Publications/merlin-05.ps
author Christian Urban <christian dot urban at kcl dot ac dot uk>
Fri, 10 Jul 2015 01:44:23 +0100
changeset 372 de46e57b7603
parent 14 680070975206
permissions -rw-r--r--
updated
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setmatrix } def
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% DG modification begin - Jan. 15, 1997
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%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
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%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
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%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
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%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
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%setmatrix } def
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/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
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CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
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div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
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d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
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% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
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% setmatrix } def
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setmatrix pop } def
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% DG/SR modification end
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/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
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def } def
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/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
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def } def
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/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
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def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
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mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
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abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
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/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
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m mul neg def } def
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/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
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ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
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def
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/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
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/NC { CC x1 y1 x2 y2 x y curveto } def
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/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
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/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
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/NAC { x2 y2 x y curveto CC x1 y1 } def
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/EAC { x2 y2 x y ArrowB curveto pop pop } def
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/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
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    n { NC } repeat EOC } ifelse } def
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/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
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/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
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pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
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/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
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CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
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repeat closepath pop pop } ifelse } def
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/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
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/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
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/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
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def
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/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
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rotate matrix concatmatrix exch findfont exch makefont setfont } def
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/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
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x1 y1 lineto closepath } def
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/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
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y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
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exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
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/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
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x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
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ifelse } def
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/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
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/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
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pop pop Rect } { OvalFrame } ifelse } def
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/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
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} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
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aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
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/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
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2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
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ifelse } def
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/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
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6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
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/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
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sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
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def
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/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
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sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
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x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
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y0 x0 2 mul x1 sub y1 ] def } def
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/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
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if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
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/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
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round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
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round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
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0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
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/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
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if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
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{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
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exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
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div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
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dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
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dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
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pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
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stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
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{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
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w add def } repeat grestore gsave n 0 gt
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% DG/SR modification begin - Nov. 7, 1997 - Patch 1
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%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
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{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
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% DG/SR modification end
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{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
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n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
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div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
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1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
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dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
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{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
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exch t L stroke /i i h add def } repeat grestore } def
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/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
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1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
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exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
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mul neg d } def
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/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
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/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
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/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
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def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
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dup a add ] cvx def } def
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/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
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cvx def } def
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/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
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/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
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/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
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/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
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abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
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div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
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h1 abs h sub dup s mul abs } ifelse } def
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/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
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mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
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{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
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exch } def
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/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
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/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
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} if } ifelse } def
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/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
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/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
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neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
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/IfVisible true def } def
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end
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% END pstricks.pro
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%%EndProcSet
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%%BeginProcSet: pst-dots.pro 0 0
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%!PS-Adobe-2.0
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%%Title: Dot Font for PSTricks
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%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
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%%Creation Date: May 7, 1993
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%% Version 97 patch 1, 99/12/16
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%% Modified by Etienne Riga <etienne.riga@skynet.be> - Dec. 16, 1999
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%% to add /Diamond, /SolidDiamond and /BoldDiamond
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10 dict dup begin
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  /FontType 3 def
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  /FontMatrix [ .001 0 0 .001 0 0 ] def
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  /FontBBox [ 0 0 0 0 ] def
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  /Encoding 256 array def
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  0 1 255 { Encoding exch /.notdef put } for
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  Encoding
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    dup (b) 0 get /Bullet put
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    dup (c) 0 get /Circle put
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    dup (C) 0 get /BoldCircle put
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    dup (u) 0 get /SolidTriangle put
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    dup (t) 0 get /Triangle put
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    dup (T) 0 get /BoldTriangle put
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    dup (r) 0 get /SolidSquare put
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    dup (s) 0 get /Square put
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    dup (S) 0 get /BoldSquare put
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    dup (q) 0 get /SolidPentagon put
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    dup (p) 0 get /Pentagon put
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    dup (P) 0 get /BoldPentagon put
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% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
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    dup (l) 0 get /SolidDiamond put
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    dup (d) 0 get /Diamond put
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        (D) 0 get /BoldDiamond put
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% DG/SR modification end
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  /Metrics 13 dict def
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  Metrics begin
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    /Bullet        1000   def
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    /Circle        1000   def
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    /BoldCircle    1000   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   338
    /SolidTriangle 1344   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   339
    /Triangle      1344   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   340
    /BoldTriangle  1344   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   341
    /SolidSquare    886   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   342
    /Square         886   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   343
    /BoldSquare     886   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   344
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   345
    /Pentagon      1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   346
    /BoldPentagon  1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   347
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   348
    /SolidDiamond  1008   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   349
    /Diamond       1008   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   350
    /BoldDiamond   1008   def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   351
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   352
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   353
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   354
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   355
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   356
    /Circle        { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   357
    /BoldCircle    /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   358
    /Bullet        /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   359
    /Triangle      { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   360
    /BoldTriangle  /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   361
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   362
