Publications/nom-tech.ps
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     1 %!PS-Adobe-2.0
       
     2 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software
       
     3 %%Title: Main-jv.dvi
       
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    61 %!
       
    62 % PostScript prologue for pstricks.tex.
       
    63 % Version 97 patch 4, 04/05/10
       
    64 % For distribution, see pstricks.tex.
       
    65 %
       
    66 /tx@Dict 200 dict def tx@Dict begin
       
    67 /ADict 25 dict def
       
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    70 /CLW /currentlinewidth load def
       
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    72 /ED { exch def } bind def
       
    73 /L /lineto load def
       
    74 /T /translate load def
       
    75 /TMatrix { } def
       
    76 /RAngle { 0 } def
       
    77 /Atan { /atan load stopped { pop pop 0 } if } def
       
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    80 /Pyth { dup mul exch dup mul add sqrt } def
       
    81 /PtoC { 2 copy cos mul 3 1 roll sin mul } def
       
    82 /PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
       
    83 def
       
    84 /PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
       
    85 } { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
       
    86 /pathforall load stopped { pop pop pop pop } if z } def
       
    87 /STP { .996264 dup scale } def
       
    88 /STV { SDict begin normalscale end STP  } def
       
    89 %
       
    90 %%-------------- DG begin patch 15 ---------------%%
       
    91 %/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
       
    92 %PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
       
    93 %mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
       
    94 %/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
       
    95 %0 } ifelse setdash stroke } def
       
    96 /DashLine {
       
    97   dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def PathLength } ifelse
       
    98   /b ED /x1 ED /y1 ED /x ED /y ED 
       
    99   /z y x add y1 add x1 add def
       
   100   /Coef b a .5 sub 2 mul y mul sub z Div round 
       
   101   z mul a .5 sub 2 mul y mul add b exch Div def 
       
   102   /y y Coef mul def /x x Coef mul def /y1 y1 Coef mul def /x1 x1 Coef mul def
       
   103   x1 0 gt y1 0 gt x 0 gt y 0 gt and { [ y x y1 x1 ] 1 a sub y mul}
       
   104   { [ 1 0] 0 } ifelse setdash stroke
       
   105 } def
       
   106 %%-------------- DG end patch 15 ---------------%%
       
   107 /DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
       
   108 { /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
       
   109 def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
       
   110 ] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
       
   111 setlinecap stroke } def
       
   112 /LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
       
   113 2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
       
   114 % DG/SR modification begin - Dec. 12, 1997 - Patch 2
       
   115 %itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
       
   116 itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
       
   117 % DG/SR modification end
       
   118 Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
       
   119 /setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
       
   120 % DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
       
   121 % a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
       
   122 % def
       
   123 a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
       
   124 pop pop } def
       
   125 % DG/SR modification end
       
   126 /BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
       
   127 exch 3 index sub exch Atan rotate newpath } def
       
   128 /EndArrow { @mtrx setmatrix CP grestore end } def
       
   129 /Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
       
   130 scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
       
   131 grestore } def
       
   132 /Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
       
   133 CLW moveto } def
       
   134 /Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
       
   135 div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
       
   136 CLW moveto } def
       
   137 /RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
       
   138 2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
       
   139 curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
       
   140 def
       
   141 /SD { 0 360 arc fill } def
       
   142 /EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
       
   143 CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
       
   144 /Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
       
   145 /closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
       
   146 load } if ] cvx newpath 3 1 roll T exec } def
       
   147 /NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
       
   148 cvi def } def
       
   149 /NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
       
   150 f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
       
   151 /Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
       
   152 def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
       
   153 /Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
       
   154 ifelse { pop } repeat a } def
       
   155 /CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
       
   156 and { pop pop /n n 1 sub def } if } def
       
   157 /Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
       
   158 repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
       
   159 ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
       
   160 moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
       
   161 Lineto pop pop closepath } ifelse } def
       
   162 /Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
       
   163 neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
       
   164 def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
       
   165 div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
       
   166 setmatrix } def
       
   167 % DG modification begin - Jan. 15, 1997
       
   168 %/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
       
   169 %pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
       
   170 %div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
       
   171 %d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
       
   172 %setmatrix } def
       
   173 /Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
       
   174 CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
       
   175 div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
       
   176 d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
       
   177 % DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
       
   178 % setmatrix } def
       
   179 setmatrix pop } def
       
   180 % DG/SR modification end
       
   181 /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
       
   182 def } def
       
   183 /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
       
   184 def } def
       
   185 /CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
       
   186 def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
       
   187 mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
       
   188 abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
       
   189 /y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
       
   190 m mul neg def } def
       
   191 /IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
       
   192 ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
       
   193 def
       
   194 /BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
       
   195 /NC { CC x1 y1 x2 y2 x y curveto } def
       
   196 /EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
       
   197 /BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
       
   198 /NAC { x2 y2 x y curveto CC x1 y1 } def
       
   199 /EAC { x2 y2 x y ArrowB curveto pop pop } def
       
   200 /OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
       
   201     n { NC } repeat EOC } ifelse } def
       
   202 /AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
       
   203 /Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
       
   204 pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
       
   205 /ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
       
   206 CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
       
   207 repeat closepath pop pop } ifelse } def
       
   208 /SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
       
   209 /ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
       
   210 /SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
       
   211 def
       
   212 /FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
       
   213 rotate matrix concatmatrix exch findfont exch makefont setfont } def
       
   214 /Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
       
   215 x1 y1 lineto closepath } def
       
   216 /OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
       
   217 y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
       
   218 exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
       
   219 /b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
       
   220 x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
       
   221 ifelse } def
       
   222 /Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
       
   223 /y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
       
   224 pop pop Rect } { OvalFrame } ifelse } def
       
   225 /BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
       
   226 } if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
       
   227 aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
       
   228 /OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
       
   229 2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
       
   230 ifelse } def
       
   231 /ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
       
   232 6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
       
   233 /BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
       
   234 sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
       
   235 def
       
   236 /Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
       
   237 sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
       
   238 x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
       
   239 y0 x0 2 mul x1 sub y1 ] def } def
       
   240 /Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
       
   241 if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
       
   242 /dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
       
   243 round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
       
   244 round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
       
   245 0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
       
   246 /Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
       
   247 if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
       
   248 { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
       
   249 exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
       
   250 div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
       
   251 dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
       
   252 dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
       
   253 pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
       
   254 stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
       
   255 {z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
       
   256 w add def } repeat grestore gsave n 0 gt
       
   257 % DG/SR modification begin - Nov. 7, 1997 - Patch 1
       
   258 %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
       
   259 { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
       
   260 % DG/SR modification end
       
   261 { 2 setlinecap } ifelse /i y1 def /f x1 dx mul
       
   262 n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
       
   263 div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
       
   264 1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
       
   265 dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
       
   266 {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
       
   267 exch t L stroke /i i h add def } repeat grestore } def
       
   268 /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
       
   269 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
       
   270 exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
       
   271 mul neg d } def
       
   272 /Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
       
   273 /Rot { CP CP translate 3 -1 roll neg rotate NET  } def
       
   274 /RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
       
   275 def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
       
   276 dup a add ] cvx def } def
       
   277 /RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
       
   278 cvx def } def
       
   279 /PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
       
   280 /PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
       
   281 /PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
       
   282 /Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
       
   283 abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
       
   284 div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
       
   285 h1 abs h sub dup s mul abs } ifelse } def
       
   286 /UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
       
   287 mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
       
   288 { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
       
   289 exch } def
       
   290 /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
       
   291 /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
       
   292 } if } ifelse } def
       
   293 /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
       
   294 /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
       
   295 neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
       
   296 /IfVisible true def } def
       
   297 end
       
   298 % END pstricks.pro
       
   299 
       
   300 %%EndProcSet
       
   301 %%BeginProcSet: pst-dots.pro 0 0
       
   302 %!PS-Adobe-2.0
       
   303 %%Title: Dot Font for PSTricks
       
   304 %%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
       
   305 %%Creation Date: May 7, 1993
       
   306 %% Version 97 patch 1, 99/12/16
       
   307 %% Modified by Etienne Riga <etienne.riga@skynet.be> - Dec. 16, 1999
       
   308 %% to add /Diamond, /SolidDiamond and /BoldDiamond
       
   309 10 dict dup begin
       
   310   /FontType 3 def
       
   311   /FontMatrix [ .001 0 0 .001 0 0 ] def
       
   312   /FontBBox [ 0 0 0 0 ] def
       
   313   /Encoding 256 array def
       
   314   0 1 255 { Encoding exch /.notdef put } for
       
   315   Encoding
       
   316     dup (b) 0 get /Bullet put
       
   317     dup (c) 0 get /Circle put
       
   318     dup (C) 0 get /BoldCircle put
       
   319     dup (u) 0 get /SolidTriangle put
       
   320     dup (t) 0 get /Triangle put
       
   321     dup (T) 0 get /BoldTriangle put
       
   322     dup (r) 0 get /SolidSquare put
       
   323     dup (s) 0 get /Square put
       
   324     dup (S) 0 get /BoldSquare put
       
   325     dup (q) 0 get /SolidPentagon put
       
   326     dup (p) 0 get /Pentagon put
       
   327     dup (P) 0 get /BoldPentagon put
       
   328 % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
       
   329     dup (l) 0 get /SolidDiamond put
       
   330     dup (d) 0 get /Diamond put
       
   331         (D) 0 get /BoldDiamond put
       
   332 % DG/SR modification end
       
   333   /Metrics 13 dict def
       
   334   Metrics begin
       
   335     /Bullet        1000   def
       
   336     /Circle        1000   def
       
   337     /BoldCircle    1000   def
       
   338     /SolidTriangle 1344   def
       
   339     /Triangle      1344   def
       
   340     /BoldTriangle  1344   def
       
   341     /SolidSquare    886   def
       
   342     /Square         886   def
       
   343     /BoldSquare     886   def
       
   344     /SolidPentagon 1093.2 def
       
   345     /Pentagon      1093.2 def
       
   346     /BoldPentagon  1093.2 def
       
   347 % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
       
   348     /SolidDiamond  1008   def
       
   349     /Diamond       1008   def
       
   350     /BoldDiamond   1008   def
       
   351 % DG/SR modification end
       
   352     /.notdef 0 def
       
   353   end
       
   354   /BBoxes 13 dict def
       
   355   BBoxes begin
       
   356     /Circle        { -550 -550 550 550 } def
       
   357     /BoldCircle    /Circle load def
       
   358     /Bullet        /Circle load def
       
   359     /Triangle      { -571.5 -330 571.5 660 } def
       
   360     /BoldTriangle  /Triangle load def
       
   361     /SolidTriangle /Triangle load def
       
   362     /Square        { -450 -450 450 450 } def
       
   363     /BoldSquare    /Square load def
       
   364     /SolidSquare   /Square load def
       
   365     /Pentagon      { -546.6 -465 546.6 574.7 } def
       
   366     /BoldPentagon  /Pentagon load def
       
   367     /SolidPentagon /Pentagon load def
       
   368 % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
       
   369     /Diamond       { -428.5 -742.5 428.5 742.5 } def
       
   370     /BoldDiamond   /Diamond load def
       
   371     /SolidDiamond  /Diamond load def
       
   372 % DG/SR modification end
       
   373     /.notdef { 0 0 0 0 } def
       
   374   end
       
   375   /CharProcs 20 dict def
       
   376   CharProcs begin
       
   377     /Adjust {
       
   378       2 copy dtransform floor .5 add exch floor .5 add exch idtransform
       
   379       3 -1 roll div 3 1 roll exch div exch scale
       
   380     } def
       
   381     /CirclePath    { 0 0 500 0 360 arc closepath } def
       
   382     /Bullet        { 500 500 Adjust CirclePath fill } def
       
   383     /Circle        { 500 500 Adjust CirclePath .9 .9 scale CirclePath
       
   384                      eofill } def
       
   385     /BoldCircle    { 500 500 Adjust CirclePath .8 .8 scale CirclePath
       
   386                      eofill } def
       
   387     /BoldCircle    { CirclePath .8 .8 scale CirclePath eofill } def
       
   388     /TrianglePath  { 0  660 moveto -571.5 -330 lineto 571.5 -330 lineto
       
   389                      closepath } def
       
   390     /SolidTriangle { TrianglePath fill } def
       
   391     /Triangle      { TrianglePath .85 .85 scale TrianglePath eofill } def
       
   392     /BoldTriangle  { TrianglePath .7 .7 scale TrianglePath eofill } def
       
   393     /SquarePath    { -450 450 moveto 450 450 lineto 450 -450 lineto
       
   394                      -450 -450 lineto closepath } def
       
   395     /SolidSquare   { SquarePath fill } def
       
   396     /Square        { SquarePath .89 .89 scale SquarePath eofill } def
       
   397     /BoldSquare    { SquarePath .78 .78 scale SquarePath eofill } def
       
   398     /PentagonPath  {
       
   399       -337.8 -465   moveto
       
   400        337.8 -465   lineto
       
   401        546.6  177.6 lineto
       
   402          0    574.7 lineto
       
   403       -546.6  177.6 lineto
       
   404       closepath
       
   405     } def
       
   406     /SolidPentagon { PentagonPath fill } def
       
   407     /Pentagon      { PentagonPath .89 .89 scale PentagonPath eofill } def
       
   408     /BoldPentagon  { PentagonPath .78 .78 scale PentagonPath eofill } def
       
   409 % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
       
   410     /DiamondPath   { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto
       
   411                      428.5 0 lineto closepath } def
       
   412     /SolidDiamond  { DiamondPath fill } def
       
   413     /Diamond       { DiamondPath .85 .85 scale DiamondPath eofill } def
       
   414     /BoldDiamond   { DiamondPath .7 .7 scale DiamondPath eofill } def
       
   415 % DG/SR modification end
       
   416     /.notdef { } def
       
   417   end
       
   418   /BuildGlyph {
       
   419     exch
       
   420     begin
       
   421       Metrics 1 index get exec 0
       
   422       BBoxes 3 index get exec
       
   423       setcachedevice
       
   424       CharProcs begin load exec end
       
   425     end
       
   426   } def
       
   427   /BuildChar {
       
   428     1 index /Encoding get exch get
       
   429     1 index /BuildGlyph get exec
       
   430   } bind def
       
   431 end
       
   432 /PSTricksDotFont exch definefont pop
       
   433 %END pst-dots.pro
       
   434 
       
   435 %%EndProcSet
       
   436 %%BeginProcSet: pst-node.pro 0 0
       
   437 %!
       
