p Fs(A)2652 3956 y(contr)2845
3968 y Fq(L)523 4195 y FA(sho)n(wn)19 b(in)i(\(1\).)e(The)h
(translations)g(are:)574 4429 y Fz(::)p Fj(A)p 751 4417
9 34 v 760 4402 38 4 v 80 w Fz(::)p Fj(A)p 532 4449 486
4 v 532 4516 a Fz(::)p Fj(A;)13 b Fz(:::)p Fj(A)p 954
4504 9 34 v 963 4489 38 4 v 1058 4461 a Fz(:)1109 4469
y Fi(L)p 523 4551 503 4 v 523 4618 a Fz(:::)p Fj(A)p
751 4606 9 34 v 760 4591 38 4 v 80 w Fz(:::)p Fj(A)1067
4563 y Fz(:)1118 4571 y Fi(R)1294 4429 y Fz(::)p Fj(A)p
1471 4417 9 34 v 1480 4402 38 4 v 80 w Fz(::)p Fj(A)p
1252 4449 486 4 v 1252 4516 a Fz(::)p Fj(A;)g Fz(:::)p
Fj(A)p 1674 4504 9 34 v 1683 4489 38 4 v 1779 4461 a
Fz(:)1830 4469 y Fi(L)p 1243 4551 503 4 v 1243 4618 a
Fz(:::)p Fj(A)p 1471 4606 9 34 v 1480 4591 38 4 v 80
w Fz(:::)p Fj(A)1787 4563 y Fz(:)1838 4571 y Fi(R)p 523
4638 1223 4 v 629 4706 a Fz(:::)p Fj(A;)h Fz(:::)p Fj(A)p
1103 4694 9 34 v 1111 4679 38 4 v 80 w Fz(:::)p Fj(A)p
Fz(^:::)p Fj(A)1787 4655 y Fz(^)1838 4663 y Fi(R)p 557
4740 1156 4 v 557 4813 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p
Fz(^:::)p Fj(A)p Fh(\))p Fj(;)f Fz(:::)p Fj(A;)h Fz(:::)p
Fj(A)p 1649 4801 9 34 v 1657 4786 38 4 v 1753 4752 a
Fz(:)1804 4760 y Fi(L)p 557 4852 1156 4 v 679 4924 a
Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^)q(:::)p Fj(A)p Fh(\))p
Fj(;)f Fz(:::)p Fj(A)p 1526 4912 9 34 v 1535 4897 38
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%%Page: 7 7
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1044 394 4 v 1861 1098 10 38 v 1871 1081 42 4 v 2125
1070 a(cut)3267 1110 y FA(\(10\))523 1287 y(By)f(induction)e
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Fs(\031)2484 1257 y Fn(\003)2481 1307 y Fl(1)2540 1287
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w(\(remember)e(we)i(omit)f(the)h(sequent-)523 1873 y(conte)o(xts\),)i
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2925 y FA(and)e Fs(\031)714 2895 y Fn(\003)711 2948 y
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%%Page: 8 8
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2124 2181 774 4 v 2124 2253 a Fz(:)p Fh(\()p Fz(::)p
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2201 y Fi(R)p 798 2292 2100 4 v 1071 2364 a Fz(:)p Fh(\()p
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9 34 v 2095 2338 38 4 v 80 w Fz(:::)p Fj(A)p Fz(^:::)p
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4 v 999 2476 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p
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Fj(A)p Fh(\))p 2633 2464 9 34 v 2641 2449 38 4 v 2737
2415 a Fz(:)2788 2423 y Fi(L)p 999 2514 1697 4 v 1257
2587 a Fz(:)p Fh(\()p Fz(:::)p Fj(A)p Fz(^:::)p Fj(A)p
Fh(\))p Fj(;)h Fz(:)p Fh(\()p Fz(::)p Fj(A)p Fz(^::)p
Fj(A)p Fh(\))p 2375 2575 9 34 v 2383 2560 38 4 v 2737
2533 a Fj(contr)2916 2541 y Fi(L)p 3402 2639 4 2271 v
523 2642 2882 4 v 523 2731 a Fw(Fig)o(.)e(1.)36 b Fx(One)h(can)g
(obtain)g(the)f(\002rst)f(proof)i(by)g(double-ne)o(gation)i
(translation)d(of)h(\(1\))f(using)h(the)f(left-)523 2822
y(translation)27 b(for)g(the)g(cut,)g(and)g(then)h(eliminating)f(all)f
(cuts)h(including)h(the)f(auxiliary)g(cuts.)g(Equally)-5
b(,)27 b(one)523 2913 y(can)21 b(\002rst)f(reduce)i(\(1\))f(to)g
(\(2\),)f(double-ne)o(gate)j(translate)e(\(2\))f(and)i(then)f
(eliminate)g(all)f(cuts.)h(Similarly)f(with)523 3005
y(the)f(second)h(proof:)f(it)f(can)h(be)g(obtained)g(by)g
(right-translating)g(the)g(cut)g(in)f(\(1\))h(and)g(then)g(eliminate)g
(all)f(cuts;)523 3096 y(or)h(by)g(double-ne)o(gate)i(translate)e(\(3\))
f(and)i(then)f(eliminate)g(all)g(cuts.)523 3364 y FA(preserving)30
b(the)i(\223structure\224)f(of)h(the)g(classical)h(proof.)e(In)g(order)
g(to)i(achie)n(v)o(e)e(this)h(one)g(needs)523 3464 y(the)19
b(property)e(that)i Fr(::)p Fm(\()p Fr(\000)p Fm(\))1328
3434 y Fq(x)p 1389 3452 10 38 v 1398 3435 42 4 v 1458
3464 a Fm(\()p Fr(\000)p Fm(\))1587 3434 y Fq(x)1649
3464 y FA(is)g(intuitionistically)f(deri)n(v)n(able.)f(Ho)n(we)n(v)o
(er)m(,)g(this)i(lea)n(v)o(es)g(us)523 3564 y(with)26
b(man)o(y)e(possible)i(double)e(ne)o(gation)f(translations\227clearly)h
(the)i(ones)f(gi)n(v)o(en)f(by)i(Gentzen,)523 3663 y(G)7
b(\250)-35 b(odel)20 b(and)f(K)m(olmogoro)o(v)e(are)j(not)g(the)g(only)
g(ones)g(that)g(satisfy)h(these)f(constraints.)523 3909
y Fu(3)99 b(Cut-Elimination)26 b(and)f(Its)g(Colour)n(ed)h(V)-9
b(ariant)523 4089 y FA(Urban)19 b(and)h(Bierman)f(ha)n(v)o(e)h(sho)n
(wn)f(in)h([19,)12 b(21])19 b(that)i(only)e(a)h(small)h(restriction)e
(on)h(the)g(standard)523 4188 y(cut-elimination)30 b(procedure)f(for)i
(classical)i(and)e(intuitionistic)h(logic)f(is)i(suf)n(\002cient)e(to)h
(obtain)523 4288 y(strongly)f(normalising)g(proof-transformations.)c
(The)32 b Ft(lo)o(gical)g(cuts)p FA(,)g(also)h(sometimes)f(called)523
4387 y Ft(k)o(e)n(y-cuts)p FA(,)c(are)h(transformed)e(by)h(this)i
(cut-elimination)d(procedure)f(in)k(a)f(completely)f(standard)523
4487 y(f)o(ashion)19 b([10].)g(F)o(or)h(e)o(xample)f(the)h(logical)g
(cut)1459 4607 y Fs(\031)1506 4619 y Fl(1)1490 4672 y
Fm(:)p 1442 4733 10 38 v 1452 4717 42 4 v 1512 4745 a
Fs(B)1696 4607 y(\031)1743 4619 y Fl(2)1727 4672 y Fm(:)p
1680 4733 10 38 v 1690 4717 42 4 v 73 x Fs(C)p 1424 4765
392 4 v 1500 4827 10 38 v 1510 4810 42 4 v 1570 4839
a(B)t Fr(^)p Fs(C)1857 4782 y Fr(^)1912 4795 y Fq(R)2145
4607 y Fs(\031)2192 4619 y Fl(3)2176 4672 y Fm(:)2110
4745 y Fs(B)p 2195 4733 10 38 v 2205 4717 42 4 v 2049
4765 276 4 v 2049 4839 a(B)t Fr(^)q Fs(C)p 2256 4827
10 38 v 2265 4810 42 4 v 2367 4778 a Fr(^)2422 4790 y
Fq(L)2468 4798 y Fc(1)p 1482 4859 844 4 v 1878 4912 10
38 v 1887 4896 42 4 v 2367 4884 a Fs(cut)3267 4924 y
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622 10 38 v 1718 605 42 4 v 1778 634 a Fs(B)1963 496
y(\031)2010 508 y Fl(3)1994 560 y Fm(:)1928 634 y Fs(B)p
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707 10 38 v 1871 691 42 4 v 2125 679 a(cut)523 872 y
FA(and)i(so)h(on)f(for)f(the)i(other)e(connecti)n(v)o(es.)g(As)i
(before)e(we)i(con)m(v)o(eniently)c(ignore)i(all)i(matters)f(to)h(do)
523 972 y(with)e(ho)n(w)f(the)h(sequent-conte)o(xts)d(should)h(be)i
(adjusted.)f(A)h(logical)f(cut)h(can)f(be)h(characterised)e(as)523
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e(are)i Ft(fr)m(eshly)h(intr)l(oduced)g FA(by)f(logical)g(rules)523
1171 y(directly)19 b(abo)o(v)o(e)g(the)h(cut.)g(F)o(or)g(e)o(xample)f
(in)h(the)g(proof)1898 1317 y Fs(\031)1912 1370 y Fm(:)p
1846 1390 156 4 v 1864 1451 10 38 v 1874 1434 42 4 v
1934 1463 a Fs(B)2042 1408 y(r)523 1648 y FA(we)33 b(say)f(the)h
(formula)e Fs(B)37 b FA(is)c(freshly)f(introduced)e(if)i(it)h(is)h
(what)e(usually)g(is)h(called)f(the)h(main)523 1748 y(formula)24
b(of)h(the)g(logical)g(inference)f(rule)h Fs(r)r FA(.)i(Consequently)-5
b(,)23 b(a)j(logical)f(cut)g(is)h(a)g(cut)g(where)e(the)523
1847 y(cut-formula)16 b(is)j(freshly)f(introduced)e(in)i(the)h(tw)o(o)g
(immediate)e(subproofs)f(of)j(the)f(cut.)g(In)g(all)h(other)523
1947 y(cases)f(we)f(ha)n(v)o(e)g(a)g Ft(commuting)f(cut)p
FA(.)h(The)g(cut)g(in)g(\(1\),)f(for)h(e)o(xample,)e(is)k(a)e
(commuting)e(cut)i(because)523 2047 y(the)26 b(cut-formula)d
Fs(A)j FA(is)h(in)f(both)f(subproofs)e(introduced)h(by)h(a)h
(contraction-rule,)c(which)j(is)i(not)523 2146 y(considered)18
b(to)j(be)f(a)h(logical)e(inference)g(rule.)648 2247
y(Gentzen)d(introduced)f(proof-transformations)e(that)k(permute)f
(commuting)f(cuts)j(upw)o(ards)e(in)523 2346 y(a)25 b(stepwise)h(f)o
(ashion)e(only)g(by)h(re)n(writing)f(neighboring)e(inference)h(rules.)i
(In)g(contrast,)f(the)h(cut-)523 2446 y(elimination)c(procedure)e(of)j
(Urban)f(and)g(Bierman)h(contains)f(proof-transformations)c(that)22
b(push)523 2545 y(commuting)15 b(cuts)i(upw)o(ards)f(in)i(a)f(single)g
(\223big\224)g(step)g(to)n(w)o(ards)g(all)g(places)g(where)g(the)g
(cut-formula)523 2645 y(w)o(as)k(introduced.)d(Consider)h(the)h(follo)n
(wing)f(picture)1106 3842 y @beginspecial 134 @llx 377
@lly 592 @urx 666 @ury 2061 @rwi @clip @setspecial
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/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
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/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
x1 y1 lineto closepath } def
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
ifelse } def
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
pop pop Rect } { OvalFrame } ifelse } def
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
ifelse } def
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
def
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
y0 x0 2 mul x1 sub y1 ] def } def
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
w add def } repeat grestore gsave n 0 gt
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
% DG/SR modification end
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
exch t L stroke /i i h add def } repeat grestore } def
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