    /Square        { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   363
    /BoldSquare    /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   364
    /SolidSquare   /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   365
    /Pentagon      { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   366
    /BoldPentagon  /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   367
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   368
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   369
    /Diamond       { -428.5 -742.5 428.5 742.5 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   370
    /BoldDiamond   /Diamond load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   371
    /SolidDiamond  /Diamond load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   372
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   373
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   374
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   375
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   376
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   377
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   378
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   379
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   380
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   381
    /CirclePath    { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   382
    /Bullet        { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   383
    /Circle        { 500 500 Adjust CirclePath .9 .9 scale CirclePath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   384
                     eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   385
    /BoldCircle    { 500 500 Adjust CirclePath .8 .8 scale CirclePath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   386
                     eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   387
    /BoldCircle    { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   388
    /TrianglePath  { 0  660 moveto -571.5 -330 lineto 571.5 -330 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   389
                     closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   390
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   391
    /Triangle      { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   392
    /BoldTriangle  { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   393
    /SquarePath    { -450 450 moveto 450 450 lineto 450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   394
                     -450 -450 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   395
    /SolidSquare   { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   396
    /Square        { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   397
    /BoldSquare    { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   398
    /PentagonPath  {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   399
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   400
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   401
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   402
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   403
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   404
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   405
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   406
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   407
    /Pentagon      { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   408
    /BoldPentagon  { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   409
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   410
    /DiamondPath   { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   411
                     428.5 0 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   412
    /SolidDiamond  { DiamondPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   413
    /Diamond       { DiamondPath .85 .85 scale DiamondPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   414
    /BoldDiamond   { DiamondPath .7 .7 scale DiamondPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   415
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   416
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   417
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   418
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   419
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   420
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   421
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   422
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   423
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   424
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   425
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   426
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   427
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   428
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   429
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   430
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   431
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   432
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   433
%END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   435
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   436
%%BeginProcSet: 8r.enc 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   437
% File 8r.enc  TeX Base 1 Encoding  Revision 2.0  2002-10-30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   438
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   439
% @@psencodingfile@{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   440
%   author    = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   441
%                W. Schmidt, P. Lehman",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   442
%   version   = "2.0",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   443
%   date      = "30 October 2002",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   444
%   filename  = "8r.enc",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   445
%   email     = "tex-fonts@@tug.org",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   446
%   docstring = "This is the encoding vector for Type1 and TrueType
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   447
%                fonts to be used with TeX.  This file is part of the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   448
%                PSNFSS bundle, version 9"
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   449
% @}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   450
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   451
% The idea is to have all the characters normally included in Type 1 fonts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   452
% available for typesetting. This is effectively the characters in Adobe
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   453
% Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   454
% MacRoman, and some extra characters from Lucida.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   455
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   456
% Character code assignments were made as follows:
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   457
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   458
% (1) the Windows ANSI characters are almost all in their Windows ANSI
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   459
% positions, because some Windows users cannot easily reencode the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   460
% fonts, and it makes no difference on other systems. The only Windows
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   461
% ANSI characters not available are those that make no sense for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   462
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   463
% (173). quotesingle and grave are moved just because it's such an
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   464
% irritation not having them in TeX positions.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   465
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   466
% (2) Remaining characters are assigned arbitrarily to the lower part
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   467
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   468
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   469
% (3) Y&Y Lucida Bright includes some extra text characters; in the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   470
% hopes that other PostScript fonts, perhaps created for public
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   471
% consumption, will include them, they are included starting at 0x12.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   472
% These are /dotlessj /ff /ffi /ffl.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   473
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   474
% (4) hyphen appears twice for compatibility with both ASCII and Windows.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   475
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   476
% (5) /Euro was assigned to 128, as in Windows ANSI
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   477
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   478
% (6) Missing characters from MacRoman encoding incorporated as follows:
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   479
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   480
%     PostScript      MacRoman        TeXBase1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   481
%     --------------  --------------  --------------
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   482
%     /notequal       173             0x16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   483
%     /infinity       176             0x17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   484
%     /lessequal      178             0x18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   485
%     /greaterequal   179             0x19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   486
%     /partialdiff    182             0x1A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   487
%     /summation      183             0x1B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   488
%     /product        184             0x1C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   489
%     /pi             185             0x1D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   490
%     /integral       186             0x81
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   491
%     /Omega          189             0x8D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   492
%     /radical        195             0x8E
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   493
%     /approxequal    197             0x8F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   494
%     /Delta          198             0x9D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   495
%     /lozenge        215             0x9E
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   496
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   497
/TeXBase1Encoding [
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   498
% 0x00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   499
 /.notdef /dotaccent /fi /fl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   500
 /fraction /hungarumlaut /Lslash /lslash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   501
 /ogonek /ring /.notdef /breve
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   502
 /minus /.notdef /Zcaron /zcaron
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   503
% 0x10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   504
 /caron /dotlessi /dotlessj /ff
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   505
 /ffi /ffl /notequal /infinity
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   506
 /lessequal /greaterequal /partialdiff /summation
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   507
 /product /pi /grave /quotesingle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   508
% 0x20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   509
 /space /exclam /quotedbl /numbersign
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   510
 /dollar /percent /ampersand /quoteright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   511
 /parenleft /parenright /asterisk /plus
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   512
 /comma /hyphen /period /slash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   513
% 0x30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   514
 /zero /one /two /three
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   515
 /four /five /six /seven
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   516
 /eight /nine /colon /semicolon
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   517
 /less /equal /greater /question
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   518
% 0x40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   519
 /at /A /B /C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   520
 /D /E /F /G
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   521
 /H /I /J /K
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   522
 /L /M /N /O
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   523
% 0x50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   524
 /P /Q /R /S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   525
 /T /U /V /W
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   526
 /X /Y /Z /bracketleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   527
 /backslash /bracketright /asciicircum /underscore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   528
% 0x60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   529
 /quoteleft /a /b /c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   530
 /d /e /f /g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   531
 /h /i /j /k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   532
 /l /m /n /o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   533
% 0x70
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   534
 /p /q /r /s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   535
 /t /u /v /w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   536
 /x /y /z /braceleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   537
 /bar /braceright /asciitilde /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   538
% 0x80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   539
 /Euro /integral /quotesinglbase /florin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   540
 /quotedblbase /ellipsis /dagger /daggerdbl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   541
 /circumflex /perthousand /Scaron /guilsinglleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   542
 /OE /Omega /radical /approxequal
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(entions)f(about)h(those)h(bound)f(v)n(ariables.)g(F)o(or)h(e)o(xam-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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-150 3887 y(ple,)g(in)g(Barendre)o(gt')-5 b(s)19 b(seminal)h(book)f([2)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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o(])h(about)g(the)g Fz(\025)p FC(-calculus:)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1735
0 TeXcolorgray -40 4067 a(2)t(.)t(1)t(.)t(1)t(2)t(.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(C)t Fw(O)t(N)t(V)t(E)t(N)t(T)t(I)t(O)t(N)r FC(.)f(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(erms)32 b(that)h(are)g Fz(\013)p FC(-congruent)-42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1738
4166 y(are)15 b(identi\002ed.)f(So)i(no)n(w)e(we)i(write)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1739
Fz(\025x:x)25 b Fv(\021)d Fz(\025y)s(:y)s FC(,)16 b(etcetera.)-40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4338 y(2)t(.)t(1)t(.)t(1)t(3)t(.)42 b(V)-6 b Fw(A)t(R)t(I)t(A)t(B)t(L)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1741
(E)43 b FC(C)t Fw(O)t(N)t(V)t(E)t(N)t(T)t(I)t(O)t(N)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1742
FC(.)d(If)f Fz(M)1388 4350 y Fu(1)1424 4338 y Fz(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1743
b(:)g(:)g(:)g(;)g(M)1690 4350 y Ft(n)-42 4438 y FC(occur)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1744
b(in)i(a)g(certain)f(mathematical)f(conte)o(xt)g(\(e.g.)h(de\002ni-)-42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1745
4537 y(tion,)h(proof\),)e(then)i(in)g(these)h(terms)f(all)h(bound)e(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1746
(ariables)-42 4637 y(are)20 b(chosen)f(to)i(be)f(dif)n(ferent)e(from)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(the)g(free)f(v)n(ariables.)p 0 TeXcolorgray -150 5002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1748
a Fs(Permission)g(to)h(mak)o(e)f(digital)h(or)f(hard)h(copies)e(of)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1749
(all)f(or)g(part)h(of)f(this)g(w)o(ork)h(for)g(personal)f(or)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1750
5068 y(classroom)d(use)f(is)h(granted)h(without)f(fee)g(pro)o(vided)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1751
(that)g(copies)e(are)h(not)h(made)f(or)g(distrib)o(uted)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1752
5135 y(for)e(pro\002t)g(or)g(commercial)g(adv)o(antage)e(and)h(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1753
(copies)f(bear)g(this)g(notice)h(and)f(the)g(full)h(citation)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1754
5201 y(on)i(the)g(\002rst)g(page.)f(T)-5 b(o)16 b(cop)o(y)h(otherwise,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1755
e(to)h(republish,)g(to)g(post)g(on)g(serv)o(ers)g(or)g(to)g(redistrib)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1756
(ute)-150 5268 y(to)f(lists,)e(requires)i(prior)h(speci\002c)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1757
(permission)g(and/or)h(a)g(fee.)-150 5351 y Fr(MERLIN'05)64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1758
b Fs(September)15 b(30,)f(2005,)h(T)-5 b(allinn,)14 b(Estonia.)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1759
5417 y(Cop)o(yright)120 5415 y(c)100 5417 y Fq(\015)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1760
Fs(2005)h(A)n(CM)f(1-59593-072-8/05/0009.)9 b(.)g(.)g($5.00.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 2141 1610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a FC(Both)23 b(con)m(v)o(entions)d(gi)n(v)o(e)i(rise)h(to)g(v)o(ery)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1763
