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2010 E001FF0FF00001FFFE00003F9FF00007FFF000000FFFF0000FFFC0000007FFF0001FFF00 |
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2011 000003FFF0003FFE00000001FFF0007FFC00000000FFF000FFF8000000007FF001FFF000 |
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2012 0000003FF003FFE0000000001FF007FFE0000000001FF007FFC0000000000FF00FFF8000 |
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2013 0000000FF00FFF800000000007F01FFF000000000007F01FFF000000000003F03FFF0000 |
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2014 00000003F03FFE000000000003F03FFE000000000003F07FFE000000000001F07FFE0000 |
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2015 00000001F07FFE000000000001F07FFC00000000000000FFFC00000000000000FFFC0000 |
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2016 0000000000FFFC00000000000000FFFC00000000000000FFFC00000000000000FFFC0000 |
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2017 0000000000FFFC00000000000000FFFC00000000000000FFFC00000000000000FFFC0000 |
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2018 0000000000FFFC00000000000000FFFC00000000000000FFFC00000000000000FFFC0000 |
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|
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4780 n(y)26 b(Munic)n(h)f(\()p Ft(urban@mathematik.uni-muenchen.)q(de)p |
|
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|
4782 Ft(tasson@dptmaths.ens-cachan.fr)p Fu(\))p Black 759 |
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4807 h(The)h(\\lo)n(w-lev)n(el")c(represen)n(tations)h(use)523 |
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4864 (w)n(ork)f(is)h(based)g(on)g(the)h(nominal)f(logic)f(w)n(ork)523 |
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4866 b(The)h(main)f(tec)n(hnical)g(no)n(v)n(elt)n(y)f(is)i(that)f(our)g(w)n |
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4870 (able)g(to)h(implemen)n(t)g(all)f(results)g(in)h(Isab)r(elle/HOL)f(and) |
|
4871 g(formalise)523 1878 y(the)41 b(simple)g(Ch)n(urc)n(h-Rosser)d(pro)r |
|
4872 (of)i(of)h(T)-7 b(ait)41 b(and)f(Martin-L\177)-42 b(of)40 |
|
4873 b(describ)r(ed)h(in)g([3],)f(and)523 1978 y(the)28 b(standard)e(T)-7 |
|
4874 b(ait-st)n(yle)26 b(strong)g(normalisation)g(pro)r(of)g(for)h(the)h |
|
4875 (simply-t)n(yp)r(ed)f(lam)n(b)r(da-)523 2077 y(calculus)g(giv)n(en,)g |
|
4876 (for)g(example,)g(in)h([7,)14 b(17].)648 2177 y(The)36 |
|
4877 b(pap)r(er)f(is)h(organised)f(as)g(follo)n(w:)g(Sec.)i(2)e(reviews)g |
|
4878 Fm(\013)p Fx(-equiv)-5 b(alence)36 b(for)g(lam)n(b)r(da-)523 |
|
4879 2276 y(terms.)d(Sec.)g(3)f(giv)n(es)g(a)h(construction)f(of)h(an)g |
|
4880 (inductiv)n(e)g(set)g(that)g(is)g(bijectiv)n(e)g(with)h(the)523 |
|
4881 2376 y Fm(\013)p Fx(-equated)29 b(lam)n(b)r(da-terms)f(and)h(adapts)f |
|
4882 (some)g(notions)h(of)g(the)g(nominal)g(logic)f(w)n(ork)g(for)523 |
|
4883 2476 y(this)21 b(construction.)e(An)i(induction)g(principle)f(for)g |
|
4884 (this)g(set)h(is)f(deriv)n(ed)f(in)i(Sec.)f(4.)g(Examples)523 |
|
4885 2575 y(of)26 b(Isab)r(elle/HOL)g(formalisations)e(are)i(giv)n(en)f(in)i |
|
4886 (Sec.)g(5.)f(Related)g(w)n(ork)f(is)h(men)n(tioned)h(in)523 |
|
4887 2675 y(Sec.)h(6,)f(and)g(Sec.)h(7)f(concludes.)523 2929 |
|
4888 y Fp(2)112 b(Preliminaries)523 3117 y Fx(In)42 b(order)d(to)j(motiv)-5 |
|
4889 b(ate)41 b(a)g(design)g(c)n(hoice)f(later)h(on,)g(w)n(e)g(b)r(egin)g |
|
4890 (with)h(a)f(review)g(of)g Fm(\013)p Fx(-)523 3217 y(equiv)-5 |
|
4891 b(alence)29 b(cast)g(in)h(terms)f(of)h(the)f(nominal)h(logic)e(w)n |
|
4892 (ork.)g(The)i(set)f(of)h(lam)n(b)r(da-terms)e(is)523 |
|
4893 3316 y(inductiv)n(ely)g(de\014ned)g(b)n(y)f(the)h(grammar:)1508 |
|
4894 3485 y Fm(\003)22 b Fx(:)107 b Fm(t)46 b Fx(::=)f Fm(a)h |
|
4895 Fl(j)h Fm(t)23 b(t)46 b Fl(j)g Fm(\025a:t)523 3653 y |
|
4896 Fx(where)22 b Fm(a)g Fx(is)g(an)h(atom)f(dra)n(wn)f(from)h(a)g(coun)n |
|
4897 (table)g(in\014nite)h(set,)f(whic)n(h)h(will)f(in)h(what)f(follo)n(ws) |
|
4898 523 3752 y(b)r(e)28 b(denoted)g(b)n(y)f Fk(A)17 b Fx(.)648 |
|
4899 3852 y(The)37 b(notion)g(of)g Fm(\013)p Fx(-equiv)-5 |
|
4900 b(alence)37 b(for)f Fm(\003)h Fx(is)h(often)f(de\014ned)h(as)e(the)i |
|
4901 (least)e(congruence)523 3952 y(of)i(the)g(equation)f |
|
4902 Fm(\025a:t)j Fx(=)1380 3964 y Fj(\013)1466 3952 y Fm(\025b:t)p |
|
4903 Fx([)p Fm(a)g Fx(:=)g Fm(b)p Fx(])d(in)n(v)n(olving)f(a)h(renaming)g |
|
4904 (substitution)h(and)f(a)523 4051 y(side-condition,)g(namely)h(that)g |
|
4905 Fm(b)g Fx(do)r(es)g(not)g(o)r(ccur)f(freely)h(in)g Fm(t)p |
|
4906 Fx(.)h(In)f(the)g(nominal)g(logic)523 4151 y(w)n(ork,)e(ho)n(w)n(ev)n |
|
4907 (er,)g(atoms)g(are)h(manipulated)g(not)h(b)n(y)f(renaming)f |
|
4908 (substitutions,)i(but)g(b)n(y)523 4251 y(p)r(erm)n(utations|bijectiv)n |
|
4909 (e)44 b(mappings)f(from)h(atoms)f(to)h(atoms.)f(While)i(p)r(erm)n |
|
4910 (utations)523 4350 y(ha)n(v)n(e)30 b(some)g(tec)n(hnical)g(adv)-5 |
|
4911 b(an)n(tages,)29 b(for)h(example)g(they)h(preserv)n(e)e |
|
4912 Fm(\013)p Fx(-equiv)-5 b(alence)30 b(whic)n(h)523 4450 |
|
4913 y(substitutions)f(do)f(not)h([18],)f(their)g(primary)g(reason)e(in)j |
|
4914 (the)g(nominal)f(logic)g(w)n(ork)f(is)i(that)523 4549 |
|
4915 y(one)j(can)g(use)h(them)g(to)f(de\014ne)h(the)g(notion)f(of)g |
|
4916 Fo(supp)l(ort)p Fx(.)h(This)f(notion)h(generalises)d(what)523 |
|
4917 4649 y(is)38 b(mean)n(t)f(b)n(y)g(the)h(set)g(of)g(free)f(atoms)g(of)g |
|
4918 (an)h(ob)5 b(ject,)37 b(whic)n(h)h(is)f(usually)g(clear)g(in)h(case)523 |
|
4919 4749 y(the)29 b(ob)5 b(ject)29 b(is)g(an)g(abstract)f(syn)n(tax)g |
|
4920 (tree,)g(but)i(less)e(so)h(if)g(the)g(ob)5 b(ject)29 |
|
4921 b(is)g(a)f(function.)i(The)523 4848 y(generalisation)i(of)i(\\free)f |
|
4922 (atoms")g(to)h(functions,)h(ho)n(w)n(ev)n(er,)d(will)i(pla)n(y)g(a)f |
|
4923 (crucial)h(r^)-42 b(ole)33 b(in)523 4948 y(our)27 b(construction)g(of)g |
|
4924 (the)h(bijectiv)n(e)g(set.)p Black Black eop end |
|
4925 %%Page: 3 3 |
|
4926 TeXDict begin 3 2 bop Black Black 648 448 a Fx(There)27 |
|
4927 b(are)g(sev)n(eral)g(w)n(a)n(ys)f(for)i(de\014ning)g(the)h(op)r |
|
4928 (eration)e(of)h(a)g(p)r(erm)n(utation)g(acting)f(on)523 |
|
4929 548 y(a)f(lam)n(b)r(da-term.)g(One)h(w)n(a)n(y)e([18])h(that)h(can)g(b) |
|
4930 r(e)g(easily)f(implemen)n(ted)h(in)g(Isab)r(elle/HOL)f(is)523 |
|
4931 648 y(to)g(represen)n(t)e(p)r(erm)n(utations)h(as)g(\014nite)h(lists)g |
|
4932 (whose)f(elemen)n(ts)g(are)g(sw)n(appings)f(\(i.e.,)i(pairs)523 |
|
4933 747 y(of)i(atoms\).)g(W)-7 b(e)28 b(write)g(suc)n(h)g(p)r(erm)n |
|
4934 (utation)f(as)h(\()p Fm(a)2117 759 y Fw(1)2168 747 y |
|
4935 Fm(b)2204 759 y Fw(1)2241 747 y Fx(\)\()p Fm(a)2349 759 |
|
4936 y Fw(2)2400 747 y Fm(b)2436 759 y Fw(2)2473 747 y Fx(\))14 |
|
4937 b Fl(\001)g(\001)g(\001)g Fx(\()p Fm(a)2706 759 y Fj(n)2765 |
|
4938 747 y Fm(b)2801 759 y Fj(n)2846 747 y Fx(\);)28 b(the)h(empt)n(y)f |
|
4939 (list)523 847 y([])34 b(stands)e(for)h(the)h(iden)n(tit)n(y)f(p)r(erm)n |
|
4940 (utation.)h(The)f(p)r(erm)n(utation)g(action,)g(written)g |
|
4941 Fm(\031)3247 856 y Fi(\001)3286 847 y Fx(\()p Fl(\000)p |
|
4942 Fx(\),)523 946 y(can)27 b(then)h(b)r(e)g(de\014ned)g(on)g(lam)n(b)r |
|
4943 (da-terms)e(as:)1135 1135 y([])1181 1144 y Fi(\001)1220 |
|
4944 1135 y Fm(a)1287 1088 y Fw(def)1298 1135 y Fx(=)34 b |
|
4945 Fm(a)798 1335 y Fx(\()p Fm(a)874 1347 y Fw(1)926 1335 |
|
4946 y Fm(a)970 1347 y Fw(2)1007 1335 y Fx(\))23 b(::)g Fm(\031)1181 |
|
4947 1344 y Fi(\001)1220 1335 y Fm(a)1287 1288 y Fw(def)1298 |
|
4948 1335 y Fx(=)1397 1165 y Fh(8)1397 1240 y(<)1397 1389 |
|
4949 y(:)1482 1235 y Fm(a)1526 1247 y Fw(2)1638 1235 y Fx(if)g |
|
4950 Fm(\031)1759 1244 y Fi(\001)1798 1235 y Fm(a)g Fx(=)f |
|
4951 Fm(a)1996 1247 y Fw(1)1482 1334 y Fm(a)1526 1346 y Fw(1)1638 |
|
4952 1334 y Fx(if)h Fm(\031)1759 1343 y Fi(\001)1798 1334 |
|
4953 y Fm(a)g Fx(=)f Fm(a)1996 1346 y Fw(2)1482 1434 y Fm(\031)1532 |
|
4954 1443 y Fi(\001)1571 1434 y Fm(a)h Fx(otherwise)2267 1222 |
|
4955 y Fm(\031)2317 1231 y Fi(\001)2356 1222 y Fx(\()p Fm(t)2418 |
|
4956 1234 y Fw(1)2478 1222 y Fm(t)2508 1234 y Fw(2)2546 1222 |
|
4957 y Fx(\))2601 1175 y Fw(def)2612 1222 y Fx(=)34 b(\()p |
|
4958 Fm(\031)2793 1231 y Fi(\001)2832 1222 y Fm(t)2862 1234 |
|
4959 y Fw(1)2922 1222 y Fm(\031)2972 1231 y Fi(\001)3011 1222 |
|
4960 y Fm(t)3041 1234 y Fw(2)3078 1222 y Fx(\))2280 1347 y |
|
4961 Fm(\031)2330 1356 y Fi(\001)2368 1347 y Fx(\()p Fm(\025a:t)p |
|
4962 Fx(\))2601 1300 y Fw(def)2612 1347 y Fx(=)g Fm(\025)p |
|
4963 Fx(\()p Fm(\031)2841 1356 y Fi(\001)2880 1347 y Fm(a)p |
|
4964 Fx(\))p Fm(:)p Fx(\()p Fm(\031)3061 1356 y Fi(\001)3100 |
|
4965 1347 y Fm(t)p Fx(\))3332 1272 y(\(1\))523 1620 y(where)g(\()p |
|
4966 Fm(a)14 b(b)p Fx(\))35 b(::)h Fm(\031)i Fx(is)c(the)i(comp)r(osition)e |
|
4967 (of)h(a)f(p)r(erm)n(utation)h(follo)n(w)n(ed)e(b)n(y)i(the)g(sw)n |
|
4968 (apping)523 1720 y(\()p Fm(a)14 b(b)p Fx(\).)30 b(The)g(comp)r(osition) |
|
4969 f(of)g Fm(\031)34 b Fx(follo)n(w)n(ed)28 b(b)n(y)i(another)e(p)r(erm)n |
|
4970 (utation)i Fm(\031)2824 1689 y Fg(0)2877 1720 y Fx(is)g(giv)n(en)f(b)n |
|
4971 (y)g(list-)523 1819 y(concatenation,)c(written)i(as)f |
|
4972 Fm(\031)1516 1789 y Fg(0)1539 1819 y Fx(@)p Fm(\031)s |
|
4973 Fx(,)h(and)f(the)h(in)n(v)n(erse)e(of)h(a)g(p)r(erm)n(utation)g(is)g |
|
4974 (giv)n(en)g(b)n(y)g(list)523 1919 y(rev)n(ersal,)f(written)j(as)f |
|
4975 Fm(\031)1294 1889 y Fg(\000)p Fw(1)1384 1919 y Fx(.)648 |
|
4976 2019 y(While)i(the)h(represen)n(tation)d(of)i(p)r(erm)n(utations)g |
|
4977 (based)f(on)h(lists)g(of)g(sw)n(appings)f(is)h(con-)523 |
|
4978 2119 y(v)n(enien)n(t)f(for)f(de\014nitions)h(lik)n(e)g(p)r(erm)n |
|
4979 (utation)g(comp)r(osition)f(and)h(the)g(in)n(v)n(erse)f(of)h(a)f(p)r |
|
4980 (erm)n(u-)523 2219 y(tation,)32 b(this)g(list-represen)n(tation)e(is)i |
|
4981 (not)g(unique;)g(for)f(example)h(the)g(p)r(erm)n(utation)g(\()p |
|
4982 Fm(a)14 b(a)p Fx(\))523 2318 y(is)24 b(\\equal")f(to)h(the)h(iden)n |
|
4983 (tit)n(y)f(p)r(erm)n(utation.)g(Therefore)f(some)h(means)g(to)g(iden)n |
|
4984 (tify)h(\\equal")523 2418 y(p)r(erm)n(utations)i(is)h(needed.)p |
|
4985 Black 523 2596 a Ff(De\014nition)j(1)p Black 41 w(\(Disagreemen)m(t)45 |
|
4986 b(Set)h(and)h(P)m(erm)m(utation)f(Equalit)m(y\).)h Fo(The)h |
|
4987 Fx(dis-)523 2696 y(agreemen)n(t)28 b(set)j Fo(of)h(two)g(p)l |
|
4988 (ermutations,)g(say)g Fm(\031)2035 2708 y Fw(1)2104 2696 |
|
4989 y Fo(and)f Fm(\031)2313 2708 y Fw(2)2351 2696 y Fo(,)h(is)g(the)f(set)g |
|
4990 (of)h(atoms)g(on)f(which)523 2820 y(the)f(p)l(ermutations)g(disagr)l(e) |
|
4991 l(e,)i(that)e(is)g Fe(ds)o Fx(\()p Fm(\031)1927 2832 |
|
4992 y Fw(1)1965 2820 y Fm(;)14 b(\031)2049 2832 y Fw(2)2086 |
|
4993 2820 y Fx(\))2142 2769 y Fd(def)2154 2820 y Fx(=)35 b |
|
4994 Fl(f)23 b Fm(a)g Fl(j)h Fm(\031)2480 2832 y Fw(1)2517 |
|
4995 2829 y Fi(\001)2555 2820 y Fm(a)g Fl(6)p Fx(=)f Fm(\031)2758 |
|
4996 2832 y Fw(2)2795 2829 y Fi(\001)2833 2820 y Fm(a)h Fl(g)p |
|
4997 Fo(.)30 b(Two)g(p)l(ermu-)523 2920 y(tations)g(ar)l(e)36 |
|
4998 b Fx(equal)p Fo(,)30 b(written)f Fm(\031)1525 2932 y |
|
4999 Fw(1)1586 2920 y Fl(\030)22 b Fm(\031)1720 2932 y Fw(2)1758 |
|
5000 2920 y Fo(,)30 b(pr)l(ovide)l(d)i Fe(ds)o Fx(\()p Fm(\031)2307 |
|
5001 2932 y Fw(1)2345 2920 y Fm(;)14 b(\031)2429 2932 y Fw(2)2466 |
|
5002 2920 y Fx(\))24 b(=)e Fk(?)p Fo(.)648 3097 y Fx(Using)d(the)i(p)r(erm)n |
|
5003 (utation)f(action)f(on)h(lam)n(b)r(da-terms,)f Fm(\013)p |
|
5004 Fx(-equiv)-5 b(alence)20 b(for)f Fm(\003)h Fx(can)g(b)r(e)g(de-)523 |
|
5005 3197 y(\014ned)26 b(in)g(a)f(syn)n(tax)g(directed)h(fashion)f(using)h |
|
5006 (the)g(relations)e(\()p Fl(\000)p Fx(\))9 b Fl(\031)g |
|
5007 Fx(\()p Fl(\000)p Fx(\))26 b(and)g(\()p Fl(\000)p Fx(\))d |
|
5008 Fl(62)h Fe(fv)o Fx(\()p Fl(\000)p Fx(\);)523 3297 y(see)j(Fig.)h(1.)f |
|
5009 (Because)g(of)g(the)h(\\asymmetric")e(rule)h Fl(\031)2234 |
|
5010 3309 y Fj(\025)p Fw(2)2310 3297 y Fx(,)h(it)g(migh)n(t)g(b)r(e)g |
|
5011 (surprising,)e(but:)p Black 523 3475 a Ff(Prop)s(osition)k(1.)p |
|
5012 Black 41 w Fl(\031)f Fo(is)h(an)g(e)l(quivalenc)l(e)h(r)l(elation.)523 |
|
5013 3653 y Fx(The)23 b(pro)r(of)g(of)g(this)g(prop)r(osition)f(is)h |
|
5014 (omitted:)h(it)f(can)g(b)r(e)h(found)f(in)g(a)g(more)f(general)g |
|
5015 (setting)523 3752 y(in)j([18].)g(\(W)-7 b(e)25 b(also)f(omit)h(a)g(pro) |
|
5016 r(of)f(sho)n(wing)g(that)h Fl(\031)g Fx(and)g(=)2408 |
|
5017 3764 y Fj(\013)2480 3752 y Fx(coincide\).)g(In)g(the)g(follo)n(wing,) |
|
5018 523 3877 y([)p Fm(t)p Fx(])599 3889 y Fj(\013)680 3877 |
|
5019 y Fx(will)34 b(stand)f(for)g(the)h Fm(\013)p Fx(-equiv)-5 |
|
5020 b(alence)34 b(class)e(of)i(the)g(lam)n(b)r(da-term)f |
|
5021 Fm(t)p Fx(,)g(that)h(is)g([)p Fm(t)p Fx(])3271 3889 y |
|
5022 Fj(\013)3351 3830 y Fw(def)3362 3877 y Fx(=)523 3977 |
|
5023 y Fl(f)23 b Fm(t)618 3947 y Fg(0)664 3977 y Fl(j)g Fm(t)740 |
|
5024 3947 y Fg(0)786 3977 y Fl(\031)g Fm(t)g Fl(g)p Fx(,)k(and)h |
|
5025 Fm(\003)1239 3992 y Fj(=)p Fg(\031)1356 3977 y Fx(for)f(the)h(set)g |
|
5026 Fm(\003)f Fx(quotien)n(t)h(b)n(y)f Fl(\031)p Fx(.)523 |
|
5027 4246 y Fp(3)112 b(The)38 b(Bijectiv)m(e)d(Set)523 4450 |
|
5028 y Fx(In)25 b(this)f(section,)g(w)n(e)h(will)f(de\014ne)h(a)f(set)g |
|
5029 Fm(\010)p Fx(;)h(inside)g(this)f(set)h(w)n(e)f(will)h(subsequen)n(tly)f |
|
5030 (iden)n(tify)523 4549 y(\(inductiv)n(ely\))g(a)g(subset,)g(called)f |
|
5031 Fm(\003)1639 4561 y Fj(\013)1686 4549 y Fx(,)h(that)g(is)f(in)h |
|
5032 (bijection)h(with)f Fm(\003)2660 4564 y Fj(=)p Fg(\031)2750 |
|
5033 4549 y Fx(.)f(In)h(order)f(to)g(obtain)523 4649 y(the)31 |
|
5034 b(bijection,)h Fm(\010)f Fx(needs)g(to)g(b)r(e)h(de\014ned)f(so)f(that) |
|
5035 i(it)f(con)n(tains)f(elemen)n(ts)h(corresp)r(onding,)523 |
|
5036 4749 y(roughly)i(sp)r(eaking,)g(to)h Fm(\013)p Fx(-equated)g(atoms,)f |
|
5037 (applications)h(and)f(lam)n(b)r(da-abstractions|)523 |
|
5038 4848 y(that)26 b(is)g(to)g([)p Fm(a)p Fx(])973 4860 y |
|
5039 Fj(\013)1021 4848 y Fx(,)g([)p Fm(t)1123 4860 y Fw(1)1160 |
|
5040 4848 y Fm(t)1190 4860 y Fw(2)1227 4848 y Fx(])1250 4860 |
|
5041 y Fj(\013)1324 4848 y Fx(and)g([)p Fm(\025a:t)p Fx(])1675 |
|
5042 4860 y Fj(\013)1723 4848 y Fx(.)g(Whereas)f(this)h(is)g(straigh)n |
|
5043 (tforw)n(ard)d(for)j(atoms)f(and)523 4948 y(applications,)f(the)i(lam)n |
|
5044 (b)r(da-abstractions)d(are)h(non-trivial:)g(for)g(them)i(w)n(e)f(shall) |
|
5045 g(use)g(some)p Black Black eop end |
|
5046 %%Page: 4 4 |
|
5047 TeXDict begin 4 3 bop Black Black Black 523 369 2915 |
|
5048 4 v 523 898 4 530 v 560 503 184 4 v 560 557 a Fr(a)21 |
|
5049 b Fc(\031)g Fr(a)760 517 y Fc(\031)820 525 y Fv(v)m(ar)956 |
|
5050 466 y Fr(t)984 474 y Fv(1)1040 466 y Fc(\031)g Fr(s)1157 |
|
5051 474 y Fv(1)1268 466 y Fr(t)1296 474 y Fv(2)1351 466 y |
|
5052 Fc(\031)h Fr(s)1469 474 y Fv(2)p 956 494 547 4 v 1033 |
|
5053 557 a Fr(t)1061 565 y Fv(1)1108 557 y Fr(t)1136 565 y |
|
5054 Fv(2)1192 557 y Fc(\031)f Fr(s)1309 565 y Fv(1)1356 557 |
|
5055 y Fr(s)1392 565 y Fv(2)1519 504 y Fc(\031)1579 512 y |
|
5056 Fv(app)1836 468 y Fr(t)h Fc(\031)f Fr(s)p 1730 488 380 |
|
5057 4 v 1730 557 a(\025a:t)g Fc(\031)g Fr(\025a:s)2125 502 |
|
5058 y Fc(\031)2185 511 y Fn(\025)p Fv(1)2305 450 y Fr(a)g |
|
5059 Fc(6)p Fu(=)g Fr(b)64 b(t)21 b Fc(\031)g Fu(\()p Fr(a)13 |
|
