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%!
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% PostScript prologue for pstricks.tex.
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% Version 97 patch 4, 04/05/10
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% For distribution, see pstricks.tex.
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%
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/tx@Dict 200 dict def tx@Dict begin
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/ADict 25 dict def
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/CM { matrix currentmatrix } bind def
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/SLW /setlinewidth load def
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/CLW /currentlinewidth load def
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/CP /currentpoint load def
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/ED { exch def } bind def
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/L /lineto load def
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/T /translate load def
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/TMatrix { } def
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/RAngle { 0 } def
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/Atan { /atan load stopped { pop pop 0 } if } def
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/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
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def
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/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
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} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
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/pathforall load stopped { pop pop pop pop } if z } def
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%
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%%-------------- DG begin patch 15 ---------------%%
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%/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
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%PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
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%mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
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%0 } ifelse setdash stroke } def
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/DashLine {
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dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def PathLength } ifelse
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/b ED /x1 ED /y1 ED /x ED /y ED
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/z y x add y1 add x1 add def
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/Coef b a .5 sub 2 mul y mul sub z Div round
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z mul a .5 sub 2 mul y mul add b exch Div def
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/y y Coef mul def /x x Coef mul def /y1 y1 Coef mul def /x1 x1 Coef mul def
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x1 0 gt y1 0 gt x 0 gt y 0 gt and { [ y x y1 x1 ] 1 a sub y mul}
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{ [ 1 0] 0 } ifelse setdash stroke
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} def
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%%-------------- DG end patch 15 ---------------%%
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/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
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{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
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def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
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] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
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setlinecap stroke } def
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/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
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2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
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% DG/SR modification begin - Dec. 12, 1997 - Patch 2
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%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
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itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
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% DG/SR modification end
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Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
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/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
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% DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis)
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% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
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% def
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a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
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pop pop } def
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% DG/SR modification end
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/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
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exch 3 index sub exch Atan rotate newpath } def
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/EndArrow { @mtrx setmatrix CP grestore end } def
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/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
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scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
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grestore } def
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/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
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CLW moveto } def
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/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
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div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
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CLW moveto } def
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/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
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2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
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curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
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def
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/SD { 0 360 arc fill } def
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/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
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/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
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/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
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load } if ] cvx newpath 3 1 roll T exec } def
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/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
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cvi def } def
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/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
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f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
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/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
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def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
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/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
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ifelse { pop } repeat a } def
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/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
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and { pop pop /n n 1 sub def } if } def
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/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
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repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
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ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
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moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
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Lineto pop pop closepath } ifelse } def
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/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
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neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
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def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
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div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
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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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225 |
/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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253 |
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
|
|
254 |
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
|
|
255 |
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
|
|
256 |
w add def } repeat grestore gsave n 0 gt
|
|
257 |
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
|
|
258 |
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
|
|
259 |
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
|
|
260 |
% DG/SR modification end
|
|
261 |
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
|
|
262 |
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
|
|
263 |
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
|
|
264 |
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
|
|
265 |
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
|
|
266 |
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
|
|
267 |
exch t L stroke /i i h add def } repeat grestore } def
|
|
268 |
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
|
|
269 |
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
|
|
270 |
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
|
|
271 |
mul neg d } def
|
|
272 |
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
|
|
273 |
/Rot { CP CP translate 3 -1 roll neg rotate NET } def
|
|
274 |
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
|
|
275 |
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
|
|
276 |
dup a add ] cvx def } def
|
|
277 |
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
|
|
278 |
cvx def } def
|
|
279 |
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
|
|
280 |
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
|
|
281 |
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
|
|
282 |
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
|
|
283 |
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
|
|
284 |
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
|
|
285 |
h1 abs h sub dup s mul abs } ifelse } def
|
|
286 |
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
|
|
287 |
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
|
|
288 |
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
|
|
289 |
exch } def
|
|
290 |
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
|
|
291 |
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
|
|
292 |
} if } ifelse } def
|
|
293 |
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
|
|
294 |
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
|
|
295 |
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
|
|
296 |
/IfVisible true def } def
|
|
297 |
end
|
|
298 |
% END pstricks.pro
|
|
299 |
|
|
300 |
%%EndProcSet
|
|
301 |
%%BeginProcSet: pst-dots.pro 0 0
|
|
302 |
%!PS-Adobe-2.0
|
|
303 |
%%Title: Dot Font for PSTricks
|
|
304 |
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
|
|
305 |
%%Creation Date: May 7, 1993
|
|
306 |
%% Version 97 patch 1, 99/12/16
|
|
307 |
%% Modified by Etienne Riga <etienne.riga@skynet.be> - Dec. 16, 1999
|
|
308 |
%% to add /Diamond, /SolidDiamond and /BoldDiamond
|
|
309 |
10 dict dup begin
|
|
310 |
/FontType 3 def
|
|
311 |
/FontMatrix [ .001 0 0 .001 0 0 ] def
|
|
312 |
/FontBBox [ 0 0 0 0 ] def
|
|
313 |
/Encoding 256 array def
|
|
314 |
0 1 255 { Encoding exch /.notdef put } for
|
|
315 |
Encoding
|
|
316 |
dup (b) 0 get /Bullet put
|
|
317 |
dup (c) 0 get /Circle put
|
|
318 |
dup (C) 0 get /BoldCircle put
|
|
319 |
dup (u) 0 get /SolidTriangle put
|
|
320 |
dup (t) 0 get /Triangle put
|
|
321 |
dup (T) 0 get /BoldTriangle put
|
|
322 |
dup (r) 0 get /SolidSquare put
|
|
323 |
dup (s) 0 get /Square put
|
|
324 |
dup (S) 0 get /BoldSquare put
|
|
325 |
dup (q) 0 get /SolidPentagon put
|
|
326 |
dup (p) 0 get /Pentagon put
|
|
327 |
dup (P) 0 get /BoldPentagon put
|
|
328 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
329 |
dup (l) 0 get /SolidDiamond put
|
|
330 |
dup (d) 0 get /Diamond put
|
|
331 |
(D) 0 get /BoldDiamond put
|
|
332 |
% DG/SR modification end
|
|
333 |
/Metrics 13 dict def
|
|
334 |
Metrics begin
|
|
335 |
/Bullet 1000 def
|
|
336 |
/Circle 1000 def
|
|
337 |
/BoldCircle 1000 def
|
|
338 |
/SolidTriangle 1344 def
|
|
339 |
/Triangle 1344 def
|
|
340 |
/BoldTriangle 1344 def
|
|
341 |
/SolidSquare 886 def
|
|
342 |
/Square 886 def
|
|
343 |
/BoldSquare 886 def
|
|
344 |
/SolidPentagon 1093.2 def
|
|
345 |
/Pentagon 1093.2 def
|
|
346 |
/BoldPentagon 1093.2 def
|
|
347 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
348 |
/SolidDiamond 1008 def
|
|
349 |
/Diamond 1008 def
|
|
350 |
/BoldDiamond 1008 def
|
|
351 |
% DG/SR modification end
|
|
352 |
/.notdef 0 def
|
|
353 |
end
|
|
354 |
/BBoxes 13 dict def
|
|
355 |
BBoxes begin
|
|
356 |
/Circle { -550 -550 550 550 } def
|
|
357 |
/BoldCircle /Circle load def
|
|
358 |
/Bullet /Circle load def
|
|
359 |
/Triangle { -571.5 -330 571.5 660 } def
|
|
360 |
/BoldTriangle /Triangle load def
|
|
361 |
/SolidTriangle /Triangle load def
|
|
362 |
/Square { -450 -450 450 450 } def
|
|
363 |
/BoldSquare /Square load def
|
|
364 |
/SolidSquare /Square load def
|
|
365 |
/Pentagon { -546.6 -465 546.6 574.7 } def
|
|
366 |
/BoldPentagon /Pentagon load def
|
|
367 |
/SolidPentagon /Pentagon load def
|
|
368 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
369 |
/Diamond { -428.5 -742.5 428.5 742.5 } def
|
|
370 |
/BoldDiamond /Diamond load def
|
|
371 |
/SolidDiamond /Diamond load def
|
|
372 |
% DG/SR modification end
|
|
373 |
/.notdef { 0 0 0 0 } def
|
|
374 |
end
|
|
375 |
/CharProcs 20 dict def
|
|
376 |
CharProcs begin
|
|
377 |
/Adjust {
|
|
378 |
2 copy dtransform floor .5 add exch floor .5 add exch idtransform
|
|
379 |
3 -1 roll div 3 1 roll exch div exch scale
|
|
380 |
} def
|
|
381 |
/CirclePath { 0 0 500 0 360 arc closepath } def
|
|
382 |
/Bullet { 500 500 Adjust CirclePath fill } def
|
|
383 |
/Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath
|
|
384 |
eofill } def
|
|
385 |
/BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath
|
|
386 |
eofill } def
|
|
387 |
/BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
|
|
388 |
/TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto
|
|
389 |
closepath } def
|
|
390 |
/SolidTriangle { TrianglePath fill } def
|
|
391 |
/Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
|
|
392 |
/BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
|
|
393 |
/SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto
|
|
394 |
-450 -450 lineto closepath } def
|
|
395 |
/SolidSquare { SquarePath fill } def
|
|
396 |
/Square { SquarePath .89 .89 scale SquarePath eofill } def
|
|
397 |
/BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
|
|
398 |
/PentagonPath {
|
|
399 |
-337.8 -465 moveto
|
|
400 |
337.8 -465 lineto
|
|
401 |
546.6 177.6 lineto
|
|
402 |
0 574.7 lineto
|
|
403 |
-546.6 177.6 lineto
|
|
404 |
closepath
|
|
405 |
} def
|
|
406 |
/SolidPentagon { PentagonPath fill } def
|
|
407 |
/Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
|
|
408 |
/BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
|
|
409 |
% DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga
|
|
410 |
/DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto
|
|
411 |
428.5 0 lineto closepath } def
|
|
412 |
/SolidDiamond { DiamondPath fill } def
|
|
413 |
/Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def
|
|
414 |
/BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def
|
|
415 |
% DG/SR modification end
|
|
416 |
/.notdef { } def
|
|
417 |
end
|
|
418 |
/BuildGlyph {
|
|
419 |
exch
|
|
420 |
begin
|
|
421 |
Metrics 1 index get exec 0
|
|
422 |
BBoxes 3 index get exec
|
|
423 |
setcachedevice
|
|
424 |
CharProcs begin load exec end
|
|
425 |
end
|
|
426 |
} def
|
|
427 |
/BuildChar {
|
|
428 |
1 index /Encoding get exch get
|
|
429 |
1 index /BuildGlyph get exec
|
|
430 |
} bind def
|
|
431 |
end
|
|
432 |
/PSTricksDotFont exch definefont pop
|
|
433 |
%END pst-dots.pro
|
|
434 |
|
|
435 |
%%EndProcSet
|
|
436 |
%%BeginProcSet: pst-node.pro 0 0
|
|
437 |
%!
|
|
438 |
% PostScript prologue for pst-node.tex.
|
|
439 |
% Version 97 patch 1, 97/05/09.
|
|
440 |
% For distribution, see pstricks.tex.
|
|
441 |
%
|
|
442 |
/tx@NodeDict 400 dict def tx@NodeDict begin
|
|
443 |
tx@Dict begin /T /translate load def end
|
|
444 |
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
|
|
445 |
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
|
|
446 |
grestore } def
|
|
447 |
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
|
|
448 |
} def
|
|
449 |
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
|
|
450 |
Sin mul } def } def
|
|
451 |
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
|
|
452 |
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
|
|
453 |
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
|
|
454 |
dup Sin mul Cos Div } ifelse } def
|
|
455 |
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
|
|
456 |
/u ED /NodePos { GetRnodePos } def } def
|
|
457 |
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
|
|
458 |
Cos mul exch Sin mul } def
|
|
459 |
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
|
|
460 |
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
|
|
461 |
} ifelse } def
|
|
462 |
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
|
|
463 |
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
|
|
464 |
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
|
|
465 |
sin def /NodePos { TriNodePos } def } def
|
|
466 |
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
|
|
467 |
hh mul Atan dup cos ww mul exch sin hh mul } def
|
|
468 |
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
|
|
469 |
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
|
|
470 |
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
|
|
471 |
{ 0 Dist Sin mul } ifelse } ifelse Do } def
|
|
472 |
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
|
|
473 |
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
|
|
474 |
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
|
|
475 |
XYPos } ifelse } def
|
|
476 |
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
|
|
477 |
roll sin mul sub exch } ifelse } def
|
|
478 |
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
|
|
479 |
AddOffset yA add /yA1 ED xA add /xA1 ED } def
|
|
480 |
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
|
|
481 |
AddOffset yB add /yB1 ED xB add /xB1 ED } def
|
|
482 |
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
|
|
483 |
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
|
|
484 |
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
|
|
485 |
} ifelse } def
|
|
486 |
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
|
|
487 |
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
|
|
488 |
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
|
|
489 |
} ifelse } def
|
|
490 |
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
|
|
491 |
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
|
|
492 |
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
|
|
493 |
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
|
|
494 |
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
|
|
495 |
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
|
|
496 |
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
|
|
497 |
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
|
|
498 |
} repeat LPutLine cleartomark } def
|
|
499 |
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
|
|
500 |
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
|
|
501 |
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
|
|
502 |
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
|
|
503 |
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
|
|
504 |
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
|
|
505 |
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
|
|
506 |
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
|
|
507 |
} def
|
|
508 |
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
|
|
509 |
/NAngle yyB yyA sub xxB xxA sub Atan def } def
|
|
510 |
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def
|
|
511 |
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
|
|
512 |
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
|
|
513 |
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd
|
|
514 |
} def
|
|
515 |
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
|
|
516 |
} def
|
|
517 |
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
|
|
518 |
/NAngle yyB yyA sub xxB xxA sub Atan def } def
|
|
519 |
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def
|
|
520 |
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
|
|
521 |
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
|
|
522 |
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
|
|
523 |
VPosEnd } def
|
|
524 |
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
|
|
525 |
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
|
|
526 |
def grestore exec /LPutVar /SaveLPutVar load def } def
|
|
527 |
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
|
|
528 |
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
|
|
529 |
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
|
|
530 |
LPutVar VPutLine } def LPutVar } def
|
|
531 |
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
|
|
532 |
end } def
|
|
533 |
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
|
|
534 |
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
|
|
535 |
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
|
|
536 |
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
|
|
537 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
538 |
ifelse } def
|
|
539 |
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
|
|
540 |
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
|
|
541 |
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
|
|
542 |
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
|
|
543 |
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
|
|
544 |
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
|
|
545 |
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
|
|
546 |
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
|
|
547 |
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
|
|
548 |
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
|
|
549 |
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
|
|
550 |
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
|
|
551 |
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
|
|
552 |
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
|
|
553 |
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
|
|
554 |
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
|
|
555 |
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
556 |
VPutLines } def } def
|
|
557 |
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
|
|
558 |
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
|
|
559 |
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
|
|
560 |
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
|
|
561 |
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
|
|
562 |
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
|
|
563 |
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
564 |
VPutLines } def } def
|
|
565 |
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
|
|
566 |
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
|
|
567 |
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
|
|
568 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
569 |
def
|
|
570 |
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
|
|
571 |
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
|
|
572 |
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
|
|
573 |
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
|
|
574 |
def
|
|
575 |
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
|
|
576 |
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
|
|
577 |
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
|
|
578 |
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
|
|
579 |
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
|
|
580 |
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
|
|
581 |
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
|
|
582 |
VPutLines } def } def
|
|
583 |
% DG/SR modification begin - May 9, 1997 - Patch 1
|
|
584 |
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
|
|
585 |
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
|
|
586 |
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
|
|
587 |
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
|
|
588 |
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
|
|
589 |
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
|
|
590 |
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
|
|
591 |
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
|
|
592 |
% DG/SR modification end
|
|
593 |
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
|
|
594 |
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
|
|
595 |
Div def /y ED /x ED } def
|
|
596 |
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
|
|
597 |
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
|
|
598 |
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
|
|
599 |
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
|
|
600 |
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
|
|
601 |
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
|
|
602 |
tx@Dict begin false Polygon end } def
|
|
603 |
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
|
|
604 |
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
|
|
605 |
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
|
|
606 |
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
|
|
607 |
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
|
|
608 |
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
|
|
609 |
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
|
|
610 |
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
|
|
611 |
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
|
|
612 |
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
|
|
613 |
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
|
|
614 |
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
|
|
615 |
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
|
|
616 |
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
|
|
617 |
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
|
|
618 |
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
|
|
619 |
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
|
|
620 |
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
|
|
621 |
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
|
|
622 |
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
|
|
623 |
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
|
|
624 |
LPutPos } def } def
|
|
625 |
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
|
|
626 |
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
|
|
627 |
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
|
|
628 |
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
|
|
629 |
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
|
|
630 |
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
|
|
631 |
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
|
|
632 |
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
|
|
633 |
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
|
|
634 |
exch X sub neg exch moveto setmatrix CP grestore } def
|
|
635 |
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
|
|
636 |
def } ifelse LPutCoor } def
|
|
637 |
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
|
|
638 |
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
|
|
639 |
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
|
|
640 |
def
|
|
641 |
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
|
|
642 |
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
|
|
643 |
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
|
|
644 |
def
|
|
645 |
end
|
|
646 |
% END pst-node.pro
|
|
647 |
|
|
648 |
%%EndProcSet
|
|
649 |
%%BeginProcSet: 8r.enc 0 0
|
|
650 |
% File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30
|
|
651 |
%
|
|
652 |
% @@psencodingfile@{
|
|
653 |
% author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,
|
|
654 |
% W. Schmidt, P. Lehman",
|
|
655 |
% version = "2.0",
|
|
656 |
% date = "30 October 2002",
|
|
657 |
% filename = "8r.enc",
|
|
658 |
% email = "tex-fonts@@tug.org",
|
|
659 |
% docstring = "This is the encoding vector for Type1 and TrueType
|
|
660 |
% fonts to be used with TeX. This file is part of the
|
|
661 |
% PSNFSS bundle, version 9"
|
|
662 |
% @}
|
|
663 |
%
|
|
664 |
% The idea is to have all the characters normally included in Type 1 fonts
|
|
665 |
% available for typesetting. This is effectively the characters in Adobe
|
|
666 |
% Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,
|
|
667 |
% MacRoman, and some extra characters from Lucida.
|
|
668 |
%
|
|
669 |
% Character code assignments were made as follows:
|
|
670 |
%
|
|
671 |
% (1) the Windows ANSI characters are almost all in their Windows ANSI
|
|
672 |
% positions, because some Windows users cannot easily reencode the
|
|
673 |
% fonts, and it makes no difference on other systems. The only Windows
|
|
674 |
% ANSI characters not available are those that make no sense for
|
|
675 |
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
|
|
676 |
% (173). quotesingle and grave are moved just because it's such an
|
|
677 |
% irritation not having them in TeX positions.
|
|
678 |
%
|
|
679 |
% (2) Remaining characters are assigned arbitrarily to the lower part
|
|
680 |
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
|
|
681 |
%
|
|
682 |
% (3) Y&Y Lucida Bright includes some extra text characters; in the
|
|
683 |
% hopes that other PostScript fonts, perhaps created for public
|
|
684 |
% consumption, will include them, they are included starting at 0x12.
|
|
685 |
% These are /dotlessj /ff /ffi /ffl.
|
|
686 |
%
|
|
687 |
% (4) hyphen appears twice for compatibility with both ASCII and Windows.
