%!PS-Adobe-2.0
%%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software
%%Title: thesis.dvi
%%Pages: 266
%%PageOrder: Ascend
%%BoundingBox: 0 0 596 842
%%EndComments
%DVIPSCommandLine: dvips thesis
%DVIPSParameters: dpi=600, compressed, comments removed
%DVIPSSource: TeX output 1999.03.15:1912
%%BeginProcSet: texc.pro
/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
/@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
/FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
/rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
/base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
.1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N
/cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
/gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1
adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}
{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2
chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage
userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook
known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
/IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
{}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
a}B /bos{/SS save N}B /eos{SS restore}B end
%%EndProcSet
%%BeginProcSet: pstricks.pro
% PostScript prologue for pstricks.tex.
% Version 97 patch 2, 97/12/12
% For copying restrictions, see pstricks.tex.
%
/tx@Dict 200 dict def tx@Dict begin
/ADict 25 dict def
/CM { matrix currentmatrix } bind def
/SLW /setlinewidth load def
/CLW /currentlinewidth load def
/CP /currentpoint load def
/ED { exch def } bind def
/L /lineto load def
/T /translate load def
/TMatrix { } def
/RAngle { 0 } def
/Atan { /atan load stopped { pop pop 0 } if } def
/Div { dup 0 eq { pop } { div } ifelse } def
/NET { neg exch neg exch T } def
/Pyth { dup mul exch dup mul add sqrt } def
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
def
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
/pathforall load stopped { pop pop pop pop } if z } def
/STP { .996264 dup scale } def
/STV { SDict begin normalscale end STP } def
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
0 } ifelse setdash stroke } def
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
setlinecap stroke } def
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
% DG/SR modification end
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
def
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
exch 3 index sub exch Atan rotate newpath } def
/EndArrow { @mtrx setmatrix CP grestore end } def
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
grestore } def
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
CLW moveto } def
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
CLW moveto } def
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
def
/SD { 0 360 arc fill } def
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
load } if ] cvx newpath 3 1 roll T exec } def
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
cvi def } def
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
ifelse { pop } repeat a } def
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
and { pop pop /n n 1 sub def } if } def
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Lineto pop pop closepath } ifelse } def
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
setmatrix } def
% DG modification begin - Jan. 15, 1997
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
%setmatrix } def
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
setmatrix } def
% DG modification end
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
def } def
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
def } def
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
m mul neg def } def
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
def
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
/NC { CC x1 y1 x2 y2 x y curveto } def
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
/NAC { x2 y2 x y curveto CC x1 y1 } def
/EAC { x2 y2 x y ArrowB curveto pop pop } def
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
n { NC } repeat EOC } ifelse } def
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
repeat closepath pop pop } ifelse } def
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
def
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
rotate matrix concatmatrix exch findfont exch makefont setfont } def
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
x1 y1 lineto closepath } def
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
ifelse } def
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
pop pop Rect } { OvalFrame } ifelse } def
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
ifelse } def
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
def
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
y0 x0 2 mul x1 sub y1 ] def } def
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
w add def } repeat grestore gsave n 0 gt
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
% DG/SR modification end
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
exch t L stroke /i i h add def } repeat grestore } def
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
mul neg d } def
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
/Rot { CP CP translate 3 -1 roll neg rotate NET } def
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
dup a add ] cvx def } def
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
cvx def } def
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
h1 abs h sub dup s mul abs } ifelse } def
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
exch } def
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
} if } ifelse } def
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
/IfVisible true def } def
end
% END pstricks.pro
%%EndProcSet
%%BeginProcSet: pst-dots.pro
10 dict dup begin
/FontType 3 def
/FontMatrix [ .001 0 0 .001 0 0 ] def
/FontBBox [ 0 0 0 0 ] def
/Encoding 256 array def
0 1 255 { Encoding exch /.notdef put } for
Encoding
dup (b) 0 get /Bullet put
dup (c) 0 get /Circle put
dup (C) 0 get /BoldCircle put
dup (u) 0 get /SolidTriangle put
dup (t) 0 get /Triangle put
dup (T) 0 get /BoldTriangle put
dup (r) 0 get /SolidSquare put
dup (s) 0 get /Square put
dup (S) 0 get /BoldSquare put
dup (q) 0 get /SolidPentagon put
dup (p) 0 get /Pentagon put
(P) 0 get /BoldPentagon put
/Metrics 13 dict def
Metrics begin
/Bullet 1000 def
/Circle 1000 def
/BoldCircle 1000 def
/SolidTriangle 1344 def
/Triangle 1344 def
/BoldTriangle 1344 def
/SolidSquare 886 def
/Square 886 def
/BoldSquare 886 def
/SolidPentagon 1093.2 def
/Pentagon 1093.2 def
/BoldPentagon 1093.2 def
/.notdef 0 def
end
/BBoxes 13 dict def
BBoxes begin
/Circle { -550 -550 550 550 } def
/BoldCircle /Circle load def
/Bullet /Circle load def
/Triangle { -571.5 -330 571.5 660 } def
/BoldTriangle /Triangle load def
/SolidTriangle /Triangle load def
/Square { -450 -450 450 450 } def
/BoldSquare /Square load def
/SolidSquare /Square load def
/Pentagon { -546.6 -465 546.6 574.7 } def
/BoldPentagon /Pentagon load def
/SolidPentagon /Pentagon load def
/.notdef { 0 0 0 0 } def
end
/CharProcs 20 dict def
CharProcs begin
/Adjust {
2 copy dtransform floor .5 add exch floor .5 add exch idtransform
3 -1 roll div 3 1 roll exch div exch scale
} def
/CirclePath { 0 0 500 0 360 arc closepath } def
/Bullet { 500 500 Adjust CirclePath fill } def
/Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
/BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
/BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
/TrianglePath {
0 660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
} def
/SolidTriangle { TrianglePath fill } def
/Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
/BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
/SquarePath {
-450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
closepath
} def
/SolidSquare { SquarePath fill } def
/Square { SquarePath .89 .89 scale SquarePath eofill } def
/BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
/PentagonPath {
-337.8 -465 moveto
337.8 -465 lineto
546.6 177.6 lineto
0 574.7 lineto
-546.6 177.6 lineto
closepath
} def
/SolidPentagon { PentagonPath fill } def
/Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
/BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
/.notdef { } def
end
/BuildGlyph {
exch
begin
Metrics 1 index get exec 0
BBoxes 3 index get exec
setcachedevice
CharProcs begin 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: <dist> PathPosition -
% Function: Searches for position of currentpath distance <dist> 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: <pathlength> <position> 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 /ang 0 N /CLIP 0 N /rwiSeen
false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B
/@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit
div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{
/CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{
10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B
/@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale
true def end /@MacSetUp{userdict /md known{userdict /md get type
/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup
length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{}
N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath
clippath mark{transform{itransform moveto}}{transform{itransform lineto}
}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{
itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{
closepath}}pathforall newpath counttomark array astore /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 magscale{DVImag dup scale}if 0 setgray}
N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict
maxlength dict begin /magscale false def normalscale currentpoint TR
/psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts
/psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx
psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy
scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR
/showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict
begin /SpecialSave save N gsave normalscale currentpoint TR
@SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial
{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath
}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{
end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin}
N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{
/SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX
SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X
/startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad
yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end
%%EndProcSet
TeXDict begin 39158280 55380996 1000 600 600 (thesis.dvi)
@start /Fa 4 111 df<EB0FF890B5FC00031480000F14E04814F0A29038F00FF8903880
03FC381E0001001814FE00101300C812FF157FA7EC7FFF010FB5FC137F48B6FC120748EB
F07F383FFC0013C048C7FC12FE5AA315FF7E5C387F8007EBE01F6CB6FCA26C147F6C13FC
6C13F0000190C7FC202B7CA92C>97 D<EB03F8EB1FFF017F13C090B57E488048803807FE
07390FF801FC9038E000FE4848137E003F143E49133F90C77E5A127EED0F80B7FCA600FC
C9FCA37E127EA2127FA26C7EA26C7E6D14806C6C1303D807FC131F01FF13FF6C90B5FC7E
6C6C14006D13FC010F13E0010190C7FC212B7DA928>101 D<12FEB3B3B3A9073F79BE16>
108 D<38FC01FF010713C0011F13F0017F13F890B512FC12FD39FFF80FFEEBE003EBC001
90388000FFA290C7127FA35AB3A9202979A82F>110 D E /Fb 39
118 df<EA0F80EA3FE0EA7FF0A2EAFFF8A213FCA3127FA2123FEA0F9CEA001C133C1338
A31378137013F0EA01E0A2EA03C0EA0780EA0F005A121C12180E1D798C1B>44
D<49B4FC010F13E0017F13FC9038FF83FE4848C67E4848EB7F804848EB3FC04848EB1FE0
A2001F15F0A24848EB0FF8A3007F15FCA500FF15FEB3007F15FCA4003F15F8A26D131F00
1F15F0A2000F15E06D133F000715C06C6CEB7F806C6CEBFF003900FF83FE6DB45A011F13
F0010190C7FC27387CB630>48 D<141E143E14FE1307133FB5FCA313CFEA000FB3B3A600
7FB61280A4213779B630>I<EB0FFC90387FFFC048B512F0000714FC390FF03FFF261F80
0F1380263F000313C05AD9C00113E0486C6C13F07FA2ED7FF8A46C5A6C5A000FC7FCC8FC
EDFFF0A216E05C16C04A138016004A5A5D4A5A4A5A4A5AEC7F8092C7FC14FEEB01F84948
1378495A495A495A013EC712F84914F05B4848130148B6FCA25A5A5A5A4815E0B7FCA425
377BB630>I<ED07C0150FA2151F153F157F15FFA25C5C5C5CA2141E5C147C5C5C495A49
5A1307495A5C131E5B137C5B5B485A485A1207485A90C7FC121E5A127C5AB81280A4C700
01EBC000AA0103B61280A429377DB630>52 D<EC0FF8ECFFFE0103EBFF8090390FF80FC0
90393FE003E090397FC001F09038FF000F48EC1FF84848133F485A120F5B121FA2003FEC
1FF0ED0FE04990C7FC127FA21408EC7FF039FFF1FFFC01F313FFD9F78013809039FF007F
C049EB3FE04914F0ED1FF85B16FCA34914FEA5127FA5123F16FCA26C7E16F8000F143F6C
6C14F0ED7FE06C6C14C03A01FF81FF806C90B51200013F13FC010F13F00101138027387C
B630>54 D<123C123EEA3FE090B71280A41700485D5E5E5EA25E007CC7EA0FC000784A5A
4BC7FC00F8147E48147C15FC4A5A4A5AC7485A5D140F4A5A143F92C8FC5C147E14FE1301
A2495AA31307A2130F5CA2131FA5133FA96D5A6D5A6D5A293A7BB830>I<49B47E010F13
F0013F13FC9038FE01FF3A01F8007F804848EB3FC04848EB1FE0150F485AED07F0121FA2
7FA27F7F01FEEB0FE0EBFF809138E01FC06CEBF03F02FC13809138FF7F006C14FC6C5C7E
6C14FE6D7F6D14C04914E048B612F0EA07F848486C13F8261FE01F13FC383FC007EB8001
007F6D13FE90C7123F48140F48140715031501A21500A216FC7E6C14016D14F86C6CEB03
F06D13076C6CEB0FE0D80FFEEB7FC00003B61200C614FC013F13F00103138027387CB630
>I<EB03FF011F13E0017F13F83901FF03FE4848C67E4848EB7F80484814C0001FEC3FE0
123F49EB1FF0127F16F8A212FF16FCA516FEA5007F143FA3123F157F6C7E000F14FF6C6C
5A3903FE03DF6CB5129F6C6C131FD91FFC13FCEB00201400A216F8D80FE0133F487E486C
14F0A216E0157F16C0EDFF80495A6C4848130090388007FE390FE01FF86CB55A6C14C0C6
91C7FCEB1FF027387CB630>I<EA0F80EA3FE0EA7FF0A2EAFFF8A5EA7FF0A2EA3FE0EA0F
80C7FCABEA0F80EA3FE0EA7FF0A2EAFFF8A5EA7FF0A2EA3FE0EA0F800D2579A41B>I<DB
3FFCEB01C00203B5EAC003021FECF00791B6EAFC0F01039039FC00FF3F4901C0EB1FFFD9
1FFEC77E49481403D97FF080494880485B48177F4849153F4890C9FC181F485A180F123F
5B1807127FA24993C7FC12FFAD127F7FF003C0123FA27F001F1707A26C6C1780180F6C6D
16006C6D5D6C173E6C6D157ED97FF85D6D6C4A5A6DB44A5A010701C0EB0FE06D01FCEBFF
80010090B548C7FC021F14F8020314E09126003FFEC8FC3A3B7BB945>67
D<B912C0A43A007FF800039338007FE0171F170F1707A21703A21701A318F0EE7800A418
00A216F8A21501150791B5FCA4ECF80715011500A21678A693C8FCADB7FCA434397DB83C
>70 D<DB3FFCEB01C00203B5EAC003021FECF00791B6EAFC0F01039039FC00FF3F4901C0
EB1FFFD91FFEC77E49481403D97FF080494880485B48177F4849153F4890C9FC181F485A
180F123F5B1807127FA24993C8FC12FFAB043FB61280A2127F7FDC0003EBC000123FA27F
121FA26C7EA26C7F6C7F6C7F7ED97FF85C6D7E6DB45C010701C05B6D01FCEBFF3F010090
B5EAFE0F021FECF8030203ECE0009126003FFEC9FC413B7BB94B>I<B6D8FC03B612F0A4
26007FF8C70001EBE000B3A391B8FCA402F8C71201B3A6B6D8FC03B612F0A444397DB84B
>I<B612FCA439007FF800B3B3ADB612FCA41E397DB824>I<B500FC0203B512F0A28080C6
6C6D90390003F0006F6E5A81017B7F13798101787F6E7E6E7E6E7F6E7FA26E7F6E7F6E7F
6E7F6F7E153F826F13806F13C06F13E06F13F06F13F88117FCEE7FFEEE3FFF7013817013
C17013E18218F17013F97013FDEF7FFF8383A28383838383187FA2183F181F01FC160FB5
00FC150718031801A244397DB84B>78 D<B8FC17F017FEEFFF8028007FF8000F13C00401
13E07013F0EF7FF8EF3FFCA2EF1FFEA218FFA818FEA2EF3FFCA2EF7FF8EFFFF04C13E004
0F13C091B7120017FC17E002F8C9FCB3A4B612FCA438397DB841>80
D<EDFFF8020FEBFF80027F14F0903A01FFE03FFC010790380007FFD91FFC010113C04948
6D7FD97FE0EC3FF049486E7E488348496E7E4890C86C7EA248486F1380A2001F18C04981
003F18E0A3007F18F04981A300FF18F8AC007F18F0A36D5D003F18E0A36C6C4B13C0A200
0FDA1FC014806C6C90267FF0071300EDFFF86C903A81F07C0FFE6C903AC3C01E1FFC6CDA
800F5BD97FE3ECBFF0D93FF36DB45AD91FFF5D010701C091C7FC01019038F01FFC6D6CB5
00F01308020F6E131C0200EBF9FC92260001FE133C9438FF80FC18FF8219F8A28319F0A2
7113E0A27113C0711380711300EF01FC3E4A7BB948>I<B712FCEEFFE017FC17FF28007F
F8000F13C004017F707F717E717EA2717EA284A760A24D5A604D5A4D5A04035B041F90C8
FC91B612FC17E0839139F8003FFCEE0FFF707F707F8284A2707FA584A51A601AF084177F
1901DD3FFE13E0B600FC011F130394390FFF87C071EBFF8005011400CBEA1FFC443A7DB8
48>I<D907FF130E013FEBE01E90B5EAF83E0003ECFE7E3A07FC01FFFE390FF0001F4848
130F48481303491301007F140090C8FC167E5A163EA27F161E7F7F6D91C7FC13FC387FFF
E014FEECFFF06C14FE6F7E6C816C15F06C816C81C681133F010F801301D9000F1480EC00
7F030F13C01503818100F0157FA3163FA27E17807E167F6C16007E6D14FE01E0495A01F8
13039039FF801FF800FC90B512E0D8F83F5CD8F00749C7FC39E0007FF02A3B7BB935>I<
003FB91280A4D9F800EBF003D87FC09238007FC049161F007EC7150FA2007C1707A20078
1703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B387DB742>I<EB3FFE0003B512
E0000F14F8391FF00FFE003FEB03FF6D6C7F6E7FA26F7EA26C5A6C5AEA0380C8FCA2EC3F
FF010FB5FC137F3901FFF87F00071380380FFE00EA3FF85B485A12FF5BA415FF6D5A127F
263FF00713F83B1FFC1FBFFFC0390FFFFE1F0003EBF80F39003FE0032A257DA42E>97
D<13FFB5FCA412077EAF4AB47E020F13F0023F13FC9138FE03FFDAF00013804AEB7FC002
80EB3FE091C713F0EE1FF8A217FC160FA217FEAA17FCA3EE1FF8A217F06E133F6EEB7FE0
6E14C0903AFDF001FF80903AF8FC07FE009039F03FFFF8D9E00F13E0D9C00390C7FC2F3A
7EB935>I<903801FFC0010F13FC017F13FFD9FF8013802603FE0013C048485AEA0FF812
1F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F123FED01E06C7E15036C6C
EB07C06C6C14806C6C131FC69038C07E006DB45A010F13F00101138023257DA42A>I<EE
7F80ED7FFFA4150381AF903801FF81010F13F1013F13FD9038FFC07F0003EB001FD807FC
1307000F8048487F5B123FA2485AA312FFAA127FA27F123FA26C6C5B000F5C6C6C5B6C6C
4913C02701FF80FD13FE39007FFFF9011F13E1010313012F3A7DB935>I<903803FF8001
1F13F0017F13FC3901FF83FE3A03FE007F804848133F484814C0001FEC1FE05B003FEC0F
F0A2485A16F8150712FFA290B6FCA301E0C8FCA4127FA36C7E1678121F6C6C14F86D14F0
00071403D801FFEB0FE06C9038C07FC06DB51200010F13FC010113E025257DA42C>I<EC
1FF0903801FFFC010713FF90391FF87F8090383FE0FFD9FFC113C0A2481381A24813016E
1380A2ED3E0092C7FCA8B6FCA4000390C8FCB3ABB512FEA4223A7DB91D>I<161FD907FE
EBFFC090387FFFE348B6EAEFE02607FE07138F260FF801131F48486C138F003F15CF4990
387FC7C0EEC000007F81A6003F5DA26D13FF001F5D6C6C4890C7FC3907FE07FE48B512F8
6D13E0261E07FEC8FC90CAFCA2123E123F7F6C7E90B512F8EDFF8016E06C15F86C816C81
5A001F81393FC0000F48C8138048157F5A163FA36C157F6C16006D5C6C6C495AD81FF0EB
07FCD807FEEB3FF00001B612C06C6C91C7FC010713F02B377DA530>I<13FFB5FCA41207
7EAFED7FC0913803FFF8020F13FE91381F03FFDA3C01138014784A7E4A14C05CA25CA291
C7FCB3A3B5D8FC3F13FFA4303A7DB935>I<EA01F0EA07FC487EA2487EA56C5AA26C5AEA
01F0C8FCA913FF127FA412077EB3A9B512F8A4153B7DBA1B>I<13FFB5FCA412077EB3B3
ACB512FCA4163A7DB91B>108 D<01FED97FE0EB0FFC00FF902601FFFC90383FFF800207
01FF90B512E0DA1F81903983F03FF0DA3C00903887801F000749DACF007F00034914DE6D
48D97FFC6D7E4A5CA24A5CA291C75BB3A3B5D8FC1FB50083B512F0A44C257DA451>I<01
FEEB7FC000FF903803FFF8020F13FE91381F03FFDA3C011380000713780003497E6D4814
C05CA25CA291C7FCB3A3B5D8FC3F13FFA430257DA435>I<903801FFC0010F13F8017F13
FFD9FF807F3A03FE003FE048486D7E48486D7E48486D7EA2003F81491303007F81A300FF
1680A9007F1600A3003F5D6D1307001F5DA26C6C495A6C6C495A6C6C495A6C6C6CB45A6C
6CB5C7FC011F13FC010113C029257DA430>I<9039FF01FF80B5000F13F0023F13FC9138
FE07FFDAF00113800003496C13C00280EB7FE091C713F0EE3FF8A2EE1FFCA3EE0FFEAA17
FC161FA217F8163F17F06E137F6E14E06EEBFFC0DAF00313809139FC07FE0091383FFFF8
020F13E0020390C7FC91C9FCACB512FCA42F357EA435>I<9038FE03F000FFEB0FFEEC3F
FF91387C7F809138F8FFC000075B6C6C5A5CA29138807F80ED3F00150C92C7FC91C8FCB3
A2B512FEA422257EA427>114 D<90383FF0383903FFFEF8000F13FF381FC00F383F0003
007E1301007C130012FC15787E7E6D130013FCEBFFE06C13FCECFF806C14C06C14F06C14
F81203C614FC131F9038007FFE140700F0130114007E157E7E157C6C14FC6C14F8EB8001
9038F007F090B512C000F8140038E01FF81F257DA426>I<130FA55BA45BA25B5BA25A12
07001FEBFFE0B6FCA3000390C7FCB21578A815F86CEB80F014816CEBC3E090383FFFC06D
1380903803FE001D357EB425>I<01FFEC3FC0B5EB3FFFA4000714016C80B3A35DA25DA2
6C5C6E4813E06CD9C03E13FF90387FFFFC011F13F00103138030257DA435>I
E /Fc 5 84 df<031EEDFFC0037E021F13F8DA01FE91B512FE0207020380021F020F1580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>66
D<943803FF80053F13E00403B512F0040F14F8167F4BB612FC5D92380FFE00DB3FE0133F
ED7F80DA01FEC7121FDA03F815F84A5A4A5A4A4815F04A5A4AC8EA3FE05C495A4AED7FC0
495A0107EEFF80495A190049485C013F5E4A4A5A017F5E4A4A5A01FF16C091C890C7FC48
93C8FCA2485AA212075BA2120F5BA2121FA25B123FA4127F5BA512FFA97FA21803F01F80
6DEE3F00007F17FF4D5A6D5E17036C6C4B5A6D5E4D5A6C6D4A5A6E4A5A6C6D4AC7FC6C01
F8EB01FE02FEEB07F86C9039FFC03FF06C91B512C06C5E6D4AC8FC6D14F8010F14C00103
49C9FC9038007FE03E587FD43F>I<EF0FFE94B512C0040714F0041F14F8047F14FC4BB6
FC4B15FE92380FF80392391FC0007FDB7F80131F03FEC7FC4A48EC0FFC4A5A4A4815F814
0F4A4815F04BEC1FE0023F16C0027FED3F004B141C02FF92C7FC5BA35BA281A281A2816D
7F15FE6D7F16E06E13FE6EEBFFF8806E5C02035C6E5C6E91C8FC4A13FCDA07FEC9FCEC1F
F0EC3F8002FFCAFCEB01FC495A495A495A495A495A137F49CBFC5B1201485A12075B120F
5B121FA2485A180C007F177E60170300FF4C5A60170F6D4B5A606D4B5A4DC7FC6D15FE6C
6C4A5A6D6CEB07F86C01E0EB1FE002FEEBFFC06C90B6C8FC6C15FC6C5D6C15E0C692C9FC
013F13F8010313803F587DD43F>69 D<031E17FF037E04071380DA01FE041F13C0020794
B512E0021F1603027F93380FFC3F02FF93393FF01FF001039438FF800F49933801FE00D9
0FCFDB07F814E0D91F0FDB1FE0130701004C48EB0FC04B02FFC71380DD01FCEC0600DD07
F891C7FCEF0FE0EF3FC0021F4ACAFC17FE9238F801F8EE07F04C5A4C5A4CCBFC023F137E
4B5AEDF1F8EDF3F0EDF7E0EDFFC0EC7FEFEDFF80A491B5FCA3EDBFC05BA2153F6F7E5BA2
14FE01076D7EA302FC7F010F1307A202F87F1503011F8014F01501013F804A7E83137F4A
6D7EA201FF6E7E5C707E5A91C76C7EA2486F6CED01C049F107E0706CED0FC0000782496E
EE1F8084000F6F6DEC3F0049157F72147E48486F6C5C716C495A72EB07F048486FB4EB1F
E0719038C1FFC048486F90B5C7FC496F14FC90CA14F000FC053F13800070DD0FF8C8FC54
587DD45B>75 D<F03FF80507B57E053F14F094B612FC040381040F81043F168093387FE0
0FDB01FEC714C0DB03F8141F4B4814074B48804B5A033F814B5AA203FF17804A90C8FC1B
004A5F62F103F04AEE01C096C8FCA282A282A2826E7F826E7F16FF6E14C06F7F6F13F86F
13FF6F14C06F80030114F86F6C13FE707F040F14C0040180706C7F717F050F7F05037F71
7F717E727F84D907C081011F707FEB7F804848C97E485AEA07F8484882121FA2484860A2
127F97C7FCA200FF4D5AA26D5F4E5A7F4E5A6D5F6C6C4C5A6D4C5A6C6D4BC8FC02E0EC01
FE6C01F8EC07F86C01FEEC3FF06C903AFFE003FFC06C91B6C9FC6C16FC6C16F0013F15C0
010F4ACAFC010314F09026003FFECBFC4A587ED448>83 D E /Fd
1 121 df<1602923801FFFC030FEBFF80033F14E092397F0007F8DA01F8EB00FCDA03E0
143E4A48140F4AC86C7E021E6F7E4A6F7E4A6F7E0270167802F016384948163C4A161C01
03171E4A160E27E00781C0150F0078010782001E010EC9FC26070F3C178026038E701603
3801CEE03800EFC0EB7F806DCA13C06D1701131E130E010C94C7FC421F7F9E47>120
D E /Fe 2 83 df<831607163F4BB47E150792381E7FE0ED7C3FDA31F87F02FF011F1340
010391380FFFC0D90F8F6D1380D91C1F903803FC000178EC01E001F091C8FC3803E00FEA
07C0000F80EA1F801407EA3F0081A2481303127EA400FE5CA25D1407EC0FE04A5A4AC9FC
EB01FC00FF13E090CBFCA37F127FA27FA26C7EA27F6C7E7F7F6C7E806C6DEC01806C01F0
14076C01FCEC1F0002FF147E6C6C01F013F86D90B55A6D15C06D5D010392C7FC010014FC
021F5B0200138032407BBC38>67 D<02FFED1F80010701E0ECFFE0011F6D01037F017F01
FC010F7F90B56C5B2701E03FFFEB3C1F2603800F6D486C7E48486CECE007000E6DEBC1C0
001E010001E3130348EDE78092387FEF00007C6EB4FC4C8000FE021F14014CECFFFC6C19
FE6F4815F06D7013C06DEF7F006C6C173C6C6C17F06D0107EC03E0001FEF0F806C6C043F
C7FC18FC6C6CED03F8000393381FFF800001047F13E004FBB57E6C4891B67EEF807F4991
38F0001F48481607491603485A4848160148C75B121C0008140FC8FC4C80A35E151F93C7
FCA24B81153E5D85013813F09026FE01E01680486C4848168348D987806E13CF4801CFC9
13FE4801FCEE7FF8484917F0263C3FF0EE3FE026701FE0EE1F8026E007C01700D84003CA
120ECC120C48407EBD4B>82 D E /Ff 5 33 df<007FB912E0BA12F0A348CA12076C170F
6D161F6D163FD8F7E0167ED8F3F016FCD8F1F8ED01F8D8F0FCED03F0017EED07E06DED0F
C06D6CEC1F806D6CEC3F006D6C147E6D6C5C6D6C495A6D6C495A027E495A6E495A6E6C48
5A6E6C48C7FC913807E07E6E6C5A913801F9F86EB45A6F5A6F5A6F5AA24B7E4B7E4B7E91
3801F9F8913803F0FC913807E07E4A487E4A486C7E4A486C7E027E6D7E4A6D7E49486D7E
49486D7E4948147E4948804948EC1F8049C8EA0FC0017EED07E049ED03F0D8F1F8ED01F8
D8F3F0ED00FCD8F7E0167EB448163F49161F90CA120F481707BAFCA36C18E03C3E7BBD47
>2 D<007FB912E0BA12F0A300F0CBFCB3B3B2BAFCA36C18E03C3E7BBD47>I<007FB912E0
BA12F0B3B3B3A66C18E03C3E7BBD47>I<00E01AE0A200701A70A26C86A26C866C86000F
1A0F6C6CF107806C6CF103C06C6CF101E06C6CF100F090BC12F81CFEA21CF801F0CD12F0
4848F101E04848F103C04848F1078048CDEA0F00000E1A0E48624862A24862A24862A257
1E79A565>26 D<1AE0A21A70A2860120151001F0033C80D801FC03FE80486C4A6C140F48
6C4A6DEB0780260FDF80902607EFC0EB03C0261F8FC090260FC7E0EB01E0263F07E09026
1F83F0EB00F0267E03F0D93F01B612F8486C6CD97E0015FE486C6C497F006090267E01F8
011F14F8C7263F87F090C812F091261FCFE0ED01E0912607FF80ED03C06E90C9EA07806E
48EE0F006E48160E03785E03305E92CAFC62A262A24F1E7BA55B>32
D E /Fg 1 50 df<1360EA01E0120F12FF12F11201B3A3387FFF80A2111C7B9B1C>49
D E /Fh 15 120 df<EB03F0EB0FF890383E1C6090387C0FF0EBF807EA01F0EA03E00007
EB03E0EA0FC0A2381F800715C0EA3F00A2140F481480127EA2141F00FE14005A1506EC3F
07EC3E0F150E147E007C141EECFE1CEB01FCD83C03133C393E07BE38391F0E1E783907FC
0FF03901F003C0202278A027>97 D<137EEA0FFE121F5B1200A35BA21201A25BA21203A2
5BA21207A2EBC3E0EBCFF8380FDC3EEBF81F497E01E01380EA1FC0138015C013005AA212
3EA2007E131F1580127CA2143F00FC14005AA2147EA25CA2387801F85C495A6C485A495A
6C48C7FCEA0FFCEA03F01A3578B323>I<14FCEB07FF90381F078090383E03C0EBFC0138
01F8033803F0073807E00F13C0120F391F80070091C7FC48C8FCA35A127EA312FE5AA400
7C14C0EC01E0A2EC03C06CEB0F80EC1F006C137C380F81F03803FFC0C648C7FC1B2278A0
23>I<ED0FC0EC03FFA21680EC001FA31600A25DA2153EA2157EA2157CA215FCA2903803
F0F8EB0FF8EB3E1DEB7C0F496C5AEA01F0EA03E000071303D80FC05BA2381F8007A2D83F
005BA2140F5A007E5CA2141F12FE4891C7FC1506EC3F075DEC3E0E147E007C141EECFE1C
EB01FCD83C03133C393E07BE38391F0E1E783907FC0FF03901F003C0223578B327>I<EB
03F8EB0FFEEB3E0F9038F807803801F003EA03E0EA07C0120FEA1F801407D83F0013005C
007E133EEB03F8387FFFE04848C7FC00FCC8FCA45AA4EC0180EC03C0A2007CEB0780EC1F
00003C133E6C13F8380F03E03807FF80D801FCC7FC1A2277A023>I<151FED7FC0EDF0E0
020113F0EC03E3A2EC07C316E0EDC1C091380FC0005DA4141F92C7FCA45C143E90381FFF
FEA3D9007EC7FC147CA414FC5CA513015CA413035CA413075CA3130FA25CA3131F91C8FC
A35B133E1238EA7E3CA2EAFE7812FC485AEA78E0EA3FC0000FC9FC244582B418>I<EB0F
C0EA03FFA25CEA001FA391C8FCA25BA2133EA2137EA2137CA213FCA29038F83F80ECFFE0
3901FBE0F09038FF80F8EC007849137C485A5B5BA2484813FC5D5BA2000F13015D1380A2
001F13035DEB0007EDC0C048ECC1E0020F13C0003E1481A2007E1483ED0380007C140716
0000FC140E151E48EB07F80070EB01F023357BB327>104 D<133FEA07FF5A13FEEA007E
A3137CA213FCA213F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121F
A21300A25AA2123EA2127EA2127C1318EAFC1C133CEAF838A21378137012F013F0EAF8E0
1279EA3FC0EA0F00103579B314>108 D<3903C007F0390FF01FFC391E787C1E391C7CF0
1F393C3DE00F26383FC01380EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F812
0000015C153E5BA20003147E157C4913FCEDF8180007153C0201133801C013F0A2000F15
78EDE070018014F016E0001FECE1C015E390C7EAFF00000E143E26227AA02B>110
D<14FCEB07FF90381F07C090383E03E09038FC01F0EA01F83903F000F8485A5B120F4848
13FCA248C7FCA214014814F8127EA2140300FE14F05AA2EC07E0A2007CEB0FC01580141F
EC3F006C137E5C381F01F0380F83E03803FF80D800FCC7FC1E2278A027>I<3903C00FC0
390FF03FF0391E78F078391C7DE03C393C3FC0FC00381380EB7F00007814F8D8707E1370
1500EAF0FC12E0EA60F812001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290
C8FC120E1E227AA020>114 D<EB03F0EB1FFCEB3C1EEB780FEBF007EA01E0140F000313
1F13C0A2EBE00414007FEBFF8014E06C13F06C13F8EB7FFC1307EB00FE147E143E123800
FC133CA3147C00F013784813F0EAF001387803E0383C0F80381FFE00EA03F818227AA01F
>I<1303EB0F80A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2B5FC3800F800A2
1201A25BA21203A25BA21207A25BA2120FA25B1460001F13F014E01300130114C0130300
1E1380EB07005BEA0F1EEA07F8EA01E015307AAE19>I<EA01F0D803FC1307D80F1E5B00
0E5C121C123C00385CD8783E133E1270A2017E137ED8F07C137CEA60FCC65A15FC00015C
5BA2140100035C13E0166002031370EDE0F0D807C014E0A20003EB07E116C09038E00FC1
EC1FC3000190383FE3809038F071E73A007FE0FF0090381F803C24227AA029>I<D801F0
1538D803FC010E13FCD80F1E131E000E143E121C123C0038027E137CD8783E137C007016
3CA2017E13FCD8F07C491338EA60FCC65A0201147800014A137013F0A2020314F0000316
E001E05BA2160117C001C013C00207EB0380A29039E00FE0071700021F130E3A01F03DF0
1E3A00F878F83C90393FF03FF090390FC00FC02E227AA033>119
D E /Fi 11 118 df<EDFFF0020313FC91380F801E91381E000F4A131F147C0278131E02
F8130C1600A2495AA690B612F8A2903803E000ED01F0A2EB07C0A2ED03E0A3130F913880
07C0A4ED0F80D91F001387A3ED1F0EA2133EED0F1C1618ED07F0ED01E04990C7FCA3EA38
78EA78F8EAF8F0A2485AEA71C0EA3F806CC9FC283581A827>12 D<90387FFF80A2903803
F0005CA21307A25CA2130FA25CA2131FA291C7FCA25BA2133EA2137EA2137CA213FCA25B
A21201A25BA21203A25BA21207B5FCA219287BA71A>73 D<017FB5FC16E0903903F003F8
9138E0007C820107143F825C1780130FEE3F005CA2011F147E167C91C75A4B5A49EB07E0
ED3F80DAFFFEC7FC15F890387E007C151F017C7F8213FCA25BA20001141FA25BA2000316
C016815B168300071680B590380FC700ED07FEC8EA01F82A297AA732>82
D<EC01F0143FA2EC03E0A21407A215C0A2140FA21580A2141F131F90387FDF003801E0FF
485A48487E380F007E48133E001E137E123E003C137C127C14FC5A5CA2130112F0ECF0E0
A3903803E1C01307127039781DE380D83C391300381FF0FE3807C03C1C2978A723>100
D<EB0FC0EB7FF0EBF0383803C01CEA0780EA0F005A121E003E1338481370EB07E0387FFF
8038FFFC0000F8C7FCA45AA214080078131C143C00381378383C01F0381E07C0380FFF00
EA03F8161B789920>I<130E131E133EA2131C1300A8EA03C0EA07F0EA0C781218123812
30EA70F8A212E0EAE1F01201EA03E0A3EA07C0A2EA0F801387A2EA1F07130EA2EA1E0C13
1CEA0E38EA0FF0EA03C010287BA615>105 D<137CEA0FFCA2EA00F8A21201A213F0A212
03A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2EA7C38A3EA
F870A21360EA78E013C0EA3F80EA0F000E297BA712>108 D<3907803F80390FE0FFC039
1CF1C1E03938F301F0EBFE00485A81495B00E1130113F01201A24848485AA34A5AEA07C0
EDC380EC0F83A23A0F801F0700A2150EEC0F0CD81F005BEC07F0000E6D5A211B7B9926>
110 D<3807807E390FE1FF80391CF383C03838F701EBFE033870FC0713F8EC0380D8E1F0
C7FCA21201A2485AA4485AA4485AA448C8FCA2120E1A1B7B991D>114
D<1338137CA25BA4485AA4485AB512C0A23803E000485AA4485AA448C7FCA4123EEB0380
A2EB0700127C130E130CEA3C1C5BEA1FE0EA07C012267AA417>116
D<EA03E0486C1370D80C7813F8EA187C0038EB01F01230127013F800E0EB03E0A2EA01F0
A2EC07C0EA03E0A33907C00F801587A3EC1F0EA2143F0003EB7F1C3901E0EF183900FF87
F890383F01E0201B7B9925>I E /Fj 34 122 df<EE3FFC4BB51280923907E007C09239
1F8001E0DB3F0013F0037E13034B1307A24A5A18E04A48EB038094C7FCA314075DA4140F
5DA3010FB7FCA25F903A001F80007EA217FE023F5C92C7FCA216015F5C147E16035FA214
FE4A13075FA30101140F5F4AECC1C0A2161F1783010316805CA2EF870013074A5CEE0F8E
EE079EEE03FC010FEC00F04A91C7FCA35C131FA2001C90CAFC127E5BEAFE3E133C137CEA
F878EA78F0EA3FE0EA0F80344C82BA2F>12 D<EE01C01603A21607160FA2161F83163FA2
167F16FF16EF150116CFED038FA2ED070FA2150E151E151C1538A203707FA2EDE007A2EC
01C014031580EC0700A2140EA25CA25C027FB5FCA291B6FC9139E00007F849481303A249
5A130791C7FC5B130E5BA25B1378137013F0EA03F8486C4A7EB56C48B512F0A3343C7BBB
3E>65 D<DB03FE130E92393FFF801E92B5EAE03C913903FE01F0913A0FF000787CDA3FC0
EB3CFC4AC7EA1FF802FE140FEB03FC49481407494815F049481403495A5C49C813E05B48
5A5B000317C0485AA2485A1880485A94C7FCA2485AA3127F5BA312FF90CBFCA41738A217
784816707E17F06C5E16015F16036C6C4A5A94C7FC001F150E6D141E000F5D6D5C6C6C49
5A6C6CEB03C0D801FEEB0F8027007F807EC8FC6DB45A010F13E0010090C9FC373D74BA3B
>67 D<0107B712FEA3903A000FF000074B1300187C021F153CA25DA2143FA25D1838147F
A292C8FCEE03804A130718004A91C7FCA201015CA24A131E163E010314FE91B5FC5EA290
3807F800167C4A1378A2130FA24A1370A2011F14F0A24A90C8FCA2133FA25CA2137FA291
CAFCA25BA25B487EB6FCA337397BB836>70 D<0103B512F8A390390007F8005DA2140FA2
5DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25CA21303A25CA21307A2
5CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FCA25C25397CB820>73
D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA2
5CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478A201
3F157017F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF0001020713
00B8FCA25E2E397BB834>76 D<0107B612F817FF1880903B000FF0003FE04BEB0FF0EF03
F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15F817074A15F0EF0FE013
01EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD907F8C9FCA25CA2130FA2
5CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512FCA337397BB838>80
D<0103B612F017FEEFFF80903B0007F8003FC04BEB0FF01707020FEC03F8EF01FC5DA202
1F15FEA25DA2143FEF03FC5DA2027FEC07F818F092C7120F18E04AEC1FC0EF3F004A14FE
EE01F80101EC0FE091B6128004FCC7FC9138FC003F0103EC0F80834A6D7E8301071403A2
5C83010F14075F5CA2011F140FA25CA2133F161F4AECE007A2017F160F180E91C7FC4902
0F131C007F01FE153CB5913807F078040313F0CAEAFFE0EF3F80383B7CB83D>82
D<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027CEB0FF84A1307
49481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7FCA2806D7E14
FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F15031501A215
00A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7FC6D133ED8F9
F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F>I<003FB53980
0FFFFEA326007F80C7EA7F8091C8EA3F00173E49153CA2491538A20001167817705BA200
0316F05F5BA2000715015F5BA2000F15035F5BA2001F150794C7FC5BA2003F5D160E5BA2
007F151E161C90C8FCA2163C4815385A16781670A216F04B5A5E1503007E4A5A4BC8FC15
0E6C143E6C6C5B15F0390FC003E03907F01FC00001B5C9FC38007FFCEB1FE0373B70B83E
>85 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120FEBC001
001F5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F1583168014
3F1587007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F000F022
2677A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EB
E7FE9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214
075A127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B38
3C03E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380F
C1E090381F0070017E13784913383901F801F83803F003120713E0120FD81FC013F091C7
FC485AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC
0F806CEB3E00380F81F83803FFE0C690C7FC1D2677A426>I<ED01F815FFA3150316F0A2
1507A216E0A2150FA216C0A2151FA21680A2153FA202F81300EB07FE90381F877F90383E
03FF017C5BEBF80112013803F00048485B120FEBC001121F5DEA3F801403127F01005BA2
14075A485CA2140FA248ECC1C0A2141F15C3ED8380143F1587007C017F1300ECFF076C48
5B9038038F8E391F0F079E3907FE03FC3901F000F0253B77B92A>I<147F903803FFC090
380FC1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0
EC1F80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003E
EB01E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>I<ED07C0
ED1FF0ED3E38ED7C3CEDF8FC15F9140115F1020313F8EDF0F0160014075DA4140F5DA414
1F5D010FB512C05B16809039003F800092C7FCA45C147EA414FE5CA413015CA413035CA4
13075CA4130F5CA3131F5CA391C8FC5B121CEA7E3EA2EAFE3C137C1378EAF8F01278EA3F
C0EA0F80264C82BA19>I<EC07C0EC3FF09138FC38E0903901F01FF0EB03E0903807C00F
EB0F80011F1307D93F0013E05B017E130F13FE4914C01201151F1203491480A2153F1207
491400A25DA249137EA215FEA25D00031301140314076C6C485A0000131FEB787BEB3FF3
90380FC3F0EB00031407A25DA2140F5D121C007E131F5D00FE49C7FC147E5C387801F838
7C07E0381FFF80D803FEC8FC24367CA426>I<EB03F0EA01FFA3EA00075CA3130F5CA313
1F5CA3133F91C8FCA35B90387E07F0EC1FFCEC783E9038FFE01F02C01380EC800F140048
5A16C05B49EB1F8012035BA2153F000715005BA25D000F147E5B15FE5D121FD98001131C
15F8163C003F01031338010013F0A216704814E0007E15F016E0EDE1C000FE903801E380
48903800FF000038143C263B7BB92A>I<EB01C0EB07E014F0130F14E01307EB038090C7
FCAB13F0EA03FCEA071EEA0E1F121CA212385B1270A25BEAF07E12E013FEC65AA212015B
1203A25B12075BA2000F13E013C013C1001F13C01381A2EB83801303EB0700A2130E6C5A
EA07F8EA01E0143879B619>I<EB03F0EA01FFA3EA00075CA3130F5CA3131F5CA3133F91
C8FCA35B017EEB0F80ED3FE015F09039FE01C1F09038FC0387EC0707140E0001011C13E0
EBF83891383003800270C7FC00035BEBF1C0EBF38001FFC8FCEA07FC7FEBFFC0EBE7F838
0FE1FCEBC07E147F80001F809039801F81C0A21583003F013F138001001303A215074815
00007E133EEC1E0E151E00FE6D5A48EB07F80038EB01E0243B7BB926>107
D<EB0FC0EA07FFA3EA001F1480A2133FA21400A25BA2137EA213FEA25BA21201A25BA212
03A25BA21207A25BA2120FA25BA2121FA25BA2123FA290C7FCA25AA2EA7E0EA212FE131E
EAFC1CA2133C133812F81378EA7870EA7CE0121FEA0F80123B79B915>I<D801E001FEEB
07F03C07F803FF801FFC3C0E3C0F07C0783E3C1E3E3C03E1E01F261C1F78D9F3C013803C
383FF001F7800F02E01400007801C013FE007018C002805B4A4848EB1F80EAF07FD8E07E
5CA200000207143F01FE1700495CA2030F5C0001177E495C18FE031F5C120349DA800113
1C18F8033F153C00070403133849020013F0A24B1570000F17E049017E15F019E003FEEC
E1C0001FEE01E34949903800FF000007C70038143C3E2679A444>I<D801E013FE3A07F8
03FF803A0E3C0F07C03A1E3E3C03E0261C1F787F39383FF00114E0007813C00070811480
4A485AEAF07FEAE07EA20000140701FE5C5BA2150F00015D5B151F5E12034990383F8380
160316070007027F130049137EA2160E000F147C49141E161C5E001FEC3C7849EB1FE000
07C7EA0780292679A42F>I<147F903803FFC090380FC1F090381F00F8017E137C5B4848
137E4848133E0007143F5B120F485AA2485A157F127F90C7FCA215FF5A4814FEA2140115
FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB3F00147E6C13F8380F83F0
3803FFC0C648C7FC202677A42A>I<9039078007C090391FE03FF090393CF0787C903938
F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01E113C15CA2D80003143FA2
5CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E013F495A6E485A5E6E48C7
FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201A25BA21203A25B1207B512
C0A3293580A42A>I<ECF803903807FE0790381F871F90383E03BF017C13FEEBF8011201
3803F000484813FC120F5B001F130115F8EA3F80A2007F1303010013F0A34813074814E0
A3140F4814C0A3141F1580143FA2007C137FECFF006C5AEB03BF381F0F7F3807FE7EEA01
F0C7FC14FE5CA313015CA313035C130748B512C0A3203577A426>I<3903C003F0390FF0
1FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F800078150000701300151CD8F0
7E90C7FCEAE0FE5BA2120012015BA312035BA312075BA3120F5BA3121F5BA3123F90C9FC
120E212679A423>I<14FE903807FF8090380F83C090383E00E04913F00178137001F813
F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13F814FE6C7F6D13807F010F13
C01300143F141F140F123E127E00FE1480A348EB1F0012E06C133E00705B6C5B381E03E0
6CB45AD801FEC7FC1C267AA422>I<EB0380EB07C0130FA4131F1480A3133F1400A35B13
7E007FB5FCA2B6FC3800FC00A312015BA312035BA312075BA3120F5BA3121FEB801CA214
3C003F1338EB0078147014F014E0EB01C0EA3E03381F0780380F0F00EA07FCEA01F01835
79B31C>I<13F8D803FEEB01C0D8078FEB03E0390E0F8007121E121C0038140F131F0078
15C01270013F131F00F0130000E015805BD8007E133FA201FE14005B5D120149137EA215
FE120349EBFC0EA20201131E161C15F813E0163CD9F003133814070001ECF07091381EF8
F03A00F83C78E090393FF03FC090390FC00F00272679A42D>I<01F0130ED803FC133FD8
071EEB7F80EA0E1F121C123C0038143F49131F0070140FA25BD8F07E140000E08013FEC6