(slick)h(informal)e(proofs:)2042 1709 y(the)30 b(\002rst)h(con)m(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ention)d(assumes)i(that)h(the)f(\223data)g(structure\224)f(o)o(v)o(er)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1765
2042 1809 y(which)j(the)g(proofs)g(are)g(done)g(is)h(not)g(that)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1766
(syntax-trees,)e(b)n(ut)2042 1909 y(of)38 b Fz(\013)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1767
FC(-equi)n(v)n(alent)f(lambda-terms)f(\(or)i Fz(\013)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1768
FC(-equi)n(v)n(alence)e(classes\).)2042 2008 y(Ho)n(we)n(v)o(er)m(,)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1769
b(the)i(claim)g(to)g(be)g(using)g Fz(\013)p FC(-equi)n(v)n(alence)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1770
(classes,)j(rather)2042 2108 y(than)g(syntax-trees,)g(is)i(often)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1771
(blurred)e(by)i(statements)g(lik)o(e)h([2)o(]:)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1772
0 TeXcolorgray 2152 2257 a(2)t(.)t(1)t(.)t(1)t(4)t(.)d(M)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1773
Fw(O)t(R)t(A)t(L)r FC(.)e(Using)f(con)m(v)o(entions)d(2.1.12)h(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1774
(2.1.13)2150 2357 y(one)k(can)h(w)o(ork)g(with)g Fz(\025)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1775
FC(-terms)h(in)f(the)g(nai)n(v)o(e)f(w)o(ay)-5 b(.)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1776
2505 y(One)18 b(adv)n(antage)d(of)i(using)g Fz(\013)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1777
FC(-equi)n(v)n(alence)f(classes)i(is)h(that)e(cap-)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1778
2605 y(ture)23 b(a)n(v)n(oiding)g(substitution)g(can)h(be)g(de\002ned)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1779
(as)h(a)h(total)f(function)2042 2705 y(satisfying)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1780
b(the)i(follo)n(wing)d(four)h(properties)g([6)o(,)i(13)o(]:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 0 TeXcolorgray 2042 2927 a Fv(\017)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1782
b Fp(var)9 b Fo(\()p Fz(a)p Fo(\)[)p Fz(a)24 b Fo(:=)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1783
Fz(N)9 b Fo(])23 b(=)g Fz(N)2042 3051 y Fv(\017)47 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1784
Fp(var)9 b Fo(\()p Fz(b)p Fo(\)[)p Fz(a)23 b Fo(:=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1785
Fz(N)9 b Fo(])23 b(=)g Fp(var)9 b Fo(\()p Fz(b)p Fo(\))104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1786
b FC(pro)o(vided)18 b Fz(b)k Fv(6)p Fo(=)h Fz(a)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1787
3174 y Fv(\017)47 b Fp(app)6 b Fo(\()p Fz(M)2376 3186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1788
y Fu(1)2413 3174 y Fz(;)14 b(M)2531 3186 y Fu(2)2568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1789
3174 y Fo(\)[)p Fz(a)23 b Fo(:=)g Fz(N)9 b Fo(])23 b(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1790
Fp(app)7 b Fo(\()p Fz(M)3256 3186 y Fu(1)3293 3174 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1791
Fo([)p Fz(a)23 b Fo(:=)g Fz(N)9 b Fo(])p Fz(;)14 b(M)3711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1792
3186 y Fu(2)3747 3174 y Fo([)p Fz(a)23 b Fo(:=)g Fz(N)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1793
b Fo(]\))2042 3297 y Fv(\017)47 b Fp(lam)6 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1794
Fz(b;)14 b(M)9 b Fo(\)[)p Fz(a)23 b Fo(:=)g Fz(N)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1795
Fo(])23 b(=)f Fp(lam)7 b Fo(\()p Fz(b;)14 b(M)9 b Fo([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1796
Fz(a)23 b Fo(:=)f Fz(N)9 b Fo(]\))2957 3397 y FC(pro)o(vided)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1797
b Fz(b)23 b Fv(6)p Fo(=)f Fz(a)f FC(and)f Fz(b)i Fv(62)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1798
Fz(F)12 b(V)18 b Fo(\()p Fz(N)9 b Fo(\))2042 3537 y FC(The)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1799
b(second)g(con)m(v)o(ention)e(assumes)j(that)g(binders)e(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1800
(al)o(w)o(ays)2042 3636 y(been)d(so)g(chosen)g(that)h(the)o(y)e(do)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1801
(not)h(clash)f(with)h(free)f(v)n(ariables.)2042 3736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1802
y(This)g(a)n(v)n(oids)f(ha)n(ving)g(to)h(rename)e(bound)g(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1803
(ariables.)h(When)g(per)n(-)2042 3836 y(forming)34 b(a)j(structural)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1804
(induction,)g(renaming)f(bound)h(v)n(ariables)2042 3935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1805
y(can)h(be)h(handled)e(by)h(switching)g(to)h(inductions)e(on)h(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1806
(size,)2042 4035 y(as)25 b(done)f(by)h(Homeier)f([7)o(].)g(When)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1807
(performing)d(rule)j(inductions,)2042 4134 y(proofs)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1808
b(typically)h(need)h(to)g(be)g(entirely)f(recast,)h(perhaps)e(by)i(pro)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1809
o(v-)2042 4234 y(ing)17 b(properties)g(in)m(v)n(olving)f(iterated)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1810
(substitutions.)g(In)h(either)g(case,)2042 4334 y(and)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1811
b(particularly)g(the)h(latter)m(,)g(the)g(mechanisation)e(can)i(hardly)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1812
f(be)2042 4433 y(seen)20 b(as)h(f)o(aithful)e(to)i(the)f(original)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1813
(presentation.)2141 4533 y(A)i(typical)f(informal)f(proof)g(making)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1814
(use)i(of)f(both)g(con)m(v)o(entions)2042 4633 y(is)h(presented)e(in)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1815
(Figure)g(1.)2141 4732 y(In)34 b(this)g(proof,)e(the)i(equational)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1816
(reasoning)g(in)i(the)f(v)n(ariable-)2042 4832 y(case)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1817
b(\(1.1.\2261.3.\))d(relies)j(on)g(the)g(f)o(act)g(that)g(substitution)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1818
g(is)h(a)f(func-)2042 4932 y(tion.)g(In)h(the)g(lambda-case,)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1819
(reasoning)e(further)h(relies)h(on)g(the)2042 5031 y(v)n(ariable)f(con)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1820
m(v)o(ention.)e(This)k(gi)n(v)o(es)f(the)h(assumption)e(that)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1821
Fz(z)j FC(satis-)2042 5131 y(\002es)20 b(freshness)f(constraints)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1822
(which)h(allo)n(w)g(the)h(substitutions)e(to)i(be)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1823
5230 y(pushed)k(under)g(the)h(binder)-5 b(.)24 b(Then)h(one)g(can)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1824
(apply)f(the)i(induction)2042 5330 y(hypothesis,)e(and)h(\002nally)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1825
(pull)h(the)f(substitutions)h(back)f(out)g(from)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1826
5430 y(under)16 b(the)h(binder)-5 b(.)16 b(In)h(the)g(absence)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1827
(the)g(v)n(ariable)f(con)m(v)o(ention,)e Fz(z)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1828
0 TeXcolorgray eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1829
%%Page: 2 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1830
TeXDict begin 2 1 bop 0 TeXcolorgray 0 TeXcolorgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1831
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4 2416 v -120 86 a FC(2)t(.)t(1)t(.)t(1)t(6)t(.)42 b(S)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1833
Fw(U)t(B)t(S)t(T)t(I)t(T)t(U)t(T)t(I)t(O)t(N)i FC(L)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1834
Fw(E)t(M)t(M)t(A)r FC(.)c(If)f Fz(x)58 b Fv(6\021)g Fz(y)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1835
b FC(and)c Fz(x)59 b Fv(62)-122 186 y Fz(F)12 b(V)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1836
b Fo(\()p Fz(L)p Fo(\))p FC(,)i(then)-33 368 y Fz(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1837
b Fo([)p Fz(x)23 b Fo(:=)f Fz(N)9 b Fo(][)p Fz(y)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1838
Fo(:=)d Fz(L)p Fo(])f Fv(\021)h Fz(M)9 b Fo([)p Fz(y)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1839
b Fo(:=)e Fz(L)p Fo(][)p Fz(x)g Fo(:=)g Fz(N)9 b Fo([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1840
Fz(y)26 b Fo(:=)c Fz(L)p Fo(]])p Fz(:)-120 551 y FC(P)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1841
Fw(R)q(O)t(O)t(F)n FC(.)i(By)d(induction)d(on)i(the)g(structure)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1842
Fz(M)9 b FC(.)-122 675 y Fn(Case)21 b(1:)f Fz(M)29 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1843
FC(is)21 b(a)g(v)n(ariable.)-72 802 y(Case)g(1.1.)h Fz(M)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1844
b Fv(\021)23 b Fz(x)p FC(.)e(Then)e(both)g(sides)i(equal)258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1845
902 y Fz(N)9 b Fo([)p Fz(y)26 b Fo(:=)d Fz(L)p Fo(])d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1846
FC(since)g Fz(x)k Fv(6\021)e Fz(y)s FC(.)-72 1049 y(Case)f(1.2.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1847
Fz(M)32 b Fv(\021)23 b Fz(y)s FC(.)d(Then)f(both)h(sides)g(equal)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1848
Fz(L)p FC(,)g(for)258 1148 y Fz(x)k Fv(62)f Fz(F)12 b(V)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1849
b Fo(\()p Fz(L)p Fo(\))i FC(implies)f Fz(L)p Fo([)p Fz(x)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1850
Fo(:=)g Fz(:)14 b(:)g(:)p Fo(])23 b Fv(\021)g Fz(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1851
FC(.)-72 1295 y(Case)e(1.3.)h Fz(M)32 b Fv(\021)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1852
Fz(z)j Fv(6\021)d Fz(x;)14 b(y)s FC(.)20 b(Then)f(both)h(sides)h(equal)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1853
e Fz(z)t FC(.)-122 1433 y Fn(Case)i(2:)g Fz(M)33 b Fv(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1854
b Fz(\025z)t(:M)560 1445 y Fu(1)596 1433 y FC(.)c(By)h(the)f(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1855
(ariable)f(con)m(v)o(ention)e(we)j(may)-122 1533 y(assume)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1856
b(that)h Fz(z)39 b Fv(6\021)c Fz(x;)14 b(y)31 b FC(and)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1857
b Fz(z)31 b FC(is)d(not)f(free)g(in)g Fz(N)t(;)14 b(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1858
FC(.)27 b(Then)f(by)-122 1632 y(induction)19 b(hypothesis)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1859
0 TeXcolorgray 0 TeXcolorgray 312 1770 a Fo(\()p Fz(\025z)t(:M)539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1860
1782 y Fu(1)576 1770 y Fo(\)[)p Fz(x)24 b Fo(:=)f Fz(N)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1861
b Fo(][)p Fz(y)25 b Fo(:=)e Fz(L)p Fo(])200 1893 y Fv(\021)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1862
b Fz(\025z)t(:)p Fo(\()p Fz(M)539 1905 y Fu(1)576 1893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1863
y Fo([)p Fz(x)24 b Fo(:=)e Fz(N)9 b Fo(][)p Fz(y)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1864
Fo(:=)d Fz(L)p Fo(]\))200 2016 y Fv(\021)47 b Fz(\025z)t(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1865
Fo(\()p Fz(M)539 2028 y Fu(1)576 2016 y Fo([)p Fz(y)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1866
b Fo(:=)c Fz(L)p Fo(][)p Fz(x)i Fo(:=)e Fz(N)9 b Fo([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1867
Fz(y)26 b Fo(:=)d Fz(L)p Fo(]]\))200 2139 y Fv(\021)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1868
b Fo(\()p Fz(\025z)t(:M)539 2151 y Fu(1)576 2139 y Fo(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1869
Fz(y)26 b Fo(:=)d Fz(L)p Fo(][)p Fz(x)g Fo(:=)g Fz(N)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1870
b Fo([)p Fz(y)25 b Fo(:=)e Fz(L)p Fo(]])p FC(.)-122 2276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1871
y Fn(Case)28 b(3:)f Fz(M)46 b Fv(\021)36 b Fz(M)483 2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1872
y Fu(1)520 2276 y Fz(M)601 2288 y Fu(2)666 2276 y FC(The)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1873
b(statement)f(follo)n(ws)g(again)g(from)-122 2376 y(the)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1874
b(induction)f(hypothesis.)p 1839 2419 V -150 2422 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1875
4 v -150 2458 1993 3 v -99 2549 a Fn(Figur)o(e)g(1.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1876
b FC(Barendre)o(gt')-5 b(s)19 b(proof)g(of)h(the)g(Substitution)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1877
(Lemma)p 0 TeXcolorgray -150 3027 a(might)g(not)g(be)g(distinct)h(from)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1878
e(the)i(free)f(v)n(ariables)g(in)g(the)h(induction)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1879
3126 y(\()p Fv(f)p Fz(x;)14 b(y)s Fv(g)21 b([)i Fz(F)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1880
b(V)19 b Fo(\()p Fz(N)t(;)14 b(L)p Fo(\))p FC(\),)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1881
b(and)g(one)g(has)h(to)f(rename)g Fz(z)k FC(a)o(w)o(ay)c(from)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1882
3226 y(that)20 b(set)h(in)g(order)e(to)h(mo)o(v)o(e)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1883
(substitutions)f(around.)-50 3326 y(The)29 b(approaches)f(reported)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1884
(in)i([6)o(,)g(14)o(,)g(7,)g(10)o(])g(sho)n(w)g(ho)n(w)f(to)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1885
3425 y(deal)24 b(with)g(the)g(\002rst)h(con)m(v)o(ention)c(in)j(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1886
(formal)f(setting,)h(and)f(do)h(so)-150 3525 y(without)f(ha)n(ving)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1887
(to)i(resort)f(to)h(a)g(nameless)f(de-Bruijn)g(or)g(HO)m(AS-)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1888
3624 y(representation)29 b(for)i Fz(\013)p FC(-equated)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1889
Fz(\025)p FC(-terms.)g(The)o(y)g(construct)f(data)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1890
3724 y(structures)e(which)h(allo)n(w)f(one)h(to)g(write)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1891
Fp(var)9 b Fo(\()p Fz(a)p Fo(\))p FC(,)30 b Fp(app)6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1892
b Fo(\()p Fz(M)1618 3736 y Fu(1)1656 3724 y Fz(;)14 b(M)1774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1893
3736 y Fu(2)1810 3724 y Fo(\))-150 3824 y FC(and)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1894
Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))25 b FC(to)h(denote)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1895
Fz(\013)p FC(-equated)g(v)n(ariables,)g(applications)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1896
3923 y(and)c(abstractions.)f(Furthermore,)e(one)j(has)g(the)h(equation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1897
429 4228 y Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1898
b(=)g Fp(lam)7 b Fo(\()p Fz(b;)14 b(N)9 b Fo(\))482 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1899
FC(\(1\))-150 4533 y(whene)n(v)o(er)13 b(the)i(corresponding)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1900
(syntax-trees)i(\223)p Fz(\025a:M)9 b FC(\224)16 b(and)e(\223)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1901
Fz(\025b:N)9 b FC(\224)-150 4633 y(are)20 b Fz(\013)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1902
FC(-equi)n(v)n(alent.)-50 4732 y(One)i(problem)f(when)h(w)o(orking)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1903
(with)i(name-carrying)c Fz(\013)p FC(-equi-)-150 4832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1904
y(v)n(alence)25 b(classes)h(is)h(that)e(con)m(v)o(enient)e(structural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1905
(induction)g(prin-)-150 4932 y(ciples)h(do)e(not)h(come)g(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1906
(\223free\224,)f(b)n(ut)h(need)g(to)g(be)g(deri)n(v)o(ed.)e(Such)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1907
5031 y(con)m(v)o(enient)g(structural)i(induction)f(principles)g(are)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1908
(deri)n(v)o(ed)e(in)i([5)o(,)-150 5131 y(14)o(,)e(13)o(].)g(These)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1909
(induction)f(principles)g(are)i(stated)g(in)g(such)f(a)h(w)o(ay)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1910
5230 y(that)c(the)o(y)g(come)f(v)o(ery)g(close)h(to)h(the)f(con)m(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1911
(enience)d(of)i(the)i(informal)-150 5330 y(reasoning)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1912
b(using)h(Barendre)o(gt')-5 b(s)15 b(v)n(ariable)g(con)m(v)o(ention.)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1913
(The)j(struc-)-150 5430 y(tural)23 b(induction)e(principle)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1914
(Urban)g(and)h(T)-7 b(asson,)22 b(for)h(e)o(xample,)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1915
0 TeXcolorgray 0 TeXcolorgray 2042 70 a(is:)2120 40 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1916
Fs(1)2105 218 y Fv(8)p Fz(x)14 b(z)t(:)44 b(P)35 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1917
Fp(var)10 b Fo(\()p Fz(z)t Fo(\)\))23 b Fz(x)2105 341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1918
y Fv(8)p Fz(x)14 b(M)21 b(N)t(:)2188 440 y Fo(\()p Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1919
Fz(y)s(:)h(P)35 b(M)c(y)s Fo(\))42 b Fv(^)g Fo(\()p Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1920
Fz(y)s(:)22 b(P)35 b(N)d(y)s Fo(\))46 b Fv(\))23 b Fz(P)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1921
b Fo(\()p Fp(app)7 b Fo(\()p Fz(M)t(;)14 b(N)9 b Fo(\)\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1922
b Fz(x)2105 564 y Fv(8)p Fz(x)14 b(z)i(M)t(:)2188 663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1923
y(z)26 b Fo(#)d Fz(x)42 b Fv(^)g Fo(\()p Fv(8)p Fz(y)s(:)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1924
b(P)35 b(M)d(y)s Fo(\))46 b Fv(\))g Fz(P)35 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1925
Fp(lam)7 b Fo(\()p Fz(z)t(;)14 b(M)9 b Fo(\)\))23 b Fz(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1926
