Publications/lfmtp-06.ps
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     2 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software
       
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    62 %
       
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    64 %   author    = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
       
    65 %                W. Schmidt, P. Lehman",
       
    66 %   version   = "2.0",
       
    67 %   date      = "30 October 2002",
       
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    75 % The idea is to have all the characters normally included in Type 1 fonts
       
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    79 % 
       
    80 % Character code assignments were made as follows:
       
    81 % 
       
    82 % (1) the Windows ANSI characters are almost all in their Windows ANSI
       
    83 % positions, because some Windows users cannot easily reencode the
       
    84 % fonts, and it makes no difference on other systems. The only Windows
       
    85 % ANSI characters not available are those that make no sense for
       
    86 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
       
    87 % (173). quotesingle and grave are moved just because it's such an
       
    88 % irritation not having them in TeX positions.
       
    89 % 
       
    90 % (2) Remaining characters are assigned arbitrarily to the lower part
       
    91 % of the range, avoiding 0, 10 and 13 in case we meet dumb software.
       
    92 % 
       
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    94 % hopes that other PostScript fonts, perhaps created for public
       
    95 % consumption, will include them, they are included starting at 0x12.
       
    96 % These are /dotlessj /ff /ffi /ffl.
       
    97 % 
       
    98 % (4) hyphen appears twice for compatibility with both ASCII and Windows.
       
    99 %
       
   100 % (5) /Euro was assigned to 128, as in Windows ANSI
       
   101 %
       
   102 % (6) Missing characters from MacRoman encoding incorporated as follows:
       
   103 %
       
   104 %     PostScript      MacRoman        TeXBase1
       
   105 %     --------------  --------------  --------------
       
   106 %     /notequal       173             0x16
       
   107 %     /infinity       176             0x17
       
   108 %     /lessequal      178             0x18
       
   109 %     /greaterequal   179             0x19
       
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   114 %     /integral       186             0x81
       
   115 %     /Omega          189             0x8D
       
   116 %     /radical        195             0x8E
       
   117 %     /approxequal    197             0x8F
       
   118 %     /Delta          198             0x9D
       
   119 %     /lozenge        215             0x9E
       
   120 %
       
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   123  /.notdef /dotaccent /fi /fl
       
   124  /fraction /hungarumlaut /Lslash /lslash
       
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   127 % 0x10
       
   128  /caron /dotlessi /dotlessj /ff
       
   129  /ffi /ffl /notequal /infinity
       
   130  /lessequal /greaterequal /partialdiff /summation
       
   131  /product /pi /grave /quotesingle
       
   132 % 0x20
       
   133  /space /exclam /quotedbl /numbersign
       
   134  /dollar /percent /ampersand /quoteright
       
   135  /parenleft /parenright /asterisk /plus
       
   136  /comma /hyphen /period /slash
       
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   138  /zero /one /two /three
       
   139  /four /five /six /seven
       
   140  /eight /nine /colon /semicolon
       
   141  /less /equal /greater /question
       
   142 % 0x40
       
   143  /at /A /B /C
       
   144  /D /E /F /G
       
   145  /H /I /J /K
       
   146  /L /M /N /O
       
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   148  /P /Q /R /S
       
   149  /T /U /V /W
       
   150  /X /Y /Z /bracketleft
       
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   153  /quoteleft /a /b /c
       
   154  /d /e /f /g
       
   155  /h /i /j /k
       
   156  /l /m /n /o
       
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   158  /p /q /r /s
       
   159  /t /u /v /w
       
   160  /x /y /z /braceleft
       
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   164  /quotedblbase /ellipsis /dagger /daggerdbl
       
   165  /circumflex /perthousand /Scaron /guilsinglleft
       
   166  /OE /Omega /radical /approxequal
       
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   170  /tilde /trademark /scaron /guilsinglright
       
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   174  /currency /yen /brokenbar /section
       
   175  /dieresis /copyright /ordfeminine /guillemotleft
       
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   178  /degree /plusminus /twosuperior /threesuperior
       
   179  /acute /mu /paragraph /periodcentered
       
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   181  /onequarter /onehalf /threequarters /questiondown
       
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   184  /Adieresis /Aring /AE /Ccedilla
       
   185  /Egrave /Eacute /Ecircumflex /Edieresis
       
   186  /Igrave /Iacute /Icircumflex /Idieresis
       
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   188  /Eth /Ntilde /Ograve /Oacute
       
   189  /Ocircumflex /Otilde /Odieresis /multiply
       
   190  /Oslash /Ugrave /Uacute /Ucircumflex
       
   191  /Udieresis /Yacute /Thorn /germandbls
       
   192 % 0xE0
       
   193  /agrave /aacute /acircumflex /atilde
       
   194  /adieresis /aring /ae /ccedilla
       
   195  /egrave /eacute /ecircumflex /edieresis
       
   196  /igrave /iacute /icircumflex /idieresis
       
   197 % 0xF0
       
   198  /eth /ntilde /ograve /oacute
       
   199  /ocircumflex /otilde /odieresis /divide
       
   200  /oslash /ugrave /uacute /ucircumflex
       
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  2718 (bene\002t)h(of)f(casual)g(users)g(of)h(theorem)291 203
       