   438 % PostScript prologue for pst-node.tex.
       
   439 % Version 97 patch 1, 97/05/09.
       
   440 % For distribution, see pstricks.tex.
       
   441 %
       
   442 /tx@NodeDict 400 dict def tx@NodeDict begin
       
   443 tx@Dict begin /T /translate load def end
       
   444 /NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
       
   445 if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
       
   446 grestore } def
       
   447 /InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
       
   448 } def
       
   449 /InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
       
   450 Sin mul } def } def
       
   451 /GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
       
   452 } ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
       
   453 ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
       
   454 dup Sin mul Cos Div } ifelse } def
       
   455 /InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
       
   456 /u ED /NodePos { GetRnodePos } def } def
       
   457 /DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
       
   458 Cos mul exch Sin mul } def
       
   459 /TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
       
   460 mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
       
   461 } ifelse } def
       
   462 /InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
       
   463 ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
       
   464 sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
       
   465 sin def /NodePos { TriNodePos } def } def
       
   466 /OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
       
   467 hh mul Atan dup cos ww mul exch sin hh mul } def
       
   468 /GetCenter { begin X Y NodeMtrx transform CM itransform end } def
       
   469 /XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
       
   470 Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
       
   471 { 0 Dist Sin mul } ifelse } ifelse Do } def
       
   472 /GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
       
   473 exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
       
   474 dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
       
   475 XYPos } ifelse } def
       
   476 /AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
       
   477 roll sin mul sub exch } ifelse } def
       
   478 /GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
       
   479 AddOffset yA add /yA1 ED xA add /xA1 ED } def
       
   480 /GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
       
   481 AddOffset yB add /yB1 ED xB add /xB1 ED } def
       
   482 /GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
       
   483 AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
       
   484 ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
       
   485 } ifelse } def
       
   486 /GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
       
   487 AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
       
   488 ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
       
   489 } ifelse } def
       
   490 /InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
       
   491 /NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
       
   492 known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
       
   493 ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
       
   494 /LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
       
   495 mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
       
   496 /LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
       
   497 { pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
       
   498 } repeat LPutLine cleartomark } def
       
   499 /BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
       
   500 /t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
       
   501 mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
       
   502 sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
       
   503 add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
       
   504 y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
       
   505 mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
       
   506 /HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
       
   507 } def
       
   508 /HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
       
   509 /NAngle yyB yyA sub xxB xxA sub Atan def } def
       
   510 /HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
       
   511 /HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
       
   512 } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
       
   513 } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
       
   514 } def
       
   515 /VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
       
   516 } def
       
   517 /VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
       
   518 /NAngle yyB yyA sub xxB xxA sub Atan def } def
       
   519 /VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
       
   520 /VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
       
   521 } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
       
   522 exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
       
   523 VPosEnd  } def
       
   524 /HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
       
   525 roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
       
   526 def grestore exec /LPutVar /SaveLPutVar load def } def
       
   527 /NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
       
   528 GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
       
   529 LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
       
   530 LPutVar VPutLine } def LPutVar } def
       
   531 /NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
       
   532 end } def
       
   533 /NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
       
   534 Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
       
   535 /AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
       
   536 yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
       
   537 LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
       
   538 ifelse } def
       
   539 /NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
       
   540 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
       
   541 GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
       
   542 begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
       
   543 cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
       
   544 } HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
       
   545 /NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
       
   546 def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
       
   547 itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
       
   548 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
       
   549 yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
       
   550 /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
       
   551 /NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
       
   552 yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
       
   553 /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
       
   554 tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
       
   555 cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
       
   556 VPutLines } def } def
       
   557 /NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
       
   558 xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
       
   559 transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
       
   560 { /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
       
   561 0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
       
   562 begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
       
   563 def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
       
   564 VPutLines } def } def
       
   565 /NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
       
   566 xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
       
   567 /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
       
   568 LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
       
   569 def
       
   570 /NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
       
   571 GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
       
   572 false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
       
   573 LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
       
   574 def
       
   575 /NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
       
   576 def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
       
   577 mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
       
   578 mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
       
   579 xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
       
   580 Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
       
   581 cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
       
   582 VPutLines } def } def
       
   583 % DG/SR modification begin - May 9, 1997 - Patch 1
       
   584 %/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
       
   585 %r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
       
   586 %exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
       
   587 %mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
       
   588 /NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
       
   589 r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
       
   590 exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
       
   591 mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
       
   592 % DG/SR modification end
       
   593 } def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
       
   594 AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
       
   595 Div def /y ED /x ED } def
       
   596 /NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
       
   597 180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
       
   598 mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
       
   599 /LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
       
   600 add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
       
   601 { xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
       
   602 tx@Dict begin false Polygon end } def
       
   603 /NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
       
   604 def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
       
   605 mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
       
   606 /x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
       
   607 def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
       
   608 GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
       
   609 sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
       
   610 eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
       
   611 { x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
       
   612 0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
       
   613 roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
       
   614 0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
       
   615 roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
       
   616 ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
       
   617 { LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
       
   618 r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
       
   619 { t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
       
   620 add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
       
   621 mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
       
   622 sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
       
   623 y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
       
   624 LPutPos } def } def
       
   625 /Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
       
   626 sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
       
   627 ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
       
   628 w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
       
   629 { LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
       
   630 { pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
       
   631 def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
       
   632 /ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
       
   633 /LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
       
   634 exch X sub neg exch moveto setmatrix CP grestore } def
       
   635 /LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
       
   636 def } ifelse LPutCoor  } def
       
   637 /HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
       
   638 if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
       
   639 ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
       
   640 def
       
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   649 %%BeginProcSet: 8r.enc 0 0
       
   650 % File 8r.enc  TeX Base 1 Encoding  Revision 2.0  2002-10-30
       
   651 %
       
   652 % @@psencodingfile@{
       
   653 %   author    = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
       
   654 %                W. Schmidt, P. Lehman",
       
   655 %   version   = "2.0",
       
   656 %   date      = "30 October 2002",
       
   657 %   filename  = "8r.enc",
       
   658 %   email     = "tex-fonts@@tug.org",
       
   659 %   docstring = "This is the encoding vector for Type1 and TrueType
       
   660 %                fonts to be used with TeX.  This file is part of the
       
   661 %                PSNFSS bundle, version 9"
       
   662 % @}
       
   663 % 
       
   664 % The idea is to have all the characters normally included in Type 1 fonts
       
   665 % available for typesetting. This is effectively the characters in Adobe
       
   666 % Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,
       
   667 % MacRoman, and some extra characters from Lucida.
       
   668 % 
       
   669 % Character code assignments were made as follows:
       
   670 % 
       
   671 % (1) the Windows ANSI characters are almost all in their Windows ANSI
       
   672 % positions, because some Windows users cannot easily reencode the
       
   673 % fonts, and it makes no difference on other systems. The only Windows
       
   674 % ANSI characters not available are those that make no sense for
       
   675 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
       
   676 % (173). quotesingle and grave are moved just because it's such an
       
   677 % irritation not having them in TeX positions.
       
   678 % 
       
   679 % (2) Remaining characters are assigned arbitrarily to the lower part
       
   680 % of the range, avoiding 0, 10 and 13 in case we meet dumb software.
       
   681 % 
       
   682 % (3) Y&Y Lucida Bright includes some extra text characters; in the
       
   683 % hopes that other PostScript fonts, perhaps created for public
       
   684 % consumption, will include them, they are included starting at 0x12.
       
   685 % These are /dotlessj /ff /ffi /ffl.
       
   686 % 
       
   687 % (4) hyphen appears twice for compatibility with both ASCII and Windows.
       
   688 %
       
   689 % (5) /Euro was assigned to 128, as in Windows ANSI
       
   690 %
       
   691 % (6) Missing characters from MacRoman encoding incorporated as follows:
       
   692 %
       
   693 %     PostScript      MacRoman        TeXBase1
       
   694 %     --------------  --------------  --------------
       
   695 %     /notequal       173             0x16
       
   696 %     /infinity       176             0x17
       
   697 %     /lessequal      178             0x18
       
   698 %     /greaterequal   179             0x19
       
   699 %     /partialdiff    182             0x1A
       
   700 %     /summation      183             0x1B
       
   701 %     /product        184             0x1C
       
   702 %     /pi             185             0x1D
       
   703 %     /integral       186             0x81
       
   704 %     /Omega          189             0x8D
       
   705 %     /radical        195             0x8E
       
   706 %     /approxequal    197             0x8F
       
   707 %     /Delta          198             0x9D
       
   708 %     /lozenge        215             0x9E
       
   709 %
       
   710 /TeXBase1Encoding [
       
   711 % 0x00
       
   712  /.notdef /dotaccent /fi /fl
       
   713  /fraction /hungarumlaut /Lslash /lslash
       
   714  /ogonek /ring /.notdef /breve
       
   715  /minus /.notdef /Zcaron /zcaron
       
   716 % 0x10
       
   717  /caron /dotlessi /dotlessj /ff
       
   718  /ffi /ffl /notequal /infinity
       
   719  /lessequal /greaterequal /partialdiff /summation
       
   720  /product /pi /grave /quotesingle
       
   721 % 0x20
       
   722  /space /exclam /quotedbl /numbersign
       
   723  /dollar /percent /ampersand /quoteright
       
   724  /parenleft /parenright /asterisk /plus
       
   725  /comma /hyphen /period /slash
       
   726 % 0x30
       
   727  /zero /one /two /three
       
   728  /four /five /six /seven
       
   729  /eight /nine /colon /semicolon
       
   730  /less /equal /greater /question
       
   731 % 0x40
       
   732  /at /A /B /C
       
   733  /D /E /F /G
       
   734  /H /I /J /K
       
   735  /L /M /N /O
       
   736 % 0x50
       
   737  /P /Q /R /S
       
   738  /T /U /V /W
       
   739  /X /Y /Z /bracketleft
       
   740  /backslash /bracketright /asciicircum /underscore
       
   741 % 0x60
       
   742  /quoteleft /a /b /c
       
   743  /d /e /f /g
       
   744  /h /i /j /k
       
   745  /l /m /n /o
       
   746 % 0x70
       
   747  /p /q /r /s
       
   748  /t /u /v /w
       
   749  /x /y /z /braceleft
       
   750  /bar /braceright /asciitilde /.notdef
       
   751 % 0x80
       
   752  /Euro /integral /quotesinglbase /florin
       
   753  /quotedblbase /ellipsis /dagger /daggerdbl
       
   754  /circumflex /perthousand /Scaron /guilsinglleft
       
   755  /OE /Omega /radical /approxequal
       
   756 % 0x90
       
   757  /.notdef /.notdef /.notdef /quotedblleft
       
   758  /quotedblright /bullet /endash /emdash
       
   759  /tilde /trademark /scaron /guilsinglright
       
   760  /oe /Delta /lozenge /Ydieresis
       
   761 % 0xA0
       
   762  /.notdef /exclamdown /cent /sterling
       
   763  /currency /yen /brokenbar /section
       
   764  /dieresis /copyright /ordfeminine /guillemotleft
       
   765  /logicalnot /hyphen /registered /macron
       
   766 % 0xD0
       
   767  /degree /plusminus /twosuperior /threesuperior
       
   768  /acute /mu /paragraph /periodcentered
       
   769  /cedilla /onesuperior /ordmasculine /guillemotright
       
   770  /onequarter /onehalf /threequarters /questiondown
       
   771 % 0xC0
       
   772  /Agrave /Aacute /Acircumflex /Atilde
       
   773  /Adieresis /Aring /AE /Ccedilla
       
   774  /Egrave /Eacute /Ecircumflex /Edieresis
       
   775  /Igrave /Iacute /Icircumflex /Idieresis
       
   776 % 0xD0
       
   777  /Eth /Ntilde /Ograve /Oacute
       
   778  /Ocircumflex /Otilde /Odieresis /multiply
       
   779  /Oslash /Ugrave /Uacute /Ucircumflex
       
   780  /Udieresis /Yacute /Thorn /germandbls
       
   781 % 0xE0
       
   782  /agrave /aacute /acircumflex /atilde
       
   783  /adieresis /aring /ae /ccedilla
       
   784  /egrave /eacute /ecircumflex /edieresis
       
   785  /igrave /iacute /icircumflex /idieresis
       
   786 % 0xF0
       
   787  /eth /ntilde /ograve /oacute
       
   788  /ocircumflex /otilde /odieresis /divide
       
   789  /oslash /ugrave /uacute /ucircumflex
       
   790  /udieresis /yacute /thorn /ydieresis
       
   791 ] def
       
   792 
       
   793 
       
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  2567 b(from)d Ft(http://isabelle.in.tum.de)q(/nom)q(ina)q(l)p
       
  2568 FF(.)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop
       
  2569 end
       
  2570 %%Page: 3 3
       
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  2573 b(o)o(v)o(er)g(the)f(bijecti)n(v)o(e)g(set.)f(Sec.)h(6)g(gi)n(v)o(es)g
       
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  2577 (lambda-calculus)h(there)f(is)f(a)h(single)g(type)g(of)g(bindable)h
       