2105 713 1867 4 v 2914 786 a(P)35 b(M)d(x)3938 886 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1927
FC(\(2\))2042 986 y(where)19 b Fz(P)32 b FC(stands)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1928
b(for)f(the)g(property)f(to)i(be)f(pro)o(v)o(ed;)f Fz(M)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1929
b FC(is)21 b(the)f(v)n(ari-)2042 1085 y(able)31 b(o)o(v)o(er)f(which)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1930
(the)g(induction)f(is)i(done,)e(and)h(the)h(v)n(ariable)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1931
Fz(x)2042 1185 y FC(for)18 b(the)h Fx(conte)n(xt)f FC(of)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1932
(induction.)d(By)k(\223the)e(conte)o(xt)g(of)g(an)h(induc-)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1933
1285 y(tion\224,)d(we)i(mean)e(all)i(free)f(v)n(ariables)f(of)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1934
(induction)f(hypothesis,)2042 1384 y(e)o(xcept)g(the)g(v)n(ariable)g(o)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1935
o(v)o(er)f(which)i(the)g(induction)e(is)i(performed.)d(In)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1936
1484 y(case)26 b(of)f(the)h(substitution)f(lemma,)g(the)h(induction)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1937
(hypothesis)h Fz(P)2042 1583 y FC(is)2170 1756 y Fz(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1938
b Fo([)p Fz(x)23 b Fo(:=)g Fz(N)9 b Fo(][)p Fz(y)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1939
Fo(:=)e Fz(L)p Fo(])g Fv(\021)f Fz(M)9 b Fo([)p Fz(y)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1940
b Fo(:=)c Fz(L)p Fo(][)p Fz(x)i Fo(:=)e Fz(N)9 b Fo([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1941
Fz(y)26 b Fo(:=)d Fz(L)p Fo(]])2042 1929 y FC(with)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1942
Fz(M)29 b FC(being)19 b(the)g(v)n(ariable)g(o)o(v)o(er)f(which)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1943
(induction)f(is)j(done.)2042 2028 y(So)j(in)h(this)g(case,)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1944
(conte)o(xt)f Fz(x)j FC(w)o(ould)e(need)f(to)i(be)g(instantiated)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1945
2128 y(with)e(the)f(tuple)g Fo(\()p Fz(x;)14 b(y)s(;)g(N)t(;)g(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1946
Fo(\))p FC(.)23 b(Then,)f(when)g(one)g(comes)g(to)h(pro)o(v)o(e)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1947
2227 y(the)d(lambda-case,)e(one)i(can)g(assume)g(in)h(\(2\))e(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1948
(the)h(binder)e Fz(z)24 b FC(in)2551 2400 y Fz(P)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1949
Fo(\()p Fp(lam)7 b Fo(\()p Fz(z)t(;)14 b(M)9 b Fo(\)\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1950
b(\()p Fz(x;)14 b(y)s(;)g(M)t(;)g(N)9 b Fo(\))2042 2573
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1951
y FC(is)29 b(fresh)e(w)-5 b(.r)g(.t.)27 b Fo(\()p Fz(x;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1952
b(y)s(;)g(N)t(;)g(L)p Fo(\))p FC(\227meaning)27 b(roughly)e(that)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1953
Fz(z)k FC(cannot)2042 2672 y(be)f(equal)h(to)g Fz(x)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1954
FC(and)g Fz(y)s FC(,)f(and)h(that)f Fz(z)36 b FC(cannot)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1955
b(be)g(a)i(free)e(v)n(ariable)2042 2772 y(in)24 b Fz(N)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1956
b FC(and)24 b Fz(L)p FC(.)g(In)g(ef)n(fect,)g(one)g(can)g(formalise)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1957
(Barendre)o(gt')-5 b(s)22 b(slick)2042 2871 y(informal)c(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1958
(without)h(dif)n(\002culties.)2141 2971 y(In)31 b(this)h(paper)e(we)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1959
(sho)n(w)g(that)g(similar)h(induction)d(principles)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1960
3071 y(can)20 b(be)g(gi)n(v)o(en)f(for)h(rule)f(inductions,)g(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1961
(vided)f(a)i(certain)g(property)2042 3170 y(holds)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1962
b(for)h(the)g(relations)f(o)o(v)o(er)g(which)h(the)g(rule)f(inductions)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1963
g(are)2042 3270 y(performed.)23 b(W)-7 b(e)27 b(illustrate)f(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1964
(technique)e(with)j(tw)o(o)f(e)o(xamples:)2042 3370 y(one)k(is)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1965
(proof)f(of)g(the)h(substituti)n(vity)g(property)d(of)j(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1966
@beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1967
 tx@Dict begin STP newpath 0.28453 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1968
} def /ArrowB { BeginArrow 1.  1.  scale false 0.6 1.4 1.5 2. Arrow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1969
0 h T gsave newpath false 0.6 1.4 1.5 2. Arrow CP grestore CP newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1970
moveto 2 copy L stroke moveto  EndArrow  } def [ 17.0 3.15005 3.0 3.15005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1971
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1972
0 setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1973
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1974
@endspecial 3933 3381 a Fm(1)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1975
3469 y FC(relation)23 b(\(this)g(is)i(one)e(part)g(of)g(the)h(simple)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1976
(Church-Rosser)f(proof)2042 3569 y(due)15 b(to)g(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1977
b(ait)17 b(and)e(Martin-L)7 b(\250)-35 b(of)13 b(for)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1978
Fz(\014)t FC(-reduction)f([2)o(]\),)h(and)g(the)h(other)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1979
3668 y(is)21 b(the)f(usual)g(proof)f(of)h(the)g(weak)o(ening)e(lemma)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1980
(for)g(simple)g(types.)2141 3768 y(The)38 b(proof)f(by)g(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1981
b(ait)39 b(and)e(Martin-L)7 b(\250)-35 b(of)37 b(does)g(not)h(sho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1982
(the)2042 3868 y(Church-Rosser)28 b(property)f(directly)i(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1983
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Fv(8)p Fz(\031)s(:)i(R)q Fo(\()p Fz(M)385 4358 y Fu(1)422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4346 y Fz(;)14 b(:)g(:)g(:)g(;)g(M)688 4358 y Ft(n)732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4346 y Fo(\))24 b Fv(\))f Fz(R)q Fo(\()p Fz(\031)1040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4355 y Fj(\001)1078 4346 y Fz(M)1159 4358 y Fu(1)1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4346 y Fz(;)14 b(:)g(:)g(:)f(;)h(\031)1430 4355 y Fj(\001)1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4346 y Fz(M)1550 4358 y Ft(n)1595 4346 y Fo(\))-150 4533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2292
y FC(Equi)n(v)n(ariance)41 b(therefore)g(is)j(the)g(property)d(that)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(a)h(relation)e(is)-150 4633 y(preserv)o(ed)e(under)g(an)o(y)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(permutation)d Fz(\031)s FC(,)k(where)e(permutations)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4732 y(are)36 b(\002nite)g(bijecti)n(v)o(e)f(mappings)f(from)h(atoms)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(to)g(atoms.)g(Beta-)-150 4832 y(reduction,)25 b(typing)h(and)h
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5131 y(ariance)36 b(is)h(v)o(ery)e(common.)f(Its)j(absence)f(w)o(ould)g
Christian Urban <urbanc@in.tum.de>
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diff changeset
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(imply)f(that)i(a)-150 5230 y(particular)26 b(relation)h(didn')o(t)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(beha)n(v)o(e)g(uniformly)f(o)o(v)o(er)h(choices)h(of)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5330 y(the)20 b(names)g(\(atoms\))f(used)h(as)h(v)n(ariables.)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5430 y(Our)e(technique)g(applies)h(to)g(all)h(equi)n(v)n(ariant)d
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2317
(es)e(a)i(v)o(ery)2042 166 y(brief)36 b(o)o(v)o(ervie)n(w)f(about)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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266 y(omitted,)17 b(b)n(ut)h(can)f(be)h(found)e(in)i([3)o(,)g(12)o(,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(14)o(].)e(Sec.)h(3)g(illustrates)h(our)2042 365 y(technique)h(by)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(deri)n(ving)g(impro)o(v)o(ed)e(principles)i(of)g(rule)h(induction)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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465 y(for)39 b(tw)o(o)h(standard)e(relations)h(from)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(literature.)e(The)i(ease-)2042 565 y(of-use)33 b(of)h(the)g(ne)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(principles)f(is)i(also)f(demonstrated.)e(Sec.)i(4)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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664 y(mentions)19 b(some)h(related)g(w)o(ork)f(and)h(Sec.)g(5)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(concludes.)2042 851 y FA(2.)99 b(Nominal)24 b(Logic)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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984 y FC(There)16 b(are)i(tw)o(o)f(central)g(notions)g(in)g(nominal)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2328
(logic:)g(permutations)2042 1083 y(and)25 b(support.)g(As)i(mentioned)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)j(the)f(pre)n(vious)f(section,)h(permu-)2042 1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2330
y(tations)k(are)h(\002nite)g(bijecti)n(v)o(e)f(mappings)f(from)h(atoms)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2331
g(to)h(atoms,)2042 1283 y(where)17 b(atoms)h(are)g(dra)o(wn)f(from)g(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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i(countably)d(in\002nite)i(set)g(denoted)2042 1382 y(by)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2333
b Fi(A)17 b FC(.)36 b(W)-7 b(e)31 b(represent)d(permutations)g(as)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(\002nite)e(lists)j(whose)d(ele-)2042 1482 y(ments)k(are)g(sw)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(appings)g(\(i.e.)g(pairs)g(of)g(atoms\).)g(W)-7 b(e)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(write)e(such)2042 1581 y(permutations)26 b(as)j Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2337
Fz(a)2678 1593 y Fu(1)2729 1581 y Fz(b)2765 1593 y Fu(1)2802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1581 y Fo(\)\()p Fz(a)2910 1593 y Fu(2)2962 1581 y Fz(b)2998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1593 y Fu(2)3034 1581 y Fo(\))14 b Fv(\001)g(\001)g(\001)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2340
Fo(\()p Fz(a)3267 1593 y Ft(n)3327 1581 y Fz(b)3363 1593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Ft(n)3407 1581 y Fo(\))p FC(;)29 b(the)g(empty)e(list)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fo([])2042 1681 y FC(stands)23 b(for)g(the)h(identity)e(permutation.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(A)j(permutation)d Fz(\031)28 b Fx(acting)2042 1781 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2344
FC(on)20 b(an)g(atom)f Fz(a)i FC(is)g(de\002ned)e(as:)2665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1932 y Fo([])2711 1941 y Fj(\001)2749 1932 y Fz(a)2876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1885 y Fs(def)2881 1932 y Fo(=)87 b Fz(a)2263 2133 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2347
Fo(\(\()p Fz(a)2371 2145 y Fu(1)2422 2133 y Fz(a)2466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2145 y Fu(2)2504 2133 y Fo(\))23 b(::)g Fz(\031)s Fo(\))2710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2142 y Fj(\001)2749 2133 y Fz(a)2876 2086 y Fs(def)2881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2133 y Fo(=)3033 1962 y Fh(8)3033 2037 y(<)3033 2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2351
y(:)3148 2032 y Fz(a)3192 2044 y Fu(2)3364 2032 y FC(if)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fz(\031)3488 2041 y Fj(\001)3526 2032 y Fz(a)g Fo(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2353
Fz(a)3725 2044 y Fu(1)3148 2132 y Fz(a)3192 2144 y Fu(1)3364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2354
2132 y FC(if)g Fz(\031)3488 2141 y Fj(\001)3526 2132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2355
y Fz(a)g Fo(=)g Fz(a)3725 2144 y Fu(2)3148 2231 y Fz(\031)3198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2356
2240 y Fj(\001)3237 2231 y Fz(a)83 b FC(otherwise)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2357
2379 y(where)30 b Fo(\()p Fz(a)14 b(b)p Fo(\))44 b(::)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2358
Fz(\031)35 b FC(is)d(the)g(composition)d(of)i(a)g(permutation)e(fol-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2359
2042 2479 y(lo)n(wed)16 b(by)h(the)g(sw)o(apping)f Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2360
Fz(a)e(b)p Fo(\))p FC(.)j(The)g(composition)e(of)i Fz(\031)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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FC(follo)n(wed)2042 2579 y(by)35 b(another)g(permutation)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2362
Fz(\031)2934 2548 y Fq(0)2994 2579 y FC(is)k(gi)n(v)o(en)e(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2363
(list-concatenation,)2042 2678 y(written)27 b(as)h Fz(\031)2452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2364
2648 y Fq(0)2475 2678 y Fo(@)p Fz(\031)s FC(,)g(and)f(the)g(in)m(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2365
(erse)f(of)h(a)h(permutation)d(is)j(gi)n(v)o(en)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2366
2778 y(by)g(list)i(re)n(v)o(ersal,)d(written)h(as)i Fz(\031)3007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2367
2748 y Fq(\000)p Fu(1)3096 2778 y FC(.)f(Our)f(representation)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2368
(per)n(-)2042 2877 y(mutations)e(as)i(lists)h(does)e(not)g(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2369
(unique)f(representati)n(v)o(es:)g(for)2042 2977 y(e)o(xample,)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2370
b(the)h(permutation)e Fo(\()p Fz(a)f(a)p Fo(\))19 b FC(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2371
(\223equal\224)e(to)i(the)g(identity)e(per)n(-)2042 3077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2372
y(mutation.)2141 3176 y(W)-7 b(e)40 b(equate)f(the)g(representations)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2373
(of)h(permutations)f(with)i(a)2042 3276 y(relation)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2374
b Fv(\030)p FC(:)p 0 TeXcolorgray 2042 3414 a Fn(De\002nition)j(1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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40 w Fx(T)-6 b(wo)22 b(permutations)f(ar)m(e)2042 3514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2377
y FC(equal)p Fx(,)33 b(written)i Fz(\031)2594 3526 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2378
Fu(1)2680 3514 y Fv(\030)48 b Fz(\031)2840 3526 y Fu(2)2877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2379
3514 y Fx(,)35 b(pr)l(o)o(vided)e Fz(\031)3305 3526 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2380
Fu(1)3342 3523 y Fj(\001)3381 3514 y Fz(x)49 b Fo(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2381
Fz(\031)3637 3526 y Fu(2)3674 3523 y Fj(\001)3713 3514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2382
y Fz(x)p Fx(,)34 b(for)h(all)2042 3613 y Fz(x)23 b Fv(2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2383
Fi(A)17 b Fx(.)2141 3751 y FC(The)27 b(action)f(of)h(a)g(permutation)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2384
(can)h(be)h(lifted)g(to)g(other)f(types)2042 3851 y(as)c(long)e(as)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2385
(the)f(action)f(on)h(the)g(ne)n(w)g(type)f(is)i Fx(sensible)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2386
FC(.)f(By)h(this)g(we)2042 3950 y(mean)d(that)i(it)g(has)f(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2387
(satisfy)h(the)f(follo)n(wing)f(three)g(properties:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2388
0 TeXcolorgray 0 TeXcolorgray 2477 4081 a Fo(\()p Fz(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2389
Fo(\))100 b([])2716 4090 y Fj(\001)2754 4081 y Fz(x)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2390
b Fo(=)g Fz(x)2448 4204 y Fo(\()p Fz(ii)p Fo(\))100 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2391
Fz(\031)2749 4216 y Fu(1)2786 4204 y Fo(@)p Fz(\031)2898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2392
4216 y Fu(2)2936 4204 y Fo(\))2968 4213 y Fj(\001)3006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2393
4204 y Fz(x)24 b Fo(=)e Fz(\031)3211 4216 y Fu(1)3249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2394
4213 y Fj(\001)3287 4204 y Fo(\()p Fz(\031)3366 4216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2395
y Fu(2)3404 4213 y Fj(\001)3442 4204 y Fz(x)p Fo(\))2420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2396
4327 y(\()p Fz(iii)p Fo(\))99 b FC(if)20 b Fz(\031)2788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2397
4339 y Fu(1)2849 4327 y Fv(\030)i Fz(\031)2983 4339 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2398
Fu(2)3042 4327 y FC(then)d Fz(\031)3252 4339 y Fu(1)3290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2399
4336 y Fj(\001)3328 4327 y Fz(x)24 b Fo(=)e Fz(\031)3533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2400
4339 y Fu(2)3571 4336 y Fj(\001)3609 4327 y Fz(x)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2401
4456 y FC(From)g(this)h(we)g(can)g(de\002ne)f(\223permutation)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2402
(sets\224)k(as)f(those)g(ha)n(ving)2042 4556 y(a)d(sensible)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2403
(permutation)d(action:)p 0 TeXcolorgray 2042 4694 a Fn(De\002nition)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2404
(2)h FC(\(PSets\))p Fn(.)p 0 TeXcolorgray 42 w Fx(A)g(set)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2405
Fz(X)30 b Fx(equipped)20 b(with)j(a)f(permutation)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2406