|
|
688 |
%
|
|
689 |
% (5) /Euro was assigned to 128, as in Windows ANSI
|
|
690 |
%
|
|
691 |
% (6) Missing characters from MacRoman encoding incorporated as follows:
|
|
692 |
%
|
|
693 |
% PostScript MacRoman TeXBase1
|
|
694 |
% -------------- -------------- --------------
|
|
695 |
% /notequal 173 0x16
|
|
696 |
% /infinity 176 0x17
|
|
697 |
% /lessequal 178 0x18
|
|
698 |
% /greaterequal 179 0x19
|
|
699 |
% /partialdiff 182 0x1A
|
|
700 |
% /summation 183 0x1B
|
|
701 |
% /product 184 0x1C
|
|
702 |
% /pi 185 0x1D
|
|
703 |
% /integral 186 0x81
|
|
704 |
% /Omega 189 0x8D
|
|
705 |
% /radical 195 0x8E
|
|
706 |
% /approxequal 197 0x8F
|
|
707 |
% /Delta 198 0x9D
|
|
708 |
% /lozenge 215 0x9E
|
|
709 |
%
|
|
710 |
/TeXBase1Encoding [
|
|
711 |
% 0x00
|
|
712 |
/.notdef /dotaccent /fi /fl
|
|
713 |
/fraction /hungarumlaut /Lslash /lslash
|
|
714 |
/ogonek /ring /.notdef /breve
|
|
715 |
/minus /.notdef /Zcaron /zcaron
|
|
716 |
% 0x10
|
|
717 |
/caron /dotlessi /dotlessj /ff
|
|
718 |
/ffi /ffl /notequal /infinity
|
|
719 |
/lessequal /greaterequal /partialdiff /summation
|
|
720 |
/product /pi /grave /quotesingle
|
|
721 |
% 0x20
|
|
722 |
/space /exclam /quotedbl /numbersign
|
|
723 |
/dollar /percent /ampersand /quoteright
|
|
724 |
/parenleft /parenright /asterisk /plus
|
|
725 |
/comma /hyphen /period /slash
|
|
726 |
% 0x30
|
|
727 |
/zero /one /two /three
|
|
728 |
/four /five /six /seven
|
|
729 |
/eight /nine /colon /semicolon
|
|
730 |
/less /equal /greater /question
|
|
731 |
% 0x40
|
|
732 |
/at /A /B /C
|
|
733 |
/D /E /F /G
|
|
734 |
/H /I /J /K
|
|
735 |
/L /M /N /O
|
|
736 |
% 0x50
|
|
737 |
/P /Q /R /S
|
|
738 |
/T /U /V /W
|
|
739 |
/X /Y /Z /bracketleft
|
|
740 |
/backslash /bracketright /asciicircum /underscore
|
|
741 |
% 0x60
|
|
742 |
/quoteleft /a /b /c
|
|
743 |
/d /e /f /g
|
|
744 |
/h /i /j /k
|
|
745 |
/l /m /n /o
|
|
746 |
% 0x70
|
|
747 |
/p /q /r /s
|
|
748 |
/t /u /v /w
|
|
749 |
/x /y /z /braceleft
|
|
750 |
/bar /braceright /asciitilde /.notdef
|
|
751 |
% 0x80
|
|
752 |
/Euro /integral /quotesinglbase /florin
|
|
753 |
/quotedblbase /ellipsis /dagger /daggerdbl
|
|
754 |
/circumflex /perthousand /Scaron /guilsinglleft
|
|
755 |
/OE /Omega /radical /approxequal
|
|
756 |
% 0x90
|
|
757 |
/.notdef /.notdef /.notdef /quotedblleft
|
|
758 |
/quotedblright /bullet /endash /emdash
|
|
759 |
/tilde /trademark /scaron /guilsinglright
|
|
760 |
/oe /Delta /lozenge /Ydieresis
|
|
761 |
% 0xA0
|
|
762 |
/.notdef /exclamdown /cent /sterling
|
|
763 |
/currency /yen /brokenbar /section
|
|
764 |
/dieresis /copyright /ordfeminine /guillemotleft
|
|
765 |
/logicalnot /hyphen /registered /macron
|
|
766 |
% 0xD0
|
|
767 |
/degree /plusminus /twosuperior /threesuperior
|
|
768 |
/acute /mu /paragraph /periodcentered
|
|
769 |
/cedilla /onesuperior /ordmasculine /guillemotright
|
|
770 |
/onequarter /onehalf /threequarters /questiondown
|
|
771 |
% 0xC0
|
|
772 |
/Agrave /Aacute /Acircumflex /Atilde
|
|
773 |
/Adieresis /Aring /AE /Ccedilla
|
|
774 |
/Egrave /Eacute /Ecircumflex /Edieresis
|
|
775 |
/Igrave /Iacute /Icircumflex /Idieresis
|
|
776 |
% 0xD0
|
|
777 |
/Eth /Ntilde /Ograve /Oacute
|
|
778 |
/Ocircumflex /Otilde /Odieresis /multiply
|
|
779 |
/Oslash /Ugrave /Uacute /Ucircumflex
|
|
780 |
/Udieresis /Yacute /Thorn /germandbls
|
|
781 |
% 0xE0
|
|
782 |
/agrave /aacute /acircumflex /atilde
|
|
783 |
/adieresis /aring /ae /ccedilla
|
|
784 |
/egrave /eacute /ecircumflex /edieresis
|
|
785 |
/igrave /iacute /icircumflex /idieresis
|
|
786 |
% 0xF0
|
|
787 |
/eth /ntilde /ograve /oacute
|
|
788 |
/ocircumflex /otilde /odieresis /divide
|
|
789 |
/oslash /ugrave /uacute /ucircumflex
|
|
790 |
/udieresis /yacute /thorn /ydieresis
|
|
791 |
] def
|
|
792 |
|
|
793 |
|
|
794 |
%%EndProcSet
|
|
795 |
%%BeginProcSet: texps.pro 0 0
|
|
796 |
%!
|
|
797 |
TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2
|
|
798 |
index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll
|
|
799 |
exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0
|
|
800 |
ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{
|
|
801 |
pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get
|
|
802 |
div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type
|
|
803 |
/nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end
|
|
804 |
definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup
|
|
805 |
sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll
|
|
806 |
mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[
|
|
807 |
exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if}
|
|
808 |
forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def
|
|
809 |
end
|
|
810 |
|
|
811 |
%%EndProcSet
|
|
812 |
%%BeginProcSet: special.pro 0 0
|
|
813 |
%!
|
|
814 |
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
|
|
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
|
|
816 |
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
|
|
817 |
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
|
|
818 |
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
|
|
819 |
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
|
|
820 |
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
|
|
821 |
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
|
|
822 |
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
|
|
823 |
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
|
|
824 |
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
|
|
825 |
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
|
|
826 |
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
|
|
827 |
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
|
|
828 |
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
|
|
829 |
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
|
|
830 |
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|
2611 |
2941 y FG(a)f Fr(=)g FG(a)2007 2953 y Fq(1)1518 3036
|
|
2612 |
y FG(a)1559 3048 y Fq(1)1668 3036 y FI(if)h FG(\031)1785
|
|
2613 |
3045 y Fo(\001)1823 3036 y FG(a)f Fr(=)g FG(a)2007 3048
|
|
2614 |
y Fq(2)1518 3132 y FG(\031)1565 3141 y Fo(\001)1603 3132
|
|
2615 |
y FG(a)j FI(otherwise)2789 2976 y(\(1\))0 3430 y(where)i
|
|
2616 |
Fr(\()p FG(a)12 b(b)p Fr(\))31 b(::)h FG(\031)27 b FI(is)e(the)g
|
|
2617 |
(composition)j(of)d(a)g(permutation)j(follo)n(wed)e(by)g(the)g(sw)o
|
|
2618 |
(apping)g Fr(\()p FG(a)13 b(b)p Fr(\))p FI(.)24 b(The)0
|
|
2619 |
3526 y(composition)d(of)e FG(\031)j FI(follo)n(wed)e(by)g(another)g
|
|
2620 |
(permutation)h FG(\031)1656 3494 y Fl(0)1698 3526 y FI(is)d(gi)n(v)o
|
|
2621 |
(en)i(by)g(list-concatenation,)g(written)0 3621 y(as)f
|
|
2622 |
FG(\031)132 3590 y Fl(0)155 3621 y Fr(@)p FG(\031)s FI(,)g(and)i(the)f
|
|
2623 |
(in)m(v)o(erse)g(of)g(a)g(permutation)i(is)d(gi)n(v)o(en)i(by)g(list)e
|
|
2624 |
(re)n(v)o(ersal,)h(written)g(as)f FG(\031)2475 3590 y
|
|
2625 |
Fl(\000)p Fq(1)2564 3621 y FI(.)125 3736 y(Our)31 b(representation)i
|
|
2626 |
(of)e(permutations)i(as)e(lists)f(does)h(not)h(gi)n(v)o(e)g(unique)g
|
|
2627 |
(representati)n(v)o(es:)g(for)0 3832 y(e)o(xample,)22
|
|
2628 |
b(the)g(permutation)h Fr(\()p FG(a)13 b(a)p Fr(\))20
|
|
2629 |
b FI(is)h(\223equal\224)h(to)g(the)g(identity)g(permutation.)h(W)-6
|
|
2630 |
b(e)22 b(equate)g(the)g(repre-)0 3927 y(sentations)e(of)h(permutations)
|
|
2631 |
g(with)f(a)g(relation)h FD(\030)p FI(:)p 0 TeXcolorgray
|
|
2632 |
0 4194 a FJ(De\002nition)f(1)p 0 TeXcolorgray 42 w(\(P)n(ermutation)25
|
|
2633 |
b(Equality\))f FE(T)-6 b(wo)24 b(permutations)h(ar)m(e)e
|
|
2634 |
FI(equal)p FE(,)h(written)g FG(\031)2471 4206 y Fq(1)2536
|
|
2635 |
4194 y FD(\030)k FG(\031)2668 4206 y Fq(2)2705 4194 y
|
|
2636 |
FE(,)c(pr)l(o-)0 4289 y(vided)d FG(\031)235 4301 y Fq(1)272
|
|
2637 |
4298 y Fo(\001)310 4289 y FG(a)g Fr(=)g FG(\031)497 4301
|
|
2638 |
y Fq(2)534 4298 y Fo(\001)572 4289 y FG(a)f FE(for)g(all)g(atoms)g
|
|
2639 |
FG(a)p FE(.)125 4536 y FI(T)-6 b(o)25 b(generalise)h(the)g(notion)g(gi)
|
|
2640 |
n(v)o(en)h(in)e(\(1\))h(of)g(a)f(permutation)j(acting)e(on)g(an)g
|
|
2641 |
(atom,)f(we)g(tak)o(e)h(ad-)0 4632 y(v)n(antage)f(of)g(the)f(o)o(v)o
|
|
2642 |
(erloading)j(mechanism)e(in)f(Isabelle)g(by)h(declaring)h(a)e
|
|
2643 |
(constant,)g(written)h(in\002x)f(as)0 4727 y Fr(\()p
|
|
2644 |
FD(\000)p Fr(\))120 4736 y Fo(\001)158 4727 y Fr(\()p
|
|
2645 |
FD(\000)p Fr(\))p FI(,)g(with)i(the)f(polymorphic)k(type)d
|
|
2646 |
Fs(name)c(prm)g FD(\))f FG(\013)h FD(\))f FG(\013)p FI(.)k(A)g
|
|
2647 |
(de\002nition)i(of)f(the)g(permutation)0 4823 y(operation)j(can)e(then)
|
|
2648 |
h(be)f(gi)n(v)o(en)i(separately)e(for)h(each)g(type-constructor;)h(for)
|
|
2649 |
f(lists,)e(products,)i(unit,)p 0 TeXcolorgray 0 TeXcolorgray
|
|
2650 |
eop end
|
|
2651 |
%%Page: 4 4
|
|
2652 |
TeXDict begin 4 3 bop 0 TeXcolorgray 0 TeXcolorgray 0
|
|
2653 |
71 2881 4 v 0 17 a FF(4)p 0 TeXcolorgray 0 228 a FI(sets,)19
|
|
2654 |
b(functions,)i(options)g(and)f(booleans)h(the)g(de\002nitions)f(are)g
|
|
2655 |
(as)g(follo)n(ws:)696 408 y FG(\013)i Fs(list)g Fr(:)350
|
|
2656 |
b FG(\031)1363 417 y Fo(\001)1401 408 y Fr([])1469 365
|
|
2657 |
y Fn(def)1476 408 y Fr(=)32 b([])1143 527 y FG(\031)1190
|
|
2658 |
536 y Fo(\001)1228 527 y Fr(\()p FG(x)20 b Fr(::)i FG(t)p
|
|
2659 |
Fr(\))1469 485 y Fn(def)1476 527 y Fr(=)32 b(\()p FG(\031)1645
|
|
2660 |
536 y Fo(\001)1683 527 y FG(x)p Fr(\))20 b(::)i(\()p
|
|
2661 |
FG(\031)1918 536 y Fo(\001)1956 527 y FG(t)p Fr(\))696
|
|
2662 |
647 y FG(\013)745 659 y Fq(1)800 647 y FD(\002)17 b FG(\013)926
|
|
2663 |
659 y Fq(2)984 647 y Fr(:)99 b FG(\031)1151 656 y Fo(\001)1188
|
|
2664 |
647 y Fr(\()p FG(x)1262 659 y Fq(1)1299 647 y FG(;)13
|
|
2665 |
b(x)1377 659 y Fq(2)1414 647 y Fr(\))1469 604 y Fn(def)1476
|
|
2666 |
647 y Fr(=)32 b(\()p FG(\031)1645 656 y Fo(\001)1683
|
|
2667 |
647 y FG(x)1727 659 y Fq(1)1763 647 y FG(;)13 b(\031)1844
|
|
2668 |
656 y Fo(\001)1882 647 y FG(x)1926 659 y Fq(2)1963 647
|
|
2669 |
y Fr(\))696 766 y Fs(unit)22 b Fr(:)404 b FG(\031)1346
|
|
2670 |
775 y Fo(\001)1384 766 y Fr(\(\))1469 723 y Fn(def)1476
|
|
2671 |
766 y Fr(=)32 b(\(\))696 886 y FG(\013)22 b Fs(set)g
|
|
2672 |
Fr(:)363 b FG(\031)1337 895 y Fo(\001)1375 886 y FG(X)1469
|
|
2673 |
843 y Fn(def)1476 886 y Fr(=)32 b FD(f)p FG(\031)1653
|
|
2674 |
895 y Fo(\001)1691 886 y FG(x)12 b FD(j)i FG(x)20 b FD(2)i
|
|
2675 |
FG(X)6 b FD(g)696 1005 y FG(\013)745 1017 y Fq(1)804
|
|
2676 |
1005 y FD(\))21 b FG(\013)951 1017 y Fq(2)1010 1005 y
|
|
2677 |
Fr(:)255 b FG(\031)1333 1014 y Fo(\001)1371 1005 y Fk(fn)1469
|
|
2678 |
962 y Fn(def)1476 1005 y Fr(=)32 b FG(\025x:\031)1725
|
|
2679 |
1014 y Fo(\001)1762 1005 y Fr(\()p Fk(fn)19 b Fr(\()p
|
|
2680 |
FG(\031)1955 973 y Fl(\000)p Fq(1)2043 1014 y Fo(\001)2081
|
|
2681 |
1005 y FG(x)p Fr(\)\))696 1124 y FG(\013)j Fs(option)h
|
|
2682 |
Fr(:)134 b FG(\031)1226 1133 y Fo(\001)1264 1124 y Fk(None)1469
|
|
2683 |
1081 y Fn(def)1476 1124 y Fr(=)32 b Fk(None)1070 1244
|
|
2684 |
y FG(\031)1117 1253 y Fo(\001)1155 1244 y Fk(Some)6 b
|
|
2685 |
Fr(\()p FG(x)p Fr(\))1469 1201 y Fn(def)1476 1244 y Fr(=)32
|
|
2686 |
b Fk(Some)6 b Fr(\()p FG(\031)1831 1253 y Fo(\001)1869
|
|
2687 |
1244 y FG(x)p Fr(\))696 1363 y Fs(bool)22 b Fr(:)431
|
|
2688 |
b FG(\031)1373 1372 y Fo(\001)1411 1363 y FG(b)1469 1320
|
|
2689 |
y Fn(def)1476 1363 y Fr(=)32 b FG(b)2789 873 y FI(\(2\))125
|
|
2690 |
1517 y(It)19 b(will)g(sa)n(v)o(e)h(much)h(w)o(ork)g(later)e(on)i(to)f
|
|
2691 |
FE(not)h FI(establish)f(properties)h(for)f(each)g(of)g(these)g
|
|
2692 |
(permutation)0 1613 y(operations)h(indi)n(vidually)-5
|
|
2693 |
b(,)21 b(b)n(ut)f(reason)g(abstractly)g(o)o(v)o(er)g(them)g(by)g
|
|
2694 |
(requiring)i(that)d(e)n(v)o(ery)i(permutation)0 1708
|
|
2695 |
y(operation)h(satis\002es)c(three)i(basic)g(properties:)p
|
|
2696 |
0 TeXcolorgray 0 1869 a FJ(De\002nition)g(2)p 0 TeXcolorgray
|
|
2697 |
42 w(\(P)n(ermutation)h(T)-6 b(ype\))21 b FE(A)f(type)g
|
|
2698 |
FG(\013)h FE(will)e(be)h(r)m(eferr)m(ed)g(to)g(as)h FI(permutation)h
|
|
2699 |
(type)p FE(,)e(written)0 1965 y FG(pt)67 1973 y Fp(\013)114
|
|
2700 |
1965 y FE(,)f(pr)l(o)o(vided)j(the)e(permutation)h(oper)o(ation)g
|
|
2701 |
(satis\002es)f(the)g(following)h(thr)m(ee)f(pr)l(operties:)p
|
|
2702 |
0 TeXcolorgray 56 2103 a(\(i\))p 0 TeXcolorgray 63 w
|
|
2703 |
Fr([])235 2112 y Fo(\001)274 2103 y FG(x)42 b Fr(=)h
|
|
2704 |
FG(x)p 0 TeXcolorgray 34 2198 a FE(\(ii\))p 0 TeXcolorgray
|
|
2705 |
63 w Fr(\()p FG(\031)267 2210 y Fq(1)304 2198 y Fr(@)p
|
|
2706 |
FG(\031)408 2210 y Fq(2)445 2198 y Fr(\))475 2207 y Fo(\001)513
|
|
2707 |
2198 y FG(x)f Fr(=)h FG(\031)746 2210 y Fq(1)783 2207
|
|
2708 |
y Fo(\001)821 2198 y Fr(\()p FG(\031)895 2210 y Fq(2)932
|
|
2709 |
2207 y Fo(\001)970 2198 y FG(x)p Fr(\))p 0 TeXcolorgray
|
|
2710 |
12 2294 a FE(\(iii\))p 0 TeXcolorgray 63 w FG(\031)237
|
|
2711 |
2306 y Fq(1)296 2294 y FD(\030)21 b FG(\031)421 2306
|
|
2712 |
y Fq(2)501 2294 y FE(implies)42 b FG(\031)815 2306 y
|
|
2713 |
Fq(1)852 2303 y Fo(\001)890 2294 y FG(x)h Fr(=)f FG(\031)1123
|
|
2714 |
2306 y Fq(2)1160 2303 y Fo(\001)1198 2294 y FG(x)0 2455
|
|
2715 |
y FI(These)25 b(properties)i(entail)e(that)g(the)h(permutations)h
|
|
2716 |
(operation)f(beha)n(v)o(es)h(o)o(v)o(er)e(permutation)j(types)d(as)0
|
|
2717 |
2551 y(one)c(e)o(xpects:)p 0 TeXcolorgray 0 2712 a FJ(Lemma)e(1)p
|
|
2718 |
0 TeXcolorgray 42 w FE(Assuming)i FG(x)e FE(and)i FG(y)h
|
|
2719 |
FE(ar)m(e)e(of)g(permutation)h(type)f(then:)p 0 TeXcolorgray
|
|
2720 |
56 2850 a(\(i\))p 0 TeXcolorgray 42 w FG(\031)219 2818
|
|
2721 |
y Fl(\000)p Fq(1)308 2859 y Fo(\001)346 2850 y Fr(\()p
|
|
2722 |
FG(\031)423 2859 y Fo(\001)460 2850 y FG(x)p Fr(\))h(=)g
|
|
2723 |
FG(x)p FE(,)p 0 TeXcolorgray 34 2945 a(\(ii\))p 0 TeXcolorgray
|
|
2724 |
42 w FG(\031)219 2954 y Fo(\001)257 2945 y FG(x)f Fr(=)i
|
|
2725 |
FG(y)g FE(if)d(and)i(only)g(if)e FG(x)i Fr(=)g FG(\031)1076
|
|
2726 |
2913 y Fl(\000)p Fq(1)1165 2954 y Fo(\001)1203 2945 y
|
|
2727 |
FG(y)s FE(,)p 0 TeXcolorgray 12 3041 a(\(iii\))p 0 TeXcolorgray
|
|
2728 |
42 w FG(\031)219 3050 y Fo(\001)257 3041 y FG(x)f Fr(=)i
|
|
2729 |
FG(\031)450 3050 y Fo(\001)487 3041 y FG(y)h FE(if)c(and)i(only)g(if)e
|
|
2730 |
FG(x)i Fr(=)g FG(y)s FE(,)e(and)p 0 TeXcolorgray 21 3136
|
|
2731 |
a(\(iv\))p 0 TeXcolorgray 42 w FG(\031)219 3145 y Fo(\001)257
|
|
2732 |
3136 y FG(x)h FD(2)i FG(\031)441 3145 y Fo(\001)479 3136
|
|
2733 |
y FG(X)k FE(if)19 b(and)i(only)g(if)f FG(x)g FD(2)i FG(X)6
|
|
2734 |
b FE(.)p 0 TeXcolorgray 0 3348 a(Pr)l(oof)p 0 TeXcolorgray
|
|
2735 |
40 w FI(The)21 b(\002rst)f(property)j(holds)f(by)f(Def.)g(2)p
|
|
2736 |
FE(\(i-iii\))g FI(since)g Fr(\()p FG(\031)1725 3316 y
|
|
2737 |
Fl(\000)p Fq(1)1814 3348 y Fr(@)p FG(\031)s Fr(\))h FD(\030)h
|
|
2738 |
Fr([])p FI(,)e(which)g(can)g(be)g(sho)n(wn)h(by)0 3444
|
|
2739 |
y(an)f(induction)i(o)o(v)o(er)e(the)g(length)h(of)f FG(\031)s
|
|
2740 |
FI(.)f(The)h(second)h(property)h(follo)n(ws)e(from)h(the)f(\002rst.)f
|
|
2741 |
(The)h(third)h(is)e(a)0 3539 y(consequence)25 b(of)e(the)g(\002rst)g
|
|
2742 |
(and)h(second.)f(F)o(or)g(the)h(fourth)g(one)g(has)f(to)g(unwind)i(the)
|
|
2743 |
e(de\002nition)h(of)f(the)0 3635 y(permutation)f(operation)g(for)e
|
|
2744 |
(sets)f(and)i(apply)g(the)f(third)h(property)-5 b(.)80
|
|
2745 |
b FD(u)-51 b(t)125 3781 y FI(Using)17 b(Isabelle')l(s)g
|
|
2746 |
FE(axiomatic)h(type-classes)g FI([37],)g(it)f(is)g(v)o(ery)h(con)m(v)o
|
|
2747 |
(enient)h(to)f(ensure)g(that)f(a)g(type)h(is)0 3876 y(a)f(permutation)j
|
|
2748 |
(type)e(because)h(most)e(of)i(the)e(routine)i(w)o(ork)g(can)f(be)g
|
|
2749 |
(performed)i(by)e(the)g(type-checking)0 3972 y(algorithm)27
|
|
2750 |
b(of)f(Isabelle:)g(one)g(only)h(has)f(to)g(establish)f(that)h(some)g
|
|
2751 |
(\223base\224)g(types,)f(such)h(as)g Fs(name)g FI(and)0
|
|
2752 |
4067 y Fs(unit)q FI(,)21 b(are)g(permutation)i(types)f(and)g(that)f
|
|
2753 |
(type-constructors,)i(such)f(as)e(products)j(and)f(lists,)e(preserv)o
|
|
2754 |
(e)0 4163 y(the)g(property)i(of)e(being)h(a)f(permutation)i(type.)e
|
|
2755 |
(More)h(formally)g(we)f(ha)n(v)o(e:)p 0 TeXcolorgray
|
|
2756 |
0 4324 a FJ(Lemma)f(2)p 0 TeXcolorgray 42 w FE(Given)30
|
|
2757 |
b FG(pt)644 4332 y Fp(\013)691 4324 y FE(,)f FG(pt)807
|
|
2758 |
4332 y Fp(\013)850 4340 y Fj(1)915 4324 y FE(and)i FG(pt)1130
|
|
2759 |
4332 y Fp(\013)1173 4340 y Fj(2)1209 4324 y FE(,)e(the)h(types)g
|
|
2760 |
Fs(name)q FE(,)f Fs(unit)q FE(,)g FG(\013)39 b Fs(list)q
|
|
2761 |
FE(,)29 b FG(\013)40 b Fs(set)p FE(,)30 b FG(\013)39
|
|
2762 |
b Fs(option)q FE(,)0 4420 y FG(\013)49 4432 y Fq(1)104
|
|
2763 |
4420 y FD(\002)16 b FG(\013)229 4432 y Fq(2)267 4420
|
|
2764 |
y FE(,)j FG(\013)355 4432 y Fq(1)414 4420 y FD(\))i FG(\013)561
|
|
2765 |
4432 y Fq(2)618 4420 y FE(and)g Fs(bool)g FE(ar)m(e)f(also)g
|
|
2766 |
(permutation)h(types.)p 0 TeXcolorgray 0 4632 a(Pr)l(oof)p
|
|
2767 |
0 TeXcolorgray 40 w FI(All)i(properties)i(follo)n(w)g(by)f(unwinding)i
|
|
2768 |
(the)e(de\002nition)h(of)g(the)f(corresponding)j(permutation)0
|
|
2769 |
4727 y(operation)c(and)f(routine)g(inductions.)g(The)g(property)h
|
|
2770 |
FG(pt)1566 4735 y Fp(\013)1609 4743 y Fj(1)1641 4735
|
|
2771 |
y Fl(\))p Fp(\013)1750 4743 y Fj(2)1807 4727 y FI(uses)e(the)g(f)o(act)
|
|
2772 |
h(that)f FG(\031)2403 4739 y Fq(1)2463 4727 y FD(\030)j
|
|
2773 |
FG(\031)2591 4739 y Fq(2)2649 4727 y FI(implies)0 4823
|
|
2774 |
y FG(\031)47 4790 y Fl(\000)p Fq(1)44 4847 y(1)157 4823
|
|
2775 |
y FD(\030)d FG(\031)285 4790 y Fl(\000)p Fq(1)282 4847
|
|
2776 |
y(2)374 4823 y FI(.)p 0 TeXcolorgray 0 TeXcolorgray eop
|
|
2777 |
end
|
|
2778 |
%%Page: 5 5
|
|
2779 |
TeXDict begin 5 4 bop 0 TeXcolorgray 0 TeXcolorgray 0
|
|
2780 |
71 2881 4 v 2848 17 a FF(5)p 0 TeXcolorgray 125 228 a
|
|
2781 |
FI(Note)20 b(that)g(the)g(permutation)i(operation)f(o)o(v)o(er)g(a)f
|
|
2782 |
(function-type,)i(say)e FG(\013)2121 240 y Fq(1)2180
|
|
2783 |
228 y FD(\))h FG(\013)2327 240 y Fq(2)2385 228 y FI(with)f
|
|
2784 |
FG(\013)2594 240 y Fq(1)2651 228 y FI(being)h(a)0 324
|
|
2785 |
y(permutation)h(type,)e(is)f(de\002ned)i(so)f(that)g(for)h(e)n(v)o(ery)
|
|
2786 |
g(function)g Fk(fn)26 b FI(we)19 b(ha)n(v)o(e)i(the)f(equation)1050
|
|
2787 |
502 y FG(\031)1097 511 y Fo(\001)1135 502 y Fr(\()p Fk(fn)e
|
|
2788 |
FG(x)p Fr(\))j(=)g(\()p FG(\031)1503 511 y Fo(\001)1540
|
|
2789 |
502 y Fk(fn)6 b Fr(\)\()p FG(\031)1720 511 y Fo(\001)1758
|
|
2790 |
502 y FG(x)p Fr(\))957 b FI(\(3\))0 682 y(in)35 b(Isabelle/HOL;)g(this)
|
|
2791 |
g(is)f(because)h(we)g(ha)n(v)o(e)g FG(\031)1456 651 y
|
|
2792 |
Fl(\000)p Fq(1)1545 691 y Fo(\001)1583 682 y Fr(\()p
|
|
2793 |
FG(\031)1660 691 y Fo(\001)1698 682 y FG(x)p Fr(\))48
|
|
2794 |
b(=)g FG(x)34 b FI(by)h(Lem.)g(1)p FE(\(i\))h FI(and)f
|
|
2795 |
FG(\031)2662 691 y Fo(\001)2700 682 y Fk(fn)54 b Fr(=)0
|
|
2796 |
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2797 |
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2798 |
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2799 |
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2800 |
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2802 |
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2803 |
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2804 |
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2805 |
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2806 |
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2807 |
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2808 |
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2809 |
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2810 |
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2811 |
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2812 |
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2813 |
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2814 |
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2815 |
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2816 |
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2817 |
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2818 |
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2820 |
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2821 |
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2822 |
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2823 |
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2824 |
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2825 |
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2826 |
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2830 |
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2831 |
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2841 |
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2842 |
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2843 |
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2847 |
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2848 |
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2855 |
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2860 |
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2861 |
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2862 |
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2863 |
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2864 |
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2865 |
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2883 |
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2885 |
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2889 |
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2890 |
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2894 |