5060 b(b)p Fu(\))2822 459 y Fi(\001)2860 450 y Fr(s)64 b(a)9 |
|
5061 b Fc(62)g Ft(fv)o Fu(\()p Fr(s)p Fu(\))p 2305 488 938 |
|
5062 4 v 2588 557 a Fr(\025a:t)21 b Fc(\031)g Fr(\025b:s)3258 |
|
5063 502 y Fc(\031)3318 511 y Fn(\025)p Fv(2)619 739 y Fr(a)g |
|
5064 Fc(6)p Fu(=)g Fr(b)p 567 773 281 4 v 567 846 a(a)9 b |
|
5065 Fc(62)g Ft(fv)o Fu(\()p Fr(b)p Fu(\))863 792 y Ft(fv)941 |
|
5066 800 y Fv(v)m(ar)1111 735 y Fr(a)g Fc(62)g Ft(fv)o Fu(\()p |
|
5067 Fr(t)1356 743 y Fv(1)1390 735 y Fu(\))64 b Fr(a)9 b Fc(62)g |
|
5068 Ft(fv)p Fu(\()p Fr(t)1730 743 y Fv(2)1764 735 y Fu(\))p |
|
5069 1111 773 683 4 v 1260 846 a Fr(a)g Fc(62)g Ft(fv)o Fu(\()p |
|
5070 Fr(t)1505 854 y Fv(1)1552 846 y Fr(t)1580 854 y Fv(2)1615 |
|
5071 846 y Fu(\))1810 787 y Ft(fv)1888 795 y Fv(app)p 2072 |
|
5072 773 382 4 v 2072 846 a Fr(a)g Fc(62)g Ft(fv)p Fu(\()p |
|
5073 Fr(\025a:t)p Fu(\))2470 792 y Ft(fv)2548 800 y Fn(\025)p |
|
5074 Fv(1)2702 735 y Fr(a)21 b Fc(6)p Fu(=)g Fr(b)64 b(a)9 |
|
5075 b Fc(62)g Ft(fv)o Fu(\()p Fr(t)p Fu(\))p 2702 773 515 |
|
5076 4 v 2772 846 a Fr(a)g Fc(62)g Ft(fv)p Fu(\()p Fr(\025b:t)p |
|
5077 Fu(\))3233 792 y Ft(fv)3311 800 y Fn(\025)p Fv(2)p 3435 |
|
5078 898 4 530 v 523 901 2915 4 v Black 976 1049 a Fs(Fig.)14 |
|
5079 b(1.)26 b Fu(Inductiv)n(e)e(de\014nitions)i(for)g(\()p |
|
5080 Fc(\000)p Fu(\))21 b Fc(\031)g Fu(\()p Fc(\000)p Fu(\))k(and)g(\()p |
|
5081 Fc(\000)p Fu(\))9 b Fc(62)g Ft(fv)o Fu(\()p Fc(\000)p |
|
5082 Fu(\).)p Black Black 523 1332 a Fo(sp)l(e)l(ci\014c)41 |
|
5083 b Fx(\\partial")34 b(functions)i(from)g Fk(A)59 b Fx(to)35 |
|
5084 b Fm(\010)i Fx(\(b)n(y)f(\\partial")e(w)n(e)h(mean)h(functions)g(that) |
|
5085 523 1432 y(return)27 b(\\error")e(for)i(unde\014ned)h(v)-5 |
|
5086 b(alues)1789 1402 y Fw(1)1826 1432 y Fx(\).)28 b(Th)n(us)f(the)h(set)g |
|
5087 Fm(\010)g Fx(is)f(de\014ned)h(b)n(y)g(the)g(grammar)1206 |
|
5088 1605 y Fm(\010)c Fx(:)106 b Fm(t)46 b Fx(::=)g Fe(er)f |
|
5089 Fl(j)h Fe(am)p Fx(\()p Fm(a)p Fx(\))g Fl(j)g Fe(pr)p |
|
5090 Fx(\()p Fm(t;)14 b(t)p Fx(\))46 b Fl(j)g Fe(se)o Fx(\()p |
|
5091 Fo(fn)7 b Fx(\))523 1779 y(where)31 b Fe(er)f Fx(stands)g(for)h |
|
5092 (\\error",)d Fm(a)j Fx(for)f(atoms)h(and)g Fo(fn)37 b |
|
5093 Fx(stands)31 b(for)f(functions)i(from)e Fk(A)54 b Fx(to)523 |
|
5094 1879 y Fm(\010)p Fx(.)601 1849 y Fw(2)666 1879 y Fx(This)27 |
|
5095 b(grammar)e(corresp)r(onds)g(to)i(the)h(inductiv)n(e)f(datat)n(yp)r(e)g |
|
5096 (that)g(one)g(migh)n(t)g(declare)523 1978 y(in)h(Isab)r(elle/HOL)e(as:) |
|
5097 p Black Black 1328 2151 a Fe(datatype)40 b(phi)22 b(=)g(er)1884 |
|
5098 2251 y Fl(j)33 b Fe(am)43 b("atom")1884 2350 y Fl(j)33 |
|
5099 b Fe(pr)43 b("phi)14 b Fl(\002)g Fe(phi")1884 2450 y |
|
5100 Fl(j)33 b Fe(se)43 b("atom)f Fl(\))h Fe(phi")523 2620 |
|
5101 y Fx(where)28 b(it)h(is)g(presupp)r(osed)f(that)h(the)g(t)n(yp)r(e)g |
|
5102 Fe(atom)e Fx(has)h(b)r(een)h(declared.)f(The)h(constructors)523 |
|
5103 2719 y Fe(am)o Fx(,)f Fe(pr)g Fx(and)g Fe(se)f Fx(will)h(b)r(e)h(used)f |
|
5104 (in)g Fm(\003)1670 2731 y Fj(\013)1745 2719 y Fx(for)g(represen)n(ting) |
|
5105 f Fm(\013)p Fx(-equated)h(atoms,)f(applications)523 2819 |
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5106 y(and)j(lam)n(b)r(da-abstractions.)d(Before)i(the)h(subset)g |
|
5107 Fm(\003)2196 2831 y Fj(\013)2273 2819 y Fx(can)g(b)r(e)g(carv)n(ed)e |
|
5108 (out)i(from)g Fm(\010)p Fx(,)g(ho)n(w-)523 2919 y(ev)n(er,)21 |
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5109 b(some)h(terminology)e(from)i(the)h(nominal)f(logic)f(w)n(ork)g(needs)h |
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5110 (to)g(b)r(e)g(adapted.)g(F)-7 b(or)22 b(this)523 3018 |
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5111 y(w)n(e)f(o)n(v)n(erload)e(the)i(notion)g(of)g(p)r(erm)n(utation)g |
|
5112 (action,)g(that)h(is)f Fm(\031)2475 3027 y Fi(\001)2513 |
|
5113 3018 y Fx(\()p Fl(\000)p Fx(\),)h(and)f(de\014ne)g(abstractly)523 |
|
5114 3118 y(sets)27 b(that)h(come)f(with)i(a)e(notion)g(of)h(p)r(erm)n |
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5115 (utation:)p Black 523 3283 a Ff(De\014nition)j(2)p Black |
|
5116 41 w(\(PSets\).)41 b Fo(A)c(set)g Fm(X)44 b Fo(e)l(quipp)l(e)l(d)38 |
|
5117 b(with)g(a)g(p)l(ermutation)f(action)i Fm(\031)3174 3292 |
|
5118 y Fi(\001)3212 3283 y Fx(\()p Fl(\000)p Fx(\))f Fo(is)523 |
|
5119 3383 y(said)32 b(to)f(b)l(e)g(a)37 b Fx(pset)p Fo(,)31 |
|
5120 b(if)h(for)g(al)t(l)f Fm(x)26 b Fl(2)f Fm(X)7 b Fo(,)31 |
|
5121 b(the)f(p)l(ermutation)h(action)h(satis\014es)f(the)g(fol)t(lowing)523 |
|
5122 3482 y(pr)l(op)l(erties:)p Black Black Black 670 3639 |
|
5123 a(\(i\))p Black 42 w Fx([])851 3648 y Fi(\001)890 3639 |
|
5124 y Fm(x)23 b Fx(=)g Fm(x)p Black 645 3737 a Fo(\(ii\))p |
|
5125 Black 42 w Fm(\031)852 3749 y Fw(1)890 3737 y Fx(@)p |
|
5126 Fm(\031)1002 3749 y Fw(2)1039 3746 y Fi(\001)1077 3737 |
|
5127 y Fm(x)h Fx(=)e Fm(\031)1282 3749 y Fw(1)1320 3746 y |
|
5128 Fi(\001)1358 3737 y Fx(\()p Fm(\031)1437 3749 y Fw(2)1475 |
|
5129 3746 y Fi(\001)1513 3737 y Fm(x)p Fx(\))p Black 619 3835 |
|
5130 a Fo(\(iii\))p Black 43 w(if)31 b Fm(\031)933 3847 y |
|
5131 Fw(1)994 3835 y Fl(\030)22 b Fm(\031)1128 3847 y Fw(2)1195 |
|
5132 3835 y Fo(then)30 b Fm(\031)1427 3847 y Fw(1)1465 3844 |
|
5133 y Fi(\001)1503 3835 y Fm(x)23 b Fx(=)g Fm(\031)1708 3847 |
|
5134 y Fw(2)1746 3844 y Fi(\001)1784 3835 y Fm(x)523 3982 |
|
5135 y Fx(The)h(informal)f(notation)g Fm(x)h Fl(2)f Fo(pset)h |
|
5136 Fx(will)g(b)r(e)h(adopted)e(whenev)n(er)g(it)h(needs)g(to)g(b)r(e)g |
|
5137 (indicated)523 4081 y(that)42 b Fm(x)g Fx(comes)e(from)h(a)g(pset.)h |
|
5138 (The)f(idea)g(b)r(ehind)i(the)e(p)r(erm)n(utation)g(action,)g(roughly) |
|
5139 523 4181 y(sp)r(eaking,)32 b(is)h(to)f(p)r(erm)n(ute)h(all)f(atoms)h |
|
5140 (in)f(a)h(giv)n(en)f(pset-elemen)n(t.)g(F)-7 b(or)32 |
|
5141 b(lists,)h(tuples)g(and)523 4281 y(sets)27 b(the)h(p)r(erm)n(utation)g |
|
5142 (action)f(is)g(therefore)g(de\014ned)h(p)r(oin)n(t-wise:)p |
|
5143 Black Black 556 4475 a(lists:)258 b Fm(\031)1031 4484 |
|
5144 y Fi(\001)1070 4475 y Fx([])1139 4428 y Fw(def)1150 4475 |
|
5145 y Fx(=)34 b([])793 4600 y Fm(\031)843 4609 y Fi(\001)882 |
|
5146 4600 y Fx(\()p Fm(x)24 b Fx(::)f Fm(t)p Fx(\))1139 4553 |
|
5147 y Fw(def)1150 4600 y Fx(=)34 b(\()p Fm(\031)1331 4609 |
|
5148 y Fi(\001)1370 4600 y Fm(x)p Fx(\))24 b(::)f(\()p Fm(\031)1624 |
|
5149 4609 y Fi(\001)1663 4600 y Fm(t)p Fx(\))1843 4475 y(tuples:)71 |
|
5150 b Fm(\031)2204 4484 y Fi(\001)2243 4475 y Fx(\()p Fm(x)2322 |
|
5151 4487 y Fw(1)2359 4475 y Fm(;)14 b(:)g(:)g(:)g(;)g(x)2591 |
|
5152 4487 y Fj(n)2637 4475 y Fx(\))2692 4428 y Fw(def)2703 |
|
5153 4475 y Fx(=)34 b(\()p Fm(\031)2884 4484 y Fi(\001)2923 |
|
5154 4475 y Fm(x)2970 4487 y Fw(1)3008 4475 y Fm(;)14 b(:)g(:)g(:)f(;)h |
|
5155 (\031)3242 4484 y Fi(\001)3281 4475 y Fm(x)3328 4487 |
|
5156 y Fj(n)3373 4475 y Fx(\))1843 4600 y(sets:)504 b Fm(\031)2555 |
|
5157 4609 y Fi(\001)2594 4600 y Fm(X)2692 4553 y Fw(def)2703 |
|
5158 4600 y Fx(=)34 b Fl(f)p Fm(\031)2894 4609 y Fi(\001)2932 |
|
5159 4600 y Fm(x)14 b Fl(j)g Fm(x)24 b Fl(2)f Fm(X)7 b Fl(g)p |
|
5160 Black 523 4680 473 4 v 546 4734 a Fv(1)606 4765 y Fu(This)26 |
|
5161 b(is)h(one)e(w)n(a)n(y)h(of)h(dealing)f(with)g(partial)h(functions)f |
|
5162 (in)f(Isab)r(elle.)546 4825 y Fv(2)606 4857 y Fu(Emplo)n(ying)e(\(on)h |
|
5163 (the)f(meta-lev)n(el\))g(a)i(lam)n(b)r(da-calculus-lik)n(e)e(notation)i |
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5164 (for)f(writing)h(suc)n(h)e(func-)606 4948 y(tions,)k(one)e(could)h(in)g |
|
5165 (this)f(grammar)g(just)h(as)h(w)n(ell)f(ha)n(v)n(e)f(written)h |
|
5166 Fr(\025a:f)34 b Fu(instead)26 b(of)h Fq(fn)5 b Fu(.)p |
|
5167 Black Black Black eop end |
|
5168 %%Page: 5 5 |
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5169 TeXDict begin 5 4 bop Black Black 523 448 a Fx(The)28 |
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5170 b(p)r(erm)n(utation)f(action)g(for)g Fm(\010)h Fx(is)g(de\014ned)g(o)n |
|
5171 (v)n(er)d(the)j(structure)f(as)g(follo)n(ws:)p Black |
|
5172 Black 852 644 a Fm(\031)902 653 y Fi(\001)940 644 y Fe(er)1159 |
|
5173 597 y Fw(def)1170 644 y Fx(=)34 b Fe(er)852 769 y Fm(\031)902 |
|
5174 778 y Fi(\001)940 769 y Fe(am)o Fx(\()p Fm(a)p Fx(\))1159 |
|
5175 722 y Fw(def)1170 769 y Fx(=)g Fe(am)o Fx(\()p Fm(\031)1438 |
|
5176 778 y Fi(\001)1477 769 y Fm(a)p Fx(\))1789 644 y Fm(\031)1839 |
|
5177 653 y Fi(\001)1877 644 y Fe(pr)p Fx(\()p Fm(t)2027 656 |
|
5178 y Fw(1)2064 644 y Fm(;)14 b(t)2131 656 y Fw(2)2168 644 |
|
5179 y Fx(\))2224 597 y Fw(def)2235 644 y Fx(=)34 b Fe(pr)o |
|
5180 Fx(\()p Fm(\031)2503 653 y Fi(\001)2542 644 y Fm(t)2572 |
|
5181 656 y Fw(1)2609 644 y Fm(;)14 b(\031)2696 653 y Fi(\001)2734 |
|
5182 644 y Fm(t)2764 656 y Fw(2)2802 644 y Fx(\))1789 769 |
|
5183 y Fm(\031)1839 778 y Fi(\001)1877 769 y Fe(se)p Fx(\()p |
|
5184 Fo(fn)6 b Fx(\))2224 722 y Fw(def)2235 769 y Fx(=)34 |
|
5185 b Fe(se)o Fx(\()p Fm(\025a:\031)2618 778 y Fi(\001)2657 |
|
5186 769 y Fx(\()p Fo(fn)c Fx(\()p Fm(\031)2873 739 y Fg(\000)p |
|
5187 Fw(1)2963 778 y Fi(\001)3001 769 y Fm(a)p Fx(\)\))523 |
|
5188 945 y(where)d(a)f(lam)n(b)r(da-term)h(\(on)g(the)h Fo(meta-level)9 |
|
5189 b Fx(!\))28 b(sp)r(eci\014es)f(ho)n(w)f(the)i(p)r(erm)n(utation)f(acts) |
|
5190 f(on)523 1070 y(the)i(function)g Fo(fn)7 b Fx(,)27 b(namely)h(as)f |
|
5191 Fm(\031)1561 1079 y Fi(\001)1599 1070 y Fo(fn)1701 1023 |
|
5192 y Fw(def)1712 1070 y Fx(=)34 b Fm(\025a:\031)1976 1079 |
|
5193 y Fi(\001)2014 1070 y Fx(\()p Fo(fn)c Fx(\()p Fm(\031)2230 |
|
5194 1040 y Fg(\000)p Fw(1)2320 1079 y Fi(\001)2358 1070 y |
|
5195 Fm(a)p Fx(\)\).)648 1169 y(When)f(reasoning)e(ab)r(out)h |
|
5196 Fm(\003)1558 1181 y Fj(\013)1634 1169 y Fx(it)i(will)f(sa)n(v)n(e)e(us) |
|
5197 h(some)h(w)n(ork,)e(if)i(w)n(e)g(sho)n(w)f(that)h(certain)523 |
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5198 1269 y(sets)e(are)g(psets)g(and)h(then)g(sho)n(w)f(prop)r(erties)f |
|
5199 (\(abstractly\))h(for)g(pset-elemen)n(ts.)p Black 523 |
|
5200 1436 a Ff(Lemma)j(1.)p Black 40 w Fx(The)25 b(follo)n(wing)f(sets)g |
|
5201 (are)f(psets:)i Fk(A)17 b Fx(,)31 b Fm(\003)p Fx(,)24 |
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5202 b Fm(\010)p Fx(,)h(and)g(ev)n(ery)e(set)i(of)f(lists)h(\(similarly)523 |
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5203 1535 y(tuples)j(and)f(sets\))h(con)n(taining)f(elemen)n(ts)g(from)g |
|
5204 (psets.)p Black 523 1702 a Fo(Pr)l(o)l(of.)p Black 43 |
|
5205 w Fx(By)g(routine)h(inductions.)1786 b Fl(u)-55 b(t)648 |
|
5206 1868 y Fx(The)31 b(most)g(imp)r(ortan)n(t)f(notion)h(of)g(a)g |
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5207 (pset-elemen)n(t)g(is)f(that)i(of)f(its)g Fo(supp)l(ort)39 |
|
5208 b Fx(\(a)31 b(set)g(of)523 1968 y(atoms\))c(and)h(deriv)n(ed)e(from)i |
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5209 (this)g(the)g(notion)f(of)g Fo(fr)l(eshness)7 b Fx([6]:)p |
|
5210 Black 523 2134 a Ff(De\014nition)31 b(3)p Black 41 w(\(Supp)s(ort)42 |
|
5211 b(and)f(F)-8 b(reshness\).)40 b Fo(Given)f(an)e Fm(x)g |
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5212 Fl(2)g Fo(pset)q(,)g(its)45 b Fx(supp)r(ort)37 b Fo(is)523 |
|
5213 2234 y(de\014ne)l(d)30 b(as:)904 2204 y Fw(3)1289 2358 |
|
5214 y Fe(supp)n Fx(\()p Fm(x)p Fx(\))1598 2307 y Fd(def)1610 |
|
5215 2358 y Fx(=)35 b Fl(f)p Fm(a)22 b Fl(j)h Fe(inf)o Fl(f)p |
|
5216 Fm(b)f Fl(j)h Fx(\()p Fm(a)14 b(b)p Fx(\))2299 2367 y |
|
5217 Fi(\001)2337 2358 y Fm(x)24 b Fl(6)p Fx(=)e Fm(x)p Fl(gg)h |
|
5218 Fm(:)523 2503 y Fo(A)n(n)i(atom)i Fm(a)f Fo(is)g(said)i(to)e(b)l(e)32 |
|
5219 b Fx(fresh)26 b Fo(for)h(such)f(an)h Fm(x)p Fo(,)g(written)e |
|
5220 Fm(a)e Fx(#)h Fm(x)p Fo(,)j(pr)l(ovide)l(d)h Fm(a)23 |
|
5221 b Fl(62)g Fe(supp)o Fx(\()p Fm(x)p Fx(\))p Fo(.)523 2669 |
|
5222 y Fx(Note)30 b(that)g(as)f(so)r(on)g(as)h(one)f(\014xes)g(the)i(p)r |
|
5223 (erm)n(utation)e(action)h(for)f(elemen)n(ts)h(of)f(a)h(set,)g(the)523 |
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5224 2769 y(notion)25 b(of)h(supp)r(ort)f(is)g(\014xed)h(as)e(w)n(ell.)i |
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5225 (That)f(means)g(that)h(Def.)g(3)f(de\014nes)h(the)f(supp)r(ort)h(for) |
|
5226 523 2869 y(lists,)k(sets)g(and)g(tuples)g(as)g(long)f(as)g(their)h |
|
5227 (elemen)n(ts)g(come)g(from)f(psets.)h(Calculating)g(the)523 |
|
5228 2968 y(supp)r(ort)21 b(for)f(terms)h(in)g Fm(\003)f Fx(is)h(simple:)g |
|
5229 Fe(supp)n Fx(\()p Fm(a)p Fx(\))j(=)e Fl(f)p Fm(a)p Fl(g)p |
|
5230 Fx(,)e Fe(supp)o Fx(\()p Fm(t)2490 2980 y Fw(1)2550 2968 |
|
5231 y Fm(t)2580 2980 y Fw(2)2618 2968 y Fx(\))j(=)g Fe(supp)n |
|
5232 Fx(\()p Fm(t)2997 2980 y Fw(1)3034 2968 y Fx(\))5 b Fl([)g |
|
5233 Fe(supp)o Fx(\()p Fm(t)3368 2980 y Fw(2)3406 2968 y Fx(\))523 |
|
5234 3068 y(and)28 b Fe(supp)n Fx(\()p Fm(\025a:t)p Fx(\))d(=)e |
|
5235 Fe(supp)o Fx(\()p Fm(t)p Fx(\))c Fl([)g(f)p Fm(a)p Fl(g)p |
|
5236 Fx(.)27 b(Because)g(of)h(the)h(functions)f(in)g Fe(se)p |
|
5237 Fx(\()p Fo(fn)6 b Fx(\),)29 b(the)f(supp)r(ort)523 3168 |
|
5238 y(for)k(terms)f(in)i Fm(\010)f Fx(is)g(more)f(subtle.)i(Ho)n(w)n(ev)n |
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5239 (er,)d(later)i(on,)f(w)n(e)h(shall)g(see)f(that)i(for)e(terms)h(of)523 |
|
5240 3267 y(the)h(subset)f Fm(\003)988 3279 y Fj(\013)1067 |
|
5241 3267 y Fx(there)g(is)g(simple)h(structural)e(c)n(haracterisation)e(for) |
|
5242 j(their)g(supp)r(ort,)g(just)523 3367 y(lik)n(e)27 b(for)g(lam)n(b)r |
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5243 (da-terms.)648 3466 y(First,)g(some)g(prop)r(erties)g(of)g(supp)r(ort)h |
|
5244 (and)f(freshness)g(are)g(established.)p Black 523 3633 |
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5245 a Ff(Lemma)j(2.)p Black 40 w Fx(F)-7 b(or)27 b(all)h |
|
5246 Fm(x)23 b Fl(2)h Fo(pset)p Fx(,)p Black Black Black 676 |
|
5247 3791 a(\(i\))p Black 42 w Fm(\031)855 3800 y Fi(\001)894 |
|
5248 3791 y Fe(supp)n Fx(\()p Fm(x)p Fx(\))g(=)f Fe(supp)n |
|
5249 Fx(\()p Fm(\031)1547 3800 y Fi(\001)1586 3791 y Fm(x)p |
|
5250 Fx(\),)28 b(and)p Black 653 3889 a(\(ii\))p Black 42 |
|
5251 w Fm(a)23 b Fx(#)h Fm(\031)1015 3898 y Fi(\001)1053 3889 |
|
5252 y Fm(x)51 b Fx(if)28 b(and)g(only)f(if)51 b Fm(\031)1720 |
|
5253 3859 y Fg(\000)p Fw(1)1810 3898 y Fi(\001)1848 3889 y |
|
5254 Fm(a)23 b Fx(#)g Fm(x)p Fx(.)p Black 523 4056 a Fo(Pr)l(o)l(of.)p |
|
5255 Black 43 w Fx(\(i\))28 b(follo)n(ws)f(from)g(the)h(calculation:)p |
|
5256 Black Black 861 4232 a Fm(\031)911 4241 y Fi(\001)950 |
|
5257 4232 y Fe(supp)n Fx(\()p Fm(x)p Fx(\))1259 4185 y Fw(def)1270 |
|
5258 4232 y Fx(=)34 b Fm(\031)1419 4241 y Fi(\001)1458 4232 |
|
5259 y Fl(f)p Fm(a)14 b Fl(j)g Fe(inf)n Fl(f)p Fm(b)g Fl(j)g |
|
5260 Fx(\()p Fm(a)g(b)p Fx(\))2012 4241 y Fi(\001)2049 4232 |
|
5261 y Fm(x)23 b Fl(6)p Fx(=)g Fm(x)p Fl(gg)1259 4311 y Fw(def)1270 |
|
5262 4358 y Fx(=)34 b Fl(f)p Fm(\031)1461 4367 y Fi(\001)1499 |
|
5263 4358 y Fm(a)14 b Fl(j)g Fe(inf)o Fl(f)p Fm(b)g Fl(j)g |
|
5264 Fx(\()p Fm(a)g(b)p Fx(\))2012 4367 y Fi(\001)2049 4358 |
|
5265 y Fm(x)23 b Fl(6)p Fx(=)g Fm(x)p Fl(gg)1270 4458 y Fx(=)34 |
|
5266 b Fl(f)p Fm(\031)1461 4467 y Fi(\001)1499 4458 y Fm(a)14 |
|
5267 b Fl(j)g Fe(inf)o Fl(f)p Fm(\031)1817 4467 y Fi(\001)1855 |
|
5268 4458 y Fm(b)g Fl(j)g Fx(\()p Fm(a)g(b)p Fx(\))2100 4467 |
|
