Publications/nom-cade-05.ps
author Christian Urban <christian dot urban at kcl dot ac dot uk>
Sun, 20 Mar 2016 18:36:07 +0000
changeset 399 4d5c31bb78c5
parent 14 680070975206
permissions -rw-r--r--
added Nominal packages

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TeXDict begin 1 0 bop Black Black Black Black 859 455
a Fz(Nominal)46 b(T)-11 b(ec)l(hniques)45 b(in)g(Isab)t(elle/HOL)3049
412 y Fy(?)1248 744 y Fx(Christian)27 b(Urban)1842 713
y Fw(1)1906 744 y Fx(and)h(Christine)f(T)-7 b(asson)2677
713 y Fw(2)608 886 y Fv(1)681 918 y Fu(Ludwig-Maximilians-Univ)n(ersit)
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1295 a Fs(Abstract.)p Black 43 w Fu(In)c(this)h(pap)r(er)h(w)n(e)f
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2951 y(concrete)22 b(names)h(for)g(binders)g(\(that)h(is)f(they)g
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3250 y(ab)r(out)22 b(substitution\))g(that)g(are)f(only)g(lo)r(osely)g
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3350 y(examples)27 b(from)g(the)h(literature:)f(V)-7
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(\(using)g(a)g(de-Bruijn)523 4160 y(represen)n(tation\))26
b(and)i(concluded)f([1,)g(p.)h(26]:)p Black Black 664
4313 a Fo(\\When)42 b(doing)i(the)f(formalization,)i(I)d(disc)l(over)l
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(with)523 1419 y(nominal)23 b(abstractions.)e(The)i(second)g(prop)r
(ert)n(y)e(ensures)i(that)g(nominal)g(abstractions)e(b)r(e-)523
1518 y(ha)n(v)n(e,)j(roughly)f(sp)r(eaking,)h(lik)n(e)g(lam)n(b)r
(da-abstractions.)e(T)-7 b(o)24 b(see)h(this)g(reconsider)e(the)i
(rules)523 1618 y Fl(\031)588 1630 y Fj(\025)p Fw(1)687
1618 y Fx(and)d Fl(\031)908 1630 y Fj(\025)p Fw(2)1006
1618 y Fx(giv)n(en)g(in)g(Fig.)h(1,)f(whic)n(h)g(can)g(b)r(e)g(used)h
(to)f(decide)g(when)h(t)n(w)n(o)e(lam)n(b)r(da-terms)523
1718 y(are)27 b Fm(\013)p Fx(-equiv)-5 b(alen)n(t.)27
b(Prop)r(ert)n(y)f(\()p Fm(ii)p Fx(\))i(paraphrases)d(these)j(rules)f
(for)g(nominal)h(abstractions.)523 1817 y(The)33 b(similarities,)f(ho)n
(w)n(ev)n(er,)f(do)h(not)h(end)g(here:)g(giv)n(en)f(a)g([)p
Fm(a)p Fx(])p Fm(:x)i Fx(with)f Fm(x)f Fl(2)g Fo(fs-pset)q
Fx(,)h(then)523 1917 y(freshness)27 b(b)r(eha)n(v)n(es)f(lik)n(e)i(\()p
Fl(\000)p Fx(\))9 b Fl(62)23 b Fe(fv)p Fx(\()p Fl(\000)p
Fx(\),)k(as)g(sho)n(wn)g(next:)p Black 523 2085 a Ff(Lemma)j(6.)p
Black 40 w Fx(Giv)n(en)e Fm(a)23 b Fl(6)p Fx(=)f Fm(b)28