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2895 |
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2896 |
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2897 |
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2898 |
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2900 |
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2904 |
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2905 |
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2908 |
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2909 |
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2910 |
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2911 |
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2912 |
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2913 |
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2915 |
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2917 |
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2918 |
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2919 |
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2920 |
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2921 |
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2922 |
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2923 |
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2924 |
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2925 |
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2926 |
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2927 |
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2928 |
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2929 |
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2930 |
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2931 |
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2932 |
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2933 |
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|
2934 |
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2935 |
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2936 |
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2937 |
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2938 |
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2939 |
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2940 |
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|
2941 |
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2942 |
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2943 |
1518 y Fq(3)2545 1550 y Fr(\))0 1716 y FI(where)18 b
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2944 |
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2945 |
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|
2946 |
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|
2947 |
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2948 |
h(number)h(of)f(elements,)0 1811 y(and)h(where)g Fr(\()p
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|
2949 |
FD(\003)411 1779 y Fq(2)448 1811 y Fr(\))f FI(holds)h(because)g
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|
2950 |
(permutations)h(preserv)o(e)f(by)g(Lem.)f(1)p FE(\(ii\))h
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|
2951 |
FI(\(in\)equalities;)g Fr(\()p FD(\003)2624 1779 y Fq(3)2662
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|
2952 |
1811 y Fr(\))f FI(holds)0 1907 y(because)g FG(\031)i
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|
2953 |
FI(commutes)e(with)g(the)f(sw)o(apping,)i(that)e(is)g
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|
2954 |
FG(\031)s Fr(@\()p FG(c)12 b(d)p Fr(\))21 b FD(\030)g
|
|
2955 |
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|
2956 |
b(\031)2075 1916 y Fo(\001)2112 1907 y FG(d)p Fr(\)@)p
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|
2957 |
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|
2958 |
y FG(d)p FI(.)k(F)o(or)h(the)g(second)h(and)f(third)h(property)h(we)e
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|
2959 |
(ha)n(v)o(e)g(by)h(Lem.)f(1)p FE(\(iv\))g FI(that)g FG(a)j
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|
2960 |
FD(2)g Fk(supp)5 b Fr(\()p FG(x)p Fr(\))21 b FI(if)h(and)h(only)g(if)0
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|
2961 |
2098 y FG(\031)47 2107 y Fo(\001)85 2098 y FG(a)e FD(2)g
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|
2962 |
FG(\031)266 2107 y Fo(\001)304 2098 y Fk(supp)6 b Fr(\()p
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|
2963 |
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|
2964 |
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|
2965 |
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2966 |
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2967 |
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2968 |
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2969 |
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2970 |
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2971 |
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|
2972 |
FG(a)i Fr(#)g FG(x)f FE(and)h FG(b)g Fr(#)g FG(x)f FE(then)g
|
|
2973 |
Fr(\()p FG(a)12 b(b)p Fr(\))2228 2643 y Fo(\001)2266
|
|
2974 |
2634 y FG(x)21 b Fr(=)g FG(x)p FE(.)p 0 TeXcolorgray
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2975 |
0 2858 a(Pr)l(oof)p 0 TeXcolorgray 40 w FI(The)30 b(case)g
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2976 |
FG(a)39 b Fr(=)g FG(b)30 b FI(is)g(clear)g(by)g(Def.)g(2)p
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|
2977 |
FE(\(i,iii\))g FI(and)h(the)f(f)o(act)h(that)f Fr(\()p
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|
2978 |
FG(a)12 b(a)p Fr(\))39 b FD(\030)h Fr([])p FI(.)30 b(In)h(the)f(other)0
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2979 |
2953 y(case,)h(the)g(assumption)i(implies)f(that)f(both)i(sets)d
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|
2980 |
FD(f)p FG(c)14 b FD(j)f Fr(\()p FG(c)g(a)p Fr(\))1705
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|
2981 |
2962 y Fo(\001)1742 2953 y FG(x)42 b FD(6)p Fr(=)g FG(x)p
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|
2982 |
FD(g)31 b FI(and)h FD(f)p FG(c)14 b FD(j)f Fr(\()p FG(c)g(b)p
|
|
2983 |
Fr(\))2446 2962 y Fo(\001)2484 2953 y FG(x)41 b FD(6)p
|
|
2984 |
Fr(=)h FG(x)p FD(g)32 b FI(are)0 3049 y(\002nite,)22
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2985 |
b(and)h(therefore)h(also)e(their)h(union)h(must)f(be)g(\002nite.)f
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2986 |
(Hence)g(the)h(corresponding)i(co-set,)e(that)f(is)0
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2987 |
3144 y FD(f)p FG(c)13 b FD(j)h Fr(\()p FG(c)e(a)p Fr(\))265
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2988 |
3153 y Fo(\001)303 3144 y FG(x)21 b Fr(=)g FG(x)8 b FD(^)g
|
|
2989 |
Fr(\()p FG(c)k(b)p Fr(\))698 3153 y Fo(\001)736 3144
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|
2990 |
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2991 |
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2992 |
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2993 |
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|
2994 |
(from)f FG(a)f FI(and)g FG(b)p FI(,)0 3335 y(one)k(has)g
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|
2995 |
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|
|
2996 |
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|
|
2997 |
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|
|
2998 |
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|
|
2999 |
3344 y Fo(\001)1563 3335 y Fr(\()p FG(c)g(b)p Fr(\))1702
|
|
3000 |
3344 y Fo(\001)1740 3335 y Fr(\()p FG(c)g(a)p Fr(\))1887
|
|
3001 |
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|
3002 |
(assumptions)0 3430 y FG(a)26 b FD(6)p Fr(=)g FG(c)p
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|
3003 |
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|
|
3004 |
FG(b)p FI(,)d(the)f(permutations)j Fr(\()p FG(c)13 b(a)p
|
|
3005 |
Fr(\)\()p FG(c)f(b)p Fr(\)\()p FG(c)h(a)p Fr(\))22 b
|
|
3006 |
FI(and)h Fr(\()p FG(a)13 b(b)p Fr(\))22 b FI(are)h(equal.)g(Therefore)h
|
|
3007 |
(one)f(can)0 3526 y(conclude)e(with)f Fr(\()p FG(a)13
|
|
3008 |
b(b)p Fr(\))611 3535 y Fo(\001)649 3526 y FG(x)21 b Fr(=)g
|
|
3009 |
FG(x)e FI(by)h(using)h(Def.)f(2)p FE(\(ii,iii\))p FI(.)79
|
|
3010 |
b FD(u)-51 b(t)125 3698 y FI(A)18 b(further)i(restriction)g(on)f
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3011 |
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3012 |
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3014 |
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3015 |
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3016 |
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3017 |
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3018 |
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|
|
3019 |
b(shall)g(write)g Fs(finite)q Fr(\()p Fk(supp)5 b Fr(\()p
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|
3020 |
FG(x)p Fr(\)\))21 b FI(to)i(indicate)f(that)g(an)g(element)h
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|
3021 |
FG(x)e FI(from)i(a)f(permutation)i(type)f(has)0 4330
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|
3022 |
y(\002nite)d(support.)h(The)f(follo)n(wing)i(holds:)p
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3023 |
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3024 |
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|
3025 |
4524 y Fp(\013)803 4532 y Fj(1)860 4504 y FE(and)h(fs)1051
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|
3026 |
4524 y Fp(\013)1094 4532 y Fj(2)1130 4504 y FE(,)f(the)h(types)f
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|
3027 |
Fs(name)q FE(,)f Fs(unit)q FE(,)h FG(\013)i Fs(list)q
|
|
3028 |
FE(,)d FG(\013)j Fs(option)q FE(,)e FG(\013)2524 4516
|
|
3029 |
y Fq(1)2579 4504 y FD(\002)d FG(\013)2705 4516 y Fq(2)2763
|
|
3030 |
4504 y FE(and)0 4599 y Fs(bool)k FE(ar)m(e)f(also)g(\002nitely)g
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|
3031 |
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3032 |
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3033 |
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3034 |
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3035 |
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3036 |
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3037 |
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3038 |
FI(The)27 b(crucial)h(property)h(entailed)f(by)g(Def.)f(4)g(is)g(that)g
|
|
3039 |
(if)g(an)h(element,)f(say)h FG(x)p FI(,)e(of)i(a)f(permutation)0
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|
3040 |
324 y(type)20 b(has)f(\002nite)f(support,)j(then)e(there)h(must)f(be)g
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|
3041 |
(a)g(fresh)g(atom)h(for)g FG(x)p FI(,)e(since)h(there)g(are)g
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|
3042 |
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3043 |
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3044 |
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|
3045 |
(support,)g(then)h(ther)m(e)e(e)n(xists)g(an)h(atom)g
|
|
3046 |
FG(a)f FE(with)0 682 y FG(a)f Fr(#)g FG(x)p FE(.)0 850
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|
3047 |
y FI(As)d(a)g(result,)h(whene)n(v)o(er)h(we)e(need)h(to)g(ha)n(v)o(e)g
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|
3048 |
(a)g(fresh)g(atom)g(for)g(an)g FG(x)f FI(of)h(permutation)i(type,)e(we)
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|
3049 |
f(ha)n(v)o(e)i(to)0 945 y(mak)o(e)j(sure)f(that)h FG(x)e
|
|
3050 |
FI(has)h(\002nite)h(support.)g(This)f(task)g(can)h(be)f(automatically)i
|
|
3051 |
(performed)g(by)f(Isabelle')l(s)0 1041 y(axiomatic)h(type-classes)f
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|
3052 |
(for)h(most)f(constructions)i(occurring)g(in)e(informal)i(proofs:)f
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3053 |
(Isabelle)f(has)h(to)0 1136 y(just)c(e)o(xamine)g(the)h(types)f(of)g
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|
3054 |
(the)g(construction)i(using)f(Lem.)f(5.)125 1232 y(Prop)i(1)g(also)g
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3055 |
(implies)g(that)g(for)g(e)n(v)o(ery)h(\002nitely)f(supported)i
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3056 |
(function)f(a)f(fresh)g(atom)g(e)o(xists.)f(Ho)n(w-)0
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3057 |
1327 y(e)n(v)o(er)m(,)e(to)g(determine)h(whether)g(a)e(function)j(has)e
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3058 |
(\002nite)g(support)h(is)e(more)i(subtle,)f(because)g(not)h(all)e
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3059 |
(func-)0 1422 y(tions)25 b(are)g(\002nitely)g(supported,)i(e)n(v)o(en)e
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3060 |
(if)g(their)g(domain)h(and)g(codomain)h(are)e(\002nitely)g(supported)h
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3061 |
(per)n(-)0 1518 y(mutation)21 b(types)g(\(see)f([27,)h(Example)g(3.4,)g
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3062 |
(P)o(age)f(470]\).)h(Introducing)i(a)d(\002nitely)h(supported)h
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3063 |
(function)0 1613 y(space)d(and)g(blending)i(it)d(well)g(into)h
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3064 |
(Isabelle')l(s)g(reasoning)h(infrastructure)g(seems)f(impractical)g
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3065 |
(for)g(rea-)0 1709 y(sons)31 b(ho)n(w)g(Isabelle)g(is)f(implemented.)i
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3066 |
(So)f(for)g(functions)h(one)g(has)e(to)h(\223manually\224)h(ensure)g
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3067 |
(\002nite)0 1804 y(support,)23 b(which)g(we)f(shall)h(do)g(in)f(Sec.)g
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3068 |
(5)g(by)h(introducing)i(a)d(weak)o(er)h(notion)h(that)e(approximates)i
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3069 |
(the)0 1900 y(support)d(of)g(an)f(element)g(from)h(\223abo)o(v)o
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3070 |
(e\224.)0 2180 y FJ(3)28 b(Constructing)20 b(a)g(Repr)o(esentation)g(f)
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3071 |
n(or)h(Alpha-Equated)f(Lambda-T)-7 b(erms)0 2368 y FI(In)32
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3072 |
b(this)f(section)h(we)f(de\002ne)h(an)g(inducti)n(v)o(e)g(set)f(that)h
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3073 |
(is)f(bijecti)n(v)o(e)h(with)f(the)h(set)f(of)g(alpha-equated)0
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3074 |
2464 y(lambda-terms.)25 b(In)f(doing)h(so)f(our)g(goal)g(is)f(to)h(gi)n
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3075 |
(v)o(e)g(in)g(Isabelle/HOL)g(a)f(formal)i(implementation)g(of)0
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3076 |
2559 y(the)j(usual)g(con)m(v)o(ention)i(\(from)g(Barendre)o(gt)f([5,)f
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3077 |
(P)o(age)g(26]\))h(emplo)o(yed)h(e)o(xplicitly)e(or)h(implicitly)g(in)0
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3078 |
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3079 |
0 TeXcolorgray 290 2811 a(C)t Fe(O)t(N)t(V)t(E)t(N)t(T)t(I)t(O)t(N)r
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3080 |
FI(.)28 b(T)-6 b(erms)34 b(that)f(are)h FG(\013)p FI(-congruent)i(are)d
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|
3081 |
(identi\002ed.)h(So)f(no)n(w)h(we)288 2906 y(write)20
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|
3082 |
b FG(\025x:x)g FD(\021)h FG(\025y)s(:y)s FI(,)e(etcetera.)125
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3083 |
3060 y(W)-6 b(e)19 b(be)o(gin)h(with)g(de\002ning)h(\223ra)o(w\224)e
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3084 |
(lambda-terms.)i(The)o(y)f(can)g(be)g(de\002ned)g(in)g(Isabelle/HOL)g
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3085 |
(with)0 3156 y(the)g(datatype)h(declaration:)844 3317
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3086 |
y Fs(datatype)i(lam)i Fr(=)g Fs(Var)43 b Fr(")p Fs(name)p
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|
3087 |
Fr(")1341 3412 y FD(j)h Fs(App)f Fr(")p Fs(lam)31 b FD(\002)f
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3088 |
Fs(lam)q Fr(")1341 3508 y FD(j)44 b Fs(Lam)f Fr(")p Fs(name)32
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3089 |
b FD(\002)d Fs(lam)q Fr(")2789 3412 y FI(\(5\))0 3667
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|
3090 |
y(Gi)n(v)o(en)20 b(the)g(follo)n(wing)i(permutation)g(operation)f(for)g
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|
3091 |
(lambda-terms)1033 3852 y FG(\031)1080 3861 y Fo(\001)1118
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|
3092 |
3852 y Fs(Var)q Fr(\()p FG(a)p Fr(\))1361 3809 y Fn(def)1368
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|
3093 |
3852 y Fr(=)32 b Fs(Var)q Fr(\()p FG(\031)1655 3861 y
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|
3094 |
Fo(\001)1692 3852 y FG(a)p Fr(\))910 3971 y FG(\031)957
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|
3095 |
3980 y Fo(\001)995 3971 y Fs(App)p Fr(\()p FG(t)1170
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|
3096 |
3983 y Fq(1)1207 3971 y FG(;)13 b(t)1269 3983 y Fq(2)1306
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|
3097 |
3971 y Fr(\))1361 3928 y Fn(def)1368 3971 y Fr(=)32 b
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|
3098 |
Fs(App)q Fr(\()p FG(\031)1655 3980 y Fo(\001)1692 3971
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|
3099 |
y FG(t)1720 3983 y Fq(1)1757 3971 y FG(;)14 b(\031)1839
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|
3100 |
3980 y Fo(\001)1876 3971 y FG(t)1904 3983 y Fq(2)1941
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|
3101 |
3971 y Fr(\))971 4090 y FG(\031)1018 4099 y Fo(\001)1056
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|
3102 |
4090 y Fs(Lam)q Fr(\()p FG(a;)e(t)p Fr(\))1361 4048 y
|
|
3103 |
Fn(def)1368 4090 y Fr(=)32 b Fs(Lam)q Fr(\()p FG(\031)1655
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|
3104 |
4099 y Fo(\001)1692 4090 y FG(a;)13 b(\031)1814 4099
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|
3105 |
y Fo(\001)1852 4090 y FG(t)p Fr(\))2789 3959 y FI(\(6\))0
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|
3106 |
4250 y(the)18 b(datatype)h Fs(lam)g FI(is)f(a)g(permutation)i(type)e
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|
3107 |
(\(routine)i(proof)g(by)f(structural)f(induction\).)i(As)e(mentioned)0
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|
3108 |
4345 y(earlier)m(,)25 b(\002xing)h(the)g(permutation)h(operation)h
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|
3109 |
(also)d(\002x)o(es)g(the)g(notion)i(of)f(support,)h(which)f(in)f(case)g
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|
3110 |
(of)0 4441 y Fs(lam)e FI(coincides)g(with)f(the)g(set)g(of)h
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|
3111 |
FE(all)f FI(atoms)g(occurring)i(in)f(a)f(lambda-term.)i(Hence)e
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|
3112 |
Fs(lam)h FI(is)e(a)h(\002nitely)0 4536 y(supported)g(permutation)g
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|
3113 |
(type.)125 4632 y(The)k(notion)i(of)f(alpha-equi)n(v)n(alence)h(for)f
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|
3114 |
Fs(lam)g FI(is)f(usually)h(de\002ned)g(as)f(the)g(least)g(congruence)i
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|
3115 |
(of)0 4727 y(the)e(equation)h Fs(Lam)q Fr(\()p FG(a;)12
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|
3116 |
b(t)p Fr(\))31 b(=)790 4735 y Fp(\013)869 4727 y Fs(Lam)q
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|
3117 |
Fr(\()p FG(b;)13 b(t)p Fr([)p FG(a)31 b Fr(:=)h FG(b)p
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|
3118 |
Fr(]\))25 b FI(in)m(v)n(olving)j(a)e(renaming)h(substitution)g(and)f(a)
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|
3119 |
g(side-)0 4823 y(condition,)k(namely)f(that)g FG(b)f
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|
3120 |
FI(does)h(not)g(occur)h(freely)f(in)f FG(t)p FI(.)g(In)h(the)g(nominal)
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|
3121 |
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3122 |
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3123 |
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3124 |
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3125 |
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3126 |
0 149 V 0 1074 4 926 v 581 244 491 4 v 581 310 a Ft(Var)q
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3127 |
Fy(\()p FA(a)p Fy(\))21 b Fz(\031)f Ft(Var)q Fy(\()p
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|
3128 |
FA(a)p Fy(\))1113 256 y Fz(\031)1168 264 y Fd(Var)1498
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|
3129 |
214 y FA(t)1523 223 y Fx(1)1578 214 y Fz(\031)g FA(s)1686
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|
3130 |
223 y Fx(1)1786 214 y FA(t)1811 223 y Fx(2)1866 214 y
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|
3131 |
Fz(\031)f FA(s)1973 223 y Fx(2)p 1387 244 733 4 v 1387
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3132 |
310 a Ft(App)p Fy(\()p FA(t)1544 319 y Fx(1)1580 310
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|
3133 |
y FA(;)11 b(t)1636 319 y Fx(2)1671 310 y Fy(\))20 b Fz(\031)g