5269 y Fi(\001)2137 4458 y Fm(x)24 b Fl(6)p Fx(=)e Fm(x)p |
|
5270 Fl(gg)531 b Fx(\()p Fl(\003)3031 4428 y Fw(1)3068 4458 |
|
5271 y Fx(\))1270 4557 y(=)34 b Fl(f)p Fm(a)14 b Fl(j)g Fe(inf)n |
|
5272 Fl(f)p Fm(b)g Fl(j)g Fx(\()p Fm(\031)1847 4527 y Fg(\000)p |
|
5273 Fw(1)1936 4566 y Fi(\001)1974 4557 y Fm(a)37 b(\031)2105 |
|
5274 4527 y Fg(\000)p Fw(1)2194 4566 y Fi(\001)2232 4557 y |
|
5275 Fm(b)p Fx(\))2300 4566 y Fi(\001)2338 4557 y Fm(x)24 |
|
5276 b Fl(6)p Fx(=)f Fm(x)p Fl(gg)1270 4657 y Fx(=)34 b Fl(f)p |
|
5277 Fm(a)14 b Fl(j)g Fe(inf)n Fl(f)p Fm(b)g Fl(j)g Fm(\031)1815 |
|
5278 4666 y Fi(\001)1852 4657 y Fx(\()p Fm(\031)1934 4627 |
|
5279 y Fg(\000)p Fw(1)2024 4666 y Fi(\001)2062 4657 y Fm(a)37 |
|
5280 b(\031)2193 4627 y Fg(\000)p Fw(1)2283 4666 y Fi(\001)2321 |
|
5281 4657 y Fm(b)p Fx(\))2389 4666 y Fi(\001)2427 4657 y Fm(x)23 |
|
5282 b Fl(6)p Fx(=)g Fm(\031)2635 4666 y Fi(\001)2674 4657 |
|
5283 y Fm(x)p Fl(gg)152 b Fx(\()p Fl(\003)3031 4627 y Fw(2)3068 |
|
5284 4657 y Fx(\))1270 4783 y(=)34 b Fl(f)p Fm(a)14 b Fl(j)g |
|
5285 Fe(inf)n Fl(f)p Fm(b)g Fl(j)g Fx(\()p Fm(a)g(b)p Fx(\))1923 |
|
5286 4792 y Fi(\001)1960 4783 y Fm(\031)2010 4792 y Fi(\001)2049 |
|
5287 4783 y Fm(x)23 b Fl(6)p Fx(=)g Fm(\031)2257 4792 y Fi(\001)2295 |
|
5288 4783 y Fm(x)p Fl(gg)2449 4736 y Fw(def)2460 4783 y Fx(=)34 |
|
5289 b Fe(supp)n Fx(\()p Fm(\031)2815 4792 y Fi(\001)2854 |
|
5290 4783 y Fm(x)p Fx(\))24 b(\()p Fl(\003)3031 4752 y Fw(3)3068 |
|
5291 4783 y Fx(\))p Black 523 4862 473 4 v 546 4916 a Fv(3)606 |
|
5292 4948 y Fu(The)i(predicate)g Ft(inf)g Fu(will)h(stand)f(for)g(a)g(set)g |
|
5293 (b)r(eing)g(in\014nite.)p Black Black Black eop end |
|
5294 %%Page: 6 6 |
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5295 TeXDict begin 6 5 bop Black Black 523 448 a Fx(where)23 |
|
5296 b(\()p Fl(\003)833 418 y Fw(1)870 448 y Fx(\))i(holds)e(b)r(ecause)h |
|
5297 (the)g(sets)g Fl(f)p Fm(b)p Fl(j)14 b Fm(:)g(:)g(:)o |
|
5298 Fl(g)23 b Fx(and)h Fl(f)p Fm(\031)2268 457 y Fi(\001)2306 |
|
5299 448 y Fm(b)p Fl(j)14 b Fm(:)g(:)g(:)o Fl(g)24 b Fx(ha)n(v)n(e)f(the)h |
|
5300 (same)f(n)n(um)n(b)r(er)h(of)523 548 y(elemen)n(ts,)f(and)f(where)h(\() |
|
5301 p Fl(\003)1347 518 y Fw(2)1384 548 y Fx(\))g(holds)f(b)r(ecause)g(p)r |
|
5302 (erm)n(utations)h(preserv)n(e)e(\(in\)equalities;)i(\()p |
|
5303 Fl(\003)3369 518 y Fw(3)3406 548 y Fx(\))523 648 y(holds)32 |
|
5304 b(b)r(ecause)g Fm(\031)j Fx(comm)n(utes)d(with)h(the)g(sw)n(apping,)e |
|
5305 (that)h(is)g Fm(\031)s Fx(@\()p Fm(a)14 b(b)p Fx(\))31 |
|
5306 b Fl(\030)f Fx(\()p Fm(\031)3019 657 y Fi(\001)3058 648 |
|
5307 y Fm(a)42 b(\031)3194 657 y Fi(\001)3232 648 y Fm(b)p |
|
5308 Fx(\)@)p Fm(\031)s Fx(.)523 747 y(\(ii\):)29 b(F)-7 b(or)28 |
|
5309 b(all)h Fm(\031)s Fx(,)g Fm(a)24 b Fl(2)h Fe(supp)o Fx(\()p |
|
5310 Fm(x)p Fx(\))k(if)h(and)e(only)g(if)h Fm(\031)2067 756 |
|
5311 y Fi(\001)2106 747 y Fm(a)24 b Fl(2)h Fm(\031)2304 756 |
|
5312 y Fi(\001)2343 747 y Fe(supp)n Fx(\()p Fm(x)p Fx(\).)30 |
|
5313 b(The)f(prop)r(ert)n(y)e(follo)n(ws)523 847 y(then)h(from)f(\(i\))i |
|
5314 (and)e Fm(x)d Fl(2)f Fo(pset)p Fx(.)1884 b Fl(u)-55 b(t)648 |
|
5315 1008 y Fx(Another)18 b(imp)r(ortan)n(t)g(prop)r(ert)n(y)f(is)h(the)h |
|
5316 (fact)g(that)f(the)h(freshness)f(of)g(t)n(w)n(o)g(atoms)f(w.r.t.)i(an) |
|
5317 523 1107 y(pset-elemen)n(t)27 b(means)g(that)g(a)g(p)r(erm)n(utation)g |
|
5318 (sw)n(apping)f(those)g(t)n(w)n(o)h(atoms)f(has)h(no)g(e\013ect:)p |
|
5319 Black 523 1268 a Ff(Lemma)j(3.)p Black 40 w Fx(F)-7 b(or)27 |
|
5320 b(all)h Fm(x)23 b Fl(2)h Fo(pset)p Fx(,)j(if)i Fm(a)22 |
|
5321 b Fx(#)i Fm(x)k Fx(and)f Fm(b)c Fx(#)g Fm(x)28 b Fx(then)g(\()p |
|
5322 Fm(a)14 b(b)p Fx(\))2628 1277 y Fi(\001)2667 1268 y Fm(x)23 |
|
5323 b Fx(=)g Fm(x)p Fx(.)p Black 523 1429 a Fo(Pr)l(o)l(of.)p |
|
5324 Black 43 w Fx(The)34 b(case)g Fm(a)g Fx(=)f Fm(b)h Fx(is)g(clear)f(b)n |
|
5325 (y)h(Def.)h(2\()p Fm(i;)14 b(iii)p Fx(\).)33 b(In)h(the)g(other)g |
|
5326 (case,)f(the)i(assump-)523 1528 y(tion)42 b(implies)g(that)h(b)r(oth)f |
|
5327 Fl(f)p Fm(c)14 b Fl(j)g Fx(\()p Fm(c)g(a)p Fx(\))1696 |
|
5328 1537 y Fi(\001)1734 1528 y Fm(x)48 b Fl(6)p Fx(=)e Fm(x)p |
|
5329 Fl(g)d Fx(and)e Fl(f)p Fm(c)14 b Fl(j)g Fx(\()p Fm(c)g(b)p |
|
5330 Fx(\))2526 1537 y Fi(\001)2564 1528 y Fm(x)47 b Fl(6)p |
|
5331 Fx(=)g Fm(x)p Fl(g)42 b Fx(are)g(\014nite,)g(and)523 |
|
5332 1628 y(therefore)34 b(also)g(their)h(union)g(m)n(ust)h(b)r(e)f |
|
5333 (\014nite.)h(Hence)f(the)g(corresp)r(onding)f(co-set,)g(that)523 |
|
5334 1728 y(is)39 b Fl(f)p Fm(c)14 b Fl(j)g Fx(\()p Fm(c)g(a)p |
|
5335 Fx(\))905 1737 y Fi(\001)943 1728 y Fm(x)43 b Fx(=)f |
|
5336 Fm(x)26 b Fl(^)h Fx(\()p Fm(c)14 b(b)p Fx(\))1445 1737 |
|
5337 y Fi(\001)1483 1728 y Fm(x)43 b Fx(=)f Fm(x)p Fl(g)p |
|
5338 Fx(,)d(is)g(in\014nite)h(\(recall)f(that)g Fk(A)62 b |
|
5339 Fx(is)40 b(in\014nite\).)g(If)f(one)523 1827 y(pic)n(ks)32 |
|
5340 b(from)g(this)g(co-set)f(one)h(elemen)n(t,)h(whic)n(h)f(is)g(from)g(no) |
|
5341 n(w)g(on)g(denoted)g(b)n(y)g Fm(c)g Fx(and)g(as-)523 |
|
5342 1927 y(sumed)g(to)f(b)r(e)h(di\013eren)n(t)g(from)g Fm(a)f |
|
5343 Fx(and)h Fm(b)p Fx(,)f(one)g(has)g(\()p Fm(c)14 b(a)p |
|
5344 Fx(\))2338 1936 y Fi(\001)2377 1927 y Fm(x)30 b Fx(=)g |
|
5345 Fm(x)i Fx(and)f(\()p Fm(c)14 b(b)p Fx(\))2943 1936 y |
|
5346 Fi(\001)2981 1927 y Fm(x)31 b Fx(=)e Fm(x)p Fx(.)j(Th)n(us)523 |
|
5347 2026 y(\()p Fm(c)14 b(a)p Fx(\))681 2035 y Fi(\001)719 |
|
5348 2026 y Fx(\()p Fm(c)g(b)p Fx(\))869 2035 y Fi(\001)908 |
|
5349 2026 y Fx(\()p Fm(c)g(a)p Fx(\))1066 2035 y Fi(\001)1104 |
|
5350 2026 y Fm(x)24 b Fx(=)f Fm(x)p Fx(.)28 b(The)g(p)r(erm)n(utations)f(\() |
|
5351 p Fm(c)14 b(a)p Fx(\)\()p Fm(c)g(b)p Fx(\)\()p Fm(c)g(a)p |
|
5352 Fx(\))28 b(and)g(\()p Fm(a)14 b(b)p Fx(\))27 b(are)g(equal,)g(since)523 |
|
5353 2126 y(they)g(ha)n(v)n(e)f(an)h(empt)n(y)g(disagreemen)n(t)e(set.)i |
|
5354 (Therefore,)f(b)n(y)h(using)f(Def.)i(2\()p Fm(ii;)14 |
|
5355 b(iii)p Fx(\),)25 b(one)h(can)523 2226 y(conclude)h(with)h(\()p |
|
5356 Fm(a)14 b(b)p Fx(\))1211 2235 y Fi(\001)1250 2226 y Fm(x)23 |
|
5357 b Fx(=)g Fm(x)p Fx(.)1905 b Fl(u)-55 b(t)648 2386 y Fx(A)25 |
|
5358 b(further)g(restriction)f(on)g(psets)h(will)g(\014lter)g(out)g(all)g |
|
5359 (psets)g(con)n(taining)f(elemen)n(ts)h(with)523 2486 |
|
5360 y(an)i(in\014nite)i(supp)r(ort.)p Black 523 2647 a Ff(De\014nition)i(4) |
|
5361 p Black 41 w(\(Fs-PSet\).)i Fo(A)e(pset)g Fm(X)37 b Fo(is)32 |
|
5362 b(said)g(to)f(b)l(e)g(an)37 b Fx(fs-pset)31 b Fo(if)h(every)g(element)f |
|
5363 (in)523 2746 y Fm(X)36 b Fo(has)31 b(\014nite)e(supp)l(ort.)p |
|
5364 Black 523 2907 a Ff(Lemma)h(4.)p Black 40 w Fx(The)c(follo)n(wing)f |
|
5365 (sets)h(are)f(fs-psets:)h Fk(A)17 b Fx(,)32 b Fm(\003)p |
|
5366 Fx(,)26 b(and)g(ev)n(ery)f(set)h(of)g(lists)g(\(similarly)523 |
|
5367 3007 y(tuples)i(and)f(\014nite)i(sets\))e(con)n(taining)g(elemen)n(ts)g |
|
5368 (from)g(fs-psets.)p Black 523 3168 a Fo(Pr)l(o)l(of.)p |
|
5369 Black 43 w Fx(The)j(supp)r(ort)f(of)h(an)f(atom)g Fm(a)h |
|
5370 Fx(is)f Fl(f)p Fm(a)p Fl(g)p Fx(.)g(The)h(supp)r(ort)f(of)h(a)f(lam)n |
|
5371 (b)r(da-term)g Fm(t)g Fx(is)h(the)523 3267 y(set)f(of)f(atoms)g(o)r |
|
5372 (ccurring)g(in)h Fm(t)p Fx(.)f(The)h(supp)r(ort)g(of)f(a)h(list)g(is)f |
|
5373 (the)h(union)g(of)g(the)g(supp)r(orts)f(of)523 3367 y(its)d(elemen)n |
|
5374 (ts,)g(and)g(th)n(us)g(\014nite)g(for)f(fs-pset-elemen)n(ts)h(\(ditto)g |
|
5375 (tuples)g(and)g(\014nite)h(sets\).)83 b Fl(u)-55 b(t)523 |
|
5376 3528 y Fx(The)34 b(set)g Fm(\010)h Fx(is)f Fo(not)42 |
|
5377 b Fx(an)34 b(fs-pset,)g(b)r(ecause)g(some)f(functions)i(from)e |
|
5378 Fk(A)58 b Fx(to)34 b Fm(\010)g Fx(ha)n(v)n(e)f(an)h(in\014-)523 |
|
5379 3627 y(nite)e(supp)r(ort.)f(Similarly)-7 b(,)30 b(some)h(in\014nite)h |
|
5380 (sets)f(ha)n(v)n(e)f(in\014nite)i(supp)r(ort,)f(ev)n(en)g(if)g(all)g |
|
5381 (their)523 3727 y(elemen)n(ts)24 b(ha)n(v)n(e)f(\014nite)i(supp)r(ort.) |
|
5382 f(On)g(the)g(other)g(hand,)g(the)g(in\014nite)h(set)g |
|
5383 Fk(A)47 b Fx(has)23 b Fo(\014nite)30 b Fx(sup-)523 3827 |
|
5384 y(p)r(ort:)d Fe(supp)o Fx(\()p Fk(A)17 b Fx(\))29 b(=)23 |
|
5385 b Fk(?)k Fx([6].)g(The)h(main)f(prop)r(ert)n(y)f(of)h(elemen)n(ts)h(of) |
|
5386 f(fs-psets)g(is)g(that)h(there)f(is)523 3926 y(alw)n(a)n(ys)f(a)h |
|
5387 (fresh)g(atom.)p Black 523 4087 a Ff(Lemma)j(5.)p Black |
|
5388 40 w Fx(F)-7 b(or)27 b(all)h Fm(x)23 b Fl(2)h Fo(fs-pset)p |
|
5389 Fx(,)k(there)f(exists)g(an)g(atom)h Fm(a)f Fx(suc)n(h)g(that)h |
|
5390 Fm(a)23 b Fx(#)h Fm(x)p Fx(.)p Black 523 4248 a Fo(Pr)l(o)l(of.)p |
|
5391 Black 43 w Fx(Since)36 b Fk(A)59 b Fx(is)35 b(an)h(in\014nite)g(set)g |
|
5392 (and)g(the)g(supp)r(ort)f(of)h Fm(x)g Fx(is)g(b)n(y)f(assumption)g |
|
5393 (\014nite,)523 4347 y(there)27 b(m)n(ust)h(b)r(e)g(an)f |
|
5394 Fm(a)c Fl(62)h Fe(supp)n Fx(\()p Fm(x)p Fx(\).)1760 b |
|
5395 Fl(u)-55 b(t)648 4508 y Fx(W)-7 b(e)35 b(men)n(tioned)g(earlier)f(that) |
|
5396 h(w)n(e)g(are)f(not)h(going)f(to)g(use)h(all)g(functions)g(from)g |
|
5397 Fk(A)58 b Fx(to)523 4608 y Fm(\010)30 b Fx(for)g(represen)n(ting)e |
|
5398 Fm(\013)p Fx(-equated)i(lam)n(b)r(da-abstractions,)e(but)i(some)f(sp)r |
|
5399 (eci\014c)h(functions.)3400 4578 y Fw(4)523 4707 y Fx(The)e(follo)n |
|
5400 (wing)e(de\014nition)i(states)f(what)h(prop)r(erties)f(these)g |
|
5401 (functions)h(need)g(to)f(satisfy)-7 b(.)p Black 523 4771 |
|
5402 473 4 v 546 4825 a Fv(4)606 4857 y Fu(This)35 b(is)f(in)g(con)n(trast)h |
|
5403 (to)f(\\w)n(eak")h(and)f(\\full")h(HO)n(AS)d([15,)15 |
|
5404 b(4])34 b(whic)n(h)g(use)g(the)g(full)h(function)606 |
|
5405 4948 y(space)26 b(for)h(represen)n(ting)f(lam)n(b)r(da-abstractions.)p |
|
5406 Black Black Black eop end |
|
5407 %%Page: 7 7 |
|
5408 TeXDict begin 7 6 bop Black Black Black 523 448 a Ff(De\014nition)31 |
|
5409 b(5)p Black 41 w(\(Nominal)26 b(Abstractions\).)j Fo(A)n(n)d(op)l(er)l |
|
5410 (ation,)j(written)e Fx([)p Fl(\000)p Fx(])p Fm(:)p Fx(\()p |
|
5411 Fl(\000)p Fx(\))p Fo(,)g(taking)523 548 y(an)h(atom)f(and)i(a)e(pset)q |
|
5412 (-element)g(is)h(said)g(to)g(b)l(e)g(a)34 b Fx(nominal)25 |
|
5413 b(abstraction)p Fo(,)h(if)j(it)e(satis\014es)h(the)523 |
|
5414 648 y(fol)t(lowing)k(pr)l(op)l(erties)f(\(wher)l(e)g |
|
5415 Fm(a)22 b Fl(6)p Fx(=)h Fm(b)p Fo(\):)p Black Black Black |
|
5416 670 807 a(\(i\))p Black 42 w Fm(\031)855 816 y Fi(\001)894 |
|
5417 807 y Fx(\([)p Fm(a)p Fx(])p Fm(:x)p Fx(\))h(=)f([)p |
|
5418 Fm(\031)1303 816 y Fi(\001)1341 807 y Fm(a)p Fx(])p Fm(:)p |
|
5419 Fx(\()p Fm(\031)1513 816 y Fi(\001)1552 807 y Fm(x)p |
|
5420 Fx(\))p Black 645 905 a Fo(\(ii\))p Black 42 w Fx([)p |
|
5421 Fm(a)p Fx(])p Fm(:x)965 917 y Fw(1)1026 905 y Fx(=)g([)p |
|
5422 Fm(b)p Fx(])p Fm(:x)1266 917 y Fw(2)1333 905 y Fo(if)31 |
|
5423 b(and)f(only)g(if)h(either:)p Black Black 1504 1033 a |
|
5424 Fm(a)23 b Fx(=)f Fm(b)c Fl(^)h Fm(x)1833 1045 y Fw(1)1894 |
|
5425 1033 y Fx(=)k Fm(x)2029 1045 y Fw(2)2066 1033 y Fo(,)30 |
|
5426 b(or)1504 1156 y Fm(a)23 b Fl(6)p Fx(=)f Fm(b)c Fl(^)h |
|
5427 Fm(x)1833 1168 y Fw(1)1894 1156 y Fx(=)k(\()p Fm(a)14 |
|
5428 b(b)p Fx(\))2140 1165 y Fi(\001)2178 1156 y Fm(x)2225 |
|
5429 1168 y Fw(2)2281 1156 y Fl(^)19 b Fm(a)k Fx(#)g Fm(x)2561 |
|
5430 1168 y Fw(2)523 1319 y Fx(The)37 b(\014rst)g(prop)r(ert)n(y)f(states)g |
|
5431 (that)h(the)h(p)r(erm)n(utation)f(action)f(needs)h(to)g(comm)n(ute)g |
|
5432 (with)523 1419 y(nominal)23 b(abstractions.)e(The)i(second)g(prop)r |
|
5433 (ert)n(y)e(ensures)i(that)g(nominal)g(abstractions)e(b)r(e-)523 |
|
5434 1518 y(ha)n(v)n(e,)j(roughly)f(sp)r(eaking,)h(lik)n(e)g(lam)n(b)r |
|
5435 (da-abstractions.)e(T)-7 b(o)24 b(see)h(this)g(reconsider)e(the)i |
|
5436 (rules)523 1618 y Fl(\031)588 1630 y Fj(\025)p Fw(1)687 |
|
5437 1618 y Fx(and)d Fl(\031)908 1630 y Fj(\025)p Fw(2)1006 |
|
5438 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 |
|
5439 (to)f(decide)g(when)h(t)n(w)n(o)e(lam)n(b)r(da-terms)523 |
|
5440 1718 y(are)27 b Fm(\013)p Fx(-equiv)-5 b(alen)n(t.)27 |
|
5441 b(Prop)r(ert)n(y)f(\()p Fm(ii)p Fx(\))i(paraphrases)d(these)j(rules)f |
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5442 (for)g(nominal)h(abstractions.)523 1817 y(The)33 b(similarities,)f(ho)n |
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5443 (w)n(ev)n(er,)f(do)h(not)h(end)g(here:)g(giv)n(en)f(a)g([)p |
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5444 Fm(a)p Fx(])p Fm(:x)i Fx(with)f Fm(x)f Fl(2)g Fo(fs-pset)q |
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5445 Fx(,)h(then)523 1917 y(freshness)27 b(b)r(eha)n(v)n(es)f(lik)n(e)i(\()p |
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5446 Fl(\000)p Fx(\))9 b Fl(62)23 b Fe(fv)p Fx(\()p Fl(\000)p |
|
5447 Fx(\),)k(as)g(sho)n(wn)g(next:)p Black 523 2085 a Ff(Lemma)j(6.)p |
|
5448 Black 40 w Fx(Giv)n(en)e Fm(a)23 b Fl(6)p Fx(=)f Fm(b)28 |
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5449 b Fx(and)f Fm(x)d Fl(2)k Fo(fs-pset)p Fx(,)g(then)p Black |
|
5450 535 2244 a(\(i\))p Black 42 w Fm(a)23 b Fx(#)g([)p Fm(b)p |
|
5451 Fx(])p Fm(:x)28 b Fx(if)g(and)g(only)f(if)h Fm(a)23 b |
|
5452 Fx(#)g Fm(x)p Fx(,)29 b(and)p Black 512 2342 a(\(ii\))p |
|
5453 Black 42 w Fm(a)23 b Fx(#)g([)p Fm(a)p Fx(])p Fm(:x)p |
|
5454 Black 523 2510 a Fo(Pr)l(o)l(of.)p Black 43 w Fx(\(i)p |
|
5455 Fl(\))p Fx(\):)38 b(Since)f Fm(x)j Fl(2)f Fo(fs-pset)p |
|
5456 Fx(,)f Fe(supp)n Fx(\([)p Fm(b)p Fx(])p Fm(:x)p Fx(\))i |
|
5457 Fl(\022)e Fe(supp)o Fx(\()p Fm(x)p Fx(\))25 b Fl([)h(f)p |
|
5458 Fm(b)p Fl(g)35 b Fx(and)i(therefore)g(the)523 2610 y(supp)r(ort)31 |
|
5459 b(of)g([)p Fm(a)p Fx(])p Fm(:x)g Fx(m)n(ust)g(b)r(e)h(\014nite.)f |
|
5460 (Hence)g(\()p Fm(a;)14 b(b;)g(x;)g Fx([)p Fm(b)p Fx(])p |
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5461 Fm(:x)p Fx(\))32 b(is)e(\014nitely)i(supp)r(orted)f(and)f(b)n(y)523 |
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5462 2709 y(Lem.)24 b(5)g(there)f(exists)h(a)g Fm(c)g Fx(with)g(\()p |
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5463 Fl(\003)p Fx(\))g Fm(c)f Fx(#)g(\()p Fm(a;)14 b(b;)g(x;)g |
|
5464 Fx([)p Fm(b)p Fx(])p Fm(:x)p Fx(\).)25 b(Using)f(the)g(assumption)g |
|
5465 Fm(a)e Fx(#)i([)p Fm(b)p Fx(])p Fm(:x)523 2809 y Fx(and)h(the)h(fact)g |
|
5466 (that)g Fm(c)d Fx(#)g([)p Fm(b)p Fx(])p Fm(:x)j Fx(\(from)f |
|
5467 Fl(\003)p Fx(\),)h(Lem.)f(3)g(and)h(Def.)g(5\()p Fm(i)p |
|
5468 Fx(\))f(giv)n(e)g([)p Fm(b)p Fx(])p Fm(:x)e Fx(=)g(\()p |
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5469 Fm(c)14 b(a)p Fx(\)[)p Fm(b)p Fx(])p Fm(:x)23 b Fx(=)523 |
|
5470 2909 y([)p Fm(b)p Fx(])p Fm(:)p Fx(\()p Fm(c)14 b(a)p |
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5471 Fx(\))786 2918 y Fi(\001)824 2909 y Fm(x)p Fx(.)37 b(Hence)g(b)n(y)f |
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5472 (Def.)h(5\()p Fm(ii)p Fx(\))e Fm(x)j Fx(=)g(\()p Fm(c)14 |
|
5473 b(a)p Fx(\))2041 2918 y Fi(\001)2079 2909 y Fm(x)p Fx(.)37 |
|
5474 b(No)n(w)f Fm(c)h Fx(#)h Fm(x)f Fx(\(from)f Fl(\003)p |
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5475 Fx(\))g(implies)h(that)523 3008 y Fm(c)c Fx(#)g(\()p |
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5476 Fm(c)14 b(a)p Fx(\))852 3017 y Fi(\001)891 3008 y Fm(x)p |
|
5477 Fx(;)34 b(and)f(mo)n(ving)g(the)g(p)r(erm)n(utation)h(to)f(the)h(other) |
|
5478 f(side)h(b)n(y)f(Lem.)g(2\(ii\))h(giv)n(es)523 3108 y |