  2719 y(pro)o(v)o(ers,)41 b(we)e(w)o(ant)g(to)f(study)g(in)h(minutiae)e
       
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  2721 324 y(lemma)24 b(using)f(this)h(formalisation)g(technique.)437
       
  2722 452 y(W)-8 b(e)30 b(assume)g(the)g(reader)h(is)e(f)o(amiliar)h(with)f
       
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  2724 573 y(using)23 b(for)i(e)o(xample)f(the)h(datatype:)p
       
  2725 Black Black 933 737 a Fn(datatype)56 b Fh(dB)65 b Fk(=)56
       
  2726 b Fh(V)-7 b(ar)38 b Fm(nat)56 b Fl(j)f Fh(App)34 b(dB)j(dB)65
       
  2727 b Fl(j)55 b Fh(L)-5 b(am)35 b(dB)291 910 y Fu(One)f(central)h(notion)e
       
  2728 (when)h(w)o(orking)f(with)h(de)h(Bruijn)f(indices)f(is)h(the)g
       
  2729 Fc(lifting)f Fu(operation,)291 1031 y(written)g Fl(")657
       
  2730 994 y Fg(n)657 1057 y(k)739 1031 y Fu(where)i Fm(n)f
       
  2731 Fu(is)g(an)h(of)n(fset)f(by)g(which)g(the)g(indices)g(greater)h(or)g
       
  2732 (equal)f(than)g Fm(k)k Fu(are)291 1151 y(incremented;)e
       
  2733 Fm(k)h Fu(is)c(the)g(upper)g(bound)g(of)g(indices)g(that)f(are)i(re)o
       
  2734 (garded)f(as)h Fc(locally)e(bound)p Fu(.)291 1271 y(This)24
       
  2735 b(operation)g(can)h(be)g(de\002ned)g(as:)p Black Black
       
  2736 1057 1550 a Fl(")1107 1514 y Fg(n)1107 1576 y(k)1181
       
  2737 1550 y Fk(\()p Fh(V)-7 b(ar)36 b Fm(i)p Fk(\))1791 1494
       
  2738 y Ff(def)1796 1550 y Fk(=)1909 1345 y Fe(8)1909 1435
       
  2739 y(<)1909 1614 y(:)2015 1482 y Fh(V)-7 b(ar)35 b Fm(i)354
       
  2740 b Fu(if)25 b Fm(i)j(<)g(k)2015 1639 y Fh(V)-7 b(ar)35
       
  2741 b Fk(\()p Fm(i)22 b Fk(+)g Fm(n)p Fk(\))100 b Fu(otherwise)1057
       
  2742 1830 y Fl(")1107 1794 y Fg(n)1107 1856 y(k)1181 1830
       
  2743 y Fk(\()p Fh(App)34 b Fm(M)1519 1845 y Fj(1)1586 1830
       
  2744 y Fm(M)1680 1845 y Fj(2)1720 1830 y Fk(\))1791 1775 y
       
  2745 Ff(def)1796 1830 y Fk(=)j Fh(App)d Fk(\()p Fl(")2203
       
  2746 1794 y Fg(n)2203 1856 y(k)2278 1830 y Fm(M)2372 1845
       
  2747 y Fj(1)2411 1830 y Fk(\))28 b(\()p Fl(")2565 1794 y Fg(n)2565
       
  2748 1856 y(k)2639 1830 y Fm(M)2733 1845 y Fj(2)2773 1830
       
  2749 y Fk(\))1057 2011 y Fl(")1107 1975 y Fg(n)1107 2037 y(k)1181
       
  2750 2011 y Fk(\()p Fh(L)-5 b(am)35 b Fm(M)1534 2026 y Fj(1)1574
       
  2751 2011 y Fk(\))1791 1955 y Ff(def)1796 2011 y Fk(=)i Fh(L)-5
       
  2752 b(am)35 b Fk(\()p Fl(")2218 1975 y Fg(n)2218 2037 y(k)r
       
  2753 Fj(+1)2378 2011 y Fm(M)2472 2026 y Fj(1)2512 2011 y Fk(\))291
       
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  2757 b Fk(])p Fu(,)29 b(can)291 2321 y(then)24 b(be)h(de\002ned)g(as)g
       
