Publications/class-tcs.ps
author Christian Urban <urbanc@in.tum.de>
Fri, 06 Jan 2012 18:37:52 +0000
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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: pst-node.pro 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   437
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   438
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   439
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   440
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   441
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   442
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   443
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   444
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   445
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   446
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   447
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   448
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   449
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   450
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   451
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   452
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   453
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   454
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   455
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   456
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   457
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   458
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   459
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   460
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   461
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   462
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   463
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   464
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   465
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   466
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   467
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   468
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   469
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   470
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   471
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   472
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   473
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   474
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   475
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   476
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   477
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   478
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   479
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   480
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   481
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   482
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   483
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   484
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   485
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   486
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   487
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   488
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   489
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   490
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   491
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   492
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   493
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   494
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   495
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   496
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   497
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   498
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   499
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   500
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   501
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   502
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   503
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   504
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   505
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   506
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   507
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   508
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   509
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   510
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   511
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   512
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   513
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   514
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   515
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   516
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   517
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   518
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   519
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   520
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   521
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   522
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   523
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   524
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   525
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   526
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   527
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   528
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   529
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   530
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   531
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   532
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   533
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   534
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   535
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   536
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   537
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   538
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   539
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   540
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   541
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   542
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   543
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   544
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   545
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   546
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   547
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   548
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   549
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   550
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   551
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   552
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   553
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   554
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   555
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   556
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   557
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   558
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   559
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   560
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   561
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   562
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   563
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   564
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   565
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   566
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   567
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   568
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   569
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   570
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   571
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   572
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   573
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   574
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   575
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   576
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   577
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   578
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   579
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   580
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   581
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   582
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   583
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   584
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   585
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   586
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   587
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   588
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   589
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   590
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   591
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   592
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   593
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   594
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   595
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   596
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   597
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   598
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   599
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   600
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   601
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   602
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   603
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   604
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   605
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   606
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   607
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   608
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   609
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   610
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   611
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   612
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   613
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   614
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   615
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   616
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   617
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   618
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   619
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   620
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   621
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   622
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   623
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   624
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   625
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   626
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   627
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   628
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   629
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   630
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   631
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   632
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   633
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   634
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   635
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   636
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   637
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   638
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   639
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   640
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   641
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   642
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   643
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   644
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   645
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   646
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   648
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   649
%%BeginProcSet: 8r.enc 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   650
% File 8r.enc  TeX Base 1 Encoding  Revision 2.0  2002-10-30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   651
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   652
% @@psencodingfile@{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   653
%   author    = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   654
%                W. Schmidt, P. Lehman",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   655
%   version   = "2.0",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   656
%   date      = "30 October 2002",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   657
%   filename  = "8r.enc",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   658
%   email     = "tex-fonts@@tug.org",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   659
%   docstring = "This is the encoding vector for Type1 and TrueType
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   660
%                fonts to be used with TeX.  This file is part of the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   661
%                PSNFSS bundle, version 9"
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   662
% @}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   663
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   664
% The idea is to have all the characters normally included in Type 1 fonts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   665
% available for typesetting. This is effectively the characters in Adobe
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   666
% Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   667
% MacRoman, and some extra characters from Lucida.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   668
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   669
% Character code assignments were made as follows:
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   670
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   671
% (1) the Windows ANSI characters are almost all in their Windows ANSI
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   672
% positions, because some Windows users cannot easily reencode the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   673
% fonts, and it makes no difference on other systems. The only Windows
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   674
% ANSI characters not available are those that make no sense for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   675
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   676
% (173). quotesingle and grave are moved just because it's such an
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   677
% irritation not having them in TeX positions.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   678
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   679
% (2) Remaining characters are assigned arbitrarily to the lower part
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   680
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   681
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   682
% (3) Y&Y Lucida Bright includes some extra text characters; in the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   683
% hopes that other PostScript fonts, perhaps created for public
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   684
% consumption, will include them, they are included starting at 0x12.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   685
% These are /dotlessj /ff /ffi /ffl.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   686
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   687
% (4) hyphen appears twice for compatibility with both ASCII and Windows.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   688
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   689
% (5) /Euro was assigned to 128, as in Windows ANSI
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   690
%
Christian Urban <urbanc@in.tum.de>
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diff changeset
   691
% (6) Missing characters from MacRoman encoding incorporated as follows:
Christian Urban <urbanc@in.tum.de>
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diff changeset
   692
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   693
%     PostScript      MacRoman        TeXBase1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   694
%     --------------  --------------  --------------
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   695
%     /notequal       173             0x16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   696
%     /infinity       176             0x17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   697
%     /lessequal      178             0x18
Christian Urban <urbanc@in.tum.de>
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diff changeset
   698
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   699
%     /partialdiff    182             0x1A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   700
%     /summation      183             0x1B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   701
%     /product        184             0x1C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   702
%     /pi             185             0x1D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   703
%     /integral       186             0x81
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   704
%     /Omega          189             0x8D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   705
%     /radical        195             0x8E
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   706
%     /approxequal    197             0x8F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   707
%     /Delta          198             0x9D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   708
%     /lozenge        215             0x9E
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   709
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   710
/TeXBase1Encoding [
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   711
% 0x00
Christian Urban <urbanc@in.tum.de>
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diff changeset
   712
 /.notdef /dotaccent /fi /fl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   713
 /fraction /hungarumlaut /Lslash /lslash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   714
 /ogonek /ring /.notdef /breve
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   715
 /minus /.notdef /Zcaron /zcaron
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   716
% 0x10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   717
 /caron /dotlessi /dotlessj /ff
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   718
 /ffi /ffl /notequal /infinity
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   719
 /lessequal /greaterequal /partialdiff /summation
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   720
 /product /pi /grave /quotesingle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   721
% 0x20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   722
 /space /exclam /quotedbl /numbersign
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   723
 /dollar /percent /ampersand /quoteright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   724
 /parenleft /parenright /asterisk /plus
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   725
 /comma /hyphen /period /slash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   726
% 0x30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   727
 /zero /one /two /three
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   728
 /four /five /six /seven
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   729
 /eight /nine /colon /semicolon
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   730
 /less /equal /greater /question
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   731
% 0x40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   732
 /at /A /B /C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   733
 /D /E /F /G
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   734
 /H /I /J /K
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   735
 /L /M /N /O
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   736
% 0x50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   737
 /P /Q /R /S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   738
 /T /U /V /W
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   739
 /X /Y /Z /bracketleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   740
 /backslash /bracketright /asciicircum /underscore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   741
% 0x60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   742
 /quoteleft /a /b /c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   743
 /d /e /f /g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   744
 /h /i /j /k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   745
 /l /m /n /o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   746
% 0x70
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   747
 /p /q /r /s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   748
 /t /u /v /w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   749
 /x /y /z /braceleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   750
 /bar /braceright /asciitilde /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   751
% 0x80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   752
 /Euro /integral /quotesinglbase /florin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   753
 /quotedblbase /ellipsis /dagger /daggerdbl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   754
 /circumflex /perthousand /Scaron /guilsinglleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   755
 /OE /Omega /radical /approxequal
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   756
% 0x90
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   757
 /.notdef /.notdef /.notdef /quotedblleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   758
 /quotedblright /bullet /endash /emdash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   759
 /tilde /trademark /scaron /guilsinglright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   760
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   761
% 0xA0
Christian Urban <urbanc@in.tum.de>
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diff changeset
   762
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   763
 /currency /yen /brokenbar /section
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   764
 /dieresis /copyright /ordfeminine /guillemotleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   765
 /logicalnot /hyphen /registered /macron
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   766
% 0xD0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   767
 /degree /plusminus /twosuperior /threesuperior
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   768
 /acute /mu /paragraph /periodcentered
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   769
 /cedilla /onesuperior /ordmasculine /guillemotright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   770
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   771
% 0xC0
Christian Urban <urbanc@in.tum.de>
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diff changeset
   772
 /Agrave /Aacute /Acircumflex /Atilde
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   773
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   774
 /Egrave /Eacute /Ecircumflex /Edieresis
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   775
 /Igrave /Iacute /Icircumflex /Idieresis
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   776
% 0xD0
Christian Urban <urbanc@in.tum.de>
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diff changeset
   777
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   778
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   779
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   780
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   781
% 0xE0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   782
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   783
 /adieresis /aring /ae /ccedilla
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   784
 /egrave /eacute /ecircumflex /edieresis
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   785
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   786
% 0xF0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   787
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   788
 /ocircumflex /otilde /odieresis /divide
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   789
 /oslash /ugrave /uacute /ucircumflex
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   790
 /udieresis /yacute /thorn /ydieresis
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   791
] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   794
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
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diff changeset
   795
%%BeginProcSet: texps.pro 0 0
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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ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   801
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   802
div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   803
/nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end
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
   807
exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   808
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   809
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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
   811
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Christian Urban <urbanc@in.tum.de>
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diff changeset
   812
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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
   815
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   816
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   817
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   818
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   819
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   821
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   823
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   824
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   825
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   826
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   827
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
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diff changeset
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curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
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>
parents:
diff changeset
   831
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   832
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   833
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   834
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   835
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   836
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   837
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   838
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   839
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
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
   844
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   845
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   846
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   847
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   848
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   849
TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def
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>
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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rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   858
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   859
lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   860
ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   861
/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{
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:
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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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  1943
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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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  1957
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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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  1970
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Christian Urban <urbanc@in.tum.de>
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  1971
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  1978
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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>
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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>
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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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497 y(Calculus)409 950 y Fy(Gianluigi)d(Bellin)1203 907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fx(a)1279 950 y Fy(Martin)f(Hyland)2006 907 y Fx(b)2086
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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950 y Fy(Edmund)f(Robinson)2999 907 y Fx(a)1313 1099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2531
y Fy(Christian)g(Urban)2100 1056 y Fx(c)977 1266 y Fw(a)1017
Christian Urban <urbanc@in.tum.de>
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diff changeset
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1299 y Fv(Queen)24 b(Mary)-5 b(,)23 b(Univer)o(sity)j(of)d(London,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(UK)1159 1412 y Fw(b)1203 1445 y Fv(Univer)o(sity)h(of)f(Cambridg)o(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(,)g(UK)936 1558 y Fw(c)971 1591 y Fv(T)-8 b(ec)o(hnical)25
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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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>
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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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(Finally)f(we)f(e)o(xplain)h(the)g(consequences)j(of)c(insisting)j(on)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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d(more)83 2658 y(f)o(amiliar)g(cate)o(gorical)h(beha)n(viour)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(.)83 2925 y Fv(K)m(e)m(y)24 b(wor)m(ds:)46 b Ft(classical)26
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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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>
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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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>
parents:
diff changeset
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4120 y(around)36 b(1970)e(Per)j(Martin-L)8 b(\250)-41
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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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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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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(so)f(we)i(get)f(a)g(whole)f(range)i(of)f(cate)o(gorical)g(models)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4602 y(of)29 b(constructi)n(v)o(e)e(proof.)h(This)g(is)g(the)g(circle)h
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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4722 y(isomorphism.)22 b(W)-8 b(e)24 b(seek)g(analogues)g(of)g(these)f
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>
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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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(the)h(sequent)f(calculus)g(\(P)o(arigot)h([16]\))g(or)g(in)f
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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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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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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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>
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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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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1528 y(general)41 b(models.)f(But)g(all)h(that)f(is)g(parasitic)h(on)f
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>
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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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2009 y(considers)c(a)h(system)e(\(as)h(in)g([5]\))h(that)f(mix)o(es)e
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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Christian Urban <urbanc@in.tum.de>
parents:
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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
  2618
(ay)f(to)h(e)o(xtract)f(models)f(for)i(our)g(system)e(from)i(cate)o
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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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>
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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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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>
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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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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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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diff changeset
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4176 y(notion)e(of)h(model)g(of)g(classical)f(proof)h(simpler)g(than)f
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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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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(sequent)f(calculus)h(reading.)83 4778 y(The)33 b(question)e(of)i(what)
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
  2651
(a)h(delicate)g(one.)83 4898 y(The)e(referee)h(rightly)d(stressed)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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b Fq(g)t Fr(,)f Fq(h)i Fr(and)f Fq(k)s Fr(,)g(where)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2792
Fq(f)40 b Fr(and)83 996 y Fq(g)33 b Fr(come)d(before)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2793
Fq(h)g Fr(and)g Fq(k)s Fr(.)f(F)o(or)h(e)o(xample)f Fq(f)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2794
b Fr(and)29 b Fq(g)k Fr(might)c(plug)g(into)f Fq(h)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2795
Fr(and)g Fq(g)j Fr(also)d(into)e Fq(k)s Fr(.)83 1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2796
y(\(There)f(are)f(essentially)f(four)h(distinct)e(cases.\))i(It)g(will)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2797
f(al)o(w)o(ays)h(be)g(possible)e(to)h(determine)h(what)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2798
1236 y(we)f(mean)g(from)g(the)f(conte)o(xt.)83 1532 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2799
Fo(2.1.2)99 b Fn(\003)p Fo(-polycate)l(gories)83 1707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2800
y Fr(Our)25 b(third)g(design)f(criterion)g(amounts)g(to)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2801
(simplifying)e(decision)h(to)g(treat)h(ne)o(gation)f(implic-)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2802
1827 y(itly)-6 b(.)24 b(In)g(proof)h(theoretic)g(terms)f(that)g(is)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2803
(to)f(tak)o(e)h(a)g(formulation)e(with)h(an)h(in)l(v)n(olutory)f(ne)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2804
(gation)1330 2043 y Fm(\()p Fn(\000)p Fm(\))1483 2002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2805
y Fl(\003)1551 2043 y Fm(:)k Fq(p)f Fn(!)h Fq(p)1859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2806
2002 y Fl(\003)1926 2043 y Fq(;)44 b(p)2046 2002 y Fl(\003)2113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2807
2043 y Fn(!)27 b Fq(p)83 2258 y Fr(on)e(atomic)f(formulae,)g(and)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2808
(xtend)f(it)h(to)f(all)g(formulae)h(by)g(setting)1141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2809
2510 y Fn(>)1218 2474 y Fl(\003)1285 2510 y Fm(=)j Fn(?)644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2810
b(?)2187 2474 y Fl(\003)2255 2510 y Fm(=)27 b Fn(>)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2811
b Fq(;)878 2690 y Fm(\()p Fq(A)22 b Fn(^)g Fq(B)5 b Fm(\))1216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2812
2654 y Fl(\003)1284 2690 y Fm(=)27 b Fq(B)1466 2654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2813
Fl(\003)1528 2690 y Fn(_)22 b Fq(A)1689 2654 y Fl(\003)1847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2814
2690 y Fm(\()p Fq(A)g Fn(_)h Fq(B)5 b Fm(\))2186 2654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2815
y Fl(\003)2253 2690 y Fm(=)28 b Fq(B)2436 2654 y Fl(\003)2497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2816
2690 y Fn(^)23 b Fq(A)2659 2654 y Fl(\003)2715 2690 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2817
Fq(;)83 2916 y Fr(that)j(is,)g(more)g(or)h(less,)f(by)g(de)g(Mor)n(gan)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2818
g(duality)-6 b(.)24 b(The)j(c)o(yclic)f(choice)g(of)h(order)f(may)g(be)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2819
h(f)o(amiliar)83 3036 y(from)37 b(non-commutati)n(v)o(e)e(linear)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2820
(logic)g(\(Ruet)g([20]\).)h(It)f(is)g(not)g(strictly)f(necessary)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2821
(here,)h(b)n(ut)83 3157 y(serv)o(es)25 b(as)f(there)h(to)g(preserv)o(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2822
g(a)g(strict)f(duality)f(at)i(the)g(le)n(v)o(el)e(of)i(proofs.)f(Exact)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2823
h(duality)e(permits)h(a)83 3277 y(purely)k(one-sided)f(sequent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2824
(calculus)f(as)h(in)g(Girard)g([8],)g(b)n(ut)g(we)g(prefer)h(to)e(k)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2825
(eep)h(both)g(sides)f(in)83 3397 y(play)e(at)g(the)f(semantic)g(le)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2826
o(el.)g(Abstractly)g(we)h(get)g(a)g Fn(\003)p Fo(-polycate)l(gory)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2827
Fr(.)p 0 TeXcolorgray 83 3572 a Fs(De\002nition)34 b(2.2.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2828
0 TeXcolorgray 46 w Fr(A)f Fo(symmetric)g Fn(\003)p Fo(-polycate)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2829
(gory)f Fr(\(henceforth)i(just)e Fn(\003)p Fo(-polycate)l(gory)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2830
Fr(\))h Fn(P)41 b Fr(con-)83 3693 y(sists)36 b(of)i(a)f(polycate)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2831
(gory)f Fn(P)46 b Fr(equipped)37 b(with)f(an)i(in)l(v)n(olutory)d(ne)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2832
(gation)h Fm(\()p Fn(\000)p Fm(\))2892 3657 y Fl(\003)2969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2833
3693 y Fr(on)h(objects)g(to-)83 3813 y(gether)f(with)f(for)h(each)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2834
Fm(\000)p Fr(,)f Fm(\001)p Fr(,)h Fq(A)p Fr(,)g(an)f(isomorphism)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2835
Fn(P)8 b Fm(\(\000;)17 b(\001)p Fq(;)g(A)p Fm(\))2537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2836
3786 y Fn(\030)2538 3817 y Fm(=)2662 3813 y Fn(P)8 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2837
Fm(\()p Fq(A)2850 3777 y Fl(\003)2890 3813 y Fq(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2838
b Fm(\000;)g(\001\))35 b Fr(coherent)83 3934 y(with)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2839
b(re-ordering)h(and)g(composition.)83 4109 y(W)l(ith)k(this)f(in)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2840
(place)g(one)g(should)f(not)g(tak)o(e)i(the)f(talk)f(of)h(input)f(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2841
(output)f(abo)o(v)o(e)g(too)h(literally:)83 4229 y(according)c(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2842
(the)g Fn(\003)p Fr(-polycate)o(gorical)f(perspecti)n(v)o(e)f(an)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2843
(input)d(wire)j(of)f(kind)f Fq(A)h Fr(is)g(ef)n(fecti)n(v)o(ely)e(an)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2844
4350 y(output)j(wire)i(of)f(type)g Fq(A)952 4313 y Fl(\003)991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2845
4350 y Fr(.)g(W)-8 b(e)28 b(shall)e(not)h(need)g(to)g(pay)g(much)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2846
(attention)g(to)h(the)g Fm(\()p Fn(\000)p Fm(\))3093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2847
4313 y Fl(\003)3160 4350 y Fr(operation)83 4470 y(which)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2848
b(tak)o(es)g(polymaps)f Fm(\000)44 b Fn(!)f Fm(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2849
Fq(;)17 b(B)39 b Fr(to)33 b(polymaps)e Fq(B)2114 4434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2850
y Fl(\003)2154 4470 y Fq(;)17 b Fm(\000)43 b Fn(!)g Fm(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2851
Fr(.)34 b(Ho)n(we)n(v)o(er)e(we)i(shall)e(need)83 4590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2852
y(notation)24 b(for)h(v)n(ariants)f(of)h(the)f(identity)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2853
Fm(id)1579 4605 y Fk(A)1664 4590 y Fm(:)j Fq(A)h Fn(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2854
Fq(A)p Fr(.)e(W)-8 b(e)26 b(write)e(these)h(as)918 4806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2855
y Fm(in)999 4821 y Fk(A)1084 4806 y Fm(:)j Fn(\000)g(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2856
Fq(A)1444 4765 y Fl(\003)1500 4806 y Fq(;)17 b(A)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2857
b Fr(and)f Fm(ev)2057 4821 y Fk(A)2141 4806 y Fm(:)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2858
b Fq(A;)17 b(A)2386 4765 y Fl(\003)2453 4806 y Fn(!)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2859
b(\000)17 b Fq(:)83 5021 y Fr(These)25 b(can)g(be)g(pictured)g(as)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2860
(follo)n(ws.)1385 5382 y @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2861
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  0.1 true 8.5359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2862
5.69046 19.91682 22.76227 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2863
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2864
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2865
@endspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2866
1550 5215 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2867
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2868
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2869
 1550 5215 a 1645 5215 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2870
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2871
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2872
 1645 5215 a 1550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2873
5314 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2874
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2875
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2876
 1550 5314 a 1645 5314 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2877
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2878
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2879
 1645 5314 a 1385 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2880
a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2881
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2882
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2883
0.0 0 0 /N@A /N@B InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2884
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2885
 1385 5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2886
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2887
 1385 5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2888
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 16.11903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2889
8.2 0.0 NAngle 90 add  Uput exch pop add a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2890
exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2891
 1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2892
5382 a 1339 5417 a Fq(A)1412 5380 y Fl(\003)1385 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2893
y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2894
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2895
 1385 5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2896
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2897
 1385 5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2898
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2899
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2900
0.0 0 0 /N@C /N@D InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2901
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2902
 1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2903
5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2904
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2905
 1385 5382 a 1385 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2906
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 8.80824
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2907
8.2 0.0 NAngle 90 add  Uput exch pop add a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2908
exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2909
 1385 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2910
a 1348 5417 a Fq(A)1385 5382 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2911
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2912
 1385 5382 a 1385 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2913
a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2914
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2915
 1385 5382 a 1462 5288 a Fm(in)1999 5382 y @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2916
@setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2917
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  0.1 true 8.5359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2918
5.69046 19.91682 22.76227 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2919
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2920
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2921
@endspecial 1975 5215 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2922
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2923
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2924
 1975 5215 a 2070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2925
5215 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2926
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2927
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2928
 2070 5215 a 1975 5314 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2929
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2930
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2931
 1975 5314 a 2070 5314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2932
a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2933
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2934
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2935
 2070 5314 a 1999 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2936
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2937
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2938
0.0 0 0 /N@A /N@B InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2939
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2940
 1999 5382 a 1999 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2941
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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 1999
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2943
5382 a 1999 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2944
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 13.5583
Christian Urban <urbanc@in.tum.de>
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8.2 0.0 NAngle 90 add  Uput exch pop add a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
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exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
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diff changeset
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 1999 5382 a 1943 5417 a Fq(A)2016
Christian Urban <urbanc@in.tum.de>
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5380 y Fl(\003)1999 5382 y
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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 1999
Christian Urban <urbanc@in.tum.de>
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5382 a 1999 5382 a
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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 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2960
 1999 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2961
a 1999 5382 a
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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diff changeset
  2964
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2965
 1999 5382 a 1962 5417 a Fq(A)1999 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2966
y
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2967
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  2968
 1999 5382 a 1999 5382 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2969
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2970
 1999 5382 a 2069 5288 a Fm(ev)p
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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Fm(\001)p Fq(;)17 b(B)37 b Fr(to)31 b Fq(f)2740 47 y
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2978
Fl(\003)2820 83 y Fm(:)40 b Fq(B)2966 47 y Fl(\003)3006
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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203 y(is)26 b(implemented)f(by)h(composition:)e(one)i(has)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2981
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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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y Fl(\003)1424 324 y Fr(,)27 b(one)g(has)g Fq(g)1858
Christian Urban <urbanc@in.tum.de>
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diff changeset
  2987
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2988
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2989
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2990
Fr(as)h(in)f(the)h(follo)n(wing)e(picture.)2100 408 y