|
5479 Fm(a)23 b Fx(#)g Fm(x)p Fx(.)g(\(i)p Fl(\()p Fx(\):)g(F)-7 |
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5480 b(rom)22 b(\()p Fl(\003)p Fx(\),)h Fm(c)g Fx(#)g([)p |
|
5481 Fm(b)p Fx(])p Fm(:x)g Fx(and)f(therefore)f(b)n(y)h(Lem.)h(2\(ii\))f(\() |
|
5482 p Fm(a)14 b(c)p Fx(\))2827 3117 y Fi(\001)2866 3108 y |
|
5483 Fm(c)23 b Fx(#)g(\()p Fm(a)14 b(c)p Fx(\))p Fm(:)p Fx(\([)p |
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5484 Fm(b)p Fx(])p Fm(:x)p Fx(\),)523 3207 y(whic)n(h)28 b(implies)h(b)n(y)f |
|
5485 (Def.)h(5\()p Fm(i)p Fx(\))f(that)h Fm(a)24 b Fx(#)g([)p |
|
5486 Fm(b)p Fx(])p Fm(:)p Fx(\(\()p Fm(a)14 b(c)p Fx(\))2137 |
|
5487 3216 y Fi(\001)2176 3207 y Fm(x)p Fx(\).)29 b(F)-7 b(rom)28 |
|
5488 b(\()p Fl(\003)p Fx(\))g Fm(c)c Fx(#)h Fm(x)j Fx(holds)g(and)g(from)523 |
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5489 3307 y(the)33 b(assumption)f(also)f Fm(a)g Fx(#)g Fm(x)p |
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5490 Fx(;)i(then)g(Lem.)g(3)e(implies)i(that)g Fm(x)e Fx(=)g(\()p |
|
5491 Fm(a)14 b(c)p Fx(\))2849 3316 y Fi(\001)2887 3307 y Fm(x)p |
|
5492 Fx(,)33 b(and)f(one)g(can)523 3407 y(conclude)27 b(with)h |
|
5493 Fm(a)23 b Fx(#)h([)p Fm(b)p Fx(])p Fm(:x)p Fx(.)523 3506 |
|
5494 y(\(ii\):)29 b(By)e Fm(c)d Fx(#)f Fm(x)29 b Fx(and)e |
|
5495 Fm(c)d Fl(6)p Fx(=)f Fm(a)k Fx(\(b)r(oth)i(from)e Fl(\003)p |
|
5496 Fx(\))h(w)n(e)g(can)f(use)h(\(i\))g(to)g(infer)g Fm(c)23 |
|
5497 b Fx(#)h([)p Fm(a)p Fx(])p Fm(:x)p Fx(.)k(F)-7 b(urther,)523 |
|
5498 3606 y(from)33 b(Lem.)h(2\(ii\))f(it)h(holds)f(that)h(\()p |
|
5499 Fm(c)14 b(a)p Fx(\))1781 3615 y Fi(\001)1819 3606 y Fm(c)33 |
|
5500 b Fx(#)g(\()p Fm(c)14 b(a)p Fx(\))2148 3615 y Fi(\001)2186 |
|
5501 3606 y Fx([)p Fm(a)p Fx(])p Fm(:x)p Fx(.)34 b(This)f(is)h |
|
5502 Fm(a)e Fx(#)h([)p Fm(c)p Fx(])p Fm(:)p Fx(\()p Fm(c)14 |
|
5503 b(a)p Fx(\))3129 3615 y Fi(\001)3168 3606 y Fm(x)33 b |
|
5504 Fx(using)523 3706 y(Def.)41 b(5\()p Fm(i)p Fx(\).)g(Since)f |
|
5505 Fm(c)45 b Fl(6)p Fx(=)f Fm(a)p Fx(,)c Fm(c)k Fx(#)h Fm(x)c |
|
5506 Fx(and)f(\()p Fm(c)14 b(a)p Fx(\))2051 3715 y Fi(\001)2090 |
|
5507 3706 y Fm(x)45 b Fx(=)f(\()p Fm(c)14 b(a)p Fx(\))2449 |
|
5508 3715 y Fi(\001)2487 3706 y Fm(x)p Fx(,)41 b(Def.)g(5\()p |
|
5509 Fm(ii)p Fx(\))f(implies)h(that)523 3805 y([)p Fm(c)p |
|
5510 Fx(])p Fm(:)p Fx(\()p Fm(c)14 b(a)p Fx(\))786 3814 y |
|
5511 Fi(\001)825 3805 y Fm(x)23 b Fx(=)g([)p Fm(a)p Fx(])p |
|
5512 Fm(:x)p Fx(.)28 b(Therefore,)e Fm(a)d Fx(#)h([)p Fm(a)p |
|
5513 Fx(])p Fm(:x)p Fx(.)1447 b Fl(u)-55 b(t)648 3973 y Fx(The)32 |
|
5514 b(functions)g(from)g Fk(A)55 b Fx(to)33 b Fm(\010)f Fx(w)n(e)g(iden)n |
|
5515 (tify)h(next)f(satisfy)g(the)h(nominal)e(abstraction)523 |
|
5516 4073 y(prop)r(erties.)c(Let)h([)p Fm(a)p Fx(])p Fm(:t)f |
|
5517 Fx(b)r(e)h(de\014ned)g(as)f(follo)n(ws)775 4274 y([)p |
|
5518 Fm(a)p Fx(])p Fm(:t)941 4227 y Fw(def)952 4274 y Fx(=)34 |
|
5519 b Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)22 b Fm(a)h Fx(=)g |
|
5520 Fm(b)g Fe(then)e Fm(t)i Fe(else)e(if)i Fm(b)f Fx(#)i |
|
5521 Fm(t)f Fe(then)e Fx(\()p Fm(a)14 b(b)p Fx(\))2732 4283 |
|
5522 y Fi(\001)2770 4274 y Fm(t)23 b Fe(else)f(er)o Fx(\))h |
|
5523 Fm(:)146 b Fx(\(2\))523 4450 y(This)25 b(op)r(eration)f(tak)n(es)g(t)n |
|
5524 (w)n(o)g(argumen)n(ts:)f(an)i Fm(a)e Fl(2)g Fk(A)48 b |
|
5525 Fx(and)25 b(a)g Fm(t)e Fl(2)g Fm(\010)p Fx(.)i(T)-7 b(o)25 |
|
5526 b(see)f(ho)n(w)h(this)g(op)r(er-)523 4549 y(ation)h(enco)r(des)f(an)h |
|
5527 Fm(\013)p Fx(-equiv)-5 b(alence)26 b(class,)f(consider)h(the)g |
|
5528 Fm(\013)p Fx(-equiv)-5 b(alence)26 b(class)f([)p Fm(\025a:)p |
|
5529 Fx(\()p Fm(a)f(b)p Fx(\)])3391 4561 y Fj(\013)523 4649 |
|
5530 y Fx(and)35 b(the)g(corresp)r(onding)d Fm(\010)p Fx(-term)j([)p |
|
5531 Fm(a)p Fx(])p Fm(:)p Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\))35 |
|
5532 b(\(for)g(the)g(momen)n(t)f(w)n(e)h(ignore)e(the)i(term)523 |
|
5533 4749 y(constructor)c Fe(se)g Fx(and)h(only)g(consider)f(the)i(function) |
|
5534 g(giv)n(en)e(b)n(y)h([)p Fm(a)p Fx(])p Fm(:)p Fe(pr)o |
|
5535 Fx(\()p Fm(a;)14 b(b)p Fx(\)\).)33 b(The)f(graph)523 |
|
5536 4848 y(of)27 b(this)h(function)g(is)f(as)g(follo)n(ws:)f(the)i(atom)f |
|
5537 Fm(a)g Fx(is)h(mapp)r(ed)f(to)h Fe(pr)o Fx(\()p Fm(a;)14 |
|
5538 b(b)p Fx(\))27 b(since)g(the)h(\014rst)f Fe(if)o Fx(-)523 |
|
5539 4948 y(condition)e(is)h(true.)f(F)-7 b(or)25 b Fm(b)p |
|
5540 Fx(,)h(the)g(\014rst)f Fe(if)o Fx(-condition)g(ob)n(viously)g(fails,)g |
|
5541 (but)h(also)f(the)h(second)p Black Black eop end |
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5542 %%Page: 8 8 |
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5543 TeXDict begin 8 7 bop Black Black 523 448 a Fx(one)31 |
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5544 b(fails,)h(b)r(ecause)f Fm(b)e Fl(2)h Fe(supp)n Fx(\()p |
|
5545 Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\)\);)32 b(therefore)f |
|
5546 Fm(b)g Fx(is)h(mapp)r(ed)f(to)h Fe(er)o Fx(.)g(F)-7 b(or)30 |
|
5547 b(all)i(other)523 548 y(atoms)d Fm(c)p Fx(,)h(w)n(e)f(ha)n(v)n(e)g |
|
5548 Fm(a)e Fl(6)p Fx(=)f Fm(c)k Fx(and)f Fm(c)e Fx(#)g Fe(pr)o |
|
5549 Fx(\()p Fm(a;)14 b(b)p Fx(\);)30 b(so)f(the)h Fm(c)p |
|
5550 Fx('s)g(are)f(mapp)r(ed)h(b)n(y)f(the)i(function)523 |
|
5551 648 y(to)c(\()p Fm(a)14 b(c)p Fx(\))782 657 y Fi(\001)820 |
|
5552 648 y Fe(pr)o Fx(\()p Fm(a;)g(b)p Fx(\),)27 b(whic)n(h)g(is)f(just)i |
|
5553 Fe(pr)o Fx(\()p Fm(c;)14 b(b)p Fx(\).)27 b(Clearly)-7 |
|
5554 b(,)26 b(the)h(function)g(returns)f Fe(er)g Fx(whenev)n(er)523 |
|
5555 747 y(the)k(corresp)r(onding)e(lam)n(b)r(da-term)h(is)h |
|
5556 Fo(not)37 b Fx(in)30 b(the)g Fm(\013)p Fx(-equiv)-5 b(alence)30 |
|
5557 b(class|in)f(this)h(exam-)523 847 y(ple)f Fm(\025b:)p |
|
5558 Fx(\()p Fm(b)24 b(b)p Fx(\))g Fl(62)h Fx([)p Fm(\025a:)p |
|
5559 Fx(\()p Fm(a)g(b)p Fx(\)])1359 859 y Fj(\013)1406 847 |
|
5560 y Fx(;)j(in)h(all)f(other)g(cases,)f(ho)n(w)n(ev)n(er,)f(it)j(returns)f |
|
5561 (an)g(appropriately)523 946 y(\\renamed")e(v)n(ersion)g(of)i |
|
5562 Fe(pr)o Fx(\()p Fm(a;)14 b(b)p Fx(\).)p Black 523 1167 |
|
5563 a Ff(Lemma)30 b(7.)p Black 40 w Fx(The)e(op)r(eration)f([)p |
|
5564 Fl(\000)p Fx(])p Fm(:)p Fx(\()p Fl(\000)p Fx(\))g(giv)n(en)g(for)g |
|
5565 Fm(\010)h Fx(in)g(\(2\))g(is)f(a)g(nominal)h(abstraction.)p |
|
5566 Black 523 1378 a Fo(Pr)l(o)l(of.)p Black 43 w Fx(Def.)h(5\()p |
|
5567 Fm(i)p Fx(\))e(follo)n(ws)g(from)g(the)h(calculation:)p |
|
5568 Black Black 643 1571 a Fm(\031)693 1580 y Fi(\001)732 |
|
5569 1571 y Fx([)p Fm(a)p Fx(])p Fm(:t)533 1650 y Fw(def)544 |
|
5570 1697 y Fx(=)34 b Fm(\031)693 1706 y Fi(\001)732 1697 |
|
5571 y Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)22 b Fm(a)h Fx(=)g |
|
5572 Fm(b)g Fe(then)e Fm(t)i Fe(else)e(if)i Fm(b)f Fx(#)i |
|
5573 Fm(t)f Fe(then)e Fx(\()p Fm(a)14 b(b)p Fx(\))2413 1706 |
|
5574 y Fi(\001)2451 1697 y Fm(t)23 b Fe(else)f(er)o Fx(\))533 |
|
5575 1776 y Fw(def)544 1823 y Fx(=)34 b Fe(se)o Fx(\()p Fm(\025b:)14 |
|
5576 b(\031)933 1832 y Fi(\001)972 1823 y Fe(if)22 b Fm(a)h |
|
5577 Fx(=)g Fm(\031)1287 1792 y Fg(\000)p Fw(1)1376 1832 y |
|
5578 Fi(\001)1415 1823 y Fm(b)f Fe(then)g Fm(t)h Fe(else)e(if)h |
|
5579 Fm(\031)2081 1792 y Fg(\000)p Fw(1)2171 1832 y Fi(\001)2209 |
|
5580 1823 y Fm(b)h Fx(#)g Fm(t)g Fe(then)e Fx(\()p Fm(a)28 |
|
5581 b(\031)2764 1792 y Fg(\000)p Fw(1)2854 1832 y Fi(\001)2892 |
|
5582 1823 y Fm(b)p Fx(\))2960 1832 y Fi(\001)2998 1823 y Fm(t)23 |
|
5583 b Fe(else)e(er)o Fx(\))544 1922 y(=)34 b Fe(se)o Fx(\()p |
|
5584 Fm(\025b:)14 b Fe(if)23 b Fm(a)g Fx(=)f Fm(\031)1198 |
|
5585 1892 y Fg(\000)p Fw(1)1288 1931 y Fi(\001)1326 1922 y |
|
5586 Fm(b)h Fe(then)e Fm(\031)1632 1931 y Fi(\001)1671 1922 |
|
5587 y Fm(t)i Fe(else)e(if)h Fm(b)h Fx(#)g Fm(\031)2232 1931 |
|
5588 y Fi(\001)2271 1922 y Fm(t)g Fe(then)e Fm(\031)2571 1931 |
|
5589 y Fi(\001)2610 1922 y Fx(\()p Fm(a)27 b(\031)2763 1892 |
|
5590 y Fg(\000)p Fw(1)2853 1931 y Fi(\001)2891 1922 y Fm(b)p |
|
5591 Fx(\))2959 1931 y Fi(\001)2997 1922 y Fm(t)c Fe(else)f(er)o |
|
5592 Fx(\))i(\()p Fl(\003)p Fx(\))544 2022 y(=)34 b Fe(se)o |
|
5593 Fx(\()p Fm(\025b:)14 b Fe(if)23 b Fm(a)g Fx(=)f Fm(\031)1198 |
|
5594 1992 y Fg(\000)p Fw(1)1288 2031 y Fi(\001)1326 2022 y |
|
5595 Fm(b)h Fe(then)e Fm(\031)1632 2031 y Fi(\001)1671 2022 |
|
5596 y Fm(t)i Fe(else)e(if)h Fm(b)h Fx(#)g Fm(\031)2232 2031 |
|
5597 y Fi(\001)2271 2022 y Fm(t)g Fe(then)e Fx(\()p Fm(\031)2603 |
|
5598 2031 y Fi(\001)2642 2022 y Fm(a)27 b(b)p Fx(\))2781 2031 |
|
5599 y Fi(\001)2820 2022 y Fm(\031)2870 2031 y Fi(\001)2908 |
|
5600 2022 y Fm(t)c Fe(else)f(er)o Fx(\))544 2121 y(=)34 b |
|
5601 Fe(se)o Fx(\()p Fm(\025b:)14 b Fe(if)23 b Fm(\031)1044 |
|
5602 2130 y Fi(\001)1082 2121 y Fm(a)g Fx(=)g Fm(b)g Fe(then)e |
|
5603 Fm(\031)1543 2130 y Fi(\001)1581 2121 y Fm(t)j Fe(else)d(if)h |
|
5604 Fm(b)h Fx(#)g Fm(\031)2143 2130 y Fi(\001)2182 2121 y |
|
5605 Fm(t)g Fe(then)e Fx(\()p Fm(\031)2514 2130 y Fi(\001)2553 |
|
5606 2121 y Fm(a)27 b(b)p Fx(\))2692 2130 y Fi(\001)2730 2121 |
|
5607 y Fm(\031)2780 2130 y Fi(\001)2819 2121 y Fm(t)c Fe(else)e(er)p |
|
5608 Fx(\))533 2200 y Fw(def)544 2247 y Fx(=)34 b([)p Fm(\031)716 |
|
5609 2256 y Fi(\001)755 2247 y Fm(a)p Fx(])p Fm(:)p Fx(\()p |
|
5610 Fm(\031)927 2256 y Fi(\001)965 2247 y Fm(t)p Fx(\))523 |
|
5611 2433 y(where)25 b(w)n(e)g(use)g(in)h(\()p Fl(\003)p Fx(\))g(the)g(fact) |
|
5612 f(that)h Fm(\031)1779 2442 y Fi(\001)1817 2433 y Fe(if)p |
|
5613 Fm(:::)p Fe(then)n Fm(:::)p Fe(else)o Fm(:::)d Fx(=)f |
|
5614 Fe(if)p Fm(:::)p Fe(then)f Fm(\031)2975 2442 y Fi(\001)3014 |
|
5615 2433 y Fm(:::)p Fe(else)g Fm(\031)3330 2442 y Fi(\001)3369 |
|
5616 2433 y Fm(:::)523 2533 y Fx(and)33 b(Lem)f(2\(ii\).)i(In)f(case)e |
|
5617 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 |
|
5618 (a)f(simple)h(calculation)f(using)g(exten-)523 2632 y(sionalit)n(y)d |
|
5619 (of)g(functions.)h(In)g(case)f Fm(a)d Fl(6)p Fx(=)g Fm(b)j |
|
5620 Fx(and)h(Def.)e(5\()p Fm(ii)22 b Fl(\))p Fx(\))q(,)29 |
|
5621 b(the)h(follo)n(wing)f(form)n(ula)f(can)523 2732 y(b)r(e)g(deriv)n(ed)f |
|
5622 (from)g(the)h(assumption)f(b)n(y)g(extensionalit)n(y:)890 |
|
5623 2933 y Fl(8)p Fm(c:)46 b Fe(if)23 b Fm(a)f Fx(=)h Fm(c)g |
|
5624 Fe(then)f Fm(t)1594 2945 y Fw(1)1654 2933 y Fe(else)f(if)h |
|
5625 Fm(c)h Fx(#)h Fm(t)2143 2945 y Fw(1)2203 2933 y Fe(then)d |
|
5626 Fx(\()p Fm(a)14 b(c)p Fx(\))2558 2942 y Fi(\001)2597 |
|
5627 2933 y Fm(t)2627 2945 y Fw(1)2687 2933 y Fe(else)21 b(er)i |
|
5628 Fx(=)1042 3033 y Fe(if)g Fm(b)f Fx(=)h Fm(c)g Fe(then)e |
|
5629 Fm(t)1585 3045 y Fw(2)1646 3033 y Fe(else)g(if)h Fm(c)h |
|
5630 Fx(#)g Fm(t)2134 3045 y Fw(2)2195 3033 y Fe(then)e Fx(\()p |
|
5631 Fm(b)14 b(c)p Fx(\))2542 3042 y Fi(\001)2580 3033 y Fm(t)2610 |
|
5632 3045 y Fw(2)2671 3033 y Fe(else)21 b(er)523 3239 y Fx(Instan)n(tiating) |
|
5633 27 b(this)h(form)n(ula)f(once)g(with)h Fm(a)f Fx(and)h(once)f(with)h |
|
5634 Fm(b)f Fx(yields)h(the)g(t)n(w)n(o)e(equations)1345 3435 |
|
5635 y Fm(t)1375 3447 y Fw(1)1436 3435 y Fx(=)c Fe(if)g Fm(a)h |
|
5636 Fx(#)h Fm(t)1823 3447 y Fw(2)1883 3435 y Fe(then)d Fx(\()p |
|
5637 Fm(b)14 b(a)p Fx(\))2238 3444 y Fi(\001)2276 3435 y Fm(t)2306 |
|
5638 3447 y Fw(2)2367 3435 y Fe(else)21 b(er)1345 3534 y Fm(t)1375 |
|
5639 3546 y Fw(2)1436 3534 y Fx(=)h Fe(if)g Fm(b)h Fx(#)g |
|
5640 Fm(t)1814 3546 y Fw(1)1875 3534 y Fe(then)e Fx(\()p Fm(a)14 |
|
5641 b(b)p Fx(\))2230 3543 y Fi(\001)2268 3534 y Fm(t)2298 |
|
5642 3546 y Fw(1)2359 3534 y Fe(else)21 b(er)523 3741 y Fx(Next,)27 |
|
5643 b(one)g(distinguishes)f(t)n(w)n(o)g(cases)g(where)g Fm(a)d |
|
5644 Fx(#)g Fm(t)2181 3753 y Fw(2)2245 3741 y Fx(and)k Fl(:)14 |
|
5645 b Fm(a)23 b Fx(#)g Fm(t)2664 3753 y Fw(2)2702 3741 y |
|
5646 Fx(,)k(resp)r(ectiv)n(ely)-7 b(.)26 b(In)h(the)523 3840 |
|
5647 y(\014rst)h(case,)g Fm(t)925 3852 y Fw(1)986 3840 y Fx(=)c(\()p |
|
5648 Fm(b)14 b(a)p Fx(\))1233 3849 y Fi(\001)1271 3840 y Fm(t)1301 |
|
5649 3852 y Fw(2)1338 3840 y Fx(,)29 b(whic)n(h)f(b)n(y)g(Lem.)h(1)f(and)g |
|
5650 (Def.)h(2\()p Fm(iii)p Fx(\))e(is)h(equal)g(to)g(\()p |
|
5651 Fm(a)14 b(b)p Fx(\))3147 3849 y Fi(\001)3186 3840 y Fm(t)3216 |
|
5652 3852 y Fw(2)3253 3840 y Fx(;)28 b(and)523 3940 y(ob)n(viously)33 |
|
5653 b Fm(a)h Fx(#)h Fm(t)1109 3952 y Fw(2)1181 3940 y Fx(b)n(y)f |
|
5654 (assumption.)g(In)g(the)h(second)f(case)f Fm(t)2519 3952 |
|
5655 y Fw(1)2591 3940 y Fx(=)h Fe(er)o Fx(.)h(This)f(substituted)523 |
|
5656 4040 y(in)n(to)k(the)g(second)f(equation)g(giv)n(es)g |
|
5657 Fm(t)1726 4052 y Fw(2)1804 4040 y Fx(=)i Fe(if)h Fm(b)f |
|
5658 Fx(#)i Fe(er)e(then)g Fx(\()p Fm(a)14 b(b)p Fx(\))2721 |
|
5659 4049 y Fi(\001)2759 4040 y Fe(er)39 b(else)g(er)o Fx(.)f(Since)523 |
|
5660 4139 y Fe(supp)n Fx(\()p Fe(er)p Fx(\))23 b(=)g Fk(?)p |
|
5661 Fx(,)g Fm(t)1101 4151 y Fw(2)1162 4139 y Fx(=)f(\()p |
|
5662 Fm(a)14 b(b)p Fx(\))1407 4148 y Fi(\001)1445 4139 y Fe(er)23 |
|
5663 b Fx(=)f Fe(er)o Fx(.)i(No)n(w)g(there)f(is)h(a)f(con)n(tradiction)f |
|
5664 (with)j(the)f(assump-)523 4239 y(tion)31 b Fl(:)14 b |
|
5665 Fm(a)29 b Fx(#)g Fm(t)967 4251 y Fw(2)1004 4239 y Fx(,)i(b)r(ecause)g |
|
5666 Fm(a)15 b Fx(#)g Fe(er)o Fx(.)32 b(Def.)c(5\()p Fm(ii)22 |
|
5667 b Fl(\()p Fx(\))31 b(for)g Fm(a)e Fl(6)p Fx(=)f Fm(b)j |
|
5668 Fx(is)g(b)n(y)g(extensionalit)n(y)f(and)g(a)523 4338 |
|
5669 y(case-analysis.)2374 b Fl(u)-55 b(t)523 4549 y Fx(Note)29 |
|
5670 b(that,)g(in)g Fo(gener)l(al)p Fx(,)h(one)e(cannot)h(decide)g(whether)f |
|
5671 (t)n(w)n(o)g(functions)i(from)e Fk(A)52 b Fx(to)29 b |
|
5672 Fm(\010)g Fx(are)523 4649 y(equal;)g(ho)n(w)n(ev)n(er)f(Def.)i(5\()p |
|
5673 Fm(ii)p Fx(\))f(pro)n(vides)g(means)g(to)g(decide)h(whether)g([)p |
|
5674 Fm(a)p Fx(])p Fm(:t)2869 4661 y Fw(1)2932 4649 y Fx(=)d([)p |
|
5675 Fm(b)p Fx(])p Fm(:t)3159 4661 y Fw(2)3225 4649 y Fx(holds:)523 |
|
5676 4749 y(one)h(just)g(has)g(to)g(consider)f(whether)h Fm(a)c |
|
5677 Fx(=)f Fm(b)28 b Fx(and)g(then)h(apply)e(the)i(appropriate)d(prop)r |
|
5678 (ert)n(y)523 4848 y(in)40 b(Def.)h(5\()p Fm(ii)p Fx(\)|just)e(lik)n(e)h |
|
5679 (deciding)f(the)i Fm(\013)p Fx(-equiv)-5 b(alence)39 |
|
5680 b(of)h(t)n(w)n(o)f(lam)n(b)r(da-terms)f(using)523 4948 |
|
5681 y(\()p Fl(\000)p Fx(\))9 b Fl(\031)g Fx(\()p Fl(\000)p |
|
5682 Fx(\).)p Black Black eop end |
|
5683 %%Page: 9 9 |
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5684 TeXDict begin 9 8 bop Black Black 648 448 a Fx(No)n(w)23 |
|
5685 b(ev)n(erything)f(is)i(in)f(place)h(for)f(de\014ning)g(the)h(subset)g |
|
5686 Fm(\003)2498 460 y Fj(\013)2545 448 y Fx(.)g(It)g(is)f(de\014ned)h |
|
5687 (inductiv)n(ely)523 548 y(b)n(y)j(the)h(rules:)p Black |
|
5688 Black 1054 704 a Fm(a)23 b Fl(2)g Fk(A)p 956 728 403 |
|
5689 4 v 956 806 a Fe(am)o Fx(\()p Fm(a)p Fx(\))g Fl(2)h Fm(\003)1311 |
|
5690 818 y Fj(\013)1598 695 y Fm(t)1628 707 y Fw(1)1688 695 |
|
5691 y Fl(2)g Fm(\003)1825 707 y Fj(\013)1955 695 y Fm(t)1985 |
|
5692 707 y Fw(2)2045 695 y Fl(2)f Fm(\003)2181 707 y Fj(\013)p |
|
5693 1598 728 631 4 v 1649 806 a Fe(pr)o Fx(\()p Fm(t)1798 |
|
5694 818 y Fw(1)1835 806 y Fm(;)14 b(t)1902 818 y Fw(2)1939 |
|
5695 806 y Fx(\))24 b Fl(2)f Fm(\003)2131 818 y Fj(\013)2469 |
|
5696 695 y Fm(a)g Fl(2)g Fk(A)106 b Fm(t)23 b Fl(2)h Fm(\003)2947 |
|
5697 707 y Fj(\013)p 2469 728 525 4 v 2556 806 a Fx([)p Fm(a)p |
|