  2758 (follo)n(ws:)p Black Black 1085 2647 a Fk(\()p Fh(V)-7
       
  2759 b(ar)36 b Fm(i)p Fk(\)[)p Fm(k)31 b Fk(:=)c Fm(N)10 b
       
  2760 Fk(])1774 2591 y Ff(def)1779 2647 y Fk(=)1892 2383 y
       
  2761 Fe(8)1892 2473 y(>)1892 2503 y(>)1892 2533 y(<)1892 2712
       
  2762 y(>)1892 2742 y(>)1892 2772 y(:)1997 2501 y Fh(V)-7 b(ar)36
       
  2763 b Fm(i)346 b Fu(if)25 b Fm(i)j(<)f(k)1997 2658 y Fl(")2047
       
  2764 2622 y Fg(k)2047 2683 y Fj(0)2117 2658 y Fm(N)373 b Fu(if)25
       
  2765 b Fm(i)j Fk(=)f Fm(k)1997 2815 y Fh(V)-7 b(ar)36 b Fk(\()p
       
  2766 Fm(i)22 b Fl(\000)h Fk(1\))99 b Fu(if)25 b Fm(i)j(>)f(k)809
       
  2767 2996 y Fk(\()p Fh(App)34 b Fm(M)1147 3011 y Fj(1)1214
       
  2768 2996 y Fm(M)1308 3011 y Fj(2)1348 2996 y Fk(\)[)p Fm(k)d
       
  2769 Fk(:=)c Fm(N)10 b Fk(])1774 2940 y Ff(def)1779 2996 y
       
  2770 Fk(=)37 b Fh(App)d Fk(\()p Fm(M)2230 3011 y Fj(1)2270
       
  2771 2996 y Fk([)p Fm(k)c Fk(:=)e Fm(N)10 b Fk(]\))28 b(\()p
       
  2772 Fm(M)2822 3011 y Fj(2)2862 2996 y Fk([)p Fm(k)j Fk(:=)d
       
  2773 Fm(N)10 b Fk(]\))984 3176 y(\()p Fh(L)-5 b(am)35 b Fm(M)10
       
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  2775 y Ff(def)1779 3176 y Fk(=)37 b Fh(L)-5 b(am)35 b Fk(\()p
       
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  2781 h(then)i(tak)o(es)f(the)g(follo)n(wing)291 3601 y(form:)p
       
  2782 Black Black 541 3780 a Fn(Substitution)35 b(Lemma)f(with)g(de)h(Bruijn)
       
  2783 g(Indices:)79 b Fu(F)o(or)33 b(all)h(indices)f Fm(i)p
       
  2784 Fu(,)j Fm(j)6 b Fu(,)541 3901 y(with)24 b Fm(i)k Fl(\024)g
       
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  2786 b Fk([)p Fm(i)29 b Fk(:=)e Fm(N)10 b Fk(][)p Fm(j)35
       
  2787 b Fk(:=)27 b Fm(L)p Fk(])i(=)e Fm(M)10 b Fk([)p Fm(j)29
       
  2788 b Fk(+)22 b(1)28 b(:=)f Fm(L)p Fk(][)p Fm(i)i Fk(:=)e
       
  2789 Fm(N)10 b Fk([)p Fm(j)29 b Fl(\000)23 b Fm(i)28 b Fk(:=)f
       
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  2798 (concrete)291 4736 y(names.)50 b(This)30 b(f)o(act)i(alone)f(already)h
       
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  2800 (this)291 4857 y(lemma.)48 b(Notice)30 b(also)g(that)h(the)f(condition)
       
  2801 g Fm(i)39 b Fl(\024)g Fm(j)e Fu(is)30 b(necessary)-6
       
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  2806 Fm(M)10 b Fu(.)71 b(Unlik)o(e)38 b(the)g(informal)f(proof,)k(ho)n(we)n
       