485B150E12015B151E0003141C5BA2153C000714385B5DA35DA24A5A140300035C6D48C7
FC0001130E3800F83CEB7FF8EB0FC0212679A426>I<01F01507D803FC903903801F80D8
071E903907C03FC0D80E1F130F121C123C0038021F131F49EC800F00701607A249133FD8
F07E168000E0ED000313FEC64849130718000001147E5B03FE5B0003160E495BA2171E00
070101141C01E05B173C1738A217781770020314F05F0003010713016D486C485A000190
391E7C07802800FC3C3E0FC7FC90393FF81FFE90390FE003F0322679A437>I<903907E0
07C090391FF81FF89039787C383C9038F03E703A01E01EE0FE3803C01F018013C0D80700
14FC481480000E1570023F1300001E91C7FC121CA2C75AA2147EA214FEA25CA21301A24A
1370A2010314F016E0001C5B007E1401010714C000FEEC0380010F1307010EEB0F003978
1CF81E9038387C3C393FF03FF03907C00FC027267CA427>I<13F0D803FCEB01C0D8071E
EB03E0D80E1F1307121C123C0038140F4914C01270A249131FD8F07E148012E013FEC648
133F160012015B5D0003147E5BA215FE00075C5BA214015DA314035D14070003130FEBF0
1F3901F87FE038007FF7EB1FC7EB000F5DA2141F003F5C48133F92C7FC147E147C007E13
FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA03F0233679A428>I
E /Fk 21 123 df<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A728927>46
D<1538157C15FCA2140115F8140315F0140715E0140F15C0141F1580143F1500A25C147E
14FE5C13015C13035C13075C130F5CA2131F5C133F91C7FC5B137E13FE5B12015B12035B
A212075B120F5B121F5B123F90C8FC5A127E12FE5AA25A12781E3A7CB327>I<121EEA7F
80A2EAFFC0A4EA7F80A2EA1E00C7FCAC121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2072
9F27>58 D<3803FFC0000F13F04813FC4813FF811380EC1FC0381F000F000480C71207A2
EB0FFF137F0003B5FC120F5A383FFC07EA7FC0130012FE5AA46C130F007F131FEBC0FF6C
B612806C15C07E000313F1C69038807F8022207C9F27>97 D<EB0FF8EB3FFE90B5128000
0314C04814E0390FFC0FF0391FE003F8EBC001D83F8013FC48C7FC127E157E12FEB612FE
A415FC00FCC8FC7E127E127F6C143C6D137E6C7E01F013FE390FFC07FC6CB5FC000114F8
6C14F0013F13C0903807FE001F207D9F27>101 D<153F90391FC0FF80D97FF313C048B6
12E05A4814EF390FF07F873A1FC01FC3C0EDC000EB800F48486C7EA66C6C485AEBC01FA2
390FF07F8090B5C7FC5C485BEB7FF0EB1FC090C9FCA27F6CB5FC15E015F84814FE4880EB
8001007EC7EA3F80007C140F00FC15C0481407A46C140F007C1580007F143F6C6CEB7F00
9038F807FF6CB55A000714F86C5CC614C0D90FFCC7FC23337EA027>103
D<EA7FE0487EA3127F1203A9147F9038F1FFC001F713F090B5FC8114C1EC01FCEBFE005B
5BA25BB03A7FFF83FFE0B500C713F0A36C018313E0242E7FAD27>I<130F497E497EA46D
5A6DC7FC90C8FCA7383FFF80487FA37EEA000FB3A4007FB512F0B6FC15F815F07E1D2F7B
AE27>I<387FFF80B57EA37EEA000FB3B2007FB512F8B612FCA36C14F81E2E7CAD27>108
D<397F07C01F3AFF9FF07FC09039FFF9FFE091B57E7E3A0FFC7FF1F89038F03FC001E013
8001C01300A3EB803EB03A7FF0FFC3FF486C01E3138001F913E701F813E36C4801C31300
2920819F27>I<387FE07F39FFF1FFC001F713F090B5FC6C80000313C1EC01FCEBFE005B
5BA25BB03A7FFF83FFE0B500C713F0A36C018313E024207F9F27>I<EB1FE0EB7FF83801
FFFE487F481480390FF03FC0391FC00FE0393F8007F0EB00034814F8007E1301A248EB00
FCA76C1301007E14F8A2007F1303393F8007F0A2391FE01FE0390FF03FC06CB512806C14
006C5B38007FF8EB1FE01E207C9F27>I<387FE0FFD8FFF313C090B512F0816C800003EB
81FE49C67E49EB3F8049131F16C049130FA216E01507A6150F16C07F151F6DEB3F80157F
6DEBFF009038FF83FEECFFFC5D5D01F313C0D9F0FEC7FC91C8FCAC387FFF80B57EA36C5B
23317F9F27>I<397FFC03FC39FFFE0FFF023F13804A13C0007F90B5FC39007FFE1F14F8
9138F00F809138E002004AC7FC5CA291C8FCA2137EAD007FB57EB67EA36C5C22207E9F27
>114 D<9038FFF3800007EBFFC0121F5A5AEB803F38FC000F5AA2EC07806C90C7FCEA7F
8013FC383FFFF06C13FC000713FF00011480D8000F13C09038003FE014070078EB03F000
FC1301A27E14036CEB07E0EBE01F90B512C01580150000FB13FC38707FF01C207B9F27>
I<133C137EA8007FB512F0B612F8A36C14F0D8007EC7FCAE1518157EA415FE6D13FC1483
ECFFF86D13F06D13E0010313C0010013001F297EA827>I<397FE01FF8486C487EA3007F
131F00031300B21401A21403EBFC0F6CB612E016F07EEB3FFE90390FF87FE024207F9F27
>I<3A7FFE07FFE000FF15F06D5A497E007F15E03A0F80001F00A36D5B0007143EA414F0
EBC1F83903E3FC7CA4EBE79EA200011478A301F713F8A2EBFF0F6C5CA3EBFE0790387C03
E024207F9F27>119 D<393FFC1FFF486C5A168016006C487E3901F807E06C6C485A4A5A
017E90C7FC6D5AEB1F7E5C6D5A13076D5A5C80497E130F497E143EEB3E3FEB7E1F90387C
0F8001F87F00016D7E3803F0033A7FFE1FFF80A2B54813C06C486C1380A222207E9F27>
I<3A7FFC0FFF80486C4813C0A36C486C13803A07E000F800000313015D13F00001130301
F85B1200A26D485A137CA290387E0F80133EA2011F90C7FC5CA2130F149E14BE130714FC
1303A25C1301A25CA213035CA213075C1208EA3E0F007F5B131FD87E7FC8FCEA7FFE6C5A
5B6C5AEA07C022317E9F27>I<001FB512FE4814FFA490380001FEEC03FCEC07F8EC0FF0
001EEB1FE0C7EA3FC0EC7F80ECFF00495A495A495AEB1FE0495A495A49C7FC485A484813
1E4848133F485A485A485A485AB7FCA46C14FE20207E9F27>I E
/Fl 4 118 df<EF0180EF03C01707A21880170FA2EF1F00A2173EA2173C177CA25F0203
B612FE023F15FF91B8FC010316FE903B0FFE0003E000EB1FE0D97F805C01FEC71207EA01
F848484A5A485A48484AC7FC5B48C8121E163E123E5E5A1678007815F8A200F84A5AA248
5D1503A24B5AA24B5AA26C92C8FC5D1278153E127C153C6C147CA26C5CA26C6C5BEBC001
EA07E06C6C485AEA01F83900FE07C0EB7F8790381FE780EB0FFF010390B612FE010016FF
143F6E15FE023EC9FCA25CA2147814F8A2495AA2495AA25C1307A2001FB812FE4817FFA2
6C17FED8001FCAFC133EA2133C137CA25BA2485AA25B1203A25B6C5A386579C847>42
D<267FFFFC91380FFFFC6E5C6E80260FC00702001380D803E06DEC3F0000006D6C143E6D
6C6C141EEBF8006D7F15786D7F6D131C9038F7801E9038F3C00F01F16D7EECE003D9F0F0
7F91387801E091383800F0023C13706E13786E7F6E6C7E02037F6F7EDA01E01380913900
F003C092387001E0ED78006F13F06F1378030E133C030F131C923807801E923803C00F92
3901E0079E0300130304F013DE93387801FEEE3C00041C137E161E70133E933807801E04
03130E17C0EE01E0EE00F017781738173C171E170F1707188EEF03CE486CED01EEEF00FE
D807FE167E387FFFF0B5163E6C171ECB120E18063E407EBD35>78
D<126012F812FE6C7EEA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FFCEB03FF010013C0EC3F
F0EC07FCEC01FF9138007FC0ED1FF0ED07FE923801FF809238007FE0EE1FF8D90780EB03
FED91FE0903800FF80496CEC3FE0496CEC0FF8EF03FEEF00FFA2EF03FEEF0FF86D48EC3F
E06D48ECFF80D90780903803FE0090C8EA1FF8EE7FE0923801FF80DB07FEC7FCED1FF0ED
7FC04A48C8FCEC07FCEC3FF0ECFFC0010390C9FCEB0FFCEB3FE0EBFF80D803FECAFCEA0F
F8EA3FE0EAFF8048CBFC12F81260383679B147>109 D<D91FE01620D9FFFC16704813FF
000714C04814F048809026E01FFE15F0273F0003FFEC01E0007E010013C00078DA3FF013
07DB0FFCEB0FC048913A07FF807F8048020190B5FC6F1500043F5B040F13F804035B0040
9238007F80CDFCA4D91FE01620D9FFFC16704813FF000714C04814F048809026E01FFE15
F0273F0003FFEC01E0007E010013C00078DA3FF01307DB0FFCEB0FC048913A07FF807F80
48020190B5FC6F1500043F5B040F13F804035B00409238007F80CDFCADBA12F019F8A219
F03D397BB447>117 D E /Fm 1 84 df<9339FF8001800307EBF003033F13FC9239FF00
7E07DA01F8EB0F0FDA07E09038079F004A486DB4FC4AC77E023E804A5D187E5C495A183C
495AA213074A1538A3130F183080A295C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC
6E13FF6E14C0020F80020314F8EC003F03077F9238007FFE160F1603707E8283A283A212
06A4000E163EA2120C177E001E167CA25F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D14
3ED87CF85CD8787E495A3AF01FC00FE0D8E007B51280010149C9FC39C0003FF039487BC5
3C>83 D E /Fn 2 104 df[<51B47E090F13F8093F13FE517F972601FF801380973A03FE
001FC05048EB07E05048EB03F05048EB1FF85048137F087F14FF50485A5213FC4F5C4F13
805213F861A21B004F16F01EE0617613C0506D1300073FEC00FC9AC7FCA34F5AA419FF62
A46062A46062A56062A46062A2047FB812FC8893B9FCA3705F93C7001F90CAFC6061A518
7F61A418FF61A45F61A45F61A55F61A45F61A45F96CBFCA45FA260A4177F60A417FF60A4
5E60A45E60A54C5BA54C5BA495CCFC5EA35FA2163FA25FA2167F5FA44C5AA35F5D5F13FC
EA07FF4801805C485C4801C091CDFCA2484A5AA25EB512804B5AA24A485A495C49495A6C
5A01E0495A6CC748CEFC6D485A391FE003FC390FFC0FF86CB512E000015C6C6C90CFFCEB
0FFC>102 187 118 272 100 102 D<953803FF80063F13F095B512FC050780051FD981
FFEB0FC0943C3FFC003F803FE0DDFFF090390FC07FF04C01C0903807E0FF0407496D7E4C
48C7EA01F14C48EC00F94C4815FD4C48157F4C5A4B496F13E05D4B5B4B90C97E4B19C04B
5A4B4882A24B4818805C4A495EA24A491800A24A604A5B644A5B1B7F5C93CA5BA291B517
FFA2494960A2625B4B60A2625B4B60A2625B4B60A262A24B60A262A299C7FC5D62A263A2
1A7FA263A26D18FFA24F5B616D5FA24F5B6D5F6F5D6D5F96B55A6E6CEC01FB023F4B5A6E
6CEC0FE7DE1FC75B6E6CEC7F876E6C903801FE0F913A03FF8007FC02009026E03FF05C6F
B512E0031FEC801F0307EBFC009226007FE092C8FC93C8FC61A262A2197FA262A219FFA2
62A26062A34E5BA2D801F85ED807FE60486C4C5B487F484D90C9FC484D5AA2B54C5A4E5A
4D5B4D5B91C8485B494B5B4D90CAFC49ED7FFCD87FF04B5A01C0020313E0D83FF0021F13
806CB46CD9FFFECBFC000790B612F8000116E0D8003F4ACCFC010114E05C857CDA61>I
E /Fo 7 84 df<DA01C0EB01FE020791381FFF80021F91B512E091267F800314F0D901FF
010F14F80107143F49DA7E0113FC49903981F8007F013CD983E0EB1FFE0120D987C0130F
0100494813074BC7FC033E1403157E494915FC5D19F85DF007F04B15E04949EC0FC01980
4BEC1F00187E18F892C7EA03F049ED1FC04AEC7F80DC03FEC7FC4AEB1FF893B5FC010F01
0314C04A4814F04B804B809126F000077F011F02007F173F4A020F138083013F6F13C04A
807113E0A2187F495A183FA349C9FC19C0A25B00011880187F491700120318FE495E0007
4C5AD9F0205DD9F1F04A5A260FF7F84A5AD9EFFEEC1F8048B56C017EC7FC01DF9038F003
FC92B512F0D83F8F15C0010792C8FCD87E0114F82678007F13C000E0D90FFCC9FC3F487D
C541>66 D<EE01FE93381FFF804BB512C01507031F14E0157F913801FE01913903F0007F
DA0FC0133F4AC7FC023E15C05C4A1580495A0103157F4948150049485C49485CA249C748
5A5B01FE4A5A5F4848EC07C094C7FC484891C8FCA212075B120FA25B121FA25B123FA312
7F5BA412FFA97FA2171E173E6C6C15FC4C5A6D4A5A5F6C6C4A5A6D4A5A6C6C4AC7FC6D14
3E6C01C013FC9138F803F06C90B55A6C15806C4AC8FC6C14F8013F13C0D907FCC9FC3348
7FC534>I<EE03FE93383FFFC04BB512E0030714F0151F037F14F8EDFC01913903F0003F
DA07C0131F4A48EB0FF04AC7FC4A15E0027E15C04AEC1F8001011600173C494891C7FC13
07A3130FA280A280806D7F15E06D13F815FF6D14FC7F6E5B021F5B6E13C0023F90C8FC02
FFC9FCEB01F8EB03E0EB0FC049CAFC133E5B13FC485A485A12075B120F485AA2123F5B12
7F173C177C00FF4B5A4C5A4C5A6D5D4C5A6D4AC7FC6D143ED87FFC14FC01FFEB01F06C90
38E00FE091B512806C92C8FC6C14FC000314E0C691C9FCEB1FF835487EC535>69
D<0403B712F8043F16FE4BB9FC1507151F157F912601FC0090C7EA07FE912603F001ED01
FCDA07C04915F0DA0F80EE0080021F1800EC3F004A495A5C5C495A4A495A5C495A6DC7FC
90C8485AA35F161FA34C5AA35F167F94B612C0A293B7FC624B93C7FC19FC04FCC7127003
0392C8FC5EA24B5AA2150F5E151F5EA24B5AA24BCBFCA215FEA24A5AA24A5AEA0180000F
495AEA1FC0486C485AD87FF05B39FFFC1F80D87FFF90CCFC14FE6C5B6C13F06C5B000313
80D800FCCDFC50477EC348>I<031FB512F00203B77E021F16F091B812FC010317FF010F
188090283FE07FC00F14C0D9FE00DA007F13E0D801F84A010F13F0D803E0160348480400
13F8000F187F484801FF153F003FF01FFC007F180F90C7FC00FE92C8FC48180712F01280
C74817F85DA21AF0190F020317E05DF11FC01A80193F020717004B157E61614E5A4A484A
5A4E5AF01F80063EC7FC4A4814FCEF07F0EF7FE09239C07FFF8091273FC1FFFEC8FC03C7
13F003CF138091267F9FFCC9FC16800380CAFC92CBFC5CA25C1301A25C1303A25C13075C
A2130F5C131FA25C133F5C91CCFC137E137C136046497EC345>80
D<031FB512FC0203B712E0021F16FC91B9FC010318C0010F8490283FE07FC00380D9FE00
EC001FD801F804037FD803E04A13004848EF3FFC000F181F4848170F003F14FF007F1807
90C7FC00FE92C8FC486112F01280C7485F190F4B5E62191F6202034CC7FC4B157E197C61
4E5A4A48EC07E0F00F80063FC8FCEF03FC4A4848B45A040F13E04C13804C48C9FC4A4848
7EA2041F7FEDC007023F6D7F824B6C7F147F717E92C7FC4A6E7EA24A141F010182170F4A
8101031907716C141F4A183E01076F6D137C4A18F8719038C001F0010F9438E003E04A6E
9038F007C0011F9438FC1F804A92397FFFFE006249486F13F091C96C13C0017C7048C7FC
0170EE03F050467EC354>82 D<EF7FF0933807FFFE043FEBFF8093B612E0030315F0030F
15F892381F800F92267E000113FC03F8EB007F4A48141F0203150F4A481407020F16F814
1F4B15F0023F16E0F00FC01900027F150895C7FCA281A281816E7E816E6C7E16E06E13F8
6E13FE6EEBFFC06E14F06E6C13FC031F7F03076D7E030180DB003F7F040F7F04037F8270
6C7E173F010E151F017C6F7E48481507485A485A48481503121F485A60127FA200FF5F4D
5AA26D5E4D5A6D4B5A6D93C7FC6C6C153E6D5D6CB44A5A02C0EB07E06C01FCEB7FC06C90
B6C8FC6C15FC6C15F0C61580013F01FCC9FC010313803E487EC53C>I
E /Fp 1 84 df<4CB46C1338041F13F0047F01FC13704BB613F0922607FE001381DB1FE0
EB1FC3DB3F80903807E7E0037EC7EA03F74A48EC01FF4A48804BED7FC04A48153F4A5A4A
5AF11F804AC9FC147EA202FEEE0F005CA30101170EA3191E191CA26E93C7FCA280816D7F
8115F86EB4FC16F06E13FF6E14E06E14FE6E6E7E020115E06E6C80031F80030180DB001F
7F04037FEE003F05077F83170085187F183FA2181FA3EA0380A3120761A290CAFC48173F
96C7FCA26D167E121F60A26D4B5A003F4C5A6D4B5A6D4B5A6D151F486C4B5A017E037EC8
FC267E3F80495A267C1FE0EB07F826F80FFEEB3FE00103B65A48C64AC9FC48011F13F848
010113C045567AD348>83 D E /Fq 8 111 df<48B512F8000714FC4814F84814F0D83C
07C7FC1270EAC006130E1200A3131E131CA2133CA35BA313F8A3485AA26C5A1E1A7D981F
>28 D<157E000349B4FC0006491380484913C048EB0F0391381C01E048EB18005C4A1360
5A5CA248484813C0A291C7FC49EB018000E01403ED0700D87006130E5D003C143CD83F0E
13F8391FEE07E06CB55A00035CC649C7FCEB1FF0013CC8FCA35BA313F8A35B5B23267C99
2C>39 D<013FB512F816FF903A01FC001FC04AEB07E0EE03F001031401A24A14F8A21307
17F04A130317E0010F1407EE0FC04AEB1F80EE7E00011F495A91B512F0A291388001FC01
3FEB007E8291C7EA1F80160F4915C0A2137EA213FEEE1F805BEE3F000001153E16FE49EB
01F84B5A0003EC1FC0B7C7FC15F82D287DA732>66 D<130E131F5BA2133E131C90C7FCA7
EA03E0487EEA0C78EA187C1230A212605B12C0A2EA01F0A3485AA2485AA2EBC180EA0F81
A2381F0300A213066C5A131CEA07F06C5A11287DA617>105 D<133EEA07FEA2EA007CA2
13FCA25BA21201A25BA21203EC07809038E01FC0EC38600007EB61E014C3EBC187EBC307
D80FC613C09038CC038001B8C7FC13E0487E13FEEB3F80EB0FC0486C7E1303003E1460A2
127EECC0C0127CECC18012FC903801E30038F800FE0070137C1B297CA723>107
D<137CEA0FFCA2EA00F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A212
1FA21300A25AA2123EA2127EA2EA7C18A3EAF830A21320EA786013C0EA3F80EA0F000E29
7EA715>I<3B07801FC007E03B0FE07FF01FF83B18F0E0F8783C3B30F1807CE03E903AFB
007D801ED860FEEB3F005B49133E00C14A133E5B1201A24848495BA35F4848485A1830EE
01F0A23C0F8003E003E060A218C0933801E180271F0007C013E3933800FF00000E6D4813
7C341B7D993B>I<3907801FC0390FE07FF03918F0E0F83930F1807CEBFB00D860FE133C
5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A23A0F8007C0C0A2EDC18091
3803C300D81F0013C7EC01FE000EEB00F8231B7D9929>I E /Fr
3 49 df<B712FEA327037A8F34>0 D<1338A50060130C00F8133E00FC137E00FE13FE38
3FBBF83807FFC000011300EA007C48B4FC000713C0383FBBF838FE38FE00FC137E00F813
3E0060130C00001300A517197B9A22>3 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F
13C0A3EA1F80A21300A25A123EA35AA3127812F8A25A12100D1E7D9F13>48
D E /Fs 8 114 df<1306130C13181330136013E0EA01C0EA0380A2EA07005A120E121E
A2121C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E120F7EEA0380A2EA01C0EA
00E0136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01
C013E0120013F0A213701378A3133CA5133E131EAB133E133CA51378A3137013F0A213E0
120113C0EA0380A2EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C700
0EC8FCB3A22B2B7DA333>43 D<EB3F803801FFF03803E0F83807803C48487E001E7F003E
1480A2003C1307007C14C0A400FC14E0AE007C14C0A36CEB0F80A36CEB1F006C131E6C6C
5A3803E0F86CB45A38003F801B277EA521>48 D<13381378EA01F8121F12FE12E01200B3
AB487EB512F8A215267BA521>I<13FF000313E0380E03F0381800F848137C48137E0078
7F12FC6CEB1F80A4127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B90
3870018013E0EA0180390300030012065A001FB5FC5A485BB5FCA219267DA521>I<B712
F0A23907F000070003EC00F816781638A21618A3160C1560A21600A215E0A2140390B5FC
A2EBF0031400A21560A21606A2ED000CA4161C16181638A21678ED01F80007EC07F0B7FC
A227287EA72D>69 D<90383F80603901FFE0E03803F0703807C019380F800D381F00075A
007E1303A2127C12FCA7127C127EA27E6C1307EB800F380FC01B3803E0733801FFE33800
7F031300A7EC07F0EC3FFEA21F257E9923>113 D E /Ft 63 122
df<B812E0A30001903880003F6C90C71207EE03F0160116001770A21730A417381718A4
1700B3B04813C0B612E0A32D397DB834>0 D<1506150FA24B7EA24B7EA24B7EA2EDDFF0
A29138018FF8A291380307FCA291380603FEA291380E01FF140CDA1C007F141802386D7E
143002706D7E146002E06D7E5C01016E7E5C01036E7E91C7FC496E7E1306010E6E7E130C
011C6E7F131801386F7E133001706F7E136001E06F7E5B170F484882170748C97F170300
06831701488383481880001FB9FC4818C0A24818E0A2BA12F0A23C3C7CBB45>I<EC0FF8
EC7FFE903901F80780903907E001C090391F8000E090383F0007017E497EA25BA2485A6F
5AED018092C8FCA9ED03F0B7FCA33901F8000F1503B3AA486C497E267FFFE0B512C0A32A
3B7FBA2E>12 D<EC0FFC91387FFF70903901F803F0903807E00790381F800FEB3F00137E
A25B150748481303ADB7FCA33901F80003B3AB486C497E267FFFE0B512C0A32A3B7FBA2E
>I<B612F8A31D037AB02A>22 D<001C131C007F137F39FF80FF80A26D13C0A3007F137F
001C131C00001300A40001130101801380A20003130301001300485B00061306000E130E
485B485B485B006013601A197DB92A>34 D<146014E0EB01C0EB0380EB0700130E131E5B
5BA25B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3121E
A2121F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E01460
135278BD20>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C13
3E131EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E
133C137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<153015
78B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43
D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A
12600A19798817>I<B512FCA516057F941C>I<121C127FEAFF80A5EA7F00121C09097988
17>I<EB03F8EB1FFF90387E0FC09038F803E03901E000F0484813780007147C48487FA2
48C77EA2481580A3007EEC0FC0A600FE15E0B3007E15C0A4007F141F6C1580A36C15006D
5B000F143EA26C6C5B6C6C5B6C6C485A6C6C485A90387E0FC0D91FFFC7FCEB03F8233A7D
B72A>48 D<EB01C013031307131F13FFB5FCA2131F1200B3B3A8497E007FB512F0A31C38
79B72A>I<EB0FF0EB7FFE48B57E3903E03FE0390F000FF0000E6D7E486D7E486D7E1230
00706D7E126012FCB4EC7F807FA56CC7FC121CC8FCEDFF00A34A5A5D14035D4A5A5D140F
4A5A4A5A92C7FC147C5C495A495A495A495A91C8FC011EEB01805B5B4913034848140048
5A485A000EC75A000FB6FC5A5A485CB6FCA321387CB72A>I<EB07F8EB3FFF4913C03901
F80FF03903C007F848486C7E380E0001000F80381FE0006D7FA56C5A6C5AC85A1401A25D
4A5AA24A5A5DEC0F80027EC7FCEB1FFCECFF809038000FE06E7EEC01FC816E7EED7F80A2
16C0A2153F16E0A2121EEA7F80487EA416C049137F007F1580007EC7FC0070ECFF006C49
5A121E390F8003F83907F00FF00001B512C06C6C90C7FCEB0FF8233A7DB72A>I<1538A2
157815F8A2140114031407A2140F141F141B14331473146314C313011483EB0303130713
06130C131C131813301370136013C01201EA038013005A120E120C5A123812305A12E0B7
12F8A3C73803F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C9038F0
03F890B5FC5D5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F803907E0
07E090388003F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300485C
12E000605C12700030495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7FC38
007FFCEB1FE0213A7CB72A>I<EC3FC0903801FFF0010713FC90380FE03E90383F800790
387E001F49EB3F804848137F485AA2485A000FEC3F0049131E001F91C7FCA2485AA3127F
90C9FCEB01FC903807FF8039FF1E07E090383801F0496C7E01607F01E0137E497FA24914
8016C0151FA290C713E0A57EA56C7E16C0A2121FED3F807F000F15006C6C5B15FE6C6C5B
6C6C485A3900FE07F090383FFFC06D90C7FCEB03FC233A7DB72A>I<12301238123E003F
B612E0A316C05A168016000070C712060060140E5D151800E01438485C5D5DC712014A5A
92C7FC5C140E140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133FA513
7FA96DC8FC131E233B7BB82A>I<EB03F8EB1FFF017F13C09038FC07F03901E001F84848
6C7E4848137C90C77E48141E000E141F001E80A3121FA27F5D01E0131E6C6C133E01FC13
3C6D5B6C6C6C5AECC1E06CEBF3C06C01FFC7FC6C5BEB3FFF6D13C081017F13F801F07F39
03E07FFE3907801FFF48486C1380481303003E6D13C0003CEB007F007C143F0078EC0FE0
00F814075A1503A21501A36C15C012781503007C15806CEC07006C5C6C6C131ED807E013
7C3903F803F0C6B55A013F1380D907FCC7FC233A7DB72A>I<121C127FEAFF80A5EA7F00
121CC7FCB2121C127FEAFF80A5EA7F00121C092479A317>58 D<007FB812F8B912FCA3CC
FCAEB912FCA36C17F836167B9F41>61 D<B712E016FC16FF0001903980007FC06C90C7EA
1FE0707E707E707EA2707EA283A75F16035F4C5A4C5A4C5A4C5AEEFF8091B500FCC7FCA2
91C7EA7F80EE1FE0EE07F0707E707E83707EA21880177F18C0A7188017FFA24C13005F16
034C5AEE1FF8486DEB7FF0B812C094C7FC16F832397DB83B>66 D<913A01FF800180020F
EBE003027F13F8903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB
00FF49C8127F01FE153F12014848151F4848150FA248481507A2485A1703123F5B007F16
01A35B00FF93C7FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C6C150E
6C6C5D00001618017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7
FC9039007FFFFC020F13F002011380313D7BBA3C>I<B812FCA30001903880000F6C90C7
1201EE007E173E171E170EA31706A317078316C0A394C7FCA31501A21503150F91B5FCA3
EC000F15031501A21500A21860A318E093C712C0A41701A3EF0380A21707A2170F173F17
7F486D903807FF00B9FCA333397DB839>69 D<B812F8A30001903880001F6C90C71201EE
00FC177C173C171CA2170CA4170E1706A2ED0180A21700A41503A21507151F91B5FCA3EC
001F15071503A21501A692C8FCAD4813C0B612C0A32F397DB836>I<B648B512FEA30001
902680000313006C90C76C5AB3A491B6FCA391C71201B3A6486D497EB648B512FEA33739
7DB83E>72 D<B612C0A3C6EBC0006D5AB3B3AD497EB612C0A31A397EB81E>I<B612E0A3
000101C0C8FC6C90C9FCB3AD1718A517381730A31770A317F0A216011603160FEE1FE048
6D13FFB8FCA32D397DB834>76 D<B5933807FFF86E5DA20001F0FC002600DFC0ED1BF8A2
D9CFE01533A3D9C7F01563A3D9C3F815C3A2D9C1FCEC0183A3D9C0FEEC0303A2027F1406
A36E6C130CA36E6C1318A26E6C1330A36E6C1360A26E6C13C0A3913901FC0180A3913900
FE0300A2ED7F06A3ED3F8CA2ED1FD8A3ED0FF0A3486C6D5A487ED80FFC6D48497EB500C0
0203B512F8A2ED018045397DB84C>I<EC03FF021F13E09138FE01FC903901F8007ED907
E0EB1F8049486D7ED93F80EB07F049C76C7E01FE6E7E48486E7E49157E0003167F4848ED
3F80A24848ED1FC0A2001F17E049150F003F17F0A3007F17F8491507A300FF17FCAC007F
17F86D150FA3003F17F0A26C6CED1FE0A36C6CED3FC0000717806D157F000317006C6C15
FEA26C6C4A5A017F4A5A6D6C495A6D6C495AD907E0EB1F80D903F8017FC7FC903900FE01
FC91381FFFE0020390C8FC363D7BBA41>79 D<B712C016F816FE000190398001FF806C90
C7EA3FC0EE0FE0EE07F0EE03F817FC17FE1601A217FFA717FEA2EE03FCA2EE07F817F0EE
0FE0EE3FC0923801FF0091B512FC16F091C9FCB3A5487FB6FCA330397DB839>I<B612FE
EDFFE016F8000190388007FE6C90C76C7EEE3FC0707E707E707EA2707EA283A65FA24C5A
A24C5A4C5AEE3F8004FFC8FCED07FC91B512E05E9138000FF0ED03F8ED00FE82707E707E
A2161F83A583A6F00180A217F8160F1803486D01071400B66D6C5A04011306933800FE0E
CAEA3FFCEF07F0393B7DB83D>82 D<D90FF813C090383FFE0190B512813903F807E33907
E000F74848137F4848133F48C7121F003E140F007E1407A2007C140312FC1501A36C1400
A37E6D14006C7E7F13F86CB47E6C13F8ECFF806C14E06C14F86C14FEC680013F14800107
14C0EB007F020713E0EC007FED3FF0151F150FED07F8A200C01403A21501A37EA216F07E
15036C15E06C14076C15C06C140F6DEB1F80D8FBF0EB3F00D8F0FE13FE39E03FFFF8010F
13E0D8C00190C7FC253D7CBA2E>I<003FB812E0A3D9C003EB001F273E0001FE130348EE
01F00078160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011F
B612E0A335397DB83C>I<EAFFF8A4EAF000B3B3B3B3A3EAFFF8A40D5378BD17>91
D<EAFFF8A4EA0078B3B3B3B3A3EAFFF8A40D537FBD17>93 D<EA01801203EA0700120E5A
12181238123012701260A212E05AA412CEEAFF8013C0A3127FA2EA3F80EA0E000A197AB9
17>96 D<EB1FE0EBFFFC3803E03F3907000F80390F8007E0486C6C7E13E06E7EA26E7E6C
5A6C5AC8FCA4147FEB07FFEB3FE0EBFE00EA03F8EA0FF0EA1FC0123F485A90C7FC160C12
FEA31401A26C13036CEB077C903980063E18383FC01E3A0FE0781FF03A03FFF00FE03A00
7F8007C026277DA52A>I<EA03F012FFA3120F1203B0EC1FE0EC7FF89038F1E03E9039F3
801F809039F7000FC001FEEB07E049EB03F049EB01F85BED00FCA216FEA2167E167FAA16
7E16FEA216FC15016D14F8ED03F07F01EEEB07E001C6EB0FC09039C7801F00903881E07E
903800FFF8C7EA1FC0283B7EB92E>I<EB03FC90381FFF8090387E03E03901F800704848
13F83907E001FC380FC003A2EA1F80123F90380001F848EB00F01500A2127E12FEAA127E
127FA26C14067F001F140E6D130C000F141C6C6C13386C6C13706C6C13E039007C07C090
381FFF00EB07F81F277DA525>I<ED0FC0EC03FFA3EC003F150FB0EB03F8EB1FFF90387E
078F9038F801EF3903F0007F4848133F4848131FA24848130F123F90C7FC5AA2127E12FE
AA127E127FA27EA26C6C131FA26C6C133F6C6C137F6C6CEBEFF03A01F801CFFF39007C07
8F90381FFE0FD907F813C0283B7DB92E>I<EB07F8EB1FFF90387C0FC03901F803E03903
F001F0D807E013F8380FC0004848137CA248C7127E153E5A153F127E12FEA3B7FCA248C8
FCA5127EA2127FA26C14037F001F14076C6C13060007140E6D131CD801F013386C6C1370
90387E03E090381FFF80903803FC0020277EA525>I<147E903803FF8090380FC1E0EB1F
8790383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387F
FFF8A31C3B7FBA19>I<ED03F090390FF00FF890393FFC3C3C9039F81F707C3901F00FE0
3903E007C03A07C003E010000FECF000A248486C7EA86C6C485AA200075C6C6C485A6D48
5A6D48C7FC38073FFC38060FF0000EC9FCA4120FA213C06CB512C015F86C14FE6CECFF80
4815C03A0F80007FE048C7EA0FF0003E140348140116F8481400A56C1401007C15F06CEC
03E0003F1407D80F80EB0F80D807E0EB3F003901FC01FC39007FFFF0010790C7FC26387E
A52A>I<EA03F012FFA3120F1203B0EC0FF0EC3FFCECF03F9039F1C01F809039F3800FC0
EBF70013FE496D7EA25BA35BB3A3486C497EB500C1B51280A3293A7EB92E>I<EA0380EA
0FE0487EA56C5AEA0380C8FCAAEA03F012FFA312071203B3AA487EB512C0A312387EB717
>I<EA03F012FFA3120F1203B1913801FFFCA39138007FC01600157C15705D4A5A4A5A4A
C7FC141E1438147814FC13F1EBF3FEEBF73F01FE7FEBF81F496C7E8114076E7E6E7E8114
00157E157F811680ED1FC0486CEB3FF0B500C0B5FCA3283A7EB92C>107
D<EA03F012FFA3120F1203B3B3AD487EB512C0A3123A7EB917>I<2703F00FF0EB1FE000
FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F802603F70013
CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B50083B5FCA3
40257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC03803F700
13FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>I<EB03FE90380FFF
8090383E03E09038F800F84848137C48487F48487F4848EB0F80001F15C090C712074815
E0A2007EEC03F0A400FE15F8A9007E15F0A2007F14076C15E0A26C6CEB0FC0000F15806D
131F6C6CEB3F006C6C137EC66C13F890387E03F090381FFFC0D903FEC7FC25277EA52A>
I<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A07F7000FC0D803FEEB07E049EB
03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13076DEB
0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB512C0A328
357EA42E>I<D903F813C090381FFE0190387E07819038FC01C33903F000E30007147748
48133749133F001F141F485A150F48C7FCA312FEAA127FA37E6D131F121F6D133F120F6C
6C137F6C6C13EF3901F801CF39007E078F90381FFE0FEB07F890C7FCABED1FE00203B5FC
A328357DA42C>I<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03E613EE
9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>I<EBFF03000313
E7380F80FF381E003F487F487F00707F12F0A2807EA27EB490C7FCEA7FE013FF6C13E06C
13F86C7F00037FC67F01071380EB007F141F00C0EB0FC01407A26C1303A37E15806C1307
7EEC0F00B4131E38F3C07C38E1FFF038C03F801A277DA521>I<1318A51338A31378A313
F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C03017E1300
6D5AEB0FFEEB01F81A347FB220>I<D803F0EB07E000FFEB01FFA3000FEB001F00031407
B3A4150FA3151F12016D133F0000EC77F86D9038E7FF8090383F03C790381FFF87903A03
FC07E00029267EA42E>I<B538803FFEA33A0FF8000FF06C48EB07E00003EC03C06D1480
00011500A26C6C1306A26D130E017E130CA26D5BA2EC8038011F1330A26D6C5AA214E001
075BA2903803F180A3D901FBC7FCA214FF6D5AA2147CA31438A227257EA32C>I<B53A1F
FFE03FFEA3260FF8009038000FF86C48017EEB03E018C00003023EEB0180A26C6C013FEB
0300A36C6CEC8006156FA2017E9038EFC00C15C7A2D93F016D5A15830281EBF038D91F83
1430150102C3EBF87090260FC6001360A2D907E66D5A02EC137CA2D903FCEB7F804A133F
A2010192C7FC4A7FA20100141E4A130E0260130C37257EA33C>I<B538807FFFA33A03FE
003FF00001EC1F80000092C7FC017E131C6D13186D6C5AECC070010F5B6D6C5AECF180EB
03FB6DB4C8FC6D5AA2147F804A7E8114CF903801C7E090380383F090380703F8EB060149
6C7E011C137E49137F01787F496D7E486C80000FEC3FF0D8FFFE90B51280A329247FA32C
>I<B538803FFEA33A0FF8000FF06C48EB07C00003EC03806C7E16007F00001406A2017E
5BA2137F6D5BA26D6C5AA2ECC070010F1360A26D6C5AA214F101035BA2D901FBC7FCA214
FF6D5AA2147CA31438A21430A214701460A25CA2EA7C0100FE5B130391C8FC1306EAFC0E
EA701C6C5AEA1FF0EA0FC027357EA32C>I E /Fu 27 107 df<007FB81280B912C0A26C
17803204799641>0 D<121C127FEAFF80A5EA7F00121C0909799917>I<0060150600F815
0F6C151F007E153F6C157E6C6C14FC6C6CEB01F86C6CEB03F06C6CEB07E06C6CEB0FC06C
6CEB1F80017EEB3F006D137E6D6C5A90380FC1F8903807E3F0903803F7E06DB45A6D5B6E
C7FCA24A7E497F903803F7E0903807E3F090380FC1F890381F80FC90383F007E017E7F49
EB1F804848EB0FC04848EB07E04848EB03F04848EB01F84848EB00FC48C8127E007E153F
48151F48150F00601506282874A841>I<EB0FE0EB7FFC497E0003EBFF803907F83FC039
0FC007E0391F8003F0393F0001F8003E130048147CA20078143C00F8143EA248141EA46C
143EA20078143C007C147CA26C14F8003F1301391F8003F0390FC007E03907F83FC06CB5
1280C6EBFE006D5AEB0FE01F207BA42A>14 D<EB0FE0EB7FFC497E0003EBFF804814C048
14E04814F04814F8A24814FCA3B612FEA86C14FCA36C14F8A26C14F06C14E06C14C06C14
80C6EBFE006D5AEB0FE01F207BA42A>I<D93F8015082601FFF0151C4813FC4813FF4880
4802E0143C393FC07FF8273E000FFC147848D903FF14F8007801009038C001F04891387F
F80F031FB512E048020714C06F148003001400EE3FFE0040ED07F0CCFCA3D93F80150826
01FFF0151C4813FC4813FF48804802E0143C393FC07FF8273E000FFC147848D903FF14F8
007801009038C001F04891387FF80F031FB512E048020714C06F148003001400EE3FFE00
40ED07F036257BA741>25 D<126012F0A37EA21278127CA27EA27E7F6C7E6C7E6C7EEA01
FC6CB4FCEB3FC0EB1FF8EB07FF010113F89039007FFFF8020F90B51280020115C05C021F
158091B500F0C7FC010301F0C8FCD90FFEC9FCEB3FF0EB7F80D801FECAFCEA03F8EA07E0
485A485A90CBFC123EA25AA2127812F8A25AA31260323179AC41>31
D<181EA4181F84A285180785727EA2727E727E85197E85F11F80F10FC0F107F0007FBA12
FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A61180F96C7
FCA260181EA4482C7BAA53>33 D<0278151EA402F8151F4A81A20101834A150701038349
486F7EA249486F7E49CA7E4983017E177E49834848EF1F804848EF0FC0D80FE0EF07F000
3FBA12FCBCFCA2003F19FCD80FE0CAEA07F0D803F0EF0FC06C6CEF1F806C6CEF3F00017E
177E6D5F6D5F6D6C4B5A6D6C4B5AA26D6C4B5A01015F6E150F010094C7FCA26E5D027815
1EA4482C7BAA53>36 D<D93F801508D9FFF0151C000313FC487F486D7E4880273FC07FF0
143C9026000FF81438007CD903FE147800786D6C14F80070903A007FC003F000F091383F
F80F48020FB512E06F14C0030114806F1400EE3FFC0040ED07F0CCFCAF007FB812F8B912
FCA26C17F836257BA641>39 D<173CA2173E171E171F717E170784717E717E717E84007F
B87EBAFC8585CBEA03E0F001F8F000FE193FF11FE0F107F8F101FFA2F107F8F11FE0F13F
0019FEF001F8F003E0BA5A6196C7FC6C5FCB5A604D5A4D5A4D5A60170F4DC8FC171E173E
173CA2482E7BAB53>41 D<DA03C0143C0207153E4B141E020F151F92C87E4A82021E1507
023E824A6F7E0278150102F88249486F7E49B87E49834983498449CAEA0FC0017E717ED8
01F8EF01F84848717ED80FE0187FD83F80F01FC0B4CCEA0FF0A2D83F80F01FC0D80FE0F0
7F00D803F018FC6C6C4D5AD8007EEF07E06D4D5A6DB95A6D95C7FC6D5F6D5FD901F0C95A
6D6C4B5A02785E027C15036E4B5A021E5E021F150F6E93C8FC6F5C0207151E6F143E0203
153C4C2E7DAB53>44 D<EE0180EE03C01607A2EE0F80A2EE1F00A2163EA25EA25EA24B5A
A24B5AA24B5A150F5E4BC7FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2
143EA25CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25B13FC5B485AA2485AA2
485AA2485AA248CAFCA2123EA25AA25AA25A12602A4E75BB00>54
D<0060161800F0163C6C167CA200781678007C16F8A2003C16F0003E1501A26CED03E0A2
6C16C06D1407A2000716806D140FA26C6CEC1F00A26CB612FEA36C5D01F8C7127CA2017C
5CA2013C5C013E1301A2011E5C011F1303A26D6C485AA201075CECC00FA2010391C7FC6E
5AA2903801F03EA20100133CECF87CA2EC7878EC7CF8A2EC3FF0A26E5AA36E5AA36E5A6E
C8FC2E3C80B92F>56 D<007FB612F0B712F8A27EC91278B3A5003FB612F85AA27EC91278
B3A5007FB612F8B7FCA26C15F0253A7CB92E>I<007FB712F8B812FCA27ECA123CB21718
2E177C9D37>I<007FB812F8B912FCA26C17F8C80078C8FCB3B3AD153036367BB541>62
D<ED03FE92381FFF80037F13C00203B5FCEC07C091381E003F4A131F14F049481480495A
495A49C7EA3F005B133E49147EA249147C000115FC495C0003EC01E0495C000791C8FC5B
120FA2485AA3123F90CAFCA35AA2127EA312FEA87E161E5E6D14F8127F6D495A4B5A6C6C
5C6D495A6C6C49C7FC01FE133C390FFF80F86CEBFFE06C1480C649C8FCEB3FF02A3D7FBA
2C>67 D<0060161800F0163CB3B26C167CA2007C16F8A26CED01F0003F15036C6CEC07E0
6C6CEC0FC0D807F0EC3F80D803FE903801FF003A00FFC00FFC6DB55A011F14E0010391C7
FC9038007FF82E347CB137>91 D<14034A7E4A7EA24A7EA34A7EA2EC7CF8A2ECF87CA2EC
F03C0101133EA249487EA249486C7EA249486C7EA2EC00034980A2013E6D7EA2496D7EA2
0178147801F8147CA2484880A2484880A24848EC0F80A2491407000F16C0A248C8EA03E0
A2003EED01F0A2003C1500007C16F8A248167CA248163C006016182E347CB137>94
D<0060161800F0163C6C167CA2007C16F8A2003C16F0003E1501A26CED03E0A26C6CEC07
C0A2000716806D140FA26C6CEC1F00A26C6C143EA26C6C5CA201781478017C14F8A26D49
5AA26D495AA26D5CEC8007A26D6C485AA26D6C48C7FCA2903801F03EA20100133CECF87C
A2EC7CF8A2EC3FF0A26E5AA36E5AA26E5A6EC8FC2E347CB137>I<126012F0B3A8B712FE
16FFA216FE00F0C9FCB3A81260283A7BB933>I<EC01F8140FEC3F80ECFC00495A495A49
5AA2130F5CB3A7131F5C133F49C7FC13FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE13
7F6D7E131F80130FB3A7801307A26D7E6D7E6D7EEC3F80EC0FF814011D537ABD2A>102
D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC
1FF0EC07F8EC1FF0EC7E00495A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA
07F0EAFFC000FCC8FC1D537ABD2A>I<14C0EB01E01303A214C01307A21480130FA2EB1F
00A2131E133EA25BA2137813F8A2485AA25B1203A25B1207A2485AA290C7FC5AA2123EA2
123C127CA2127812F8A41278127CA2123C123EA27EA27E7FA26C7EA212037FA212017FA2
6C7EA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A21301EB00C01352
78BD20>I<126012F07EA21278127CA2123C123EA27EA27E7FA26C7EA212037FA26C7EA2
12007FA21378137CA27FA2131E131FA2EB0F80A2130714C0A2130314E0A414C01307A214
80130FA2EB1F00A2131E133EA25BA2137813F8A25B1201A2485AA25B1207A2485AA290C7
FC5AA2123EA2123C127CA2127812F8A25A126013527CBD20>I<126012F0B3B3B3B3A912
60045377BD17>I E /Fv 55 127 df<EC3FC0ECFFF8903807E07C90380F801FD93F00EB
800C017E130F49903807C01C4848ECE018485A484801031338000FEDF0305B001F167048
48156017E0007F16C090C713F1178016F34816004815F716FE5EA2485D5EA4007E140715
0F003E91381DF818003F14796C02E11338270F8007C013303B07E03F007CF02601FFF8EB
3FC026003FC0EB0F802E267DA435>11 D<ED01FC923807FF8092381E07C092387801E003
E013F0913901C000F84A5A4AC7FC020E14FC5C5C143002701301026014F814E05C010114
034A14F0130391C7EA07E017C049140F0106EC1F809238FFBF00D90E0113FC90390C0381
F8EDFFFE0200131F011C010013800118140F17C0A213380130EC07E0A2160F13701360A3
01E0141F4915C0A30001153F1780A2EE7F00120316FE5E6D495AD80760495AD80670495A
6D495A6D495AD80E0E49C7FC390C0780FC903801FFF09038007F80001C90C9FC1218A312
381230A312701260A312E05AA22E4B7EBA2F>I<EB07F0D91FFC1430D97FFE147090B514
6000036E13E0486E13C0D9F01F1301270F8007E01380261F00011303001CD900F0130048
EC7007160648EC300E0060EC380CED181C00E01518C8121CED0C3816301670166016E05E
A2150D5EA2150F93C7FCA2150EA3150CA3151CA215181538A45DA45DA44A5AA35D2C377F
A42B>I<1403EC3FF891387FFF80D901E313C014800103133F9138001F80ED070092C7FC
80A280A2808013018080130080147F81143F8149B47E130790380F8FF0EB3E0F496C7E13
F83801F003D803E07F1207380FC0011380121FEA3F0014005A127EA212FE5D481301A35D
A24813035D6C13075D127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0D8003FC8
FC223D7DBB25>I<133F14C0EB07F06D7E801301A26D7EA3147FA36E7EA36E7EA36E7EA3
6E7EA36E7EA36E7EA26E7EA214014A7E5C4A7E91381E3F80143C14784A6C7E1301EB03E0
49486C7EEB0F80EB1F00496D7E137E5B48486D7E485A485A000F6E7E485A485A48C87E12
FE167F4816800070151F293B7CB930>21 D<027FB512C00103B612E0130F5B017F15C090
26FF81FEC7FC3901FC007E48487F485A497F484880485AA248C7FCA2127EA2153F00FE92
C7FC5AA25D157E5A5DA24A5AA24A5A007C495A5D003C495A003E013FC8FC6C137C380F81
F83803FFE0C66CC9FC2B257DA32F>27 D<013FB512FE90B7FC5A5A4815FE260F801CC7FC
EA1E005A00385B5A5A481378C7FC147014F0A4495AA31303A3495AA3130FA25C131FA313
3FA291C8FC131E28257EA324>I<EC3FE0903801FFFC010713FF011F148090397F801FC0
9038F80007D801E01303484890C7FC48C9FCA21206A5000713F038039FFF3801FF0F14FF
38039FFE48C9FC120E5A5A123012701260A212E0A215080060141C00701418007814386C
14F0391F8007E06CB55A6C91C7FC000113FC38003FE022287FA527>34
D<EE1F8001C0EC7FE00001913801FFF848484913FC90C75A4891380FC07E000691381F00
3E000E023E131E000C023C130F001C4A13070018147000385C00305C4A5AA2007049C712
0612601406170E170C00E049141CA2173800605B0070167017E00078013813016CED03C0
003E0130EB0F806C0170EB1F0001C0147ED80FF0495A3A07FFF01FF8000190B512E06C15
80013F49C7FC010F13F8010113C0D903E0C8FCA25C1307A4495AA3131FA349C9FCA3131E
30377DA436>39 D<121C127FEAFF80A5EA7F00121C0909798817>58
D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A
12600A19798817>I<EF0180EF07C0171F177F933801FF00EE07FCEE1FF0EE7FC04B48C7
FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC0
4848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0
EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238
007FC0EE1FF0EE07FCEE01FF9338007FC0171F1707EF0180323279AD41>I<1760177017
F01601A21603A21607160FA24C7EA216331673166316C3A2ED0183A2ED0303150683150C
160115181530A21560A215C014011580DA03007FA202061300140E140C5C021FB5FC5CA2
0260C7FC5C83495A8349C8FC1306A25BA25B13385B01F01680487E000716FFB56C013F13
FF5EA2383C7DBB3E>65 D<0103B77E4916F018FC903B0007F80003FE4BEB00FFF07F8002
0FED3FC0181F4B15E0A2141FA25DA2143F19C04B143F1980027F157F190092C812FE4D5A
4A4A5AEF0FF04AEC1FC005FFC7FC49B612FC5F02FCC7B4FCEF3FC00103ED0FE0717E5C71
7E1307844A1401A2130F17035CA2131F4D5A5C4D5A133F4D5A4A4A5A4D5A017F4BC7FC4C
5A91C7EA07FC49EC3FF0B812C094C8FC16F83B397DB83F>I<9339FF8001C0030F13E003
7F9038F80380913A01FF807E07913A07F8000F0FDA1FE0EB079FDA3F80903803BF0002FF
C76CB4FCD901FC80495A4948157E495A495A4948153E017F163C49C9FC5B120148481638
5B1207485A1830121F4993C7FCA2485AA3127F5BA312FF90CCFCA41703A25F1706A26C16
0E170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C6C6C5C6CB4
495A90393FE00FC0010FB5C9FC010313FC9038007FC03A3D7CBA3B>I<0103B7FC4916E0
18F8903B0007F80007FE4BEB00FFF03F80020FED1FC0180F4B15E0F007F0021F1503A24B
15F81801143F19FC5DA2147FA292C8FCA25C18035CA2130119F84A1507A2130319F04A15
0FA2010717E0181F4A16C0A2010FEE3F80A24AED7F00187E011F16FE4D5A4A5D4D5A013F
4B5A4D5A4A4A5A057FC7FC017F15FEEE03FC91C7EA0FF049EC7FC0B8C8FC16FC16C03E39
7DB845>I<0103B812F05BA290260007F8C7123F4B1407F003E0020F150118005DA2141F
A25D19C0143FA24B1330A2027F1470190092C7126017E05C16014A495A160F49B6FCA25F
9138FC000F01031407A24A6DC8FCA201075C18034A130660010F160693C7FC4A150E180C
011F161C18184A1538A2013F5E18F04A4A5AA2017F15074D5A91C8123F49913803FF80B9
FCA295C7FC3C397DB83D>I<DCFF8013E0030F13F0037F9038FC01C0913A01FF803E0391
3A07FC000F07DA0FE0EB038FDA3FC0903801DF804AC812FFEB01FED903F8157F4948ED3F
00495A495A494881017F161E49C9FC5B12014848161C5B1207485A1818121F4993C7FCA2
485AA3127F5BA312FF90CCFC93387FFFFE93B5FCA29338007FC0715A177F95C7FCA27E5F
5F7F123F16016C7E5F6C6C14036D14071207D803FCEC1EF86C6C143C26007F80EBF07890
393FF007E0010FB5EA8030010349C9FC9038003FE03B3D7DBA41>71
D<0103B5D8F803B512F8495DA290260007F8C73807F8004B5DA2020F150F615DA2021F15
1F615DA2023F153F615DA2027F157F96C7FC92C8FCA24A5D605CA249B7FC60A202FCC712
0101031503605CA201071507605CA2010F150F605CA2011F151F605CA2013F153F605CA2
017F157F95C8FC91C8FC496C4A7EB690B6FCA345397DB845>I<0107B512FCA216F89039
0007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25C
A21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FC
A326397DB824>I<0203B512FCA3DA000113006F5AA215015EA315035EA315075EA3150F
5EA3151F5EA3153F5EA3157F93C7FCA35D5DA31401A25DA21403120FD83F805B127FEBC0
07D8FF805BA24A5AEB001F00FC5C00E0495A006049C8FC007013FE383801F8381E07F038
07FFC0D801FEC9FC2E3B7AB82E>I<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA214
1FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA2130718404A15
C0A2010F150118804A1403A2011F16005F4A1406170E013F151E171C4A143C177C017F5D
160391C7120F49EC7FF0B8FCA25F32397DB839>76 D<902603FFF891381FFFF8496D5CA2
D90007030113006FEC007C02061678DA0EFF157081020C6D1460A2DA1C3F15E0705CEC18
1F82023815016F6C5C1430150702706D1303030392C7FC02607FA2DAE0015C701306ECC0
