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load exec end + end + } def + /BuildChar { + 1 index /Encoding get exch get + 1 index /BuildGlyph get exec + } bind def +end +/PSTricksDotFont exch definefont pop +% END pst-dots.pro +%%EndProcSet +%%BeginProcSet: pst-grad.pro +% PostScript prologue for pst-grad.tex. +% Version 97, 93/05/12 +% For copying restrictions, see pstricks.tex. +% +% For the PSTricks gradient fillstyle. +% +% Based on some EPS files by leeweyr!bill@nuchat.sccsi.com (W. R. Lee). +% +% Syntax: +% R0 G0 B0 R1 G1 B1 NumLines MidPoint Angle GradientFill + +/tx@GradientDict 40 dict def +tx@GradientDict begin +/GradientFill { + rotate + /MidPoint ED + /NumLines ED + /LastBlue ED + /LastGreen ED + /LastRed ED + /FirstBlue ED + /FirstGreen ED + /FirstRed ED + % This avoids gaps due to rounding errors: + clip + pathbbox %leave llx,lly,urx,ury on stack + /y ED /x ED + 2 copy translate + y sub neg /y ED + x sub neg /x ED + % This avoids gaps due to rounding errors: + LastRed FirstRed add 2 div + LastGreen FirstGreen add 2 div + LastBlue FirstBlue add 2 div + setrgbcolor + fill + /YSizePerLine y NumLines div def + /CurrentY 0 def + /MidLine NumLines 1 MidPoint sub mul abs cvi def + MidLine NumLines 2 sub gt + { /MidLine NumLines def } + { MidLine 2 lt { /MidLine 0 def } if } + ifelse + MidLine 0 gt + { + /Red FirstRed def + /Green FirstGreen def + /Blue FirstBlue def + /RedIncrement LastRed FirstRed sub MidLine 1 sub div def + /GreenIncrement LastGreen FirstGreen sub MidLine 1 sub div def + /BlueIncrement LastBlue FirstBlue sub MidLine 1 sub div def + MidLine { GradientLoop } repeat + } if + MidLine NumLines lt + { + /Red LastRed def + /Green LastGreen def + /Blue LastBlue def + /RedIncrement FirstRed LastRed sub NumLines MidLine sub 1 sub div def + /GreenIncrement FirstGreen LastGreen sub NumLines MidLine sub 1 sub div def + /BlueIncrement FirstBlue LastBlue sub NumLines MidLine sub 1 sub div def + NumLines MidLine sub { GradientLoop } repeat + } if +} def +/GradientLoop { + 0 CurrentY moveto + x 0 rlineto + 0 YSizePerLine rlineto + x neg 0 rlineto + closepath + Red Green Blue setrgbcolor fill + /CurrentY CurrentY YSizePerLine add def + /Blue Blue BlueIncrement add def + /Green Green GreenIncrement add def + /Red Red RedIncrement add def +} def + +end +% END pst-grad.pro +%%EndProcSet +%%BeginProcSet: pst-coil.pro +% PostScript prologue for pst-coil.tex. +% Version 97, 93/03/12. +% For copying restrictions, see pstricks.tex. +% +/tx@CoilDict 40 dict def tx@CoilDict begin +/CoilLoop { /t ED t sin AspectSin mul t 180 div AspectCos mul add t cos +lineto } def +/Coil { /Inc ED dup sin /AspectSin ED cos /AspectCos ED /ArmB ED /ArmA ED +/h ED /w ED /y1 ED /x1 ED /y0 ED /x0 ED x0 y0 translate y1 y0 sub x1 x0 +sub 2 copy Pyth /TotalLength ED Atan rotate /BeginAngle ArmA AspectCos +Div w h mul Div 360 mul def /EndAngle TotalLength ArmB sub AspectCos Div +w h mul Div 360 mul def 1 0 0 0 ArrowA ArmA 0 lineto /mtrx CM def w h +mul 2 Div w 2 Div scale BeginAngle Inc EndAngle { CoilLoop } for +EndAngle CoilLoop mtrx setmatrix TotalLength ArmB sub 0 lineto CP +TotalLength 0 ArrowB lineto } def +/AltCoil { /Inc ED dup sin /AspectSin ED cos /AspectCos ED /h ED /w ED +/EndAngle ED /BeginAngle ED /mtrx CM def w h mul 2 Div w 2 Div scale +BeginAngle sin AspectSin mul BeginAngle 180 div AspectCos mul add +BeginAngle cos /lineto load stopped { moveto } if BeginAngle Inc +EndAngle { CoilLoop } for EndAngle CoilLoop mtrx setmatrix } def +/ZigZag { /ArmB ED /ArmA ED 2 div /w ED w mul /h ED /y1 ED /x1 ED /y0 ED +/x0 ED x1 y1 translate y0 y1 sub x0 x1 sub 2 copy Pyth /TotalLength ED +Atan rotate TotalLength ArmA sub ArmB sub dup h div cvi /n ED n h mul +sub 2 div dup ArmA add /ArmA ED ArmB add /ArmB ED /x ArmB h 2 div add +def mark 0 0 ArmB 0 n { x w /w w neg def /x x h add def } repeat +TotalLength ArmA sub 0 TotalLength 0 } def +end +% END pst-coil.pro +%%EndProcSet +%%BeginProcSet: pst-text.pro +% PostScript header file pst-text.pro +% Version 97, 94/04/20 +% For copying restrictions, see pstricks.tex. + +/tx@TextPathDict 40 dict def +tx@TextPathDict begin + +% Syntax: PathPosition - +% Function: Searches for position of currentpath distance from +% beginning. Sets (X,Y)=position, and Angle=tangent. +/PathPosition +{ /targetdist exch def + /pathdist 0 def + /continue true def + /X { newx } def /Y { newy } def /Angle 0 def + gsave + flattenpath + { movetoproc } { linetoproc } { } { firstx firsty linetoproc } + /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if + grestore +} def + +/movetoproc { continue { @movetoproc } { pop pop } ifelse } def + +/@movetoproc +{ /newy exch def /newx exch def + /firstx newx def /firsty newy def +} def + +/linetoproc { continue { @linetoproc } { pop pop } ifelse } def + +/@linetoproc +{ + /oldx newx def /oldy newy def + /newy exch def /newx exch def + /dx newx oldx sub def + /dy newy oldy sub def + /dist dx dup mul dy dup mul add sqrt def + /pathdist pathdist dist add def + pathdist targetdist ge + { pathdist targetdist sub dist div dup + dy mul neg newy add /Y exch def + dx mul neg newx add /X exch def + /Angle dy dx atan def + /continue false def + } if +} def + +/TextPathShow +{ /String exch def + /CharCount 0 def + String length + { String CharCount 1 getinterval ShowChar + /CharCount CharCount 1 add def + } repeat +} def + +% Syntax: InitTextPath - +/InitTextPath +{ gsave + currentpoint /Y exch def /X exch def + exch X Hoffset sub sub mul + Voffset Hoffset sub add + neg X add /Hoffset exch def + /Voffset Y def + grestore +} def + +/Transform +{ PathPosition + dup + Angle cos mul Y add exch + Angle sin mul neg X add exch + translate + Angle rotate +} def + +/ShowChar +{ /Char exch def + gsave + Char end stringwidth + tx@TextPathDict begin + 2 div /Sy exch def 2 div /Sx exch def + currentpoint + Voffset sub Sy add exch + Hoffset sub Sx add + Transform + Sx neg Sy neg moveto + Char end tx@TextPathSavedShow + tx@TextPathDict begin + grestore + Sx 2 mul Sy 2 mul rmoveto +} def + +end +% END pst-text.pro +%%EndProcSet +%%BeginProcSet: pst-node.pro +% PostScript prologue for pst-node.tex. +% Version 97 patch 1, 97/05/09. +% For copying restrictions, see pstricks.tex. +% +/tx@NodeDict 400 dict def tx@NodeDict begin +tx@Dict begin /T /translate load def end +/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def } +if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end +grestore } def +/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def +} def +/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch +Sin mul } def } def +/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def +} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def } +ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx +dup Sin mul Cos Div } ifelse } def +/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub +/u ED /NodePos { GetRnodePos } def } def +/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup +Cos mul exch Sin mul } def +/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h +mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul +} ifelse } def +/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d +ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d +sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan +sin def /NodePos { TriNodePos } def } def +/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos +hh mul Atan dup cos ww mul exch sin hh mul } def +/GetCenter { begin X Y NodeMtrx transform CM itransform end } def +/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist +Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg } +{ 0 Dist Sin mul } ifelse } ifelse Do } def +/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform +exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx +dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop +XYPos } ifelse } def +/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1 +roll sin mul sub exch } ifelse } def +/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA +AddOffset yA add /yA1 ED xA add /xA1 ED } def +/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB +AddOffset yB add /yB1 ED xB add /xB1 ED } def +/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA +AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED +ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED +} ifelse } def +/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB +AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED +ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED +} ifelse } def +/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED +/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b +known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA +ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def +/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub +mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def +/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt +{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop +} repeat LPutLine cleartomark } def +/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED +/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 +mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx +sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul +add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add +y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul +mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def +/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def +} def +/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def +/NAngle yyB yyA sub xxB xxA sub Atan def } def +/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def +/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def +} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit +} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd +} def +/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def +} def +/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def +/NAngle yyB yyA sub xxB xxA sub Atan def } def +/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def +/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def +} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check { +exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark +VPosEnd } def +/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2 +roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx +def grestore exec /LPutVar /SaveLPutVar load def } def +/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def +GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos { +LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos { +LPutVar VPutLine } def LPutVar } def +/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop +end } def +/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub +Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan +/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1 +yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +ifelse } def +/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1 +roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA +GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict +begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ] +cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines +} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def +/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate +def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx +itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2 +yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 +yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def +/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def +/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2 +yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform +/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1 +tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ] +cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def +xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx +transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def } +{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB +0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict +begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx +def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if +xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end +/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +def +/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED +GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin +false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { +LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } +def +/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate +def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2 +mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3 +mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 +xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false +Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ] +cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { +VPutLines } def } def +% DG/SR modification begin - May 9, 1997 - Patch 1 +%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r +%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add +%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 +%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED +/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r +r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add +exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 +mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED +% DG/SR modification end +} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add +AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r +Div def /y ED /x ED } def +/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB +180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos +mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def +/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy +add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos +{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar +tx@Dict begin false Polygon end } def +/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan +def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2 +mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def +/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div +def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def +GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul +sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0 +eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn } +{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def +0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2 +roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0 +0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2 +roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat } +ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos +{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le { +r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC } +{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add +add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub +mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t +sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse +y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos { +LPutPos } def } def +/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB +sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1 +ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB +w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos +{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop } +{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine } +def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin +/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def +/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg +exch X sub neg exch moveto setmatrix CP grestore } def +/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0 +def } ifelse LPutCoor } def +/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg } +if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop } +ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED } +def +/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if +r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop } +ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED } +def +end +% END pst-node.pro +%%EndProcSet +%%BeginProcSet: special.pro +TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N +/vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N 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/gc xdf pop ct 39 +0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N +/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 +scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get +ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip +not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 +TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR +pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 +-1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg +TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg +sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr +0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add +2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp +{pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 +div VResolution 72 div neg scale 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FN(g)i FT(to)g(denote)g(the)378 5366 y(substitution)30 +b(whic)m(h)h(maps)g FP(x)1439 5380 y FO(i)1500 5366 y +FT(to)i FP(t)1646 5380 y FO(i)1706 5366 y FT(for)f FP(i)c +FN(2)g(f)p FT(1)p FP(;)15 b(:)g(:)g(:)j(;)d(n)p FN(g)32 +b FT(and)g FP(y)j FT(to)e(itself)e(for)g FP(y)42 b(=)-55 +b FN(2)27 b(f)p FP(x)3416 5380 y FL(1)3456 5366 y FP(;)15 +b(:)g(:)g(:)i(;)e(x)3710 