mul neg d } def
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
/Rot { CP CP translate 3 -1 roll neg rotate NET } def
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
dup a add ] cvx def } def
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
cvx def } def
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
h1 abs h sub dup s mul abs } ifelse } def
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
exch } def
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
} if } ifelse } def
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
/IfVisible true def } def
end
% END pstricks.pro
%%EndProcSet
%%BeginProcSet: pst-dots.pro 0 0
%!PS-Adobe-2.0
%%Title: Dot Font for PSTricks
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
%%Creation Date: May 7, 1993
%% Version 97 patch 1, 99/12/16
%% Modified by Etienne Riga <etienne.riga@skynet.be> - Dec. 16, 1999
%% to add /Diamond, /SolidDiamond and /BoldDiamond
10 dict dup begin
/FontType 3 def
/FontMatrix [ .001 0 0 .001 0 0 ] def
/FontBBox [ 0 0 0 0 ] def
/Encoding 256 array def
0 1 255 { Encoding exch /.notdef put } for
Encoding
dup (b) 0 get /Bullet put
dup (c) 0 get /Circle put
dup (C) 0 get /BoldCircle put
dup (u) 0 get /SolidTriangle put
dup (t) 0 get /Triangle put
dup (T) 0 get /BoldTriangle put
dup (r) 0 get /SolidSquare put
dup (s) 0 get /Square put
dup (S) 0 get /BoldSquare put
dup (q) 0 get /SolidPentagon put
dup (p) 0 get /Pentagon put
dup (P) 0 get /BoldPentagon put
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
dup (l) 0 get /SolidDiamond put
dup (d) 0 get /Diamond put
(D) 0 get /BoldDiamond put
% DG/SR modification end
/Metrics 13 dict def
Metrics begin
/Bullet 1000 def
/Circle 1000 def
/BoldCircle 1000 def
/SolidTriangle 1344 def
/Triangle 1344 def
/BoldTriangle 1344 def
/SolidSquare 886 def
/Square 886 def
/BoldSquare 886 def
/SolidPentagon 1093.2 def
/Pentagon 1093.2 def
/BoldPentagon 1093.2 def
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
/SolidDiamond 1008 def
/Diamond 1008 def
/BoldDiamond 1008 def
% DG/SR modification end
/.notdef 0 def
end
/BBoxes 13 dict def
BBoxes begin
/Circle { -550 -550 550 550 } def
/BoldCircle /Circle load def
/Bullet /Circle load def
/Triangle { -571.5 -330 571.5 660 } def
/BoldTriangle /Triangle load def
/SolidTriangle /Triangle load def
/Square { -450 -450 450 450 } def
/BoldSquare /Square load def
/SolidSquare /Square load def
/Pentagon { -546.6 -465 546.6 574.7 } def
/BoldPentagon /Pentagon load def
/SolidPentagon /Pentagon load def
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
/Diamond { -428.5 -742.5 428.5 742.5 } def
/BoldDiamond /Diamond load def
/SolidDiamond /Diamond load def
% DG/SR modification end
/.notdef { 0 0 0 0 } def
end
/CharProcs 20 dict def
CharProcs begin
/Adjust {
2 copy dtransform floor .5 add exch floor .5 add exch idtransform
3 -1 roll div 3 1 roll exch div exch scale
} def
/CirclePath { 0 0 500 0 360 arc closepath } def
/Bullet { 500 500 Adjust CirclePath fill } def
/Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath
eofill } def
/BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath
eofill } def
/BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
/TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto
closepath } def
/SolidTriangle { TrianglePath fill } def
/Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
/BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
/SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto
-450 -450 lineto closepath } def
/SolidSquare { SquarePath fill } def
/Square { SquarePath .89 .89 scale SquarePath eofill } def
/BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
/PentagonPath {
-337.8 -465 moveto
337.8 -465 lineto
546.6 177.6 lineto
0 574.7 lineto
-546.6 177.6 lineto
closepath
} def
/SolidPentagon { PentagonPath fill } def
/Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
/BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
/DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto
428.5 0 lineto closepath } def
/SolidDiamond { DiamondPath fill } def
/Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def
/BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def
% DG/SR modification end
/.notdef { } def
end
/BuildGlyph {
exch
begin
Metrics 1 index get exec 0
BBoxes 3 index get exec
setcachedevice
CharProcs begin load exec end
end
} def
/BuildChar {
1 index /Encoding get exch get
1 index /BuildGlyph get exec
} bind def
end
/PSTricksDotFont exch definefont pop
%END pst-dots.pro
%%EndProcSet
%%BeginProcSet: pst-node.pro 0 0
%!
% PostScript prologue for pst-node.tex.
% Version 97 patch 1, 97/05/09.
% For distribution, see pstricks.tex.
%
/tx@NodeDict 400 dict def tx@NodeDict begin
tx@Dict begin /T /translate load def end
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
grestore } def
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
} def
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Sin mul } def } def
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
dup Sin mul Cos Div } ifelse } def
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
/u ED /NodePos { GetRnodePos } def } def
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Cos mul exch Sin mul } def
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
} ifelse } def
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
sin def /NodePos { TriNodePos } def } def
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
hh mul Atan dup cos ww mul exch sin hh mul } def
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
{ 0 Dist Sin mul } ifelse } ifelse Do } def
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
XYPos } ifelse } def
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
roll sin mul sub exch } ifelse } def
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
AddOffset yA add /yA1 ED xA add /xA1 ED } def
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
AddOffset yB add /yB1 ED xB add /xB1 ED } def
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
} ifelse } def
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
} ifelse } def
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
} repeat LPutLine cleartomark } def
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
} def
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
/NAngle yyB yyA sub xxB xxA sub Atan def } def
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd
} def
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
} def
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
/NAngle yyB yyA sub xxB xxA sub Atan def } def
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
VPosEnd } def
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
def grestore exec /LPutVar /SaveLPutVar load def } def
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
LPutVar VPutLine } def LPutVar } def
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
end } def
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
ifelse } def
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
VPutLines } def } def
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
VPutLines } def } def
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
def
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
def
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
VPutLines } def } def
% DG/SR modification begin - May 9, 1997 - Patch 1
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
% DG/SR modification end
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Div def /y ED /x ED } def
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
tx@Dict begin false Polygon end } def
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
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3764 y Fs(A)2610 3599 y Fr(^)2665 3612 y Fq(R)p 2020
3792 538 4 v 2071 3901 a(\()2073 3974 y Fs(A)p 2179 3954
10 38 v 2188 3938 42 4 v 2274 3853 a Fn(\000)g(\000)g(\000)g(\000)g
Fq(*)2271 3901 y(\()2273 3974 y Fs(A)2361 3966 y Fr(^)2439
3901 y Fq(\()2441 3974 y Fs(A)2598 3812 y(contr)2791
3824 y Fq(L)p 1150 3994 1356 4 v 1529 4055 a(\()g Fn(\000)g(\000)g
(\000)g(\000)1527 4102 y Fq(\()1529 4176 y Fs(A)1616
4168 y Fr(_)1694 4102 y Fq(\()1696 4176 y Fs(A)p 1802
4156 10 38 v 1812 4139 42 4 v 1897 4055 a Fn(\000)g(\000)g(\000)g(\000)
g Fq(*)1895 4102 y(\()1897 4176 y Fs(A)1984 4168 y Fr(^)2062
4102 y Fq(\()2064 4176 y Fs(A)2547 4019 y(cut)523 4361
y FA(If)26 b(the)f(normalform)e(\(3\))i(is)i(to)f(be)f(reached,)g(then)
g(accordingly)e(we)j(need)f(to)h(orient)f(the)h(colour)523
4461 y(of)e(the)h(cut-formula)d(to)j(the)g(right.)f(Note)g(ho)n(we)n(v)
o(er)f(that)i(choosing)e(colours)g(has)i(nothing)e(to)i(do)523
4560 y(with)j(imposing)e(a)i(strate)o(gy)f(for)g(cut-elimination:)f(it)
i(is)h(not)e(cuts)h(that)g(are)f(selected)h(by)f(them,)523
4660 y(b)n(ut)f(rather)f(the)h(w)o(ay)h(ho)n(w)e(cuts)h(are)g(reduced.)