008201016E130EEF800C5C163F0103EDC01C041F131891C713E0160F49EDF03818300106
140717F8010E02031370EFFC60130CEE01FE011C16E004005B011815FF177F1338600130
153FA20170151F95C8FC01F081EA07FCB512E01706A245397DB843>78
D<0103B7FC4916E018F8903B0007F80007FC4BEB00FE187F020FED3F80F01FC05DA2021F
16E0A25DA2143FF03FC05DA2027FED7F80A292C8130018FE4A4A5A604AEC07F04D5A0101
ED3FC04CB4C7FC91B612FC17E0D903FCCAFCA25CA21307A25CA2130FA25CA2131FA25CA2
133FA25CA2137FA291CBFC497EB6FCA33B397DB835>80 D<4BB4FC031F13F09238FE01FC
913903F0007EDA07C0EB1F80DA1F80EB0FC0023EC7EA07E002FCEC03F0495A4948EC01F8
495A4948EC00FC495A013F16FE49C9FC13FE187F485A12035B12075B120F4916FF121FA2
485AA34848ED01FEA448C9EA03FCA3EF07F8A218F0170F18E0171F18C0EF3F807EEF7F00
17FEDA07C05B6C90391FF001F8903980383803001F496C485A9139E00C0FE0260FC0C0EB
1F80D807E1D90E3FC7FC0280137ED803F1EB07F8D801F95C3A007FC00FC0903A3FE07F00
03903807FFFE0100018F5BDA000F1306170E171E705A177CEEC1F816FF5FA25F5F6F5B6F
48C7FCED00F8384B7CBA42>I<0103B612F849EDFF8018E0903B0007F8001FF84BEB03FC
EF00FE020F157FA24BEC3F80A2021F16C0A25DA2143FF07F805DA2027FEDFF006092C748
5A4D5A4A4A5A4D5A4AEC1F80057FC7FC0101EC07F891B612E094C8FC9139FC000FC00103
EC03F0707E4A6D7E831307177E5C177F010F5D5F5CA2011F1401A25CA2133F16034A4A13
60A2017F17E019C091C71401496C01011480B61503933900FE0700EF7E0ECAEA1FFCEF07
F03B3B7DB83F>I<003FB56C48B51280485DA226007F80C7381FF00091C8EA07C0604993
C7FCA2491506A20001160E170C5BA20003161C17185BA20007163817305BA2000F167017
605BA2001F16E05F5BA2003F15015F5BA2007F150394C8FC90C8FCA25E4815065A160E16
0C161C161816385E127E5E4B5A6C4A5A4BC9FC6C6C131E6C6C5B6C6C13F83903F807E06C
B55A6C6C48CAFCEB0FF0393B7BB839>85 D<267FFFFC91383FFFC0B55DA2000390C83807
FC006C48ED03E06060000094C7FC5F17065FA25F6D5DA26D5D17E05F4C5AA24CC8FC6E13
06A2013F5C161C16185EA25E6E5BA2011F495A150393C9FC1506A25D6E5AA2010F5B1570
15605DA2ECE18002E3CAFC14F3EB07F614FE5C5CA25C5CA26D5AA25C91CBFC3A3B7CB830
>I<277FFFFC01B500F890B51280B5FC60000390C7D807FCC7380FF80001FC4BEC03E000
016204035E98C7FC621A0604075DA2040F5DA2041B5D6216336D02735D1663000003C34A
5A83DB01834AC8FC04815CDB0301140603075D1506030C5DA203185D1970033015606115
606D01E04A5A15C090267F01804AC9FC17FEDA030014060400130E0206150C020E5D140C
4A5DA24A5D18E04A5D715A5C4A92CAFCA26DC85AA2013E157C1778133C17701338013015
60513B7CB84E>I<49B500F890387FFFF095B5FC1AE0D90003018090380FFC004BC713E0
0201ED07804EC7FC6E6C140E606F5C705B606F6C485A4D5A031F91C8FCEEE0065F6F6C5A
5F03075B705A16F96FB45A94C9FC6F5AA36F7EA34B7FED037F9238063FC0150E4B6C7E15
38ED700F03E07F15C04A486C7EEC0300020613034A805C4A6D7E14704A1300494880495A
49C86C7E130E011E153F017E4B7ED803FF4B7E007F01E0011FEBFFC0B5FC6144397EB845
>I<B500FC91383FFFE0A3000390C83807FC006CEE03E06C5F4D5A95C7FC6D6C140E5F5F
6D6C14305F5F6D6C495A4CC8FC010F5C6E130E160C01075C6E5B5E6D6C5B15014B5AD901
FE90C9FC15065D6D6C5A5D15706E5A5D5D6ECAFC5CA3147E14FEA35C1301A35C1303A35C
1307A2130F000FB512F0A25D3B397DB830>I<91B712FCA25B9239E00007F84AC7EA0FF0
D903F8EC1FE04AEC3FC04AEC7F804A150049485C91C7485A4C5A010E4A5A4C5A010C4A5A
011C4A5A01185D167F4CC7FC90C7485A4B5A4B5A4B5A5E151F4B5A4B5A4BC8FC4A5A4A5A
4A5A5D140F4A5A4A5A4A48130C4AC7FC495A4A141C01031518495A494814384948143049
481470495A49C812F0495D000115014848140348484A5A4848140F4848141F4848EC7F80
4848EB07FF90B7FCB8FC94C7FC36397BB839>I<147E903803FF8090390FC1C38090391F
00EFC0017E137F49133F485A4848EB1F8012075B000F143F48481400A2485A5D007F147E
90C7FCA215FE485C5AA214015D48150CA21403EDF01C16181407007C1538007E010F1330
003E131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F0026267DA42C
>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0FCEBE3FF9038
E707C0390FFE03E09038F801F001F013F8EBE000485A15FC5BA2123F90C7FCA214015A12
7EA2140312FE4814F8A2140715F05AEC0FE0A215C0EC1F80143F00781400007C137E5C38
3C01F86C485A380F07C06CB4C7FCEA01FC1E3B7CB924>I<EC3FC0903801FFF0903807E0
3C90380F800E90383F0007017E131F49137F484813FF485A485A120F4913FE001F143848
481300A2127F90C8FCA35A5AA45AA315031507007E1406150E003E143C003F14706C14E0
390F8007C03907C03F003801FFF838003FC020267DA424>I<163FED1FFFA3ED007F167E
A216FEA216FCA21501A216F8A21503A216F0A21507A2027E13E0903803FF8790380FC1CF
90381F00EF017EEB7FC049133F485A4848131F000715805B000F143F485A1600485A5D12
7F90C7127EA215FE5A485CA21401A248ECF80CA21403161CEDF0181407007C1538007E01
0F1330003E131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F00283B
7DB92B>I<EC3FC0903801FFF0903807E07890381F801C90387E001E49130E485A485A12
07485A49131E001F141C153C484813F8EC03E0007FEB3FC09038FFFE0014E090C8FC5A5A
A7007E140315071506003E140E153C6C14706C6C13E0EC07C03903E03F003801FFF83800
3FC020267DA427>I<16F8ED03FEED0F8792381F0F80ED3E3F167F157CA215FC1700161C
4A48C7FCA414035DA414075DA20107B512F0A39026000FE0C7FC5DA4141F5DA4143F92C8
FCA45C147EA514FE5CA413015CA4495AA45C1307A25C121E123F387F8F80A200FF90C9FC
131E12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA29>I<EC07E0EC1FF891387C1C389039
01F80EFC903803F007903807E003EB0FC090381F8001D93F0013F85B017E130313FE16F0
485A150712034914E0A2150F12074914C0A2151FA2491480A2153FA2160000035C6D5B00
015B4A5A3900F8077E90387C1EFEEB1FF8903807E0FC90C7FC1401A25DA21403001E5C12
3F387F80075D00FF495A49485A4849C7FC007C137E383C01F8381FFFE0000390C8FC2636
7FA428>I<EB03F0EA01FFA3EA00075CA3130F5CA3131F5CA3133F91C9FCA35B90387E03
F8EC0FFF91383C0F809039FEF007C0D9FDC07FEBFF80EC0003485A5BA249130712035BA2
150F00075D5BA2151F000F5D5B153F93C7FC121F4990387F0180157EEDFE03003F02FC13
0090C7FC5EEDF80648150E007E150C161C5E00FEEC787048EC3FE00038EC0F80293B7CB9
30>I<14E0EB03F8A21307A314F0EB01C090C7FCAB13F8EA03FEEA070F000E1380121C12
1812381230EA701F1260133F00E0130012C05BEA007EA213FE5B1201A25B12035BA20007
131813E01438000F133013C01470EB806014E014C01381EB838038078700EA03FEEA00F8
15397EB71D>I<EB03F0EA01FFA3EA00075CA3130F5CA3131F5CA3133F91C8FCA35B017E
EB07C0ED1FF0ED783801FEEBE0F89039FC01C1FCEC0383EC07070001130ED9F81C13F891
383803F091387001E0000349C7FCEBF1C0EBF38001F7C8FCEA07FEA2EBFFE0EBE7F8380F
E0FEEBC07F6E7E141F001F80D9800F1330A21670003F011F136001001380A216E04815C0
007E1481020F1380158300FE903807870048EB03FE0038EB00F8263B7CB92B>107
D<EB0FC0EA03FF5AA2EA001F1480A2133FA21400A25BA2137EA213FEA25BA21201A25BA2
1203A25BA21207A25BA2120FA25BA2121FA25BA2123FA290C7FCA25AA2EA7E03A2EAFE07
130612FCA2130E130C131C1318EA7C38EA3C70EA1FE0EA0780123B7DB919>I<D803E001
7F14FE3D07F801FFE003FFC03D0E3C0781F00F03E03D1C3E1E00F83C01F026383F38D9FC
707F00304914E04A90387DC000007049EB7F8000604991C7FCA200E090C700FE1301485A
017E5CA200000201140301FE5F495CA203031407000160495C180F03075D1203494A011F
13601980030F023F13E00007F000C0495C1901031F023E1380000F1803494A150061033F
150E001FEF1E1C4991C7EA0FF80007C7000EEC03E043267EA449>I<D803E0137F3A07F8
01FFE03A0E3C0781F03A1C3E1E00F826383F387F00305B4A137C00705B00605BA200E090
C712FC485A137EA20000140101FE5C5BA2150300015D5B15075E120349010F133016C003
1F13700007ED80605B17E0EE00C0000F15014915801603EE0700001FEC0F0E49EB07FC00
07C7EA01F02C267EA432>I<90390F8003F090391FE00FFC903939F03C1F903A70F8700F
80903AE0FDE007C09038C0FF80030013E00001491303018015F05CEA038113015CA2D800
031407A25CA20107140FA24A14E0A2010F141F17C05CEE3F80131FEE7F004A137E16FE01
3F5C6E485A4B5A6E485A90397F700F80DA383FC7FC90387E1FFCEC07E001FEC9FCA25BA2
1201A25BA21203A25B1207B512C0A32C3583A42A>112 D<02FC13C0903803FF0190380F
838390383F01C790397E00EF8049137F485A4848133F000715005B485A001F5C157E485A
A2007F14FE90C75AA3481301485CA31403485CA314075D140F127C141F007E495A003E13
7F381F01EF380F839F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5C13019038
7FFFF0A322357DA425>I<3903E001F83907F807FE390E3C1E07391C3E381F3A183F703F
800038EBE07F0030EBC0FF00705B00601500EC007E153CD8E07F90C7FCEAC07EA2120013
FE5BA312015BA312035BA312075BA3120F5BA3121F5B0007C9FC21267EA425>I<14FF01
0313C090380F80F090383E00380178131C153C4913FC0001130113E0A33903F000F06D13
007F3801FFE014FC14FF6C14806D13C0011F13E013039038003FF014071403001E130112
7FA24814E0A348EB03C012F800E0EB07800070EB0F006C133E001E13F83807FFE0000190
C7FC1E267CA427>I<EB01C0497E1307A4130F5CA3131F5CA3133F91C7FC007FB51280A2
B6FCD8007EC7FCA313FE5BA312015BA312035BA312075BA3120FEBC006A2140E001F130C
EB801C141814385C146014E0380F81C038078780D803FEC7FCEA00F819357EB31E>I<01
F8EB03C0D803FEEB07E0D8070F130F000E018013F0121C12180038140700301403D8701F
130112601500D8E03F14E000C090C7FC5BEA007E16C013FE5B1501000115805B15031600
1203495B1506150E150C151C151815385D00015C6D485A6C6C485AD97E0FC7FCEB1FFEEB
07F024267EA428>118 D<903907E001F090391FF807FC9039783E0E0F9039E01F1C1FD8
01C09038383F803A03800FF07F0100EBE0FF5A000E4A1300000C157E021F133C001C4AC7
FC1218A2C7123FA292C8FCA25CA2147EA214FEA24A130CA20101141C001E1518003F5BD8
7F81143801835C00FF1560010714E03AFE0E7C01C0D87C1C495A2778383E0FC7FC391FF0
0FFC3907C003F029267EA42F>120 D<13F8D803FE1470D8070F14F8000EEB8001121C12
1800381403003015F0EA701F1260013F130700E0010013E012C05BD8007E130F16C013FE
5B151F000115805BA2153F000315005BA25D157EA315FE5D1401000113033800F8079038
7C1FF8EB3FF9EB0FE1EB00035DA2000E1307D83F805B007F495AA24A5A92C7FCEB003E00
7C5B00705B6C485A381E07C06CB4C8FCEA01FC25367EA429>I<D901E01360D90FF813E0
496C13C090383FFE0190397FFF038090B5EA07009038F81FFF3901E003FE9038C0001C49
5B5DC85A4A5A4A5A4AC7FC140E5C5C14F0495AEB038049C8FC130E5B4913035B495B4848
13064848130E48C75AD80FFC137C391FFF81F8381E0FFFD838075B486C5B00605CD8E001
90C7FC38C0007C23267DA427>I<1504151E151FA2ED0F8016C0ED07E0007FB612F0B712
F8A26C15F0C8EA1FC0ED3F00157E5D5D5D1560251271BB2A>126
D E /Fw 76 123 df<121C127FEAFF80B1EA7F00AF123EC7FCA8121C127FA2EAFF80A3EA
7F00A2121C09346FB32C>33 D<003C131E007F137F481480A66C1400A6007E7FA6003E13
3EA3003C131E001C131C191977B32C>I<EB0FC0EB3FE0497E497E80EA01F8EBF07C147E
0003133E13E0A5147E147C9138FC3FF89039F0F87FFCEA01F1EBF3F001F7EB3FF89138E0
1F009038FFC03F6CEB803EA2EC007E49137C485A486C13FC00075CEBFF01D80FDF5B381F
9F81383F8F8390380FC3E0387E07E75D38FC03F7EB01FF5D6D1410ED007C80A26CEBFF80
D87E0113C0D87F03EBE0FC3A3F87F7F1F89038FFE3FF6C01C113F06C13806C9038007FC0
D801FCEB1F8026357EB32C>38 D<EA0F80EA1FC0EA3FE013F0A213F8A2121F120F1200A4
120113F0A2120313E01207EA0FC0121FEA3F80EA7F0012FE5A5A12700D1B71B22C>I<14
3814FC13011303EB07F8EB0FF0EB1FC0EB3F80EB7F0013FE485A485A5B12075B120F5B48
5AA2123F90C7FCA25A127EA312FE5AAC7E127EA3127F7EA27F121FA26C7E7F12077F1203
7F6C7E6C7E137FEB3F80EB1FC0EB0FF0EB07F8EB03FC130113001438164272B92C>I<12
7012FC7E7E6C7E6C7EEA0FE06C7E6C7E6C7E6C7E137F7F1480131F14C0130FEB07E0A214
F01303A214F81301A314FC1300AC130114F8A3130314F0A2130714E0A2EB0FC0131F1480
133F14005B13FE485A485A485A485AEA3FC0485A48C7FC5A5A1270164279B92C>I<EB03
80497EA60020140800F8143E00FE14FE00FF13C1EBC7C7EBE7CF003FB512F8000F14E000
0314806C140038007FFCA248B5FC481480000F14E0003F14F839FFE7CFFEEBC7C7EB07C1
00FE13C000F8143E0020140800001400A66D5A1F247AAA2C>I<147814FCAF007FB612F0
B712F8A46C15F0C700FCC7FCAF147825267DAB2C>I<EA0F80EA1FE0EA3FF0EA7FF8A213
FCA3123F121F120F120013F8A21201EA03F01207EA1FE0EA7FC0EAFF80130012FC12700E
17718A2C>I<007FB6FCB71280A46C150021067B9B2C>I<121FEA3F80EA7FC0EAFFE0A5EA
7FC0EA3F80EA1F000B0B708A2C>I<EB03F8EB0FFE90383FFF80497F90B57E3901FE0FF0
3903F803F848486C7EEBE0004848137EA248487FA248C7EA1F80A2003E140F007E15C0A3
007C140700FC15E0AC6C140F007E15C0A46CEC1F80A36C6CEB3F00A26C6C137E6D13FE00
075CEBF0016C6C485A3901FE0FF06CB55A6D5B6D5BD90FFEC7FCEB03F823357CB32C>48
D<1307497EA2131FA2133F137F13FF5A1207127FB5FC13DF139FEA7C1F1200B3AE007FB5
12E0B612F0A36C14E01C3477B32C>I<EB0FF890387FFF8048B512E00007804814FC391F
F80FFE393FE001FF903880007F48C7EA3F80007E141F00FE15C0150F6C15E01507A3127E
123CC8FCA2150F16C0151F1680153F16005D15FE4A5A14034A5A4A5A4A5A4A5AECFF8049
48C7FC495A495A495AEB3FE0EB7F8049C8FC485A4848EB03C04848EB07E0EA1FE0485A48
B6FCB7FCA36C15C023347CB32C>I<EB0FFC90387FFF8048B512E0000714F84880391FF8
07FEEBC0004848137F6D7F1680151FA26C5A6CC7FCC8FC153F16005D15FE14014A5AEC1F
F890381FFFF0495BA215F86D7F90380007FEEC00FF81ED3F80ED1FC0150FA216E01507A2
123C127EB4FC150F16C0A248141F007FEC3F806DEB7F006C6C5B391FF807FE6CB55A6C5C
6C14E0C66C1380D90FFCC7FC23357CB32C>I<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F
80EA1F00C7FCAE121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B2470A32C>58
D<EA0F80EA1FC0EA3FE0EA7FF0A5EA3FE0EA1FC0EA0F80C7FCAEEA0F80EA1FC0EA3FE012
7F13F0A4123F121F120F1201120313E01207EA0FC0A2EA3F80EA7F005A5A12F812700C30
71A32C>I<1507ED1F80153F15FF14034A1300EC1FFC4A5AECFFE0491380010790C7FCEB
0FFCEB3FF8EB7FE048485A4890C8FCEA0FFEEA1FF8EA7FF0EAFFC05BA27FEA7FF0EA1FF8
EA0FFEEA03FF6C13C06C6C7EEB3FF8EB0FFC6DB4FC01017F6D13E0EC3FF86E7EEC07FF6E
13801400153F151FED0700212A7BAD2C>I<003FB612E04815F0B712F8A36C15F0CAFCA8
007FB612F0B712F8A36C15F06C15E025147DA22C>I<127012FC7E6C7E13E06C7EEA1FFC
6C7E3803FF80C67FEB7FF0EB1FF8EB0FFEEB03FF6D13C06D6C7EEC3FF8EC0FFC6EB4FC02
01138080A25C02071300EC0FFCEC3FF8EC7FE049485A4990C7FCEB0FFEEB1FF8EB7FF0EB
FFC000035BD80FFEC8FC485AEA7FF0485A138048C9FC5A1270212A7BAD2C>I<14FE497E
A4497FA214EFA2130781A214C7A2010F7FA314C390381F83F0A590383F01F8A490387E00
FCA549137E90B512FEA34880A29038F8003FA34848EB1F80A4000715C049130FD87FFEEB
FFFC6D5AB514FE6C15FC497E27347EB32C>65 D<007FB512E015F8B612FE6C8016C03903
F0003FED0FE0ED07F01503A2ED01F8A6ED03F0A21507ED0FE0ED1FC0EDFF8090B612005D
5D15FF16C09039F0001FE0ED07F0ED03F81501ED00FCA216FE167EA616FE16FC1501ED03
F8150FED3FF0007FB612E016C0B712806CECFE0015F027337FB22C>I<02FF13700107EB
E0F84913F9013F13FD4913FFEBFF813901FE007F4848131FD807F0130F1507485A491303
485A150148C7FCA25A007EEC00F01600A212FE5AAB7E127EA3007F15F06CEC01F8A26C7E
A26C6C13036D14F06C6C130716E0D803FC131F6C6CEB3FC03A00FF81FF806DB512006D5B
010F5B6D13F00100138025357DB32C>I<007FB5FCB612C015F0816C803907E003FEEC00
FFED7F80153FED1FC0ED0FE0A2150716F0150316F81501A4ED00FCACED01F8A3150316F0
A2150716E0150FED1FC0153FED7F80EDFF00EC03FE007FB55AB65A5D15C06C91C7FC2633
7EB22C>I<007FB612F0B712F8A37E3903F00001A7ED00F01600A4EC01E04A7EA490B5FC
A5EBF003A46E5A91C8FCA5163C167EA8007FB612FEB7FCA36C15FC27337EB22C>I<007F
B612F8B712FCA37ED803F0C7FCA716781600A515F04A7EA490B5FCA5EBF001A46E5A92C7
FCAD387FFFE0B5FC805C7E26337EB22C>I<903901FC038090390FFF87C04913EF017F13
FF90B6FC4813073803FC01497E4848137F4848133F49131F121F5B003F140F90C7FCA212
7EED078092C7FCA212FE5AA8913803FFF84A13FCA27E007E6D13F89138000FC0A36C141F
A27F121F6D133F120F6D137F6C7E6C6C13FF6D5A3801FF076C90B5FC6D13EF011F13CF6D
EB0780D901FCC7FC26357DB32C>I<D87FFEEBFFFCB54813FEA36C486C13FCD807E0EB0F
C0B190B6FCA59038E0000FB3D87FFEEBFFFCB54813FEA36C486C13FC27337EB22C>I<00
7FB512F8B612FCA36C14F839000FC000B3B3A5007FB512F8B612FCA36C14F81E3379B22C
>I<0107B512804914C0A36D148090390003F000B3AF1218127EA2B4FCA24A5A48130F00
7F131F9038C07FC06CB55A6C91C7FC6C5B000313F838007FC022347BB22C>I<D87FFCEB
7FF8486CEBFFFCA36C48EB7FF8D807C0EB1F80153FED7F00157E5D4A5A14034A5A5D4A5A
4A5A143F4AC7FC147E5CEBC1F813C3EBC7FCA2EBCFFEEBDFBEEBFFBF141F01FE7F496C7E
13F86E7EEBF00301E07FEBC001816E7EA2157E153E153F811680ED0FC0A2ED07E0D87FFC
EB1FFC486CEB3FFEA36C48EB1FFC27337EB22C>I<387FFFE0B57EA36C5BD803F0C8FCB3
AE16F0ED01F8A8007FB6FCB7FCA36C15F025337DB22C>I<D87FE0EB0FFC486CEB1FFEA2
6D133F007F15FC000F15E001BC137BA4019E13F3A3EB9F01A2018F13E3A21483A2018713
C314C7A201831383A214EFA201811303A214FFEB80FEA3147C14381400ACD87FF0EB1FFC
486CEB3FFEA36C48EB1FFC27337EB22C>I<D87FF0EB7FFC486CEBFFFEA27F007FEC7FFC
D807FEEB07C013DEA213DF13CFA2148013C714C0A213C314E0A213C114F0A213C014F8A2
147CA3143EA2141E141FA2140F1587A2140715C7A2140315E71401A215F71400A215FFD8
7FFC137F487E153FA26C48EB1F8027337EB22C>I<EB7FFF0003B512E0000F14F8488048
80EBE003EB800048C7127FA2007E80A300FE158048141FB3A86C143FA2007E1500A3007F
5CA26C6C13FEEBF00790B5FC6C5C6C5C000314E0C66C90C7FC21357BB32C>I<007FB512
C0B612F88115FF6C15802603F00013C0153FED0FE0ED07F0A2150316F81501A6150316F0
1507A2ED0FE0ED3FC015FF90B61280160015FC5D15C001F0C8FCB0387FFF80B57EA36C5B
25337EB22C>I<EB7FFF0003B512E0000F14F848804880EBF007EB800048C7127FA2007E
80A300FE158048141FB3A7EB01F0EB03F800FE143F267E01FC1300A2EB00FE007F5C147F
D83F8013FEEBF03F90B5FC6C5C6C5C000314E0C67E90380007F0A26E7EA26E7EA26E7EA2
157FA2153E21407BB32C>I<387FFFFCB67E15E015F86C803907E007FE1401EC007F6F7E
151FA26F7EA64B5AA2153F4BC7FCEC01FE140790B55A5D15E081819038E007FCEC01FE14
00157F81A8160FEE1F80A5D87FFEEB1FBFB5ECFF00815E6C486D5AC8EA01F029347EB22C
>I<90381FF80790B5EA0F804814CF000714FF5A381FF01F383FC003497E48C7FC007E14
7F00FE143F5A151FA46CEC0F00007E91C7FC127F7FEA3FE0EA1FFCEBFFC06C13FC0003EB
FFC06C14F06C6C7F01077F9038007FFEEC07FF02001380153FED1FC0A2ED0FE0A2007814
0712FCA56CEC0FC0A26CEC1F806D133F01E0EB7F009038FE01FF90B55A5D00F914F0D8F8
3F13C0D8700790C7FC23357CB32C>I<007FB612FCB712FEA43AFC007E007EA70078153C
C71400B3AF90383FFFFCA2497F6D5BA227337EB22C>I<3B7FFF803FFFC0B56C4813E0A3
6C496C13C03B03F00001F800B3AF6D130300015DA26D130700005D6D130F017F495A6D6C
485AECE0FF6DB5C7FC6D5B010313F86D5B9038003F802B3480B22C>I<D87FFCEB7FFC48
6CEBFFFEA36C48EB7FFCD80FC0EB07E06D130F000715C0A36D131F00031580A36D133F00
011500A36D5B0000147EA4017E5BA46D485AA490381F83F0A4010F5B14C7A301075BA214
EFA201035BA214FFA26D90C7FCA46D5A27347EB22C>I<D87FF0EB07FF486C491380A36C
486D1300001FC8127CA46C6C5CA76C6C495AA4143E147FA33A03E0FF83E0A214F7A201E1
13C3A3000101E35BA201F113C701F313E7A314C1A200005DA201F713F71480A301FF13FF
017F91C7FC4A7EA4013E133E29347FB22C>I<3A3FFF03FFE0484913F0148714076C6D13
E03A01F800FE007F0000495A13FE017E5BEB7F03013F5B1487011F5B14CF010F5B14FF6D
5BA26D90C7FCA26D5AA26D5AA2497EA2497EA2497F81EB0FCF81EB1FC7EC87F0EB3F83EC
03F8EB7F01017E7FEBFE00497F0001147E49137F000380491480151FD87FFEEBFFFC6D5A
B514FE6C15FC497E27337EB22C>I<D87FFCEB7FFC486CEBFFFEA36C48EB7FFCD807F0EB
0FC0151F000315806D133F12016DEB7F0012006D137E017E13FE017F5BEB3F01EC81F813
1FEC83F0EB0FC314C7903807E7E0A201035B14EF6DB45AA292C7FC7F5C147EB0903807FF
E0497FA36D5B27337EB22C>I<003FB612C04815E0A4007EC7EA1FC0ED3F80A2ED7F0015
7E15FE4A5A003C5CC712034A5AA24A5A4A5AA24A5A4AC7FCA214FE495AA2495A495AA249
5A495AA2495A49C8FCA213FE485AA24848EB03C049EB07E01207485A5B121F485AA248C7
FCB7FCA46C15C023337CB22C>I<387FFFFCB512FEA314FC00FCC7FCB3B3B3B512FC14FE
A36C13FC17416FB92C>I<387FFFFCB512FEA37EC7127EB3B3B3387FFFFEB5FCA36C13FC
17417DB92C>93 D<007FB6FCB71280A46C150021067B7D2C>95 D<137013F812011203EA
07F0EA0FE0EA1FC01380EA3F00123E127E127CA212FC5AA4EAFF8013C013E0A2127FA212
3FEA1FC0EA0F800D1B71B82C>I<3801FFF0000713FE001F6D7E15E048809038C01FF814
07EC01FC381F80000006C77EC8127EA3ECFFFE131F90B5FC1203120F48EB807E383FF800
EA7FC090C7FC12FE5AA47E007F14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001EBFE
1F3A003FF007FC27247CA32C>I<EA7FF0487EA3127F1201AAEC1FE0ECFFF801FB13FE90
B6FC16809138F07FC09138801FE091380007F049EB03F85BED01FC491300A216FE167EA8
16FE6D14FCA2ED01F86D13036DEB07F0150F9138801FE09138E07FC091B51280160001FB
5B01F813F83900F03FC027337FB22C>I<903803FFE0011F13F8017F13FE48B5FC488048
48C6FCEA0FF0485A49137E4848131890C9FC5A127EA25AA8127EA2127F6C140F6DEB1F80
6C7E6D133F6C6CEB7F003907FE03FF6CB55A6C5C6C6C5B011F13E0010390C7FC21247AA3
2C>I<EC0FFE4A7EA380EC003FAAEB07F8EB3FFE90B512BF4814FF5A3807FC0F380FF003
48487E497E48487F90C7FC007E80A212FE5AA87E007E5CA2007F5C6C7E5C6C6C5A380FF0
073807FC1F6CB612FC6CECBFFE6C143FEB3FFC90390FF01FFC27337DB22C>I<EB03FE90
381FFFC0017F13F048B57E48803907FE03FE390FF800FFD81FE0EB3F805B4848EB1FC090
C7120F5A007E15E015075AB7FCA416C000FCC9FC7E127EA2127F6CEC03C06DEB07E06C7E
D80FF0130F6C6CEB3FC001FF13FF000190B512806C1500013F13FC010F13F00101138023
247CA32C>I<EC0FF8EC3FFE91B5FC4914805B903807FC7F14F090390FE03F0014C092C7
FCA6007FB512FEB7FCA36C5C26000FC0C7FCB3A8003FB512F04880A36C5C21337DB22C>
I<ED03F8903907F80FFC90391FFE3FFE017FB6FC48B7FC48ECFE7F9038FC0FF82607F003
133E3A0FE001FC1CD9C0001300001F8049137EA66D13FE000F5CEBE0016C6C485A3903FC
0FF048B5FC5D481480D99FFEC7FCEB87F80180C8FCA37F6C7E90B512F06C14FE48ECFF80
4815E04815F03A3FC0001FF848C7EA03FC007E1400007C157C00FC157E48153EA46C157E
007E15FCD87F801303D83FE0EB0FF8D81FFCEB7FF06CB612E0000315806C1500D8003F13
F8010713C028387EA42C>I<EA7FF0487EA3127F1201AAEC1FE0EC7FFC9038F9FFFE01FB
7F90B6FC9138F03F80ECC01F02807FEC000F5B5BA25BB3267FFFE0B5FCB500F11480A36C
01E0140029337FB22C>I<1307EB1FC0A2497EA36D5AA20107C7FC90C8FCA7387FFFC080
B5FC7EA2EA0007B3A8007FB512FCB612FEA36C14FC1F3479B32C>I<140EEC3F80A2EC7F
C0A3EC3F80A2EC0E0091C7FCA748B512804814C0A37EC7120FB3B3A2141F003C1480007E
133FB414005CEB01FEEBFFFC6C5B5C001F5B000790C7FC1A467CB32C>I<EA7FE0487EA3
127F1201AA91381FFFF04A13F8A36E13F0913800FE004A5A4A5A4A5A4A5A4A5A4A5A4AC7
FC14FEEBF1FC13F3EBF7FE90B5FCA2EC9F80EC0FC001FE7FEBFC07496C7E496C7E811400
157E811680151F3A7FFFC0FFFCB500E113FEA36C01C013FC27337EB22C>I<387FFFE0B5
7EA37EEA0003B3B3A5007FB61280B712C0A36C158022337BB22C>I<3A7F83F007E09039
CFFC1FF83AFFDFFE3FFCD87FFF13FF91B57E3A07FE1FFC3E01FCEBF83F496C487E01F013
E001E013C0A301C01380B33B7FFC3FF87FF0027F13FFD8FFFE6D13F8D87FFC4913F0023F
137F2D2481A32C>I<397FF01FE039FFF87FFC9038F9FFFE01FB7F6CB6FC00019038F03F
80ECC01F02807FEC000F5B5BA25BB3267FFFE0B5FCB500F11480A36C01E0140029247FA3
2C>I<EB07FCEB1FFF017F13C048B512F048803907FC07FC390FF001FE48486C7E018013
3F003F158090C7121F007EEC0FC0A348EC07E0A76C140F007E15C0A2007F141F6C15806D
133F6C6CEB7F006D5B6C6C485A3907FC07FC6CB55A6C5C6C6C13C0011F90C7FCEB07FC23
247CA32C>I<397FF01FE039FFF8FFF801FB13FE90B6FC6C158000019038F07FC0913880
1FE091380007F049EB03F85BED01FC491300A216FE167EA816FE6D14FCA2ED01F86D1303
6DEB07F0150F9138801FE09138E07FC091B51280160001FB5B01F813F8EC3FC091C8FCAD
387FFFE0B57EA36C5B27367FA32C>I<903903FC078090391FFF0FC0017F13CF48B512EF
4814FF3807FE07380FF00148487E49137F4848133F90C7FC48141F127E150F5AA87E007E
141FA26C143F7F6C6C137F6D13FF380FF0033807FC0F6CB6FC6C14EF6C6C138F6D130FEB
07F890C7FCAD0203B5FC4A1480A36E140029367DA32C>I<D87FFEEB3FC0B53801FFF002
0713F8021F13FC6C5B39003F7FE1ECFF019138FC00F84A13704A13005CA25C5CA391C8FC
AF007FB512E0B67EA36C5C26247EA32C>I<90387FF8700003B512F8120F5A5A387FC00F
387E00034813015AA36CEB00F0007F140013F0383FFFC06C13FE6CEBFF80000314E0C66C
13F8010113FCEB0007EC00FE0078147F00FC143F151F7EA26C143F6D133E6D13FE9038F0
07FC90B5FC15F815E000F8148039701FFC0020247AA32C>I<131E133FA9007FB6FCB712
80A36C1500D8003FC8FCB1ED03C0ED07E0A5EC800F011FEB1FC0ECE07F6DB51280160001
035B6D13F89038003FE0232E7EAD2C>I<3A7FF003FF80486C487FA3007F7F0001EB000F
B3A3151FA2153F6D137F3900FE03FF90B7FC6D15807F6D13CF902603FE07130029247FA3
2C>I<3A7FFF01FFFCB514FE148314016C15FC3A03E0000F80A26D131F00011500A26D5B
0000143EA26D137E017C137CA2017E13FC013E5BA2EB3F01011F5BA21483010F5BA214C7
01075BA214EF01035BA214FF6D90C7FCA26D5A147C27247EA32C>I<D87FFFEB7FFF6EB5
FCB515806C16004A7ED807C0EB01F0A66C6C495AA3143E147FA2D801F0495AECFF87A214
F7A201F113C700005D9038F9E3CFA201FB13EFA3D97BC190C7FC017F13FFA21480A2013F
5B90381F007C29247FA32C>I<3A3FFF03FFF048018713F8A36C010313F03A00FC007E00
5D90387E01F8013F5BEB1F83EC87E090380FCFC0903807EF80EB03FF6D90C7FC5C6D5A14
7C14FE130180903803EF80903807CFC0EB0FC7EC83E090381F01F0013F7FEB7E00017C13
7C49137E0001803A7FFF01FFFC1483B514FE6C15FC140127247EA32C>I<3A7FFF01FFFC
B5008113FE148314816C010113FC3A03E0000F806C7E151F6D140012005D6D133E137C01
7E137E013E137CA2013F13FC6D5BA2EB0F815DA2EB07C1ECC3E0A2EB03E3ECE7C0130114
F75DEB00FFA292C7FC80A2143EA2147E147CA214FC5CA2EA0C01003F5BEA7F83EB87E0EA
7E0F495A387FFF806C90C8FC6C5A6C5AEA07E027367EA32C>I<003FB612E04815F0A400
7EC7EA1FE0ED3FC0ED7F80EDFF004A5A003C495AC7485A4A5A4A5A4A5A4A5A4AC7FCEB01
FC495AEB0FF0495A495A495A49C8FC4848EB01E04848EB03F0485A485A485A485A485AB7
FCA46C15E024247DA32C>I E /Fx 19 112 df<B6FCA600FCC7FCB3B3B3B3B3B3B3B3B3
B3B3A8B6FCA618DA6A8330>20 D<B6FCA6C7123FB3B3B3B3B3B3B3B3B3B3B3A8B6FCA618
DA7F8330>I[<177CEE01FC1607160F163FEE7FF0EEFFE04B1380030713004B5A4B5A5E4B
5A4B5A4B5A5E5C4A90C7FC5D14075D140F5DA2141F5DA3143F5DB3B3B3B3A6147F5DA44A
5AA34990C8FCA2495AA2495AA2495AA2495A495A5C137F495A4890C9FC485A485AEA0FF0
EA3FE0485A48CAFC5AA27EEA7FC06C7EEA0FF0EA07FC6C7E6C7E6C7F6D7E133F806D7E6D
7EA26D7EA26D7EA26D7EA26D7FA36E7EA481143FB3B3B3B3A681141FA381140FA2811407
811403816E7F80826F7E6F7E6F7E826F7E6F7E030113806F13E0EE7FF0EE3FFC160F1607
1601EE007C>46 272 115 131 73 40 D<EF07C0170FEF1F80A2EF3F00177E5F16015F4C
5A16075F160F4C5A5F163F4CC7FCA216FE4B5AA24B5AA24B5A150F5E151F5E153F5E157F
93C8FC5D5D1401A24A5AA24A5AA25D140FA24A5AA2143F5DA24A5AA214FF92C9FCA25B5C
A21303A25C1307A25C130FA35C131FA25C133FA4495AA4495AA45A91CAFCA45A5BA41207
A35BA3120FA45BA3121FA55BA3123FA85BA3127FAE5BA212FFB3A732A4668250>48
D<12F87E127EA27E6C7E6C7E7F12076C7E7F12017F6C7E137E137F6D7EA26D7E6D7EA26D
7EA26D7E801301801300808081143F81141F81A26E7EA26E7EA2140381A26E7EA28180A2
6F7EA282153FA282151FA282A2150F82A2150782A3150382A2150182A46F7FA4707EA483
163FA483161FA483A3160FA383A41607A383A51603A383A882A31880AE82A218C0B3A732
A47D8250>I<157CEC01FC1403140F141FEC7FF8ECFFE04913C0491380491300495A495A
495A495A5C13FF485B5C5A4890C7FCA2485AA25B121FA2485AA3127F5BA412FF5BB3B3AB
1E525D7E51>56 D<12F812FE6C7E7F13F0EA3FF86C7E6C7E6C7E000113C07E6D7E806D7E
6D7E6D7EA26D7E6D1380A26D13C0A26D13E0A2EC7FF0A215F8143FA3EC1FFCA415FE140F
B3B3AB1F52717E51>I<EAFFE0B3B3AB7F127FA47F123FA36C7EA2120F7FA26C7EA26C7F
7E806C7F137F806D7E6D7E6D7E6D7E6D13806D13C06D13E0EC7FF8EC1FFC140F14031401
EC007C1E525D8051>I<EC0FFEB3B3AB141F15FCA4EC3FF8A3147F15F0A2ECFFE0A24913
C0A2491380A2491300495AA2495A495A495A5C495A5A000790C7FC485A485A485AB45A13
C05B48C8FC12F81F52718051>I<EC0FFEB3B3ACEC1FFCA5EC3FF8A315F0147FA215E014
FF15C05B15805B1500495AA2495A495AA2495A495A495A485B91C7FC485AEA0FFC485AEA
3FE0485A48C8FC5AA27EEA7FC06C7EEA1FF86C7EEA03FE6C7E806C7F6D7E6D7E6D7EA26D
7E6D7EA26D7E15807F15C07F15E0147F15F0A2143F15F8A3EC1FFCA5EC0FFEB3B3AC1FA6
718051>I<EAFFE0B3B3AC6C7EA56C7EA3121F7FA2120F7F1207A26C7EA26C7FA26C7F6D
7EA26D7E6D7E130F6D7E806D7E6D13809038007FC0EC3FE0EC1FF0EC0FF8EC03FC1401A2
1403EC0FF8EC1FF0EC3FE0EC7FC0903801FF80491300495A5C495A131F495A495AA2495A
485BA24890C7FCA2485AA2120F5B121FA25B123FA3485AA5485AB3B3AC1EA65D8051>I<
EAFFC0B3A7127FA27FAE123FA37FA8121FA37FA5120FA37FA41207A37FA31203A47F7EA4
807EA46D7EA46D7EA4131F80A2130F80A3130780A2130380A21301A2807FA281147FA26E
7EA281141FA26E7EA2140781A26E7EA26E7EA21400818182153F82151F82150F8215076F
7EA26F7EA26F7E167FA2707E161F83707E1607831603707E831600177E83EF1F80A2EF0F
C0170732A4668450>64 D<EFFFC0B3A71880A25EAE1800A35EA85FA31607A55FA3160FA4
5FA3161FA35FA4163F5FA4167F5FA44C5AA44B90C7FCA45E1503A25E1507A35E150FA25E
151FA25EA2153F5EA2157F5EA24BC8FCA25C5DA24A5AA25D1407A24A5AA24A5AA25D143F
5D147F92C9FC5C5C13015C13035C495AA2495AA2495A495AA249CAFC137E13FE485A5B12
035B485A120F5B485A48CBFC127EA25A5A32A47D8450>I<007C181FA200FEF03F80B3B3
B3A86C187FA26C19006D5FA2003F606D1601A26C6C4C5A6D1607000F606D160F6C6C4C5A
6C6C4C5A6C6C6CEDFFC06E5C6C01F002075B6D6C4A90C7FC6DB4EC7FFE6D9039F007FFFC
010790B612F06D5E010016806E92C8FC020F14F8020114C09126001FFCC9FC415B7B7F4C
>83 D<ED01E015034B7EA24B7EA34B7EA34B7EA34B7EA28203FF7FA24A486C7EA2EDFC1F
020380A2EDF80F020780A2EDF007020F80A2EDE003021F80A2EDC001023F80A2ED800002
7F80A292C77E4A81A24A143F010182A249486E7EA24A140F010782A24A1407010F82A24A
1403011F82A24A1401013F82A24A1400017F82A291C97E4983A249163F000184A249161F
000384A249160F000784A24848707EA2491603001F84A2491601003F84A2491600007F84
A290CB7E481980A248183FA2007CF01F0085415B7B7F4C>86 D<003C1B0F007EF31F80B4
F33FC0B3B3B3B3AF6D1A7F007F1C80A46D1AFF003F1C006D61A26C6C4F5AA26C6C4F5AA2
6C6C4F5AA26C6C4F5A6D193F6C6D4E5A6E18FF6C6D4D5B6D6C4D5B6D6C4D90C7FC6D6C4D
5A6DB4EF3FFC6D6D4C5A6D01E04B485A6D01F803075B6D01FE031F5B91267FFFC091B55A
6E01FE011F91C8FC020F90B712FC6E5F020117E06E6C1680031F4BC9FC030315F0DB007F
1480040301F0CAFC5A7F7B7F65>91 D<171E173F4D7EA34D7EA34C7FA34C7FA34C7FA34C
7F17F3041F7FA217E1043F7F17C0A2047F80EF807FA204FF80EF003FA24B814C131FA203
03814C130FA20307814C1307A2030F814C1303A2031F814C1301A2033F814C7F037F82A2
4C147F03FF8293C8123FA24A834B151FA20203834B150FA20207834B1507A2020F834B15
03A2021F834B1501A2023F834B81A2027F844B167FA202FF8492CA123F4985A24A171F01
03854A170FA20107854A1707A2010F854A1703A2011F854A1701A2013F854A83A2017F86
4A187FA201FF8691CC123FA2488749191FA200038749190F000787A2491907000F874919
03A2001F87491901A2003F874985A2007F1C80491A7FA200FF1CC090CE123FA3007EF31F
80003CF30F005A7F7B7F65>94 D<160F167FED01FF1507ED1FFCED7FE0EDFF804A1300EC
03FC4A5A4A5A4A5A4A5A5D147F92C7FCA25C5CB3B3AA13015CA213035C13075C495A131F
495A495A49C8FCEA03FC485AEA1FE0EA7FC048C9FC12FCB4FCEA7FC0EA1FE0EA07F86C7E
C6B4FC6D7E6D7E6D7E130F6D7E801303801301A2801300B3B3AA8080A281143F816E7E6E
7E6E7E6E7E6EB4FC6E1380ED7FE0ED1FFCED07FF1501ED007F160F28A376833D>110
D<12F012FE6C7E13E0EA3FF8EA0FFCEA03FEC66C7E6D7E6D7E131F6D7E6D7E1303801301
80A21300B3B3AA8080A281143F816E7E140F6E7E81EC01FC6EB4FCED7F80ED1FE0ED0FF8
ED03FEED00FF163F16FFED03FEED0FF8ED1FE0ED7F80EDFF00EC01FCEC07F85D4A5A141F
4A5A5D147F92C7FCA25C5CB3B3AA1301A25C13035C1307495A495A133F495A495AD803FE
C8FCEA0FFCEA3FF8EAFFE0138048C9FC12F028A376833D>I E /Fy
8 122 df<91380FF0039138FFFE06903903F8070E90390FC0019E013FC712FE017C147C
5B4848143C485A48481438485A121F90C8FC481530123E007E1500A25AA4913803FFFCA2
91380007C0A2ED0F80127CA27EED1F007E6C6C5BD807E0136F3903F803C6C6B51202D91F
F8C7FC28247CA22F>71 D<903AFFFE07FFF0A2903A07C0003E00A249485BA449C75AA401
3E495AA3013FB5FC495C90387C0003A349495AA44848495AA4484849C7FCA300075C3AFF
FE07FFF0A22C227CA132>I<D97FFFEB7FFCA2D903F8EB1F800101EC1E0016186D6C5B5E
027E5B4B5A6E48C7FC1506EC1F8C15B8EC0FF05D1407A281140FEC1DF814384A7EECE07C
903801C07E903803803E903807003F010E6D7E1318496D7E5BD801E06D7E1207D87FF8EB
3FFF00FF5C2E227EA132>88 D<D8FFFE903803FF80495BD807E0903800F80016E06C6CEB
01804BC7FC6C6C13065D6C6C5B1538017E5B5D6D485A4A5AD91F87C8FC148E149CEB0FF8
5C6D5A5CA35C130FA391C9FC5BA35B3807FFF0A229227DA124>I<000F017E13FC3A1F81
FF83FF3B31C383C707803A61EE03CC039026EC01F813C0D8C1F813F013F001E013E00003
903903C0078013C0A2EE0F003907800780A2EE1E041706270F000F00130C163C1718A200
1E011EEB1C70EE1FE0000C010CEB07802F177D9536>109 D<000F13FC381FC3FF3931C7
07803861EC0301F813C0EAC1F0A213E03903C00780A3EC0F00EA0780A2EC1E041506D80F
00130C143C15181538001EEB1C70EC1FE0000CEB07801F177D9526>I<3801F01E3907FC
7F80390E1CE1C038180F8100301383007013071260EC0380D8001EC7FCA45BA215800030
14C0397878018012F8EC030038F0FC0638E19C1C387F0FF8381E03E01A177D9523>120
D<EA07C0380FE0033918F0078012300060EB0F0012C0A2EAC1E00001131EEA03C0A34848
5AA45CA214F813813803C3F0EA01FFEA00FC1300495A121E383E03C05CD83C07C7FCEA30
0EEA383CEA1FF8EA07C019217D9520>I E /Fz 9 107 df<127012FCB4FCEA7FC0EA1FF0
EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF913800
7FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FEA2EE07FCEE1FF0EE7FC092
3801FF00ED07FCED1FF0ED7FC04A48C7FCEC07FCEC1FF0EC7FC04948C8FCEB07FCEB1FF0
EB7FC04848C9FCEA07FCEA1FF0EA7FC048CAFC12FC1270CBFCAC007FB712FCB812FEA26C
16FC2F3E7AB03C>21 D<17F0A283177C173C173E83A2717E717E717E007FB87EB97E846C
17FFCBEA1F80F00FE0F003F0F001FE9538007F80F11FF0A2F17F80953801FE00F003F0F0
0FE0F01F80007FB9C7FCB912FC606C5FCAEA03E04D5A4D5A4DC8FCA2173E173C177C5F5F
A2442A7CA74D>41 D<91383FFFF849B512FC1307011F14F8D93FE0C7FC01FFC8FCEA01FC
EA03F0485A485A5B48C9FC5A123E5AA21278A212F8A25AB712F816FCA216F800F0C9FC7E
A21278A2127CA27E123F7E6C7E7F6C7E6C7EEA01FC6CB4FCEB3FE06DB512F8010714FC13
01D9003F13F8262E7AA933>50 D<0060ED018000F0ED03C06C1507A200781680007C150F
A2003C1600003E5DA26C153EA26C153C6D147CA26C6C5CA200035D90B6FCA26C5DA29038
F000036C6C495AA201785C017C130FA2013C91C7FC013E5BA26D133EA26D133CEC807CA2
01071378ECC0F8A2903803E1F0A201015B14F3A26DB45AA26E5AA36EC8FCA3141E140C2A
3680B32B>56 D<007FB61280B712C0A27EC81203B3A2003FB6FC5AA27EC81203B3A2007F
B6FCB7FCA26C158022347CB32B>I<0270EC03F0D901F0EC1FF80107157F011FEC01F101
7FEC07C0D9F7E0EB1F00D801C7143CD8000702F013F0010F494813C0DB03801300030FC8
FC151E1538ECC0F0ECC1E0ECC3C090381FC78002CFC9FC149E14BE14BCEB3FFC147CA214
7E137F137E143E143F13FCA26E7E120101F87F140FA248486C7EA2813807E003811401D8
0FC07F020015186F14784848017F14F01701DB3F8013E048C7391FC003C09338E0078000
3E91390FF00F00007E913807FC3C007C913803FFF8486E13E000E06E6CC7FC35377EB43A
>75 D<EC07E0143FECFE00EB03F8495A495A5C131F5CB3A5133F91C7FC137E5BEA03F8EA
7FE048C8FCEA7FE0EA03F8C67E137E7F80131FB3A580130F806D7E6D7EEB00FEEC3FE014
071B4B7BB726>102 D<12FCEAFFC0EA07F0EA01FC6C7E137F7F80131FB3A580130F6D7E
6D7EEB01FC9038007FC0EC1FE0EC7FC0903801FC00EB03F0495A495A131F5CB3A5133F91
C7FC5B13FE485AEA07F0EAFFC000FCC8FC1B4B7BB726>I<126012F0B3B3B3B31260044B
78B715>106 D E /FA 13 122 df<EC7FE0903801FFFC903803C7FEEB0701EB0600153C
010E1300130FA26D7EA280A2806D7E801301806D7E8080903803FF80010F13C0EB3F3FEB
7C1F01F813E0EA01F03803E00F1207390FC007F01380121F123F010013E0A25A127EA315
C05A140FA2007C1480141F1500A26C133E5C6C13786C6C5A3807C1E03801FFC0D8007EC7
FC1F367EB422>14 D<123C127E12FFA4127E123C08087A8715>58
D<123C127EB4FCA21380A2127F123D1201A412031300A25A1206120E120C121C5A5A1260
09177A8715>I<1530157815F8A215F01401A215E01403A215C01407A21580140FA21500
5CA2143EA2143C147CA2147814F8A25C1301A25C1303A25C1307A2495AA291C7FC5BA213
1E133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C8FC5AA2121E
123EA2123C127CA2127812F8A25A12601D4B7CB726>61 D<DB0FF81318DB7FFF13380203
B5EAC078913A0FFC03E0F091393FC000F102FFC7123BD901FC141FD907F815E04948140F
49481407EB3F80017F16C049C81203485A5B00031780485A485AA248481600A2485A94C7
FC127F5BA312FF90CBFCA45AA21738A21730A21770176017E0007F4B5A5F16036C6C4AC7
FC160E6C6C5C6C6C5C6C6C5C6C6C495A6C6CEB07C02700FF803FC8FC90383FFFFC010F13
F00101138035377CB437>67 D<010FB612F017FE83903B003FC0007FC0EF1FE0EF07F05D
EF03F8147FA292C713FCA25CEF07F85CA2010116F0170F4A15E0EF1FC00103ED3F80EF7F
004A14FEEE03FC0107EC1FF091B612C04CC7FC02F0C9FC130FA25CA2131FA25CA2133FA2
5CA2137FA291CAFCA25BA25B1201B512FCA336337DB231>80 D<03FF13180207EBE03802
1FEBF87891397F00FCF802FCEB1FF0D901F0130F4948130749481303494814E0A249C712
01A2013E15C0A3137E1780A2017F91C7FC8080EB3FF014FF15F06D13FE6D6D7E6D806D80
010080020F7F1400150F6F7E150315011500A2120CA2001C5D1218A2150100385D003C14
035E4B5A007E4A5A007F141F6D49C7FCD87BE0137C39F9FC03F839F07FFFE0D8E01F1380
26C003FEC8FC2D377CB42F>83 D<0103B539C007FFFC5BA29026000FFCC713804BECFC00
020715F0606E6C495A4D5A02014AC7FC6F130E5F6E6C5B5F92387F80605F92383F818004
C3C8FC16C6ED1FEC16F86F5AA2150782A282150FED1DFE153915704B7E4A5A4A486C7E15
0002066D7E5C4A131F4A805C4A6D7E495A49C76C7E1306010E1403013C81137CD803FE4A
7EB500C090387FFFFCA2603E337EB23F>88 D<ED0F80ED3FE0ED7870913801F0F815E314
03A2020713F0A291380FC1C0EDC000A4141F5DA4143F92C7FC011FB512805B1600D9007F
C7FC147EA414FE5CA513015CA413035CA413075CA4130F5CA45C131FA291C8FC121CEA7F
1E133EA2EAFE3C5B1278EA70F0EA3FC0EA0F8025457CB425>102
D<D801E001FEEB07F03C07F803FF801FFC3C0E3E0F07E0783F001C903B3C03F1E01F803C
181F7001F3800F003801E0EBF70026303FC001FE14C01270006001805B02005B0303141F
D8E07F4A1480EA407E12000307143F01FE1700495C60030F147E1201494A13FEF0FC0403
1F150E00030401130C49028013F8191C033F01031318000717F04902001438050113704B
15E0000F933800F1C049017EEC7F80D80380011CEC1E003F227EA044>109
D<D801E013FE3A07F803FF803A0E3E0F07E0001C90383C03F039181F7001003813E02630
3FC07F12700060138014001503D8E07F5CEA407E1200150701FE5C5B150F5E120149131F
EE8080EE81C00003023F13804914011603037F13000007147E495CED3E0E5E000FEC1E38
49EB0FF0D80380EB03C02A227EA02E>I<90391F801F8090397FE07FE09039E0F0E0703A
01C0F9C0F83903807D833807007F000E1403000C15F0001C137E0018EC01C002FEC7FC00
385B1210C7FC13015CA31303A25C1640010714E016C0001C5B007E1401010F148000FE14
03011FEB0700011B130E39F839F01C397070F878393FE07FE0390F801F8025227EA02C>
120 D<13F0D803FCEB01C0D8071EEB03E0D80E1F1307121C003813800030140F013F14C0
00701300126049131FD8E07E14801240EA00FE49133F000115005BA25D0003147E5BA215
FE5D5BA214015DEBF00314070001130F3900F83FF0EB3FFBEB0FC3EB00075DA20007130F
D81F805B003F495AA24AC7FCEB007E003E137C00385B381803F0381E07C0D807FFC8FCEA
01F823317EA026>I E /FB 64 128 df<91393FE00FE0903A01FFF83FF8903A07E01EF8
3C903A1F800FF07E903A3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812
C0A33B03F0001F8000B3A7486C497EB50083B5FCA32F357FB42D>11
D<EC1FE0ECFFFC903803F01E90390FC00780EB1F8090393F000FC0017E131F5BA2485AED
0F8092C7FCA9ED0FC0B7FCA33901F8001F150FB3A6486CEB1FE0267FFFC1B5FCA328357F
B42B>I<EC1FF891B512C0903803F00F90380FC01FEB1F80EB3F00137E5B150F485AACB7
FCA33901F8000FB3A7486CEB1FE0267FFFE3B5FCA328357FB42B>I<DA1FE013FF9126FF
FC0713E0903B03F01E1F80F0903B0FC0077E003CD91F805B90273F001FF8137E017E4A13
FE495CA248485C030F147C95C7FCA9187EB912FEA33B01F8000FC000187EB3A6486C496C
13FF297FFFC1FFFE0F13F8A33D357FB440>I<123C127EB4FCA21380A2127F123D1201A4
12031300A25A1206120E120C121C5A5A126009177AB315>39 D<14C01301EB0380EB0F00
130E5B133C5B5BA2485A485AA212075B120F90C7FC5AA2121E123EA3123C127CA55AB012
7CA5123C123EA3121E121FA27E7F12077F1203A26C7E6C7EA213787F131C7F130FEB0380
EB01C01300124A79B71E>I<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F13
78137C133C133EA2131E131FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C
137C137813F85BA2485A485AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<123C
127EB4FCA21380A2127F123D1201A412031300A25A1206120E120C121C5A5A126009177A
8715>44 D<B512F0A514057F921A>I<123C127E12FFA4127E123C08087A8715>I<EB0FE0
EB7FFCEBF83E3903E00F803907C007C0EB8003000F14E0391F0001F0A24814F8A2003E13
00007E14FCA500FE14FEB2007E14FCA56CEB01F8A36C14F0A2390F8003E03907C007C0A2
3903E00F803900F83E00EB7FFCEB0FE01F347DB126>48 D<13075B5B137FEA07FFB5FC13
BFEAF83F1200B3B3A2497E007FB51280A319327AB126>I<EB3FC0EBFFF0000313FC380F
80FF391E007F80001CEB3FC048EB1FE048130F15F00060130712FC6C14F87E1403A3007E
1307123CC7FC15F0A2140F15E0EC1FC0A2EC3F801500147E5C495A5C495A495A495A49C7
FC133E133C4913185B485A48481330485A48C7FC001C1470001FB512F05A5AB612E0A31D
327CB126>I<EB1FE0EBFFFC4813FF3907E03F80390F001FC0001EEB0FE0001CEB07F012
3F018013F8140313C01380A2381F0007C7FC15F0A2EC0FE015C0141FEC3F80EC7E00EB01
F8EB7FE014FCEB003FEC1FC0EC0FE0EC07F015F8140315FC140115FEA3127EB4FCA415FC
48130312780070EB07F86C14F0003C130F001FEB1FE0390FE03F800003B51200C613FCEB
1FE01F347DB126>I<EC01C0A214031407A2140F141FA2143F147F146F14CF1301EB038F
140F1307130E130C131C13381330137013E013C0EA0180120313001206120E120C5A1238
12305A12E0B71280A3C7380FC000A94A7E0107B51280A321337EB226>I<000C14C0380F
C00F90B5128015005C5C14F014C0D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F
9038001F80EC0FC0120E000CEB07E0A2C713F01403A215F8A41218127E12FEA315F01407
12F8006014E01270EC0FC06C131F003C14806CEB7F00380F80FE3807FFF8000113E03800
3F801D347CB126>I<14FE903807FF80011F13E090383F00F0017C13703901F801F8EBF0
03EA03E01207EA0FC0EC01F04848C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038
700F809038E007C039FFC003E0018013F0EC01F8130015FC1400A24814FEA5127EA4127F