5380 y FO(n)3757 5366 y FN(g)p +FT(.)378 5479 y(The)27 b(expression)e FP(A\022)30 b FT(where)d +FP(\022)i FT(is)d(a)h(substitution)e(represen)m(ts)i(the)h(result)e(of) +h(replacing)f(ev)m(ery)i(free)378 5591 y(v)-5 b(ariable)24 +b FP(x)i FT(in)e FP(A)i FT(with)f FP(\022)s FT(\()p FP(x)p +FT(\),)h(with)f(the)h(con)m(v)m(en)m(tion)g(that)h(w)m(e)f(alw)m(a)m +(ys)g(mak)m(e)h(a)f(suitable)e(renaming)378 5704 y(of)42 +b(v)-5 b(ariables)40 b(to)j(prev)m(en)m(t)f(free)g(v)-5 +b(ariables)40 b(in)h(the)h(range)g(of)g FP(\022)i FT(b)s(ecoming)c(b)s +(ound)g(in)h FP(A\022)s FT(.)74 b(W)-8 b(e)p eop +%%Page: 4 14 +4 13 bop 378 5 a FF(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)2031 +b FT(4)378 396 y(abbreviate)36 b(\()p FN(\001)15 b(\001)g(\001)i +FT(\()p FP(A\022)1134 410 y FL(1)1174 396 y FT(\))e FN(\001)g(\001)g +(\001)h FT(\))p FP(\022)1423 410 y FO(n)1507 396 y FT(b)m(y)37 +b FP(A\022)1751 410 y FL(1)1805 396 y FN(\001)15 b(\001)g(\001)h +FP(\022)1969 410 y FO(n)2016 396 y FT(.)60 b(W)-8 b(e)37 +b(write)f FP(A)p FT([)p FP(s)2638 410 y FL(1)2678 396 +y FP(;)15 b(:)g(:)g(:)h(;)f(s)2922 410 y FO(n)2969 396 +y FT(])37 b(to)h(indicate)d(that)j(the)378 509 y(expression)d +FP(A)h FT(con)m(tains)g(the)g(free)g(sub)s(expressions)d +FP(s)2292 523 y FL(1)2331 509 y FT(,)j FP(:)15 b(:)g(:)31 +b FT(,)38 b FP(s)2634 523 y FO(n)2680 509 y FT(,)g(and)d(denote)h(b)m +(y)g FP(A)p FT([)p FP(t)3481 523 y FL(1)3521 509 y FP(;)15 +b(:)g(:)g(:)h(;)f(t)3755 523 y FO(n)3803 509 y FT(])378 +622 y(the)35 b(result)f(of)h(replacing)f(these)h(particular)f(o)s +(ccurrences)h(of)g FP(s)2605 636 y FO(i)2668 622 y FT(in)e +FP(A)i FT(b)m(y)g 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b(di\016cult)m(y)e(in)h(formalising)f +(mathematics,)j(ho)m(w)m(ev)m(er,)i(turned)c(out)h(to)h(b)s(e)e(its)h +(prac-)378 3179 y(tical)33 b(infeasibilit)m(y)-8 b(,)31 +b(rather)i(than)g(the)h(imp)s(ossibilit)m(y)28 b(of)34 +b(formalising)d(all)h(mathematical)i(truths.)378 3292 +y(It)29 b(is)f(b)s(eliev)m(ed)g(b)m(y)h(most,)g(if)f(not)i(all,)e +(mathematicians)h(that)g(one)h(can)f(in)f(theory)h(formalise)f(most)378 +3405 y(of)33 b(presen)m(t)f(da)m(y)h(mathematics)g(using)e(a)h +(su\016cien)m(tly)f(strong)i(axiomatisation)f(suc)m(h)g(as)h(ZF)m(C)f +(set)378 3518 y(theory)-8 b(.)78 b(The)42 b(v)-5 b(alid)41 +b(statemen)m(ts)j(whic)m(h)e(cannot)h(b)s(e)f(deriv)m(ed)f(in)h(suc)m +(h)g(a)h(strong)g(system)f(are)378 3631 y(probably)34 +b(unin)m(teresting)h(statemen)m(ts)j(whic)m(h)d(w)m(ould)g(not)h(o)s +(ccur)g(in)f(the)i(mathematical)f(litera-)378 3744 y(ture.)k(Despite)27 +b(the)h(results)f(of)h(G\177)-45 b(odel)27 b(and)g(T)-8 +b(arski,)28 b(a)g(group)f(of)h(F)-8 b(renc)m(h)29 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y(to)h(de\014ne)e(a)i(set)g FA(X)k FB(=)25 b Fz(f)p +FA(x)p Fz(j)p FA(x)35 b(=)-47 b Fz(2)27 b FA(x)p Fz(g)p +FB(,)i(and)f(as)h(a)g(result)g(b)r(oth)f FA(X)33 b Fz(2)26 +b FA(X)32 b Fz(\))26 b FA(X)41 b(=)-46 b Fz(2)26 b FA(X)35 +b FB(and)28 b FA(X)41 b(=)-47 b Fz(2)27 b FA(X)32 b Fz(\))26 +b FA(X)32 b Fz(2)27 b FA(X)34 b FB(can)29 b(b)r(e)378 +5529 y(deriv)n(ed.)p eop +%%Page: 10 20 +10 19 bop 378 5 a FF(CHAPTER)30 b(2.)71 b(ON)30 b(THE)g(MECHANISA)-8 +b(TION)30 b(OF)h(MA)-8 b(THEMA)g(TICAL)31 b(PR)m(OOFS)165 +b FT(10)378 396 y FH(2.3)135 b(The)45 b(Mec)l(hanisation)h(of)f +(Mathematics)378 599 y FT(The)c(term)g(\\mec)m(hanisation)g(of)h +(mathematics")g(refers)f(to)h(the)f(use)g(of)h(mac)m(hines)e(to)i(p)s +(erform)378 712 y(mathematical)35 b(tasks.)53 b(This)33 +b(includes)f(for)i(instance)g(the)g(use)h(of)f(computers)g(to)i +(calculate)e(sp)s(e-)378 825 y(ci\014c)e(n)m(umeric)g(expressions,)h +(as)g(w)m(ell)f(as)h(in)e(manipulating)f(sym)m(b)s(olic)i(terms)h 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b(et)i(al.)49 b(1997\).)j(In)378 4312 y(this)27 +b(system,)j(resolution)d(and)h(natural)f(deduction)h(pro)s(ofs)f(are)i +(\014rst)f(abstracted)h(in)m(to)g FI(assertion-)378 4425 +y(level)44 b FT(pro)s(ofs)35 b(where)g(steps)g(are)g(justi\014ed)f(b)m +(y)h(high-lev)m(el)f(inferences)g(called)h(assertions)f(\(Huang)378 +4538 y(1994\).)40 b(These)24 b(usually)d(consist)i(of)g(the)h +(application)e(of)h(some)h(theorem)g(or)f(de\014nition.)36 +b(Assertion-)378 4650 y(lev)m(el)24 b(pro)s(ofs)g(are)h(then)f +(transformed)g(in)m(to)g(natural)g(language)h(pro)s(ofs)f(\(Huang)h +(and)f(Fiedler)f(1996;)378 4763 y(Huang)30 b(and)g(Fiedler)f(1997\).) +519 4876 y(Researc)m(h)h(in)e(this)g(area)i(suggests)g(that)g(readable) +f(pro)s(of)f(accoun)m(ts)j(need)e(to)g(b)s(e)g(presen)m(ted)g(at)378 +4989 y(quite)j(a)i(high)d(lev)m(el)i(of)g(abstraction)g(when)f +(compared)h(to)h(their)e(mac)m(hine)g(orien)m(ted)h(represen)m(ta-)378 +5102 y(tion.)40 b(The)29 b(dev)m(elopmen)m(t)g(of)h(readable)f(mac)m 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b(T)-8 b(actic-based)37 +b(pro)s(of)f(steps)g(can)g(b)s(e)f(annotated)i(with)e(explanations)g +(of)h(the)g(e\013ect)378 4214 y(of)e(the)g(application)e(of)i(eac)m(h)h +(tactic.)52 b(This)32 b(approac)m(h)j(can)f(b)s(e)f(e\013ectiv)m(e)i +(in)e(the)h(explanation)f(of)378 4327 y(ho)m(w)28 b(short)f(pro)s(ofs)g +(deriv)m(e)g(particular)f(goals.)40 b(It)28 b(ma)m(y)g(not)g(b)s(e)f +(applicable)f(to)i(long)g(tactic)h(pro)s(ofs,)378 4440 +y(though,)24 b(b)s(ecause)e(of)g(the)h(di\013erence)f(b)s(et)m(w)m(een) +g(the)h(t)m(yp)s(e)f(of)h(inferences)e(pro)m(vided)g(b)m(y)h(HOL)g +(tactics)378 4553 y(and)30 b(those)h(usually)d(found)h(in)g(informal)f +(mathematics.)378 4793 y FQ(Literate)34 b(Pro)s(of)i(Programming)378 +4965 y FT(Literate)j(programming)f(\(Kn)m(uth)g(1992\))j(in)m(v)m(olv)m +(es)e(the)g(use)f(of)h(a)g(programming)f(language)h(for)378 +5078 y(the)28 b(implemen)m(tation)e(of)h(algorithms)g(together)i(with)d +(a)i(t)m(yp)s(esetting)f(language)h(for)g(explanation.)378 +5191 y(T)-8 b(o)s(ols)26 b(based)g(on)g(Kn)m(uth's)f +FM(WEB)g FT(system)i(can)f(b)s(e)g(used)f(to)i(extract)h(a)e(readable)g +(t)m(yp)s(eset)h(do)s(cumen)m(t)378 5304 y(from)e(a)g(literate)g +(source)g(co)s(de.)39 b(The)25 b(tec)m(hniques)f(used)h(in)e(literate)i +(programming)f(can)i(b)s(e)e(used)g(in)378 5416 y(the)j(implemen)m +(tation)f(of)i(pro)s(of)e(scripts.)g(W)-8 b(ong)29 b(\(1994\))h(has)d +(implemen)m(ted)e(simple)h FM(WEB)g FT(to)s(ols)h(for)378 +5529 y(the)34 b(literate)g(dev)m(elopmen)m(t)g(of)g(HOL)g(pro)s(ofs,)g +(and)g(Bailey)f(\(1998\))k(used)c(literate)h(tec)m(hniques)f(in)378 +5642 y(the)c(formalisation)f(of)h(algebra)h(in)e(LEGO.)h(Simons)e +(\(1996\))32 b(dev)m(elop)s(ed)d FM(WEB)f FT(to)s(ols)h(for)g(the)h +(pro)s(of)p eop +%%Page: 25 35 +25 34 bop 378 5 a FF(CHAPTER)30 b(2.)71 b(ON)30 b(THE)g(MECHANISA)-8 +b(TION)30 b(OF)h(MA)-8 b(THEMA)g(TICAL)31 b(PR)m(OOFS)165 +b FT(25)378 396 y(language)31 b(Dev)-5 b(a)33 b(\(W)-8 +b(eb)s(er,)32 b(Simons,)d(and)i(Lafon)m(taine)g(1993\))i(and)e(for)f +(the)i(Isab)s(elle)d(system,)i(and)378 509 y(illustrates)h(their)i(use) +g(in)e(a)j(n)m(um)m(b)s(er)e(of)h(examples.)52 b(The)34 +b(pro)s(ofs)f(implemen)m(ted)g(in)g(his)g(systems)378 +622 y(are)g(presen)m(ted)g(in)f(a)h(hierarc)m(hical)e(format)i(and)g +(calculational)f(pro)s(ofs)g(are)h(used)f(in)f(the)j(b)s(ottom)378 +735 y(lev)m(el)h(justi\014cations.)54 b(He)36 b(also)g(implemen)m(ted)e +(a)h(n)m(um)m(b)s(er)f(of)i(Isab)s(elle)e(tactics)i(and)f(tacticals)h +(to)378 848 y(allo)m(w)30 b(calculational)f(st)m(yle)i(reasoning)f +(during)e(pro)s(of)h(dev)m(elopmen)m(t.)378 1088 y FQ(Appro)m(ximating) +35 b(the)f(Informal)g(Language)h(of)g(Mathematics)378 +1260 y FT(Apart)i(from)g(implemen)m(ting)f(to)s(ols)h(to)h(aid)e(the)i +(explanation)e(of)i(mec)m(hanised)f(argumen)m(ts,)j(one)378 +1373 y(can)33 b(in)m(v)m(estigate)i(ho)m(w)e(to)h(de\014ne)e(a)i +(formal)e(pro)s(of)h(language)g(in)f(order)h(to)h(appro)m(ximate)f +(that)h(of)378 1486 y(informal)f(mathematics.)55 b(In)34 +b(section)i(2.3.2)h(w)m(e)e(men)m(tioned)g(that)g(substan)m(tial)f +(e\013ort)i(has)f(b)s(een)378 1599 y(put)c(in)g(the)h(dev)m(elopmen)m +(t)h(of)f(the)g(Mizar)g(language)g(in)f(order)h(to)g(mak)m(e)h(it)f +(similar)d(to)k(that)g(used)378 1711 y(b)m(y)i(mathematicians.)56 +b(The)35 b(researc)m(h)h(presen)m(ted)f(in)f(this)g(thesis)h(deals)g +(with)f(issues)g(concerned)378 1824 y(with)25 b(minimising)d(the)27 +b(di\013erence)f(b)s(et)m(w)m(een)h(mec)m(hanised)f(and)g(informal)e +(pro)s(ofs,)i(and)g(the)h(simple)378 1937 y(pro)s(of)38 +b(language)h(SPL)f(describ)s(ed)f(in)g(c)m(hapter)i(4)g(is)f(based)h +(on)f(Mizar.)66 b(The)38 b(theorem)h(used)f(in)378 2050 +y(section)30 b(2.4.3)j(to)e(illustrate)d(a)j(n)m(um)m(b)s(er)e(of)i +(HOL)f(pro)s(ofs)f(can)i(b)s(e)f(deriv)m(ed)f(in)g(SPL)h(b)m(y:)473 +2238 y FM(theorem)46 b(example:)g("\(A)h FN(\))g FM(B\))g +FN(\))h FM(\(B)f FN(\))h FM(C\))f FN(\))h FM(\(A)f FN(\))h +FM(C\)")473 2351 y(proof)569 2464 y(assume)e(A_B:)h("A)g +FN(\))g FM(B")712 2577 y(and)g(B_C:)g("B)g FN(\))g FM(C")569 +2802 y(hence)f("A)h FN(\))h FM(C")f(by)g(A_B,)g(B_C;)473 +2915 y(qed;)378 3103 y FT(Although)41 b(all)f(the)i(constructs)g(in)f +(the)h(ab)s(o)m(v)m(e)g(formal)f(pro)s(of)g(ha)m(v)m(e)i(a)f(precise)g +(meaning,)i(it)d(is)378 3216 y(easier)k(to)h(follo)m(w)e(this)g(pro)s +(of)g(rather)h(than)g(those)g(giv)m(en)g(in)f(section)h(2.4.3.)87 +b(The)44 b(syn)m(tax)i(of)378 3329 y(Mizar)37 b(and)f(similar)e +(languages)j(is)e(expressiv)m(e)i(enough)f(to)h(allo)m(w)g(a)g(hierarc) +m(hical)e(presen)m(tation)378 3442 y(of)g(pro)s(ofs.)55 +b(The)35 b(Mizar)g(pro)s(ofs)g(of)g(a)h(n)m(um)m(b)s(er)e(of)i(prop)s +(erties)d(equiv)-5 b(alen)m(t)35 b(to)h(w)m(ell-foundedness)378 +3555 y(b)m(y)h(Rudnic)m(ki)e(and)h(T)-8 b(rybulec)36 +b(\(1997\))j(are)f(examples)e(of)h(non-trivial)e(mac)m(hine)i(c)m(hec)m +(k)m(ed)i(pro)s(ofs)378 3667 y(presen)m(ted)30 b(hierarc)m(hically)-8 +b(.)519 3780 y(The)36 b(Mizar)h(language)f(has)h(also)f(inspired)d +(other)k(w)m(ork.)59 b(F)-8 b(or)38 b(instance,)g(Harrison)d(\(1996b\)) +378 3893 y(dev)m(elop)s(ed)40 b(a)g(Mizar)g(mo)s(de)g(in)f(the)i(HOL)f +(system)g(whic)m(h)f(can)i(b)s(e)e(used)h(to)h(implemen)m(t)e(read-)378 +4006 y(able)e(pro)s(ofs)f(in)m(teractiv)m(ely)h(in)f(a)h(goal)h +(directed)f(fashion.)59 b(Syme)37 b(\(1997a\))j(dev)m(elop)s(ed)c(a)i +(Mizar)378 4119 y(lik)m(e)f(language,)k(DECLARE,)d(for)g(soft)m(w)m +(are)h(v)m(eri\014cation,)h(and)d(used)h(it)f(in)g(v)m(erifying)f(the)j +(t)m(yp)s(e)378 4232 y(correctness)31 b(of)g(Ja)m(v)-5 +b(a)31 b(\(Syme)f(1997b\))i(\(see)g(also)e(\(Syme)g(1998\)\).)519 +4345 y(The)35 b(Mizar)g(system)g(is)f(often)i(describ)s(ed)d(as)i(supp) +s(orting)e(a)j FI(de)-5 b(clar)g(ative)39 b(pr)-5 b(o)g(of)39 +b(style)j FT(as)36 b(op-)378 4458 y(p)s(osed)k(to)h(the)f(more)h +FI(pr)-5 b(o)g(c)g(e)g(dur)g(al)53 b FT(ones)41 b(often)g(supp)s(orted) +d(b)m(y)j(other)f(systems.)71 b(Although)40 b(the)378 +4571 y(di\013erence)31 b(b)s(et)m(w)m(een)h(a)g(declarativ)m(e)g(and)f +(pro)s(cedural)f(st)m(yle)h(is)g(somewhat)h(v)-5 b(ague,)33 +b(a)e(declarativ)m(e)378 4684 y(approac)m(h)26 b(puts)f(more)h +(emphasis)e(on)i FI(what)36 b FT(is)24 b(required,)h(rather)h(than)g +(on)f FI(how)37 b FT(to)26 b(obtain)g(it.)38 b(The)378 +4797 y(statemen)m(ts)31 b(deriv)m(ed)d(b)m(y)h(Mizar)g(pro)s(of)f +(steps)h(are)g(stated)h(explicitly)-8 b(.)38 b(F)-8 b(urthermore,)30 +b(pro)s(of)e(steps)378 4909 y(are)j(justi\014ed)e(simply)g(b)m(y)i(a)g +(list)f(of)h(premises,)f(rather)h(than)f(b)m(y)h(a)h(sequence)f(of)g +(inferences.)42 b(This)378 5022 y(lac)m(k)25 b(of)h(pro)s(cedural)d +(information)g(increases)i(the)g(readabilit)m(y)e(of)j(the)f(pro)s +(ofs,)g(but)g(it)f(implies)e(that)378 5135 y(more)k(w)m(ork)g(is)f +(required)f(b)m(y)i(the)g(pro)s(of)f(c)m(hec)m(k)m(er)j(to)f(v)-5 +b(alidate)25 b(Mizar)h(scripts.)38 b(One)26 b(m)m(ust)g(ho)m(w)m(ev)m +(er)378 5248 y(b)s(e)k(careful)g(to)i(c)m(ho)s(ose)f(the)g(righ)m(t)g +(lev)m(el)f(of)h(automation)g(supp)s(orted)e(b)m(y)i(the)g(pro)s(of)f +(c)m(hec)m(k)m(er.)43 b(T)-8 b(o)s(o)378 5361 y(m)m(uc)m(h)28 +b(automation)h(results)e(in)g(pro)s(ofs)g(that)i(are)f(not)h(detailed)e +(enough)h(to)h(b)s(e)f(follo)m(w)m(ed)g(easily)f(or)378 +5474 y(mac)m(hine)33 b(c)m(hec)m(k)m(ed)i(e\016cien)m(tly)-8 +b(.)50 b(T)-8 b(o)s(o)34 b(little)e(automation)i(results)e(in)g(to)s(o) +i(detailed)f(pro)s(ofs)f(whic)m(h)378 5587 y(are)g(generally)f(tedious) +g(to)h(implemen)m(t)e(and)h(hard)g(to)h(follo)m(w.)43 +b(This)30 b(giv)m(es)i(rise)f(to)h(the)g(notion)f(of)378 +5700 y(an)g(ob)m(vious)g(inference)g(\(Da)m(vis)h(1981;)i(Rudnic)m(ki) +29 b(1987\))34 b(|)d(one)h(whic)m(h)e(is)h(simple)e(enough)i(to)i(b)s +(e)p eop +%%Page: 26 36 +26 35 bop 378 5 a FF(CHAPTER)30 b(2.)71 b(ON)30 b(THE)g(MECHANISA)-8 +b(TION)30 b(OF)h(MA)-8 b(THEMA)g(TICAL)31 b(PR)m(OOFS)165 +b FT(26)378 396 y(easily)36 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STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-0 /N@T-0-0-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 1558 997 a 1358 1274 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 48.2307 24.11534 3.30017 +} false /N@T-0-0-0-0 16 {InitRnode } NewNode end end + 1358 +1274 a FN(h)p FP(v)1437 1288 y FL(1)1477 1274 y FN(i)26 +b(6)p FT(=)f FN(h)p FP(f)10 b FN(i)1558 1246 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-0-0 /N@T-0-0-0-0 InitNC { NCLine } if end gsave 0.8 +SLW 0. setgray 0 setlinecap stroke grestore grestore end + 1558 1246 +a 1523 1407 a + tx@Dict begin tx@NodeDict begin {6.3875 0.9125 8.5167 4.25835 3.30017 +} false /N@T-0-0-0-0-0 16 {InitRnode } NewNode end end + 1523 1407 a FN(\002)1558 1379 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-0-0-0 /N@T-0-0-0-0-0 InitNC { NCLine } if end grestore +end + 1558 1379 +a 2069 776 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 58.70663 29.35332 +3.30017 } false /N@T-0-1 16 {InitRnode } NewNode end end + 2069 776 a FP(p)p FT(\()p FP(e;)15 b(v)2276 +790 y FL(1)2317 776 y FT(\))25 b(=)g FP(v)2517 790 y +FL(1)2313 748 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0 /N@T-0-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2313 748 a 1885 1025 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 34.48392 17.24196 +3.30017 } false /N@T-0-1-0 16 {InitRnode } NewNode end end + 1885 1025 a FN(:)p +FP(G)p FT(\()p FP(v)2097 1039 y FL(2)2136 1025 y FT(\))2028 +997 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1 /N@T-0-1-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 2028 997 a 1936 1274 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 22.22472 11.11235 +3.30017 } false /N@T-0-1-0-0 16 {InitRnode } NewNode end end + 1936 1274 a FP(G)p FT(\()p +FP(e)p FT(\))2028 1246 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-0 /N@T-0-1-0-0 InitNC { NCLine } if end gsave 0.8 +SLW 0. setgray 0 setlinecap stroke grestore grestore end + 2028 1246 a 1834 1523 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 46.78969 23.39484 +3.30017 } false /N@T-0-1-0-0-0 16 {InitRnode } NewNode end end + 1834 +1523 a FN(h)p FP(v)1913 1537 y FL(2)1953 1523 y FN(i)g(6)p +FT(=)g FN(h)p FP(e)p FN(i)2028 1496 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-0-0 /N@T-0-1-0-0-0 InitNC { NCLine } if end gsave +0.8 SLW 0. setgray 0 setlinecap stroke grestore grestore end + 2028 1496 a 1993 +1656 a + tx@Dict begin tx@NodeDict begin {6.3875 0.9125 8.5167 4.25835 3.30017 +} false /N@T-0-1-0-0-0-0 16 {InitRnode } NewNode end end + 1993 1656 a FN(\002)2028 1628 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-0-0-0 /N@T-0-1-0-0-0-0 InitNC { NCLine } if end +grestore end + 2028 1628 a 2348 +1025 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 60.14764 30.07382 +3.30017 } false /N@T-0-1-1 16 {InitRnode } NewNode end end + 2348 1025 a FP(p)p FT(\()p FP(v)2473 1039 y FL(2)2513 +1025 y FP(;)15 b(f)10 b FT(\))25 b(=)g FP(v)2808 1039 +y FL(2)2598 997 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1 /N@T-0-1-1 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 2598 997 a 2489 1274 a + tx@Dict begin tx@NodeDict begin {7.60416 2.12917 26.23813 13.11906 +3.30017 } false /N@T-0-1-1-0 16 {InitRnode } NewNode end end + 2489 1274 a +FP(e)h FN(6)p FT(=)f FP(f)2598 1246 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-1 /N@T-0-1-1-0 InitNC { NCLine } if end gsave 0.8 +SLW 0. setgray 0 setlinecap stroke grestore grestore end + 2598 1246 a 2563 +1407 a + tx@Dict begin tx@NodeDict begin {6.3875 0.9125 8.5167 4.25835 3.30017 +} false /N@T-0-1-1-0-0 16 {InitRnode } NewNode end end + 2563 1407 a FN(\002)2598 1379 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 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+b FT(are)31 b(form)m(ulae)f(and)f FP(X)38 b FT(and)30 +b FP(Y)50 b FT(are)31 b(structured)e(expressions.)453 +b Ff(\003)378 3811 y FQ(De\014nition)35 b(6.4)h(\(Justi\014cation)f(b)m +(y)g(Structured)g(Expressions\))46 b FT(F)-8 b(or)49 +b(ev)m(ery)h(form)m(ula)d FP(C)378 3924 y FT(and)30 b(structured)f +(expression)g FP(X)7 b FT(,)31 b(w)m(e)g(sa)m(y)g(that)g +FP(X)38 b FT(justi\014es)29 b FP(C)36 b FT(if)30 b(and)g(only)f(if)g +FP(X)k Ff( )25 b FP(C)7 b FT(.)288 b Ff(\003)378 4135 +y FQ(Example)34 b(6.1)46 b FT(As)36 b(an)h(example,)h(w)m(e)f(sho)m(w)g +(that)g(the)g(follo)m(wing)e(conclusion)h(is)f(justi\014ed)g(cor-)378 +4248 y(rectly:)473 4435 y FM(")p FN(9)p FM(c.)15 b(C\(a,c\)")46 +b(by)h(")p FN(8)p FM(x,y,z.)14 b(A\(x,y\))46 b FN(\))h +FM(B\(y,z\))f FN(\))i FM(C\(x,z\)")1237 4548 y(on)f(")p +FN(8)p FM(x.)p FN(9)p FM(c.)14 b(B\(x,c\)")46 b(and)g("A\(a,b\)";)378 +4735 y FT(First)30 b(of)g(all,)g(it)g(is)f(the)i(case)g(that)1057 +4939 y Ft(\()p Fu(8)p Fv(x)p Fw(.)p Fu(9)p Fv(c)p 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Fv(a;)14 b(b)p Ft(\))q FT(;)514 5705 y FN(\017)46 +b Ft(\()p Fu(9)p Fv(c)p Fw(.)15 b Fv(B)t Ft(\()p Fv(b;)f(c)p +Ft(\)\))44 b Fu(^)g Ft(\()p Fv(A)p Ft(\()p Fv(a;)14 b(b)p +Ft(\)\))26 b Ff(\032)1627 5672 y FK(\003)1691 5705 y +Fu(9)p Fv(c)p Fw(.)15 b Ft(\()p Fv(A)p Ft(\()p Fv(a;)f(b)p +Ft(\))19 b Fu(^)62 b Fv(B)t Ft(\()p Fv(b;)14 b(c)p Ft(\)\))p +FT(.)p eop +%%Page: 110 120 +110 119 bop 378 