  2807 (v)o(er)l(,)f(the)e(induc-)291 5347 y(tion)27 b(hypothesis)e(needs)j
       
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  3497 end
       
  3498  990 444
       
  3499 a 0 0 1 TeXcolorrgb 990 444 a
       
  3500 SDict begin H.S end
       
  3501  990 444 a 1006 408 a Fj(4)1062
       
  3502 444 y
       
  3503 SDict begin 14.5 H.L end
       
  3504  1062 444 a 1062 444 a
       
  3505 SDict begin [ /Subtype /Link /Dest (Hfootnote.2) cvn /H /I /Border
       
  3506 [0 0 0] /Color [1 0 0] H.B /ANN pdfmark end
       
  3507  1062 444 a Black 42 w Fu(When)28
       
  3508 b(it)g(then)g(comes)g(to)g(establishing)f(the)h Fh(L)-5
       
  3509 b(am)7 b Fu(-case,)30 b(we)f(can)g(as-)291 565 y(sume)d(that)h(the)g
       
  3510 (binder)g Fm(z)32 b Fu(is)27 b(fresh)h(for)f Fk(\()p
       
  3511 Fm(x;)17 b(y)t(;)g(N)5 b(;)17 b(L)p Fk(\))p Fu(,)27 b(that)g(means)g
       
  3512 (is)f(not)h(equal)g(to)g Fm(x)h Fu(and)f Fm(y)t Fu(,)291
       
  3513 685 y(and)33 b(is)g(not)f(free)j(in)d Fm(N)44 b Fu(and)34
       
  3514 b Fm(L)p Fu(.)56 b(As)33 b(a)h(result,)h(the)e(induction)f(in)h(the)g
       
  3515 (substitution)d(lemma)291 805 y(will)j(go)i(through)f(smoothly)-6
       
  3516 b(,)34 b(just)g(lik)o(e)g(in)h(Barendre)o(gt')-5 b(s)34
       
  3517 b(informal)g(proof.)60 b(If)36 b(the)e(nomi-)291 926
       
  3518 y(nal)26 b(datatype)f(package)i(had)f Fc(not)i Fu(pro)o(vided)d(such)g
       
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  3520 y(w)o(ould)e(be)i(quite)f(incon)l(v)o(enient:)31 b(one)25
       
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  3525 (proof)f(idea,)291 1416 y(for)k(the)f(formalisation)f(of)i(the)g
       
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  3527 (that)291 1536 y(are)k(left)f(out)f(in)h(his)g(notes.)34
       
  3528 b(F)o(or)26 b(e)o(xample)f(in)h(Case)h(1.2)f(the)g(details)f(are)i
       
  3529 (left)f(out)g(for)g(ho)n(w)g(to)291 1656 y(pro)o(v)o(e)d(the)i
       
  3530 (property)f(of)291 1678 y
       
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  3532  291 1678 a 291 1678 a
       
  3533 SDict begin 14.5 H.A end
       
  3534  291 1678
       
  3535 a 291 1678 a
       
  3536 SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark
       
  3537 end
       
  3538  291 1678 a Black Black 1312 1815 a Fm(x)k
       
  3539 Fk(#)g Fm(L)e Fu(implies)d(that)h Fm(L)p Fk([)p Fm(x)29
       
  3540 b Fk(:=)f Fm(P)14 b Fk(])27 b(=)g Fm(L)h(:)p Black -2460
       
  3541 w Fu(\(8\))p Black 291 1982 a(where)19 b Fm(x)28 b Fk(#)g
       
  3542 Fm(L)20 b Fu(stands)e(for)i Fm(x)28 b Fl(62)g Fm(F)14
       
  3543 b(V)21 b Fk(\()p Fm(L)p Fk(\))p Fu(.)29 b(This)19 b(f)o(act)g(can)h(be)
       