4794 y(action)32 b Fz(\031)2333 4803 y Fj(\001)2372 4794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2407
y Fo(\()p Fv(\000)p Fo(\))i Fx(is)g(said)g(to)f(be)h(a)f(pset,)h(if)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2408
(for)g(all)g Fz(x)48 b Fv(2)g Fz(X)7 b Fx(,)33 b(the)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2409
4893 y(permutation)18 b(action)i(satis\002es)g(the)h(pr)l(operties)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2410
Fo(\()p Fz(i)p Fo(\))p Fx(-)p Fo(\()p Fz(iii)p Fo(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2411
Fx(.)2141 5031 y FC(The)f(informal)e(notation)g Fz(x)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2412
b Fv(2)h Fx(pset)18 b FC(will)i(be)e(adopted)f(whene)n(v)o(er)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2413
5131 y(it)f(needs)f(to)h(be)g(indicated)e(that)i Fz(x)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2414
FC(comes)e(from)f(a)i Fx(pset)p FC(.)g(T)-7 b(ypical)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2415
b(per)n(-)2042 5230 y(mutation)21 b(actions)g(permute)g(all)i(atoms)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2416
(in)g(a)h(gi)n(v)o(en)d Fx(pset)p FC(-element.)2042 5330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2417
y(F)o(or)c(e)o(xample,)e(lists,)j(tuples)g(and)e(sets)j(can)e(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2418
(seen)g(as)h Fx(pset)p FC(s)g(if)f(their)2042 5430 y(respecti)n(v)o(e)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2419
(permutation)f(actions)i(are)g(de\002ned)f(point-wise:)p
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2420
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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220 y Fo(\()p Fz(x)23 b Fo(::)h Fz(t)p Fo(\))911 173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2427
y Fs(def)916 220 y Fo(=)52 b(\()p Fz(\031)1115 229 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2428
Fj(\001)1153 220 y Fz(x)p Fo(\))24 b(::)f(\()p Fz(\031)1407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2429
229 y Fj(\001)1446 220 y Fz(t)p Fo(\))57 345 y FC(tuples:)70
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2430
b Fz(\031)399 354 y Fj(\001)437 345 y Fo(\()p Fz(x)516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2431
357 y Fu(1)554 345 y Fz(;)14 b(:)g(:)g(:)g(;)g(x)786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2432
357 y Ft(n)832 345 y Fo(\))911 298 y Fs(def)916 345 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2433
Fo(=)52 b(\()p Fz(\031)1115 354 y Fj(\001)1153 345 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2434
Fz(x)1200 357 y Fu(1)1238 345 y Fz(;)14 b(:)g(:)g(:)g(;)g(\031)1473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2435
354 y Fj(\001)1511 345 y Fz(x)1558 357 y Ft(n)1604 345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2436
y Fo(\))57 470 y FC(sets:)496 b Fz(\031)750 479 y Fj(\001)789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2437
470 y Fz(X)911 423 y Fs(def)916 470 y Fo(=)52 b Fv(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2438
Fz(\031)1125 479 y Fj(\001)1163 470 y Fz(x)14 b Fv(j)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2439
Fz(x)23 b Fv(2)h Fz(X)7 b Fv(g)-150 602 y FC(On)16 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2440
Fz(\013)p FC(-equated)f Fz(\025)p FC(-terms)h(the)h(permutation)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2441
(action)h(is)i(de\002ned)e(such)-150 701 y(that)20 b(it)h(satis\002es:)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2442
401 847 y Fz(\031)451 856 y Fj(\001)489 847 y Fp(var)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2443
b Fo(\()p Fz(a)p Fo(\))48 b(=)f Fp(var)9 b Fo(\()p Fz(\031)1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2444
856 y Fj(\001)1127 847 y Fz(a)p Fo(\))164 970 y Fz(\031)214
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2445
979 y Fj(\001)253 970 y Fp(app)d Fo(\()p Fz(M)498 982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2446
y Fu(1)536 970 y Fz(;)14 b(M)654 982 y Fu(2)690 970 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2447
Fo(\))48 b(=)f Fp(app)6 b Fo(\()p Fz(\031)1096 979 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2448
Fj(\001)1135 970 y Fz(M)1216 982 y Fu(1)1253 970 y Fz(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2449
b(\031)1340 979 y Fj(\001)1378 970 y Fz(M)1459 982 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2450
Fu(2)1496 970 y Fo(\))261 1094 y Fz(\031)311 1103 y Fj(\001)350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2451
1094 y Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))47 b(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2452
Fp(lam)6 b Fo(\()p Fz(\031)1101 1103 y Fj(\001)1140 1094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2453
y Fz(a;)14 b(\031)1271 1103 y Fj(\001)1310 1094 y Fz(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2454
b Fo(\))1746 971 y FC(\(4\))-150 1255 y(W)-7 b(e)21 b(note)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(1.)p 0 TeXcolorgray 48 w FC(The)f(follo)n(wing)f(sets)j(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2457
Fx(pset)p FC(s:)h Fi(A)17 b FC(,)38 b(the)31 b(set)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2458
Fz(\013)p FC(-)-150 1495 y(equated)16 b(lambda-terms,)e(and)j(e)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2459
(ery)e(set)j(of)f(lists)h(\(similarly)e(tuples)-150 1595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2460
y(and)k(sets\))h(containing)d(elements)i(from)f Fx(pset)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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FC(s.)-50 1735 y(One)k(interesting)g(consequence)f(of)i(nominal)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2462
(logic)i([3)o(])g(is)h(that)-150 1834 y(as)j(soon)e(as)h(one)g(\002x)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2463
(es)g(the)f(notion)g(of)h(permutation)d(action)i(for)h(a)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2464
1934 y Fx(pset)p FC(,)17 b(then)f(the)h(notion)e(of)h(support,)f(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2465
(ery)h(roughly)e(speaking)i(its)h(set)-150 2033 y(of)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2466
b(free)h(atoms,)f(is)h(\002x)o(ed)f(as)i(well.)f(The)f(support)f(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2467
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Fn(.)p 0 TeXcolorgray 45 w Fx(Given)g Fz(x)36 b Fv(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2471
Fx(pset,)27 b(its)-150 2373 y FC(support)19 b Fx(is)i(de\002ned)d(as:)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2472
54 2566 y Fg(supp)o Fo(\()p Fz(x)p Fo(\))364 2514 y Ff(def)367
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2473
2566 y Fo(=)26 b Fv(f)p Fz(a)c Fv(j)h Fg(infinite)m Fv(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2474
Fz(b)f Fv(j)h Fo(\()p Fz(a)14 b(b)p Fo(\))1265 2575 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2475
Fj(\001)1304 2566 y Fz(x)23 b Fv(6)p Fo(=)g Fz(x)p Fv(gg)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2476
Fz(:)-150 2736 y Fx(An)h(atom)h Fz(a)g Fx(is)g(said)g(to)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2477
FC(fresh)f Fx(for)h(suc)o(h)f(an)g Fz(x)p Fx(,)h(written)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2478
Fz(a)k Fo(#)g Fz(x)p Fx(,)-150 2836 y(pr)l(o)o(vided)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2479
b Fz(a)k Fv(62)h Fg(supp)n Fo(\()p Fz(x)p Fo(\))p Fx(.)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2480
2976 y FC(W)m(ith)d(these)g(notions)f(in)h(place)f(we)i(can)e(mak)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2481
(the)f(equation)g(in)h(\(1\))-150 3076 y(precise,)f(stating)g(when)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(tw)o(o)i Fz(\013)p FC(-equated)e(abstractions)g(are)h(equal:)190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2483
3238 y Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))23 b(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2484
Fp(lam)7 b Fo(\()p Fz(b;)14 b(N)9 b Fo(\))22 b Fv(,)356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2485
3350 y Fo(\()p Fz(a)h Fo(=)g Fz(b)18 b Fv(^)h Fz(M)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fo(=)23 b Fz(N)9 b Fo(\))23 b Fv(_)356 3450 y Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2487
Fz(a)g Fv(6)p Fo(=)g Fz(b)18 b Fv(^)h Fz(M)31 b Fo(=)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2488
b(\()p Fz(a)14 b(b)p Fo(\))1029 3459 y Fj(\001)1067 3450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2489
y Fz(N)27 b Fv(^)19 b Fz(a)k Fo(#)g Fz(N)9 b Fo(\))1746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2490
3345 y FC(\(5\))-150 3611 y(In)22 b(what)g(follo)n(ws)g(we)g(will)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(often)e(mak)o(e)h(use)g(the)g(follo)n(wing)f(prop-)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3711 y(erties)g(of)e Fx(pset)p FC(s:)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2493
-150 3851 a Fn(Lemma)i(2.)p 0 TeXcolorgray 41 w FC(F)o(or)e(all)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2494
Fz(x)j Fv(2)f Fx(pset)p FC(,)57 3942 y Fo(\()p Fz(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2495
Fo(\))99 b Fz(a)23 b Fo(#)h Fz(x)d FC(implies)f Fz(\031)792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2496
3951 y Fj(\001)831 3942 y Fz(a)j Fo(#)g Fz(\031)1040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2497
3951 y Fj(\001)1078 3942 y Fz(x)28 4042 y Fo(\()p Fz(ii)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2498
Fo(\))99 b FC(if)21 b Fz(a)i Fo(#)g Fz(x)e FC(and)f Fz(b)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2499
Fo(#)g Fz(x)p FC(,)e(then)f Fo(\()p Fz(a)14 b(b)p Fo(\))1251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2500
4051 y Fj(\001)1289 4042 y Fz(x)23 b Fo(=)g Fz(x)-1 4141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2501
y Fo(\()p Fz(iii)p Fo(\))99 b Fz(a)23 b Fo(#)h(\()p Fz(x;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2502
b(y)s Fo(\))21 b FC(if)f(and)g(only)f(if)i Fz(a)i Fo(#)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2503
Fz(x)e FC(and)f Fz(a)j Fo(#)g Fz(y)-50 4272 y FC(A)f(further)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2504
(restriction)g(on)h Fx(pset)p FC(s)h(\002lters)g(out)f(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2505
Fx(pset)o FC(s)g(contain-)-150 4372 y(ing)d(elements)g(with)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in\002nite)g(support:)p 0 TeXcolorgray -150 4512 a Fn(De\002nition)g
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2508
41 w Fx(A)i(pset)h Fz(X)27 b Fx(is)20 b(said)g(to)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2509
4611 y(be)g(an)g(fs-pset)g(if)h(e)o(very)g(element)f(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2510
Fz(X)27 b Fx(has)20 b(\002nite)g(support.)-50 4751 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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FC(W)-7 b(e)21 b(note)f(the)g(follo)n(wing:)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
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diff changeset
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-150 4891 a Fn(Lemma)25 b(3.)p 0 TeXcolorgray 43 w Fx(The)f(following)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2513
(sets)h(ar)m(e)f(fs-psets:)h Fi(A)17 b Fx(,)30 b(the)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2514
b(set)h(of)f Fz(\013)p Fx(-)-150 4991 y(equated)d Fz(\025)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2515
Fx(-terms,)j(and)d(e)o(very)i(set)h(of)e(lists)i(\(similarly)f(tuples)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2516
(and)-150 5090 y(\002nite)d(sets\))h(containing)d(elements)i(fr)l(om)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2517
(fs-psets.)-50 5230 y FC(Since)28 b(the)h(set)g(of)f(atoms)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2518
Fi(A)52 b FC(is)30 b(in\002nite,)e(the)g(most)h(important)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2519
5330 y(property)21 b(of)i Fx(fs-pset)o FC(s)h(is)g(that)f(for)g(each)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2520
(element)h(one)f(can)h(choose)-150 5430 y(a)e(fresh)e(atom.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2521
0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 2042 66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a Fn(Lemma)24 b(4.)p 0 TeXcolorgray 43 w Fx(F)-9 b(or)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2523
b(all)g Fz(x)30 b Fv(2)g Fx(fs-pset)o(,)24 b(ther)m(e)g(e)n(xists)h(an)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2524
e(atom)g Fz(a)h Fx(suc)o(h)2042 166 y(that)c Fz(a)j Fo(#)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2525
Fz(x)p Fx(.)2141 323 y FC(Unwinding)34 b(the)h(de\002nitions)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2526
(permutation)e(actions)h(and)2042 423 y(support)15 b(one)i(can)f(often)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2527
g(easily)i(calculate)e(the)h(support)f(for)g Fx(fs-pset)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2528
FC(-)2042 523 y(elements:)p 0 TeXcolorgray 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2529
2042 754 a(atoms:)216 b Fg(supp)o Fo(\()p Fz(a)p Fo(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2530
b(=)g Fv(f)p Fz(a)p Fv(g)2042 877 y FC(tuples:)216 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2531
Fg(supp)o Fo(\()p Fz(x)2734 889 y Fu(1)2772 877 y Fz(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2532
b(:)g(:)g(:)f(;)h(x)3003 889 y Ft(n)3049 877 y Fo(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2533
b(=)g Fg(supp)n Fo(\()p Fz(x)3445 889 y Fu(1)3483 877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2534
y Fo(\))c Fv([)g Fz(:)14 b(:)g(:)k Fv([)h Fg(supp)n Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2535
Fz(x)4050 889 y Ft(n)4096 877 y Fo(\))2042 1000 y FC(lists:)282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2536
b Fg(supp)o Fo(\([]\))24 b(=)e Fi(?)2480 1100 y Fg(supp)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2537
Fo(\()p Fz(x)i Fo(::)f Fz(xs)p Fo(\))h(=)e Fg(supp)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2538
Fo(\()p Fz(x)p Fo(\))e Fv([)f Fg(supp)n Fo(\()p Fz(xs)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2539
Fo(\))2042 1223 y FC(\002nite)h(sets:)100 b Fg(supp)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2540
Fo(\()p Fz(X)7 b Fo(\))23 b(=)2905 1161 y Fh(S)2975 1248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2541
y Ft(x)p Fq(2)p Ft(X)3134 1223 y Fg(supp)n Fo(\()p Fz(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2542
Fo(\))2042 1346 y Fz(\013)p FC(-equated)c(lambda-terms:)2480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2543
1470 y Fg(supp)o Fo(\()p Fp(var)10 b Fo(\()p Fz(a)p Fo(\)\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2544
b(=)g Fv(f)p Fz(a)p Fv(g)2480 1569 y Fg(supp)o Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2545
Fp(app)7 b Fo(\()p Fz(M)t(;)14 b(N)9 b Fo(\)\))23 b(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2546
Fg(supp)n Fo(\()p Fz(M)9 b Fo(\))19 b Fv([)g Fg(supp)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2547
Fo(\()p Fz(N)9 b Fo(\))2480 1669 y Fg(supp)o Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2548
Fp(lam)e Fo(\()p Fz(a;)14 b(M)9 b Fo(\)\))23 b(=)g Fg(supp)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2549
Fo(\()p Fz(M)9 b Fo(\))19 b Fv(\000)f(f)p Fz(a)p Fv(g)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2550
1817 y FC(The)40 b(last)h(three)f(equations)f(sho)n(w)g(that)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2551
(support)e(of)i Fz(\013)p FC(-)2042 1917 y(equated)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2552
b(lambda-terms)f(coincides)i(with)g(the)h(usual)f(notion)f(of)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2553
2017 y(free)23 b(v)n(ariables.)g(In)h(turn,)f Fz(a)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2554
b Fo(#)h Fz(M)j FC(with)24 b Fz(M)33 b FC(being)23 b(an)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2555
Fz(\013)p FC(-equated)2042 2116 y(lambda-term)17 b(coincides)i(with)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2556
Fz(a)g FC(not)g(being)f(free)g(in)h Fz(M)9 b FC(.)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2557
b(If)f Fz(b)h FC(is)h(an)2042 2216 y(atom,)e(then)h Fz(a)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2558
Fo(#)g Fz(b)e FC(coincides)e(with)h Fz(a)j Fv(6)p Fo(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2559
Fz(b)p FC(.)2042 2464 y FA(3.)99 b(Rule)25 b(Inductions)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2560
2597 y FC(Inductions)40 b(o)o(v)o(er)i(inducti)n(v)o(ely-de\002ned)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2561
(relations,)k(also)h(called)2042 2696 y(rule)f(inductions,)f(are)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2562
(important)e(reasoning)g(tools)i(in)g(the)f Fz(\025)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2563
FC(-)2042 2796 y(calculus)26 b(and)g(programming)d(languages)h([1].)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2564
(Here)g(we)h(pro)o(vide)2042 2895 y(tw)o(o)18 b(small,)g(b)n(ut)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2565
(typical,)g(e)o(xamples)f(of)h(such)f(rule)h(inductions,)e(and)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2566
2995 y(illustrate)k(our)g(technique)e(on)i(both.)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2567
3194 y Fn(3.1)82 b(W)-5 b(eak)o(ening)19 b(f)n(or)h(Simple)h(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2568
b(ypes)2042 3326 y FC(T)g(erms)26 b(of)g(the)h Fz(\025)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2569
FC(-calculus)f(can)g(be)h(gi)n(v)o(en)e(types)h(with)h(respect)f(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2570
2042 3426 y(conte)o(xts)c(\(for)h(e)o(xample)f(\002nite)h(sets)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2571
(name-type)e(pairs\).)i(T)-7 b(ypes)2042 3526 y(are)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2572
b(of)g(the)g(form)2618 3741 y Fp(ty)30 b Fo(:)106 b Fz(\034)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2573
b Fo(::=)46 b Fz(X)29 b Fv(j)23 b Fz(\034)33 b Fv(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2574
b Fz(\034)2042 3956 y FC(Conte)o(xts)d(are)g Fx(valid)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2575
FC(if)e(no)g(v)n(ariable)f(occurs)g(twice:)p 2548 4180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2576