5 a FF(CHAPTER)30 b(6.)71 b(STR)m(UCTURED)30 +b(STRAIGHTF)m(OR)-10 b(W)g(ARD)31 b(JUSTIFICA)-8 b(TIONS)243 +b FT(110)378 396 y(It)30 b(is)g(also)g(the)h(case)g(that)853 +601 y Fu(8)p Fv(x;)14 b(y)s(;)g(z)t Fw(.)f Fv(A)p Ft(\()p +Fv(x;)h(y)s Ft(\))44 b Fu(\))g Fv(B)t Ft(\()p Fv(y)s(;)14 +b(z)t Ft(\))43 b Fu(\))g Fv(C)6 b Ft(\()p Fv(x;)14 b(z)t +Ft(\))1681 739 y Ff( )56 b Ft(\()p Fu(9)p Fv(c)p Fw(.)14 +b Fv(A)p Ft(\()p Fv(a;)g(b)p Ft(\))44 b Fu(^)g Fv(B)t +Ft(\()p Fv(b;)14 b(c)p Ft(\)\))44 b Fu(\))g Ft(\()p Fu(9)p +Fv(c)p Fw(.)14 b Fv(C)6 b Ft(\()p Fv(a;)14 b(c)p Ft(\)\))433 +b FT(\(2\))378 943 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y(given)g("x:'a";)689 2146 y(there)g(is)h(some)e("y:'a")h(such)g +(that)907 2246 y(Rxy:)g("R)h(x)g(y")g(by)g(R_ex;)776 +2345 y(so)g(Ryx:)f("R)h(y)g(x")g(by)g(R_symm)e(on)h(Rxy;)689 +2445 y(hence)g("R)h(x)g(x")f(by)h(R_trans)e(on)i(Rxy)f(and)g(Ryx;)602 +2644 y(qed;)602 2843 y(theorem)f(R_equiv:)f("Equivalence)f(R")864 +2943 y()f(by)43 b(R_refl)e(and)h(R_symm)f(and)h(R_trans;) +515 3142 y(end;)809 3537 y FT(Figure)30 b(14:)42 b(An)30 +b(SPL)f(Pro)s(of)h(Script)f(using)g(Structured)g(Justi\014cations.)p +3829 3707 V 378 3710 3453 4 v 378 4067 a(Chapter)e(9)h(illustrates)e +(the)h(mec)m(hanisation)h(of)f(a)h(n)m(um)m(b)s(er)f(of)g(results)g(in) +f(group)h(theory)h(in)e(whic)m(h)378 4180 y(most)31 b(of)f(the)h +(results)e(are)i(justi\014ed)d(b)m(y)i(means)h(of)f(structured)g +(justi\014cations.)519 4293 y(Figure)j(14)i(giv)m(es)f(an)g(example)g +(of)g(a)g(simple)e(SPL)h(script)g(whic)m(h)f(uses)i(structured)f +(justi\014ca-)378 4406 y(tions.)63 b(The)37 b(same)i(results)d(giv)m +(en)i(in)f(this)g(example)h(are)g(deriv)m(ed)f(using)f(unstructured)g +(justi\014-)378 4519 y(cations)e(in)f(the)h(pro)s(of)g(script)f(in)g +(\014gure)g(5,)j(page)f(56.)53 b(Since)33 b(structured)g +(justi\014cations)g(con)m(tain)378 4632 y(more)27 b(information)f(whic) +m(h)g(is)g(relev)-5 b(an)m(t)28 b(to)g(the)f(understanding)e(of)i(the)h +(pro)s(of,)f(they)g(are)h(easier)f(to)378 4745 y(follo)m(w)33 +b(than)h(unstructured)e(ones.)53 b(Since)33 b(this)g(information)f(can) +j(also)f(b)s(e)f(used)h(to)h(restrict)f(the)378 4858 +y(searc)m(h)f(space)g(during)d(pro)s(of)h(c)m(hec)m(king,)j(they)e(can) +h(also)f(b)s(e)g(mac)m(hine)g(c)m(hec)m(k)m(ed)i(more)e(e\016cien)m +(tly)-8 b(.)378 4971 y(F)g(urthermore,)30 b(the)f(implemen)m(tation)f +(of)h(structured)f(justi\014cations)g(during)f(pro)s(of)h(dev)m +(elopmen)m(t)378 5084 y(do)s(es)e(not)g(need)h(m)m(uc)m(h)f(more)g +(e\013ort)h(than)f(the)h(implemen)m(tation)e(of)h(unstructured)f(ones)h +(since)g(the)378 5197 y(detailed)c(inferences)f(whic)m(h)h(w)m(ould)f +(mak)m(e)j(the)f(justi\014cation)e(tedious)h(to)h(implemen)m(t)f(are)h +(omitted.)519 5309 y(One)29 b(problem)e(with)g(the)i(use)g(of)g +(structured)f(justi\014cations)f(is)h(that)i(there)f(is)f(no)g(straigh) +m(tfor-)378 5422 y(w)m(ard)d(w)m(a)m(y)i(of)f(using)e(the)h(last)h +(deriv)m(ed)f(result)f(implicitly)e(in)i(the)i(curren)m(t)f +(justi\014cation.)38 b(In)25 b(Mizar)378 5535 y(one)34 +b(can)h(use)e(the)i Fw(then)d FT(construct)i(to)h(sho)m(w)f(that)h(the) +f(previous)f(result)g(is)g(used)g(automatically)378 5648 +y(in)c(the)i(curren)m(t)f(justi\014cation.)39 b(F)-8 +b(or)31 b(example,)g(one)f(can)h(implemen)m(t)e(the)h(pro)s(of:)p +eop +%%Page: 116 126 +116 125 bop 378 5 a FF(CHAPTER)30 b(6.)71 b(STR)m(UCTURED)30 +b(STRAIGHTF)m(OR)-10 b(W)g(ARD)31 b(JUSTIFICA)-8 b(TIONS)243 +b FT(116)807 396 y FM("R)48 b(x)f(y")g(by)g(R_ex;)569 +509 y(then)f("R)i(y)f(x")g(by)g(R_symm;)569 622 y(then)f("R)i(x)f(x")g +(by)g(R_trans,)f(Rxy;)378 798 y FT(in)23 b(whic)m(h)h(the)h(result)f +Fw("R)43 b(x)g(y")24 b FT(is)g(used)g(implicitly)d(as)k(a)g(premise)f +(in)f(the)i(justi\014cation)f(of)h Fw("R)42 b(y)i(x")o +FT(,)378 911 y(and)31 b(similarly)-8 b(,)29 b Fw("R)43 +b(y)g(x")31 b FT(is)g(used)g(automatically)g(in)g(the)h +(justi\014cation)e(of)i Fw("R)43 b(x)g(x")o FT(.)i(In)31 +b(general,)378 1024 y(suc)m(h)d(a)g(mec)m(hanism)f(cannot)h(b)s(e)g +(used)f(with)f(structured)h(justi\014cations)g(b)s(ecause)h(one)g(is)f +(required)378 1137 y(to)f(giv)m(e)f(some)h(information)d(on)i(ho)m(w)g +(the)g(premises)f(are)i(b)s(eing)d(used.)38 b(In)25 b(the)g(SPL)f +(language)i(used)378 1250 y(in)h(the)h(case)i(study)d(describ)s(ed)f +(in)h(c)m(hapter)i(9,)g(an)f(exclamation)h(mark)f(\()p +Fw(!)p FT(\))g(is)f(used)h(to)h(denote)g(the)378 1363 +y(last)24 b(deriv)m(ed)g(result,)g(and)g(statemen)m(ts)i(lik)m(e)e +Fw(then)n FT(,)i Fw(hence)n FT(,)g Fw(therefore)21 b +FT(and)i Fw(so)h FT(are)h(ignored)e(during)378 1476 y(pro)s(of)30 +b(c)m(hec)m(king.)41 b(The)30 b(ab)s(o)m(v)m(e)i(pro)s(of)d(fragmen)m +(t)i(can)g(b)s(e)f(implemen)m(ted)f(as)h(follo)m(ws:)807 +1651 y FM("R)48 b(x)f(y")g(by)g(R_ex;)569 1764 y(then)f("R)i(y)f(x")g +(by)g(R_symm)g(on)g(!;)521 1877 y(hence)f("R)i(x)f(x")g(by)g(R_trans)f +(on)h(Rxy)g(and)g(!;)378 2053 y FT(Although)38 b(structured)f +(justi\014cations)h(can)g(b)s(e)h(more)f(readable)g(than)h +(unstructured)e(ones,)k(the)378 2166 y(inabilit)m(y)33 +b(to)k(use)f(the)g(last)g(deriv)m(ed)g(result)f(automatically)h(ma)m(y) +h(reduce)e(their)h(readabilit)m(y)-8 b(.)57 b(In)378 +2279 y(\014gures)27 b(15)i(and)e(16)i(w)m(e)g(giv)m(e)f(t)m(w)m(o)i +(SPL)d(pro)s(ofs)g(of)h(the)g Fw(nonobv)e FT(theorem.)40 +b(The)28 b(pro)s(of)f(in)g(\014gure)g(15)378 2392 y(uses)44 +b(unstructured)e(justi\014cations)h(in)g(whic)m(h)f(the)j +Fw(then)d FT(and)i Fw(hence)e FT(statemen)m(ts)k(denote)f(the)378 +2505 y(fact)33 b(that)g(the)f(previously)e(deriv)m(ed)i(result)f(is)g +(used)g(implicitly)e(in)i(the)h(curren)m(t)g(one.)47 +b(The)32 b(pro)s(of)378 2618 y(in)38 b(\014gure)i(16)g(uses)f +(structured)g(justi\014cations)f(in)h(whic)m(h)f(an)i(exclamation)g +(mark)g(denotes)g(the)378 2731 y(previously)21 b(deriv)m(ed)h(result.) +37 b(F)-8 b(or)24 b(completeness,)h(\014gure)e(17)h(sho)m(ws)f(a)h(pro) +s(of)e(of)h(the)h(same)f(theorem)378 2843 y(using)34 +b(structured)g(justi\014cations)g(without)g Fw(!)o FT(.)56 +b(It)35 b(can)g(b)s(e)g(noted)g(that)h(most)g(of)f(the)g(use)g(of)h +(the)378 2956 y(exclamation)31 b(mark)f(in)f(the)h(pro)s(of)g(in)f +(\014gure)h(16)h(is)e(of)i(the)f(form:)473 3132 y FP(:)15 +b(:)g(:)49 b FM(by)e FP(:)15 b(:)g(:)49 b FM(on)e(!)378 +3308 y FT(This)31 b(is)g(also)i(observ)m(ed)g(in)e(the)i(pro)s(ofs)f +(implemen)m(ted)f(in)g(the)i(mec)m(hanisation)f(of)h(group)f(theory)-8 +b(,)378 3421 y(and)30 b(therefore)h(one)f(can)h(de\014ne)f(the)g +Fw(then)f FT(construct)h(suc)m(h)h(that:)473 3597 y FM(then)47 +b FP(C)54 b FM(by)47 b FI(exp)378 3773 y FT(is)29 b(an)i(abbreviation)e +(of)712 3949 y FP(C)54 b FM(by)47 b(\()p FI(exp)6 b FM(\))48 +b(on)f(!)519 4124 y FT(W)-8 b(e)23 b(will)d(see)j(in)e(section)h(8.2.4) +i(that)f(the)f(problem)f(of)h(c)m(hec)m(king)h(the)g(v)-5 +b(alidit)m(y)20 b(of)i(the)h(structured)378 4237 y(justi\014cations)h +(de\014ned)g(in)h(this)f(c)m(hapter)j(is)d(undecidable.)37 +b(In)25 b(particular,)g(c)m(hec)m(king)i(whether)e(t)m(w)m(o)378 +4350 y(form)m(ulae)36 b(are)g(implicitly)c(deriv)-5 b(able)34 +b(from)i(eac)m(h)h(other)f(\(i.e.,)16 b(whether)35 b +FP(A)g Ff(\032)3143 4317 y FK(\003)3217 4350 y FP(B)40 +b FT(for)c(arbitrary)378 4463 y(form)m(ulae)c FP(A)g +FT(and)g FP(B)5 b FT(\))32 b(is)f(undecidable.)43 b(This)31 +b(suggests)i(that)f(the)h(implicit)c(deriv)-5 b(abilit)m(y)29 +b(de\014ned)378 4576 y(in)e(section)i(6.4.1)i(is)c(to)s(o)j(strong)e +(and)h(therefore)g(cannot)g(in)e(general)i(b)s(e)f(considered)f(to)j +(represen)m(t)378 4689 y(trivial)h(deriv)-5 b(ations.)49 +b(Most)34 b(of)g(the)f(structured)g(justi\014cations)f(that)i(w)m(ere)g +(implemen)m(ted)e(in)g(the)378 4802 y(case)41 b(study)d(\(c)m(hapter)j +(9\))f(are)g(rather)g(easy)g(to)g(mac)m(hine)f(c)m(hec)m(k,)44 +b(and)39 b(probably)f(only)g(a)i(small)378 4915 y(\(p)s(ossibly)h +(decidable\))i(subset)g(of)g(the)h(implicit)d(deriv)-5 +b(ations)42 b(are)i(actually)f(used)g(in)f(practice.)378 +5028 y(Section)f(8.5)i(illustrates)c(ho)m(w)j(the)f(searc)m(h)i(space)f +(considered)e(during)f(pro)s(of)i(c)m(hec)m(king)h(of)g(the)378 +5141 y(scripts)33 b(implemen)m(ted)f(in)h(the)h(case)h(study)e(is)g +(restricted)h(to)h(a)f(\014nite)f(one.)52 b(As)34 b(a)g(result,)g(only) +f(a)378 5253 y(decidable)g(subset)i(of)g(the)g(explicit)e(deriv)-5 +b(ations)34 b(discussed)f(in)g(this)h(c)m(hapter)i(could)e(b)s(e)g(c)m +(hec)m(k)m(ed)378 5366 y(e\013ectiv)m(ely)-8 b(.)58 b(Alternativ)m(e)36 +b(de\014nitions)e(of)i(implicit)d(and)i(explicit)f(inferences)h(in)f +(the)j(pure)d(\014rst-)378 5479 y(order)c(logic)g(ma)m(y)h(b)s(e)f +(considered)f(in)g(future.)519 5592 y(One)45 b(of)g(the)g(motiv)-5 +b(ations)45 b(for)g(the)g(de\014nition)e(and)h(use)h(of)g(structured)g +(justi\014cations)e(in)378 5705 y(a)e(declarativ)m(e)h(language)f(is)f +(to)i(explore)e(whether)g(simple)f(results)h(can)h(b)s(e)f(deriv)m(ed)g +(b)m(y)h(a)h(less)p eop +%%Page: 117 127 +117 126 bop 378 5 a FF(CHAPTER)30 b(6.)71 b(STR)m(UCTURED)30 +b(STRAIGHTF)m(OR)-10 b(W)g(ARD)31 b(JUSTIFICA)-8 b(TIONS)243 +b FT(117)p 378 1264 3453 4 v 376 4754 4 3491 v 515 1500 +a Fw(assume)41 b(sr:)h(")p Fu(8)p Fw(x)g(y.)h(P\(x,y\))e +Fu(_)j Fw(Q\(x,y\)")820 1599 y(sq:)e(")p Fu(8)p Fw(x)g(y.)h(Q\(x,y\))e +Fu(\))j Fw(Q\(y,x\)")820 1699 y(tp:)e(")p Fu(8)p Fw(x)g(y)h(z.)g +(P\(x,y\))e Fu(^)j Fw(P\(y,z\))d Fu(\))j Fw(P\(x,z\)")820 +1798 y(tq:)e(")p Fu(8)p Fw(x)g(y)h(z.)g(Q\(x,y\))e Fu(^)j +Fw(Q\(y,z\))d Fu(\))j Fw(Q\(x,z\)";)515 1998 y(theorem)d(nonobv:)f("\() +p Fu(8)p Fw(x)i(y.)h(P\(x,y\)\))d Fu(_)k Fw(\()p Fu(8)p +Fw(x)e(y.)h(Q\(x,y\)\)")515 2097 y(proof)602 2197 y(given)f("a:'a")f +(and)h("b:'a";)602 2297 y(assume)f(1:)i(")p Fu(:)p Fw(P\(a,b\)";)689 +2396 y(then)f(2:)h("Q\(b,a\)")d(by)j(sr,)f(sq;)602 2595 +y(given)g("x:'a")f(and)h("y:'a";)602 2695 y(auxstep:)e(")p +Fu(8)p Fw(z.)i(Q\(a,z\)")602 2795 y(proof)689 2894 y(given)g("z:'a";) +689 2994 y(")p Fu(:)p Fw(P\(z,b\))f Fu(\))j Fw(Q\(a,z\)")689 +3094 y(proof)776 3193 y(assume)e(")p Fu(:)p Fw(P\(z,b\)";)864 +3293 y(then)g("Q\(z,a\)")e(by)j(sr,)f(tq,)g(2;)820 3392 +y(hence)g("Q\(a,z\)")e(by)j(sq;)689 3492 y(end;)689 3592 +y(hence)f("Q\(a,z\)")e(by)j(sr,)f(tp,)h(1;)602 3691 y(end;)602 +3891 y("Q\(x,a\)")d(by)j(auxstep,)d(sq;)602 3990 y(hence)i("Q\(x,y\)")e +(by)j(auxstep,)d(tq)515 4189 y(qed;)681 4585 y FT(Figure)30 +b(15:)42 b(An)30 b(SPL)f(Pro)s(of)h(of)h FM(nonobv)d +FT(using)h(Unstructured)g(Justi\014cations.)p 3829 4754 +V 378 4758 3453 4 v eop +%%Page: 118 128 +118 127 bop 378 5 a FF(CHAPTER)30 b(6.)71 b(STR)m(UCTURED)30 +b(STRAIGHTF)m(OR)-10 b(W)g(ARD)31 b(JUSTIFICA)-8 b(TIONS)243 +b FT(118)p 378 1264 3453 4 v 376 4754 4 3491 v 515 1500 +a Fw(assume)41 b(sr:)h(")p Fu(8)p Fw(x)g(y.)h(P\(x,y\))e +Fu(_)j Fw(Q\(x,y\)")820 1599 y(sq:)e(")p Fu(8)p Fw(x)g(y.)h(Q\(x,y\))e +Fu(\))j Fw(Q\(y,x\)")820 1699 y(tp:)e(")p Fu(8)p Fw(x)g(y)h(z.)g +(P\(x,y\))e Fu(^)j Fw(P\(y,z\))d Fu(\))j Fw(P\(x,z\)")820 +1798 y(tq:)e(")p Fu(8)p Fw(x)g(y)h(z.)g(Q\(x,y\))e Fu(^)j +Fw(Q\(y,z\))d Fu(\))j Fw(Q\(x,z\)";)515 1998 y(theorem)d(nonobv:)f("\() +p Fu(8)p Fw(x)i(y.)h(P\(x,y\)\))d Fu(_)k Fw(\()p Fu(8)p +Fw(x)e(y.)h(Q\(x,y\)\)")515 2097 y(proof)602 2197 y(given)f("a:'a")f +(and)h("b:'a";)602 2297 y(assume)f(1:)i(")p Fu(:)p Fw(P\(a,b\)";)689 +2396 y(then)f(2:)h("Q\(b,a\)")d(by)j(sr)g(then)f(sq)g(on)h(!;)602 +2595 y(given)f("x:'a")f(and)h("y:'a";)602 2695 y(auxstep:)e(")p +Fu(8)p Fw(z.)i(Q\(a,z\)")602 2795 y(proof)689 2894 y(given)g("z:'a";) +689 2994 y(")p Fu(:)p Fw(P\(z,b\))f Fu(\))j Fw(Q\(a,z\)")689 +3094 y(proof)776 3193 y(assume)e(")p Fu(:)p Fw(P\(z,b\)";)864 +3293 y(then)g("Q\(z,a\)")e(by)j(sr)f(then)g(\(tq)h(on)f(2\))h(on)g(!;) +820 3392 y(hence)f("Q\(a,z\)")e(by)j(sq)f(on)h(!;)689 +3492 y(end;)689 3592 y(hence)f("Q\(a,z\)")e(by)j(\(sr)f(and)h(!\))f(on) +h(\(tp)f(on)h(1\);)602 3691 y(end;)602 3891 y("Q\(x,a\)")d(by)j(sq)g +(on)g(auxstep;)602 3990 y(hence)f("Q\(x,y\)")e(by)j(tq)f(on)h(auxstep)e +(and)h(!;)515 4189 y(qed;)733 4585 y FT(Figure)30 b(16:)42 +b(An)30 b(SPL)f(Pro)s(of)h(of)h FM(nonobv)d FT(using)h(Structured)g +(Justi\014cations.)p 3829 4754 V 378 4758 3453 4 v eop +%%Page: 119 129 +119 128 bop 378 5 a FF(CHAPTER)30 b(6.)71 b(STR)m(UCTURED)30 +b(STRAIGHTF)m(OR)-10 b(W)g(ARD)31 b(JUSTIFICA)-8 b(TIONS)243 +b FT(119)p 378 1264 3453 4 v 376 4754 4 3491 v 515 1500 +a Fw(assume)41 b(sr:)h(")p Fu(8)p Fw(x)g(y.)h(P\(x,y\))e +Fu(_)j Fw(Q\(x,y\)")820 1599 y(sq:)e(")p Fu(8)p Fw(x)g(y.)h(Q\(x,y\))e +Fu(\))j Fw(Q\(y,x\)")820 1699 y(tp:)e(")p Fu(8)p Fw(x)g(y)h(z.)g +(P\(x,y\))e Fu(^)j Fw(P\(y,z\))d Fu(\))j Fw(P\(x,z\)")820 +1798 y(tq:)e(")p Fu(8)p Fw(x)g(y)h(z.)g(Q\(x,y\))e Fu(^)j +Fw(Q\(y,z\))d Fu(\))j Fw(Q\(x,z\)";)515 1998 y(theorem)d(nonobv:)f("\() +p Fu(8)p Fw(x)i(y.)h(P\(x,y\)\))d Fu(_)k Fw(\()p Fu(8)p +Fw(x)e(y.)h(Q\(x,y\)\)")515 2097 y(proof)602 2197 y(given)f("a:'a")f +(and)h("b:'a";)602 2297 y(assume)f(1:)i(")p Fu(:)p Fw(P\(a,b\)";)689 +2396 y(then)f(2:)h("Q\(b,a\)")d(by)j(sr)g(then)f(sq)g(on)h(1;)602 +2595 y(given)f("x:'a")f(and)h("y:'a";)602 2695 y(auxstep:)e(")p +Fu(8)p Fw(z.)i(Q\(a,z\)")602 2795 y(proof)689 2894 y(given)g("z:'a";) +689 2994 y(auxstep_1:)e(")p Fu(:)p Fw(P\(z,b\))h Fu(\))i +Fw(Q\(a,z\)")689 3094 y(proof)776 3193 y(assume)f(auxstep_1_1:)c(")p +Fu(:)p Fw(P\(z,b\)";)864 3293 y(then)k(auxstep_1_2:)c("Q\(z,a\)")i(by)j +(sr)g(then)f(\(tq)g(on)h(2\))g(on)g(auxstep_1_1;)820 +3392 y(hence)f("Q\(a,z\)")e(by)j(sq)f(on)h(auxstep_1_2;)689 +3492 y(end;)689 3592 y(hence)f("Q\(a,z\)")e(by)j(\(sr)f(and)h +(auxstep_1\))c(on)k(\(tp)f(on)h(1\);)602 3691 y(end;)602 +3891 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NewNode end end + 2001 3155 a FN(:)p FP(Y)2218 3155 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 17.23105 8.61552 3.74124 +} false /N@b2 16 {InitRnode } NewNode end end + 2218 3155 a FN(:)p +FP(X)2027 3268 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 9.93103 4.96552 3.74124 +} false /N@b1 16 {InitRnode } NewNode end end + 2027 3268 a FP(X)2361 3083 y Fx(\025)2409 +3211 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@a1 /N@a2 InitNC { /AngleA 45. def /AngleB 135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2409 3211 a 2409 3211 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@b1 /N@b2 InitNC { /AngleA -15. def /AngleB -105. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore 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{InitRnode } NewNode end end + 1605 +614 a FN(:)p FP(A)1734 581 y FO(i)1852 614 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118 +} false /N@y 16 {InitRnode } NewNode end end + 1852 614 +a FP(A)1920 581 y FO(j)1597 727 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 20.65147 10.32573 4.7646 +} false /N@z 16 {InitRnode } NewNode end end + 1597 727 a FN(:)p FP(A)1726 +694 y FO(k)1956 543 y Fx(\025)197 b(\024)2249 614 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118 +} false /N@a 16 {InitRnode } NewNode end end + 2249 +614 a FP(A)2317 581 y FO(j)2437 614 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 20.65147 10.32573 4.7646 +} false /N@b 16 {InitRnode } NewNode end end + 2437 614 a FN(:)p +FP(A)2566 581 y FO(k)2255 727 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 11.34447 5.67223 4.7646 +} false /N@c 16 {InitRnode } NewNode end end + 2255 727 a FP(A)2323 +694 y FO(l)2609 543 y Fx(\025)2657 671 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@x /N@y InitNC { /AngleA 45. def /AngleB 135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2657 671 a 2657 +671 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@z /N@y InitNC { /AngleA -45. def /AngleB -100. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2657 671 a 2657 671 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@a /N@b InitNC { /AngleA 45. def /AngleB 135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2657 671 a 2657 671 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@c /N@b InitNC { /AngleA -45. def /AngleB -100. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2657 +671 a 378 892 a FT(and)28 b(furthermore)h(\()p FP(X)1166 +906 y FL(1)1206 892 y FP(;)15 b(X)1321 906 y FL(2)1361 +892 y FT(\))29 b(satis\014es)g(the)g(other)h(conditions)d(\(i.e.,)16 +b(1)30 b(and)e(3\))i(of)f(de\014nition)e(7.24,)378 1005 +y(and)37 b(is)g(th)m(us)g(a)i FN(K)q FT(-in)m(terp)s(olan)m(t)e(for)h +(\()p FP(S)1765 1019 y FL(1)1804 1005 y FP(;)15 b(S)1900 +1019 y FL(2)1940 1005 y FT(\).)63 b(Ho)m(w)m(ev)m(er,)42 +b(the)c(set)h FP(S)2834 1019 y FL(1)2898 1005 y FN([)25 +b FP(S)3040 1019 y FL(2)3117 1005 y FT(is)37 b FN(K)q +FT(-consisten)m(t)i(as)378 1118 y(illustrated)28 b(b)m(y)i(the)h(follo) +m(wing)e(matrix.)1875 1242 y Fx(\024)1931 1314 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 18.90648 9.45323 4.63118 +} false /N@a 16 {InitRnode } NewNode end end + 1931 +1314 a FN(:)p FP(A)2060 1281 y FO(i)2178 1314 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118 +} false /N@b 16 {InitRnode } NewNode end end + 2178 +1314 a FP(A)2246 1281 y FO(j)1923 1427 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 20.65147 10.32573 4.7646 +} false /N@c 16 {InitRnode } NewNode end end + 1923 1427 a +FN(:)p FP(A)2052 1394 y FO(k)2183 1427 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 11.34447 5.67223 4.7646 +} false /N@d 16 {InitRnode } NewNode end end + 2183 1427 a +FP(A)2251 1394 y FO(l)2283 1242 y Fx(\025)2331 1371 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@a /N@b InitNC { /AngleA 45. def /AngleB 135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + +2331 1371 a 2331 1371 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@c /N@d InitNC { /AngleA -45. def /AngleB -135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2331 1371 a 2331 1371 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@c /N@b InitNC { /AngleA 10. def /AngleB -135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2331 +1371 a 378 1640 a FT(Note)j(that)f(the)f(path)g FN(f:)p +FP(A)1337 1607 y FO(i)1366 1640 y FP(;)15 b(A)1474 1607 +y FO(l)1500 1640 y FN(g)31 b FT(is)e(not)i FN(K)q FT(-inconsisten)m(t)f +(as)h FP(i)26 b FN(6\030)2668 1654 y FK(K)2751 1640 y +FP(l)r FT(.)952 b Ff(\003)519 1795 y FT(In)39 b(order)g(that)i(the)f +(set)g FP(S)1469 1809 y FL(1)1534 1795 y FN([)27 b FP(S)1678 +1809 y FL(2)1756 1795 y FT(is)39 b FN(K)q 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b(TIONS)650 b FT(158)378 +396 y(\(implemen)m(tation-based\))29 b(b)s(ounds)e(on)j(an)m(y)f(pro)s +(of)g(searc)m(h)h(required)e(to)i(c)m(hec)m(k)h(structured)e(justi-)378 +509 y(\014cations.)42 b(This)29 b(issue)g(is)h(discussed)f(in)g +(section)i(8.5,)i(whic)m(h)c(describ)s(es)g(the)i(mec)m(hanism)f(used)g +(in)378 622 y(c)m(hec)m(king)25 b(the)g(structured)f(justi\014cations)f +(implemen)m(ted)g(in)g(the)i(mec)m(hanisation)f(of)h(group)f(theory)378 +735 y(illustrated)k(in)h(c)m(hapter)i(9.)378 1018 y FH(8.3)135 +b(F)-11 b(rom)45 b(Structured)f(Justi\014cations)i(to)f(Coloured)g +(Problems)378 1224 y FG(8.3.1)112 b(A)52 b(Restricted)f(Pro)s(of)g +(Searc)m(h)i(for)f(Chec)m(king)g(Structured)g(Justi\014ca-)720 +1340 y(tions)378 1512 y FT(The)33 b(previous)e(section)j(illustrated)d +(ho)m(w)i(implicit)d(inferences)j(are)g(equiv)-5 b(alen)m(t)33 +b(to)h(the)g(inconsis-)378 1625 y(tency)27 b(of)g(coloured)f +(\014rst-order)g(problems.)38 b(In)26 b(this)g(section,)i(w)m(e)f(sho)m 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{InitRnode } NewNode end end + 2263 941 a FP(S)g FN([)2486 941 +y + tx@Dict begin tx@NodeDict begin {9.26236 2.73749 44.3564 22.17819 +3.26244 } false /N@and2 16 {InitRnode } NewNode end end + 2486 941 a FN(f)2531 941 y + tx@Dict begin tx@NodeDict begin {6.83331 0.0 9.06943 4.53471 3.41666 +} false /N@Xand 16 {InitRnode } NewNode end end + 2531 941 a Fv(X)2606 908 +y FO(i)2635 941 y FP(;)2705 941 y + tx@Dict begin tx@NodeDict begin {6.83331 0.0 8.02779 4.01389 3.41666 +} false /N@Yand 16 {InitRnode } NewNode end end + 2705 941 a Fv(Y)2772 +908 y FO(j)2808 941 y FN(g)2854 941 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 2.0 2.0 +0 0 /N@on1 /N@on2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 2854 941 a 2854 +941 a + tx@Dict begin tx@NodeDict begin /t 0.7 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 2854 941 a 2848 968 a FL(c)2854 941 y + tx@Dict begin PutEnd end + 2854 941 +a 2854 941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 2.0 2.0 +0 0 /N@and1 /N@and2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 2854 941 a 2854 941 a + tx@Dict begin tx@NodeDict begin /t 0.7 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 2854 941 a 2848 968 +a FL(c)2854 941 y + tx@Dict begin PutEnd end + 2854 941 a 2854 941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 2.0 +2.0 0 0 /N@Son1 /N@on InitNC { /AngleA 45. def /AngleB 135. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2854 941 a 2854 +941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.0 +1.0 0 0 /N@Son2 /N@Xon InitNC { /AngleA 45. def /AngleB 135. def +0.67 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap +stroke grestore grestore end + 2854 941 a 2854 941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@Xon /N@Yon InitNC { /AngleA 65. def /AngleB 115. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2854 941 a 2854 941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 2.0 +2.0 0 0 /N@Sand1 /N@and InitNC { /AngleA 45. def /AngleB 135. def +0.67 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap +stroke grestore grestore end + 2854 +941 a 2854 941 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto 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NewNode end end + 2785 1086 a FT(\()p FP(S)2876 1100 y FL(6)2916 +1086 y FP(;)15 b FN(K)3025 1100 y FL(6)3065 1086 y FT(\))3100 +1086 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@LT1 /N@LB1 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 1086 a 3094 1113 +a FL(c)3100 1086 y + tx@Dict begin PutEnd end + 3100 1086 a 3100 1086 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@RT1 /N@RB1 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 +a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 1086 a 3094 1113 a FL(c)3100 1086 +y + tx@Dict begin PutEnd end + 3100 1086 a 3100 1086 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@LT2 /N@LM2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 +1086 a 3094 1113 a FL(c)3100 1086 y + tx@Dict begin PutEnd end + 3100 1086 a 3100 +1086 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@RT2 /N@RM2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 1086 a 3094 1113 +a FL(c)3100 1086 y + tx@Dict begin PutEnd end + 3100 1086 a 3100 1086 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@LM2 /N@LB2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 +a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 1086 a 3094 1113 a FL(c)3100 1086 +y + tx@Dict begin PutEnd end + 3100 1086 a 3100 1086 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@RM2 /N@RB2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray +0 setlinecap stroke grestore grestore end + 3100 1086 a 3100 1086 a + tx@Dict begin tx@NodeDict begin /t 0.67 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 0.71027 def +/r 0.71027 def /h 0.14427 def /d 3.30017 def /flag true def HPutAdjust +LPutCoor end PutBegin end + 3100 +1086 a 3094 1113 a FL(c)3100 1086 y + tx@Dict begin PutEnd end + 3100 1086 a 869 1491 +a FQ(Fig.)j(19.)41 b FT(The)30 b(Relation)g FN(!)1880 +1505 y FL(c)1945 1491 y FT(is)g(Strongly)f(Con\015uen)m(t)h(Mo)s(dulo)f +Fl(u)3224 1505 y FL(rc)3287 1491 y FT(.)378 1884 y FI(then)46 +b(either)g FT(\()p FP(S)956 1898 y FL(2)995 1884 y FP(;)15 +b FN(K)1104 1898 y FL(2)1145 1884 y FT(\))48 b Fl(u)1299 +1898 y FL(rc)1411 1884 y FT(\()p FP(S)1502 1898 y FL(4)1541 +1884 y 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moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@C InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2596 2364 a 2596 2364 +a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@B InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2596 2364 a 378 2568 a FT(and)30 b(is)f FP(k)g FN($)c +FP(i)g FN($)g FP(j)5 b FT(-inconsisten)m(t)31 b(if)e(and)h(only)f(if)h +(the)g(sets)1421 2773 y FN(f)1466 2773 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 11.60646 5.80322 4.90492 +} false /N@A 16 {InitRnode } NewNode end end + 1466 2773 a +FP(A)1534 2735 y FO(i)1563 2773 y FP(;)106 b FN(:)1755 +2773 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 1755 2773 a FP(C)1827 2735 y FO(j)1863 2773 y +FP(;)1994 2773 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 10.81165 5.40582 5.03836 +} false /N@I 16 {InitRnode } NewNode end end + 1994 2773 a FP(I)2041 2735 y FO(k)2084 +2773 y FN(g)182 b(f)2356 2773 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 13.99399 6.997 5.03836 +} false /N@B 16 {InitRnode } NewNode end end + 2356 2773 a FP(B)2430 +2735 y FO(k)2472 2773 y FP(;)107 b FN(:)2665 2773 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 9.06665 4.53333 4.90492 +} false /N@nI 16 {InitRnode } NewNode end end + 2665 +2773 a FP(I)2712 2735 y FO(i)2740 2773 y FN(g)2785 2773 +y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@C InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2785 2773 a 2785 2773 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@I InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2785 2773 a 2785 2773 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@nI InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2785 +2773 a 378 2977 a FT(are)36 b(for)g(some)g(sen)m(tence)h +FP(I)7 b FT(.)57 b(The)35 b(curv)m(es)h(connecting)g(the)g(coloured)f +(sen)m(tences)i(corresp)s(ond)d(to)378 3090 y(the)h(w)m(a)m(y)g(the)g +(colours)f(in)f(the)i(ab)s(o)m(v)m(e)h(sets)f(relate)g(with)e(eac)m(h)j +(other)f(according)g(to)g(the)g(relation)378 3203 y FP(k)28 +b FN($)e FP(i)f FN($)g FP(j)5 b FT(.)519 3316 y(F)-8 +b(or)32 b(the)f(general)h(case)g(where)f FP(P)39 b FT(=)27 +b Fv(X)49 b Fw(on)43 b Fv(Y)50 b FT(for)31 b(some)h(structured)e +(expressions)g FP(X)38 b FT(and)31 b FP(Y)20 b FT(,)378 +3428 y(w)m(e)31 b(ha)m(v)m(e)1157 3541 y(\()p Fv(C)50 +b Fw(by)43 b Fv(X)50 b Fw(on)42 b Fv(Y)19 b FT(\))26 +b FN(!)1900 3504 y FK(\003)1900 3564 y FL(c)1964 3541 +y FT(\()p FN(f)p FP(X)2126 3504 y FO(i)2156 3541 y FP(;)15 +b(Y)2269 3504 y FO(k)2312 3541 y FP(;)g FN(:)p FP(C)2485 +3504 y FO(j)2521 3541 y FN(g)p FP(;)g(k)29 b FN($)c FP(i)h +FN($)f FP(j)5 b FT(\))378 3708 y(and)30 b(although)g(w)m(e)g(can)h +(partition)e FN(f)p FP(X)1740 3675 y FO(i)1769 3708 y +FP(;)15 b(Y)1883 3675 y FO(k)1925 3708 y FP(;)g FN(:)p +FP(C)2098 3675 y FO(j)2134 3708 y FN(g)31 b FT(in)m(to)1602 +3912 y(\()p FN(f)1682 3912 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.32498 6.66249 4.90492 +} false /N@X 16 {InitRnode } NewNode end end + 1682 3912 a FP(X)1764 3875 +y FO(i)1794 3912 y FP(;)106 b FN(:)1986 3912 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 1986 3912 +a FP(C)2058 3875 y FO(j)2094 3912 y FN(g)p FP(;)198 b +FN(f)2407 3912 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 13.92937 6.96468 5.03836 +} false /N@Y 16 {InitRnode } NewNode end end + 2407 3912 a FP(Y)2480 3875 y FO(k)2523 +3912 y FN(g)p FT(\))378 4117 y(w)m(e)33 b(cannot)f(use)g(theorem)h(7.8) +g(to)g(sho)m(w)f(that)h(it)f(is)f FP(k)h FN($)c FP(i)h +FN($)f FP(j)5 b FT(-inconsisten)m(t)32 b(if)g(and)f(only)h(if)f(the)378 +4230 y(sets)1414 4343 y FN(f)1459 4343 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.32498 6.66249 4.90492 +} false /N@X 16 {InitRnode } NewNode end end + 1459 4343 a +FP(X)1541 4305 y FO(i)1570 4343 y FP(;)106 b FN(:)1762 +4343 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 1762 4343 a FP(C)1834 4305 y FO(j)1870 4343 y +FP(;)2001 4343 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 10.81165 5.40582 5.03836 +} false /N@I 16 {InitRnode } NewNode end end + 2001 4343 a FP(I)2048 4305 y FO(k)2091 +4343 y FN(g)182 b(f)2363 4343 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 13.92937 6.96468 5.03836 +} false /N@Y 16 {InitRnode } NewNode end end + 2363 4343 a FP(Y)2437 +4305 y FO(k)2479 4343 y FP(;)107 b FN(:)2672 4343 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 9.06665 4.53333 4.90492 +} false /N@nI 16 {InitRnode } NewNode end end + 2672 +4343 a FP(I)2719 4305 y FO(i)2747 4343 y FN(g)378 4509 +y FT(are)36 b(for)g(some)g FP(I)7 b FT(,)38 b(since)d(the)h(structured) +f(expressions)f FP(X)43 b FT(and)36 b FP(Y)55 b FT(ma)m(y)37 +b(not)f(b)s(e)f(\(unstructured\))378 4622 y(sen)m(tences.)519 +4735 y(W)-8 b(e)32 b(can)e(apply)f(the)i(relation)e FN(!)1673 +4749 y FL(c)1739 4735 y FT(on)h(the)h(coloured)f(structured)f(problem) +1561 4939 y(\()p FN(f)p FP(X)1723 4902 y FO(i)1752 4939 +y FP(;)15 b(Y)1865 4902 y FO(k)1908 4939 y FP(;)g FN(:)p +FP(C)2081 4902 y FO(j)2117 4939 y FN(g)p FP(;)g(k)29 +b FN($)d FP(i)f FN($)g FP(j)5 b FT(\))378 5144 y(as)31 +b(follo)m(ws)967 5348 y(\()p FN(f)p FP(X)1129 5310 y +FO(i)1159 5348 y FP(;)15 b(Y)1272 5310 y FO(k)1315 5348 +y FP(;)g FN(:)p FP(C)1488 5310 y FO(j)1524 5348 y FN(g)p +FP(;)g(k)29 b FN($)c FP(i)h FN($)f FP(j)5 b FT(\))26 +b FN(!)2168 5310 y FK(\003)2168 5370 y FL(c)2233 5348 +y FT(\()p FP(S)2324 5362 y FO(X)2412 5348 y FN([)19 b(f)p +FP(Y)2611 5310 y FO(k)2654 5348 y FP(;)c FN(:)p FP(C)2827 +5310 y FO(j)2863 5348 y FN(g)p FP(;)g FN(K)3017 5362 +y FO(X)5 b(Y)3142 5348 y FT(\))2077 5486 y FN(!)2168 +5448 y FK(\003)2168 5508 y FL(c)2233 5486 y FT(\()p FP(S)2324 +5500 y FO(X)2412 5486 y FN([)19 b FP(S)2548 5500 y FO(Y)2629 +5486 y FN([)h(f:)p FP(C)2888 5448 y FO(j)2924 5486 y +FN(g)p FP(;)15 b FN(K)3079 5448 y FK(0)3078 5508 y FO(X)5 +b(Y)3203 5486 y FT(\))p eop +%%Page: 171 181 +171 180 bop 378 5 a FF(CHAPTER)30 b(8.)122 b(CHECKING)29 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b(f:)p FP(I)2714 1719 y FO(i)2743 +1756 y FN(g)378 1961 y FT(are)31 b(for)f(some)h(sen)m(tence)g +FP(I)7 b FT(.)519 2074 y(An)21 b(imp)s(ortan)m(t)f(result)g(whic)m(h)g +(is)g(required)f(to)j(p)s(erform)e(this)g(step)h(is)f(giv)m(en)h(in)f +(prop)s(osition)e(8.11)378 2186 y(\(and)38 b(illustrated)e(in)g +(example)i(8.4\))i(and)d(allo)m(ws)h(us)f(to)i(sho)m(w)f(that)h(the)f +(subsets)g FP(S)3395 2200 y FO(X)3462 2186 y FT(,)i FP(S)3583 +2200 y FO(Y)3681 2186 y FT(and)378 2299 y FN(f:)p FP(C)556 +2266 y FO(j)592 2299 y FN(g)32 b FT(are)h(connected)f(\(b)m(y)h(the)f +(relation)f FN(\031)1976 2319 y FK(K)2030 2296 y FD(0)2030 +2342 y Fy(X)t(Y)2140 2299 y FT(,)i(see)f(de\014nition)e(7.9)j(on)f +(page)h(125\))g(with)e(eac)m(h)378 2429 y(other)k(according)g(to)g +FN(K)1219 2396 y FK(0)1218 2456 y FO(X)5 b(Y)1377 2429 +y FT(in)33 b(the)i(same)g(w)m(a)m(y)h(that)f(the)g(sen)m(tences)h +FP(A)2901 2396 y FO(i)2929 2429 y FT(,)g FP(B)3064 2396 +y FO(k)3141 2429 y FT(and)e FN(:)p FP(C)3455 2396 y FO(j)3525 +2429 y FT(connect)378 2542 y(with)29 b(eac)m(h)j(other)e(according)h +(to)g FP(k)d FN($)d FP(i)h FN($)f FP(j)5 b FT(.)41 b(More)31 +b(precisely)-8 b(,)980 2746 y FP(S)1036 2760 y FO(X)1129 +2746 y FN(\031)1200 2765 y FK(K)1254 2743 y FD(0)1254 +2789 y Fy(X)t(Y)1390 2746 y FP(S)1446 2760 y FO(Y)1688 +2746 y FP(S)1744 2760 y FO(X)1836 2746 y FN(\031)1907 +2765 y FK(K)1961 2743 y FD(0)1961 2789 y Fy(X)t(Y)2097 +2746 y FN(f:)p FP(C)2275 2709 y FO(j)2311 2746 y FN(g)182 +b FP(S)2594 2760 y FO(Y)2680 2746 y FN(6\031)2751 2765 +y FK(K)2805 2743 y FD(0)2805 2789 y Fy(X)t(Y)2941 2746 +y FN(f:)p FP(C)3119 2709 y FO(j)3155 2746 y FN(g)p FP(;)378 +2950 y FT(or)30 b(as)h(sho)m(wn)f(in)f(the)h(follo)m(wing)f(diagram.) +1631 3155 y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374 +} false /N@SX 16 {InitRnode } NewNode end end + 1631 3155 a FP(S)1687 3169 y FO(X)1835 3155 +y FN([)1977 3155 y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374 +} false /N@SY 16 {InitRnode } NewNode end end + 1977 3155 a FP(S)2033 3169 y FO(Y)2174 +3155 y FN([)80 b(f:)2421 3155 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 2421 3155 a FP(C)2493 +3117 y FO(j)2529 3155 y FN(g)2574 3155 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@SX /N@C InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2574 3155 a +2574 3155 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@SX /N@SY InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2574 3155 a 378 3359 a FT(F)-8 b(urthermore,)514 +3545 y FN(\017)46 b FT(the)36 b(sets)g FP(S)1006 3559 +y FO(X)1109 3545 y FT(and)f FP(S)1347 3559 y FO(Y)1443 +3545 y FT(ha)m(v)m(e)i(no)f(colour)f(in)g(common,)i(and)f(no)f(one)h +(whic)m(h)f(is)g(equal)g(to)h FP(j)605 3658 y FT(\(whic)m(h)30 +b(is)f(the)i(only)e(colour)h(in)f FN(f:)1833 3658 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118 +} false /N@C 16 {InitRnode } NewNode end end + 1833 +3658 a FP(C)1905 3625 y FO(j)1941 3658 y FN(g)p FT(\);)514 +3845 y FN(\017)46 b FT(the)30 b(colours)f(in)f FP(S)1230 +3859 y FO(X)1326 3845 y FT(that)i(relate)g(with)e(the)h(colours)g(in)f +FP(S)2604 3859 y FO(Y)2694 3845 y FT(relate)h(also)h(with)e(the)h +(colour)g FP(j)5 b FT(,)605 3997 y(that)31 b(is)f(\()p +FP(S)985 4011 y FO(X)1077 3931 y FK(K)1131 3908 y FD(0)1131 +3954 y Fy(X)t(Y)1112 3997 y FN(!)60 b FP(S)1319 4011 +y FO(Y)1380 3997 y FT(\))25 b(=)g(\()p FP(S)1627 4011 +y FO(X)1720 3931 y FK(K)1774 3908 y FD(0)1774 3954 y +Fy(X)t(Y)1755 3997 y FN(!)60 b(f:)p FP(C)2084 3964 y +FO(j)2120 3997 y FN(g)p FT(\);)514 4229 y FN(\017)46 +b FT(all)29 b(the)i(colours)f(in)f FP(S)1359 4243 y FO(X)1451 +4163 y FK(K)1505 4139 y FD(0)1505 4185 y Fy(X)t(Y)1486 +4229 y FN(!)60 b FP(S)1693 4243 y FO(Y)1784 4229 y FT(relate)31 +b(with)e(all)g(the)i(colours)e(in)h FP(S)2998 4243 y +FO(Y)3083 4163 y FK(K)3137 4139 y FD(0)3137 4185 y Fy(X)t(Y)3118 +4229 y FN(!)60 b FP(S)3325 4243 y FO(X)3392 4229 y FT(.)378 +4415 y(These)25 b(prop)s(erties)f(allo)m(w)i(us)f(to)h(use)f(the)h +(sets)g FP(S)2041 4429 y FO(X)2108 4415 y FT(,)h FP(S)2216 +4429 y FO(Y)2302 4415 y FT(and)e FN(f:)p FP(C)2652 4382 +y FO(j)2688 4415 y FN(g)h FT(in)f(a)h(similar)d(fashion)h(that)i(w)m(e) +378 4528 y(use)33 b(the)h(sen)m(tences)g FP(A)1166 4495 +y FO(i)1195 4528 y FT(,)g FP(B)1328 4495 y FO(k)1404 +4528 y FT(and)e FN(:)p FP(C)1716 4495 y FO(j)1785 4528 +y FT(in)h(the)g(pro)s(of)g(of)h(prop)s(osition)d(8.8,)k(and)e(are)h +(generalised)378 4641 y(in)m(to)24 b(the)g(de\014nition)d(of)j(w)m +(ell-coloured)e(partitions)h(giv)m(en)g(in)g(section)g(8.4.2)j(b)s(elo) +m(w.)38 b(This)22 b(notion)h(of)378 4754 y(w)m(ell-coloured)28 +b(partitions)f(is)g(also)i(a)g(generalisation)f(of)h(the)g(notion)f(of) +h(w)m(ell-coloured)e(partitions)378 4867 y(\(for)k(partitions)f(of)h(t) +m(w)m(o)h(elemen)m(ts\))g(giv)m(en)f(in)e(de\014nition)g(7.27.)44 +b(F)-8 b(or)32 b(completeness,)f(w)m(e)h(no)m(w)f(can)378 +4980 y(partition)e FP(S)813 4994 y FO(X)900 4980 y FN([)20 +b FP(S)1037 4994 y FO(Y)1118 4980 y FN([)g(f:)p FP(C)1377 +4947 y FO(j)1413 4980 y FN(g)30 b FT(in)m(to)1550 5184 +y(\()p FN(f)1630 5184 y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374 +} false /N@A 16 {InitRnode } NewNode end end + 1630 5184 a FP(S)1686 5198 y +FO(X)1835 5184 y FN([)81 b(f:)2083 5184 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 2083 5184 a +FP(C)2155 5146 y FO(j)2191 5184 y FN(g)p FP(;)2458 5184 +y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374 +} false /N@B 16 {InitRnode } NewNode end end + 2458 5184 a FP(S)2514 5198 y FO(Y)2575 5184 y FN(g)p +FT(\))2655 5184 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@C InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2655 5184 a 2655 5184 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@B InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2655 5184 a +378 5388 a FT(whic)m(h)33 b(is)g(w)m(ell-coloured)g(with)g(resp)s(ect)h +(to)h FN(K)2000 5355 y FK(0)1999 5415 y FO(X)5 b(Y)2158 +5388 y FT(and)33 b(therefore,)j(b)m(y)e(theorem)h(7.8,)h(it)e(is)f +FN(K)3674 5355 y FK(0)3673 5415 y FO(X)5 b(Y)3798 5388 +y FT(-)378 5501 y(inconsisten)m(t)29 b(if)h(and)g(only)f(if)g(the)i +(sets)1316 5705 y FN(f)1361 5705 y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374 +} false /N@A 16 {InitRnode } NewNode end end + 1361 5705 a FP(S)1417 +5719 y FO(X)1566 5705 y FN([)80 b(f:)1813 5705 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492 +} false /N@C 16 {InitRnode } NewNode end end + 1813 +5705 a FP(C)1885 5668 y FO(j)1921 5705 y FN(g)p FP(;)2098 +5705 y + tx@Dict begin tx@NodeDict begin {10.07672 0.0 10.81165 5.40582 5.03836 +} false /N@I 16 {InitRnode } NewNode end end + 2098 5705 a FP(I)2145 5668 y FO(k)2188 5705 y +FN(g)182 b(f)2460 5705 y + tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374 +} false /N@B 16 {InitRnode } NewNode end end + 2460 5705 a FP(S)2516 5719 y +FO(Y)2577 5705 y FP(;)106 b FN(:)2769 5705 y + tx@Dict begin tx@NodeDict begin {9.80986 0.0 9.06665 4.53333 4.90492 +} false /N@nI 16 {InitRnode } NewNode end end + 2769 5705 +a FP(I)2816 5668 y FO(i)2844 5705 y FN(g)2889 5705 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@C InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + +2889 5705 a 2889 5705 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@I InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2889 5705 a 2889 5705 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@nI InitNC { /AngleA 30. def /AngleB 150. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2889 +5705 a eop +%%Page: 172 182 +172 181 bop 378 5 a FF(CHAPTER)30 b(8.)122 b(CHECKING)29 +b(STR)m(UCTURED)h(JUSTIFICA)-8 b(TIONS)650 b FT(172)378 +396 y(are)32 b(for)f(some)h(sen)m(tence)h FP(I)7 b FT(.)45 +b(This)29 b(sequence)j(of)g(steps)g(is)e(rep)s(eated)i(in)e(more)i +(detail)f(in)f(the)i(pro)s(of)378 509 y(of)d(theorem)g(8.7.)42 +b(In)28 b(the)h(follo)m(wing)e(section)i(w)m(e)g(de\014ne)f(the)h +(notion)g(of)g(w)m(ell-coloured)e(partitions)378 622 +y(of)i(more)h(than)e(t)m(w)m(o)j(elemen)m(ts,)f(whic)m(h,)e(as)i +(suggested)g(in)d(this)h(section,)i(pla)m(ys)f(an)g(imp)s(ortan)m(t)f +(role)378 735 y(in)h(the)i(pro)s(of)e(of)i(theorem)g(8.7.)378 +979 y FG(8.4.2)112 b(On)38 b(W)-9 b(ell-Coloured)35 b(P)m(artitions)378 +1150 y FT(In)d(this)h(section)g(w)m(e)h(generalise)f(the)g(notion)g(of) 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y(and)39 b(let)g(the)h(set)g FP(S)46 b FT(=)40 b +FP(S)1290 410 y FL(1)1356 396 y FN([)26 b FP(S)1499 410 +y FL(2)1564 396 y FN([)g FP(S)1707 410 y FL(3)1746 396 +y FT(.)68 b(The)40 b(w)m(a)m(y)g(the)g(colours)f(in)f +FP(S)45 b FT(relate)40 b(with)e(eac)m(h)j(other)378 509 +y(according)30 b(to)h FN(K)h FT(can)f(b)s(e)f(illustrated)e(b)m(y)i +(the)h(follo)m(wing)d(diagram:)1147 851 y FN(f)1223 851 +y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 11.60646 5.80322 4.63118 +} false /N@A 16 {InitRnode } NewNode end end + 1223 851 a FP(A)1291 818 y FO(i)1319 851 y FP(;)1450 +851 y + tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.25363 6.62682 4.63118 +} false /N@B 16 {InitRnode } NewNode end end + 1450 851 a FP(B)1524 818 y FO(j)1560 851 y FP(;)1783 +851 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.74835 6.87418 4.7646 +} false /N@C 16 {InitRnode } NewNode end end + 1783 851 a FP(C)1855 818 y FO(k)1897 851 y FP(;)2028 +851 y + tx@Dict begin tx@NodeDict begin {9.52922 0.0 12.50182 6.2509 4.7646 +} false /N@D 16 {InitRnode } NewNode end end + 2028 851 a FP(D)2106 818 y FO(l)2132 851 y FP(;)2263 +851 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 16.73299 8.36649 3.74124 +} false /N@E 16 {InitRnode } NewNode end end + 2263 851 a FP(E)2335 818 y FO(m)2402 851 y FP(;)2624 +851 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 14.21977 7.10988 3.74124 +} false /N@F 16 {InitRnode } NewNode end end + 2624 851 a FP(F)2695 818 y FO(n)2742 851 y FP(;)2874 +851 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 13.21782 6.6089 3.74124 +} false /N@G 16 {InitRnode } NewNode end end + 2874 851 a FP(G)2946 818 y FO(o)3014 851 y FN(g)3059 +851 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@A /N@B InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 851 a 3059 851 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@C InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 851 a 3059 851 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@D InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 +851 a 3059 851 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@F InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 851 a 3059 851 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@B /N@G InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 851 a 3059 +851 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@D /N@E InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 3059 851 a 378 1056 a FT(W)-8 b(e)40 b(can)f(also)g(illustrate)e +(whic)m(h)g(subsets)h(of)h(sets)g(in)f FN(f)p FP(S)2407 +1070 y FL(1)2446 1056 y FP(;)15 b(S)2542 1070 y FL(2)2582 +1056 y FP(;)g(S)2678 1070 y FL(3)2718 1056 y FN(g)39 +b FT(ha)m(v)m(e)h(colours)e(whic)m(h)f(relate)378 1169 +y(with)29 b(eac)m(h)j(other)e(b)m(y)g(the)h(follo)m(wing)e(diagram:) +1778 1373 y + tx@Dict begin tx@NodeDict begin {7.48248 1.64249 11.46454 5.73227 +2.92 } false /N@S1 16 {InitRnode } NewNode end end + 1778 1373 a FP(S)1834 1387 y FL(1)2055 1373 +y + tx@Dict begin tx@NodeDict begin {7.48248 1.64249 11.46454 5.73227 +2.92 } false /N@S2 16 {InitRnode } NewNode end end + 2055 1373 a FP(S)2111 1387 y FL(2)2332 1373 y + tx@Dict begin tx@NodeDict begin {7.48248 1.64249 11.46454 5.73227 +2.92 } false /N@S3 16 {InitRnode } NewNode end end + 2332 +1373 a FP(S)2388 1387 y FL(3)2428 1373 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@S1 /N@S2 InitNC { /AngleA 35. def /AngleB 145. def 0.67 + 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke + grestore grestore end + 2428 1373 a +2428 1373 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 +0.0 0 0 /N@S1 /N@S3 InitNC { /AngleA 35. def /AngleB 145. 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b(b)26 b FN(2)f FP(G)p +FT(.)39 b(This)25 b(rule)h(is)g(represen)m(ted)i(b)m(y)f(the)378 +4799 y(HOL)j(theorem)473 4987 y FN(`)48 b(8)p FP(p)f(G)p +FM(.)g(Group)f FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\))g +FT(\()p FN(8)p FP(H)7 b FM(.)47 b(Subset)f FP(H)55 b(G)47 +b FN(\))712 5100 y FT(\()p FN(8)p FP(a)p FM(.)g FP(G)h(a)f +FN(\))h FT(\()p FN(8)p FP(b)p FM(.)f FP(G)h(b)f FN(\))903 +5213 y FT(\()p FM(RightCoset)e FT(\()p FM(RightCoset)h +FT(\()p FP(H)7 b(;)15 b(p)p FT(\))48 b FP(a;)15 b(p)p +FT(\))48 b FP(b)73 b FT(=)1285 5326 y FM(RightCoset)45 +b FT(\()p FP(H)7 b(;)15 b(p)p FT(\))48 b(\()p FP(p)g(a)g(b)p +FT(\)\)\)\)\))378 5513 y(The)32 b(simpli\014er)e(whic)m(h)h(normalises) +g(terms)i(constructed)g(using)f(the)h(subset)f(constructing)g(func-)378 +5626 y(tions)40 b(is)g(named)g Fw(cos)n FT(.)72 b(Eac)m(h)41 +b(rule)e(is)h(applied)e(only)i(if)g(all)f(its)h(conditions)f(are)i +(automatically)p eop +%%Page: 204 214 +204 213 bop 378 5 a FF(CHAPTER)30 b(9.)61 b(A)30 b(MECHANISA)-8 +b(TION)30 b(OF)h(GR)m(OUP)g(THEOR)-8 b(Y)847 b FT(204)p +378 416 3453 4 v 376 1872 4 1456 v 928 892 a + tx@Dict begin tx@NodeDict begin {7.5 2.5 26.84492 13.42245 3.01385 +} false /N@M-1-1-1 16 {InitRnode } NewNode end end + 928 892 +a Ft(\()p Fv(H)7 b(a)p Ft(\))p Fv(e)1383 1158 y + tx@Dict begin tx@NodeDict begin {7.5 2.5 26.84492 13.42245 3.01385 +} false /N@M-1-2-2 16 {InitRnode } NewNode end end + 1383 +1158 a Fv(H)g Ft(\()p Fv(ae)p Ft(\))979 1424 y + tx@Dict begin tx@NodeDict begin {6.83331 0.0 14.41086 7.20543 3.01385 +} false /N@M-1-3-1 16 {InitRnode } NewNode end end + 979 1424 +a Fv(H)g(a)1099 1424 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-1-1 /N@M-1-3-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 1099 1424 a 1099 1424 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-1-1 /N@M-1-2-2 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 1099 1424 +a 1099 1424 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-2-2 /N@M-1-3-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 5.0 3.0 -2 0 add DashLine grestore grestore end + 1099 1424 a 1099 1424 a + tx@Dict begin tx@NodeDict begin /t 0.5 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 15.39174 +def /r 15.39174 def /h 1.29169 def /d 3.01385 def /flag false def HPutAdjust +LPutCoor end PutBegin end + 1099 1424 a 971 +1449 a Fv(ae)23 b Fu(!)g Fv(a)1099 1424 y + tx@Dict begin PutEnd end + 1099 1424 a +2408 606 a + tx@Dict begin tx@NodeDict begin {7.5 2.5 38.58568 19.29283 3.01385 +} false /N@M-1-1-2 16 {InitRnode } NewNode end end + 2408 606 a Ft(\(\()p Fv(H)7 b(a)p Ft(\))p +Fv(b)p Ft(\))p Fv(c)1855 872 y + tx@Dict begin tx@NodeDict begin {7.5 2.5 38.58568 19.29283 3.01385 +} false /N@M-1-2-1 16 {InitRnode } NewNode end end + 1855 872 a Ft(\()p Fv(H)g(a)p +Ft(\)\()p Fv(bc)p Ft(\))2961 872 y + tx@Dict begin tx@NodeDict begin {7.5 2.5 38.58568 19.29283 3.01385 +} false /N@M-1-2-3 16 {InitRnode } NewNode end end + 2961 872 a Ft(\()p +Fv(H)g Ft(\()p Fv(ab)p Ft(\)\))p Fv(c)2961 1137 y + tx@Dict begin tx@NodeDict begin {7.5 2.5 38.58568 19.29283 3.01385 +} false /N@M-1-3-3 16 {InitRnode } NewNode end end + 2961 +1137 a Fv(H)g Ft(\(\()p Fv(ab)p Ft(\))p Fv(c)p Ft(\))2408 +1403 y + tx@Dict begin tx@NodeDict begin {7.5 2.5 38.58568 19.29283 3.01385 +} false /N@M-1-4-2 16 {InitRnode } NewNode end end + 2408 1403 a Fv(H)g Ft(\()p Fv(a)p Ft(\()p Fv(bc)p +Ft(\)\))2728 1403 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-1-2 /N@M-1-2-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2728 1403 a 2728 1403 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-2-1 /N@M-1-4-2 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2728 1403 +a 2728 1403 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-1-2 /N@M-1-2-3 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2728 1403 a 2728 1403 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-2-3 /N@M-1-3-3 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2728 1403 a 2728 +1403 a + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow + EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 0.0 0.0 +0 0 /N@M-1-3-3 /N@M-1-4-2 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 5.0 3.0 -2 0 add DashLine grestore grestore end + 2728 1403 a 2728 1403 a + tx@Dict begin tx@NodeDict begin /t 0.5 def tx@NodeDict /HPutPos known +{ HPutPos } { CP /Y ED /X ED /NAngle 0 def /NCLW 0 def } ifelse /Sin +NAngle sin def /Cos NAngle cos def /s 5.0 NCLW add def /l 29.46062 +def /r 29.46062 def /h 4.48615 def /d 5.51385 def /flag false def HPutAdjust +LPutCoor end PutBegin end + 2728 1403 a 2483 1428 +a Ft(\()p Fv(ab)p Ft(\))p Fv(c)24 b Fu(!)f Fv(a)p Ft(\()p +Fv(bc)p Ft(\))2728 1403 y + tx@Dict begin PutEnd end + 2728 1403 a 485 1703 a FT(Figure)30 +b(25:)41 b(The)30 b(Need)h(for)f(the)h(Group)e(Elemen)m(t)i(Normaliser) +e(in)g(Normalising)f(Subsets.)p 3829 1872 4 1456 v 378 +1876 3453 4 v 378 2233 a(deriv)m(ed)39 b(b)m(y)h(appropriate)f(queries) +g(to)i(the)g(kno)m(wledge)f(database.)71 b(The)39 b(follo)m(wing)g +(additional)378 2346 y(theorem)31 b(is)e(used)h(b)m(y)g(the)h +(simpli\014er)26 b(to)31 b(rewrite)f(terms)g(in)m(v)m(olving)f(the)i +(function)e Fw(LRCoset)m FT(:)473 2533 y FN(`)48 b(8)p +FP(p)f(H)54 b(a)48 b(b)p FM(.)g(LRCoset)d FP(a)j FT(\()p +FP(H)7 b(;)15 b(p)p FT(\))49 b FP(b)e FT(=)1237 2646 +y FM(LeftCoset)e FP(a)j FT(\()p FM(RightCoset)d FT(\()p +FP(H)7 b(;)15 b(p)p FT(\))49 b FP(b;)15 b(p)p FT(\)\))378 +2834 y(The)35 b(examples)g(giv)m(en)h(in)f(the)h(next)f(section)h(sho)m +(w)g(ho)m(w)g(a)g(n)m(um)m(b)s(er)e(of)i(SPL)f(pro)s(ofs)g(using)f +(this)378 2947 y(simpli\014er)27 b(are)j(quite)g(similar)e(to)j(those)g +(found)e(in)g(the)i(literature.)378 3233 y FH(9.4)135 +b(F)-11 b(urther)44 b(Results)378 3436 y FT(This)27 b(section)h +(illustrates)f(a)i(n)m(um)m(b)s(er)e(of)h(in)m(teresting)g(results)f +(in)g(group)h(theory)h(whic)m(h)e(are)i(mec)m(h-)378 +3549 y(anised)38 b(as)h(SPL)f(pro)s(ofs.)66 b(In)39 b(particular,)g +(normal)f(subgroups)g(are)h(de\014ned)f(and)g(sho)m(wn)h(to)h(b)s(e)378 +3662 y(exactly)k(those)h(subgroups)d(whose)h(left)h(cosets)h(are)f +(equal)f(to)i(their)d(righ)m(t)i(cosets.)82 b(Quotien)m(t)378 +3775 y(groups,)33 b(whic)m(h)e(are)i(groups)f(whose)h(elemen)m(ts)g +(are)g(cosets)h(and)e(whose)g(pro)s(duct)g(elemen)m(t)h(is)f(the)378 +3888 y(pro)s(duct)40 b(of)i(subsets,)h(are)f(also)f(de\014ned.)73 +b(Section)41 b(9.4.2)i(giv)m(es)f(the)f(de\014nition)e(of)j(homomor-) +378 4001 y(phisms)24 b(and)h(isomorphisms,)g(as)h(w)m(ell)f(as)i(a)f(n) +m(um)m(b)s(er)f(of)h(results)f(including)e(the)j(t)m(w)m(o)i +(isomorphism)378 4113 y(theorems.)378 4357 y FG(9.4.1)112 +b(Normal)36 b(Subgroups)j(and)f(Quotien)m(t)f(Groups)378 +4529 y FT(Although,)29 b(in)e(general,)j(the)f(left)f(cosets)j(and)d +(righ)m(t)g(cosets)j(of)e(a)g(subgroup)e(are)i(di\013eren)m(t,)g +(Galois)378 4641 y(iden)m(ti\014ed)j(the)i(particular)e(criterion)h +(whic)m(h)g(a)h(subgroup)e(m)m(ust)i(satisfy)f(so)h(that)h(its)e(left)h +(cosets)378 4754 y(are)e(equal)f(to)h(its)e(righ)m(t)h(cosets.)45 +b(This)30 b(prop)s(ert)m(y)h(is)f(called)g(normalit)m(y)-8 +b(,)32 b(and)f(a)g(normal)g(subgroup)378 4867 y(is)e(de\014ned)g(as)i +(follo)m(ws:)473 5055 y FN(`)529 5070 y FE(def)686 5055 +y FM(NormalSG)46 b FT(\()p FP(G)p FM(:'a)g FN(!)i FM(bool)p +FP(;)f(p)p FT(\))h FP(N)57 b FN(\021)807 5168 y FT(\()p +FM(SubGroup)46 b FP(p)i(N)58 b(G)47 b FN(^)855 5281 y(8)p +FP(g)s FM(.)h FP(G)f(g)k FN(\))d(8)p FP(n)p FM(.)f FP(N)57 +b(n)47 b FN(\))h FP(N)58 b FT(\()p FP(p)48 b(g)j FT(\()p +FP(p)d(n)f FT(\()p FM(InvG)g FT(\()p FP(G;)15 b(p)p FT(\))48 +b FP(g)s FT(\)\)\)\))378 5468 y(that)42 b(is,)i(a)d(subgroup)f +FP(N)52 b FT(of)41 b FP(G)g FT(is)g(normal)f(if)g(for)i(ev)m(ery)g +FP(g)47 b FN(2)d FP(G)d FT(and)f FP(n)k FN(2)f FP(N)10 +b FT(,)45 b FP(g)s(ng)3477 5435 y FK(\000)p FL(1)3615 +5468 y FN(2)f FP(N)10 b FT(.)378 5581 y(Equiv)-5 b(alen)m(tly)d(,)39 +b FP(N)48 b FT(is)37 b(normal)g(if)g FP(g)s(N)10 b(g)1734 +5548 y FK(\000)p FL(1)1867 5581 y FT(=)38 b FP(N)48 b +FT(for)38 b(ev)m(ery)h FP(g)j FN(2)37 b FP(G)p FT(,)j(as)e(giv)m(en)g +(b)m(y)g(the)g(follo)m(wing)378 5694 y(theorem:)p eop +%%Page: 205 215 +205 214 bop 378 5 a FF(CHAPTER)30 b(9.)71 b(A)30 b(MECHANISA)-8 +b(TION)31 b(OF)f(GR)m(OUP)h(THEOR)-8 b(Y)837 b FT(205)473 +396 y FN(`)48 b(8)p FP(G)e(N)58 b(p)p FM(.)47 b(Group)g +FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\))903 509 y FT(\()p +FM(NormalSG)e FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(N)58 +b FT(=)951 622 y FM(SubGroup)45 b FP(p)j(N)57 b(G)48 +b FN(^)951 735 y FT(\()p FN(8)p FP(g)s FM(.)g FP(G)f(g)k +FN(\))d FT(\()p FM(LRCoset)e FP(g)51 b FT(\()p FP(N)10 +b(;)15 b(p)p FT(\))49 b(\()p FM(InvG)e FT(\()p FP(G;)15 +b(p)p FT(\))48 b FP(g)s FT(\))74 b(=)f FP(N)10 b FT(\)\)\))378 +919 y(Giv)m(en)30 b(this)f(result,)g(it)g(can)h(b)s(e)g(sho)m(wn)f +(that)h(if)f FP(N)40 b FT(is)29 b(a)h(normal)f(subgroup)f(of)i +FP(G)p FT(,)g(then)g 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y(considered)e(essen)m(tial)g +(to)i(the)f(understanding)d(of)j(a)h(pro)s(of)e(giv)m(en)h(in)e(the)j +(early)e(stages)i(of)f(a)h(theory)378 1751 y(are)37 b(often)h(omitted)f +(in)e(the)i(pro)s(ofs)f(found)g(in)g(later)h(stages)h(of)f(the)g(same)g +(theory)-8 b(.)62 b(In)36 b(order)g(to)378 1864 y(ac)m(hiev)m(e)c(the)f +(same)g(e\013ect)i(in)c(mec)m(hanised)h(pro)s(ofs,)h(the)g(deductiv)m +(e)f(p)s(o)m(w)m(er)h(of)g(the)g(pro)s(of)f(c)m(hec)m(k)m(er)378 +1977 y(should)e(v)-5 b(ary)31 b(during)d(the)i(mec)m(hanisation)g(of)h +(a)g(theory)-8 b(.)519 2090 y(One)30 b(metho)s(d)f(of)h(mo)s(difying)d +(the)k(deductiv)m(e)e(p)s(o)m(w)m(er)h(of)g(the)h(SPL)e(pro)s(of)g(c)m +(hec)m(k)m(er)j(during)c(the)378 2203 y(mec)m(hanisation)21 +b(of)g(a)h(theory)g(is)e(b)m(y)h(the)h(use)f(of)g(a)h(database)g(of)g +(trivial)d(kno)m(wledge.)38 b(This)20 b(database,)378 +2316 y(whic)m(h)25 b(is)g(describ)s(ed)f(in)h(section)i(4.4.1,)i(can)e 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b(the)g(user)e(can)i(extend:)514 4074 +y FN(\017)46 b FT(pro)s(of)30 b(pro)s(cedures)f(used)h(to)h(justify)d +(the)j(pro)s(of)f(statemen)m(ts;)514 4258 y FN(\017)46 +b FT(simpli\014ers,)27 b(whic)m(h)i(are)i(used)e(to)j(normalise)c +(terms)j(in)m(to)f(canonical)g(forms;)514 4441 y FN(\017)46 +b FT(inference)29 b(rules,)f(whic)m(h)g(are)i(used)e(to)i(deriv)m(e)f +(facts)h(in)e(a)i(forw)m(ard)f(\(and)g(somewhat)h(pro)s(ce-)605 +4554 y(dural\))f(manner;)514 4737 y FN(\017)46 b FT(the)22 +b(syn)m(tax)g(and)f(seman)m(tics)h(of)g(the)f(SPL)g(language)h +(constructs)g(b)m(y)f(up)s(dating)f(the)h(language)605 +4850 y(parser)30 b(and)g(other)g(comp)s(onen)m(ts)h(of)f(the)h(pro)s +(of)f(c)m(hec)m(k)m(er.)378 5028 y(It)i(should)e(b)s(e)h(noted)h(that)h +(not)f(all)f(the)h(ab)s(o)m(v)m(e)h(p)s(ossible)c(w)m(a)m(ys)k(of)f +(extending)g(the)g(pro)s(of)f(c)m(hec)m(k)m(er)378 5141 +y(w)m(ere)h(used)e(during)g(the)h(case)i(study)d(describ)s(ed)f(in)h(c) +m(hapter)i(9.)45 b(The)30 b(mec)m(hanisation)h(p)s(erformed)378 +5253 y(during)41 b(the)j(case)h(study)e(made)h(use)f(of)h(sev)m(eral)g +(database)g(query)f(functions)g(and)g(simpli\014ers)378 +5366 y(whic)m(h)26 b(w)m(ere)i(implemen)m(ted)f(and)g(extended)g +(during)f(the)h(dev)m(elopmen)m(t)h(of)g(the)g(theory)-8 +b(.)41 b(Ho)m(w)m(ev)m(er,)378 5479 y(no)30 b(c)m(hanges)h(w)m(ere)f +(made)g(to)h(pro)s(of)e(pro)s(cedures,)g(the)h(forw)m(ard)f(inference)g +(rules,)g(and)g(the)h(syn)m(tax)378 5592 y(and)23 b(seman)m(tics)h(of)g +(the)g(language)g(constructs.)39 b(In)23 b(particular,)h(it)f(is)g +(suggested)i(that)f(the)g(frequen)m(t)378 5705 y(use)30 +b(of)h(forw)m(ard)e(inference)h(rules)f(should)f(b)s(e)i(a)m(v)m(oided) +h(b)s(ecause)f(of)h(their)e(pro)s(cedural)g(nature.)p +eop +%%Page: 217 227 +217 226 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969 +b FT(217)519 396 y(W)-8 b(e)35 b(remark)e(that)h(it)g(w)m(as)g(p)s +(ossible)d(to)j(implemen)m(t)e(the)i(SPL)f(pro)s(of)g(c)m(hec)m(k)m(er) 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1526 y(applied)j +FI(b)-5 b(efor)g(e)35 b