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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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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702 a
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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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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16 {InitRnode } NewNode end end
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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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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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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(we)g(consider)g(only)g Fn(>)p Fr(,)h Fn(^)p Fr(,)f(and)g(their)g(de)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(Mor)n(gan)e(duals,)h Fn(?)p Fr(,)g Fn(_)p Fr(.)h(Ne)o(gation)83
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>
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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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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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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
  3416
b(.\))23 b(The)h(problem)83 203 y(which)35 b(we)f(will)g(come)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)g(4.3)g(is)h(that)f(tak)o(en)g(together)g(with)g(contraction)g(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3418
h(weak)o(ening)83 324 y(this)22 b(naturality)g(equation)g(identi\002es)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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g(proofs)h(with)f(essentially)f(dif)n(ferent)i(collections)e(of)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(normal)83 444 y(forms.)83 624 y(Ho)n(we)n(v)o(er)c(there)h(are)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(cases)f(where)h(that)e(cannot)h(happen;)f(and)h(it)g(does)f(seem)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(reasonable)g(to)g(allo)n(w)83 745 y(some)30 b(maps)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3423
Fq(u)h Fr(and)g Fq(v)j Fr(to)d(slip)f(harmlessly)g(past)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(imagined)f(box)g(around)h Fm(\()p Fq(f)37 b Fn(\001)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3425
b Fq(g)t Fm(\))p Fr(.)30 b(After)g(all)83 865 y(we)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3426
b(ine)n(vitably)e(ha)n(v)o(e)966 1096 y Fn(f)p Fm(id)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3427
Fq(;)17 b Fm(id)o Fn(g)p Fm(;)g(\()p Fq(f)33 b Fn(\001)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3428
b Fq(g)t Fm(\))27 b(=)h Fq(f)33 b Fn(\001)22 b Fq(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3429
b Fm(=)c(\(id)o(;)17 b Fq(f)11 b Fm(\))22 b Fn(\001)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3430
Fm(\(id)o(;)17 b Fq(g)t Fm(\))26 b Fq(:)83 1326 y Fr(So)k(we)f(adopt)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3431
(a)h(restricted)g(form)g(of)g(an)g(idea)g(from)f([11].)h(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3432
b(e)29 b(call)g(maps)g Fq(u)p Fr(,)f Fq(v)33 b Fr(for)c(which)f(both)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3433
1446 y(the)d Fn(^)g Fr(equations)99 1677 y Fq(u)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3434
b(\()p Fq(f)32 b Fn(\001)22 b Fq(g)t Fm(\))27 b(=)g(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3435
Fq(u)p Fm(;)17 b Fq(f)11 b Fm(\))21 b Fn(\001)h Fq(g)e(;)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3436
b(v)t Fm(;)17 b(\()p Fq(f)33 b Fn(\001)21 b Fq(g)t Fm(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3437
b(=)h(\()p Fq(f)11 b Fm(\))22 b Fn(\001)f Fm(\()p Fq(v)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3438
Fm(;)c Fq(g)t Fm(\))g Fq(;)43 b Fr(and)25 b(so)i Fn(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3439
Fq(u;)17 b(v)t Fn(g)p Fm(;)g(\()p Fq(f)31 b Fn(\001)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3440
b Fq(g)t Fm(\))27 b(=)g(\()p Fq(u)p Fm(;)17 b Fq(f)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3441
b Fm(\))21 b Fn(\001)h Fm(\()p Fq(v)t Fm(;)17 b Fq(g)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3442
Fm(\))83 1907 y Fr(and)31 b(the)f(dual)g(equations)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3443
Fn(_)h Fr(hold)e Fo(linear)p Fr(.)h(\(This)f(de\002nition)h(does)g(mak)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3444
o(e)g(sense!\))g(W)-8 b(e)31 b(ha)n(v)o(e)83 2028 y(the)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3445
b(follo)n(wing.)83 2168 y Fs(Additional)38 b(assumption)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3446
Fo(Linear)f(maps)g(ar)l(e)g(closed)g(under)g(the)g(lo)o(gical)g(oper)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3447
(ations)e(intr)l(o-)83 2288 y(duced)25 b(abo)o(ve)o(.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3448
2429 y Fr(In)36 b(vie)n(w)e(of)i(the)f(other)g(naturalities,)f(the)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3449
(essential)e(assumption)f(is)i(that)g Fq(?)g Fr(is)g(linear)h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3450
(that)83 2549 y(linear)25 b(maps)f(are)i(closed)e(under)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3451
Fn(\000)e(\001)f(\000)p Fr(.)83 2865 y Fo(2.2.3)99 b(Commutation:)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3452
b(lo)o(gical)f(rules)83 3045 y Fr(The)33 b(polycate)o(gorical)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3453
(perspecti)n(v)o(e)g(supports)g(equalities)g(arising)g(from)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3454
(commuting)e(con-)83 3166 y(v)o(ersions)d(in)h(sequent)g(calculus.)f(W)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3455
-8 b(e)30 b(sk)o(etch,)f(again)f(using)g(our)h(imprecise)g(notation,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3456
(the)h(basic)83 3286 y(phenomena)c(for)g(the)f(binary)h(operators.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3466 y(First)g(gi)n(v)o(en)e(proofs)597 3697 y Fq(f)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3458
b Fm(:)27 b(\000)799 3712 y Fw(1)866 3697 y Fn(!)h Fm(\001)1075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3712 y Fw(1)1115 3697 y Fq(;)17 b(A;)g(B)32 b(;)116 b(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3460
b Fm(:)d(\000)1719 3712 y Fw(2)1786 3697 y Fn(!)f Fm(\001)1994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3461
3712 y Fw(2)2034 3697 y Fq(;)17 b(C)34 b(;)117 b(h)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3462
b Fm(:)h(\000)2525 3712 y Fw(3)2592 3697 y Fn(!)g Fm(\001)2801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3463
3712 y Fw(3)2840 3697 y Fq(;)17 b(D)30 b(;)83 3927 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3464
Fr(we)25 b(ha)n(v)o(e)g(\(perhaps)g(modulo)e(e)o(xchange\))i(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(equality)f(of)h(the)f(form)861 4157 y Fm(\()p Fq(f)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fn(\001)22 b Fq(g)t Fm(\))f Fn(\001)h Fq(h)28 b Fm(=)g(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(f)k Fn(\001)22 b Fq(h)p Fm(\))g Fn(\001)g Fq(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3468
b Fm(:)d(\000)g Fn(!)f Fm(\001)p Fq(;)17 b(A)22 b Fn(^)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(C)r(;)17 b(B)26 b Fn(^)d Fq(D)83 4388 y Fr(\(with)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3470
b Fm(\000)p Fr(,)h Fm(\001)p Fr(,)g(the)f(sum)g(of)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\000)1240 4403 y Fk(i)1297 4388 y Fr(and)g Fm(\001)1550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4403 y Fk(i)1608 4388 y Fr(respecti)n(v)o(ely\).)f(Of)i(course)f(there)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3473
h(are)g(other)g(v)o(ersions)83 4508 y(obtained)24 b(by)h(duality)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3474
4689 y(Secondly)g(gi)n(v)o(en)e(proofs)940 4919 y Fq(f)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fm(:)28 b Fq(A;)17 b(B)5 b(;)17 b Fm(\000)1382 4934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fw(1)1449 4919 y Fn(!)27 b Fm(\001)1657 4934 y Fw(1)1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4919 y Fq(;)17 b(C)34 b(;)117 b(g)31 b Fm(:)c(\000)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4934 y Fw(2)2249 4919 y Fn(!)h Fm(\001)2458 4934 y Fw(2)2497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4919 y Fq(;)17 b(D)31 b(;)83 5149 y Fr(we)25 b(ha)n(v)o(e)g(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(equality)f(of)h(form)p 1063 5301 59 4 v 1063 5380 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(f)32 b Fn(\001)22 b Fq(g)31 b Fm(=)p 1375 5301 181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4 v 28 w Fq(f)i Fn(\001)21 b Fq(g)31 b Fm(:)d Fq(A)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fn(^)h Fq(B)5 b(;)17 b Fm(\000)27 b Fn(!)h Fm(\001)p
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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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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b(B)5 b(;)17 b Fm(\000)28 b Fn(!)f Fm(\001)p Fq(;)17
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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1182 561 104 4 v 24 w Fm(\()i(\))e Fr(in)g(tw)o(o)f(dif)n(ferent)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(orders)h(getting)f(an)h(equality)f(of)h(the)f(form)p
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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1347 783 V 1347 800 V 27 w Fq(f)g Fm(:)28 b Fq(A)22 b
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Fn(^)g Fq(B)5 b(;)17 b Fm(\000)28 b Fn(!)f Fm(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(;)17 b(C)29 b Fn(_)23 b Fq(D)30 b(:)83 1097 y Fr(There)c(are)f(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ariants)f(by)h(duality)-6 b(.)23 b(The)h(picture)h(is)f(as)h(follo)n
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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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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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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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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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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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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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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3607
b Fq(D)1692 1413 y
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>
parents:
diff changeset
  3611
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3612
a 83 1584 a Fr(So)j(f)o(ar)g(we)g(ha)n(v)o(e)f(only)g(considered)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3613
g(binary)g(operators.)g(There)h(are)g(man)o(y)e(similar)h(e)o(xamples)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3614
83 1704 y(in)l(v)n(olving)e(also)i(the)f(rules)h(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3615
Fn(>)g Fr(which)g(we)g(merely)f(list.)763 1931 y Fq(f)822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3616
1890 y Fw(+)903 1931 y Fn(\001)e Fq(g)31 b Fm(=)d(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3617
Fq(f)k Fn(\001)22 b Fq(g)t Fm(\))1392 1890 y Fw(+)1467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3618
1931 y Fq(;)p 1710 1852 59 4 v 216 w(f)1769 1871 y Fw(+)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3619
1931 y Fm(=)p 1959 1852 118 4 v 28 w Fq(f)2018 1902 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3620
Fw(+)2093 1931 y Fq(;)216 b(f)2395 1890 y Fw(++)2537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3621
1931 y Fm(=)27 b Fq(f)2699 1890 y Fw(++)2829 1931 y Fq(:)83
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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n(ferent)g(orders)g(of)h(applying)e(rules)h(to)g(obtain)g(a)h(proof)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3624
2269 y(of)d Fn(>)p Fq(;)17 b Fm(\000)28 b Fn(`)g Fm(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3625
Fq(;)17 b Fn(?)p Fr(.\))25 b(W)-8 b(e)25 b(are)h(happ)o(y)e(to)h(adopt)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3626
f(all)g(these)h(equalities.)83 2564 y Fo(2.2.4)99 b(Reduction)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3627
2740 y Fr(Most)30 b(of)g(our)h(equalities)e(on)i(proofs)f(k)o(eep)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3628
(track)h(of)g(inessential)e(re)n(writings,)g(b)n(ut)h(in)g(itself)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3629
(that)83 2860 y(is)e(dull.)f(The)h(critical)f(equalities)g(tak)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3630
(account)g(of)g(meaning)g(preserving)f(reductions.)g(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3631
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3632
83 3148 y(Suppose)j(that)f Fq(f)42 b Fm(:)31 b Fq(A)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3633
Fn(`)f Fq(C)7 b Fr(,)27 b Fq(g)34 b Fm(:)e Fq(B)k Fn(`)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3634
b Fq(D)f Fr(and)c Fq(k)35 b Fm(:)c Fq(C)r(;)17 b(D)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3635
b Fn(`)f Fq(E)g Fr(are)c(proofs.)e(\(Again)g(we)h(suppress)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3261 y(further)e(conte)o(xts.\))f(W)-8 b(e)25 b(can)g(form)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3637
(proof)1090 3472 y Fj(A)g Fh(`)1239 3439 y Fk(f)1309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3638
3472 y Fj(C)98 b(B)30 b Fh(`)1627 3439 y Fk(g)1692 3472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3639
y Fj(D)p 1090 3492 680 4 v 1160 3572 a(A;)15 b(B)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3640
Fh(`)25 b Fj(C)h Fh(^)20 b Fj(D)1872 3454 y(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3641
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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3492 429 4 v 1860 3572 a(C)e Fh(^)20 b Fj(D)28 b Fh(`)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3643
Fj(E)p 1160 3609 1129 4 v 1544 3689 a(A;)15 b(B)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3644
Fh(`)25 b Fj(E)2330 3640 y Fi(CUT)83 3886 y Fr(which)g(reduces)g(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3645
1083 3975 y Fj(A)h Fh(`)1233 3942 y Fk(f)1303 3975 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3646
Fj(C)97 b(B)30 b Fh(`)1620 3942 y Fk(g)1685 3975 y Fj(D)94
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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Fj(E)p 1083 4013 1177 4 v 1491 4093 a(A;)15 b(B)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3649
Fh(`)24 b Fj(E)2301 4044 y Fi(CUTs)83 4264 y Fr(where)h(by)f(associati)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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n(vity)e(we)j(write)f(the)g(tw)o(o)g(Cuts)g(together)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(.)24 b(This)f(gi)n(v)o(es)g(a)i(simple)e(equation)g(for)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3652
4384 y(our)i(polycate)o(gory:)1367 4504 y Fm(\()p Fq(f)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3653
b Fn(\001)22 b Fq(g)t Fm(\);)p 1669 4425 55 4 v 17 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3654
Fq(k)30 b Fm(=)d Fn(f)p Fq(f)5 b(;)17 b(g)t Fn(g)p Fm(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3655
Fq(k)29 b(:)83 4644 y Fr(Similarly)24 b(suppose)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3656
Fq(f)39 b Fm(:)28 b Fq(A)f Fn(`)h Fq(B)i Fr(is)24 b(a)h(proof.)g(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3657
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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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4805 y Fk(f)1903 4838 y Fj(B)p 1651 4857 359 4 v 1651
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3660
4937 a Fh(>)p Fj(;)15 b(A)25 b Fh(`)g Fj(B)p 1369 4975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3661
641 4 v 1565 5055 a(A)h Fh(`)f Fj(B)2051 5006 y Fi(CUT)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5259 y Fr(and)g(this)f(reduces)h(outright)f(to)1626 5380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3663
y Fq(A)j Fn(`)1787 5339 y Fk(f)1860 5380 y Fq(B)33 b(:)p
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>
parents:
diff changeset
  3670
b Fq(f)1734 257 y Fw(+)1820 298 y Fm(=)28 b Fq(f)38 b(:)83
Christian Urban <urbanc@in.tum.de>
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diff changeset
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514 y Fr(These)23 b(equations)e(\(and)h(their)g(duals\))g(constitute)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fo(the)h(r)l(eduction)g(principle)f(for)h(lo)o(gical)f(cuts)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3673
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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
  3675
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fr(The)27 b(structural)g(rule)g(of)g(Exchange)g(is)g(implicit)e(in)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(our)g(notion)e(of)j(symmetric)d Fn(\003)p Fr(-polycate)o(gory)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3678
b(,)83 1502 y(b)n(ut)25 b(we)g(need)g(to)f(consider)h(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3679
b(eak)o(ening)24 b(and)h(Contraction.)294 1690 y Fi(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3680
Fh(`)g Fi(\001)p 240 1710 348 4 v 240 1790 a Fj(A;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3681
b Fi(\000)26 b Fh(`)e Fi(\001)629 1740 y Ft(W)-6 b(-L)885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3682
1790 y Fj(;)1074 1690 y Fi(\000)25 b Fh(`)f Fi(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3683
1017 1710 353 4 v 1017 1790 a(\000)h Fh(`)g Fi(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3684
Fj(;)15 b(B)1411 1740 y Ft(W)-6 b(-R)1672 1790 y Fj(;)1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3685
1673 y(A;)15 b(A;)g Fi(\000)26 b Fh(`)f Fi(\001)p 1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3686
1710 456 4 v 1858 1790 a Fj(A;)15 b Fi(\000)26 b Fh(`)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3687
Fi(\001)2301 1741 y Ft(C-L)2538 1790 y Fj(;)2670 1673
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3688
y Fi(\000)g Fh(`)g Fi(\001)p Fj(;)15 b(B)5 b(;)15 b(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3689
2670 1710 467 4 v 2727 1790 a Fi(\000)25 b Fh(`)g Fi(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3690
Fj(;)15 b(B)3178 1741 y Ft(C-R)3355 1790 y Fj(:)83 1988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3691
y Fr(Naturalities)35 b(implicit)e(in)i(proof)g(nets)g(in)g(tandem)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3692
(with)g(our)g(reduction)g(principle)g(for)g(logical)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3693
2108 y(cuts)25 b(suggest)f(a)h(nice)g(w)o(ay)f(to)h(represent)g(these)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3694
(in)f(our)h Fn(\003)p Fr(-polycate)o(gory)-6 b(.)83 2276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3695
y(W)e(e)34 b(treat)f(contraction)f(\002rst.)h(F)o(or)g(all)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3696
Fq(A)p Fr(,)h(`generic')h(instances)e(of)h(contraction)f(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3697
(maps)g Fq(d)42 b Fm(:)83 2389 y Fq(A)28 b Fn(!)f Fq(A)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3698
Fn(^)f Fq(A)j Fr(and)g Fq(m)j Fm(:)g Fq(A)22 b Fn(_)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3699
Fq(A)k Fn(!)h Fq(A)d Fr(arising)f(from)g(the)h(proofs)816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3700
2559 y Fj(A)h Fh(`)f Fj(A)91 b(A)25 b Fh(`)g Fj(A)p 816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3701
2579 576 4 v 844 2659 a(A;)15 b(A)26 b Fh(`)f Fj(A)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3702
b Fh(^)g Fj(A)1434 2609 y Fh(^)p Ft(-R)p 844 2696 521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3703
4 v 898 2776 a Fj(A)26 b Fh(`)f Fj(A)20 b Fh(^)g Fj(A)1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3704
2727 y Ft(C-L)1767 2776 y Fj(;)1989 2559 y(A)26 b Fh(`)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3705
Fj(A)92 b(A)25 b Fh(`)g Fj(A)p 1989 2579 576 4 v 2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3706
2659 a(A)20 b Fh(_)g Fj(A)26 b Fh(`)e Fj(A;)15 b(A)2606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3707
2609 y Fh(_)p Ft(-L)p 2017 2696 521 4 v 2071 2776 a Fj(A)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3708
b Fh(_)e Fj(A)26 b Fh(`)f Fj(A)2579 2727 y Ft(C-R)2778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3709
2776 y Fj(:)83 2974 y Fr(These)h(are)h(ob)o(viously)d(constructed)h(as)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3710
h(de)g(Mor)n(gan)f(duals,)g(so)h(we)g(assume)f(that)h(the)o(y)f(are)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3711
(inter)n(-)83 3094 y(changed)e(by)g(the)f(duality)g(in)g(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3712
Fn(\003)p Fr(-polycate)o(gory)-6 b(,)23 b(that)i(is,)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3713
3310 y Fm(\()p Fq(d)1265 3325 y Fk(A)1321 3310 y Fm(\))1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3714
3268 y Fl(\003)1426 3310 y Fm(=)j Fq(m)1615 3325 y Fk(A)1668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3715
3306 y Fg(\003)1725 3310 y Fq(;)116 b Fm(\()p Fq(m)1991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3716
3325 y Fk(A)2048 3310 y Fm(\))2086 3268 y Fl(\003)2153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3717
3310 y Fm(=)28 b Fq(d)2308 3325 y Fk(A)2361 3306 y Fg(\003)2417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3718
3310 y Fq(:)83 3525 y Fr(It)c(is)f(consonant)g(with)g(earlier)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3719
(assumptions)d(to)i(suppose)g(that)g(we)h(can)g(implement)e(the)h(C-L)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3720
(rule)83 3645 y(by)31 b(composition)e(with)h(its)h(`generic')g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3721
(instance)g Fq(d)p Fr(:)g(that)f(is,)h(we)g(form)p 2605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3722
3566 59 4 v 31 w Fq(f)50 b Fm(:)40 b Fq(A)27 b Fn(^)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3723
Fq(A;)17 b Fm(\000)39 b Fn(`)g Fm(\001)32 b Fr(and)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3724
3766 y(then)25 b(compose)f(with)g Fq(d)h Fr(to)f(gi)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3725
Fq(d)p Fm(;)p 1326 3687 V 17 w Fq(f)38 b Fm(:)28 b Fq(A;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3726
b Fm(\000)27 b Fn(`)h Fm(\001)d Fr(as)g(in)g(the)f(follo)n(wing)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3727
(picture.)2046 4079 y @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3728
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3729
5.69046 17.07181 22.76227 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3730
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3731
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3732
@endspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3733
2111 3990 a Fq(f)1946 3989 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3734
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3735
} false /N@A 16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3736
 1946 3989 a 20 w @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3737
@setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3738
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3739
0.0 add 2 div 2 copy 0.0 sub 4 2 roll Pyth 0.28453 add CLW 2 div add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3740
0 360 arc closepath gsave 0.85358 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3741
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3742
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3743
@endspecial Fn(^)2093 3914 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3744
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3745
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3746
 2093 3914 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3747
2093 4009 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3748
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3749
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3750
 2093 4009 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3751
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3752
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3753
0.0 0 0 /N@A /N@B InitNC { /AngleA 10.  def /AngleB 180.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3754
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3755
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3756
 2046 4079 a 2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3757
4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3758
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3759
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3760
0.0 0 0 /N@A /N@C InitNC { /AngleA -10.  def /AngleB 180.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3761
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3762
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3763
 2046 4079 a 1621 3961 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3764
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@D
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3765
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3766
 1621 3961 a 2046 4079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3767
a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3768
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3769
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3770
0.0 0 0 /N@A /N@D InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3771
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3772