  2578 (names,)f(here)g(denoted)h(by)f Fs(name)q FI(,)f(whose)0
       
  2579 1083 y(elements)26 b(in)h(the)f(tradition)i(of)e(the)h(nominal)h(logic)
       
  2580 e(w)o(ork)i(we)e(call)g FE(atoms)p FI(.)g(While)h(the)f(structure)h(of)
       
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  2582 (hold)g(for)g(the)g(type)g Fs(name)q FI(:)f(one)h(has)f(to)g(be)h(able)
       
  2583 g(to)0 1274 y(distinguishing)33 b(dif)n(ferent)f(atoms)e(and)i(one)f
       
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  2586 (Isabelle/HOL)g(by)h(implementing)g(the)f(type)h Fs(name)f
       
  2587 FI(as)0 1464 y(natural)21 b(numbers)g(or)g(strings.)125
       
  2588 1579 y(Permutations)h(are)f(\002nite)g(bijecti)n(v)o(e)g(mappings)i
       
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  2592 (ws)h(the)f(type-)0 1770 y(abbre)n(viation)j Fs(name)d(prm)h
       
  2593 FI(will)f(stand)h(for)g(the)g(type)g(of)g(permutations,)h(that)f(is)e
       
  2594 Fr(\()p Fs(name)d FD(\002)f Fs(name)p Fr(\))22 b Fs(list)q
       
  2595 FI(,)0 1866 y(and)f(we)e(will)h(write)g(permutations)h(as)1044
       
  2596 2147 y Fr(\()p FG(a)1115 2159 y Fq(1)1165 2147 y FG(b)1198
       
  2597 2159 y Fq(1)1235 2147 y Fr(\)\()p FG(a)1336 2159 y Fq(2)1385
       
  2598 2147 y FG(b)1418 2159 y Fq(2)1455 2147 y Fr(\))13 b FD(\001)g(\001)g
       
  2599 (\001)g Fr(\()p FG(a)1671 2155 y Fp(n)1729 2147 y FG(b)1762
       
  2600 2155 y Fp(n)1807 2147 y Fr(\))0 2448 y FI(with)22 b(the)h(empty)g(list)
       
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  2602 (operation)h(of)f(a)f(permutation)0 2544 y FG(\031)g
       
  2603 FE(acting)f FI(on)g(an)f(atom)g FG(a)g FI(is)f(de\002ned)i(as:)1192
       
  2604 2846 y Fr([])1234 2855 y Fo(\001)1272 2846 y FG(a)1338
       
  2605 2803 y Fn(def)1345 2846 y Fr(=)32 b FG(a)814 3036 y Fr(\(\()p
       
  2606 FG(a)915 3048 y Fq(1)965 3036 y FG(a)1006 3048 y Fq(2)1043
       
  2607 3036 y Fr(\))21 b(::)h FG(\031)s Fr(\))1235 3045 y Fo(\001)1272
       
  2608 3036 y FG(a)1338 2993 y Fn(def)1345 3036 y Fr(=)1437
       
  2609 2883 y Fm(8)1437 2950 y(<)1437 3085 y(:)1518 2941 y FG(a)1559
       
  2610 2953 y Fq(2)1668 2941 y FI(if)g FG(\031)1785 2950 y Fo(\001)1823
       
  2611 2941 y FG(a)f Fr(=)g FG(a)2007 2953 y Fq(1)1518 3036
       
  2612 y FG(a)1559 3048 y Fq(1)1668 3036 y FI(if)h FG(\031)1785
       
  2613 3045 y Fo(\001)1823 3036 y FG(a)f Fr(=)g FG(a)2007 3048
       
  2614 y Fq(2)1518 3132 y FG(\031)1565 3141 y Fo(\001)1603 3132
       
  2615 y FG(a)j FI(otherwise)2789 2976 y(\(1\))0 3430 y(where)i
       
  2616 Fr(\()p FG(a)12 b(b)p Fr(\))31 b(::)h FG(\031)27 b FI(is)e(the)g
       
  2617 (composition)j(of)d(a)g(permutation)j(follo)n(wed)e(by)g(the)g(sw)o
       
  2618 (apping)g Fr(\()p FG(a)13 b(b)p Fr(\))p FI(.)24 b(The)0
       
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  2621 (en)i(by)g(list-concatenation,)g(written)0 3621 y(as)f
       
  2622 FG(\031)132 3590 y Fl(0)155 3621 y Fr(@)p FG(\031)s FI(,)g(and)i(the)f
       
  2623 (in)m(v)o(erse)g(of)g(a)g(permutation)i(is)d(gi)n(v)o(en)i(by)g(list)e
       
  2624 (re)n(v)o(ersal,)h(written)g(as)f FG(\031)2475 3590 y
       
  2625 Fl(\000)p Fq(1)2564 3621 y FI(.)125 3736 y(Our)31 b(representation)i
       
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  2628 b(the)g(permutation)h Fr(\()p FG(a)13 b(a)p Fr(\))20
       
  2629 b FI(is)h(\223equal\224)h(to)g(the)g(identity)g(permutation.)h(W)-6
       
  2630 b(e)22 b(equate)g(the)g(repre-)0 3927 y(sentations)e(of)h(permutations)
       
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  2633 b(Equality\))f FE(T)-6 b(wo)24 b(permutations)h(ar)m(e)e
       
  2634 FI(equal)p FE(,)h(written)g FG(\031)2471 4206 y Fq(1)2536
       
  2635 4194 y FD(\030)k FG(\031)2668 4206 y Fq(2)2705 4194 y
       
  2636 FE(,)c(pr)l(o-)0 4289 y(vided)d FG(\031)235 4301 y Fq(1)272
       
  2637 4298 y Fo(\001)310 4289 y FG(a)g Fr(=)g FG(\031)497 4301
       
  2638 y Fq(2)534 4298 y Fo(\001)572 4289 y FG(a)f FE(for)g(all)g(atoms)g
       
  2639 FG(a)p FE(.)125 4536 y FI(T)-6 b(o)25 b(generalise)h(the)g(notion)g(gi)
       
  2640 n(v)o(en)h(in)e(\(1\))h(of)g(a)f(permutation)j(acting)e(on)g(an)g
       
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  2643 (constant,)g(written)h(in\002x)f(as)0 4727 y Fr(\()p
       
  2644 FD(\000)p Fr(\))120 4736 y Fo(\001)158 4727 y Fr(\()p
       
  2645 FD(\000)p Fr(\))p FI(,)g(with)i(the)f(polymorphic)k(type)d
       
  2646 Fs(name)c(prm)g FD(\))f FG(\013)h FD(\))f FG(\013)p FI(.)k(A)g
       
  2647 (de\002nition)i(of)f(the)g(permutation)0 4823 y(operation)j(can)e(then)
       
  2648 h(be)f(gi)n(v)o(en)i(separately)e(for)h(each)g(type-constructor;)h(for)
       
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  2650 eop end
       
  2651 %%Page: 4 4
       
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  2654 b(functions,)i(options)g(and)f(booleans)h(the)g(de\002nitions)f(are)g
       
  2655 (as)g(follo)n(ws:)696 408 y FG(\013)i Fs(list)g Fr(:)350
       
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  2657 y Fn(def)1476 408 y Fr(=)32 b([])1143 527 y FG(\031)1190
       
  2658 536 y Fo(\001)1228 527 y Fr(\()p FG(x)20 b Fr(::)i FG(t)p
       
  2659 Fr(\))1469 485 y Fn(def)1476 527 y Fr(=)32 b(\()p FG(\031)1645
       
  2660 536 y Fo(\001)1683 527 y FG(x)p Fr(\))20 b(::)i(\()p
       
  2661 FG(\031)1918 536 y Fo(\001)1956 527 y FG(t)p Fr(\))696
       
  2662 647 y FG(\013)745 659 y Fq(1)800 647 y FD(\002)17 b FG(\013)926
       
  2663 659 y Fq(2)984 647 y Fr(:)99 b FG(\031)1151 656 y Fo(\001)1188
       
  2664 647 y Fr(\()p FG(x)1262 659 y Fq(1)1299 647 y FG(;)13
       
  2665 b(x)1377 659 y Fq(2)1414 647 y Fr(\))1469 604 y Fn(def)1476
       
  2666 647 y Fr(=)32 b(\()p FG(\031)1645 656 y Fo(\001)1683
       
  2667 647 y FG(x)1727 659 y Fq(1)1763 647 y FG(;)13 b(\031)1844
       
  2668 656 y Fo(\001)1882 647 y FG(x)1926 659 y Fq(2)1963 647
       
  2669 y Fr(\))696 766 y Fs(unit)22 b Fr(:)404 b FG(\031)1346
       
  2670 775 y Fo(\001)1384 766 y Fr(\(\))1469 723 y Fn(def)1476
       
  2671 766 y Fr(=)32 b(\(\))696 886 y FG(\013)22 b Fs(set)g
       
  2672 Fr(:)363 b FG(\031)1337 895 y Fo(\001)1375 886 y FG(X)1469
       
  2673 843 y Fn(def)1476 886 y Fr(=)32 b FD(f)p FG(\031)1653
       
  2674 895 y Fo(\001)1691 886 y FG(x)12 b FD(j)i FG(x)20 b FD(2)i
       
  2675 FG(X)6 b FD(g)696 1005 y FG(\013)745 1017 y Fq(1)804
       
  2676 1005 y FD(\))21 b FG(\013)951 1017 y Fq(2)1010 1005 y
       
  2677 Fr(:)255 b FG(\031)1333 1014 y Fo(\001)1371 1005 y Fk(fn)1469
       
  2678 962 y Fn(def)1476 1005 y Fr(=)32 b FG(\025x:\031)1725
       
  2679 1014 y Fo(\001)1762 1005 y Fr(\()p Fk(fn)19 b Fr(\()p
       
  2680 FG(\031)1955 973 y Fl(\000)p Fq(1)2043 1014 y Fo(\001)2081
       
  2681 1005 y FG(x)p Fr(\)\))696 1124 y FG(\013)j Fs(option)h
       
  2682 Fr(:)134 b FG(\031)1226 1133 y Fo(\001)1264 1124 y Fk(None)1469
       
  2683 1081 y Fn(def)1476 1124 y Fr(=)32 b Fk(None)1070 1244
       
  2684 y FG(\031)1117 1253 y Fo(\001)1155 1244 y Fk(Some)6 b
       
  2685 Fr(\()p FG(x)p Fr(\))1469 1201 y Fn(def)1476 1244 y Fr(=)32
       
  2686 b Fk(Some)6 b Fr(\()p FG(\031)1831 1253 y Fo(\001)1869
       
  2687 1244 y FG(x)p Fr(\))696 1363 y Fs(bool)22 b Fr(:)431
       
  2688 b FG(\031)1373 1372 y Fo(\001)1411 1363 y FG(b)1469 1320
       
  2689 y Fn(def)1476 1363 y Fr(=)32 b FG(b)2789 873 y FI(\(2\))125
       
  2690 1517 y(It)19 b(will)g(sa)n(v)o(e)h(much)h(w)o(ork)g(later)e(on)i(to)f
       
  2691 FE(not)h FI(establish)f(properties)h(for)f(each)g(of)g(these)g
       
  2692 (permutation)0 1613 y(operations)h(indi)n(vidually)-5
       
  2693 b(,)21 b(b)n(ut)f(reason)g(abstractly)g(o)o(v)o(er)g(them)g(by)g
       
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  2697 42 w(\(P)n(ermutation)h(T)-6 b(ype\))21 b FE(A)f(type)g
       
  2698 FG(\013)h FE(will)e(be)h(r)m(eferr)m(ed)g(to)g(as)h FI(permutation)h
       
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  2715 y FI(These)25 b(properties)i(entail)e(that)g(the)h(permutations)h
       
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  2721 y Fl(\000)p Fq(1)308 2859 y Fo(\001)346 2850 y Fr(\()p
       
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  2739 y(an)f(induction)i(o)o(v)o(er)e(the)g(length)h(of)f FG(\031)s
       
  2740 FI(.)f(The)h(second)h(property)h(follo)n(ws)e(from)h(the)f(\002rst.)f
       
  2741 (The)h(third)h(is)e(a)0 3539 y(consequence)25 b(of)e(the)g(\002rst)g
       
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  2744 (sets)f(and)i(apply)g(the)f(third)h(property)-5 b(.)80
       
  2745 b FD(u)-51 b(t)125 3781 y FI(Using)17 b(Isabelle')l(s)g
       
  2746 FE(axiomatic)h(type-classes)g FI([37],)g(it)f(is)g(v)o(ery)h(con)m(v)o
       
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  2751 (\223base\224)g(types,)f(such)h(as)g Fs(name)g FI(and)0
       
  2752 4067 y Fs(unit)q FI(,)21 b(are)g(permutation)i(types)f(and)g(that)f
       
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  2777 end
       
  2778 %%Page: 5 5
       
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  2889 %%Page: 6 6
       
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  3371 b FG(a)g Fr(=)g FG(\031)831 491 y Fl(\000)p Fq(1)920
       