FT(the)28 b FN(C)5 b(B)s(S)i(E)35 +b FT(rule)27 b(\(or)h(other)g(pro)m(v)m(ers\))h(are)f(used)f(to)h(c)m +(hec)m(k)i(the)e(pro)s(of)f(statemen)m(ts.)378 1638 y(More)36 +b(e\013ectiv)m(e)h(results)e(can)g(b)s(e)g(ac)m(hiev)m(ed)i(if)d(the)i +(simpli\014ers)31 b(and)k(other)h(decision)e(pro)s(cedures)378 +1751 y(are)h(incorp)s(orated)f(in)f(the)i(\014rst-order)f(pro)m(v)m(er) +h(as)g(studied,)f(for)h(instance,)g(b)m(y)g(Bj\034rner,)h(Stic)m(k)m +(el,)378 1864 y(and)j(Urib)s(e)e(\(1997\).)70 b(The)38 +b(incorp)s(oration)g(of)h(the)g(kno)m(wledge)g(database)i(with)c(the)j +(\014rst-order)378 1977 y(\(or)31 b(higher-order\))f(logic)g(pro)m(v)m +(er,)i(so)f(that)g(trivial)e(facts)i(can)h(b)s(e)e(automatically)g +(deriv)m(ed)g(b)m(y)h(the)378 2090 y(pro)m(v)m(er,)k(can)f(also)g +(impro)m(v)m(e)g(the)g(deductiv)m(e)f(p)s(o)m(w)m(er)h(of)g(the)g(pro)s +(of)f(c)m(hec)m(k)m(er.)53 b(This)32 b(will)f(o\013er)k(the)378 +2203 y(p)s(ossibilit)m(y)30 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y(require)d(inductiv)m(e)g +(reasoning.)519 3106 y(An)e(imp)s(ortan)m(t)f(area)i(of)f(researc)m(h)h +(whic)m(h)e(has)h(not)g(b)s(een)g(considered)f(in)f(this)i(thesis)f +(concerns)378 3219 y(the)31 b(feedbac)m(k)h(giv)m(en)f(b)m(y)g(the)g +(pro)s(of)f(c)m(hec)m(k)m(er)j(in)d(case)i(of)f(failure.)41 +b(The)31 b(SPL)f(pro)s(of)g(c)m(hec)m(k)m(er)j(do)s(es)378 +3332 y(not)j(pro)m(vide)g(an)m(y)g(p)s(ositiv)m(e)f(feedbac)m(k)i(when) +e(a)i(conclusion)d(cannot)j(b)s(e)e(justi\014ed)g(b)m(y)h(the)g(giv)m +(en)378 3445 y(justi\014cation.)67 b(It)39 b(is)g(desirable)f(that)i +(in)e(suc)m(h)h(cases)i(the)e(pro)s(of)g(c)m(hec)m(k)m(er)j(giv)m(es)e +(a)g(useful)d(error)378 3558 y(message)31 b(whic)m(h)f(helps)e(in)h +(understanding)f(wh)m(y)i(the)h(pro)s(of)e(c)m(hec)m(king)i(pro)s(cess) +f(failed.)519 3671 y(The)24 b(dev)m(elopmen)m(t)h(of)f(user-in)m +(terfaces)h(whic)m(h)e(pro)m(vide)g(the)i(in)m(teractiv)m(e)g(disco)m +(v)m(ery)g(of)f(declar-)378 3784 y(ativ)m(e)e(pro)s(ofs)e(is)g(also)g 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+%%Page: 229 239 +229 238 bop 378 5 a FF(APPENDIX)31 b(B.)122 b(T)-8 b(ABLEA)m(UX)31 +b(F)m(OR)g(FIRST-ORDER)f(LOGIC)844 b FT(229)2143 561 +y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.6094 4.3047 3.30017 +} false /N@T-0 16 {InitRnode } NewNode end end + 2143 561 a 2143 561 a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.6094 4.3047 3.74124 +} false /N@C 16 {InitRnode } NewNode end end + 2143 561 a FP(C)1928 810 y + tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.21251 4.10625 3.30017 +} false /N@T-0-0 16 {InitRnode } NewNode end end + 1928 +810 a FP(A)1962 783 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0 /N@T-0-0 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 1962 783 a 1743 1060 a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 15.51253 7.75626 3.30017 +} false /N@T-0-0-0 16 {InitRnode } NewNode end end + 1743 1060 +a FN(:)p FP(A)1808 1032 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-0 /N@T-0-0-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 1808 1032 a 2049 1060 a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 15.90942 7.95471 3.30017 +} false /N@T-0-0-1 16 {InitRnode } NewNode end end + 2049 +1060 a FN(:)p FP(C)2115 1032 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-0 /N@T-0-0-1 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 2115 1032 a 2328 810 a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 16.15506 8.07753 3.30017 +} false /N@T-0-1 16 {InitRnode } NewNode end end + +2328 810 a FN(:)p FP(B)2395 783 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0 /N@T-0-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2395 783 a 2358 1060 +a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.85504 4.42752 3.30017 +} false /N@T-0-1-0 16 {InitRnode } NewNode end end + 2358 1060 a 2358 1060 a + tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.85504 4.42752 3.74124 +} false /N@B 16 {InitRnode } NewNode end end + 2358 1060 a FP(B)2395 1032 +y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1 /N@T-0-1-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 2395 1032 a 1338 1314 a FT(Figure)30 b(27:)41 b(An)30 +b(Example)g(of)g(a)h(T)-8 b(ableau.)378 1707 y(F)g(or)31 +b(example)f(the)h(tableau)f(giv)m(en)g(in)f(\014gure)h(27)h(represen)m +(ts)g(the)f(form)m(ula)1392 1911 y FP(C)d 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setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1 /N@T-0-1-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + 1864 997 a 1743 1274 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 29.10628 14.55313 +3.30017 } false /N@T-0-1-0-0 16 {InitRnode } NewNode end end + 1743 1274 a FN(:)p FP(P)g +FT(\()p FP(c)p FT(\))1864 1246 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-0 /N@T-0-1-0-0 InitNC { NCLine } if end gsave 0.8 +SLW 0. setgray 0 setlinecap stroke grestore grestore end + 1864 1246 a 1744 1382 +a + tx@Dict begin tx@NodeDict begin {6.3875 11.8325 28.98734 14.49367 +3.30017 } false /N@T-0-1-0-0-0 16 {InitRnode } NewNode end end + 1744 1382 a 85 w FN(\002)1744 1454 y FK(f)p FO(v)1813 +1463 y FC(2)1848 1454 y FK(!)p FO(c)p FK(g)1864 1354 +y + tx@Dict begin gsave STV 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tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1-1 /N@T-0-1-1-0 InitNC { NCLine } if end grestore +end + 2302 1105 a +2596 776 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 34.2733 17.13664 3.30017 +} false /N@T-0-2 16 {InitRnode } NewNode end end + 2596 776 a FN(:)p FP(R)q FT(\()p FP(v)2806 +790 y FL(1)2846 776 y FT(\))2739 748 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0 /N@T-0-2 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2739 748 a 2649 +1025 a + tx@Dict begin tx@NodeDict begin {8.2125 2.73749 21.65417 10.82709 +3.30017 } false /N@T-0-2-0 16 {InitRnode } NewNode end end + 2649 1025 a FP(R)q FT(\()p FP(c)p FT(\))2739 997 +y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { 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/ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0 /N@T-0-1 InitNC { NCLine } if end gsave 0.8 SLW 0. +setgray 0 setlinecap stroke grestore grestore end + 2363 789 a 2185 1066 a + tx@Dict begin tx@NodeDict begin {9.12923 2.73749 42.76768 21.38383 +3.30017 } false /N@T-0-1-0 16 {InitRnode } NewNode end end + 2185 1066 +a 2185 1066 a + tx@Dict begin tx@NodeDict begin {9.12923 2.73749 42.76768 21.38383 +3.19586 } false /N@Qfa 16 {InitRnode } NewNode end end + 2185 1066 a FP(Q)p FT(\()p FP(f)2347 1033 +y FL(3)2386 1066 y FT(\()p FP(a)p FT(\)\))2363 1038 y + tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { moveto +} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0 +4.0 0 0 /N@T-0-1 /N@T-0-1-0 InitNC { NCLine } if end gsave 0.8 SLW +0. setgray 0 setlinecap stroke grestore grestore end + +2363 1038 a 887 1342 a FT(Figure)30 b(32:)41 b(T)-8 b(ableau)30 +b(Branc)m(hes)h(with)e(Di\013eren)m(t)i(Rigid)e(Equations.)378 +1736 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4468 y(rigid)j(paramo)s +(dulation.)g(In)h(B.)h(Neb)s(el)f(and)g(L.)h(Dresc)m(hler-Fisc)m(her)g +(\(Eds.\),)g FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)32 b(of)610 +4581 y(the)g(18th)h(German)f(A)n(nnual)f(Confer)-5 b(enc)g(e)32 +b(on)g(A)n(rti\014cial)g(Intel)5 b(ligenc)-5 b(e)31 b(:)41 +b(KI-94:)g(A)-5 b(dvanc)g(es)610 4694 y(in)33 b(A)n(rti\014cial)f +(Intelel)5 b(ligenc)-5 b(e)p FT(,)31 b(V)-8 b(olume)30 +b(861)i(of)e FI(LNAI)p FT(,)g(Berlin,)f(pp.)h(319{330.)k(Springer.)474 +4843 y(Bec)m(k)m(ert,)f(B.)e(\(1997,)i(F)-8 b(ebruary\).)31 +b(Seman)m(tic)f(tableaux)g(with)f(equalit)m(y)-8 b(.)31 +b FI(Journal)j(of)f(L)-5 b(o)g(gic)33 b(and)610 4956 +y(Computation)40 b(7)12 b FT(\(1\),)32 b(39{58.)474 5105 +y(Bec)m(k)m(ert,)42 b(B.)c(and)f(R.)g(H\177)-45 b(ahnle)37 +b(\(1992,)k(June\).)c(An)g(impro)m(v)m(ed)g(metho)s(d)g(for)g(adding)f +(equalit)m(y)610 5217 y(to)41 b(free)g(v)-5 b(ariable)39 +b(seman)m(tic)h(tableaux.)h(In)e(D.)i(Kapur)e(\(Ed.\),)k +FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)44 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b(on)g(A)n(utomate)-5 +b(d)26 b(de)-5 b(duction)p FT(,)25 b(V)-8 b(olume)610 +2653 y(1249)32 b(of)f FI(LNAI)p FT(,)f(Berlin,)f(pp.)g(101{115.)34 +b(Springer.)474 2803 y(Blac)m(k,)j(P)-8 b(.)35 b(E.)g(and)f(P)-8 +b(.)35 b(J.)g(Windley)e(\(1995,)38 b(Septem)m(b)s(er\).)d +(Automatically)g(syn)m(thesized)f(term)610 2916 y(denotation)24 +b(predicates:)37 b(A)23 b(pro)s(of)g(aid.)f(See)i(Sc)m(h)m(ub)s(ert,)g +(Windley)-8 b(,)24 b(and)f(Alv)m(es-F)-8 b(oss)24 b(\(1995\),)610 +3029 y(pp.)30 b(46{57.)474 3179 y(Bo)s(ole,)45 b(G.)d(\(1848\).)j(The)c +(calculus)f(of)i(logic.)f FI(The)j(Cambridge)f(and)h(Dublin)f +(Mathematic)-5 b(al)610 3292 y(Journal)41 b(3)p FT(,)31 +b(183{198.)474 3443 y(Boulton,)g(R.)g(J.)g(\(1993,)i(August\).)e(Lazy)h +(tec)m(hniques)e(for)h(fully)e(expansiv)m(e)h(theorem)h(pro)m(ving.)610 +3555 y FI(F)-7 b(ormal)35 b(Metho)-5 b(ds)34 b(in)f(System)g(Design)k +(3)12 b FT(\(1/2\),)33 b(25{47.)474 3706 y(Camilleri,)22 +b(J.)i(and)g(T.)g(Melham)f(\(1992,)28 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b(56{68.)474 +4871 y(Comon,)26 b(H.)g(\(1990\).)i(Solving)c(sym)m(b)s(olic)f +(ordering)h(constrain)m(ts.)i FI(IJF)n(CS:)i(International)i(Jour-)610 +4984 y(nal)k(of)e(F)-7 b(oundations)35 b(of)e(Computer)h(Scienc)-5 +b(e)37 b(1)12 b FT(\(4\),)33 b(387{411.)474 5135 y(Constable)27 +b(et)h(al.,)g(R.)f(L.)g(\(1986\).)j FI(Implementing)h(mathematics)h +(with)f(the)f(Nuprl)g(pr)-5 b(o)g(of)32 b(devel-)610 +5247 y(opment)i(system)p FT(.)e(Pren)m(tice)e(Hall.)474 +5398 y(Co)s(quand,)37 b(T.)g(and)f(G.)i(Huet)f(\(1986,)k(Ma)m(y\).)e +(The)d(calculus)g(of)h(constructions.)g(Rapp)s(ort)f(de)610 +5511 y(Rec)m(herc)m(he)c(530,)g(INRIA,)f(Ro)s(cquencourt,)f(F)-8 +b(rance.)p eop +%%Page: 249 259 +249 258 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(249)474 +396 y(Cosco)m(y)-8 b(,)49 b(Y.)44 b(\(1997,)49 b(Septem)m(b)s(er\).)44 +b(A)g(natural)e(language)i(explanation)f(for)h(formal)f(pro)s(ofs.)610 +509 y(In)d(C.)h(Retor)m(\023)-43 b(e)43 b(\(Ed.\),)h +FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)44 b(of)e(the)h(1st)g(International)h +(Confer)-5 b(enc)g(e)44 b(on)e(L)-5 b(o)g(gic)g(al)610 +622 y(Asp)g(e)g(cts)35 b(of)g(Computational)i(Linguistics)d(\(LA)n +(CL-96\))p FT(,)f(V)-8 b(olume)32 b(1328)i(of)e FI(LNAI)p +FT(,)f(Berlin,)610 735 y(pp.)f(149{167.)k(Springer.)474 +881 y(Cosco)m(y)-8 b(,)42 b(Y.,)f(G.)f(Hahn,)g(and)e(L.)g(Th)m(\023)-43 +b(ery)39 b(\(1997,)k(April\).)37 b(Extracting)h(text)i(from)e(pro)s +(ofs.)g(In)610 993 y FI(T)-7 b(yp)i(e)g(d)51 b(L)-5 b(amb)g(da)52 +b(Calculus)e(and)h(Applic)-5 b(ations)51 b(\(Edinbur)-5 +b(gh\))p FT(,)55 b(V)-8 b(olume)49 b(902)h(of)f FI(LNCS)p +FT(.)610 1106 y(Springer-V)-8 b(erlag.)474 1252 y(Craig,)25 +b(W.)h(\(1957\).)h(A)e(new)f(form)g(of)h(the)g(Herbrand-Gen)m(tzen)g +(theorem.)g FI(Journal)j(of)g(Symb)-5 b(olic)610 1365 +y(L)g(o)g(gic)36 b(22)p FT(,)c(250{268.)474 1510 y(Cutland,)i(N.)h(J.)f +(\(1980\).)j FI(Computability:)51 b(A)n(n)36 b(intr)-5 +b(o)g(duction)39 b(to)e(r)-5 b(e)g(cursive)36 b(function)h(the)-5 +b(ory)p FT(.)610 1623 y(Cam)m(bridge:)40 b(Cam)m(bridge)29 +b(Univ.)h(Press.)474 1768 y(Cyrluk,)h(D.,)j(P)-8 b(.)33 +b(Lincoln,)e(and)h(N.)h(Shank)-5 b(ar)31 b(\(1996\).)36 +b(On)31 b(Shostak's)i(decision)e(pro)s(cedure)g(for)610 +1881 y(com)m(binations)25 b(of)g(theories.)g(In)f(M.)i(A.)g(McRobbie)e +(and)h(J.)g(K.)g(Slaney)f(\(Eds.\),)j FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings) +610 1994 y(of)34 b(the)g(13th)h(International)h(Confer)-5 +b(enc)g(e)35 b(on)f(A)n(utomate)-5 b(d)35 b(De)-5 b(duction,)34 +b(\(New)g(Brunswick,)610 2107 y(NJ\))p FT(,)20 b(V)-8 +b(olume)20 b(1104)i(of)f FI(L)-5 b(e)g(ctur)g(e)24 b(Notes)g(in)f(A)n +(rti\014cial)g(Intel)5 b(ligenc)-5 b(e)p FT(,)23 b(pp.)c(463{477.)24 +b(Springer-)610 2220 y(V)-8 b(erlag.)474 2365 y(Da)m(vis,)27 +b(M.)e(\(1965\).)j FI(The)g(Unde)-5 b(cidable.)29 b(Basic)f(p)-5 +b(ap)g(ers)30 b(on)e(unde)-5 b(cidable)29 b(pr)-5 b(op)g(ositions,)32 +b(unsolv-)610 2478 y(able)h(pr)-5 b(oblems)35 b(and)e(c)-5 +b(omputable)35 b(functions)p FT(.)30 b(Ra)m(v)m(en)i(Press,)e(Hewlett,) +h(N.Y.)474 2623 y(Da)m(vis,)44 b(M.)d(\(1981,)46 b(24{28)d(August\).)e +(Ob)m(vious)e(logical)h(inferences.)g(In)g(P)-8 b(.)41 +b(J.)g(Ha)m(y)m(es)h(\(Ed.\),)610 2736 y FI(Pr)-5 b(o)g(c)g(e)g(e)g +(dings)45 b(of)f(the)f(7th)h(International)i(Joint)d(Confer)-5 +b(enc)g(e)44 b(on)g(A)n(rti\014cial)f(Intel)5 b(ligenc)-5 +b(e)610 2849 y(\(IJCAI)32 b('81\))p FT(,)g(Los)e(Altos,)h(CA,)f(pp.)g +(530{531.)k(William)28 b(Kaufmann.)474 2994 y(de)55 b(Bruijn,)60 +b(N.)55 b(G.)h(\(1970\).)i(The)d(mathematical)g(language)h(A)m(UTOMA)-8 +b(TH,)56 b(its)f(usage,)610 3107 y(and)67 b(some)g(of)h(its)e +(extensions.)h(In)f(M.)i(Laudet,)76 b(D.)68 b(Lacom)m(b)s(e,)77 +b(L.)67 b(Nolin,)75 b(and)610 3220 y(M.)30 b(Sc)m(h)s(\177)-48 +b(utzen)m(b)s(erger)29 b(\(Eds.\),)h FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings) +34 b(Symp)-5 b(osium)33 b(on)g(A)n(utomatic)f(Demonstr)-5 +b(ation,)610 3333 y(V)e(ersail)5 b(les,)39 b(F)-7 b(r)i(anc)g(e,)40 +b(De)-5 b(c)37 b(1968)p FT(,)i(V)-8 b(olume)36 b(125)g(of)g +FI(L)-5 b(e)g(ctur)g(e)38 b(Notes)g(in)f(Mathematics)p +FT(,)i(pp.)610 3446 y(29{61.)33 b(Berlin:)39 b(Springer-V)-8 +b(erlag.)474 3591 y(de)29 b(Bruijn,)f(N.)h(G.)h(\(1980\).)i(A)d(surv)m +(ey)g(of)g(the)h(pro)5 b(ject)29 b(A)m(UTOMA)-8 b(TH.)31 +b(In)d(J.)h(R.)g(Hindley)e(and)610 3704 y(J.)k(P)-8 b(.)31 +b(Seldin)d(\(Eds.\),)j FI(Essays)i(on)h(Combinatory)h(L)-5 +b(o)g(gic,)33 b(L)-5 b(amb)g(da)35 b(Calculus)e(and)h(F)-7 +b(ormal-)610 3817 y(ism)p FT(,)31 b(pp.)f(580{606.)k(London:)39 +b(Academic)31 b(Press.)474 3962 y(Degt)m(y)m(arev,)h(A.)e(and)e(A.)i(V) +-8 b(oronk)m(o)m(v)31 b(\(1996,)g(Octob)s(er\).)f(The)f(undecidabilit)m +(y)c(of)k(sim)m(ultaneous)610 4075 y(rigid)g(E-uni\014cation.)g +FI(The)-5 b(or)g(etic)g(al)35 b(Computer)f(Scienc)-5 +b(e)37 b(166)12 b FT(\(1-2\),)34 b(291{300.)474 4221 +y(Degt)m(y)m(arev,)c(A.)25 b(and)g(A.)h(V)-8 b(oronk)m(o)m(v)28 +b(\(1998\).)g(What)e(y)m(ou)g(alw)m(a)m(ys)g(w)m(an)m(ted)g(to)g(kno)m +(w)g(ab)s(out)f(rigid)610 4334 y FP(E)5 b FT(-uni\014cation.)30 +b FI(Journal)j(of)g(A)n(utomate)-5 b(d)34 b(R)-5 b(e)g(asoning)40 +b(20)12 b FT(\(1\),)33 b(47{80.)474 4479 y(F)-8 b(a)m(y)g(,)47 +b(M.)42 b(\(1979,)k(F)-8 b(ebruary\).)42 b(First-order)f(uni\014cation) +f(in)g(an)h(equational)g(theory)-8 b(.)43 b(In)e FI(Pr)-5 +b(o-)610 4592 y(c)g(e)g(e)g(dings)44 b(of)g(the)f(F)-7 +b(ourth)45 b(Workshop)g(on)f(A)n(utomate)-5 b(d)44 b(De)-5 +b(duction)p FT(,)45 b(Austin,)f(T)-8 b(exas,)46 b(pp.)610 +4705 y(161{167.)474 4850 y(Fitting,)32 b(M.)g(\(1972\).)i(T)-8 +b(ableau)31 b(metho)s(ds)g(of)h(pro)s(of)f(for)g(mo)s(dal)g(logics.)g +FI(Notr)-5 b(e)35 b(Dame)f(Journal)610 4963 y(of)f(F)-7 +b(ormal)35 b(L)-5 b(o)g(gic)36 b(13)12 b FT(\(2\),)33 +b(237{247.)474 5108 y(Fitting,)c(M.)g(C.)g(\(1996\).)i +FI(First-Or)-5 b(der)33 b(L)-5 b(o)g(gic)32 b(and)g(A)n(utomate)-5 +b(d)32 b(The)-5 b(or)g(em)33 b(Pr)-5 b(oving)38 b FT(\(2nd)28 +b(ed.\).)610 5221 y(Graduate)j(T)-8 b(exts)31 b(in)e(Computer)h +(Science.)g(Berlin:)39 b(Springer-V)-8 b(erlag.)30 b(1st)h(ed.,)f +(1990.)474 5366 y(F)-8 b(rege,)58 b(G.)52 b(\(1879\).)i +FI(Be)-5 b(gri\013sschrift,)58 b(eine)51 b(der)h(A)n(rithmetischen)h +(Nachgebildete)f(F)-7 b(ormel-)610 5479 y(spr)i(ache)37 +b(des)e(R)-5 b(einen)36 b(Denkens)p FT(.)c(Halle.)h(English)d +(translation)i(in)g FI(Fr)-5 b(om)37 b(Fr)-5 b(e)g(ge)35 +b(to)h(G\177)-46 b(odel,)610 5592 y(a)37 b(Sour)-5 b(c)g(e)38 +b(Bo)-5 b(ok)38 b(in)f(Mathematic)-5 b(al)39 b(Lo)-5 +b(gic)35 b FT(\(J.)g(v)-5 b(an)35 b(Heijeno)s(ort,)h(Editor\),)g(Harv) +-5 b(ard)35 b(Uni-)610 5705 y(v)m(ersit)m(y)c(Press,)f(Cam)m(bridge,)g +(1967,)i(pp.)e(1{82.)p eop +%%Page: 250 260 +250 259 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(250)474 +396 y(Gallier,)28 b(J.,)i(P)-8 b(.)30 b(Narendran,)f(D.)h(Plaisted,)e +(S.)h(Raatz,)j(and)d(W.)g(Sn)m(yder)g(\(1993,)j(Jan)m(uary\).)d(An)610 +509 y(algorithm)36 b(for)g(\014nding)e(canonical)j(sets)g(of)f(ground)g +(rewrite)g(rules)f(in)g(p)s(olynomial)f(time.)610 622 +y FI(Journal)g(of)f(the)g(A)n(CM)44 b(40)12 b FT(\(1\),)33 +b(1{16.)474 773 y(Gallier,)27 b(J.,)i(P)-8 b(.)28 b(Narendran,)g(D.)h +(Plaisted,)e(and)h(W.)g(Sn)m(yder)f(\(1990,)k(July/August\).)c(Rigid)f +FP(E)5 b FT(-)610 885 y(uni\014cation:)35 b(NP-completeness)23 +b(and)e(applications)f(to)j(equational)f(matings.)g FI(Information)610 +998 y(and)34 b(Computation)40 b(87)12 b FT(\(1/2\),)33 +b(129{195.)474 1149 y(Gallier,)e(J.)h(H.,)i(S.)d(Raatz,)k(and)c(W.)i +(Sn)m(yder)e(\(1987,)k(22{25)f(June\).)e(Theorem)g(pro)m(ving)f(using) +610 1262 y(rigid)39 b FP(E)5 b FT(-uni\014cation)39 b(equational)h +(matings.)g(In)f FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings,)46 +b(Symp)-5 b(osium)44 b(on)f(L)-5 b(o)g(gic)42 b(in)610 +1374 y(Computer)34 b(Scienc)-5 b(e)p FT(,)30 b(Ithaca,)i(New)e(Y)-8 +b(ork,)31 b(pp.)e(338{346.)34 b(The)c(Computer)f(So)s(ciet)m(y)h(of)h +(the)610 1487 y(IEEE.)474 1638 y(Girard,)47 b(J.