5698 Fx(])p Fm(:t)f Fl(2)h Fm(\003)2859 818 y Fj(\013)523 |
|
5699 972 y Fx(using)j(in)h(the)g(third)g(inference)f(rule)h(the)g(op)r |
|
5700 (eration)e(de\014ned)i(in)g(\(2\).)g(F)-7 b(or)27 b Fm(\003)2964 |
|
5701 984 y Fj(\013)3038 972 y Fx(w)n(e)h(ha)n(v)n(e:)p Black |
|
5702 523 1146 a Ff(Lemma)i(8.)p Black 68 w Fm(\003)1063 1158 |
|
5703 y Fj(\013)1138 1146 y Fx(is:)p Black Black Black 676 |
|
5704 1269 a(\(i\))p Black 42 w(an)e(fs-pset,)f(and)p Black |
|
5705 653 1368 a(\(ii\))p Black 42 w(closed)g(under)h(p)r(erm)n(utations,)f |
|
5706 (that)h(is)f(if)h Fm(x)c Fl(2)f Fm(\003)2363 1380 y Fj(\013)2438 |
|
5707 1368 y Fx(then)28 b Fm(\031)2677 1377 y Fi(\001)2715 |
|
5708 1368 y Fm(x)c Fl(2)f Fm(\003)2922 1380 y Fj(\013)2969 |
|
5709 1368 y Fx(.)p Black 523 1541 a Fo(Pr)l(o)l(of.)p Black |
|
5710 43 w Fx(\(i\):)34 b(The)f(pset-prop)r(erties)e(of)i Fm(\010)g |
|
5711 Fx(carry)e(o)n(v)n(er)g(to)i Fm(\003)2430 1553 y Fj(\013)2477 |
|
5712 1541 y Fx(.)g(The)g(fs-pset)g(prop)r(ert)n(y)e(fol-)523 |
|
5713 1641 y(lo)n(ws)g(b)n(y)g(a)g(routine)g(induction)h(on)f(the)h |
|
5714 (de\014nition)g(of)f Fm(\003)2360 1653 y Fj(\013)2439 |
|
5715 1641 y Fx(using)g(the)h(fact)g(deriv)n(ed)e(from)523 |
|
5716 1740 y(Lem.)d(6\(i,ii\))g(that)g(for)f Fm(x)d Fl(2)g |
|
5717 Fo(fs-pset)q Fx(,)j Fe(supp)o Fx(\([)p Fm(a)p Fx(])p |
|
5718 Fm(:x)p Fx(\))e(=)f Fe(supp)n Fx(\()p Fm(x)p Fx(\))17 |
|
5719 b Fl(\000)f(f)p Fm(a)p Fl(g)p Fx(.)26 b(\(ii\))h(Routine)g(induc-)523 |
|
5720 1840 y(tion)h(o)n(v)n(er)e(the)h(de\014nition)h(of)g |
|
5721 Fm(\003)1536 1852 y Fj(\013)1583 1840 y Fx(.)1777 b Fl(u)-55 |
|
5722 b(t)523 2013 y Fx(T)-7 b(aking)31 b(Lem.)h(8\(i\))g(and)g(Lem.)g(6)f |
|
5723 (together)g(giv)n(es)g(us)g(a)h(simple)g(c)n(haracterisation)d(of)j |
|
5724 (the)523 2113 y(supp)r(ort)38 b(of)h(elemen)n(ts)f(in)h |
|
5725 Fm(\003)1463 2125 y Fj(\013)1510 2113 y Fx(:)g Fe(supp)n |
|
5726 Fx(\()p Fe(am)o Fx(\()p Fm(a)p Fx(\)\))j(=)f Fl(f)p Fm(a)p |
|
5727 Fl(g)p Fx(,)d Fe(supp)n Fx(\()p Fe(pr)p Fx(\()p Fm(t)2698 |
|
5728 2125 y Fw(1)2735 2113 y Fm(;)14 b(t)2802 2125 y Fw(2)2839 |
|
5729 2113 y Fx(\)\))42 b(=)f Fe(supp)n Fx(\()p Fm(t)3287 2125 |
|
5730 y Fw(1)3325 2113 y Fx(\))26 b Fl([)523 2213 y Fe(supp)n |
|
5731 Fx(\()p Fm(t)759 2225 y Fw(2)797 2213 y Fx(\))j(and)g |
|
5732 Fe(supp)n Fx(\([)p Fm(a)p Fx(])p Fm(:t)p Fx(\))d(=)f |
|
5733 Fe(supp)n Fx(\()p Fm(t)p Fx(\))20 b Fl(\000)f(f)p Fm(a)p |
|
5734 Fl(g)p Fx(.)28 b(In)h(other)f(w)n(ords)f(it)j(coincides)e(with)h(what) |
|
5735 523 2312 y(one)e(usually)g(means)h(b)n(y)f(the)h(free)f(v)-5 |
|
5736 b(ariables)26 b(of)i(a)f(lam)n(b)r(da-term.)648 2412 |
|
5737 y(Next,)j(one)f(of)h(the)h(main)f(p)r(oin)n(ts)f(of)h(this)g(pap)r(er:) |
|
5738 g(there)g(is)g(a)f(bijection)h(b)r(et)n(w)n(een)g Fm(\003)3348 |
|
5739 2427 y Fj(=)p Fg(\031)523 2511 y Fx(and)d Fm(\003)742 |
|
5740 2523 y Fj(\013)789 2511 y Fx(.)h(This)g(is)f(sho)n(wn)g(b)n(y)g(using)h |
|
5741 (the)g(follo)n(wing)e(mapping)h(from)h Fm(\003)f Fx(to)g |
|
5742 Fm(\003)2973 2523 y Fj(\013)3020 2511 y Fx(:)797 2714 |
|
5743 y Fm(q)s Fx(\()p Fm(a)p Fx(\))969 2666 y Fw(def)980 2714 |
|
5744 y Fx(=)33 b Fe(am)p Fx(\()p Fm(a)p Fx(\))166 b Fm(q)s |
|
5745 Fx(\()p Fm(t)1542 2726 y Fw(1)1603 2714 y Fm(t)1633 2726 |
|
5746 y Fw(2)1670 2714 y Fx(\))1725 2666 y Fw(def)1736 2714 |
|
5747 y Fx(=)34 b Fe(pr)o Fx(\()p Fm(q)s Fx(\()p Fm(t)2056 |
|
5748 2726 y Fw(1)2094 2714 y Fx(\))p Fm(;)14 b(q)s Fx(\()p |
|
5749 Fm(t)2265 2726 y Fw(2)2303 2714 y Fx(\)\))166 b Fm(q)s |
|
5750 Fx(\()p Fm(\025a:t)p Fx(\))2807 2666 y Fw(def)2818 2714 |
|
5751 y Fx(=)33 b([)p Fm(a)p Fx(])p Fm(:q)s Fx(\()p Fm(t)p |
|
5752 Fx(\))523 2895 y(and)27 b(the)h(follo)n(wing)f(lemma:)p |
|
5753 Black 523 3068 a Ff(Lemma)j(9.)p Black 40 w Fm(t)1007 |
|
5754 3080 y Fw(1)1068 3068 y Fl(\031)22 b Fm(t)1185 3080 y |
|
5755 Fw(2)1250 3068 y Fx(if)28 b(and)g(only)f(if)h Fm(q)s |
|
5756 Fx(\()p Fm(t)1848 3080 y Fw(1)1886 3068 y Fx(\))23 b(=)g |
|
5757 Fm(q)s Fx(\()p Fm(t)2131 3080 y Fw(2)2168 3068 y Fx(\).)p |
|
5758 Black 523 3242 a Fo(Pr)l(o)l(of.)p Black 43 w Fx(By)k(routine)h |
|
5759 (induction)g(o)n(v)n(er)d(de\014nition)j(of)g Fm(\003)2268 |
|
5760 3254 y Fj(\013)2315 3242 y Fx(.)1045 b Fl(u)-55 b(t)p |
|
5761 Black 523 3415 a Ff(Theorem)30 b(1.)p Black 41 w Fx(There)d(is)h(a)f |
|
5762 (bijection)h(b)r(et)n(w)n(een)g Fm(\003)2155 3430 y Fj(=)p |
|
5763 Fg(\031)2272 3415 y Fx(and)f Fm(\003)2491 3427 y Fj(\013)2538 |
|
5764 3415 y Fx(.)p Black 523 3588 a Fo(Pr)l(o)l(of.)p Black |
|
5765 43 w Fx(The)k(mapping)f Fm(q)k Fx(needs)c(to)g(b)r(e)h(lifted)h(to)e |
|
5766 Fm(\013)p Fx(-equiv)-5 b(alence)30 b(classes)f(\(see)i([14]\).)f(F)-7 |
|
5767 b(or)523 3688 y(this)31 b(de\014ne)g Fm(q)971 3658 y |
|
5768 Fg(0)995 3688 y Fx(\([)p Fm(t)p Fx(])1103 3700 y Fj(\013)1150 |
|
5769 3688 y Fx(\))h(as)e(follo)n(ws:)g(apply)g Fm(q)k Fx(to)d(ev)n(ery)f |
|
5770 (elemen)n(t)h(of)f(the)i(set)e([)p Fm(t)p Fx(])3010 3700 |
|
5771 y Fj(\013)3089 3688 y Fx(and)h(build)523 3787 y(the)d(union)f(of)h(the) |
|
5772 g(results.)f(By)g(Lem.)h(9)f(this)h(m)n(ust)f(yield)h(a)f(singleton)g |
|
5773 (set.)g(The)h(result)f(of)523 3887 y Fm(q)563 3857 y |
|
5774 Fg(0)586 3887 y Fx(\([)p Fm(t)p Fx(])694 3899 y Fj(\013)742 |
|
5775 3887 y Fx(\))33 b(is)g(then)g(the)g(singleton.)f(Surjectivit)n(y)h(of)f |
|
5776 Fm(q)2209 3857 y Fg(0)2265 3887 y Fx(is)h(sho)n(wn)f(b)n(y)g(a)g |
|
5777 (routine)h(induction)523 3987 y(o)n(v)n(er)26 b(the)j(de\014nition)f |
|
5778 (of)g Fm(\003)1367 3999 y Fj(\013)1414 3987 y Fx(.)h(Injectivit)n(y)f |
|
5779 (of)g Fm(q)2000 3957 y Fg(0)2052 3987 y Fx(follo)n(ws)f(from)g(Lem.)i |
|
5780 (9)e(since)h([)p Fm(t)3057 3999 y Fw(1)3095 3987 y Fx(])3118 |
|
5781 3999 y Fj(\013)3189 3987 y Fx(=)23 b([)p Fm(t)3330 3999 |
|
5782 y Fw(2)3368 3987 y Fx(])3391 3999 y Fj(\013)523 4086 |
|
5783 y Fx(for)k(all)g Fm(t)795 4098 y Fw(1)856 4086 y Fl(\031)22 |
|
5784 b Fm(t)973 4098 y Fw(2)1011 4086 y Fx(.)2349 b Fl(u)-55 |
|
5785 b(t)523 4351 y Fp(4)112 b(Structural)36 b(Induction)h(Principle)523 |
|
5786 4549 y Fx(The)c(de\014nition)h(of)f Fm(\003)1232 4561 |
|
5787 y Fj(\013)1313 4549 y Fx(pro)n(vides)e(an)i(induction)h(principle)f |
|
5788 (for)g(free.)g(Ho)n(w)n(ev)n(er,)f(this)h(in-)523 4649 |
|
5789 y(duction)h(principle)f(is)h(not)g(v)n(ery)e(con)n(v)n(enien)n(t)h(in)h |
|
5790 (practice.)f(Consider)f(Fig.)i(2)f(sho)n(wing)f(a)523 |
|
5791 4749 y(t)n(ypical)k(informal)f(pro)r(of)h(in)n(v)n(olving)e(lam)n(b)r |
|
5792 (da-terms|it)i(is)g(Barendregt's)e(pro)r(of)i(of)g(the)523 |
|
5793 4848 y(substitution)28 b(lemma)f(tak)n(en)g(from)g([3].)g(This)g |
|
5794 (informal)g(pro)r(of)g(considers)f(in)h(the)h(lam)n(b)r(da-)523 |
|
5795 4948 y(case)e(only)g(binders)g Fm(z)j Fx(that)e(ha)n(v)n(e)e(suitable)i |
|
5796 (prop)r(erties)e(\(namely)h(b)r(eing)h(fresh)f(for)g |
|
5797 Fm(x)p Fx(,)h Fm(y)s Fx(,)f Fm(N)p Black Black eop end |
|
5798 %%Page: 10 10 |
|
5799 TeXDict begin 10 9 bop Black Black Black 523 369 2915 |
|
5800 4 v 523 2049 4 1681 v 563 464 a Fs(Substitution)32 b(Lemma:)25 |
|
5801 b Fu(If)g Fr(x)c Fc(6\021)g Fr(y)28 b Fu(and)e Fr(x)20 |
|
5802 b Fc(62)i Fr(F)11 b(V)17 b Fu(\()p Fr(L)p Fu(\),)25 b(then)1035 |
|
5803 600 y Fr(M)8 b Fu([)p Fr(x)22 b Fu(:=)f Fr(N)8 b Fu(][)p |
|
5804 Fr(y)25 b Fu(:=)d Fr(L)p Fu(])f Fc(\021)h Fr(M)8 b Fu([)p |
|
5805 Fr(y)24 b Fu(:=)d Fr(L)p Fu(][)p Fr(x)h Fu(:=)g Fr(N)8 |
|
5806 b Fu([)p Fr(y)24 b Fu(:=)e Fr(L)p Fu(]].)563 759 y Fs(Pro)r(of:)27 |
|
5807 b Fu(By)e(induction)h(on)f(the)h(structure)f(of)h Fr(M)8 |
|
5808 b Fu(.)563 860 y Fs(Case)29 b(1:)c Fr(M)34 b Fu(is)26 |
|
5809 b(a)g(v)l(ariable.)681 960 y(Case)h(1.1.)e Fr(M)30 b |
|
5810 Fc(\021)21 b Fr(x)p Fu(.)k(Then)h(b)r(oth)f(sides)h(equal)g |
|
5811 Fr(N)8 b Fu([)p Fr(y)25 b Fu(:=)c Fr(L)p Fu(])26 b(since)g |
|
5812 Fr(x)21 b Fc(6\021)g Fr(y)s Fu(.)681 1061 y(Case)27 b(1.2.)e |
|
5813 Fr(M)30 b Fc(\021)21 b Fr(y)s Fu(.)k(Then)h(b)r(oth)f(sides)h(equal)g |
|
5814 Fr(L)p Fu(,)g(for)h Fr(x)21 b Fc(62)g Fr(F)11 b(V)17 |
|
5815 b Fu(\()p Fr(L)p Fu(\))25 b(implies)1008 1161 y Fr(L)p |
|
5816 Fu([)p Fr(x)c Fu(:=)h Fr(:)13 b(:)g(:)p Fu(])22 b Fc(\021)f |
|
5817 Fr(L)p Fu(.)681 1262 y(Case)27 b(1.3.)e Fr(M)30 b Fc(\021)21 |
|
5818 b Fr(z)j Fc(6\021)d Fr(x;)13 b(y)s Fu(.)25 b(Then)h(b)r(oth)f(sides)h |
|
5819 (equal)g Fr(z)s Fu(.)563 1362 y Fs(Case)i(2:)c Fr(M)30 |
|
5820 b Fc(\021)21 b Fr(\025z)s(:M)1231 1370 y Fv(1)1266 1362 |
|
5821 y Fu(.)k(By)g(the)f(v)l(ariable)i(con)n(v)n(en)n(tion)e(w)n(e)h(ma)n(y) |
|
5822 f(assume)h(that)f Fr(z)h Fc(6\021)c Fr(x;)12 b(y)28 b |
|
5823 Fu(and)563 1453 y Fr(z)g Fu(is)f(not)e(free)i(in)e Fr(N)t(;)14 |
|
5824 b(L)p Fu(.)26 b(Then)f(b)n(y)g(induction)g(h)n(yp)r(othesis)799 |
|
5825 1577 y(\()p Fr(\025z)s(:M)1008 1585 y Fv(1)1043 1577 |
|
5826 y Fu(\)[)p Fr(x)c Fu(:=)g Fr(N)8 b Fu(][)p Fr(y)25 b |
|
5827 Fu(:=)d Fr(L)p Fu(])47 b Fc(\021)g Fr(\025z)s(:)p Fu(\()p |
|
5828 Fr(M)1974 1585 y Fv(1)2009 1577 y Fu([)p Fr(x)21 b Fu(:=)h |
|
5829 Fr(N)8 b Fu(][)p Fr(y)25 b Fu(:=)c Fr(L)p Fu(]\))1658 |
|
5830 1678 y Fc(\021)47 b Fr(\025z)s(:)p Fu(\()p Fr(M)1974 |
|
5831 1686 y Fv(1)2009 1678 y Fu([)p Fr(y)24 b Fu(:=)e Fr(L)p |
|
5832 Fu(][)p Fr(x)f Fu(:=)h Fr(N)8 b Fu([)p Fr(y)25 b Fu(:=)c |
|
5833 Fr(L)p Fu(]]\))1658 1778 y Fc(\021)47 b Fu(\()p Fr(\025z)s(:M)1974 |
|
5834 1786 y Fv(1)2009 1778 y Fu(\)[)p Fr(y)24 b Fu(:=)e Fr(L)p |
|
5835 Fu(][)p Fr(x)f Fu(:=)h Fr(N)8 b Fu([)p Fr(y)24 b Fu(:=)e |
|
5836 Fr(L)p Fu(]].)563 1902 y Fs(Case)31 b(3:)d Fr(M)33 b |
|
5837 Fc(\021)25 b Fr(M)1140 1910 y Fv(1)1175 1902 y Fr(M)1249 |
|
5838 1910 y Fv(2)1284 1902 y Fu(.)j(The)g(statemen)n(t)f(follo)n(ws)k(again) |
|
5839 e(from)f(the)f(induction)h(h)n(yp)r(othesis.)3291 1994 |
|
5840 y Fb(\003)p 3435 2049 V 523 2052 2915 4 v 523 2199 a |
|
5841 Fs(Fig.)15 b(2.)29 b Fu(The)g(informal)h(pro)r(of)h(of)f(the)f |
|
5842 (substitution)g(lemma)g(copied)g(from)g([3].)i(In)e(the)g(lam)n(b)r |
|
5843 (da-)523 2291 y(case,)35 b(the)f(v)l(ariable)g(con)n(v)n(en)n(tion)f |
|
5844 (allo)n(ws)j(Barendregt)f(to)f(mo)n(v)n(e)f(the)g(substitutions)h |
|
5845 (under)f(the)523 2382 y(binder,)25 b(to)h(apply)g(the)f(induction)g(h)n |
|
5846 (yp)r(othesis)h(and)f(then)g(to)h(pull)f(out)h(the)f(substitutions.)p |
|
5847 Black 523 2718 a Fx(and)d Fm(L)p Fx(\).)f(If)h(w)n(e)g(w)n(ould)f(pro)n |
|
5848 (v)n(e)f(the)i(substitution)g(lemma)g(b)n(y)g(induction)g(o)n(v)n(er)e |
|
5849 (the)i(de\014nition)523 2817 y(of)30 b Fm(\003)678 2829 |
|
5850 y Fj(\013)725 2817 y Fx(,)g(then)h(w)n(e)e(w)n(ould)h(need)g(to)g(sho)n |
|
5851 (w)f(the)i(lam)n(b)r(da-case)d(for)h Fo(al)t(l)40 b Fm(z)t |
|
5852 Fx(,)30 b(not)g(just)g(the)h(ones)523 2917 y(b)r(eing)f(suitably)f |
|
5853 (fresh.)h(This)f(w)n(ould)g(mean)h(w)n(e)f(ha)n(v)n(e)f(to)i(rename)f |
|
5854 (binders)g(and)g(establish)523 3016 y(a)d(n)n(um)n(b)r(er)g(of)g |
|
5855 (auxiliary)f(lemmas)h(concerning)f(suc)n(h)i(renamings.)e(In)h(this)h |
|
5856 (section)f(w)n(e)g(will)523 3116 y(deriv)n(e)33 b(an)g(induction)h |
|
5857 (principle)g(whic)n(h)f(allo)n(ws)g(a)g(similar)g(con)n(v)n(enien)n(t)f |
|
5858 (reasoning)g(as)h(in)523 3216 y(Barendregt's)26 b(informal)h(pro)r(of.) |
|
5859 648 3340 y(F)-7 b(or)39 b(this)h(w)n(e)f(only)h(consider)e(induction)i |
|
5860 (h)n(yp)r(otheses)f(of)h(the)g(form)g Fm(P)55 b(t)43 |
|
5861 b(x)p Fx(,)e(where)523 3439 y Fm(P)49 b Fx(is)36 b(the)h(prop)r(ert)n |
|
5862 (y)e(to)i(b)r(e)g(pro)n(v)n(ed;)e Fm(P)48 b Fx(dep)r(ends)37 |
|
5863 b(on)f(a)h(v)-5 b(ariable)35 b Fm(t)j Fl(2)g Fm(\003)2925 |
|
5864 3451 y Fj(\013)3009 3439 y Fx(\(o)n(v)n(er)d(whic)n(h)523 |
|
5865 3539 y(the)i(induction)h(is)f(done\),)g(and)g(a)g(v)-5 |
|
5866 b(ariable)36 b Fm(x)h Fx(standing)g(for)f(the)i(\\other")d(v)-5 |
|
5867 b(ariables)36 b(or)523 3639 y Fo(c)l(ontext)31 b Fx(of)24 |
|
5868 b(the)h(induction.)g(Since)f Fm(x)h Fx(is)g(allo)n(w)n(ed)d(to)j(b)r(e) |
|
5869 f(a)g(tuple,)h(sev)n(eral)e(v)-5 b(ariables)23 b(can)h(b)r(e)523 |
|
5870 3738 y(enco)r(ded.)g(In)g(case)f(of)h(the)h(substitution)f(lemma)g(in)g |
|
5871 (Fig.)g(2)g(the)g(notation)f Fm(P)35 b(t)23 b(x)i Fx(should)f(b)r(e)523 |
|
5872 3838 y(understo)r(o)r(d)31 b(as)g(follo)n(ws:)f(the)i(induction)g(v)-5 |
|
5873 b(ariable)30 b Fm(t)i Fx(is)f Fm(M)9 b Fx(,)32 b(the)f(con)n(text)g |
|
5874 Fm(x)h Fx(is)g(the)g(tuple)523 3938 y(\()p Fm(x;)14 b(y)s(;)g(N)t(;)g |
|
5875 (L)p Fx(\))28 b(and)f(the)h(induction)g(h)n(yp)r(othesis)f |
|
5876 Fm(P)40 b Fx(is)755 4169 y Fm(\025M)t(:)47 b(\025)p Fx(\()p |
|
5877 Fm(x;)14 b(y)s(;)g(N)t(;)g(L)p Fx(\))p Fm(:)47 b(M)9 |
|
5878 b Fx([)p Fm(x)23 b Fx(:=)f Fm(N)9 b Fx(][)p Fm(y)26 b |
|
5879 Fx(:=)d Fm(L)p Fx(])f Fl(\021)h Fm(M)9 b Fx([)p Fm(y)25 |
|
5880 b Fx(:=)e Fm(L)p Fx(][)p Fm(x)g Fx(:=)g Fm(N)9 b Fx([)p |
|
5881 Fm(y)25 b Fx(:=)e Fm(L)p Fx(]])523 4425 y(where)h(w)n(e)h(use)f(Isab)r |
|
5882 (elle's)g(con)n(v)n(enien)n(t)g(tuple-notation)g(for)g(the)h(second)g |
|
5883 (lam)n(b)r(da-abstrac-)523 4525 y(tion)30 b([11].)f(So)h(b)n(y)f |
|
5884 (writing)g Fm(P)39 b(t)27 b(x)j Fx(w)n(e)g(just)g(mak)n(e)f(explicit)h |
|
5885 (all)g(the)g(v)-5 b(ariables)28 b(in)n(v)n(olv)n(ed)h(in)523 |
|
5886 4625 y(the)f(induction.)648 4749 y(F)-7 b(rom)23 b(the)i(inductiv)n(e)f |
|
5887 (de\014nition)g(of)g Fm(\003)1872 4761 y Fj(\013)1943 |
|
5888 4749 y Fx(w)n(e)g(can)f(deriv)n(e)g(a)h(structural)f(induction)h(prin-) |
|
5889 523 4848 y(ciple)36 b(that)g(requires)f(to)h(pro)n(v)n(e)e(the)j(lam)n |
|
5890 (b)r(da-case)d(for)i(binders)f(that)h(are)f(fresh)h(for)g(the)523 |
|
5891 4948 y(con)n(text)27 b Fm(x)p Fx(|this)i(is)e(what)h(the)g(v)-5 |
|
5892 b(ariable)26 b(con)n(v)n(en)n(tion)g(assumes.)p Black |
|
5893 Black eop end |
|
5894 %%Page: 11 11 |
|
5895 TeXDict begin 11 10 bop Black Black Black 523 448 a Ff(Lemma)30 |
|
5896 b(10)p Black 41 w(\(Induction)f(Principle\).)g Fx(Giv)n(en)c(an)h |
|
5897 (induction)f(h)n(yp)r(othesis)h Fm(P)34 b(t)23 b(x)k |
|
5898 Fx(with)523 548 y Fm(t)c Fl(2)g Fm(\003)712 560 y Fj(\013)787 |
|
5899 548 y Fx(and)28 b Fm(x)23 b Fl(2)g Fo(fs-pset)q Fx(,)k(then)i(pro)n |
|
5900 (ving)d(the)i(follo)n(wing:)p Black 581 751 a Fl(\017)p |
|
5901 Black 41 w(8)p Fm(x)23 b(a:)g(P)35 b Fe(am)o Fx(\()p |
|
5902 Fm(a)p Fx(\))23 b Fm(x)p Black 581 859 a Fl(\017)p Black |
|
5903 41 w(8)p Fm(x)g(t)811 871 y Fw(1)871 859 y Fm(t)901 871 |
|
5904 y Fw(2)938 859 y Fm(:)g(P)35 b(t)1102 871 y Fw(1)1163 |
|
5905 859 y Fm(x)19 b Fl(^)f Fm(P)35 b(t)1420 871 y Fw(2)1480 |
|
5906 859 y Fm(x)47 b Fl(\))23 b Fm(P)35 b Fe(pr)o Fx(\()p |
|
5907 Fm(t)1917 871 y Fw(1)1955 859 y Fm(;)14 b(t)2022 871 |
|
5908 y Fw(2)2059 859 y Fx(\))23 b Fm(x)p Black 581 968 a Fl(\017)p |
|
5909 Black 41 w(8)p Fm(x)g(a:)g(a)g Fx(#)g Fm(x)47 b Fl(\))f |
|
5910 Fx(\()p Fl(8)p Fm(t:)22 b(P)35 b(t)23 b(x)47 b Fl(\))f |
|
5911 Fm(P)35 b Fx([)p Fm(a)p Fx(])p Fm(:t)23 b(x)p Fx(\))523 |
|
5912 1162 y(giv)n(es)j Fl(8)p Fm(t)d(x:)g(P)35 b(t)23 b(x)p |
|
5913 Fx(.)p Black 523 1373 a Fo(Pr)l(o)l(of.)p Black 43 w |
|
5914 Fx(By)38 b(induction)h(o)n(v)n(er)d(the)i(de\014nition)h(of)f |
|
5915 Fm(\003)2188 1385 y Fj(\013)2235 1373 y Fx(.)g(W)-7 b(e)39 |
|
5916 b(need)f(to)g(strengthen)g(the)g(in-)523 1472 y(duction)h(h)n(yp)r |
|
5917 (othesis)g(to)g Fl(8)p Fm(t)i(\031)k(x:)e(P)53 b Fx(\()p |
|
5918 Fm(\031)1879 1481 y Fi(\001)1918 1472 y Fm(t)p Fx(\))43 |
|