  3372 532 y Fo(\001)958 523 y FG(b)h Fs(then)g Fk(Some)6 b
       
  3373 Fr(\()p FG(t)p Fr(\))457 597 y Fs(else)22 b(if)g FG(\031)782
       
  3374 565 y Fl(\000)p Fq(1)870 606 y Fo(\001)909 597 y FG(b)f
       
  3375 Fr(#)g FG(t)g Fs(then)i Fk(Some)6 b Fr(\(\()p FG(a)25
       
  3376 b(\031)1635 565 y Fl(\000)p Fq(1)1723 606 y Fo(\001)1762
       
  3377 597 y FG(b)p Fr(\))1825 606 y Fo(\001)1862 597 y FG(t)p
       
  3378 Fr(\))c Fs(else)i Fk(None)254 695 y Fr(=)31 b FG(\025b:)13
       
  3379 b Fs(if)21 b FG(\031)603 704 y Fo(\001)641 695 y Fr(\()p
       
  3380 FG(a)g Fr(=)g FG(\031)861 663 y Fl(\000)p Fq(1)950 704
       
  3381 y Fo(\001)988 695 y FG(b)p Fr(\))g Fs(then)i Fk(Some)6
       
  3382 b Fr(\()p FG(\031)1514 704 y Fo(\001)1551 695 y FG(t)p
       
  3383 Fr(\))457 769 y Fs(else)22 b(if)g FG(\031)782 778 y Fo(\001)820
       
  3384 769 y Fr(\()p FG(\031)897 737 y Fl(\000)p Fq(1)985 778
       
  3385 y Fo(\001)1023 769 y FG(b)g Fr(#)f FG(t)p Fr(\))g Fs(then)h
       
  3386 Fk(Some)6 b Fr(\()p FG(\031)1683 778 y Fo(\001)1721 769
       
  3387 y Fr(\()p FG(a)25 b(\031)1864 737 y Fl(\000)p Fq(1)1953
       
  3388 778 y Fo(\001)1991 769 y FG(b)p Fr(\))2054 778 y Fo(\001)2092
       
  3389 769 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506 695 y FI(\()p
       
  3390 FD(\003)2570 663 y Fq(1)2608 695 y FI(\))254 867 y Fr(=)31
       
  3391 b FG(\025b:)13 b Fs(if)21 b FG(\031)603 876 y Fo(\001)641
       
  3392 867 y Fr(\()p FG(a)g Fr(=)g FG(\031)861 835 y Fl(\000)p
       
  3393 Fq(1)950 876 y Fo(\001)988 867 y FG(b)p Fr(\))g Fs(then)i
       
  3394 Fk(Some)6 b Fr(\()p FG(\031)1514 876 y Fo(\001)1551 867
       
  3395 y FG(t)p Fr(\))457 941 y Fs(else)22 b(if)g FG(\031)782
       
  3396 950 y Fo(\001)820 941 y Fr(\()p FG(\031)897 909 y Fl(\000)p
       
  3397 Fq(1)985 950 y Fo(\001)1023 941 y FG(b)g Fr(#)f FG(t)p
       
  3398 Fr(\))g Fs(then)h Fk(Some)6 b Fr(\(\()p FG(\031)1713
       
  3399 950 y Fo(\001)1751 941 y FG(a)25 b(b)p Fr(\))1880 950
       
  3400 y Fo(\001)1918 941 y FG(\031)1965 950 y Fo(\001)2003
       
  3401 941 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506 867 y FI(\()p
       
  3402 FD(\003)2570 835 y Fq(2)2608 867 y FI(\))254 1039 y Fr(=)31
       
  3403 b FG(\025b:)13 b Fs(if)21 b FG(\031)603 1048 y Fo(\001)641
       
  3404 1039 y FG(a)g Fr(=)g FG(b)h Fs(then)g Fk(Some)6 b Fr(\()p
       
  3405 FG(\031)1280 1048 y Fo(\001)1318 1039 y FG(t)p Fr(\))457
       
  3406 1111 y Fs(else)22 b(if)g FG(b)f Fr(#)g FG(\031)921 1120
       
  3407 y Fo(\001)959 1111 y FG(t)g Fs(then)h Fk(Some)6 b Fr(\(\()p
       
  3408 FG(\031)1479 1120 y Fo(\001)1517 1111 y FG(a)25 b(b)p
       
  3409 Fr(\))1646 1120 y Fo(\001)1684 1111 y FG(\031)1731 1120
       
  3410 y Fo(\001)1769 1111 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506
       
  3411 1039 y FI(\()p FD(\003)2570 1007 y Fq(3)2608 1039 y FI(\))247
       
  3412 1187 y Fn(def)254 1230 y Fr(=)31 b([)p FG(\031)413 1239
       
  3413 y Fo(\001)451 1230 y FG(a)p Fr(])p FG(:)p Fr(\()p FG(\031)611
       
  3414 1239 y Fo(\001)649 1230 y FG(t)p Fr(\))0 1395 y FI(where)20
       
  3415 b(we)g(use)g(in)g(\()p FD(\003)594 1364 y Fq(1)632 1395
       
  3416 y FI(\))g(the)g(f)o(act)g(that)608 1563 y FG(\031)655
       
  3417 1572 y Fo(\001)692 1563 y Fs(if)q FG(:::)p Fs(then)r
       
  3418 FG(:::)p Fs(else)r FG(:::)i Fr(=)f Fs(if)13 b FG(\031)1517
       
  3419 1572 y Fo(\001)1555 1563 y FG(:::)p Fs(then)23 b FG(\031)1844
       
  3420 1572 y Fo(\001)1882 1563 y FG(:::)p Fs(else)g FG(\031)2171
       
  3421 1572 y Fo(\001)2209 1563 y FG(:::)477 b FI(\(10\))0 1732
       
  3422 y(and)27 b(in)g Fr(\()p FD(\003)296 1700 y Fq(2)334 1732
       
  3423 y Fr(\))f FI(that)h FG(\031)s Fr(@\()p FG(a)45 b(\031)805
       
  3424 1700 y Fl(\000)p Fq(1)894 1741 y Fo(\001)932 1732 y FG(b)p
       
  3425 Fr(\))33 b FD(\030)g Fr(\()p FG(\031)1198 1741 y Fo(\001)1236
       
  3426 1732 y FG(a)46 b(b)p Fr(\)@)p FG(\031)s FI(;)26 b(for)h
       
  3427 Fr(\()p FD(\003)1727 1700 y Fq(3)1765 1732 y Fr(\))f
       
  3428 FI(the)h(f)o(acts)f(that)h FG(\031)2310 1741 y Fo(\001)2348
       
  3429 1732 y Fr(\()p FG(a)33 b Fr(=)g FG(\031)2592 1700 y Fl(\000)p
       
  3430 Fq(1)2681 1741 y Fo(\001)2719 1732 y FG(b)p Fr(\))26
       
  3431 b FI(if)n(f)0 1827 y FG(\031)47 1836 y Fo(\001)85 1827
       
  3432 y FG(a)i Fr(=)f FG(b)d FI(and)g FG(\031)482 1836 y Fo(\001)520
       
  3433 1827 y Fr(\()p FG(\031)597 1795 y Fl(\000)p Fq(1)685
       
  3434 1836 y Fo(\001)724 1827 y FG(b)j Fr(#)h FG(t)p Fr(\))c
       
  3435 FI(if)n(f)f FG(b)28 b Fr(#)g FG(\031)1253 1836 y Fo(\001)1291
       
  3436 1827 y FG(t)p FI(,)23 b(which)h(can)g(be)g(easily)f(deri)n(v)o(ed)i
       
  3437 (from)g(Lemmas)f(1)p FE(\(ii\))0 1923 y FI(and)d(3)p
       
  3438 FE(\(ii\))f FI(and)h(the)f(permutation)i(operation)f(on)g
       
  3439 Fs(bool)q FI(.)125 2018 y(F)o(or)d(the)h(second)g(property)i(the)d
       
  3440 (case)g FG(a)k Fr(=)f FG(b)d FI(is)g(by)h(a)f(simple)h(calculation)g
       
  3441 (using)g(e)o(xtensionality)h(of)0 2114 y(functions.)h(In)f(case)f
       
  3442 FG(a)i FD(6)p Fr(=)g FG(b)f FI(we)f(sho)n(w)h(\002rst)f(the)h
       
  3443 FD(\))p FI(-direction:)h(the)f(follo)n(wing)h(formula)g(holds)g(then)f
       
  3444 (by)0 2209 y(e)o(xtensionality)h(of)f(functions:)586
       
  3445 2377 y FD(8)p FG(c:)89 b Fs(if)22 b FG(a)f Fr(=)g FG(c)h
       
  3446 Fs(then)g Fk(Some)6 b Fr(\()p FG(t)1492 2389 y Fq(1)1529
       
  3447 2377 y Fr(\))772 2473 y Fs(else)22 b(if)g FG(c)g Fr(#)f
       
  3448 FG(t)1218 2485 y Fq(1)1276 2473 y Fs(then)i Fk(Some)6
       
  3449 b Fr(\(\()p FG(a)12 b(c)p Fr(\))1817 2482 y Fo(\001)1855
       
  3450 2473 y FG(t)1883 2485 y Fq(1)1920 2473 y Fr(\))21 b Fs(else)h
       
  3451 Fk(None)687 2592 y Fr(=)j Fs(if)d FG(b)f Fr(=)g FG(c)h
       
  3452 Fs(then)g Fk(Some)6 b Fr(\()p FG(t)1484 2604 y Fq(2)1521
       
  3453 2592 y Fr(\))772 2687 y Fs(else)22 b(if)g FG(c)g Fr(#)f
       
  3454 FG(t)1218 2699 y Fq(2)1276 2687 y Fs(then)i Fk(Some)6
       
  3455 b Fr(\(\()p FG(b)12 b(c)p Fr(\))1809 2696 y Fo(\001)1847
       
  3456 2687 y FG(t)1875 2699 y Fq(2)1912 2687 y Fr(\))21 b Fs(else)i
       
  3457 Fk(None)0 2851 y FI(Instantiating)e(this)f(formula)h(with)f
       
  3458 FG(a)g FI(yields)g(the)g(equation)537 3019 y Fk(Some)6
       
  3459 b Fr(\()p FG(t)781 3031 y Fq(1)818 3019 y Fr(\))21 b(=)g
       
  3460 Fs(if)h FG(a)f Fr(#)g FG(t)1225 3031 y Fq(2)1283 3019
       
  3461 y Fs(then)h Fk(Some)6 b Fr(\(\()p FG(b)13 b(a)p Fr(\))1824
       
  3462 3028 y Fo(\001)1862 3019 y FG(t)1890 3031 y Fq(2)1927
       
  3463 3019 y Fr(\))21 b Fs(else)h Fk(None)28 b FG(:)0 3187
       
  3464 y FI(Ne)o(xt,)20 b(one)i(distinguishes)f(the)g(cases)f(where)h
       
  3465 FG(a)i Fr(#)g FG(t)1455 3199 y Fq(2)1512 3187 y FI(and)e
       
  3466 FD(:)13 b FG(a)23 b Fr(#)g FG(t)1889 3199 y Fq(2)1926
       
  3467 3187 y FI(,)d(respecti)n(v)o(ely)-5 b(.)21 b(In)g(the)g(\002rst)f
       
  3468 (case,)0 3283 y Fk(Some)6 b Fr(\()p FG(t)244 3295 y Fq(1)281
       
  3469 3283 y Fr(\))22 b(=)f Fk(Some)6 b Fr(\(\()p FG(b)13 b(a)p
       
  3470 Fr(\))777 3292 y Fo(\001)814 3283 y FG(t)842 3295 y Fq(2)879
       
  3471 3283 y Fr(\))p FI(,)20 b(which)h(by)g(Def.)f(2)p Fr(\()p
       
  3472 FG(iii)p Fr(\))h FI(implies)f FG(t)1897 3295 y Fq(1)1956
       
  3473 3283 y Fr(=)h(\()p FG(a)13 b(b)p Fr(\))2184 3292 y Fo(\001)2222
       
  3474 3283 y FG(t)2250 3295 y Fq(2)2307 3283 y FI(since)20
       
  3475 b Fr(\()p FG(a)12 b(b)p Fr(\))d FD(\030)g Fr(\()p FG(b)j(a)p
       
  3476 Fr(\))p FI(;)0 3378 y(and)26 b(ob)o(viously)i FG(a)j
       
  3477 Fr(#)h FG(t)671 3390 y Fq(2)733 3378 y FI(by)26 b(assumption.)h(In)f
       
  3478 (the)g(second)g(case)f Fk(Some)6 b Fr(\()p FG(t)2104
       
  3479 3390 y Fq(1)2141 3378 y Fr(\))31 b(=)h Fk(None)f FI(which)c(gi)n(v)o
       
  3480 (es)0 3474 y(a)h(contradiction.)j(The)e FD(\()p FI(-direction)h(for)f
       
  3481 (the)g(case)f FG(a)37 b FD(6)p Fr(=)g FG(b)28 b FI(is)g(similarly)h(by)
       
  3482 h(e)o(xtensionality)f(and)h(a)0 3569 y(case-analysis.)79
       
  3483 b FD(u)-51 b(t)0 3741 y FI(Note)26 b(that,)f(in)g FE(g)o(ener)o(al)p
       
  3484 FI(,)h(one)h(cannot)f(decide)g(whether)h(tw)o(o)e(functions)i(from)g
       
  3485 Fs(name)f FI(to)f Fs(phi)d(option)0 3836 y FI(are)28
       
  3486 b(equal;)h(ho)n(we)n(v)o(er)g(for)g(the)f(abstraction)h(functions)g
       
  3487 (Lem.)f(6)p FE(\(ii\))h FI(pro)o(vides)g(the)f(means)h(to)f(decide)0
       
  3488 3932 y(whether)g Fr([)p FG(a)p Fr(])p FG(:t)413 3944
       
  3489 y Fq(1)486 3932 y Fr(=)35 b([)p FG(b)p Fr(])p FG(:t)705
       
  3490 3944 y Fq(2)771 3932 y FI(holds:)28 b(one)g(just)f(has)h(to)g(consider)
       
  3491 g(whether)h FG(a)35 b Fr(=)g FG(b)p FI(,)27 b(which)h(is)f(just)g(lik)o
       
  3492 (e)0 4027 y(deciding)f(the)f(alpha-equi)n(v)n(alence)i(of)e(tw)o(o)g
       
  3493 (lambda-terms)h(using)g(the)e(relation)i Fr(\()p FD(\000)p
       
  3494 Fr(\))9 b FD(\031)g Fr(\()p FD(\000)p Fr(\))22 b FI(gi)n(v)o(en)k(in)0
       
  3495 4123 y(Fig.)e(2.)h(No)n(w)g(it)f(is)g(also)h(clear)g(why)f(abstraction)
       