  3544 f(pro)o(v)o(ed)f(by)h(an)g(induction)f(o)o(v)o(er)g Fm(L)291
       
  3545 2102 y Fu(using)k(the)i(strong)f(induction)f(principle.)29
       
  3546 b(F)o(or)24 b(this)f(we)g(mak)o(e)h(the)f(follo)n(wing)f
       
  3547 (instantiations:)p Black Black 1116 2297 a Fm(P)46 b
       
  3548 Fl(7!)33 b Fm(\025L:\025)p Fk(\()p Fm(x;)17 b(P)d Fk(\))p
       
  3549 Fm(:)55 b(x)29 b Fk(#)e Fm(L)56 b Fl(\))f Fm(L)p Fk([)p
       
  3550 Fm(x)29 b Fk(:=)e Fm(P)14 b Fk(])28 b(=)f Fm(L)1087 2454
       
  3551 y(M)44 b Fl(7!)33 b Fm(L)1150 2611 y(c)g Fl(7!)g Fk(\()p
       
  3552 Fm(x;)17 b(P)d Fk(\))291 2788 y Fu(As)21 b(a)g(result,)g(the)g(v)n
       
  3553 (ariable)g(and)g(application)f(case)i(are)g(completely)e(routine.)29
       
  3554 b(In)21 b(the)g(lambda-)291 2908 y(case)39 b(we)g(ha)n(v)o(e)g(to)f
       
  3555 (sho)n(w)g(that)g Fm(x)55 b Fk(#)f(\()p Fh(L)-5 b(am)46
       
  3556 b Fm(z)d(L)2100 2923 y Fj(1)2140 2908 y Fk(\))c Fu(implies)f
       
  3557 Fk(\()p Fh(L)-5 b(am)46 b Fm(z)d(L)2974 2923 y Fj(1)3014
       
  3558 2908 y Fk(\)[)p Fm(x)54 b Fk(:=)g Fm(P)14 b Fk(])53 b(=)291
       
  3559 3028 y(\()p Fh(L)-5 b(am)32 b Fm(z)d(L)687 3043 y Fj(1)727
       
  3560 3028 y Fk(\))c Fu(with)f(the)h(assumption)d(that)j Fm(z)32
       
  3561 b Fk(#)c(\()p Fm(x;)17 b(P)d Fk(\))24 b Fu(and)h(the)g(induction)e
       
  3562 (hypothesis)p Black Black 1343 3187 a Fl(8)p Fm(x;)17
       
  3563 b(L:)29 b(x)f Fk(#)g Fm(L)53 b Fl(\))27 b Fm(L)p Fk([)p
       
  3564 Fm(x)i Fk(:=)f Fm(P)14 b Fk(])27 b(=)g Fm(L)h(:)291 3354
       
  3565 y Fu(From)g(the)h(assumption)e(that)i Fm(z)k Fu(is)c(not)f(equal)h(to)g
       
  3566 Fm(x)g Fu(and)g(not)f(free)i(in)f Fm(L)p Fu(,)h(we)f(can)g(infer)h
       
  3567 (from)291 3474 y Fm(x)k Fk(#)g(\()p Fh(L)-5 b(am)35 b
       
  3568 Fm(z)e(L)898 3489 y Fj(1)938 3474 y Fk(\))28 b Fu(that)g
       
  3569 Fm(x)34 b Fk(#)h Fm(L)1453 3489 y Fj(1)1521 3474 y Fu(holds)27
       
  3570 b(and)h(by)g(applying)f(the)h(de\002nition)f(of)h(substitution)291
       
  3571 3595 y(that)f Fk(\()p Fh(L)-5 b(am)35 b Fm(z)e(L)871
       
  3572 3610 y Fj(1)911 3595 y Fk(\)[)p Fm(x)h Fk(:=)f Fm(P)14
       
  3573 b Fk(])33 b(=)h Fh(L)-5 b(am)35 b Fm(z)e Fk(\()p Fm(L)1851
       
  3574 3610 y Fj(1)1891 3595 y Fk([)p Fm(x)h Fk(:=)f Fm(P)14
       
  3575 b Fk(]\))p Fu(.)40 b(No)n(w)27 b(we)h(just)f(need)h(to)g(apply)g(the)
       
  3576 291 3715 y(induction)23 b(hypothesis)f(and)j(are)h(done.)437
       
  3577 3844 y(Although)k(not)h(ob)o(vious)f(from)i(\002rst)f(glance,)j(also)d
       
  3578 (in)h(Case)g(2,)h(in)f(the)f(last)g(step)h(of)g(the)291
       
  3579 3964 y(calculation)f(where)i(the)f(substitution)d(is)j(pulled)f(back)h
       
  3580 (from)g(under)h(the)f(binder)f Fm(\025z)t Fu(,)k(there)291
       
  3581 4085 y(are)e(some)f(details)g(missing)f(from)h(Barendre)o(gt')-5
       
  3582 b(s)32 b(informal)g(proof.)54 b(In)33 b(order)g(to)f(get)h(from)291
       
  3583 4205 y Fh(L)-5 b(am)35 b Fm(z)e Fk(\()p Fm(M)722 4220
       
  3584 y Fj(1)762 4205 y Fk([)p Fm(y)k Fk(:=)d Fm(L)p Fk(][)p
       
  3585 Fm(x)h Fk(:=)f Fm(N)10 b Fk([)p Fm(y)38 b Fk(:=)c Fm(L)p
       
  3586 Fk(]]\))29 b Fu(to)f Fk(\()p Fh(L)-5 b(am)35 b Fm(z)f(M)2421
       
  3587 4220 y Fj(1)2460 4205 y Fk(\)[)p Fm(y)k Fk(:=)c Fm(L)p
       
  3588 Fk(][)p Fm(x)h Fk(:=)f Fm(N)10 b Fk([)p Fm(y)38 b Fk(:=)c
       
  3589 Fm(L)p Fk(]])p Fu(,)291 4325 y(we)25 b(need)g(the)f(property)h(that:)
       
  3590 1362 4325 y
       
  3591 SDict begin H.S end
       
  3592  1362 4325 a 1362 4325 a
       
  3593 SDict begin 14.5 H.A end
       
  3594  1362 4325 a 1362
       
  3595 4325 a
       
  3596 SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark
       
  3597 end
       
  3598  1362 4325 a Black Black 1155 4484 a Fu(if)g Fm(z)33
       
  3599 b Fk(#)28 b Fm(N)35 b Fu(and)25 b Fm(z)32 b Fk(#)c Fm(L)d
       
  3600 Fu(then)g Fm(z)32 b Fk(#)c(\()p Fm(N)10 b Fk([)p Fm(y)32
       
  3601 b Fk(:=)27 b Fm(L)p Fk(]\))17 b Fm(:)p Black -2617 w
       
  3602 Fu(\(9\))p Black 291 4651 a(where)24 b(the)f(preconditions)f(are)i(gi)n
       
  3603 (v)o(en)e(by)i(his)f(use)g(of)h(the)f(v)n(ariable)g(con)l(v)o(ention.)
       