6C14FCA26C1301018013F8000F14F0EBC0030007EB07E03903E00FC03901F81F806CB512
00EB3FFCEB0FE01F347DB126>I<EB0FE0EB7FFC90B5FC3903F01F803907C007C0390F00
03E0000EEB01F0001E1300001C14F8003C1478A3123EA2003F14F86D13F0EBC001D81FF0
13E09038F803C0390FFE07803907FF0F006C13DE6C13F87EEB3FFE8001F713C0D803E313
E0D8078013F0390F007FF8001E131F003EEB07FC003C1303481301EC007E12F848143EA2
151EA37E153C1278007C14787E6C14F0390F8003E03907F01FC00001B5120038007FFCEB
1FE01F347DB126>56 D<EB0FE0EB7FF8EBFFFE3803F83F3907E00F80390FC007C0D81F80
13E0EC03F0EA3F0048EB01F8127EA200FE14FC1400A415FEA5007E1301A2127F7E1403EA
1F80000F13073807C00E3803E01C3801F03838007FF090381FC0FC90C7FC1401A215F8A2
15F01403001F14E0383F800715C0140FEC1F809038003F00001C137E381F01FC380FFFF0
000313C0C690C7FC1F347DB126>I<123C127E12FFA4127E123C1200B0123C127E12FE12
FFA3127F123F1203A412071206A3120E120C121C1238123012701260082F7A9F15>59
D<007FB812C0B912E0A26C17C0CCFCAC007FB812C0B912E0A26C17C033147C9C3C>61
D<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E0
7FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA249800118C77E
A24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A33336
7DB53A>65 D<B7FC16E016F83A03FC0003FE0001EC00FFEE7F80EE3FC0161F17E0160F17
F0A617E0161F17C0EE3F80EE7F0016FEED03FC90B612F05E9039FC0007FCED00FEEE3F80
EE1FC0EE0FE017F0160717F8160317FCA617F81607A2EE0FF0EE1FE0163FEE7FC0000391
3803FF00B75A16F816C02E337DB236>I<DA03FE130C91393FFF801C91B512E0903A03FE
01F83C903A0FF0003C7CD91FC0EB0EFCD97F80130701FEC7120348481401000315005B48
48157C485A173C485A171C123F5B007F160CA390C9FC481600AB7E6D150CA3123F7F001F
161C17186C7E17386C6C15306C6C15706D15E012016C6CEC01C0D97F80EB0380D91FC0EB
0F00D90FF0131ED903FE13FC0100B512F0023F13C0DA03FEC7FC2E377CB437>I<B812C0
A3D803FCC7127F0001150FEE03E01601A21600A21760A403061330A41700150EA2151E15
7E90B512FEA39038FC007E151E150EA21506170CA3171892C7FCA41738A21770A217F016
01160316070003157FB812E0A32E337DB234>69 D<B81280A3D803FCC7FC0001151FEE07
C01603A21601A21600A41760150CA31700A2151CA2153C15FC90B5FCA3EBFC00153C151C
A2150CA592C8FCAB487EB512FEA32B337DB232>I<B5D8FE03B512F8A3000190C73807FC
006C486E5AB390B7FCA349C71203B3A3486C4A7EB5D8FE03B512F8A335337EB23A>72
D<B512FEA3000113006C5AB3B3A7487EB512FEA317337EB21C>I<B512FEA3D803FEC9FC
6C5AB3A9EE0180A416031700A45EA25E5E5E5E16FE00031407B7FCA329337DB230>76
D<D8FFFC923801FFF86D5DA20003EFFE00D801BFED06FCA3D99F80140CA2D98FC01418A3
D987E01430A2D983F01460A3D981F814C0A3D980FCEB0180A2027EEB0300A36E1306A26E
6C5AA36E6C5AA36E6C5AA26E6C5AA36E6C5AA3913800FD80A2037FC7FCA3486C133ED80F
F04B7EB5011C90387FFFF8A33D337CB246>I<D8FFFE91381FFFF87F80C6030013006E14
3CD9DFE01418EBCFF0A2EBC7F8EBC3FCA2EBC1FEEBC0FF6E7EA26E7E6E7EA26E7E6E7E6E
7EA26E7E6E7EA2ED7F80ED3FC0ED1FE0A2ED0FF0ED07F8A2ED03FCED01FEED00FFA2EE7F
98EE3FD8A2EE1FF8160F1607A216031601A2486C1400D807F81578B500C01438A2171835
337EB23A>I<EC07FC91387FFFC0903901FC07F0903907E000FCD90F80133E013FC76C7E
017E6E7E496E7E48486E7E48486E7EA248486E7E000F8249157E001F167FA24848ED3F80
A2007F17C0A290C9121FA24817E0AB6C17C06D153FA3003F17806D157FA2001F17006D5D
000F5E6C6C4A5AA26C6C4A5A00015E6C6C4A5A017E4A5A6D4A5AD91FC0017FC7FCD907E0
13FC903901FC07F09039007FFFC0DA07FCC8FC33377CB43C>I<B612FCEDFF8016F03A01
FE0007FC0000EC01FEED007F707E707E83160F83A65FA24C5AA24C5A047EC7FC4B5AED0F
F090B612C093C8FC9039FE001FC0ED07F06F7E6F7E150082167E167FA583A5180C17C0A2
043F131C486C1618B500FEEB1FE0040F1338933807F070C93801FFE09338003F8036357E
B239>82 D<90381FE00390387FFC0748B5FC3907F01FCF390F8003FF48C7FC003E808148
80A200788000F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F06C13FF6C14C06C14
F0C680013F7F01037F9038003FFF140302001380157F153FED1FC0150F12C0A21507A37E
A26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B512E0011F138026C0
03FEC7FC22377CB42B>I<007FB712FEA390398007F001D87C00EC003E0078161E007016
0EA20060160600E01607A3481603A6C71500B3AB4A7E011FB512FCA330337DB237>I<B5
D8F007B539800FFFF0A3000390C7273FF000011300D801FC6E48EB007C1A386D140F0000
1930836D020715706D1860A26E496C14E0013F60A26ED919FC1301011F60A26ED930FE13
03010F95C7FCA26ED9607F5B01071706A26E9039C03F800E0103170CA2913BFC01801FC0
1C01011718A2913BFE03000FE03801001730A2DAFF06EB07F0027F5EA2038CEB03F8023F
5EA203D8EB01FC021FEDFD80A203F0EB00FF020F93C8FCA24B800207157EA24B143E0203
153CA24B141C020115184C357FB24F>87 D<EB7F803803FFF0380F80FC381C003E003F13
3F6D6C7E6E7EA26E7EEA1F00C7FCA4EB01FF131FEBFF873803FC07EA0FF0EA1FC0EA3F80
127F13004815C05AA3140FA26C131F6C133B3A3F8071F180391FC1E1FF2607FFC0130039
00FE003C22237DA126>97 D<EA03F012FFA312071203AEEC3F80ECFFE09038F3C0F89038
F7007E01FE7F49EB1F8049EB0FC05BED07E016F0A2150316F8AA16F0150716E0A2ED0FC0
7F6DEB1F8001ECEB3F0001CF137C90388381F8903801FFE0C76CC7FC25357EB32B>I<EB
07F8EB3FFF9038FC07C03901F000E03903E003F03807C007120FEA1F80123F90380003E0
4890C7FCA2127E12FEAA127FA26C14187F001F14386D1330000F14706C6C13E03903F001
C03900FC0F8090383FFE00EB07F01D237EA122>I<153FEC0FFFA3EC007F81AEEB07F0EB
3FFCEBFC0F3901F003BF3907E001FF48487E48487F8148C7FCA25A127E12FEAA127E127F
A27E6C6C5BA26C6C5B6C6C4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357D
B32B>I<EB0FE0EB7FFCEBF83F3903F00F80D807E013C0390FC007E0381F800315F0EA3F
0014014814F8127EA212FEA2B6FCA248C8FCA5127E127FA26C1418A26C6C1338000F1430
6D13706C6C13E03901F003C03900FC0F00EB3FFEEB07F01D237EA122>I<EB01FCEB07FF
90381F078090383E0FC0EB7C1F13FCEA01F8A20003EB070049C7FCACB512F0A3D803F0C7
FCB3A7487E387FFFE0A31A357FB417>I<151F90391FC07F809039FFF8E3C03901F07FC7
3907E03F033A0FC01F83809039800F8000001F80EB00074880A66C5CEB800F000F5CEBC0
1F6C6C48C7FCEBF07C380EFFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14
F06C14FC4880381F0001003EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C
390FC001F83903F007E0C6B51280D91FFCC7FC22337EA126>I<EA03F012FFA312071203
AEEC1FC0EC7FF09038F1E0FC9038F3807C9038F7007E13FE497FA25BA25BB3486CEB7F80
B538C7FFFCA326347EB32B>I<EA0780EA0FC0EA1FE0A4EA0FC0EA0780C7FCAAEA07E012
FFA3120F1207B3A6EA0FF0B5FCA310337EB215>I<EB03C0EB07E0EB0FF0A4EB07E0EB03
C090C7FCAAEB03F013FFA313071303B3B01238127C00FE13E0130714C0130F007C138038
381F00EA1FFCEA07F0144384B217>I<EA03F012FFA312071203AF913803FFE0A36E1300
EC00F8EC01E05D4A5A020FC7FC141C5C5C14F0EBF3F8EBF7FC13FEEBFC7EEBF87F496C7E
141F6E7E8114076E7E8114016E7E81486CEBFF80B500C313F0A324347EB329>I<EA07E0
12FFA3120F1207B3B3A7EA0FF0B5FCA310347EB315>I<2703F01FE013FF00FF90267FF8
0313C0903BF1E07C0F03E0903BF3803E1C01F02807F7003F387FD803FE1470496D486C7E
A2495CA2495CB3486C496C487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FF
EB7FF09038F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7
FFFCA326217EA02B>I<EB07F0EB3FFE9038FC1F803901F007C03903C001E00007804848
6C7E48C7127CA248147E003E143E007E143FA300FE1580A8007E1500A36C147EA26C147C
6D13FC6C6C485A00075C3903F007E03900FC1F80D93FFEC7FCEB07F021237EA126>I<39
03F03F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0
ED07F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381
F89038F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<903807F00390383F
FC07EBFC0F3901F8038F3807E001000F14DF48486CB4FC497F123F90C77E5AA25A5AA912
7FA36C6C5B121F6D5B000F5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE090C7FCAA
ED7F8091380FFFFCA326307DA029>I<3803E07C38FFE1FF9038E38F809038E71FC0EA07
EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA01E>I<EBFF06000713CE
381F00FE003C133E48131E140E5A1406A27EA200FE90C7FC6C7EEA7FFC383FFFC014F000
0F7F6C7FC67FEB0FFF1300EC3F8000C0131F140F6C1307A37E15006C5B6C130E6C5B38F7
807838E1FFE038C07F8019237EA11E>I<1330A51370A313F0A21201A212031207381FFF
FEB5FCA23803F000AF1403A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD
1E>I<D803F0133F00FFEB0FFFA30007EB007F000380B35DA35D12016D48138000009038
03BFFC90387E073FEB1FFED907F8130026227EA02B>I<B5EBFFF0A3D80FF0EB3F800007
EC1F000003140E150C6D131C00011418A26C6C5BA26D1370017E1360137F6D5BA290381F
8180A214C3010F90C7FCA2EB07E6A214FE6D5AA26D5AA36D5AA2146024217E9F29>I<B5
3A1FFF81FFF0A33C07F801FC003F8001F049EB1E0000030100141C816C6C017C1318A26D
017E1338000002FE1330A290267E01FF5B159F168090263F030F5BA216C0903A1F8607C1
80A202C613E390260FCC0390C7FCA2D907FC13F6ECF80116FE6D486C5AA36D481378A36D
48133034217F9F37>I<B53801FFF8A32603FE0013806C48EB7C0000001478017E137001
7F5B90383F81C090381F8380D90FC3C7FCEB07E614FE6D5A6D5A6D7E80805B9038039F80
9038071FC09038060FE0EB0C0790381C03F0496C7E01707FEBF000000180000FECFF8026
FFFC0313FCA326207F9F29>I<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC0700
15066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E180
A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC03
C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2EB000F00
3C14E00038EB1FC00030EB3F800070137F1500006013FE495A13035CC6485A495AA2495A
495A49C7FC153013FE485A12035B48481370485A001F14604913E0485A387F000348130F
90B5FCA21C207E9F22>I<BC12F0A24C0280944D>124 D<001C1370387F01FC00FF13FEA4
007F13FC381C0070170879B226>127 D E /FC 6 55 df<13E01201120712FF12F91201
B3A7487EB512C0A212217AA01E>49 D<EA01FC3807FF80381C0FC0383003E0386001F0EB
00F812F86C13FCA2147C1278003013FCC7FC14F8A2EB01F0EB03E014C0EB0780EB0F0013
1E13385B5B3801C00CEA0380380600185A5A383FFFF85AB512F0A216217CA01E>I<13FF
000313C0380F03E0381C00F014F8003E13FC147CA2001E13FC120CC712F8A2EB01F0EB03
E0EB0FC03801FF00A2380003E0EB00F01478147C143E143F1230127812FCA2143E48137E
0060137C003813F8381E03F0380FFFC00001130018227DA01E>I<14E01301A213031307
A2130D131D13391331136113E113C1EA01811203EA07011206120C121C12181230127012
E0B6FCA2380001E0A6EB03F0EB3FFFA218227DA11E>I<00101330381E01F0381FFFE014
C01480EBFE00EA1BF00018C7FCA513FE381BFF80381F03C0381C01E0381800F014F8C712
78A2147CA21230127812F8A214784813F8006013F0387001E01238381E07803807FF00EA
01F816227CA01E>I<EB0FC0EB7FF03801F0383803C0183807803C380F007C121E001C13
38003C1300A2127C1278EB7FC038F9FFE038FB80F038FE0038143C48131EA248131FA412
78A36C131EA2001C133C001E13386C1370380781E03801FFC038007F0018227DA01E>I
E /FD 4 76 df<136013701360A20040132000E0137038F861F0387E67E0381FFF803807
FE00EA00F0EA07FE381FFF80387E67E038F861F038E060700040132000001300A2137013
6014157B9620>3 D<ED01C01507151FED7F00EC01FCEC07F0EC1FC0027FC7FC14FCEB03
F0EB0FC0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000FEC9FC12F812FEEA3F80EA0FE0EA03
F8EA00FEEB3F80EB0FC0EB03F0EB00FC147FEC1FC0EC07F0EC01FCEC007FED1FC0150715
0192C7FCA9007FB61280B712C0A2222F7AA230>20 D<EA01E0EA03F0A4EA07E0A213C012
0FA21380A2EA1F00A2121EA2123E123CA25AA3127012F05A12600C1A7E9B12>48
D<013814F801F8EB03FC0003141F0007EC7C3C000E14F00000903803C030020FC7FC141C
EBF078000113E0EBF1C0EBF38001F7C8FCA25B120380A213C700077FA2EB83E0120FEB81
F080130048137C147E001E013E130E003E6D131C003C1480007C90380FC038913807E0F0
0078903803FFE0486D138048903800FC0027247DA22F>75 D E /FE
17 119 df<ED3FF8913801FFFE913903F00F8091390FC003C0EC1F00160F143EA2147E02
7CEB070093C7FCA214FC5CA5017FB512FEA2903901F0007E167CA213034A13FC5EA30107
130102C05BA31503010F5C1480A2923807E18016C3131FA2140016C7EE87005B013E148E
ED03DEED01FC6F5A017E91C7FC137CA3EA38F812FCA25B12FDEAF1E0EAF3C0EA7F80001E
CAFC2A3D81AE28>12 D<16E01501821503A21507150FA2151FA2153B157B157315E382EC
01C114031581EC0701A2140EA2141C143C143802707F15005C13015C49B5FCA249C7FCA2
130E131E131C4980167E5B13F0485AA21203D80FF014FFD8FFFC011F13F0A22C2F7CAE35
>65 D<90381FFFF8A2903800FE00A25CA21301A25CA21303A25CA21307A25CA2130FA25C
A2131FA25CA2133FA291C7FCA25BA2137EA213FEA25BA21201A25BA21203B512C0A21D2D
7CAC1B>73 D<011FB512FCEEFF80903A00FE000FE0EE03F04AEB00F8A20101157CA25C17
7E130317FC5CA20107EC01F8A24AEB03F017E0010FEC07C0EE0F804AEB3F00ED01FC91B5
12F04991C7FC0280C8FCA3133F91C9FCA35B137EA313FE5BA312015BA21203B512C0A22F
2D7CAC30>80 D<011FB512E016FC903900FE003FEE0FC04AEB07E016030101EC01F0A24A
14F8A21303EE03F05CA20107EC07E017C04AEB0F80EE1F00010F143E16FC9138C007F091
B512805B9138C00FE091388003F06F7E133F6F7E91C7FCA2491301A2017E5CA201FE1303
A2495C17080001163C17384914E0EEF07800031670B5D8C00113E09238007FC0C9EA1F00
2E2E7BAC34>82 D<13F8121FA21201A25BA21203A25BA21207A25BA2120FEBC7C0EB9FF0
EBF878381FF03CEBE03EEBC01EEB801FEA3F00A2123EA2007E133FA2127CA2147F00FC13
7E5AA214FCA214F8130114F0EB03E0EA780714C0383C0F80381E3E00EA0FF8EA03E0182F
78AD21>98 D<153EEC07FEA2EC007EA2157CA215FCA215F8A21401A215F0A21403EB07C3
90381FF3E0EB7C3BEBF81FEA01F03903E00FC0EA07C0120FEA1F801580EA3F00141F5A00
7E1400A25C12FE48133EA2EC7E18153848137CA214FCD878011378397C03F870A2393C0F
78E0381E1E3D390FF81FC03903E00F001F2F79AD24>100 D<EB03F8EB0FFEEB3E0FEBF8
073901F00380EA03E0EA07C0000F1307D81F8013005C383F001E5C387F03F8EBFFE049C7
FC007EC8FC12FE5AA4127CEC0180EC03C0EC07806CEB0F00141E6C137C380F83F03803FF
C0C648C7FC1A1F799D21>I<EC01F0EC07FCEC0F9EEC1F1EEC1E3EEC3E7EA3EC7C381500
A314FC5CA590387FFFF0A2903801F000A313035CA413075CA4130F5CA4131F91C7FCA45B
133EA4137E137CA3EA38F812FCA25B12FDEAF1E0EAF3C0EA7F80001EC8FC1F3D81AE16>
I<130E131FEB3F80A2EB1F00130E90C7FCA9EA03E0EA0FF0EA1E78EA1C7C123812781270
13FCEAF0F812E012E1EAC1F0120112035B12075BA2120F13831387121F13075BEA3F0E12
3EEA1E1C133C1338EA0FF0EA03C0112E7AAC16>105 D<137CEA0FFCA21200A213F8A212
01A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127E
A2127CA2EAFC30137012F8A213F013E012F012F113C012FBEA7F80EA1E000E2F7AAD12>
108 D<3907801FC0391FE07FF0393DF1E0F83938F3C0783978FF007CEA70FEA2EAF1FCEA
E1F8A25B00C314FC00035C5BA2000713015D13C01403000FECE0C015E1EB800715C1001F
14C3020F13800100138391380787005A158E003EEB03FC001CEB00F0221F7A9D28>110
D<EB03F8EB0FFE90383E0F809038FC07C03801F003D803E013E01207390FC001F0138012
1FEA3F0014035A127EA2140700FE14E05AA2EC0FC0A2EC1F80A2007CEB3F00143E5C6C5B
381E01F0380F07C06CB4C7FCEA01FC1C1F799D24>I<3807803E391FE0FF80393CF3C1C0
3938F781E03878FF07EA70FE13FC12F139E1F8038091C7FC5B12C312035BA21207A25BA2
120FA25BA2121FA290C8FCA25AA2123E121C1B1F7A9D1E>114 D<131C133EA2137EA213
7CA213FCA25BA21201A2B512E0A23803F000A25BA21207A25BA2120FA25BA2121FA290C7
FCA24813C01301123E130314801307003C1300130E131E6C5AEA0FF0EA07C0132B7AA918
>116 D<EA03C0D80FF01338D81E78137CD81C7C13FC003814F812781270EBFC01D8F0F8
13F012E012E138C1F003000114E0120313E01407000714C013C0A2EC0FC3000F14871380
A2141F158F0007EB3F0E147F01C0131C3903E1E7BC3901FF83F839007E01E0201F7A9D26
>I<3903C001C0390FF003E0391E7807F0EA1C7C1238007813030070130113FCD8F0F813
E012E000E1130038C1F001000114C0120313E014030007148013C0A2EC0700120F138014
0EA25C12076D5A00035B6D5AC6B45A013FC7FC1C1F7A9D21>I E
/FF 36 90 df<007FB5FCA2B512FEA418067C961E>45 D<121EEA3F80EA7FC012FFA413
80EA7F00123C0A0A788919>I<EC07F8EC3FFF9138FC0FC0903903F003E0903907C001F0
D90F8013F849C7FC013E14FC017E147C017C147E13FC485AA20003157F5B1207A2120F5B
A2121F16FF5BA2123FA44848EB01FEA648C7EA03FCA5ED07F8A25A16F0A2150F16E0A3ED
1FC0A21680007E143F1600157E123E003F5C4A5AD81F805B000FEB07E06C6C485A2603F0
3FC7FC3800FFFCEB1FE0283F79BC2D>48 D<157015F014011407143F903803FFE0137FEB
FFCFEBF80F1300141F15C0A5143F1580A5147F1500A55C5CA513015CA513035CA513075C
A5130F5CA3131F497EB612F8A31D3D78BC2D>I<EC01FE91380FFFE0023F13F89138FC07
FC903901E001FE903907C000FF49C7EA7F80011E15C0163F4915E05B0170141F13FF80A3
5A163FA26C90C7FC137E0118EC7FC090C8FCEEFF80A24B1300A24B5A5E4B5A4B5A4B5A5E
4B5A4BC7FC15FEEC01F84A5A4A5A4A5A4AC8FC143E5C5CEB01E04948130E49485B49C7FC
131E495C13705B48485C484814F0000FB6FC5A485D5AB7FC5EA22B3D7CBC2D>I<EC07FC
91383FFF809138F80FE0903903C007F09039078003FC90380F0001011C14FE013C14FF13
7F1480EBFFC0A31480A291380003FE137E90C7FCED07FC16F8150F16F0ED1FE016C0ED3F
80ED7E005DEC07F0903803FF8015F090380001FC6E7EED7F80ED3FC0A2ED1FE016F0A316
F8A4120EEA3F80486C133F16F012FFA216E0157F5B48C7EAFFC000F01580007049130012
786C495A003EEB07F86C495A390FE03FE00003B51280C649C7FCEB1FE0283F7ABC2D>I<
161C163C167CA216FCED01F815031507150FA2151DED3BF0157315E315C31401EC038391
380707E0140E141CA2143814709138E00FC0EB01C014801303EB0700130E49EB1F805B13
3013705B485A4848EB3F0090C7FC5A120E5A5A48147E1260B8FCA3C73801FE00A25DA414
03A25DA314074A7E0107B512F8A3283E7BBD2D>I<01061403D90780131F90390FF801FE
91B512FC16F816F016E0168049EBFE0015F890381C7FC091C8FCA3133C1338A513781370
A2EC1FE0ECFFF8903873E03E9038FF001F01FCEB0F804914C049EB07E04914F049130390
C7FC16F8A61507A21206EA3F80487EA2150F00FF15F0A24914E090C7121F00FC15C000F0
143F00701580ED7F0012786C14FE4A5A6C495A390F800FE03907E03FC06CB5C7FCC613FC
EB1FE0283F7ABC2D>I<ED7F80913803FFE091380FC0F091383E003802FC131C495A4948
13FE903807E003EB0FC090381F8007133FD97F0013FC01FE1303ED01F0484890C7FC1203
A2485AA2120F5BA2001FEB3F809038E0FFE0393FE3C0F89038E7007C01EE7F01FC133F48
48EB1F80A24914C05B16E0A2485AA216F05BA2ED3FE0A290C7FCA4157F16C0A316804814
FF007E1500007F5C14016C5C4A5A6C6C485A4A5A6C6C485A2607E07FC7FC3803FFFEC613
F8EB3FC0273F78BC2D>I<EA0380120713E090B712805AA217005E485D5E001EC85A484A
5A00385D150300784A5A00704AC7FC151E5D485CC8127015F04A5A4A5A4A5A4AC8FC140E
141E5C147C14785C1301495AA213075C130F495AA2133F91C9FC5BA25B5B1201A312035B
A21207A3485AA5121F5BA26C5AEA0780294074BD2D>I<EC03FC91381FFF8091387C07E0
903901F001F0903903C000F84948137C49C7123E131E013E141F133C137C137813F8A316
3F486C143E167E6D147C6C6C14FC6E13F89138C001F09138F003E090397FF807C09138FC
0F0090383FFF3E6D13F86D13E06D7F01017F8101077F90391F1FFF80D93E0F13C0EBF807
2601F00113E048486C13F04848137F4848131F001FEC0FF890C71207003E1403A2481401
A300FC15F05AA3ED03E0A26CEC07C0007C1580007E140F003EEC1F00003F143E6C6C5B6C
6C485A3907F00FE00001B512806C6C48C7FCEB0FF0283F7ABC2D>I<EC07F8EC3FFE9138
FC0F80903901F007C0903907E003E0D90FC013F090381F8001013F14F8EB7F004914FC48
481300A24848EB01FEA21207A3485AA41503121F5BA31507A2000F15FC150FA2151F1207
153F000315F86C6C137F000014EF90387C01CF90393E078FF090380FFE1FEB03F890C713
E0A2ED3FC0A3ED7F8016005D003F5C487E4A5A00FF495A5D4A5A49485A48495A007049C7
FC0078137E383E03FC381FFFF06C13C0D801FEC8FC273F79BC2D>I<17E0160116038316
07A2160FA2161F83163FA2167F167716F7EEE7FCED01E316C3150316831507EE03FEED0F
01150E151E151C153C03387FED7800157015F05D4A4880177F4A5AA24AC7FCA2020E8117
3F5C021FB6FC5CA20270C7EA3FE0171F5CA2495AA2494881170F49C8FCA2130EA2498201
3C1507A2137CD801FE4B7E2607FF80EC3FFEB500F00107B512FC19F85E3E417DC044>65
D<013FB7FC18E018FC903B007FE00007FE6E48903801FF809438007FC05DF03FE0F01FF0
A3027F16F892C8FCA54A16F04A153F19E0187F19C0F0FF8001014B13004A4A5A4D5AEF1F
F04D5ADC03FFC7FC49B612F8EFFF8002F8C7EA3FE0EF0FF0EF07FC717E010715014A8171
1380A319C0130F5CA5011F4B13805C19005F601707013F4B5A4A4A5A4D5A4D5A017F9138
01FF8001FF020F90C7FCB812FC17F094C8FC3D3E7DBD40>I<DCFFC01338030F01F01378
037F01FC13F0913A01FF803F01913A07FC000781DA1FE0EB03C3DA7FC0EB01E74AC812FF
4948ED7FE0D907FC153F495A4948151F495A4948150F494816C018074890C9FC485AA248
5A000F1880491603121FA248481607A295C7FC485AA412FF5BA75BA2181C183C1838A27F
007F1778187018F0003F5F6D150160001F16036C6C4B5A95C7FC6C6C5D6C6C151E6C6C5D
6C6C15F86D6C495A6D6CEB07C0D91FF0EB1F80D907FE01FEC8FC0101B512F86D6C13E0DA
07FEC9FC3D4276BF42>I<013FB7FC18E018F8903B007FF0000FFE6E48EB01FF9438007F
C04B6E7E180F85727E727E147F4B6E7EA2727EA302FF178092C9FCA54918C05CA41A8013
034A5DA41A0013074A5DA261A24E5A130F4A5E180F61181F61011F4C5A5C4E5A4EC7FC4D
5A4D5A013F4B5A4A4A5AEF3FE0EF7F80017F4A48C8FC01FFEC1FFCB812F0178004FCC9FC
423E7DBD45>I<013FB812F8A39026007FF0C7127F6E48140F18034B14011800A3197814
7F4B1570A502FF143892C7FCA3190017784915704A14F016011603160F91B6FC495DA291
38FC001F16071603160101075D5CA2197019F019E0010F4A5A4A90C7120119C0A2180319
80011F16075CF00F00A260181E013F163E4A157E4D5A1703017F150F01FFEDFFF8B9FCA2
603D3E7DBD3E>I<013FB812E0A3903A007FF000016E48EB003F180F4B14071803A31801
147F4B15C0A514FF92C71270A395C7FC17F0495D5C160116031607161F49B65AA39138FC
003F160F160701075D4A1303A5010F4AC8FC5C93C9FCA4131F5CA5133F5CA3137FEBFFF0
B612F8A33B3E7DBD3B>I<4BB46C1370031F01F013F0037F9038FC01E0913A03FF807E03
913A0FF8000F83DA1FE0EB07C7DA7F80EB01EF4AC812FFD903FE16C04948157F4948153F
495A4948151F495A4948168091C9120F5A485AA2485A000F18004982121FA248485EA295
C7FC485AA412FF5BA6043FB512E05BA29339001FFC00715AA2607F127FA2171F123F6D5E
A2121F7F000F163F6C7E6C6C4B5A7F6C6C15FF6C6DEB01EFD93FC0EB07C7D91FF0EB1F87
D907FE9038FE03800101B5EAF8016D6C01E0C8FCDA07FEC9FC3C4276BF47>I<013FB5D8
F807B6FC04F015FEA29026007FF0C7380FFE006E486E5AA24B5DA4180F147F4B5DA4181F
14FF92C85BA4183F5B4A5EA491B8FC5B6102FCC8127FA318FF13074A93C7FCA45F130F4A
5DA41703131F4A5DA41707133F4A5DA3017F150F496C4A7EB6D8E01FB512FC6115C0483E
7DBD44>I<011FB512FC5BA29039003FF8006E5AA25DA5143F5DA5147F5DA514FF92C7FC
A55B5CA513035CA513075CA5130F5CA5131F5CA3133F497E007FB512F0A2B6FC263E7EBD
21>I<021FB512FCA3DA000713006F5AA25EA41507A25EA4150FA25EA4151FA25EA4153F
A25EA4157FA25EA415FFA293C7FCA45C121FD87F805BEAFFC0A214035D13804A5AEAFE00
00F8495A48495A00705C6C495A6C01FEC8FC380F81FC3803FFE0C690C9FC2E407ABD2F>
I<013FB500F8010FB5FC4C5BA29026007FF0C7000313E06E486E130019FC4B15F04E5A4E
5A4E5A061EC7FC027F5D4B5C4D5A4D5AEF07804DC8FC02FF141E92C7127C5FEE01E04C5A
4C5A49021FC9FC4A5B5E4C7E5D03077F01035B9139FC1F3FE0153C4B6C7E15F09139FFE0
0FF84913C092380007FC5C4A6D7E5C707E130F4A6D7F84177F717EA2011F6F7E5C717EA2
717EA2013F6F7E5C84A2017F83496C4A13E0B600E0017F13FFA24B90B6FC483E7DBD47>
I<013FB512FEA25E9026007FF8C8FCEC3FE0A25DA5147F5DA514FF92C9FCA55B5CA51303
5CA513075CA21838A21870130F5CA218E0A3011F15014A15C01703A21707EF0F80013F15
1F4A143F177FEFFF00017F140301FF143FB9FC5FA2353E7DBD39>I<90263FFFF0933807
FFFE5013FC629026007FF8EFFC00023F4D5AA2023BEF77F0A2DA39FC16E7A2F101CF0279
EE038FDA70FE5FF1070FA2190E1A1FDAF07F151C02E060193819706F7EF1E03F130102C0
DB01C05BA26F6CEB0380A2953807007F0103160E4A6C6C93C7FC60A2606201076D6C5B02
005F60A26F6C485A94380380015B010EDB07005BA2923801FC0EA24D1303131E011C6D6C
485C5FA25F1907013CEC7FC0013860013C5D137C01FE6EC7120F2607FF80013E4A7EB500
FC031FB512F8043C5E4A131C573E7DBD53>I<90263FFFE0023FB5FC6F16FEA29026003F
F8020313C0021F030013004A6C157C023B163C6F15381439810238167802787FDA707F15
7082153F82031F15F002F07FDAE00F5D8215078203031401010180DAC0015D8281178004
7F1303010315C04A013F5C17E0161F17F0040F1307010715F891C7000791C7FC17FC1603
17FE04015B4915FF010E6E130E188E177F18CEEF3FDE011E16FE011C6F5AA2170FA21707
133C01386F5A133C017C150113FE2607FF801400B512FC18705C483E7DBD44>I<923803
FF80031F13F09238FE01FE913903F0003FDA0FC0EB1FC0DA3F80EB07E0027EC76C7E4948
6E7E49488149486E7E4948157F495A013F17804948ED3FC049C9FCA24848EE1FE012035B
000718F05B120FA2485A19F8123F5BA2127FA219F04848163FA5F07FE0A35BF0FFC0A219
805F19007F4D5A127F4D5A60003F160F6D5E001F4C5A4D5A6C6C4B5A95C7FC6C6C15FE00
034B5A6C6C4A5A6C6C4A5A017FEC1FC06D6C495AD90FE001FEC8FC903903F807F80100B5
12C0DA0FFCC9FC3D4276BF47>I<013FB612FEEFFFE018F8903B007FF0000FFC6E48EB01
FF7113804BEC7FC0183F19E0F01FF0A2147F5D19F8A402FFED3FF092C8FCA219E0A2F07F
C05B4AEDFF8019004D5A4D5AEF0FF80103ED3FE04A903801FF8091B648C7FC17F002FCCA
FCA213075CA5130F5CA5131F5CA5133F5CA3137F497EB612E0A25D3D3E7DBD3E>I<013F
B612F017FF18E0903B007FF0003FF86E48EB07FCEF01FE4B6D7EF07F8019C0183F19E014
7F4B15F0A502FFED7FE092C8FCA219C0F0FF80A2494B13004A5D4D5AEF0FF04D5AEF7F80
0103DA07FEC7FC91B612F017809139FC0007E0EE03F8EE00FC0107814A147F717EA284A2
130F5CA484011F157F5CA41902013F17075CA2F0F00F017F170E496C143FB600E0011F13
1C94380FF83C4B01071378CA3801FFE09438003F8040407DBD43>82
D<9239FF8003800207EBF007021F9038FC0F0091387F00FE02FCEB1F1FD903F0EB07BF49
486DB4FC49487F4A6D5A49C8FC49157E133E137E173E49153CA57F1738A26D92C7FC8080
80EB7FFEECFFE06D13FEEDFFC06D14F06D14FC010380010080143F020380DA003F7F1503
1500707E163F161FA2160F121CA31607160F003C5EA35F003E151F94C7FC007E5D007F15
3E6D5C16FC01E0495AD87DF0495AD8FCFCEB0FC03AF87F803F8027F01FFFFEC8FCD8E007
13F839C0007FC031427BBF33>I<0007B912F0A33C0FFE000FF8003F01F0160F01C04A13
034848160190C7FC121EF000E048141F5E1238A212781270153F5E5AA3C81600157F5EA5
15FF93C9FCA55C5DA514035DA514075DA5140F5DA3141FEC7FFC0003B7FCA33C3D76BC42
>I<B600E090B512FC4B15F8A2000101C0C7000F13006C49EC03FCEF01F091C9FC60A317
015A495EA417031203495EA4170712074993C7FCA45F120F49150EA4171E121F49151CA4
173C123F491538A31778177017F05F001F15015F16036D4A5A000F93C8FC5E6C6C141E6C
6C5C000115F86C6C495A017FEB07C090393FC03F8090260FFFFEC9FC010313F89038007F
C03E4073BD44>I<B6020FB5FC19FEA2000301E0020113E06C01809138007F8091C9EA7E
006C173C18386E15781870017F16F0604D5A804D5A133F4DC7FCA26E140E171E011F151C
173C17386E1478010F15705FA24C5A8001074A5AA24CC8FC5E6E130E0103141E161C163C
16386E5B13015EA24B5A14FF6D495AA24BC9FC5D158EEC7F9E159C15B8A215F0143F5DA2
5DA26E5AA292CAFCA2140E404074BD44>I<B6017FB5D88007B512804A1A00A2000701C0
010101E0C713F06C90C80180EC3FC06C48735A99C7FC057F150E1B1E6D191C6C1A3C1B38
05FF15787214705E636EEB03BF017F4E5AEE073F505A040E7F051F4AC8FC161C6E170E01
3F143862167804706D5BEEF00F04E05D90381FE00104C015F003035E0480140106F85B92
26070007130302F05F010F010E150797C9FC5D190E4BEB03FC616E5A01075F5D61DAF9C0
14FE05015BECFB8002FF6F5A7F92C75CA24A93CAFC835C606D5A605C604A157818705940
74BD5D>I<010FB500F090B512F85B5FD9003F902680003F1300DA0FFEC7EA1FF84BEC0F
E00207168096C7FC6E6C141E181C6E6C143C606E6D5B4D5ADB7FC05B4D5A92383FE0074D
C8FC92381FF01E171C6F6C5A5F923807FCF0EEFDE06FB45A5F6F90C9FCA26F7FA2707EA2
16FF4B7FED03DF9238079FF0ED0F1F92380E0FF8151C92383C07FC15784B6C7EEC01E04B
6C7EEC038002076D7F4AC7FC021E6E7E5C02386E7E5C02F06E7E495A49486E7E13074948
6E7E497E017F4B7E2603FFF091383FFF80007F01FC49B512FEB55CA2453E7EBD44>I<B6
6C0103B51280A3000101F0C8EBF0006C49ED7FC06D486FC7FC6E153E013F163C606D6C5D
606D6C4A5A17036D6C4A5A95C8FC6E140E0103151E5F6D6C14385F6D6D13F04C5ADA7FC0
5B4C5AEDE007023F49C9FC161E91381FF01C5E91380FF8785E6E6C5AEDFDC015FF6E5B93
CAFC6E5AA35DA21403A45DA21407A45DA2140FA4141F4A7E013FB512F0A3413E75BD44>
I E /FG 60 122 df<ED0FFF4AB512C0020F14F0027F80903A01FFF803FC499038C000FE
010FEB00034948497E49485B5C495A4C138001FF6E13005CA3705AEE01F893C8FCA74BB5
1280B9FCA5C69038E00003B3B0007FD9FFC1B6FCA538467EC53E>12
D<ED0FFF4AB5EAEF80020F14FF147F903901FFF807491380010F495A495A495A5C495A82
13FF4A7FADB9FCA5C69038E00003B3B0007FD9FFC1B6FCA538467EC53E>I<B612F8A91D
097F9A25>45 D<EA07C0EA1FF0EA3FF8EA7FFCEAFFFEA7EA7FFCEA3FF8EA1FF0EA07C00F
0F788E1F>I<EC03C01407141F147FEB03FF133FB6FCA413C3EA0003B3B3ADB712FCA526
4177C038>49 D<ECFFE0010F13FE013F6D7E90B612E0000315F82607FC0313FE3A0FE000
7FFFD81F806D138048C7000F13C0488001C015E001F07F00FF6E13F07F17F881A46C5A6C
5A6C5AC9FC17F05DA217E05D17C04B13804B1300A2ED1FFC4B5A5E4B5A4B5A4A90C7FC4A
5A4A5AEC0FF04A5AEC3F804AC7127814FE495A494814F8D907E014F0495A495A49C8FC01
7C140149140348B7FC4816E05A5A5A5A5AB8FC17C0A42D417BC038>I<ECFFF0010713FF
011F14C0017F14F049C66C7ED803F8EB3FFED807E06D7E81D80FF86D138013FE001F16C0
7FA66C5A6C4815806C485BC814005D5E4B5A4B5A4B5A4A5B020F1380902607FFFEC7FC15
F815FF16C090C713F0ED3FFCED0FFEEEFF80816F13C017E0A26F13F0A217F8A3EA0FC0EA
3FF0487EA2487EA217F0A25D17E06C5A494913C05BD83F80491380D81FF0491300D80FFE
EBFFFE6CB612F800015D6C6C14C0011F49C7FC010113E02D427BC038>I<163FA25E5E5D
5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8EB
01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A12
FCB91280A5C8000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FFEB
07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C714
C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8FC
6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0123E
003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01FC
C7FC010113C02D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F809039
0FFC001FD93FF014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE7F
80EE1E00003F92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9F0
1FFC9039FBC007FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127FA5
123FA217F07F121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038E0
1FFC6DB55A011F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC90
B712FEA45A17FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D15014B
5A00F84A5A484A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA214
7F5D14FFA25BA35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>I<EC7FF00103B5FC
010F14C0013F14F090397F801FFC3A01FC0003FE48486D7E497F4848EC7F80163F484815
C0A2001F151FA27FA27F7F01FE143F6D158002C0137F02F014006C01FC5B6E485A6C9038
FF83FCEDE7F86CECFFE06C5D6C92C7FC6D14C06D80010F14F882013F8090B7FC48013F14
802607FC0F14C0260FF80314E04848C6FC496D13F0003F141F48481307496D13F8150000
FF157F90C8123F161F160FA21607A36D15F0127F160F6D15E06C6C141F6DEC3FC06C6CEC
7F80D80FFE903801FF003A07FFC00FFE6C90B55AC615F0013F14C0010F91C7FC010013F0
2D427BC038>I<EC7FF0903807FFFE011F6D7E017F14E09039FFE03FF0489038800FF848
496C7E48488048486D7E001F80003F1680A2484815C08117E0A212FF17F0A617F8A45D12
7FA3003F5CA26C7E5D6C6C5B12076C6C131E6CEBC07C6CEBFFF8013F5B010F01C013F001
01130090C8FCA217E05DA2EA03C0D80FF015C0487E486C491380A217004B5A150F5E4949
5A6C48495A01C0EBFFE0260FF0035B6CB65A6C4AC7FC6C14F86C6C13E0D907FEC8FC2D42
7BC038>I<EE1F80A24C7EA24C7EA34C7EA24B7FA34B7FA24B7FA34B7F169F031F80161F
82033F80ED3E07037E80157C8203FC804B7E02018115F0820203814B137F0207815D173F
020F814B7F021F8292C77EA24A82023E80027E82027FB7FCA291B87EA2498302F0C8FCA2
0103834A157F0107834A153FA249488284011F8491C97E4984133E017E82B6020FB612F0
A54C457CC455>65 D<B9FC18F018FE727E19E026003FFCC700077F05017F716C7E727E72
7EA2721380A37213C0A74E1380A24E1300A24E5A4E5A4E5A4D5B05075B94B5128091B700
FCC7FC18F018FF19E002FCC7000113F8716C7EF01FFE727E7213801AC07213E0A27213F0
A31AF8A71AF0A2601AE0604E13C0604E138095B5120005075BBA12F86119C04EC7FC18E0
45447CC350>I<DCFFF01470031F01FF14F04AB6EAE0010207EDF803023FEDFE0791B539
E001FF0F4949C7EA3F9F010701F0EC0FFF4901C0804990C87E4948814948814948167F48
49163F4849161F5A4A160F485B19074890CAFC19035A5BA2007F1801A34994C7FC12FFAE
127F7F1AF0A2123FA27F6C18011AE06C7F19036C6D17C06E16077E6C6DEE0F806C6DEE1F
006D6C5E6D6C167E6D6C6C5D6D6D4A5A6D01F0EC07F0010101FEEC1FE06D903AFFF001FF
80023F90B6C7FC020715FC020115F0DA001F1480030001F8C8FC44467AC451>I<B9FC18
F018FE727E19E026003FFEC7001F13F805017F9438003FFF060F7F727F727F727F84737E
737EA2737EA2737EA21B80A2851BC0A51BE0AD1BC0A51B8061A21B006162193F624F5A19
FF624E5B06075B4E5B063F90C7FC4DB45A050F13F8BA5A19C04EC8FC18F095C9FC4B447C
C356>I<BA12F8A485D8001F90C71201EF003F180F180318011800A2197E193EA3191EA2
1778A285A405F890C7FCA316011603161F92B5FCA5ED001F160316011600A2F101E01778
A2F103C0A494C7FC1907A21A80A2190FA2191FA2193FF17F0061601807181F4DB5FCBBFC
61A443447DC34A>I<BA1280A419C026003FFEC7121F1701EF007F183F181F180F180719
E01803A31801A3EE01E0F000F0A419001603A31607160F167F91B6FCA59138FE007F160F
16071603A31601A693C9FCAFB712F0A53C447CC346>I<DCFFF01470031F01FF14F04AB6
EAE0010207EDF803023FEDFE0791B539E001FF0F4949C7EA3F9F010701F0EC0FFF4901C0
804990C87E4948814948814948167F4849163F4849161F5A4A160F485B19074890CAFC19
035A5BA2007F1801A34994C8FC12FFAD057FB612F0127F7FA3003FDC0001EBF000A27F7E
A26C7FA26C7F807E6C7F6C7F6D7E6D6C5D6D6C7E6D6D5C6D01F05C010101FE143F6D903A
FFF001FF9F023F90B6120F0207EDFC030201EDF000DA001F02C01330030001FCC9FC4C46
7AC458>I<B7D88003B612FEA526003FFEC9EBF800B3A791B9FCA54AC9FCB3AAB7D88003
B612FEA54F447CC358>I<B712E0A5D8001F90C7FCB3B3B3A4B712E0A523447DC32A>I<01
07B7FCA590C7001F1300B3B3A9EA1FE0487E487EA2487EA44B5AA26C48495A495C6C4813
FF6C48485B260FFC0713C06CB65A6C4AC7FCC66C13F8010F138030457DC33A>I<B76C01
03B512F8A526003FFEC93807E0004F5A4F5A077EC7FC614E5A4E5A4E5AF01F804EC8FC18
7E604D5AEF07F0EF0FC04D5A4DC9FC177E4C5AEE03F04C5A4C5A4C7EEE7FF04C7E5D4B7F
4B7F4B7FED3F3FDB7E1F7F03FC806E486C7F4B7E4B6C7F0380804B6C7F4A7F717E84717F
83717F85717F83717F85717F187F727E86727F84727F86727F84B76C90B612FCA54E447C
C358>I<B712F0A526003FFECAFCB3B1F00780A4180F1900A460A360A2187EA218FE1701
17031707171F177FEE03FFB95AA539447CC343>I<B500FE067FB512806E95B6FCA26F5E
A2D8003F50C7FC013D6DEE03DFA2013C6DEE079FA26E6CEE0F1FA26E6C161EA26E6C163C
A36E6C1678A26E6C16F0A26E6DEC01E0A26E6DEC03C0A36E6DEC0780A26F6CEC0F00A26F
6C141EA26F6C5CA36F6C5CA26F6C5CA26F6D485AA26F6D485AA26F6D485AA3706C48C7FC
A293383FF81EA2706C5AA2706C5AA3706C5AA2705BA2705BA2705BA2B6057FB6128071C7
FCA2173E171C61447CC36A>I<B64BB512FE8181A281D8003F6D91C7EA780081013D7F81
133C6E7E6E7F6E7F6E7F6E7F82806E7F6E7F6F7E6F7F83816F7F6F7F6F7F6F7F6F7F8382
707F707F707F707F8482707F707F717E7113807113C019E0837113F07113F87113FC7113
FE19FF847213F884848484A28484197F193F191FA2190F1907B61603190119001A78A24F
447CC358>I<923807FFC092B512FE0207ECFFC0021F15F091267FFE0013FC902601FFF0
EB1FFF01070180010313C04990C76C7FD91FFC6E6C7E49486F7E49486F7E01FF8348496F
7E48496F1380A248496F13C0A24890C96C13E0A24819F04982003F19F8A3007F19FC4917
7FA400FF19FEAD007F19FC6D17FFA3003F19F8A26D5E6C19F0A26E5D6C19E0A26C6D4B13
C06C19806E5D6C6D4B13006C6D4B5A6D6C4B5A6D6C4B5A6D6C4A5B6D01C001075B6D01F0
011F5B010101FE90B5C7FC6D90B65A023F15F8020715C002004AC8FC030713C047467AC4
54>I<B9FC18F018FE727E19E0D8001F90C7000F7F05017F716C7E727E727E721380A21A
C084A21AE0A91AC0A24E1380A21A00604E5A4E5A4D485A050F5B92B712C096C7FC18FC18
C092CBFCB3A7B712E0A543447DC34D>I<923807FFC092B512FE0207ECFFC0021F15F091
267FFE0013FC902601FFF0EB1FFF010701C0010713C04990C700017F49486E7F49486F7E
49486F7E49486F7E48496F7E48496F1380A248496F13C0A24819E091C97E4819F0A24848
7013F8A3007F19FCA249177FA300FF19FEAD007F19FCA36D17FF003F19F8A3001F19F06D
5EA26C19E06E01FE5B6C912603FF8014C06C6D486D4813804B13E06C9028E01F83F00F13
006C903BF01E00F81FFE90267FF83E90387C3FFC90263FFC3C6D485AD91FFE91381EFFF0
D90FFF021F5B6D01FE5D010194C7FC6D6D6CB45A023F90B512F8020703E0130202006F13
07030713C792C7EA07F8716C130F72131F9538FF80FF96B5FC7114FEA3831AFCA27213F8
1AF0847213E07213C0721300F001FC48587AC454>I<B812F8EFFFC018F818FE727ED800
1F90C7003F13E005037F05007F727E727E727EA28684A286A762A24E90C7FCA24E5A6118
7F943801FFF005075B053F138092B7C8FC18F818E018F892C77FEF3FFF050F7F717F717F
A2717FA2717FA785A61B0F85A2187F73131F72141EB700E06DEB803E72EBE0FC72EBFFF8
060114F0726C13E0CC0007138050457DC354>I<DAFFE0131C010701FE133C013F9038FF
807C90B6EAE0FC4815F9489038801FFF3907FC00014848EB007F4848143F4848140F4914
07007F15035B1601160012FF177CA27FA26D153C7F7F6D92C7FC6C7EEBFFE014FE6CEBFF
F015FF6C15E016FC6C816C6F7E6C826C826C6C81011F810107811300020F80140003077F
ED007F82040F1380828212F082A282A27EA218007EA26C5D6C5E6D14036D5D6D140701F8
4A5A01FFEC3FF002F8EBFFE0486CB65AD8FC1F92C7FCD8F80714FC48C614F04801071380
31467AC43E>I<003FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607
A2007E1803007C1801A300781800A400F819F8481978A5C81700B3B3A20107B8FCA54543
7CC24E>I<B76C010FB512F8A526003FFEC93803E000B3B3A9011F17076280190F6D606F
151F6D95C7FC6D6D5D197E6D6D5D6D6D1403DA7FFC4A5A6EB4EC3FF0020F9039F003FFE0
6E90B61280020193C8FC6E6C14FC030F14E09226007FFEC9FC4D457CC356>I<B792B6FC
A526003FFECAEAFC00806D606F15016D608119036D606F15076D606F150F6D6081191F6D
6D93C7FC61027F163E6F157E023F167C8119FC6E6D5C18016E5E7013036E5E8218076E6D
5C180F6E5E70131F6E93C8FC705B037F143E82187E033F147C7013FC6F5C17816F5C17C1
17C36F5C17E76F5C17FF6F5CA36F91C9FCA2705AA2705AA3705AA2705AA2705AA250457E
C355>I<B600FE017FB691B512FEA526007FFCC8D83FFEC9EA7C006E82013F7017788074
15F86D705F6F7014016D705FA26F7014036D64814E6D14076D646F70140F6D041E94C7FC
A26F023E6D5C6DDC3C7F151E81027F037C6D5CF0783F6F70147C023F4B6C1578A26F0101
6F13F86E4B6C5D16806E02036F485A4E7E04C0EEE0036E4A486C5DA2DCE00FEDF0076E4B
6C5D16F06E4A6F48C8FC051E7F04F8705A6E4A027F131EA2DCFC7CEDFE3E037F0178023F
133C04FE16FF033F01F85E4D8004FF17F86F496E5BA36F496E5BA26F604D80A26F90C86C
5BA36F486F90C9FCA26F48167EA30478163C6F457EC374>I<903801FFE0011F13FE017F
6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00
F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE0048
5A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EB
FFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 D<EB7FC0B5FCA512037EB1ED0F
F892B57E02C314E002CF14F89139DFC03FFC9139FF000FFE02FCEB03FF4A6D13804A15C0
4A6D13E05CEF7FF0A218F8173FA318FCAC18F8A2177F18F0A3EFFFE06E15C06E5B6E4913
80027C491300496C495A903AFC1FC07FFC496CB512F0D9F00314C049C691C7FCC8EA1FF0
36467DC43E>I<EC3FFC49B512C0010F14F0013F14FC90397FF003FE9039FFC001FF0003
495A48494813805B120F485AA2485A6F1300007F6E5AED00784991C7FCA212FFAC6C7EA3
123F6DEC03C0A26C6C1407000F16806D140F6C6DEB1F006C6D133E6C01F05B3A007FFC03
F86DB55A010F14C0010391C7FC9038003FF82A2F7CAD32>I<EE03FEED07FFA5ED001F16
0FB1EC3FE0903803FFFC010FEBFF8F013F14CF9039FFF807FF48EBC00148903880007F48
90C7123F4848141F49140F121F485AA3127F5BA212FFAC127FA37F123FA26C6C141FA26C
6C143F0007157F6C6C91B5FC6CD9C00314FC6C9038F01FEF6DB5128F011FEBFE0F010713
F89026007FC0EBF80036467CC43E>I<EC3FF80103B57E010F14E0013F8090397FF83FF8
9039FFC007FC48496C7E48496C7E48486D1380485A001FED7FC05B003FED3FE0A2127F5B
17F0161F12FFA290B7FCA401F0C9FCA5127FA27FA2123F17F06C7E16016C6C15E06C6C14
036C6DEB07C06C6DEB0F806C01F0EB3F0090397FFE01FE011FB55A010714F0010114C090
26001FFEC7FC2C2F7DAD33>I<EDFF80020F13E0027F13F049B512F849EB8FFC90390FFE
0FFE90381FFC1F14F8133FEB7FF0A2ED0FFCEBFFE0ED03F0ED00C01600ABB612F8A5C601
E0C7FCB3B0007FEBFFE0A527467DC522>I<DAFFE0137E010F9039FE03FF80013FEBFF8F
90B812C048D9C07F133F489038001FF84848EB0FFC4848903907FE1F80001F9238FF0F00
496D90C7FCA2003F82A8001F93C7FCA26D5B000F5D6C6C495A6C6C495A6C9038C07FF048
90B55A1680D8078F49C8FC018013E0000F90CAFCA47F7F7F90B612C016FC6CEDFF8017E0
6C826C16FC7E000382000F82D81FF0C77ED83FC014074848020113808248C9FC177FA46D
15FF007F17006D5C6C6C4A5A6C6C4A5AD80FFEEC3FF83B07FFC001FFF0000190B612C06C
6C92C7FC010F14F8D9007F90C8FC32427DAC38>I<EB7FC0B5FCA512037EB1ED07FE9238
3FFF8092B512E002C114F89139C7F03FFC9138CF801F9139DF000FFE14DE14FC4A6D7E5C
A25CA35CB3A7B60083B512FEA537457CC43E>I<137C48B4FC4813804813C0A24813E0A5
6C13C0A26C13806C1300EA007C90C7FCAAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC5
20>I<EB7FC0B5FCA512037EB293387FFFE0A593380FE0004C5A4CC7FC167E5EED03F8ED
07E04B5A4B5A037FC8FC15FEECC1FCECC3FE14C7ECDFFF91B57E82A202F97F02E17F02C0
7FEC807F6F7E826F7E816F7F836F7F816F7F83707E163FB60003B512F8A535457DC43B>
107 D<EB7FC0B5FCA512037EB3B3B3A3B61280A519457CC420>I<90277F8007FEEC0FFC
B590263FFFC090387FFF8092B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D
8FC00FFE1F801FFC0003D99F009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7E
A24A5D4A5DA34A5DB3A7B60081B60003B512FEA5572D7CAC5E>I<90397F8007FEB59038
3FFF8092B512E0028114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02
F86D7E5CA25CA35CB3A7B60083B512FEA5372D7CAC3E>I<EC1FFC49B512C0010714F001
1F14FC90397FF80FFF9026FFC0017F48496C7F4848C7EA3FE000078248486E7E49140F00
1F82A2003F82491407007F82A400FF1780AA007F1700A46C6C4A5AA2001F5E6D141F000F
5E6C6C4A5AA26C6C6CEBFFE06C6D485B27007FF80F90C7FC6DB55A010F14F8010114C090
26001FFCC8FC312F7DAD38>I<90397FC00FF8B590B57E02C314E002CF14F89139DFC03F
FC9139FF001FFE000301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF
3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07F