5919 b Fm(x)p Fx(,)c(that)g(means)g(considering)e Fm(t)j Fx(under)e(all)523 |
|
5920 1572 y(p)r(erm)n(utations)30 b Fm(\031)s Fx(.)h(Only)f(the)h(case)e |
|
5921 (for)h(terms)g(of)h(the)f(form)g([)p Fm(a)p Fx(])p Fm(:t)h |
|
5922 Fx(will)g(b)r(e)g(explained.)f(W)-7 b(e)523 1672 y(need)28 |
|
5923 b(to)h(sho)n(w)e(that)h Fm(P)36 b Fx(\()p Fm(\031)1377 |
|
5924 1681 y Fi(\001)1416 1672 y Fx([)p Fm(a)p Fx(])p Fm(:t)p |
|
5925 Fx(\))25 b Fm(x)p Fx(,)k(where)e Fm(\031)2005 1681 y |
|
5926 Fi(\001)2044 1672 y Fx([)p Fm(a)p Fx(])p Fm(:t)d Fx(=)g([)p |
|
5927 Fm(\031)2373 1681 y Fi(\001)2411 1672 y Fm(a)p Fx(])p |
|
5928 Fm(:)p Fx(\()p Fm(\031)2583 1681 y Fi(\001)2622 1672 |
|
5929 y Fm(t)p Fx(\))29 b(b)n(y)f(Def.)h(5\()p Fm(i)p Fx(\).)f(By)g(IH,)523 |
|
5930 1771 y(\()p Fl(\003)597 1741 y Fw(1)634 1771 y Fx(\))33 |
|
5931 b Fl(8)p Fm(\031)h(x:)e(P)43 b Fx(\()p Fm(\031)1107 1780 |
|
5932 y Fi(\001)1146 1771 y Fm(t)p Fx(\))31 b Fm(x)j Fx(holds.)e(Since)h |
|
5933 Fm(x;)14 b(\031)1921 1780 y Fi(\001)1960 1771 y Fm(t;)g(\031)2077 |
|
5934 1780 y Fi(\001)2115 1771 y Fm(a)31 b Fl(2)h Fo(fs-pset)h |
|
5935 Fx(holds,)f(one)g(can)h(deriv)n(e)e(b)n(y)523 1871 y(Lem.)j(5)g(that)g |
|
5936 (there)f(is)h(a)g Fm(c)g Fx(suc)n(h)f(that)h(\()p Fl(\003)1907 |
|
5937 1841 y Fw(2)1944 1871 y Fx(\))h Fm(c)e Fx(#)h(\()p Fm(x;)14 |
|
5938 b(\031)2349 1880 y Fi(\001)2388 1871 y Fm(t;)g(\031)2505 |
|
5939 1880 y Fi(\001)2544 1871 y Fm(a)p Fx(\).)34 b(F)-7 b(rom)33 |
|
5940 b Fm(c)h Fx(#)g Fm(x)g Fx(and)g(the)523 1970 y(assumption,)29 |
|
5941 b(one)f(can)h(further)f(deriv)n(e)g(\()p Fl(8)p Fm(t:)d(P)37 |
|
5942 b(t)26 b(x)51 b Fl(\))f Fm(P)37 b Fx([)p Fm(c)p Fx(])p |
|
5943 Fm(:t)26 b(x)p Fx(\).)k(Giv)n(en)e(\()p Fl(\003)3052 |
|
5944 1940 y Fw(1)3089 1970 y Fx(\))i(w)n(e)e(ha)n(v)n(e)523 |
|
5945 2070 y(that)j Fm(P)39 b Fx(\(\()p Fm(c)j(\031)990 2079 |
|
5946 y Fi(\001)1028 2070 y Fm(a)p Fx(\))14 b(::)g Fm(\031)1242 |
|
5947 2079 y Fi(\001)1294 2070 y Fm(t)p Fx(\))28 b Fm(x)i Fx(holds)g(and)g |
|
5948 (th)n(us)g(also)g Fm(P)39 b Fx(\([)p Fm(c)p Fx(])p Fm(:)p |
|
5949 Fx(\(\()p Fm(c)j(\031)2621 2079 y Fi(\001)2660 2070 y |
|
5950 Fm(a)p Fx(\))14 b(::)g Fm(\031)2873 2079 y Fi(\001)2925 |
|
5951 2070 y Fm(t)p Fx(\)\))28 b Fm(x)p Fx(.)j(Because)523 |
|
5952 2170 y(of)d(\()p Fl(\003)692 2140 y Fw(2)729 2170 y Fx(\))h |
|
5953 Fm(c)24 b Fl(6)p Fx(=)f Fm(\031)988 2179 y Fi(\001)1027 |
|
5954 2170 y Fm(a)28 b Fx(and)g Fm(c)c Fx(#)g Fm(\031)1464 |
|
5955 2179 y Fi(\001)1502 2170 y Fm(t)p Fx(,)29 b(and)f(b)n(y)f(Def.)i(5\()p |
|
5956 Fm(ii)p Fx(\))f(w)n(e)g(ha)n(v)n(e)f(that)h([)p Fm(c)p |
|
5957 Fx(])p Fm(:)p Fx(\(\()p Fm(c)42 b(\031)3022 2179 y Fi(\001)3061 |
|
5958 2170 y Fm(a)p Fx(\))10 b(::)g Fm(\031)3267 2179 y Fi(\001)3319 |
|
5959 2170 y Fm(t)24 b Fx(=)523 2269 y([)p Fm(\031)596 2278 |
|
5960 y Fi(\001)635 2269 y Fm(a)p Fx(])p Fm(:)p Fx(\()p Fm(\031)807 |
|
5961 2278 y Fi(\001)845 2269 y Fm(t)p Fx(\).)k(Therefore)f(w)n(e)g(can)g |
|
5962 (conclude)g(with)i Fm(P)34 b Fx(\()p Fm(\031)2309 2278 |
|
5963 y Fi(\001)2348 2269 y Fx([)p Fm(a)p Fx(])p Fm(:t)p Fx(\))24 |
|
5964 b Fm(x)p Fx(.)766 b Fl(u)-55 b(t)648 2480 y Fx(With)33 |
|
5965 b(this)f(w)n(e)g(ha)n(v)n(e)f(ac)n(hiev)n(ed)g(what)i(w)n(e)f(set)g |
|
5966 (out)g(in)h(the)f(in)n(tro)r(duction:)g(w)n(e)g(ha)n(v)n(e)f(a)523 |
|
5967 2580 y(represen)n(tation)17 b(for)h Fm(\013)p Fx(-equiv)-5 |
|
5968 b(alen)n(t)19 b(lam)n(b)r(da-terms)e(based)h(on)g(names)h(\(for)f |
|
5969 (example)g([)p Fm(\025a:t)p Fx(])3390 2592 y Fj(\013)523 |
|
5970 2680 y Fx(is)23 b(represen)n(ted)f(b)n(y)g([)p Fm(a)p |
|
5971 Fx(])p Fm(:t)p Fx(\))i(and)f(w)n(e)f(ha)n(v)n(e)g(an)h(induction)g |
|
5972 (principle)g(where)f(the)i(lam)n(b)r(da-case)523 2779 |
|
5973 y(needs)31 b(to)g(b)r(e)h(pro)n(v)n(ed)d(for)i(binders)g(that)g(are)f |
|
5974 (fresh)h(w.r.t.)g(the)h(v)-5 b(ariables)30 b(in)h(the)g(con)n(text)523 |
|
5975 2879 y(of)26 b(the)g(induction,)g(i.e.,)g(w)n(e)g(can)f(reason)g(as)g |
|
5976 (if)h(w)n(e)g(had)f(emplo)n(y)n(ed)g(a)h(v)-5 b(ariable)24 |
|
5977 b(con)n(v)n(en)n(tion.)523 3190 y Fp(5)112 b(Examples)523 |
|
5978 3435 y Fx(It)35 b(is)f(reasonably)f(straigh)n(tforw)n(ard)f(to)i |
|
5979 (implemen)n(t)h(the)g(results)f(from)g(Sec.)h(3)f(and)h(4)f(in)523 |
|
5980 3535 y(Isab)r(elle/HOL:)25 b(the)h(set)h Fm(\010)f Fx(is)g(an)g |
|
5981 (inductiv)n(e)g(datat)n(yp)r(e,)g(the)g(pset)g(and)g(fs-pset)g(prop)r |
|
5982 (erties)523 3635 y(can)20 b(b)r(e)g(form)n(ulated)f(as)h(axiomatic)f(t) |
|
5983 n(yp)r(e-classes)f([20],)h(and)h(the)h(subset)f Fm(\003)2863 |
|
5984 3647 y Fj(\013)2930 3635 y Fx(can)f(b)r(e)i(de\014ned)523 |
|
5985 3734 y(using)31 b(the)h(Isab)r(elle's)f Fe(typedef)n |
|
5986 Fx(-mec)n(hanism.)f(This)i(section)f(fo)r(cuses)h(on)f(ho)n(w)g |
|
5987 (reasoning)523 3834 y(o)n(v)n(er)26 b Fm(\003)759 3846 |
|
5988 y Fj(\013)833 3834 y Fx(pans)i(out)f(in)h(practice.)648 |
|
5989 3943 y(The)38 b(\014rst)h(obstacle)f(is)h(that)g(so)f(far)g(Isab)r |
|
5990 (elle's)g(datat)n(yp)r(e)h(pac)n(k)-5 b(age)37 b(is)i(not)f(general)523 |
|
5991 4042 y(enough)33 b(to)h(allo)n(w)f(a)h(direct)g(de\014nition)g(of)g |
|
5992 (functions)g(o)n(v)n(er)f Fm(\003)2545 4054 y Fj(\013)2592 |
|
5993 4042 y Fx(:)h(although)f Fm(\003)3059 4054 y Fj(\013)3140 |
|
5994 4042 y Fx(con)n(tains)523 4142 y(only)i(terms)h(of)f(the)h(form)g |
|
5995 Fe(am)o Fx(\()p Fm(a)p Fx(\),)g Fe(pr)o Fx(\()p Fm(t)1814 |
|
5996 4154 y Fw(1)1852 4142 y Fm(;)14 b(t)1919 4154 y Fw(2)1956 |
|
5997 4142 y Fx(\))36 b(and)f([)p Fm(a)p Fx(])p Fm(:t)p Fx(,)h(pattern-matc)n |
|
5998 (hing)f(in)g(Isab)r(elle)523 4241 y(requires)h(the)j(injectivit)n(y)f |
|
5999 (of)g(term-constructors.)d(But)j(clearly)-7 b(,)37 b([)p |
|
6000 Fm(a)p Fx(])p Fm(:t)h Fx(is)f Fo(not)46 b Fx(injectiv)n(e.)523 |
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6001 4341 y(F)-7 b(ortunately)g(,)37 b(one)f(can)h(w)n(ork)e(around)h(this)h |
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6002 (obstacle)f(b)n(y)-7 b(,)37 b(roughly)f(sp)r(eaking,)g(de\014ning)523 |
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6003 4441 y(functions)h(as)g(inductiv)n(e)g(relations)e(and)i(then)h(use)f |
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6004 (the)g(de\014nite)g(description)g(op)r(erator)523 4540 |
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6005 y Fo(THE)h Fx(of)28 b(Isab)r(elle)f(to)g(turn)h(the)g(relations)f(in)n |
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6006 (to)g(functions.)648 4649 y(W)-7 b(e)30 b(giv)n(e)f(an)h(example:)f |
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6007 (capture-a)n(v)n(oiding)e(substitution)k(can)e(b)r(e)i(de\014ned)f(as)f |
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6008 (a)h(four-)523 4749 y(place)21 b(relation)g(\(the)i(\014rst)e(argumen)n |
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6009 (t)g(con)n(tains)g(the)h(term)g(in)n(to)f(whic)n(h)h(something)f(is)h |
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6010 (b)r(eing)523 4848 y(substituted,)35 b(the)g(second)e(the)i(v)-5 |
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6011 b(ariable)33 b(that)i(is)f(substituted)h(for,)f(the)g(third)h(the)f |
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6012 (term)523 4948 y(that)28 b(is)f(substituted,)i(and)e(the)h(last)g(con)n |
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6013 (tains)e(the)i(result)g(of)f(the)h(substitution\):)p |
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6014 Black Black eop end |
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6015 %%Page: 12 12 |
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6016 TeXDict begin 12 11 bop Black Black Black Black 535 526 |
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6017 a Fe(consts)41 b(Subst)g(::)i("\()p Fm(\003)1378 538 |
|
6018 y Fj(\013)1443 526 y Fl(\002)18 b Fk(A)41 b Fl(\002)18 |
|
6019 b Fm(\003)1745 538 y Fj(\013)1810 526 y Fl(\002)h Fm(\003)1952 |
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6020 538 y Fj(\013)1998 526 y Fe(\))44 b(set")535 626 y(inductive)c(Subst) |
|
6021 535 726 y(intros)535 825 y(s1:)27 b("\(am)o(\(a\),a,t',t'\))p |
|
6022 Fl(2)p Fe(Subst)o(")535 925 y(s2:)g("a)p Fl(6)p Fx(=)p |
|
6023 Fe(b)42 b Fx(=)-14 b Fl(\))44 b Fe(\(am)n(\(b\),a,t',am)l(\(b\)\))p |
|
6024 Fl(2)p Fe(Subst")535 1024 y(s3:)27 b(")p Fa(J)p Fe(\(s)859 |
|
6025 1036 y Fw(1)896 1024 y Fe(,a,t',s)1204 1036 y Fw(1)1224 |
|
6026 1024 y Fe('\))p Fl(2)p Fe(Subst;)41 b(\(s)1760 1036 y |
|
6027 Fw(2)1796 1024 y Fe(,a,t',s)2104 1036 y Fw(2)2125 1024 |
|
6028 y Fe('\))p Fl(2)p Fe(Subst)p Fa(K)1834 1124 y Fx(=)-14 |
|
6029 b Fl(\))43 b Fe(\(pr)o(\(s)2230 1136 y Fw(1)2266 1124 |
|
6030 y Fe(,s)2354 1136 y Fw(2)2391 1124 y Fe(\),a,t',pr)l(\(s)2871 |
|
6031 1136 y Fw(1)2894 1124 y Fe(',s)3026 1136 y Fw(2)3048 |
|
6032 1124 y Fe('\)\))p Fl(2)p Fe(Subst")535 1224 y(s4:)27 |
|
6033 b(")p Fa(J)p Fe(b)p Fx(#)p Fe(\(a,t'\);\(s,a,t',s')o(\))p |
|
6034 Fl(2)p Fe(Su)o(bst)o Fa(K)38 b Fx(=)-14 b Fl(\))43 b |
|
6035 Fe(\([b].s,a,t',[b].s)o('\))p Fl(2)p Fe(Su)o(bs)o(t")523 |
|
6036 1375 y Fx(While)32 b(on)g(\014rst)f(sigh)n(t)g(this)h(relation)f(lo)r |
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6037 (oks)g(as)g(if)h(it)g(de\014ned)g(a)f(non-total)g(function,)h(one)523 |
|
6038 1475 y(should)22 b(b)r(e)g(careful!)f(Clearly)-7 b(,)21 |
|
6039 b(the)h(lam)n(b)r(da-case)e(\(i.e.)44 b Fe(\([b].s,a,t',[b].s)o('\))16 |
|
6040 b Fl(2)24 b Fe(Subst)n Fx(\))523 1575 y(holds)34 b(only)f(under)h(the)g |
|
6041 (precondition)f Fe(b)p Fx(#)p Fe(\(a,s\))p Fx(|roughly)e(meaning)i |
|
6042 (that)h Fm(a)g Fl(6)p Fx(=)f Fm(b)g Fx(and)523 1674 y |
|
6043 Fm(b)c Fx(cannot)f(o)r(ccur)g(freely)h(in)g Fm(s)p Fx(.)g(Ho)n(w)n(ev)n |
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6044 (er,)e Fe(Subst)g Fo(do)l(es)37 b Fx(de\014ne)29 b(a)f(total)h |
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6045 (function,)g(b)r(ecause)523 1774 y Fe(Subst)i Fx(is)h(de\014ned)h(o)n |
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6046 (v)n(er)e Fm(\013)p Fx(-equiv)-5 b(alen)n(t)32 b(lam)n(b)r(da-terms)g |
|
6047 (\(more)g(precisely)g Fm(\003)2996 1786 y Fj(\013)3043 |
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6048 1774 y Fx(\),)h Fo(not)40 b Fx(o)n(v)n(er)523 1873 y(lam)n(b)r |
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6049 (da-terms.)27 b(W)-7 b(e)28 b(can)f(indeed)h(sho)n(w)e(\\totalit)n(y":) |
|
6050 p Black 523 2020 a Ff(Lemma)k(11.)p Black 40 w Fo(F)-6 |
|
6051 b(or)30 b(al)t(l)h Fe(t)1343 2032 y Fw(1)1380 2020 y |
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6052 Fo(,)f Fe(a)p Fo(,)g Fe(t)1578 2032 y Fw(2)1614 2020 |
|
6053 y Fo(,)h Fl(9)p Fe(t)1760 2032 y Fw(3)1797 2020 y Fm(:)23 |
|
6054 b Fx(\()p Fe(t)1919 2045 y Fw(1)1956 2020 y Fm(;)14 b |
|
6055 Fe(a)p Fm(;)g Fe(t)2117 2041 y Fw(2)2154 2020 y Fm(;)g |
|
6056 Fe(t)2235 2041 y Fw(3)2272 2020 y Fx(\))23 b Fl(2)h Fe(Subst)c |
|
6057 Fm(:)p Black 523 2167 a Fo(Pr)l(o)l(of.)p Black 43 w |
|
6058 Fx(The)f(pro)r(of)f(in)h(Isab)r(elle/HOL)e(uses)h(the)h(induction)g |
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6059 (principle)f(deriv)n(ed)g(in)h(Thm.)g(10.)523 2267 y(It)28 |
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6060 b(is)f(as)g(follo)n(ws:)p Black Black 546 2384 a Fe(proof)42 |
|
6061 b(\(nominal)p 1165 2384 27 4 v 28 w(induct)65 b(t)1561 |
|
6062 2396 y Fw(1)1598 2384 y Fe(\))664 2484 y(case)42 b(\(1)h(b\))g(\(*)f |
|
6063 (variable)f(case)h(*\))664 2583 y(show)g(")p Fl(9)p Fe(t)1016 |
|
6064 2595 y Fw(3)1053 2583 y Fe(.)h(\(am\(b\),a,t)1580 2595 |
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6065 y Fw(2)1613 2583 y Fe(,t)1701 2595 y Fw(3)1737 2583 y |
|
6066 Fe(\))p Fl(2)p Fe(Subst")e(by)i(\(cases)e("b=a"\))g(\(force+\))546 |
|
6067 2683 y(next)664 2783 y(case)h(\(2)h(s)1057 2795 y Fw(1)1137 |
|
6068 2783 y Fe(s)1181 2795 y Fw(2)1218 2783 y Fe(\))g(\(*)g(application)c |
|
6069 (case)j(*\))664 2882 y(thus)g(")p Fl(9)p Fe(t)1016 2894 |
|
6070 y Fw(3)1053 2882 y Fe(.)h(\(pr\(s)1360 2894 y Fw(1)1395 |
|
6071 2882 y Fe(,s)1483 2894 y Fw(2)1519 2882 y Fe(\),a,t)1739 |
|
6072 2894 y Fw(2)1775 2882 y Fe(,t)1863 2894 y Fw(3)1899 2882 |
|
6073 y Fe(\))p Fl(2)p Fe(Subst")e(by)i(force)546 2982 y(next)664 |
|
6074 3081 y(case)f(\(3)h(b)g(s\))g(\(*)g(lambda)e(case)h(*\))664 |
|
6075 3181 y(thus)g(")p Fl(9)p Fe(t)1016 3193 y Fw(3)1053 3181 |
|
6076 y Fe(.)h(\([b].s,a,t)1580 3193 y Fw(2)1613 3181 y Fe(,t)1701 |
|
6077 3193 y Fw(3)1737 3181 y Fe(\))p Fl(2)p Fe(Subst")e(by)i(force)546 |
|
6078 3281 y(qed)523 3421 y Fx(The)21 b(induction)g(metho)r(d)g |
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6079 Fe(nominal)p 1657 3421 V 29 w(induct)d Fx(brings)i(the)h(induction)g(h) |
|
6080 n(yp)r(othesis)f(automat-)523 3521 y(ically)27 b(in)n(to)g(the)h(form) |
|
6081 1031 3676 y(\()p Fm(\025)p Fe(t)1155 3688 y Fw(1)1207 |
|
6082 3676 y Fm(\025)p Fx(\()p Fe(a)p Fm(;)14 b Fe(t)1412 3688 |
|
6083 y Fw(2)1449 3676 y Fx(\))p Fm(:)g Fl(9)p Fe(t)1608 3688 |
|
6084 y Fw(3)1645 3676 y Fm(:)p Fx(\()p Fe(t)1744 3688 y Fw(1)1781 |
|
6085 3676 y Fm(;)g Fe(a)p Fm(;)g Fe(t)1943 3688 y Fw(2)1979 |
|
6086 3676 y Fm(;)g Fe(t)2060 3688 y Fw(3)2097 3676 y Fx(\))23 |
|
6087 b Fl(2)h Fe(Subst)m Fx(\))1031 3732 y Fh(|)p 1068 3732 |
|
6088 651 10 v 651 w({z)p 1793 3732 V 651 w(})1730 3810 y Fj(P)2552 |
|
6089 3676 y Fe(t)2596 3688 y Fw(1)2518 3723 y Fh(|{z})2580 |
|
6090 3798 y Fj(t)2704 3676 y Fx(\()p Fe(a)p Fm(;)14 b Fe(t)2861 |
|
6091 3688 y Fw(2)2898 3676 y Fx(\))2704 3732 y Fh(|)p 2741 |
|
6092 3732 39 10 v 39 w({z)p 2854 3732 V 39 w(})2798 3806 y |
|
6093 Fj(x)523 3940 y Fx(b)n(y)37 b(collecting)g(all)h(free)f(v)-5 |
|
6094 b(ariables)37 b(in)h(the)g(goal,)e(and)i(then)g(it)g(applies)f(Thm.)h |
|
6095 (10.)f(This)523 4040 y(results)29 b(in)h(three)f(cases)g(to)g(b)r(e)h |
|
6096 (pro)n(v)n(ed|v)-5 b(ariable)27 b(case,)i(application)g(case)f(and)i |
|
6097 (lam)n(b)r(da-)523 4139 y(case.)c(The)h(requiremen)n(t)f(that)i(the)f |
|
6098 (con)n(text)g(\()p Fe(a)p Fm(;)14 b Fe(t)2126 4151 y |
|
6099 Fw(2)2162 4139 y Fx(\))28 b(is)e(a)h Fo(fs-pset)p Fx(-elemen)n(t)g(is)g |
|
6100 (enforced)f(b)n(y)523 4239 y(using)k(axiomatic)f(t)n(yp)r(e-classes)f |
|
6101 (and)i(relying)f(on)h(Isab)r(elle's)f(t)n(yp)r(e-system.)h(Note)g(that) |
|
6102 g(in)523 4339 y(the)f(lam)n(b)r(da-case)f(it)h(is)g(imp)r(ortan)n(t)g |
|
6103 (to)g(kno)n(w)f(that)h(the)h(binder)f Fe(b)f Fx(is)h(fresh)g(for)g |
|
6104 Fe(a)f Fx(and)h Fe(t)3378 4351 y Fw(2)3415 4339 y Fx(.)523 |
|
6105 4438 y(The)f(pro)r(of)f(obligation)f(in)i(this)g(case)f(is:)938 |
|
6106 4593 y Fe(b)c Fx(#)g(\()p Fe(a)p Fm(;)14 b Fe(t)1254 |
|
6107 4614 y Fw(2)1291 4593 y Fx(\))19 b Fl(^)f(9)p Fe(t)1505 |
|
6108 4605 y Fw(3)1542 4593 y Fm(:)p Fx(\()p Fe(s)p Fm(;)c |
|
6109 Fe(a)p Fm(;)g Fe(t)1802 4614 y Fw(2)1839 4593 y Fm(;)g |
|
6110 Fe(t)1920 4605 y Fw(3)1957 4593 y Fx(\))47 b(implies)f |
|
6111 Fl(9)p Fe(t)2426 4605 y Fw(3)2463 4593 y Fm(:)p Fx(\([)p |
|
6112 Fe(b)p Fx(])p Fm(:)p Fe(s)p Fm(;)14 b Fe(a)p Fm(;)g Fe(t)2835 |
|
6113 4618 y Fw(2)2873 4593 y Fm(;)g Fe(t)2954 4605 y Fw(3)2990 |
|
6114 4593 y Fx(\))523 4749 y(whic)n(h)29 b(can)f(b)r(e)i(easily)e(b)r(e)h |
|
6115 (sho)n(wn)g(b)n(y)f(rule)h Fe(s4)p Fx(.)f(As)h(a)g(result,)f(the)i |
|
6116 (only)e(case)g(in)i(whic)n(h)e(w)n(e)523 4848 y(really)f(need)i(to)f |