  3604 29 b(This)22 b(prop-)291 4771 y(erty)-6 b(,)27 b(too,)h(can)g(be)g
       
  3605 (easily)g(pro)o(v)o(ed)e(by)i(strong)e(induction)h(o)o(v)o(er)g(the)g
       
  3606 (structure)h(of)f Fm(N)10 b Fu(.)40 b(In)28 b(this)291
       
  3607 4892 y(induction)c(we)i(instantiate)e(the)i(induction)e(conte)o(xt)h
       
  3608 (with)g Fm(c)30 b Fl(7!)f Fk(\()p Fm(z)t(;)17 b(y)t(;)g(L)p
       
  3609 Fk(\))p Fu(,)25 b(because)i(then)e(we)291 5012 y(can)i(in)g(the)f
       
  3610 Fh(L)-5 b(am)8 b Fu(-case,)28 b(say)f(instantiated)e(as)i
       
  3611 Fk(\()p Fh(L)-5 b(am)34 b Fm(x)28 b(N)2335 5027 y Fj(1)2374
       
  3612 5012 y Fk(\))p Fu(,)g(mo)o(v)o(e)d(the)i(substitution)d(under)p
       
  3613 Black 291 5156 299 4 v 291 5217 a Ft(4)p 0 TeXcolorgray
       
  3614 374 5247 a
       
  3615 SDict begin H.S end
       
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  3618 SDict begin H.R end
       
  3619  374 5247 a 374 5247 a
       
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  3621 end
       
  3622  374 5247 a Black Fs(An)f(aspect)h(we)f(do)
       
  3623 g(not)g(dwell)h(on)f(here)g(is)h(the)g(f)o(act)f(that)h(the)f
       
  3624 (induction)f(conte)o(xt)g(must)i(al)o(w)o(ays)g(be)f(\002nitely)291
       
  3625 5347 y(supported,)18 b(i.e.)i(mentions)f(only)g(\002nitely)h(man)o(y)f
       
  3626 (free)h(names,)g(see)g([)p 0 0 1 TeXcolorrgb 2293 5348
       
  3627 a
       
  3628 SDict begin H.S end
       
  3629  2293 5348 a 0 0 1 TeXcolorrgb -1 x Fs(10)p 0 0 1 TeXcolorrgb
       
  3630 2376 5290 a
       
  3631 SDict begin H.R end
       
  3632  2376 5290 a 2376 5347 a
       
  3633 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
       
  3634 (cite.Pitts03) cvn H.B /ANN pdfmark end
       
  3635  2376 5347 a Black
       
  3636 Fs(,)p 0 0 1 TeXcolorrgb 2397 5348 a
       
  3637 SDict begin H.S end
       
  3638  2397 5348 a 0 0 1
       
  3639 TeXcolorrgb -1 x Fs(15)p 0 0 1 TeXcolorrgb 2480 5290
       
  3640 a
       
  3641 SDict begin H.R end
       
  3642  2480 5290 a 2480 5347 a
       
  3643 SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
       
  3644 (cite.UrbanTasson05) cvn H.B /ANN pdfmark end
       
  3645  2480 5347 a Black Fs(].)p Black
       
  3646 Black 1897 5513 a Fu(8)p Black eop end
       
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  3649 SDict begin /product where{pop product(Distiller)search{pop pop pop
       
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  3652 grestore}if}{pop}ifelse}{pop}ifelse}if end
       