308 4 v 2548 4259 a Fp(valid)11 b Fo(\()p Fi(?)p Fo(\))2939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2577
4139 y Fz(a)23 b Fo(#)g(\000)83 b Fp(valid)10 b Fo(\(\000\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2578
2939 4180 589 4 v 2990 4259 a Fp(valid)h Fo(\()p Fz(a)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2579
b Fo(:)g Fz(\034)5 b(;)14 b Fo(\000\))2042 4465 y FC(The)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2580
b(relation)f(associating)h(terms)g(and)g(types)g(is)h(straightforw)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2581
(ard)2042 4564 y(to)20 b(de\002ne:)2198 4747 y Fp(valid)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2582
b Fo(\(\000\))84 b(\()p Fz(a)23 b Fo(:)g Fz(\034)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2583
Fo(\))24 b Fv(2)f Fo(\000)p 2198 4788 755 4 v 2327 4867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2584
a(\000)g Fv(`)f Fp(var)10 b Fo(\()p Fz(a)p Fo(\))23 b(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2585
Fz(\034)3035 4752 y(a)f Fo(#)g(\000)83 b Fz(a)23 b Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2586
Fz(\034)5 b(;)14 b Fo(\000)23 b Fv(`)g Fz(M)32 b Fo(:)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2587
b Fz(\033)p 3035 4788 844 4 v 3049 4867 a Fo(\000)g Fv(`)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2588
Fp(lam)6 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))23 b(:)h Fz(\034)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2589
b Fv(!)23 b Fz(\033)2560 5025 y Fo(\000)g Fv(`)g Fz(M)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2590
b Fo(:)23 b Fz(\034)33 b Fv(!)23 b Fz(\033)86 b Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2591
b Fv(`)g Fz(N)32 b Fo(:)23 b Fz(\034)p 2560 5045 956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2592
4 v 2706 5124 a Fo(\000)g Fv(`)g Fp(app)7 b Fo(\()p Fz(M)t(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2593
b(N)9 b Fo(\))23 b(:)g Fz(\033)2042 5330 y FC(Making)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2594
b(such)h(a)h(de\002nition)e(also)i(results)g(in)f(the)h(proof)e(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2595
(the)2042 5430 y(associated)20 b(induction)f(principle,)g(with)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2596
Fz(P)33 b FC(a)21 b(three-place,)d(curried)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2597
0 TeXcolorgray eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2598
%%Page: 5 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2599
TeXDict begin 5 4 bop 0 TeXcolorgray 0 TeXcolorgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2600
TeXcolorgray -150 66 a FC(predicate:)-49 216 y Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2601
Fo(\000)14 b Fz(a)g(\034)5 b(:)45 b Fp(valid)11 b Fo(\(\000\))18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2602
b Fv(^)h Fo(\()p Fz(a)24 b Fo(:)f Fz(\034)9 b Fo(\))24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2603
b Fv(2)f Fo(\000)g Fv(\))g Fz(P)35 b Fo(\000)23 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2604
Fp(var)10 b Fo(\()p Fz(a)p Fo(\)\))24 b Fz(\034)-49 339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2605
y Fv(8)p Fo(\000)14 b Fz(M)22 b(N)g(\034)i(\033)n(:)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2606
b Fo(\000)23 b Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\034)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2607
b Fv(!)23 b Fz(\033)f Fv(^)d Fz(P)35 b Fo(\000)23 b Fz(M)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2608
b Fo(\()p Fz(\034)i Fv(!)23 b Fz(\033)s Fo(\))h Fv(^)431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2609
439 y Fo(\000)f Fv(`)g Fz(N)31 b Fo(:)24 b Fz(\034)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2610
Fv(^)18 b Fz(P)35 b Fo(\000)23 b Fz(N)32 b(\034)h Fv(\))431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2611
538 y Fz(P)i Fo(\000)23 b(\()p Fp(app)6 b Fo(\()p Fz(M)t(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2612
b(N)9 b Fo(\)\))24 b Fz(\033)-49 662 y Fv(8)p Fo(\000)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2613
b Fz(a)g(M)21 b(\034)j(\033)n(:)47 b(a)23 b Fo(#)g(\000)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2614
b Fv(^)h Fo(\()p Fz(a)24 b Fo(:)f Fz(\034)5 b(;)14 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2615
Fo(\000\))23 b Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\033)k Fv(^)399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2616
761 y Fz(P)35 b Fo(\()p Fz(a)23 b Fo(:)g Fz(\034)5 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2617
b Fo(\000\))23 b Fz(M)32 b(\033)27 b Fv(\))399 861 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2618
Fz(P)35 b Fo(\000)23 b(\()p Fp(lam)7 b Fo(\()p Fz(a;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2619
b(M)9 b Fo(\)\))23 b(\()p Fz(\034)33 b Fv(!)23 b Fz(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2620
Fo(\))p -90 911 1873 4 v 421 985 a(\000)g Fv(`)g Fz(M)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2621
b Fo(:)23 b Fz(\034)56 b Fv(\))46 b Fz(P)35 b Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2622
b Fz(M)31 b(\034)1746 1084 y FC(\(6\))-50 1184 y(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2623
b(e)20 b(wish)f(to)g(pro)o(v)o(e)e(the)i(follo)n(wing)e(property)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2624
b(,)16 b(where)i(a)h(conte)o(xt)-150 1284 y Fo(\000)-98
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2625
1296 y Fu(2)-39 1284 y FC(is)i(weak)o(er)f(than)h Fo(\000)514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2626
1296 y Fu(1)572 1284 y FC(\(written)g Fo(\000)909 1296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2627
y Fu(1)970 1284 y Fv(\022)j Fo(\000)1111 1296 y Fu(2)1148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2628
1284 y FC(\),)d(if)g(e)n(v)o(ery)e(name-type)-150 1383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2629
y(pair)h(in)g Fo(\000)137 1395 y Fu(1)195 1383 y FC(also)h(appears)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2630
(in)h Fo(\000)761 1395 y Fu(2)798 1383 y FC(:)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2631
-150 1522 a Fn(Lemma)f(5)f FC(\(W)-7 b(eak)o(ening)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2632
b(Lemma\))p Fn(.)p 0 TeXcolorgray 36 w Fx(If)i Fo(\000)1078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2633
1534 y Fu(1)1138 1522 y Fv(`)23 b Fz(M)32 b Fo(:)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2634
Fz(\034)28 b Fx(is)19 b(derivable)o(,)-150 1622 y(and)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2635
Fo(\000)50 1634 y Fu(1)116 1622 y Fv(\022)29 b Fo(\000)262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2636
1634 y Fu(2)322 1622 y Fx(with)24 b Fo(\000)541 1634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2637
y Fu(2)602 1622 y Fx(being)f(valid,)f(then)h Fo(\000)1240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2638
1634 y Fu(2)1306 1622 y Fv(`)28 b Fz(M)38 b Fo(:)28 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2639
Fz(\034)34 b Fx(is)24 b(also)-150 1721 y(derivable)o(.)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2640
1860 y FC(Proofs)e(of)h(this)g(lemma)f(are)h(often)f(claimed)h(to)g(be)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2641
g(straightfor)n(-)-150 1959 y(w)o(ard)18 b(\(e.g.)f([11)n(]\).)h(The)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2642
(informal)e(proof)h(usually)g(goes)h(as)g(follo)n(ws:)-148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2643
2099 y(I)t Fw(N)t(F)t(O)t(R)t(M)t(A)t(L)25 b FC(P)t Fw(R)q(O)t(O)t(F)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2644
(O)t(F)e(T)t(H)t(E)g FC(W)t Fw(E)t(A)t(K)t(E)t(N)t(I)t(N)t(G)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2645
FC(L)t Fw(E)t(M)t(M)t(A)r FC(.)c(By)g(rule)-150 2198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2646
y(induction)d(o)o(v)o(er)h Fo(\000)402 2210 y Fu(1)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2647
2198 y Fv(`)k Fz(M)32 b Fo(:)23 b Fz(\034)9 b FC(.)-48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2648
2298 y(C)t Fw(A)t(S)t(E)26 b FC(1)t(:)d Fo(\000)315 2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2649
y Fu(1)378 2298 y Fv(`)h Fz(M)34 b Fo(:)25 b Fz(\034)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2650
b FC(is)22 b Fo(\000)813 2310 y Fu(1)875 2298 y Fv(`)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2651
Fp(var)10 b Fo(\()p Fz(a)p Fo(\))25 b(:)g Fz(\034)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2652
FC(.)23 b(By)e(assumption)-150 2398 y(we)j(kno)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2653
Fz(v)s(al)r(id)p Fo(\(\000)448 2410 y Fu(2)484 2398 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2654
Fo(\))p FC(,)h Fo(\()p Fz(a)30 b Fo(:)f Fz(\034)9 b Fo(\))31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2655
b Fv(2)e Fo(\000)963 2410 y Fu(1)1025 2398 y FC(and)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2656
b Fo(\000)1221 2410 y Fu(1)1287 2398 y Fv(\022)29 b Fo(\000)1433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2657
2410 y Fu(2)1470 2398 y FC(.)24 b(Therefore)-150 2497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2658
y(we)d(can)f(use)g(the)g(typing)f(rules)h(to)h(deri)n(v)o(e)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2659
Fo(\000)1135 2509 y Fu(2)1195 2497 y Fv(`)k Fp(var)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2660
b Fo(\()p Fz(a)p Fo(\))24 b(:)f Fz(\034)9 b FC(.)-48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2661
2597 y(C)t Fw(A)t(S)t(E)31 b FC(2)t(:)d Fo(\000)325 2609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2662
y Fu(1)397 2597 y Fv(`)34 b Fz(M)43 b Fo(:)34 b Fz(\034)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2663
FC(is)27 b Fo(\000)870 2609 y Fu(1)941 2597 y Fv(`)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2664
b Fp(app)6 b Fo(\()p Fz(M)1272 2609 y Fu(1)1309 2597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2665
y Fz(;)14 b(M)1427 2609 y Fu(2)1464 2597 y Fo(\))34 b(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2666
Fz(\034)9 b FC(.)27 b(Case)-150 2696 y(follo)n(ws)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2667
b(from)f(the)h(induction)f(hypotheses)f(and)h(the)i(typing)e(rules.)-48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2668
2796 y(C)t Fw(A)t(S)t(E)33 b FC(3)t(:)e Fo(\000)330 2808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2669
y Fu(1)406 2796 y Fv(`)39 b Fz(M)47 b Fo(:)39 b Fz(\034)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2670
FC(is)30 b Fo(\000)898 2808 y Fu(1)974 2796 y Fv(`)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2671
b Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)1395 2808 y Fu(1)1432
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2672
2796 y Fo(\))39 b(:)g Fz(\034)49 b Fv(!)38 b Fz(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2673
FC(.)-150 2896 y(Although,)33 b(one)h(has)h(to)g(pro)o(v)o(e)d(this)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2674
(case)f(for)f(all)h Fz(a)p FC(,)g(using)f(the)-150 2995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2675
y(v)n(ariable)22 b(con)m(v)o(ention)e(we)j(assume)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2676
Fz(a)h FC(does)f(not)f(occur)g(in)i Fo(\000)1785 3007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2677
y Fu(2)1822 2995 y FC(.)-150 3095 y(Then)17 b(we)h(kno)n(w)f(by)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2678
(induction)e(hypothesis)g(that)i Fo(\()p Fz(a)23 b Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2679
Fz(\034)5 b(;)14 b Fo(\000)1699 3107 y Fu(2)1736 3095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2680
y Fo(\))24 b Fv(`)-150 3195 y Fz(M)-69 3207 y Fu(1)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2681
3195 y Fo(:)h Fz(\033)g FC(holds.)c(Hence)f(also)i Fo(\000)776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2682
3207 y Fu(2)837 3195 y Fv(`)j Fp(lam)7 b Fo(\()p Fz(a;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2683
b(M)1245 3207 y Fu(1)1282 3195 y Fo(\))25 b(:)f Fz(\034)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2684
b Fv(!)24 b Fz(\033)h FC(by)c(the)-150 3294 y(typing)e(rules.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2685
1725 3294 4 57 v 1729 3241 50 4 v 1729 3294 V 1778 3294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2686
4 57 v 1413 w(\223)90 b(\224)-50 3433 y(Because)23 b(of)g(the)g(ar)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2687
(guably)e(questionable)h(use)h(of)g(the)g(v)n(ariable)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2688
3533 y(con)m(v)o(ention)j(in)j(the)g(third)f(case,)h(this)h(informal)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2689
(proof)g(is)j(painful)-150 3633 y(to)g(formalise)f(using)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2690
(original)e(induction)g(principle)h(\(6\))n(,)i(see)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2691
3732 y(for)25 b(e)o(xample)f([4)o(].)i(In)f(particular)m(,)e(the)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2692
(abstraction)e(case)i(in)g(a)g(for)n(-)-150 3832 y(mal)k(proof)e(will)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2693
(typically)e(require)h(the)g(abstraction)f(to)i(ha)n(v)o(e)f(its)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2694
3932 y(bound)e(v)n(ariable)g(renamed)g(to)i(be)g(suitably)f(fresh.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2695
(The)g(proof)f(of)-150 4031 y(the)20 b(lemma)g(also)g(then)g(requires)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2696
(an)h(equi)n(v)n(ariance)e(result)252 4188 y Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2697
b Fv(`)g Fz(M)32 b Fo(:)23 b Fz(\034)56 b Fv(\))46 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2698
Fo(\()p Fz(\031)863 4197 y Fj(\001)902 4188 y Fo(\000\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2699
b Fv(`)g Fo(\()p Fz(\031)1165 4197 y Fj(\001)1203 4188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2700
y Fz(M)9 b Fo(\))23 b(:)g Fz(\034)316 b FC(\(7\))-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2701
4345 y(to)26 b(be)g(sho)n(wn.)283 4315 y Fs(2)341 4345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2702
y FC(This)f(lemma)h(is)g(generally)f(useful,)g(and)g(because)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2703
4445 y(of)18 b(the)h(e)o(xistence)f(of)g(in)m(v)o(erses)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2704
(permutations)f(can)h(be)h(recast)f(as)252 4602 y Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2705
Fz(\031)334 4611 y Fj(\001)373 4602 y Fo(\000\))23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2706
Fv(`)g Fo(\()p Fz(\031)636 4611 y Fj(\001)675 4602 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2707
Fz(M)9 b Fo(\))23 b(:)g Fz(\034)56 b Fv(,)46 b Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2708
b Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\034)-150 4759 y FC(or)d(e)n(v)o(en)208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2709
4916 y Fo(\()p Fz(\031)290 4925 y Fj(\001)329 4916 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2710
Fo(\000\))j Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\034)56 b Fv(,)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2711
b Fo(\000)23 b Fv(`)g Fo(\()p Fz(\031)1120 4882 y Fq(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2712
Fu(1)1210 4925 y Fj(\001)1248 4916 y Fz(M)9 b Fo(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2713
b(:)g Fz(\034)-150 5074 y FC(which)d(is)i(a)g(useful)e(theorem)f(to)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2714
(re)n(write)g(with)g(because)f(it)i(collects)-150 5173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2715
y(all)34 b(of)f(the)h(permutations)e(in)h(a)h(typing)f(judgement)e(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2716
i(mo)o(v)o(es)-150 5273 y(them)20 b(so)g(that)h(the)o(y)e(apply)g(only)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2717
h(to)g(the)g(term)g(ar)o(gument.)p 0 TeXcolorgray -150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2718
5355 997 3 v -150 5406 a Fk(2)-113 5430 y Fw(The)d(proof)h(is)f(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2719
(easy)h(induction)h(using)f(\(6\).)p 0 TeXcolorgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2720
TeXcolorgray 0 TeXcolorgray 2141 66 a FC(\(Another)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2721
b(possibility)h(when)g(formalising)f(results)i(such)f(as)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2722
166 y(these)34 b(is)h(to)f(sho)n(w)g(that)g(the)h(relation)e(is)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2723
(preserv)o(ed)d(under)h(iter)n(-)2042 266 y(ated)f(v)n(ariable-for)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2724
(-v)n(ariable)c(substitutions.)k(This)h(result)f(applies)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2725
365 y(when)f(the)h(body)f(of)g(an)h(abstraction)f(acquires)g(an)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2726
(xtra)f(substi-)2042 465 y(tution)g(through)f(renaming.)g(Because)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2727
(substitutions)e(are)h(harder)2042 565 y(to)e(reason)g(with)g(than)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2728
(permutations,)e(this)j(approach)d(is)j(usually)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2729
664 y(less)d(attracti)n(v)o(e)f(than)g(performing)e(the)i(body')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2730
b(s)27 b(renaming)f(with)h(a)2042 764 y(permutation.\))2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2731
863 y(The)22 b(painful)f(requirement)e(to)j(rename)f(the)h(bound)e(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2732
(ariable)h(in)2042 963 y(the)15 b(rule)g(induction)f(tends)h(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2733
(happen)e(e)n(v)o(ery)g(time)i(a)g(rule)f(induction)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2734
1063 y(with)22 b(\(6\))g(is)h(performed)d(\(though)g(not)i(in)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2735
(proof)f(of)h(\(7\))n(,)h(pleas-)2042 1162 y(antly\).)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2736
b(W)-7 b(e)20 b(can)e(a)n(v)n(oid)g(this)h(by)f(pro)o(ving)e(a)i(more)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2737
(useful)g(induction)2042 1262 y(hypothesis)g(once)i(and)f(for)h(all:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2738
0 TeXcolorgray 2042 1401 a Fn(Theor)o(em)15 b(1.)p 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2739