f(Important)f(for)h(our)g(discussion)h(is)h(the)f(f)o(act)g(that)p
523 4746 473 4 v 558 4801 a Fo(3)606 4833 y Fx(A)15 b(less)g(na)n(\250)
-23 b(\021v)o(e)16 b(method,)g(which)g(only)g(tries)f(out)g(all)g
(possible)h(reduction)g(for)g(outermost)g(cuts,)f(is)g(described)606
4924 y(in)k([19].)p eop end
%%Page: 11 11
TeXDict begin 11 10 bop 523 448 a FA(one)21 b(is,)i(ho)n(we)n(v)o(er)m
(,)c(not)i(completely)g(free)g(about)g(ho)n(w)g(to)h(annotate)f
(colours)f(to)i(a)g(sequent)f(proof.)523 548 y(In)k(f)o(act)g(once)g
(the)g(colour)f(`)p Fs(\()p FA(')h(is)h(chosen)e(for)h(the)g
(cut-formula)d Fs(A)k FA(in)g(\(1\),)e(all)i(occurrences)d(of)523
648 y Fs(A)j FA(must)g(ha)n(v)o(e)f(this)h(colour)-5
b(.)24 b(The)i(only)e(\223free\224)h(choices)g(in)h(this)g(proof)e(are)
h(the)h(colours)f(for)f(the)523 747 y(formulae)16 b Fs(A)p
Fr(_)q Fs(A)i FA(and)f Fs(A)p Fr(^)q Fs(A)p FA(\227for)g(them)g(we)h
(can)g(mak)o(e)f(an)o(y)g(choice,)g(b)n(ut)g(it)i(has)e(to)h(be)g
(consistent)523 847 y(throughout)k(the)j(proof.)f(Danos)h(et)h(al.)f
(state)h(this)g(consistenc)o(y)e(requirement)f(using)h(the)i(notion)523
946 y(of)j(an)g Ft(identity)f(class)i FA(in)f(a)h(proof)d(\(the)i
(follo)n(wing)e(de\002nition)h(is)i(slightly)f(adapted)f(from)g([16,)
523 1046 y(P)o(age)20 b(107]\):)523 1205 y Fb(De\002nition)g(1.)41
b Ft(Occurr)m(ences)18 b(of)g(\(sub\)formulae)e(in)j(a)f(pr)l(oof)g(ar)
m(e)g FA(identi\002ed)f Ft(whene)o(ver)h(the)n(y)f(ar)m(e)523
1304 y(the)j(corr)m(esponding)e(occurr)m(ences)i(of)g(the)g(same)h
(\(sub\)formula)d(in)581 1455 y Fr(\017)41 b Ft(the)20
b(two)h(formulae)f(in)g(an)g(axiom,)581 1551 y Fr(\017)41
b Ft(the)20 b(cut-formulae)f(in)h(a)h(cut)f(and)581 1648
y Fr(\017)41 b Ft(the)26 b(up)f(and)g(down)g(occurr)m(ences)g(of)h(a)f
(formula)h(in)f(an)h(infer)m(ence)f(rule)h(\(this)g(includes)e(the)664
1747 y(contr)o(acted)19 b(occurr)m(ences)g(in)i(contr)o(actions)d
(rules\).)523 1901 y(An)32 b FA(identity)f(class)i Ft(in)f(a)f(pr)l
(oof)h(is)h(the)f(r)m(e\003e)n(xive)o(,)f(symmetric)i(and)e(tr)o
(ansitive)h(closur)m(e)f(of)h(the)523 2000 y(identi\002cation)18
b(r)m(elation.)2086 b Fr(u)-55 b(t)523 2159 y FA(The)25
b(consistenc)o(y)e(requirement)g(can)i(then)f(be)h(stated)g(as)h(follo)
n(ws:)f(Whene)n(v)o(er)e(colours)h(are)h(an-)523 2259
y(notated)18 b(to)h(a)g(proof,)e(then)i(e)n(v)o(ery)e(formula)h(in)h
(an)g(identity)f(class)i(must)f(recei)n(v)o(e)f(the)h(same)g(colour)-5
b(.)648 2358 y(The)21 b(interesting)g(point)g(of)h(colour)n
(-annotations)d(is)j(the)g(f)o(act)h(that)f(the)o(y)f(determine)f
(uniquely)523 2458 y(a)27 b(normalform.)d(In)j(light)g(of)g(this,)g(it)
h(seems)f(reasonable)f(to)h(re)o(gard)e(as)j(the)f(collection)f(of)g
(nor)n(-)523 2557 y(malforms)17 b(reachable)f(from)h(a)i(classical)f
(proof)f(all)h(those)g(for)g(which)f(a)h(colour)f(annotation)f(e)o
(xists)523 2657 y(that)28 b(mak)o(es)h(them)e(reachable.)g(Then)h(the)g
(question)f(arises:)i(Can)g(we)f(\002nd)g(for)g(e)n(v)o(ery)f(normal-)
523 2757 y(form)17 b(reachable)f(by)i(the)g(\(un-coloured\))c
(cut-elimination)h(procedure)h(of)h(Urban)g(and)h(Bierman)f(a)523
2856 y(colour)n(-annotation)12 b(that)17 b(mak)o(es)f(them)g(reachable)
f(by)h(the)g(cut-elimination)e(procedure)g(of)i(Danos)523
2956 y(et)21 b(al.?)f(The)g(answer)g(is)h(no)f(and)f(for)h(deep)f
(reasons!)h(Consider)f(the)i(follo)n(wing)d(classical)j(proof)1213
3178 y Fm(\(1\))1254 3251 y(:)1042 3325 y Fs(A)p Fr(_)p
Fs(A)p 1240 3313 10 38 v 1250 3296 42 4 v 89 w(A)p Fr(^)p
Fs(A)1598 3106 y(A)p Fr(^)p Fs(A)p 1796 3094 10 38 v
1806 3078 42 4 v 89 w(A)p Fr(^)p Fs(A)84 b(A)p Fr(^)p
Fs(A)p 2327 3094 10 38 v 2336 3078 42 4 v 88 w(A)p Fr(^)q
Fs(A)p 1573 3126 1029 4 v 1573 3205 a(A)p Fr(^)p Fs(A;)14
b(A)p Fr(^)q Fs(A)p 1988 3193 10 38 v 1997 3176 42 4
v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p
Fs(A)p Fr(^)q Fs(A)p Fm(\))2643 3143 y Fr(^)2698 3156
y Fq(R)p 1573 3246 1029 4 v 1681 3325 a Fs(A)p Fr(^)q
Fs(A)p 1879 3313 10 38 v 1889 3296 42 4 v 88 w Fm(\()p
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
Fs(A)p Fm(\))2643 3265 y Fs(contr)2836 3277 y Fq(L)p
1042 3365 1452 4 v 1361 3444 a Fs(A)p Fr(_)q Fs(A)p 1560
3432 10 38 v 1569 3416 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q
Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2535
3391 y Fs(cut)3267 3444 y FA(\(12\))523 3611 y(where)k(we)g(cut)h(the)f
(proof)e(from)i(\(1\))f(against)h(a)g(proof)f(whose)h(cut-formula,)e
Fs(A)p Fr(^)p Fs(A)p FA(,)j(is)g(contracted)523 3711
y(in)k(the)g(right-subproof.)18 b(W)-7 b(e)24 b(can)e(reduce)g(the)g
(lo)n(wer)h(cut)f(so)h(that)g(we)g(obtain)f(tw)o(o)h(copies)f(of)g(the)
523 3810 y(proof)d(\(1\):)1503 3899 y Fm(\(1\))1545 3972
y(:)1332 4045 y Fs(A)p Fr(_)q Fs(A)p 1531 4033 10 38
v 1540 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)2034 3899 y Fm(\(1\))2075
3972 y(:)1863 4045 y Fs(A)p Fr(_)q Fs(A)p 2061 4033 10
38 v 2071 4017 42 4 v 88 w(A)p Fr(^)q Fs(A)p 1307 4065
1029 4 v 1307 4144 a(A)p Fr(_)q Fs(A;)14 b(A)p Fr(_)q
Fs(A)p 1722 4132 10 38 v 1732 4116 42 4 v 88 w Fm(\()p
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
Fs(A)p Fm(\))2378 4082 y Fr(^)2433 4095 y Fq(R)p 1307
4185 1029 4 v 1416 4264 a Fs(A)p Fr(_)p Fs(A)p 1614 4252
10 38 v 1623 4235 42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p
Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378
4204 y Fs(contr)2571 4216 y Fq(L)3267 4264 y FA(\(13\))523
4404 y(W)m(ithout)20 b(colours,)f(we)h(can)g(then)g(reduce)f(each)h
(cop)o(y)f(completely)g(independently)e(as)k(follo)n(ws:)1503
4560 y Fm(\(2\))1545 4633 y(:)1332 4706 y Fs(A)p Fr(_)q
Fs(A)p 1531 4694 10 38 v 1540 4677 42 4 v 88 w(A)p Fr(^)q
Fs(A)2034 4560 y Fm(\(3\))2075 4633 y(:)1863 4706 y Fs(A)p
Fr(_)q Fs(A)p 2061 4694 10 38 v 2071 4677 42 4 v 88 w(A)p
Fr(^)q Fs(A)p 1307 4726 1029 4 v 1307 4805 a(A)p Fr(_)q
Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1722 4793 10 38 v 1732 4776
42 4 v 88 w Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q
Fm(\()p Fs(A)p Fr(^)q Fs(A)p Fm(\))2378 4743 y Fr(^)2433
4755 y Fq(R)p 1307 4845 1029 4 v 1416 4924 a Fs(A)p Fr(_)p
Fs(A)p 1614 4912 10 38 v 1623 4896 42 4 v 88 w Fm(\()p
Fs(A)p Fr(^)q Fs(A)p Fm(\))p Fr(^)q Fm(\()p Fs(A)p Fr(^)q
Fs(A)p Fm(\))2378 4865 y Fs(contr)2571 4877 y Fq(L)3267
4924 y FA(\(14\))p eop end
%%Page: 12 12
TeXDict begin 12 11 bop 523 448 a FA(Such)24 b(a)g(beha)n(viour)e
(cannot)h(be)g(achie)n(v)o(ed)g(by)g(using)h(colours:)e(the)i(colours)f
(must)h(be)g(annotated)523 548 y(before)17 b(cut-elimination)f
(commences)g(and)i(is)h(in)m(v)n(ariant)d(under)h(cut-reductions.)e
(Consequently)-5 b(,)523 648 y(whene)n(v)o(er)22 b(a)i(cut)g(is)h
(duplicated)e(in)h(a)g(reduction)e(sequence)h(\(as)i(in)f(the)g
(reduction)e(\(12\))p Fr(!)p FA(\(13\)\),)523 747 y(the)29
b(colour)n(-annotation)d(pre)n(v)o(ents)h(both)i(instances)g(from)f
(reducing)g(dif)n(ferently)-5 b(.)26 b(\(The)j(deeper)523
847 y(reason)e(mentioned)g(earlier)g(is)j(that)e(one)f(just)i(cannot)e
(pre-determine)f(the)i(choices)f(in)i(a)f(com-)523 946
y(pletely)20 b(non-deterministic)d(reduction)i(system.\))648
1061 y(Comparing)26 b(the)j(cut-elimination)e(procedure)f(of)j(Urban)e
(and)i(Bierman)f(with)h(the)g(one)f(of)523 1161 y(Danos)16
b(et)i(al.,)f(tw)o(o)g(points)f(stand)g(out:)h(Both)g(cut-elimination)d
(procedures)h(are)i(strongly)e(normal-)523 1261 y(ising)685
1230 y Fp(4)742 1261 y FA(and)22 b(also)i(determine)e(a)h(collection)f