  3496 i(functions)g(represent)g(alpha-equi)n(v)n(alence)h(classes:)0
       
  3497 4218 y(the)19 b(condition)i(we)e(deri)n(v)o(ed)h(for)f(the)h(equality)f
       
  3498 (between)h(abstraction)g(functions)g(paraphrase)g(the)g(rules)0
       
  3499 4314 y FD(\031)60 4326 y Fc(Lam)o Fq(1)209 4314 y FI(and)h
       
  3500 FD(\031)403 4326 y Fc(Lam)o Fq(2)552 4314 y FI(de\002ning)g(alpha-equi)
       
  3501 n(v)n(alence)i(for)e Fs(lam)p FI(.)125 4409 y(The)g(properties)i(in)e
       
  3502 (Lem.)g(6)h(also)f(help)h(us)f(to)g(calculate)g(the)h(support)h(for)e
       
  3503 (abstraction)i(functions,)0 4505 y(pro)o(vided)f(the)o(y)e
       
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  3508 0 TeXcolorgray 56 4823 a(\(i\))p 0 TeXcolorgray 42 w
       
  3509 FG(a)g Fr(#)g([)p FG(b)p Fr(])p FG(:t)f FE(if)g(and)h(only)g(if)e
       
  3510 FG(a)i Fr(#)h FG(t)p FE(,)d(and)p 0 TeXcolorgray 0 TeXcolorgray
       
  3511 eop end
       
  3512 %%Page: 11 11
       
  3513 TeXDict begin 11 10 bop 0 TeXcolorgray 0 TeXcolorgray
       
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  3519 Fr(\))26 b FD(\022)g Fk(supp)6 b Fr(\()p FG(b;)13 b(t)p
       
  3520 Fr(\))21 b FI(because)i(for)g(all)g FG(c)f FI(and)0 585
       
  3521 y FG(d)28 b FI(we)h(ha)n(v)o(e)h FD(f)p FG(d)13 b FD(j)g
       
  3522 Fr(\()p FG(c)g(d)p Fr(\))635 594 y Fo(\001)673 585 y
       
  3523 Fr([)p FG(b)p Fr(])p FG(:t)38 b FD(6)p Fr(=)f([)p FG(b)p
       
  3524 Fr(])p FG(:t)p FD(g)h(\022)f(f)p FG(d)13 b FD(j)g Fr(\()p
       
  3525 FG(c)g(d)p Fr(\))1500 594 y Fo(\001)1538 585 y Fr(\()p
       
  3526 FG(b;)g(t)p Fr(\))37 b FD(6)p Fr(=)g(\()p FG(b;)13 b(t)p
       
  3527 Fr(\))p FD(g)p FI(.)28 b(Since)h FG(b)g FI(and)g FG(t)f
       
  3528 FI(are)h(\002nitely)0 681 y(supported,)c Fr([)p FG(b)p
       
  3529 Fr(])p FG(:t)e FI(must)g(be)h(\002nitely)f(supported.)h(Hence)g
       
  3530 Fr(\()p FG(a;)12 b(b;)h(t;)g Fr([)p FG(b)p Fr(])p FG(:t)p
       
  3531 Fr(\))24 b FI(is)e(\002nitely)h(supported)i(and)f(by)0
       
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  3533 FI(with)g Fr(\()p FD(\003)p Fr(\))g FG(c)h Fr(#)g(\()p
       
  3534 FG(a;)13 b(b;)g(t;)g Fr([)p FG(b)p Fr(])p FG(:t)p Fr(\))p
       
  3535 FI(.)125 878 y(No)n(w)30 b(we)f(sho)n(w)i(the)f(direction)h
       
  3536 FE(\(i)f FD(\))p FE(\))p FI(:)g(using)h(the)f(assumption)h
       
  3537 FG(a)39 b Fr(#)h([)p FG(b)p Fr(])p FG(:t)30 b FI(and)h(the)f(f)o(act)h
       
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  3539 FI(\(from)h FD(\003)p FI(\),)f(Lem.)g(4)g(and)h(6)p FE(\(i\))f
       
  3540 FI(gi)n(v)o(e)h Fr([)p FG(b)p Fr(])p FG(:t)k Fr(=)f(\()p
       
  3541 FG(c)13 b(a)p Fr(\))1684 983 y Fo(\001)1721 974 y Fr([)p
       
  3542 FG(b)p Fr(])p FG(:t)29 b Fr(=)f([\()p FG(c)14 b(a)p Fr(\))2131
       
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  3546 Fr([)p FG(b)p Fr(])p FG(:)p Fr(\(\()p FG(c)13 b(a)p Fr(\))700
       
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  3554 FG(c)37 b Fr(#)g(\()p FG(c)12 b(a)p Fr(\))2793 1174 y
       
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  3566 FI(that)g FG(a)i Fr(#)h([)p FG(b)p Fr(])p FG(:)p Fr(\(\()p
       
  3567 FG(a)13 b(c)p Fr(\))2198 1460 y Fo(\001)2236 1451 y FG(t)p
       
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  3569 1547 y(ha)n(v)o(e)e FG(c)h Fr(#)h FG(t)c FI(and)i(from)g(the)f
       
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  3898 b(t)1712 4289 y Fq(1)1749 4277 y Fr(\)\))p 771 4325 1340
       
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  3910 %%Page: 14 14
       
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  4010 4068 y Fo(\001)660 4059 y FG(t)p Fr(\))12 b FG(c)p FI(.)24
       
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  4017 y Fr(\()p FG(a;)12 b(t)1388 4166 y Fq(1)1425 4154 y Fr(\)\))h
       
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  4020 FG(a;)13 b(t)2170 4166 y Fq(1)2207 4154 y Fr(\))25 b(=)g
       
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  4034 4450 y Fo(\001)395 4441 y FG(a)28 b FI(and)g FG(b)36
       
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  4038 4441 y Fr(\()p FG(b;)13 b Fr(\()p FG(b)38 b(\031)2399
       
  4039 4450 y Fo(\001)2437 4441 y FG(a)p Fr(\))2508 4450 y Fo(\001)2546
       
  4040 4441 y Fr(\()p FG(\031)2623 4450 y Fo(\001)2661 4441
       
  4041 y FG(t)2689 4453 y Fq(1)2726 4441 y Fr(\)\))d(=)0 4536
       
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  4043 4545 y Fo(\001)280 4536 y FG(a;)12 b(\031)401 4545 y
       
  4044 Fo(\001)439 4536 y FG(t)467 4548 y Fq(1)504 4536 y Fr(\))p
       
  4045 FI(.)26 b(From)g(the)g(induction)i(hypothesis,)e(which)h(is)e
       
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  4047 4545 y Fo(\001)2258 4536 y FG(t)2286 4548 y Fq(1)2323
       
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  4050 FG(b)38 b(\031)536 4641 y Fo(\001)574 4632 y FG(a)p Fr(\)@)p
       
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  4052 4644 y Fq(1)884 4632 y Fr(\))12 b FG(c)p FI(.)28 b(Then)h(we)f(can)g
       
  4053 (use)g(the)g(f)o(act)h FG(b)36 b Fr(#)g FG(f)21 b(c)28
       
  4054 b FI(and)h FE(\(iv\))p FI(,)f(and)h(infer)g(that)0 4727
       
  4055 y FG(P)23 b Fr(\()p Fs(Lam)220 4735 y Fp(\013)267 4727
       
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  4058 4736 y Fo(\001)825 4727 y FG(t)853 4739 y Fq(1)890 4727
       
  4059 y Fr(\)\))13 b FG(c)22 b FI(holds.)g(Moreo)o(v)o(er)h(this)f(is)f(by)i
       
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  4064 4823 y Fr(\()p FG(\031)736 4832 y Fo(\001)774 4823 y
       
  4065 FG(t)802 4835 y Fq(1)839 4823 y Fr(\)\)\))12 b FG(c)p
       
  4066 FI(.)20 b(By)f Fr(\()p FD(\003)p Fr(\))h FI(we)g(can)g(conclude)i(with)
       
  4067 e FG(P)j Fr(\()p Fs(Lam)2169 4831 y Fp(\013)2216 4823
       
  4068 y Fr(\()p FG(\031)2293 4832 y Fo(\001)2331 4823 y FG(a;)12
       
  4069 b(\031)2452 4832 y Fo(\001)2490 4823 y FG(t)2518 4835
       
  4070 y Fq(1)2555 4823 y Fr(\)\))h FG(c)p FI(.)79 b FD(u)-51
       
  4071 b(t)p 0 TeXcolorgray 0 TeXcolorgray eop end
       
  4072 %%Page: 15 15
       
  4073 TeXDict begin 15 14 bop 0 TeXcolorgray 0 TeXcolorgray
       
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  4075 a FI(If)22 b(we)f(set)g(in)h(Thm.)g(2)f FG(f)30 b FI(to)22
       
  4076 b(the)f(identity-function)k(and)d(require)h(that)e FG(c)h
       
  4077 FI(has)f(\002nitely)h(supported)h(type,)0 324 y(we)c(can)g(dischar)o
       
  4078 (ge)h(condition)h FE(\(i\))e FI(in)g(and)h(obtain)g(the)g(structural)f
       
  4079 (induction)i(principle)g(stated)d(in)i(\(21\).)0 419
       
  4080 y(The)f(adv)n(antage)i(of)e(\(21\))h(is)e(that)h(Isabelle')l(s)g
       
  4081 (axiomatic)h(type)f(classes)f(can)h(be)g(used)g(to)g(ensure)h(that)e
       
  4082 (the)0 515 y(induction)h(conte)o(xt)f(is)f(a)g(\002nitely)g(supported)i
       
  4083 (type,)f(while)f(the)g(induction)i(principle)g(pro)o(v)o(ed)f(in)g
       
  4084 (Thm.)f(2)0 610 y(requires)23 b(manual)g(reasoning)g(to)f(ensure)h(the)
       
  4085 f(\002nite)g(support)i(property)-5 b(.)23 b(Ho)n(we)n(v)o(er)m(,)g(we)e
       
  4086 (will)h(need)g(the)0 706 y(more)c(general)h(induction)g(principle)g(in)
       
  4087 f(the)g(ne)o(xt)g(section)g(where)g(we)f(deri)n(v)o(e)h(a)g(recursion)h
       
  4088 (combinator)0 801 y(for)i Fs(lam)229 809 y Fp(\013)277
       
  4089 801 y FI(.)0 1082 y FJ(5)28 b(A)20 b(Recursion)f(Combinator)0
       
  4090 1271 y FI(Before)24 b(we)f(can)g(formalise)h(Barendre)o(gt')l(s)g
       
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  4096 (a)f(function.)0 1557 y(This)k(has)h(been)g(done)g(in)g(Urban)g(and)g
       
  4097 (T)-6 b(asson)23 b([36].)g(Ho)n(we)n(v)o(er)m(,)g(this)f(w)o(ay)h(is)e
       
  4098 (rather)j(inele)o(gant.)e(More)0 1653 y(ele)o(gant)e(is)f(a)h
       
  4099 (de\002nition)h(by)g(structural)f(recursion.)125 1748
       
  4100 y(It)32 b(turns)h(out)g(that)g(de\002ning)h(functions)g(by)f(recursion)
       
  4101 h(o)o(v)o(er)f(the)g(structure)g(of)g(alpha-equated)0
       
  4102 1844 y(lambda-terms)18 b(is)e(rather)h(subtle.)g(Let)g(us)f(assume)h
       
  4103 (we)f(w)o(ant)h(to)g(de\002ne)g(capture-a)n(v)n(oiding)k(substitution)0
       
  4104 1939 y(by)g(the)f(follo)n(wing)h(three)g(clauses)p 0
       
  4105 TeXcolorgray 0 TeXcolorgray 469 2096 a Fs(Var)586 2104
       
  4106 y Fp(\013)634 2096 y Fr(\()p FG(x)p Fr(\)[)p FG(y)i Fr(:=)e
       
  4107 FG(t)950 2064 y Fl(0)973 2096 y Fr(])j(=)g(\()p FI(if)d
       
  4108 FG(x)g Fr(=)g FG(y)j FI(then)e FG(t)1594 2064 y Fl(0)1638
       
  4109 2096 y FI(else)f Fs(Var)1900 2104 y Fp(\013)1947 2096
       
  4110 y Fr(\()p FG(x)p Fr(\)\))348 2215 y Fs(App)466 2236 y
       
  4111 Fp(\013)513 2215 y Fr(\()p FG(t)571 2227 y Fq(1)608 2215
       
  4112 y FG(;)13 b(t)670 2227 y Fq(2)707 2215 y Fr(\)[)p FG(y)24
       
  4113 b Fr(:=)d FG(t)950 2184 y Fl(0)973 2215 y Fr(])j(=)g
       
  4114 Fs(App)1219 2236 y Fp(\013)1267 2215 y Fr(\()p FG(t)1325
       
  4115 2227 y Fq(1)1362 2215 y Fr([)p FG(y)g Fr(:=)d FG(t)1575
       
  4116 2184 y Fl(0)1598 2215 y Fr(])p FG(;)14 b(t)1682 2227
       
  4117 y Fq(2)1719 2215 y Fr([)p FG(y)24 b Fr(:=)d FG(t)1932
       
  4118 2184 y Fl(0)1955 2215 y Fr(]\))407 2334 y Fs(Lam)524
       
  4119 2342 y Fp(\013)572 2334 y Fr(\()p FG(x;)12 b(t)p Fr(\)[)p
       
  4120 FG(y)24 b Fr(:=)d FG(t)950 2303 y Fl(0)973 2334 y Fr(])j(=)g
       
  4121 Fs(Lam)1219 2342 y Fp(\013)1267 2334 y Fr(\()p FG(x;)12
       
  4122 b(t)p Fr([)p FG(y)24 b Fr(:=)d FG(t)1615 2303 y Fl(0)1639
       
  4123 2334 y Fr(]\))208 b FI(pro)o(vided)22 b FG(x)f Fr(#)h(\()p
       
  4124 FG(y)s(;)12 b(t)2480 2303 y Fl(0)2503 2334 y Fr(\))0
       
  4125 2490 y FI(where)18 b(the)f(side-condition)j(in)d(the)h(lambda-case)g
       
  4126 (amounts)h(to)e(the)h(usual)g(condition)h(about)f FG(x)j
       
  4127 FD(6)p Fr(=)g FG(y)f FI(and)0 2585 y FG(x)k FI(not)i(being)g(a)f(free)g
       
  4128 (atom)h(in)f FG(t)898 2554 y Fl(0)921 2585 y FI(.)f(Then)i(de\002ning)h
       
  4129 (it)d(o)o(v)o(er)i Fs(lam)1784 2593 y Fp(\013)1857 2585
       
  4130 y FI(results)e(in)h(a)g(total)g(function,)i(while)0 2681
       
  4131 y(de\002ning)e(it)e(o)o(v)o(er)g(\223ra)o(w\224)h(lambda-terms)g(of)g
       
  4132 (type)g Fs(lam)g FI(results)f(in)g(a)h(partial)f(function.)i
       
  4133 (Furthermore,)0 2776 y(attempting)30 b(to)f(de\002ne)g(the)h(functions)
       