TeXcolorgray 33 w Fx(F)-9 b(or)16 b(all)g(typing)f(conte)n(xts)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2740
Fo(\000)p Fx(,)h(all)g Fz(\013)p Fx(-equated)e Fz(\025)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2741
Fx(-terms)2042 1501 y Fz(M)9 b Fx(,)28 b(all)g Fz(\034)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2742
b Fv(2)38 b Fp(ty)d Fx(and)28 b(all)g(conte)n(xts)g Fz(x)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2743
b Fv(2)g Fx(fs-pset,)28 b(the)h(following)2042 1601 y(implication)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2744
b(holds:)2100 1758 y Fv(8)p Fz(x)14 b Fo(\000)g Fz(a)g(\034)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2745
b(:)46 b Fp(valid)10 b Fo(\(\000\))19 b Fv(^)g Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2746
Fz(a)k Fo(:)g Fz(\034)9 b Fo(\))24 b Fv(2)f Fo(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2747
Fv(\))g Fz(P)35 b Fo(\000)23 b(\()p Fp(var)10 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2748
Fz(a)p Fo(\)\))24 b Fz(\034)33 b(x)2100 1881 y Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2749
Fz(x)14 b Fo(\000)g Fz(M)22 b(N)h(\034)g(\033)n(:)2183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2750
1981 y Fo(\000)g Fv(`)g Fz(M)32 b Fo(:)23 b Fz(\034)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2751
b Fv(!)23 b Fz(\033)f Fv(^)c Fo(\()p Fv(8)p Fz(z)t(:)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2752
b(P)35 b Fo(\000)23 b Fz(M)32 b Fo(\()p Fz(\034)h Fv(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2753
b Fz(\033)s Fo(\))h Fz(z)t Fo(\))e Fv(^)2183 2080 y Fo(\000)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2754
Fv(`)g Fz(N)32 b Fo(:)23 b Fz(\034)28 b Fv(^)19 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2755
Fv(8)p Fz(z)t(:)45 b(P)35 b Fo(\000)23 b Fz(N)32 b(\034)g(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2756
Fo(\))46 b Fv(\))2183 2180 y Fz(P)35 b Fo(\000)23 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2757
Fp(app)7 b Fo(\()p Fz(M)t(;)14 b(N)9 b Fo(\)\))24 b Fz(\033)i(x)2100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2758
2303 y Fv(8)p Fz(x)14 b(a)g Fo(\000)g Fz(M)22 b(\034)h(\033)n(:)2183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2759
2403 y(a)g Fo(#)h Fz(x)42 b Fv(^)18 b Fz(a)23 b Fo(#)h(\000)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2760
b Fv(^)h Fo(\()p Fz(a)k Fo(:)g Fz(\034)5 b(;)14 b Fo(\000\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2761
b Fv(`)g Fz(M)32 b Fo(:)23 b Fz(\033)j Fv(^)2183 2502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2762
y Fo(\()p Fv(8)p Fz(z)t(:)c(P)35 b Fo(\()p Fz(a)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2763
Fo(:)h Fz(\034)5 b(;)14 b Fo(\000\))23 b Fz(M)31 b(\033)c(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2764
Fo(\))46 b Fv(\))2183 2602 y Fz(P)35 b Fo(\000)23 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2765
Fp(lam)8 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\)\))23 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2766
Fz(\034)33 b Fv(!)23 b Fz(\033)s Fo(\))h Fz(x)p 2059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2767
2652 1959 4 v 2578 2726 a Fo(\000)f Fv(`)g Fz(M)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2768
Fo(:)23 b Fz(\034)56 b Fv(\))46 b Fz(P)35 b Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2769
b Fz(M)32 b(\034)h(x)p 0 TeXcolorgray 2042 2892 a Fx(Pr)l(oof.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2770
0 TeXcolorgray 28 w FC(The)d(proof)g(uses)h(the)g(original)e(induction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2771
g(principle)h(\(6\))o(.)2042 2991 y(W)-7 b(e)21 b(strengthen)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2772
(goal)g(by)f(aiming)h(to)g(pro)o(v)o(e)2062 3157 y Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2773
Fz(\031)26 b Fo(\000)14 b Fz(M)22 b(\034)i Fo(\()p Fz(x)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2774
Fv(2)h Fx(fs-pset)p Fo(\))p Fz(:)46 b Fo(\()23 b Fz(:)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2775
b(:)g(:)23 b Fo(\))h Fv(\))f Fz(P)35 b Fo(\()p Fz(\031)3417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2776
3166 y Fj(\001)3455 3157 y Fo(\000\))24 b(\()p Fz(\031)3645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2777
3166 y Fj(\001)3683 3157 y Fz(M)9 b Fo(\))23 b Fz(\034)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2778
b(x)47 b(:)2042 3323 y FC(In)20 b(the)g(v)n(ariable-case)f(we)h(need)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2779
(to)g(pro)o(v)o(e)2617 3488 y Fz(P)35 b Fo(\()p Fz(\031)2787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2780
3497 y Fj(\001)2826 3488 y Fo(\000\))23 b(\()p Fp(var)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2781
b Fo(\()p Fz(\031)3172 3497 y Fj(\001)3211 3488 y Fz(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2782
Fo(\)\))23 b Fz(\034)33 b(x)2042 3654 y FC(while)22 b(kno)n(wing)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2783
(that)i Fp(valid)11 b Fo(\(\000\))23 b FC(and)e Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2784
Fz(a)27 b Fo(:)f Fz(\034)9 b Fo(\))28 b Fv(2)f Fo(\000)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2785
b FC(hold.)f(V)-9 b(alidity)2042 3754 y(of)37 b(conte)o(xts)f(is)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2786
(preserv)o(ed)c(under)h(permutations,)f(so)j(we)f(ha)n(v)o(e)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2787
3853 y Fp(valid)10 b Fo(\()p Fz(\031)2302 3862 y Fj(\001)2341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2788
3853 y Fo(\000\))p FC(.)18 b(From)f Fo(\()p Fz(a)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2789
Fo(:)h Fz(\034)9 b Fo(\))24 b Fv(2)f Fo(\000)18 b FC(we)g(can)g(infer)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2790
Fo(\()p Fz(\031)3563 3862 y Fj(\001)3601 3853 y Fz(a)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2791
b Fo(:)g Fz(\034)9 b Fo(\))25 b Fv(2)e Fz(\031)3944 3862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2792
y Fj(\001)3982 3853 y Fo(\000)2042 3953 y FC(and)c(hence)h(we)g(can)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2793
(use)h(the)f(assumed)g(implication)2153 4119 y Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2794
Fz(x)14 b Fo(\000)g Fz(a)g(\034)5 b(:)46 b Fp(valid)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2795
b Fo(\(\000\))19 b Fv(^)g Fo(\()p Fz(a)k Fo(:)g Fz(\034)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2796
b Fo(\))24 b Fv(2)f Fo(\000)g Fv(\))g Fz(P)35 b Fo(\000)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2797
b Fp(var)10 b Fo(\()p Fz(a)p Fo(\))23 b Fz(\034)33 b(x)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2798
4284 y FC(to)20 b(obtain)f Fz(P)35 b Fo(\()p Fz(\031)2525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2799
4293 y Fj(\001)2564 4284 y Fo(\000\))23 b(\()p Fp(var)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2800
b Fo(\()p Fz(\031)2910 4293 y Fj(\001)2949 4284 y Fz(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2801
Fo(\)\))23 b Fz(\034)33 b(x)p FC(.)2141 4384 y(The)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2802
b(application-case)e(is)j(routine.)e(The)h(interesting)f(case)i(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2803
(the)2042 4484 y(lambda-case.)h(In)i(this)h(case)f(we)h(need)f(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2804
(pro)o(v)o(e)2381 4649 y Fz(P)35 b Fo(\()p Fz(\031)2551
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2805
4658 y Fj(\001)2590 4649 y Fo(\000\))23 b(\()p Fp(lam)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2806
b Fo(\()p Fz(\031)2949 4658 y Fj(\001)2988 4649 y Fz(a;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2807
b(\031)3119 4658 y Fj(\001)3157 4649 y Fz(M)9 b Fo(\)\))24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2808
b(\()p Fz(\034)33 b Fv(!)23 b Fz(\033)s Fo(\))g Fz(x)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2809
4815 y FC(under)c(the)h(assumption)f(that)2398 4973 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2810
Fo(\()p Fz(i)p Fo(\))83 b Fz(a)23 b Fo(#)g(\000)2369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2811
5073 y(\()p Fz(ii)p Fo(\))83 b Fz(a)23 b Fo(:)g Fz(\034)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2812
b(;)14 b Fo(\000)23 b Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\033)2340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2813
5172 y Fo(\()p Fz(iii)p Fo(\))83 b Fv(8)p Fz(\031)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2814
b(x:)24 b(P)34 b Fo(\()p Fz(\031)2947 5181 y Fj(\001)2986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2815
5172 y Fo(\()p Fz(a)23 b Fo(:)h Fz(\034)5 b(;)14 b Fo(\000\)\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2816
b(\()p Fz(\031)3431 5181 y Fj(\001)3470 5172 y Fz(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2817
b Fo(\))23 b Fz(\033)j(x)3938 5073 y FC(\(8\))2042 5330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2818
y(Since)j(atoms,)h Fz(\025)p FC(-terms)f(and)g(typing)g(conte)o(xts)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2819
(are)i(\002nitely)f(sup-)2042 5430 y(ported)22 b(and)h(by)h(assumption)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2820
e(also)i Fz(x)p FC(,)h(there)e(e)o(xists)h(by)g(Lem.)f(4)h(an)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2821
0 TeXcolorgray 0 TeXcolorgray eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2822
%%Page: 6 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2823
TeXDict begin 6 5 bop 0 TeXcolorgray 0 TeXcolorgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2824
TeXcolorgray -150 66 a FC(atom)29 b Fz(c)g FC(with)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2825
Fz(c)39 b Fo(#)h(\()p Fz(\031)555 75 y Fj(\001)594 66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2826
y Fz(a;)14 b(\031)725 75 y Fj(\001)763 66 y Fz(M)t(;)g(\031)935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2827
75 y Fj(\001)974 66 y Fo(\000)p Fz(;)g(x)p Fo(\))p FC(.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2828
b(Using)f Fo(\()p Fz(iii)p Fo(\))g FC(we)g(can)-150 166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2829
y(infer)1 314 y Fv(8)p Fz(x:)23 b(P)35 b Fo(\(\()p Fz(c)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2830
b(\031)407 323 y Fj(\001)445 314 y Fz(a)p Fo(\))521 323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2831
y Fj(\001)560 314 y Fz(\031)610 323 y Fj(\001)648 314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2832
y Fo(\()p Fz(a)23 b Fo(:)g Fz(\034)5 b(;)14 b Fo(\000\)\))24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2833
b(\(\()p Fz(c)k(\031)1189 323 y Fj(\001)1228 314 y Fz(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2834
Fo(\))1304 323 y Fj(\001)1342 314 y Fz(\031)1392 323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2835
y Fj(\001)1430 314 y Fz(M)9 b Fo(\))23 b Fz(\034)33 b(x)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2836
463 y FC(which)20 b(is)-11 611 y Fv(8)p Fz(x:)j(P)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2837
b Fo(\()p Fz(c)24 b Fo(:)f Fz(\034)5 b(;)14 b Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2838
Fz(c)28 b(\031)578 620 y Fj(\001)616 611 y Fz(a)p Fo(\))692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2839
620 y Fj(\001)731 611 y Fz(\031)781 620 y Fj(\001)819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2840
611 y Fo(\000\))23 b(\(\()p Fz(c)28 b(\031)1104 620 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2841
Fj(\001)1143 611 y Fz(a)p Fo(\))1219 620 y Fj(\001)1257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2842
611 y Fz(\031)1307 620 y Fj(\001)1346 611 y Fz(M)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2843
Fo(\))23 b Fz(\034)32 b(x)140 b FC(\(9\))-150 759 y(From)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2844
b Fo(\()p Fz(i)p Fo(\))h FC(and)f(Lem.)g(2)p Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2845
Fz(i)p Fo(\))h FC(follo)n(ws)f(that)h Fz(\031)1134 768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2846
y Fj(\001)1172 759 y Fz(a)k Fo(#)h Fz(\031)1390 768 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2847
Fj(\001)1429 759 y Fo(\000)p FC(.)22 b(W)m(ith)h Fz(c)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2848
Fo(#)-150 859 y Fz(\031)-100 868 y Fj(\001)-62 859 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2849
Fo(\000)17 b FC(we)f(can)g(infer)f(using)g(Lem.)h(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2850
Fo(\()p Fz(ii)p Fo(\))g FC(that)g Fo(\()p Fz(c)27 b(\031)1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2851
868 y Fj(\001)1316 859 y Fz(a)p Fo(\))1392 868 y Fj(\001)1430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2852
859 y Fz(\031)1480 868 y Fj(\001)1519 859 y Fo(\000)c(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2853
Fz(\031)1731 868 y Fj(\001)1770 859 y Fo(\000)p FC(,)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2854
958 y(and)e(hence)f(simplify)h(\(9\))f(further)g(to)187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2855
1106 y Fv(8)p Fz(x:)j(P)35 b Fo(\()p Fz(c)23 b Fo(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2856
Fz(\034)5 b(;)14 b(\031)680 1115 y Fj(\001)718 1106 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2857
Fo(\000\))23 b(\(\()p Fz(c)28 b(\031)1003 1115 y Fj(\001)1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2858
1106 y Fz(a)p Fo(\))1118 1115 y Fj(\001)1156 1106 y Fz(\031)1206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2859
1115 y Fj(\001)1245 1106 y Fz(M)9 b Fo(\))23 b Fz(\034)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2860
b(x)199 b FC(\(10\))-150 1255 y(By)28 b(equi)n(v)n(ariance)c(of)j(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2861
(typing)f(relation)h(\(7\))n(,)h(we)f(can)g(use)h Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2862
Fz(ii)p Fo(\))-150 1354 y FC(and)20 b(infer)152 1503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2863
y Fo(\()p Fz(c)28 b(\031)298 1512 y Fj(\001)337 1503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2864
y Fz(a)p Fo(\))413 1512 y Fj(\001)451 1503 y Fz(\031)501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2865
1512 y Fj(\001)539 1503 y Fo(\()p Fz(a)c Fo(:)f Fz(\034)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2866
b(;)14 b Fo(\000\))23 b Fv(`)g Fo(\()p Fz(c)28 b(\031)1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2867
1512 y Fj(\001)1128 1503 y Fz(a)p Fo(\))1204 1512 y Fj(\001)1243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2868
1503 y Fz(\031)1293 1512 y Fj(\001)1331 1503 y Fz(M)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2869
Fo(:)23 b Fz(\033)-150 1651 y FC(which)d(is)306 1750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2870
y Fo(\()p Fz(c)j Fo(:)g Fz(\034)5 b(;)14 b(\031)571 1759
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2871
y Fj(\001)609 1750 y Fo(\000\))24 b Fv(`)e Fo(\()p Fz(c)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2872
b(\031)936 1759 y Fj(\001)975 1750 y Fz(a)p Fo(\))1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2873
1759 y Fj(\001)1089 1750 y Fz(\031)1139 1759 y Fj(\001)1178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2874
1750 y Fz(M)j Fo(:)24 b Fz(\033)320 b FC(\(11\))-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2875
1879 y(No)n(w)38 b(we)h(can)f(use)g Fz(c)56 b Fo(#)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2876
Fz(x)p FC(,)39 b(\(10\))n(,)g(\(11\))e(and)g(infer)h(from)f(the)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2877
1979 y(assumed)20 b(implication)f(in)h(the)g(lambda-case)f(that)86
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2878
2127 y Fz(P)35 b Fo(\()p Fz(\031)256 2136 y Fj(\001)294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2879
2127 y Fo(\000\))24 b(\()p Fz(l)r(am)p Fo(\()p Fz(c;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2880
b Fo(\()p Fz(c)27 b(\031)828 2136 y Fj(\001)866 2127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2881
y Fz(a)p Fo(\))942 2136 y Fj(\001)981 2127 y Fz(\031)1031
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2882
2136 y Fj(\001)1069 2127 y Fz(M)9 b Fo(\)\))24 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2883
Fz(\034)33 b Fv(!)23 b Fz(\033)s Fo(\))g Fz(x)-150 2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2884
y FC(Because)f Fz(c)j Fo(#)g Fz(\031)354 2284 y Fj(\001)393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2885
2275 y Fz(a)d FC(and)e Fz(c)26 b Fo(#)f Fz(\031)806 2284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2886
y Fj(\001)845 2275 y Fz(M)9 b FC(,)21 b(we)h(kno)n(w)e(by)h(\(5\))o(,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2887
(ho)n(we)n(v)o(er)m(,)-150 2375 y(that)135 2523 y Fz(l)r(am)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2888
Fo(\()p Fz(c;)14 b Fo(\()p Fz(c)27 b(\031)529 2532 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2889
Fj(\001)567 2523 y Fz(a)p Fo(\))643 2532 y Fj(\001)682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2890
2523 y Fz(\031)732 2532 y Fj(\001)770 2523 y Fz(M)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2891
Fo(\))23 b(=)g Fz(l)r(am)p Fo(\()p Fz(\031)1229 2532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2892
y Fj(\001)1267 2523 y Fz(a;)14 b(\031)1398 2532 y Fj(\001)1436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2893
2523 y Fz(M)9 b Fo(\))-150 2671 y FC(and)20 b(we)g(are)g(done.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2894
1782 2671 4 57 v 1786 2619 50 4 v 1786 2671 V 1835 2671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2895
4 57 v -50 2831 a(W)m(ith)g(this)h(induction)d(principle)h(at)i(our)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2896
(disposal,)f(the)i(proof)d(of)-150 2930 y(the)i(weak)o(ening)f(lemma)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2897
(is)h(simple.)p 0 TeXcolorgray -150 3089 a Fx(Pr)l(oof)f(of)g(the)h(W)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2898
-8 b(eak)o(ening)19 b(Lemma.)p 0 TeXcolorgray 28 w FC(Perform)49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2899
b(a)h(rule)g(induction)-150 3189 y(o)o(v)o(er)19 b Fo(\000)68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2900
3201 y Fu(1)128 3189 y Fv(`)k Fz(M)31 b Fo(:)24 b Fz(\034)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2901
b FC(using)20 b(the)g(induction)e(hypothesis)211 3337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2902
y Fo(\000)263 3349 y Fu(1)324 3337 y Fv(\022)k Fo(\000)463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2903