(of)h(normalforms)e(reachable)h(from)g(a)i(sequent-proof)523
1360 y(in)c(classical)g(logic.)f(As)h(sho)n(wn)f(by)g(e)o(xample,)f
(these)i(collections)f(contain)f(in)i(general)e(more)h(than)523
1460 y(one)g(element.)g(Also)i(as)f(sho)n(wn)f(by)h(e)o(xample,)e(the)i
(collection)f(determined)f(by)h(the)h(procedure)e(of)523
1559 y(Danos)k(et)g(al.)g(is)h(generally)e(a)h(proper)f(subset)h(of)f
(the)h(collection)f(determined)f(by)i(the)g(procedure)523
1659 y(of)f(Urban)g(and)g(Bierman.)g(The)g(colour)n(-annotations)d(in)k
(the)f(procedure)e(of)i(Danos)h(et)g(al.)g(cannot)523
1759 y(fully)e(account)f(for)g(the)h(non-determinism)d(present)j(in)g
(classical)h(logic.)648 1873 y(Both)f(cut-elimination)d(procedures)h
(can)i(also)g(be)g(used)f(for)h(reducing)e(intuitionistic)h(proofs.)523
1973 y(Because)k(of)g(the)h(restrictions)e(imposed)g(upon)g
(intuitionistic)h(sequents,)f(non-deterministic)f(re-)523
2073 y(duction)e(sequences)h(such)g(as)i(\(12\))p Fr(!)p
FA(\(13\))p Fr(!)p FA(\(14\))16 b(cannot)j(be)i(constructed.)d(But)k
(still)f(the)g(proce-)523 2172 y(dure)d(of)g(Urban)g(and)g(Bierman)g
(is)h Ft(not)h FA(Church-Rosser)d(in)i(the)f(intuitionistic)g(case,)h
(and)f(also)h(dif-)523 2272 y(ferent)f(colour)n(-annotations)d(of)j(an)
g(intuitionistic)g(proof)f(might)h(lead)g(to)h(dif)n(ferent)d
(normalforms.)523 2372 y(Ho)n(we)n(v)o(er)m(,)31 b(as)i(mentioned)e
(earlier)m(,)h(we)h(re)o(gard)e(the)i(dif)n(ferences)e(between)h(the)h
(normalforms)523 2471 y(reachable)19 b(from)g(an)h(intuitionistic)f
(sequent-proof)e(as)k(inessential)f(and)g(re)o(gard)e(cut-elimination)
523 2571 y(as)23 b(morally)e(Church-Rosser)-5 b(.)21
b(That)h(in)h(turn)e(means)h(that)h(in)f(the)g(intuitionistic)g(case)h
(there)f(is)h(no)523 2670 y(dif)n(ference)18 b(between)i(coloured)e
(and)h(un-coloured)e(cut-elimination\227at)h(least)j(morally)-5
b(.)523 3012 y Fu(4)99 b(Conjectur)n(e)523 3287 y FA(W)-7
b(e)19 b(ha)n(v)o(e)e(already)g(seen)h(that)f(by)h(translating)f(the)g
(cut)h(in)g(\(1\))f(using)g(a)h(left-)g(and)f(right-translation,)523
3387 y(we)k(can)f(simulate)g(the)h(reductions)e(\(1\))p
Fr(!)p FA(\(2\))f(and)i(\(1\))p Fr(!)p FA(\(3\))f(by)h(double-ne)o
(gations.)15 b(Ho)n(we)n(v)o(er)k(in)523 3486 y(general,)g(a)h(left-)g
(or)g(right-translation)e(of)i(a)g(cut)h(is)g(not)f(suf)n(\002cient)f
(to)h(simulate)h(all)f(cut-reduction)523 3586 y(sequences)f(in)i
(classical)g(logic.)e(Consider)h(the)g(follo)n(wing)f(instance)g(of)h
(a)h(logical)f(cut:)1520 3790 y Fs(\031)1567 3802 y Fl(1)1550
3855 y Fm(:)p 1503 3916 10 38 v 1512 3900 42 4 v 1572
3928 a Fs(B)p 1424 3948 276 4 v 1442 4010 10 38 v 1452
3993 42 4 v 1512 4022 a(B)t Fr(_)p Fs(C)1741 3961 y Fr(_)1797
3974 y Fq(R)1847 3982 y Fc(1)2001 3790 y Fs(\031)2048
3802 y Fl(2)2032 3855 y Fm(:)1966 3928 y Fs(B)p 2052
3916 10 38 v 2061 3900 42 4 v 2239 3790 a(\031)2286 3802
y Fl(3)2269 3855 y Fm(:)2204 3928 y Fs(C)p 2288 3916
10 38 v 2298 3900 42 4 v 1966 3948 392 4 v 2024 4022
a(B)t Fr(_)p Fs(C)p 2230 4010 10 38 v 2240 3993 42 4
v 2399 3965 a Fr(_)2454 3978 y Fq(L)p 1424 4042 876 4
v 1836 4095 10 38 v 1846 4079 42 4 v 2341 4067 a Fs(cut)3267
4107 y FA(\(15\))523 4318 y(W)-7 b(e)21 b(can)f(reduce)f(this)i(cut)f
(to)1725 4408 y Fs(\031)1772 4420 y Fl(1)1756 4473 y
Fm(:)p 1708 4534 10 38 v 1718 4518 42 4 v 1778 4546 a
Fs(B)1963 4408 y(\031)2010 4420 y Fl(2)1994 4473 y Fm(:)1928
4546 y Fs(B)p 2014 4534 10 38 v 2023 4518 42 4 v 1690
4566 394 4 v 1861 4620 10 38 v 1871 4603 42 4 v 2125
4592 a(cut)p 523 4746 473 4 v 558 4801 a Fo(4)606 4833
y Fx(Danos)29 b(et)g(al.)f(sho)n(wed)i(strong)f(normalisation)g(of)g
(their)f(cut-elimination)h(procedure)h(by)f(translating)606
4924 y(reduction)20 b(sequences)h(in)d(classical)h(logic)g(to)g
(reduction)h(sequences)h(of)e(proof-nets)g(in)g(linear)g(logic.)p
eop end
%%Page: 13 13
TeXDict begin 13 12 bop 523 448 a FA(and)28 b(assuming)g(that)h(the)f
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Fs(\031)2821 460 y Fl(1)2859 448 y FA(,)g(we)g(can)f(further)523
548 y(permute)19 b Fs(\031)863 560 y Fl(2)921 548 y FA(inside)h
Fs(\031)1187 560 y Fl(1)1225 548 y FA(.)g(This)h(beha)n(viour)d
(correspond)f(to)k(the)f(colour)n(-annotation)1508 786
y Fs(\031)1555 798 y Fl(1)1539 851 y Fm(:)p 1480 952
10 38 v 1489 935 42 4 v 1575 898 a Fq(\()1572 972 y Fs(B)p
1389 991 322 4 v 1408 1092 10 38 v 1417 1075 42 4 v 1503
1038 a Fq(\()1500 1112 y Fs(B)1591 1104 y Fr(_)p Fs(C)1753
1004 y Fr(_)1808 1016 y Fq(R)1858 1024 y Fc(1)2024 786
y Fs(\031)2071 798 y Fl(2)2055 851 y Fm(:)1980 898 y
Fq(\()1978 972 y Fs(B)p 2086 952 10 38 v 2096 935 42
4 v 2273 825 a(\031)2320 837 y Fl(3)2304 890 y Fm(:)2239
964 y Fs(C)p 2323 952 10 38 v 2332 935 42 4 v 1978 991
415 4 v 2038 1038 a Fq(\()2035 1112 y Fs(B)2126 1104
y Fr(_)p Fs(C)p 2265 1092 10 38 v 2274 1075 42 4 v 2434
1008 a Fr(_)2489 1020 y Fq(L)p 1389 1132 945 4 v 1836
1185 10 38 v 1846 1169 42 4 v 2376 1157 a Fs(cut)3267
1197 y FA(\(16\))523 1519 y(where)d(we)h(lea)n(v)o(e)g(the)g(colour)e
(annotation)g(for)h Fs(C)25 b FA(and)17 b Fs(B)t Fr(_)p
Fs(C)25 b FA(unspeci\002ed,)16 b(since)i(it)h(is)f(not)g(impor)n(-)523
1619 y(tant)25 b(for)f(the)i(ar)o(gument)c(at)k(hand.)d(The)i(beha)n
(viour)e(of)i(\(16\))f(can)g(be)h(simulated)g(by)f(the)h(double-)523
1719 y(ne)o(gation)18 b(translation)h Fm(\()p Fr(\000)p
Fm(\))1328 1688 y Fn(\003)1367 1719 y FA(.)h(The)g(double-ne)o(gated)c
(v)o(ersion)j(of)h(\(15\))f(is)i(as)g(follo)n(ws:)1364
2163 y Fs(\031)1414 2133 y Fn(\003)1411 2184 y Fl(1)1397
2236 y Fm(:)1284 2310 y Fr(:)p Fs(B)1406 2280 y Fn(\003)p
1463 2298 10 38 v 1473 2281 42 4 v 1177 2330 463 4 v
1177 2404 a Fr(:)p Fs(B)1299 2373 y Fn(\003)1338 2404
y Fr(^:)p Fs(C)1513 2373 y Fn(\003)p 1571 2392 10 38
v 1580 2375 42 4 v 1682 2342 a Fr(^)1737 2355 y Fq(L)1783
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2472 y Fn(\003)1454 2502 y Fr(^:)p Fs(C)1629 2472 y Fn(\003)1668
2502 y Fm(\))1742 2435 y Fr(:)1797 2447 y Fq(R)p 1090
2543 639 4 v 1090 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)1354
2592 y Fn(\003)1393 2622 y Fr(^)q(:)p Fs(C)1569 2592
y Fn(\003)1607 2622 y Fm(\))p 1658 2610 10 38 v 1668
2593 42 4 v 1769 2555 a Fr(:)1824 2567 y Fq(L)2037 2069
y Fs(\031)2087 2039 y Fn(\003)2084 2090 y Fl(2)2070 2142
y Fm(:)1985 2216 y Fs(B)2052 2186 y Fn(\003)p 2109 2204
10 38 v 2118 2188 42 4 v 1957 2236 249 4 v 1976 2298
10 38 v 1985 2281 42 4 v 2045 2310 a Fr(:)p Fs(B)2167
2280 y Fn(\003)2247 2248 y Fr(:)2302 2260 y Fq(R)2519
2069 y Fs(\031)2569 2039 y Fn(\003)2566 2090 y Fl(3)2552
2142 y Fm(:)2468 2216 y Fs(C)2533 2186 y Fn(\003)p 2590
2204 10 38 v 2599 2188 42 4 v 2440 2236 247 4 v 2459
2298 10 38 v 2468 2281 42 4 v 2528 2310 a Fr(:)p Fs(C)2648
2280 y Fn(\003)2728 2248 y Fr(:)2783 2260 y Fq(R)p 1957
2330 730 4 v 2109 2392 10 38 v 2119 2375 42 4 v 2179
2404 a Fr(:)p Fs(B)2301 2373 y Fn(\003)2339 2404 y Fr(^)q(:)p
Fs(C)2515 2373 y Fn(\003)2728 2347 y Fr(^)2784 2359 y
Fq(R)p 2031 2423 583 4 v 2031 2502 a Fr(:)p Fm(\()p Fr(:)p
Fs(B)2240 2472 y Fn(\003)2279 2502 y Fr(^:)p Fs(C)2454
2472 y Fn(\003)2493 2502 y Fm(\))p 2544 2490 10 38 v
2553 2474 42 4 v 2655 2435 a Fr(:)2710 2447 y Fq(L)p
2003 2543 639 4 v 2022 2610 10 38 v 2031 2593 42 4 v
2091 2622 a Fr(::)p Fm(\()p Fr(:)p Fs(B)2355 2592 y Fn(\003)2395
2622 y Fr(^:)p Fs(C)2570 2592 y Fn(\003)2609 2622 y Fm(\))2683
2555 y Fr(:)2738 2567 y Fq(R)p 1090 2663 1552 4 v 1840