FC02CFB512F002C314C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>I<DA3FE013
1E902603FFFC133E010F01FF137E013F1480903AFFF80FE0FE489038E003F148EBC00148
90388000FB4890C7127F49143F001F151F485A160F5B127FA3485AAC6C7EA46C7EA26C6C
141F163F6C6C147F6C15FF6C6D5A6C9038E003EF6C9038F01FCF6DB5128F011FEBFE0F01
0313F89038007FC091C7FCAD0307B512FCA536407CAC3B>I<90387F807FB53881FFE002
8313F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED
01E092C7FCA35CB3A5B612E0A5272D7DAC2E>I<90391FFC038090B51287000314FF120F
381FF003383FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387F
FFF014FF6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E0140000
F0143FA26C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A
00FC5CD8F03F13E026E007FEC7FC232F7CAD2C>I<EB01E0A51303A41307A2130FA2131F
A2133F137F13FF1203000F90B51280B7FCA4C601E0C7FCB3A3ED01E0A9150302F013C013
7F150790393FF80F8090391FFC1F006DB5FC6D13FC01015B9038003FE023407EBE2C>I<
D97FC049B4FCB50103B5FCA50003EC000F6C81B3A85EA25EA25E7E6E491380017FD901F7
13FE9138F807E76DB512C7010F1407010313FE9026007FF0EBFC00372E7CAC3E>I<B690
3803FFFCA5000101E09038003E006C163C80017F5D8017F8013F5D6E1301011F5D6E1303
010F5D6E13076D5DED800F6D92C7FC15C05E6DEBE01E163E6D143CEDF07C027F1378EDF8
F8023F5B15FD021F5B15FF6E5BA36E5BA26E90C8FCA26E5AA26E5AA21578362C7EAB3B>
I<B5D8FE1FB539801FFFF0A500019027C0003FE0C7EA7C007114786E17F86C6F6C5C6E16
01017F6E6C5CA26E011F1403013F6F5C6E013F1407011F6F5CA26E0179140F010F048090
C7FC6E01F95C6D02F0EBC01E15806D902681E07F5B18E003C3157C6D9139C03FF07815E7
6DDA801F5B18F803FF14F96E9039000FFDE018FF6E486D5BA36E486D5BA26E486D90C8FC
A24B7F02075DA26E48147C4B143C4C2C7EAB51>I<B500FE90383FFFF0A5C601F0903803
E0006D6C495A013F4A5A6D6C49C7FC6E5B6D6C137E6DEB807C6D6D5A6DEBC1F0EDE3E06D
EBF7C06EB45A806E90C8FC5D6E7E6E7F6E7FA24A7F4A7F8291381F3FFCEC3E1F027C7F4A
6C7E49486C7F01036D7F49487E02C08049486C7F49C76C7E013E6E7E017E141FB500E090
B512FCA5362C7EAB3B>I<B6903803FFFCA5000101E09038003E006C163C80017F5D8017
F8013F5D6E1301011F5D6E1303010F5D6E13076D5DED800F6D92C7FC15C05E6DEBE01E16
3E6D143CEDF07C027F1378EDF8F8023F5B15FD021F5B15FF6E5BA36E5BA26E90C8FCA26E
5AA26E5AA21578A215F85D14015D001F1303D83F805B387FC007D8FFE05B140F92C9FC5C
143E495A387FC1F8EB07F06CB45A6C5B000790CAFCEA01FC36407EAB3B>I
E /FH 57 122 df<EEFFFC031FEBFF804AB612E0020781021F9038C00FF8913A7FFE0003
FCDAFFF0EB00FE4949EB03FF4901805B4990C7487F49485CA2495A4D7F013F6F5B5CA371
90C7FC715AEF01F894C9FCA90403B512C0BAFCA526003FFCC7120783B3B3A6003FB5D8FC
03B612C0A542547DD34B>12 D<EA07F0EA1FF8487E487E7FB5FC1480A314C0A37EA27E7E
EA07F3EA0003A213071480A3130F1400A25B131E133E133C137C5BA2485A485A485A485A
48C7FC121E120C1228769025>44 D<B712F0AB240B7F9F2D>I<EA07F0487E487E487E48
7EB51280A76C13006C5A6C5A6C5A6C5A1111769025>I<913803FFC0023F13FC91B6FC01
0315C0010F018113F0903A1FFC003FF849486D7E49486D7E49486D7E48496D138048496D
13C0A24817E04890C813F0A34817F8A24817FC49157FA3007F17FEA600FF17FFB3A5007F
17FEA6003F17FCA26D15FFA26C17F8A36C17F0A26C6D4913E0A26C6D4913C06C17806E5B
6C6D4913006D6C495AD91FFCEB3FF8903A0FFF81FFF06D90B55A01011580D9003F01FCC7
FC020313C0384F7BCD43>48 D<157815FC14031407141F14FF130F0007B5FCB6FCA2147F
13F0EAF800C7FCB3B3B3A6007FB712FEA52F4E76CD43>I<EC3FFE0103B512E0010F14FC
013F14FF90B712C048D9C07F7F2703FE000F13F8D807F801037FD80FE06D7F48486D7F48
488001F01680486C6E13C07F486C6E13E07FA27013F0A56C5AA26C5AEA0FF0EA03C0C914
E05EA218C05E1880A24C13005F4C5A4B5B5F4B5B5F4B5B4B90C7FC4B5A5E4B5AED7FE04B
5A4A5B4A48C8FC4A5A5D4A48EB01F04A5AEC3F804AC7FC02FEEC03E0495A495A495A495A
D91F80140749C8FC013E150F017FB7FC90B812C05A5A5A5A5A5A5AB9FC1880A4344E79CD
43>I<91380FFFC091B512FC0107ECFF80011F15E090263FF8077F9026FF800113FC4848
C76C7ED803F86E7E491680D807FC8048B416C080486D15E0A4805CA36C17C06C5B6C90C7
5AD801FC1680C9FC4C13005FA24C5A4B5B4B5B4B13C04B5BDBFFFEC7FC91B512F816E016
FCEEFF80DA000713E0030113F89238007FFE707E7013807013C018E07013F0A218F8A270
13FCA218FEA2EA03E0EA0FF8487E487E487EB57EA318FCA25E18F891C7FC6C17F0495C6C
4816E001F04A13C06C484A1380D80FF84A13006CB44A5A6CD9F0075BC690B612F06D5D01
1F1580010302FCC7FCD9001F1380374F7ACD43>I<177C17FEA2160116031607160FA216
1F163F167FA216FF5D5DA25D5DED1FBFED3F3F153E157C15FCEC01F815F0EC03E01407EC
0FC01580EC1F005C147E147C5C1301495A495A5C495A131F49C7FC133E5B13FC485A5B48
5A1207485A485A90C8FC123E127E5ABA12C0A5C96C48C7FCAF020FB712C0A53A4F7CCE43
>I<D80380150ED807E0157E01FEEC03FED9FFF0137F91B65A5F5F5F5F5F94C7FC5E5E16
F016C093C8FC15F801E190C9FC01E0CAFCABEC0FFF027F13F001E3B512FE01E76E7E9026
FFF8077FDAC0017F49C713F8496E7E49143F4981496E7E6C481680C9FC18C08218E0A418
F0A3EA0FE0487E487E487E487EA418E0A35B6C484A13C05B491680003EC85A003F17006C
6C4A5A6D5D6C6C4A5AD807F8495BD803FE01075B2701FFC03F5B6C90B65A013F4AC7FC6D
14F8010314C09026007FF8C8FC344F79CD43>I<ED0FFF92B512E0020780021F14FC9139
7FFE03FE903A01FFF0007F4901C0EB3F804990C7121F4948EC7FC0494814FF49484913E0
49485B01FF5C485BA2485B5AA2486F13C04A6D1380486F1300177E94C7FC5AA291CAFC5A
A21508913801FFF8020713FFB54814C04A14F04AC66C7E023C6D7E4A6D7E4A6D7E701380
4A15C0A24A15E07013F05C18F8A491C714FCA37EA67EA46C17F880A27E18F06C5D18E06C
6D15C07E6E4913806C6D15006D6C495A6D6CEB7FFC6DB448485A6D90B55A010315C00100
92C7FC023F13FC020713C0364F7ACD43>I<121F7F7FEBFF8091B81280A45A1900606060
A2606060485F0180C86CC7FC007EC95A4C5A007C4B5A5F4C5A160F4C5A484B5A4C5A94C8
FC16FEC812014B5A5E4B5A150F4B5AA24B5AA24B5A15FFA24A90C9FCA25C5D1407A2140F
A25D141FA2143FA4147F5DA314FFA55BAC6D5BA2EC3FC06E5A395279D043>I<913807FF
C0027F13FC0103B67E010F15E090261FFC0113F8903A3FE0003FFCD97F80EB0FFE49C76C
7E48488048486E1380000717C04980120F18E0177FA2121F7FA27F7F6E14FF02E015C014
F802FE4913806C7FDBC00313009238F007FE6C02F85B9238FE1FF86C9138FFBFF06CEDFF
E017806C4BC7FC6D806D81010F15E06D81010115FC010781011F81491680EBFFE7480181
15C048D9007F14E04848011F14F048487F48481303030014F8484880161F4848020713FC
1601824848157F173FA2171FA2170FA218F8A27F007F17F06D151FA26C6CED3FE0001F17
C06D157F6C6CEDFF806C6C6C010313006C01E0EB0FFE6C01FCEBFFFC6C6CB612F06D5D01
0F1580010102FCC7FCD9000F13C0364F7ACD43>I<91380FFF8091B512F8010314FE010F
6E7E4901037F90267FF8007F4948EB3FF048496D7E484980486F7E484980824817805A91
C714C05A7013E0A218F0B5FCA318F8A618FCA46C5DA37EA25E6C7F6C5DA26C5D6C7F6C6D
137B6C6D13F390387FF803011FB512E36D14C30103028313F89039007FFE03EC00401500
A218F05EA3D801F816E0487E486C16C0487E486D491380A218005E5F4C5A91C7FC6C484A
5A494A5A49495B6C48495BD803FC010F5B9027FF807FFEC7FC6C90B55A6C6C14F06D14C0
010F49C8FC010013F0364F7ACD43>I<171F4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C
7FA24C7FA34C8083047F80167E8304FE804C7E03018116F8830303814C7E03078116E083
030F814C7E031F81168083033F8293C77E4B82157E8403FE824B800201835D840203834B
800207835D844AB87EA24A83A3DA3F80C88092C97E4A84A2027E8202FE844A82010185A2
4A820103854A82010785A24A82010F855C011F717FEBFFFCB600F8020FB712E0A55B547B
D366>65 D<BA12C019FEF1FFC01AF01AFCD8000701F0C7000313FFDE007F7F737F070F7F
737F878587858785A287A84F5BA263616361634F5B4F5B077F90C7FC4E485A060713F892
B812E097C8FC861AF003F0C7000313FE9539003FFF80070F13E0737F07017F87737F747E
1C807413C0A27413E0A31CF0A386A362A31CE0A2621CC0A250138097B5FC1C004F5B1907
4F5B073F13F04EB55ABC128098C7FC1AF81AC007F8C8FC54527CD160>I<932601FFFCEC
01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA803F0203DAE003EBC07F02
0F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848C9FC4901F8824949824949
824949824949824990CA7E494883A2484983485B1B7F485B481A3FA24849181FA3485B1B
0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C7FA21B0F6C6D1980A26C1A1F6C7F
1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D6D4C5A6D6D4C5A6D01FC4C5A6D6D
EE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1FF80203903AFFE001FFF0020091B6
12C0033F93C8FC030715FCDB007F14E0040101FCC9FC525479D261>I<BA7E19FCF1FF80
1AF01AFCD8000701F0C7000F13FF060014C0071F7F070713F807017F737F747E747F747F
86747F747F8886888688A2757EA31D8087A21DC0A51DE0A387A963A31DC0A51D80A2631D
00A3515AA2646264505B6264505B505B5090C7FCF2FFFE4F5B07075B071F5B96B512C006
0F91C8FCBB5A1AF01AC007FCC9FC19805B527CD167>I<BC1280A5D8000701F8C7000114
C0F0001F19071901851A7F1A3F1A1FA2F20FE0A21A07A31A03A318F81BF01A01A497C7FC
1701A317031707170F177F92B6FCA59238F8007F170F170717031701A317001B3EA31B7C
A395C8FCA21BFCA21BF8A21A01A31A031BF01A071A0FA21A1F1A3FF27FE0F101FF190719
1F0603B5FCBCFCA21BC0A34F517CD058>I<BB12FEA5D8000701F8C700077FF0007F191F
190785858586861B80A21A1FA31A0FA41BC006F81307A497C7FCA31701A317031707170F
177F92B6FCA59238F8007F170F170717031701A31700A795C9FCB3B812F8A54A517CD055
>I<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA803F02
03DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848C9FC49
01F8824949824949824949824949824990CA7E494883A2484983485B1B7F485B481A3FA2
4849181FA3485B1B0FA25AA298C8FC5CA2B5FCAE6C057FB712E0A280A36C94C7003FEBC0
00A36C7FA36C7FA27E6C7FA26C7F6C7FA26D7E6D7F6D7F6D6D5E6D7F6D01FC93B5FC6D13
FF6D6C6D5C6E01F0EC07FB020F01FEEC1FF10203903AFFF001FFE0020091B6EAC07F033F
EE001F030703FC1307DB007F02E01301040149CAFC5B5479D26A>I<B8D8C003B8FCA5D8
000701F8C9001FEBE000B3AE92BAFCA503F8C9121FB3B1B8D8C003B8FCA560527CD169>
I<B812C0A5D8000701F8C7FCB3B3B3B2B812C0A52A527CD132>I<027FB71280A591C76C
90C7FCB3B3B3EA07F0EA1FFC487E487EA2B57EA44C5AA34A485B7E49495BD83FF8495BD8
1FE05DD80FFC011F5B2707FF807F90C8FC000190B512FC6C6C14F0011F14C0010101F8C9
FC39537DD145>I<B812F8A5D8000701F8CAFCB3B3A91A7CA41AFC1AF8A51901A31903A2
19071AF0190FA2191F193F197F19FF180360183F4DB5FCBB12E0A546527CD151>76
D<B600FC073FB512FE6F61A26F96B6FCA2D80007F5C00070EF01EFA202EF6DEF03CFA202
E76DEF078FA202E36DEF0F0FA202E16D171EA302E06D173CA26F6C1778A26F6C17F0A26F
6DED01E0A26F6DED03C0A36F6DED0780A26F6DED0F00A26F6D151EA26F6D5DA3706C5DA2
706C5DA2706D495AA2706D495AA2706D495AA3706D49C7FCA2706D131EA2706D5BA2716C
5BA3716C5BA271EB81E0A271EBC3C0A271EBE780A27101FFC8FCA3715BA2715BA2725AA2
725AA2D93FFC6F5AB74DB712FEA2725AA2725A77527CD180>I<93380FFFC00303B6FC03
1F15E092B712FC0203D9FC0013FF020F01C0010F13C0023F90C7000313F0DA7FFC02007F
494848ED7FFE4901E0ED1FFF49496F7F49496F7F4990C96C7F49854948707F4948707FA2
4849717E48864A83481B804A83481BC0A2481BE04A83A2481BF0A348497113F8A5B51AFC
AF6C1BF86E5FA46C1BF0A26E5F6C1BE0A36C6D4D13C0A26C6D4D1380A26C1B006C6D4D5A
6E5E6C626D6C4C5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B90C7FC6D6D4B5A6D01
FF02035B023F01E0011F13F0020F01FC90B512C0020390B7C8FC020016FC031F15E00303
92C9FCDB001F13E0565479D265>79 D<BAFC19F819FF1AE086D8000701F0C7001F13FC06
0113FF726C13807313C0070F13E01BF0857313F81BFCA27313FEA41BFFA81BFEA31BFC61
A21BF84F13F04F13E0614F13C04F13004E485A061F5B92B812F01AC04FC7FC19E003F8CB
FCB3AEB812C0A550527CD15C>I<B912F0F0FF8019F819FF1AC0D8000701F0C714F0060F
7F060113FE727F737F737F85737F87A2737FA387A863A2616363A24F5B4F5B4F90C8FC4F
5A06035B060F13F095B512C092B8C9FC19F819E019F89226F0000313FE9439007FFF8072
7F727F727F727F727F8684A28684A787A71D1C75133EA38575137E73157C7513FC731401
B86C6D9038F803F807039038FE07F07390B512E0736C14C0080F1400CEEA7FFC5F537CD1
64>82 D<91260FFF80130791B500F85B010702FF5B011FEDC03F49EDF07F9026FFFC006D
5A4801E0EB0FFD4801800101B5FC4848C87E48488149150F001F824981123F4981007F82
A28412FF84A27FA26D82A27F7F6D93C7FC14C06C13F014FF15F86CECFF8016FC6CEDFFC0
17F06C16FC6C16FF6C17C06C836C836D826D82010F821303010082021F16801400030F15
C0ED007F040714E01600173F050F13F08383A200788200F882A3187FA27EA219E07EA26C
EFFFC0A27F6D4B13806D17006D5D01FC4B5A01FF4B5A02C04A5A02F8EC7FF0903B1FFFC0
03FFE0486C90B65AD8FC0393C7FC48C66C14FC48010F14F048D9007F90C8FC3C5479D24B
>I<003FBC1280A59126C0003F9038C0007F49C71607D87FF8060113C001E08449197F49
193F90C8171FA2007E1A0FA3007C1A07A500FC1BE0481A03A6C994C7FCB3B3AC91B912F0
A553517BD05E>I<B800C00103B612FCA5D8000701F8CAEBF000F31F80B3B3B11B3FA26D
97C7FC81637F1B7E6D6D17FE505A6E7E505A6E6D15076E4D5A6E6D4B5A6E6D4B5A6E01F8
4B5A6E6DDA03FFC8FC6E6CB46CEB0FFE6F9039F001FFF8030F90B65A030316C0DB007F92
C9FC040F14F8DC007F13805E537CD167>I<B700FC017FB600FE91B612F0A5D8003F01C0
C8001F01E0C9EBF8006F71EE0FC06D7161876F1C1F6D7196C7FC6F8373606D1E3E6F836D
7160876F1CFC6D666F4B801F016D66704A806E525A88704A17076E059F5F70021F80080F
160F6E6570023F806EDC3E074CC8FC8870027E5F6EDC7C03163E7002FC804F6C167E6E1C
7C700101814F6C16FC6E745B70010317016E4C6D5D060716C00580496D14036F63DDC00F
16E04F6D14076F07F05BDDE01F170F6F92C76C5D1DF8DDF03E6E141F6F98C9FCDDF87E16
FC067C6E5C6FF1FE3EDDFCFC177E6F4A6E147C1DFFDDFFF06E14FC6F62A24E816F62A270
496F5BA24E817061A295C97E7061A270487090CAFCA37048705AA24D1601040360A27048
705A84537DD18B>87 D<EC7FFF0107B512F0013F14FE90B77E48D9E00F7F2703FE000113
F0486C6D7F6EEB3FFC48826E131F83707FA36C496D7FA26C90C7FC6C5AC9FCA6037FB5FC
020FB6FC91B7FC01071487013FEBF0074913803901FFFC004813F0485B485B485B4890C7
FC5A5BA2485AA45EA26D5C007F151D163D6C6C02797F6C6D01F113F86C9026C003E1EBFF
E06C9026F81FC014F06C90B5487EC6ED001F011F01FC010713E0010101E090C8FC3C387C
B641>97 D<EB3FF0B5FCA51203C6FCB3A4923801FFE0030F13FE033FEBFFC092B612F002
F301017F913AF7F8003FFEDAFFE0EB0FFF03806D7F92C76C7F4A6E7F4A824A6E7FA2727E
A285A28584A31A80AC1A00A44E5AA36118FF616E4A5BA26E4A5B6E4A5B6F495BDACFC049
90C7FCDA87F0EB7FFC913A03FE03FFF849C6B612E0496D148049011F01FCC8FC90C70003
13C041547BD24B>I<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A
1FFE0001FFD97FFC491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F13
00705A4892C8FC5BA312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F
806C6DEC3F006C6D147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A0100158002
3F49C7FC020113E033387CB63C>I<4DB47E0407B5FCA5EE001F1707B3A4913801FFE002
1F13FC91B6FC010315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F484914
3F484980485B83485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E
6C6D5C6C6D49B5FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101
ECFE0FD9003F13F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC0107
15C04901817F903A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18
C0488191C7FC4817E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218
E06CEE01F06E14037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91F
FEEB03FE903A0FFFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>I<
ED3FFC0203B5FC020F14C0023F14E09139FFF81FF0499038C03FF849EB807F49903800FF
FC495A495AA2495AA2EE7FF8495AEE3FF0EE0FC093C7FCAEB712E0A526007FF8C8FCB3B3
A7007FB512FEA52E547CD329>I<DA3FFF14FF0103B5D8F00713C0010FDAFC1F13E0013F
ECFF7F90267FFC0F9038FF9FF09026FFE001EBF83F48496C13E0484990387FF01F4890C7
D83FF813E0489338FC0FC0F0078048486E6CC7FCA2003F82A9001F5EA26C6C4A5AA26C5E
6C6D495A6C6D495A6C6D485BDAFC0F5B4890B6C8FCD803EF14FC01C314F02607C03F90C9
FC91CBFCA2120FA37FA213F813FE90B7FC6C16F817FF18C06C836C836C836D828448B9FC
12074848C700031480D81FF8EC003F4848150748486F13C083485A83A56D5D007F18806D
5D003F18006C6C4B5AD80FFEED1FFC6C6C6CEC7FF86C01E049485A6C01FE011F5B6C6CB7
1280010F03FCC7FC010115E0D9000F01FCC8FC3C4F7CB543>I<EB3FF0B5FCA51203C6FC
B3A4EE1FFC93B512C0030314F0030F8092391FE07FFC92393F001FFE037C8003F07FDAF1
E081ECF3C0DAF7807F8502FFC7FC5CA25CA45CB3ACB6D8F807B612C0A542537BD24B>I<
137F497E000313E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512
017EB3B3A6B612E0A51B547BD325>I<EB3FF0B5FCA51203C6FCB3A54CB512F8A5933900
3FFE00EF1FF0EF3FC04D5A4DC7FCEE03FEEE07F84C5A4C5AEE7FC04CC8FC4B5A4B5AED0F
F8ED1FE04B7E4B7EECF1FF02F37F02F77F91B6FC83159F030F7F02FE80DAF8077F4A7E6F
7F6F7F83707E82707F84707F707F82707F84707F177F717E4D13C0B6D8F003B6FCA54053
7CD247>107 D<EB3FF0B5FCA512017EB3B3B3B1B612F0A51C537BD225>I<D93FF0D91FFC
EDFFE0B591B500C0010713FE030302F0011F6D7E030F6E017F8092271FE07FFCD9FF037F
922A3F001FFE01F8007F0003027C9126FF03E080C602F06DD90780137FDAF1E0038FC77F
DAF3C0159EDAF7806D01BC143F07FC8102FFC75C4A5EA24A5EA44A5EB3ACB6D8F807B6D8
C03FB512FEA567367BB570>I<D93FF0EB1FFCB591B512C0030314F0030F8092391FE07F
FC92393F001FFE0003027C80C602F07FDAF1E081ECF3C0DAF7807F8502FFC7FC5CA25CA4
5CB3ACB6D8F807B612C0A542367BB54B>I<913801FFE0021F13FE91B612C0010315F001
0F9038807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F48496D7F4A147F4883
4890C86C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A3003F18006D5DA2
6C5FA26C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE011F90C7FC903A
0FFF807FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I<903A3FF001FFE0
B5010F13FE033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9FFE0EB1FFFC602
806D7F92C76C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A0060A36118FFA2
615F616E4A5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FBFE075B02F8B612
E06F1480031F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I<912601FFE0EB07
80021F01F8130F91B500FE131F0103ECFF80010F9039F03FC03F499039800FE07F903A7F
FE0003F04948903801F8FF4849EB00FD4849147F4A805A4849805A4A805AA291C87E5AA3
5B12FFAC6C7EA37EA2806C5EA26C6D5CA26C6D5C6C6D5C6C93B5FC6C6D5B6D6C5B6DB4EB
0FEF010F9038C07FCF6D90B5120F010114FED9003F13F80203138091C8FCB1040FB61280
A5414D7CB547>I<90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F8
3FFC0003D9E3E013FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90
C7FCA55CB3AAB612FCA52F367CB537>I<903903FFF00F013FEBFE1F90B7FC120348EB00
3FD80FF81307D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEB
FFC06C13FF15F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714
E0EC003F030713F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6D
EC3F8001F8EC7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8
C7FC2C387CB635>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A00
0F90B6FCB8FCA426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E
6DEBF0FC6DEBFFF86D6C5B021F5B020313802A4D7ECB34>I<D93FF8913801FFC0B50207
B5FCA50003ED001FC61607B3AE5FA35FA2017F5D173B177B6D6C14F3DC01E313F06D6CD9
07C3EBFFC0903A0FFFC03F836D90B51203010114FE6D6C13F8020701E091C7FC42377BB5
4B>I<B600F00107B5FCA5000101F8C8EA7FE06C6DED3F00A2017F163E6E157E013F167C
6E15FC6D5E6F13016D5E8117036D5E6F13076D5E6F130F6D5E6F131F6D93C7FC815F6E6C
133E177E023F147C6F13FC6E5C16816E5C16C3A26EEBE3E016E76E5C16FF6E5CA26E91C8
FCA26F5AA36F5AA26F5AA26F5AA26F5A6F5A40367DB447>I<B6D8E07FB5D8C003B512C0
A5000101F0C701F0C7381FF8006E027FED07E06C715DA26E023F150F017F705DA26E181F
013F4B6C92C7FC6E606D70143E94B5FC6F177E6D4A6E137C03C001F315FC6D715B160303
E001E114016D020702E05B03F013C06D71485A160F03F8D9807F13076D05F85B93381F00
3F03FC160F027F4902FC5BDBFE3E011F131F023F04FE90C8FC167EDBFF7C010F5B6E01FC
ECFF3E4C6D137E6E5FA24C7F6E5F4C7F6E5FA24C7F6E5F4C147FA26E5F93C8123F6F5EA2
033E6FC9FC5A367DB461>I<007FB500F090387FFFFEA5C66C48C7000F90C7FC6D6CEC07
F86D6D5C6D6D495A6D4B5A6F495A6D6D91C8FC6D6D137E6D6D5B91387FFE014C5A6E6C48
5A6EEB8FE06EEBCFC06EEBFF806E91C9FCA26E5B6E5B6F7E6F7EA26F7F834B7F4B7F92B5
FCDA01FD7F03F87F4A486C7E4A486C7E020F7FDA1FC0804A486C7F4A486C7F02FE6D7F4A
6D7F495A49486D7F01076F7E49486E7E49486E7FEBFFF0B500FE49B612C0A542357EB447
>I<B600F00107B5FCA5C601F8C8EA7FE06EED3F00A26D6C153E187E013F167C6E15FC6D
5E6F13016D5E6F13036D5E8117076D6D5C170F6D6D5C171F6D93C7FC6F5B027F143E6F13
7E023F147C6F13FCA26E6D5A16816EEBC1F016C36E5C16E76E5C16FF6E5CA26E91C8FCA3
6F5AA26F5AA26F5AA26F5AA26F5AA35E150F5E151F93C9FC5DD81FC0133E486C137E486C
137C486C13FC5D14015D14034A5A6C48485A49485A263FC07FCAFCEB81FE6CB45A6C13F0
00035BC690CBFC404D7DB447>I E /FI 73 128 df<DC0FF0EB0F80DC7FFEEB3FE09226
01FC0FEBF878923B03F00381F03C923B07C007C3E07C923B0F801FC7E1FC031F013F13C3
18CFED3F001983069F13F8037E90390E1F80E005001400183F03FE92C7FC5DA360020115
7E5DA318FE02035D0103B9FCA26190290003F00001FCC7FC14074B5CA41703020F5D5DA3
1707021F5D5DA3170F023F5D92C7FCA3171F4A5D147EA3173F02FE92C8FC5CA35F010115
7E5CA35F495AA34C5A495AA2001E02F05B007F9038C3F803010F5D00FF018713075F0207
495AD8FE1F4948C9FCD8F81EEBC01E3A703C03803C3A787801E0F83A1FF000FFE0D807C0
EB3F80465383BF38>11 D<933807FF80043F13E09338FE00F8DB01F0133EDB07E0130E4B
48131F4C137F031F14FF4BC7FCA218FE157E1878180015FE5DA31401A25DA414030103B7
12F0A218E0903A0003F000070207140F4B14C0A3171F020F15805DA2173F1800141F5D5F
177EA2143F92C712FE5FA34A1301027EECF81CA3160302FEECF03C4A1538A21878187013
014A010113F018E0933800F1C0EF7F804948EC1F0094C7FCA35C1307A2001E5B127F130F
00FF5BA249CAFC12FEEAF81EEA703CEA7878EA1FF0EA07C0385383BF33>I<EE07FC9339
3FFF87F09338FC07C7923A03F001E7E0DB07C013F792390F8007FF031F4913C016005DA2
037E1580EF039FEF001F183F4B1500A3600201157E5DA218FE6014035D0103B7FC60A290
3A0007F000014B130360A31707020F5D5DA2170F60141F5D171F60A2143F92C7123F95C7
FCA34A5C027EEC7E07A317FE02FE4A5A4A150EA2181E181C13014AEC7C3C1838EF3C70EF
1FE04948EC07C094C8FCA3495AA3001E5BEA7F0FA200FF5BA249CBFC12FEEAF83EEA703C
EA7878EA1FF0EA07C03C5383BF35>I<DC07FCEC7FF893273FFF8003B5FC933CFC03C00F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>I<EB0380
EB07C0130F131F133FEB7F0013FE485A485AEA07E0485A485A48C7FC127C5A5A5A121162
BE2E>19 D<EA01E0EA07F8120F13FC121FA4120FEA03D8EA0018A2133813301370136013
E013C01201EA038013005A120E5A5A5A5A5A0E1C6DBE1C>39 D<ED01801507ED0F00151C
5D5D5D14014A5A4A5A4AC7FC141E143E5C14785C1301495AA2495A5C130F49C8FCA2133E
A25BA213FC5B12015BA212035B1207A25B120FA25BA2121FA290C9FCA25AA2123EA2127E
A2127CA65AAB1278A67EA47EA2120E120F7EA26C7EA26C7E6C7E1360215A73C325>I<14
031580A2EC01C0EC00E0A21570A215781538153CA3151EA4151FA2150FA7151FA9153FA2
153EA3157EA2157CA215FCA215F8A21401A215F0A2140315E0A2140715C0A2EC0F80A214
1F15005C143EA25CA25CA2495A5C1303495A5C130F49C7FC131E5B137C5B5B485A485A48
5A48C8FC121E5A12705A5A205A7FC325>I<EA01E0EA07F8120FA2EA1FFCA4EA0FF8EA07
98EA001813381330A21370136013E013C01201EA0380EA07001206120E5A5A5A5A5A0E1C
7A891C>44 D<387FFFFEA3B5FCA21705799521>I<120FEA3FC0127FA212FFA31380EA7F
00123C0A0A77891C>I<1838187CA218F8170118F0EF03E0A2EF07C0A2EF0F80171F1800
173EA25FA25F16015F4C5AA24C5AA24C5A161F94C7FC163EA25EA25E15015E4B5AA24B5A
A24B5A151F93C8FC153EA25DA25D14015D4A5AA24A5AA24A5A141F92C9FC143EA25CA25C
13015C495AA2495AA2495A131F91CAFC133EA25BA25B12015B485AA2485AA2485A121F90
CBFC123EA25AA25AA25A5A365B7FC32E>I<15FE913807FF8091381F07C091387C01F0EC
F000494813F8494813780107147C495A49C7FC167E133E137EA25BA2485AA2000315FEA2
5B000715FCA2491301120FA34848EB03F8A44848EB07F0A448C7EA0FE0A316C0007E141F
12FE1680153FA2481500A2157EA25DA25D4813015D6C495A127C4A5A4A5A6C49C7FC143E
6C5B380FC1F03803FFC0C648C8FC273F76BC2E>I<15031507150F151F151E153E157EEC
01FEEC03FC1407141FEB01FF90380FFBF8EB1FC3EB0E07130015F0A2140FA215E0A2141F
A215C0A2143FA21580A2147FA21500A25CA25CA21301A25CA21303A25CA21307A25CA213
0FA25CA2131FA25CEB7FE0B612F0A215E0203D77BC2E>I<15FE913803FFC091380F01F0
91383C00F84A137C4A7F4948133F49487F4A148049C7FC5BEB0E0C011E15C0EB1C0EEB3C
06133813781370020E133FD9F00C148013E0141C0218137F00011600EBC0384A13FEEC60
0102E05B3A00E3C003F89039FF0007F0013C495A90C7485A5E037FC7FC15FC4A5A4A5AEC
0FC04AC8FC147E14F8EB03E0495A011FC9FC133E49141801F0143C48481438485A167848
5A48C85A120E001E4A5AD83FE0130301FF495A397C3FF01FD8780FB55AD8700391C7FCD8
F0015B486C6C5A6E5AEC07C02A3F79BC2E>I<ED7F80913803FFE091380F80F891383C00
7C02F87FD901E07F494814804948130F49C7FC010E15C0131EEB1C18EB3C1CEB380C0178
141F17801370A2021C133F6D4814004A5BD91FE0137ED90F805B90C8FC4B5A4B5A4B5AED
1F8003FFC7FCECFFFC15F0A2EC00FC153E153F8182150F82A4151FA2121E127F153F485D
A3484AC7FC12F800E014FE5D14016C495A0070495A0078EB0FC00038495A6C017EC8FC38
0F01F83803FFE0C690C9FC2A3F78BC2E>I<1638167E16FE16FCA3150116F8A3150316F0
A2150716E0A2ED0FC0A3ED1F80A216005DA2157EA2157C15FC5D14015D14035D4A5AA24A
5AA24AC7FC143EED038091387C0FC014F8ECF01F01011480EB03E014C0903807803F010F
1400EB1F00133E495B49137E485A485A484813FE48B46C5A4813F04813FE267C00FF1308
00F090380FFFFC00601301C714E0913803F8005DA314075DA3140F5DA3141F5DA3020EC7
FC274F7DBC2E>I<02C0EB018002F0130FD901FEEB7F0091B512FE5E5E4914E016804BC7
FCECBFF8D90780C8FC91C9FCA35B130EA3131E131CA3133C9038381FC0ECFFF090383BE0
7C90387F003E017E133F017C7F0178805B498090C7FCA6153FA4001F147F486C5C487EA2
4913FF00FF92C7FC90C7FC48495A12E04A5A5D6C495A140F00705C0078495A6C495A003E
01FEC8FC381F03FC380FFFF0000313C0C648C9FC293F77BC2E>I<ED0FE0ED7FF8913801
F01C913807C00E91381F800F91383E00074A131F4A137F494813FF1303495A494813FE13
1F4948137891C8FC5B13FEA2485AA212035B0007EB1FC0EC7FF0390FF1E07C9038F3803E
EBF700D81FFE7F4914805B49EB0FC0123F5B151F4914E0127FA25BED3FC0A248C7FCA215
7F16805AA215FF1600A24A5AA2485C007C495AA2007E495A4A5A003E495A003F495A261F
807EC7FC380FC1FC6CB45A6C13E0C66CC8FC283F76BC2E>I<D9703FEB01C09138FF8003
01F3158001EFEBC00790B5EB0F0048151E14832603FE015B01F8147C2607F0005B49EBE3
F04848137F496D5A48C71201001E4A5A1507485D0038140F007892C7FC00705C00F0143E
5A5DC8FC5DA24A5AA214035D14074A5AA2141F5DA24AC8FCA25C147E14FEA2495AA3495A
A313075C130FA25C131FA35C133FA35C91C9FC131E2A3F73BC2E>I<157F913801FFE091
3807C0F091381F007C023C133C4A133E4A131F1301495A5C1307A2495AA2163F011F143E
A2167E6E137C16F8ECE00102F013F09138F803E09138FC07C090390FFE0F00ECFFBE6D13
F86D5B7F6D7F8101037F90380F9FFFD91F0F1380D97C0713C0497E48486C13E03903E000
7F4848133F4848131F001F140F90C7FC003E1407A2127E127CA200FC15C05AA2ED0F80A2
ED1F00153E007C143C157C007E5C6CEB03F0391F8007C0390FE03F802607FFFEC7FC0001
13F838003FC0283F78BC2E>I<15FF020713C091381F81E091383E00F002FC13F8494813
7C495A4948137E010F143E495A133F4A133F017F147F91C7FC5BA2485AA216FF12035B16
FE150112075B1503A216FC491307A20003140F16F8151F12016D133F0000EC7FF015EF90
387C01CF90393E079FE090380FFE1FD903F813C090C7123FA21680157F160015FEA24A5A
001C5C007F1303485C4A5A4A5A4A5A4849C7FC00F8137E00E05B6C485A387C07E0383FFF
C06C90C8FCEA03F8283F77BC2E>I<131EEB3F80137FEBFFC05AA214806C13005B133C90
C7FCB3120FEA3FC0127FA212FFA35B6CC7FC123C122777A61C>I<EB03C0EB07F0130FEB
1FF8133FA214F0EB1FE014C0EB078090C7FCB3EA01E0EA07F0487EA2121FA46C5AEA07B0
EA003013701360A213E05B12015B120348C7FC1206120E5A5A123012705A5A15397AA61C
>I<171C173C177CA217FCA216011603A21607A24C7EA2161DA216391679167116E1A2ED
01C1A2ED038115071601150EA2031C7FA24B7EA25D15F05D4A5AA24A5AA24AC7FC5C140E
5C021FB6FC4A81A20270C7127FA25C13015C495AA249C8FCA2130E131E131C133C5B01F8
82487ED807FEEC01FFB500E0017FEBFF80A25C39417BC044>65 D<49B712C018F818FE90
3B0003FC0001FF9438007F804BEC3FC0A2F01FE014074B15F0180FA2140F5D181FA2021F
16E05D183F19C0023FED7F804B14FF19004D5A027F4A5A92C7EA07F0EF1FE0EF7F804AD9
03FEC7FC92B512F017FE4AC7EA3F800101ED1FE04A6E7E17078401036F7E5CA30107825C
A3010F5E4A1407A260011F150F5C4D5A60013F153F4A4A5A4D5A017F4A90C7FC4C5A91C7
EA0FF849EC3FF0B812C094C8FC16F83C3E7BBD40>I<9339FF8001C0030F13E0033F9038
F803809239FF807E07913A03FC001F0FDA0FF0EB071FDA1FC0ECBF00DA7F806DB4FC4AC7
7E495AD903F86E5A495A130F4948157E4948157C495A13FF91C9FC4848167812035B1207
491670120FA2485A95C7FC485AA3127F5BA312FF5BA490CCFCA2170FA2170EA2171E171C
173C173817786C16706D15F04C5A003F5E6D1403001F4B5A6D4AC8FC000F151E6C6C5C6C
6C14F86C6C495A6C6CEB07C090397FC03F8090261FFFFEC9FC010713F0010013803A4272
BF41>I<49B712C018F818FE903B0003FE0003FF9438007F804BEC1FC0F00FE0F007F014
074BEC03F8F001FCA2140F4BEC00FEA3141F4B15FFA3143F5DA3027F5D5DA219FE14FF92
C81203A34917FC4A1507A219F813034A150F19F0A20107EE1FE05CF03FC0A2010FEE7F80
4A16006060011F4B5A4A4A5A4D5AA2013F4B5A4AEC3FC04DC7FC017F15FEEE03FC4AEB0F
F001FFEC7FE0B8128004FCC8FC16E0403E7BBD45>I<49B812F8A390260003FEC7121F18
074B14031801F000F014075DA3140F5D19E0A2141F4B1338A2EF7801023F027013C04B91
C7FCA217F0027F5CED80011603160F91B65AA3ED001F49EC07805CA3010392C8FC5CF003
804C13070107020E14005C93C75A180E010F161E4A151C183CA2011F5E5C60A2013F1501
4A4A5A1707017F150F4D5A4A147F01FF913807FF80B9FCA295C7FC3D3E7BBD3E>I<49B8
12F0A390260003FEC7123F180F4B1403A2F001E014075DA3140F5D19C0A2141F5D1770EF
F003023F02E013804B91C7FCA21601027F5CED8003A2160702FFEB1F8092B5FCA349D900
3FC8FC4A7F82A20103140E5CA2161E0107141C5CA293C9FC130F5CA3131F5CA3133F5CA2
137FA25C497EB612E0A33C3E7BBD3B>I<DCFF8013E0030713F0033F9038FC01C09239FF
C03E03913A03FC000F07DA0FF0EB078F4A48903803DF80DA7F80EB01FF4AC8FC495A4948
ED7F00495A495A4948814948153E495A13FF91C9FC4848163C12035B1207491638120FA2
485A95C7FC485AA3127F5BA312FF5BA34BB512FE90C7FCA292C71380A295C7FCA25EA25F
A216037E6D5DA2003F15077F001F5E6D140F6C6C141F0007153F6C6CEC7BF0D801FE14F1
6C6CEB03E090393FE01F806DB5EA0060010701F890C8FC9038007FC03B4273BF46>I<49
B648B6FC495DA2D9000390C7000313004B5D4B5DA2180714074B5DA2180F140F4B5DA218
1F141F4B5DA2183F143F4B5DA2187F147F4B5DA218FF91B8FC96C7FCA292C712015B4A5D
A2170313034A5DA2170713074A5DA2170F130F4A5DA2171F131F4A5DA2173F133F4A5DA2
017F157FA24A5D496C4A7EB66CB67EA3483E7BBD44>I<49B6FC5BA2D9000313005D5DA3
14075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C7FCA35B5CA313035CA313075C
A3130F5CA3131F5CA3133F5CA2137FA25C497EB67EA3283E7BBD23>I<4AB61280A21800
91C713C0167F5FA216FF94C7FCA35D5EA315035EA315075EA3150F5EA3151F5EA3153F5E
A3157FA25EA215FFA293C8FCA25CA25DA2380F8003EA3FC0D87FE05BA21407D8FFC05B14
0F01805B49485A12FC0070495A4A5A6C01FEC9FC383C01FC380F07F03807FFC0C648CAFC
314079BD30>I<49B6903807FFFE605ED9000390C7000113E04B6E13004B15FC4E5A19E0
02074B5A4BEC0F804EC7FC183C020F5D4B5C4D5AEF07C0021F4AC8FC4B131E5F5F023F5C
9238C003E0EE07804CC9FC027F5B4B5AEEFF801581ECFF834B7FED0F7FED1E3F49017C7F
ECFEF89138FFE01F03C07F491380ED000F4A805C010714074A80A21603010F815C160183
131F4A6D7FA2177F013F825C173F017F82A24A81496C4A7EB6D8800FB512C0A261473E7B
BD46>I<49B612C0A25FD9000390C8FC5D5DA314075DA3140F5DA3141F5DA3143F5DA314
7F5DA314FF92C9FCA35B5CA313035C18C0EF01E0010716C05C17031880130F4A14071800
5F131F4A141EA2173E013F5D4A14FC1601017F4A5A16074A131F01FFECFFF0B8FCA25F33
3E7BBD39>I<49B5933807FFFC496062D90003F0FC00505ADBBF805E1A771AEF1407033F
923801CFE0A2F1039F020FEE071F020E606F6C140E1A3F021E161C021C04385BA2F1707F
143C023804E090C7FCF001C0629126780FE0495A02705FF00700F00E0114F002E0031C5B
A2F03803010116704A6C6C5D18E019070103ED01C00280DA03805BA2943807000F130702
00020E5C5FDB03F8141F495D010E4B5CA24D133F131E011CDAF9C05CEEFB80197F013C6D
B4C7FC013895C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F04948B512FE16E01500
563E7BBD52>I<902601FFFE020FB5FC496D5CA2D900016D010013C04AEE3F00193E7014
1C193CEC07BFDB3FE01438151F1978020F7FDA0E0F15708219F0EC1E07021C6D5CA20303
1401023C7FDA38015DA2701303EC7800027002805BA2047F130702F014C04A013F91C7FC
A2715A0101141F4AECF00EA2040F131E010315F84A151C1607EFFC3C0107140391C71438
17FE040113784915FF010E16708218F0131E011C6F5AA2173F133C01385E171F137813F8
486C6F5AEA07FEB500F01407A295C8FC483E7BBD44>I<EEFFC0030713F892383F80FE92
38FC003FDA03F0EB0F804A486D7EDA1F80804AC76C7E027E6E7E4A814948140049488113
07495A4948157F133F5C49C9FC4917805B1201485AA212075B000F17FFA25B121F190048
485DA448484B5AA34D5AA25B4D5A12FF60171F60007F163F604D5AA24DC7FC5F003F1501
4C5A6D5D001F4B5A4C5A6C6C4A5A4C5A6C6C4AC8FC000315FC6C6C495A6C6CEB07E0017F
EB1F8090261FC07EC9FC903807FFF801001380394273BF46>I<49B77E18F018FC903B00
03FE0003FEEF00FF4BEC7F80F03FC00207151F19E05DA2020F16F0A25DA2141FF03FE05D
A2023F16C0187F4B1580A2027FEDFF00604B495A4D5A02FF4A5A4D5A92C7EA3FC04CB4C7
FC4990B512FC17E04ACAFCA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C
A2137FA25C497EB67EA33C3E7BBD3E>I<49B612FCEFFF8018F0903B0003FE000FF8EF03
FE4BEB00FF8419800207ED3FC05DA219E0140F5DA3021FED7FC05DA2F0FF80143F4B1500
4D5A60027F4A5A4B495A4D5AEF3F8002FF02FEC7FC92380007F892B512E0178049903800
0FE04A6D7E707E707E0103814A130083A213075CA25E130F5C5F1603131F5CA3013F0207
14404A16E05F017F160119C04A01031303496C1680B6D8800113079438FE0F009338007E
1ECAEA3FFCEF07F03B407BBD42>82 D<92390FF001C0ED7FFE4AB5EA0380913907F80FC7
91390FC003EF91391F8001FF4AC71300027E805C495A4948143EA2495AA2010F153C5CA3
011F1538A38094C7FC80A214FC6DB4FC15F015FE6DEBFFC06D14F06D14FC6D80143F020F
7F020180EC001F150303007F167F163FA2161FA212075A5F120EA2001E153F94C7FCA216
3E003E157E167C003F15FC4B5A486C5C4B5A6D495AD87DE0EB1F80D8F8F849C8FC017F13
FE39F03FFFF8D8E00F13E048C690C9FC32427ABF33>I<48B9FCA25A903AFE001FF00101
F89138E0007FD807E0163E49013F141E5B48C75BA2001E147FA2001C4B131C123C003814
FFA2007892C7FC12704A153C00F01738485CC716001403A25DA21407A25DA2140FA25DA2
141FA25DA2143FA25DA2147FA25DA214FFA292C9FCA25BA25CA21303A25CEB0FFE003FB6
7E5AA2383D71BC41>I<001FB500F090B512F0485DA226003FF0C7380FFC004AEC03F04A
5D715A017F1503A24A5DA201FF150795C7FC91C8FCA2485E170E5BA20003161E171C5BA2
0007163C17385BA2000F167817705BA2001F16F05F5BA2003F1501A2495DA2007F1503A2
495DA2160794C8FC48C8FC5E160E161E6C151C163C5E5E5E6C6C13014B5A001F4A5A6C6C
011FC9FC6D133E6C6C13F83903FC07F0C6B512C0013F90CAFCEB07F83C406FBD44>I<B5
00FE91387FFFE094B5FC19C00003018091380FFC0049C8EA07F000015F606095C7FC170E
A25F173C17386D5DA26C5E16015F4C5AA24CC8FC5E160E5E805E137F5E5EA24B5AA24B5A
150793C9FCECC00EA2013F5B153C15385DA25D14C15DECC38014E302E7CAFCEB1FEF14EE
14FCA25CA25CA25C5C130F5CA291CBFC130E3B406DBD44>I<277FFFFE01B500FC90B512
E0B5FCA20003902680000790C7380FFC006C90C701FCEC07F049725A04035EA26350C7FC
A20407150EA2040F5D1A3C041F153862163B6216734F5A6D14E303014B5A6C15C303034B
C8FC1683DB0703140E191E030E151C61031C7F61ED380161157003F04A5A15E002014B5A
15C0DA03804AC9FC60DA0700140E60140E605C029C5D14B8D97FF85D5C715A5C4A5DA24A
92CAFC5F91C7FC705A137E5F137C5F137801705D53406EBD5B>I<147E49B47E903907C1
C38090391F80EFC090383F00FF017E137F4914804848133F485AA248481400120F5B001F
5C157E485AA215FE007F5C90C7FCA21401485C5AA21403EDF0385AA21407EDE078020F13
70127C021F13F0007E013F13E0003E137FECF3E1261F01E313C03A0F8781E3803A03FF00
FF00D800FC133E252977A72E>97 D<EB1FC0EA0FFF5CA2EA003FA291C7FCA25BA2137EA2
13FEA25BA21201A25BA21203A25B147E3907F1FF809038F783E09038EF01F013FE390FF8
00F8A24913FC49137C485A157E5B15FE123FA290C7FCA248130115FC127EA2140300FE14
F85AA2EC07F0A215E048130F15C0141F15800078EB3F00127C147E003C5B383E01F8381E
03E06C485A6CB4C7FCEA01F81F4076BE2A>I<EC1FE0ECFFF8903803F03E903807C00F90
381F8007D93F001380017E131F49137F485A485A000715005B000F147E484890C7FCA248
5AA3127F90C9FCA35A5AA6481403007E5C5D151E003E5C5D6C5CEC03E0390F800F802603
E07EC7FC3801FFF838003FC0212977A72A>I<EE3F80ED1FFF1700A2ED007FA2167EA216
FEA25EA21501A25EA21503A25EA21507A25E147E903801FF8F903807C1CF90391F80EFC0
90383F00FF017E137F5B48486D5A485AA2485A000F92C7FC5B001F5CA24848137EA215FE
127F90C75AA214015A485CA2140316384814F0A21407167891380FE070127C021F13F000
7E013F5B003E137FECF3E1261F01E35B3A0F8781E3802703FF00FFC7FCD800FC133E2940
77BE2E>I<EC3F80903801FFE0903807E0F890381F803CEB3E0001FC131E485A485A1207
4848133E49133C121F4848137C15F8EC03F0397F000FE0ECFF80B5EAFC0014C048C8FCA4
5AA61506150E151E007C143C15786C14F0EC01E06CEB07C0390F801F003807C0FC3801FF
F038007F801F2976A72A>I<167C4BB4FC923807C78092380F83C0ED1F87161FED3F3FA2
157EA21780EE0E004BC7FCA414015DA414035DA30103B512F8A390260007E0C7FCA3140F
5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495AA4495A121E127F
5C12FF49C9FCA2EAFE1EEAF83C1270EA7878EA3FE0EA0F802A5383BF1C>I<EC03F0EC0F
FC91383E0E1C9138FC077E903901F003FE1303903807E001D90FC013FCEB1F80A2EB3F00
4914F8137E01FE1303A2484814F0A2150712034914E0A2150F12074914C0A2151FA21680
5B153F1203ED7F006D5BA200015B0000495A9038F80F7E90387C1EFEEB1FF8903807E0FC
90C7FC1401A25DA21403A25D001C1307007F5C48130F5D4A5A4AC7FC48137E00F85B387C
03F0381FFFC0D803FEC8FC273B7CA72A>I<EB01FC13FF5CA21303A25CA21307A25CA213
0FA25CA2131FA25CA2133FA291C8FCEC03F890387F0FFE91383E0F80D97E7813C0ECE007
D9FFC013E014801400A2485A5BA25B0003140F16C05BA20007141F16805BA2000F143F16
005B5D001F147EEDFE074913FCA2003F0101130FEDF80E1300161E48ECF01CA2007E1538
A200FE1570020013E048EC7FC00038EC1F0028407ABE2E>I<1478EB01FCA21303A314F8
EB00E01400AD137C48B4FC38038F80EA0707000E13C0121E121CEA3C0F1238A2EA781F00
701380A2EAF03F140012005B137E13FE5BA212015BA212035B1438120713E0000F1378EB
C070A214F0EB80E0A2EB81C01383148038078700EA03FEEA00F8163E79BC1C>I<1507ED
1FC0A2153FA31680ED0E0092C7FCADEC07C0EC3FF0EC78F8ECE07CEB01C01303EC807EEB
0700A2010E13FE5D131E131CEB3C01A201005BA21403A25DA21407A25DA2140FA25DA214
1FA25DA2143FA292C7FCA25CA2147EA214FEA25CA213015CA2121C387F03F012FF495A5C
495A4848C8FCEAF83EEA707CEA3FF0EA0FC0225083BC1C>I<EB01FC13FF5CA21303A25C
A21307A25CA2130FA25CA2131FA25CA2133FA291C8FCED03E049EB0FF8ED3C3C017EEB70
7CEDE1FC9038FE01C1EC03839038FC0703140E0001011C13F891383800E0494813001460
000313E0EBF9C0EBF78001FEC8FC1207EBFFE0EBE7F8EBE0FE000F137F6E7EEBC01F6E7E
121F16701380A2003F15F0021F13E001001380A248148116C0007EEB0F83168000FE1487
9138078F0048EB03FE0038EB00F826407ABE2A>I<EB07F0EA03FF14E0A2EA000FA214C0
A2131FA21480A2133FA21400A25BA2137EA213FEA25BA21201A25BA21203A25BA21207A2
5BA2120FA25BA2121FA25BA2123FA290C7FCA25A1307127EA2EAFE0F130E12FCA2131E13
1CA2EA7C381378EA3C70EA1FE0EA0780144079BE17>I<D801F0D93F80137F3D07FC01FF
E003FFC03D0F3E07C1F80F83F03D0E1F0F00FC1E01F8001E011C90387C3800001C49D97E
707F003C01F05C0038157F4A5C26783FC05C12704A91C7FC91C7127E00F003FE1301494A
5CEA007EA20301140301FE5F495CA203031407000160495C180F03075D0003051F13E049
4A1480A2030FEC3F810007F001C0495CA2031F91383E0380120F494AEC0700A2033F150E
001FEF1E1C4991C7EA0FF80007C7000EEC03E0432979A74A>I<D801F0EB3F803A07FC01
FFE03A0F3E07C1F83A0E1F0F00FC001E011C137C001C49137E003C13F012385C38783FC0
12705C91C7FC00F015FE495CEA007EA2150101FE5C5BA2150300015D5B15075E0003020F
13704914C0A2031F13F00007ED80E05B1681EE01C0120F49EC0380A2EE0700001FEC0F0E
49EB07FC0007C7EA01F02C2979A733>I<EC1FC0ECFFF8903803F07C90380FC01FEB1F80
90393F000F80017E14C0491307484814E0485A12075B000F15F0485AA2485AA2ED0FE012
7F90C7FCA2151F4815C05AA2ED3F80A2ED7F00A248147E007C5C007E13015D4A5A003E49
5A6C495A4A5A260F803EC7FC3807C0FC3801FFF038003F80242977A72E>I<903903E001
F890390FF807FE903A1E7C1E0F80903A1C3E3C07C0013C137801389038E003E0EB783F01
7001C013F0ED80019038F07F0001E015F8147E1603000113FEA2C75AA20101140717F05C
A20103140F17E05CA20107EC1FC0A24A1480163F010F15005E167E5E131F4B5A6E485A4B
5A90393FB80F80DA9C1FC7FCEC0FFCEC03E049C9FCA2137EA213FEA25BA21201A25BA212
03A2387FFFE0B5FCA22D3A80A72E>I<027E1360903901FF81E0903807C1C390391F80E7
C090383F00F7017E137F5B4848EB3F80485AA2485A000F15005B121F5D4848137EA3007F
14FE90C75AA3481301485CA31403485CA314074A5A127C141F007E133F003E495A14FF38
1F01EF380F879F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5CA21301130390
B512F05AA2233A77A72A>I<D801F013FC3A07FC07FF803A0F3E0F03C0260E1F1C13E000
1EEB380F001C1370003CEBE01F123814C0D8783F14C00070903880070092C7FC91C8FC12
F05BEA007EA313FE5BA312015BA312035BA312075BA3120F5BA3121F5B0007C9FC232979
A726>I<EC7F80903801FFE0903807C0F890381F003C013E131C013C131E017C133E4913
7E15FEA2000114FCA215706D13007FEBFFC014FC6C13FF15806D13C06D13E0010F13F013
00140F14071403120C123F387F80011403D8FF0013E0A300FCEB07C000F0EB0F80127000
78EB1F006C133C381F01F83807FFE0C690C7FC1F297AA725>I<EB01C0EB03F01307A25C
A2130FA25CA2131FA25CA2133FA291C7FCA2007FB51280B6FC1500D8007EC7FC13FEA25B
A21201A25BA21203A25BA21207A25BA2120FA25BA2121F141C1380A2003F133C1438EB00
78147014F05C495AEA1F03495A6C48C7FCEA07FCEA01F0193A78B81E>I<137C48B4141C