-Y.)e(\(1972\).)j +FI(Interpr)n(\023)-44 b(etation)48 b(fonctionel)5 b(le)46 +b(et)d(\023)-44 b(elimination)48 b(des)e(c)-5 b(oupur)g(es)47 +b(dans)610 1751 y(l'arith)n(\023)-44 b(etique)34 b(d'or)-5 +b(dr)g(e)35 b(sup)n(\023)-44 b(erieur)p FT(.)31 b(Ph.)e(D.)i(thesis,)f +(Univ)m(ersit)m(\023)-43 b(e)31 b(P)m(aris)f(VI)s(I.)474 +1901 y(G\177)-45 b(odel,)36 b(K.)e(\(1931\).)1222 1878 +y(\177)1210 1901 y(Ub)s(er)g(formal)g(unen)m(tsc)m(heidbare)f(s\177)-45 +b(atze)36 b(der)e FI(principia)k(matematic)-5 b(a)36 +b FT(und)610 2014 y(v)m(erw)m(andter)26 b(systeme)g(I.)f +FI(Monatshefte)k(f)q(\177)-48 b(ur)28 b(Matematik)h(und)f(Physik)35 +b(38)p FT(,)27 b(173{98.)h(English)610 2127 y(T)-8 b(ranslation)29 +b(in)g(\(Da)m(vis)j(1965\),)g(pp.)e(4{38.)474 2277 y(Gordon,)39 +b(M.)f(\(1985\).)i(Wh)m(y)e(higher-order)d(logic)j(is)e(a)i(go)s(o)s(d) +f(formalism)e(for)i(sp)s(ecifying)e(and)610 2390 y(v)m(erifying)23 +b(hardw)m(are.)i(T)-8 b(ec)m(hnical)24 b(Rep)s(ort)g(77,)j(Univ)m +(ersit)m(y)c(of)i(Cam)m(bridge)e(Computer)h(Lab-)610 +2503 y(oratory)-8 b(,)32 b(New)f(Museums)e(Site,)h(P)m(em)m(brok)m(e)i +(Street,)f(Cam)m(bridge,)f(CB2)h(3QG,)g(UK.)474 2653 +y(Gordon,)g(M.)h(\(1996,)h(August\).)f(Set)f(theory)-8 +b(,)32 b(higher)e(order)g(logic)h(or)g(b)s(oth?)42 b(See)31 +b(v)m(on)g(W)-8 b(righ)m(t,)610 2766 y(Grundy)g(,)30 +b(and)f(Harrison)h(\(1996\),)j(pp.)c(191{201.)474 2916 +y(Gordon,)g(M.)h(J.,)g(A.)f(J.)g(Milner,)f(and)h(C.)f(P)-8 +b(.)30 b(W)-8 b(adsw)m(orth)30 b(\(1979\).)i FI(Edinbur)-5 +b(gh)32 b(LCF:)g(A)e(Me)-5 b(ch-)610 3029 y(anise)g(d)38 +b(L)-5 b(o)g(gic)37 b(of)g(Computation)p FT(,)h(V)-8 +b(olume)34 b(78)h(of)g FI(L)-5 b(e)g(ctur)g(e)37 b(Notes)g(in)f +(Computer)i(Scienc)-5 b(e)p FT(.)610 3142 y(Springer-V)d(erlag.)474 +3292 y(Gordon,)24 b(M.)g(J.)f(C.)g(and)f(T.)h(F.)h(Melham)f(\(1993\).)i +FI(Intr)-5 b(o)g(duction)29 b(to)d(HOL:)f(A)h(The)-5 +b(or)g(em)28 b(Pr)-5 b(oving)610 3405 y(Envir)g(onment)34 +b(for)f(Higher)g(Or)-5 b(der)33 b(L)-5 b(o)g(gic)p FT(.)31 +b(Cam)m(bridge)f(Univ)m(ersit)m(y)f(Press.)474 3555 y(Goubault,)23 +b(J.)f(\(1993,)k(August\).)c(A)g(rule-based)f(algorithm)g(for)h(rigid)e +FP(e)p FT(-uni\014cation.)h(In)g(G.)i(Got-)610 3668 y(tlob,)g(A.)e +(Leitsc)m(h,)i(and)d(D.)i(Mundici)c(\(Eds.\),)23 b FI(3r)-5 +b(d)26 b(Kurt)e(G\177)-46 b(odel)25 b(Col)5 b(lo)-5 b(quium)24 +b(\(K)n(GC\))p FT(,)e(LNCS)610 3781 y(713,)32 b(Brno,)f(Czec)m(h)g +(Republic,)d(pp.)i(202{210.)j(Springer.)474 3932 y(Gries,)28 +b(D.)g(and)f(F.)h(B.)g(Sc)m(hneider)e(\(1995\).)k(T)-8 +b(eac)m(hing)28 b(math)g(more)f(e\013ectiv)m(ely)-8 b(,)30 +b(through)d(calcu-)610 4044 y(lational)j(pro)s(ofs.)f +FI(A)n(meric)-5 b(an)33 b(Mathematic)-5 b(al)35 b(Monthly)k(102)p +FT(,)32 b(691{697.)474 4195 y(Grundy)-8 b(,)52 b(J.)c(\(1996,)55 +b(Ma)m(y\).)50 b(T)-8 b(ransformational)47 b(hierarc)m(hical)g +(reasoning.)h FI(The)h(Computer)610 4308 y(Journal)41 +b(39)12 b FT(\(4\),)33 b(291{302.)474 4458 y(Grundy)-8 +b(,)35 b(J.)f(and)g(T.)g(L)-11 b(\027)-57 b(angbac)m(k)-5 +b(a)36 b(\(1997,)i(Decem)m(b)s(er\).)e(Recording)e(HOL)g(pro)s(ofs)g +(in)f(a)i(struc-)610 4571 y(tured)23 b(bro)m(wsable)f(format.)i(In)e +(M.)i(Johnson)e(\(Ed.\),)j FI(A)n(lgebr)-5 b(aic)25 b(Metho)-5 +b(dolo)g(gy)29 b(and)e(Softwar)-5 b(e)610 4684 y(T)e(e)i(chnolo)g(gy:) +58 b(6th)41 b(International)h(Confer)-5 b(enc)g(e,)43 +b(AMAST'97)p FT(,)d(V)-8 b(olume)38 b(1349)i(of)e FI(L)-5 +b(e)g(ctur)g(e)610 4797 y(Notes)33 b(in)g(Computer)h(Scienc)-5 +b(e)p FT(,)30 b(Sydney)-8 b(,)30 b(Australia,)f(pp.)h(567{571.)j +(Springer-V)-8 b(erlag.)474 4947 y(Gun)m(ter,)27 b(E.)f(\(1990,)k +(Octob)s(er\).)c(Doing)g(algebra)g(in)f(higher)g(order)g(logic.)i(In)e +FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)31 b(of)d(the)610 5060 +y(Thir)-5 b(d)45 b(HOL)e(Users)h(Me)-5 b(eting)p FT(,)45 +b(Computer)c(Science)h(Departmen)m(t,)47 b(Aarh)m(us)42 +b(Univ)m(ersit)m(y)-8 b(,)610 5173 y(Ny)40 b(Munk)m(egade,)i(Building) +37 b(540,)43 b(DK-8000)f(Aarh)m(us)c(C,)i(Denmark.)f(T)-8 +b(ec)m(hnical)39 b(Rep)s(ort)610 5286 y(D)m(AIMI)32 b(PB)e({)h(340)h +(\(Decem)m(b)s(er)f(1990\).)474 5436 y(H\177)-45 b(ahnle,)29 +b(R.)h(and)e(P)-8 b(.)30 b(H.)g(Sc)m(hmitt)f(\(1994,)j(Octob)s(er\).)e +(The)f(lib)s(eralized)d FP(\016)s FT(-rule)j(in)g(free)g(v)-5 +b(ariable)610 5549 y(seman)m(tic)31 b(tableaux.)g FI(Journal)i(of)g(A)n +(utomate)-5 b(d)34 b(R)-5 b(e)g(asoning,)32 b(13)12 b +FT(\(2\),)33 b(211{222.)p eop +%%Page: 251 261 +251 260 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(251)474 +396 y(Hak)m(en,)49 b(A.)c(\(1985,)50 b(August\).)45 b(The)f(in)m +(tractabilit)m(y)f(of)h(resolution.)f FI(The)-5 b(or)g(etic)g(al)48 +b(Computer)610 509 y(Scienc)-5 b(e)37 b(39)12 b FT(\(2{3\),)34 +b(297{308.)474 660 y(Halmos,)46 b(P)-8 b(.)44 b(\(1983\).)i(Ho)m(w)e +(to)g(write)e(mathematics.)i(In)e(D.)i(E.)f(Sarason)g(and)g(L.)g +(Gillman)610 773 y(\(Eds.\),)31 b FI(Sele)-5 b(cta)33 +b(Exp)-5 b(ository)35 b(Writing)p FT(,)c(pp.)e(157{186.)34 +b(Springer-V)-8 b(erlag.)474 923 y(Hanna,)28 b(F.)h(K.)e(and)g(N.)h +(Daec)m(he)i(\(1985\).)h(Sp)s(eci\014cation)26 b(and)h(v)m +(eri\014cation)g(using)f(higher-order)610 1036 y(logic.)k(In)e(C.)i(J.) +f(Ko)s(omen)g(and)g(T.)g(Moto-ok)-5 b(a)32 b(\(Eds.\),)e +FI(Computer)j(Har)-5 b(dwar)g(e)34 b(Description)610 +1149 y(L)-5 b(anguages)p FT(,)32 b(pp.)d(418{433.)34 +b(Elsevier)29 b(Science)h(Publishers,)d(North-Holland.)474 +1299 y(Harrison,)i(J.)i(\(1995a,)i(August\).)e(HOL)f(done)g(righ)m(t.)g +(Unpublished)c(Draft.)474 1449 y(Harrison,)j(J.)g(\(1995b,)j(Septem)m +(b)s(er\).)e(Inductiv)m(e)e(de\014nitions:)38 b(Automation)30 +b(and)f(application.)610 1562 y(See)i(Sc)m(h)m(ub)s(ert,)f(Windley)-8 +b(,)29 b(and)h(Alv)m(es-F)-8 b(oss)31 b(\(1995\),)j(pp.)29 +b(200{213.)474 1712 y(Harrison,)24 b(J.)g(\(1996a\).)j(F)-8 +b(ormalized)24 b(mathematics.)h(T)-8 b(ec)m(hnical)23 +b(Rep)s(ort)h(36,)i(T)-8 b(urku)23 b(Cen)m(tre)h(for)610 +1825 y(Computer)29 b(Science)h(\(TUCS\),)g(Lemmink\177)-45 +b(aisenk)-5 b(atu)28 b(14)i(A,)g(FIN-20520)j(T)-8 b(urku,)29 +b(Finland.)474 1976 y(Harrison,)g(J.)h(\(1996b,)i(August\).)e(A)g +(Mizar)g(mo)s(de)g(for)f(HOL.)h(See)g(v)m(on)h(W)-8 b(righ)m(t,)30 +b(Grundy)-8 b(,)29 b(and)610 2088 y(Harrison)h(\(1996\),)j(pp.)c +(203{220.)474 2239 y(Harrison,)37 b(J.)f(\(1996c,)41 +b(July30)35 b(August{3)i(\).)g(Optimizing)d(pro)s(of)i(searc)m(h)h(in)e +(mo)s(del)g(elimina-)610 2352 y(tion.)f(In)f(M.)h(A.)g(McRobbie)f(and)g +(J.)h(K.)f(Slaney)g(\(Eds.\),)i FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)37 +b(of)f(the)g(Thirte)-5 b(enth)610 2465 y(International)40 +b(Confer)-5 b(enc)g(e)39 b(on)f(A)n(utomate)-5 b(d)39 +b(De)-5 b(duction)39 b(\(CADE-96\))p FT(,)e(V)-8 b(olume)36 +b(1104)i(of)610 2577 y FI(LNAI)p FT(,)30 b(Berlin,)f(pp.)h(313{327.)j +(Springer.)474 2728 y(Harrison,)25 b(J.)f(\(1997\).)j(Pro)s(of)d(st)m +(yle.)g(T)-8 b(ec)m(hnical)24 b(Rep)s(ort)g(410,)j(Univ)m(ersit)m(y)d +(of)g(Cam)m(bridge)g(Com-)610 2841 y(puter)35 b(Lab)s(oratory)-8 +b(,)37 b(New)e(Museums)f(Site,)i(P)m(em)m(brok)m(e)g(Street,)h(Cam)m +(bridge,)f(CB2)f(3QG,)610 2954 y(UK.)474 3104 y(Herstein,)30 +b(I.)g(\(1975\).)k FI(T)-7 b(opics)33 b(in)f(A)n(lgebr)-5 +b(a)38 b FT(\(2nd)30 b(ed.\).)h(New)g(Y)-8 b(ork:)41 +b(John)29 b(Wiley)h(&)g(Sons.)474 3254 y(Huang,)64 b(X.)58 +b(\(1994,)66 b(June/July\).)56 b(Reconstructing)h(pro)s(ofs)f(at)i(the) +g(assertion)e(lev)m(el.)i(In)610 3367 y(A.)32 b(Bundy)f(\(Ed.\),)h +FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)36 b(of)e(the)g(12th)i(International)g +(Confer)-5 b(enc)g(e)34 b(on)h(A)n(utomate)-5 b(d)610 +3480 y(De)g(duction)p FT(,)31 b(V)-8 b(olume)31 b(814)g(of)g +FI(LNAI)p FT(,)f(Berlin,)f(pp.)g(738{752.)34 b(Springer.)474 +3630 y(Huang,)24 b(X.)f(and)f(A.)h(Fiedler)e(\(1996,)26 +b(July30)c(August{3)h(\).)g(Presen)m(ting)f(mac)m(hine-found)g(pro)s +(ofs.)610 3743 y(In)29 b(M.)i(A.)f(McRobbie)f(and)h(J.)f(K.)h(Slaney)f +(\(Eds.\),)h FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)34 b(of)f(the)f(Thirte)-5 +b(enth)34 b(Inter-)610 3856 y(national)c(Confer)-5 b(enc)g(e)29 +b(on)f(A)n(utomate)-5 b(d)30 b(De)-5 b(duction)28 b(\(CADE-96\))p +FT(,)f(V)-8 b(olume)25 b(1104)i(of)e FI(LNAI)p FT(,)610 +3969 y(Berlin,)k(pp.)h(221{225.)k(Springer.)474 4119 +y(Huang,)39 b(X.)e(and)f(A.)h(Fiedler)f(\(1997\).)j(Pro)s(of)e(presen)m +(tation)f(as)i(an)e(application)f(of)i(NLG.)h(In)610 +4232 y FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)40 b(of)e(the)h(15th)g +(International)i(Joint)d(Confer)-5 b(enc)g(e)39 b(on)f(A)n(rti\014cial) +g(Intel)5 b(ligenc)-5 b(e)610 4345 y(\(IJCAI\))p FT(,)31 +b(Nago)m(y)m(a,)i(Japan.)474 4495 y(Hullot,)e(J.-M.)i(\(1980\).)h +(Canonical)c(forms)h(and)g(uni\014cation.)f(In)h(W.)i(Bib)s(el)d(and)h +(R.)g(Ko)m(w)m(alski)610 4608 y(\(Eds.\),)i FI(Pr)-5 +b(o)g(c)g(e)g(e)g(dings)37 b(of)d(the)h(Fifth)g(Confer)-5 +b(enc)g(e)35 b(on)g(A)n(utomate)-5 b(d)36 b(De)-5 b(duction)p +FT(,)33 b(V)-8 b(olume)32 b(87)610 4721 y(of)f FI(L)-5 +b(e)g(ctur)g(e)33 b(Notes)g(in)f(Computer)i(Scienc)-5 +b(e)p FT(,)31 b(pp.)e(318{334.)34 b(Les)c(Arc:)41 b(Springer.)474 +4871 y(Hutter,)35 b(D.)f(\(1997,)j(June\).)c(Coloring)f(terms)h(to)i +(con)m(trol)f(equational)f(reasoning.)g FI(Journal)k(of)610 +4984 y(A)n(utomate)-5 b(d)34 b(R)-5 b(e)g(asoning)40 +b(18)12 b FT(\(3\),)33 b(399{442.)474 5135 y(Hutter,)23 +b(D.)e(and)f(M.)h(Kohlhase)e(\(1997,)25 b(July13{17)c(\).)g(A)f +(colored)h(v)m(ersion)f(of)g(the)h FP(\025)p FT(-Calculus.)e(In)610 +5247 y(W.)k(McCune)e(\(Ed.\),)j FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)27 +b(of)e(the)g(14th)h(International)h(Confer)-5 b(enc)g(e)25 +b(on)h(A)n(utomate)-5 b(d)610 5360 y(de)g(duction)p FT(,)32 +b(V)-8 b(olume)30 b(1249)j(of)d FI(LNAI)p FT(,)g(Berlin,)f(pp.)g +(291{305.)34 b(Springer.)474 5511 y(Jac)m(kson,)29 b(P)-8 +b(.)29 b(B.)f(\(1995,)j(Jan)m(uary\).)e FI(Enhancing)h(the)h(Nuprl)g +(Pr)-5 b(o)g(of)32 b(Development)f(System)g(and)610 5624 +y(Applying)i(it)g(to)f(Computational)k(A)n(bstr)-5 b(act)33 +b(A)n(lgebr)-5 b(a)p FT(.)30 b(Ph.)g(D.)g(thesis,)g(Cornell)e(Univ)m +(ersit)m(y)-8 b(.)p eop +%%Page: 252 262 +252 261 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(252)474 +396 y(Jacobs,)31 b(B.)f(and)g(T.)g(F.)h(Melham)e(\(1993\).)k(T)-8 +b(ranslating)29 b(dep)s(enden)m(t)g(t)m(yp)s(e)i(theory)f(in)m(to)g +(higher)610 509 y(order)f(logic.)g(In)f FI(TLCA)j('93)h(International)i +(Confer)-5 b(enc)g(e)33 b(on)e(T)-7 b(yp)i(e)g(d)33 b(L)-5 +b(amb)g(da)34 b(Calculi)e(and)610 622 y(Applic)-5 b(ations,)38 +b(Utr)-5 b(e)g(cht,)37 b(16{18)g(Mar)-5 b(ch)36 b(1993)p +FT(,)g(V)-8 b(olume)33 b(664)h(of)f FI(L)-5 b(e)g(ctur)g(e)36 +b(Notes)g(in)f(Com-)610 735 y(puter)e(Scienc)-5 b(e)p +FT(,)31 b(pp.)e(209{229.)34 b(Springer-V)-8 b(erlag.)474 +885 y(Je\013rey)g(,)28 b(R.)f(C.)f(\(1967\).)j FI(F)-7 +b(ormal)31 b(L)-5 b(o)g(gic:)41 b(Its)29 b(Sc)-5 b(op)g(e)31 +b(and)f(Limits)p FT(.)d(New)g(Y)-8 b(ork,)28 b(N.Y.:)39 +b(McGra)m(w-)610 998 y(Hill)29 b(Bo)s(ok)i(Co.)474 1149 +y(Jouannaud,)g(J.-P)-8 b(.)33 b(and)f(C.)g(Kirc)m(hner)e(\(1991\).)35 +b(Solving)c(equations)h(in)e(abstract)k(algebras:)44 +b(A)610 1262 y(rule-based)28 b(surv)m(ey)h(of)h(uni\014cation.)e(In)g +(J.-L.)i(Lassez)g(and)f(G.)h(Plotkin)e(\(Eds.\),)i FI(Computa-)610 +1374 y(tional)k(L)-5 b(o)g(gic:)43 b(Essays)33 b(in)f(Honor)i(of)f(A)n +(lan)f(R)-5 b(obinson)p FT(.)33 b(MIT-Press.)474 1525 +y(Jo)m(yce,)k(J.)d(J.)g(and)g(C.-J.)h(H.)f(Seger)h(\(Eds.\))g(\(1993,)i +(August\).)e FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)38 b(of)f(the)g(6th)g +(Inter-)610 1638 y(national)e(Workshop)g(on)e(Higher)f(Or)-5 +b(der)34 b(L)-5 b(o)g(gic)33 b(The)-5 b(or)g(em)35 b(Pr)-5 +b(oving)33 b(and)h(its)f(Applic)-5 b(ations)610 1751 +y(\(HUG'93\))p FT(,)35 b(V)-8 b(olume)33 b(780)i(of)e +FI(L)-5 b(e)g(ctur)g(e)36 b(Notes)f(in)g(Computer)i(Scienc)-5 +b(e)p FT(,)34 b(V)-8 b(ancouv)m(er,)35 b(B.C.,)610 1863 +y(Canada.)c(Springer-V)-8 b(erlag,)29 b(1994.)474 2014 +y(Kalv)-5 b(ala,)31 b(S.)f(\(1994\).)k(Annotations)c(in)g(formal)g(sp)s +(eci\014cations)g(and)g(pro)s(ofs.)g FI(F)-7 b(ormal)36 +b(Metho)-5 b(ds)610 2127 y(in)33 b(System)g(Design)k(5)p +FT(,)31 b(119{144.)474 2277 y(Kamin,)d(S.)h(and)f(J.-J.)h(L)m(\023)-43 +b(evy)31 b(\(1980\).)g(Tw)m(o)f(generalizations)e(of)h(the)h(recursiv)m +(e)e(path)h(ordering.)610 2390 y(Unpublished)e(man)m(uscript.)474 +2540 y(Kamm)s(\177)-48 b(uller,)26 b(F.)h(\(1997\).)j(F)-8 +b(ormal)28 b(pro)s(of)e(of)h(Sylo)m(w's)g(theorem.)h(Submitted)d(to)j +(the)f(Journal)f(of)610 2653 y(Automated)32 b(Reasoning.)474 +2803 y(Kapur,)e(D.)i(\(1997\).)i(Shostak's)e(congruence)g(closure)e(as) +i(completion.)f(In)f FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)36 +b(of)d(the)610 2916 y(8th)44 b(International)h(Confer)-5 +b(enc)g(e)43 b(on)g(R)-5 b(ewriting)44 b(T)-7 b(e)i(chniques)42 +b(and)i(Applic)-5 b(ations)44 b(\(R)-7 b(T)g(A-)610 3029 +y(97\))p FT(,)32 b(V)-8 b(olume)30 b(1232)i(of)f FI(LNCS)p +FT(,)f(Berlin,)f(pp.)h(23{37.)j(Springer-V)-8 b(erlag.)474 +3179 y(Kerb)s(er,)46 b(M.)e(\(1990\).)i(Ho)m(w)e(to)g(pro)m(v)m(e)h +(higher)d(order)h(theorems)g(in)g(\014rst)f(order)h(logic.)h(Seki)610 +3292 y(Rep)s(ort)30 b(SR-90-19,)j(F)-8 b(ac)m(h)m(b)s(ereic)m(h)31 +b(Informatik,)f(Univ)m(ersit\177)-45 b(at)30 b(Kaiserslautern,)f +(German)m(y)-8 b(.)474 3443 y(Kleiner,)27 b(I.)i(and)f(N.)i(Mo)m(vsho)m +(vitz-Hadar)g(\(1994,)i(Decem)m(b)s(er\).)e(The)e(role)h(of)g(parado)m +(xes)g(in)f(the)610 3555 y(ev)m(olution)i(of)h(mathematics.)g +FI(A)n(meric)-5 b(an)33 b(Mathematic)-5 b(al)35 b(Monthly)k(101)12 +b FT(\(10\),)34 b(963{974.)474 3706 y(Klop,)40 b(J.)e(W.)h(\(1992\).)j +(T)-8 b(erm)38 b(rewriting)f(systems.)h(In)g(S.)h(Abramsky)-8 +b(,)40 b(D.)f(M.)h(Gabba)m(y)-8 b(,)41 b(and)610 3819 +y(T.)g(S.)g(E.)g(Maibaum)f(\(Eds.\),)k FI(Handb)-5 b(o)g(ok)44 +b(of)e(L)-5 b(o)g(gic)43 b(in)f(Computer)i(Scienc)-5 +b(e)p FT(,)43 b(V)-8 b(olume)41 b(2,)610 3932 y(Chapter)30 +b(1,)h(pp.)f(1{116.)i(Oxford:)40 b(Oxford)29 b(Univ)m(ersit)m(y)h +(Press.)474 4082 y(Kn)m(uth,)k(D.)g(E.)f(\(1992\).)k +FI(Liter)-5 b(ate)36 b(Pr)-5 b(o)g(gr)g(amming)p FT(.)37 +b(CSLI)32 b(Lecture)i(Notes)h(Num)m(b)s(er)d(27.)j(Stan-)610 +4195 y(ford,)30 b(CA,)f(USA:)h(Stanford)f(Univ)m(ersit)m(y)g(Cen)m(ter) +h(for)f(the)h(Study)f(of)h(Language)g(and)f(Infor-)610 +4308 y(mation.)i(Distributed)d(b)m(y)i(the)h(Univ)m(ersit)m(y)e(of)i +(Chicago)f(Press.)474 4458 y(Kn)m(uth,)35 b(D.)h(E.)f(and)g(P)-8 +b(.)35 b(E.)h(Bendix)e(\(1970\).)k(Simple)32 b(w)m(ord)j(problems)e(in) +h(univ)m(ersal)g(algebra.)610 4571 y(In)39 b(J.)g(Leec)m(h)h(\(Ed.\),)i +FI(Computational)i(Pr)-5 b(oblems)41 b(in)g(A)n(bstr)-5 +b(act)41 b(A)n(lgebr)-5 b(a,)43 b(Pr)-5 b(o)g(c)g(e)g(e)g(dings)43 +b(of)610 4684 y(a)37 b(Confer)-5 b(enc)g(e)36 b(Held)g(at)h(Oxfor)-5 +b(d)37 b(Under)f(the)g(A)n(uspic)-5 b(es)36 b(of)g(the)g(Scienc)-5 +b(e)36 b(R)-5 b(ese)g(ar)g(ch)38 b(Coun-)610 4797 y(cil,)e(A)n(tlas)f +(Computer)h(L)-5 b(ab)g(or)g(atory,)40 b(29.)35 b(A)n(ug.)f(to)i(2.)f +(Sept.)h(1967)p FT(,)f(Oxford,)e(pp.)g(263{297.)610 4910 +y(P)m(ergamon)f(Press.)474 5060 y(Ko)s(etsier,)49 b(T.)c(\(1991\).)j +FI(L)-5 b(akatos')48 b(Philosophy)h(of)d(Mathematics,)51 +b(A)46 b(Historic)-5 b(al)48 b(Appr)-5 b(o)g(ach)p FT(.)610 +5173 y(Amsterdam:)41 b(North-Holland.)474 5323 y(Kogel,)66 +b(E.)59 b(D.)g(\(1995,)68 b(Ma)m(y\).)61 b(Rigid)c(E-uni\014cation)g +(simpli\014ed.)e(In)j(P)-8 b(.)59 b(Baumgartner,)610 +5436 y(R.)43 b(H\177)-45 b(ahnle,)44 b(and)e(J.)g(P)m(osegga)j +(\(Eds.\),)g FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)46 b(of)e(the)g(4th)g +(International)i(Work-)610 5549 y(shop)39 b(on)g(The)-5 +b(or)g(em)39 b(Pr)-5 b(oving)38 b(with)h(A)n(nalytic)e(T)-7 +b(able)i(aux)39 b(and)g(R)-5 b(elate)g(d)39 b(Metho)-5 +b(ds)p FT(,)39 b(V)-8 b(olume)610 5662 y(918)32 b(of)e +FI(LNAI)p FT(,)g(Berlin,)f(pp.)h(17{30.)j(Springer.)p +eop +%%Page: 253 263 +253 262 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(253)474 +396 y(Kohlhase,)41 b(M.)f(\(1995,)j(Ma)m(y\).)f(Higher-order)c +(tableaux.)h(In)g(P)-8 b(.)40 b(Baumgartner,)i(R.)d(H\177)-45 +b(ahnle,)610 509 y(and)24 b(J.)g(P)m(osegga)j(\(Eds.\),)f +FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)29 b(of)e(the)h(4th)g(International)h 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b(\(1976\).)k FI(Pr)-5 b(o)g(ofs)46 +b(and)g(R)-5 b(efutations:)68 b(The)46 b(lo)-5 b(gic)45 +b(of)g(Mathematic)-5 b(al)47 b(Disc)-5 b(overy)p FT(.)610 +1788 y(Cam)m(bridge)30 b(Univ)m(ersit)m(y)f(Press.)h(Edited)g(b)m(y)g +(John)f(W)-8 b(orrall)30 b(and)g(Elie)f(G.)i(Zahar.)474 +1938 y(Lamp)s(ort,)e(L.)g(\(1995,)j(August/Septem)m(b)s(er\).)e(Ho)m(w) +g(to)g(write)f(a)g(pro)s(of.)g FI(A)n(meric)-5 b(an)32 +b(Mathemat-)610 2051 y(ic)-5 b(al)34 b(Monthly)39 b(102)12 +b FT(\(7\),)33 b(600{608.)474 2201 y(Lecat,)f(M.)f(\(1935\).)i +FI(Err)-5 b(eurs)33 b(de)g(Math)n(\023)-44 b(ematiciens)p +FT(.)32 b(Brussels.)474 2352 y(Letz,)k(R.)d(\(1993,)k(June\).)c +FI(First-Or)-5 b(der)37 b(Calculi)f(and)h(Pr)-5 b(o)g(of)37 +b(Pr)-5 b(o)g(c)g(e)g(dur)g(es)38 b(for)e(A)n(utomate)-5 +b(d)37 b(De-)610 2465 y(duction)p FT(.)32 b(Ph.)d(D.)i(thesis,)f(T)-8 +b(ec)m(hnisc)m(he)31 b(Ho)s(c)m(hsc)m(h)m(ule)g(Darmstadt.)474 +2615 y(Lo)m(v)m(eland,)41 b(D.)e(W.)g(\(1968,)j(April\).)37 +b(Mec)m(hanical)i(theorem-pro)m(ving)f(b)m(y)g(mo)s(del)f(elimination.) +610 2728 y FI(Journal)d(of)f(the)g(A)n(CM)44 b(15)12 +b FT(\(2\),)33 b(236{251.)474 2878 y(Luo,)c(Z.)f(and)g(R.)h(P)m(ollac)m +(k)g(\(1992,)i(Ma)m(y\).)g(The)d(LEGO)g(pro)s(of)g(dev)m(elopmen)m(t)h +(system:)40 b(A)28 b(user's)610 2991 y(man)m(ual.)i(T)-8 +b(ec)m(hnical)30 b(Rep)s(ort)g(ECS-LF)m(CS-92-211,)j(Univ)m(ersit)m(y)d +(of)g(Edin)m(burgh.)474 3141 y(MacKenzie,)35 b(D.)e(\(1995,)j(F)-8 +b(all\).)33 b(The)g(automation)g(of)g(pro)s(of:)45 b(an)33 +b(historical)e(and)h(so)s(ciological)610 3254 y(exploration.)e +FI(IEEE)i(A)n(nnals)g(of)h(the)g(History)h(of)f(Computing)39 +b(17)12 b FT(\(3\),)33 b(7{29.)474 3404 y(Martin-L\177)-45 +b(of,)60 b(P)-8 b(.)55 b(\(1984\).)i FI(Intuitionistic)e(T)-7 +b(yp)i(e)55 b(The)-5 b(ory)p FT(.)57 b(Nap)s(oli:)87 +b(Bibioplois.)51 b(Notes)56 b(of)610 3517 y(Gio)m(w)m(anni)30 +b(Sam)m(bin)f(on)h(a)h(series)e(of)i(lectues)f(giv)m(en)h(in)e(P)m(ado) +m(v)-5 b(a.)474 3668 y(McCune,)35 b(W.)g(\(1997,)j(Decem)m(b)s(er\).)d +(Solution)e(of)h(the)h(Robbins)d(problem.)h FI(Journal)k(of)g(A)n(uto-) +610 3780 y(mate)-5 b(d)34 b(R)-5 b(e)g(asoning)41 b(19)12 +b FT(\(3\),)33 b(263{276.)474 3931 y(Melham,)44 b(T.)c(F.)i(\(1988,)k +(July\).)40 b(Using)g(recursiv)m(e)h(t)m(yp)s(es)g(to)h(reason)f(ab)s +(out)g(hardw)m(are)f(and)610 4044 y(higher)29 b(order)h(logic.)g(In)f +(G.J.)i(Milne)e(\(Ed.