2716 10 38 v 1849 2700 42 4 v 2683 2688 a Fs(cut)523
3064 y FA(which)f(reduces)f(in)h(three)g(steps)h(to)f(the)g(proof)1676
3508 y Fs(\031)1726 3478 y Fn(\003)1723 3529 y Fl(1)1709
3581 y Fm(:)1596 3655 y Fr(:)p Fs(B)1718 3625 y Fn(\003)p
1775 3643 10 38 v 1785 3626 42 4 v 2008 3414 a Fs(\031)2058
3384 y Fn(\003)2055 3435 y Fl(2)2041 3487 y Fm(:)1956
3561 y Fs(B)2023 3531 y Fn(\003)p 2079 3549 10 38 v 2089
3532 42 4 v 1928 3581 249 4 v 1946 3643 10 38 v 1956
3626 42 4 v 2016 3655 a Fr(:)p Fs(B)2138 3625 y Fn(\003)2218
3593 y Fr(:)2273 3605 y Fq(R)p 1596 3675 581 4 v 1861
3728 10 38 v 1871 3712 42 4 v 2218 3700 a Fs(cut)523
4076 y FA(Because)f(the)g Fr(:)995 4088 y Fq(R)1049 4076
y FA(-rule)f(introduces)f(freshly)h(the)h(cut-formula)d
Fr(:)p Fs(B)2494 4046 y Fn(\003)2533 4076 y FA(,)j(the)f(cut)h(is)h
(\223block)o(ed\224)d(from)523 4175 y(reducing)h(to)j(the)f(right.)f
(It)i(must)f(\002rst)h(reduce)e(to)h(the)g(left)g(just)h(as)g(the)f
(colour)f(annotation)f(in)j(\(16\))523 4275 y(prescribed.)d(If)i(we)h
(w)o(anted)f(to)g(simulated)g(the)g(opposite)f(colouring)f(for)h
Fs(B)t FA(,)i(namely)1508 4513 y Fs(\031)1555 4525 y
Fl(1)1539 4578 y Fm(:)p 1480 4679 10 38 v 1489 4662 42
4 v 1575 4625 a Fq(*)1572 4699 y Fs(B)p 1389 4718 322
4 v 1408 4819 10 38 v 1417 4802 42 4 v 1503 4765 a Fq(*)1500
4839 y Fs(B)1591 4831 y Fr(_)p Fs(C)1753 4731 y Fr(_)1808
4743 y Fq(R)1858 4751 y Fc(1)2024 4513 y Fs(\031)2071
4525 y Fl(2)2055 4578 y Fm(:)1980 4625 y Fq(*)1978 4699
y Fs(B)p 2086 4679 10 38 v 2096 4662 42 4 v 2273 4552
a(\031)2320 4564 y Fl(3)2304 4617 y Fm(:)2239 4691 y
Fs(C)p 2323 4679 10 38 v 2332 4662 42 4 v 1978 4718 415
4 v 2038 4765 a Fq(*)2035 4839 y Fs(B)2126 4831 y Fr(_)p
Fs(C)p 2265 4819 10 38 v 2274 4802 42 4 v 2434 4735 a
Fr(_)2489 4747 y Fq(L)p 1389 4859 945 4 v 1836 4912 10
38 v 1846 4896 42 4 v 2376 4884 a Fs(cut)3267 4924 y
FA(\(17\))p eop end
%%Page: 14 14
TeXDict begin 14 13 bop 523 448 a FA(it)29 b(turns)g(out)f(we)h(ha)n(v)
o(e)f(to)h(double-ne)o(gate)c(translate)j(\(15\))g(using)g(the)h
(translation)e Fm(\()p Fr(\000)p Fm(\))3156 418 y Fn(\016)3224
448 y FA(gi)n(v)o(en)523 548 y(in)20 b(\(6\).)g(The)g(resulting)f
(intuitionistic)g(proof)g(is:)1436 721 y Fs(\031)1486
691 y Fn(\016)1483 741 y Fl(1)1469 794 y Fm(:)1356 867
y Fr(:)p Fs(B)1478 837 y Fn(\016)p 1535 855 10 38 v 1545
839 42 4 v 1328 887 304 4 v 1347 949 10 38 v 1356 932
42 4 v 1416 961 a Fr(::)p Fs(B)1593 931 y Fn(\016)1674
899 y Fr(:)1729 911 y Fq(R)p 1328 981 304 4 v 1402 1043
10 38 v 1412 1026 42 4 v 1472 1055 a Fs(B)1539 1025 y
Fn(\016)1674 993 y Fr(::)1784 1005 y Fq(R)p 1304 1075
353 4 v 1323 1136 10 38 v 1332 1120 42 4 v 1392 1148
a Fs(B)1459 1118 y Fn(\016)1498 1148 y Fr(_)p Fs(C)1618
1118 y Fn(\016)1698 1087 y Fr(_)1753 1100 y Fq(R)1803
1108 y Fc(1)p 1244 1168 472 4 v 1244 1247 a Fr(:)p Fm(\()p
Fs(B)1398 1217 y Fn(\016)1437 1247 y Fr(_)q Fs(C)1558
1217 y Fn(\016)1596 1247 y Fm(\))p 1647 1235 10 38 v
1656 1219 42 4 v 1758 1180 a Fr(:)1813 1192 y Fq(L)p
1217 1288 528 4 v 1235 1355 10 38 v 1245 1338 42 4 v
1305 1367 a Fr(::)p Fm(\()p Fs(B)1514 1337 y Fn(\016)1553
1367 y Fr(_)p Fs(C)1673 1337 y Fn(\016)1712 1367 y Fm(\))1786
1300 y Fr(:)1841 1312 y Fq(R)p 1189 1407 583 4 v 1189
1486 a Fr(:::)p Fm(\()p Fs(B)1453 1456 y Fn(\016)1493
1486 y Fr(_)p Fs(C)1613 1456 y Fn(\016)1651 1486 y Fm(\))p
1702 1474 10 38 v 1712 1458 42 4 v 1813 1419 a Fr(:)1868
1431 y Fq(L)2111 908 y Fs(\031)2161 878 y Fn(\016)2158
929 y Fl(2)2144 981 y Fm(:)2059 1055 y Fs(B)2126 1025
y Fn(\016)p 2183 1043 10 38 v 2192 1026 42 4 v 2387 908
a Fs(\031)2437 878 y Fn(\016)2434 929 y Fl(3)2419 981
y Fm(:)2335 1055 y Fs(C)2400 1025 y Fn(\016)p 2457 1043
10 38 v 2467 1026 42 4 v 2059 1075 468 4 v 2117 1148
a Fs(B)2184 1118 y Fn(\016)2222 1148 y Fr(_)p Fs(C)2342
1118 y Fn(\016)p 2399 1136 10 38 v 2409 1120 42 4 v 2568
1092 a Fr(_)2623 1104 y Fq(L)p 2057 1168 472 4 v 2075
1235 10 38 v 2085 1219 42 4 v 2145 1247 a Fr(:)p Fm(\()p
Fs(B)2299 1217 y Fn(\016)2338 1247 y Fr(_)p Fs(C)2458
1217 y Fn(\016)2496 1247 y Fm(\))2570 1180 y Fr(:)2625
1192 y Fq(R)p 2029 1288 528 4 v 2029 1367 a Fr(::)p Fm(\()p
Fs(B)2238 1337 y Fn(\016)2277 1367 y Fr(_)q Fs(C)2398
1337 y Fn(\016)2436 1367 y Fm(\))p 2487 1355 10 38 v
2496 1338 42 4 v 2598 1300 a Fr(:)2653 1312 y Fq(L)p
2001 1407 583 4 v 2020 1474 10 38 v 2029 1458 42 4 v
2089 1486 a Fr(:::)p Fm(\()p Fs(B)2353 1456 y Fn(\016)2393
1486 y Fr(_)p Fs(C)2513 1456 y Fn(\016)2552 1486 y Fm(\))2625
1419 y Fr(:)2680 1431 y Fq(R)p 1189 1527 1395 4 v 1861
1581 10 38 v 1871 1564 42 4 v 2625 1553 a Fs(cut)523
1785 y FA(which)h(after)g(four)f(steps)h(reduces)g(to)g(the)g(proof)
1601 1958 y Fs(\031)1651 1927 y Fn(\016)1648 1978 y Fl(1)1634
2030 y Fm(:)1521 2104 y Fr(:)p Fs(B)1643 2074 y Fn(\016)p
1700 2092 10 38 v 1709 2076 42 4 v 1493 2124 304 4 v
1512 2186 10 38 v 1521 2169 42 4 v 1581 2198 a Fr(::)p
Fs(B)1758 2168 y Fn(\016)1839 2136 y Fr(:)1894 2148 y
Fq(R)p 1493 2218 304 4 v 1567 2280 10 38 v 1576 2263
42 4 v 1636 2292 a Fs(B)1703 2262 y Fn(\016)1839 2230
y Fr(::)1949 2242 y Fq(R)2139 2145 y Fs(\031)2189 2115
y Fn(\016)2186 2166 y Fl(2)2172 2218 y Fm(:)2087 2292
y Fs(B)2154 2262 y Fn(\016)p 2210 2280 10 38 v 2220 2263
42 4 v 1548 2312 732 4 v 1889 2365 10 38 v 1898 2349
42 4 v 2321 2337 a Fs(cut)3267 2377 y FA(\(18\))523 2569
y(What)g(happens)e(ne)o(xt,)g(ho)n(we)n(v)o(er)m(,)f(is)j(not)f(clear)g
(at)h(\002rst)g(sight.)g(If)f(we)g(e)o(xpand)f(the)h
Fr(::)3010 2581 y Fq(R)3065 2569 y FA(-rule)g(to)h(an)523
2669 y(auxiliary)e(cut,)i(then)f(the)g(cut)h(can)f(reduce)g(into)g
(both)g(directions.)f(If)i(we)g(re)o(gard)d(the)j Fr(::)3104
2681 y Fq(R)3159 2669 y FA(-rule)f(as)523 2769 y(an)k(inference)f(rule)
g(in)i(its)g(o)n(wn)e(right,)h(then)f(the)h(cut-formula)e
Fs(B)2441 2738 y Fn(\016)2503 2769 y FA(is)j(freshly)e(introduced)f(in)
i(the)523 2868 y(subproof)h(on)i(the)g(left-hand)f(side)h(and)g
(therefore)e(the)j(cut)f(can)g(only)g(mo)o(v)o(e)e(to)j(the)f
(right\227just)523 2968 y(as)21 b(prescribed)d(by)i(the)g(colour)n
(-annotation)d(in)j(\(17\).)f(Although)f(we)j(do)f(not)f(ha)n(v)o(e)h
(a)h(proof)d(of)i(this)523 3067 y(f)o(act,)i(e)o(xperiments)f(with)h
([17])g(ha)n(v)o(e)g(con)m(vinced)d(us)k(that)g(when)e(cut-elimination)
g(is)i(concerned,)523 3167 y(we)18 b(can)g(indeed)e(re)o(gard)g(the)i
Fr(::)1470 3179 y Fq(R)1525 3167 y FA(-rule)f(as)h(a)h(proper)d
(inference)g(rule)h(with)h(the)g(consequence)d(that)523
3267 y(in)j(the)g(proof)e(abo)o(v)o(e)g Fs(B)1205 3237
y Fn(\016)1262 3267 y FA(is)j(freshly)e(introduced.)e(\(Roughly)h
(speaking,)g(if)j(we)f(had)f(e)o(xpanded)e(the)523 3366
y Fr(::)633 3378 y Fq(R)688 3366 y FA(-rule)26 b(to)g(an)g(auxiliary)f
(cut)h(in)h(the)f(proof)f(\(18\),)f(then)i(mo)o(ving)e(the)j(cut)f(to)g
(the)g(left)h(means)523 3466 y(it)d(cannot)e(mo)o(v)o(e)g(v)o(ery)g(f)o
(ar)m(,)g(namely)g(only)h(to)g(the)g(place)g(where)g
Fs(B)2492 3436 y Fn(\016)2554 3466 y FA(is)h(introduced)d(in)i(the)g
(proof)523 3566 y(of)k(the)h(sequent)f Fr(::)p Fs(B)1209
3535 y Fn(\016)p 1266 3554 10 38 v 1276 3537 42 4 v 1336
3566 a Fs(B)1403 3535 y Fn(\016)1469 3566 y FA(and)g(then)g(the)h(cut)g
(has)f(to)h(mo)o(v)o(e)e(right.)h(In)g(ef)n(fect)g(we)h(obtain)f(a)523
3665 y(beha)n(viour)18 b(which)i(is)h(almost)f(identical)g(to)g(mo)o
(ving)e(the)j(cut)f(to)g(the)g(right)g(in)g(the)g(\002rst)h(place.\))
648 3768 y(Since)29 b(the)h(colour)n(-protocol)c(of)k(Danos)f(et)i(al.)