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|
3134 |
Ft(App)q Fy(\()p FA(s)1959 319 y Fx(1)1993 310 y FA(;)12
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|
3135 |
b(s)2058 319 y Fx(2)2092 310 y Fy(\))2161 251 y Fz(\031)2216
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3136 |
259 y Fd(App)655 446 y FA(t)20 b Fz(\031)g FA(s)p 426
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3137 |
467 612 4 v 426 533 a Ft(Lam)q Fy(\()p FA(a;)12 b(t)p
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3138 |
Fy(\))20 b Fz(\031)g Ft(Lam)q Fy(\()p FA(a;)12 b(s)p
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3139 |
Fy(\))1079 479 y Fz(\031)1134 488 y Fd(Lam)q Fx(1)1382
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|
3140 |
430 y FA(a)20 b Fz(6)p Fy(=)g FA(b)59 b(t)20 b Fz(\031)f
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|
3141 |
Fy(\()p FA(a)13 b(b)p Fy(\))1856 439 y Fo(\001)1895 430
|
|
3142 |
y FA(s)58 b(a)8 b Fz(62)g Ft(fv)q Fy(\()p FA(s)p Fy(\))p
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|
3143 |
1382 467 863 4 v 1511 533 a Ft(Lam)q Fy(\()p FA(a;)13
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3144 |
b(t)p Fy(\))20 b Fz(\031)g Ft(Lam)p Fy(\()p FA(b;)12
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3145 |
b(s)p Fy(\))2286 479 y Fz(\031)2341 488 y Fd(Lam)q Fx(2)781
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|
3146 |
704 y FA(a)21 b Fz(6)p Fy(=)e FA(b)p 654 737 417 4 v
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|
3147 |
654 803 a(a)8 b Fz(62)g Ft(fv)q Fy(\()p Ft(Var)q Fy(\()p
|
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3148 |
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3150 |
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3151 |
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3153 |
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3154 |
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3155 |
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3156 |
Ft(fv)q Fy(\()p Ft(Lam)q Fy(\()p FA(a;)k(t)p Fy(\)\))1192
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3157 |
976 y Ft(fv)1263 985 y Fd(Lam)q Fx(1)1512 923 y FA(a)20
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3158 |
b Fz(6)p Fy(=)g FA(b)59 b(a)8 b Fz(62)g Ft(fv)q Fy(\()p
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3159 |
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3160 |
Ft(fv)q Fy(\()p Ft(Lam)r Fy(\()p FA(b;)k(t)p Fy(\)\))2026
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3161 |
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3162 |
v 0 1077 2881 4 v 0 1194 a FB(Fig)o(.)k(2)34 b FF(Inducti)n(v)o(e)20
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3163 |
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3164 |
Fy(\()p Fz(\000)p Fy(\))d FF(and)h Fy(\()p Fz(\000)p
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3165 |
Fy(\))8 b Fz(62)g Ft(fv)q Fy(\()p Fz(\000)p Fy(\))p FF(.)p
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3166 |
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3167 |
(renaming)g(substitutions,)f(b)n(ut)h(by)f(permutations.)h(This)f(has)g
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3168 |
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3169 |
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3170 |
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3171 |
FE(et)f(al)g FI([35]\),)i(because)f(permutations)h(are)e(bijections)h
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3172 |
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3175 |
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3176 |
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3177 |
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3178 |
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3179 |
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3180 |
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3181 |
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3182 |
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3183 |
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3184 |
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3185 |
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3186 |
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3187 |
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3188 |
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3189 |
(2)p 0 TeXcolorgray 42 w FE(The)g(r)m(elation)h FD(\031)e
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3190 |
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3192 |
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3193 |
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|
3194 |
e FD(\031)f FI(and)i Fr(=)1792 2767 y Fp(\013)1864 2759
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3195 |
y FI(coincide\).)g(In)g(the)f(follo)n(wing,)h Fr([)p
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3196 |
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3197 |
(the)f(alpha-equi)n(v)n(alence)j(class)c(of)i(the)f(lambda-term)i
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3198 |
FG(t)p FI(,)d(that)h(is)g Fr([)p FG(t)p Fr(])2281 2884
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3199 |
y Fp(\013)2350 2834 y Fn(def)2357 2876 y Fr(=)29 b FD(f)21
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3200 |
b FG(t)2533 2845 y Fl(0)2578 2876 y FD(j)g FG(t)2648
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3201 |
2845 y Fl(0)2693 2876 y FD(\031)g FG(t)g FD(g)p FI(,)0
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3202 |
2972 y(and)g Fs(lam)251 2990 y Fp(=)p Fl(\031)361 2972
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3203 |
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3204 |
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3205 |
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3206 |
(identify)g(\(inducti)n(v)o(ely\))0 3167 y(a)29 b(subset,)f(called)h
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3207 |
Fs(lam)644 3175 y Fp(\013)691 3167 y FI(,)g(that)f(is)h(in)g(bijection)
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|
3208 |
g(with)g Fs(lam)1649 3184 y Fp(=)p Fl(\031)1739 3167
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3209 |
y FI(.)f(Since)h(Isabelle/HOL)g(supports)h(sub-)0 3262
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|
3210 |
y(set)h(types,)g(we)g(can)h(later)f(turn)i Fs(lam)1047
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|
3211 |
3270 y Fp(\013)1126 3262 y FI(into)f(a)f(ne)n(w)h(type.)f(In)h(order)h
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3212 |
(to)e(obtain)i(the)e(bijection,)h Fs(phi)0 3357 y FI(needs)22
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3213 |
b(to)g(be)h(de\002ned)g(so)f(that)g(it)f(contains)i(elements)f
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3214 |
(corresponding,)j(roughly)f(speaking,)f(to)f(alpha-)0
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3215 |
3453 y(equated)c(v)n(ariables,)f(applications)h(and)f
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|
3216 |
(lambda-abstractions\227that)i(is)d(to)h Fr([)p Fs(Var)q
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|
3217 |
Fr(\()p FG(a)p Fr(\)])2366 3461 y Fp(\013)2413 3453 y
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|
3218 |
FI(,)f Fr([)p Fs(App)q Fr(\()p FG(t)2646 3465 y Fq(1)2683
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|
3219 |
3453 y FG(;)d(t)2745 3465 y Fq(2)2782 3453 y Fr(\)])2833
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3220 |
3461 y Fp(\013)0 3548 y FI(and)20 b Fr([)p Fs(Lam)q Fr(\()p
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3221 |
FG(a;)12 b(t)p Fr(\)])455 3556 y Fp(\013)502 3548 y FI(.)19
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3222 |
b(Whereas)g(this)g(is)f(straightforw)o(ard)j(for)f(v)n(ariables)f(and)h
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3223 |
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3224 |
b(are)f(non-tri)n(vial:)i(for)g(them)e(we)g(shall)h(use)f(some)h
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3225 |
FE(speci\002c)f FI(\223partial\224)h(functions)h(from)0
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3226 |
3739 y Fs(name)19 b FI(to)g Fs(phi)g FI(\(by)g(\223partial\224)g(we)f
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3227 |
(mean)h(here)g(functions)g(that)g(return)g Fk(None)25
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3228 |
b FI(for)19 b(unde\002ned)h(v)n(alues)f(and)0 3835 y
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3229 |
Fk(Some)6 b Fr(\()p FG(x)p Fr(\))19 b FI(for)i(de\002ned)g(ones)817
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3230 |
3803 y Fv(4)849 3835 y FI(\).)f(W)-6 b(e)19 b(therefore)j(de\002ne)e
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3231 |
Fs(phi)h FI(as)e(the)h(Isabelle/HOL)g(datatype:)705 4022
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3232 |
y Fs(datatype)j(phi)j Fr(=)e Fs(Am)44 b Fr(")p Fs(name)p
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3233 |
Fr(")1202 4117 y FD(j)g Fs(Pr)g Fr(")p Fs(phi)30 b FD(\002)g
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3234 |
Fs(phi)p Fr(")1202 4213 y FD(j)44 b Fs(Se)g Fr(")p Fs(name)22
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3235 |
b FD(\))f Fr(\()p Fs(phi)h(option)p Fr(\)")2789 4117
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3236 |
y FI(\(7\))0 4400 y(where)27 b Fs(Am)g FI(will)e(be)i(used)g(to)f
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3237 |
(encode)i(atoms;)e Fs(Pr)h FI(to)g(encode)g(applications,)g(which)g
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3238 |
(are)g(b)n(uilt)g(up)g(by)0 4496 y(a)g(pair)g(of)h(terms;)f(and)h
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3239 |
Fs(Se)f FI(to)g(encode)h(an)g(alpha-equi)n(v)n(alence)h(class)d(\(that)
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3240 |
i(is)e(a)h(set\))g(of)g(terms.)g(The)p 0 TeXcolorgray
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3241 |
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3242 |
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3243 |
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3245 |
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3247 |
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3249 |
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3250 |
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3251 |
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3252 |
(structure)h(as)f(follo)n(ws:)1072 477 y FG(\031)1119
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3253 |
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3254 |
434 y Fn(def)1368 477 y Fr(=)32 b Fs(Am)p Fr(\()p FG(\031)1615
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3255 |
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3256 |
629 y Fo(\001)1034 620 y Fs(Pr)p Fr(\()p FG(t)1170 632
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3257 |
y Fq(1)1207 620 y FG(;)13 b(t)1269 632 y Fq(2)1306 620
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3258 |
y Fr(\))1361 577 y Fn(def)1368 620 y Fr(=)32 b Fs(Pr)p
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3259 |
Fr(\()p FG(\031)1615 629 y Fo(\001)1653 620 y FG(t)1681
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3260 |
632 y Fq(1)1718 620 y FG(;)13 b(\031)1799 629 y Fo(\001)1837
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3261 |
620 y FG(t)1865 632 y Fq(2)1902 620 y Fr(\))1040 763
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3262 |
y FG(\031)1087 772 y Fo(\001)1125 763 y Fs(Se)q Fr(\()p
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3263 |
Fk(fn)5 b Fr(\))1361 720 y Fn(def)1368 763 y Fr(=)32
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3264 |
b Fs(Se)p Fr(\()p FG(\031)1615 772 y Fo(\001)1653 763
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3265 |
y Fk(fn)6 b Fr(\))2789 607 y FI(\(8\))0 996 y(using)24
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3266 |
b(in)f(the)g(last)g(clause)g(the)g(permutations)i(operation)g(for)f
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3267 |
(functions)g(gi)n(v)o(en)g(in)g(\(2\).)f(It)g(is)g(not)h(hard)0
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3268 |
1092 y(to)c(sho)n(w)g(that)g Fs(phi)h FI(is)e(a)h(permutation)i(type)e
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3269 |
(\(routine)i(induction)f(o)o(v)o(er)g(the)f(structure)h(of)f
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3270 |
Fs(phi)q FI(-terms\).)125 1197 y(W)-6 b(e)16 b(mentioned)j(earlier)d
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3271 |
(that)h(we)g(are)f(not)i(going)g(to)f(use)f(all)h(functions)h(from)f
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3272 |
Fs(name)h FI(to)e Fs(phi)22 b(option)0 1293 y FI(for)34
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3273 |
b(representing)g(alpha-equated)h(lambda-abstractions,)g(b)n(ut)f(some)f
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3274 |
(speci\002c)g(functions.)2627 1261 y Fv(5)2693 1293 y
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3275 |
FI(These)0 1388 y(functions)21 b(are)f(of)h(the)f(form:)519
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|
3276 |
1629 y Fr([)p FG(a)p Fr(])p FG(:t)672 1586 y Fn(def)680
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3277 |
1629 y Fr(=)28 b FG(\025b:)13 b Fs(if)22 b FG(a)f Fr(=)g
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3278 |
FG(b)g Fs(then)h Fk(Some)6 b Fr(\()p FG(t)p Fr(\))880
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3279 |
1757 y Fs(else)22 b(if)g FG(b)f Fr(#)g FG(t)g Fs(then)i
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3280 |
Fk(Some)6 b Fr(\(\()p FG(a)12 b(b)p Fr(\))1887 1766 y
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|
3281 |
Fo(\001)1925 1757 y FG(t)p Fr(\))21 b Fs(else)h Fk(None)2789
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3282 |
1684 y FI(\(9\))0 2005 y(and)g(we)f(will)g(refer)h(to)f(them)h(as)f
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3283 |
FE(abstr)o(action)h(functions)p FI(;)g(their)g(parameters)g(are)f(an)h
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3284 |
(atom)g(and)g(a)f Fs(phi)q FI(-)0 2100 y(term.)125 2206
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|
3285 |
y(W)-6 b(e)20 b(claim)h(that)f(these)h(functions)g(represent)h
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3286 |
(alpha-equi)n(v)n(alence)h(classes.)c(T)-6 b(o)21 b(see)f(this,)g
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|
3287 |
(consider)0 2302 y Fr([)p Fs(Lam)q Fr(\()p FG(a;)13 b
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|
3288 |
Fs(App)p Fr(\()p Fs(Var)q Fr(\()p FG(a)p Fr(\))p FG(;)f
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|
3289 |
Fs(Var)q Fr(\()p FG(b)p Fr(\)\)\)])935 2310 y Fp(\013)1015
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|
3290 |
2302 y FI(and)34 b(the)f(corresponding)j Fs(phi)p FI(-term)e
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|
3291 |
Fs(Se)p Fr(\([)p FG(a)p Fr(])p FG(:)p Fs(Pr)q Fr(\()p
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|
3292 |
Fs(Am)q Fr(\()p FG(a)p Fr(\))p FG(;)12 b Fs(Am)q Fr(\()p
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|
3293 |
FG(b)p Fr(\)\)\))p FI(.)0 2397 y(The)34 b(graph)h(of)f(the)g
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|
3294 |
(abstraction)g(function)h(is)e(as)g(follo)n(ws:)h(the)g(atom)g
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|
3295 |
FG(a)f FI(is)g(mapped)i(to)f(the)f(term)0 2492 y Fk(Some)6
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|
3296 |
b Fr(\()p Fs(Pr)p Fr(\()p Fs(Am)q Fr(\()p FG(a)p Fr(\))p
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|
3297 |
FG(;)12 b Fs(Am)q Fr(\()p FG(b)p Fr(\)\)\))i FI(since)i(the)g(\002rst)f
|
|
3298 |
Fs(if)q FI(-condition)i(is)e(true.)h(F)o(or)g FG(b)p
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|
3299 |
FI(,)f(the)h(\002rst)f Fs(if)q FI(-condition)j(ob)o(vi-)0
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|
3300 |
2588 y(ously)f(f)o(ails,)f(b)n(ut)i(also)f(the)f(second)i(one)f(f)o
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|
3301 |
(ails,)f(because)h Fk(supp)6 b Fr(\()p Fs(Pr)p Fr(\()p
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|
3302 |
Fs(Am)q Fr(\()p FG(a)p Fr(\))p FG(;)12 b Fs(Am)q Fr(\()p
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|
3303 |
FG(b)p Fr(\)\)\))20 b(=)h FD(f)p FG(a;)14 b(b)p FD(g)p
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|
3304 |
FI(;)i(therefore)0 2683 y FG(b)i FI(is)g(mapped)i(to)f
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|
3305 |
Fk(None)6 b FI(.)18 b(F)o(or)h(all)f(other)i(atoms)e
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|
3306 |
FG(c)p FI(,)h(we)f(ha)n(v)o(e)i FG(a)h FD(6)p Fr(=)g
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|
3307 |
FG(c)d FI(and)i FG(c)h Fr(#)g Fs(Pr)q Fr(\()p Fs(Am)p
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|
3308 |
Fr(\()p FG(a)p Fr(\))p FG(;)13 b Fs(Am)p Fr(\()p FG(b)p
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|
3309 |
Fr(\)\))p FI(;)18 b(conse-)0 2779 y(quently)25 b(these)e
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|
3310 |
FG(c)p FI(')l(s)g(are)h(mapped)h(by)f(the)f(abstraction)i(function)g
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|
3311 |
(to)f Fk(Some)5 b Fr(\(\()p FG(a)13 b(c)p Fr(\))2271
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|
3312 |
2788 y Fo(\001)2309 2779 y Fs(Pr)p Fr(\()p Fs(Am)q Fr(\()p
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|
3313 |
FG(a)p Fr(\))p FG(;)f Fs(Am)q Fr(\()p FG(b)p Fr(\)\)\))p
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|
3314 |
FI(,)0 2874 y(which)35 b(is)f Fk(Some)6 b Fr(\()p Fs(Pr)p
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|
3315 |
Fr(\()p Fs(Am)p Fr(\()p FG(c)p Fr(\))p FG(;)13 b Fs(Am)q
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|
3316 |
Fr(\()p FG(b)p Fr(\)\)\))p FI(.)33 b(Clearly)-5 b(,)34
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|
3317 |
b(the)g(abstraction)h(function)h(returns)f Fk(None)41
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|
3318 |
b FI(when-)0 2970 y(e)n(v)o(er)23 b(the)f(corresponding)j(lambda-term)f
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|
3319 |
(is)e FE(not)i FI(in)e(the)h(alpha-equi)n(v)n(alence)h(class\227in)e
|
|
3320 |
(this)g(e)o(xample)0 3065 y(the)f(lambda-term)h Fs(Lam)q
|
|
3321 |
Fr(\()p FG(b;)13 b Fs(App)p Fr(\()p Fs(Var)q Fr(\()p
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|
3322 |
FG(b)p Fr(\))p FG(;)g Fs(Var)p Fr(\()p FG(b)p Fr(\)\)\))22
|
|
3323 |
b FD(62)h Fr([)p Fs(Lam)q Fr(\()p FG(a;)13 b Fs(App)q
|
|
3324 |
Fr(\()p Fs(Var)p Fr(\()p FG(a)p Fr(\))p FG(;)g Fs(Var)p
|
|
3325 |
Fr(\()p FG(b)p Fr(\)\)\)])2448 3073 y Fp(\013)2495 3065
|
|
3326 |
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3327 |
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3328 |
h Fs(Pr)p Fr(\()p Fs(Am)q Fr(\()p FG(a)p Fr(\))p FG(;)12
|
|
3329 |
b Fs(Am)q Fr(\()p FG(b)p Fr(\)\))p FI(.)125 3267 y(T)-6
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3330 |
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3333 |
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3337 |
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3338 |
FG(\031)219 3864 y Fo(\001)257 3855 y Fr(\([)p FG(a)p
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3339 |
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3340 |
3855 y FG(a)p Fr(])p FG(:)p Fr(\()p FG(\031)817 3864
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3341 |
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3342 |
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3343 |
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3344 |
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3345 |
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3346 |
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3347 |
4178 y FE(or)158 b FG(a)21 b FD(6)p Fr(=)g FG(b)c FD(^)g
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3348 |
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3349 |
b(b)p Fr(\))2032 4187 y Fo(\001)2070 4178 y FG(t)2098
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3350 |
4190 y Fq(2)2152 4178 y FD(^)k FG(a)k Fr(#)g FG(t)2395
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3351 |
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3352 |
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3354 |
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3355 |
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3356 |
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3357 |
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3360 |
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3361 |
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3362 |
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3363 |
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|
3364 |
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3365 |
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3366 |
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3367 |
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3368 |
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3369 |
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|
|
3370 |
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|
|
3371 |
b FG(a)g Fr(=)g FG(\031)831 491 y Fl(\000)p Fq(1)920