\),)i FI(International)j(Workshop)h(on)e(Higher)f +(Or)-5 b(der)610 4157 y(L)g(o)g(gic)34 b(The)-5 b(or)g(em)35 +b(Pr)-5 b(oving)33 b(and)h(its)f(Applic)-5 b(ations)p +FT(,)33 b(Glasgo)m(w,)f(Scotland,)f(pp.)f(27{50.)j(IFIP)610 +4269 y(W)m(G)f(10.2:)42 b(North-Holland.)474 4420 y(Melham,)32 +b(T.)g(F.)h(\(1991,)i(August\).)d(A)h(pac)m(k)-5 b(age)34 +b(for)e(inductiv)m(e)e(relation)i(de\014nitions)d(in)i(HOL.)610 +4533 y(In)f(M.)h(Arc)m(her,)g(J.)g(J.)f(Jo)m(yce,)j(K.)d(N.)h(Levitt,)g +(and)f(P)-8 b(.)31 b(J.)g(Windley)e(\(Eds.\),)i FI(Pr)-5 +b(o)g(c)g(e)g(e)g(dings)35 b(of)610 4646 y(the)k(1991)h(International)h +(Workshop)g(on)d(the)h(HOL)f(The)-5 b(or)g(em)40 b(Pr)-5 +b(oving)39 b(System)g(and)h(its)610 4758 y(Applic)-5 +b(ations)p FT(,)28 b(Da)m(vis,)e(California,)d(USA,)i(pp.)e(350{357.)28 +b(IEEE)23 b(Computer)h(So)s(ciet)m(y)g(Press,)610 4871 +y(1992.)474 5022 y(Melham,)30 b(T.)g(F.)h(\(1992,)i(Septem)m(b)s(er\).) +d(The)g(HOL)g(logic)g(extended)h(with)e(quan)m(ti\014cation)h(o)m(v)m +(er)610 5135 y(t)m(yp)s(e)22 b(v)-5 b(ariables.)21 b(In)h(L.)g(J.)f(M.) +i(Claesen)e(and)h(M.)g(J.)g(C.)g(Gordon)g(\(Eds.\),)i +FI(Higher)g(Or)-5 b(der)26 b(L)-5 b(o)g(gic)610 5247 +y(The)g(or)g(em)42 b(Pr)-5 b(oving)41 b(and)g(its)g(Applic)-5 +b(ations:)58 b(Pr)-5 b(o)g(c)g(e)g(e)g(dings)43 b(of)d(the)h(IFIP)f +(TC10/WG10.2)610 5360 y(Workshop)p FT(,)28 b(V)-8 b(olume)25 +b(A-20)h(of)f FI(IFIP)i(T)-7 b(r)i(ansactions)p FT(,)29 +b(Leuv)m(en,)d(Belgium,)f(pp.)f(3{18.)j(North-)610 5473 +y(Holland/Elsevier.)p eop +%%Page: 254 264 +254 263 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(254)474 +396 y(M.J.C.)40 b(Gordon)g(\(1988\).)j(HOL:)d(A)g(pro)s(of)f +(generating)h(system)g(for)g(higher-order)f(logic.)h(In)610 +509 y(G.M.)30 b(Birt)m(wistle)d(and)g(P)-8 b(.A.)29 b(Subrahman)m(y)m +(am)f(\(Eds.\),)h FI(VLSI)h(Sp)-5 b(e)g(ci\014c)g(ation,)33 +b(V)-7 b(eri\014c)i(ation)610 622 y(and)34 b(Synthesis)p +FT(,)d(pp.)f(73{128.)j(Boston:)42 b(Klu)m(w)m(er)29 b(Academic)i +(Publishers.)474 769 y(Naur,)f(P)-8 b(.)31 b(\(1994\).)i(Pro)s(of)d(v)m +(ersus)g(formalization.)g FI(BIT:)i(BIT)43 b(34)p FT(,)31 +b(148{164.)474 915 y(Nelson,)37 b(G.)f(and)f(D.)i(C.)e(Opp)s(en)f +(\(1980,)39 b(April\).)34 b(F)-8 b(ast)38 b(decision)c(pro)s(cedures)g +(based)i(on)f(con-)610 1028 y(gruence)c(closure.)f FI(Journal)k(of)e +(the)h(A)n(CM)44 b(27)12 b FT(\(2\),)33 b(356{364.)474 +1174 y(Newman,)40 b(M.)f(H.)f(A.)h(\(1942\).)i(On)c(theories)h(with)f +(a)h(com)m(binatorial)g(de\014nition)e(of)i(`equiv)-5 +b(a-)610 1287 y(lence'.)31 b FI(A)n(nnals)i(of)g(Mathematics)39 +b(43)12 b FT(\(2\),)33 b(223{243.)474 1433 y(Nieu)m(w)m(enh)m(uis,)h +(R.)g(\(1993,)j(August\).)e(Simple)d(LPO)h(constrain)m(t)h(solving)f +(metho)s(ds.)h FI(Informa-)610 1546 y(tion)f(Pr)-5 b(o)g(c)g(essing)34 +b(L)-5 b(etters)39 b(47)12 b FT(\(2\),)33 b(65{69.)474 +1692 y(Nieu)m(w)m(enh)m(uis,)45 b(R.)e(and)g(A.)h(Rubio)d(\(1995,)49 +b(Ma)m(y\).)c(Theorem)e(pro)m(ving)g(with)f(ordering)f(and)610 +1805 y(equalit)m(y)30 b(constrained)g(clauses.)g FI(Journal)k(of)f +(Symb)-5 b(olic)33 b(Computation)40 b(19)12 b FT(\(4\),)33 +b(321{351.)474 1951 y(Nordstr\177)-45 b(om,)37 b(B.,)g(K.)f(P)m +(etersson,)h(and)e(J.)h(M.)g(Smith)d(\(1990\).)39 b FI(Pr)-5 +b(o)g(gr)g(amming)39 b(in)f(Martin-L\177)-46 b(of)610 +2064 y(typ)-5 b(e)34 b(the)-5 b(ory:)43 b(an)33 b(intr)-5 +b(o)g(duction)p FT(.)33 b(Clarendon.)474 2210 y(P)m(aren)m(t,)g(C.)e +(\(1993,)j(Ma)m(y\).)f(Dev)m(eloping)e(certi\014ed)f(programs)h(in)f +(the)i(system)f(Co)s(q)g(-)g(the)h(Pro-)610 2323 y(gram)e(tactic.)h(In) +f(H.)g(Barendregt)g(and)f(T.)h(Nipk)m(o)m(w)g(\(Eds.\),)g +FI(International)k(Workshop)g(on)610 2436 y(T)-7 b(yp)i(es)26 +b(for)g(Pr)-5 b(o)g(ofs)26 b(and)g(Pr)-5 b(o)g(gr)g(ams)p +FT(,)27 b(V)-8 b(olume)22 b(806)h(of)f FI(L)-5 b(e)g(ctur)g(e)26 +b(Notes)f(in)g(Computer)h(Scienc)-5 b(e)p FT(,)610 2549 +y(pp.)30 b(291{312.)k(Springer-V)-8 b(erlag.)474 2695 +y(P)m(aulin-Mohring,)40 b(C.)f(\(1989,)44 b(Jan)m(uary\).)c(Extracting) +g FP(F)2519 2709 y FO(!)2570 2695 y FT('s)f(programs)h(from)f(pro)s +(ofs)f(in)h(the)610 2808 y(Calculus)32 b(of)i(Constructions.)f(In)g(A.) +h(for)g(Computing)e(Mac)m(hinery)i(\(Ed.\),)h FI(Sixte)-5 +b(enth)37 b(A)n(n-)610 2921 y(nual)c(A)n(CM)f(Symp)-5 +b(osium)34 b(on)f(Principles)g(of)g(Pr)-5 b(o)g(gr)g(amming)35 +b(L)-5 b(anguages)p FT(,)32 b(Austin.)474 3067 y(P)m(aulin-Mohring,)21 +b(C.)g(and)f(B.)h(W)-8 b(erner)22 b(\(1993,)j(??\).)c(Syn)m(thesis)f +(of)h(ML)g(programs)g(in)f(the)h(system)610 3180 y(Co)s(q.)30 +b FI(Journal)k(of)f(Symb)-5 b(olic)34 b(Computation)39 +b(15)12 b FT(\(5-6\),)34 b(607{640.)474 3327 y(P)m(aulson,)45 +b(L.)d(C.)g(\(1994\).)j FI(Isab)-5 b(el)5 b(le:)65 b(a)44 +b(generic)f(the)-5 b(or)g(em)46 b(pr)-5 b(over)p FT(,)47 +b(V)-8 b(olume)42 b(828)i(of)e FI(L)-5 b(e)g(ctur)g(e)610 +3440 y(Notes)33 b(in)g(Computer)h(Scienc)-5 b(e)p FT(.)30 +b(New)g(Y)-8 b(ork,)32 b(NY,)e(USA:)h(Springer-V)-8 b(erlag)29 +b(Inc.)474 3586 y(P)m(eano,)j(G.)f(\(1895{97\).)j FI(F)-7 +b(ormulair)i(e)35 b(de)e(Math)n(\023)-44 b(ematiques)p +FT(.)474 3732 y(Plaisted,)24 b(D.)h(A.)g(\(1993a\).)i(Equational)c +(reasoning)g(and)h(term)g(rewriting)e(systems.)i(In)g(D.)g(Gab-)610 +3845 y(ba)m(y)-8 b(,)40 b(C.)c(Hogger,)k(J.)d(A.)g(Robinson,)g(and)f +(J.)h(Siekmann)e(\(Eds.\),)k FI(Handb)-5 b(o)g(ok)40 +b(of)f(L)-5 b(o)g(gic)39 b(in)610 3958 y(A)n(rti\014cial)30 +b(Intel)5 b(ligenc)-5 b(e)29 b(and)h(L)-5 b(o)g(gic)30 +b(Pr)-5 b(o)g(gr)g(amming)p FT(,)30 b(V)-8 b(olume)26 +b(1,)i(Chapter)f(5,)h(pp.)e(273{364.)610 4071 y(Oxford:)40 +b(Oxford)29 b(Univ)m(ersit)m(y)h(Press.)474 4217 y(Plaisted,)21 +b(D.)g(A.)g(\(1993b\).)i(P)m(olynomial)c(time)h(termination)f(and)h +(constrain)m(t)h(satisfaction)f(tests.)610 4330 y(In)h(C.)g(Kirc)m +(hner)f(\(Ed.\),)k FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)27 +b(of)d(the)h(5th)h(International)h(Confer)-5 b(enc)g(e)25 +b(on)g(R)-5 b(ewriting)610 4443 y(T)e(e)i(chniques)25 +b(and)h(Applic)-5 b(ations)28 b(\(R)-7 b(T)g(A-93\))p +FT(,)24 b(V)-8 b(olume)23 b(690)g(of)f FI(LNCS)p FT(,)h(Berlin,)f(pp.)g +(405{420.)610 4556 y(Springer-V)-8 b(erlag.)474 4702 +y(Plaisted,)25 b(D.)h(A.)f(\(1995\).)j(Sp)s(ecial)23 +b(cases)j(and)f(substitutes)e(for)i(rigid)e FP(E)5 b +FT(-uni\014cation.)24 b(T)-8 b(ec)m(hnical)610 4815 y(Rep)s(ort)30 +b(MPI-I-95-2-010,)35 b(Max-Planc)m(k-Institut)30 b(f)s(\177)-48 +b(ur)30 b(Informatik,)f(Saarbr)s(\177)-48 b(uc)m(k)m(en.)474 +4961 y(Praset)m(y)m(a,)34 b(I.)d(S.)h(W.)g(B.)g(\(1993,)i(August\).)e +(On)f(the)h(st)m(yle)g(of)f(mec)m(hanical)h(pro)m(ving.)f(See)h(Jo)m +(yce)610 5074 y(and)e(Seger)h(\(1993\),)i(pp.)c(475{488.)474 +5220 y(Putnam,)40 b(H.)e(\(1979\).)j(Philosoph)m(y)36 +b(of)i(mathematics:)57 b(A)38 b(rep)s(ort.)g(In)f FI(Curr)-5 +b(ent)41 b(R)-5 b(ese)g(ar)g(ch)42 b(in)610 5333 y(Philosophy)29 +b(of)d(Scienc)-5 b(e)p FT(,)25 b(pp.)e(386{398.)k(East)d(Lansing)e(Mic) +m(higan:)37 b(Philosoph)m(y)21 b(of)j(Science)610 5446 +y(Asso)s(ciation.)474 5592 y(Robinson,)37 b(J.)g(A.)g(\(1965,)k(Jan)m +(uary\).)c(A)g(mac)m(hine-orien)m(ted)g(logic)g(based)f(on)h(the)g +(resolution)610 5705 y(principle.)28 b FI(Journal)33 +b(of)g(the)g(A)n(CM)44 b(12)12 b FT(\(1\),)33 b(23{41.)p +eop +%%Page: 255 265 +255 264 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(255)474 +396 y(Robinson,)28 b(J.)h(A.)h(\(1971\).)i(Computational)c(logic:)40 +b(The)29 b(uni\014cation)e(computation.)j FI(Machine)610 +509 y(Intel)5 b(ligenc)-5 b(e)37 b(6)p FT(,)31 b(63{72.)474 +658 y(Robinson,)d(P)-8 b(.)29 b(J.)f(and)g(J.)h(Staples)f(\(1993,)j(F) +-8 b(ebruary\).)29 b(F)-8 b(ormalizing)28 b(a)h(hierarc)m(hical)e +(structure)610 771 y(of)g(practical)g(mathematical)g(reasoning.)f +FI(Journal)k(of)g(L)-5 b(o)g(gic)30 b(and)g(Computation)36 +b(3)12 b FT(\(1\),)30 b(47{)610 884 y(61.)474 1032 y(Ro)m(xas,)d(R.)d +(E.)h(O.)f(\(1993,)k(August\).)d(A)f(HOL)g(pac)m(k)-5 +b(age)27 b(for)d(reasoning)g(ab)s(out)g(relations)f(de\014ned)610 +1145 y(b)m(y)31 b(m)m(utual)e(induction.)f(See)j(Jo)m(yce)h(and)e +(Seger)g(\(1993\),)j(pp.)d(129{140.)474 1293 y(Rudnic)m(ki,)38 +b(P)-8 b(.)38 b(\(1987,)k(Decem)m(b)s(er\).)d(Ob)m(vious)d(inferences.) +i FI(Journal)i(of)g(A)n(utomate)-5 b(d)40 b(R)-5 b(e)g(ason-)610 +1406 y(ing)38 b(3)12 b FT(\(4\),)33 b(383{394.)474 1554 +y(Rudnic)m(ki,)26 b(P)-8 b(.)29 b(\(1992,)i(June\).)d(An)g(o)m(v)m +(erview)g(of)h(the)f(MIZAR)g(pro)5 b(ject.)29 b(Av)-5 +b(ailable)27 b(b)m(y)h(ftp)f(from)610 1667 y FM(menaik.cs.ualberta.ca)e +FT(as)30 b FM(pub/Mizar/Mizar_Over.tar.Z)o FT(.)474 1815 +y(Rudnic)m(ki,)35 b(P)-8 b(.)37 b(and)f(A.)g(T)-8 b(rybulec)35 +b(\(1997,)40 b(Jan)m(uary\).)c(On)f(equiv)-5 b(alen)m(ts)36 +b(of)g(w)m(ell-foundedness.)610 1928 y(Av)-5 b(ailable)29 +b(on)i(the)f(w)m(eb)g(at)i FM(http://www.cs.ualberta.)o(ca/)o(~pio)o +(tr/M)o(iza)o(r/Wf)o(nd/)p FT(.)474 2077 y(Sc)m(h)m(ub)s(ert,)c(E.)h +(T.,)g(P)-8 b(.)29 b(J.)g(Windley)-8 b(,)28 b(and)g(J.)g(Alv)m(es-F)-8 +b(oss)30 b(\(Eds.\))f(\(1995,)j(Septem)m(b)s(er\).)c +FI(Pr)-5 b(o)g(c)g(e)g(e)g(d-)610 2190 y(ings)35 b(of)g(the)h(8th)g +(International)h(Workshop)g(on)e(Higher)g(Or)-5 b(der)36 +b(L)-5 b(o)g(gic)36 b(The)-5 b(or)g(em)37 b(Pr)-5 b(oving)610 +2303 y(and)39 b(Its)e(Applic)-5 b(ations)p FT(,)40 b(V)-8 +b(olume)35 b(971)i(of)f FI(L)-5 b(e)g(ctur)g(e)38 b(Notes)g(in)f +(Computer)i(Scienc)-5 b(e)p FT(,)37 b(Asp)s(en)610 2415 +y(Gro)m(v)m(e,)c(UT,)d(USA.)g(Springer-V)-8 b(erlag.)474 +2564 y(Shank)j(ar,)30 b(N.,)h(S.)g(Owre,)f(and)g(J.)h(M.)g(Rush)m(b)m +(y)f(\(1993,)j(F)-8 b(ebruary\).)31 b FI(The)i(PVS)f(Pr)-5 +b(o)g(of)35 b(Che)-5 b(cker:)610 2677 y(A)32 b(R)-5 b(efer)g(enc)g(e)34 +b(Manual)p FT(.)d(Menlo)f(P)m(ark,)h(CA:)f(Computer)g(Science)g(Lab)s +(oratory)-8 b(,)31 b(SRI)e(In)m(ter-)610 2790 y(national.)474 +2938 y(Shostak,)g(R.)f(E.)h(\(1978,)i(July\).)c(An)h(algorithm)f(for)h +(reasoning)g(ab)s(out)g(equalit)m(y)-8 b(.)29 b FI(Communic)-5 +b(a-)610 3051 y(tions)34 b(of)e(the)i(A)n(CM)44 b(21)12 +b FT(\(7\),)32 b(583{585.)474 3199 y(Siekmann,)e(J.)h(H.)h(\(1989,)i +(Marc)m(h{April\).)d(Uni\014cation)f(theory)-8 b(.)33 +b FI(Journal)h(of)g(Symb)-5 b(olic)35 b(Com-)610 3312 +y(putation)k(7)12 b FT(\(3-4\),)33 b(207{274.)474 3460 +y(Simons,)44 b(M.)f(\(1996,)48 b(Decem)m(b)s(er\).)c +FI(The)g(Pr)-5 b(esentation)45 b(of)f(F)-7 b(ormal)46 +b(Pr)-5 b(o)g(ofs)p FT(.)44 b(Ph.)e(D.)h(thesis,)610 +3573 y(T)-8 b(ec)m(hnisc)m(he)31 b(Univ)m(ersit\177)-45 +b(at)30 b(Berlin.)474 3722 y(Slind,)41 b(K.)f(\(1991,)46 +b(No)m(v)m(em)m(b)s(er\).)d(Ob)5 b(ject)41 b(language)g(em)m(b)s +(edding)e(in)g(Standard)h(ML)h(of)g(New)610 3834 y(Jersey)-8 +b(.)31 b(In)f FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)35 b(of)e(the)g(Se)-5 +b(c)g(ond)34 b(ML)e(Workshop)i(held)g(at)f(Carne)-5 b(gie)33 +b(Mel)5 b(lon)33 b(Uni-)610 3947 y(versity,)43 b(Pittsbugh,)e +(Pennsylvania,)i(Septermb)-5 b(er)42 b(26-27,)h(1991,)h(CMU)39 +b(SCS)i(T)-7 b(e)i(chnic)g(al)610 4060 y(R)g(ep)g(ort)p +FT(.)474 4209 y(Slind,)32 b(K.)i(\(1996,)k(August\).)c(F)-8 +b(unction)34 b(de\014nition)e(in)h(higher-order)f(logic.)i(See)h(v)m +(on)f(W)-8 b(righ)m(t,)610 4322 y(Grundy)g(,)30 b(and)f(Harrison)h +(\(1996\),)j(pp.)c(381{397.)474 4470 y(Sm)m(ully)m(an,)21 +b(R.)g(M.)g(\(1995\).)j FI(First-Or)-5 b(der)25 b(L)-5 +b(o)g(gic)27 b FT(\(Second)21 b(corrected)h(ed.\).)f(Do)m(v)m(er)i +(Publications,)610 4583 y(New)31 b(Y)-8 b(ork.)31 b(First)f(published)c +(1968)32 b(b)m(y)f(Springer-V)-8 b(erlag.)474 4731 y(Sommerhalder,)32 +b(R.)g(and)g(S.)h(v)-5 b(an)32 b(W)-8 b(estrhenen)33 +b(\(1988\).)j FI(The)e(the)-5 b(ory)37 b(of)e(c)-5 b(omputability:)48 +b(pr)-5 b(o-)610 4844 y(gr)g(ams,)47 b(machines,)f(e\013e)-5 +b(ctiveness)43 b(and)h(fe)-5 b(asibility)p FT(.)42 b(Addison-W)-8 +b(esley)41 b(publishing)c(com-)610 4957 y(pan)m(y)-8 +b(.)474 5105 y(Syme,)27 b(D.)g(\(1997a\).)j(DECLARE:)d(A)g(protot)m(yp) +s(e)g(declarativ)m(e)g(pro)s(of)f(system)h(for)f(higher)f(order)610 +5218 y(logic.)j(T)-8 b(ec)m(hnical)27 b(Rep)s(ort)g(416,)j(Univ)m +(ersit)m(y)d(of)g(Cam)m(bridge)g(Computer)g(Lab)s(oratory)-8 +b(,)29 b(New)610 5331 y(Museums)h(Site,)g(P)m(em)m(brok)m(e)i(Street,)f +(Cam)m(bridge,)e(CB2)i(3QG,)g(UK.)474 5479 y(Syme,)40 +b(D.)g(\(1997b\).)g(Pro)m(ving)e(Ja)m(v)-5 b(a)40 b(t)m(yp)s(e)f +(soundness.)e(T)-8 b(ec)m(hnical)38 b(Rep)s(ort)g(427,)43 +b(Univ)m(ersit)m(y)610 5592 y(of)29 b(Cam)m(bridge)e(Computer)h(Lab)s +(oratory)-8 b(,)29 b(New)g(Museums)e(Site,)i(P)m(em)m(brok)m(e)g +(Street,)h(Cam-)610 5705 y(bridge,)g(CB2)g(3QG,)h(UK.)p +eop +%%Page: 256 266 +256 265 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(256)474 +396 y(Syme,)23 b(D.)e(\(1998\).)j FI(De)-5 b(clar)g(ative)25 +b(The)-5 b(or)g(em)26 b(Pr)-5 b(oving)25 b(for)f(Op)-5 +b(er)g(ating)26 b(Semantics)p FT(.)c(Ph.)f(D.)g(thesis,)610 +509 y(Univ)m(ersit)m(y)30 b(of)g(Cam)m(bridge.)g(Submitted)f(for)h +(Examination.)474 660 y(T)-8 b(arski,)27 b(A.)h(\(1936\).)i(Der)d(w)m +(ahrheitsb)s(egri\013)e(in)h(den)g(formalisierten)f(sprac)m(h)m(ten.)j +FI(Studia)j(Philo-)610 773 y(sophic)-5 b(a)40 b(1)p FT(,)31 +b(261{405.)474 923 y(Thompson,)23 b(S.)f(\(1991\).)i +FI(T)-7 b(yp)i(e)26 b(The)-5 b(ory)27 b(and)f(F)-7 b(unctional)26 +b(Pr)-5 b(o)g(gr)g(amming)p FT(.)25 b(Reading,)f(MA,)f(USA:)610 +1036 y(Addison-W)-8 b(esley)g(.)474 1186 y(Th)m(urston)34 +b(\(1994,)39 b(April\).)33 b(On)i(pro)s(of)f(and)h(progress)g(in)f +(mathematics.)i FI(BAMS:)g(Bul)5 b(letin)37 b(of)610 +1299 y(the)c(A)n(meric)-5 b(an)33 b(Mathematic)-5 b(al)35 +b(So)-5 b(ciety)39 b(30)12 b FT(\(2\),)33 b(161{177.)474 +1449 y(T)-8 b(ourlakis,)29 b(G.)i(\(1984\).)i FI(Computability)p +FT(.)f(Reston)f(Publishing)26 b(Compan)m(y)-8 b(.)474 +1599 y(T)g(rybulec,)29 b(A.)h(\(1978\).)j(The)c(Mizar-QC/6000)j(logic)d +(information)g(language.)h FI(Bul)5 b(letin)32 b(of)g(the)610 +1712 y(Asso)-5 b(ciation)34 b(for)g(Liter)-5 b(ary)34 +b(and)f(Linguistic)f(Computing)40 b(6)p FT(,)31 b(136{140.)474 +1863 y(T)-8 b(uring,)28 b(A.)i(M.)h(\(1936\).)h(On)d(computable)g(n)m +(um)m(b)s(ers,)g(with)f(an)i(application)e(to)j(the)f(En)m(tsc)m(hei-) +610 1976 y(dungsproblem.)e FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)35 +b(of)e(the)g(L)-5 b(ondon)34 b(Mathematic)-5 b(al)35 +b(So)-5 b(ciety)39 b(42)12 b FT(\(2\),)33 b(230{265.)474 +2126 y(v)-5 b(an)27 b(Gasteren,)i(A.)e(J.)g(M.)h(\(1990\).)i +FI(On)f(the)h(shap)-5 b(e)32 b(of)e(mathematic)-5 b(al)32 +b(ar)-5 b(guments)p FT(,)30 b(V)-8 b(olume)27 b(445)610 +2239 y(of)i FI(L)-5 b(e)g(ctur)g(e)31 b(Notes)f(in)h(Computer)h(Scienc) +-5 b(e)p FT(.)28 b(New)g(Y)-8 b(ork,)29 b(NY,)g(USA:)f(Springer-V)-8 +b(erlag)27 b(Inc.)474 2389 y(V)-8 b(eanes,)38 b(M.)e(\(1997\).)i(The)c +(undecidabilit)m(y)e(of)k(sim)m(ultaneous)e(rigid)f(E-uni\014cation)h +(with)g(t)m(w)m(o)610 2502 y(v)-5 b(ariables.)30 b(In)f +FI(5th)34 b(Kurt)f(G\177)-46 b(odel)33 b(Col)5 b(lo)-5 +b(quium)34 b(\(K)n(GC\))p FT(,)d(LNCS)e(1289,)k(pp.)c(305{318.)474 +2652 y(V)-8 b(o)s(da,)39 b(P)-8 b(.)37 b(J.)f(and)g(J.)g(Komara)h +(\(1995,)k(July\).)35 b(On)h(Herbrand)f(sk)m(eletons.)i(T)-8 +b(ec)m(hnical)36 b(rep)s(ort,)610 2765 y(Institute)30 +b(of)h(Informatics,)e(Comenius)g(Univ)m(ersit)m(y)h(Bratisla)m(v)-5 +b(a.)31 b(Revised)e(Jan)m(uary)h(1996.)474 2915 y(v)m(on)36 +b(W)-8 b(righ)m(t,)39 b(J.)d(\(1992\).)j(Doing)d(lattice)g(theory)h(in) +e(higher)g(order)g(logic.)h(T)-8 b(ec)m(hnical)36 b(Rep)s(ort)610 +3028 y(136,)802 3012 y(\027)802 3028 y(Ab)s(o)30 b(Ak)-5 +b(ademi,)30 b(T)-8 b(urku,)29 b(Finland.)474 3179 y(v)m(on)41 +b(W)-8 b(righ)m(t,)44 b(J.,)f(J.)d(Grundy)-8 b(,)43 b(and)d(J.)g +(Harrison)f(\(Eds.\))i(\(1996,)46 b(August\).)41 b FI(Pr)-5 +b(o)g(c)g(e)g(e)g(dings)44 b(of)610 3291 y(the)i(9th)h(International)h +(Confer)-5 b(enc)g(e)47 b(on)f(The)-5 b(or)g(em)47 b(Pr)-5 +b(oving)46 b(in)g(Higher)f(Or)-5 b(der)47 b(L)-5 b(o)g(gics)610 +3404 y(\(TPHOLs'96\))p FT(,)36 b(V)-8 b(olume)35 b(1125)h(of)f +FI(L)-5 b(e)g(ctur)g(e)37 b(Notes)g(in)f(Computer)i(Scienc)-5 +b(e)p FT(,)35 b(T)-8 b(urku,)35 b(Fin-)610 3517 y(land.)30 +b(Springer.)474 3668 y(W)-8 b(eb)s(er,)29 b(M.,)h(M.)f(Simons,)f(and)g +(C.)g(Lafon)m(taine)h(\(1993\).)i FI(The)h(generic)e(development)i +(language)610 3780 y(Deva:)51 b(pr)-5 b(esentation)40 +b(and)e(c)-5 b(ase)38 b(studies)p FT(,)g(V)-8 b(olume)35 +b(738)i(of)f FI(L)-5 b(e)g(ctur)g(e)38 b(Notes)f(in)g(Computer)610 +3893 y(Scienc)-5 b(e)p FT(.)31 b(New)f(Y)-8 b(ork,)31 +b(NY,)g(USA:)g(Springer-V)-8 b(erlag)29 b(Inc.)474 4044 +y(Whitehead,)k(A.)g(N.)f(and)g(B.)h(Russell)d(\(1910\).)36 +b FI(Principia)e(Mathematic)-5 b(a)p FT(.)35 b(Cam)m(bridge:)44 +b(Cam-)610 4157 y(bridge)29 b(Univ)m(ersit)m(y)h(Press.)474 +4307 y(Windley)-8 b(,)27 b(P)-8 b(.)28 b(J.)g(\(1994,)i(Septem)m(b)s +(er\).)e(Sp)s(ecifying)d(instruction-set)i(arc)m(hitectures)h(in)e +(HOL:)i(A)610 4420 y(primer.)h(In)h(T.)g(F.)h(Melham)f(and)f(J.)h +(Camilleri)e(\(Eds.\),)i FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)35 +b(of)e(the)g(7th)g(Interna-)610 4533 y(tional)41 b(Workshop)g(on)f +(Higher)f(Or)-5 b(der)40 b(L)-5 b(o)g(gic)40 b(The)-5 +b(or)g(em)41 b(Pr)-5 b(oving)40 b(and)g(Its)g(Applic)-5 +b(ations)p FT(,)610 4646 y(V)d(olume)31 b(859)h(of)e +FI(L)-5 b(e)g(ctur)g(e)34 b(Notes)f(in)g(Computer)h(Scienc)-5 +b(e)p FT(,)30 b(V)-8 b(alletta,)32 b(Malta,)g(pp.)e(440{455.)610 +4758 y(Springer-V)-8 b(erlag.)474 4909 y(W)g(ong,)51 +b(W.)46 b(\(1994\).)i FM(mweb)p FT(:)69 b(Pro)s(of)45 +b(script)g(managemen)m(t)i(utilities.)c(Man)m(ual)i(of)h(the)f(HOL)610 +5022 y FM(contrib)29 b FT(pac)m(k)-5 b(age.)474 5172 +y(Zammit,)36 b(V.)g(\(1996,)k(August\).)c(A)g(mec)m(hanisation)f(of)h +(computabilit)m(y)e(theory)i(in)e(HOL.)i(See)610 5285 +y(v)m(on)31 b(W)-8 b(righ)m(t,)31 b(Grundy)-8 b(,)29 +b(and)h(Harrison)f(\(1996\),)34 b(pp.)29 b(431{446.)474 +5435 y(Zammit,)24 b(V.)h(\(1997,)i(Marc)m(h\).)e(A)f(pro)s(of)f(of)h +(the)g FP(S)2190 5402 y FO(m)2185 5458 y(n)2281 5435 +y FT(theorem)g(in)e(Co)s(q.)i(T)-8 b(ec)m(hnical)24 b(Rep)s(ort)f +(9-97,)610 5548 y(The)30 b(Computing)f(Lab)s(oratory)-8 +b(,)31 b(The)f(Univ)m(ersit)m(y)g(of)g(Ken)m(t,)h(Can)m(terbury)-8 +b(,)30 b(Ken)m(t,)h(UK.)p eop +%%Trailer +end +userdict /end-hook known{end-hook}if +%%EOF