f(allo)n(ws)g(us)g(to)g(annotate)e(in)i(man)o(y)f(circum-)523
3868 y(stances)f(either)f(colour)g(`)p Fs(\()p FA(')g(or)h(`)p
Fs(*)p FA(')f(to)h(the)g(formulae)e(in)i(a)g(classical)g(proof,)e(we)i
(need)f(ho)n(w-)523 3967 y(e)n(v)o(er)21 b(to)h(depart)g(from)f(the)h
(traditional)f(double-ne)o(gation)c(technique)k(that)h(translates)g(a)h
(classical)523 4067 y(proof)k(uniformly)e(using)j(a)g(single)g
(double-ne)o(gation)c(translation.)j(T)-7 b(o)28 b(simulate)g(the)g
(coloured)523 4166 y(cut-elimination)14 b(procedure)h(in)h(a)h
(meaningful)e(w)o(ay)-5 b(,)15 b(we)i(need)f(to)h(allo)n(w)f(more)g
(than)g(one)g(double-)523 4266 y(ne)o(gation)f(translation.)h(Let)h(us)
h(e)o(xplain)e(this)h(f)o(act)h(with)f(a)g(classical)h(proof)e
Fs(\031)21 b FA(ending)15 b(in)j(a)f(cut)g(with)523 4366
y(the)j(cut-formula)1815 4429 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g
(\000)1815 4476 y Fq(*)1812 4550 y Fs(B)1902 4542 y Fr(_)1984
4476 y Fq(\()1981 4550 y Fs(C)2093 4542 y(:)523 4703
y FA(W)k(e)27 b(will)f(sho)n(w)f(that)g(the)g(beha)n(viour)f(of)h(this)
h(\(coloured\))c(cut)k(can)f(be)g(simulated)g(by)g(a)g(double-)523
4825 y(ne)o(gation)30 b(translation)g(with)i(the)g(clause)g
Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2028 4795 y Fn(\017)2111
4778 y Fp(def)2116 4825 y Fm(=)48 b Fr(::)p Fm(\()p Fs(B)2438
4795 y Fn(\017)2478 4825 y Fr(_::)p Fs(C)2708 4795 y
Fn(\017)2747 4825 y Fm(\))p FA(.)33 b(T)-7 b(o)31 b(sho)n(w)h(this)g
(we)523 4924 y(analyse)26 b(all)h(cases)g(ho)n(w)f(the)h(cut)f(in)h
Fs(\031)j FA(could)c(ha)n(v)o(e)f(arisen.)h(Consider)g(\002rst)h(the)g
(case)g(where)f Fs(\031)p eop end
%%Page: 15 15
TeXDict begin 15 14 bop 523 448 a FA(ends)20 b(with)g(the)h(follo)n
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666 y Fm(:)p 1468 767 10 38 v 1477 751 42 4 v 1563 714
a Fq(*)1560 787 y Fs(B)p 1365 807 346 4 v 1384 969 10
38 v 1393 952 42 4 v 1480 868 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g
(\000)1479 915 y Fq(*)1476 989 y Fs(B)1567 981 y Fr(_)1648
915 y Fq(\()1645 989 y Fs(C)1753 820 y Fr(_)1808 832
y Fq(R)1858 840 y Fc(1)2024 602 y Fs(\031)2071 614 y
Fl(2)2055 666 y Fm(:)1980 714 y Fq(*)1978 787 y Fs(B)p
2086 767 10 38 v 2096 751 42 4 v 2285 602 a(\031)2332
614 y Fl(3)2316 666 y Fm(:)2242 714 y Fq(\()2239 787
y Fs(C)p 2347 767 10 38 v 2356 751 42 4 v 1978 807 439
4 v 2027 868 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026
915 y Fq(*)2024 989 y Fs(B)2114 981 y Fr(_)2196 915 y
Fq(\()2193 989 y Fs(C)p 2300 969 10 38 v 2310 952 42
4 v 2458 824 a Fr(_)2513 836 y Fq(L)p 1365 1008 1005
4 v 1842 1062 10 38 v 1852 1045 42 4 v 2411 1034 a Fs(cut)3267
1074 y FA(\(19\))523 1269 y(which)20 b(can)g(reduce)f(to)1714
1327 y Fs(\031)1761 1339 y Fl(1)1744 1391 y Fm(:)p 1685
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1764 y(The)h Fm(\()p Fr(\000)p Fm(\))802 1733 y Fn(\017)840
1764 y FA(-translated)f(v)o(ersion)g(of)h Fs(\031)1381
1941 y(\031)1431 1911 y Fn(\017)1428 1962 y Fl(1)1414
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1480 2076 10 38 v 1489 2059 42 4 v 1273 2108 304 4 v
1292 2169 10 38 v 1301 2153 42 4 v 1361 2181 a Fr(::)p
Fs(B)1538 2151 y Fn(\017)1619 2119 y Fr(:)1674 2131 y
Fq(R)p 1273 2201 304 4 v 1347 2263 10 38 v 1356 2247
42 4 v 1416 2275 a Fs(B)1483 2245 y Fn(\017)1619 2213
y Fr(::)1729 2225 y Fq(R)p 1194 2295 463 4 v 1212 2357
10 38 v 1222 2340 42 4 v 1282 2369 a Fs(B)1349 2339 y
Fn(\017)1387 2369 y Fr(_::)p Fs(C)1617 2339 y Fn(\017)1698
2308 y Fr(_)1753 2320 y Fq(R)1803 2328 y Fc(1)p 1134
2389 583 4 v 1134 2468 a Fr(:)p Fm(\()p Fs(B)1288 2438
y Fn(\017)1327 2468 y Fr(_::)p Fs(C)1557 2438 y Fn(\017)1596
2468 y Fm(\))p 1647 2456 10 38 v 1656 2439 42 4 v 1758
2400 a Fr(:)1813 2412 y Fq(L)p 1106 2508 639 4 v 1124
2575 10 38 v 1134 2559 42 4 v 1194 2587 a Fr(::)p Fm(\()p
Fs(B)1403 2557 y Fn(\017)1442 2587 y Fr(_)q(::)p Fs(C)1673
2557 y Fn(\017)1712 2587 y Fm(\))1786 2520 y Fr(:)1841
2532 y Fq(R)p 1078 2628 694 4 v 1078 2707 a Fr(:::)p
Fm(\()p Fs(B)1342 2677 y Fn(\017)1382 2707 y Fr(_::)p
Fs(C)1612 2677 y Fn(\017)1651 2707 y Fm(\))p 1702 2695
10 38 v 1712 2678 42 4 v 1813 2640 a Fr(:)1868 2652 y
Fq(L)2111 2128 y Fs(\031)2161 2098 y Fn(\017)2158 2149
y Fl(2)2144 2201 y Fm(:)2059 2275 y Fs(B)2126 2245 y
Fn(\017)p 2183 2263 10 38 v 2192 2247 42 4 v 2442 1941
a Fs(\031)2492 1911 y Fn(\017)2489 1962 y Fl(3)2475 2014
y Fm(:)2390 2088 y Fs(C)2455 2058 y Fn(\017)p 2512 2076
10 38 v 2522 2059 42 4 v 2363 2108 247 4 v 2381 2169
10 38 v 2391 2153 42 4 v 2451 2181 a Fr(:)p Fs(C)2571
2151 y Fn(\017)2651 2119 y Fr(:)2706 2131 y Fq(R)p 2335
2201 303 4 v 2335 2275 a Fr(::)p Fs(C)2510 2245 y Fn(\017)p
2568 2263 10 38 v 2577 2247 42 4 v 2679 2213 a Fr(:)2734
2225 y Fq(L)p 2059 2295 579 4 v 2117 2369 a Fs(B)2184
2339 y Fn(\017)2222 2369 y Fr(_::)p Fs(C)2452 2339 y
Fn(\017)p 2510 2357 10 38 v 2519 2340 42 4 v 2679 2312
a Fr(_)2734 2324 y Fq(L)p 2057 2389 583 4 v 2075 2456
10 38 v 2085 2439 42 4 v 2145 2468 a Fr(:)p Fm(\()p Fs(B)2299
2438 y Fn(\017)2338 2468 y Fr(_::)p Fs(C)2568 2438 y
Fn(\017)2607 2468 y Fm(\))2681 2400 y Fr(:)2736 2412
y Fq(R)p 2029 2508 639 4 v 2029 2587 a Fr(::)p Fm(\()p
Fs(B)2238 2557 y Fn(\017)2277 2587 y Fr(_)q(::)p Fs(C)2508
2557 y Fn(\017)2547 2587 y Fm(\))p 2597 2575 10 38 v
2607 2559 42 4 v 2709 2520 a Fr(:)2764 2532 y Fq(L)p
2001 2628 694 4 v 2020 2695 10 38 v 2029 2678 42 4 v
2089 2707 a Fr(:::)p Fm(\()p Fs(B)2353 2677 y Fn(\017)2393
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Fm(\))2736 2640 y Fr(:)2791 2652 y Fq(R)p 1078 2747 1617
4 v 1861 2801 10 38 v 1871 2785 42 4 v 2736 2773 a Fs(cut)523
3008 y FA(reduces)f(to)1601 3087 y Fs(\031)1651 3057
y Fn(\017)1648 3108 y Fl(1)1634 3160 y Fm(:)1521 3234
y Fr(:)p Fs(B)1643 3204 y Fn(\017)p 1700 3222 10 38 v
1709 3205 42 4 v 1493 3254 304 4 v 1512 3315 10 38 v
1521 3299 42 4 v 1581 3327 a Fr(::)p Fs(B)1758 3297 y
Fn(\017)1839 3265 y Fr(:)1894 3277 y Fq(R)p 1493 3347
304 4 v 1567 3409 10 38 v 1576 3393 42 4 v 1636 3421
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y Fq(R)2139 3274 y Fs(\031)2189 3244 y Fn(\017)2186 3295
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Fn(\017)p 2210 3409 10 38 v 2220 3393 42 4 v 1548 3441
732 4 v 1889 3495 10 38 v 1898 3478 42 4 v 2321 3467
a Fs(cut)523 3673 y FA(where)g(\(remember)f(we)j(re)o(gard)d(the)i
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Fs(\031)995 3742 y Fn(\017)992 3793 y Fl(2)1054 3772