  4134 g(that)f(return)h(the)f(set)f(of)h(bound)i(names)e(and)h(the)f
       
  4135 (immediate)0 2872 y(subterms)21 b(by)f(the)g(clauses)516
       
  4136 3042 y FE(bn)q Fr(\()p Fs(Var)742 3050 y Fp(\013)790
       
  4137 3042 y Fr(\()p FG(x)p Fr(\)\))34 b(=)h Fh(?)395 3197
       
  4138 y FE(bn)q Fr(\()p Fs(App)622 3218 y Fp(\013)669 3197
       
  4139 y Fr(\()p FG(t)727 3209 y Fq(1)764 3197 y FG(;)13 b(t)826
       
  4140 3209 y Fq(2)863 3197 y Fr(\)\))35 b(=)g FE(bn)q Fr(\()p
       
  4141 FG(t)1190 3209 y Fq(1)1227 3197 y Fr(\))t FD([)t FI(bn)q
       
  4142 Fr(\()p FG(t)1453 3209 y Fq(2)1490 3197 y Fr(\))454 3351
       
  4143 y FE(bn)q Fr(\()p Fs(Lam)680 3359 y Fp(\013)728 3351
       
  4144 y Fr(\()p FG(x;)12 b(t)p Fr(\)\))35 b(=)g FE(bn)q Fr(\()p
       
  4145 FG(t)p Fr(\))t FD([)t(f)p FG(x)p FD(g)1712 3042 y FE(ist)o
       
  4146 Fr(\()p Fs(Var)1934 3050 y Fp(\013)1981 3042 y Fr(\()p
       
  4147 FG(x)p Fr(\)\))g(=)g Fh(?)1591 3197 y FE(ist)p Fr(\()p
       
  4148 Fs(App)1813 3218 y Fp(\013)1860 3197 y Fr(\()p FG(t)1918
       
  4149 3209 y Fq(1)1955 3197 y FG(;)13 b(t)2017 3209 y Fq(2)2054
       
  4150 3197 y Fr(\)\))36 b(=)f FD(f)p FG(t)2311 3209 y Fq(1)2348
       
  4151 3197 y FG(;)13 b(t)2410 3209 y Fq(2)2447 3197 y FD(g)1650
       
  4152 3351 y FE(ist)o Fr(\()p Fs(Lam)1872 3359 y Fp(\013)1919
       
  4153 3351 y Fr(\()p FG(x;)g(t)p Fr(\)\))35 b(=)g FD(f)p FG(t)p
       
  4154 FD(g)2749 3220 y FI(\(22\))0 3535 y(results)15 b(in)h(an)g
       
  4155 (inconsistenc)o(y)h(when)f(de\002ned)g(o)o(v)o(er)h Fs(lam)1536
       
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  4157 (without)f(problems)0 3631 y(o)o(v)o(er)27 b Fs(lam)q
       
  4158 FI(.)f(The)h(inconsistenc)o(y)g(with)g FE(bn)g FI(and)g
       
  4159 FE(ist)f FI(arises)g(by)h(the)g(principle)g(of)g(HOL)g(stating)f(that)h
       
  4160 (a)0 3726 y(function)22 b(has)d(to)i(return)g(the)f(\223same)f
       
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  4163 3917 y Fr(\()p FG(x;)12 b Fs(Var)1212 3925 y Fp(\013)1259
       
  4164 3917 y Fr(\()p FG(x)p Fr(\)\))21 b(=)g Fs(Lam)1612 3925
       
  4165 y Fp(\013)1660 3917 y Fr(\()p FG(y)s(;)12 b Fs(Var)1882
       
  4166 3925 y Fp(\013)1929 3917 y Fr(\()p FG(y)s Fr(\)\))0 4059
       
  4167 y FI(for)k(all)f FG(x)f FI(and)i FG(y)s FI(,)e(we)h(can)h(assume)f
       
  4168 (that)g(this)g(equation)i(holds)f(for)g FG(x)21 b FD(6)p
       
  4169 Fr(=)g FG(y)s FI(.)14 b(Then)i FE(bn)q Fr(\()p Fs(Lam)2408
       
  4170 4067 y Fp(\013)2456 4059 y Fr(\()p FG(x;)c Fs(Var)2681
       
  4171 4067 y Fp(\013)2728 4059 y Fr(\()p FG(x)p Fr(\)\)\))0
       
  4172 4154 y FI(must)20 b(be)f(equal)h(to)g FE(bn)q Fr(\()p
       
  4173 Fs(Lam)763 4162 y Fp(\013)810 4154 y Fr(\()p FG(y)s(;)13
       
  4174 b Fs(Var)1033 4162 y Fp(\013)1080 4154 y Fr(\()p FG(y)s
       
  4175 Fr(\)\)\))p FI(,)18 b(which)i(implies)f(by)h(the)g(clauses)f(in)g
       
  4176 (\(22\))i(that)e FG(x)g FI(must)g(be)0 4250 y(equal)i(to)f
       
  4177 FG(y)i FI(gi)n(ving)f(a)f(contradiction)i(with)e(the)g(assumption)h
       
  4178 FG(x)g FD(6)p Fr(=)g FG(y)s FI(\227similar)e(with)h(the)g(function)i
       
  4179 FE(ist)o FI(.)125 4345 y(One)16 b(w)o(ay)g(around)i(the)e(problem)i
       
  4180 (with)e(the)g(inconsistencies)h(is)e(to)i(deri)n(v)o(e)f(a)g(recursion)
       
  4181 i(combinator)0 4441 y(for)27 b Fs(lam)236 4449 y Fp(\013)309
       
  4182 4441 y FI(that)f(includes)h(certain)g(preconditions)h(for)f(binders)g
       
  4183 (ensuring)h(no)f(inconsistenc)o(y)g(can)f(be)0 4536 y(deri)n(v)o(ed.)f
       
  4184 (F)o(or)g(this)f(we)g(will)g(adapt)h(w)o(ork)g(by)g(Pitts)f([27])h(who)
       
  4185 g(introduced)i(such)d(preconditions.)j(W)-6 b(e)0 4632
       
  4186 y(will)22 b(also)g(adapt)h(his)f(proof)i(establishing)g(the)e(e)o
       
  4187 (xistence)h(of)f(a)h(structural)g(recursion)g(combinator)i(for)0
       
  4188 4727 y Fs(lam)118 4735 y Fp(\013)165 4727 y FI(.)20 b(The)g(main)h(dif)
       
  4189 n(ference)h(of)e(our)h(proof)h(is)e(that)g(we)g(gi)n(v)o(e)h(here)f(a)g
       
  4190 (direct)h(proof)h(for)f(the)f(e)o(xistence,)0 4823 y(because)31
       
  4191 b(in)g(our)g(implementation)i(we)d(do)i(not)f(use)f(an)o(ywhere)i(the)f
       
  4192 (type)g Fs(lam)g FI(\(Pitts)f(uses)g Fs(lam)i FI(to)p
       
  4193 0 TeXcolorgray 0 TeXcolorgray eop end
       
  4194 %%Page: 16 16
       
  4195 TeXDict begin 16 15 bop 0 TeXcolorgray 0 TeXcolorgray
       
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  4197 b(a)e(structural)i(induction)g(principle\).)g(Another)g(dif)n(ference)g
       
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  4199 y(combinator)c(without)f(deri)n(ving)g(an)g(iteration)f(combinator)i
       
  4200 (\002rst.)1819 292 y Fv(6)125 419 y FI(While)i(in)g(\223e)n(v)o
       
  4201 (ery-day\224)i(formalisation,)f(Lem.)f(4)g(is)f(suf)n(\002cient)i(in)f
       
  4202 (nearly)g(all)g(situations)g(to)g(\002nd)0 515 y(out)19
       
  4203 b(when)f(an)h(object)f(has)g(\002nite)g(support,)h(the)g(reasoning)g
       
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  4205 610 y(se)n(v)o(eral)24 b(places)h(proof)h(obligations)f(about)h
       
  4206 (ensuring)g(that)e(functions)i(ha)n(v)o(e)g(\002nite)e(support.)i(And)f
       
  4207 (for)0 706 y(functions)e(one)e(cannot)i(\002nd)e(out)h(whether)g(the)o
       
  4208 (y)g(ha)n(v)o(e)g(\002nite)f(support)i(by)f(just)f(looking)i(at)e
       
  4209 (their)g(type.)0 801 y(In)f(order)i(to)e(automate)g(such)h(proof)g
       
  4210 (obligations)g(we)f(use)g(the)g(auxiliary)h(notion)g(of)g
       
  4211 FE(supports)f FI([11].)p 0 TeXcolorgray 0 947 a FJ(De\002nition)g(5)p
       
  4212 0 TeXcolorgray 42 w FE(A)31 b(set)h FG(S)j FE(of)d(atoms)g
       
  4213 FI(supports)h FE(an)f FG(x)f FE(of)h(permutation)h(type)o(,)f(written)f
       
  4214 FG(S)17 b Fk(supp)l(orts)22 b FG(x)p FE(,)0 1043 y(pr)l(o)o(vided:)822
       
  4215 1138 y FD(8)12 b FG(a)h(b:)21 b(a)g FD(62)h FG(S)34 b
       
  4216 FD(^)c FG(b)21 b FD(62)h FG(S)46 b FD(\))d Fr(\()p FG(a)12
       
  4217 b(b)p Fr(\))1789 1147 y Fo(\001)1827 1138 y FG(x)21 b
       
  4218 Fr(=)g FG(x)g(:)0 1284 y FI(This)28 b(notion)h(allo)n(ws)e(us)h(to)g
       
  4219 (approximate)i(the)d(support)j(of)e(an)g FG(x)f FI(from)i(\223abo)o(v)o
       
  4220 (e\224,)f(because)g(we)f(can)0 1380 y(sho)n(w)20 b(that:)p
       
  4221 0 TeXcolorgray 0 1526 a FJ(Lemma)f(11)p 0 TeXcolorgray
       
  4222 43 w FE(If)g(a)i(set)e FG(S)24 b FE(is)19 b(\002nite)i(and)g
       
  4223 FG(S)c Fk(supp)l(orts)k FG(x)p FE(,)e(then)h Fk(supp)6
       
  4224 b Fr(\()p FG(x)p Fr(\))21 b FD(\022)g FG(S)t FE(.)p 0
       
  4225 TeXcolorgray 0 1709 a(Pr)l(oof)p 0 TeXcolorgray 40 w
       
  4226 FI(By)k(contradiction)k(we)d(assume)g Fk(supp)5 b Fr(\()p
       
  4227 FG(x)p Fr(\))32 b FD(6\022)g FG(S)t FI(,)26 b(then)h(there)g(e)o(xists)
       
  4228 e(an)h(atom)h FG(a)32 b FD(2)h Fk(supp)6 b Fr(\()p FG(x)p
       
  4229 Fr(\))0 1805 y FI(and)20 b FG(a)h FD(62)h FG(S)t FI(.)d(From)h
       
  4230 FG(S)d Fk(supp)l(orts)k FG(x)e FI(follo)n(ws)h(that)g(for)g(all)f
       
  4231 FG(b)i FD(62)h FG(S)h FI(we)d(ha)n(v)o(e)g Fr(\()p FG(a)12
       
  4232 b(b)p Fr(\))2190 1814 y Fo(\001)2228 1805 y FG(x)21 b
       
  4233 Fr(=)g FG(x)p FI(.)e(Hence)h(the)f(set)0 1900 y FD(f)p
       
  4234 FG(b)j FD(j)f Fr(\()p FG(a)13 b(b)p Fr(\))282 1909 y
       
  4235 Fo(\001)320 1900 y FG(x)20 b FD(6)p Fr(=)i FG(x)p FD(g)16
       
  4236 b FI(is)h(a)g(subset)g(of)h FG(S)t FI(,)f(and)h(since)f
       
  4237 FG(S)k FI(is)16 b(\002nite)h(by)h(assumption,)g(also)f
       
  4238 FD(f)p FG(b)22 b FD(j)g Fr(\()p FG(a)12 b(b)p Fr(\))2615
       
  4239 1909 y Fo(\001)2653 1900 y FG(x)21 b FD(6)p Fr(=)g FG(x)p
       
  4240 FD(g)0 1996 y FI(must)f(be)g(\002nite.)g(But)g(this)f(implies)h(that)g
       
  4241 FG(a)h FD(62)h Fk(supp)5 b Fr(\()p FG(x)p Fr(\))19 b
       
  4242 FI(which)i(gi)n(v)o(es)f(the)g(contradiction.)81 b FD(u)-51
       
  4243 b(t)0 2156 y FI(Lem.)23 b(11)g(gi)n(v)o(es)g(us)f(some)h(means)g(to)f
       
  4244 (decide)i(relati)n(v)o(ely)f(easily)f(whether)h(a)g(function)h(has)f
       
  4245 (\002nite)f(sup-)0 2252 y(port:)j(one)h(only)f(needs)g(to)g(\002nd)g(a)
       