 2046 4079 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3773
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3774
 2046 4079 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3775
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 30.94975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3776
8.2 0.0 NAngle 90 sub  Uput exch pop add a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3777
exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3778
 2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3779
4079 a 1918 4114 a Fq(A)f Fn(^)g Fq(A)2046 4079 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3780
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3781
 2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3782
4079 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3783
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3784
 2046 4079 a 1548 3990 a Fq(d)2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3785
4079 y @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3786
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3787
5.69046 -51.21501 22.76227 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3788
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3790
@endspecial 1526 3961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3792
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3793
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3794
 1526 3961 a 1385 3961 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3795
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3796
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3797
 1385 3961 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3798
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3799
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0.0 0 0 /N@E /N@F InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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 2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3803
4079 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3804
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3805
 2046 4079 a 2046 4079 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3806
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3807
8.2 0.0 NAngle 90 sub  Uput exch pop add a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3808
exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3809
 2046 4079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3810
a 2009 4114 a Fq(A)2046 4079 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3811
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3812
 2046 4079 a 2046 4079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3813
a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3814
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3815
 2046 4079 a 83 4248 a Fr(Dually)f Fq(m)28 b Fm(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3816
Fq(A)11 b Fn(_)g Fq(A)28 b Fn(!)g Fq(A)22 b Fr(implements)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3817
(contraction)j(on)f(the)h(right:)e(contracting)h Fq(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3818
b Fm(:)d Fq(B)33 b Fn(!)27 b Fq(D)s(;)17 b(D)83 4368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3819
y Fr(on)25 b(the)g(right)f(is)p 659 4315 51 4 v 24 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3820
Fq(g)s Fm(;)17 b Fq(m)p Fr(.)83 4536 y(Similarly)23 b(we)h(ha)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3821
(a)g(w)o(ay)g(to)g(implement)e(weak)o(ening.)h(In)h(our)g(polycate)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3822
(gory)f(we)h(should)f(ha)n(v)o(e)83 4649 y(maps)i Fq(t)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3823
Fm(:)h Fq(A)g Fn(!)f(>)f Fr(and)f Fq(u)i Fm(:)g Fn(?)i(!)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3824
Fq(A)e Fr(arising)f(from)h(the)f(proofs)p 1096 4787 243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3825
4 v 1187 4867 a Fh(`)h(>)1380 4819 y(>)p Ft(-R)p 1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3826
4887 245 4 v 1095 4967 a Fj(A)g Fh(`)g(>)1382 4917 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3827
Ft(W)-6 b(-L)1729 4967 y Fj(;)p 1953 4787 243 4 v 1953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3828
4867 a Fh(?)24 b(`)2236 4819 y(?)p Ft(-L)p 1951 4887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3829
245 4 v 1951 4967 a Fh(?)h(`)g Fj(A)2238 4917 y Ft(W)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3830
b(-R)2499 4967 y Fj(:)83 5164 y Fr(Again)24 b(these)h(are)g(de)g(Mor)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3831
(gan)f(duals)h(and)f(should)g(be)h(interchanged)f(by)h(duality:)1221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3832
5380 y Fm(\()p Fq(t)1294 5395 y Fk(A)1351 5380 y Fm(\))1389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3833
5339 y Fl(\003)1456 5380 y Fm(=)j Fq(u)1616 5395 y Fk(A)1669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3834
5376 y Fg(\003)1725 5380 y Fq(;)116 b Fm(\()p Fq(u)1962
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3835
5395 y Fk(A)2018 5380 y Fm(\))2056 5339 y Fl(\003)2124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3836
5380 y Fm(=)27 b Fq(t)2262 5395 y Fk(A)2315 5376 y Fg(\003)2372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3837
5380 y Fq(:)p 0 TeXcolorgray 1773 5712 a Fr(9)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%Page: 10 10
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
  3841
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3842
f(a)h(proof)g Fq(f)44 b Fm(:)33 b(\000)g Fn(`)g Fm(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3843
Fr(,)28 b(and)g(we)g(wish)f(to)g(weak)o(en)h(on)g(the)g(left.)f(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3844
b(e)83 203 y(form)28 b Fq(f)364 167 y Fw(+)455 203 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3845
Fm(:)k Fn(>)p Fq(;)17 b Fm(\000)33 b Fn(`)f Fm(\001)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3846
Fr(and)f(compose)g(with)f Fq(t)i Fr(to)f(gi)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3847
Fq(t)p Fm(;)17 b Fq(f)2189 167 y Fw(+)2280 203 y Fm(:)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3848
b Fq(A;)17 b Fm(\000)32 b Fn(`)g Fm(\001)p Fr(.)c(Thus)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3849
Fq(t)g Fr(can)h(be)g(used)83 324 y(to)e(implement)e(weak)o(ening)i(on)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3850
(the)h(left.)g(Dually)f Fq(u)g Fr(can)i(be)f(used)f(to)h(implement)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3851
(weak)o(ening)i(on)83 444 y(the)f(right:)f(in)g(that)h(case)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3852
Fq(g)j Fr(is)c(weak)o(ened)i(to)e Fq(g)1686 408 y Fw(+)1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3853
444 y Fm(;)17 b Fq(u)p Fr(.)83 740 y Fo(2.3.2)99 b(Commuting)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3854
b(con)l(ver)o(sions)83 915 y Fr(Implementing)29 b(rules)i(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3855
(composition)e(with)i(generic)h(instances)f(tak)o(es)h(care)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3856
(naturality)f(is-)83 1035 y(sues;)d(and)h(some)f(commuting)e(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3857
(ersions)h(are)j(an)e(immediate)g(consequence)g(of)h(the)g(associa-)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3858
1156 y(ti)n(vity)23 b(of)i(composition)e(in)h(a)h(polycate)o(gory)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3859
b(.)23 b(Ho)n(we)n(v)o(er)h(there)h(are)g(more)g(such.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3860
1331 y(W)-8 b(e)28 b(e)o(xpect)g(C-L)g(to)f(enjo)o(y)h(the)f(same)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3861
(commuting)e(possibilities)e(as)k Fn(^)p Fr(-L.)g(This)f(requires)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3862
(equa-)83 1451 y(tions)c(of)h(the)g(form)660 1660 y Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3863
Fq(d)p Fm(;)17 b Fq(f)11 b Fm(\))21 b Fn(\001)h Fq(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3864
b Fm(=)d Fq(d)p Fm(;)17 b(\()p Fq(f)32 b Fn(\001)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3865
Fq(g)t Fm(\))27 b Fq(;)116 b(d)p Fm(;)p 1760 1580 59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3866
4 v 17 w Fq(f)38 b Fm(=)p 1949 1580 154 4 v 27 w Fq(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3867
Fm(;)17 b Fq(f)38 b(;)116 b(d)p Fm(;)17 b Fq(f)2427 1618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3868
y Fw(+)2513 1660 y Fm(=)28 b(\()p Fq(d)p Fm(;)17 b Fq(f)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3869
b Fm(\))2847 1618 y Fw(+)2932 1660 y Fq(:)83 1868 y Fr(\(In)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3870
b(the)g(second)f(equation,)g(the)h(typing)e(should)h(gi)n(v)o(e)f(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3871
(commuting)e(con)l(v)o(ersion)h(in)g Fq(d)p Fm(;)p 3224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3872
1789 59 4 v 17 w Fq(f)10 b Fr(,)29 b(not)f(a)83 1988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3873
y(logical)c(cut.\))h(Similar)f(considerations)g(for)h(W)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3874
b(-L)25 b(and)f Fn(>)p Fr(-L)i(gi)n(v)o(e)d(the)i(equations)707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3875
2197 y Fm(\()p Fq(t)p Fm(;)17 b Fq(f)11 b Fm(\))21 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3876
Fn(\001)h Fq(g)31 b Fm(=)d Fq(t)p Fm(;)17 b(\()p Fq(f)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3877
b Fn(\001)22 b Fq(g)t Fm(\))27 b Fq(;)116 b(t)p Fm(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3878
1759 2118 V 17 w Fq(f)39 b Fm(=)p 1949 2118 138 4 v 27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3879
w Fq(t)p Fm(;)17 b Fq(f)39 b(;)116 b(t)p Fm(;)17 b Fq(f)2396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3880
2156 y Fw(+)2482 2197 y Fm(=)28 b(\()p Fq(t)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3881
b Fq(f)11 b Fm(\))2800 2156 y Fw(+)2886 2197 y Fq(:)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3882
2405 y Fr(\(In)38 b(the)f(last)g(equation)f(the)h(typing)g(should)f(gi)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3883
n(v)o(e)g(a)h(commuting)e(con)l(v)o(ersion)i(in)f Fq(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3884
Fm(;)17 b Fq(f)3206 2369 y Fw(+)3265 2405 y Fr(,)37 b(not)g(a)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3885
2526 y(logical)24 b(cut.\))h(W)-8 b(e)25 b(tak)o(e)g(all)g(these.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3886
2821 y Fo(2.3.3)99 b(Corr)l(ectness)25 b(equations)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3887
2996 y Fr(There)20 b(are)g(further)g(issues)f(to)g(consider)g(arising)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3888
(from)g(the)g(decision)g(to)g(implement)f(the)h(structural)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3889
3117 y(rules.)i(W)-8 b(e)21 b(implement)e(contraction)g(via)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3890
(composition)d(with)i Fq(d)27 b Fm(:)h Fq(A)g Fn(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3891
Fq(A)7 b Fn(^)g Fq(A)21 b Fr(and)f Fq(m)28 b Fm(:)g Fq(A)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3892
b Fn(_)g Fq(A)28 b Fn(!)83 3237 y Fq(A)p Fr(.)e(But)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3893
Fq(d)g Fr(and)h Fq(m)f Fr(are)h(themselv)o(es)e(produced)h(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3894
(contractions)g(on)g(proofs)g Fm(id)2866 3252 y Fk(A)2945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3895
3237 y Fn(\001)e Fm(id)3077 3252 y Fk(A)3162 3237 y Fm(:)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3896
b Fq(A;)17 b(A)29 b Fn(!)83 3357 y Fq(A)14 b Fn(^)g Fq(A)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3897
b Fr(and)g Fm(id)594 3372 y Fk(A)665 3357 y Fn(\001)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3898
b Fm(id)788 3372 y Fk(A)872 3357 y Fm(:)28 b Fq(A)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3899
b Fn(_)g Fq(A)28 b Fn(!)f Fq(A;)17 b(A)23 b Fr(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3900
b(.)21 b(So)h(we)h(need)g(to)f(mak)o(e)h(these)f(agree.)h(This)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3901
3478 y(gi)n(v)o(es)h(us)g(equations:)962 3686 y Fq(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3902
Fm(;)p 1057 3602 425 4 v 17 w(\(id)1175 3701 y Fk(A)1254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3903
3686 y Fn(\001)e Fm(id)1386 3701 y Fk(A)1443 3686 y Fm(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3904
b(=)h Fq(d)f(;)p 1833 3602 V 116 w Fm(\(id)1953 3701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3905
y Fk(A)2032 3686 y Fn(\001)22 b Fm(id)2163 3701 y Fk(A)2220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3906
3686 y Fm(\);)17 b Fq(m)27 b Fm(=)h Fq(m)g(:)83 3895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3907
y Fr(Similarly)-6 b(,)25 b(we)i(implement)e(weak)o(ening)h(via)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3908
(composition)e(with)i Fq(t)31 b Fm(:)g Fq(A)g Fn(!)f(>)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3909
Fr(and)g Fq(u)j Fm(:)h Fn(?)g(!)f Fq(A)p Fr(.)83 4015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3910
y(But)40 b(again)f(these)h(are)h(themselv)o(es)d(produced)h(by)h(weak)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3911
(ening)f(proofs)h Fq(?)55 b Fm(:)g(\()p Fn(\000)p Fm(\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3912
Fn(!)f(>)41 b Fr(and)83 4136 y Fq(?)28 b Fm(:)g Fn(?)g(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3913
Fm(\()p Fn(\000)p Fm(\))e Fr(respecti)n(v)o(ely)-6 b(.)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3914
b(Making)h(these)h(agree)g(gi)n(v)o(es)e(us)i(equations)1324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3915
4344 y Fq(t)p Fm(;)17 b Fq(?)1452 4303 y Fw(+)1538 4344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3916
y Fm(=)27 b Fq(t)h(;)117 b(?)1897 4303 y Fw(+)1956 4344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3917
y Fm(;)17 b Fq(u)26 b Fm(=)i Fq(u)f(:)83 4600 y Fr(There)j(is)f(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3918
(further)f(delicate)h(point)e(which)h(we)g(mention)f(here.)i(Gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3919
(en)e Fm(\000)36 b Fn(`)2816 4564 y Fk(f)2898 4600 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3920
Fm(\001)29 b Fr(there)h(are)g(tw)o(o)83 4713 y(distinct)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3921
b(w)o(ays)g(to)h(introduce)f Fn(>)h Fr(on)g(the)f(left:)1164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3922
4862 y Fi(\000)h Fh(`)g Fi(\001)p 1109 4882 350 4 v 1109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3923
4962 a Fh(>)p Fj(;)15 b Fi(\000)25 b Fh(`)f Fi(\001)1500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3924
4914 y Fh(>)p Ft(-L)1682 4962 y Fj(;)1959 4862 y Fi(\000)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3925
Fh(`)g Fi(\001)p 1904 4882 V 1904 4962 a Fh(>)p Fj(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3926
b Fi(\000)25 b Fh(`)g Fi(\001)2295 4912 y Ft(W)-6 b(-L)2486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3927
4962 y Fj(:)83 5139 y Fr(In)33 b(our)f(notation)f(these)h(are)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3928
Fq(f)1173 5103 y Fw(+)1264 5139 y Fr(and)g Fq(t)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3929
b Fq(f)1579 5103 y Fw(+)1670 5139 y Fr(respecti)n(v)o(ely:)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3930
b(the)o(y)i(are)h(not)f(tak)o(en)g(as)g(equal.)g(This)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3931
5259 y(decision)25 b(arises)g(from)g(an)h(austere)f(vie)n(w)g(of)g(cut)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3932
h(reductions)e(where)i(a)g(last)e(rule)i(is)f(structural.)f(In)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3933
5380 y(this)g(paper)h(we)g(mak)o(e)g(no)g(equality)f(assumptions)e(in)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3934
(such)f(circumstances.)p 0 TeXcolorgray 1748 5712 a(10)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3935
0 TeXcolorgray eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3936
%%Page: 11 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3937
TeXDict begin 11 10 bop 0 TeXcolorgray 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3938
0 TeXcolorgray 83 83 a Fo(2.3.4)99 b(Structur)o(al)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3939
b(congruence)83 258 y Fr(In)j(the)g(interests)f(of)g(simplicity)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3940
b(,)23 b(we)j(subject)f(the)h(structural)f(rules)g(to)h(structural)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3941
(congruence)h(in)83 379 y(a)g(sense)e(popular)g(in)h(concurrenc)o(y)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3942
(theory)-6 b(.)83 546 y(Consider)25 b(the)g(process)f(of)h(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3943
b(eak)o(ening)25 b(only)f(immediately)f(to)h(Contract:)1611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3944
727 y Fj(A;)15 b Fi(\000)26 b Fh(`)e Fi(\001)p 1557 765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3945
456 4 v 1557 845 a Fj(A;)15 b(A;)g Fi(\000)26 b Fh(`)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3946
Fi(\001)p 1557 882 V 1611 962 a Fj(A;)15 b Fi(\000)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3947
b Fh(`)e Fi(\001)80 b Fj(:)83 1171 y Fr(That)27 b(seems)g(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3948
(pointless)d(a)j(detour)f(as)g(a)h(logical)e(Cut,)i(and)f(we)g(allo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3949
(w)g(it)g(to)f(be)i(deleted.)f(Gi)n(v)o(en)83 1291 y(the)e(analysis)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3950
(abo)o(v)o(e)g(we)h(can)g(e)o(xpress)f(this)g(by)h(the)f(equation:)1504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3951
1510 y Fq(d)p Fm(;)p 1599 1425 273 4 v 17 w(\()p Fq(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3952
Fm(;)17 b Fq(f)1775 1481 y Fw(+)1833 1510 y Fm(\))28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3953
b(=)f Fq(f)39 b(:)83 1729 y Fr(Similarly)26 b(it)h(seems)f(willful)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3954
(to)h(distinguish)d(between)j(the)g(v)n(arious)f(w)o(ays)h(in)g(which)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3955
(a)h(series)g(of)83 1849 y(contractions)d(may)h(be)g(performed.)f(This)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3956
g(pro)o(vides)g(the)h(seemingly)e(pointless)g(equation)1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3957
2080 y Fq(d)p Fm(;)17 b(\()p 1527 1985 154 4 v Fq(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3958
Fm(;)p 1622 2001 59 4 v 17 w Fq(f)9 b Fm(\))28 b(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3959
Fq(d)p Fm(;)17 b(\()p 1981 1985 154 4 v Fq(d)p Fm(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3960
2076 2001 59 4 v 17 w Fq(f)10 b Fm(\))27 b Fq(;)83 2299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3961
y Fr(which)32 b(properly)f(inde)o(x)o(ed)f(is)i(a)g(v)o(ersion)f(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3962
(associati)n(vity)-6 b(.)29 b(Finally)j(there)g(is)f(an)h(issue)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3963
(relating)83 2420 y(contraction)j(to)g(e)o(xchange:)g(one)h(can)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3964
(xchange)f(before)h(contracting)f(tw)o(o)g(copies)g(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3965
Fq(A)p Fr(.)f(One)83 2540 y(may)21 b(as)h(well)e(identify)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3966
(proofs.)f(Write)i Fm(\()p Fn(\000)p Fm(\))1750 2504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3967
y Fk(s)1808 2540 y Fr(to)f(indicate)g(a)g(use)g(of)h(symmetry)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3968
b(.)19 b(Then)i(modulo)83 2660 y(elimination)i(of)i(logical)f(cuts)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3969
(we)g(can)g(e)o(xpress)f(this)g(by)1137 2879 y Fq(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3970
Fm(;)p 1232 2800 349 4 v 17 w(id)1313 2894 y Fk(A)1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3971
2879 y Fn(\001)e Fm(id)1523 2894 y Fk(A)1580 2820 y(s)1645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3972
2879 y Fm(=)27 b Fq(d)h Fm(:)g Fq(A)f Fn(\000)-16 b(!)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3973
b Fq(A)22 b Fn(^)g Fq(A)28 b(:)83 3098 y Fr(Thus)36 b(structural)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3974
(congruence)g(gi)n(v)o(es)f(us)h(identity)-6 b(,)34 b(associati)n(vity)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3975
g(and)j(commutati)n(vity)c(condi-)83 3219 y(tions.)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3976
b(W)-8 b(e)25 b(assume)f(these)h(in)f(the)h(interests)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3977
(mathematical)f(ele)o(gance.)83 3574 y Fs(3)100 b(Categorical)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3978
b(f)n(ormulation)83 3869 y Fr(In)e(section)f(2)g(we)h(surv)o(e)o(yed)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3979
(all)g(the)g(structure)h(on)f(a)h Fn(\003)p Fr(-polycate)o(gory)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  3980
(needed)i(to)f(model)g(classical)83 3990 y(proofs,)j(and)g(we)g(ga)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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o(e)f(the)h(equations)f(which)h(we)g(think)f(should)g(hold.)g(This)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(gi)n(v)o(es)g(us)h(a)g(genuine)83 4110 y(though)34 b(unwieldy)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(notion)g(of)h(model.)f(W)-8 b(e)35 b(shall)f(not)h(spell)f(it)g(out.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(Instead)f(we)i(shall)e(e)o(xtract)83 4230 y(from)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(\003)p Fr(-polycate)o(gorical)e(formulation)h(structure)g(on)h(its)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(underlying)g(cate)o(gory)g(gi)n(ving)f(an)83 4351 y(equi)n(v)n(alent)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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23 b(notion)h(of)h(cate)o(gorical)f(model.)83 4526 y(Before)g(we)e(get)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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g(do)n(wn)f(to)g(w)o(ork,)h(we)g(note)g(that)g(the)g(in)l(v)n(olutary)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ne)o(gation)h Fm(\()p Fn(\000)p Fm(\))2840 4478 y Fl(\003)2901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4526 y Fr(e)o(xtends)g(to)h(maps)83 4646 y(as)j(we)g(ha)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(\(for)g(e)o(xample\))f(natural)h(isomorphisms)1040 4865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fn(C)6 b Fm(\()p Fq(A)p Fm(;)17 b Fq(B)5 b Fm(\))1398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4838 y Fn(\030)1399 4869 y Fm(=)1503 4865 y Fn(C)h Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(\000)p Fm(;)17 b Fq(A)1793 4824 y Fl(\003)1833 4865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fq(;)g(B)5 b Fm(\))2022 4838 y Fn(\030)2022 4869 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(=)2127 4865 y Fn(C)h Fm(\()p Fq(B)2302 4824 y Fl(\003)2342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4865 y Fm(;)17 b Fq(A)2459 4824 y Fl(\003)2498 4865 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\))g Fq(:)83 5084 y Fr(It)25 b(is)f(easy)h(to)g(see)g(that)p
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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TeXcolorgray 37 w Fo(The)g(oper)o(ation)e Fm(\()p Fn(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\))1515 5223 y Fl(\003)1583 5259 y Fm(:)27 b Fn(C)1695
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5223 y Fk(op)1797 5259 y Fn(!)g(C)i Fo(is)21 b(a)h(strict)e(functorial)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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g(self-duality)g(on)i(our)83 5380 y(cate)l(gory)j Fn(C)6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fo(.)p 0 TeXcolorgray 1748 5712 a Fr(11)p 0 TeXcolorgray
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>
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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n(v)o(e)h(its)f(dual.)83 545 y Fo(3.1)100 b(Cate)l(gorical)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(Pr)l(eliminaries)83 849 y Fr(W)-8 b(e)28 b(start)g(by)f(introducing)f
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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(gories)g Fn(C)34 b Fr(equipped)83 969 y(with)e(a)g(special)g(class)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(of)h Fn(C)1070 984 y Fw(id)1166 969 y Fr(of)g(idempotents,)e(which)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(we)i(shall)e(call)h Fo(linear)f(idempotents)p Fr(.)f(In)83
Christian Urban <urbanc@in.tum.de>
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diff changeset
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1089 y(our)24 b(application)f(these)h(will)f(be)h(idempotents)e(\(maps)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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h Fq(e)h Fr(with)f Fq(e)p Fm(;)17 b Fq(e)28 b Fm(=)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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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(nothing)f(be)o(yond)h(the)g(ob)o(vious)f(require-)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1330 y(ment)c(that)f(e)n(v)o(ery)g(identity)f(is)i(in)f(the)h(class.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(W)-8 b(e)26 b(call)e(such)h(data)g(a)g Fo(guar)l(ded)f(cate)l(gory)p
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Fr(.)3168 1294 y Fw(2)p 0 TeXcolorgray 83 1514 a Fs(De\002nition)k
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>
parents:
diff changeset
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(gories)f(consists)f(of)83 1634 y(the)g(usual)g(data)g(for)h(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(functor)f(such)g(that)f Fq(F)39 b Fr(maps)25 b(linear)g(idempotents)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(to)g(linear)i(idempotents;)83 1755 y(and)f(whene)n(v)o(er)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4030