y FA(just)h(lik)o(e)f(the)g(beha)n(viour)f(of)g(\(20\).)g(If)h
Fs(\031)k FA(ends)c(with)h(the)f(logical)g(cut)1496 3928
y Fs(\031)1543 3940 y Fl(1)1527 3993 y Fm(:)p 1468 4094
10 38 v 1478 4077 42 4 v 1564 4040 a Fq(\()1561 4114
y Fs(C)p 1365 4134 346 4 v 1384 4295 10 38 v 1393 4279
42 4 v 1480 4194 a Fq(\()-11 b Fn(\000)g(\000)g(\000)g(\000)1479
4242 y Fq(*)1476 4315 y Fs(B)1567 4307 y Fr(_)1648 4242
y Fq(\()1645 4315 y Fs(C)1753 4146 y Fr(_)1808 4159 y
Fq(R)1858 4167 y Fc(2)2024 3928 y Fs(\031)2071 3940 y
Fl(2)2055 3993 y Fm(:)1980 4040 y Fq(*)1978 4114 y Fs(B)p
2086 4094 10 38 v 2096 4077 42 4 v 2285 3928 a(\031)2332
3940 y Fl(3)2316 3993 y Fm(:)2242 4040 y Fq(\()2239 4114
y Fs(C)p 2347 4094 10 38 v 2356 4077 42 4 v 1978 4134
439 4 v 2027 4194 a Fq(\()g Fn(\000)g(\000)g(\000)g(\000)2026
4242 y Fq(*)2024 4315 y Fs(B)2114 4307 y Fr(_)2196 4242
y Fq(\()2193 4315 y Fs(C)p 2300 4295 10 38 v 2310 4279
42 4 v 2458 4150 a Fr(_)2513 4163 y Fq(L)p 1365 4335
1005 4 v 1842 4389 10 38 v 1852 4372 42 4 v 2411 4360
a Fs(cut)3267 4401 y FA(\(21\))523 4596 y(which)20 b(can)g(reduce)f(to)
1771 4653 y Fs(\031)1818 4665 y Fl(1)1802 4718 y Fm(:)p
1743 4819 10 38 v 1753 4802 42 4 v 1838 4765 a Fq(\()1836
4839 y Fs(C)2031 4653 y(\031)2078 4665 y Fl(3)2062 4718
y Fm(:)1988 4765 y Fq(\()1985 4839 y Fs(C)p 2092 4819
10 38 v 2102 4802 42 4 v 1725 4859 438 4 v 1918 4912
10 38 v 1927 4896 42 4 v 3267 4924 a FA(\(22\))p eop
end
%%Page: 16 16
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816 y Fn(\017)p 1479 834 10 38 v 1488 817 42 4 v 1274
866 303 4 v 1292 927 10 38 v 1302 911 42 4 v 1362 939
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1077 y Fq(L)p 1106 1173 639 4 v 1124 1239 10 38 v 1134
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1304 a Fr(:)1868 1316 y Fq(L)2111 793 y Fs(\031)2161
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y Fs(B)2126 909 y Fn(\017)p 2183 927 10 38 v 2192 911
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y Fl(3)2475 678 y Fm(:)2390 752 y Fs(C)2455 722 y Fn(\017)p
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866 303 4 v 2335 939 a Fr(::)p Fs(C)2510 909 y Fn(\017)p
2568 927 10 38 v 2577 911 42 4 v 2679 877 a Fr(:)2734
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2510 1021 10 38 v 2519 1005 42 4 v 2679 976 a Fr(_)2734
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1103 42 4 v 2145 1132 a Fr(:)p Fm(\()p Fs(B)2299 1102
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1132 y Fm(\))2681 1065 y Fr(:)2736 1077 y Fq(R)p 2029
1173 639 4 v 2029 1251 a Fr(::)p Fm(\()p Fs(B)2238 1221
y Fn(\017)2277 1251 y Fr(_)q(::)p Fs(C)2508 1221 y Fn(\017)2547
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1184 a Fr(:)2764 1196 y Fq(L)p 2001 1292 694 4 v 2020
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1316 y Fq(R)p 1078 1412 1617 4 v 1861 1466 10 38 v 1871
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1808 y Fs(\031)1727 1778 y Fn(\017)1724 1829 y Fl(1)1710
1881 y Fm(:)1598 1955 y Fr(:)p Fs(C)1718 1925 y Fn(\017)p
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2028 10 38 v 1871 2012 42 4 v 2216 2000 a Fs(cut)523
2183 y FA(where)h(proof)f Fs(\031)998 2152 y Fn(\017)995
2203 y Fl(3)1057 2183 y FA(has)i(to)f(mo)o(v)o(e)f(inside)h
Fs(\031)1746 2152 y Fn(\017)1743 2203 y Fl(1)1806 2183
y FA(just)h(lik)o(e)f(in)h(the)f(proof)f(\(22\).)g(The)h(only)g(case)h
(we)f(still)523 2282 y(need)g(to)g(consider)f(is)i(when)f
Fs(\031)k FA(ends)c(in)g(a)h(commuting)d(cut)i(of)g(the)g(form)1630
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2642 10 38 v 1527 2625 42 4 v 1613 2541 a Fq(\()-11 b
Fn(\000)g(\000)g(\000)g(\000)1613 2588 y Fq(*)1610 2662
y Fs(B)1700 2654 y Fr(_)1782 2588 y Fq(\()1779 2662 y
Fs(C)2059 2415 y(\031)2106 2427 y Fl(2)2089 2480 y Fm(:)1931
2541 y Fq(\()g Fn(\000)g(\000)g(\000)g(\000)1931 2588
y Fq(*)1928 2662 y Fs(B)2018 2654 y Fr(_)2100 2588 y
Fq(\()2097 2662 y Fs(C)p 2204 2642 10 38 v 2214 2625
42 4 v 1499 2681 775 4 v 1861 2735 10 38 v 1871 2718
42 4 v 2315 2707 a(cut)3267 2747 y FA(\(23\))523 2917
y(The)25 b(beha)n(viour)e(of)i(this)h(cut)g(is)g(determined)e(by)h(the)
g(outermost)f(colour)g(`)p Fs(\()p FA('.)h(This)h(beha)n(viour)523
3017 y(can)21 b(be)h(simulated)f(by)g(the)h Fm(\()p Fr(\000)p
Fm(\))1464 2987 y Fn(\017)1502 3017 y FA(-translation,)e(pro)o(vided)f
(we)j(use)g(a)g(left-translation)e(for)h(the)g(cut)523
3117 y(in)f(\(23\).)f(The)h(translated)g(proof)e(is)j(then)f(as)h
(follo)n(ws:)1454 3390 y Fs(\031)1504 3360 y Fn(\017)1501
3411 y Fl(1)1487 3463 y Fm(:)1152 3542 y Fr(:::)p Fm(\()p
Fs(B)1416 3512 y Fn(\017)1455 3542 y Fr(_)q(::)p Fs(C)1686
3512 y Fn(\017)1725 3542 y Fm(\))p 1775 3530 10 38 v
1785 3514 42 4 v 2230 3271 a Fs(\031)2280 3241 y Fn(\017)2277
3291 y Fl(2)2263 3344 y Fm(:)1956 3423 y Fr(::)p Fm(\()p
Fs(B)2165 3393 y Fn(\017)2204 3423 y Fr(_::)p Fs(C)2434
3393 y Fn(\017)2473 3423 y Fm(\))p 2524 3411 10 38 v
2534 3394 42 4 v 1928 3463 694 4 v 1946 3530 10 38 v
1956 3514 42 4 v 2016 3542 a Fr(:::)p Fm(\()p Fs(B)2280
3512 y Fn(\017)2320 3542 y Fr(_::)p Fs(C)2550 3512 y
Fn(\017)2589 3542 y Fm(\))2663 3475 y Fr(:)2718 3487
y Fq(R)p 1152 3583 1470 4 v 1861 3637 10 38 v 1871 3620
42 4 v 2663 3608 a Fs(cut)523 3819 y FA(No)n(w)31 b Fs(\031)764
3789 y Fn(\017)761 3840 y Fl(1)834 3819 y FA(will)h(freshly)e
(introduce)g(the)h(cut-formula)d Fr(:::)p Fm(\()p Fs(B)2423
3789 y Fn(\017)2463 3819 y Fr(_::)p Fs(C)2693 3789 y
Fn(\017)2732 3819 y Fm(\))k FA(only)f(if)g Fs(\031)3105
3831 y Fl(1)3174 3819 y FA(freshly)523 3919 y(introduces)20
b(the)i(formula)f Fs(B)t Fr(_)p Fs(C)29 b FA(\(recall)22
b(that)g(double-ne)o(gation)17 b(translations)22 b(need)f(to)h(preserv)
o(e)523 4018 y(the)g(structure)f(of)g(a)i(classical)f(proof\).)e
(Consequently)-5 b(,)19 b(the)j(translated)f(proof)f(simulates)i(e)o
(xactly)523 4118 y(the)e(beha)n(viour)e(of)i(\(23\).)648
4218 y(What)32 b(this)g(e)o(xample)f(sho)n(ws)h(is)g(that)g(the)g
Fm(\()p Fr(\000)p Fm(\))2070 4187 y Fn(\017)2109 4218
y FA(-translation)e(of)i Fs(\031)k FA(can)31 b(simulate)h(the)g(be-)523
4317 y(ha)n(viour)19 b(where)g(the)i(cut-formula)c(is)k(annotated)e
(with)h(the)g(colours)1815 4442 y Fq(\()-11 b Fn(\000)g(\000)g(\000)g
(\000)1815 4489 y Fq(*)1812 4563 y Fs(B)1902 4555 y Fr(_)1984
4489 y Fq(\()1981 4563 y Fs(C)2093 4555 y(:)523 4725
y FA(Note)27 b(that)g(there)f(are)h(other)f(double)g(ne)o(gation)e
(translation)i(which)h(can)f(be)h(used)g(for)f(a)h(similar)523