|
6117 (man)n(ually)g(\\in)n(terfere")f(is)h(in)h(the)g(v)-5 |
|
6118 b(ariable)27 b(case)g(where)h(w)n(e)g(ha)n(v)n(e)g(to)g(giv)n(e)523 |
|
6119 4948 y(Isab)r(elle)f(the)h(hin)n(t)g(to)g(distinguish)f(the)h(cases)f |
|
6120 Fm(b)c Fx(=)f Fm(a)28 b Fx(and)f Fm(b)c Fl(6)p Fx(=)f |
|
6121 Fm(a)p Fx(.)778 b Fl(u)-55 b(t)p Black Black eop end |
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6122 %%Page: 13 13 |
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6123 TeXDict begin 13 12 bop Black Black 523 448 a Fx(T)-7 |
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6124 b(ogether)27 b(with)h(a)f(uniqueness-lemma)g(\(whose)g(pro)r(of)g(w)n |
|
6125 (e)g(omit\))h(asserting)e(that)837 621 y Fl(8)p Fe(s)928 |
|
6126 633 y Fw(1)964 621 y Fe(s)1008 633 y Fw(2)1045 621 y |
|
6127 Fm(:)p Fx(\()p Fe(t)1144 633 y Fw(1)1181 621 y Fm(;)14 |
|
6128 b Fe(a)p Fm(;)g Fe(t)1343 633 y Fw(2)1380 621 y Fm(;)g |
|
6129 Fe(s)1461 633 y Fw(1)1497 621 y Fx(\))24 b Fl(2)f Fe(Subst)17 |
|
6130 b Fl(^)h Fx(\()p Fe(t)2017 633 y Fw(1)2054 621 y Fm(;)c |
|
6131 Fe(a)p Fm(;)g Fe(t)2216 633 y Fw(2)2253 621 y Fm(;)g |
|
6132 Fe(s)2334 633 y Fw(2)2370 621 y Fx(\))24 b Fl(2)f Fe(Subst)e |
|
6133 Fl(\))i Fe(s)2895 633 y Fw(1)2955 621 y Fx(=)g Fe(s)3087 |
|
6134 633 y Fw(2)3332 621 y Fx(\(3\))523 795 y(one)k(can)g(pro)n(v)n(e)f(the) |
|
6135 i(stronger)e(totalit)n(y-prop)r(ert)n(y)-7 b(,)26 b(namely)h(for)g(all) |
|
6136 h Fe(t)2744 807 y Fw(1)2780 795 y Fx(,)g Fe(a)p Fx(,)f |
|
6137 Fe(t)2969 807 y Fw(2)3006 795 y Fx(:)1469 968 y Fl(9)p |
|
6138 Fx(!)p Fe(t)1582 980 y Fw(3)1619 968 y Fm(:)c Fx(\()p |
|
6139 Fe(t)1741 993 y Fw(1)1778 968 y Fm(;)14 b Fe(a)p Fm(;)g |
|
6140 Fe(t)1939 988 y Fw(2)1976 968 y Fm(;)g Fe(t)2057 988 |
|
6141 y Fw(3)2094 968 y Fx(\))24 b Fl(2)f Fe(Subst)e Fm(:)840 |
|
6142 b Fx(\(4\))523 1141 y(Ha)n(ving)34 b(this)h(at)g(our)f(disp)r(osal,)g |
|
6143 (w)n(e)g(can)g(use)h(Isab)r(elle's)f(de\014nite)i(description)e(op)r |
|
6144 (erator)523 1241 y Fm(T)12 b(H)7 b(E)30 b Fx(and)c(turn)g(capture-a)n |
|
6145 (v)n(oiding)d(substitution)k(in)n(to)f(a)f(function;)i(w)n(e)f(write)f |
|
6146 (this)i(func-)523 1340 y(tion)h(as)f(\()p Fl(\000)p Fx(\)[\()p |
|
6147 Fl(\000)p Fx(\))c(:=)g(\()p Fl(\000)p Fx(\)],)28 b(and)f(establish)g |
|
6148 (the)h(equations:)951 1591 y Fe(am)p Fx(\()p Fe(a)p Fx(\)[)p |
|
6149 Fe(a)22 b Fx(:=)h Fe(t)p Fx(])g(=)g Fe(t)951 1691 y(am)p |
|
6150 Fx(\()p Fe(b)p Fx(\)[)p Fe(a)f Fx(:=)h Fe(t)p Fx(])g(=)g |
|
6151 Fe(am)o Fx(\()p Fe(b)p Fx(\))760 b(pro)n(vided)22 b Fe(a)h |
|
6152 Fl(6)p Fx(=)f Fe(b)796 1791 y(pr)p Fx(\()p Fe(s)960 1803 |
|
6153 y Fw(1)997 1791 y Fm(;)14 b Fe(s)1078 1803 y Fw(2)1114 |
|
6154 1791 y Fx(\)[)p Fe(a)23 b Fx(:=)g Fe(t)p Fx(])g(=)g Fe(pr)o |
|
6155 Fx(\()p Fe(s)1688 1803 y Fw(1)1725 1791 y Fx([)p Fe(a)g |
|
6156 Fx(:=)g Fe(t)p Fx(])p Fm(;)14 b Fe(s)2074 1803 y Fw(2)2110 |
|
6157 1791 y Fx([)p Fe(a)23 b Fx(:=)f Fe(t)p Fx(]\))926 1890 |
|
6158 y(\([)p Fe(b)p Fx(])p Fm(:)p Fe(s)p Fx(\)[)p Fe(a)g Fx(:=)h |
|
6159 Fe(t)p Fx(])g(=)g([)p Fe(b)p Fx(])p Fm(:)p Fx(\()p Fe(s)p |
|
6160 Fx([)p Fe(a)f Fx(:=)h Fe(t)p Fx(]\))467 b(pro)n(vided)22 |
|
6161 b Fe(b)h Fx(#)g(\()p Fe(a)p Fm(;)14 b Fe(t)p Fx(\))3332 |
|
6162 1742 y(\(5\))523 2060 y(These)24 b(equations)f(can)h(b)r(e)g(supplied)g |
|
6163 (to)g(Isab)r(elle's)g(simpli\014er)g(and)f(one)h(can)g(reason)e(ab)r |
|
6164 (out)523 2159 y(substitution)30 b(\\just)g(lik)n(e)f(on)h(pap)r(er".)f |
|
6165 (F)-7 b(or)29 b(this)h(w)n(e)f(giv)n(e)g(in)h(Fig.)g(3)f(one)g |
|
6166 Fo(simple)38 b Fx(example)523 2259 y(as)25 b(evidence|giving)g(the)h |
|
6167 (whole)g(formalised)f(Ch)n(urc)n(h-Rosser)e(pro)r(of)j(from)f([3,)j(p.) |
|
6168 f(60{62])523 2359 y(w)n(ould)36 b(b)r(e)g(b)r(ey)n(ond)f(the)i(space)e |
|
6169 (constrain)n(ts)f(of)i(this)g(pap)r(er.)g(The)g(complete)g(formalisa-) |
|
6170 523 2458 y(tions)c(of)h(all)f(the)h(results,)g(the)g(Ch)n(urc)n |
|
6171 (h-Rosser)d(and)i(strong)g(normalisation)f(pro)r(of)h(is)g(at)523 |
|
6172 2558 y Fe(http://www.mathe)o(mat)o(ik)o(.un)o(i-)o(mu)o(enc)o(he)o(n.d) |
|
6173 o(e/)-11 b Fl(\030)9 b Fe(urban/nominal/)18 b Fx(.)523 |
|
6174 2816 y Fp(6)112 b(Related)37 b(W)-9 b(ork)523 3008 y |
|
6175 Fx(There)21 b(are)f(man)n(y)g(approac)n(hes)f(to)i(formal)f(treatmen)n |
|
6176 (ts)h(of)g(binders;)g(this)g(section)g(describ)r(es)523 |
|
6177 3107 y(the)28 b(ones)f(from)g(whic)n(h)h(w)n(e)f(ha)n(v)n(e)f(dra)n(wn) |
|
6178 h(inspiration.)648 3207 y(Our)g(w)n(ork)h(uses)g(man)n(y)g(ideas)g |
|
6179 (from)g(the)h(nominal)g(logic)e(w)n(ork)h(b)n(y)g(Pitts)h |
|
6180 Fo(et)h(al)f Fx([16,)13 b(6].)523 3306 y(The)32 b(main)g(di\013erence)g |
|
6181 (is)f(that)i(b)n(y)e(constructing,)g(so)g(to)h(sa)n(y)-7 |
|
6182 b(,)31 b(an)h(explicit)g(mo)r(del)g(of)g(the)523 3406 |
|
6183 y Fm(\013)p Fx(-equated)22 b(lam)n(b)r(da-terms)f(based)g(on)h |
|
6184 (functions,)g(w)n(e)g(ha)n(v)n(e)e(no)i(problem)f(with)i(the)f(axiom-) |
|
6185 523 3506 y(of-c)n(hoice.)29 b(This)i(is)f(imp)r(ortan)n(t.)g(F)-7 |
|
6186 b(or)30 b(consider)g(the)g(alternativ)n(e:)g(if)h(the)g(axiom-of-c)n |
|
6187 (hoice)523 3605 y(causes)e(inconsistencies,)f(then)i(one)f(cannot)g |
|
6188 (build)h(a)f(framew)n(ork)f(for)h(binding)g(on)h(top)f(of)523 |
|
6189 3705 y(Isab)r(elle/HOL)37 b(with)h(its)g(ric)n(h)f(reasoning)f |
|
6190 (infrastructure.)h(One)g(w)n(ould)h(ha)n(v)n(e)e(to)i(in)n(ter-)523 |
|
6191 3805 y(face)31 b(on)h(a)f(lo)n(w)n(er)f(lev)n(el)h(and)g(has)g(to)g |
|
6192 (redo)g(the)h(e\013ort)f(that)h(has)f(b)r(een)h(sp)r(end)f(to)h(dev)n |
|
6193 (elop)523 3904 y(Isab)r(elle/HOL.)27 b(This)g(w)n(as)g(attempted)h(in)g |
|
6194 ([5],)f(but)h(the)g(attempt)h(w)n(as)d(later)h(abandoned.)648 |
|
6195 4004 y(Closely)d(related)g(to)h(our)f(w)n(ork)f(is)i([9])g(b)n(y)f |
|
6196 (Gordon)g(and)h(Melham;)g(it)g(has)g(b)r(een)g(applied)523 |
|
6197 4104 y(and)c(further)h(dev)n(elop)r(ed)f(b)n(y)g(Norrish)g([13].)f |
|
6198 (This)i(w)n(ork)e(states)h(\014v)n(e)g(axioms)g(c)n(haracterising)523 |
|
6199 4203 y Fm(\013)p Fx(-equiv)-5 b(alence)19 b(and)g(then)h(sho)n(ws)e |
|
6200 (that)i(a)f(mo)r(del)g(based)g(on)g(de-Bruijn)g(indices)g(satis\014es)g |
|
6201 (the)523 4303 y(axioms.)28 b(This)g(is)h(somewhat)f(similar)g(to)g(our) |
|
6202 g(approac)n(h)f(where)h(w)n(e)h(construct)f(explicitly)523 |
|
6203 4402 y(the)22 b(set)f Fm(\003)841 4414 y Fj(\013)888 |
|
6204 4402 y Fx(.)g(In)g([9])g(they)g(giv)n(e)f(an)h(induction)h(principle)f |
|
6205 (that)g(requires)f(in)h(the)h(lam)n(b)r(da-case)523 4502 |
|
6206 y(to)28 b(pro)n(v)n(e)d(\(using)j(their)g(notation\))1128 |
|
6207 4675 y Fl(8)14 b Fm(x)g(t:)22 b Fx(\()p Fl(8)14 b Fm(v)s(:)23 |
|
6208 b(P)i Fx(\()p Fm(t)p Fx([)p Fm(x)f Fx(:=)f Fo(V)-8 b(AR)25 |
|
6209 b Fm(v)s Fx(]\)\))f(=)-14 b Fl(\))23 b Fm(P)i Fx(\()p |
|
6210 Fo(LAM)37 b Fm(x)24 b(t)p Fx(\))523 4848 y(That)30 b(means)g(they)g(ha) |
|
6211 n(v)n(e)f(to)i(pro)n(v)n(e)d Fm(P)12 b Fx(\()p Fo(LAM)41 |
|
6212 b Fm(x)28 b(t)p Fx(\))j(for)f(a)f(v)-5 b(ariable)30 b |
|
6213 Fm(x)g Fx(for)g(whic)n(h)g(nothing)523 4948 y(can)40 |
|
6214 b(b)r(e)i(assumed;)e(explicit)h Fm(\013)p Fx(-renamings)e(are)h(then)h |
|
6215 (necessary)e(in)i(order)f(to)g(get)h(the)p Black Black |
|
6216 eop end |
|
6217 %%Page: 14 14 |
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6218 TeXDict begin 14 13 bop Black Black Black 523 789 2915 |
|
6219 4 v 523 3451 4 2662 v 587 884 a Ft(lemma)40 b(substitution)p |
|
6220 1295 884 24 4 v 31 w(lemma:)587 985 y(assumes)h(a1:)f("x)13 |
|
6221 b Fc(6)p Fu(=)f Ft(y")747 1085 y(and)40 b(a2:)g("x)14 |
|
6222 b Fu(#)e Ft(L")587 1186 y(shows)40 b("M[x:=N][y:=L])j(=)c |
|
6223 (M[y:=L][x:=N[y:=L]]")587 1286 y(proof)h(\(nominal)p |
|
6224 1139 1286 V 30 w(induct)h(M\))634 1387 y(case)f(\(1)g(z\))g(\(*)f(case) |
|
6225 h(1:)g(variables)h(*\))634 1487 y(have)f("z)p Fu(=)p |
|
6226 Ft(x)g Fc(_)f Ft(\(z)p Fc(6)p Fu(=)p Ft(x)h Fc(^)f Ft(z)p |
|
6227 Fu(=)p Ft(y\))h Fc(_)f Ft(\(z)p Fc(6)p Fu(=)p Ft(x)h |
|
6228 Fc(^)f Ft(z)p Fc(6)p Fu(=)p Ft(y\)")h(by)g(force)634 |
|
6229 1588 y(thus)g("am)q(\(z\)[x:=N][y:=L])j(=)c(am\(z\)[y:=L][x:=N[y:=L]]") |
|
6230 705 1688 y(using)h(a1)g(a2)g(forget)g(by)g(force)587 |
|
6231 1789 y(next)634 1889 y(case)g(\(2)g(z)f(M)1065 1897 y |
|
6232 Fv(1)1100 1889 y Ft(\))h(\(*)f(case)h(2:)g(lambdas)h(*\))634 |
|
6233 1990 y(assume)g(ih:)f("M)1144 1998 y Fv(1)1179 1990 y |
|
6234 Ft([x:=N][y:=L])i(=)d(M)1806 1998 y Fv(1)1841 1990 y |
|
6235 Ft([y:=L][x:=N[y:=L]]")634 2090 y(assume)i(f1:)f("z)13 |
|
6236 b Fu(#)g Ft(\(L,N,x,y\)")634 2190 y(from)40 b(f1)g(fresh)p |
|
6237 1148 2190 V 29 w(fact1)h(have)f(f2:)g("z)13 b Fu(#)g |
|
6238 Ft(N[y:=L]")41 b(by)e(simp)634 2291 y(show)h("\([z].M)1103 |
|
6239 2299 y Fv(1)1139 2291 y Ft(\)[x:=N][y:=L]=\([z].M)1919 |
|
6240 2299 y Fv(1)1959 2291 y Ft(\)[y:=][x:=N[y:=L]]")j(\(is)d("?LHS=?RHS"\)) |
|
6241 634 2391 y(proof)g(-)705 2492 y(have)g("?LHS)g(=)g([z].\(M)1449 |
|
6242 2500 y Fv(1)1485 2492 y Ft([x:=N][y:=L]\)")i(using)f(f1)e(by)h(simp)705 |
|
6243 2592 y(also)g(have)g("...=)h([z].\(M)1567 2600 y Fv(1)1603 |
|
6244 2592 y Ft([y:=L][x:=N[y:=L]]\)")i(using)e(ih)e(by)h(simp)705 |
|
6245 2693 y(also)g(have)g("...=)h(\([z].\(M)1606 2701 y Fv(1)1642 |
|
6246 2693 y Ft([y:=L]\)\)[x:=N[y:=L]]")j(using)c(f1)g(f2)f(by)h(simp)705 |
|
6247 2793 y(also)g(have)g("...=)h(?RHS")f(using)g(f1)g(by)g(simp)705 |
|
6248 2894 y(finally)h(show)f("?LHS)g(=)g(?RHS")g(by)g(simp)634 |
|
6249 2994 y(qed)587 3095 y(next)634 3195 y(case)g(\(3)g(M)987 |
|
6250 3203 y Fv(1)1061 3195 y Ft(M)1100 3203 y Fv(2)1135 3195 |
|
6251 y Ft(\))f(\(*)h(case)g(3:)g(applications)i(*\))634 3295 |
|
6252 y(thus)e("pr)q(\(M)1026 3303 y Fv(1)1061 3295 y Ft(,M)1139 |
|
6253 3303 y Fv(2)1174 3295 y Ft(\)[x:=N][y:=L]=pr)t(\(M)1880 |
|
6254 3303 y Fv(1)1915 3295 y Ft(,M)1993 3303 y Fv(2)2028 3295 |
|
6255 y Ft(\)[y:=L][x:=N[y:=L]]")j(by)d(simp)587 3396 y(qed)p |
|
6256 3435 3451 4 2662 v 523 3454 2915 4 v 523 3597 a Fs(Fig.)15 |
|
6257 b(3.)27 b Fu(An)g(Isab)r(elle)i(pro)r(of)f(using)g(the)f(Isar)h |
|
6258 (language)h(for)g(the)e(substitution)h(lemma)e(sho)n(wn)i(in)523 |
|
6259 3689 y(Fig.)j(2.)g(It)e(uses)i(the)e(follo)n(wing)k(auxiliary)d |
|
6260 (lemmas:)g Ft(forget)h Fu(whic)n(h)f(states)h(that)f |
|
6261 Ft(x)13 b Fu(#)g Ft(L)30 b Fu(implies)523 3780 y Ft(L[x:=T]=L)p |
|
6262 Fu(,)23 b(needed)d(in)h(the)f(v)l(ariable)i(case.)f(This)h(case)g(pro)r |
|
6263 (ceeds)f(b)n(y)f(stating)h(the)f(three)h(sub)r(cases)523 |
|
6264 3871 y(to)g(b)r(e)g(considered)g(and)f(then)g(pro)n(ving)g(them)g |
|
6265 (automatically)h(using)g(the)f(assumptions)h Ft(a1)g |
|
6266 Fu(and)f Ft(a2)p Fu(.)523 3963 y(The)f(lemma)e Ft(fresh)p |
|
6267 1114 3963 24 4 v 29 w(fact1)j Fu(in)e(the)g(lam)n(b)r(da-case)g(sho)n |
|
6268 (ws)h(from)f Ft(z)13 b Fu(#)g Ft(\(L,N,x,y\))21 b Fu(that)d |
|
6269 Ft(z)13 b Fu(#)f Ft(N[x:=L])523 4054 y Fu(holds.)27 b(This)g(lemma)f |
|
6270 (is)h(not)f(explicitly)h(men)n(tioned)e(in)i(Barendregt's)h(informal)f |
|
6271 (pro)r(of,)h(but)e(it)g(is)523 4145 y(necessary)k(to)g(pull)f(out)g |
|
6272 (the)h(substitution)f(from)g(under)g(the)g(binder)g Ft(z)p |
|
6273 Fu(.)h(This)g(case)g(pro)r(ceeds)g(as)523 4237 y(follo)n(ws:)38 |
|
6274 b(the)c(substitutions)h(on)g(left-hand)g(side)g(of)h(the)e(equation)h |
|
6275 (can)g(b)r(e)g(mo)n(v)n(ed)e(under)h(the)523 4328 y(binder)28 |
|
6276 b Ft(z)p Fu(;)g(then)g(one)g(can)h(apply)f(the)f(induction)h(h)n(yp)r |
|
6277 (othesis;)h(after)g(this)f(one)h(can)f(pull)g(out)g(the)523 |
|
6278 4419 y(second)e(substitution)f(using)h Ft(z)13 b Fu(#)g |
|
6279 Ft(N[y:=L])27 b Fu(and)e(\014nally)h(mo)n(v)n(e)e(out)i(the)f(\014rst)g |
|
6280 (substitution)h(using)523 4511 y Ft(z)13 b Fu(#)g Ft(\(L,N,x,y\))p |
|
6281 Fu(.)28 b(This)e(giv)n(es)g(the)g(righ)n(t-hand)e(side)i(of)h(the)e |
|
6282 (equation.)p Black Black Black eop end |
|
6283 %%Page: 15 15 |
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6284 TeXDict begin 15 14 bop Black Black 523 448 a Fx(pro)r(of)21 |
|
6285 b(through.)h(This)f(incon)n(v)n(enience)g(has)g(b)r(een)i(alleviated)e |
|
6286 (b)n(y)h(the)g(v)n(ersion)e(of)i(structural)523 548 y(induction)28 |
|
6287 b(giv)n(en)f(in)h([8])f(and)g([12],)g(whic)n(h)h(is)f(as)g(follo)n(ws) |
|
6288 984 756 y Fl(9)p Fm(X)r(:)22 b Fe(FINITE)f Fm(X)k Fl(^)19 |
|
6289 b Fx(\()p Fl(8)j Fm(x)i(t:)f(x)g Fl(62)h Fm(X)g Fl(^)19 |
|
6290 b Fm(P)35 b(t)23 b Fx(=)-14 b Fl(\))23 b Fm(P)35 b Fx(\()p |
|
6291 Fm(LAM)c(x)24 b(t)p Fx(\)\))523 915 y(F)-7 b(or)35 b(this)h(principle)g |
|
6292 (one)f(has)g(to)h(pro)n(vide)f(a)g(\014nite)h(set)g Fm(X)42 |
|
6293 b Fx(and)36 b(then)g(has)f(to)h(sho)n(w)f(the)523 1014 |
|
6294 y(lam)n(b)r(da-case)28 b(for)i(all)f(binders)h(not)g(in)g(this)g(set.)g |
|
6295 (This)g(is)g(v)n(ery)f(similar)g(to)h(our)f(induction)523 |
|
6296 1114 y(principle,)34 b(but)g(w)n(e)f(claim)g(that)h(our)f(v)n(ersion)f |
|
6297 (based)h(on)g(freshness)f(\014ts)i(b)r(etter)g(with)g(in-)523 |
|
6298 1214 y(formal)f(practise)h(and)f(can)h(mak)n(e)f(use)h(of)g(the)h |
|
6299 (infrastructure)e(of)h(Isab)r(elle)g(\(namely)g(the)523 |
|
6300 1313 y(axiomatic)27 b(t)n(yp)r(e-classes)f(enforce)h(the)h |
|
6301 (\014nite-supp)r(ort)f(prop)r(ert)n(y\).)648 1415 y(Lik)n(e)22 |
|
6302 b(our)h Fm(\003)1026 1427 y Fj(\013)1073 1415 y Fx(,)g(HO)n(AS)h(uses)f |
|
6303 (functions)g(to)g(enco)r(de)g(lam)n(b)r(da-abstractions;)e(it)j(comes)f |
|
6304 (in)523 1515 y(t)n(w)n(o)18 b(\015a)n(v)n(ours:)f Fo(we)l(ak)28 |
|
6305 b Fx(HO)n(AS)19 b([4])g(and)f Fo(ful)t(l)29 b Fx(HO)n(AS)18 |
|
6306 b([15].)h(The)f(adv)-5 b(an)n(tage)18 b(of)h(full)g(HO)n(AS)g(o)n(v)n |
|
6307 (er)523 1614 y(our)24 b(w)n(ork)g(is)h(that)g(notions)f(suc)n(h)h(as)f |
|
6308 (capture-a)n(v)n(oiding)e(substitution)k(come)e(for)h(free.)g(W)-7 |
|
6309 b(e,)523 1714 y(on)23 b(the)g(other)g(hand,)g(load)f(the)h(w)n(ork)f |
|
6310 (of)h(suc)n(h)f(de\014nitions)i(on)n(to)e(the)h(user.)g(The)g(adv)-5 |
|
6311 b(an)n(tage)523 1814 y(of)36 b(our)g(w)n(ork)f(is)h(that)h(w)n(e)f(ha)n |
|
6312 (v)n(e)f(no)h(di\016culties)h(with)g(notions)f(suc)n(h)g(as)f(sim)n |
|
6313 (ultaneous-)523 1913 y(substitution)29 b(\(a)f(crucial)f(notion)h(in)g |
|
6314 (the)h(usual)f(strong)f(normalisation)f(pro)r(of)6 b(\),)29 |
|
6315 b(whic)n(h)f(in)523 2013 y(full)i(HO)n(AS)g(seem)g(rather)e(di\016cult) |
|
6316 j(to)f(enco)r(de.)f(Another)h(adv)-5 b(an)n(tage)28 b(w)n(e)h(see)h(is) |
|
6317 f(that)h(b)n(y)523 2112 y(inductiv)n(ely)f(de\014ning)g |
|
6318 Fm(\003)1324 2124 y Fj(\013)1400 2112 y Fx(one)f(has)g(induction)i(for) |
|
6319 e(\\free",)g(whereas)f(induction)i(requires)523 2212 |
|
6320 y(considerable)37 b(e\013ort)i(in)g(full)g(HO)n(AS.)g(The)g(main)g |
|
6321 (di\013erence)g(of)f(our)g(w)n(ork)g(with)h(w)n(eak)523 |
|
6322 2312 y(HO)n(AS)32 b(is)g(that)g(w)n(e)g(use)g Fo(some)39 |
|
6323 b Fx(sp)r(eci\014c)32 b(functions)h(to)f(represen)n(t)e(lam)n(b)r |
|
6324 (da-abstractions;)523 2411 y(in)22 b(con)n(trast,)e(w)n(eak)h(HO)n(AS)g |
|
6325 (uses)h(the)g Fo(ful)t(l)31 b Fx(function)22 b(space.)f(This)g(causes)g |
|
6326 (problems)g(kno)n(wn)523 2511 y(b)n(y)27 b(the)h(term)g(\\exotic)f |
|
6327 (terms"|essen)n(tially)e(junk)j(in)g(the)g(mo)r(del.)523 |
|
6328 2788 y Fp(7)112 b(Conclusion)523 2998 y Fx(The)37 b(pap)r(er)f([2],)g |
|
6329 (whic)n(h)h(sets)f(out)h(some)f(c)n(hallenges)f(for)h(automated)g(pro)r |