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|
3372 |
532 y Fo(\001)958 523 y FG(b)h Fs(then)g Fk(Some)6 b
|
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3373 |
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3374 |
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3375 |
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3376 |
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3377 |
597 y FG(b)p Fr(\))1825 606 y Fo(\001)1862 597 y FG(t)p
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3378 |
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|
|
3379 |
b Fs(if)21 b FG(\031)603 704 y Fo(\001)641 695 y Fr(\()p
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3380 |
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|
3381 |
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|
3382 |
b Fr(\()p FG(\031)1514 704 y Fo(\001)1551 695 y FG(t)p
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3383 |
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|
3384 |
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|
3385 |
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|
3386 |
Fk(Some)6 b Fr(\()p FG(\031)1683 778 y Fo(\001)1721 769
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|
3387 |
y Fr(\()p FG(a)25 b(\031)1864 737 y Fl(\000)p Fq(1)1953
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|
3388 |
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3389 |
769 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506 695 y FI(\()p
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3390 |
FD(\003)2570 663 y Fq(1)2608 695 y FI(\))254 867 y Fr(=)31
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|
3391 |
b FG(\025b:)13 b Fs(if)21 b FG(\031)603 876 y Fo(\001)641
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3392 |
867 y Fr(\()p FG(a)g Fr(=)g FG(\031)861 835 y Fl(\000)p
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|
3393 |
Fq(1)950 876 y Fo(\001)988 867 y FG(b)p Fr(\))g Fs(then)i
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3394 |
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3395 |
y FG(t)p Fr(\))457 941 y Fs(else)22 b(if)g FG(\031)782
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3396 |
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3397 |
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|
3398 |
Fr(\))g Fs(then)h Fk(Some)6 b Fr(\(\()p FG(\031)1713
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3399 |
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3400 |
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3401 |
941 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506 867 y FI(\()p
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|
3402 |
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|
3403 |
b FG(\025b:)13 b Fs(if)21 b FG(\031)603 1048 y Fo(\001)641
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|
3404 |
1039 y FG(a)g Fr(=)g FG(b)h Fs(then)g Fk(Some)6 b Fr(\()p
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|
3405 |
FG(\031)1280 1048 y Fo(\001)1318 1039 y FG(t)p Fr(\))457
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|
3406 |
1111 y Fs(else)22 b(if)g FG(b)f Fr(#)g FG(\031)921 1120
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|
3407 |
y Fo(\001)959 1111 y FG(t)g Fs(then)h Fk(Some)6 b Fr(\(\()p
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|
3408 |
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|
3409 |
Fr(\))1646 1120 y Fo(\001)1684 1111 y FG(\031)1731 1120
|
|
3410 |
y Fo(\001)1769 1111 y FG(t)p Fr(\))c Fs(else)h Fk(None)2506
|
|
3411 |
1039 y FI(\()p FD(\003)2570 1007 y Fq(3)2608 1039 y FI(\))247
|
|
3412 |
1187 y Fn(def)254 1230 y Fr(=)31 b([)p FG(\031)413 1239
|
|
3413 |
y Fo(\001)451 1230 y FG(a)p Fr(])p FG(:)p Fr(\()p FG(\031)611
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|
3414 |
1239 y Fo(\001)649 1230 y FG(t)p Fr(\))0 1395 y FI(where)20
|
|
3415 |
b(we)g(use)g(in)g(\()p FD(\003)594 1364 y Fq(1)632 1395
|
|
3416 |
y FI(\))g(the)g(f)o(act)g(that)608 1563 y FG(\031)655
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|
3417 |
1572 y Fo(\001)692 1563 y Fs(if)q FG(:::)p Fs(then)r
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|
3418 |
FG(:::)p Fs(else)r FG(:::)i Fr(=)f Fs(if)13 b FG(\031)1517
|
|
3419 |
1572 y Fo(\001)1555 1563 y FG(:::)p Fs(then)23 b FG(\031)1844
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|
3420 |
1572 y Fo(\001)1882 1563 y FG(:::)p Fs(else)g FG(\031)2171
|
|
3421 |
1572 y Fo(\001)2209 1563 y FG(:::)477 b FI(\(10\))0 1732
|
|
3422 |
y(and)27 b(in)g Fr(\()p FD(\003)296 1700 y Fq(2)334 1732
|
|
3423 |
y Fr(\))f FI(that)h FG(\031)s Fr(@\()p FG(a)45 b(\031)805
|
|
3424 |
1700 y Fl(\000)p Fq(1)894 1741 y Fo(\001)932 1732 y FG(b)p
|
|
3425 |
Fr(\))33 b FD(\030)g Fr(\()p FG(\031)1198 1741 y Fo(\001)1236
|
|
3426 |
1732 y FG(a)46 b(b)p Fr(\)@)p FG(\031)s FI(;)26 b(for)h
|
|
3427 |
Fr(\()p FD(\003)1727 1700 y Fq(3)1765 1732 y Fr(\))f
|
|
3428 |
FI(the)h(f)o(acts)f(that)h FG(\031)2310 1741 y Fo(\001)2348
|
|
3429 |
1732 y Fr(\()p FG(a)33 b Fr(=)g FG(\031)2592 1700 y Fl(\000)p
|
|
3430 |
Fq(1)2681 1741 y Fo(\001)2719 1732 y FG(b)p Fr(\))26
|
|
3431 |
b FI(if)n(f)0 1827 y FG(\031)47 1836 y Fo(\001)85 1827
|
|
3432 |
y FG(a)i Fr(=)f FG(b)d FI(and)g FG(\031)482 1836 y Fo(\001)520
|
|
3433 |
1827 y Fr(\()p FG(\031)597 1795 y Fl(\000)p Fq(1)685
|
|
3434 |
1836 y Fo(\001)724 1827 y FG(b)j Fr(#)h FG(t)p Fr(\))c
|
|
3435 |
FI(if)n(f)f FG(b)28 b Fr(#)g FG(\031)1253 1836 y Fo(\001)1291
|
|
3436 |
1827 y FG(t)p FI(,)23 b(which)h(can)g(be)g(easily)f(deri)n(v)o(ed)i
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|
3437 |
(from)g(Lemmas)f(1)p FE(\(ii\))0 1923 y FI(and)d(3)p
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|
3438 |
FE(\(ii\))f FI(and)h(the)f(permutation)i(operation)f(on)g
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|
3439 |
Fs(bool)q FI(.)125 2018 y(F)o(or)d(the)h(second)g(property)i(the)d
|
|
3440 |
(case)g FG(a)k Fr(=)f FG(b)d FI(is)g(by)h(a)f(simple)h(calculation)g
|
|
3441 |
(using)g(e)o(xtensionality)h(of)0 2114 y(functions.)h(In)f(case)f
|
|
3442 |
FG(a)i FD(6)p Fr(=)g FG(b)f FI(we)f(sho)n(w)h(\002rst)f(the)h
|
|
3443 |
FD(\))p FI(-direction:)h(the)f(follo)n(wing)h(formula)g(holds)g(then)f
|
|
3444 |
(by)0 2209 y(e)o(xtensionality)h(of)f(functions:)586
|
|
3445 |
2377 y FD(8)p FG(c:)89 b Fs(if)22 b FG(a)f Fr(=)g FG(c)h
|
|
3446 |
Fs(then)g Fk(Some)6 b Fr(\()p FG(t)1492 2389 y Fq(1)1529
|
|
3447 |
2377 y Fr(\))772 2473 y Fs(else)22 b(if)g FG(c)g Fr(#)f
|
|
3448 |
FG(t)1218 2485 y Fq(1)1276 2473 y Fs(then)i Fk(Some)6
|
|
3449 |
b Fr(\(\()p FG(a)12 b(c)p Fr(\))1817 2482 y Fo(\001)1855
|
|
3450 |
2473 y FG(t)1883 2485 y Fq(1)1920 2473 y Fr(\))21 b Fs(else)h
|
|
3451 |
Fk(None)687 2592 y Fr(=)j Fs(if)d FG(b)f Fr(=)g FG(c)h
|
|
3452 |
Fs(then)g Fk(Some)6 b Fr(\()p FG(t)1484 2604 y Fq(2)1521
|
|
3453 |
2592 y Fr(\))772 2687 y Fs(else)22 b(if)g FG(c)g Fr(#)f
|
|
3454 |
FG(t)1218 2699 y Fq(2)1276 2687 y Fs(then)i Fk(Some)6
|
|
3455 |
b Fr(\(\()p FG(b)12 b(c)p Fr(\))1809 2696 y Fo(\001)1847
|
|
3456 |
2687 y FG(t)1875 2699 y Fq(2)1912 2687 y Fr(\))21 b Fs(else)i
|
|
3457 |
Fk(None)0 2851 y FI(Instantiating)e(this)f(formula)h(with)f
|
|
3458 |
FG(a)g FI(yields)g(the)g(equation)537 3019 y Fk(Some)6
|
|
3459 |
b Fr(\()p FG(t)781 3031 y Fq(1)818 3019 y Fr(\))21 b(=)g
|
|
3460 |
Fs(if)h FG(a)f Fr(#)g FG(t)1225 3031 y Fq(2)1283 3019
|
|
3461 |
y Fs(then)h Fk(Some)6 b Fr(\(\()p FG(b)13 b(a)p Fr(\))1824
|
|
3462 |
3028 y Fo(\001)1862 3019 y FG(t)1890 3031 y Fq(2)1927
|
|
3463 |
3019 y Fr(\))21 b Fs(else)h Fk(None)28 b FG(:)0 3187
|
|
3464 |
y FI(Ne)o(xt,)20 b(one)i(distinguishes)f(the)g(cases)f(where)h
|
|
3465 |
FG(a)i Fr(#)g FG(t)1455 3199 y Fq(2)1512 3187 y FI(and)e
|
|
3466 |
FD(:)13 b FG(a)23 b Fr(#)g FG(t)1889 3199 y Fq(2)1926
|
|
3467 |
3187 y FI(,)d(respecti)n(v)o(ely)-5 b(.)21 b(In)g(the)g(\002rst)f
|
|
3468 |
(case,)0 3283 y Fk(Some)6 b Fr(\()p FG(t)244 3295 y Fq(1)281
|
|
3469 |
3283 y Fr(\))22 b(=)f Fk(Some)6 b Fr(\(\()p FG(b)13 b(a)p
|
|
3470 |
Fr(\))777 3292 y Fo(\001)814 3283 y FG(t)842 3295 y Fq(2)879
|
|
3471 |
3283 y Fr(\))p FI(,)20 b(which)h(by)g(Def.)f(2)p Fr(\()p
|
|
3472 |
FG(iii)p Fr(\))h FI(implies)f FG(t)1897 3295 y Fq(1)1956
|
|
3473 |
3283 y Fr(=)h(\()p FG(a)13 b(b)p Fr(\))2184 3292 y Fo(\001)2222
|
|
3474 |
3283 y FG(t)2250 3295 y Fq(2)2307 3283 y FI(since)20
|
|
3475 |
b Fr(\()p FG(a)12 b(b)p Fr(\))d FD(\030)g Fr(\()p FG(b)j(a)p
|
|
3476 |
Fr(\))p FI(;)0 3378 y(and)26 b(ob)o(viously)i FG(a)j
|
|
3477 |
Fr(#)h FG(t)671 3390 y Fq(2)733 3378 y FI(by)26 b(assumption.)h(In)f
|
|
3478 |
(the)g(second)g(case)f Fk(Some)6 b Fr(\()p FG(t)2104
|
|
3479 |
3390 y Fq(1)2141 3378 y Fr(\))31 b(=)h Fk(None)f FI(which)c(gi)n(v)o
|
|
3480 |
(es)0 3474 y(a)h(contradiction.)j(The)e FD(\()p FI(-direction)h(for)f
|
|
3481 |
(the)g(case)f FG(a)37 b FD(6)p Fr(=)g FG(b)28 b FI(is)g(similarly)h(by)
|
|
3482 |
h(e)o(xtensionality)f(and)h(a)0 3569 y(case-analysis.)79
|
|
3483 |
b FD(u)-51 b(t)0 3741 y FI(Note)26 b(that,)f(in)g FE(g)o(ener)o(al)p
|
|
3484 |
FI(,)h(one)h(cannot)f(decide)g(whether)h(tw)o(o)e(functions)i(from)g
|
|
3485 |
Fs(name)f FI(to)f Fs(phi)d(option)0 3836 y FI(are)28
|
|
3486 |
b(equal;)h(ho)n(we)n(v)o(er)g(for)g(the)f(abstraction)h(functions)g
|
|
3487 |
(Lem.)f(6)p FE(\(ii\))h FI(pro)o(vides)g(the)f(means)h(to)f(decide)0
|
|
3488 |
3932 y(whether)g Fr([)p FG(a)p Fr(])p FG(:t)413 3944
|
|
3489 |
y Fq(1)486 3932 y Fr(=)35 b([)p FG(b)p Fr(])p FG(:t)705
|
|
3490 |
3944 y Fq(2)771 3932 y FI(holds:)28 b(one)g(just)f(has)h(to)g(consider)
|
|
3491 |
g(whether)h FG(a)35 b Fr(=)g FG(b)p FI(,)27 b(which)h(is)f(just)g(lik)o
|
|
3492 |
(e)0 4027 y(deciding)f(the)f(alpha-equi)n(v)n(alence)i(of)e(tw)o(o)g
|
|
3493 |
(lambda-terms)h(using)g(the)e(relation)i Fr(\()p FD(\000)p
|
|
3494 |
Fr(\))9 b FD(\031)g Fr(\()p FD(\000)p Fr(\))22 b FI(gi)n(v)o(en)k(in)0
|
|
3495 |
4123 y(Fig.)e(2.)h(No)n(w)g(it)f(is)g(also)h(clear)g(why)f(abstraction)
|
|
3496 |
i(functions)g(represent)g(alpha-equi)n(v)n(alence)h(classes:)0
|
|
3497 |
4218 y(the)19 b(condition)i(we)e(deri)n(v)o(ed)h(for)f(the)h(equality)f
|
|
3498 |
(between)h(abstraction)g(functions)g(paraphrase)g(the)g(rules)0
|
|
3499 |
4314 y FD(\031)60 4326 y Fc(Lam)o Fq(1)209 4314 y FI(and)h
|
|
3500 |
FD(\031)403 4326 y Fc(Lam)o Fq(2)552 4314 y FI(de\002ning)g(alpha-equi)
|
|
3501 |
n(v)n(alence)i(for)e Fs(lam)p FI(.)125 4409 y(The)g(properties)i(in)e
|
|
3502 |
(Lem.)g(6)h(also)f(help)h(us)f(to)g(calculate)g(the)h(support)h(for)e
|
|
3503 |
(abstraction)i(functions,)0 4505 y(pro)o(vided)f(the)o(y)e
|
|
3504 |
(\223abstract\224)g(o)o(v)o(er)h(a)f(\002nitely)g(supported)i
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|
3505 |
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|
3506 |
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|
3507 |
FE(and)h FG(t)e FE(being)i(\002nitely)g(supported,)g(then)p
|
|
3508 |
0 TeXcolorgray 56 4823 a(\(i\))p 0 TeXcolorgray 42 w
|
|
3509 |
FG(a)g Fr(#)g([)p FG(b)p Fr(])p FG(:t)f FE(if)g(and)h(only)g(if)e
|
|
3510 |
FG(a)i Fr(#)h FG(t)p FE(,)d(and)p 0 TeXcolorgray 0 TeXcolorgray
|
|
3511 |
eop end
|
|
3512 |
%%Page: 11 11
|
|
3513 |
TeXDict begin 11 10 bop 0 TeXcolorgray 0 TeXcolorgray
|
|
3514 |
0 71 2881 4 v 2814 17 a FF(11)p 0 TeXcolorgray 0 TeXcolorgray
|
|
3515 |
34 228 a FE(\(ii\))p 0 TeXcolorgray 42 w FG(a)21 b Fr(#)g([)p
|
|
3516 |
FG(a)p Fr(])p FG(:t)p 0 TeXcolorgray 0 490 a FE(Pr)l(oof)p
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|
3517 |
0 TeXcolorgray 40 w FI(By)h(a)g(simple)h(calculations)g(we)f(ha)n(v)o
|
|
3518 |
(e)h(that)g Fk(supp)5 b Fr(\([)p FG(b)p Fr(])p FG(:t)p
|
|
3519 |
Fr(\))26 b FD(\022)g Fk(supp)6 b Fr(\()p FG(b;)13 b(t)p
|
|
3520 |
Fr(\))21 b FI(because)i(for)g(all)g FG(c)f FI(and)0 585
|
|
3521 |
y FG(d)28 b FI(we)h(ha)n(v)o(e)h FD(f)p FG(d)13 b FD(j)g
|
|
3522 |
Fr(\()p FG(c)g(d)p Fr(\))635 594 y Fo(\001)673 585 y
|
|
3523 |
Fr([)p FG(b)p Fr(])p FG(:t)38 b FD(6)p Fr(=)f([)p FG(b)p
|
|
3524 |
Fr(])p FG(:t)p FD(g)h(\022)f(f)p FG(d)13 b FD(j)g Fr(\()p
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|
3525 |
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|
|
3526 |
FG(b;)g(t)p Fr(\))37 b FD(6)p Fr(=)g(\()p FG(b;)13 b(t)p
|
|
3527 |
Fr(\))p FD(g)p FI(.)28 b(Since)h FG(b)g FI(and)g FG(t)f
|
|
3528 |
FI(are)h(\002nitely)0 681 y(supported,)c Fr([)p FG(b)p
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|
3529 |
Fr(])p FG(:t)e FI(must)g(be)h(\002nitely)f(supported.)h(Hence)g
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|
3530 |
Fr(\()p FG(a;)12 b(b;)h(t;)g Fr([)p FG(b)p Fr(])p FG(:t)p
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|
3531 |
Fr(\))24 b FI(is)e(\002nitely)h(supported)i(and)f(by)0
|
|
3532 |
776 y(Prop.)c(1)h(there)f(e)o(xists)f(an)h(atom)h FG(c)f
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|
3533 |
FI(with)g Fr(\()p FD(\003)p Fr(\))g FG(c)h Fr(#)g(\()p
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|
3534 |
FG(a;)13 b(b;)g(t;)g Fr([)p FG(b)p Fr(])p FG(:t)p Fr(\))p
|
|
3535 |
FI(.)125 878 y(No)n(w)30 b(we)f(sho)n(w)i(the)f(direction)h
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|
3536 |
FE(\(i)f FD(\))p FE(\))p FI(:)g(using)h(the)f(assumption)h
|
|
3537 |
FG(a)39 b Fr(#)h([)p FG(b)p Fr(])p FG(:t)30 b FI(and)h(the)f(f)o(act)h
|
|
3538 |
(that)0 974 y FG(c)e Fr(#)f([)p FG(b)p Fr(])p FG(:t)c
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|
3539 |
FI(\(from)h FD(\003)p FI(\),)f(Lem.)g(4)g(and)h(6)p FE(\(i\))f
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|
3540 |
FI(gi)n(v)o(e)h Fr([)p FG(b)p Fr(])p FG(:t)k Fr(=)f(\()p
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|
3541 |
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|
3542 |
FG(b)p Fr(])p FG(:t)29 b Fr(=)f([\()p FG(c)14 b(a)p Fr(\))2131
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|
3543 |
983 y Fo(\001)2168 974 y FG(b)p Fr(])p FG(:)p Fr(\(\()p
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|
3544 |
FG(c)g(a)p Fr(\))2421 983 y Fo(\001)2458 974 y FG(t)p
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|
3545 |
Fr(\))p FI(.)23 b(The)i(right-)0 1069 y(hand)33 b(side)e(is)g
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|
3546 |
Fr([)p FG(b)p Fr(])p FG(:)p Fr(\(\()p FG(c)13 b(a)p Fr(\))700
|
|
3547 |
1078 y Fo(\001)738 1069 y FG(t)p Fr(\))31 b FI(because)h
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|
3548 |
FG(c)42 b FD(6)p Fr(=)g FG(b)31 b FI(\(from)i FD(\003)p
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|
3549 |
FI(\))e(and)h FG(a)42 b FD(6)p Fr(=)g FG(b)31 b FI(by)h(assumption.)g
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|
3550 |
(Hence)g(by)0 1165 y(Lem.)d(6)p FE(\(ii\))g FI(we)f(can)h(infer)g(that)
|
|
3551 |
g FG(t)36 b Fr(=)h(\()p FG(c)13 b(a)p Fr(\))1247 1174
|
|
3552 |
y Fo(\001)1285 1165 y FG(t)p FI(.)28 b(No)n(w)g FG(c)37
|
|
3553 |
b Fr(#)g FG(t)28 b FI(\(from)i FD(\003)p FI(\))f(implies)g(that)g
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|
3554 |
FG(c)37 b Fr(#)g(\()p FG(c)12 b(a)p Fr(\))2793 1174 y
|
|
3555 |
Fo(\001)2831 1165 y FG(t)p FI(;)0 1260 y(and)31 b(mo)o(ving)h(the)f
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|
3556 |
(permutation)i(to)d(the)h(other)g(side)g(by)g(Lem.)f(3)p
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|
3557 |
FE(\(ii\))i FI(gi)n(v)o(es)e FG(a)40 b Fr(#)h FG(t)p
|
|
3558 |
FI(.)30 b(The)g(direction)0 1356 y FE(\(i)i FD(\()p FE(\))f
|
|
3559 |
FI(is)g(as)g(follo)n(ws:)h(from)g(\()p FD(\003)p FI(\),)g(we)g(ha)n(v)o
|
|
3560 |
(e)g(that)g FG(c)42 b Fr(#)h([)p FG(b)p Fr(])p FG(:t)32
|
|
3561 |
b FI(and)g(therefore)h(by)f(Lem.)g(3)p FE(\(iii\))g FI(also)0
|
|
3562 |
1451 y Fr(\()p FG(a)12 b(c)p Fr(\))146 1460 y Fo(\001)185
|
|
3563 |
1451 y FG(c)25 b Fr(#)f(\()p FG(a)13 b(c)p Fr(\))478
|
|
3564 |
1460 y Fo(\001)516 1451 y Fr(\([)p FG(b)p Fr(])p FG(:t)p
|
|
3565 |
Fr(\))p FI(,)21 b(which)i(implies)f(by)g(Lem.)g(6)p FE(\(i\))g
|
|
3566 |
FI(that)g FG(a)i Fr(#)h([)p FG(b)p Fr(])p FG(:)p Fr(\(\()p
|
|
3567 |
FG(a)13 b(c)p Fr(\))2198 1460 y Fo(\001)2236 1451 y FG(t)p
|
|
3568 |
Fr(\))p FI(.)21 b(From)i(\()p FD(\003)p FI(\))f(we)g(also)0
|
|
3569 |
1547 y(ha)n(v)o(e)e FG(c)h Fr(#)h FG(t)c FI(and)i(from)g(the)f
|
|
3570 |
(assumption)h FG(a)h Fr(#)g FG(t)p FI(;)d(then)i(Lem.)f(4)g(implies)f
|
|
3571 |
(that)h FG(t)i Fr(=)h(\()p FG(a)12 b(c)p Fr(\))2424 1556
|
|
3572 |
y Fo(\001)2462 1547 y FG(t)p FI(,)18 b(and)i(we)e(can)0
|
|
3573 |
1642 y(conclude)j(with)f FG(a)h Fr(#)h([)p FG(b)p Fr(])p
|
|
3574 |
FG(:t)p FI(.)125 1744 y(The)e(second)h(property)i(follo)n(ws)d(from)i
|
|
3575 |
(the)e(\002rst:)g(we)g(ha)n(v)o(e)h FG(c)i Fr(#)f FG(t)e
|
|
3576 |
FI(and)h FG(c)h FD(6)p Fr(=)g FG(a)d FI(\(both)j(from)f
|
|
3577 |
FD(\003)p FI(\),)g(and)0 1840 y(can)c(use)g FE(\(i\))g
|
|
3578 |
FI(to)f(infer)i FG(c)j Fr(#)h([)p FG(a)p Fr(])p FG(:t)p
|
|
3579 |
FI(.)16 b(Further)m(,)i(from)f(Lem.)g(3)p FE(\(iii\))g
|
|
3580 |
FI(it)g(holds)g(that)g Fr(\()p FG(c)c(a)p Fr(\))2210
|
|
3581 |
1849 y Fo(\001)2247 1840 y FG(c)22 b Fr(#)f(\()p FG(c)13
|
|
3582 |
b(a)p Fr(\))2534 1849 y Fo(\001)2572 1840 y Fr([)p FG(a)p
|
|
3583 |
Fr(])p FG(:t)p FI(.)k(This)0 1935 y(is)k FG(a)j Fr(#)h([)p
|
|
3584 |
FG(c)p Fr(])p FG(:)p Fr(\()p FG(c)14 b(a)p Fr(\))472
|
|
3585 |
1944 y Fo(\001)509 1935 y FG(t)22 b FI(by)g(Lem.)g(6)p
|
|
3586 |
FE(\(i\))p FI(.)g(Since)f FG(c)k FD(6)p Fr(=)f FG(a)d
|
|
3587 |
FI(and)i FG(c)h Fr(#)h FG(t)p FI(,)c(Lem.)g(6)p FE(\(ii\))i
|
|
3588 |
FI(implies)e(that)h Fr([)p FG(c)p Fr(])p FG(:)p Fr(\()p
|
|
3589 |
FG(c)14 b(a)p Fr(\))2731 1944 y Fo(\001)2769 1935 y FG(t)24
|
|
3590 |
b Fr(=)0 2031 y([)p FG(a)p Fr(])p FG(:t)p FI(.)c(Therefore,)h
|
|
3591 |
FG(a)g Fr(#)h([)p FG(a)p Fr(])p FG(:t)p FI(.)79 b FD(u)-51
|
|
3592 |
b(t)0 2228 y FI(Note)18 b(that)g(taking)h(both)g(f)o(acts)f(of)g(Lem.)g
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|
3593 |
(7)g(together)i(implies)e(the)g(follo)n(wing)h(equation)g(for)g(the)f
|
|
3594 |
(support)0 2324 y(of)i(abstraction)h(functions)988 2461
|
|
3595 |
y Fk(supp)6 b Fr(\([)p FG(a)p Fr(])p FG(:t)p Fr(\))21
|
|
3596 |
b(=)g Fk(supp)6 b Fr(\()p FG(t)p Fr(\))16 b FD(\000)h(f)p
|
|
3597 |
FG(a)p FD(g)857 b FI(\(11\))0 2638 y(pro)o(vided)22 b
|
|
3598 |
FG(t)d FI(is)h(\002nitely)g(supported.)125 2740 y(No)n(w)g(e)n(v)o
|
|
3599 |
(erything)i(is)e(in)g(place)h(for)g(de\002ning)h(the)e(subset)g
|
|
3600 |
Fs(lam)1833 2748 y Fp(\013)1880 2740 y FI(.)g(It)h(is)e(de\002ned)j
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|
3601 |
(inducti)n(v)o(ely)g(by)e(the)0 2835 y(three)g(rules:)p
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|
3602 |
375 3034 438 4 v 375 3107 a Fs(Am)q Fr(\()p FG(a)p Fr(\))g
|
|
3603 |
FD(2)i Fs(lam)765 3115 y Fp(\013)1025 3002 y FG(t)1053
|
|
3604 |
3014 y Fq(1)1112 3002 y FD(2)f Fs(lam)1302 3010 y Fp(\013)1428
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|
3605 |
3002 y FG(t)1456 3014 y Fq(2)1514 3002 y FD(2)h Fs(lam)1705
|
|
3606 |
3010 y Fp(\013)p 1025 3034 727 4 v 1108 3107 a Fs(Pr)q
|
|
3607 |
Fr(\()p FG(t)1245 3119 y Fq(1)1281 3107 y FG(;)14 b(t)1344
|
|
3608 |
3119 y Fq(2)1381 3107 y Fr(\))21 b FD(2)g Fs(lam)1622
|
|
3609 |
3115 y Fp(\013)2086 3006 y FG(t)g FD(2)h Fs(lam)2325
|
|
3610 |
3014 y Fp(\013)p 1965 3034 530 4 v 1965 3107 a Fs(Se)p
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|
3611 |
Fr(\([)p FG(a)p Fr(])p FG(:t)p Fr(\))f FD(2)h Fs(lam)2447
|
|
3612 |
3115 y Fp(\013)2749 3065 y FI(\(12\))0 3316 y(using)f(in)f(the)g(third)
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|
3613 |
h(rule)f(the)g(abstraction)h(functions)g(gi)n(v)o(en)g(in)f(\(9\).)g(W)
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|
3614 |
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|
3615 |
0 TeXcolorgray 42 w FE(F)-8 b(or)20 b(the)g(set)f Fs(lam)830
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3616 |
3529 y Fp(\013)897 3521 y FE(we)g(have)i(that:)p 0 TeXcolorgray
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|
3617 |
56 3687 a(\(i\))p 0 TeXcolorgray 42 w(all)f(its)f(elements)h(ar)m(e)g
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|
3618 |
(\002nitely)g(supported,)h(and)p 0 TeXcolorgray 34 3782
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|
3619 |
a(\(ii\))p 0 TeXcolorgray 42 w(it)e(is)h(closed)g(under)h
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|
3620 |
(permutations,)f(that)h(is)e FG(t)i FD(2)h Fs(lam)1642
|
|
3621 |
3790 y Fp(\013)1709 3782 y FE(implies)e FG(\031)2004
|
|
3622 |
3791 y Fo(\001)2042 3782 y FG(t)h FD(2)g Fs(lam)2281
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|
3623 |
3790 y Fp(\013)2328 3782 y FE(.)p 0 TeXcolorgray 0 3980
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|
3624 |
a(Pr)l(oof)p 0 TeXcolorgray 40 w FI(Both)e(properties)i(follo)n(w)f(by)
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|
3625 |
g(routine)g(inductions)h(o)o(v)o(er)f(the)f(de\002nition)i(of)e
|
|
3626 |
Fs(lam)2429 3988 y Fp(\013)2476 3980 y FI(.)g(F)o(or)h(the)f(\002rst)0
|
|
3627 |
4075 y(induction)j(we)d(use)h(the)g(equations)927 4279
|
|
3628 |
y Fk(supp)6 b Fr(\()p Fs(Am)p Fr(\()p FG(a)p Fr(\)\))24
|
|
3629 |
b(=)h FD(f)p FG(a)p FD(g)804 4398 y Fk(supp)5 b Fr(\()p
|
|
3630 |
Fs(Pr)q Fr(\()p FG(t)1126 4410 y Fq(1)1163 4398 y FG(;)13
|
|
3631 |
b(t)1225 4410 y Fq(2)1262 4398 y Fr(\)\))24 b(=)h Fk(supp)6
|
|
3632 |
b Fr(\()p FG(t)1645 4410 y Fq(1)1681 4398 y Fr(\))17
|
|
3633 |
b FD([)h Fk(supp)5 b Fr(\()p FG(t)2010 4410 y Fq(2)2047
|
|
3634 |
4398 y Fr(\))835 4518 y Fk(supp)h Fr(\()p Fs(Se)p Fr(\([)p
|
|
3635 |