  4246 g(\002nite)f(set)g(of)i(atoms)e(and)i(then)f(v)o(erify)h(whether)f
       
  4247 (this)g(set)f(supports)0 2347 y(the)c(function.)125 2443
       
  4248 y(If)29 b(the)g(function)i(is)e(gi)n(v)o(en)h(as)e(a)h(lambda-term)i
       
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  4250 2538 y(set)d(we)h(use)g(the)g(heuristic)g(of)h(considering)h(the)e
       
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  4252 2634 y(This)20 b(is)e(a)i(heuristic,)f(because)h(it)f(cannot)i(be)f
       
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  4258 0 3006 a FE(Example)h(1)p 0 TeXcolorgray 42 w FI(Gi)n(v)o(en)30
       
  4259 b(a)h(function)h Fk(fn)1068 2963 y Fn(def)1075 3006 y
       
  4260 Fr(=)47 b FG(f)1219 3018 y Fq(1)1270 3006 y FG(c)30 b
       
  4261 FI(where)h FG(f)1593 3018 y Fq(1)1661 3006 y FI(is)f(a)g(function)i(of)
       
  4262 f(type)g Fs(name)41 b FD(\))f FG(\013)p FI(.)31 b(W)-6
       
  4263 b(e)0 3101 y(also)25 b(assume)g(that)g FG(f)589 3113
       
  4264 y Fq(1)652 3101 y FI(has)g(\002nite)g(support.)h(The)g(question)g(is)f
       
  4265 (whether)h Fk(fn)k FI(has)c(\002nite)f(support?)h(The)0
       
  4266 3197 y(free)31 b(v)n(ariables)g(of)g Fk(fn)37 b FI(are)30
       
  4267 b FG(f)829 3209 y Fq(1)897 3197 y FI(and)i FG(c)p FI(.)e(According)j
       
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  4269 3292 y Fk(supp)6 b Fr(\()p FG(f)223 3304 y Fq(1)260 3292
       
  4270 y FG(;)13 b(c)p Fr(\))g Fk(supp)l(orts)21 b(fn)6 b FI(,)19
       
  4271 b(which)i(amounts)g(to)f(sho)n(wing)h(that)518 3483 y
       
  4272 FD(8)p FG(a)13 b(b:)21 b(a)g FD(62)h Fk(supp)5 b Fr(\()p
       
  4273 FG(f)1047 3495 y Fq(1)1085 3483 y FG(;)13 b(c)p Fr(\))30
       
  4274 b FD(^)g FG(b)21 b FD(62)h Fk(supp)5 b Fr(\()p FG(f)1642
       
  4275 3495 y Fq(1)1680 3483 y FG(;)13 b(c)p Fr(\))34 b FD(\))43
       
  4276 b Fr(\()p FG(a)12 b(b)p Fr(\))2077 3492 y Fo(\001)2115
       
  4277 3483 y Fk(fn)27 b Fr(=)21 b Fk(fn)0 3618 y FI(T)-6 b(o)26
       
  4278 b(do)h(so)f(we)g(can)h(assume)f(by)h(the)f(de\002nition)i(of)f
       
  4279 (freshness)f(\(Def.)g(3\))h(that)f FG(a)33 b Fr(#)g(\()p
       
  4280 FG(f)2451 3630 y Fq(1)2488 3618 y FG(;)13 b(c)p Fr(\))26
       
  4281 b FI(and)h FG(b)33 b Fr(#)0 3714 y(\()p FG(f)67 3726
       
  4282 y Fq(1)104 3714 y FG(;)14 b(c)p Fr(\))21 b FI(and)g(sho)n(w)h(that)f
       
  4283 Fr(\()p FG(a)12 b(b)p Fr(\))828 3723 y Fo(\001)866 3714
       
  4284 y Fk(fn)30 b Fr(=)23 b Fk(fn)5 b FI(.)21 b(This)g(equation)i(follo)n
       
  4285 (ws)e(from)i(the)e(calculation)h(that)f(pushes)0 3809
       
  4286 y(the)f(sw)o(apping)h Fr(\()p FG(a)13 b(b)p Fr(\))19
       
  4287 b FI(inside)h Fk(fn)6 b FI(:)p 0 TeXcolorgray 0 TeXcolorgray
       
  4288 460 3989 a Fr(\()p FG(a)12 b(b)p Fr(\))606 3998 y Fo(\001)644
       
  4289 3989 y Fk(fn)738 3946 y Fn(def)745 3989 y Fr(=)28 b(\()p
       
  4290 FG(a)13 b(b)p Fr(\))980 3998 y Fo(\001)1018 3989 y Fr(\()p
       
  4291 FG(f)1085 4001 y Fq(1)1135 3989 y FG(c)p Fr(\))1219 3941
       
  4292 y Fv(by)g(\(3\))1256 3989 y Fr(=)58 b(\(\()p FG(a)12
       
  4293 b(b)p Fr(\))1550 3998 y Fo(\001)1588 3989 y FG(f)1625
       
  4294 4001 y Fq(1)1663 3989 y Fr(\))g(\(\()p FG(a)h(b)p Fr(\))1882
       
  4295 3998 y Fo(\001)1920 3989 y FG(c)p Fr(\))2004 3939 y Fq(\()p
       
  4296 Fl(\003)p Fq(\))2017 3989 y Fr(=)34 b FG(f)2148 4001
       
  4297 y Fq(1)2199 3989 y FG(c)2253 3946 y Fn(def)2260 3989
       
  4298 y Fr(=)29 b Fk(fn)0 4133 y FI(where)23 b Fr(\()p FD(\003)p
       
  4299 Fr(\))f FI(follo)n(ws)h(because)g(we)g(kno)n(w)g(that)g
       
  4300 FG(a)i Fr(#)h FG(f)1508 4145 y Fq(1)1568 4133 y FI(and)d
       
  4301 FG(b)j Fr(#)g FG(f)1890 4145 y Fq(1)1928 4133 y FI(,)c(and)h(therefore)
       
  4302 h(by)f(Lem.)g(4)g(that)0 4228 y Fr(\()p FG(a)12 b(b)p
       
  4303 Fr(\))146 4237 y Fo(\001)184 4228 y FG(f)221 4240 y Fq(1)280
       
  4304 4228 y Fr(=)21 b FG(f)398 4240 y Fq(1)455 4228 y FI(\(similarly)g(for)g
       
  4305 FG(c)p FI(\).)125 4324 y(W)-6 b(e)28 b(can)i(conclude)g(that)f
       
  4306 Fk(supp)6 b Fr(\()p Fk(fn)f Fr(\))29 b FI(is)f(a)h(subset)g(of)g
       
  4307 Fk(supp)6 b Fr(\()p FG(f)1860 4336 y Fq(1)1897 4324 y
       
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  4309 0 4419 y(\(since)21 b FG(f)246 4431 y Fq(1)303 4419 y
       
  4310 FI(has)g(\002nite)f(support)i(by)f(assumption)g(and)g
       
  4311 FG(c)g FI(is)f(\002nitely)g(supported)j(because)d(the)h(type)g
       
  4312 Fs(name)0 4515 y FI(is)e(a)h(\002nitely)g(supported)i(type\).)f(So)f
       
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  4315 4 v 62 4646 a Fu(6)125 4669 y FF(The)17 b(dif)n(ference)j(between)f(a)f
       
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  4321 eop end
       
  4322 %%Page: 17 17
       
  4323 TeXDict begin 17 16 bop 0 TeXcolorgray 0 TeXcolorgray
       
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  4327 b FG(\025x:)12 b FI(if)22 b FG(x)f Fr(=)g FG(y)k FI(then)d
       
  4328 FG(t)1301 204 y Fl(0)1346 236 y FI(else)f Fr(\()p Fs(Var)1637
       
  4329 244 y Fp(\013)1685 236 y Fr(\()p FG(x)p Fr(\)\))p FI(\227where)e
       
  4330 FG(x)g FI(and)i FG(y)i FI(are)d(of)g(type)h Fs(name)0
       
  4331 332 y FI(and)f FG(t)161 300 y Fl(0)202 332 y FI(a)f Fs(lam)374
       
  4332 340 y Fp(\013)421 332 y FI(-term.)g(The)g(free)h(v)n(ariables)f(of)g
       
  4333 (this)g(HOL-function)i(are)e FG(y)i FI(and)f FG(t)2236
       
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  4343 582 y Fr(\()p FG(x)p Fr(\)\))268 658 y Fn(def)275 701
       
  4344 y Fr(=)107 b FG(\025x:)21 b Fr(\()p FG(a)12 b(b)p Fr(\))719
       
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  4346 b(b)p Fr(\))1003 669 y Fl(\000)p Fq(1)1092 710 y Fo(\001)1130
       
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  4596 y Fq(1)497 3547 y FG(;)13 b(f)568 3559 y Fq(2)606 3547
       
  4597 y FG(;)g(f)677 3559 y Fq(3)715 3547 y Fr(\))19 b FI(and)i(the)g
       
  4598 (induction)h(conte)o(xt)e FG(c)h FI(to)f Fs(unit)q FI(.)1895
       
  4599 3515 y Fv(8)1946 3547 y FI(Condition)i FE(\(i\))e FI(of)h(Thm.)g(2)f
       
  4600 (holds)0 3643 y(because)i(by)g(assumption)g FG(f)788
       
  4601 3655 y Fq(1)826 3643 y FI(,)e FG(f)903 3655 y Fq(2)962
       
  4602 3643 y FI(and)i FG(f)1134 3655 y Fq(3)1193 3643 y FI(ha)n(v)o(e)g
       
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  4604 b(then)i(is)0 3738 y(the)j(lambda-case)g(with)g(sho)n(wing)g(that)g
       
  4605 FD(9)p Fr(!)p FG(r)n(:)13 b Fr(\()p Fs(Lam)1414 3746
       
  4606 y Fp(\013)1461 3738 y Fr(\()p FG(a;)g(t)p Fr(\))p FG(;)g(r)r
       
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  4608 3761 y Fj(1)1999 3753 y Fp(f)2031 3761 y Fj(2)2064 3753
       
  4609 y Fp(f)2096 3761 y Fj(3)2157 3738 y FI(holds.)25 b(This)f(is)g(dif)n
       
  4610 (\002cult,)0 3834 y(because)g(for)g(lambdas)h(we)e(do)h(not)h(ha)n(v)o
       
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  4612 (case)h(proceeds)0 3929 y(as)19 b(follo)n(ws.)125 4025
       
  4613 y(The)28 b(induction)i(principle)f(allo)n(ws)f(us)g(to)g(assume)g(that)
       
  4614 g FG(a)36 b Fr(#)g(\()p FG(f)1978 4037 y Fq(1)2015 4025
       
  4615 y FG(;)13 b(f)2086 4037 y Fq(2)2124 4025 y FG(;)g(f)2195
       
  4616 4037 y Fq(3)2232 4025 y Fr(\))p FI(,)28 b(therefore)h(the)f(\223e)o(x-)
       
  4617 0 4120 y(istential\224)f(part)g(of)h(the)f(lemma)h(is)e(immediate.)i
       
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  4620 4216 y Fr(\()p FG(a;)13 b(t)p Fr(\))p FG(;)f(f)505 4228
       
  4621 y Fq(3)556 4216 y FG(a)g(t)h(r)r Fr(\))37 b FD(2)h Fk(r)l(e)l(c)942
       
  4622 4231 y Fp(f)974 4239 y Fj(1)1007 4231 y Fp(f)1039 4239
       
  4623 y Fj(2)1072 4231 y Fp(f)1104 4239 y Fj(3)1170 4216 y
       
  4624 FI(and)29 b(also)g Fr(\()p Fs(Lam)1616 4224 y Fp(\013)1663
       
  4625 4216 y Fr(\()p FG(b;)13 b(t)1788 4184 y Fl(0)1811 4216
       
  4626 y Fr(\))p FG(;)g(f)1912 4228 y Fq(3)1963 4216 y FG(b)f(t)2036
       
  4627 4184 y Fl(0)2072 4216 y FG(r)2109 4184 y Fl(0)2132 4216
       
  4628 y Fr(\))38 b FD(2)g Fk(r)l(e)l(c)2388 4231 y Fp(f)2420
       
  4629 4239 y Fj(1)2453 4231 y Fp(f)2485 4239 y Fj(2)2518 4231
       
  4630 y Fp(f)2550 4239 y Fj(3)2615 4216 y FI(with)29 b(the)0
       
  4631 4311 y(equation)23 b Fs(Lam)411 4319 y Fp(\013)458 4311
       
  4632 y Fr(\()p FG(a;)13 b(t)p Fr(\))24 b(=)h Fs(Lam)847 4319
       
  4633 y Fp(\013)895 4311 y Fr(\()p FG(b;)13 b(t)1020 4279 y
       
  4634 Fl(0)1043 4311 y Fr(\))p FI(,)21 b(then)h FG(f)1308 4323
       
  4635 y Fq(3)1358 4311 y FG(a)13 b(t)f(r)27 b Fr(=)d FG(f)1635
       
  4636 4323 y Fq(3)1686 4311 y FG(b)12 b(t)1759 4279 y Fl(0)1795
       
  4637 4311 y FG(r)1832 4279 y Fl(0)1877 4311 y FI(holds.)22
       
  4638 b(By)g(rule)g(in)m(v)o(ersion)h(we)e(can)0 4406 y(assume)f(that)g
       
  4639 FG(b)i Fr(#)g(\()p FG(f)598 4418 y Fq(1)635 4406 y FG(;)13
       
  4640 b(f)706 4418 y Fq(2)744 4406 y FG(;)g(f)815 4418 y Fq(3)852
       
  4641 4406 y Fr(\))20 b FI(and)h(that)f(there)h(e)o(xists)e(an)h
       
  4642 FG(r)1681 4375 y Fl(0)1724 4406 y FI(such)h(that)f Fr(\()p
       
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  4644 y Fl(0)2202 4406 y Fr(\))22 b FD(2)g Fk(r)l(e)l(c)2426
       