4825 y(simulation)g(of)h(this)g(particular)f(colour)n(-annotation.)d
(From)j(our)g(discussion)h(it)g(seems)h(reason-)523 4924
y(able)e(to)g(e)o(xpect)f(that)g(one)h(can)f(\002nd)h(corresponding)c
(double-ne)o(gation)f(translations)27 b(for)f(e)n(v)o(ery)p
eop end
%%Page: 17 17
TeXDict begin 17 16 bop 523 448 a FA(possible)20 b(colour)n
(-annotation.)15 b(Since)20 b(the)g(formulae)e Fs(B)25
b FA(and)19 b Fs(C)27 b FA(can)19 b(be)h(compound)d(with)j(further)523
548 y(colour)n(-annotations)j(inside,)i(we)i(need)e(some)h(\003e)o
(xibility)g(of)f(ho)n(w)h(to)g(double-ne)o(gate)d(translate)523
669 y(formulae.)30 b(One)h(has)h(to)g(be)g(able)g(to)g(b)n(uild)f(into)
h(the)f(clause)h Fm(\()p Fs(B)t Fr(_)q Fs(C)6 b Fm(\))2653
639 y Fn(\017)2736 622 y Fp(def)2741 669 y Fm(=)49 b
Fr(::)p Fm(\()p Fs(B)3064 639 y Fn(\017)3103 669 y Fr(_)q(::)p
Fs(C)3334 639 y Fn(\017)3373 669 y Fm(\))523 769 y FA(that)26
b(the)g(translations)f(of)g Fs(B)31 b FA(and)25 b Fs(C)32
b FA(might)26 b(follo)n(w)f(a)h(completely)e(dif)n(ferent)g(double-ne)o
(gation)523 868 y(scheme.)k(Ho)n(w)g(to)h(do)f(this)h(ele)o(gantly)d
(is)k(not)e(kno)n(wn)f(to)h(us.)h(On)g(the)f(other)g(hand,)f(we)i
(cannot)523 968 y(e)o(xpect)23 b(complete)h(\223freedom\224)e(in)j(a)f
(double-ne)o(gation)c(translation)k(as)h(we)g(ha)n(v)o(e)e(to)i(mak)o
(e)f(sure,)523 1068 y(roughly)15 b(speaking,)g(that)i(dif)n(ferent)e
(double-ne)o(gation)d(translation)k(still)h(\002t)h(together)d(in)i
(the)g(trans-)523 1167 y(lated)j(proof.)e(This)i(means)g(we)h(ha)n(v)o
(e)e(to)h(mak)o(e)g(sure)g(that)g(double-ne)o(gation)15
b(translations)20 b(respect)523 1267 y(the)j(identity-class)g
(constraint)g(from)f(the)h(colour)n(-annotations.)d(F)o(or)j(e)o
(xample)f(we)i(cannot)e(ha)n(v)o(e)523 1367 y(an)e(axiom)f(in)i(a)f
(translated)g(proof)f(where)g(the)h(double-ne)o(gation)c(translations)k
(disagree)f(as)i(in)1815 1550 y Fs(B)1882 1520 y Fn(\003)p
1938 1538 10 38 v 1948 1522 42 4 v 2008 1550 a Fs(B)2075
1520 y Fn(\016)523 1735 y FA(This)c(point)g(about)f(\002tting)h
(double-ne)o(gation)12 b(translations)17 b(together)e(and)i(the)g
(identity-class)f(con-)523 1834 y(straint)22 b(of)g(colours)e(we)j(tak)
o(e)e(as)i(a)f(further)e(e)n(vidence)h(that)h(colours)f(and)g(double)f
(ne)o(gation)g(trans-)523 1934 y(lation)g(must)g(ha)n(v)o(e)g
(something)e(to)j(do)f(with)g(each)g(other)-5 b(.)648
2034 y(From)22 b(the)i(observ)n(ations)e(made)g(abo)o(v)o(e)g(we)i
(conjecture)e(that)h(e)n(v)o(ery)f(colour)n(-annotation)e(de-)523
2134 y(termining)31 b(a)i(single)f(normalform)e(of)i(a)h(classical)g
(proof)e(can)h(be)g(equally)g(determined)e(by)i(a)523
2233 y(double)24 b(ne)o(gation)g(translation,)g(and)h(e)n(v)o(ery)g
(double-ne)o(gation)c(translation)k(determining)e(a)j(nor)n(-)523
2333 y(malform)17 b(can)i(be)g(equally)e(determined)g(by)i(a)g(colour)n
(-annotation.)14 b(In)19 b(ef)n(fect)f(we)h(conjecture)e(that)523
2433 y(double-ne)o(gation)f(translations)j(can)h(be)g(simulated)g(by)g
(colours)f(and)h(vice)g(v)o(ersa.)648 2533 y(While)h(establishing)f
(the)h(simulation)f(properties)g(is)h(a)h(\002rst)f(step)h(for)e
(understanding)e(the)j(re-)523 2632 y(lation)d(between)f(double-ne)o
(gation)c(translations)18 b(and)f(colour)n(-annotations,)d(we)k
(consider)f(this)i(as)523 2732 y(not)k(yet)h(gi)n(ving)f(the)g
(complete)g(\223picture\224.)f(F)o(or)h(this)i(consider)d(the)i
(collection)f(of)g(normalforms)523 2832 y(of)30 b(a)g(classical)g
(proof)e(determined)g(by)i(double-ne)o(gation)25 b(translations)k(and)g
(by)g(colour)g(anno-)523 2931 y(tations.)e(W)-7 b(e)28
b(conjecture)d(that)i(both)f(are)h(the)g(\223same\224)g(collection,)e
(whereby)h(one)g(needs)g(a)i(\(yet)523 3031 y(unkno)n(wn\))c(v)o(ery)i
(cle)n(v)o(er)g(notion)g(of)h(\223sameness\224.)g(Clearly)-5
b(,)26 b(there)h(are)g(more)f(double-ne)o(gation)523
3131 y(translations)17 b(of)h(a)h(classical)f(proof)f(than)g(there)h
(are)g(colour)n(-annotations)c(\(there)k(are)f(only)h(\002nitely)523
3230 y(man)o(y)k(colour)n(-annotations,)e(b)n(ut)j(there)g(are)h
(in\002nitely)f(man)o(y)f(double)g(ne)o(gation)f(translations)i(as)523
3330 y(already)j(the)h(translations)g(of)g(atomic)f(formulae)g(as)i
Fr(::)p Fs(A)p FA(,)g Fr(::::)p Fs(A;)14 b(:)g(:)g(:)29
b FA(indicate\).)d(F)o(or)h(us)h(it)523 3429 y(is,)19
b(ho)n(we)n(v)o(er)m(,)d(clear)i(that)h(one)e(can)i(group)d(double-ne)o
(gation)e(translations)k(into)g(dif)n(ferent)f(classes,)523
3529 y(where)j(each)f(class)j(corresponds)c(to)i(a)g(colour)n
(-annotation.)648 3629 y(Let)32 b(us)h(consider)f(what)h(is)g
(necessary)f(to)h(turn)f(these)h(conjectures)e(into)h(theorems.)g
(First)523 3729 y(we)e(ha)n(v)o(e)f(to)h(mak)o(e)g(precise)f(what)h(we)
g(mean)f(by)h Ft(all)g FA(double-ne)o(gation)25 b(translation.)k(As)h
(seen)523 3828 y(abo)o(v)o(e,)22 b(the)h(notion)f(of)h(double-ne)o
(gation)c(translation)k(has)g(to)h(be)f(a)h(generalised)e(v)o(ersion)g
(of)h(the)523 3928 y(traditional)g(notion\227lik)o(e)f(the)h(ones)h(gi)
n(v)o(en)e(by)h(Gentzen,)g(G)7 b(\250)-35 b(odel)23 b(and)g(K)m
(olmogoro)o(v\227because)523 4028 y(one)g(needs)g(to)g(tak)o(e)h(into)f
(account)f(the)i(dif)n(ferent)d(colour)n(-annotations)f(by)j(v)n
(arying)f(the)h(double-)523 4127 y(ne)o(gations)g(translation)h(when)g
(inducti)n(v)o(ely)e(descending)h(a)i(formula.)e(Further)m(,)h(one)g(w)
o(ould)g(ide-)523 4227 y(ally)g(lik)o(e)f(to)h(ha)n(v)o(e)f(a)h(rather)
f(general)f(notion)g(of)i(double-ne)o(gation)19 b(translation)j(so)i
(that)g(one)f(can)523 4327 y(meaningful)h(state)k(properties)d(for)h
(all)h Ft(possible)g FA(double-ne)o(gation)22 b(translation.)k(Ho)n(we)
n(v)o(er)m(,)e(we)523 4426 y(ha)n(v)o(e)f(been)f(unable)h(to)g
(formulate)f(such)h(a)g(general)g(notion.)f(Then)g(we)i(ha)n(v)o(e)e
(to)i(cate)o(gorise)e(ho)n(w)523 4526 y(the)k(double-ne)o(gation)c
(translations)k(beha)n(v)o(e)f(under)f(cut-elimination.)g(Finally)-5
b(,)26 b(one)f(has)i(to)f(es-)523 4625 y(tablish)e(the)h
(\223simulation-property\224\227which)19 b(unfortunately)i(f)o(ails)k
(if)f(one)g(tak)o(es)h(a)f(v)o(ery)f(na)n(\250)-26 b(\021v)o(e)523
4725 y(vie)n(w)17 b(on)g(sequent-proofs)d(and)i(cut-elimination.)f(One)
i(problem)e(is)j(that)f(double-ne)o(gation)c(trans-)523
4825 y(lations)29 b(might)g(introduce)e(auxiliary)h(cuts.)h(Such)g
(auxiliary)f(cuts)h(only)g(occur)f(in)h(the)g(double-)523
4924 y(ne)o(gated)22 b(proofs)g(and)i(are)f(thus)h(not)g(tak)o(en)f
(account)g(of)g(by)g(colour)n(-annotations.)d(Therefore)i(we)p
eop end
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