FG(a)p Fr(])p FG(:t)p Fr(\)\))25 b(=)g Fk(supp)6 b Fr(\()p
|
|
3636 |
FG(t)p Fr(\))16 b FD(\000)h(f)p FG(a)p FD(g)2749 4398
|
|
3637 |
y FI(\(13\))0 4727 y(where)27 b(the)g(last)f(follo)n(ws)h(from)h
|
|
3638 |
(\(11\)\227)p FG(t)g FI(is)e(\002nitely)h(supported)i(by)e(induction)i
|
|
3639 |
(hypothesis;)e(for)g(the)0 4823 y(second)21 b(we)e(use)h(Lem.)g(6)p
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|
3640 |
FE(\(i\))p FI(.)80 b FD(u)-51 b(t)p 0 TeXcolorgray 0
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|
3641 |
TeXcolorgray eop end
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|
3642 |
%%Page: 12 12
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|
3643 |
TeXDict begin 12 11 bop 0 TeXcolorgray 0 TeXcolorgray
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|
3644 |
0 71 2881 4 v 0 17 a FF(12)p 0 TeXcolorgray 125 228 a
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|
3645 |
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|
|
3646 |
(there)g(is)f(a)g(bijection)h(between)g Fs(lam)2456 246
|
|
3647 |
y Fp(=)p Fl(\031)2564 228 y FI(and)g Fs(lam)2814 236
|
|
3648 |
y Fp(\013)2861 228 y FI(.)0 324 y(This)h(is)f(sho)n(wn)i(using)g(the)f
|
|
3649 |
(follo)n(wing)h(mapping)h(from)f Fs(lam)f FI(to)g Fs(lam)1892
|
|
3650 |
332 y Fp(\013)1939 324 y FI(:)1035 510 y FG(q)s Fr(\()p
|
|
3651 |
Fs(Var)p Fr(\()p FG(a)p Fr(\)\))1375 467 y Fn(def)1382
|
|
3652 |
510 y Fr(=)32 b Fs(Am)p Fr(\()p FG(a)p Fr(\))911 653
|
|
3653 |
y FG(q)s Fr(\()p Fs(App)q Fr(\()p FG(t)1154 665 y Fq(1)1191
|
|
3654 |
653 y FG(;)13 b(t)1253 665 y Fq(2)1290 653 y Fr(\)\))1375
|
|
3655 |
610 y Fn(def)1382 653 y Fr(=)32 b Fs(Pr)p Fr(\()p FG(q)s
|
|
3656 |
Fr(\()p FG(t)1677 665 y Fq(1)1714 653 y Fr(\))p FG(;)13
|
|
3657 |
b(q)s Fr(\()p FG(t)1873 665 y Fq(2)1910 653 y Fr(\)\))973
|
|
3658 |
796 y FG(q)s Fr(\()p Fs(Lam)q Fr(\()p FG(a;)f(t)p Fr(\)\))1375
|
|
3659 |
753 y Fn(def)1382 796 y Fr(=)32 b Fs(Se)p Fr(\([)p FG(a)p
|
|
3660 |
Fr(])p FG(:q)s Fr(\()p FG(t)p Fr(\)\))0 957 y FI(and)21
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|
3661 |
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|
3662 |
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|
|
3663 |
FG(t)554 1138 y Fq(2)611 1126 y FE(if)f(and)h(only)f(if)g
|
|
3664 |
FG(q)s Fr(\()p FG(t)1127 1138 y Fq(1)1164 1126 y Fr(\))h(=)g
|
|
3665 |
FG(q)s Fr(\()p FG(t)1391 1138 y Fq(2)1428 1126 y Fr(\))p
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|
3666 |
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|
3667 |
40 w FI(By)e(routine)i(induction)h(o)o(v)o(er)f(de\002nition)g(of)f
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|
3668 |
Fs(lam)1570 1353 y Fp(\013)1617 1345 y FI(.)79 b FD(u)-51
|
|
3669 |
b(t)p 0 TeXcolorgray 0 1514 a FJ(Theor)o(em)19 b(1)p
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|
3670 |
0 TeXcolorgray 42 w FE(Ther)m(e)h(is)f(a)h(bijection)h(between)f
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|
3671 |
Fs(lam)1430 1532 y Fp(=)p Fl(\031)1540 1514 y FE(and)h
|
|
3672 |
Fs(lam)1795 1522 y Fp(\013)1843 1514 y FE(.)p 0 TeXcolorgray
|
|
3673 |
0 1733 a(Pr)l(oof)p 0 TeXcolorgray 40 w FI(The)d(mapping)j
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|
3674 |
FG(q)g FI(needs)e(to)f(be)h(lifted)g(to)f(alpha-equi)n(v)n(alence)k
|
|
3675 |
(classes)17 b(\(see)h(P)o(aulson)h([24]\).)g(F)o(or)0
|
|
3676 |
1828 y(this)24 b(de\002ne)g FG(q)391 1796 y Fl(0)414
|
|
3677 |
1828 y Fr(\([)p FG(t)p Fr(])514 1836 y Fp(\013)562 1828
|
|
3678 |
y Fr(\))f FI(as)g(follo)n(ws:)i(apply)f FG(q)j FI(to)d(e)n(v)o(ery)h
|
|
3679 |
(element)f(of)g(the)g(set)f Fr([)p FG(t)p Fr(])2188 1836
|
|
3680 |
y Fp(\013)2260 1828 y FI(and)h(b)n(uild)h(the)f(union)0
|
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3887 |
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3888 |
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3899 |
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3907 |
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3908 |
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3909 |
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3910 |
%%Page: 14 14
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3911 |
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3912 |
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3916 |
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3923 |
b(P)32 b Fr(\()p Fs(Var)896 773 y Fp(\013)944 765 y Fr(\()p
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3924 |
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3925 |
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3926 |
b Fr(\()p FD(8)p FG(d:)21 b(P)32 b(t)1034 896 y Fq(1)1092
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3927 |
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3928 |
b(P)32 b(t)1554 896 y Fq(2)1612 884 y FG(d)p Fr(\))42
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3929 |
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3930 |
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3931 |
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3932 |
1003 y FD(8)p FG(c)13 b(a)f(t)644 1015 y Fq(1)681 1003
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3933 |
y FG(:)43 b(a)21 b Fr(#)h FG(c)38 b FD(^)h Fr(\()p FD(8)p
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3934 |
FG(d:)21 b(P)32 b(t)1318 1015 y Fq(1)1376 1003 y FG(d)p
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3935 |
Fr(\))43 b FD(\))f FG(P)24 b Fr(\()p Fs(Lam)1828 1011
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3936 |
y Fp(\013)1875 1003 y Fr(\()p FG(a;)13 b(t)2008 1015
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3937 |
y Fq(1)2045 1003 y Fr(\)\))g FG(c)p 474 1051 1933 4 v
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3938 |
1367 1119 a(P)24 b(t)12 b(c)1236 b FI(\(21\))0 1277 y(where)26
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3939 |
b(the)f(v)n(ariable)h FG(t)f FI(in)g(the)h(conclusion)g(stands)g(for)g
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3940 |
(a)f Fs(lam)1760 1285 y Fp(\013)1808 1277 y FI(-term)g(o)o(v)o(er)h
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3941 |
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3942 |
(ariable)h FG(c)f FI(stands)f(for)i(the)f FE(conte)n(xt)i
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3943 |
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3944 |
0 1468 y(we)k(mean)g(all)g(free)g(v)n(ariables)g(of)g(the)g(lemma)h(to)
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3946 |
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3947 |
b(e)24 b(also)g(assume)g(that)g(the)g(conte)o(xt)h(is)e(of)i
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3948 |
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3949 |
(substitution)h(lemma)f(from)h(Fig.)f(1,)f(for)i(e)o(xample,)g(we)e(ha)
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3950 |
n(v)o(e)639 1816 y FG(M)8 b Fr([)p FG(x)21 b Fr(:=)h
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3951 |
FG(N)8 b Fr(][)p FG(y)25 b Fr(:=)c FG(L)p Fr(])h FD(\021)f
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3952 |
FG(M)8 b Fr([)p FG(y)24 b Fr(:=)e FG(L)p Fr(][)p FG(x)g
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3953 |
Fr(:=)f FG(N)8 b Fr([)p FG(y)25 b Fr(:=)c FG(L)p Fr(]])0
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|
3954 |
1974 y FI(with)34 b FG(M)41 b FI(being)35 b(the)f(v)n(ariable)g(o)o(v)o
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3955 |
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3956 |
(the)h(con-)0 2069 y(te)o(xt)27 b FG(c)f FI(w)o(ould)i(be)f
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3957 |
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3958 |
(lemma,)h(namely)g(the)g(tuple)0 2165 y Fr(\()p FG(x;)13
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|
3959 |
b(y)s(;)f(N)t(;)h(L)p Fr(\))p FI(\227which)22 b(is)f(of)h(\002nitely)g
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3960 |
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3961 |
(lambda-case,)0 2260 y(that)e(is)973 2356 y FG(P)32 b
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|
3962 |
Fr(\()p Fs(Lam)1202 2364 y Fp(\013)1249 2356 y Fr(\()p
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|
3963 |
FG(z)s(;)13 b(M)1426 2368 y Fq(1)1463 2356 y Fr(\)\))21
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|
3964 |
b(\()p FG(x;)13 b(y)s(;)f(N)t(;)i(L)p Fr(\))0 2490 y
|
|
3965 |
FI(one)24 b(can)f(assume)g(in)g(\(21\))h(that)f(the)g(binder)h
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|
3966 |
FG(z)h FI(is)e(fresh)g(for)g Fr(\()p FG(x;)13 b(y)s(;)g(N)t(;)g(L)p
|
|
3967 |
Fr(\))p FI(\227which)23 b(is)g(equi)n(v)n(alent)h(to)f
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|
3968 |
FG(z)0 2585 y FI(not)18 b(being)h(equal)f(to)f FG(x)g
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|
3969 |
FI(and)h FG(y)s FI(,)f(and)h(not)g(free)g(in)g FG(N)25
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|
3970 |
b FI(and)18 b FG(L)p FI(.)g(As)f(we)g(shall)g(see)g(later)m(,)g(with)h
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3971 |
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3972 |
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3973 |
(\002culties.)125 2776 y(In)f(the)g(follo)n(wing)i(we)d(shall)h
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3974 |
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3975 |
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|
3976 |
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3977 |
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3978 |
b(b)n(ut)g(not)g(necessarily)f(has)g(\002nitely)g(supported)i(type.)p
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3979 |
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3980 |
42 w(\(Str)o(ong)27 b(Induction)f(Principle\))g FE(A)h(pr)l(operty)g
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|
3981 |
FG(P)44 b(t)33 b(c)26 b FE(holds)h(for)g(all)f FG(t)g
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|
3982 |
FE(terms)g(of)g(type)0 3222 y Fs(lam)118 3230 y Fp(\013)165
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3983 |
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3985 |
w FD(8)p FG(c:)43 b Fk(\014nite)7 b Fr(\()p Fk(supp)e
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3987 |
34 3454 a(\(ii\))p 0 TeXcolorgray 84 w FD(8)p FG(c)13
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3988 |
b(a:)43 b(P)32 b Fr(\()p Fs(Var)636 3462 y Fp(\013)684
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|
3989 |
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3990 |
12 3549 a(\(iii\))p 0 TeXcolorgray 84 w FD(8)p FG(c)13
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3991 |
b(t)331 3561 y Fq(1)381 3549 y FG(t)409 3561 y Fq(2)446
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3992 |
3549 y FG(:)43 b Fr(\()p FD(8)p FG(d:)21 b(P)32 b(t)774
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3993 |
3561 y Fq(1)832 3549 y FG(d)p Fr(\))38 b FD(^)h Fr(\()p
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|
3994 |
FD(8)p FG(d:)21 b(P)32 b(t)1294 3561 y Fq(2)1352 3549
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|
3995 |
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|
3996 |
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3997 |
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4000 |
Fq(1)422 3645 y FG(:)43 b(a)21 b Fr(#)g FG(f)g(c)39 b
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|
4001 |
FD(^)f Fr(\()p FD(8)p FG(d:)21 b(P)32 b(t)1116 3657 y
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|
4002 |
Fq(1)1175 3645 y FG(d)p Fr(\))42 b FD(\))g FG(P)33 b
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4003 |
Fr(\()p Fs(Lam)1635 3653 y Fp(\013)1682 3645 y Fr(\()p
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4004 |
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4005 |
b FG(c)0 3781 y FE(hold.)p 0 TeXcolorgray 0 3963 a(Pr)l(oof)p
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4006 |
0 TeXcolorgray 40 w FI(By)27 b(induction)i(o)o(v)o(er)f
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4007 |
FG(t)f FI(using)i(\(20\).)f(W)-6 b(e)27 b(strengthen)i(the)f(induction)
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|
4008 |
h(hypothesis)g(by)f(aiming)0 4059 y(to)c(pro)o(v)o(e)i
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|
4009 |
FD(8)p FG(\031)15 b(c:)30 b(P)23 b Fr(\()p FG(\031)622
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|
4010 |
4068 y Fo(\001)660 4059 y FG(t)p Fr(\))12 b FG(c)p FI(.)24
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|
4011 |
b(The)h(cases)f(for)h Fs(Var)1378 4067 y Fp(\013)1450
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4012 |
4059 y FI(and)g Fs(App)1705 4080 y Fp(\013)1777 4059
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|
4013 |
y FI(are)f(routine.)i(The)e(interesting)h(case)f(is)0
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4014 |
4154 y Fs(Lam)118 4162 y Fp(\013)165 4154 y FI(:)e(we)g(need)g(to)h
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4015 |
(sho)n(w)f(that)g FG(P)i Fr(\()p FG(\031)1053 4163 y
|
|
4016 |
Fo(\001)1091 4154 y Fs(Lam)1208 4162 y Fp(\013)1256 4154
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|
4017 |
y Fr(\()p FG(a;)12 b(t)1388 4166 y Fq(1)1425 4154 y Fr(\)\))h
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|
4018 |
FG(c)p FI(,)22 b(where)g FG(\031)1834 4163 y Fo(\001)1872
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4019 |
4154 y Fs(Lam)1990 4162 y Fp(\013)2037 4154 y Fr(\()p
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4020 |
FG(a;)13 b(t)2170 4166 y Fq(1)2207 4154 y Fr(\))25 b(=)g
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4021 |
Fs(Lam)2464 4162 y Fp(\013)2512 4154 y Fr(\()p FG(\031)2589
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|
4022 |
4163 y Fo(\001)2626 4154 y FG(a;)13 b(\031)2748 4163
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|
4023 |
y Fo(\001)2786 4154 y FG(t)2814 4166 y Fq(1)2851 4154
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|
4024 |
y Fr(\))0 4250 y FI(by)35 b(\(17\).)f(Since)g(by)h FE(\(i\))f
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|
4025 |
FG(f)8 b(c)35 b FI(is)e(\002nitely)h(supported,)i(and)e(by)h(Lemmas)f
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|
4026 |
(4)h(and)f(10)h(also)f FG(\031)2655 4259 y Fo(\001)2693
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4027 |
4250 y FG(a)f FI(and)0 4345 y FG(\031)47 4354 y Fo(\001)85
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|
4028 |
4345 y FG(t)113 4357 y Fq(1)150 4345 y FI(,)26 b(we)g(can)g(use)h
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|
4029 |
(Prop.)f(1)h(to)f(obtain)i(a)e FG(b)g FI(with)g FG(b)33
|
|
4030 |
b Fr(#)g(\()p FG(f)21 b(c;)13 b(\031)1803 4354 y Fo(\001)1841
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|
4031 |
4345 y FG(a;)g(\031)1963 4354 y Fo(\001)2001 4345 y FG(t)2029
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4032 |
4357 y Fq(1)2066 4345 y Fr(\))p FI(.)25 b(From)i(this)f(we)g(can)h
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4033 |
(infer)0 4441 y(that)h FG(b)36 b FD(6)p Fr(=)f FG(\031)357
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|
4034 |
4450 y Fo(\001)395 4441 y FG(a)28 b FI(and)g FG(b)36
|
|
4035 |
b Fr(#)g FG(\031)821 4450 y Fo(\001)859 4441 y FG(t)887
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|
4036 |
4453 y Fq(1)924 4441 y FI(,)27 b(which)i(implies)f(by)g(\(18\))h(that)f
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|
4037 |
Fr(\()p FD(\003)p Fr(\))g Fs(Lam)2107 4449 y Fp(\013)2154
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|
4038 |
4441 y Fr(\()p FG(b;)13 b Fr(\()p FG(b)38 b(\031)2399
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|
4039 |
4450 y Fo(\001)2437 4441 y FG(a)p Fr(\))2508 4450 y Fo(\001)2546
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|
4040 |
4441 y Fr(\()p FG(\031)2623 4450 y Fo(\001)2661 4441
|
|
4041 |
y FG(t)2689 4453 y Fq(1)2726 4441 y Fr(\)\))d(=)0 4536
|
|
4042 |
y Fs(Lam)118 4544 y Fp(\013)165 4536 y Fr(\()p FG(\031)242
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|
4043 |
4545 y Fo(\001)280 4536 y FG(a;)12 b(\031)401 4545 y
|
|
4044 |
Fo(\001)439 4536 y FG(t)467 4548 y Fq(1)504 4536 y Fr(\))p
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|
4045 |
FI(.)26 b(From)g(the)g(induction)i(hypothesis,)e(which)h(is)e
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|
4046 |
FD(8)p FG(\031)15 b(c:)f(P)23 b Fr(\()p FG(\031)2220
|
|
4047 |
4545 y Fo(\001)2258 4536 y FG(t)2286 4548 y Fq(1)2323
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|
4048 |
4536 y Fr(\))12 b FG(c)p FI(,)26 b(we)g(obtain)g(the)0
|
|
4049 |
4632 y(f)o(act)i FD(8)p FG(c:)13 b(P)24 b Fr(\(\(\()p
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|
4050 |
FG(b)38 b(\031)536 4641 y Fo(\001)574 4632 y FG(a)p Fr(\)@)p
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|
4051 |
FG(\031)s Fr(\))782 4641 y Fo(\001)819 4632 y FG(t)847
|
|
4052 |
4644 y Fq(1)884 4632 y Fr(\))12 b FG(c)p FI(.)28 b(Then)h(we)f(can)g
|
|
4053 |
(use)g(the)g(f)o(act)h FG(b)36 b Fr(#)g FG(f)21 b(c)28
|
|
4054 |
b FI(and)h FE(\(iv\))p FI(,)f(and)h(infer)g(that)0 4727
|
|
4055 |
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|
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4056 |
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4057 |
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4058 |
4736 y Fo(\001)825 4727 y FG(t)853 4739 y Fq(1)890 4727
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4059 |
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4060 |
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4061 |
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|
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4062 |
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4063 |
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4064 |
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4065 |
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4066 |
FI(.)20 b(By)f Fr(\()p FD(\003)p Fr(\))h FI(we)g(can)g(conclude)i(with)
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|
4067 |
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4068 |
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4069 |
b(\031)2452 4832 y Fo(\001)2490 4823 y FG(t)2518 4835
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4070 |
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4071 |
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4073 |
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4074 |
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4075 |
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|
4076 |
b(the)f(identity-function)k(and)d(require)h(that)e FG(c)h
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|
4077 |
FI(has)f(\002nitely)h(supported)h(type,)0 324 y(we)c(can)g(dischar)o
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4078 |
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4079 |
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4080 |
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4081 |
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4082 |
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4083 |
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4084 |
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4085 |
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4086 |
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4087 |
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4088 |
(combinator)0 801 y(for)i Fs(lam)229 809 y Fp(\013)277
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4089 |
801 y FI(.)0 1082 y FJ(5)28 b(A)20 b(Recursion)f(Combinator)0
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4090 |
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4093 |
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4095 |
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4096 |
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4097 |
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4098 |
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4099 |
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4101 |
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4107 |
FG(t)950 2064 y Fl(0)973 2096 y Fr(])j(=)g(\()p FI(if)d
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4108 |
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4109 |
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4110 |
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4111 |
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4112 |
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4119 |
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4120 |
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4121 |
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4122 |
b(t)p Fr([)p FG(y)24 b Fr(:=)d FG(t)1615 2303 y Fl(0)1639
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4123 |
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4124 |
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4126 |
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4130 |
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4133 |
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4136 |
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4137 |
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4138 |
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4142 |