Fq(e)h Fr(and)g Fq(e)944 1718 y Fl(0)993 1755 y Fr(are)g(linear)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4031
(idempotents,)e(then)856 1992 y Fq(F)14 b Fm(\()p Fq(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\);)j Fq(F)d Fm(\()p Fq(f)d Fm(\);)17 b Fq(F)d Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4033
Fq(g)t Fm(\);)j Fq(F)d Fm(\()p Fq(e)1762 1950 y Fl(0)1783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4034
1992 y Fm(\))27 b(=)h Fq(F)14 b Fm(\()p Fq(e)p Fm(\);)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4035
Fq(F)d Fm(\()p Fq(f)d Fm(;)17 b Fq(g)t Fm(\);)g Fq(F)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\()p Fq(e)2705 1950 y Fl(0)2725 1992 y Fm(\))83 2229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fr(W)-8 b(e)27 b(say)g(that)f(a)h(guarded)f(functor)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(F)41 b Fr(is)26 b Fo(domain)f(absorbing)g Fr(when)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(F)14 b Fm(\()p Fq(e)p Fm(\);)j Fq(F)d Fm(\()p Fq(f)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4040
Fm(\))30 b(=)g Fq(F)14 b Fm(\()p Fq(e)p Fm(;)j Fq(f)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4041
b Fm(\))26 b Fr(for)83 2349 y(linear)j(idempotents)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4042
Fq(e)p Fr(;)i(it)f(is)h Fo(codomain)f(absorbing)f Fr(when)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4043
Fq(F)14 b Fm(\()p Fq(f)d Fm(\);)17 b Fq(F)d Fm(\()p Fq(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4044
Fm(\))34 b(=)h Fq(F)14 b Fm(\()p Fq(f)d Fm(;)17 b Fq(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4045
Fm(\))28 b Fr(for)i(linear)83 2469 y(idempotents)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4046
Fq(e)p Fr(.)83 2653 y(W)-8 b(e)30 b(should)e(interpret)h(this)f(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(the)h(case)h Fn(C)36 b Fr(is)28 b(the)h(tri)n(vial)f(one)h(object)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4048
(cate)o(gory)g Fm(1)g Fr(with)f(its)g(only)83 2773 y(choice)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4049
b(of)g(linear)g(idempotents.)e(A)i(guarded)f(functor)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4050
Fq(D)31 b Fm(:)c(1)h Fn(!)f(D)c Fr(is)c(a)h(choice)g(of)g(object)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4051
Fq(D)31 b Fn(2)d(D)83 2894 y Fr(and)d(linear)g(idempotent)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4052
Fq(e)1022 2909 y Fk(D)1114 2894 y Fm(:)28 b Fq(D)i Fn(!)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4053
Fq(D)s Fr(.)e(W)-8 b(e)25 b(call)g(this)f(a)h Fo(guar)l(ded)f(object)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4054
Fr(.)83 3077 y(W)-8 b(e)26 b(also)e(need)h(some)f(notion)g(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(2)p Fr(-cell)f(between)h(guarded)g(functors)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
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diff changeset
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83 3261 a Fs(De\002nition)31 b(3.3.)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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43 w Fr(Let)e Fq(F)s(;)17 b(G)36 b Fm(:)g Fn(C)42 b(!)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4058
b(D)c Fr(be)d(guarded)h(functors.)e(A)i Fo(guar)l(ded)f(tr)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4059
(ansformation)c Fr(or)83 3382 y(simply)f Fo(tr)o(ansformation)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4060
Fr(consists)i(of)i(data)g Fq(\013)1702 3397 y Fk(A)1787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3382 y Fm(:)j Fq(F)14 b(A)27 b Fn(!)g Fq(GA)e Fr(satisfying)1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3619 y Fq(F)14 b Fm(\(id)1231 3634 y Fk(A)1288 3619 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\);)j Fq(F)d Fm(\()p Fq(u)p Fm(\);)j Fq(\013)1685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4064
3634 y Fk(B)1772 3619 y Fm(=)27 b Fq(\013)1937 3634 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(A)1994 3619 y Fm(;)17 b Fq(G)p Fm(\()p Fq(u)p Fm(\);)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4066
Fq(G)p Fm(\(id)2486 3634 y Fk(B)2547 3619 y Fm(\))83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3856 y Fr(for)25 b(all)g Fq(u)i Fm(:)h Fq(A)g Fn(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4068
Fq(B)j Fr(in)24 b Fn(C)6 b Fr(.)83 4039 y(W)-8 b(e)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(do)e(not)g(spell)g(out)h(here)g(the)g(consequences)f(of)h(these)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(de\002nitions,)e(b)n(ut)i(note)f(the)h(follo)n(wing.)p
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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TeXcolorgray 47 w Fo(Guar)l(ded)e(cate)l(gories,)f(guar)l(ded)h
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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Fm(2)p Fo(-)83 4343 y(cate)l(gory)-5 b(,)24 b(the)h Fr(guarded)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(2)p Fr(-cate)o(gory)p Fo(.)83 4527 y Fr(The)30 b(only)e(subtle)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(point)f(is)h(the)g(composition)e(of)j Fm(2)p Fr(-cells)f(along)g(a)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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(additionally)e(with)h(maps)h(of)g(the)g(form)g Fq(GF)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4079
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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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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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5266 y(idempotents)30 b(or)d(guards.)h(There)f(is)g(no)g(stronger)i
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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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>
parents:
diff changeset
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Fo(in)l(ver)o(se)f(to)g Fq(\014)6 b Fr(\))36 b(just)g(when)g
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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339 y Fk(A)336 324 y Fm(\))25 b Fr(and)g Fq(\014)623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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339 y Fk(A)680 324 y Fm(;)17 b Fq(\013)786 339 y Fk(A)870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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324 y Fm(=)28 b Fq(G)p Fm(\(id)1170 339 y Fk(A)1227 324
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
  4105
(of)h Fm(2)p Fr(-cells)g(in)f(the)h(guarded)f Fm(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4106
Fr(-cate)o(gory)-6 b(.)83 807 y Fo(3.2)100 b(Lo)o(gical)24
Christian Urban <urbanc@in.tum.de>
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diff changeset
  4107
b(oper)o(ator)o(s)83 1104 y(3.2.1)99 b(Extension)24 b(to)g(maps)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4108
1274 y Fr(Clearly)34 b Fn(C)39 b Fr(must)32 b(be)h(equipped)g(on)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4109
(objects)f(with)g(the)h(structure,)g Fo(true)p Fr(,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4110
Fo(and)p Fr(,)h Fo(false)p Fr(,)f Fo(or)p Fr(,)h Fo(not)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4111
Fr(of)83 1387 y(classical)39 b(logic:)f(we)i(write)f(this)f(structure)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4112
(as)g Fm(1)p Fr(,)g Fn(^)p Fr(,)h Fm(0)p Fr(,)f Fn(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4113
Fr(.)g(There)h(is)f(a)g(compelling)f(w)o(ay)h(to)83 1500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4114
y(e)o(xtend)24 b(the)h(propositional)e(operators)h(to)h(maps.)f(Gi)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4115
o(en)f(proofs)i Fq(A)i Fn(`)2533 1464 y Fk(f)2606 1500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4116
y Fq(B)j Fr(and)25 b Fq(C)35 b Fn(`)3045 1464 y Fk(g)3112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4117
1500 y Fq(D)s Fr(,)25 b(there)g(is)83 1613 y(a)h(canonical)e(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4118
Fq(A)d Fn(^)h Fq(C)34 b Fn(`)1149 1577 y Fk(f)7 b Fl(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4119
Fk(g)1305 1613 y Fq(B)28 b Fn(^)22 b Fq(D)28 b Fr(gi)n(v)o(en)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4120
b(by)i(the)f(follo)n(wing)1513 1803 y Fj(A)h Fh(`)g Fj(B)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4121
b(C)31 b Fh(`)25 b Fj(D)p 1513 1823 594 4 v 1541 1903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4122
a(A;)15 b(C)32 b Fh(`)25 b Fj(B)g Fh(^)19 b Fj(D)p 1510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4123
1940 600 4 v 1510 2020 a(A)i Fh(^)f Fj(C)31 b Fh(`)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4124
b Fj(B)g Fh(^)20 b Fj(D)83 2238 y Fr(Similarly)h(we)h(ha)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4125
Fq(f)g Fn(_)10 b Fq(g)26 b Fr(a)21 b(proof)h(of)g Fq(A)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4126
b Fn(_)g Fq(C)35 b Fn(`)28 b Fq(B)15 b Fn(_)10 b Fq(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4127
Fr(.)22 b(So)g(in)f(terms)g(of)h(our)f(algebraic)h(notation)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4128
2358 y(we)j(should)f(de\002ne)1059 2482 y Fq(f)33 b Fn(^)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4129
b Fq(g)31 b Fm(=)p 1410 2398 257 4 v 28 w(\()p Fq(f)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4130
Fn(\001)21 b Fq(g)t Fm(\))99 b Fq(;)117 b(f)32 b Fn(_)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4131
b Fq(g)31 b Fm(=)p 2260 2398 V 27 w(\()p Fq(f)i Fn(\001)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4132
b Fq(g)t Fm(\))16 b Fq(:)83 2636 y Fr(Thus)35 b Fn(C)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4133
b Fr(is)35 b(equipped)g(with)f(operations)h Fn(^)h Fr(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4134
Fn(_)h Fr(on)f(maps.)g(It)h(turns)e(out)h(that)g(the)o(y)g(are)h(not)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4135
2756 y(functorial,)28 b(b)n(ut)f(in)h(a)g(suitable)f(sense)h(guarded)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4136
(functorial.)f(T)-8 b(o)28 b(mak)o(e)f(sense)h(of)g(that)g(we)g(need)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4137
(a)83 2876 y(collection)c(of)h(linear)g(idempotents.)e(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4138
b(e)25 b(identify)f(that)g(class)h(as)g(follo)n(ws.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4139
3053 y(W)-8 b(e)29 b(\002rst)e(note)h(a)g(useful)f(computation)f(in)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4140
(our)h Fn(\003)p Fr(-polycate)o(gories)e(for)i(classical)g(logic.)f(W)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4141
-8 b(e)28 b(gi)n(v)o(e)83 3174 y(just)c(the)h(v)o(ersion)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4142
(conjunction)e(as)i(that)f(for)h(disjunction)e(is)h(dual)h(to)f(it.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4143
0 TeXcolorgray 83 3351 a Fs(Pr)n(oposition)k(3.6.)p 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4144
TeXcolorgray 43 w Fo(Suppose)f(that)g Fq(f)44 b Fm(:)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4145
b Fq(A)h Fn(!)f Fq(C)7 b Fo(,)27 b Fq(g)37 b Fm(:)c Fq(B)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4146
b Fn(!)33 b Fq(D)s Fo(,)27 b Fq(h)34 b Fm(:)f Fq(C)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4147
b Fn(!)33 b Fq(E)h Fo(and)27 b Fq(k)36 b Fm(:)e Fq(D)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4148
Fn(!)d Fq(F)83 3471 y Fo(ar)l(e)25 b(maps.)g(Then)g Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4149
Fq(f)32 b Fn(^)23 b Fq(g)t Fm(\);)17 b(\()p Fq(h)k Fn(^)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4150
Fq(k)s Fm(\))k(=)p 1489 3387 549 4 v 28 w Fn(f)p Fq(f)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4151
b(;)17 b(g)t Fn(g)p Fm(;)g(\()p Fq(h)k Fn(\001)h Fq(k)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4152
Fm(\))p Fo(.)83 3648 y Fr(Using)i(also)h(the)f(Additional)f(Assumption)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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g(of)i(2.2.2)f(we)h(deduce)g(at)g(once)g(the)g(follo)n(wing.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4154
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4155
TeXcolorgray 43 w Fo(If)g Fq(e)929 3840 y Fk(A)1020 3825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4156
y Fm(:)33 b Fq(A)h Fn(!)f Fq(A)28 b Fo(and)g Fq(e)1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4157
3840 y Fk(B)1738 3825 y Fm(:)34 b Fq(B)k Fn(!)c Fq(B)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4158
Fo(ar)l(e)28 b(linear)f(and)h(idempotent,)e(then)i(so)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4159
3946 y(ar)l(e)d Fq(e)282 3961 y Fk(A)362 3946 y Fn(^)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4160
Fq(e)495 3961 y Fk(B)581 3946 y Fo(and)i Fq(e)800 3961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4161
y Fk(A)880 3946 y Fn(_)e Fq(e)1013 3961 y Fk(B)1074 3946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4162
y Fo(.)83 4123 y Fr(W)-8 b(e)21 b(no)n(w)e(associate)g(with)g(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4163
(cate)o(gorical)g(model)f Fn(C)26 b Fr(a)20 b(class)g(of)g(linear)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4164
(idempotents.)e(W)-8 b(e)20 b(simply)83 4243 y(close)k(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(collection)e(of)i(identity)f(maps)g(under)h(the)f(logical)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4166
(operations.)g(\(W)-8 b(e)25 b(mak)o(e)e(clear)i(what)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4167
4364 y(that)30 b(means)f(in)g(case)i(of)f Fn(>)g Fr(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(?)p Fr(.\))g(W)-8 b(e)30 b(introduce)f(some)h(notation)e(for)i(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4169
Fo(canonical)e(linear)83 4484 y(idempotents)c Fr(which)g(we)h(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4170
(e)g(identi\002ed.)f(W)-8 b(e)25 b(write)510 4708 y Fq(e)555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4171
4723 y Fk(A;B)716 4708 y Fm(=)j Fq(e)865 4723 y Fk(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4172
Fl(^)p Fk(B)1053 4708 y Fm(=)g(id)1238 4723 y Fk(A)1317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4173
4708 y Fn(^)23 b Fm(id)1487 4723 y Fk(B)1575 4708 y Fq(;)415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4174
b(e)2062 4723 y Fk(A;B)2223 4708 y Fm(=)28 b Fq(e)2372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4175
4723 y Fk(A)p Fl(_)p Fk(B)2560 4708 y Fm(=)g(id)2745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4176
4723 y Fk(A)2824 4708 y Fn(_)23 b Fm(id)2994 4723 y Fk(B)3083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4177
4708 y Fq(:)83 4932 y Fr(W)-8 b(e)26 b(also)e(tak)o(e)h(a)g(nullary)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4178
(v)o(ersion)g(of)h(these,)f(setting)1254 5156 y Fq(e)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4179
5171 y Fl(>)1385 5156 y Fm(=)k Fq(?)1538 5115 y Fw(+)2023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4180
5156 y Fq(e)2068 5171 y Fl(?)2155 5156 y Fm(=)f Fq(?)2307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4181
5115 y Fw(+)83 5380 y Fr(with)d(the)h(ob)o(vious)e(interpretation)h(in)
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>
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diff changeset
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%%Page: 14 14
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>
parents:
diff changeset
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b(3.8.)p 0 TeXcolorgray 41 w Fo(\(i\))e Fn(>)g Fo(with)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(e)1148 98 y Fl(>)1232 83 y Fo(and)f(dually)g Fn(?)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4189
Fo(with)e Fq(e)2024 98 y Fl(?)2108 83 y Fo(ar)l(e)h(guar)l(ded)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4190
(objects.)83 203 y(\(ii\))i(The)h(oper)o(ator)d Fn(^)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4191
b Fm(:)f Fn(C)e(\002)24 b(C)39 b(!)31 b(C)j Fo(is)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4192
b(a)h(domain)f(absorbing)g(guar)l(ded)g(functor)-11 b(,)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4193
b(while)h(dually)83 324 y Fn(_)h Fm(:)g Fn(C)h(\002)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4194
b(C)34 b(!)28 b(C)j Fo(is)24 b(a)h(codomain)f(absorbing)f(guar)l(ded)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4195
(functor)-11 b(.)83 637 y(3.2.2)99 b(Coher)l(ence)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4196
817 y Fr(When)28 b(we)g(come)f(to)g(reconstruct)g(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4197
Fn(\003)p Fr(-polycate)o(gory)f(from)g(our)g(cate)o(gory)g(we)h(need)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4198
(to)f(observ)o(e)83 937 y(some)h(relations)g(between)g(our)g(canonical)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4199
g(linear)h(idempotents.)d(W)-8 b(e)29 b(illustrate)e(the)h(point)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4200
(here.)83 1057 y(Concentrating)e(on)f(conjunction)g(we)h(ha)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4201
(on)h(the)f(one)h(hand)g(the)f(idempotent)682 1287 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4202
Fq(e)727 1302 y Fk(A;B)s Fl(^)p Fk(C)990 1287 y Fm(=)k(id)1175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4203
1302 y Fk(A)1254 1287 y Fn(^)23 b Fm(id)1424 1302 y Fk(B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4204
Fl(^)p Fk(C)1614 1287 y Fm(:)28 b Fq(A)22 b Fn(^)h Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4205
Fq(B)k Fn(^)c Fq(C)7 b Fm(\))27 b Fn(\000)-16 b(!)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4206
b Fq(A)c Fn(^)f Fm(\()p Fq(B)27 b Fn(^)c Fq(C)7 b Fm(\))83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4207
1516 y Fr(and)25 b(on)g(the)f(other)556 1745 y Fq(e)601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4208
1760 y Fk(A;B)s(;C)836 1745 y Fm(=)k(id)1021 1760 y Fk(A)1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4209
1745 y Fn(^)23 b Fm(\(id)1308 1760 y Fk(B)1391 1745 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4210
Fn(^)g Fm(id)1561 1760 y Fk(C)1620 1745 y Fm(\))28 b(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4211
Fq(A)c Fn(^)f Fm(\()p Fq(B)27 b Fn(^)c Fq(C)7 b Fm(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4212
b Fn(\000)-16 b(!)28 b Fq(A)22 b Fn(^)h Fm(\()p Fq(B)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4213
Fn(^)22 b Fq(C)7 b Fm(\))28 b Fq(:)83 1974 y Fr(Intuiti)n(v)o(ely)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4214
b(the)i(second)f(decomposes)h(things)e(more)i(than)g(the)g(\002rst,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4215
(and)g(this)f(is)h(re\003ected)h(in)f(the)83 2094 y(f)o(act)i(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4216
(the)h(second)f(absorbs)g(the)h(\002rst)g(in)f(the)h(sense)g(that)650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4217
2323 y Fq(e)695 2338 y Fk(A;B)s(;C)903 2323 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4218
b Fq(e)992 2338 y Fk(A;B)s Fl(^)p Fk(C)1254 2323 y Fm(=)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4219
b Fq(e)1403 2338 y Fk(A;B)s(;C)1711 2323 y Fr(and)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4220
b Fq(e)1999 2338 y Fk(A;B)s Fl(^)p Fk(C)2234 2323 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4221
b Fq(e)2323 2338 y Fk(A;B)s(;C)2559 2323 y Fm(=)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4222
Fq(e)2707 2338 y Fk(A;B)s(;C)2943 2323 y Fq(:)83 2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4223
y Fr(The)e(\002rst)f(calculation)f(depends)h(on)g(the)g(linearity)g(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4224
g Fm(id)2069 2567 y Fk(B)2150 2552 y Fn(\001)19 b Fm(id)2279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4225
2567 y Fk(C)2362 2552 y Fr(from)24 b(the)g(Additional)f(Assump-)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4226
2673 y(tion)h(of)h(2.2.2.)83 2852 y(Generally)h(the)g(situation)e(is)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4227
(as)g(follo)n(ws.)e(Gi)n(v)o(en)g(propositions)g Fq(A)2425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4228
2867 y Fk(i)2479 2852 y Fr(we)j(ha)n(v)o(e)e(man)o(y)g(brack)o(etings)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4229
83 2973 y(to)d(gi)n(v)o(e)f(a)h(conjunction)932 2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4230
y Ff(V)1018 2973 y Fq(A)1091 2988 y Fk(i)1119 2973 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4231
Fr(.)g(Gi)n(v)o(en)f(one)h(such)g(we)g(ha)n(v)o(e)g(a)g(v)n(ariety)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4232
(of)g(idempotents)e(depending)83 3093 y(on)26 b(ho)n(w)e(deeply)i(we)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4233
(`analyse)f(the)h(brack)o(etings'.)f(The)g(shallo)n(west)f(analysis)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(yields)f Fm(id)3200 3059 y Ff(V)3281 3117 y Fk(A)3334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3127 y Fe(i)3364 3093 y Fr(,)i(the)83 3214 y(deepest)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4236
Fq(e)452 3179 y Ff(V)533 3237 y Fk(A)586 3247 y Fe(i)644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4237
3214 y Fm(=)748 3147 y Ff(V)834 3214 y Fm(id)915 3229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4238
y Fk(A)968 3239 y Fe(i)998 3214 y Fr(.)g(The)g(coherence)h(of)e(these)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(idempotents)e(is)h(the)h(follo)n(wing)e(f)o(act.)p 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXcolorgray 83 3393 a Fs(Pr)n(oposition)g(3.9.)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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38 w Fo(Suppose)f(in)h(the)f(given)h(situation)e(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4242
Fq(e)2219 3408 y Fw(1)2281 3393 y Fo(is)h(an)f(idempotent)g(corr)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4243
(esponding)83 3514 y(to)j(a)g(deeper)f(analysis)g(than)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4244
Fq(e)1155 3529 y Fw(2)1195 3514 y Fo(.)g(Then)i Fq(e)1514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4245
3529 y Fw(1)1553 3514 y Fm(;)17 b Fq(e)1642 3529 y Fw(2)1709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4246
3514 y Fm(=)28 b Fq(e)1858 3529 y Fw(1)1925 3514 y Fm(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4247
Fq(e)2074 3529 y Fw(2)2113 3514 y Fm(;)17 b Fq(e)2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4248
3529 y Fw(1)2242 3514 y Fo(.)83 3693 y Fr(W)-8 b(e)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4249
b(note)e(the)h(nullary)f(v)o(ersion)g(of)g(the)h(proposition:)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4250
Fq(e)2012 3708 y Fl(>)2071 3693 y Fm(;)17 b(id)2196 3708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4251
y Fl(>)2283 3693 y Fm(=)27 b Fq(e)2431 3708 y Fl(>)2518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4252
3693 y Fm(=)h(id)2703 3708 y Fl(>)2762 3693 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4253
b Fq(e)2851 3708 y Fl(>)2910 3693 y Fr(.)83 4013 y Fo(3.3)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4254
b(Structur)l(e)83 4313 y(3.3.1)f(Units)24 b(and)g(associator)o(s)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4255
4493 y Fr(Our)36 b(logical)f(operations)f(are)i(only)f(guarded)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4256
(functorial,)e(b)n(ut)h(the)o(y)g(do)g(come)g(equipped)g(with)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4257
4613 y(structure)29 b(f)o(amiliar)g(in)g(the)f(case)i(of)f(tensor)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4258
(products.)f(W)-8 b(e)30 b(concentrate)f(on)g(the)g(case)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4259
Fn(>)h Fr(and)83 4734 y Fn(^)p Fr(;)25 b(the)g(case)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4260
Fn(?)h Fr(and)e Fn(_)i Fr(follo)n(ws)d(by)i(duality)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4261
b(.)83 4913 y(First)25 b(we)g(can)g(de\002ne)h(maps)776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4262
5178 y Fq(l)k Fm(=)e Fq(?)22 b Fn(\001)f Fm(id)1141 5193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4263
y Fk(A)1225 5178 y Fm(:)28 b Fq(A)g Fn(!)f(>)c(^)f Fq(A)1968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4264
5152 y Fm(~)1969 5178 y Fq(l)30 b Fm(=)p 2131 5085 141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4265
4 v 27 w(id)2213 5135 y Fw(+)2213 5203 y Fk(A)2299 5178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4266
y Fm(:)e Fn(>)23 b(^)f Fq(A)28 b Fn(!)f Fq(A)705 5359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4267
y(r)j Fm(=)d(id)964 5374 y Fk(A)1043 5359 y Fn(\001)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4268
b Fq(?)27 b Fm(:)h Fq(A)g Fn(!)f Fq(A)22 b Fn(^)h(>)204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4269
b Fm(~)-53 b Fq(r)30 b Fm(=)p 2090 5266 253 4 v 27 w(\(id)2210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4270
5316 y Fw(+)2210 5383 y Fk(A)2269 5359 y Fm(\))2307 5330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4271
y Fk(s)2371 5359 y Fm(:)e Fq(A)22 b Fn(^)h(>)28 b(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(A)p 0 TeXcolorgray 1748 5712 a Fr(14)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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%%Page: 15 15
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 83 83 a Fr(where)35 b(the)f(superscript)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(s)h Fr(indicates)e(a)i(tacit)f(use)g(of)h(e)o(xchange.)f(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4278
b(e)35 b(also)f(ha)n(v)o(e)g(associati)n(vity)83 203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4279