b Fx(and)f Fm(x)d Fl(2)k Fo(fs-pset)p Fx(,)g(then)p Black
535 2244 a(\(i\))p Black 42 w Fm(a)23 b Fx(#)g([)p Fm(b)p
Fx(])p Fm(:x)28 b Fx(if)g(and)g(only)f(if)h Fm(a)23 b
Fx(#)g Fm(x)p Fx(,)29 b(and)p Black 512 2342 a(\(ii\))p
Black 42 w Fm(a)23 b Fx(#)g([)p Fm(a)p Fx(])p Fm(:x)p
Black 523 2510 a Fo(Pr)l(o)l(of.)p Black 43 w Fx(\(i)p
Fl(\))p Fx(\):)38 b(Since)f Fm(x)j Fl(2)f Fo(fs-pset)p
Fx(,)f Fe(supp)n Fx(\([)p Fm(b)p Fx(])p Fm(:x)p Fx(\))i
Fl(\022)e Fe(supp)o Fx(\()p Fm(x)p Fx(\))25 b Fl([)h(f)p
Fm(b)p Fl(g)35 b Fx(and)i(therefore)g(the)523 2610 y(supp)r(ort)31
b(of)g([)p Fm(a)p Fx(])p Fm(:x)g Fx(m)n(ust)g(b)r(e)h(\014nite.)f
(Hence)g(\()p Fm(a;)14 b(b;)g(x;)g Fx([)p Fm(b)p Fx(])p
Fm(:x)p Fx(\))32 b(is)e(\014nitely)i(supp)r(orted)f(and)f(b)n(y)523
2709 y(Lem.)24 b(5)g(there)f(exists)h(a)g Fm(c)g Fx(with)g(\()p
Fl(\003)p Fx(\))g Fm(c)f Fx(#)g(\()p Fm(a;)14 b(b;)g(x;)g
Fx([)p Fm(b)p Fx(])p Fm(:x)p Fx(\).)25 b(Using)f(the)g(assumption)g
Fm(a)e Fx(#)i([)p Fm(b)p Fx(])p Fm(:x)523 2809 y Fx(and)h(the)h(fact)g
(that)g Fm(c)d Fx(#)g([)p Fm(b)p Fx(])p Fm(:x)j Fx(\(from)f
Fl(\003)p Fx(\),)h(Lem.)f(3)g(and)h(Def.)g(5\()p Fm(i)p
Fx(\))f(giv)n(e)g([)p Fm(b)p Fx(])p Fm(:x)e Fx(=)g(\()p
Fm(c)14 b(a)p Fx(\)[)p Fm(b)p Fx(])p Fm(:x)23 b Fx(=)523
2909 y([)p Fm(b)p Fx(])p Fm(:)p Fx(\()p Fm(c)14 b(a)p
Fx(\))786 2918 y Fi(\001)824 2909 y Fm(x)p Fx(.)37 b(Hence)g(b)n(y)f
(Def.)h(5\()p Fm(ii)p Fx(\))e Fm(x)j Fx(=)g(\()p Fm(c)14
b(a)p Fx(\))2041 2918 y Fi(\001)2079 2909 y Fm(x)p Fx(.)37
b(No)n(w)f Fm(c)h Fx(#)h Fm(x)f Fx(\(from)f Fl(\003)p
Fx(\))g(implies)h(that)523 3008 y Fm(c)c Fx(#)g(\()p
Fm(c)14 b(a)p Fx(\))852 3017 y Fi(\001)891 3008 y Fm(x)p
Fx(;)34 b(and)f(mo)n(ving)g(the)g(p)r(erm)n(utation)h(to)f(the)h(other)
f(side)h(b)n(y)f(Lem.)g(2\(ii\))h(giv)n(es)523 3108 y
Fm(a)23 b Fx(#)g Fm(x)p Fx(.)g(\(i)p Fl(\()p Fx(\):)g(F)-7
b(rom)22 b(\()p Fl(\003)p Fx(\),)h Fm(c)g Fx(#)g([)p
Fm(b)p Fx(])p Fm(:x)g Fx(and)f(therefore)f(b)n(y)h(Lem.)h(2\(ii\))f(\()
p Fm(a)14 b(c)p Fx(\))2827 3117 y Fi(\001)2866 3108 y
Fm(c)23 b Fx(#)g(\()p Fm(a)14 b(c)p Fx(\))p Fm(:)p Fx(\([)p
Fm(b)p Fx(])p Fm(:x)p Fx(\),)523 3207 y(whic)n(h)28 b(implies)h(b)n(y)f
(Def.)h(5\()p Fm(i)p Fx(\))f(that)h Fm(a)24 b Fx(#)g([)p
Fm(b)p Fx(])p Fm(:)p Fx(\(\()p Fm(a)14 b(c)p Fx(\))2137
3216 y Fi(\001)2176 3207 y Fm(x)p Fx(\).)29 b(F)-7 b(rom)28