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4143 |
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4144 |
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4146 |
Fr(\()p Fs(Var)1934 3050 y Fp(\013)1981 3042 y Fr(\()p
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4147 |
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4148 |
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4149 |
3209 y Fq(1)1955 3197 y FG(;)13 b(t)2017 3209 y Fq(2)2054
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4150 |
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4151 |
3197 y FG(;)13 b(t)2410 3209 y Fq(2)2447 3197 y FD(g)1650
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4152 |
3351 y FE(ist)o Fr(\()p Fs(Lam)1872 3359 y Fp(\013)1919
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4153 |
3351 y Fr(\()p FG(x;)g(t)p Fr(\)\))35 b(=)g FD(f)p FG(t)p
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4154 |
FD(g)2749 3220 y FI(\(22\))0 3535 y(results)15 b(in)h(an)g
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4155 |
(inconsistenc)o(y)h(when)f(de\002ned)g(o)o(v)o(er)h Fs(lam)1536
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4156 |
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4157 |
(without)f(problems)0 3631 y(o)o(v)o(er)27 b Fs(lam)q
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4158 |
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4159 |
FE(ist)f FI(arises)g(by)h(the)g(principle)g(of)g(HOL)g(stating)f(that)h
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(we)e(ha)n(v)o(e)822 3917 y Fs(Lam)939 3925 y Fp(\013)987
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4163 |
3917 y Fr(\()p FG(x;)12 b Fs(Var)1212 3925 y Fp(\013)1259
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4164 |
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4165 |
y Fp(\013)1660 3917 y Fr(\()p FG(y)s(;)12 b Fs(Var)1882
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4166 |
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4167 |
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4067 y Fp(\013)2456 4059 y Fr(\()p FG(x;)c Fs(Var)2681
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4171 |
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FG(y)i FI(gi)n(ving)f(a)f(contradiction)i(with)e(the)g(assumption)h
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FG(x)g FD(6)p Fr(=)g FG(y)s FI(\227similar)e(with)h(the)g(function)i
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4179 |
FE(ist)o FI(.)125 4345 y(One)16 b(w)o(ay)g(around)i(the)e(problem)i
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4180 |
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4181 |
i(combinator)0 4441 y(for)27 b Fs(lam)236 4449 y Fp(\013)309
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4182 |
4441 y FI(that)f(includes)h(certain)g(preconditions)h(for)f(binders)g
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4187 |
(xistence)h(of)f(a)h(structural)g(recursion)g(combinator)i(for)0
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4188 |
4727 y Fs(lam)118 4735 y Fp(\013)165 4727 y FI(.)20 b(The)g(main)h(dif)
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4189 |
n(ference)h(of)e(our)h(proof)h(is)e(that)g(we)g(gi)n(v)o(e)h(here)f(a)g
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4192 |
(type)g Fs(lam)g FI(\(Pitts)f(uses)g Fs(lam)i FI(to)p
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4193 |
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4197 |
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4198 |
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4199 |
y(combinator)c(without)f(deri)n(ving)g(an)g(iteration)f(combinator)i
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4200 |
(\002rst.)1819 292 y Fv(6)125 419 y FI(While)i(in)g(\223e)n(v)o
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4201 |
(ery-day\224)i(formalisation,)f(Lem.)f(4)g(is)f(suf)n(\002cient)i(in)f
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4202 |
(nearly)g(all)g(situations)g(to)g(\002nd)0 515 y(out)19
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4203 |
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4204 |
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4205 |
610 y(se)n(v)o(eral)24 b(places)h(proof)h(obligations)f(about)h
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4206 |
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4210 |
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4213 |
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4214 |
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4215 |
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4216 |
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4217 |
b(b)p Fr(\))1789 1147 y Fo(\001)1827 1138 y FG(x)21 b
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4218 |
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4219 |
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4224 |
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4229 |
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4230 |
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|
|
4231 |
FG(b)i FD(62)h FG(S)h FI(we)d(ha)n(v)o(e)g Fr(\()p FG(a)12
|
|
4232 |
b(b)p Fr(\))2190 1814 y Fo(\001)2228 1805 y FG(x)21 b
|
|
4233 |
Fr(=)g FG(x)p FI(.)e(Hence)h(the)f(set)0 1900 y FD(f)p
|
|
4234 |
FG(b)j FD(j)f Fr(\()p FG(a)13 b(b)p Fr(\))282 1909 y
|
|
4235 |
Fo(\001)320 1900 y FG(x)20 b FD(6)p Fr(=)i FG(x)p FD(g)16
|
|
4236 |
b FI(is)h(a)g(subset)g(of)h FG(S)t FI(,)f(and)h(since)f
|
|
4237 |
FG(S)k FI(is)16 b(\002nite)h(by)h(assumption,)g(also)f
|
|
4238 |
FD(f)p FG(b)22 b FD(j)g Fr(\()p FG(a)12 b(b)p Fr(\))2615
|
|
4239 |
1909 y Fo(\001)2653 1900 y FG(x)21 b FD(6)p Fr(=)g FG(x)p
|
|
4240 |
FD(g)0 1996 y FI(must)f(be)g(\002nite.)g(But)g(this)f(implies)h(that)g
|
|
4241 |
FG(a)h FD(62)h Fk(supp)5 b Fr(\()p FG(x)p Fr(\))19 b
|
|
4242 |
FI(which)i(gi)n(v)o(es)f(the)g(contradiction.)81 b FD(u)-51
|
|
4243 |
b(t)0 2156 y FI(Lem.)23 b(11)g(gi)n(v)o(es)g(us)f(some)h(means)g(to)f
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|
4244 |
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4245 |
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|
4246 |
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4247 |
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4248 |
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4249 |
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4253 |
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4255 |
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4258 |
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4259 |
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|
4260 |
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4261 |
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4262 |
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4263 |
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4264 |
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4266 |
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4269 |
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4270 |
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4271 |
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4272 |
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4273 |
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4274 |
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4276 |
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4277 |
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4280 |
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4281 |
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4282 |
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4283 |
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4284 |
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4286 |
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4288 |
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4289 |
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4290 |
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4291 |
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4292 |
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4293 |
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4294 |
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4295 |
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4296 |
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4297 |
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4298 |
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4299 |
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4300 |
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4301 |
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4302 |
h(by)f(Lem.)g(4)g(that)0 4228 y Fr(\()p FG(a)12 b(b)p
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4303 |
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4304 |
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4305 |
FG(c)p FI(\).)125 4324 y(W)-6 b(e)28 b(can)i(conclude)g(that)f
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4306 |
Fk(supp)6 b Fr(\()p Fk(fn)f Fr(\))29 b FI(is)f(a)h(subset)g(of)g
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4307 |
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4308 |
FG(;)13 b(c)p Fr(\))p FI(,)29 b(because)g(the)h(latter)e(is)h(\002nite)
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4309 |
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4310 |
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4311 |
FG(c)g FI(is)f(\002nitely)g(supported)j(because)d(the)h(type)g
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4312 |
Fs(name)0 4515 y FI(is)e(a)h(\002nitely)g(supported)i(type\).)f(So)f
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4313 |
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4314 |
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eop end
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4322 |
%%Page: 17 17
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4323 |
TeXDict begin 17 16 bop 0 TeXcolorgray 0 TeXcolorgray
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4327 |
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4328 |
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4329 |
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4330 |
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4331 |
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4332 |
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4333 |
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4334 |
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4335 |
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4336 |
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4337 |
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4339 |
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4340 |
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4341 |
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4342 |
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4343 |
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4344 |
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4345 |
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4346 |
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4347 |
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4348 |
y Fl(0)1567 701 y FI(else)f Fs(Var)1829 709 y Fp(\013)1876
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4349 |
701 y Fr(\(\()p FG(a)13 b(b)p Fr(\))2053 669 y Fl(\000)p
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4350 |
Fq(1)2142 710 y Fo(\001)2180 701 y FG(x)p Fr(\)\))275
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4351 |
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4352 |
FG(a)12 b(b)p Fr(\))926 806 y Fo(\001)964 797 y FG(y)24
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4353 |
b FI(then)e Fr(\()p FG(a)13 b(b)p Fr(\))1330 806 y Fo(\001)1368
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4354 |
797 y FG(t)1396 765 y Fl(0)1440 797 y FI(then)22 b Fs(Var)1715
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4355 |
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4356 |
b(\(10\))275 892 y Fr(=)107 b FG(\025x:)21 b FI(if)h
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4357 |
FG(x)e Fr(=)i FG(y)h FI(then)f FG(t)1035 860 y Fl(0)1080
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|
4358 |
892 y FI(else)f Fs(Var)1342 900 y Fp(\013)1389 892 y
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4359 |
Fr(\()p FG(x)p Fr(\))1021 b(\()p FD(\003)p Fr(\))0 1053
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4360 |
y FI(where)25 b Fr(\()p FD(\003)p Fr(\))g FI(follo)n(ws)g(by)g(Lem.)g
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4361 |
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4362 |
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4363 |
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4364 |
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4365 |
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4366 |
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4367 |
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4374 |
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4376 |
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4377 |
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4378 |
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4379 |
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4380 |
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4381 |
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4382 |
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|
4383 |
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|
4384 |
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4385 |
(application-)g(and)g(lambda-case,)f(respec-)0 2087 y(ti)n(v)o(ely)-5
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4389 |
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|
4390 |
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|
4391 |
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|
4392 |
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|
4393 |
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|
4394 |
2422 y FD(\))f FG(\013)h FD(\))f FG(\013)0 2565 y FI(with)f
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|
4395 |
FG(\013)h FI(being)g(a)f(permutation)i(type.)f(Then)g(the)g(\002rst)e
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|
4396 |
(condition)j(Pitts)e(introduced)i(in)f([27])g(states)f(that)0
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4397 |
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|
4398 |
g(lambda)g(case\227needs)f(to)h(satisfy)f(the)g FE(fr)m(eshness)g
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|
4399 |
(condition)i(for)e(binder)o(s)p FI(,)0 2756 y(or)f(short)h
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|
4400 |
FE(FCB)p FI(.)e(W)-6 b(e)20 b(formulate)i(this)d(condition)j(as:)1440
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|
4401 |
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4402 |
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4403 |
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|
4404 |
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|
4405 |
y FD(\))g FG(\013)g FD(\))h FG(\013)e FE(satis\002es)f(the)h
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|
4406 |
FI(FCB)f FE(pr)l(o)o(vided:)666 3196 y FD(8)p FG(a)12
|
|
4407 |
b(t)h(r)n(:)21 b(a)g Fr(#)g FG(f)47 b FD(^)39 b Fk(\014nite)6
|
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4408 |
b Fr(\()p Fk(supp)f Fr(\()p FG(r)r Fr(\)\))42 b FD(\))h
|
|
4409 |
FG(a)21 b Fr(#)g FG(f)8 b(a)13 b(t)f(r)24 b(:)0 3369
|
|
4410 |
y FI(As)19 b(we)h(shall)g(see)g(later)g(on,)g(this)g(condition)i
|
|
4411 |
(ensures)e(that)g(the)h(result)f(of)g FG(f)2067 3381
|
|
4412 |
y Fq(3)2125 3369 y FI(is)f(independent)k(of)d(which)0
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|
4413 |
3465 y(particular)30 b(fresh)g(name)g(one)g(chooses)g(for)g(the)f
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4414 |
(binder)i FG(a)p FI(.)d(The)i(second)g(condition)h(states)d(that)h(the)
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4415 |
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|
4416 |
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4417 |
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4418 |
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|
4419 |
3656 y(Prop.)e(1)h(when)f(choosing)i(a)d(fresh)i(name)f(for)h(the)f
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|
4420 |
FG(f)8 b FI(s.)125 3752 y(W)m(ith)22 b(these)f(tw)o(o)h(conditions)h
|
|
4421 |
(we)f(can)g(deri)n(v)o(e)g(a)g(recursion)h(combinator)m(,)g(we)f(call)f
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4422 |
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4423 |
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4424 |
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4425 |
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4427 |
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4428 |
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4429 |
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4430 |
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4431 |
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4433 |
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4437 |
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4438 |
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4439 |
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4442 |
4379 y Fr(\()p Fs(App)867 4400 y Fp(\013)915 4379 y Fr(\()p
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4444 |
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4450 |
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4452 |
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4454 |
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4455 |
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4456 |
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4457 |
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4458 |
4495 y Fp(f)1753 4503 y Fj(2)1786 4495 y Fp(f)1818 4503
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4459 |
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4460 |
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4461 |
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4466 |
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4470 |
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4471 |
0 71 2881 4 v 0 17 a FF(18)p 0 TeXcolorgray 0 228 a FI(T)-6
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4472 |
b(o)15 b(gi)n(v)o(e)h(a)f(proof)i(of)e(this)g(theorem)i(we)e(start)g
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4473 |
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4474 |
Fk(r)l(e)l(c)2690 243 y Fp(f)2722 251 y Fj(1)2755 243
|
|
4475 |
y Fp(f)2787 251 y Fj(2)2820 243 y Fp(f)2852 251 y Fj(3)0
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|
4476 |
324 y FI(and)i(which)g(has)f(type)h Fr(\()p Fs(lam)760
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|
4477 |
332 y Fp(\013)811 324 y FD(\002)t FG(\013)p Fr(\))c Fs(set)k
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|
4478 |
FI(where,)f(lik)o(e)h(abo)o(v)o(e,)g FG(\013)f FI(is)g(assumed)h(to)f
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|
4479 |
(be)h(a)f(permutation)i(type:)p 268 581 879 4 v 268 653
|
|
4480 |
a Fr(\()p Fs(Var)416 661 y Fp(\013)463 653 y Fr(\()p
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4481 |
FG(a)p Fr(\))p FG(;)13 b(f)635 665 y Fq(1)685 653 y FG(a)p
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|
4482 |
Fr(\))21 b FD(2)h Fk(r)l(e)l(c)949 668 y Fp(f)981 676
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4483 |
y Fj(1)1014 668 y Fp(f)1046 676 y Fj(2)1078 668 y Fp(f)1110
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4484 |
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|
4485 |
535 y FG(;)13 b(r)1453 547 y Fq(1)1490 535 y Fr(\))21
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|
4486 |
b FD(2)h Fk(r)l(e)l(c)1713 550 y Fp(f)1745 558 y Fj(1)1778
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|
4487 |
550 y Fp(f)1810 558 y Fj(2)1842 550 y Fp(f)1874 558 y
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|
4488 |
Fj(3)1990 535 y Fr(\()p FG(t)2048 547 y Fq(2)2085 535
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4489 |
y FG(;)13 b(r)2154 547 y Fq(2)2191 535 y Fr(\))21 b FD(2)h
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4490 |
Fk(r)l(e)l(c)2414 550 y Fp(f)2446 558 y Fj(1)2479 550
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4491 |
y Fp(f)2511 558 y Fj(2)2543 550 y Fp(f)2575 558 y Fj(3)p
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4492 |
1289 581 1324 4 v 1314 653 a Fr(\()p Fs(App)1461 674
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4493 |
y Fp(\013)1509 653 y Fr(\()p FG(t)1567 665 y Fq(1)1604
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4494 |
653 y FG(;)13 b(t)1666 665 y Fq(2)1703 653 y Fr(\))p
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4495 |
FG(;)g(f)1804 665 y Fq(2)1854 653 y FG(t)1882 665 y Fq(1)1932
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|
4496 |
653 y FG(t)1960 665 y Fq(2)2010 653 y FG(r)2045 665 y
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|
4497 |
Fq(1)2094 653 y FG(r)2129 665 y Fq(2)2166 653 y Fr(\))21
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|
4498 |