26038F80137EEA0707000E7F001E15FE121CD83C0F5C12381501EA781F007001805BA2D8
F03F1303140000005D5B017E1307A201FE5C5B150F1201495CA2151F0003EDC1C0491481
A2153F1683EE0380A2ED7F07000102FF13005C01F8EBDF0F00009038079F0E90397C0F0F
1C90391FFC07F8903907F001F02A2979A731>I<017CEB01C048B4EB07F038038F80EA07
07000E01C013F8121E001C1403EA3C0F0038EC01F0A2D8781F130000705BA2EAF03F91C7
12E012005B017E130116C013FE5B1503000115805BA2ED07001203495B150EA25DA25D15
78000114706D5B0000495A6D485AD97E0FC7FCEB1FFEEB03F0252979A72A>I<017C1670
48B491387001FC3A038F8001F8EA0707000E01C015FE001E1403001CEDF000EA3C0F0038
177C1507D8781F4A133C00701380A2D8F03F130F020049133812005B017E011F14784C13
7013FE5B033F14F0000192C712E05BA2170100034A14C049137E17031880A2EF070015FE
170E00010101141E01F86D131C0000D9039F5BD9FC076D5A903A3E0F07C1E0903A1FFC03
FFC0902703F0007FC7FC372979A73C>I<903903F001F890390FFC07FE90393C1E0E0F90
26780F1C138001F0EBB83FD801E013F89039C007F07FEA0380000714E0D9000F14004815
1C000E4AC7FCA2001E131FA2C75BA2143F92C8FCA35C147EA314FE4A131CA30101143C00
1E1538003F491378D87F811470018314F000FF5D9039077801C039FE0F7C033A7C0E3C07
8027783C1E1EC7FC391FF80FFC3907E003F029297CA72A>I<137C48B4143826038F8013
FCEA0707000E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F13
07020013E012005B017E130F16C013FE5B151F1201491480A2153F000315005BA25D157E
A315FE5D00011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA21407000E5C
EA3F80007F495AA24A5AD8FF0090C7FC143E007C137E00705B387801F0383803E0381E0F
C06CB4C8FCEA03F8263B79A72C>I<B8FCA2280278982E>123 D<000E131E383F807F007F
EBFF8012FFA215005B007E5B003C1338190968BD2E>127 D E /FJ
23 121 df<EE3FFF0307B512F8033F14FF4AB712E0020716F8021F16FE4AD9F8077F91B5
D8C00014C04991C7003F7F4901FC020F7F49496E7F49496E7F49496E7F49496E7F4B8149
8590B5C96C7FA24849707FA24886A248864A824886A34886A448864A82A4481B80A8B51A
C0B3AA6C1B80A86C1B006E5EA46C62A36C62A36C6D4C5BA36C62A26C6D4C5BA26C6E4B5B
6D616F92B5FC6D96C7FC6D6D4A5B6D6D4A5B6D6D4A5B6D6D4A5B6D01FF023F5B6D02C090
B55A6ED9F8075C021F90B648C8FC020716F8020116E06E6C1580030702F8C9FCDB003F90
CAFC527379F061>48 D<EE01F0EE07F8160F163F167FED01FF150F153F4AB5FC143F010F
B6FCB8FCA54A7E14C0EBF000C8FCB3B3B3B3AE007FBA12F0A8447171F061>I<92380FFF
E04AB67E020F15F0027F15FE49B87E4917E0010F17F8013F8349D9C01F14FF9027FFFC00
01814801E06D6C80480180021F804890C86C8048486F8048486F8001FF6F804801C06E80
02F081486D18806E816E18C0B5821BE06E81A37214F0A56C5BA36C5B6C5B6C5B000313C0
C690C9FC90CA15E060A34E14C0A21B80601B0060626295B55A5F624D5C624D5C4D91C7FC
614D5B4D13F04D5B6194B55A4C49C8FC4C5B4C5B4C13E04C5B604C90C9FCEE7FFC4C5A4B
5B4B5B4B0180EC0FF04B90C8FC4B5A4B5A4B48ED1FE0EDFFE04A5B4A5B4A90C9FC4A4816
3F4A5ADA3FF017C05D4A48167F4A5A4990CA12FFD903FC160749BAFC5B4919805B5B90BB
FC5A5A5A5A481A005A5ABCFCA462A44C7176F061>I<923801FFFE033FEBFFF84AB7FC02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>I<F10FF04F7E193F
A2197F19FF60A260606060A2606095B5FCA25F5F5FA25F5F5F5F18BFEFFF3F5EEE03FE17
FCEE07F8160FEE1FF0EE3FE017C0167FEEFF804B13005E4B5A15074B5A4B5A5E153F4B5A
4B5A93C7FC4A5A14034A5A5D4A5A141F4A5A4A5A5D4AC8FC5B495A5C495A130F495A495A
5C137F495A4890C9FC5B485A1207485A485A5B123F485A485A90BC12FCA8CB02F8C7FCB3
A20307B912FCA856727BF161>I<0170187001FEEF01F86D6C160F02F8167FDAFF80EC07
FF03FE49B5FC92B85A6262A26297C7FC61616119E061614EC8FC18F86018C095C9FC17F8
17C0020701F8CAFC91CDFCB0923801FFFC031FEBFFE092B612FC020315FF020F16C04A16
F0027FD9003F7FDAFFF0010F13FE038001037F4AC76C8002F86E804A6F7F4A6F7F4A834A
6F7F91C980137E017C707F90CAFC1B80A21BC0A2841BE0A51BF0A313FE3803FF80000F7F
4813F0487F5A80B5FCA41BE0A44E14C05C7E4A18805C4A5D6C90C9150001E0606C6C5E6D
606C6C4C5B7F000794B55A6C6C6C4A5C6C6D4A5C6E4A5C26007FF8021F49C7FC6DB4027F
5B6DD9F007B55A6D90B712E0010317806D4CC8FC6D6C15F8021F15C002034AC9FCDA003F
13804C7376F061>I<94381FFF800403B512F8043F14FE4BB77E030782031F16F0037F82
92B5D8FC017F02039139C0001FFE4A49C7EA07FF021F01F8804A496E13804A01C0140F91
B548023F13C04991C85A494992B5FC49494A14E0495B495E5D5B495BA290B55A5A5D4871
14C0A24891C91480731300735A48F00FF896C8FC485BA45AA44849903807FFE0041F13FE
047FEBFFC04BB612F84B81030F15FFB590261FF8038092273FC0007F13E04C011F7F037E
C76C7F4B6E7F02FD6F7F4B6E7FDAFFF017804B6E14C01BE05D7313F05D1BF8A292C914FC
85A24A18FEA41BFFA26C5BA87EA4807EA21BFE7EA37E1BFC6E5E6C1AF8A27E6F17F06C95
B512E06D7F1BC06D6D4A14806D4C1400816D6D4A5B6D6D4A5B6D01FF4A13F001006E017F
5B6ED9F007B55A6E90B7C7FC020F5E020316F86E16E0DA003F1580030702FCC8FCDB007F
1380507378F061>I<EA03FCA2487E7F14C0ECFFF092BA12C0A45AA31C801C0063A24862
63636363A26398C7FC48616249CAEA0FF801F0171F494D5A4F5A49604F5A007F4D90C8FC
60494C5A4E5A614E5A4E5A48CA127F4E5A4D5B96C9FCCA485A4D5A170F4D5A60173F4D5A
6017FF4C5BA25E4C90CAFCA24C5A161FA24C5AA2167FA24C5AA25DA24B5BA25DA25DA25F
5DA25DA35DA392B5FCA25FA25CA45CA75CAD6E5CA26E91CBFCA26E5BED3FF8ED0FE05277
75F461>I<93B57E031F14FC92B77E020316F0020F16FC023F16FF4A8349B5D8800314E0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>I<93B5FC031F14F092
B612FE02076F7E021F16E04A16F891B87E49DAF00713FF0107DA0001804901FC6D6C7F49
496E7F49496E7F49496E7F90B5486E7F484A8048854891C86C7FA2487114805C481AC0A2
487213E0A2484918F0A31BF8A2B5FCA27313FCA51BFEA71BFF61A27EA46C95B6FCA2806C
5FA27E606C7F607E6C6E5C6CEF1FBF6D6DEC3F3F6D7F6D6D14FE6D6DEB01FC6D01FE1307
01019039FFC01FF86D91B500F014FE023F15C06E15800203ECFE00DA007F13F8030713C0
92C9FC4F13FCA41BF8A31BF0D91FF093B5FCEB7FFC496C18E0487F486E17C06048801B80
4E1400A26260624E5B4B5C626C91C8485B4A4B5B4A92B55A6C01F04A91C7FC02804A5B6C
01E0020F5B6D6C023F13F002FE91B55A90273FFFE00F5C6D90B7C8FC010716FC6D16F001
0016C0023F92C9FC020714F09126007FFECAFC507378F061>I<F103F84F7E4F7EA24F7E
A34F7FA24F7FA396B57EA24E80A34E80A24E80A34E80A24E80A34E81A24E81A219BFDEFF
9F80191F4D6D80A218FE05036D8018FC05076D80A218F8050F6D8018F0051F6D80A26005
3F6E8060057F6E80A26005FF6E8095C7FC4C6F80A25F04036F805F04076F80A25F040F6F
805F041F6F80A25F043F70805F047F7080A25F04FF708094C9FC4B7180A25E030371805E
4BBB7EA34B86A24B86A3DB3FE0CA6C805E037F7280A25E03FF7280A24A90CB6C80A25D02
037380A24A487280A25D020F7380A24B84021F885D023F7480A24B85027F895D902607FF
FC7380B86C031FB912E0A8837979F892>65 D<BDFC1CFEF4FFC01DF81DFF1EC01EF08AC7
003F49C9000F14FE090180756C800A1F807680768076807680A27680A2777FA2208089A3
20C0A289A565A22080A4531400A29AB55AA2525C6764525C525C525C525C5249C7FC51B5
5A090714F0093F14C00807B6C8FC93BA12F81DC0651DFCF5FF801EF04CCA14FC0A3F13FF
0A0F800A0314E076807614FC777F777F2080897714C020E0A27714F0A220F88920FCA477
14FEA96520FCA45314F8A26520F06520E05314C0659AB61280521500525C1C0F5214F899
B65A09075DC05A9CC7FC1EFC1EF01EC053C8FC1DE00AF8C9FC777679F58A>I<96267FFF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>I<92383FFF
F80207B612E0027F15FC49B87E010717E0011F83499026F0007F13FC4948C7000F7F90B5
02036D7E486E6D806F6D80727F486E6E7F8486727FA28684A26C5C72806C5C6D90C8FC6D
5AEB0FF8EB03E090CAFCA70507B6FC041FB7FC0303B8FC157F0203B9FC021FECFE0391B6
12800103ECF800010F14C04991C7FC017F13FC90B512F04814C0485C4891C8FC485B5A48
5B5C5A5CA2B5FC5CA360A36E5DA26C5F6E5D187E6C6D846E4A48806C6D4A4814FC6C6ED9
0FF0ECFFFC6C02E090263FE07F14FE00019139FC03FFC06C91B6487E013F4B487E010F4B
1307010303F01301D9003F0280D9003F13FC020101F8CBFC57507ACE5E>97
D<97380FFFE00607B6FCA8F00003190086B3AD93383FFF800307B512F8033F14FF4AB712
C0020716F0021F16FC027F9039FE007FFE91B500F0EB0FFF01030280010190B5FC4949C8
7E49498149498149498149498190B548814884484A8192CAFC5AA2485BA25A5C5AA35A5C
A4B5FCAF7EA4807EA37EA2807EA26C7F616C6E5D6C606C80616D6D5D6D6D5D6D6D92B67E
6D6D4A15FC010301FF0207EDFFFE6D02C0EB3FFE6D6C9039FC01FFF86E90B65A020F16C0
02031600DA007F14FC030F14E09226007FFEC749C7FC5F797AF76C>100
D<93387FFF80030FB512FC037FECFF804AB712E0020716F8021F16FE027FD9F8077F49B5
D8C000804991C7003F13E04901FC020F7F49496E7F49498049496E7F49496E7F90B55A48
727E92C914804884485B1BC048841BE0485BA27313F05AA25C5AA21BF885A2B5FCA391BA
FCA41BF002F8CCFCA67EA3807EA47E806CF103F0F207F86C7F1A0F6C6E17F06C191F6F17
E06C6E163F6D6DEE7FC06D6D16FF6D6D4B13806D6D4B13006D6D6CEC0FFE6D02E0EC3FFC
6D02F8ECFFF86D9126FFC00F5B023F91B65A020F178002034CC7FC020016F8031F15E003
0392C8FCDB000F13E04D507BCE58>I<903801FFFCB6FCA8C67E131F7FB3AD95380FFFE0
95B512FE05036E7E050F15E0053F15F84D81932701FFF01F7F4CD900077FDC07FC6D80DC
0FF06D80DC1FC07F4C48824CC8FC047E6F7F5EEDFDF85E03FF707F5EA25EA25EA293C9FC
A45DB3B3A6B8D8E003B81280A8617879F76C>104 D<EB01FCEB07FF011F13C0497F497F
90B57EA24880A24880A76C5CA26C5CA26D5B6D5B6D5B010790C8FCEB01FC90CAFCB29038
01FFFC007FB5FCA8C67E131F7FB3B3B3A5B81280A8297979F835>I<902601FFF891380F
FFE0B692B512FE05036E7E050F15E0053F15F84D81932701FFF01F7F4CD900077FDC07FC
6D80C66CDA0FF06D80011FDA1FC07F6D4A48824CC8FC047E6F7F5EEDF9F85E03FB707F5E
15FF5EA25EA293C9FCA45DB3B3A6B8D8E003B81280A8614E79CD6C>110
D<902601FFFCEC7FFEB6020FB512F0057F14FE4CB712C0040716F0041F82047F16FE93B5
C66C7F92B500F0010F14C0C66C0380010380011F4AC76C806D4A6E8004F06F7F4C6F7F4C
6F7F4C8193C915804B7014C0861DE0A27414F0A27414F8A47513FCA57513FEAF5113FCA5
98B512F8A31DF0621DE0621DC0621D806F5E701800704B5B505B704B5B7092B55A04FC4A
5C704A5C706C010F5C05E0013F49C7FC9227FE7FFC01B55A70B712F0040F16C0040393C8
FC040015F8053F14C0050301F0C9FC94CCFCB3A6B812E0A85F6F7ACD6C>112
D<902601FFF8EB07FEB691383FFFC094B512F00403804C14FE4C8093261FFC3F13809326
3FE07F13C0DC7F80B5FCC66C5D011FDAFE0114E06DEBF9FC16F815FB16F016E015FF16C0
7114C05E72138095381FFE0093C76C5AF001E095C8FCA25DA65DB3B3A2B812F8A8434E7A
CD4F>114 D<15FFA75CA55CA45CA25CA25CA25CA25C91B5FCA25B5B5B131F5B90B9FC12
0FBAFCA6D8000791C9FCB3B3A3F01FE0AE183F7014C07F187F7014806D16FF826D4B1300
6E6D485AEEFE0F6E90B55A020F5D6E5D020115C06E6C5C031F49C7FC030113F03B6E7CEC
4B>116 D<007FB7023FB612F0A8D8000302C0020191C7FC6D6E9138007FF0705E6D4E5A
6E6D4A5B6E6D4A90C8FC6E6D5C704A5A6E4C5A6E6E5C6E6E495A6E6E495A7113FF6E6E48
5B6F4A5B6F6D4890C9FC6F01FE5B71485A6FEC9FF86F14BF6FECFFF06F5D616F5D7091CA
FC705B828470808270807080854C805E4C80854C804C81EE7FE7DCFFE3804B01C1804B01
80804D804B487F4B486D7F031F6E7F4B486D7F4B48824B487F4C6D804A496D804A90C880
4A844A48814A486F7F4A486F7F4B6F7F4A48844A486F80010F01F881B76C91B712FEA85F
4D7DCC66>120 D E /FK 24 107 df<B812C0A32A037A9137>0 D<123C127E12FFA4127E
123C08087A9414>I<006015C000E014016C14030078EC07806CEC0F006C141E6C5C6C6C
5B6C6C5B6C6C485A6C6C485A90387807806D48C7FCEB1E1E6D5AEB07F86D5A6D5A497E49
7EEB0F3CEB1E1E497E496C7E496C7E48486C7E48486C7E4848137848C77E001E80488048
EC078048EC03C048140100601400222376A137>I<130C131EA50060EB01800078130739
FC0C0FC0007FEB3F80393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FF
E03807CCF8383F8C7F397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C
1A1D7C9E23>I<EE01C01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0
EC1FC0027FC8FCEB01FCEB07F0EB1FC0017FC9FCEA01FCEA07F0EA1FC0007FCAFC12FCA2
127FEA1FC0EA07F0EA01FCEA007FEB1FC0EB07F0EB01FCEB007FEC1FC0EC07F0EC01FCEC
007FED1FC0ED07F0ED01FCED007FEE1FC01607160193C7FCAC007FB71280B812C0A22A39
7AAB37>20 D<01FE150C3803FFC0000F7F4813F8D983FC141C383E007E00786D6C133800
70EB0FC0DA07F0137848903903F801F0913900FF07E048EC7FFF031F13C06F1300ED01FC
CBFCA201FE150C3803FFC0000F7F4813F8D983FC141C383E007E00786D6C13380070EB0F
C0DA07F0137848903903F801F0913900FF07E048EC7FFF031F13C06F1300ED01FC2E207C
A137>25 D<12C07EA37E1270A212787E121C121E7E6C7EEA03E06C7EEA00FCEB3F80EB1F
F0EB07FF010013FE021FB512C01403143F902601FFFCC7FCD90FFEC8FCEB3FE049C9FCEA
01F8EA03E0485A48CAFC121E5A123812781270A212F05AA42A2A7AA437>31
D<EB01C0A313035C130791CBFC5B130E131E5B5B13F8485AEA03C0EA0F80007FB912FCBA
FC7ED80F80CBFCEA03C0EA01F06C7E13787F7F130E130F7F801303801301A33E237CA147
>I<170EA3170F8384170384170184717E1878187C84180FF007C0BA12F819FC19F8CBEA
07C0F00F00183E601878604D5A60170360170795C7FC5F170EA33E237CA147>I<01FE15
0C3803FF804813E0487F381F83FC263E00FE141C003C133F486D6C13380070D907E01378
486D6C13F0913801FC0148903900FF07E092383FFFC06F138003071300ED01FCCBFCAD00
7FB712FCB8FCA22E207C9F37>39 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0F
E0A313C0121F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48
D<B712FCA3C9121CAF160826137C972F>58 D<173017F0160116031607A2160FA2161F16
1B163B1633167316E3A2ED01C316831503EE03F81507150EA2ED1C011538A2157015E0A2
EC01C0EC0380A2DA07007F140E92B5FC141F5C5C0270C7FC4A801301382003C038700780
D8780FC8127FEAFE3FD8FFFE160449169C49ED3FF86C4816E06C4816C06C48ED1F000007
CBFC36337EAF38>65 D<023FB512F80107B6FC011F15E0017F15C09026F8001EC7FCD803
C05B4848137C000F147890C712F8121E0018495AC7FCA24A5AA34A5AA3140F5DA3141F92
C8FCA35C143EA35CA2147814F85C010114064A131C0103143C4A5B49485B000FB612C000
3F5D484AC7FCB612F02D2D81AC27>73 D<D90380147E0107EC03FF011F140F017FEC3E1F
D801FFECF80FD803DF903803E00ED8021F90380F800C0000021EC7FCEC00785DEC03C049
485A020EC8FC5CEB3E385C5C137F137DA28013FD13F913F8801201497E147C147E3803E0
3E143F80D807C07F140F81380F8007816E6C1307D81F00150F6E6C130E001E6D6C131E00
3E6E131C037F13784891383FC1F092381FFFC000706E130048EC03F8302F7EAD36>75
D<91383FFFFC0107B612C0011F15F0017F15FC2701F83E007FD803C0EC0FFFD80F001403
001E017E0100138048167F007C163F127848017C141F5AC7160014FC171E173E4A143C17
7C177801015D4A495A4C5A4CC7FC0103141E4A137CED03F09138E1FFC0D907E790C8FCEC
CFF8ECDF8002C0C9FC495AA349CAFCA3133EA2133C137CA25BA25B485A138031307EAC31
>80 D<013E150748B4150E0007163E121FD83C7F157CEA703FEA003E17F8A25BEE01F0A2
49140317E01607485AEE0FC05B0003151F49EC3F801207167F484814FF170090C75A485C
5E003EEC07BEED0F3EED1E7E007EEC3E7C007C143C1578EDF0FC00FCEB01E04A485AEC07
80EC0F006CEB1E0114386C13F0D983C0EBFC40267FFF80EBFFC0D9FE001400D83FF85CD8
1FC0EB00F0302E81AC2C>85 D<14301478A214FCA2497E14CEA2EB03CF148701077F1403
010F7FEB0E01011E7FEB1C00013C7F013813700178137801701338A201F0133C49131C00
01141E49130E0003140F497F0007158090C712034815C0000E1401001E15E0001C1400A2
003C15F000381570007815780070153800F0153C48151C160C26297CA72F>94
D<126012E0B3A2B7128016C0A200E0C9FCB3A31260222E7CAD2B>96
D<387FFF80B5FCA200E0C7FCB3B3B3A91260114374B11F>100 D<B51280A3EA0003B3B3
B3A9130111437FB11F>I<141F14FFEB03F0EB0FC0EB1F8014005B133EB3A2137E137C13
FC485A485AEA7FC048C7FCEA7FC0EA03F06C7E6C7E137C137E133EB3A2133F7F1480EB0F
C0EB03F0EB00FF141F18437BB123>I<12FCB47EEA0FE0EA01F0EA00FC137C137E133EB3
A37F1480130FEB07E0EB01FEEB007FEB01FEEB07E0EB0F80131F1400133EB3A3137E137C
13FCEA01F0EA0FE0EAFF8000FCC7FC18437BB123>I<12E0B3B3B3AD034378B114>106
D E /FL 27 122 df<B712F0A23903F8000700011401ED00F816781638A21618A3161C16
0CA31600B3A9487EB512F8A2262D7EAC2C>0 D<B6FCA318037BA623>22
D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35AA45A
A512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F01370133813
1C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2
EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2EA01
E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I<EC0380B3A4B812FCA3C7
D80380C7FCB3A42E2F7CA737>43 D<EB3FC0EBFFF03803E07C48487E48487E497E001EEB
0780A2003E14C0A248EB03E0A500FC14F0B0007C14E0A3007E1307003E14C0A36CEB0F80
6C14006D5A3807C03E3803F0FC3800FFF0EB3FC01C2D7DAB23>48
D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>I<EB7F80
3801FFF0380780FC380E003F48EB1F8048EB0FC05A0060EB07E012F000FC14F07E1403A3
007C1307C7FCA215E0140F15C0141F1580EC3F00147E147C5C495A495A495A495A011EC7
FC5B5B4913305B485A4848136048C7FC000E14E0001FB5FC5A4814C0B6FCA21C2C7DAB23
>I<EB3FC03801FFF03807C0FC380E007E487FEC1F80003F14C0A2EB800F1300A2000C13
1FC7FC1580A2EC3F00143E5C5CEB03F0EBFFC014F0EB00FC143FEC1F8015C0140F15E0A2
EC07F0A21238127C12FEA3EC0FE012F8006014C00070131F6C1480001EEB3F00380780FC
3801FFF038007FC01C2D7DAB23>I<140EA2141E143EA2147E14FEA2EB01BE1303143E13
06130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812305A
12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90
B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380
000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C0007013
0F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>I<EB03F8EB0FFE
90383E0780EBF8013901F007C03803E00FEA07C0EA0F80A2391F00078091C7FC123EA212
7EA2127CEB0FC038FC3FF0EBF07C38FDC01EB4487E01001380EC07C04814E0A214034814
F0A4127CA3127EA2003E14E01407121E001F14C06CEB0F803907801F003803C03E6C6C5A
38007FF0EB1FC01C2D7DAB23>I<123C127E12FFA4127E123C1200AD123C127E12FFA412
7E123C081D7A9C14>58 D<48B4FC000713E0381E01F8383800FC48137E0060133E00F813
3F7EA40030137FC7127E14FCEB01F8EB03F014E0EB0780EB0F00A2131E131CA213181338
1330A790C7FCA7137813FC487EA46C5A1378182F7CAE21>63 D<B712FEA23903F8000100
01EC003E828282A282A3178016011518A293C7FCA31538157815F890B5FCA2EBF8001578
15381518A21760A392C712C0A4160117801603A21607160F163F0003913801FF00B8FCA2
2B2D7EAC30>69 D<007FB712F8A29039000FC003007C150000701638A200601618A200E0
161CA248160CA5C71500B3A94A7E011FB512E0A22E2D7EAC33>84
D<EAFFE0A3EAE000B3B3B3A7EAFFE0A30B4379B114>91 D<EAFFE0A31200B3B3B3A712FF
A30B437FB114>93 D<EB1FE0EB7FFC3801F01E3803E0073907C01F80EA0F80EA1F005A00
3EEB0F00007E90C7FCA2127C12FCA9127EA215C07E6C130101801380380FC0033907E007
003801F03E38007FF8EB1FC01A207E9E1F>99 D<EB1F80EBFFF03803E0783807C03E380F
801E381F001FEC0F80123E007E130715C0127C12FCA3B6FCA200FCC8FCA5127EA2003E14
C0123F6C1301390F80038001C013003803E00F3801F03C38007FF8EB1FC01A207E9E1F>
101 D<EA07C012FFA2120F1207B3B3A3EA0FE0EAFFFEA20F2E7EAD14>108
D<EB1FE0EB7FF83801F03E3803C00F3907800780390F0003C04814E0003EEB01F0A248EB
00F8A300FC14FCA9007C14F8A26CEB01F0A26CEB03E0A2390F8007C03907C00F803901F0
3E0038007FF8EB1FE01E207E9E23>111 D<3807C0FE39FFC7FF809038CF03E0390FDC01
F03907F800FC49137E49133E49133FED1F80A3ED0FC0A8151F1680A2ED3F00A26D137E6D
137C5D9038FC01F09038CE07E09038C7FF80D9C1FCC7FC01C0C8FCA9487EEAFFFEA2222B
7E9D27>I<90380FE01890387FF8383801F81C3903E00E783807C007390F8003F8001F13
01EA3F00A2007E1300A212FE5AA8127EA36C13017EEB8003380FC0073803E00E3801F03C
38007FF0EB1FC090C7FCA94A7E91381FFFC0A2222B7E9D25>I<380781F838FF87FEEB8E
3FEA0F9CEA07B813B0EBF01EEBE000A45BB0487EB5FCA2181E7E9D1C>I<3AFFFC07FF80
A23A0FF003FC000003EB01F0000114C06D485A000091C7FCEB7C06EB3E0E6D5A14B8EB0F
B0EB07E013036D7E497E1307EB067C497EEB1C1F01387FEB700F496C7E6E7ED803C07F00
076D7E391FE003FC3AFFF007FFC0A2221D7F9C25>120 D<3AFFFC01FFC0A23A0FE0007E
000007147C1538000314306D137000011460A26C6C5BA2EBFC01017C5BEB7E03013E90C7
FCA2EB1F06A2148EEB0F8CA2EB07D8A2EB03F0A36D5AA26D5AA2495AA2130391C8FC1278
EAFC06A25B131CEA7838EA7070EA3FE0EA0F80222B7F9C25>I E
/FM 78 127 df<121C127FEAFF80B3EA7F00B2123EC7FCA8121C127FA2EAFF80A3EA7F00
A2121C09396DB830>33 D<00101304007C131F00FEEB3F80A26C137FA248133FB2007E14
00007C7F003C131E00101304191C75B830>I<EB07E0EB1FF8497E137F497E803801FC7F
497E810003131F13F0A6143F92C8FC91387F0FFF9026F87E1F1380000113FEEBF9FC13FB
4A6C1300D9FFF013C06C13E0151F02C05BEB7F809038FF003F4892C7FC485C48EB807E5A
15FE391FDFC0FC383F8FE014E1397F07F1F8EB03F300FEEBFBF0EB01FF5D7FEDC006027F
130F91393F801F8015C06C137F6CEBFFE049EBF83F018701FC1300263FFFFBB5FC6C01F1
5B14E06C9038C03FFC00039038001FF8D801FCEB07E0293A7DB830>38
D<EA07C0EA0FF0EA1FF8A213FCA213FE120F1207EA007EA513FE13FCA2120113F81203EA
07F0120FEA1FE0127FEAFFC013801300127C12380F1D70B730>I<141E147F14FF5BEB03
FEEB07FCEB0FF0EB1FE0EB3FC0EB7F80EBFF00485A5B12035B485A120F5BA2485AA2123F
5BA2127F90C7FCA412FEAD127FA47F123FA27F121FA26C7EA27F12076C7E7F12017F6C7E
EB7F80EB3FC0EB1FE0EB0FF0EB07FCEB03FEEB01FF7F147F141E184771BE30>I<127812
FE7E7F6C7E6C7EEA0FF06C7E6C7E6C7E6C7EEB7F80133F14C0131FEB0FE014F01307A2EB
03F8A214FC1301A214FE1300A4147FAD14FEA4130114FCA2130314F8A2EB07F0A2130F14
E0EB1FC0133F1480137FEBFF00485A485A485A485AEA3FE0485A485A90C7FC5A12781847
78BE30>I<14E0497E497EA60038EC0380007EEC0FC0D8FF83EB3FE001C3137F9038F3F9
FF267FFBFB13C06CB61280000FECFE00000314F86C5C6C6C13C0011F90C7FC017F13C048
B512F04880000F14FE003FECFF80267FFBFB13C026FFF3F913E09038C3F87F0183133FD8
7E03EB0FC00038EC0380000091C7FCA66D5A6D5A23277AAE30>I<143EA2147FAF007FB7
FCA2B81280A36C1600A2C76CC8FCAF143EA229297DAF30>I<EA03E0EA0FF0EA1FF813FC
EA3FFEA213FFA27EA27E1203EA007FA2137E13FEEA01FC1203EA07F8EA3FF0127FEAFFE0
EA7F801300123C1019708B30>I<007FB612F0A2B712F8A36C15F0A225077B9E30>I<120F
EA3FC0EA7FE0A2EAFFF0A4EA7FE0A2EA3FC0EA0F000C0C6E8B30>I<16F01501ED03F8A2
1507A2ED0FF0A2ED1FE0A2ED3FC0A2ED7F80A2EDFF00A24A5AA25D1403A24A5AA24A5AA2
4A5AA24A5AA24A5AA24AC7FCA2495AA25C1303A2495AA2495AA2495AA2495AA2495AA249
C8FCA2485AA25B1203A2485AA2485AA2485AA2485AA2485AA248C9FCA25AA2127CA22547
7BBE30>I<14FE903807FFC0497F013F13F8497F90B57E48EB83FF4848C6138049137F48
48EB3FC04848EB1FE049130F001F15F0491307A24848EB03F8A290C712014815FCA400FE
EC00FEAD6C14016C15FCA36D1303003F15F8A26D1307001F15F0A26D130F6C6CEB1FE0A2
6C6CEB3FC06C6CEB7F806D13FF2601FF8313006CEBFFFE6D5B6D5B010F13E06D5BD900FE
C7FC273A7CB830>I<EB03C0497EA2130FA2131FA2133F137F13FF1203123FB5FCA213EF
138FEA7E0F1200B3B0003FB512F84814FCB612FEA26C14FC6C14F81F3977B830>I<EB07
FC90383FFFC090B512F00003804814FE4880261FF80F1380263FE00113C09038C0007F48
48EB3FE090C7121FED0FF04814075A6C15F81503A3127E1218C8FCA2150716F0150F16E0
151F16C0153FED7F8015FF4A13005DEC07FC4A5A4A5A4A5A4A5A4A5A4990C7FC495A495A
EB0FF0EB3FE0495A495A4890C8FC4848EB01F04848EB03F8485AEA1FE048B6FCB7FCA37E
6C15F025397BB830>I<120FEA3FC0EA7FE0A2EAFFF0A4EA7FE0A2EA3FC0EA0F00C7FCAF
120FEA3FC0EA7FE0A2EAFFF0A4EA7FE0A2EA3FC0EA0F000C276EA630>58
D<EA03C0EA0FF0EA1FF8A2EA3FFCA4EA1FF8A2EA0FF0EA03C0C7FCAFEA03C0EA0FF0121F
13F8123F13FCA3121FA2120F12031200120113F8120313F01207EA1FE0123FEA7FC0EAFF
80EA7F00127E12380E3470A630>I<16F01503ED07F8151F157FEDFFF014034A13C0021F
138091383FFE00ECFFF8495B010713C0495BD93FFEC7FC495A3801FFF0485B000F138048
90C8FCEA7FFC5BEAFFE05B7FEA7FF87FEA1FFF6C7F000313E06C7F38007FFC6D7E90380F
FF806D7F010113F06D7FEC3FFE91381FFF80020713C06E13F01400ED7FF8151F1507ED03
F01500252F7BB230>I<007FB7FCA2B81280A36C16006C5DCBFCA7003FB612FE4881B812
80A36C1600A229157DA530>I<1278127EB4FC13C07FEA7FF813FEEA1FFF6C13C000037F
6C13F86C6C7EEB1FFF6D7F010313E06D7F9038007FFC6E7E91380FFF806E13C0020113F0
80ED3FF8151F153FEDFFF05C020713C04A138091383FFE004A5A903801FFF0495B010F13
804990C7FCEB7FFC48485A4813E0000F5B4890C8FCEA7FFE13F8EAFFE05B90C9FC127E12
78252F7BB230>I<147F4A7EA2497FA4497F14F7A401077F14E3A3010F7FA314C1A2011F
7FA490383F80FEA590387F007FA4498049133F90B6FCA34881A39038FC001F0003814913
0FA4000781491307A2D87FFFEB7FFFB56CB51280A46C496C130029397DB830>65
D<007FB512F0B612FE6F7E82826C813A03F8001FF815076F7E1501A26F7EA615015EA24B
5A1507ED1FF0ED7FE090B65A5E4BC7FC6F7E16E0829039F8000FF8ED03FC6F7E1500167F
A3EE3F80A6167F1700A25E4B5A1503ED1FFC007FB6FCB75A5E16C05E6C02FCC7FC29387E
B730>I<91387F803C903903FFF03E49EBFC7E011F13FE49EBFFFE5B9038FFE07F48EB80
1F3903FE000F484813075B48481303A2484813015B123F491300A2127F90C8FC167C1600
5A5AAC7E7EA2167C6D14FE123FA27F121F6D13016C6C14FCA26C6CEB03F86D13076C6CEB
0FF03901FF801F6C9038E07FE06DB512C06D14806D1400010713FC6D13F09038007FC027
3A7CB830>I<003FB512E04814FCB67E6F7E6C816C813A03F8007FF0ED1FF8150F6F7E6F
7E15016F7EA2EE7F80A2163F17C0161FA4EE0FE0AC161F17C0A3163F1780A2167F17005E
4B5A15034B5A150F4B5AED7FF0003FB65A485DB75A93C7FC6C14FC6C14E02B387FB730>
I<007FB7FCB81280A47ED803F8C7123FA8EE1F0093C7FCA4157C15FEA490B5FCA6EBF800
A4157C92C8FCA5EE07C0EE0FE0A9007FB7FCB8FCA46C16C02B387EB730>I<003FB71280
4816C0B8FCA27E7ED801FCC7121FA8EE0F8093C7FCA5153E157FA490B6FCA69038FC007F
A4153E92C8FCAE383FFFF8487FB5FCA27E6C5B2A387EB730>I<02FF13F00103EBC0F801
0F13F1013F13FD4913FF90B6FC4813C1EC007F4848133F4848131F49130F485A49130712
1F5B123F491303A2127F90C7FC6F5A92C8FC5A5AA892B5FC4A14805CA26C7F6C6D1400ED
03F8A27F003F1407A27F121F6D130F120F7F6C6C131FA2D803FE133F6C6C137FECC1FF6C
90B5FC7F6D13FB010F13F30103EBC1F0010090C8FC293A7DB830>I<3B3FFF800FFFE048
6D4813F0B56C4813F8A26C496C13F06C496C13E0D803F8C7EAFE00B290B6FCA601F8C7FC
B3A23B3FFF800FFFE0486D4813F0B56C4813F8A26C496C13F06C496C13E02D387FB730>
I<007FB6FCB71280A46C1500260007F0C7FCB3B3A8007FB6FCB71280A46C1500213879B7
30>I<49B512F04914F85BA27F6D14F090C7EAFE00B3B3123C127EB4FCA24A5A1403EB80
07397FF01FF86CB55A5D6C5C00075C000149C7FC38003FF025397AB730>I<D83FFF9038
0FFF80486D4813C0B56C5AA26C497E6C496C1380D803F0903803F8004B5A4B5A151F4B5A
5E4BC7FC15FE14014A5A5D4A5A4A5A141F5D4A5A4AC8FC5C13F18101F37F13F790B57E14
EFECC7F01483EC03F8140101FE7F496C7E5B157F497F82151F82150F826F7EA26F7E1501
821500D83FFF903803FFC0486D4813E0B56C5AA26C497E6C496C13C02B387FB730>I<38
3FFFF8487FB57EA26C5B6C5BD801FCC9FCB3B0EE0F80EE1FC0A9003FB7FC5AB8FCA27E6C
16802A387EB730>I<D83FF8ECFFE0486C4913F0486C4913F8A2007F16F06C6C4913E000
07160001EF14BFEC800FA39039E7C01F3FA4ECE03F01E3133EA2ECF07EA201E1137CA2EC
F8FCA201E013F8A214FDEC7DF0A3147FEC3FE0A3EC1FC0A2EC070091C7FCADD83FFC9038
01FFE0486C4913F0B54913F8A26C486D13F06C486D13E02D387FB730>I<D83FFC90381F
FF80486C4913C0B54913E0A26C6D6C13C06C6E13800003913801F800EBF7C0A3EBF3E0A3
14F013F1A214F8A213F014FCA2147C147EA2143E143FA2141FA21581A2140F15C1A21407
15E1A2140315F1A21401A215F91400A3157DA3153FEA3FFF481380B5EAC01FA26CEB800F
6C496C5A2B387EB730>I<90383FFFE048B512FC000714FF4815804815C04815E0EBF800
01E0133FD87F80EB0FF0A290C71207A44815F8481403B3A96C1407A26C15F0A36D130FA2
6D131F6C6CEB3FE001F813FF90B6FC6C15C06C15806C1500000114FCD8003F13E0253A7B
B830>I<007FB512F0B612FE6F7E16E0826C813903F8003FED0FFCED03FE15016F7EA282
1780163FA6167F17005EA24B5A1503ED0FFCED3FF890B6FC5E5E16804BC7FC15F001F8C9
FCB0387FFFC0B57EA46C5B29387EB730>I<90383FFFE048B512FC000714FF4815804815
C04815E0EBF80001E0133F4848EB1FF049130F90C71207A44815F8481403B3A8147E14FE
6CEBFF076C15F0EC7F87A2EC3FC7018013CF9038C01FFFD83FE014E0EBF80F90B6FC6C15
C06C15806C1500000114FCD8003F7FEB00016E7EA21680157F16C0153F16E0151F16F015
0FED07E025467BB830>I<003FB57E4814F0B612FC15FF6C816C812603F8017F9138003F
F0151F6F7E15071503821501A515035E1507150F4B5A153F4AB45A90B65A5E93C7FC5D81
82D9F8007FED3FE0151F150F821507A817F8EEF1FCA53A3FFF8003FB4801C0EBFFF8B56C
7E17F06C496C13E06C49EB7FC0C9EA1F002E397FB730>I<90390FF803C0D97FFF13E048
B512C74814F74814FF5A381FF80F383FE001497E4848137F90C7123F5A48141FA2150FA3
7EED07C06C91C7FC7F7FEA3FF0EA1FFEEBFFF06C13FF6C14E0000114F86C80011F13FF01
031480D9003F13C014019138007FE0151FED0FF0A2ED07F8A2007C140312FEA56C140716
F07F6DEB0FE06D131F01F8EB3FC001FF13FF91B51280160000FD5CD8FC7F13F8D8F81F5B
D878011380253A7BB830>I<003FB712C04816E0B8FCA43AFE003F800FA8007CED07C0C7
91C7FCB3B1011FB5FC4980A46D91C7FC2B387EB730>I<3B7FFFC007FFFCB56C4813FEA4
6C496C13FCD803F8C7EA3F80B3B16D147F00011600A36C6C14FE6D13016D5CEC80039039
3FE00FF890391FF83FF06DB55A6D5C6D5C6D91C7FC9038007FFCEC1FF02F3980B730>I<
D87FFE90380FFFC0B54913E06E5AA24A7E6C486D13C0D807F0903801FC00A26D13030003
5DA46C6C495AA46C6C495AA46D131F6D5CA3EC803F013F5CA46D6C48C7FCA490380FE0FE
A401075B14F1A301035BA314FB01015BA314FFA26D5BA46E5A6E5A2B397EB730>I<D83F
FC903801FFE0486C4913F000FF16F8A2007F16F06C486D13E0D81FC09038001FC0000F16
80A76D143F00071600A7000390380F803E9039F01FC07EEC3FE0A3EC7FF0A2147D000115
7CA29039F8FDF8FCA314F8A300005D01F913FCA2ECF07CA201FD137DA2017D5CECE03DA3
017F133FA2ECC01FA2013F5CA2EC800F6D486C5A2D397FB730>I<3A3FFF01FFF8480183
7F02C77FA202835B6C01015B3A01FC007F806D91C7FC00005C6D5BEB7F01EC81FCEB3F83
14C3011F5B14E7010F5B14FF6D5BA26D5BA26D5BA26D90C8FCA4497FA2497FA2815B81EB
0FE781EB1FC381EB3F8181EB7F0081497F49800001143F49800003141F49800007140FD8
7FFEEB7FFFB590B5128080A25C6C486D130029387DB730>I<D87FFF90381FFFC0B56C48
13E0A46C496C13C0D803F8903803F8006D1307A26C6C495AA26C6C5C151F6D5CEC803F01
3F5CECC07F011F91C7FCA290380FE0FEA214F101075BA2903803FBF8A201015B14FF6D5B
A26E5AA36E5AB1903803FFF8497F497FA26D5B6D5B2B387EB730>I<001FB612FC4815FE
5AA490C7EA03FCED07F816F0150FED1FE016C0153FED7F80003E1500C85A4A5A5D14034A
5A5D140F4A5A5D143F4A5A92C7FC5C495A5C1303495A5C130F495A5C133F495A91C8FC5B
4848147C4914FE1203485A5B120F485A5B123F485A90B6FCB7FCA46C15FC27387CB730>
I<007FB5FCB61280A4150048C8FCB3B3B3A5B6FC1580A46C140019476DBE30>I<007FB5
FCB61280A47EC7123FB3B3B3A5007FB5FCB6FCA46C140019477DBE30>93
D<007FB612F0A2B712F8A36C15F0A225077B7D30>95 D<1338137CEA01FE12031207EA0F
FC13F0EA1FE013C0EA3F8013005A127EA212FE5AA5EAFFC013E013F0127FA2123FA2EA1F
E0EA07C00F1D70BE30>I<EB3FFC48B57E4814E04880488048809038F00FFE9038E001FF
806F7E6C48133F6C4880C8121FA491B5FC130F137F48B6FC12075A48EBC01F383FFC00EA
7FE0138048C7FC5AA46C143FA26C6C137F9038C001FF263FF80FEBFFC06CB712E0A20007
14F76C14C3C6020013C0D93FF090C7FC2B2A7CA830>I<EA3FFC487E12FFA2127F123F12
00AAEC03FE91381FFF80027F13E091B57E90B612FC82ECFE079138F001FF4A6C13804A13
7F4AEB3FC091C7121F4915E0160FA217F01607A8160FA217E07F161F6EEB3FC0A26EEB7F
806E13FFDAF00313009138FC0FFE91B55A5E495CD97E7F13C0D93C1F90C7FC90380003FC
2C3980B730>I<ECFFE0010713FC011F7F017F7F90B612804815C048EB807F3907FC003F
485A485A49EB1F804848EB0F004990C7FC127F90C9FCA25A5AA87E7EA27F003FEC07C06D
EB0FE06C7E6D131F6C6C14C0D807FE133F9039FFC0FF806C90B5FCC615006D5B011F13F8
01075B01011380232A7AA830>I<913801FFE04A7F5CA28080EC0007AAEB03FE90381FFF
874913E790B6FC5A5A481303380FFC00D81FF0133F49131F485A150F4848130790C7FCA2
5AA25AA87E6C140FA27F003F141F6D133F6C7E6D137F390FF801FF2607FE07EBFFC06CB7
12E06C16F06C14F76D01C713E0011F010313C0D907FCC8FC2C397DB730>I<49B4FC0107
13E0011F13F8017F7F90B57E488048018113803A07FC007FC04848133FD81FE0EB1FE015
0F484814F0491307127F90C7FCED03F85A5AB7FCA516F048C9FC7E7EA27F003FEC01F06D
EB03F86C7E6C7E6D1307D807FEEB1FF03A03FFC07FE06C90B5FC6C15C0013F14806DEBFE
00010713F8010013C0252A7CA830>I<EDFF80020713E0021F13F05C4A13F891B5FC4913
87903803FE079138FC03F0903907F800C04A1300A8003FB612C04815E0B7FCA36C15C026
0007F0C7FCB3A9003FB512FE4880B71280A26C15006C5C25397DB830>I<D903FC13FF90
261FFF8713C04913DF90B712E05A5A2607FE07138F903AF801FE07C048486C6CC7FCA249
7F001F8149133FA56D137F000F92C7FC6D5BA26C6C485AEBFE0790B55A5D485C15C001DF
5BD9C3FCC8FC01C0C9FCA37F7F6CB512F015FF6C15C04815F0488148813A3FE0001FFE01
80130148C8127F007E8100FE168048151FA56C153F007FED7F006D5C6C6C495A01F01307
6CB4EB7FFC6C90B55A6C5D000115C06C6C91C7FC011F13FC010113C02B3E7DA730>I<EA
3FFC487E12FFA2127F123F1200AAEC01FE91380FFF80023F13E091B57E90B67EA29138FE
07FCECF8039138E001FE14C0EC8000A291C7FCA25BB3A23B3FFFF81FFFF8486D4813FCB5
00FE14FEA26C01FC14FC6C496C13F82F3880B730>I<14E0EB03F8A2497EA36D5AA2EB00
E091C8FCA9381FFFF8487F5AA27E7EEA0001B3A9003FB612C04815E0B7FCA27E6C15C023
397AB830>I<EC01C0EC07F0A2EC0FF8A3EC07F0A2EC01C091C7FCA990B512F04814F8A4
7EEB0003B3B3A5EC07F0A2123C007EEB0FE0B4131FEC3FC0147F90B512806C14005C6C5B
000F13F0000313C01D4E7CB830>I<EA7FF8487EA4127F1200AB0203B512804A14C017E0
A217C06E14809139001FE0004B5A4B5A4BC7FC4A5A4A5AEC0FF84A5A4A5A4A5A4A5A01FD
7F90B57E8114F7ECE3F8ECC1FCEC81FEEC00FF497F496D7E6F7E826F7E15076F7E6F7E3B
7FFFF81FFFE0B56C4813F017F8A217F06C496C13E02D387FB730>I<387FFFF8B57EA47E
EA0001B3B3A8007FB612F0B712F8A46C15F025387BB730>I<02FC137E3B7FC3FF01FF80
D8FFEF01877F90B500CF7F15DF92B57E6C010F13872607FE07EB03F801FC13FE9039F803
FC01A201F013F8A301E013F0B3A23C7FFE0FFF07FF80B548018F13C0A46C486C01071380
322881A730>I<EC01FE3A3FFC0FFF80267FFE3F13E000FF90B57E90B67E7E6C9038FE07
FCC6EBF8039138E001FE14C0EC8000A291C7FCA25BB3A23B3FFFF81FFFF8486D4813FCB5
00FE14FEA26C01FC14FC6C496C13F82F2880A730>I<49B4FC010F13E0013F13F8497F90
B57E0003ECFF8014013A07FC007FC04848EB3FE0D81FE0EB0FF0A24848EB07F849130300
7F15FC90C71201A300FEEC00FEA86C14016C15FCA26D1303003F15F86D13076D130F6C6C
EB1FF06C6CEB3FE06D137F3A07FF01FFC06C90B512806C15006C6C13FC6D5B010F13E001
0190C7FC272A7CA830>I<EC03FE3A3FFC1FFF80267FFE7F13E000FF90B57E90B612FC6C
816CEBFE07C69038F001FF4A6C13804A137F4AEB3FC091C7121F4915E0160FA217F01607
A8160FA217E07F161F6EEB3FC0A26EEB7F806E13FFDAF00313009138FC0FFE91B55A5E49
5C6E13C0021F90C7FCEC03FC91C9FCAD383FFFF8487FB57EA26C5B6C5B2C3C80A730>I<
49B413F8010FEBC1FC013F13F14913FD48B6FC5A481381390FFC007F49131F4848130F49
1307485A491303127F90C7FC15015A5AA77E7E15037FA26C6C1307150F6C6C131F6C6C13
3F01FC137F3907FF01FF6C90B5FC6C14FD6C14F9013F13F1010F13C1903803FE0190C7FC
AD92B512F84A14FCA46E14F82E3C7DA730>I<ED07F83A3FFF803FFF486DB51280B512C3
02CF14C06C13DF6C9038FFFC3FD8001F13E09238801F809238000F004A90C7FC5C5C5CA2
5CA45CAF003FB512FC4880B7FCA26C5C6C5C2A287EA730>I<90381FFC1E48B5129F0007
14FF5A5A5A387FF007EB800100FEC7FC4880A46C143E007F91C7FC13E06CB4FC6C13FC6C
EBFF806C14E0000114F86C6C7F01037F9038000FFF02001380007C147F00FEEC1FC0A215
0F7EA27F151F6DEB3F806D137F9039FC03FF0090B6FC5D5D00FC14F0D8F83F13C026780F
FEC7FC222A79A830>I<EB0780497E131FA9003FB612E04815F0B7FCA36C15E026001FC0
C7FCB216F8ED01FCA5ECE003010FEB07F814F09138FC1FF06DB512E06D14C016806D1400
9038007FFCEC1FF026337EB130>I<D83FFCEB3FFC486C497E00FF14FFA2007F147F003F
143F00001400B3A41501A2150315076D130F903A7FC07FFFF891B612FC6D15FE7F6D4913
FC6D9038F87FF8010001C0C7FC2F2880A630>I<3B3FFFC07FFF80486DB512C0B515E0A2
6C16C06C496C13803B01F80003F000A26D130700005DA26D130F017E5CA2017F131F6D5C
A2EC803F011F91C7FCA26E5A010F137EA2ECE0FE01075BA214F101035BA3903801FBF0A3
14FF6D5BA36E5A6E5A2B277EA630>I<3B3FFFC01FFFE0486D4813F0B515F8A26C16F06C
496C13E0D807E0C7EA3F00A26D5C0003157EA56D14FE00015DEC0F80EC1FC0EC3FE0A33A
00FC7FF1F8A2147DA2ECFDF9017C5C14F8A3017E13FBA290393FF07FE0A3ECE03FA2011F
5C90390F800F802D277FA630>I<3A3FFF81FFFC4801C37FB580A26C5D6C01815BC648C6
6CC7FC137FEC80FE90383F81FC90381FC3F8EB0FE3ECE7F06DB45A6D5B7F6D5B92C8FC14
7E147F5C497F81903803F7E0EB07E790380FE3F0ECC1F890381F81FC90383F80FE90387F
007E017E137F01FE6D7E48486D7E267FFF80B5FCB500C1148014E3A214C16C0180140029
277DA630>I<3B3FFFC07FFF80486DB512C0B515E0A26C16C06C496C13803B01FC0003F0
00A2000014076D5C137E150F017F5C7F151FD91F805BA214C0010F49C7FCA214E0010713
7EA2EB03F0157C15FCEB01F85DA2EB00F9ECFDF0147D147FA26E5AA36E5AA35DA2143F92
C8FCA25C147EA2000F13FE486C5AEA3FC1EBC3F81387EB8FF0EBFFE06C5B5C6C90C9FC6C
5AEA01F02B3C7EA630>I<001FB612FC4815FE5AA316FC90C7EA0FF8ED1FF0ED3FE0ED7F
C0EDFF80003E491300C7485A4A5A4A5A4A5A4A5A4A5A4A5A4990C7FC495A495A495A495A
495A495A4948133E4890C7127F485A485A485A485A485A48B7FCB8FCA46C15FE28277DA6
30>I<127CA212FEB3B3B3AD127CA207476CBE30>124 D<017C133848B4137C48EB80FE48
13C14813C348EBEFFC397FEFFFF0D8FF8713E0010713C0486C1380D87C0113003838007C
1F0C78B730>126 D E /FN 59 107 df<007FB812FEBAFCA26C17FE3804799847>0
D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A799B19>I<0060166000F816F06C1501
007E15036CED07E06C6CEC0FC06C6CEC1F806C6CEC3F006C6C147E6C6C5C6C6C495A017E
495A6D495A6D6C485A6D6C485A6D6C48C7FC903803F07E6D6C5A903800FDF8EC7FF06E5A
6E5AA24A7E4A7EECFDF8903801F8FC903803F07E49487E49486C7E49486C7E49486C7E01
7E6D7E496D7E48486D7E4848147E4848804848EC1F804848EC0FC048C8EA07E0007EED03
F0481501481500006016602C2C73AC47>I<EB03C0A2805CA600F0140F00FC143F00FE14
7F00FF14FF393FC3C3FC390FE187F03903F18FC03900FDBF00EB3FFCEB0FF0EB03C0EB0F
F0EB3FFCEBFDBF3903F18FC0390FE187F0393FC3C3FC39FF03C0FF00FE147F00FC143F00
F0140F00001400A6805CA220277AA92D>I<EB0FFCEB3FFF90B512C0000314F03907F807
F8390FE001FC391F80007E48C77E003E8048EC0F80A20078140700F815C0A2481403A66C
1407A200781580007C140FA26CEC1F00003F5C6C6C137E390FE001FC3907F807F86CB55A
C614C0013F90C7FCEB0FFC22227BA72D>14 D<EB0FFCEB3FFF90B512C0000314F0488048
8048804880A2481580A3B712C0AA6C1580A36C1500A26C5C6C5C6C5C6C5CC614C0013F90
C7FCEB0FFC22227BA72D>I<007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A2
6C18E0CDFCAE007FB912E0BA12F0A26C18E03C287BAA47>17 D<0203B612FE023F15FF91
B8FC010316FED90FFEC9FCEB1FE0EB7F8001FECAFCEA01F8485A485A485A5B48CBFCA212
3EA25AA21278A212F8A25AA77EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FE
EB7F80EB1FE0EB0FFE0103B712FE010016FF143F020315FE91CAFCAE001FB812FE4817FF
A26C17FE384779B947>I<180E183F18FFEF03FEEF0FF8EF3FE0EFFF80933803FE00EE1F
F8EE7FE0923801FF80DB07FEC7FCED1FF0ED7FC04A48C8FCEC07FCEC3FF0ECFFC0010390
C9FCEB0FFCEB3FE0EBFF80D803FECAFCEA0FF8EA3FE0EAFF8048CBFC6C7EEA7FE0EA0FF8
EA03FEC66C7EEB3FE0EB0FFCEB03FF010013C0EC3FF0EC07FCEC01FF9138007FC0ED1FF0
ED07FE923801FF809238007FE0EE1FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FF
183F180E1800AE007FB812FEBAFCA26C17FE384779B947>20 D<127012FCB4FCEA7FC0EA
1FF0EA07FCEA01FF38007FC0EB1FF8EB07FE903801FF809038007FE0EC0FF8EC03FE9138
00FF80ED3FE0ED0FFCED03FF030013C0EE3FF0EE07FCEE01FF9338007FC0EF1FF0EF07FC
EF01FFEF007FEF01FFEF07FEEF1FF0EF7FC0933801FF00EE07FCEE3FF0EEFFC0030390C7
FCED0FFCED3FE0EDFF80DA03FEC8FCEC0FF8EC7FE0903801FF80D907FEC9FCEB1FF8EB7F
C04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB812FEBAFCA26C17FE38
4779B947>I<126012F0A37EA21278A2127CA27E123F7E6C7E7F6C7E6C7EEA01FC6C7EEB
3FC0EB1FF0EB07FE903803FFE09038007FFF021FEBFF800203ECFFFEDA003F14FFA20203
B612FE021FEC8000027F90C8FC903801FFE0D907FEC9FCEB1FF0EB3FC001FFCAFCEA01FC
EA03F0485A485A485A90CBFC123EA25AA2127812F8A25AA31260CCFCAE007FB812FEBAFC
A26C17FE384779B947>23 D<D91FE01620D97FF816703801FFFE486D7E48804814F09038
E01FF8271F8007FC15F0273E0001FE15E0003CD9007F1401007CDA3FC013030078DA0FE0
14C00070DA07F8130700F0DA03FEEB1F8048913A01FF807F006F90B5FC043F5B705B0407
5B040113E000409238007F803C157BA047>I<D91FE01620D9FFFC16704813FF000714C0
4814F048809026E01FFE15F0273F0003FFEC01E0007E010013C00078DA3FF01307DB0FFC
EB0FC048913A07FF807F8048020190B5FC6F1500043F5B040F13F804035B00409238007F
80CDFCA4D91FE01620D9FFFC16704813FF000714C04814F048809026E01FFE15F0273F00
03FFEC01E0007E010013C00078DA3FF01307DB0FFCEB0FC048913A07FF807F8048020190
B5FC6F1500043F5B040F13F804035B00409238007F803C287BAB47>I<0203B612FE023F
15FF91B8FC010316FED90FFEC9FCEB1FE0EB7F8001FECAFCEA01F8485A485A485A5B48CB
FCA2123EA25AA21278A212F8A25AA87EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7E
EA00FEEB7F80EB1FE0EB0FFE0103B712FE010016FF143F020315FE383679B147>I<1260
12F0A37EA21278A2127CA27E123F7E6C7E7F6C7E6C7EEA01FC6C7EEB3FC0EB1FF0EB07FE
903803FFE09038007FFF021FEBFF800203ECFFFEDA003F14FFA20203B612FE021FEC8000
027F90C8FC903801FFE0D907FEC9FCEB1FF0EB3FC001FFCAFCEA01FCEA03F0485A485A48
5A90CBFC123EA25AA2127812F8A25AA31260383579B047>31 D<140C141EA2143E143CA2
147C1478A214F8495AA2495A495AA2495A49CDFC133E137EEA01F8485AEA0FE0003FBB12
FEBDFCA2003F1AFED80FE0CDFCEA03F06C7EEA007E133E7F6D7E6D7EA26D7E6D7EA26D7E
1478A2147C143CA2143E141EA2140C50307BAE5B>I<19301978A2197C193CA2193E191E
A2191F737EA2737E737EA2737E737E1A7C1A7EF21F80F20FC0F207F0007FBB12FCBDFCA2
6C1AFCCDEA07F0F20FC0F21F80F27E001A7C624F5A4F5AA24F5A4F5AA24FC7FC191EA219
3E193CA2197C1978A2193050307BAE5B>I<140CA3141EA3143FA34A7EA24A7E497F497F
A2903807DEF890380F9E7C90383F1E3F017E6D7ED801FCEB0FE0D807F8EB07F8D83FE0EB
01FFD8FF809038007FC00100143F00F815070040ED0080C71500B3B3B2140C2A517FBE2D
>I<140C141EB3B3B20040168000F8ED07C0B4153F0180147FD83FE0903801FF00D807F8
EB07F8D801FCEB0FE0D8007EEB1F80013F49C7FC90380F9E7C903807DEF86DB45AA26D5B
6D5B6E5AA26EC8FCA3141EA3140CA32A517FBE2D>I<020C1630021E1678A2023E167C02
3C163CA2027C163E0278161EA202F8161F4948707EA24948707E4948707EA24948707E49
CB7E013E187C017E187ED801F8F01F804848F00FC0D80FE0F007F0003FBB12FCBDFCA200