b(\()p Fl(\003)p Fx(\))g Fm(c)c Fx(#)h Fm(x)j Fx(holds)g(and)g(from)523
3307 y(the)33 b(assumption)f(also)f Fm(a)g Fx(#)g Fm(x)p
Fx(;)i(then)g(Lem.)g(3)e(implies)i(that)g Fm(x)e Fx(=)g(\()p
Fm(a)14 b(c)p Fx(\))2849 3316 y Fi(\001)2887 3307 y Fm(x)p
Fx(,)33 b(and)f(one)g(can)523 3407 y(conclude)27 b(with)h
Fm(a)23 b Fx(#)h([)p Fm(b)p Fx(])p Fm(:x)p Fx(.)523 3506
y(\(ii\):)29 b(By)e Fm(c)d Fx(#)f Fm(x)29 b Fx(and)e
Fm(c)d Fl(6)p Fx(=)f Fm(a)k Fx(\(b)r(oth)i(from)e Fl(\003)p
Fx(\))h(w)n(e)g(can)f(use)h(\(i\))g(to)g(infer)g Fm(c)23
b Fx(#)h([)p Fm(a)p Fx(])p Fm(:x)p Fx(.)k(F)-7 b(urther,)523
3606 y(from)33 b(Lem.)h(2\(ii\))f(it)h(holds)f(that)h(\()p
Fm(c)14 b(a)p Fx(\))1781 3615 y Fi(\001)1819 3606 y Fm(c)33
b Fx(#)g(\()p Fm(c)14 b(a)p Fx(\))2148 3615 y Fi(\001)2186
3606 y Fx([)p Fm(a)p Fx(])p Fm(:x)p Fx(.)34 b(This)f(is)h
Fm(a)e Fx(#)h([)p Fm(c)p Fx(])p Fm(:)p Fx(\()p Fm(c)14
b(a)p Fx(\))3129 3615 y Fi(\001)3168 3606 y Fm(x)33 b
Fx(using)523 3706 y(Def.)41 b(5\()p Fm(i)p Fx(\).)g(Since)f
Fm(c)45 b Fl(6)p Fx(=)f Fm(a)p Fx(,)c Fm(c)k Fx(#)h Fm(x)c
Fx(and)f(\()p Fm(c)14 b(a)p Fx(\))2051 3715 y Fi(\001)2090
3706 y Fm(x)45 b Fx(=)f(\()p Fm(c)14 b(a)p Fx(\))2449
3715 y Fi(\001)2487 3706 y Fm(x)p Fx(,)41 b(Def.)g(5\()p
Fm(ii)p Fx(\))f(implies)h(that)523 3805 y([)p Fm(c)p
Fx(])p Fm(:)p Fx(\()p Fm(c)14 b(a)p Fx(\))786 3814 y
Fi(\001)825 3805 y Fm(x)23 b Fx(=)g([)p Fm(a)p Fx(])p
Fm(:x)p Fx(.)28 b(Therefore,)e Fm(a)d Fx(#)h([)p Fm(a)p
Fx(])p Fm(:x)p Fx(.)1447 b Fl(u)-55 b(t)648 3973 y Fx(The)32
b(functions)g(from)g Fk(A)55 b Fx(to)33 b Fm(\010)f Fx(w)n(e)g(iden)n
(tify)h(next)f(satisfy)g(the)h(nominal)e(abstraction)523
4073 y(prop)r(erties.)c(Let)h([)p Fm(a)p Fx(])p Fm(:t)f
Fx(b)r(e)h(de\014ned)g(as)f(follo)n(ws)775 4274 y([)p
Fm(a)p Fx(])p Fm(:t)941 4227 y Fw(def)952 4274 y Fx(=)34
b Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)22 b Fm(a)h Fx(=)g
Fm(b)g Fe(then)e Fm(t)i Fe(else)e(if)i Fm(b)f Fx(#)i
Fm(t)f Fe(then)e Fx(\()p Fm(a)14 b(b)p Fx(\))2732 4283
y Fi(\001)2770 4274 y Fm(t)23 b Fe(else)f(er)o Fx(\))h
Fm(:)146 b Fx(\(2\))523 4450 y(This)25 b(op)r(eration)f(tak)n(es)g(t)n
(w)n(o)g(argumen)n(ts:)f(an)i Fm(a)e Fl(2)g Fk(A)48 b
Fx(and)25 b(a)g Fm(t)e Fl(2)g Fm(\010)p Fx(.)i(T)-7 b(o)25
b(see)f(ho)n(w)h(this)g(op)r(er-)523 4549 y(ation)h(enco)r(des)f(an)h
Fm(\013)p Fx(-equiv)-5 b(alence)26 b(class,)f(consider)h(the)g
Fm(\013)p Fx(-equiv)-5 b(alence)26 b(class)f([)p Fm(\025a:)p