3349 y Fu(2)523 3337 y Fv(\))i Fp(valid)10 b Fo(\(\000)892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2904
3349 y Fu(2)929 3337 y Fo(\))24 b Fv(\))f Fo(\000)1143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2905
3349 y Fu(2)1203 3337 y Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\034)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2906
3485 y FC(That)d(is,)h(we)f(instantiate)g(Thm.)g(1)g(with)-96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2907
3620 y Fz(P)107 b Fo(=)83 b Fz(\025)p Fo(\000)312 3632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2908
y Fu(1)363 3620 y Fz(M)23 b(\034)g Fo(\000)578 3632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2909
Fu(2)615 3620 y Fz(:)46 b Fo(\000)736 3632 y Fu(1)797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2910
3620 y Fv(\022)22 b Fo(\000)936 3632 y Fu(2)996 3620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2911
y Fv(\))i Fp(valid)10 b Fo(\(\000)1365 3632 y Fu(2)1402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2912
3620 y Fo(\))24 b Fv(\))f Fo(\000)1616 3632 y Fu(2)1676
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2913
3620 y Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\034)-90 3719 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2914
Fo(\000)102 b(=)83 b(\000)264 3731 y Fu(1)-108 3819 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2915
Fz(M)91 b Fo(=)83 b Fz(M)-87 3919 y(\034)115 b Fo(=)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2916
b Fz(\034)-87 4018 y(x)104 b Fo(=)83 b(\000)264 4030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2917
y Fu(2)-150 4158 y FC(where)16 b Fo(\000)122 4170 y Fu(2)182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2918
4158 y Fv(2)24 b Fx(fs-pset)17 b FC(by)f(Lem.)h(3.)f(This)h(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2919
(es)g(the)f(follo)n(wing)g(three)-150 4257 y(sub-goals:)-150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2920
4398 y Fo(\(1\))47 b Fp(valid)11 b Fo(\(\000)266 4368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2921
y Fq(0)266 4418 y Fu(1)303 4398 y Fo(\))42 b Fv(^)g Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2922
Fz(a)23 b Fo(:)g Fz(\034)9 b Fo(\))24 b Fv(2)g Fo(\000)851
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2923
4368 y Fq(0)851 4418 y Fu(1)929 4398 y Fv(^)42 b Fo(\000)1078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2924
4368 y Fq(0)1078 4418 y Fu(1)1138 4398 y Fv(\022)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2925
Fo(\000)1278 4368 y Fq(0)1278 4418 y Fu(2)1356 4398 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2926
Fv(^)42 b Fp(valid)11 b Fo(\(\000)1716 4368 y Fq(0)1716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2927
4418 y Fu(2)1753 4398 y Fo(\))3 4497 y Fv(\))23 b Fo(\000)161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2928
4467 y Fq(0)161 4518 y Fu(2)222 4497 y Fv(`)f Fp(var)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2929
b Fo(\()p Fz(a)p Fo(\))23 b(:)h Fz(\034)-150 4621 y Fo(\(2\))47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2930
b(\000)55 4591 y Fq(0)55 4641 y Fu(1)116 4621 y Fv(`)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2931
b Fz(M)270 4633 y Fu(1)330 4621 y Fo(:)h Fz(\034)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2932
Fv(!)23 b Fz(\033)45 b Fv(^)d Fo(\000)792 4591 y Fq(0)792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2933
4641 y Fu(1)852 4621 y Fv(`)23 b Fz(M)1007 4633 y Fu(2)1067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2934
4621 y Fo(:)g Fz(\034)33 b Fv(^)3 4720 y Fo(\()p Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2935
Fo(\000)134 4690 y Fq(0)134 4741 y Fu(3)171 4720 y Fz(:)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2936
b Fo(\000)269 4690 y Fq(0)269 4741 y Fu(1)329 4720 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2937
Fv(\022)g Fo(\000)469 4690 y Fq(0)469 4741 y Fu(3)529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2938
4720 y Fv(\))g Fp(valid)10 b Fo(\(\000)897 4690 y Fq(0)897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2939
4741 y Fu(3)935 4720 y Fo(\))23 b Fv(\))g Fo(\000)1148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2940
4690 y Fq(0)1148 4741 y Fu(3)1208 4720 y Fv(`)g Fz(M)1363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2941
4732 y Fu(1)1423 4720 y Fo(:)g Fz(\034)33 b Fv(!)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2942
Fz(\033)s Fo(\))h Fv(^)3 4820 y Fo(\()p Fv(8)p Fo(\000)134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2943
4790 y Fq(0)134 4840 y Fu(3)171 4820 y Fz(:)f Fo(\000)269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2944
4790 y Fq(0)269 4840 y Fu(1)329 4820 y Fv(\022)g Fo(\000)469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2945
4790 y Fq(0)469 4840 y Fu(3)529 4820 y Fv(\))g Fp(valid)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2946
b Fo(\(\000)897 4790 y Fq(0)897 4840 y Fu(3)935 4820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2947
y Fo(\))23 b Fv(\))g Fo(\000)1148 4790 y Fq(0)1148 4840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2948
y Fu(3)1208 4820 y Fv(`)g Fz(M)1363 4832 y Fu(2)1423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2949
4820 y Fo(:)g Fz(\034)9 b Fo(\))24 b Fv(^)3 4920 y Fo(\000)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2950
4889 y Fq(0)55 4940 y Fu(1)116 4920 y Fv(\022)e Fo(\000)255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2951
4889 y Fq(0)255 4940 y Fu(2)334 4920 y Fv(^)42 b Fp(valid)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2952
b Fo(\(\000)693 4889 y Fq(0)693 4940 y Fu(2)730 4920
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2953
y Fo(\))3 5019 y Fv(\))23 b Fo(\000)161 4989 y Fq(0)161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2954
5040 y Fu(2)222 5019 y Fv(`)f Fp(app)7 b Fo(\()p Fz(M)541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2955
5031 y Fu(1)578 5019 y Fz(;)14 b(M)696 5031 y Fu(2)733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2956
5019 y Fo(\))23 b(:)g Fz(\033)-150 5142 y Fo(\(3\))47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2957
b Fz(a)23 b Fo(#)h(\000)215 5112 y Fq(0)215 5163 y Fu(2)293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2958
5142 y Fv(^)42 b Fz(a)23 b Fo(#)g(\000)601 5112 y Fq(0)601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2959
5163 y Fu(1)680 5142 y Fv(^)42 b Fo(\()p Fz(a)23 b Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2960
Fz(\034)5 b(;)14 b Fo(\000)1052 5112 y Fq(0)1052 5163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2961
y Fu(1)1089 5142 y Fo(\))24 b Fv(`)e Fz(M)32 b Fo(:)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2962
b Fz(\033)k Fv(^)3 5242 y Fo(\()p Fv(8)p Fo(\000)134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2963
5212 y Fq(0)134 5263 y Fu(3)171 5242 y Fz(:)c Fo(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2964
Fz(a)g Fo(:)g Fz(\034)5 b(;)14 b Fo(\000)492 5212 y Fq(0)492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2965
5263 y Fu(1)529 5242 y Fo(\))24 b Fv(\022)e Fo(\000)724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2966
5212 y Fq(0)724 5263 y Fu(3)808 5242 y Fv(\))h Fp(valid)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2967
b Fo(\(\000)1176 5212 y Fq(0)1176 5263 y Fu(3)1214 5242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2968
y Fo(\))46 b Fv(\))g Fo(\000)1473 5212 y Fq(0)1473 5263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2969
y Fu(3)1533 5242 y Fv(`)23 b Fz(M)32 b Fo(:)23 b Fz(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2970
Fo(\))h Fv(^)3 5342 y Fo(\000)55 5312 y Fq(0)55 5362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2971
y Fu(1)116 5342 y Fv(\022)e Fo(\000)255 5312 y Fq(0)255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2972
5362 y Fu(2)334 5342 y Fv(^)d Fp(valid)10 b Fo(\(\000)670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2973
5312 y Fq(0)670 5362 y Fu(2)707 5342 y Fo(\))3 5441 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2974
Fv(\))23 b Fo(\000)161 5411 y Fq(0)161 5462 y Fu(2)222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2975
5441 y Fv(`)f Fp(lam)7 b Fo(\()p Fz(a;)14 b(M)9 b Fo(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2976
b(:)g Fz(\034)33 b Fv(!)23 b Fz(\033)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2977
0 TeXcolorgray 2042 66 a FC(where)31 b(the)h(\002rst)g(tw)o(o)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2978
(tri)n(vial.)f(F)o(or)h Fo(\(3\))g FC(we)g(instantiate)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2979
Fv(8)p Fo(\000)3998 36 y Fq(0)3998 87 y Fu(3)2042 166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2980
y FC(with)24 b Fo(\()p Fz(a)30 b Fo(:)g Fz(\034)5 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2981
b Fo(\000)2503 136 y Fq(0)2503 187 y Fu(2)2541 166 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2982
Fo(\))p FC(.)24 b(The)g(f)o(act)g Fo(\()p Fz(a)30 b Fo(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2983
Fz(\034)5 b(;)14 b Fo(\000)3210 136 y Fq(0)3210 187 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2984
Fu(1)3247 166 y Fo(\))30 b Fv(\022)g Fo(\()p Fz(a)g Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2985
Fz(\034)5 b(;)14 b Fo(\000)3693 136 y Fq(0)3693 187 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2986
Fu(2)3730 166 y Fo(\))25 b FC(follo)n(ws)2042 266 y(from)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2987
b Fo(\000)2288 236 y Fq(0)2288 286 y Fu(1)2371 266 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2988
Fv(\022)45 b Fo(\000)2533 236 y Fq(0)2533 286 y Fu(2)2570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2989
266 y FC(;)34 b Fp(valid)10 b Fo(\()p Fz(a)46 b Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2990
Fz(\034)5 b(;)14 b Fo(\000)3126 236 y Fq(0)3126 286 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2991
Fu(2)3163 266 y Fo(\))33 b FC(follo)n(ws)f(from)f Fp(valid)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2992
b Fo(\(\000)3965 236 y Fq(0)3965 286 y Fu(2)4002 266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2993
y Fo(\))2042 365 y FC(and)32 b Fz(a)45 b Fo(#)h(\000)2451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2994
335 y Fq(0)2451 386 y Fu(2)2488 365 y FC(.)33 b(This)g(gi)n(v)o(es)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2995
(us)g Fo(\()p Fz(a)46 b Fo(:)g Fz(\034)5 b(;)14 b Fo(\000)3355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2996
335 y Fq(0)3355 386 y Fu(2)3392 365 y Fo(\))46 b Fv(`)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2997
Fz(M)54 b Fo(:)46 b Fz(\033)s FC(.)33 b(No)n(w)2042 465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2998
y(we)22 b(immediately)f(obtain)h Fo(\000)2882 435 y Fq(0)2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2999
486 y Fu(2)2946 465 y Fv(`)27 b Fp(lam)7 b Fo(\()p Fz(a;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3000
b(M)9 b Fo(\))27 b(:)g Fz(\034)37 b Fv(!)27 b Fz(\033)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3001
FC(using)c(the)2042 565 y(de\002nition)15 b(of)h(the)g(typing)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3002
(relation)h(and)f(the)i(f)o(act)f(that)h Fz(a)23 b Fo(#)g(\000)3829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3003
534 y Fq(0)3829 585 y Fu(2)3866 565 y FC(.)p 3974 565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3004
V 3978 512 50 4 v 3978 565 V 4027 565 4 57 v 2141 724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3005
a(This)17 b(e)o(xample)e(also)i(sho)n(ws)g(why)e(the)i(ne)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3006
(induction)f(principle)2042 823 y(needs)25 b(to)h(ha)n(v)o(e)e(uni)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3007
o(ersal)h(quanti\002cations)f(o)o(v)o(er)g(the)h(conte)o(xts)g(in)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3008
923 y(the)e(premises.)g(F)o(or)g(e)o(xample,)f(in)i(the)f(lambda-case)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3009
(the)h(assumed)2042 1022 y(implication)c(has)h(the)g(premise)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3010
Fo(\()p Fv(8)p Fz(z)t(:)i(P)35 b Fo(\()p Fz(a)23 b Fo(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3011
Fz(\034)5 b(;)14 b Fo(\000\))23 b Fz(M)32 b(\033)26 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3012
Fo(\))p FC(:)2104 1170 y Fv(8)p Fz(x)14 b(a)g Fo(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3013
Fz(M)22 b(\034)h(\033)n(:)47 b(a)23 b Fo(#)g Fz(x)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3014
b Fv(^)19 b Fz(a)k Fo(#)g(\000)c Fv(^)f Fo(\()p Fz(a)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3015
b Fo(:)f Fz(\034)5 b(;)14 b Fo(\000\))23 b Fv(`)g Fz(M)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3016
b Fo(:)23 b Fz(\033)k Fv(^)2613 1270 y Fo(\()p Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3017
Fz(z)t(:)22 b(P)35 b Fo(\()p Fz(a)23 b Fo(:)g Fz(\034)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3018
b(;)14 b Fo(\000\))23 b Fz(M)32 b(\033)26 b(z)t Fo(\))46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3019
b Fv(\))2613 1369 y Fz(P)35 b Fo(\000)23 b(\()p Fp(lam)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3020
b Fo(\()p Fz(a;)14 b(M)9 b Fo(\)\))23 b(\()p Fz(\034)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3021
b Fv(!)24 b Fz(\033)s Fo(\))f Fz(x)2042 1508 y FC(If)g(we)h(had)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3022
(stated)i(the)f(induction)f(principle)f(using)i(the)h(follo)n(wing)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3023
1608 y(simpler)c(implication)2692 1577 y Fs(3)2104 1755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3024
y Fv(8)p Fz(x)14 b(a)g Fo(\000)g Fz(M)22 b(\034)h(\033)n(:)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3025
b(a)23 b Fo(#)g Fz(x)42 b Fv(^)19 b Fz(a)k Fo(#)g(\000)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3026
Fv(^)f Fo(\()p Fz(a)24 b Fo(:)f Fz(\034)5 b(;)14 b Fo(\000\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3027
b Fv(`)g Fz(M)31 b Fo(:)23 b Fz(\033)k Fv(^)2613 1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3028
y Fz(P)35 b Fo(\()p Fz(a)23 b Fo(:)g Fz(\034)5 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3029
b Fo(\000\))24 b Fz(M)31 b(\033)c(x)46 b Fv(\))2613 1954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3030
y Fz(P)35 b Fo(\000)23 b(\()p Fp(lam)7 b Fo(\()p Fz(a;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3031
b(M)9 b Fo(\)\))23 b(\()p Fz(\034)33 b Fv(!)24 b Fz(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3032
Fo(\))f Fz(x)2042 2093 y FC(then)37 b(the)g(induction)e(hypothesis)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3033
(with)i(which)e(we)i(pro)o(v)o(ed)d(the)2042 2193 y(weak)o(ening)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3034
b(lemma,)i(namely)2403 2340 y Fo(\000)2455 2352 y Fu(1)2515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3035
2340 y Fv(\022)j Fo(\000)2655 2352 y Fu(2)2715 2340 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3036
Fv(\))g Fp(valid)11 b Fo(\(\000)3084 2352 y Fu(2)3121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3037
2340 y Fo(\))23 b Fv(\))g Fo(\000)3334 2352 y Fu(2)3395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3038
2340 y Fv(`)f Fz(M)32 b Fo(:)23 b Fz(\034)2042 2488 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3039
FC(w)o(ould)17 b(not)g(ha)n(v)o(e)h(been)f(strong)g(enough.)e(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3040
b(e)19 b(w)o(ould)e(ha)n(v)o(e)g(to)h(mak)o(e)2042 2588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3041
y(it)j(stronger)e(as)h(follo)n(ws)2312 2735 y Fv(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3042
Fo(\000)2411 2747 y Fu(2)2448 2735 y Fz(:)j Fo(\000)2546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3043
2747 y Fu(1)2606 2735 y Fv(\022)g Fo(\000)2746 2747 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3044
Fu(2)2806 2735 y Fv(\))g Fp(valid)10 b Fo(\(\000)3174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3045
2747 y Fu(2)3212 2735 y Fo(\))23 b Fv(\))g Fo(\000)3425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3046
2747 y Fu(2)3485 2735 y Fv(`)g Fz(M)32 b Fo(:)23 b Fz(\034)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3047
2883 y FC(b)n(ut)28 b(then)f(we)h(w)o(ould)g(not)f(be)h(able)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3048
(use)g(the)g(assumption)f Fz(a)37 b Fo(#)2042 2982 y(\000)2094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3049
2994 y Fu(2)2131 2982 y FC(,)h(which)g(w)o(as)g(vital)h(in)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3050
(weak)o(ening)e(lemma)i(to)g(get)g(the)2042 3082 y(lambda-case)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3051
b(through.)f(In)i(ef)n(fect,)g(we)h(w)o(ould)e(ha)n(v)o(e)h(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3052
(perform)2042 3182 y(renamings.)d(W)m(ith)i(the)f(v)o(ersion)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3053
(the)h(rule)f(induction)f(we)i(ha)n(v)o(e)2042 3281 y(gi)n(v)o(en,)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3054
b(this)j(is)g(unnecessary)-5 b(.)2042 3433 y Fn(3.2)82
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3055
b(Substituti)o(vity)19 b(of)h(One-Reduction)2042 3566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3056
y FC(The)34 b(central)h(lemma)f(in)h(proof)e(gi)n(v)o(en)h(in)h([2)o(])
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3057
g(for)f(the)h(Church-)2042 3666 y(Rosser)22 b(property)d(of)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3058
(beta-reduction)c(is)23 b(the)f(substituti)n(vity)f(of)g(the)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3059
3765 y @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3060
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3061
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3062
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3063
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  3064
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3065
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3067
@endspecial 2106 3776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3068
a Fm(1)2208 3765 y FC(-reduction)29 b(sho)n(wn)i(in)h(\(3\))o(.)g(The)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3069
(induction)f(principle)h(that)2042 3865 y(comes)20 b(with)g(this)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4600 y(nience:)g(we)h(can)f(relati)n(v)o(ely)g(easily)h(deri)n(v)o(e,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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