b FD(2)h Fk(r)l(e)l(c)2389 668 y Fp(f)2421 676 y Fj(1)2454
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|
4499 |
668 y Fp(f)2486 676 y Fj(2)2519 668 y Fp(f)2551 676 y
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4500 |
Fj(3)877 821 y FG(a)f Fr(#)g(\()p FG(f)1091 833 y Fq(1)1128
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4501 |
821 y FG(;)13 b(f)1199 833 y Fq(2)1237 821 y FG(;)g(f)1308
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|
4502 |
833 y Fq(3)1346 821 y Fr(\))78 b(\()p FG(t;)13 b(r)r
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|
4503 |
Fr(\))21 b FD(2)h Fk(r)l(e)l(c)1806 836 y Fp(f)1838 844
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|
4504 |
y Fj(1)1871 836 y Fp(f)1903 844 y Fj(2)1935 836 y Fp(f)1967
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4505 |
844 y Fj(3)p 877 867 1128 4 v 925 939 a Fr(\()p Fs(Lam)1073
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|
4506 |
947 y Fp(\013)1120 939 y Fr(\()p FG(a;)13 b(t)p Fr(\))p
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|
4507 |
FG(;)f(f)1353 951 y Fq(3)1404 939 y FG(a)g(t)h(r)r Fr(\))21
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|
4508 |
b FD(2)g Fk(r)l(e)l(c)1757 954 y Fp(f)1789 962 y Fj(1)1822
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|
4509 |
954 y Fp(f)1854 962 y Fj(2)1887 954 y Fp(f)1919 962 y
|
|
4510 |
Fj(3)2749 742 y FI(\(23\))125 1081 y(W)-6 b(e)26 b(shall)g(sho)n(w)h
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|
4511 |
(ne)o(xt)f(that)h(the)f(relation)h Fk(r)l(e)l(c)1413
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|
4512 |
1096 y Fp(f)1445 1104 y Fj(1)1478 1096 y Fp(f)1510 1104
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|
4513 |
y Fj(2)1543 1096 y Fp(f)1575 1104 y Fj(3)1638 1081 y
|
|
4514 |
FI(de\002nes)g(a)f(function)i(in)e(the)h(sense)e(that)i(for)0
|
|
4515 |
1177 y(all)k(lambda-terms)h FG(t)f FI(there)h(e)o(xists)e(a)h(unique)i
|
|
4516 |
FG(r)f FI(so)f(that)h Fr(\()p FG(t;)12 b(r)r Fr(\))42
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|
4517 |
b FD(2)f Fk(r)l(e)l(c)2053 1192 y Fp(f)2085 1200 y Fj(1)2117
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|
4518 |
1192 y Fp(f)2149 1200 y Fj(2)2182 1192 y Fp(f)2214 1200
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|
4519 |
y Fj(3)2251 1177 y FI(.)31 b(From)h(this)e(we)h(can)0
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|
4520 |
1272 y(again)19 b(use)g(standard)i(techniques)f(of)f(HOL)g(to)h(obtain)
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|
4521 |
g(a)f(function)h(from)h Fs(lam)2143 1280 y Fp(\013)2210
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|
4522 |
1272 y FI(to)e FG(\013)g FI(\(see)g(for)h(e)o(xample)0
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|
4523 |
1368 y(Slind)28 b([28]\).)h(W)-6 b(e)27 b(\002rst)g(sho)n(w)h(that)g
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|
4524 |
(in)g Fk(r)l(e)l(c)1205 1383 y Fp(f)1237 1391 y Fj(1)1270
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|
4525 |
1383 y Fp(f)1302 1391 y Fj(2)1335 1383 y Fp(f)1367 1391
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|
4526 |
y Fj(3)1431 1368 y FI(the)g(\223result\224)g FG(r)h FI(has)e(\002nite)h
|
|
4527 |
(support)h(pro)o(vided)g(the)0 1463 y(functions)21 b
|
|
4528 |
FG(f)350 1475 y Fq(1)388 1463 y FI(,)e FG(f)464 1475
|
|
4529 |
y Fq(2)521 1463 y FI(and)i FG(f)692 1475 y Fq(3)749 1463
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|
4530 |
y FI(ha)n(v)o(e)g(\002nite)f(support.)p 0 TeXcolorgray
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|
4531 |
0 1609 a FJ(Lemma)f(12)p 0 TeXcolorgray 43 w(\(Finite)25
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|
4532 |
b(Support\))g FE(If)g FG(f)1084 1621 y Fq(1)1122 1609
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|
4533 |
y FE(,)g FG(f)1204 1621 y Fq(2)1266 1609 y FE(and)h FG(f)1446
|
|
4534 |
1621 y Fq(3)1509 1609 y FE(have)g(\002nite)g(support,)f(then)h
|
|
4535 |
Fr(\()p FG(t;)13 b(r)r Fr(\))30 b FD(2)h Fk(r)l(e)l(c)2682
|
|
4536 |
1624 y Fp(f)2714 1632 y Fj(1)2747 1624 y Fp(f)2779 1632
|
|
4537 |
y Fj(2)2812 1624 y Fp(f)2844 1632 y Fj(3)0 1704 y FE(implies)20
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|
4538 |
b(that)g FG(r)i FE(has)e(\002nite)g(support.)p 0 TeXcolorgray
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|
4539 |
0 1870 a(Pr)l(oof)p 0 TeXcolorgray 40 w FI(By)i(induction)j(o)o(v)o(er)
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|
4540 |
e(the)g(relation)h(de\002ned)f(in)g(\(23\).)h(In)f(the)g(v)n
|
|
4541 |
(ariable-case)h(we)f(ha)n(v)o(e)g(to)g(sho)n(w)0 1965
|
|
4542 |
y(that)f FG(f)177 1977 y Fq(1)227 1965 y FG(a)f FI(has)h(\002nite)f
|
|
4543 |
(support,)i(which)f(we)g(inferred)h(in)e(Example)i(1)f(using)g(our)h
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|
4544 |
(heuristic.)f(The)g(appli-)0 2061 y(cation)e(and)h(lambda-case)g(are)f
|
|
4545 |
(by)h(similar)e(calculations.)80 b FD(u)-51 b(t)0 2206
|
|
4546 |
y FI(In)20 b(the)f(proof)h(of)g(Thm)g(3,)f(we)g(need)g(the)h(follo)n
|
|
4547 |
(wing)g(lemma)g(establishing)g(that)f Fk(r)l(e)l(c)2303
|
|
4548 |
2221 y Fp(f)2335 2229 y Fj(1)2368 2221 y Fp(f)2400 2229
|
|
4549 |
y Fj(2)2432 2221 y Fp(f)2464 2229 y Fj(3)2520 2206 y
|
|
4550 |
FI(is)g FE(equivari-)0 2302 y(ant)j FI(\(see)e(Pitts)f([26]\).)p
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|
4551 |
0 TeXcolorgray 0 2447 a FJ(Lemma)g(13)p 0 TeXcolorgray
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|
4552 |
43 w(\(Equi)o(v)o(ariance\))29 b FE(If)g Fr(\()p FG(t;)12
|
|
4553 |
b(r)r Fr(\))37 b FD(2)g Fk(r)l(e)l(c)1388 2462 y Fp(f)1420
|
|
4554 |
2470 y Fj(1)1453 2462 y Fp(f)1485 2470 y Fj(2)1518 2462
|
|
4555 |
y Fp(f)1550 2470 y Fj(3)1615 2447 y FE(holds)29 b(then)g(for)g(all)f
|
|
4556 |
FG(\031)s FE(,)g(also)h Fr(\()p FG(\031)2542 2456 y Fo(\001)2579
|
|
4557 |
2447 y FG(t;)13 b(\031)2688 2456 y Fo(\001)2726 2447
|
|
4558 |
y FG(r)r Fr(\))37 b FD(2)0 2543 y FG(r)r(ec)106 2564
|
|
4559 |
y Fq(\()p Fp(\031)173 2573 y Fo(\001)211 2564 y Fp(f)243
|
|
4560 |
2572 y Fj(1)276 2564 y Fq(\)\()p Fp(\031)369 2573 y Fo(\001)407
|
|
4561 |
2564 y Fp(f)439 2572 y Fj(2)471 2564 y Fq(\)\()p Fp(\031)564
|
|
4562 |
2573 y Fo(\001)602 2564 y Fp(f)634 2572 y Fj(3)667 2564
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|
4563 |
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|
4564 |
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|
4565 |
(the)g(rules)g(gi)n(v)o(en)h(in)f(\(23\).)h(All)e(cases)g(are)h
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|
4566 |
(routine)h(by)g(pushing)g(the)f(permu-)0 2804 y(tation)j
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|
4567 |
FG(\031)h FI(into)f FG(t)f FI(and)h FG(r)r FI(,)e(e)o(xcept)i(in)f(the)
|
|
4568 |
h(lambda-case)g(where)f(we)g(ha)n(v)o(e)h(to)g(apply)g(Lem.)f(3)p
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|
4569 |
FE(\(iii\))h FI(in)f(order)0 2899 y(to)g(infer)h FG(\031)297
|
|
4570 |
2908 y Fo(\001)335 2899 y FG(a)g Fr(#)g(\()p FG(\031)559
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|
4571 |
2908 y Fo(\001)597 2899 y FG(f)634 2911 y Fq(1)671 2899
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|
4572 |
y FG(;)13 b(\031)752 2908 y Fo(\001)790 2899 y FG(f)827
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|
4573 |
2911 y Fq(2)865 2899 y FG(;)g(\031)946 2908 y Fo(\001)983
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|
4574 |
2899 y FG(f)1020 2911 y Fq(3)1058 2899 y Fr(\))20 b FI(from)h
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|
4575 |
FG(a)g Fr(#)g(\()p FG(f)1495 2911 y Fq(1)1532 2899 y
|
|
4576 |
FG(;)13 b(f)1603 2911 y Fq(2)1641 2899 y FG(;)g(f)1712
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|
4577 |
2911 y Fq(3)1750 2899 y Fr(\))p FI(.)78 b FD(u)-51 b(t)0
|
|
4578 |
3045 y FI(Ne)o(xt)20 b(we)g(can)g(sho)n(w)g(the)g(crucial)g(lemma)h
|
|
4579 |
(about)g Fk(r)l(e)l(c)1476 3060 y Fp(f)1508 3068 y Fj(1)1541
|
|
4580 |
3060 y Fp(f)1573 3068 y Fj(2)1606 3060 y Fp(f)1638 3068
|
|
4581 |
y Fj(3)1694 3045 y FI(being)g(a)f(\223function\224.)p
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|
4582 |
0 TeXcolorgray 0 3191 a FJ(Lemma)f(14)p 0 TeXcolorgray
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4583 |
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4680 |
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4682 |
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4685 |
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4699 |
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4702 |
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4703 |
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4710 |
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4711 |
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4721 |
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4723 |
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|
4758 |
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4759 |
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4760 |
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4761 |
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4762 |
172 1661 y(by)g(Lem.)g(11)h(that)f FG(c)i Fr(#)f FG(f)868
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4763 |
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4764 |
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4765 |
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4766 |
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4769 |
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4770 |
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4771 |
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4772 |
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4773 |
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|
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4775 |
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4777 |
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|
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4778 |
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4779 |
1852 y FG(r)2458 1820 y Fl(0)2481 1852 y Fr(\))19 b FI(holds.)i(This)
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4780 |
172 1947 y(in)e(turn)h(is)f(by)h(\(3\))g(equi)n(v)n(alent)g(to)f
|
|
4781 |
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|
|
4782 |
FG(a)f(c)p Fr(\))1412 1956 y Fo(\001)1450 1947 y FG(t)p
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|
4783 |
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4784 |
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4785 |
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|
|
4786 |
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|
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4788 |
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4789 |
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4791 |
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4792 |
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4793 |
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4794 |
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4796 |
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4797 |
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4798 |
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4799 |
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4800 |
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4801 |
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4803 |
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4804 |
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4805 |
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4806 |
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4807 |
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4808 |
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4809 |
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4810 |
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4811 |
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4814 |
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4815 |
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4816 |
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4817 |
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4818 |
3003 y(vided:)600 3099 y FD(9)p FG(a:)21 b(a)g Fr(#)g
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4819 |
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4820 |
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4821 |
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4822 |
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4823 |
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4827 |
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4831 |
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4832 |
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4833 |
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4834 |
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4835 |
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4836 |
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4837 |
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4838 |
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4839 |
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4840 |
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4841 |
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4842 |
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4843 |
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4844 |
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4845 |
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4846 |
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4847 |
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4848 |
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4849 |
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4850 |
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4851 |
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4852 |
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4853 |
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4854 |
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4855 |
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4856 |
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4857 |
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4858 |
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4859 |
FI(we)f(can)h(apply)h(on)f(both)g(sides)f(of)h Fr(#)f
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4860 |
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4861 |
4194 y Fl(0)469 4226 y Fr(\))19 b FI(and)i(obtain)699
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4862 |
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4863 |
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4864 |
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4865 |
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4866 |
4373 y FG(t)p Fr(\))g(\(\()p FG(a)h(a)1720 4337 y Fl(0)1743
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4867 |
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4868 |
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4869 |
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4870 |
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4871 |
FI(is)e(equi)n(v)n(alent)i(to)f FG(a)h Fr(#)h FG(f)8
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4872 |
b(a)13 b(t)f(r)r FI(\227the)20 b(f)o(act)g(we)g(had)g(to)g(sho)n(w)-5
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4873 |
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4874 |
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4875 |
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4876 |
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4877 |
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4878 |
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4879 |
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4880 |
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4881 |
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4883 |
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4884 |
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4886 |
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4887 |
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4888 |
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4889 |
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4890 |
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4891 |
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4892 |
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4893 |
b(,)30 b(we)g(can)g(start)f(to)h(formalise)h(Barendre)o(gt')l(s)g
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4894 |
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4895 |
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4896 |
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4897 |
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4898 |
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4899 |
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4900 |
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|
4901 |
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4902 |
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4903 |
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4906 |
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4907 |
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|
4908 |
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|
4909 |
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|
4910 |
1814 y FD(\002)17 b Fs(lam)2168 1822 y Fp(\013)2215 1814
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|
4911 |
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|
4912 |
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4913 |
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4914 |
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4915 |
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4916 |
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4917 |
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4918 |
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4919 |
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4920 |
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4921 |
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4922 |
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4923 |
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4924 |
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4925 |
(5)h FE(without)h FI(an)o(y)f(user)g(interference.)h(Furthermore,)h
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4926 |
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4927 |
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4928 |
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4929 |
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4930 |
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4931 |
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4932 |
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5053 |
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5056 |
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5057 |
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5058 |
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5059 |
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5060 |
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5061 |
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5063 |
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5064 |
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5065 |
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5066 |
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5068 |
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5069 |
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5070 |
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5071 |
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5074 |
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5075 |
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5076 |
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5077 |
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5079 |
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5080 |
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5081 |
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5083 |
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5084 |
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5085 |
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5086 |
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5087 |
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5088 |
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5089 |
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5090 |
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5091 |
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5093 |
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5208 |
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5213 |
y FE(et)20 b(al)g FI([2])g(and)h(Homeier)g([16].)125
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5216 |
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5229 |
%%Page: 23 23
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5230 |
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Fy(])-12 b(])35 b(=)-12 b Fz(\))35 b Ft(M)1064 2531 y
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