Fx(\()p Fm(a)f(b)p Fx(\)])3391 4561 y Fj(\013)523 4649
y Fx(and)35 b(the)g(corresp)r(onding)d Fm(\010)p Fx(-term)j([)p
Fm(a)p Fx(])p Fm(:)p Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\))35
b(\(for)g(the)g(momen)n(t)f(w)n(e)h(ignore)e(the)i(term)523
4749 y(constructor)c Fe(se)g Fx(and)h(only)g(consider)f(the)i(function)
g(giv)n(en)e(b)n(y)h([)p Fm(a)p Fx(])p Fm(:)p Fe(pr)o
Fx(\()p Fm(a;)14 b(b)p Fx(\)\).)33 b(The)f(graph)523
4848 y(of)27 b(this)h(function)g(is)f(as)g(follo)n(ws:)f(the)i(atom)f
Fm(a)g Fx(is)h(mapp)r(ed)f(to)h Fe(pr)o Fx(\()p Fm(a;)14
b(b)p Fx(\))27 b(since)g(the)h(\014rst)f Fe(if)o Fx(-)523
4948 y(condition)e(is)h(true.)f(F)-7 b(or)25 b Fm(b)p
Fx(,)h(the)g(\014rst)f Fe(if)o Fx(-condition)g(ob)n(viously)g(fails,)g
(but)h(also)f(the)h(second)p Black Black eop end
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b(fails,)h(b)r(ecause)f Fm(b)e Fl(2)h Fe(supp)n Fx(\()p
Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\)\);)32 b(therefore)f
Fm(b)g Fx(is)h(mapp)r(ed)f(to)h Fe(er)o Fx(.)g(F)-7 b(or)30
b(all)i(other)523 548 y(atoms)d Fm(c)p Fx(,)h(w)n(e)f(ha)n(v)n(e)g
Fm(a)e Fl(6)p Fx(=)f Fm(c)k Fx(and)f Fm(c)e Fx(#)g Fe(pr)o
Fx(\()p Fm(a;)14 b(b)p Fx(\);)30 b(so)f(the)h Fm(c)p
Fx('s)g(are)f(mapp)r(ed)h(b)n(y)f(the)i(function)523
648 y(to)c(\()p Fm(a)14 b(c)p Fx(\))782 657 y Fi(\001)820
648 y Fe(pr)o Fx(\()p Fm(a;)g(b)p Fx(\),)27 b(whic)n(h)g(is)f(just)i
Fe(pr)o Fx(\()p Fm(c;)14 b(b)p Fx(\).)27 b(Clearly)-7
b(,)26 b(the)h(function)g(returns)f Fe(er)g Fx(whenev)n(er)523
747 y(the)k(corresp)r(onding)e(lam)n(b)r(da-term)h(is)h
Fo(not)37 b Fx(in)30 b(the)g Fm(\013)p Fx(-equiv)-5 b(alence)30
b(class|in)f(this)h(exam-)523 847 y(ple)f Fm(\025b:)p
Fx(\()p Fm(b)24 b(b)p Fx(\))g Fl(62)h Fx([)p Fm(\025a:)p
Fx(\()p Fm(a)g(b)p Fx(\)])1359 859 y Fj(\013)1406 847
y Fx(;)j(in)h(all)f(other)g(cases,)f(ho)n(w)n(ev)n(er,)f(it)j(returns)f
(an)g(appropriately)523 946 y(\\renamed")e(v)n(ersion)g(of)i
Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\).)p Black 523 1167
a Ff(Lemma)30 b(7.)p Black 40 w Fx(The)e(op)r(eration)f([)p
Fl(\000)p Fx(])p Fm(:)p Fx(\()p Fl(\000)p Fx(\))g(giv)n(en)g(for)g
Fm(\010)h Fx(in)g(\(2\))g(is)f(a)g(nominal)h(abstraction.)p
Black 523 1378 a Fo(Pr)l(o)l(of.)p Black 43 w Fx(Def.)h(5\()p
Fm(i)p Fx(\))e(follo)n(ws)g(from)g(the)h(calculation:)p
Black Black 643 1571 a Fm(\031)693 1580 y Fi(\001)732