3F1AFCD80FE0CBEA07F0D803F0F00FC06C6CF01F80D8007EF07E00013E187C6D606D6C4C
5A6D6C4C5AA26D6C4C5A6D6C4C5AA26D6C4CC7FC0278161EA2027C163E023C163CA2023E
167C021E1678A2020C163050307BAE5B>I<D91FE01620D97FFC167048B5FC4814C04880
4814F8271FE01FFC15F0273F8003FE15E0003EC76C6C130148DA3FC013030078DA1FF0EB
07C00070DA07FC131F00F0913A03FF807F80486E90B512006F6C5B705B040F5B040313E0
00409238007F80CDFCB0007FB912E0BA12F0A26C18E03C277BA947>39
D<EF01E0841700841878187C84A284727E727E851803727E007FB912FCBA7E856C85CCEA
07E0737EF101FCF1007FF21FC0F20FF8F203FFA2F20FF8F21FC0F27F00F101FCF103F04F
5A007FBA1280BBC7FC616C60CBEA01F04E5A1807614E5A4EC8FC183EA260187818F86017
016050327BAF5B>41 D<496C1360496C13F0B3B3AB00C017C000F0160300FC160F00FF16
3FD83F83ED7F00D80FC315FCD803F3ECF3F0D801FBECF7E0D800FFECFFC0013F92C7FC01
1F5C010F5C01075C6D6C485A01015CECF0036D6C485A91387C0F80023C90C8FCEC3E1FEC
1E1E6E5AA2EC07F8A26E5AA36E5AA36E5AA232517DBE38>43 D<03F015F0A20201824B15
780203167C4B153C0207163E4A488192C97E4A83023E707E023C1603027C834A707E49B9
7E498449844984013FCBEA0FC0017E727E49727ED803F8F001FCD80FE0F0007FD83FC0F1
3FC0B4CDEA0FF0A2D83FC0F13FC0D80FE0F17F00D803F8F001FCC66CF003F0017E4E5A6D
4E5A010FBAC7FC6D606D606D60D900F8C9EA01F0027C4C5A023C5F023E16076E4C5A6E94
C8FC6F5D6E6C153E0203163C6F157C020116786F15F802005EA254327DAF5B>I<0203B5
12F8023F14FC91B6FC010315F8D90FFEC8FCEB1FE0EB7F8001FEC9FCEA01F8485A485A48
5A5B48CAFCA2123EA25AA21278A212F8A25AA2B812F817FCA217F800F0CAFCA27EA21278
A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FE0EB0FFE0103B612F80100
15FC143F020314F82E3679B13D>50 D<1718173C177CA217F8A2EE01F0A2EE03E0A2EE07
C0160F1780EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153E157E
157C5DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25C13015C495AA2495AA249
5AA249C9FCA2133EA25BA25BA2485AA2485AA2485A120F5B48CAFCA2123EA25AA25AA25A
12602E5474C000>54 D<126012F0AE12FCA412F0AE126006227BA700>I<0060EE018000
F0EE03C06C1607A200781780007C160FA2003C1700003E5EA26C163EA26C163C6D157CA2
000716786D15F8A26C6C4A5AA200015E6D140390B7FC6C5EA3017CC7EA0F80A2013C92C7
FC013E5CA2011E141E011F143EA26D6C5BA2010714786E13F8A26D6C485AA201015CECF0
03A201005CECF807A291387C0F80A2023C90C8FCEC3E1FA2EC1E1EEC1F3EA2EC0FFCA26E
5AA36E5AA36E5A6E5A324180BE33>I<007FB612FEB8FCA27EC9120FB3A7001FB7FC127F
A3C9120FB3A8007FB7FCB8FCA26C15FE283F7BBE33>I<007FB81280B912C0A27ECA1203
B3A232187B9F3D>I<007FB912E0BA12F0A26C18E0C8000FC9FCB3B3B315063C3C7BBB47>
62 D<1506150FB3B3B3007FB912E0BA12F0A26C18E03C3C7BBB47>I<4E7EF007C0180F18
1F183F187FA218FFA25FA25F18BF1707183F170F170E171E171C173C17381778177017F0
17E01601EE03C0A2EE0780A2EE0F005E161E5EA25E16F85E4B5A854B5A15075E4BC7121F
5D151E033FB6FC5DA292B7FC4A82DA03E0C7121FA24A5A4A48140F0010131F003091C87F
143E00785B007C13FC26FE01F86F7E38FF87F0D9FFE0171CF1FE7C4A923803FFF86C4917
E091C914C06C487013006C48EE00FCD80FF094C7FCEA03C046477EC149>65
D<020EEC7FC0023E903803FFF802FE011F7F0103027F7F010F49B6FC011F903803F81F01
3F90260FC0031380903A79FC1F00010101013E7F5D4B147F903803FDF002FF16005D5D18
7E4B14FE4990C85A604A4A5A4D5A4A4A5AEF1F80010F037EC7FC4A495AEE0FF04AEB7FC0
DB03FFC8FC011F011F13E04A4813F84B13FE92B6FC4AC66C7F013F020F7F04037F4A1300
717E173F49C86C7EA2170FA201FE1507A448485EA3495E0003160F605B00074C5A4993C7
FCD9E180143E260FE7C05CD9DFE05C48B46CEB03F0D9BFFCEB0FC09139FF80FF80D83F1F
D9FFFEC8FC6D14F8D87E0714E0D8780191C9FC39E0003FF039427DBF3C>I<EE3FE09238
03FFF8031F13FC037F13FE4AB5FC913807FC0F91380FC00391383F0001147C4A14FC495A
494814F8495A4948EB03F0131F49C7EA07E0133E017EEC0FC05BEE1F8048481500163E48
48143893C7FC485AA2120F5B121FA3485AA4127F90CAFCA45AA87F17E01603EE07C06C6C
EC0F80161F6DEC3F00163E6C6C5C6D5C6C6C495A01FFEB07E06C9038E03F806C90B5C7FC
6C14FC6C14F06C6C1380D90FFCC8FC2F427FBF30>I<4AB512FC023FECFFE049B712FC01
07EEFF80011F8390277FE1FC0114F02601FC01D9000F7FD803F003017FD807C09238003F
FE260F80036F7ED81F001607487113804883007E4A6E13C012FE48187F00F019E000C001
07163FC7FC5D191FA3140F5DA21AC0A24A5AA2F13F80A24A5A1A0061197E4AC9FC61A202
7E4B5A02FE5E18034A4B5A01015F4E5A4A4BC7FC0103163E604A5D0107ED03F04AEC07C0
EF1F80010F037EC8FC4A495A011FEC0FF04AEB7FC0DB0FFFC9FC49B512FC90B612E04892
CAFC4814F84891CBFC433E7EBD46>I<EEFFE0030713FC033F13FE92B6FC1403913807F0
0F91381F8001023EC7FC5C4A14FE1301494814FC17F84948EB01F017C0010F91C7FCA213
1FA380A280806DB4FC15C06D13F86DEBFFC07F6D6C5B021F90C7FCEC7FFC4948C8FCEB03
F0EB0FC049C9FC133E5B5B485A1203485A5B120F485AA2123F90CAFC5AEE01C0160748ED
0F80EE1F005E6D147E167C6D5C6D495A6C6C495A01FCEB0FC0273FFF807FC7FC6CEBFFFE
6C14F86C14C0000191C8FC38003FF830427EBF30>I<047FB612FC0307B8FC031F178015
7F4AB9FC912903F80FE000011300DA0FC0ED007EDA1F00167C023E17604A011F92C7FC02
FC5C495AA213034A495A495A5C0106C7FC90C848CAFCA3167E16FEA34B5AA35E150393B6
12F0A24B5D614B92C8FC04E0CAFC5E151F5EA24BCBFCA25D157E15FE5DA24A5AA24A5AA2
4A5AA20003495A121F486C485A127F486C48CCFCEBE03E387FFC7CEBFFF86C5B6C13C06C
5BD801FCCDFC49417FBD41>I<033FB612F00207B7FC023F16E091B812800103EEFE0090
280FFC0007C0C7FCD91F80130F013EC7485A4992C8FC01FC5C48485C167E484814FE01C0
5C90C8FCC812015E1503A34B5AA35E150FA34B5AA44B5AA44BC9FCA415FEA35D1401A25D
14035DA24A5A18704A48EB01F04D5A4A48130792C7485A023E5D4A023FC7FC0007B712FE
001F16F8485E481680B700FCC8FC3C3E83BD32>73 D<020EED03F0023EED1FF802FEED7F
FC0103ED01FF010F923807F0FE011F92381FC07E013FED7F00017903FC133E0101DA03F0
133CDC07C01338DC1F8013004A013EC8FC16FCED01F04B5A0103495A031FC9FCECF83E5D
5DECF9F0903807FBE04A5AA2ECFF80A2EB0FEFA381EB1FCFA214C781EB3F87A2EC83F014
03137F6E7E137E13FE6E7EA248487F157E157F497F000381151F49800007140F496E143C
0307157C000F81496D6C14F8001F810301EC01F0496D6CEB03E0003F037FEB07C0943880
1F8090C8393FE07E00486FB45A007E6F13F000786F13C000E0DB01FEC7FC3F427DBF45>
75 D<F41F80F301FF1B07631B3F51130098B5FCED01C0DB07E04B5B030F95C7FC031F17
F0F203C0705E1A0798C8FC82621A0E153F70151E1A1CA2DB3BFE153C1539037916388203
7816781A7003707F047F15F015F0715CEDE03F1901020180041F5DA203C06D1303160F02
035F03807F04071407020794C9FC83DB00035CA24A6E130E020E130171131E141E70141C
021C1580057F133C023C15C002381638173F4AEDE078171F02F015F04A1670EF0FF80101
17F04AEC07FCD81003705A0038491403D83C0716FFD87F0FC87E01FF6F5B485A187F5B72
5A6C48041ECAFC6C4893CBFC6C5AEA0780594C82C64B>78 D<4AB6FC023F15F849B712FE
0107EEFF80011F17E090287FE1FC007F13F02601FC01020313F8D803F0030013FC2607C0
03ED3FFED80F80160FD81F00160748EF03FF484A80127E12FE488300F0130712C0C74915
FEA319FC020F15014B15F8A2F003F0A2021FED07E04B15C0F00F80F01F00183E4A485C4D
5AEF03E0EF0FC04AC7007FC7FCEE0FFE923807FFF8DA7E1F13C0DAFE3F90C8FCED7FF84B
C9FC4948CAFCA35C1303A25C1307A25C130F5CA2131F5C133FA291CBFC5B137EA25B13F0
13C040437EBD3F>80 D<4AB612C0023F15FE49B812C0010717F0011F8390287FE1FC001F
7F2601FC0102007FD803F0161FD807C0EE07FF260F800381D81F00825A4883007E5C12FE
486012F000C01307C75F4B140161611803020F4B5A4B5D4E5A4EC8FC183E4A4814FCEF01
F0EF0FE0EFFF8091263F803F90C9FCEEFFFC038113E015834A487F1500EE3FF8027E131F
02FE6D7EA24A6D7E130116034A80010380845C01078072EB01804A027F140F010F70EB1F
00624A6E6C137E011F187C4A6E6C5B72485A013F92390FFF0FE091C8ECFF80496F91C7FC
017E6F13FC01786F13E001E06F6CC8FC49407EBD4D>82 D<EE0FFE93387FFFC00303B512
F8030F14FC033F14FE4BC6FCDA01F8EB1FFFDA03E013034A487F4A5A4AC8FC4A15FE5C02
7E15FC02FE15F8EF01F018C0010192C7FC80A3816D7F8181EC7FFC6EB4FC6E13C06E13F0
6E13FC020113FF6E6C7F031F13E003077F03017F6F6C7E707E160F707E8201786E1380EA
01F8D807F080EA0FC04848157F123F90C9FC481700A2177E5A5F7F4C5A6D5D6D4A5A6C6C
4A5A6D4A5AD83FFE023FC7FC6C6C6C13FC6C9038F00FF86C90B512E06C1580C64AC8FC01
3F13F0010790C9FC38427EBF37>I<1A071A1F1A7E023FB812FC49B912F8010718E0011F
18C0017FEFFE0090B912F0D801F0C76CC9FCD807E05C120F48485B003F5D5B127F150348
C75B5A5A00F01407C85BA3150F5EA3151F5EA3153F5EA3157F5EA315FF93CAFCA35C5DA3
14035DA314075DA34A5AA34A5AA25D143FA24A5AA292CBFC14FE5C495A14C048477DC032
>I<D903F81607D91FFC161FD97FFE167ED801FF17FE1207D80FC317FCD81F011601121C
000049ED03F8A219F04A1507130319E04A150F13074A151F19C0010F163F5C011FEE7F80
5C18FF49C913005F5B017E5D01FE4B5A5B170F48484B5AA20003163F49157F00075F4915
FFEE01FB000F16F3494A485A001F1507EE0FC7491587003F4B485A163E167E48C812FC4C
485AED01F0ED03E048EC07C092380F803FED1F00033E5C6D5B5DEC03F09026C007C0137F
397FE01F80D9F87FC7138ED9FFFC15FE6C01F05D6C01C05D6C90C813E0D803F86F5ACA00
1EC7FC404182BD39>I<DC3FF8EB01C00303B5EAF007031F91B51280037F160092B75A02
035E913807E00791271F80000F5B4AC8EA07F04A5E02FE4B5A01014C5A4A4BC7FC494815
7E4A15FE02C04A5A0102C8485A90C95B4D5A4D5A4D5A4DC8FC177E5F4C5A4C5A92387FFF
FE92B67E4A81A291C7EA7E1F9338FC0780922601F802C8FC4B48C9FC4B5A4B5A4B5A033E
CAFC5D5DEC03F04A5A4A5A4A4815704AC8EA01F0027E15074A4B5AD901F0151F495A4948
4B5AD91F805E49C9FC017E4CC7FC01F816FE48B500C05C489139FFF801F8000F92B512E0
485F485F484CC8FC26F8003F14F800C0D9000713C0423E7CBD42>90
D<0060EE018000F0EE03C0B3B3A36C1607A200781780007C160FA26CEE1F00003F5E6C6C
157E6C6C5DD807F0EC03F8D803FCEC0FF06CB4EC3FE03B007FF003FF80011FB548C7FC01
0714F8010114E09026001FFEC8FC32397BB63D>I<EC1FFE49B512E0010714F8011F14FE
903A7FF003FF804848C7EA3FE0D803FCEC0FF0D807F0EC03F8D80FC0EC00FC4848157E48
C97E003E8248EE0F80A20078160700F817C0A2481603B3B3A30060EE018032397BB63D>
I<15C04A7E4A7EA24A7EA34A7EA2EC1F3EA2EC3E1FA2EC3C0F027C7FA24A6C7EA249486C
7EA2ECE001010380A249486C7EA24948137CA249C77EA2011E141E013E141FA2496E7EA2
496E7EA2491403000182A248486E7EA248486E7EA2491578000F167CA248C97EA2003E82
A2003C82007C1780A248EE07C0A24816030060EE018032397BB63D>94
D<0060EE018000F0EE03C06C1607A2007CEE0F80A2003C1700003E5EA26C163EA26C6C5D
A2000716786D15F8A26C6C4A5AA26C6C4A5AA200005E6D1407A2017C4A5AA26D4AC7FCA2
011E141E011F143EA26D6C5BA26D6C5BA26D6C485AA201015CECF003A26D6C485AA29138
7C0F80A2023C90C8FCEC3E1FA2EC1F3EA2EC0FFCA26E5AA36E5AA26E5A6E5A32397BB63D
>I<126012F0B3AAB812F017F8A300F0CAFCB3AB12602D3F7BBE38>I<387FFFFCB5FCA300
F0C7FCB3B3B3B3AD1260165A71C328>100 D<B512F814FCA3C7123CB3B3B3B3AD141816
5A7EC328>I<153FEC03FFEC0FE0EC3F80EC7E00495A5C495AA2495AB3AA130F5C131F49
5A91C7FC13FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE133F806D7E130F801307B3AA
6D7EA26D7E80EB007EEC3F80EC0FE0EC03FFEC003F205B7AC32D>I<12FCEAFFC0EA07F0
EA01FCEA007E6D7E131F6D7EA26D7EB3AA801303806D7E1300147FEC1FC0EC07FEEC00FF
EC07FEEC1FC0EC7F0014FC1301495A5C13075CB3AA495AA2495A133F017EC7FC485AEA07
F0EAFFC000FCC8FC205B7AC32D>I<146014F01301A214E01303A214C01307A2EB0F80A2
14005BA2131E133EA25BA2137813F8A25B1201A25B1203A2485AA25B120FA290C7FC5AA2
123EA2123C127CA2127812F8A41278127CA2123C123EA27EA27E7FA212077FA26C7EA212
017FA212007FA21378137CA27FA2131E131FA27F1480A2EB07C0A2130314E0A2130114F0
A213001460145A77C323>I<126012F07EA21278127CA2123C123EA27EA27E7FA212077F
A26C7EA212017FA212007FA21378137CA27FA2131E131FA27F1480A2EB07C0A2130314E0
A2130114F0A414E01303A214C01307A2EB0F80A214005BA2131E133EA25BA2137813F8A2
5B1201A25B1203A2485AA25B120FA290C7FC5AA2123EA2123C127CA2127812F8A25A1260
145A7BC323>I<126012F0B3B3B3B3B11260045B76C319>I E /FO
41 123 df<EB01FCEB0FFF90383F07C090387C03E03A01F801F00CEA03F03907E000F848
48141C49EBFC18001F147C48C713381630481570007E156016E016C000FE147D48EC7F80
1600157EA2157CA2127C15FE0203130C6CEB073E6C011E131C3A0F81F81F383A03FFE00F
F03A00FE0003C0261F7D9D2D>11 D<0103B512F0131F137F90B612E03A01FC1F80003903
F00FC03807C00748486C7E121F1300123EA25AA2140700FC5C5AA2140F5D141F92C7FC14
3E0078133C147C007C5B383C01E0381F07C0D807FFC8FCEA01F8241E7D9C28>27
D<160E486C143F120348C813801206000E151F000C150F001C160000188112380030130C
141E007015061260143E023C130E00E0150C5A0238131C6C01785B14705E02F013F06C48
6C485A010313033A7C0FFC0FC03A7FFFBFFF80023F90C7FC393FFC1FFE391FF80FF83907
E007E0291F7F9D2C>33 D<123C127E12FFA4127E123C08087A8714>58
D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009157A
8714>I<1780EE03C0160FEE3F80EEFE00ED03F8ED0FE0ED3F8003FEC7FCEC03F8EC0FE0
EC3F8002FEC8FCEB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F8000FECAFC12F812
FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00
FEED3F80ED0FE0ED03F8ED00FEEE3F80EE0FC01603EE00802A2B7AA537>I<013FB6FC17
C0903A00FE0007F0EE01F84AEB00FC177E1301177F5CA21303177E4A14FEA20107EC01FC
17F84AEB03F0EE07E0010FEC1FC0EE7F009138C003FC91B55A4914FE9139C0003F804AEB
0FC017E0013F140717F091C7FC16035BA2017E1407A201FE15E0160F4915C0161F0001ED
3F80EE7F004914FEED03F80003EC0FF0B712C003FCC7FC302D7CAC35>66
D<92387FC003913903FFF80791391FC03E0F91397E00071FD901F8EB03BF4948EB01FED9
0FC013004948147E49C8FC017E157C49153C485A120348481538485AA2485A1730484815
00A2127F90CAFCA35A5AA5EE018016031700A2007E5D1606160E6C5D5E6C6C5C000F5D6C
6C495A6C6CEB0780D801F8011EC7FCD8007E13F890381FFFE0010390C8FC302F7CAD32>
I<013FB71280A2D900FEC7127F170F4A1407A20101150318005CA21303A25C1630010714
7094C7FC4A136016E0130F15019138C007C091B5FC5BECC0074A6C5AA2133FA20200EB00
0CA249151C92C71218017E1538173001FE15705F5B4C5A000115034C5A49140F161F0003
4AB4C7FCB8FC5E312D7DAC34>69 D<013FB7FCA2D900FEC7127F171F4A140FA201011507
17065CA21303A25C16300107147017004A136016E0130F15019138C007C091B5FC5BECC0
074A6C5AA2133FA2020090C7FCA25B92C8FC137EA213FEA25BA21201A25BA21203B512F0
A2302D7DAC2D>I<92387FC003913903FFF80791391FC03E0F91397E00071FD901F8EB03
BF4948EB01FED90FC013004948147E49C8FC017E157C49153C485A120348481538485AA2
485A173048481500A2127F90CAFCA35A5AA292381FFFFCA29238003FC0EE1F80163F1700
A2127E5E167E7EA26C6C14FE000F4A5A6C7E6C6C1307D801F8EB1E3CD8007EEBFC389039
1FFFE018010390C8FC302F7CAD37>I<90273FFFFC0FB5FCA2D900FEC7EA3F80A24A1500
A201015D177E5CA2010315FE5F5CA2010714015F5CA2010F14035F5C91B6FC5B9139C000
07E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7FC5BA200015D167E5BA2
000315FEB539E03FFFF8A2382D7CAC3A>I<90383FFFFEA2010090C8FC5C5CA21301A25C
A21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7EA0180A2491403170001
7E5C160601FE140EA2495C163C12015E49EB01F84B5A0003141FB7FC5E292D7DAC30>76
D<013FB512F816FF903A00FE001FC0EE07E04A6D7E707E01016E7EA24A80A213034C5A5C
A201074A5A5F4A495A4C5A010F4A5A047EC7FC9138C003F891B512E04991C8FC9138C007
C04A6C7E6F7E013F80150091C77EA2491301A2017E5CA201FE1303A25BA20001EE038018
005B5F0003913801FC0EB5D8E000133CEE7FF0C9EA0FC0312E7CAC35>82
D<913807F00691383FFE0E9138F80F9E903903E001FE903807800049C7127C131E49143C
A2491438A313F81630A26D1400A27FEB7F8014F86DB47E15F06D13FC01077F01007F141F
02011380EC003F151F150FA215071218A3150F00381500A2151EA2007C5C007E5C007F5C
397B8003E039F1F00F8026E07FFEC7FC38C00FF0272F7CAD2B>I<000FB8FCA23B1FC003
F8003F0100151F001C4A130E123C003801071406123000704A130EA20060010F140C12E0
485CA2141FC715005DA2143FA292C8FCA25CA2147EA214FEA25CA21301A25CA21303A25C
A21307A25C130F131F001FB512F0A2302D7FAC29>I<90260FFFFCEB7FFFA29026007FC0
EB0FF06E48148018006E6C131E1718020F5C6F5B02075C6F485A020349C7FCEDF8065E6E
6C5A5E6E6C5A5EED7F8093C8FC6F7EA26F7E153F156FEDCFE0EC018791380307F0EC0703
020E7F141C4A6C7E14704A6C7E495A4948137F49C7FC010E6E7E5B496E7E5BD801F081D8
07F8143FD8FFFE0103B5FCA2382D7EAC3A>88 D<B500C090380FFFC0A2D807F8C73801FC
00000316F05F6C6CEC038094C7FC16066C6C5C5E017F5C16706D5C6E485A4B5A6D6C48C8
FC15066D6C5A5D5D6D6C5A5D903803F98002FBC9FC14FF6D5A5C5CA35C1303A35C1307A3
5C130FA3131F0007B5FCA2322D7DAC29>I<0107B612F8A2903A0FFC0007F002E0EB0FE0
0280EB1FC049C71380011E143F011CEC7F004914FE4B5A0130495A0170495A0160495A4B
5A4B5A90C748C7FCA215FE4A5A4A5A4A5A4A5A4A5A4A5A4AC8FC14FE5C13014948130649
5A4948130E4948130C495A49C7121C01FE141848481438485A5E484814F0484813014848
13034848495A48C7127FB7FC5E2D2D7CAC30>I<EB07E0EB1FF890387C1CE0EBF80D3801
F00F3803E007EA07C0120FD81F8013C0A2EA3F00140F481480127EA2141F00FE14005AA2
EC3F02EC3E06A25AEC7E0E007CEBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0F
F03903F003C01F1F7D9D25>97 D<EB01F8EB0FFE90383E0780EB7C01D801F813C03803F0
073807E00FEA0FC001801380121F48C8FCA25A127EA312FE5AA51560007C14E0EC01C0EC
03806CEB0F00001E131C380F81F83807FFE0C648C7FC1B1F7D9D1F>99
D<151FEC03FFA2EC003FA2153EA2157EA2157CA215FCA215F8A21401EB07E190381FF9F0
EB7C1DEBF80FEA01F03903E007E0EA07C0120FEA1F8015C0EA3F00140F5A007E1480A214
1F12FE481400A2EC3F021506143E5AEC7E0E007CEBFE0C14FC0101131C393E07BE18391F
0E1E38390FFC0FF03903F003C0202F7DAD24>I<EB03F8EB0FFE90383E0780EBF803D801
F013C03803E001EA07C0000F1303D81F8013801407393F000F00141E387F01FCEBFFF091
C7FC007EC8FC12FE5AA4127C156015E0EC01C06CEB0380EC0F006C131C380F81F83803FF
E0C648C7FC1B1F7D9D21>I<157C4AB4FC913807C380EC0F87150FEC1F1FA391383E0E00
92C7FCA3147E147CA414FC90383FFFF8A2D900F8C7FCA313015CA413035CA413075CA513
0F5CA4131F91C8FCA4133EA3EA383C12FC5BA25B12F0EAE1E0EA7FC0001FC9FC213D7CAE
22>I<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E1238
12301270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F1380
1381381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105
D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C013
30EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA2
1301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B
81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F0
03FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13
F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038
ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA21200A213
F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123E
A2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E
2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01
263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A4914800081021F
130300015F491400A200034A13076049133E170F0007027EEC8080188149017C131F1801
000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB
03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B801F3870
7F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF80816
18EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E
143E251F7E9D2B>I<EB01F8EB0FFF90383F078090387C03C0D801F813E03903F001F0EA
07E0D80FC013F8EB8000121F48C7FC14015A127EA2140300FE14F05AA2EC07E0A2EC0FC0
A2007CEB1F801500143E6C5B6C485A380F83E03803FF80D800FCC7FC1D1F7D9D22>I<90
387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C0000713F000069038E003E0
EB0FC0000E1380120CA2D8081F130712001400A249130F16C0133EA2017EEB1F80A2017C
14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FFC7FCEBF0FC000390C8FC
A25BA21207A25BA2120FA2EAFFFCA2232B829D24>I<903807E03090381FF87090387C1C
F0EBF80D3801F00F3903E007E0EA07C0000F1303381F800715C0EA3F00A248130F007E14
80A300FE131F481400A35C143E5A147E007C13FE5C1301EA3E07EA1F0E380FFCF8EA03F0
C7FC13015CA313035CA21307A2EBFFFEA21C2B7D9D20>I<3807C01F390FF07FC0391CF8
E0E0383879C138307B8738707F07EA607E13FC00E0EB03804848C7FCA2128112015BA212
03A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B1F7E9D20>I<EB07E0EB3FF8EB
781EEBF0063801E0073803C00F141FA20007131E140CEBE00013F8EBFF806C13E06C13F0
6C13F8EB3FFC13011300147C0078133C12FCA2147C48137800E013F814F0386001E03878
07C0381FFF00EA07F8181F7C9D21>I<130E131FA25BA2133EA2137EA2137CA213FCA2B5
12F8A23801F800A25BA21203A25BA21207A25BA2120FA25BA2001F131014301300147014
6014E0381E01C0EB0380381F0700EA0F0EEA07FCEA01F0152B7EA919>I<EA01E0D807F8
130ED80E3C131FD81C3E5B0038143E12301270D8607E137ED8E07C137C12C013FC484813
FC00005C12015B140100035C13E0A2020313200007ECE06013C0A216E0020713C00003EB
0FC09038E01FC191383BE1803901F071E33A007FE0FF0090381F803C231F7E9D29>I<D8
01E01370D807F813F8380E3C01D81C3E13FC1238003013000070147CEA607ED8E07C1338
12C013FC485A0000143012015B1570000314605B15E015C01207EBC00115801403EC0700
00031306EBE00E00015BEBF07838007FE0EB1F801E1F7E9D22>I<013F137C9038FFC1FF
3A01C1E383803A0380F703C0390700F60F000E13FE4813FC12180038EC0700003049C7FC
A2EA200100005BA313035CA301075B5D14C000385CD87C0F130600FC140E011F130C011B
131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E9D28>120
D<EA01E0D807F8130ED80E3C131FD81C3E133F0038143E12301270D8607E137ED8E07C13
7C12C013FC484813FC000014F812015B1401000314F013E0A21403000714E013C0A21407
15C00003130FEBE01F143F3901F07F8038007FEFEB1F8FEB001F1500A2003E133EA2007E
5B5C387C01F0387003E0383007C0383C0F80D80FFEC7FCEA03F0202C7E9D23>I<011E13
30EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F80903800070048130EC75A5C
5C5C495A495A49C7FC131E13385B491340484813C0485A38070001000EEB0380380FE007
391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F7D9D21>I
E /FP 76 127 df<EC07F8EC3FFF9138FC0F80903903F003E090270FC001F01370D91F80
6D13F0017FC714E001FE804848147E000316014916C04848143F000FEE03805B001F1607
003F1700495D170E007F161E49151C173C5F00FF167090C813F05F5F5FA294C7FC7EA25E
6C5D4B7F6C6C9039079F81C0ED0F1F6C6C90383C0F836C6C01F014803C01F80FC007C700
3B007FFF0003FED91FF0EB00F834297DA73A>11 D<EE1FE0EEFFFC923803F03E92390F80
0F8092381E0007033814C003F0EB03E04A5A4A5A4A4814F092C7FC140E141E021C14074A
15E01478147002F0140F4A15C013014AEC1F80EF3F0013034A147E5F010790381FF9F091
39007FFFE09238700FC092387FFFE04990381FF1F0010E90380001F8707E83011E157E01
1C157FA3133C1338A301785D1370A301F05C5F5BA200014B5AA25F1607486C5D4C5A161F
6D5DD807B84A5A011C4AC7FC011E14FC6D495A3A0F078007E03A0E01F01F809026007FFE
C8FCEC0FF0001E90CAFC121CA3123C1238A312781270A312F05AA434527EBF33>I<EB03
F8D90FFE140F90383FFF8090B56C131E4880486E131C2607FC1F143C270FE003F8133826
1F8000147890C7007C1370003E023C13F0003C023E13E048EC1E010070020E13C000F0EC
0F034816801507C81307EE8700168FED038E169E169CA216BC16B8A216F85EA25EA35EA3
5EA31507A293C7FC5DA4151EA3153E153CA45DA31570A21560303C7FA72F>I<EC01C0EC
0FFF023F13E0027913F0ECE03F903801C00FEC8007ED01E0010390C7FCA280130180A280
A26D7E80A2147E147F6E7EA26E7E81140FEC7FF0903801F7F8EB07C790381F83FCEB3E03
EB7C0101F87F12013803F000485AA24848137E485AA2123F90C7FCA25A127EA300FE147C
5A15FCA25D1401A25D007C13035DA26C495A4A5A6C91C7FC6C131E6C6C5A3803E0783800
FFE0EB3F8024427CC028>I<D801F0EB0FE0D803FCEB7FFC3A071F01F03E3A0E0F03801F
001E90398F000F80001C139E003C01FC14C000385B5C38781FE012705CA200F049131F01
3F1580000090C7FCA2163F5B017E1500A25E13FE49147EA216FE1201495CA21501120349
5CA215031207495CA21507120F495CEA0380C8120FA25EA2151FA25EA2153FA293C7FCA2
5DA2157EA3157C15382A3C7EA72D>17 D<15FCEC03FF91380F87C091383E03E0EC7C0102
F813F01301903903F000F8495A010F14FC5C495A133F91C7FC4914FE13FEA212015B1203
4913011207A25B000F15FC1503121F5BA21507003F15F890B6FCA33A7FC0000FF05BA215
1F16E048C7FCA2ED3FC0A2481580157F1600A215FEA24A5AA24A5A007E5C14075D4A5A00
3E5C141F4AC7FC6C137E5C380F81F03807C3E03801FF80D8007EC8FC27417DBF2B>I<13
38137C13FCA312015BA312035BA312075BA3120F5BA2121F5BA2123F90C7FCA248130E12
7EA200FE131E48131C143C147848137014F0EB01E0EB03C0EB0F00EA7C3EEA3FF8EA0FC0
17297BA720>I<133F14E0EB07F0EB03FC13016D7EA3147FA26E7EA36E7EA36E7EA36E7E
A36E7EA26E7EA36E7EA3157FA36F7E157F15FF4A7F5C913807CFE0EC0F8FEC1F0F91383E
07F0147C14FC49486C7EEB03F0EB07E049486C7EEB1F80EB3F00496D7E13FE4848147F48
5A485A4848EC3F80485A123F4848EC1FC048C8FC4816E048150F48ED07F0007015032C40
7BBE35>21 D<EB01C0496C14E00107EC03F0A3010F1407A24A14E0A2011F140FA24A14C0
A2013F141FA291C71380A249143FA2017E1500A201FE5CA249147EA2000115FE17074914
FCA215010003160F923803F80EA200070207131E030F131C6D131F033C1338486C137090
3AFF81E07C70903AC7FF803FE0903AC1FE000F80D81FC0C9FCA25BA2123FA290CAFCA25A
A2127EA212FEA25AA35A1270303C7EA737>I<EC01C014035DA5EDFFF04A13F891383FE0
3891B512F8903901FC3FE0D907F8C7FC495A495A495A137F5C13FF91C8FCA25A5BA61200
7F7F90383F9FFC6DB47E903807E00E90380FFFFE90383E3FF80178C8FC5B485A485A485A
48C9FCA2121E5AA25AA312F85AA37EA27E7E7E6C7E13E0EA3FF86CB4FC6C13C06C13F800
0113FE39007FFFC0010F13F001037F9038007FFCEC1FFE14031400157E153EA3153C010C
137C010F1378903803C0F0903800FFC0023FC7FC25527EBE28>24
D<ED1FC0EDFFF0913803E07C91380F803E4A487E023E14804AEB0FC05C494814E0130349
5A5C130F494814F0A2133F91C7FCEE1FE05B137EA201FE143F17C05BA20001ED7F80A249
15005E00035D4B5AA24B5A486C495A4B5A6D5C01EEEB3F80D80FE7017EC7FC9038E3C1F8
9038C1FFE0D9C07FC8FC001F90C9FCA25BA2123FA290CAFCA25AA2127EA212FEA25AA35A
12702C3C7EA72F>26 D<020FB512FE027F14FF49B7FC1307011F15FE903A3FE03FE00090
387F000F01FE6D7E4848130348488048481301485A5B121F5B123F90C7FC5A127EA21503
00FE5D5AA24B5AA2150F5E4B5AA2007C4AC7FC157E157C6C5C001E495A001FEB07E0390F
800F802603E07EC8FC3800FFF8EB3FC030287DA634>I<011FB612C090B7FC5A5A481680
260FC007C8FC48C65A123E003C130E48131E5A5AA2C75AA3147CA2147814F8A4495AA313
03A25CA21307A3495AA3131FA25C6DC9FC2A287DA628>I<1678A21670A216F0A25EA215
01A25EA21503A25EA21507A293C7FCA25DA2150EEDFFC0020F13FC91383F9E3F903A01F8
1C0FC0D903E0EB03E0903A0FC03C01F0D91F00EB00F8017E0138137C5B48480178133E48
5A48480170133F120F4901F0131F485A5D48C7FC0201143F5A007E5CA20203147F00FE16
7E485C17FE020714FC1601007C020013F8EE03F0007E49EB07E0A2003E010EEB0FC0003F
ED1F806C011EEB3F00D80F80147C3A07C01C01F8D803E0EB03E03A01F03C1F80D8007E01
FEC7FC90381FFFF801011380D90078C8FCA21470A214F0A25CA21301A25CA21303A25CA2
1307A230527CBE36>30 D<13FE2603FF80157026078FE015F0260F07F01401000E6D15E0
0103ED03C0000C6DEC0780D80001ED0F006E141E01005D5F027F5C4C5A91383F80035F4C
5A6E6C48C7FC161E5E6E6C5A5EEDE1E0913807E3C015F75E6EB4C8FC5D5D5D6E7EA21403
14074A7EA2141EEC3C7F147814F049486C7EEB03C0EB078049486C7EA2131E496D7E5B49
8048481307485A48486D7E48C7FC48EDFC03001E0201EB07804803FE1300486E6C5A48ED
7F1E0060ED1FFCC9EA03F0343B7EA739>I<EE01C0A21603A25FA21607A294C7FCA25EA2
160EA2161EA2161CA2163CA21638A21678017C167048B491387001FC2603C7C0EC03FED8
070314F0000F7F000E15E0121C010701011301003816004C137ED8780F163E0070EBC003
181ED8F01F5C0280151C00001407133F020090C7123C1838495B137E030E147801FE1670
49011E14F018E0031C13010001EE03C049013C148017070338EB0F006C6C151E03785B5F
017E01705B4C5A6D9038F003C0D91F80010FC7FC90390FE0E03E903903FCE1F89039007F
FFE0020790C8FCEC01C0A21403A25DA21407A292C9FCA25CA2140EA2141EA2141CA23752
7EBE3B>I<EC1FF0903801FFFC010713FF011F14C090397FC01FE09038FC0007D801F013
01D803C090C7FC485AA290C9FC5A120EA2120F7E7F3803CFFF6CB57E023FC7FC3803DFFE
380780E0000EC9FC5A123C5A127012F05AA515066C140E5D00785C007C14F0393F8007E0
6CB55A000791C7FC000113FC38007FE0232B7DA82A>34 D<16FCED03FF92380F07809238
1E03C0153C037813E0EDF00103E013F0140115C01403158017F8A214071500A415800203
130315C0140115E02701F000F013F0D803FCEB7807D8071F133CD80E0F131F001E903880
0787001CEC01FF003C9039C0007FE00038151FEE0FFED8781FEC1FFF00704914C0A3D8F0
3F143F1780000090C7FCA249EC7F00A2017E147E16FE13FE5E4913015E150300015D4949
5AA24B5AA24B5A93C7FC153E5D12006D5B90387E01E06D485AD90FFFC8FCEB01FC30417E
BF36>I<EE03F801E0EC0FFE0001ED3FFF4991B5128000034A14C04848903903F80FE090
C73807E0034891390FC001F0000E1500001E021E1300001C5C003C177000385C15700078
14F000705C140100F04A14F018E048495A17014AC7FC18C01703020EEC07807EEF0F006C
011E141E0078163E007C011C5C6C5E003F013CEB03F0D81F804A5AD80FE0EC1FC02607FC
78EBFF802803FFF807FEC7FC6C90B55A6C6C14F0011F14C0010749C8FC010013F0D901F0
C9FCA3495AA31307A25CA2130FA3495AA491CAFC130E343C7CA73B>39
D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919>58 D<121EEA7F8012FF13C0A2
13E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C
798919>I<1806181F187FEF01FFEF07FCEF1FF0EF7FC0933801FF00EE07FCEE3FF0EEFF
C0030390C7FCED0FFCED3FE0EDFF80DA03FEC8FCEC0FF8EC7FE0903801FF80D907FEC9FC
EB1FF8EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA07FCEA01FF38
007FC0EB1FF8EB07FE903801FF809038007FE0EC0FF8EC03FE913800FF80ED3FE0ED0FFC
ED03FF030013C0EE3FF0EE07FCEE01FF9338007FC0EF1FF0EF07FCEF01FFEF007F181F18
06383679B147>I<ED0180ED03C01507A21680150FA216005DA2151E153EA2153C157CA2
157815F8A25D1401A25D1403A25D1407A25D140FA24AC7FCA2141E143EA2143C147CA214
7814F8A25C1301A25C1303A25C1307A25C130FA291C8FC5BA2131E133EA25BA2137813F8
A25B1201A25B1203A25B1207A25B120FA290C9FC5AA2121E123EA2123C127CA2127812F8
A25A1260225B7BC32D>I<126012F812FE6C7EEA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0F
FCEB03FF010013C0EC3FF0EC07FCEC01FF9138007FC0ED1FF0ED07FE923801FF80923800
7FE0EE1FF8EE03FE933800FF80EF3FE0EF0FF8EF03FEEF00FFA2EF03FEEF0FF8EF3FE0EF
FF80933803FE00EE1FF8EE7FE0923801FF80DB07FEC7FCED1FF0ED7FC04A48C8FCEC07FC
EC3FF0ECFFC0010390C9FCEB0FFCEB3FE0EBFF80D803FECAFCEA0FF8EA3FE0EAFF8048CB
FC12F81260383679B147>I<17075F84171FA2173F177FA217FFA25E5EA24C6C7EA2EE0E
3F161E161C1638A21670A216E0ED01C084ED0380171FED07005D150E5DA25D157815705D
844A5A170F4A5A4AC7FC92B6FC5CA2021CC7120F143C14384A81A24A140713015C495AA2
49C8FC5B130E131E4982137C13FED807FFED1FFEB500F00107B512FCA219F83E417DC044
>65 D<49B712F818FF19E090260001FEC7EA3FF0F007F84B6E7E727E850203815D1A80A2
0207167F4B15FFA3020F17004B5C611803021F5E4B4A5A180FF01FE0023F4B5A4B4A5ADD
01FEC7FCEF07F8027FEC7FE092B6C8FC18E092C7EA07F84AEC01FE4A6E7E727E727E1301
4A82181FA213034A82A301075F4A153FA261010F167F4A5E18FF4D90C7FC011F5E4A1403
4D5A013FED1FF04D5A4AECFFC0017F020790C8FCB812FC17F094C9FC413E7DBD45>I<DC
1FF81307923801FFFE030F9038FF800E923A7FF007E01E4A48C7EAF03EDA03FCEC787EDA
0FF0EC3CFCDA3FC0141F4A48140F4AC8FC4948ED07F8EB07F849481503131F4A16F04948
1501495A13FF4890C913E05B1203485A19C0485AA2485A95C7FC123F5BA2127F5BA312FF
5BA590CCFC183CA21838A21878187018F06C6C5E17014D5A003F5F6D15074DC7FC001F16
1E6C6C5D6D5D6C6C5D00034B5AD801FEEC07C06C6C4AC8FCD97FC0137E90391FF803F801
07B512E0010114809026001FF8C9FC40427BBF41>I<49B712F818FF19C0D9000190C7EA
3FF0F00FF84BEC03FCF000FE197F0203EE3F805DF11FC0A20207EE0FE05D1AF0A2020F16
075DA21AF8141F5DA2190F143F5DA21AF0147F4B151FA302FF17E092C9123FA21AC04917
7F5C1A8019FF010318005C4E5A61010716034A5E4E5A180F010F4C5A4A5E4E5A4EC7FC01
1F16FE4A4A5AEF07F8013FED0FE0EF3FC04A49B4C8FC017FEC0FFCB812F017C004FCC9FC
453E7DBD4B>I<49B912C0A3D9000190C71201F0003F4B151F190F1A80020316075DA314
075D1A00A2140F4BEB0380A205075B021FED000E4B92C7FC5FA2023F141E5D173EEE01FE
4AB55AA3ED800102FF6D5A92C71278A34915705C191C05F0133C01034B13384A167894C7
1270A2010717F04A5E180161010F16034A4B5AA2180F011F4CC7FC4A5D187E013F16FE4D
5A4A140F017F15FFB95AA260423E7DBD43>I<49B9FCA3D9000190C7120718004B157F19
3F191E14035DA314075D191CA2140F5D17074D133C021F020E13384B1500A2171E023F14
1C4B133C177C17FC027FEB03F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101
035D5CA293C9FC13075CA3130F5CA3131F5CA2133FA25C497EB612F8A3403E7DBD3A>I<
DC3FF0130E923803FFFE031F9038FF801C923A7FF00FC03C913B01FF0001E07CDA07FC90
3800F0FCDA0FF0EC79F8DA3FC0143F4A48141F4AC8120FD903FC16F0495A49481507495A
013F17E04A1503495A49C9FC4818C05B1203485A1980485AA2485A95C7FC123F5BA2127F
5BA312FF5BA3043FB512E0A290C8FC9339001FF800170F60A2171FA260A26C6C153FA260
6C7E177F121F6D5E6C6C15FF00075D6C6C5C6C6CDA07BFC7FC6CB4EC1F1FD97FC0EB7E0F
90391FF803F80107B5EAE00601010280C8FC9026001FF8C9FC3F427BBF48>I<49B6D8C0
3FB512F81BF01780D900010180C7383FF00093C85B4B5EA2197F14034B5EA219FF14074B
93C7FCA260140F4B5DA21803141F4B5DA21807143F4B5DA2180F4AB7FC61A20380C7121F
14FF92C85BA2183F5B4A5EA2187F13034A5EA218FF13074A93C8FCA25F130F4A5DA21703
131F4A5DA2013F1507A24A5D496C4A7EB6D8E01FB512FCA2614D3E7DBD4C>I<49B612C0
5BA2D90001EB800093C7FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5D
A314FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612E0
A32A3E7DBD28>I<92B612E0A39239003FF000161F5FA2163F5FA3167F5FA316FF94C7FC
A35D5EA315035EA315075EA3150F5EA3151FA25EA2153FA25EA2157FA25EA2D80F8013FF
EA3FC0486C91C8FCA25CD8FFC05B140301805B49485A00FC5C0070495A0078495A003849
5A001E017EC9FC380F81FC3803FFE0C690CAFC33407ABD32>I<49B600C090387FFFF896
B5FC5FD900010180C7000F130093C813F84B16E01A804FC7FC0203163C4B15F84E5AF003
C002074B5A4B021FC8FC183E1878020F5D4BEB03E0EF07804DC9FC021F143E4B5B17F04C
5A023F1307EDC00F4C7E163F027FEBFFF8ED81EFED83CF92388F87FC9138FF9F0792383C
03FE15784B6C7E4913E0158092C77F5C01036F7E5C717EA213074A6E7EA2717E130F4A6E
7EA284011F15035C717E133F855C496C4A13E0B600E0017F13FFA34D3E7DBD4D>I<49B6
12F0A3D900010180C7FC93C8FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA314
7F5DA314FF92C9FCA35B5C180C181E0103161C5C183C183813074A1578187018F0130F4A
EC01E0A21703011FED07C04A140F171F013FED3F8017FF4A1303017F021F1300B9FCA25F
373E7DBD3E>I<49B56C49B512F81BF0A290C76D9039000FFE004AEE03F0705D735A03DF
150302037F038F5E82190791380787FC030793C7FC1503705C140F91260E01FF140EA26F
151E021E80021C017F141C83193C023C6D7E02381638161F711378147802706D6C1370A2
040714F002F0804A01035C8318010101EC01FF4A5E82188313034A91387FC380A2EF3FC7
010716E791C8001F90C8FC18F718FF4981010E5E1707A2131E011C6F5AA2013C1501137C
01FE6F5AEA03FFB512FC187818704D3E7DBD49>78 D<49B712F018FF19C0D9000190C76C
7EF00FF84BEC03FC1801020382727E5DA214071A805DA2140F4E13005DA2021F5E18034B
5D1807023F5E4E5A4B4A5A4E5A027F4B5A06FEC7FC4BEB03FCEF3FF091B712C005FCC8FC
92CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C497EB612
E0A3413E7DBD3A>80 D<EE3FF00303B5FC92391FC03FC092397E0007E0DA01F8EB01F8DA
07E06D7E4A48147EDA3F8080027EC813804AED1FC0EB03F84948ED0FE0130F494816F04A
1507494816F8137F49C9FC485AA2484817FCA2485A120FA2485AA25B123F19F84848160F
A44848EE1FF0A3F03FE0A290CAFCF07FC0A2198018FF19004D5AA24D5A606C16074D5A6D
01F85C003FD903FE495ADA0F07495A271FC01C0349C7FC9139380180FE260FE030EB81FC
EEC3F82607F070EBC7E03B03F86000CFC0D801FCECFF80D800FE4AC8FC90393FF003F890
270FF81FE0130C0103B5FC9026007FF1141CDA00011418183882607013F017039338FC0F
E093B5FC6060A26F91C7FC5F705AEE3FF0EE0FC03E527BBF48>I<49B77E18F818FFD900
01D900017F9438003FE04BEC0FF0727E727E14034B6E7EA30207825DA3020F4B5A5DA24E
5A141F4B4A5A614E5A023F4B5A4B4A5A06FEC7FCEF03FC027FEC0FF04BEBFF8092B500FC
C8FC5F9139FF8001FE92C7EA7F80EF1FC084496F7E4A1407A28413035CA2170F13075C60
171F130F5CA3011F033F5B4AEE038018E0013F17071A004A021F5B496C160EB600E09038
0FF01E05075B716C5ACBEAFFE0F03F8041407DBD45>I<DB07FC131C92383FFF8092B5EA
E038913A03F803F078913A0FE000F8F8DA1F80133D4AC7EA1FF0027E140F5C494814074A
15E0130349481403A2010F16C05CA3011F1680A38094C7FC808014FE90380FFFC015FC6D
EBFFC016F86D14FE6D806D81023F800207801400030F7F1500163F707E160F1607A21603
12075A5F120EA2001E15075FA24C5A123E003F4B5AA26D4AC7FC007F157E6D5C6D495AD8
7DF0495AD8F8FCEB0FE090393F803F8027F01FFFFEC8FCD8E00713F839C0007FC036427B
BF38>I<48B912FCA25A913A0003FE000F01F84A1301D807E0EE00F8491307491778000F
5D90C7FC001E140FA2001C4B1470123C0038141FA200785D1270033F15F000F018E0485D
C81600157FA25EA215FFA293C9FCA25CA25DA21403A25DA21407A25DA2140FA25DA2141F
A25DA2143FA25DA2147FA214FF497F001FB612FCA25E3E3D7FBC35>I<007FB500F09038
7FFFFE19FC5D26007FE0C7000313804A913800FC004A5D187001FF16F0A291C95AA24816
01605BA200031603605BA20007160795C7FC5BA2000F5E170E5BA2001F161E171C5BA200
3F163C17385BA2007F1678A2491570A200FF16F0A290C95AA216015F5A16035F16074CC8
FC160E161E5E007F5D5E6C4A5A6D495A6C6C495A6C6C011FC9FC6C6C137E3903FC03F8C6
B512E0013F1380D907FCCAFC3F407ABD3E>I<B6020FB5FCA219FE000301C0020013E06C
90C9EA7F00183E183C6C5F187060A24D5A17036E5D4DC7FC017F5D170E5FA25F17786E14
705F133F4C5A4C5AA24CC8FC5E6E130E5EA2011F5C167816705E15015E6E485AA2010F49
C9FC5D150E5DA25D6E5AA201075B14F95DECFB80A202FFCAFC5CA25C13035C5CA25CA25C
40407BBD35>I<B6017FB5D88007B51280A24A1A0000030180010101E0C7EA7FF049C801
80EC1FC0000194C85B99C7FC1B1E1B1C63A2634C7F634C150163DC077F4A5A160F6D020E
4BC8FC161C6C190E1638620470153C04F0153804E05DED01C062DB03807F4F5A92260700
3F130362030E4BC9FC151EDA801C150E5D017F5F5D614B1578028116704B5DEC8380F0C1
C00287C713E1F0E380028EEC1FE796CAFC029C15EE14BC02F815FC5C013F5E5C605C604A
5D91C8FC60133E95CBFC013C81170E59407BBD56>I<027FB5D88007B512C091B6FCA202
0101F8C7EBF8009126007FE0EC7F804C92C7FC033F157C701478616F6C495A4E5A6F6C49
5A4EC8FC180E6F6C5B606F6C5B6017016F6C485A4D5A6F018FC9FC179E17BCEE7FF85F70
5AA3707EA283163F167FEEF7FCED01E7EEC3FEED0383ED070392380E01FF151E4B6C7F5D
5D4A486D7E4A5A4A486D7E92C7FC140E4A6E7E5C4A6E7E14F0495A49486E7E1307D91F80
6E7ED97FC014072603FFE0EC1FFF007F01FC49B512FEB55CA24A3E7EBD4B>I<B66C0103
B51280A3000101E0C8387FF0006C49ED3F80017F94C7FC183C606D6C1570606D6C4A5A17
034D5A6D6C4AC8FC170E5F6D6C5C17785F6D6C495A5F6E495A6D4AC9FC160E6DEB801E5E
5E91387FC0705EEDC1C0EC3FE3EDE78003FFCAFC6E5A5D6E5AA25DA25D141FA35D143FA3
5D147FA392CBFC5CA3495AA3497E0007B512FEA3413E7DBD35>I<027FB712F0A3DAFFFC
C7EA3FE003E0EC7FC092C8EAFF8049484A13004A4A5A5C4A4A5A49484A5A4A4A5A4D5A49
484A5A4D5A91C74890C7FC5B010E4A5A4C5A4C5A011E4A5A90C8485A4C5A4C5A4B90C8FC
A24B5A4B5A4B5A4B5A4B5A4B5A4B5AA24A90C9FC4A5A4A5A4A5A4A4814704A4814F04A48
5C14FF5D4990C7120149485D49481403495A49485D49481407495A4DC7FC49485C4890C8
FC48485D4848157E484815FE484814034848EC0FFC16FF48B7FCB8FC5F3C3E7BBD3E>I<
1438A5120C121CAC143B143F14FF1307131F90B5FC121F5AB512FC14F8143813F813E013
0012FC12DC121CB3A5143B143F14FF1307131F90B5FC121F5AB512FC14F8143813F813E0
130012FC12DC121CA614301400A518557BC023>93 D<EC1F80ECFFE0903903F070709039
0FC039F890381F801D90383F000F017E5C5B00011407485A48485CA2485A001F140F5E48
5AA2151F007F5D5BA2153F00FF92C7FC90C7FCA25D92387E03805AA215FEEDFC07007E01
01140014035E6C0107130E140E3A1F801C7C1C000F13783A07C1F03E383A01FFC01FF03A
007F0007C029297DA730>97 D<EB1FC0EA0FFF5CA2EA003FA291C8FCA25BA2137EA213FE
A25BA21201A25BA21203A25BEC3F800007EBFFE09038F3C1F849C67E01FE137E4848133E
49133F5B491480001F141F5B5BED3FC0123FA290C7FCA248147F1680127EA215FF00FE15
005AA24A5AA25D1403485C1407007C5C4A5A5D003C495A003E49C7FC001E137E6C13F838
0783F03803FFC0C648C8FC22407CBE27>I<EC07F0EC7FFE903801FC0F903907E0038090
390FC001C0D93F8013E090387F000701FE131F485A485A16C0485A000F15804990C7FC12
1F485AA3127F5BA312FF90C9FCA6007E1560007F15E01501ED03C06CEC07806DEB0F0000
1F141E6C6C137C3907E001F03901F01FC06CB5C7FCEB1FF023297DA727>I<EE07F0ED03
FF17E0A2ED000FA217C0A2161FA21780A2163FA21700A25EA2167EA216FEA25EEC1F80EC
FFE1903803F07190390FC039F890381F801D90383F000F137E495C00011407485A485A5E
485A001F140FA248485CA2151F127F495CA2153F12FF90C790C7FCA25DEE038048147EA2
15FE1607007ED901FC130014035E6C0107130E140E3A1F801C7C1C000F13783A07C1F03E
383A01FFC01FF03A007F0007C02C407DBE2F>I<EC1FE0ECFFFC903803F01E90380FC00F
90393F800780D97E0013C0491303EA03F8120749130748481480121F49130F003FEC1F00
153E397F8001FCEC1FF0B6128002F8C7FC90C9FCA45AA616C01501007E1403ED07806CEC
0F00151E6C5C6C6C13F83907C003E03903E03F802600FFFEC7FCEB3FE022297CA72A>I<
163EEEFFC0923803E1E0923807C0F0ED0F811687ED1F8F160F153FA217E092387E038093
C7FCA45DA514015DA30103B512FCA390260003F0C7FCA314075DA4140F5DA5141F5DA414
3F92C8FCA45C147EA414FE5CA413015CA4495AA35CEA1E07127F5C12FF495AA200FE90C9
FCEAF81EEA703EEA7878EA1FF0EA07C02C537CBF2D>I<EC01F8EC0FFE91383F07879139
FC03DF80903801F801903903F000FFEB0FE04948EB7F005C133F49C7FC49147E5B000115
FEA248485CA215011207495CA21503120F495CA21507A25E5B0007140FA24B5A6D133F00
03147F000114FF6D485B0000EB03DF90387E0F3FEB1FFCD907F090C7FC90C7FC5DA2157E
A215FEA25D001C1301007F5C4813035D4A5A4A5A48495A00F8017EC8FC387E01FC381FFF
E0000390C9FC293B7FA72B>I<EB01FC13FF5CA21303A25CA21307A25CA2130FA25CA213
1FA25CA2133FA291C9FC15FE90397F07FFC091381F03E090397E3801F09138F000F8EBFF
E04A7F5C91C7FC485AA25BA2484813015E5BA2000714035E5B1507120F5E49130F5E121F
031F1370491480A2003F023F13F0EE00E090C7FC160148023E13C01603007E1680EE0700