y(maps)25 b(de\002ned)g(as)g(follo)n(ws)695 471 y Fq(a)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(=)p 878 370 641 4 v 878 386 V 28 w(\(id)997 486 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4281
Fk(A)1076 471 y Fn(\001)22 b Fm(id)1207 486 y Fk(B)1268
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4282
471 y Fm(\))g Fn(\001)g Fm(id)1459 486 y Fk(C)1546 471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4283
y Fm(:)28 b Fq(A)22 b Fn(^)h Fm(\()p Fq(B)k Fn(^)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4284
Fq(C)7 b Fm(\))28 b Fn(\000)-16 b(!)27 b Fm(\()p Fq(A)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4285
b Fn(^)h Fq(B)5 b Fm(\))22 b Fn(^)h Fq(C)34 b(;)697 651
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4286
y Fm(~)-51 b Fq(a)28 b Fm(=)p 878 550 V 878 567 V 28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4287
w(id)959 666 y Fk(A)1038 651 y Fn(\001)22 b Fm(\(id)1207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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666 y Fk(B)1290 651 y Fn(\001)g Fm(id)1421 666 y Fk(C)1480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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651 y Fm(\))28 b(:)g(\()p Fq(A)22 b Fn(^)h Fq(B)5 b Fm(\))22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4290
b Fn(^)g Fq(C)35 b Fn(\000)-16 b(!)27 b Fq(A)22 b Fn(^)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4291
Fm(\()p Fq(B)k Fn(^)c Fq(C)7 b Fm(\))27 b Fq(:)83 893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4292
y Fr(\(There)c(is)e(only)g(one)h(sensible)f(w)o(ay)h(to)g(read)g(those)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4293
g(de\002nitions!\))f(W)-8 b(e)22 b(note)g(at)f(once)i(that)e(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4294
(these)83 1013 y(structural)j(maps)f(are)h(linear)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(.)83 1194 y(By)26 b(direct)e(computation)f(we)i(sho)n(w)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(follo)n(wing.)p 0 TeXcolorgray 83 1375 a Fs(Theor)n(em)k(3.10.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 42 w Fo(The)e(pair)o(s)f(of)g(maps)g Fq(l)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4298
Fo(and)1720 1349 y Fm(~)1721 1375 y Fq(l)r Fo(,)d Fq(r)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4299
Fo(and)g Fm(~)-53 b Fq(r)s Fo(,)26 b Fq(a)h Fo(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4300
Fm(~)-50 b Fq(a)p Fo(,)27 b(ar)l(e)g(in)f(eac)o(h)h(case)g(mutually)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1496 y(in)l(ver)o(se)e(guar)l(ded)f(tr)o(ansformations.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1676 y Fr(Note)36 b(that)g(the)g(equations)f(gi)n(v)o(en)g(by)g(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4303
(de\002nitions)f(are)i(not)f(quite)f(the)h(f)o(amiliar)g(ones.)g(F)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4304
(or)83 1797 y(e)o(xample)24 b(since)h Fn(^)g Fr(is)g(domain)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4305
(absorbing)h(we)h(do)g(ha)n(v)o(e)950 2028 y Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4306
Fq(f)33 b Fn(^)23 b Fq(g)t Fm(\))e Fn(^)h Fq(h)p Fm(;)d(~)-51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4307
b Fq(a)28 b Fm(=)h(~)-50 b Fq(a)p Fm(;)17 b Fq(f)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4308
Fn(^)22 b Fm(\()p Fq(g)j Fn(^)e Fq(h)p Fm(\);)17 b Fq(e)2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4309
2044 y Fk(A)2338 2025 y Fg(0)2360 2044 y Fl(^)p Fw(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4310
Fk(B)2490 2025 y Fg(0)2514 2044 y Fl(^)p Fk(C)2616 2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4311
y Fg(0)2638 2044 y Fw(\))83 2260 y Fr(b)n(ut)25 b(we)g(only)f(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4312
(e)g(the)h(more)g(f)o(amiliar)1187 2492 y Fm(\()p Fq(f)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4313
b Fn(^)22 b Fq(g)t Fm(\))g Fn(^)g Fq(h)p Fm(;)c(~)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4314
b Fq(a)28 b Fm(=)h(~)-51 b Fq(a)q Fm(;)17 b Fq(f)32 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4315
Fn(^)23 b Fm(\()p Fq(g)i Fn(^)e Fq(h)p Fm(\))83 2723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4316
y Fr(when)i Fq(f)11 b Fr(,)25 b Fq(g)j Fr(and)d Fq(h)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4317
Fr(are)g(linear)-5 b(.)83 2904 y(Perhaps)30 b(surprisingly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4318
b(,)27 b(it)i(is)g(automatic)g(that)g(our)g(associati)n(vities)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4319
(satisfy)i(the)g(Mac)h(Lane)f(pen-)83 3024 y(tagon)j(condition)f(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4320
(the)f(usual)g(unit)g(conditions)f(on)h(the)g(nose.)g(The)h(diagrams)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4321
(are)h(f)o(amiliar)83 3145 y(and)25 b(we)g(do)g(not)f(e)o(xhibit)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4322
(them)h(here.)p 0 TeXcolorgray 83 3326 a Fs(Theor)n(em)j(3.11.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4323
0 TeXcolorgray 41 w Fo(The)e(Mac)g(Lane)g(penta)o(gon)f(and)g(unit)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4324
(conditions)621 3557 y Fq(a)672 3572 y Fk(A;B)s(;C)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4325
b Fl(^)p Fk(D)987 3557 y Fm(;)17 b Fq(a)1082 3572 y Fk(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4326
Fl(^)p Fk(B)s(;C)q(;D)1420 3557 y Fm(=)28 b(id)1605 3572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4327
y Fk(A)1684 3557 y Fn(^)23 b Fq(a)1824 3572 y Fk(B)s(;C)q(;D)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4328
3557 y Fm(;)17 b Fq(a)2130 3572 y Fk(A;B)s Fl(^)p Fk(C)q(;D)2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4329
3557 y Fm(;)g Fq(a)2535 3572 y Fk(A;B)s(;C)2765 3557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4330
y Fn(^)22 b Fm(id)2935 3572 y Fk(D)1257 3800 y Fq(a)1308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4331
3815 y Fk(A;I)5 b(;C)1495 3800 y Fm(;)17 b Fq(r)1583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4332
3815 y Fk(A)1662 3800 y Fn(^)23 b Fm(id)1832 3815 y Fk(B)1920
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4333
3800 y Fm(=)28 b(id)2105 3815 y Fk(A)2184 3800 y Fn(^)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4334
b Fq(l)2302 3815 y Fk(B)83 3961 y Fo(both)h(hold.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4335
4142 y Fr(Of)g(course)g(man)o(y)e(other)h(v)o(ersion)g(of)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4336
(diagrams)f(\(e.g.)g(in)l(v)n(olving)g Fm(~)-50 b Fq(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4337
Fr(,)2623 4116 y Fm(~)2624 4142 y Fq(l)r Fr(,)28 b Fm(~)-54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4338
b Fq(r)s Fr(\))24 b(hold.)e(Ho)n(we)n(v)o(er)h(the)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4339
4263 y(information)28 b(contained)g(in)h(the)g(coherence)h(diagrams)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4340
(is)h(quite)f(subtle.)g(One)i(needs)f(to)f(bear)i(in)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4341
4383 y(mind)23 b(that)h(e.g.)g Fq(a)705 4398 y Fk(A;B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4342
Fl(^)p Fk(C)q(;D)1040 4383 y Fr(is)g(not)g(guarded)g(natural)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4343
Fq(B)30 b Fr(and)24 b Fq(C)7 b Fr(.)24 b(Let)g(us)g(say)g(that)g(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4344
Fo(mixed)g(path)83 4503 y Fr(in)29 b(the)g(pentagon)g(is)g(one)g(which)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4345
g(in)l(v)n(olv)o(es)e(both)i Fq(a)g Fr(and)i Fm(~)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4346
b Fq(a)p Fr(.)29 b(Man)o(y)f(b)n(ut)h(by)g(no)g(means)g(all)g(mix)o(ed)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4347
83 4624 y(paths)c(are)g(equal.)g(F)o(or)g(e)o(xample,)e(the)i(tw)o(o)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4348
(maps)838 4855 y Fq(a)k Fm(:)g Fq(A)22 b Fn(^)h Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4349
Fq(B)k Fn(^)c Fm(\()p Fq(C)29 b Fn(^)22 b Fq(D)s Fm(\)\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4350
b Fn(\000)-16 b(!)28 b Fm(\()p Fq(A)22 b Fn(^)g Fq(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4351
b Fm(\))23 b Fn(^)f Fm(\()p Fq(C)29 b Fn(^)23 b Fq(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4352
Fm(\))83 5087 y Fr(and)444 5219 y Fm(id)525 5234 y Fk(A)604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4353
5219 y Fn(^)g Fq(a)p Fm(;)17 b Fq(a)p Fm(;)g Fq(a)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4354
b Fn(^)g Fm(id)1126 5234 y Fk(D)1190 5219 y Fm(;)17 b(~)-49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4355
b Fq(a)27 b Fm(:)h Fq(A)22 b Fn(^)h Fm(\()p Fq(B)k Fn(^)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4356
b Fm(\()p Fq(C)30 b Fn(^)22 b Fq(D)s Fm(\)\))27 b Fn(\000)-16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4357
b(!)27 b Fm(\()p Fq(A)c Fn(^)f Fq(B)5 b Fm(\))22 b Fn(^)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4358
Fm(\()p Fq(C)29 b Fn(^)23 b Fq(D)s Fm(\))83 5380 y Fr(are)j(not)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4359
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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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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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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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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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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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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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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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16 {InitRnode } NewNode end end
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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16 {InitRnode } NewNode end end
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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1231 a 1739 995 a
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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16 {InitRnode } NewNode end end
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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16 {InitRnode } NewNode end end
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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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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg -1.13809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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-1.13809 0 0 /N@A /N@C InitNC { /AngleA 130.  def /AngleB 0.  def 0.67
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>
parents:
diff changeset
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1113 a
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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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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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a
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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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>
parents:
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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>
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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:
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exch } PutCoor PutBegin  end
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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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>
parents:
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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>
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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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1113 a
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:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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exch } PutCoor PutBegin  end
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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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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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>
parents:
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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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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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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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\()p Fq(B)30 b Fn(^)c Fq(C)7 b Fm(\))34 b Fn(!)h Fm(\()p
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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2985 y Fr(cannot)i(be)h(de\002ned)f(in)g(the)g(usual)g(w)o(ay)g(from)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4677
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4678
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4679
0.0 0 0 /N@B /N@CR InitNC { /AngleA 170.  def /AngleB 0.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4680
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4681
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4682
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4683
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4684
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4685
0.0 0 0 /N@B /N@BR InitNC { /AngleA 190.  def /AngleB 0.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4686
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4687
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4688
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4689
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4690
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4691
0.0 0 0 /N@C /N@CLL InitNC { /AngleA 10.  def /AngleB 180.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4692
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4693
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4694
 771 691 a 771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4695
691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4696
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4697
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4698
0.0 0 0 /N@C /N@A InitNC { /AngleA -10.  def /AngleB 180.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4699
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4700
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4701
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4702
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4703
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4704
0.0 0 0 /N@CL /N@CLL InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4705
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4706
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4707
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4708
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4709
0.0 0 0 /N@D /N@B InitNC { /AngleA 170.  def /AngleB 0.  def 0.67 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4710
0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4711
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4712
 771 691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4713
a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4714
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4715
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4716
0.0 0 0 /N@D /N@ARR InitNC { /AngleA 190.  def /AngleB 0.  def 0.67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4717
 0.67  NCCurve  } if end gsave 0.8 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4718
 grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4719
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4720
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4721
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4722
0.0 0 0 /N@AR /N@ARR InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4723
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4724
 771 691 a 1515 484 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4725
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@E
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4726
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4728
1515 484 a 192 402 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4729
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4730
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4731
 192 402 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4732
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4733
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4734
0.0 0 0 /N@D /N@E InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4735
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4736
 771 691 a 771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4737
691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4738
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4739
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4740
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 51.22697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4741
8.2125 2.73749 NAngle 90 add  Uput exch pop add a PtoC h1 add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4742
w1 add exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4743
 771 691 a 558 714 a Fi(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4744
Fj(A)5 b Fh(^)g Fj(B)g Fi(\))g Fh(_)g Fj(C)771 691 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4745
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4747
771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4748
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4749
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4750
 tx@Dict begin gsave STV newpath 0.8 SLW 0  setgray  /ArrowA { moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4751
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4752
0.0 0 0 /N@C /N@F InitNC { NCLine  } if end gsave 0.8 SLW 0  setgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4753
0 setlinecap stroke  grestore  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4754
 771 691 a 771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4755
691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4756
 tx@Dict begin tx@NodeDict begin /t .5 def LPut end PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4757
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4758
 tx@Dict begin { 0.85358 tx@Dict /NCLW known { NCLW add } if 51.22697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4759
8.2125 2.73749 NAngle 90 sub  Uput exch pop add a PtoC h1 add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4760
w1 add exch } PutCoor PutBegin  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4761
 771 691 a 558 714 a Fj(A)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4762
Fh(^)g Fi(\()p Fj(B)10 b Fh(_)5 b Fj(C)i Fi(\))771 691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4763
y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4764
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4765
 771 691 a 771 691 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4766
 tx@Dict begin PutEnd  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4767
 771 691 a 1488 w @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4768
@setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4769
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  0.1 true 3.9833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4770
1.99179 15.93352 17.92503 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4771
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4772
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4773
@endspecial 2297 628 a Fd(id)2338 639 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4774
Fc(C)2259 691 y @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4775
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  0.1 true 3.9833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4776
21.90863 15.93352 37.84187 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4777
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4778
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4779
@endspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4780
2296 463 a Fd(id)2337 474 y Fc(B)2259 691 y @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4781
@setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4782
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  0.1 true 3.9833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4783
41.82547 15.93352 57.75871 .5 Frame  gsave 0.85358 SLW 0  setgray 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4784
setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4785
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4786
@endspecial 2297 298 a Fd(id)2338 309 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4787
Fc(A)2292 278 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4788
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@AL
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4789
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4790
 2292 278 a 2292 443 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4791
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@BL
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4792
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4793
 2292 443 a 2292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4794
608 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4795
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@CL
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4796
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4797
 2292 608 a 2159 608 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4798
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@CLL
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4799
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4800
 2159 608 a 2391 278 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4801
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@AR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4802
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4803
 2391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4804
278 a 2523 278 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4805
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@ARR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4806
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4807
 2523 278 a 2391 443 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4808
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@BR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4809
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4810
 2391 443 a 2391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4811
608 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4812
 tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@CR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4813
16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4814
 2391 608 a 2139 388 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4815
 tx@Dict begin tx@NodeDict begin {9.74438 3.0777 12.82208 6.41104 3.33334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4816
} false /N@A 16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4817
 2139 388 a 20 w @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4818
@setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4819
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  8.00002 2 div 6.66669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4820
0.0 add 2 div 2 copy 0.0 sub 4 2 roll Pyth 0.28453 add CLW 2 div add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4821
0 360 arc closepath gsave 0.85358 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4822
 grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4823
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4824
@endspecial Fn(_)2437 553 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4825
 tx@Dict begin tx@NodeDict begin {9.74438 3.0777 12.82208 6.41104 3.33334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4826
} false /N@B 16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4827
 2437 553 a 20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4828
w @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4829
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  8.00002 2 div 6.66669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4830
0.0 add 2 div 2 copy 0.0 sub 4 2 roll Pyth 0.28453 add CLW 2 div add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4831
0 360 arc closepath gsave 0.85358 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4832
 grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4833
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4834
@endspecial Fn(^)1991 512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4835
y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4836
 tx@Dict begin tx@NodeDict begin {9.74438 3.0777 12.82208 6.41104 3.33334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4837
} false /N@C 16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4838
 1991 512 a 20 w @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4839
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  8.00002 2 div 6.66669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4840
0.0 add 2 div 2 copy 0.0 sub 4 2 roll Pyth 0.28453 add CLW 2 div add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4841
0 360 arc closepath gsave 0.85358 SLW 0  setgray 0 setlinecap stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4842
 grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4843