1571 y Fx([)p Fm(a)p Fx(])p Fm(:t)533 1650 y Fw(def)544
1697 y Fx(=)34 b Fm(\031)693 1706 y Fi(\001)732 1697
y Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)22 b Fm(a)h Fx(=)g
Fm(b)g Fe(then)e Fm(t)i Fe(else)e(if)i Fm(b)f Fx(#)i
Fm(t)f Fe(then)e Fx(\()p Fm(a)14 b(b)p Fx(\))2413 1706
y Fi(\001)2451 1697 y Fm(t)23 b Fe(else)f(er)o Fx(\))533
1776 y Fw(def)544 1823 y Fx(=)34 b Fe(se)o Fx(\()p Fm(\025b:)14
b(\031)933 1832 y Fi(\001)972 1823 y Fe(if)22 b Fm(a)h
Fx(=)g Fm(\031)1287 1792 y Fg(\000)p Fw(1)1376 1832 y
Fi(\001)1415 1823 y Fm(b)f Fe(then)g Fm(t)h Fe(else)e(if)h
Fm(\031)2081 1792 y Fg(\000)p Fw(1)2171 1832 y Fi(\001)2209
1823 y Fm(b)h Fx(#)g Fm(t)g Fe(then)e Fx(\()p Fm(a)28
b(\031)2764 1792 y Fg(\000)p Fw(1)2854 1832 y Fi(\001)2892
1823 y Fm(b)p Fx(\))2960 1832 y Fi(\001)2998 1823 y Fm(t)23
b Fe(else)e(er)o Fx(\))544 1922 y(=)34 b Fe(se)o Fx(\()p
Fm(\025b:)14 b Fe(if)23 b Fm(a)g Fx(=)f Fm(\031)1198
1892 y Fg(\000)p Fw(1)1288 1931 y Fi(\001)1326 1922 y
Fm(b)h Fe(then)e Fm(\031)1632 1931 y Fi(\001)1671 1922
y Fm(t)i Fe(else)e(if)h Fm(b)h Fx(#)g Fm(\031)2232 1931
y Fi(\001)2271 1922 y Fm(t)g Fe(then)e Fm(\031)2571 1931
y Fi(\001)2610 1922 y Fx(\()p Fm(a)27 b(\031)2763 1892
y Fg(\000)p Fw(1)2853 1931 y Fi(\001)2891 1922 y Fm(b)p
Fx(\))2959 1931 y Fi(\001)2997 1922 y Fm(t)c Fe(else)f(er)o
Fx(\))i(\()p Fl(\003)p Fx(\))544 2022 y(=)34 b Fe(se)o
Fx(\()p Fm(\025b:)14 b Fe(if)23 b Fm(a)g Fx(=)f Fm(\031)1198
1992 y Fg(\000)p Fw(1)1288 2031 y Fi(\001)1326 2022 y
Fm(b)h Fe(then)e Fm(\031)1632 2031 y Fi(\001)1671 2022
y Fm(t)i Fe(else)e(if)h Fm(b)h Fx(#)g Fm(\031)2232 2031
y Fi(\001)2271 2022 y Fm(t)g Fe(then)e Fx(\()p Fm(\031)2603
2031 y Fi(\001)2642 2022 y Fm(a)27 b(b)p Fx(\))2781 2031
y Fi(\001)2820 2022 y Fm(\031)2870 2031 y Fi(\001)2908
2022 y Fm(t)c Fe(else)f(er)o Fx(\))544 2121 y(=)34 b
Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)23 b Fm(\031)1044
2130 y Fi(\001)1082 2121 y Fm(a)g Fx(=)g Fm(b)g Fe(then)e
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Fm(b)h Fx(#)g Fm(\031)2143 2130 y Fi(\001)2182 2121 y
Fm(t)g Fe(then)e Fx(\()p Fm(\031)2514 2130 y Fi(\001)2553
2121 y Fm(a)27 b(b)p Fx(\))2692 2130 y Fi(\001)2730 2121
y Fm(\031)2780 2130 y Fi(\001)2819 2121 y Fm(t)c Fe(else)e(er)p
Fx(\))533 2200 y Fw(def)544 2247 y Fx(=)34 b([)p Fm(\031)716
2256 y Fi(\001)755 2247 y Fm(a)p Fx(])p Fm(:)p Fx(\()p
Fm(\031)927 2256 y Fi(\001)965 2247 y Fm(t)p Fx(\))523