|
6330 (of)g(assistan)n(ts,)523 3098 y(claims)18 b(that)h(theorem)g(pro)n |
|
6331 (ving)e(tec)n(hnologies)g(ha)n(v)n(e)h(almost)g(reac)n(hed)f(the)i |
|
6332 (threshold)g(where)523 3198 y(they)36 b(can)f(b)r(e)i(used)e |
|
6333 Fo(by)j(the)f(masses)43 b Fx(for)36 b(formal)f(reasoning)e(ab)r(out)j |
|
6334 (programming)e(lan-)523 3297 y(guages.)d(W)-7 b(e)34 |
|
6335 b(hop)r(e)f(to)f(ha)n(v)n(e)g(pushed)h(with)h(this)f(pap)r(er)f(the)i |
|
6336 (b)r(oundary)e(of)h(the)g(state-of-)523 3397 y(the-art)i(in)h(formal)f |
|
6337 (reasoning)f(closer)h(to)h(this)g(threshold.)f(W)-7 b(e)36 |
|
6338 b(sho)n(w)n(ed)f(all)h(our)f(results)523 3497 y(for)e(the)h(lam)n(b)r |
|
6339 (da-calculus.)f(But)h(the)g(lam)n(b)r(da-calculus)e(is)h(only)h |
|
6340 Fo(one)40 b Fx(example.)33 b(W)-7 b(e)34 b(en-)523 3596 |
|
6341 y(visage)e(no)g(problems)g(generalising)f(our)h(results)h(to)g(other)f |
|
6342 (term-calculi.)g(In)h(fact,)g(there)523 3696 y(is)e(already)e(w)n(ork)g |
|
6343 (b)n(y)i(Bengtson)f(adapting)g(our)g(results)g(to)h(the)g |
|
6344 Fm(\031)s Fx(-calculus.)f(W)-7 b(e)32 b(also)d(do)523 |
|
6345 3795 y(not)k(en)n(visage)e(problems)i(with)g(pro)n(viding)f(a)h |
|
6346 (general)e(framew)n(ork)g(for)i(reasoning)e(ab)r(out)523 |
|
6347 3895 y(binders)26 b(based)g(on)h(our)f(results.)g(The)h(real)e |
|
6348 (\(implemen)n(tation\))j(c)n(hallenge)d(is)i(to)f(in)n(tegrate)523 |
|
6349 3995 y(these)f(results)f(in)n(to)h(Isab)r(elle's)g(datat)n(yp)r(e)f |
|
6350 (pac)n(k)-5 b(age)24 b(so)g(that)h(the)g(user)g(do)r(es)f(not)h(see)g |
|
6351 (an)n(y)f(of)523 4094 y(the)j(tedious)e(details)h(through)g(whic)n(h)g |
|
6352 (w)n(e)f(had)h(to)g(go.)g(F)-7 b(or)25 b(example)h(one)f(w)n(ould)h |
|
6353 (lik)n(e)g(that)523 4194 y(the)31 b(subset)f(construction)g(from)g(a)h |
|
6354 (bigger)e(set)h(is)h(done)f(completely)g(b)r(ehind)i(the)f(scenes.)523 |
|
6355 4294 y(Deriving)24 b(an)g(induction)h(principle)f(should)g(also)f(b)r |
|
6356 (e)i(done)f(automatically)-7 b(.)24 b(Ideally)-7 b(,)24 |
|
6357 b(a)g(user)523 4393 y(just)33 b(de\014nes)f(an)g(inductiv)n(e)g(datat)n |
|
6358 (yp)r(e)g(and)g(indicates)g(where)f(binders)h(are|the)f(rest)h(of)523 |
|
6359 4493 y(the)24 b(infrastructure)e(should)i(b)r(e)f(pro)n(vided)g(b)n(y)g |
|
6360 (the)h(theorem)f(pro)n(v)n(er.)e(This)i(is)h(future)f(w)n(ork.)523 |
|
6361 4649 y Ff(Ac)m(kno)m(wledgemen)m(ts:)39 b Fx(The)i(\014rst)g(author)f |
|
6362 (is)h(v)n(ery)f(grateful)g(to)h(Andrew)g(Pitts)g(and)523 |
|
6363 4749 y(Mic)n(hael)25 b(Norrish)g(for)g(the)h(man)n(y)f(discussions)g |
|
6364 (with)h(them)g(on)g(the)g(sub)5 b(ject)26 b(of)f(the)h(pap)r(er.)523 |
|
6365 4848 y(W)-7 b(e)36 b(thank)g(James)f(Cheney)-7 b(,)36 |
|
6366 b(Aaron)f(Bohannon,)g(Daniel)h(W)-7 b(ang)35 b(and)h(one)g(anon)n |
|
6367 (ymous)523 4948 y(referee)27 b(for)g(their)h(suggestions.)e(The)i |
|
6368 (\014rst)g(author's)f(in)n(terest)g(in)h(this)g(w)n(ork)f(w)n(as)g |
|
6369 (spark)n(ed)p Black Black eop end |
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6370 %%Page: 16 16 |
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6371 TeXDict begin 16 15 bop Black Black 523 448 a Fx(b)n(y)30 |
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6372 b(an)g(email-discussion)e(with)j(F)-7 b(rank)29 b(Pfenning)h(and)g(b)n |
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6373 (y)g(a)f(question)h(from)g(Neil)g(Ghani)523 548 y(at)e(the)g(spring)e |
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6374 (sc)n(ho)r(ol)h(of)g(Midland)h(Graduate)f(Sc)n(ho)r(ol.)g(The)h |
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6375 (Alexander-v)n(on-Hum)n(b)r(oldt)523 648 y(F)-7 b(oundation)27 |
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6376 b(funded)i(the)f(\014rst)f(author.)523 913 y Fp(References)p |
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6377 Black 561 1104 a Fu(1.)p Black 43 w(T.)e(Altenkirc)n(h.)32 |
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6378 b(A)25 b(Formalization)h(of)f(the)g(Strong)f(Normalisation)i(Pro)r(of)g |
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6379 (for)g(System)d(F)i(in)663 1196 y(LEGO.)34 b(In)25 b |
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6380 Fq(Pr)l(o)l(c.)j(of)f(TLCA)p Fu(,)f(v)n(olume)f(664)i(of)f |
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6381 Fq(LNCS)p Fu(,)g(pages)h(13{28,)h(1993.)p Black 561 1287 |
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6382 a(2.)p Black 43 w(B.)i(E.)f(Aydemir,)f(A.)h(Bohannon,)h(M.)g(F)-6 |
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6383 b(airbairn,)30 b(J.)g(N.)g(F)-6 b(oster,)30 b(B.)g(C.)g(Pierce,)g(P)-6 |
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6384 b(.)30 b(Sew)n(ell,)663 1378 y(D.)21 b(Vytiniotis,)h(G.)g(W)-6 |
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6385 b(ash)n(burn,)20 b(S.)h(W)-6 b(eiric)n(h,)22 b(and)f(S.)g(Zdancewic.)29 |
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6386 b(Mec)n(hanized)22 b(Metatheory)663 1469 y(for)k(the)f(Masses:)j(The)e |
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6387 (PoplMark)h(Challenge.)g(accepted)f(at)g(tphol)f(05.)p |
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6388 Black 561 1561 a(3.)p Black 43 w(H.)33 b(Barendregt.)58 |
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6389 b Fq(The)35 b(Lamb)l(da)g(Calculus:)g(Its)f(Syntax)j(and)e(Semantics)p |
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6390 Fu(,)f(v)n(olume)f(103)h(of)663 1652 y Fq(Studies)28 |
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6391 b(in)f(L)l(o)l(gic)h(and)g(the)h(F)-6 b(oundations)29 |
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6392 b(of)e(Mathematics)p Fu(.)36 b(North-Holland,)26 b(1981.)p |
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6393 Black 561 1743 a(4.)p Black 43 w(J.)21 b(Desp)r(eyroux,)f(A.)g(F)-6 |
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6394 b(elt)n(y)g(,)20 b(and)g(A.)g(Hirsc)n(ho)n(witz.)27 b(Higher-Order)20 |
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6395 b(Abstract)g(Syn)n(tax)f(in)h(Co)r(q.)663 1835 y(In)25 |
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6396 b Fq(Pr)l(o)l(c.)j(of)f(TLCA)p Fu(,)f(v)n(olume)e(902)j(of)f |
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6397 Fq(LNCS)p Fu(,)g(pages)h(124{138,)i(1995.)p Black 561 |
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6398 1926 a(5.)p Black 43 w(M.)e(J.)g(Gabba)n(y)-6 b(.)37 |
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6399 b Fq(A)28 b(The)l(ory)h(of)f(Inductive)i(De\014nitions)f(With)f |
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6400 Fr(\013)p Fq(-e)l(quivalenc)l(e)p Fu(.)39 b(PhD)26 b(thesis,)663 |
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6401 2017 y(Univ)n(ersit)n(y)e(of)j(Cam)n(bridge,)f(2000.)p |
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6402 Black 561 2109 a(6.)p Black 43 w(M.)d(J.)h(Gabba)n(y)f(and)f(A.)h(M.)g |
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6403 (Pitts.)31 b(A)22 b(New)h(Approac)n(h)g(to)g(Abstract)g(Syn)n(tax)e |
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6404 (with)i(Variable)663 2200 y(Binding.)34 b Fq(F)-6 b(ormal)28 |
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6405 b(Asp)l(e)l(cts)i(of)d(Computing)p Fu(,)g(13:341{363,)i(2001.)p |
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6406 Black 561 2291 a(7.)p Black 43 w(J.-Y.)f(Girard,)g(Y.)g(Lafon)n(t,)h |
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6407 (and)e(P)-6 b(.)28 b(T)-6 b(a)n(ylor.)41 b Fq(Pr)l(o)l(ofs)30 |
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6408 b(and)g(Typ)l(es)p Fu(,)f(v)n(olume)e(7)h(of)g Fq(Cambridge)663 |
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6409 2383 y(T)-6 b(r)l(acts)29 b(in)e(The)l(or)l(etic)l(al)i(Computer)g |
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6410 (Scienc)l(e)p Fu(.)35 b(Cam)n(bridge)26 b(Univ)n(ersit)n(y)e(Press,)j |
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6411 (1989.)p Black 561 2474 a(8.)p Black 43 w(A.)21 b(D.)g(Gordon.)28 |
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6412 b(A)21 b(Mec)n(hanisation)i(of)f(Name-Carrying)f(Syn)n(tax)f(up)g(to)i |
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6413 (Alpha-Con)n(v)n(ersion.)663 2565 y(In)28 b Fq(Pr)l(o)l(c.)i(of)g |
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6414 (Higher-or)l(der)j(lo)l(gic)d(the)l(or)l(em)i(pr)l(oving)f(and)g(its)f |
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6415 (applic)l(ations)p Fu(,)g(v)n(olume)d(780)j(of)663 2657 |
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6416 y Fq(LNCS)p Fu(,)c(pages)g(414{426,)j(1993.)p Black 561 |
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6417 2748 a(9.)p Black 43 w(A.)35 b(D.)h(Gordon)g(and)g(T.)g(Melham.)64 |
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6418 b(Fiv)n(e)36 b(Axioms)f(of)i(Alpha-Con)n(v)n(ersion.)64 |
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6419 b(In)35 b Fq(Pr)l(o)l(c.)i(of)663 2839 y(TPHOL)p Fu(,)25 |
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6420 b(v)n(olume)g(1125)i(of)g Fq(LNCS)p Fu(,)f(pages)g(173{190,)j(1996.)p |
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6421 Black 523 2931 a(10.)p Black 43 w(D.)e(Hirsc)n(hk)n(o\013.)40 |
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6422 b(A)28 b(Full)g(F)-6 b(ormalisation)29 b(of)f Fr(\031)s |
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6423 Fu(-Calculus)g(Theory)g(in)g(the)f(Calculus)i(of)g(Con-)663 |
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6424 3022 y(structions.)35 b(In)25 b Fq(Pr)l(o)l(c.)j(of)f(TPHOL)p |
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6425 Fu(,)f(v)n(olume)e(1275)j(of)g Fq(LNCS)p Fu(,)f(pages)g(153{169,)j |
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6426 (1997.)p Black 523 3113 a(11.)p Black 43 w(T.)k(Nipk)n(o)n(w,)f(L.)h |
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6427 (C.)g(P)n(aulson,)g(and)f(M.)h(W)-6 b(enzel.)54 b Fq(Isab)l(el)t(le)34 |
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6428 b(HOL:)f(A)g(Pr)l(o)l(of)h(Assistant)i(for)663 3205 y(Higher-Or)l(der) |
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6429 29 b(Lo)l(gic)p Fu(,)e(v)n(olume)d(2283)j(of)g Fq(LNCS)p |
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6430 Fu(.)34 b(Springer-V)-6 b(erlag,)26 b(2002.)p Black 523 |
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6431 3296 a(12.)p Black 43 w(M.)f(Norrish.)34 b(Mec)n(hanising)26 |
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6432 b Fr(\025)p Fu(-calculus)f(using)h(a)f(Classical)j(First)e(Order)e |
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6433 (Theory)h(of)h(Terms)663 3387 y(with)g(Perm)n(utations,)f(forthcoming.) |
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6434 p Black 523 3479 a(13.)p Black 43 w(M.)34 b(Norrish.)60 |
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6435 b(Recursiv)n(e)34 b(Function)g(De\014nition)g(for)h(Typ)r(es)f(with)g |
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6436 (Binders.)60 b(In)34 b Fq(Pr)l(o)l(c.)h(of)663 3570 y(TPHOL)p |
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6437 Fu(,)25 b(v)n(olume)g(3223)i(of)g Fq(LNCS)p Fu(,)f(pages)g(241{256,)j |
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6438 (2004.)p Black 523 3661 a(14.)p Black 43 w(L.)c(P)n(aulson.)35 |
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6439 b(De\014ning)25 b(F)-6 b(unctions)26 b(on)f(Equiv)l(alence)h(Classes.) |
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6440 36 b(T)-6 b(o)26 b(app)r(ear)g(in)f(A)n(CM)h(T)-6 b(rans-)663 |
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6441 3753 y(actions)26 b(on)g(Computational)g(Logic.)p Black |
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6442 523 3844 a(15.)p Black 43 w(F.)k(Pfenning)g(and)f(C.)h(Elliott.)47 |
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6443 b(Higher-Order)29 b(Abstract)h(Syn)n(tax.)44 b(In)29 |
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6444 b Fq(Pr)l(o)l(c.)j(of)e(the)i(A)n(CM)663 3935 y(SIGPLAN)27 |
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6445 b(Confer)l(enc)l(e)i(PLDI)p Fu(,)d(pages)g(199{208.)j(A)n(CM)d(Press,)h |
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6446 (1989.)p Black 523 4027 a(16.)p Black 43 w(A.)g(M.)i(Pitts.)42 |
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6447 b(Nominal)27 b(Logic,)i(A)f(First)g(Order)g(Theory)g(of)h(Names)e(and)g |
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6448 (Binding.)42 b Fq(Infor-)663 4118 y(mation)27 b(and)h(Computation)p |
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6449 Fu(,)g(186:165{193,)i(2003.)p Black 523 4209 a(17.)p |
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6450 Black 43 w(A.)18 b(S.)h(T)-6 b(ro)r(elstra)21 b(and)e(H.)f(Sc)n(h)n |
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6451 (wic)n(h)n(ten)n(b)r(erg.)23 b Fq(Basic)f(Pr)l(o)l(of)g(The)l(ory)p |
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6452 Fu(,)e(v)n(olume)e(43)i(of)f Fq(Cambridge)663 4301 y(T)-6 |
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6453 b(r)l(acts)29 b(in)e(The)l(or)l(etic)l(al)i(Computer)g(Scienc)l(e)p |
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6454 Fu(.)35 b(Cam)n(bridge)26 b(Univ)n(ersit)n(y)e(Press,)j(2000.)p |
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6455 Black 523 4392 a(18.)p Black 43 w(C.)f(Urban,)g(A.)g(M.)h(Pitts,)g(and) |
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6456 f(M.)g(J.)h(Gabba)n(y)-6 b(.)35 b(Nominal)26 b(Uni\014cation.)35 |
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6457 b Fq(The)l(or)l(etic)l(al)30 b(Com-)663 4483 y(puter)f(Scienc)l(e)p |
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6458 Fu(,)e(323\(1-2\):473{497,)j(2004.)p Black 523 4575 a(19.)p |
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6459 Black 43 w(M.)21 b(V)-6 b(anIn)n(w)n(egen.)25 b Fq(The)f |
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6460 (Machine-Assiste)l(d)h(Pr)l(o)l(of)e(of)g(Pr)l(o)l(gr)l(amming)h |
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6461 (Language)g(Pr)l(op)l(erties)p Fu(.)663 4666 y(PhD)h(thesis,)h(Univ)n |
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6462 (ersit)n(y)f(of)i(P)n(ennsylv)l(ania,)e(1996.)36 b(Av)l(ailable)26 |
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6463 b(as)h(MS-CIS-96-31.)p Black 523 4757 a(20.)p Black 43 |
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6464 w(M.)38 b(W)-6 b(enzel.)69 b Fq(Using)38 b(Axiomatic)h(Typ)l(e)g |
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6465 (Classes)g(in)f(Isab)l(el)t(le)p Fu(.)70 b(Man)n(ual)38 |
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6466 b(in)f(the)g(Isab)r(elle)663 4848 y(distribution.)p Black |
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6467 Black eop end |
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6468 %%Trailer |
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6469 |
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6470 userdict /end-hook known{end-hook}if |
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6471 %%EOF |