 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4844
@endspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4845
Fn(^)2586 429 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4846
 tx@Dict begin tx@NodeDict begin {9.74438 3.0777 12.82208 6.41104 3.33334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4847
} false /N@D 16 {InitRnode } NewNode end end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4848
 2586 429 a 20 w @beginspecial @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4849
 tx@Dict begin STP newpath 0.85358 SLW 0  setgray  8.00002 2 div 6.66669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4850
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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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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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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:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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:
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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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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} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0
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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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>
parents:
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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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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>
parents:
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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>
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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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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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>
parents:
diff changeset
  5004
(Coherence')i(where)83 4095 y(one)f(\002nds)g(canonical)f(idempotents)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5005
(\(identities)h(in)g(the)h Fm(2)p Fr(-cate)o(gory)f(of)h(guarded)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5006
(functors\).)83 4403 y Fo(3.3.4)99 b(Duality)83 4581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5007
y Fr(A)30 b Fn(\003)p Fr(-polycate)o(gory)f(supports)g(polymaps)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5008
Fm(in)1684 4596 y Fk(A)1778 4581 y Fm(:)37 b Fn(\000)h(!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(A)2167 4545 y Fl(\003)2206 4581 y Fq(;)17 b(A)30 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5010
Fr(and)g Fm(ev)2623 4596 y Fk(A)2718 4581 y Fm(:)37 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5011
Fq(A;)17 b(A)2972 4545 y Fl(\003)3048 4581 y Fn(!)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5012
b(\000)31 b Fr(which)83 4701 y(enable)25 b(us)g(to)f(de\002ne)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5013
(something)d(lik)o(e)h(a)h(unit)1053 4928 y Fq(\021)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5014
4886 y Fk(A)1101 4952 y(B)1189 4928 y Fm(=)p 1293 4848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5015
353 4 v 28 w(in)1374 4943 y Fk(A)1453 4928 y Fn(\001)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5016
Fm(id)1584 4943 y Fk(B)1672 4928 y Fm(:)28 b Fq(B)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5017
b Fn(!)27 b Fq(A)2034 4886 y Fl(\003)2096 4928 y Fn(_)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5018
Fm(\()p Fq(A)f Fn(^)g Fq(B)5 b Fm(\))17 b Fq(;)83 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5019
y Fr(and)25 b(something)e(lik)o(e)i(a)g(counit)1046 5380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5020
y Fq(")1092 5339 y Fk(A)1092 5404 y(B)1180 5380 y Fm(=)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5021
1284 5301 368 4 v 28 w(ev)1380 5395 y Fk(A)1459 5380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5022
y Fn(\001)d Fm(id)1590 5395 y Fk(B)1679 5380 y Fm(:)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5023
b Fq(A)c Fn(^)f Fm(\()p Fq(A)2028 5339 y Fl(\003)2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5024
5380 y Fn(_)g Fq(B)5 b Fm(\))28 b Fn(!)g Fq(B)21 b(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 1748 5712 a Fr(17)p 0 TeXcolorgray eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%Page: 18 18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5029
0 TeXcolorgray 83 83 a Fr(One)34 b(e)o(xpects)e(that)h(the)g(unit)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(and)h(counit)f(are)i(interchanged)f(by)f(the)h(self-duality)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5031
b(,)32 b(though)g(our)83 203 y(con)l(v)o(entions)f(on)i(duality)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5032
(require)i(mediating)e(symmetries.)g(\(W)l(ith)i(the)f(other)h(choice)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(of)f(con-)83 324 y(v)o(ention,)24 b(the)g(problem)g(emer)n(ges)h(else)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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n(where!\))p 0 TeXcolorgray 83 518 a Fs(Pr)n(oposition)g(3.19.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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0 TeXcolorgray 41 w Fo(The)g Fq(\021)k Fo(and)24 b Fq(")h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fo(ar)l(e)g(dual)f(in)g(the)h(sense)g(that)e(the)i(equations)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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639 y Fq(c)125 654 y Fk(A)178 635 y Fg(\003)214 654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(;B)290 635 y Fg(\003)353 639 y Fn(^)d Fm(id)522 654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5039
y Fk(A)579 639 y Fm(;)17 b(\()p Fq(\021)713 603 y Fk(A)709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5040
663 y(B)770 639 y Fm(\))808 603 y Fl(\003)875 639 y Fm(=)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5041
b Fq(c)1020 654 y Fk(A)1073 635 y Fg(\003)1109 654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5042
Fl(_)p Fk(B)1212 635 y Fg(\003)1249 654 y Fk(;A)1325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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639 y Fm(;)17 b Fq(")1415 603 y Fk(A)1415 663 y(B)1471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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644 y Fg(\003)1536 639 y Fo(and)25 b Fm(\()p Fq(")1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5045
603 y Fk(A)1795 663 y(B)1855 639 y Fm(\))1893 603 y Fl(\003)1932
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5046
639 y Fm(;)17 b Fq(c)2018 654 y Fk(A)2071 635 y Fg(\003)2107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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654 y Fk(;B)2183 635 y Fg(\003)2219 654 y Fl(^)p Fk(A)2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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635 y Fg(\003)2387 639 y Fm(=)27 b Fq(\021)2542 603 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(A)2538 663 y(B)2594 644 y Fg(\003)2635 639 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(id)2760 654 y Fk(A)2813 635 y Fg(\003)2875 639 y Fn(_)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5051
b Fq(c)3006 654 y Fk(B)3062 635 y Fg(\003)3098 654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5052
Fk(;A)3199 639 y Fo(hold.)83 833 y Fr(Finally)i(we)g(get)f(triangle)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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83 1028 a Fs(Theor)n(em)i(3.20.)p 0 TeXcolorgray 41 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fo(W)-9 b(e)25 b(have)g Fm(id)1172 1043 y Fk(A)1251 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5056
y Fn(^)e Fq(\021)1392 991 y Fk(A)1388 1052 y(B)1448 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5057
y Fm(;)17 b Fq(")1538 991 y Fk(A)1538 1052 y(A)p Fl(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5058
Fk(B)1726 1028 y Fm(=)27 b Fq(e)1874 1043 y Fk(A)p Fl(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5059
Fk(B)2060 1028 y Fo(and)d Fq(\021)2286 991 y Fk(A)2282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5060
1052 y(A)2335 1033 y Fg(\003)2371 1052 y Fl(_)p Fk(B)2479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1028 y Fm(;)17 b(id)2604 1043 y Fk(A)2657 1024 y Fg(\003)2719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1028 y Fn(_)23 b Fq(")2854 991 y Fk(A)2854 1052 y(B)2942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1028 y Fm(=)k Fq(e)3090 1043 y Fk(A)3143 1024 y Fg(\003)3179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5064
1043 y Fl(_)p Fk(B)3287 1028 y Fo(.)83 1222 y Fr(This)d(essentially)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5065
(gi)n(v)o(es)f(an)i(adjunction)f(in)g(the)h(guarded)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5066
Fm(2)p Fr(-cate)o(gory)-6 b(.)83 1593 y Fo(3.3.5)99 b(Alg)o(ebr)o(as)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5067
b(and)i(coalg)o(ebr)o(as)83 1788 y Fr(W)-8 b(e)26 b(consider)e(no)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5068
(the)h(structural)f(maps)613 2082 y Fq(d)k Fm(:)f Fq(A)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5069
Fn(!)g Fq(A)22 b Fn(^)g Fq(A)34 b(t)28 b Fm(:)f Fq(A)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5070
Fn(!)f(>)190 b Fq(m)28 b Fm(:)g Fq(A)22 b Fn(_)g Fq(A)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5071
b Fn(!)g Fq(A)33 b(u)27 b Fm(:)h Fn(?)g(!)f Fq(A)83 2351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5072
y Fr(The)33 b(de)h(Mor)n(gan)e(duality)g(of)h(the)g(proof)g(rules)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5073
(sho)n(ws)e(that)i(these)g(structures)f(are)i(dual)f(to)g(one)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5074
2471 y(another:)759 2631 y Fq(d)810 2589 y Fl(\003)810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5075
2655 y Fk(A)894 2631 y Fm(=)28 b Fq(m)1083 2646 y Fk(A)1136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5076
2627 y Fg(\003)1193 2631 y Fq(;)116 b(m)1421 2589 y Fl(\003)1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5077
2655 y Fk(A)1506 2631 y Fm(=)28 b Fq(d)1661 2646 y Fk(A)1714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5078
2627 y Fg(\003)1770 2631 y Fq(;)116 b(t)1948 2589 y Fl(\003)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5079
2655 y Fk(A)2033 2631 y Fm(=)28 b Fq(u)2193 2646 y Fk(A)2246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5080
2627 y Fg(\003)2302 2631 y Fq(;)116 b(u)2501 2589 y Fl(\003)2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5081
2655 y Fk(A)2585 2631 y Fm(=)28 b Fq(t)2724 2646 y Fk(A)2777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5082
2627 y Fg(\003)2834 2631 y Fq(:)83 2893 y Fr(In)i(f)o(amiliar)e(cate)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5083
(gory)h(theoretic)g(settings)f(maps)g(of)h(these)g(kinds)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(usually)g(associated)f(with)83 3013 y(product)f(and)g(coproduct)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5085
(structure;)g(b)n(ut)g(here)h(we)f(do)g(not)g(e)n(v)o(en)f(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5086
(guarded)h(naturality)-6 b(.)25 b(But)83 3134 y(the)e(correctness)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5087
(equations)g(of)h(2.3.3)f(gi)n(v)o(e)f(at)h(once)h(the)f(follo)n(wing)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5088
(relation)h(to)g(canonical)h(linear)83 3254 y(idempotents.)p
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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0 TeXcolorgray 44 w Fo(The)g(maps)f Fq(d)p Fo(,)g Fq(t)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fo(ar)l(e)g(codomain)f(absorbing)f(while)i Fq(m)g Fo(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5092
Fq(u)h Fo(ar)l(e)g(domain)83 3569 y(absorbing)23 b(in)i(the)f(sense)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(that)f(following)f(equations)h(hold.)574 3828 y Fq(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5094
Fm(;)17 b Fq(e)714 3843 y Fk(A)p Fl(^)p Fk(A)898 3828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5095
y Fm(=)27 b Fq(d)17 b(;)116 b(t)p Fm(;)17 b Fq(e)1336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5096
3843 y Fl(>)1423 3828 y Fm(=)27 b Fq(t)100 b Fr(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5097
Fq(e)1950 3843 y Fk(A)p Fl(_)p Fk(A)2107 3828 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5098
b Fq(m)27 b Fm(=)h Fq(m)17 b(;)116 b(e)2657 3843 y Fl(?)2717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5099
3828 y Fm(;)17 b Fq(u)26 b Fm(=)i Fq(u)17 b(:)83 4161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5100
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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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0 TeXcolorgray 41 w Fo(The)g(structur)l(e)e Fm(\()p Fq(A;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5104
Fk(A)1799 4355 y Fm(\))26 b Fo(forms)g(a)g(commutative)f(comonoid,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5105
(while)i(the)83 4475 y(structur)l(e)d Fm(\()p Fq(A;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5106
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4475 y Fm(\))25 b Fo(forms)e(a)i(commutative)f(monoid.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4670 y Fr(W)-8 b(e)26 b(list)d(the)i(equations)f(in)l(v)n(olv)o(ed)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5110
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(;)e(~)-53 b Fq(r)30 b Fm(=)e(id)1661 4980 y Fk(A)1917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5112
4965 y Fq(d)p Fm(;)17 b(id)2093 4980 y Fk(A)2172 4965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5113
y Fn(^)23 b Fq(t)p Fm(;)2339 4938 y(~)2340 4965 y Fq(l)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5114
b Fm(=)f(id)2583 4980 y Fk(A)696 5145 y Fq(d)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5115
b(id)872 5160 y Fk(A)951 5145 y Fn(^)22 b Fq(d)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5116
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5117
5160 y Fk(A)1910 5145 y Fq(d)p Fm(;)17 b Fq(d)k Fn(^)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5118
Fm(id)2247 5160 y Fk(A)2304 5145 y Fm(;)17 b(~)-49 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(a)27 b Fm(=)h Fq(d)p Fm(;)17 b(id)2705 5160 y Fk(A)2785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5120
5145 y Fn(^)22 b Fq(d)1629 5326 y(d)p Fm(;)17 b Fq(c)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5121
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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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%%Page: 19 19
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>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5127
(xplain)f(a)i(notion)e(of)i(cate)o(gorical)f(model)g(for)h(classical)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5128
(proof.)83 499 y(In)j(the)f(de\002nition)g(one)g(should)f(think)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5129
(the)g(hom-sets)g Fn(C)6 b Fm(\()p Fq(A)p Fm(;)17 b Fq(B)5
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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619 y(classical)24 b(proofs)f(of)h Fq(A)k Fn(`)g Fq(B)5
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>
parents:
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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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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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Fk(D)1863 1903 y Fm(=)27 b Fq(e)2011 1918 y Fk(A)p Fl(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(B)2172 1903 y Fm(;)17 b Fq(g)t Fm(;)g Fq(e)2356 1918
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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Fm(\()p Fq(A)22 b Fn(^)h Fq(B)5 b Fm(;)17 b Fq(C)29 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5260
Fn(_)22 b Fq(D)s Fm(\))p Fr(.)83 2303 y(So)39 b(in)f(outline)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5261
(construction)g(of)h(the)g(polycate)o(gorical)f(model)h(is)f(as)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5262
(follo)n(ws.)d(W)-8 b(e)39 b(mak)o(e)83 2424 y(a)i(choice)g(of)g(brack)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5263
o(etings)e(of)i(both)f Fm(\000)h Fr(and)f Fm(\001)p Fr(.)h(This)f(gi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5264
(v)o(es)f(us)h(hom-sets)f Fn(C)6 b Fm(\()2956 2357 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5265
Ff(V)3042 2424 y Fm(\000)p Fq(;)3147 2357 y Ff(W)3233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5266
2424 y Fm(\001\))41 b Fr(and)83 2544 y(canonical)j(idempotents)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5267
Fq(e)1085 2510 y Ff(V)1166 2568 y Fw(\000)1258 2544 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5268
Fr(and)h Fq(e)1490 2510 y Ff(W)1571 2568 y Fw(\001)1634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5269
2544 y Fr(.)h(W)-8 b(e)44 b(can)g(then)f(tak)o(e)h Fn(C)6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5270
b Fm(\(\000;)17 b(\001\))43 b Fr(to)h(consist)e(of)i(the)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5271
2676 y Fq(f)54 b Fn(2)44 b(C)6 b Fm(\()391 2610 y Ff(V)477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5272
2676 y Fm(\000)p Fq(;)582 2610 y Ff(W)668 2676 y Fm(\001\))33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5273
b Fr(such)g(that)g Fq(e)1264 2642 y Ff(V)1345 2700 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5274
Fw(\000)1393 2676 y Fm(;)17 b Fq(f)11 b Fm(;)17 b Fq(e)1585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2642 y Ff(W)1666 2700 y Fw(\001)1772 2676 y Fm(=)43 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5276
Fq(f)11 b Fr(.)33 b(Finally)g(we)g(ha)n(v)o(e)g(a)h(series)f(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(\002ddly)g(b)n(ut)83 2796 y(routine)25 b(tasks.)p 0
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(e)34 b(sho)n(w)d(that)i Fn(C)6 b Fm(\(\000;)17 b(\001\))33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5280
b Fr(is)f(essentially)g(independent)g(of)h(the)f(brack)o(eting)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5281
(chosen.)f(This)289 3094 y(follo)n(ws)24 b(from)g(the)h(coherence)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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e(the)h(canonical)g(linear)f(idempotents.)p 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
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diff changeset
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123 3214 a(\(2\))p 0 TeXcolorgray 50 w(W)-8 b(e)23 b(sho)n(w)f(ho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(to)g(de\002ne)h(composition)e(on)h(the)h(sets)f(of)h(polymaps.)e(This)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5285
h(combines)g(point)289 3334 y(\(1\))36 b(with)f(hea)n(vy)h(use)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5286
(the)g(linear)f(distrib)n(uti)n(vities.)d(And)k(we)g(sho)n(w)f(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(the)h(result)f(is)289 3455 y(indeed)25 b(a)g Fn(\003)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5288
Fr(-polycate)o(gory)-6 b(.)p 0 TeXcolorgray 123 3575
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a(\(3\))p 0 TeXcolorgray 50 w(W)e(e)37 b(de\002ne)f(the)g(logical)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5290
(operations)g(on)g(the)h(collections)f(of)h(polymaps)e(and)i(deri)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5291
(e)f(the)289 3695 y(man)o(y)24 b(equations.)g(This)g(is)g(pretty)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5292
(much)h(routine.)83 4063 y Fs(4)100 b(Explanation)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5293
b(and)h(comparison)83 4360 y Fo(4.1)100 b(Repr)l(esentable)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5294
b(polycate)l(gories)83 4658 y Fr(W)-8 b(e)27 b(recall)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(relationship)f(between)h Fn(\003)p Fr(-polycate)o(gories)f(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(\003)p Fr(-autonomous)d(cate)o(gories)i(\(see)83 4778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y([2])e(or)f([11])g(for)g(e)o(xample\).)f(T)-8 b(ak)o(e)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5298
b(the)g(ob)o(vious)e Fm(2)p Fr(-cate)o(gories)38 b Fn(\003)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5299
b Fb(P)m(oly)23 b Fr(of)g Fn(\003)p Fr(-polycate)o(gories)f(and)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5300
4898 y Fn(\003)p Fb(Aut)29 b Fr(of)g Fn(\003)p Fr(-autonomous)f(cate)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5301
(gories:)g(all)h Fm(2)p Fr(-cells)g(are)h(in)l(v)o(ertible)e(so)h(we)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5302
(are)g(in)f(the)g(groupoid)83 5019 y(enriched)38 b(setting.)e(An)o(y)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(\003)p Fr(-autonomous)e(cate)o(gory)i(determines)g(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(\003)p Fr(-polycate)o(gory)-6 b(,)36 b(with)h(the)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5139 y(linear)c(tensor)f(and)h(par)g(representing)f(polymaps;)f(so)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(one)f(sees)h(that)f(there)h(is)g(a)g(groupoid)e(en-)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5259 y(riched)h(for)n(getful)g(functor)f Fq(S)6 b(P)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(ol)r(y)43 b Fm(:)d Fn(\003)p Fb(Aut)g Fn(!)g(\003)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fb(P)m(oly)p Fr(.)32 b(On)f(the)h(other)f(hand)g(one)h(can)g(freely)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5380 y(construct)e(a)h Fn(\003)p Fr(-autonomous)e(cate)o(gory)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>
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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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0 TeXcolorgray 83 83 a Fr(identi\002cations.)27 b(This)g(gi)n(v)o(es)g
Christian Urban <urbanc@in.tum.de>
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diff changeset
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(a)h(groupoid)f(enriched)i(functor)f Fq(S)6 b(Aut)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fm(:)h Fn(\003)p Fb(P)m(oly)g Fn(!)g(\003)p Fb(Aut)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5319
b Fr(and)h(a)83 203 y(groupoid)i(enriched)i(adjunction)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5320
Fq(S)6 b(Aut)39 b Fn(a)h Fq(S)6 b(P)14 b(ol)r(y)r Fr(.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(The)g(basic)g(conserv)n(ati)n(vity)e(result)i(pro)o(v)o(ed)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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324 y(by)22 b(direct)g(syntactic)f(considerations)g(in)h([2])g
Christian Urban <urbanc@in.tum.de>
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diff changeset
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(\(though)f(see)h([11])h(for)f(an)g(indication)f(of)h(a)g(semantic)83
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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83 620 a Fs(Theor)n(em)j(4.1.)p 0 TeXcolorgray 41 w Fo(In)e(the)f(gr)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(oupoid)f(enric)o(hed)h(adjunction)f Fq(S)6 b(Aut)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5327
b Fn(a)f Fq(S)6 b(P)14 b(ol)r(y)s Fo(,)24 b(the)h(unit)1366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5328
840 y Fn(P)37 b(!)27 b Fq(S)6 b(P)14 b(ol)r(y)r(S)6 b(Aut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5329
Fm(\()p Fn(P)i Fm(\))83 1060 y Fo(is)25 b(full)f(and)g(faithful)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5330
(any)h Fn(\003)p Fo(-polycate)l(gory)f Fn(P)8 b Fo(.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5331
1236 y Fr(When)29 b(does)f(a)h Fn(\003)p Fr(-polycate)o(gory)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5332
Fn(P)37 b Fr(arise)29 b(from)f(a)h Fn(\003)p Fr(-autonomous)d(cate)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5333
(gory)-6 b(,)28 b(that)g(is)g(when)g(is)g(it)83 1356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5334
y(in)d(the)f(essential)g(image)h(of)g Fq(S)6 b(P)14 b(ol)r(y)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5335
Fr(?)23 b(This)h(occurs)h(just)f(when)h(there)g(are)h(maps)305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5336
1612 y Fq(i)338 1627 y Fk(A;B)499 1612 y Fm(:)h Fq(A;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5337
b(B)33 b Fn(!)27 b Fq(A)c Fn(^)f Fq(B)39 b(i)1234 1627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5338
y Fl(>)1321 1612 y Fm(:)27 b Fn(\000)i(!)e(>)237 b Fq(i)1955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5339
1627 y Fk(C)q(;D)2117 1612 y Fm(:)28 b Fq(C)h Fn(_)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5340
b Fq(D)30 b Fn(!)d Fq(C)r(;)17 b(D)35 b(i)2863 1627 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5341
Fl(?)2950 1612 y Fm(:)28 b Fn(?)g(!)g(\000)83 1843 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5342
Fr(composition)23 b(with)h(which)g(induces)h(isomorphisms)525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5343
2099 y Fn(P)8 b Fm(\()p Fq(A)22 b Fn(^)h Fq(B)5 b(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5344
b Fm(\000;)g(\001\))1198 2072 y Fn(\030)1199 2103 y Fm(=)1304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5345
2099 y Fn(P)8 b Fm(\()p Fq(A;)17 b(B)5 b(;)17 b Fm(\000;)g(\001\))241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5346
b Fn(P)8 b Fm(\()p Fn(>)p Fq(;)17 b Fm(\000;)g(\001\))2613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5347
2072 y Fn(\030)2613 2103 y Fm(=)2718 2099 y Fn(P)8 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5348