2433 y(where)25 b(w)n(e)g(use)g(in)h(\()p Fl(\003)p Fx(\))g(the)g(fact)
f(that)h Fm(\031)1779 2442 y Fi(\001)1817 2433 y Fe(if)p
Fm(:::)p Fe(then)n Fm(:::)p Fe(else)o Fm(:::)d Fx(=)f
Fe(if)p Fm(:::)p Fe(then)f Fm(\031)2975 2442 y Fi(\001)3014
2433 y Fm(:::)p Fe(else)g Fm(\031)3330 2442 y Fi(\001)3369
2433 y Fm(:::)523 2533 y Fx(and)33 b(Lem)f(2\(ii\).)i(In)f(case)e
Fm(a)h Fx(=)f Fm(b)p Fx(,)i(Def.)g(5\()p Fm(ii)p Fx(\))g(is)f(b)n(y)h
(a)f(simple)h(calculation)f(using)g(exten-)523 2632 y(sionalit)n(y)d
(of)g(functions.)h(In)g(case)f Fm(a)d Fl(6)p Fx(=)g Fm(b)j
Fx(and)h(Def.)e(5\()p Fm(ii)22 b Fl(\))p Fx(\))q(,)29
b(the)h(follo)n(wing)f(form)n(ula)f(can)523 2732 y(b)r(e)g(deriv)n(ed)f
(from)g(the)h(assumption)f(b)n(y)g(extensionalit)n(y:)890
2933 y Fl(8)p Fm(c:)46 b Fe(if)23 b Fm(a)f Fx(=)h Fm(c)g
Fe(then)f Fm(t)1594 2945 y Fw(1)1654 2933 y Fe(else)f(if)h
Fm(c)h Fx(#)h Fm(t)2143 2945 y Fw(1)2203 2933 y Fe(then)d
Fx(\()p Fm(a)14 b(c)p Fx(\))2558 2942 y Fi(\001)2597
2933 y Fm(t)2627 2945 y Fw(1)2687 2933 y Fe(else)21 b(er)i
Fx(=)1042 3033 y Fe(if)g Fm(b)f Fx(=)h Fm(c)g Fe(then)e
Fm(t)1585 3045 y Fw(2)1646 3033 y Fe(else)g(if)h Fm(c)h
Fx(#)g Fm(t)2134 3045 y Fw(2)2195 3033 y Fe(then)e Fx(\()p
Fm(b)14 b(c)p Fx(\))2542 3042 y Fi(\001)2580 3033 y Fm(t)2610
3045 y Fw(2)2671 3033 y Fe(else)21 b(er)523 3239 y Fx(Instan)n(tiating)
27 b(this)h(form)n(ula)f(once)g(with)h Fm(a)f Fx(and)h(once)f(with)h
Fm(b)f Fx(yields)h(the)g(t)n(w)n(o)e(equations)1345 3435
y Fm(t)1375 3447 y Fw(1)1436 3435 y Fx(=)c Fe(if)g Fm(a)h
Fx(#)h Fm(t)1823 3447 y Fw(2)1883 3435 y Fe(then)d Fx(\()p
Fm(b)14 b(a)p Fx(\))2238 3444 y Fi(\001)2276 3435 y Fm(t)2306
3447 y Fw(2)2367 3435 y Fe(else)21 b(er)1345 3534 y Fm(t)1375
3546 y Fw(2)1436 3534 y Fx(=)h Fe(if)g Fm(b)h Fx(#)g
Fm(t)1814 3546 y Fw(1)1875 3534 y Fe(then)e Fx(\()p Fm(a)14
b(b)p Fx(\))2230 3543 y Fi(\001)2268 3534 y Fm(t)2298
3546 y Fw(1)2359 3534 y Fe(else)21 b(er)523 3741 y Fx(Next,)27
b(one)g(distinguishes)f(t)n(w)n(o)g(cases)g(where)g Fm(a)d
Fx(#)g Fm(t)2181 3753 y Fw(2)2245 3741 y Fx(and)k Fl(:)14
b Fm(a)23 b Fx(#)g Fm(t)2664 3753 y Fw(2)2702 3741 y
Fx(,)k(resp)r(ectiv)n(ely)-7 b(.)26 b(In)h(the)523 3840
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Black Black eop end
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userdict /end-hook known{end-hook}if
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