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  4646 y Fj(2)2555 4421 y Fp(f)2587 4429 y Fj(3)2624 4406 y
       
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  4650 4525 y Fj(1)2146 4517 y Fp(f)2178 4525 y Fj(2)2211 4517
       
  4651 y Fp(f)2243 4525 y Fj(3)2280 4502 y FI(.)d(No)n(w)g(we)h(sho)n(w)f(the)
       
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  4653 0 4658 898 4 v 62 4722 a Fu(8)125 4746 y FF(F)o(or)e(this)g(induction)j
       
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  4656 4823 y(tions)d(do)f(not)g(ha)o(v)o(e)h(\002nitely)h(supported)f(type.)p
       
  4657 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end
       
  4658 %%Page: 19 19
       
  4659 TeXDict begin 19 18 bop 0 TeXcolorgray 0 TeXcolorgray
       
  4660 0 71 2881 4 v 2814 17 a FF(19)p 0 TeXcolorgray 0 TeXcolorgray
       
  4661 59 228 a FI(\(i\))p 0 TeXcolorgray 39 w(From)33 b Fr(\()p
       
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  4663 y Fp(f)805 251 y Fj(1)838 243 y Fp(f)870 251 y Fj(2)902
       
  4664 243 y Fp(f)934 251 y Fj(3)1004 228 y FI(and)33 b Fr(\()p
       
  4665 FG(t)1208 197 y Fl(0)1231 228 y FG(;)13 b(r)1302 197
       
  4666 y Fl(0)1325 228 y Fr(\))45 b FD(2)f Fk(r)l(e)l(c)1595
       
  4667 243 y Fp(f)1627 251 y Fj(1)1659 243 y Fp(f)1691 251 y
       
  4668 Fj(2)1724 243 y Fp(f)1756 251 y Fj(3)1826 228 y FI(we)32
       
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  4670 y FI(and)25 b FG(r)347 292 y Fl(0)395 324 y FI(are)g(\002nitely)g
       
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  4672 (a)f FG(c)g FI(with)f FG(c)31 b Fr(#)172 419 y(\()p FG(f)239
       
  4673 431 y Fq(1)276 419 y FG(;)13 b(f)347 431 y Fq(2)385 419
       
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  4680 515 y Fr(\()p FG(a;)13 b(t)p Fr(\))23 b FI(and)i FG(b)k
       
  4681 Fr(#)f Fs(Lam)1872 523 y Fp(\013)1919 515 y Fr(\()p FG(b;)13
       
  4682 b(t)2044 483 y Fl(0)2067 515 y Fr(\))p FI(.)24 b(W)m(ith)g(\(i\))g(we)g
       
  4683 (can)g(further)172 610 y(infer)30 b(that)g FG(c)39 b
       
  4684 Fr(#)g Fs(Lam)791 618 y Fp(\013)838 610 y Fr(\()p FG(a;)13
       
  4685 b(t)p Fr(\))29 b FI(and)h FG(c)39 b Fr(#)g Fs(Lam)1466
       
  4686 618 y Fp(\013)1513 610 y Fr(\()p FG(b;)13 b(t)1638 579
       
  4687 y Fl(0)1661 610 y Fr(\))p FI(.)29 b(From)h(the)g(assumption)h
       
  4688 Fs(Lam)2573 618 y Fp(\013)2620 610 y Fr(\()p FG(a;)13
       
  4689 b(t)p Fr(\))38 b(=)172 706 y Fs(Lam)289 714 y Fp(\013)337
       
  4690 706 y Fr(\()p FG(b;)13 b(t)462 674 y Fl(0)485 706 y Fr(\))p
       
  4691 FI(,)28 b(we)h(can)g(then)h(use)e(Lem.)i(4)f(to)g(deri)n(v)o(e)h
       
  4692 Fr(\()p FG(a)12 b(c)p Fr(\))1841 715 y Fo(\001)1879 706
       
  4693 y Fs(Lam)1997 714 y Fp(\013)2044 706 y Fr(\()p FG(a;)h(t)p
       
  4694 Fr(\))37 b(=)g(\()p FG(b)13 b(c)p Fr(\))2480 715 y Fo(\001)2518
       
  4695 706 y Fs(Lam)2636 714 y Fp(\013)2683 706 y Fr(\()p FG(b;)g(t)2808
       
  4696 674 y Fl(0)2831 706 y Fr(\))p FI(,)172 801 y(which)24
       
  4697 b(implies)f(that)g Fs(Lam)902 809 y Fp(\013)949 801 y
       
  4698 Fr(\()p FG(c;)14 b Fr(\()p FG(a)e(c)p Fr(\))1193 810
       
  4699 y Fo(\001)1231 801 y FG(t)p Fr(\))27 b(=)h Fs(Lam)1521
       
  4700 809 y Fp(\013)1569 801 y Fr(\()p FG(c;)13 b Fr(\()p FG(a)f(c)p
       
  4701 Fr(\))1812 810 y Fo(\001)1850 801 y FG(t)1878 770 y Fl(0)1901
       
  4702 801 y Fr(\))p FI(;)23 b(hence)h(by)g(\(18\))h(that)e
       
  4703 Fr(\()p FG(a)13 b(c)p Fr(\))2728 810 y Fo(\001)2766 801
       
  4704 y FG(t)27 b Fr(=)172 897 y(\()p FG(b)12 b(c)p Fr(\))310
       
  4705 906 y Fo(\001)349 897 y FG(t)377 865 y Fl(0)400 897 y
       
  4706 FI(.)p 0 TeXcolorgray 15 992 a(\(iii\))p 0 TeXcolorgray
       
  4707 39 w(From)22 b Fr(\()p FG(t;)13 b(r)r Fr(\))24 b FD(2)h
       
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  4709 y Fp(f)819 1015 y Fj(2)852 1007 y Fp(f)884 1015 y Fj(3)921
       
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  4711 FG(;)13 b(r)1114 960 y Fl(0)1137 992 y Fr(\))25 b FD(2)g
       
  4712 Fk(r)l(e)l(c)1367 1007 y Fp(f)1399 1015 y Fj(1)1432 1007
       
  4713 y Fp(f)1464 1015 y Fj(2)1497 1007 y Fp(f)1529 1015 y
       
  4714 Fj(3)1587 992 y FG(a)f Fr(#)h(\()p FG(f)1808 1004 y Fq(1)1846
       
  4715 992 y FG(;)13 b(f)1917 1004 y Fq(2)1954 992 y FG(;)g(f)2025
       
  4716 1004 y Fq(3)2063 992 y Fr(\))21 b FI(and)i FG(b)h Fr(#)h(\()p
       
  4717 FG(f)2463 1004 y Fq(1)2500 992 y FG(;)14 b(f)2572 1004
       
  4718 y Fq(2)2609 992 y FG(;)f(f)2680 1004 y Fq(3)2718 992
       
  4719 y Fr(\))p FI(,)21 b(we)172 1088 y(can)f(infer)g(by)g(Lem.)f(4)h(and)g
       
  4720 (13)g(that)g Fr(\(\()p FG(a)12 b(c)p Fr(\))1354 1097
       
  4721 y Fo(\001)1393 1088 y FG(t;)g Fr(\()p FG(a)h(c)p Fr(\))1601
       
  4722 1097 y Fo(\001)1639 1088 y FG(r)r Fr(\))21 b FD(2)h Fk(r)l(e)l(c)1899
       
  4723 1103 y Fp(f)1931 1111 y Fj(1)1963 1103 y Fp(f)1995 1111
       
  4724 y Fj(2)2028 1103 y Fp(f)2060 1111 y Fj(3)2116 1088 y
       
  4725 FI(and)f Fr(\(\()p FG(b)12 b(c)p Fr(\))2418 1097 y Fo(\001)2456
       
  4726 1088 y FG(t)2484 1056 y Fl(0)2507 1088 y FG(;)i Fr(\()p
       
  4727 FG(b)e(c)p Fr(\))2680 1097 y Fo(\001)2718 1088 y FG(r)2755
       
  4728 1056 y Fl(0)2778 1088 y Fr(\))22 b FD(2)172 1183 y Fk(r)l(e)l(c)271
       
  4729 1198 y Fp(f)303 1206 y Fj(1)336 1198 y Fp(f)368 1206
       
  4730 y Fj(2)401 1198 y Fp(f)433 1206 y Fj(3)469 1183 y FI(.)h(Since)h(by)g
       
  4731 (induction)h(hypothesis)f FD(9)p Fr(!)p FG(r)n(:)12 b
       
  4732 Fr(\()p FG(t;)h(r)r Fr(\))27 b FD(2)h Fk(r)l(e)l(c)1988
       
  4733 1198 y Fp(f)2020 1206 y Fj(1)2053 1198 y Fp(f)2085 1206
       
  4734 y Fj(2)2118 1198 y Fp(f)2150 1206 y Fj(3)2210 1183 y
       
  4735 FI(we)23 b(also)g(ha)n(v)o(e)h(the)f(f)o(act)172 1279
       
  4736 y(that)d FD(9)p Fr(!)p FG(r)n(:)12 b Fr(\(\()p FG(a)h(c)p
       
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  4738 b FD(2)g Fk(r)l(e)l(c)977 1294 y Fp(f)1009 1302 y Fj(1)1041
       
  4739 1294 y Fp(f)1073 1302 y Fj(2)1106 1294 y Fp(f)1138 1302
       
  4740 y Fj(3)1175 1279 y FI(.)e(Thus)i(we)f(can)g(use)g(\(ii\))g(to)g(infer)g
       
  4741 (that)g Fr(\()p FG(a)13 b(c)p Fr(\))2408 1288 y Fo(\001)2446
       
  4742 1279 y FG(r)23 b Fr(=)e(\()p FG(b)13 b(c)p Fr(\))2724
       
  4743 1288 y Fo(\001)2762 1279 y FG(r)2799 1247 y Fl(0)2822
       
  4744 1279 y FI(.)p 0 TeXcolorgray 21 1374 a(\(i)n(v\))p 0
       
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  4747 b Fr(#)g FG(f)1689 1386 y Fq(3)1752 1374 y FI(and)27
       
  4748 b FG(b)32 b Fr(#)g FG(f)2090 1386 y Fq(3)2153 1374 y
       
  4749 FI(as)25 b(well)h(as)f FG(r)i FI(and)g FG(r)2736 1342
       
  4750 y Fl(0)2785 1374 y FI(are)172 1470 y(\002nitely)20 b(supported)i
       
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  4752 Fr(#)g FG(f)1817 1482 y Fq(3)1867 1470 y FG(a)13 b(t)f(r)22
       
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  4754 y FG(b)13 b(t)2452 1438 y Fl(0)2488 1470 y FG(r)2525
       
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  4757 b Fr(\()p FG(f)589 1577 y Fq(3)626 1565 y FG(;)13 b(a;)g(t;)g(r)r
       
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  4759 1565 y FG(a)13 b(t)f(r)r Fr(\))19 b FI(and)i(since)e
       
  4760 FG(c)i Fr(#)h(\()p FG(f)1964 1577 y Fq(3)2001 1565 y
       
  4761 FG(;)13 b(a;)g(t;)g(r)r Fr(\))19 b FI(\(from)i(\(i\)\),)e(we)h(kno)n(w)
       
  4762 172 1661 y(by)g(Lem.)g(11)h(that)f FG(c)i Fr(#)f FG(f)868
       
  4763 1673 y Fq(3)918 1661 y FG(a)13 b(t)f(r)22 b FI(holds.)e(Similarly)g(we)
       
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  4765 Fq(3)2367 1661 y FG(b)13 b(t)2441 1629 y Fl(0)2477 1661
       
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  4770 1768 y Fq(3)1551 1756 y FG(b)13 b(t)1625 1724 y Fl(0)1661
       
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  4773 1852 y(f)o(acts)19 b(deri)n(v)o(ed)i(in)f(\(i)n(v\))h(and)f(\(v\))h(to)
       
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  4775 y Fo(\001)1637 1852 y Fr(\()p FG(f)1704 1864 y Fq(3)1754
       
  4776 1852 y FG(a)h(t)f(r)r Fr(\))21 b(=)g(\()p FG(b)13 b(c)p
       
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  4781 FG(f)1139 1959 y Fq(3)1190 1947 y FG(c)13 b Fr(\(\()p
       
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  4785 1947 y FG(c)13 b Fr(\(\()p FG(b)f(c)p Fr(\))2206 1956
       
  4786 y Fo(\001)2244 1947 y FG(t)2272 1915 y Fl(0)2295 1947
       
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  4789 b(By)h(the)172 2042 y(f)o(acts)h(deri)n(v)o(ed)h(in)f(\(ii\))g(and)g
       
  4790 (\(iii\))h(we)e(ha)n(v)o(e)i(that)f(these)g(terms)g(are)g(indeed)h
       
  4791 (equal.)79 b FD(u)-51 b(t)0 2189 y FI(T)-6 b(o)18 b(pro)o(v)o(e)h(our)g
       
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  5735 %%Trailer
       
  5736 
       
  5737 userdict /end-hook known{end-hook}if
       
  5738 %%EOF