Fm(\(\000;)17 b(\001\))g Fq(;)519 2280 y Fn(P)8 b Fm(\(\000;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5349
b(\001)p Fq(;)g(C)29 b Fn(_)23 b Fq(D)s Fm(\))1201 2252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5350
y Fn(\030)1202 2284 y Fm(=)1306 2280 y Fn(P)8 b Fm(\(\000;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5351
b(\001)p Fq(;)g(C)r(;)g(D)s Fm(\))249 b Fn(P)8 b Fm(\(\000;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5352
b(\001)g Fn(?)p Fm(\))2599 2252 y Fn(\030)2600 2284 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5353
Fm(=)2704 2280 y Fn(P)8 b Fm(\(\000;)17 b(\001\))g Fq(:)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5354
2517 y Fr(In)34 b(particular)f(for)g(an)o(y)g Fm(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5355
Fr(,)g Fm(\001)h Fr(we)g(ha)n(v)o(e)f(isomorphisms)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5356
Fn(C)6 b Fm(\(\000;)17 b(\001\))2498 2489 y Fn(\030)2498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5357
2521 y Fm(=)2618 2517 y Fn(C)6 b Fm(\()2714 2451 y Ff(V)2800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5358
2517 y Fm(\000;)2905 2451 y Ff(W)2991 2517 y Fm(\001\))33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5359
b Fr(where)h(we)83 2637 y(write)322 2571 y Ff(V)408 2637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5360
y Fm(\000)g Fr(and)681 2571 y Ff(W)767 2637 y Fm(\001)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5361
Fr(for)g(a)h(conjunction)d(and)i(disjunction)e(according)i(to)g(some)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5362
(brack)o(etings.)83 2758 y(In)i(these)g(circumstances)f(we)i(say)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5363
(that)h Fq(i)1577 2773 y Fk(A;B)1710 2758 y Fr(,)g Fq(i)1803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5364
2773 y Fl(>)1862 2758 y Fr(,)g Fq(i)1955 2773 y Fk(C)q(;D)2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5365
2758 y Fr(and)g Fq(i)2337 2773 y Fl(?)2431 2758 y Fo(pr)l(o)o(vide)f(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5366
h(r)l(epr)l(esentation)f(of)83 2878 y(polymaps)p Fr(,)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5367
b(or)h(more)g(loosely)e(that)i Fn(^)p Fo(,)g Fn(>)p Fo(,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5368
Fn(_)p Fo(,)g Fn(?)h Fo(r)l(epr)l(esent)f(polymaps)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5369
Fr(.)83 3175 y Fo(4.2)100 b(Repr)l(esentability)23 b(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5370
(functoriality)83 3471 y Fr(Consider)j(no)n(w)g(a)g Fn(\003)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5371
Fr(-polycate)o(gorical)f(model)h Fn(C)33 b Fr(for)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5372
b(classical)e(proof:)h(it)g(comes)g(equipped)f(with)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5373
3591 y(structure)805 3712 y Fq(i)838 3727 y Fk(A;B)999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5374
3712 y Fm(=)i Fq(i)1136 3727 y Fk(A)p Fl(^)p Fk(B)1313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5375
3712 y Fq(;)117 b(i)1490 3727 y Fl(>)1577 3712 y Fm(=)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5376
b Fq(?)17 b(;)116 b(i)1922 3727 y Fk(C)q(;D)2084 3712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5377
y Fm(=)28 b Fq(i)2221 3727 y Fk(C)5 b Fl(_)p Fk(C)2399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5378
3712 y Fq(;)116 b(i)2575 3727 y Fl(?)2662 3712 y Fm(=)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5379
b Fq(?)83 3862 y Fr(\(using)c(earlier)i(notation\))d(potentially)g(pro)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5380
o(viding)g(a)i(representation)g(of)g(polymaps.)83 4037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5381
y(From)i(our)f(outline)g(of)h(the)f(reconstruction)g(of)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5382
Fn(\003)p Fr(-polycate)o(gory)-6 b(,)25 b(we)i(see)g(that)f(we)h(ha)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5383
(v)o(e)f(rep-)83 4157 y(resentability)e(just)g(when)g(the)h(canonical)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5384
(linear)f(idempotents)600 4378 y Fq(e)645 4393 y Fk(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5385
Fl(^)p Fk(B)834 4378 y Fm(=)p 937 4302 167 4 v 27 w Fq(i)970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5386
4393 y Fk(A;B)1120 4378 y Fq(;)117 b(e)1309 4393 y Fl(>)1396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5387
4378 y Fm(=)27 b(\()p Fq(i)1570 4393 y Fl(>)1629 4378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5388
y Fm(\))1667 4337 y Fw(+)1743 4378 y Fq(;)116 b(e)1931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5389
4393 y Fk(C)5 b Fl(_)p Fk(D)2125 4378 y Fm(=)p 2229 4302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5390
168 4 v 28 w Fq(i)2262 4393 y Fk(C)q(;D)2413 4378 y Fq(;)117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5391
b(e)2602 4393 y Fl(?)2689 4378 y Fm(=)27 b(\()p Fq(i)2863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5392
4393 y Fl(?)2922 4378 y Fm(\))2960 4337 y Fw(+)83 4598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5393
y Fr(are)f(in)e(f)o(act)h(identities.)e(By)i(duality)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5394
b(,)23 b(we)i(only)f(need)h(half)g(of)g(this)e(so)i(representability)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5395
(is)i(equi)n(v)n(a-)83 4719 y(lent)g(to)f(the)h(conditions)1047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5396
4939 y Fq(e)1092 4954 y Fl(>)1178 4939 y Fm(=)j(id)1363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5397
4954 y Fl(>)1522 4939 y Fr(and)99 b Fm(id)1847 4954 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5398
Fk(A)1926 4939 y Fn(^)22 b Fm(id)2096 4954 y Fk(B)2184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5399
4939 y Fm(=)28 b(id)2369 4954 y Fk(A)p Fl(^)p Fk(B)2546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5400
4939 y Fq(:)83 5159 y Fr(Ne)o(xt)c(note)h(that,)f(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5401
Fn(^)g Fr(is)g(guarded)g(domain)e(absorbing,)h(we)h(ha)n(v)o(e)779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5402
5380 y Fq(f)33 b Fn(^)23 b Fq(g)t Fm(;)17 b Fq(h)k Fn(^)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5403
Fq(k)s Fm(;)17 b(id)1389 5395 y Fk(E)1471 5380 y Fn(^)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5404
b Fm(id)1640 5395 y Fk(F)1727 5380 y Fm(=)27 b(\()p Fq(f)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5405
b Fm(;)17 b Fq(h)p Fm(\))22 b Fn(^)g Fm(\()p Fq(h)p Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5406
b Fq(k)s Fm(\);)g(id)2530 5395 y Fk(E)2612 5380 y Fn(^)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5407
b Fm(id)2782 5395 y Fk(F)p 0 TeXcolorgray 1748 5712 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5408
Fr(21)p 0 TeXcolorgray eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%Page: 22 22
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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y Fk(E)534 83 y Fn(^)f Fm(id)704 98 y Fk(F)790 83 y Fm(=)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5413
b(id)975 98 y Fk(E)t Fl(^)p Fk(F)1161 83 y Fr(gi)n(v)o(es)1215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5414
303 y Fq(f)k Fn(^)23 b Fq(g)t Fm(;)17 b Fq(h)k Fn(^)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5415
Fq(k)31 b Fm(=)c(\()p Fq(f)11 b Fm(;)17 b Fq(h)p Fm(\))22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5416
b Fn(^)g Fm(\()p Fq(h)p Fm(;)17 b Fq(k)s Fm(\))83 524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5417
y Fr(which)22 b(is)f(functoriality)g(of)g Fn(^)p Fr(.)i(One)f(should)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5418
(re)o(gard)i Fq(e)1965 539 y Fl(>)2052 524 y Fm(=)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5419
b(id)2236 539 y Fl(>)2317 524 y Fr(as)22 b(functoriality)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5420
Fn(>)p Fr(.)g(Then)g(one)83 644 y(can)k(summarise)d(the)i(discussion)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)h(the)h(follo)n(wing.)p 0 TeXcolorgray 83 820 a Fs(Theor)n(em)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(4.2.)p 0 TeXcolorgray 41 w Fo(Let)e Fn(C)31 b Fo(be)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(a)g(model)f(for)g(classical)g(pr)l(oof)o(.)f(Then)i(the)g(following)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ar)l(e)i(equivalent.)p 0 TeXcolorgray 123 995 a(\(1\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5425
0 TeXcolorgray 50 w(The)h(identity)d(conditions)g Fm(id)1309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5426
1010 y Fk(A)1388 995 y Fn(^)f Fm(id)1558 1010 y Fk(B)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5427
995 y Fm(=)28 b(id)1831 1010 y Fk(A)p Fl(^)p Fk(B)2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5428
995 y Fo(and)d Fq(e)2236 1010 y Fl(>)2323 995 y Fm(=)i(id)2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5429
1010 y Fl(>)2567 995 y Fo(.)p 0 TeXcolorgray 123 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5430
a(\(2\))p 0 TeXcolorgray 50 w(Full)d(functoriality)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5431
Fn(^)p Fo(,)h(and)g Fn(>)p Fo(.)p 0 TeXcolorgray 123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5432
1236 a(\(3\))p 0 TeXcolorgray 50 w(Repr)l(esentability)f(of)g(polymaps)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5433
g(by)h Fn(^)p Fo(,)g Fn(>)g Fo(and)g Fn(_)p Fo(,)g Fn(?)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5434
Fo(.)83 1411 y Fr(This)39 b(mak)o(es)g(clear)g(the)g(o)o(v)o(ersight)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5435
(in)i([11].)g(There)h(linear)f(maps)f(were)i(assumed)f(to)f(form)h(a)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5436
1531 y Fn(\003)p Fr(-autonomous)29 b(cate)o(gory;)h(b)n(ut)g(that)h(gi)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5437
n(v)o(es)e Fm(id)1739 1546 y Fk(A)1823 1531 y Fn(^)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5438
Fm(id)1997 1546 y Fk(B)2096 1531 y Fm(=)39 b(id)2292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5439
1546 y Fk(A)p Fl(^)p Fk(B)2483 1531 y Fr(and)31 b(so)f(functoriality)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5440
(of)h(the)83 1652 y(logical)21 b(operators.)f(Note)h(also)g(that)f(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5441
h(condition)e Fq(f)g Fn(^)8 b Fq(g)t Fm(;)17 b Fq(h)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5442
b Fn(^)g Fq(k)30 b Fm(=)e(\()p Fq(f)11 b Fm(;)17 b Fq(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5443
Fm(\))8 b Fn(^)g Fm(\()p Fq(h)p Fm(;)17 b Fq(k)s Fm(\))j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5444
Fr(follo)n(ws)g(from)83 1772 y(that)g(naturality)e(of)i(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5445
Fn(^)p Fr(-R)h(rule)f(which)f(we)h(did)f(not)g(adopt.)g(Ho)n(we)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5446
(er)g(that)g(condition)f(is)i(weak)o(er)83 1892 y(than)k(full)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5447
(functoriality)-6 b(.)22 b(It)h(is)h(easy)g(to)f(\002nd)h(models)f(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5448
(which)g(it)g(holds)g(b)n(ut)g Fm(id)2862 1907 y Fk(A)2937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5449
1892 y Fn(^)18 b Fm(id)3103 1907 y Fk(B)3191 1892 y Fm(=)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5450
b(id)3376 1907 y Fk(A)p Fl(^)p Fk(B)83 2013 y Fr(f)o(ails.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5451
2309 y Fo(4.3)100 b(Why)24 b(functoriality)f(should)h(fail)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5452
2605 y Fr(As)h(we)g(shall)f(see)h(the)f(assumption)f(of)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5453
(representability)e(pro)o(vides)g(a)i(substantial)e(simpli\002cation)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5454
2726 y(of)i(the)g(notion)e(of)i(cate)o(gorical)g(model.)f(So)h(it)f(is)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5455
g(time)h(to)f(e)o(xplain)g(why)g(we)h(do)f(not)g(adopt)h(it.)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5456
2901 y(First)g(we)g(ar)n(gue)g(against)f(the)h(tempting)e(naturality)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5457
(of)h Fn(^)p Fr(-R)1184 3121 y Fn(f)p Fq(u;)17 b(v)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5458
Fn(g)p Fm(;)g(\()p Fq(f)31 b Fn(\001)22 b Fq(g)t Fm(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5459
b(=)h(\()p Fq(u)p Fm(;)17 b Fq(f)11 b Fm(\))20 b Fn(\001)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5460
Fm(\()p Fq(v)t Fm(;)17 b Fq(g)t Fm(\))g Fq(:)83 3334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5461
y Fr(Consider)29 b(\002rst)g Fn(f)p Fq(m;)17 b Fm(id)916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5462
3349 y Fk(B)977 3334 y Fn(g)p Fm(;)g(\(id)1190 3349 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5463
Fk(A)1247 3334 y Fq(;)g Fm(id)1372 3349 y Fk(B)1432 3334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5464
y Fm(\))p Fr(.)29 b(Composing)f(with)g Fm(id)2300 3349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5465
y Fk(B)2390 3334 y Fr(does)g(nothing)g(so)g(this)g(is)h(equal)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5466
3447 y(to)c Fq(m)p Fm(;)17 b(\(id)434 3462 y Fk(A)491
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5467
3447 y Fq(;)g Fm(id)616 3462 y Fk(B)677 3447 y Fm(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5468
Fr(,)24 b(which)h(is)f(represented)h(by)g(the)f(proof)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5469
3635 y Fj(A)i Fh(`)f Fj(A)91 b(A)25 b Fh(`)g Fj(A)p 1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5470
3655 576 4 v 1211 3735 a(A)20 b Fh(_)g Fj(A)26 b Fh(`)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5471
Fj(A;)15 b(A)p 1211 3773 521 4 v 1265 3852 a(A)21 b Fh(_)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5472
Fj(A)25 b Fh(`)g Fj(A)1850 3753 y(A)g Fh(`)g Fj(A)91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5473
b(B)30 b Fh(`)25 b Fj(B)p 1850 3773 587 4 v 1878 3852
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5474
a(A;)15 b(B)30 b Fh(`)25 b Fj(A)20 b Fh(^)g Fj(B)p 1265
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5475
3890 1144 4 v 1487 3970 a(A)g Fh(_)g Fj(A;)15 b(B)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5476
b Fh(`)24 b Fj(A)d Fh(^)f Fj(B)1247 b Ft(\(1\))83 4175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5477
y Fr(There)26 b(are)f(tw)o(o)f(distinct)g(w)o(ays)g(to)h(eliminate)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5478
(the)g(Cut.)h(One)g(results)f(in)g(the)h(normal)f(form)1350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5479
4359 y Fj(A)h Fh(`)g Fj(A)91 b(A)26 b Fh(`)f Fj(A)p 1350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5480
4379 576 4 v 1378 4459 a(A)20 b Fh(_)g Fj(A)25 b Fh(`)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5481
Fj(A;)15 b(A)p 1378 4497 521 4 v 1432 4576 a(A)20 b Fh(_)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5482
Fj(A)26 b Fh(`)f Fj(A)173 b(B)29 b Fh(`)c Fj(B)p 1432
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5483
4596 838 4 v 1501 4676 a(A)20 b Fh(_)g Fj(A;)15 b(B)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5484
b Fh(`)25 b Fj(A)c Fh(^)e Fj(B)1234 b Ft(\(2\))83 4881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5485
y Fr(and)25 b(the)g(other)f(in)h(the)f(normal)h(form)1178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5486
5045 y Fj(A)g Fh(`)g Fj(A)91 b(B)30 b Fh(`)25 b Fj(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5487
1178 5065 587 4 v 1206 5145 a(A;)15 b(B)30 b Fh(`)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5488
b Fj(A)20 b Fh(^)g Fj(B)1855 5045 y(A)26 b Fh(`)f Fj(A)91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5489
b(B)29 b Fh(`)c Fj(B)p 1855 5065 V 1883 5145 a(A;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5490
b(B)30 b Fh(`)25 b Fj(A)c Fh(^)e Fj(B)p 1206 5183 1209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5491
4 v 1261 5262 a(A)i Fh(_)f Fj(A;)15 b(B)5 b(;)15 b(B)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5492
b Fh(`)25 b Fj(A)20 b Fh(^)g Fj(B)5 b(;)15 b(A)20 b Fh(^)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5493
Fj(B)p 1261 5300 1098 4 v 1460 5380 a(A)g Fh(_)g Fj(A;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5494
b(B)30 b Fh(`)25 b Fj(A)c Fh(^)e Fj(B)1275 b Ft(\(3\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5495
0 TeXcolorgray 1748 5712 a Fr(22)p 0 TeXcolorgray eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5496
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5497
%%Page: 23 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXDict begin 23 22 bop 0 TeXcolorgray 0 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5499
0 TeXcolorgray 83 83 a Fr(No)n(w)22 b(consider)g Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5500
Fq(m)p Fm(;)17 b(id)905 98 y Fk(A)962 83 y Fm(\))12 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5501
Fn(\001)g Fm(\(id)1171 98 y Fk(B)1232 83 y Fm(;)17 b(id)1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5502
98 y Fk(B)1417 83 y Fm(\))p Fr(.)22 b(This)g(is)f(clearly)i(equal)f(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5503
f Fq(m)12 b Fn(\001)g Fm(id)2640 98 y Fk(B)2723 83 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5504
Fr(which)22 b(is)f(represented)83 203 y(by)i(the)f(\002rst)g(of)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5505
(abo)o(v)o(e)g(tw)o(o)g(normal)g(forms.)f(There)i(is)f(no)h(w)o(ay)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5506
(to)g(get)g(at)h(the)f(second)g(\(though)83 324 y(that)i(is)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5507
(normal)f(form)h(in)g(which)f Fq(m)i Fr(has)e(done)h(its)f(intended)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5508
(job)h(of)g(cop)o(ying\).)f(No)n(w)g(we)h(tak)o(e)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5509
444 y(the)k(vie)n(w)e(that)h(f)o(ailure)h(to)f(ha)n(v)o(e)g(the)g(same)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5510
h(normal)e(forms)h(\(e)n(v)o(en)g(modulo)f(ob)o(vious)g(re)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5511
(writings\))83 565 y(is)f(a)g(clear)g(sign)f(of)h(non-identity)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5512
b(.)23 b(W)-8 b(e)25 b(conclude)f(that)h(the)f(naturality)g(equation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5513
914 786 y Fn(f)p Fq(m;)17 b Fm(id)1174 801 y Fk(B)1235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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786 y Fn(g)p Fm(;)g(\(id)1447 801 y Fk(A)1526 786 y Fn(\001)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5515
b Fm(id)1658 801 y Fk(B)1718 786 y Fm(\))28 b(=)f(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5516
Fq(m)p Fm(;)17 b(id)2136 801 y Fk(A)2193 786 y Fm(\))22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5517
b Fn(\001)g Fm(\(id)2422 801 y Fk(B)2482 786 y Fm(;)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5518
b(id)2607 801 y Fk(B)2668 786 y Fm(\))83 1008 y Fr(is)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(not)f(f)o(aithful)g(to)h(the)f(notion)g(of)h(proof)f(encapsulated)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)f(the)h(sequent)f(calculus.)83 1184 y(W)-8 b(e)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(e)o(xplain)e(the)i(signi\002cance)f(of)h(this)e(for)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(functoriality)f(of)h Fn(^)p Fr(.)h(Consider)g Fm(id)2982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1199 y Fk(A)3061 1184 y Fn(^)d Fm(id)3232 1199 y Fk(B)3292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1184 y Fr(.)j(Note)83 1304 y(that)271 1526 y Fm(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(m)d Fn(^)f Fm(id)586 1541 y Fk(B)647 1526 y Fm(\);)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5526
b(\(id)848 1541 y Fk(A)927 1526 y Fn(^)23 b Fm(id)1097
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1541 y Fk(B)1158 1526 y Fm(\))k(=)p 1327 1441 843 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5528
28 w Fn(f)p Fq(m;)17 b Fm(id)1587 1541 y Fk(B)1647 1526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5529
y Fn(g)p Fm(;)g(\(id)1860 1541 y Fk(A)1939 1526 y Fn(\001)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5530
b Fm(id)2070 1541 y Fk(B)2131 1526 y Fm(\))100 b(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5531
Fq(m)23 b Fn(^)f Fm(id)2803 1541 y Fk(B)2891 1526 y Fm(=)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5532
2995 1447 300 4 v 28 w Fq(m)g Fn(\001)g Fm(id)3233 1541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5533
y Fk(B)3322 1526 y Fq(:)83 1748 y Fr(No)n(w)i(we)i(just)d(ar)n(gued)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5534
(that)e(we)h(should)f(not)g(ha)n(v)o(e)1146 1970 y Fn(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5535
Fq(m;)17 b Fm(id)1406 1985 y Fk(B)1467 1970 y Fn(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5536
Fm(;)g(\(id)1680 1985 y Fk(A)1759 1970 y Fn(\001)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5537
Fm(id)1890 1985 y Fk(B)1951 1970 y Fm(\))27 b(=)h Fq(m)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5538
b Fn(\001)g Fm(id)2358 1985 y Fk(B)2447 1970 y Fq(:)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5539
2191 y Fr(But)j(as)g Fq(i)393 2206 y Fk(A)p Fl(^)p Fk(B)554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5540
2191 y Fm(;)p 598 2112 57 4 v 17 w Fq(h)i Fm(=)h Fq(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fr(,)d(the)g(operation)p 1439 2107 104 4 v 24 w Fm(\()i(\))e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fr(is)g(injecti)n(v)o(e.)e(So)i(we)g(cannot)f(ha)n(v)o(e)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5543
(equation)1086 2413 y Fm(\()p Fq(m)e Fn(^)h Fm(id)1401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2428 y Fk(B)1462 2413 y Fm(\);)17 b(\(id)1662 2428 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(A)1742 2413 y Fn(^)22 b Fm(id)1911 2428 y Fk(B)1972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2413 y Fm(\))28 b(=)f Fq(m)c Fn(^)f Fm(id)2419 2428 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fk(B)2507 2413 y Fq(:)83 2635 y Fr(The)32 b(general)g(point)e(seems)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(to)g(be)h(this.)e(If)i(we)g(cut)f(a)h(classical)f(proof)g(e)n(v)o(en)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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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
  5553
83 3176 y Fo(4.4)100 b(F)377 3177 y(\250)369 3176 y(uhrmann-Pym)23
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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Fn(\003)p Fr(-polycate)o(gory)e(in)h(which)g(the)g(polymaps)f(are)i
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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b(.)25 b(If)i(one)f(has)g(a)g(model)f(for)i(classical)e(proof)h(of)83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3713 y(this)e(kind)g(the)h(structure)f(simpli\002es)g(drastically)-6
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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g(pr)l(oof)f(in)h(whic)o(h)h Fn(^)p Fo(,)g Fn(>)h Fo(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fn(_)p Fo(,)h Fn(?)g Fo(r)l(epr)l(esent)83 4009 y(polymaps)24
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>
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>
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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>
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>
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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>
parents:
diff changeset
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(that)f Fn(C)28 b Fr(is)22 b(a)g(model)83 5380 y(for)f(classical)f
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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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>
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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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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324 y Fm(=)k Fq(e)1794 339 y Fk(A)1867 324 y Fn(_)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fq(e)1994 339 y Fk(B)2055 324 y Fr(.)23 b(\(Implicitly)f(we)h(ha)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fq(e)3477 339 y Fl(?)83 444 y Fr(as)i(we)g(found)g(them.\))f(Then)h(we)
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>
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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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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657 y Fm(=)27 b Fq(f)11 b Fn(g)27 b Fq(:)83 870 y Fr(The)e(quotient)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(functor)g(is)h(gi)n(v)o(en)e(by)992 1083 y Fn(C)6 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Fm(\()p Fq(A;)17 b(B)5 b Fm(\))27 b Fn(\000)-16 b(!)1556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1057 y Fm(^)1538 1083 y Fn(C)6 b Fm(\()p Fq(A;)17 b(B)5
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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1098 y Fk(B)2601 1083 y Fq(:)83 1296 y Fr(No)n(w)22 b(it)f(is)h(easy)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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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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b(,)23 b(then)3497 1445 y Fm(^)3478 1471 y Fn(C)83 1591
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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1591 y(uhrmann-Pym)f(sense)o(.)83 1881 y(4.5)100 b(Semantic)24
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>
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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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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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