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  3658 SDict begin H.R end
       
  3659  291 -166 a 291 -166 a
       
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  3661 end
       
  3662  291 -166 a Black 1173 w
       
  3663 Fo(Ber)n(ghofer)33 b(and)e(Urban)p Black 291 83 a Fu(the)i(binder)g
       
  3664 Fm(x)i Fu(and)e(also)h(infer)f(from)h(the)f(assumption)f
       
  3665 Fm(z)49 b Fk(#)44 b(\()p Fh(L)-5 b(am)41 b Fm(x)34 b(N)2939
       
  3666 98 y Fj(1)2979 83 y Fk(\))g Fu(that)f Fm(z)39 b Fu(is)33
       
  3667 b(also)291 203 y(fresh)c(for)h Fm(N)743 218 y Fj(1)812
       
  3668 203 y Fu(\(this)e(reasoning)h(step)g(depends)g(on)g Fm(z)41
       
  3669 b Fl(6)p Fk(=)36 b Fm(x)p Fu(\).)45 b(Consequently)29
       
  3670 b(we)g(can)h(apply)291 324 y(the)c(induction)f(hypothesis)g(and)h
       
  3671 (infer)h(that)g Fm(z)35 b Fk(#)d Fm(N)2128 339 y Fj(1)2167
       
  3672 324 y Fk([)p Fm(y)j Fk(:=)c Fm(L)p Fk(])c Fu(holds.)36
       
  3673 b(Again)25 b(since)i Fm(z)36 b Fl(6)p Fk(=)31 b Fm(x)p
       
  3674 Fu(,)291 444 y(also)24 b Fm(z)32 b Fk(#)c(\()p Fh(L)-5
       
  3675 b(am)32 b Fm(x)26 b(N)1077 459 y Fj(1)1116 444 y Fk([)p
       
  3676 Fm(y)31 b Fk(:=)d Fm(L)p Fk(]\))d Fu(holds)f(and)h(we)g(are)g(done.)437
       
  3677 573 y(The)g(formalisation)e(of)i(the)f(substitution)e(lemma)p
       
  3678 Black Black 541 752 a Fn(Substitution)k(Lemma)f(with)h(Names:)52
       
  3679 b Fu(If)25 b Fm(x)j Fl(6)p Fk(=)g Fm(y)g Fu(and)d Fm(x)j
       
  3680 Fk(#)g Fm(L)d Fu(then)777 932 y Fm(M)10 b Fk([)p Fm(x)29
       
  3681 b Fk(:=)f Fm(N)10 b Fk(][)p Fm(y)31 b Fk(:=)d Fm(L)p
       
  3682 Fk(])g(=)f Fm(M)10 b Fk([)p Fm(y)32 b Fk(:=)c Fm(L)p
       
  3683 Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 b Fk([)p Fm(y)31 b Fk(:=)d
       
  3684 Fm(L)p Fk(]])g Fu(.)291 1106 y(no)n(w)k(follo)n(ws)g(almost)h(to)g(the)
       
  3685 g(w)o(ord)g(Barendre)o(gt')-5 b(s)33 b(informal)g(proof.)57
       
  3686 b(The)33 b(v)n(ariable-case,)291 1226 y(say)f(with)f(the)h
       
  3687 (instantiation)e Fk(\()p Fh(V)-7 b(ar)43 b Fm(z)t Fk(\))p
       
  3688 Fu(,)34 b(proceeds)e(by)g(a)h(case-analysis)f(with)f
       
  3689 Fm(z)46 b Fk(=)41 b Fm(x)p Fu(,)34 b Fm(z)46 b Fl(6)p
       
  3690 Fk(=)291 1347 y Fm(x)29 b Fl(^)h Fm(z)50 b Fk(=)c Fm(y)37
       
  3691 b Fu(and)e Fm(z)50 b Fl(6)p Fk(=)c Fm(x)29 b Fl(^)h Fm(z)50
       
  3692 b Fl(6)p Fk(=)c Fm(y)t Fu(.)59 b(The)34 b(calculations)g(in)l(v)n(olv)o
       
  3693 (ed)e(are)j(routine)f(using)g(in)291 1467 y(the)24 b(second)h(case)g
       
  3694 (the)g(property)f(in)g(\()p 0 0 1 TeXcolorrgb 1579 1468
       
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  3697  1579 1468 a 0 0 1 TeXcolorrgb -1 x Fu(8)p 0 0 1 TeXcolorrgb
       
  3698 1629 1399 a
       
  3699 SDict begin H.R end
       
  3700  1629 1399 a 1629 1467 a
       
  3701 SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest
       
  3702 (equation.8) cvn H.B /ANN pdfmark end
       
  3703  1629 1467 a Black
       
  3704 Fu(\).)31 b(The)25 b(application)f(case)h(does)f(not)g(need)h(an)o(y)g
       
  3705 (special)291 1588 y(attention.)49 b(The)31 b(lambda-case,)i(too,)g(is)e
       
  3706 (relati)n(v)o(ely)e(easy:)44 b(by)31 b(instantiating)f(the)h(induction)
       
  3707 291 1708 y(conte)o(xt)d(with)h Fm(c)36 b Fl(7!)g Fk(\()p
       
  3708 Fm(x;)17 b(y)t(;)g(N)5 b(;)17 b(L)p Fk(\))p Fu(,)30 b(the)g(strong)e
       
  3709 (induction)g(principle)h(allo)n(ws)g(us)g(to)g(assume)291
       
  3710 1828 y(that)23 b(the)h(binder)g(is)f(not)h(equal)g(to)g
       
  3711 Fm(x)g Fu(and)g Fm(y)t Fu(,)g(and)g(is)f(not)h(free)h(in)e
       
  3712 Fm(N)10 b Fu(and)25 b Fm(L)p Fu(.)31 b(Consequently)23
       
  3713 b(we)291 1949 y(can)i(reason)g(lik)o(e)f(Barendre)o(gt:)p
       
  3714 Black Black 495 2148 a Fk(\()p Fh(L)-5 b(am)32 b Fm(z)e(M)920
       
  3715 2163 y Fj(1)960 2148 y Fk(\)[)p Fm(x)e Fk(:=)g Fm(N)10
       
  3716 b Fk(][)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(])47 b(=)h Fh(L)-5
       
  3717 b(am)32 b Fm(z)d Fk(\()p Fm(M)2279 2163 y Fj(1)2319 2148
       
  3718 y Fk([)p Fm(x)f Fk(:=)g Fm(N)10 b Fk(][)p Fm(y)31 b Fk(:=)d
       
  3719 Fm(L)p Fk(]\))1731 2329 y(=)48 b Fh(L)-5 b(am)32 b Fm(z)d
       
  3720 Fk(\()p Fm(M)2279 2344 y Fj(1)2319 2329 y Fk([)p Fm(y)i
       
  3721 Fk(:=)d Fm(L)p Fk(][)p Fm(x)g Fk(:=)g Fm(N)10 b Fk([)p
       
  3722 Fm(y)31 b Fk(:=)d Fm(L)p Fk(]]\))1731 2509 y(=)48 b(\()p
       
  3723 Fh(L)-5 b(am)32 b Fm(z)d(M)2279 2524 y Fj(1)2319 2509
       
  3724 y Fk(\)[)p Fm(y)i Fk(:=)c Fm(L)p Fk(][)p Fm(x)i Fk(:=)f
       
  3725 Fm(N)10 b Fk([)p Fm(y)31 b Fk(:=)d Fm(L)p Fk(]])291 2683
       
  3726 y Fu(where,)d(as)g(mentioned)e(earlier)l(,)i(in)g(the)g(last)f
       
  3727 (equation)g(we)h(mak)o(e)g(use)f(the)h(property)f(in)h(\()p
       
  3728 0 0 1 TeXcolorrgb 3401 2685 a
       
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  3730  3401 2685 a 0 0 1 TeXcolorrgb
       
  3731 -2 x Fu(9)p 0 0 1 TeXcolorrgb 3451 2615 a
       
  3732 SDict begin H.R end
       
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  3738 (formalised)h(proof)g(is)g(quite)g(simple:)29 b(one)24
       
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  3760 2301 2932 a Black Fu(\))g(to)g(the)f(automatic)g(pro)o(ving)f(tools)291
       
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  3763 3173 y(lemmas\).)63 b(W)-8 b(e)37 b(tak)o(e)f(this)f(as)h(an)h
       
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  3765 3293 y(\223simpler\224)24 b(than)g(the)h(one)g(based)f(on)h(de)g
       
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  3771 3469 a 291 3469 a
       
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  3776 y Fu(Another)i(proof)h(where)h(we)f(can)h(compare)f(names)g(and)g(de)h
       
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  3794  3260 4309 a Black Fu(])g(by)g(the)291 4429
       
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  3803  291 4605 a 107
       
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  3805 (P)-8 b(ac)n(ka)o(g)o(e)291 4903 y Fu(Using)23 b(the)i(nominal)f
       
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  3807 Black Black 440 5061 a Fn(nominal)p 795 5061 30 4 v 35
       
  3808 w(datatype)57 b Fh(ty)c Fk(=)44 b Fh(Tvar)38 b Fm(name)57
       
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  3810 b(ty)j(ty)65 b Fl(j)27 b Fh(A)n(l)5 b(l)38 b(ty)3169
       
  3811 5048 y Fb(h)-11 b(h)3209 5061 y Fh(name)3446 5048 y Fb(i)g(i)3486
       
  3812 5061 y Fh(ty)291 5226 y Fu(with)30 b(typing)g(conte)o(xts)g(being)h
       
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  3816 Fu(,)j(written)e Fk(\000)42 b Fl(`)g Fm(T)14 b Fu(,)34
       
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  3906 Fk(\(\001@\()p Fm(X)r(;)17 b(Q)p Fk(\)@\000\))28 b Fl(`)f
       
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  3909 Fk(\(\001@\()p Fm(X)r(;)17 b(P)d Fk(\)@\000\))27 b Fl(`)h
       
  3910 Fm(M)38 b(<)p Fk(:)28 b Fm(N)38 b Fu(.)291 3214 y(About)24
       
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  3912 Black Black 390 3413 a Fc(\223)f(The)c(two)f(parts)f(ar)l(e)h(pr)l(o)o
       
  3913 (ved)g(simultaneously)-5 b(,)18 b(by)i(induction)f(on)h(the)g(size)f
       
  3914 (of)h Fm(Q)p Fc(.)30 b(The)470 3534 y(ar)l(gument)g(for)g(part)g(\(2\))
       
  3915 h(assumes)e(that)h(part)g(\(1\))h(has)f(been)g(established)f(alr)l
       
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  3918 Fm(Q)p Fc(.)-14 b(\224)291 3828 y Fu(The)20 b(main)g(point)g(we)h(w)o
       
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  3929 4430 y Fu(a)36 b(strong)g(v)o(ersion)f(of)i(the)f(induction)f
       
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