00FE5DED1F1E48EC0FF80038EC03E02C407CBE34>I<143C14FEA21301A314FCEB007014
00AD137E3801FF803803C7C0EA0703000F13E0120E121C13071238A2EA780F007013C0A2
EAF01F14801200133F14005B137EA213FE5BA212015B0003130E13F0A20007131EEBE01C
A2143CEBC0381478147014E013C13803E3C03801FF00EA007C173E7EBC1F>I<ED01C0ED
07F0A2150FA316E0ED038092C7FCADEC03E0EC0FF8EC3C3EEC701EECE01FEB01C0010314
80EB0780140049133F010E1400131E131C013C5BA290C7127EA215FEA25DA21401A25DA2
1403A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C147EA2001C13FE00
7F5BEAFF015C495A495A48485A38F81F80D8783EC8FCEA3FF8EA0FE0245081BC25>I<EB
01FC13FF5CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C9FC16FC49EB
03FE92380F0780017EEB3C0FED703F01FE13E0913801C07F9038FC0380EC07000001010E
14004A131C494890C7FC5C00035BEBF9C0495A01FFC9FC5A14F0EBE3FE9038E07F80000F
EB1FC06E7EEBC00781001F1303160E1380A2003F151E0207131C010013E0A2485DA2007E
01031378167000FE01015B15F1489038007F800038023EC7FC29407CBE2F>I<EB07F0EA
03FF14E0A2EA000FA214C0A2131FA21480A2133FA21400A25BA2137EA213FEA25BA21201
A25BA21203A25BA21207A25BA2120FA25BA2121FA25BA2123FA290C7FCA25AEB0380127E
A212FE130700FC1300A25B130EA2EA7C1C133CEA3E38EA1FF0EA07C014407DBE1B>I<D8
01F0D90FF0EB03F8D807FCD93FFEEB1FFFD80F1FD9F01F90387C0F80000E903C03C00F80
E007C0271E0F87009039C3C003E0001C018E903807C780003C01DCDAEF007F003801F814
EE4A14FCD8781F5D00705B4A5CA200F04949481307013F60000090C75BA2041F140F4960
017E5D191F043F5D13FE4992C7123F97C7FC5E000195387F01C049027E147EA204FEECFE
03000306FC1380495C1A07030103F81300000761494A150E620303163C000FF07C78494A
EC3FE0D80380D900E0EC0F804A297EA750>I<D801F0EB0FF0D807FCEB3FFED80F1FEBF0
1F000E903903C00F80271E0F87007F001C018E1307003C01DC80003813F85CEA781F0070
5B5CA200F049130F013F5D000090C7FCA2161F495D137E163F94C7FC13FE495C167EA200
019238FE03804914FCA203011307000303F813005B5FEEF00E0007161E49151C5F177800
0F6E6C5A49EC7FC0D80380021FC7FC31297EA737>I<EC07F8EC7FFE903901FC0F809039
07E007E090390FC003F090393F8001F8EB7F0001FEEB00FC485A484814FEA2485A120F5B
001F15FF485AA2ED01FE127F5BA2150300FF15FC90C7FCA2ED07F8A2ED0FF0A2007E15E0
007FEC1FC0ED3F80A26CEC7F006C6C13FC4A5A6C6C485A3907E00FC02601F03FC7FC3800
FFFCEB1FE028297DA72C>I<D907C013FE903A0FF003FF80903A1C7C0F07E0903A383C1C
03F0903A783E7801F80170EBF0009026F03FE013FC01E05B4B13FE0001017F147E01C090
C7FC147E17FF000313FEA2C75AA201015C17FE5CA20103140317FC5CA20107EC07F8A24A
14F0160F010F15E0161F17C0EE3F80011F15006E137E5E9138B801F890393FBC03E09138
9E0FC0DA07FFC7FCEC01F849C9FCA2137EA213FEA25BA21201A25BA21203A2B512E0A330
3A84A72E>I<91381F800C9138FFE01C903903F0707C90390FC0387890391F801CF89038
3F000F137E4914F000011407485A485A16E0485A121F150F484814C0A3007F141F491480
A300FF143F90C71300A35D48147EA315FE007E495A1403A26C13074A5A381F801D000F13
793807C1F33901FFC3F038007F03130014075DA3140F5DA3141F5DA2143F147F90381FFF
FE5BA2263A7DA729>I<D801F0EB3F80D807FCEBFFE03A0F1F03C0F0000E90380F00F839
1E0F9E03001C13BC003CEBF807003813F0A226781FE013F000709038C001C092C7FC5C12
F0133F000090C8FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA038025
297EA729>I<EC1FC0ECFFF8903803E03C903807800E90381E0007168049130F49131F15
3FA201F81400A2151C6D90C7FC7FEBFFE014FE90387FFFC06D7F6D13F86D7F1303903800
1FFE14031400157E000C143E123F487EA248C7123CA25D12FC00F05C0070495A0078495A
6C495A260F803EC7FC3803FFF838007FC021297CA72B>I<147014FC1301A25CA21303A2
5CA21307A25CA2130FA25CA2007FB512F0B6FC15E039001F8000133FA291C7FCA25BA213
7EA213FEA25BA21201A25BA21203A25BA21207EC01C013E01403000F1480A2EBC0071500
140E141E5C000713385C3803E1E03801FF80D8003EC7FC1C3A7EB821>I<137C48B4EC03
802603C7C0EB0FC0EA0703000F7F000E151F121C010715801238163FEA780F0070491400
A2D8F01F5C5C0000157E133F91C712FEA2495C137E150113FE495CA215030001161C4914
F0A21507173CEEE038150F031F1378000016706D133F017C017313F0017E01E313E0903A
3F03C1F1C0903A0FFF007F80D901FCEB1F002E297EA734>I<017E147848B4EB01FC2603
C7C013FED807031303000F13E0120E121C0107130100381400167ED8780F143E00705B16
1EEAF01F4A131C1200133F91C7123C16385B137E167801FE14705B16F016E0120149EB01
C0A2ED0380A2ED0700A20000140E5D6D133C017C5B6D5B90381F03C0903807FF80D901FC
C7FC27297EA72C>I<017CEE038048B40207EB0FE02603C7C090391F801FF0EA0703000F
7F000E153F001C16000107160F003817074C1303D8780F027E130100705B1800D8F01F14
FE4A4914E01200133FDA000114014C14C05B137E0303140301FE4A14805BA2F007000001
1407494A5B180EA260A2030F5C12006D011F5C017C496C5B017E0139495A6D903870F803
90281F81E07C0FC7FC903A07FFC01FFE010090380007F03C297EA741>I<D901F8133FD9
07FEEBFFE0903A1E0F83C0F0903A3807C780F890397003CF0301E013FED801C0EBFC0712
03018013F8D8070015F0EE01C0000E4AC7FCA2001E1307A2C75BA2140F5DA3141F5DA314
3F92380001C0A34A1303001E1680003F017E1307267F80FE14005ED8FF81141ED901DF13
1CD8FE035C3A7C078F80F03A3C0F07C1E03A1FFC03FF802707F0007EC7FC2D297EA734>
I<137C48B4EC03802603C7C0EB0FC0EA0703000F7F000E151F001C168013071238163FD8
780F150000705BA2D8F01F5C4A137E1200133F91C712FE5E5B137E150113FE495CA21503
00015D5BA215075EA2150F151F00005D6D133F017C137F017E13FF90393F03DF8090380F
FF1FEB01FC90C7123F93C7FCA25DD80380137ED80FE013FE001F5C4A5AA24848485A4A5A
6CC6485A001C495A001E49C8FC000E137C380781F03803FFC0C648C9FC2A3B7EA72D>I<
02F8130ED903FE131ED90FFF131C49EB803C49EBC0784914F090397E07F1E09038F800FF
49EB1FC049EB07800001EC0F006C48131E90C75A5D5D4A5A4A5A4A5A4AC7FC143E14785C
495A495A495A49C8FC011E14E05B5B4913014848EB03C0485AD807F8EB078048B4131F3A
1F87E07F00391E03FFFE486C5B00785CD870005B00F0EB7FC048011FC7FC27297DA72A>
I<ED01C016E0A216F0A2ED00F816FC007FB612FEB8FCA26C15FEC8EA01F8ED03F0ED07E0
ED0FC0ED1F801600150E281270C02D>126 D E /FQ 65 122 df<EDFFF8020F13FF027F
8049B612E001079038C01FF090390FFE0007D91FF8497ED93FE0131F4948497E13FF5C5A
91C7FCA2705A705AEE03C093C8FCA6EE03FCB8FCA50001903880001F160FB3AB007FD9FE
03B512F0A534407EBF3A>12 D<EDFFFC021FEBFFFC147F49B6FC0107EBC01F90390FFE00
3FEB1FF8EB3FE04948137F01FF143F5C5A91C7121F160FABB8FCA50001903880000FB3AC
007FD9FE03B512F0A534407EBF3A>I<EC0780140F141FEC3E0014FE495A495A5C495A13
0F495A495AA249C7FC5B5B1201485AA212075BA2120F5B121FA3485AA4127F5BA512FFB0
127FA57F123FA46C7EA3120F7F1207A27F1203A26C7E12007F7F6D7EA26D7E6D7E13076D
7E806D7E6D7E143EEC1F80140F1407195A77C329>40 D<127012F8127C7EEA3F806C7E6C
7E12076C7E7F6C7E6C7EA2137F80133F806D7EA280130FA280130780A36D7EA4807FA515
80B01500A55B5CA4495AA35C130F5CA2131F5CA2495A5C137F91C7FC13FEA2485A485A5B
485A120F485A485A003EC8FC5A5A1270195A7AC329>I<EA0FC0EA1FE0EA3FF0EA7FF8EA
FFFCA313FEA3127F123F121FEA0FDEEA001EA2133E133CA2137C1378A213F8EA01F0A2EA
03E0EA07C0EA0F80121FEA3F00121E120C0F20798D1D>44 D<B612E0A91B097F9823>I<
EA0FC0EA1FE0EA3FF0EA7FF8EAFFFCA6EA7FF8EA3FF0EA1FE0EA0FC00E0E798D1D>I<EC
FFE0010713FC011F13FF017F14C0D9FFE07F489038803FF03A03FE000FF848486D7EA248
486D7E001F81A348486D1380A3007F16C0A500FF16E0B3A2007F16C0A5003F16806D5BA2
001F1600A2000F5D6D13076C6C495A6C6C495A6C6D485A6C9038E0FFE06DB55A011F91C7
FC010713FC010013E02B3D7CBB34>48 D<140F143F5C495A130F48B5FCB6FCA313F7EAFE
071200B3B3A8B712F0A5243C78BB34>I<903803FF80013F13F890B512FE00036E7E4881
260FF80F7F261FC0037F4848C67F486C6D7E6D6D7E487E6D6D7EA26F1380A46C5A6C5A6C
5A0007C7FCC8FC4B1300A25E153F5E4B5AA24B5A5E4A5B4A5B4A48C7FC5D4A5AEC1FE04A
5A4A5A9139FF000F80EB01FC495A4948EB1F00495AEB1F8049C7FC017E5C5B48B7FC485D
5A5A5A5A5AB7FC5EA4293C7BBB34>I<903801FFE0010F13FE013F6D7E90B612E0480181
7F3A03FC007FF8D807F06D7E82D80FFC131F6D80121F7FA56C5A5E6C48133FD801F05CC8
FC4B5A5E4B5A4A5B020F5B902607FFFEC7FC15F815FEEDFFC0D9000113F06E6C7E6F7E6F
7E6F7E1780A26F13C0A217E0EA0FC0487E487E487E487EA317C0A25D491580127F494913
00D83FC0495A6C6C495A3A0FFE01FFF86CB65A6C5DC61580013F49C7FC010313E02B3D7C
BB34>I<ED01F815031507A2150F151F153FA2157F15FF5C5CA25C5CEC1FBFEC3F3F143E
147C14FCEB01F814F0EB03E01307EB0FC0EB1F801400133E137E5B485A5B485A1207485A
5B48C7FC5A127E5AB812F8A5C8387FF800AA49B612F8A52D3C7DBB34>I<00071538D80F
E0EB01F801FE133F90B6FC5E5E5E5E93C7FC5D15F85D15C04AC8FC0180C9FCA9ECFFC001
8713FC019F13FF90B67E020113E09039F8007FF0496D7E01C06D7E5B6CC77FC8120F82A3
1780A21207EA1FC0487E487E12FF7FA21700A25B4B5A6C5A01805C6CC7123F6D495AD81F
E0495A260FFC075B6CB65A6C92C7FCC614FC013F13F0010790C8FC293D7BBB34>I<EC07
FF023F13C049B512F001078049EB03FC90383FF80090397FE001FE9038FFC0034849487E
48495AA2485A120FA2485A6F5A003F6E5A6F5A92C8FC485AA21402EC3FFE00FF496C7E01
F9B512E001FB809138E03FF89039FF800FFC4A6C7E825B6F13804915C0A317E05BA4127F
A5123FA26D15C0121FA2000F4A13806D150012076C6C495A6C6D485A6C9038E07FF86DB5
5A6D5C6D1480010749C7FC010013F02B3D7CBB34>I<121F7F13F890B712F0A45A17E017
C0178017005E5E5A007EC7EA01F84B5A007C4A5A4B5A4B5A93C7FC485C157E5DC7485A4A
5AA24A5A140F5D141F143F5D147FA214FF92C8FC5BA25BA3495AA3130FA5131FAA6D5A6D
5A6D5A2C3F7ABD34>I<ECFFF0010713FE011F6D7E017F14E09039FFC07FF03A01FE001F
F848486D7E48486D7E1503485A8281121FA27F7F7F6D5B02C05B14F06C6D485A9138FE0F
F89138FF9FF06CECFFE06C5D5E6C92C7FC6C816D14E0011F80498090B67E48812607FE3F
7F48486C1480381FF807D9F00114C048486C7E007F8049010F13E0150348487F81167FA2
163FA36D15C0127FEE7F807F6C6CECFF006C6C5B01FEEB07FE3A0FFFC03FFC6C90B55A00
0115E06C6C5C011F49C7FC010113F02B3D7CBB34>I<903801FFE0010F13FC013F13FF90
B612C04801E07F489038003FF048486D7E000F6E7E485A6F7E123F48488081178012FFA2
17C0A517E0A4007F5CA4003F5C6C7E5D6C7E00075C3903FF80FB6C13FF6C6C13F36D13C3
010F018313C090380008031400A24B1380EA03F0487E486C1500487E4B5AA25E151F4B5A
495C6C48EBFFE049485B2607FC0F5B6CB6C7FC6C14FC6C14F06D13C0D90FFEC8FC2B3D7C
BB34>I<EA0FC0EA1FE0EA3FF0EA7FF8EAFFFCA6EA7FF8EA3FF0EA1FE0EA0FC0C7FCACEA
0FC0EA1FE0EA3FF0EA7FF8EAFFFCA6EA7FF8EA3FF0EA1FE0EA0FC00E2879A71D>I<16FC
A24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC7FEDF87FA2020180EDF03F
0203804B7E02078115C082020F814B7E021F811500824A81023E7F027E81027C7FA202FC
814A147F49B77EA34982A2D907E0C7001F7F4A80010F835C83011F8391C87E4983133E83
017E83017C81B500FC91B612FCA5463F7CBE4F>65 D<B812F8EFFF8018F018FC8426003F
FCC7EA3FFF050F13807113C07113E08319F0A27113F8A719F05FA24D13E019C04D13804D
1300EF3FFE933801FFF891B712E0188018F818FE02FCC7380FFF80050313C07113E07113
F019F8F07FFCA2F03FFEA219FFA38460A419FE187FA2F0FFFC4D13F85F4D13F0053F13E0
BA12C0190018FC18F095C7FC403E7DBD4A>I<922607FFC0130E92B500FC131E020702FF
133E023FEDC07E91B7EAE1FE01039138803FFB499039F80003FF4901C01300013F90C812
7F4948151FD9FFF8150F48491507485B4A1503481701485B18004890CAFC197E5A5B193E
127FA349170012FFAC127F7F193EA2123FA27F6C187E197C6C7F19FC6C6D16F86C6D1501
19F06C6D15036C6DED07E0D97FFEED0FC06D6CED3F80010F01C0ECFF006D01F8EB03FE6D
9039FF801FFC010091B55A023F15E002071580020002FCC7FC030713C03F407ABE4C>I<
B812F8EFFF8018F018FC18FF26003FFCC76C13C005077F05017F716C7E727E727E727E72
1380A27213C0A27213E0A21AF084A21AF8A41AFCA5197FA319FFA51AF8A41AF0A2601AE0
A24E13C0A24E13804E1300604E5A4E5A4D485A050713E0057F5BBA5A4EC7FC18F818C005
F8C8FC463E7DBD50>I<BAFCA4198026003FFEC7123F1707170183183FA2181FF00FC0A3
1807EE07C0A3F003E0A3160F95C7FC161F163F16FF91B6FCA54AC6FC163F161F040F147C
A2160719F8A593C71201A219F01803A21807A2180FF01FE0183F18FF1703173FBAFCA219
C0A33E3D7DBC45>I<B912FEA48426003FFEC77E170F1703170084A284F01F80A3180FA2
EE07C0A2F007C0A4040F90C7FCA2161F163F16FF91B6FCA54AC6FC163F161F160FA21607
A693C9FCACB712E0A53A3D7DBC42>I<922607FFC0130E92B500FC131E020702FF133E02
3FEDC07E91B7EAE1FE01039138803FFB499039F80003FF4901C01300013F90C8127F4948
151FD9FFF8150F48491507485B4A1503481701485B18004890CAFC197E5A5B193E127FA3
4994C7FC12FFAB0407B612FC127F7FA3003F92C7383FFE00A27F7EA26C7FA26C7F6C7FA2
6C7F6C7FD97FFE157F6D6C7E010F01E014FF6D01F813036D9038FF801F010091B512F302
3F15C00207ED803E02009138FE000E030701E090C7FC46407ABE52>I<B7D8803FB612E0
A526003FFEC8000FEB8000B3A491B9FCA54AC8120FB3A7B7D8803FB612E0A54B3E7DBD52
>I<B71280A526003FFEC7FCB3B3B0B71280A5213E7DBD28>I<0103B612F8A590C7383FFC
00B3B3A4EA1FE0487E487EA2487EA3157F5EA26C48495A495C263FE0035B261FF80F5B6C
B6C7FC000314FCC614F0011F90C8FC2D3F7EBD36>I<B712E0A526003FFEC9FCB3AD183E
A4187E187CA418FCA21701A2EF03F8A21707170F171F177FEE01FF160FB9FC18F0A4373E
7DBD3F>76 D<B6051FB512C06F5EA26F5EA2D8003F97C7FC6F16F7A26E6CED01E7A26E6C
ED03C7A36E6CED0787A26E6CED0F07A26E6C151EA36E6D143CA26E6D1478A26E6D14F0A2
6F6CEB01E0A36F6CEB03C0A26F6CEB0780A26F6CEB0F00A36F6C131EA26F6D5AA26F6D5A
A26F6D5AA393387FF1E0A293383FFBC0A270B45AA37090C7FCA2705AA2705AB600C0031F
B612C0A2705AA2705A5A3E7CBD63>I<B6037FB512E0A2818181D8003F6D9139001F8000
81A281816E7E6E7F6E7F80826E7F6E7F6E7F6E7F157F826F7F6F7F6F7F6F7F81836F7F6F
7F707E701380A27013C07013E07013F07013F87013FCA27013FEEF7FFF71139F7113DF83
19FF8383838384A28484848484A284B600C080197F193F191FA24B3E7DBD52>I<ED3FFF
0203B512F0021F14FE027F6E7E902701FFF80713E00107D9C00013F84990C7EA3FFCD93F
FCEC0FFF49486E7F49486E7F48496E7F4A80488448496F7EA24890C96C7E4884A249161F
003F84A34848701380A400FF19C0AD007F19806D5EA3003F1900A26D5E6C60A26C6D4B5A
A26C6D4B5A6C6D4A5BA26C6D4A5B6C6D4A5B6D6C4A5B6DB4023F90C7FC6D01C0EBFFFE01
07D9F80713F8010190B612E06D5E021F4AC8FC020314F0DA003F90C9FC42407ABE4F>I<
B812F017FF18C018F018FC26003FFCC77FEF1FFF7113807113C07113E0A27113F0A319F8
A819F0A34D13E019C05F4D1380053F1300EFFFFE91B712F860188005FCC7FC4ACAFCB3A4
B77EA53D3E7DBD47>I<B87E17FCEFFF8018F08428003FFC000113FE9338003FFF050F7F
717F717FA2858385A761A25F61614D5B4D90C8FCEF3FFE4CB45A91B712F018C04DC9FC71
7E9126FC000F7F040113F0707F717EA2717EA2717EA685A6F207C019C0A271140F07E013
80B76DEBF01F719038FC3F007190B5FC716C5B061F13F8CB000113E04A3F7DBD4E>82
D<903A03FFC001C0011FEBF803017FEBFE0748B6128F4815DF48010013FFD80FF8130F48
481303497F4848EB007F127F49143F161F12FF160FA27F1607A27F7F01FC91C7FCEBFF80
6C13F8ECFFC06C14FCEDFF806C15E016F86C816C816C816C16806C6C15C07F010715E0EB
007F020714F0EC003F1503030013F8167F163F127800F8151FA2160FA27EA217F07E161F
6C16E06D143F01E015C001F8EC7F8001FEEB01FF9026FFE00713004890B55A486C14F8D8
F81F5CD8F00314C027E0003FFEC7FC2D407ABE3A>I<003FB912FCA5903BFE003FFE003F
D87FF0EE0FFE01C0160349160190C71500197E127EA2007C183EA400FC183F48181FA5C8
1600B3AF010FB712F8A5403D7CBC49>I<B76C90B61280A526003FFEC9003EC7FCB3B3A4
197E011F177C80A26D17FC616D6D14014E5A6D6D4A5A6D6D140F6D01F8EC3FC0DA7FFEEC
FF8091273FFFC00F90C8FC020F90B512FC02035D020015E0031F1480030101F8C9FC493F
7DBD50>I<B600FC020FB512C0A5C66C48C9381F8000013F95C7FC80616D173E6F157E6D
177C6F15FC6D5F8118016D6D5D18036D5F6F14076D5F6F140F027F5E81181F023F93C8FC
6F5C6E153E70137E6E157C8218FC6E6D5B17016E5DEEF0036E5DEEF8076E5D16FC170F03
7F5CEEFE1F033F91C9FC705A6F133E17BE17FE6F5BA26F5BA26F5BA26F5BA36F5BA2705A
A270CAFCA24A3F7EBD4F>I<B6D8FC03B600F090B512FEA5C601FCC7000301F0C8EA7E00
017F6F177C856E6E17FC013F63856D6C037F4B5AA26F4A6C14036D634D7F6F18076D634D
806F02EF150F6D636F01076E131F6D04C793C7FC050F806F02835D6D1A3E051F806F0201
157E027F197C6F013F6E13FC023FDA3E005D057E806F017C017F13016E6105FC14FE7048
013F13036E6104C1EDFF076E4A6D5C04C31687DCE3E06D138F6E6104E716CFDCF7C06D13
DF6E96C8FC04FF16FF6E4A6D5BA294C77E6F5FA24C80033F5FA26F486F5AA24C153F030F
5FA24C151F03075FA26F486F5A673F7EBD6C>I<003FB812E0A59126E0001F13C091C714
8001FC5C01F04A1300495D4914FF4848495B5F90C75A4B5B007E5E5D4B5B007C5E5D4B90
C7FC5E15FFC7485B5E4A5B5C5E4A5B5C5E4A90C8FC5C5D4A5A5B4BEB01F0495B5B495B5D
491503494914E092C7FC5B495A4A14075A4849140F5C48161F4849143F4A147F4816FF48
495B91C7000713C048157FB9FCA5343E7ABD40>90 D<903807FFC0013F13F848B6FC4881
2607FE037F260FF8007F6DEB3FF0486C806F7EA36F7EA26C5A6C5AEA01E0C8FC153F91B5
FC130F137F3901FFFE0F4813E0000F1380381FFE00485A5B485A12FF5BA4151F7F007F14
3F6D90387BFF806C6C01FB13FE391FFF07F36CEBFFE100031480C6EC003FD91FF890C7FC
2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE0020713FC021FEBFF80027F80DAFF8113
F09139FC003FF802F06D7E4A6D7E4A13074A80701380A218C082A318E0AA18C0A25E1880
A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF01FFE0496CB55AD9F01F91C7FCD9E0
0713FCC7000113C033407DBE3A>I<EC7FF00107B5FC011F14C0017F14E09039FFF01FF0
489038800FF848EB001F4848EB3FFC120F485AA2485AA2007FEC1FF849EB0FF0ED03C000
FF91C7FCAB127F7FA3003F153E7F001F157E6C6C147C6C6C14FC91388001F86C9038C003
F0C69038F81FE06DB512C0011F14800107EBFE009038007FF0272B7DA92E>I<EE07F8ED
07FFA5ED003F161FAFEC7FF0903807FFFE011FEBFF9F017F14DF9039FFF01FFF48EBC003
48EB00014848EB007F485A001F153F5B123FA2127F5BA212FFAA127FA37F123FA26C6C14
7F120F6D14FF6C6C01037F6C6D48EBFFE06CEBF03F6C6CB512BF6D143F010713FC010001
E0EBE00033407DBE3A>I<ECFFF0010713FE011F6D7E017F809039FFE07FE0489038801F
F048496C7E48486D7E48486D7E121F491301003F81A2485A6F1380A212FFA290B7FCA401
F0C9FCA5127FA27F123FEE0F806C7E161F6C6C15006C6C5C6C6D137E6C9038E001FC6C90
38F80FF8013FB55A6D14C0010391C7FC9038007FF8292B7DA930>I<EC07FE91387FFF80
49B512C0010714E090390FFE3FF0EB1FF090393FE07FF8EB7FC013FF1480A2489038003F
F0ED1FE0ED0FC092C7FCAAB612E0A500010180C7FCB3AC007FEBFF80A525407DBF20>I<
903A03FF8007F0013F9038F83FF8499038FCFFFC48B712FE48018313F93A07FC007FC348
48EB3FE1001FEDF1FC4990381FF0F81700003F81A7001F5DA26D133F000F5D6C6C495A3A
03FF83FF8091B5C7FC4814FC01BF5BD80F03138090CAFCA2487EA27F13F06CB6FC16F016
FC6C15FF17806C16C06C16E01207001F16F0393FE000034848EB003F49EC1FF800FF150F
90C81207A56C6CEC0FF06D141F003F16E001F0147FD81FFC903801FFC02707FF800F1300
6C90B55AC615F8013F14E0010101FCC7FC2F3D7DA834>I<13FFB5FCA512077EAFED1FF8
EDFFFE02036D7E4A80DA0FE07F91381F007F023C805C4A6D7E5CA25CA35CB3A4B5D8FE0F
B512E0A5333F7CBE3A>I<EA01F8487E487E487E481380A66C13006C5A6C5A6C5AC8FCA9
13FFB5FCA512077EB3ABB512F8A515407CBF1D>I<EC0FC0EC1FE0EC3FF0EC7FF8ECFFFC
A6EC7FF8EC3FF0EC1FE0EC0FC091C7FCA9EC03FCEB07FFA5EB001F140FB3B3A2EA1F80EA
3FC0EA7FE0EAFFF0EC1FF8A3EC3FF015E0397FE07FC0393FC1FF806CB512006C5B6C13F8
C613801E5386BF20>I<13FFB5FCA512077EB092380FFFFEA5DB01FEC7FC4B5AED07F0ED
1FE04B5A4B5A4BC8FCEC03FC4A5A4A5A141FEC7FF84A7EA2818102E77F02C37F14810200
7F826F7E6F7E151F6F7E826F7F6F7F816F7FB5D8FC07EBFFC0A5323F7DBE37>I<13FFB5
FCA512077EB3B3AFB512FCA5163F7CBE1D>I<01FFD91FF8ECFFC0B590B5010713F80203
DAC01F13FE4A6E487FDA0FE09026F07F077F91261F003FEBF8010007013EDAF9F0806C01
78ECFBC04A6DB4486C7FA24A92C7FC4A5CA34A5CB3A4B5D8FE07B5D8F03FEBFF80A55129
7CA858>I<01FFEB1FF8B5EBFFFE02036D7E4A80DA0FE07F91381F007F0007013C806C5B
4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A533297CA83A>I<EC7FF0903803FFFE011FEB
FFC0017F14F09039FFE03FF8489038800FFC3A03FE0003FE48486D7E000F168048486D13
C0A2003F16E049147F007F16F0A400FF16F8AA007F16F0A46C6CECFFE0A2001F16C06C6C
491380A26C6C4913003A03FF800FFE6C9038E03FFC6C6CB512F0011F14C0010791C7FC90
38007FF02D2B7DA934>I<01FFEBFFE0B5000713FC021FEBFF80027F80DAFF8113F09139
FC007FF8000301F06D7E4A6D7E4A130F4A6D7E1880A27013C0A38218E0AA4C13C0A31880
5E18005E6E5C6E495A6E495A02FCEBFFF0DAFF035B92B55A029F91C7FC028713FC028113
C00280C9FCACB512FEA5333B7DA83A>I<DA7FE01378902607FFFC13F8011FEBFF01017F
14819039FFF81FC3489038E007E74890388003F74890380001FF48487F001F157F5B003F
153F5B127F161FA2485AAA127F7FA36C6C143F167F121F6C6C14FF6D5B6C6D5A6CEBC00F
6CEBF03F6C6CB512BF6DEBFE3F010713F8010013C091C7FCAC030FB512E0A5333B7DA837
>I<3901FE01FE00FF903807FF804A13E04A13F0EC3F1F91387C3FF8000713F8000313F0
EBFFE0A29138C01FF0ED0FE091388007C092C7FCA391C8FCB3A2B6FCA525297DA82B>I<
90383FFC1E48B512BE000714FE5A381FF00F383F800148C7FC007E147EA200FE143EA27E
7F6D90C7FC13F8EBFFE06C13FF15C06C14F06C806C806C806C80C61580131F1300020713
C014000078147F00F8143F151F7EA27E16806C143F6D140001E013FF9038F803FE90B55A
15F0D8F87F13C026E00FFEC7FC222B7DA929>I<EB07C0A5130FA4131FA3133F137FA213
FF5A1207001FEBFFFEB6FCA40001EBC000B3151FA96CEBE03EA2017F137EECF8FC90383F
FFF86D13F0010713E001001380203B7EB929>I<D9FF80EB0FF8B5EB0FFFA50007EC007F
6C153FB3A5167FA316FF6C5C4B7F6C903AC007DFFFE09138F01F9F6DB5121F6D13FE010F
13F8010101E0EBE000332A7CA83A>I<B500FC90383FFFC0A5000101C0903803E0006E13
07A26C5E6E130F017F5D6E131F013F92C7FC6E5B011F143E6E137E010F147C6E13FCA26D
5C15816D5C15C36D5C15E76D5C15FF6E5BA36E90C8FCA26E5AA26E5AA26E5AA26E5AA232
287EA737>I<B53CFC3FFFFC03FFFEA50003D980009039C0000F806E161F6C037F15006E
496C5B6C183E836E48157E017F177C6E486D13FC013F02EF5C83DAFC071401011F02C75C
DAFE0FEBFE03010F02835C17FFDAFF1F14076D02015C03BF148F6DD9BE005C18CF03FE14
DF6D49017F90C7FC18FF6D496D5AA36E486D5AA26E486D5AA36E486D5AA26E486D5A4728
7EA74C>I<B5D8FC03B51280A5C69026E0007FC7FC6E13FE6D6C5B6D6C485A6D6C485A01
0F13076D6C485AED9FC06DEBFF806D91C8FC6D5B6E5AA2143F6E7E140F814A7F4A7F4A7F
02FE7F903801FC7F49486C7E02F07F49486C7E49486C7E011F7F49486C7FD97F008001FE
6D7FB5D8C007EBFFC0A532287EA737>I<B500FC90383FFFC0A5000101C0903803E0006E
1307A26C5E6E130F017F5D6E131F013F92C7FC6E5B011F143E6E137E010F147C6E13FCA2
6D5C15816D5C15C36D5C15E76D5C15FF6E5BA36E90C8FCA26E5AA26E5AA26E5AA26E5AA3
5D14075D000E130FD83F805B387FC01FD8FFE090C9FC5C143E147E5CEBC1F8387FC3F038
7E0FE06CB45A6C5B6C48CAFCEA03F8323B7EA737>I E /FR 44 122
df[<96380FFFFE060FB612E04DB712FC051F16FF94B912C0040784041F18F8047F9126FC
001F7F4BB6008001017F030702F8C8EA3FFF4B02E0030F7F033F02804B7F4B49C9127F92
B54893B57E4A02F05D4A4A4B804A4A5D4A4A84634A91C9FC4A5BA24A5B5180755C91B5FC
5EA3755CA2755C755C755CE23FFEC8FCF40FF899CAFCAF083FB612FCBFFCA9C702FCC912
038787B3B3B3B2003FB800F0013FB812F0A9>116 144 123 271
129 12 D<BA12F0B33C137DB74D>45 D[<F31FF0517E517EA2517EA3507FA25080A25080
A35080A25080A35080A25080A397B67EA24F81A34F82A24F82A34F82A24F821AFD1AF9DF
3FF881A24F4881871AE007FF6D80A24E01C081871A804E6E81A24E0100828761060F6E81
A24E48838761063F6E81A24E4883886106FF6F80A24D498388614D7081A24D90C8828860
050F7081A24D48858860053F7081A24D4885896005FF7180A24C498589604C7281A24C90
CA82895F040F728194BDFC4C88A34C88A24C88A3DCFFE0CB003F80A24B49878A5F4B7481
A24B90CC828A5E030F7481A24B48898A5E033F7481A24B48898B5E03FF7580A24A497480
A25E4A7681A24A90CE6C81A25D91261FFF8074810103B512FEB900C0041FBA12FEA9>
159 145 120 272 176 65 D[<BFFC1EFEF6FFE01FFCF7FF8020F020FC20FF8DC7000392
CA001F15E00B0181776C800C1F807814FF0C03818A78818E79808B8E8B8E8B8EA27980A4
791580AB551500A4555CA26A676A676A555C9CB65AA2545D5492C7FC545C5414F80C3F5C
545C53B612800B0F4AC8FC0B7F14F80A1FB612E095BBC9FC1FF81F801FF8F7FF8020F820
FE95CBECFFC00C1F14F00C07807814FE0C006E7E7980798079807980798079808E791580
23C08B23E08C23F0A223F88CA223FCA38C23FEAB5614FCA55614F8A39DB612F0A25515E0
6723C055158067551500555C555C9CB6FC545D0C075D0C1F15C09BB75A0B0F93C7FCC212
FC6921E021800EFCC8FC20F020800DF0C9FC0CFCCAFC>143 142
120 269 165 I[<0803B500C0EE01F00703B600FEEE03F8077FDBFFE015070607B800FC
150F063F05FF151F4DBA00E0143F050F07F8147F053F07FE14FF94BC5B04039326F8000F
ECC003040F4BC86CEBF007043F03C0030F6D5A93B648C900036D5A4B03F09339007FFF3F
030703C0051F90B5FC4B92CB7E033F02FC18034B02F08492B648844A0380193F4A92CD7E
4A4A864A4A864A02F0864A4A864A8991B65A494B874992CF7E4C885B494A885E498B494A
88A2495C8D90B65A8D5A5E48217FA24892D1FC223FA25A5DA248211FA3485CFA0FF09FC7
FCA25AA45DA3B6FCB27EA381A47EA46C80FA07F0FA0FF87EA2817EA36C6F1D1F23F07E82
7E223F6D6E1EE0A26D6E1D7F23C06D6E1DFF7F705213806D806D55130070646D6F646D6F
515A6E6E1B1F6E6E515A6E6E515A6E6E1BFF6E6E505B6E6E505B6E6F4F5B6E03E04F90C7
FC6F6EF13FFE6F02FC4F5A030F02FF4E485A6F03C005075B030103F0051F5B6F03FE057F
1380043FDAFFE00303B5C8FC040F03FE033F13FC0403DBFFF80107B55A040093B812E005
3F1A80050F4FC9FC050119F8DD003F18C0060795CAFCDE007F16F0070393CBFCDF000314
C0>141 146 115 271 168 I[<BE12FEF5FFFCF6FFC01FFCF7FF8020E020FC20FF21C0C7
00030380C8000116F0E2000F810B0015FE0C1F800C0315C00C00810D3F8079800D0714FE
79807981796C808C7A807A808F7A807A808C8F7A818DA17E8DA17E8DA17EA27B80A2A17E
8DA17EA28DA17EA3A113808DA3A113C0A57B15E0A6A113F0B3A2A113E0A569A113C0A5A1
1380A269A2A11300A3575CA2A15AA269A15A69A15AA2575CA15A69A15A9EB6FC5692C7FC
6B565C68565C565C565C565C9DB65A5592C8FC0D075C555C0D3F5C9CB65A0C0315C00C0F
5D0C7F92C9FC0B07B612FC52B712F0C212C09ECAFC20FC20F020800DFCCBFC1FE00CFCCC
FC53CDFC>156 142 120 269 178 I[<C212F8A48DA5C7000303C0C9123FF5007F1E0F0C
0180787E1F1F8B8B1F018BA27A7E8C8CA28CA28CA28C8EA28CA38CA38EA2E21FF0157FA6
7B7EA40A3F93C8FCA41C7FA21CFFA26363631B1F1B7F0807B5FC95B8FCA99538C00007F2
007F1B1F1B07878787A21C7FA21C3FA3FA7FC01C1FA2FAFF80A6571300A299CAFCA369A2
6AA22107A3210F6AA2211FA2213FA2575AA221FFA26868565BA26868207F9DB5FC555C1F
071F1F1F7F0C03B6FC1E3F0B1FB7FCC35AA66AA3>138 141 120
268 153 I[<C21280A421C0A5C7000303C0C81201F40007F5007F0C1F14E01E071E0178
7E8B8B8B8B7913F0A28B8BA2207FA3203F21F8201FA4200FA321FC2007A4F47FC0A3F803
FEA49DC7FCA31CFFA463A263A26363631B7F50B5FC1A1F95B8FCA99538C0001F1A01747E
1B1F878787A287A287A41C7FAA99CBFCB3AFBC12F8A9>127 141
120 268 146 I[<0803B500C0EE01F00703B600FE4C7E077FDBFFE015070607B800FC15
0F063F05FF151F4DBA00E0143F050F07F8147F053F07FE14FF94BC5B04039326F8000FEC
C003040F4BC86CEBF007043F03C0030F6D5A93B648C900036D5A4B03F09339007FFF3F03
0703C0051F90B5FC4B92CB7E033F02FC18034B02F08492B648844A0380193F4A92CD7E4A
4A864A4A864A02F0864A4A864A8991B65A494B874992CF7E4C885B494A885E498B494A88
A2495C8D90B65A8D5A5E48217FA24892D1FC223FA25A5DA248211FA3485C7C5A9FC9FCA2
5AA45DA3B6FCB27EA381A20A0FBB12F8A27EA46C80A36C98C96C02F8C7FCA2817EA36C81
A27E827E827FA26D80A26D806D80A26D806D80A26D816D816E806E806E806E6E97B6FC6E
806E806E03C0606E816F02F8606F02FE60030F6E606F03E0173F030103F85F6F03FF9338
01FFFC043F03E00307497E040F03FF033F497E040304FC0107B5EAE00F040093B8487E05
3FF20001050F07FCEB007F050107F0141FDD003F06C01407060795C81201DE007F04F8ED
00700703048093C8FCDF000302E0CDFC>157 146 115 271 183
I[<BC0207BB12F8A9C7000303C0CC001F4AC8FCB3B3B395BEFCA906C0CC121FB3B3B3A6
BC0207BB12F8A9>165 142 120 269 182 I[<BC12C0A9C7000103E0C8FCB3B3B3B3B3B3
B0BC12C0A9>74 142 122 269 87 I[<0307BB12F0A992C96C02F0C7FCB3B3B3B3B3A4EB
0FC0EB7FF83801FFFE487F4880488048804880A24880A2B67EA36496B6FCA364A26C4A4A
5DA24B95C8FC6C5F4B5F6C4A4A5C4B4A5C6C91C8FC6C01FC4B5C6C01F04B5C6C6D4B5C6C
01FE4AB6C9FC90267FFFE001075C6D01FE013F14F8010F90B85A6D18C0010195CAFCD900
3F16F8020F16C0020003FCCBFC030791CCFC>100 144 123 269
120 I[<BC12F8A9C7000303C0CEFCB3B3B3B3A5F8FF80A4672100A667A368A21F07A41F
0FA3555AA21F3FA21F7FA21FFFA2666668666666666653B5FC65650B1F5C1D7F0A03B6FC
1C1F0903B7FCC1FCA468A5>121 142 120 269 140 76 D[<B900C00C7FB812E0729BB9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>203
142 120 269 220 I[<97B512F0077FECFFE00607B712FE067FEEFFE00503B912FC051F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>148 146
115 271 175 79 D[<BE12F8F5FFF01EFF1FE01FFCF7FF8020E020F820FEC700030380C8
000781E2003F15C00B03810B00810C3F8078800C07807880788178818E8B8E8B8E8B8EA2
8EA28B8EA42380AC2300A46A67A26AA26A676A676A9CB65A6A665492C7FC545C0C1F5C54
5C9BB612E00B075D0B3F5D0A07B648C8FC95BB12F820E0208055C9FC1FF09CCAFC1EF00B
F8CBFC06C0D0FCB3B3B2BCFCA9>137 142 120 269 159 I[<BD12FCF4FFFCF5FFE01EFC
F6FFC01FF01FFE797E20E0C700030380C86C15F80A0181E2003F14FF0B07810B0115E077
6C8078807880788078808A78818E7881A28E8B8EA37980A48EAA6AA3676AA26AA29CB65A
A26A545D9FCAFC66545C545C545C545C9BB612C0535D0B074ACBFC0B3F5C52B612F00A7F
15C095BBCCFC1FF81FC054CDFC66F6FF801FE00680C7000315F8E1003F14FE0A0F800A03
15C07681766C807780778077808C7781898D898D898DA38A8DAD8DACA1EB1FC0A1EB3FE0
8D8AA37880A1137FA114C078817818FF7C1580785F786F1500BB00FE6F6F5B796E495A79
02FEEB1FFC799139FFC07FF80D0792B55A0D015F796C5E0E1F5E0E034BC7FCD4001F14F8
E7003F13C0>163 144 120 269 173 82 D[<93260FFFF8163E4BB600E0153F031F03FE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>102 146 115 271 129 I[<000FC312F8A6488EA304C0C7001F4AC7120103
F8C8F0000F03C01C0192C9737E02FC1E1F4A1E0702E08A4A8A4A8A4890CA757EA249203F
49201FA349200FA2492007A4492003007F8EA4498CA848487A1380A6CC99C7FCB3B3B3B3
AA030FBD12FCA9>145 140 120 267 162 I[<BB6C010FBA00FC0307B812F8A9C76C02FC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>229 144 123 269 240 87 D<93B512FC037FECFFF00207B8FC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>97 D[<ED1FF0017FB5FCB7FCA9EA003F1307A27FB3B296383FFFC00607B512FE063F
ECFFE04DB712F8050716FF051F17C0057F17F094B5D8C00F8004F301FCC714FE04F701E0
023F7F93B50080020F804DC86C14E005F80301804D6F804D707F05808294CA804C717F4C
7180A24C71808BA27680A28B88A28BA28BA3888BA52080B02000A56764A267A3676467A2
525CA267647062704D91C7FC704D5BA2714C5B7193B55A05F04B5CDCBFF84B5CDC1FFC03
0F5C4B6CB44B91C8FC7001C0027F5B4B6C01F00103B55A4BC601FF013F14F04B6D90B712
C04B011F94C9FC4B6D16FC4B010316F092C86C15804A030F02F8CAFC90CB49CBFC>113
144 121 270 129 I<94387FFFF0041FB612E093B712FE0307707E031F17F092B97E4A18
FE020784021F9126F8000F14804A0280010014C04A49C74814E049B500F85C494A17F049
4A5C495C494A4A14F84991C8FC5D495B90B5FC5D5A485C7314F05A4B6F14E05A7314C048
7214804B93383FFE00F20FF84896C8FCA4485CA5B6FCB07EA281A37EA36C80A37E6F18FE
6CF201FFA26C6E5F1CFE6C801B076C6EEF0FFC6D7F70EE1FF86DF13FF06D6E167F6D6EEE
FFE06D02F84B13C06D6E5D6D02FF030F13806D03C0023F1300023F02F0903801FFFC6E91
26FF801F5B020792B65A6E18C0020060033F4CC7FC030716F8030016C0041F4AC8FCDC00
7F13C0585F78DD67>I[<F53FE098B6FC4FB7FCA996C77E1B0FA287B3B294383FFF80040F
B512FC93B71280030716E0031F16F8037F16FE4AB9128702074AC66C13C7021F02E00107
13F74A91C890B6FC4A01FC153F49B548150F4902E081494A81494A814991CA7E495B8749
498390B548835A5D5AA2485CA25A5D5AA35AA25D5AA5B6FCB07EA57E81A37EA27EA2817E
A26C80A26C626C6E5F636D7F6D6D94B6FC6D606D6D1607705D6D6E4B81010102F0157F6D
6E92B712FE6E01FE020301EF91B512806E6D6C011F13CF020FDAF801B5120F020391B612
FE6E17F86E6C16E0030F16800301EDFC00DB003F14E0040049C74AC8FC>113
144 120 270 129 I<94387FFFC0040FB6FC93B712E0030716FC031F16FF037F17C04AB9
12F00207DAF80380021F912680003F13FE4A49C7000F7F4A01F802038049B5486E804902
C06E6C7F494A6F7F4991C9FC49727F4949707F4B84498490B548707F5A4B198048855D48
1CC086481CE05D5A871DF05AA25D5AA21DF887A2B6FCA392BBFCA51DF00380CDFCA77EA4
817EA37EA2817EA26CF307F06FF00FF87E816C1B1F6F19F06C1B3F6D6DF07FE06D7FF4FF
C06D6E4C13806D6E5E6D02F04C13006D6EEE1FFE6D6E4C5A6D6C01FFEEFFF86E02E00203
5B6E02FC021F5B02079126FFC003B55A6E92B7C7FC020060033F17F8030F17E003011780
DB003F03FCC8FC040315C0DC000F01F8C9FC5D5F7ADD6A>I[<95383FFF80050FB512F094
B612FE040781041F16C0047F824BB87E0307DAF8077F031FDAC00F7F4B49C6487F4B495B
92B500F0814A4A5B4A5C4A93B612805F4A91C7FC5C5E5C5E5C731400A24C6E5B91B56F5B
A2735B070313E00700138097C8FCB3A4BA12F8A9C702FCCBFCB3B3B3B3A2003FB9FCA9>
81 144 121 271 71 I<F5FFC093260FFFFC030F13F04BB600E0027F7F031F03FE49B512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>I[<ED1FF0017FB5FCB7FCA9EA00
3F1307A27FB3B2963803FFFC073FEBFFE096B612F8060715FE061F6F7E4E16E095B87E4D
D9FC03804DD9C000804D48C76C7FDD0FF880DD1FE0824D486E804D5A05FEC881DCF1FC81
5F04F385EEF7F04D81EEFFC0A24D84A294C9FCA25EA35EA45EB3B3AFB9D8E001B912C0A9
>114 143 119 270 129 I[<EC3FC0ECFFF0010313FC497F497F498049804980A290B67E
A24881A86C5DA26D5CA26D5C6D5C6D91C8FC6D5B6D5B010013F0EC3FC091CAFCB3A3ED1F
F0017FB5FCB7FCA9EA003F1307A27FB3B3B3B0B91280A9>49 144
119 271 65 I[<ED1FF0017FB5FCB7FCA9EA003F1307A27FB3B3083FB712C0A908014AC7
FCE0003F13C09AC8FC515A515A505B5013E0080F5B505B5090C9FCF27FFC4F485A4F5B4F
5B4F5B4F90CAFCF17FFE4F5A4E5B4E5B4E13C0061F5B4E90CBFC4E5AF0FFF805037F5F4D
7F4D7F4D8094B67E16F104F38104F78193B77EA2868605F18017E04D814D6C804D6C8004
FC6D805E4C6D804C6D807280A27280728173808588738073807380A2738073807381A274
80748074808689748074807480A274818A5015F0B96C017F92B5FCA9>112
143 121 270 123 107 D[<ED1FF0017FB5FCB7FCA9EA003F1307A27FB3B3B3B3B3B3AC
B912C0A9>50 143 119 270 65 I<DB3FE0912601FFFC943801FFFC017FB5031FD9FFE0
041FEBFFE0B792B600FC93B612FC060303FF030315FF060F04C0020F16C0063F04F0023F
16F095B86C91B87E4DD9FC036E49D9FC03804DD9C0006E49D9C000804D48C7003F6D4948
C7003F7FDD0FF86EDB0FF880D8003F4B48714848830107DB3FC06E9126C03FC06E804D48
4E5A6D4BC86F48C881DCE1FE6FDAE1FE814D61DCE3F8DEF3F884DCE7F0F0F7F04D6F4B81
DCEFC0F0FFC0A2DCFF804F84A294C993C9FCA24C61A34C61A44C61B3B3AFB900E090B900
E090B912E0A9B35D77DCC2>I<DB3FE0913803FFFC017FB5033FEBFFE0B792B612F80607
15FE061F6F7E4E16E095B87E4DD9FC03804DD9C000804D48C76C7FDD0FF880D8003FDB1F
E08201074B486E804D5A6D03FEC881DCE1FC815F04E385EEE7F04D81EEEFC0A2DCFF8084
A294C9FCA25EA35EA45EB3B3AFB9D8E001B912C0A9725D77DC81>I<94381FFFF00407B6
12C0047F15FC0303B87E030F17E0037F17FC4ABAFC4A9126FC007F80020F02C0010714E0
4A49C880027F01F8033F13FC91B5486F7F4902C003077F494A6F804991C96C8049497080
4949717F49874949717FA290B548717F48884B83481D80A2481DC04B83481DE0A2481DF0
A3484A7114F8A4481DFCA5B61BFEAF6C1DFCA56C6E4D14F8A36C1DF0A36C1DE06F5F6C1D
C0A26C6E4D1480A26C1D006F5F6C646D6D4D5B6F94B5FC6D636D6D4C5C6D6E4B5C6D6E4B
5C6D02F0031F5C6D6E4B91C7FC6D6C01FE92B512FC6ED9FFC001075C6E02FC017F5C0207
91B812C0020196C8FC6E6C17FC031F17F003031780DB007F03FCC9FC040715C0DC001F01
F0CAFC675F7ADD74>I<DB1FF091381FFFC0017FB50203B6FCB7021F15E095B712FC0503
16FF050F17C0053F17F094B912FC04F1DAC01F8004F79026FC00018093B500E06D6C14C0
D8003F93C86C8001074B030F8005F86F806D03E06F804D6F804D8194CA6C7F4C864C7180
5E7680A27680A27680A28B88A28BA288A28BA4882080B0200064A467A26467A3525CA267
64676467647062704D91C7FC7094B55AA2714B5C714B5C714B5C05F84B5C71033F5C05FF
4B91C8FC06C049B55A04FB01F001075C04F801FF017F14F07190B712C0051F94C9FC7116
FC050316F0DD007F1580060F02F8CAFC060049CBFC96CDFCB3ACB912E0A9718579DC81>
I<DD7FFFEE1FE0040FB500F0153F93B600FE157F03076F7E031F04E014FF92B800F85B02
03834A715B021F923A007FFF80074A02F0010F13C091B600C001036D5A4992C86D5A4902
FCED3FF8494A031F5B494A6F6C5A494AED07FE494A6FB6FC494A8193C9FC90B682484A83
A2484A83A2484A83A2484A83A25A4B83A25AA25D5AA5B65AB07E81A47EA3817EA26C80A3
6C62816C62816C626C6E5F98B6FC6D7F6D6E5D6D606D6E5D6D6E151F6D6E5D6D6E5D6D02
FE913801FFEF6E6D020713CF6E02C0011F138F020F913AFC01FFFE0F020391B612FC0200
17F0033F16C0030F1600030115FCDB003F14E0040049C7FC94C9FCB3AC0703B91280A971
8578DC7B>I<DB7FC049B47E90B6021F13F8B7027F13FE4DB67E4D15E04D814D814D0107
7F94263FF00F7F94387FC01F4D48487FD8003F16000107DAC1FE491480EEC3FC6D5DEEC7
F05F16CF5F16DF4D6D1400A204FFC76C5BA2735B4C6E5B735B070013C04C92C8FCA45EA6
5EB3B3AAB912FCA9515D79DC5F>I<92261FFFF814F80203B638C001FC023FEDFC0791B8
121F010317FF130F013F9038F8001F4990C8FCD9FFF8153F4801E0150F48491503484981
4890CAFC197F4848173F191F485AA2007F180FA31907487EA27FA28002E0705A6E93C8FC
14FC14FF15F06CECFF8016FCEEFFF06CEEFF8018F06C17FE727E6C18E0856C18FC6C846C
727E6C856D84011F846D841303010084023F83140F020183EC001FDB007F16801603DC00
0F15C01700183F060F14E0007F1703486C82727E857F85857FA2857F1BC07FA27F1B806D
5F7F1B006E5E6E5F6E163F6E4C5A02FC4C5A6E03035B6E6C4A5B03F0023F5B03FF0107B5
5A01F991B7C7FCD9F07F16FCD9E01F16F0D9800716C0D9000193C8FC48D9003F14F8007C
020349C9FC4B5F78DD5C>I[<ED03FEA81507A5150FA4151FA3153FA2157FA215FFA25CA2
5C5CA25C5C5C5C91B5FC13035B131F017F91B712F00007BAFCBBFCA7C74AC9FCB3B3AAF1
01FFB1616E17FE82A219076E17FC836EEE0FF871131F6E6EEB3FF071137F6E6EEBFFE06E
DAFF0313C06E92B512806E1700033F5D6F5D03075D030015E0041F1480040001FCC7FC>
72 132 124 258 90 I<DB0FF8F01FF0017FB594B6FCB74BB7FCA9D8003F94C77E010719
0FA26D85B3B3B063A463A263A27F6398B6FCA26DF001FB7015036EEF07F3E00FE3806E6D
151FE07FC314FF6E6D6CDAFF83EDFFC06E6E010313036E02FCEB3FFE6E91B612FC020017
F86F16E0031F16800303EDFE00DB007F14F8040102C093C8FC725E77DC81>I<B90303B7
FCA9D8000702F8CA000FEBFE006D6E050013E0666D6E6164826D5090C7FC836E4F5AA26E
6E4C5AA26E6E4C5AA26E6E5F1C3F836E4F5A836E4F5AA26E6E4B5BA26E6E4B90C8FCA26F
6E5D1B07846F4D5A846F4D5AA26F6E4A5AA26F6E4A5AA26F6E5D1BFF846F4C5B846F4C90
C9FCA2706E485AA27002C05B1A0F7002E05B1A1F19F0704B5A19F8704B5AA2706E485AA2
706E5B96B5FC7093CAFCA3715CA2715CA2715CA2715CA3715CA2715CA2715CA27191CBFC
A2725AA3725A725A725A705D7BDB7B>I<B800FE017FB700F8023FB612F8A9D8000F02F0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>I<007F
B86C49B712FEA9C792C9000F02C0C7FC6E6E030101F0C8FC715F6E6E4B5B6E6E4B5B6E4E
90C9FC6E6E5E71151F6E6E4B5A6E6E4B5A6E4E5A6F6E495B72495B6F6E495B6F806F6E49
90CAFC6F4C5A72495A6F6E495A6F6E495A6F03815B705E7014C307E75B7091B5CBFC705D
705D705D6282705D715C8386718071807180837180864D814D815F4D81874D814D81DDFF
F3804C13E14C01C1804C0180814E6C804C6E804C487F4C48824C486D804C486D804B496D
804B497F73804B49834B90C86C804B486F804B48814B486F804B48844C6F804A71804A49
6F804A49814A90CA814A487180023F7280010FB500E07080B8031FB812E0A9735C7CDB7B
>I<007FB800C04AB71280A9D800034ACA000791C7FC6D080013F0775A6D6E4E5AA26E6E
6064836E4F90C8FC836E4F5A836E4F5AA26E6E4C5AA26E6E5F1C3F6E6E5F1C7F836E4F5A
846F4D5B846F4D90C9FCA26F6E4A5AA26F6E5D1B0F846F4D5A846F4D5A846F4D5AA26F6E
4A5AA2706E5C627002C091CAFC6219E0704B5A19F0704B5AA2706E485AA2706E485AA270
02FE5B1A7F19FF704B5AA2715DA27192CBFCA2715CA2715CA3715CA2715CA2715CA2715C
A2725BA27290CCFCA3725AA2725AA24E5AA24E5AA261187FA24E5AA24D5B13FE2603FF80
4A90CDFC000F13E0486D4A5A487F486D4A5AA260B56C141F4D5AA24D5A17FF604C5B4A49
90CEFC6C5D4C5A6C49EB3FFC4A495A6C4948485A9026FE80075B270FFFC03F5B6C90B6CF
FC6C5D6C15F86C6C5C011F14C0010749D0FC9038007FE071857CDB7B>I
E /FS 8 122 df<90BB1280A25AA29129F0001FF8000F13004890C749130101FC023F80
498448484B805B49027F153E120F495DA248C812FFA2001E4C143C123E5D123C007C93C8
FC0078197C4B167812F8485DA21507C894C7FC5EA2150FA25EA2151FA25EA2153FA25EA2
157FA25EA215FFA25EA25CA293CBFCA25CA25DA21407A25DA2140FA25DA2141FA25DA214
3FA25DA2147F14FF01037F003FB7FCA449516BD054>84 D<EC01FCEC07FF021FEBC0E091
397F03E3F8903901FC01F3903903F800FF4948137F49485C4948133F495A137F49C76C5A
5B1201485A5F120749143F120F495D121F167F123F4992C7FCA25E127F495CA21501A248
485CA21503EF03C016F890C7FC150717076CDA0FF01380A2031F130F033F14006C6CEB7F
E003FF5B001FD901F7131E9026C003E7133E000F903907C3F03C2707E00F83137C3B03F0
3F01F8F83B01FFFC00FFF026007FF06D5AD91FC0EB1F80323574B33C>97
D<EF1F80EF7FF0933801FFF8933803F0FC933807E07E93380FC1FE161F1783163FA2EF03
FC047F13F8EF00F0180016FEA415015EA415035EA415075EA4021FB512FE5CA39126000F
E0C7FC151F5EA4153F5EA4157F93C8FCA45DA25DA414015DA414035DA414075DA4140F5D
A3141F5DA4143F5DA3147F92C9FCA3147E14FEA25C121EEA3F81007F5B12FFEB83F0A2EB
03E0495A38FC0F80D87E1FCAFCEA3FFE6C5AEA03F0376C83D324>102
D<EC01E0EC03F0EC07F8140FA315F015E0EC038091C7FCB2EB07C0EB3FF0EB7FFCEBF87C
3801E07E3803C03E3807803F120F495A5A121E123E003C5B5C127CEA78015CEAF8035C12
0013075CA2130F5CA2131F5C133F5CA2137F91C7FCA24913785BA2000114F84913F01203
EBF80115E0A2EC03C013F0EC0780140F15000001131EEBF87C6CB45AEB7FE0EB1F801D50
77CE24>105 D<ECFF8090B5FC15005AA2EA00016D5AA21301A25CA21303A25CA21307A2
5CA2130FA25CA2131FA25CA2133FA25CA2137FA291C7FCA25BA25BA21201A25BA21203A2
5BA21207A25BA2120FA25BA2121FA25BA2123FA25BA2127F14781300A24813F85C5AA213
015C12FC13035C1307007E5B130F6C48C7FCEA1FFE6C5AEA03F0195475D21E>108
D<013EDA1FE0EC7F80D9FFC0D9FFFC903803FFF048D9E0036D010F7F3F03E3F00FE07F80
3F81FE2907C1F81F001F90387C007ED80F81017EDAC1F8137F902601FCF890260FC3E07F
001FD9FDF0DAE7C080001E4A5D02FF15EF003E4A02FFC7FC003C4A5C92C75B5B007C495D
12784A5DD8F807031F157F4A4B92C7FC12004A5D010F033F5D634A5DA2011F037F140163
4A92C7FC1A03013F4B5DA24A4A140763017F1401080FEB078091C74915E0A2490203031F
130F09C01300494B143F515A00010307171EA2494B5E1B000003030F5F1CF8494B5E9738
1F81E0F387C0494B6EB45A7448C7FCD801C06EC912F8593577B360>I<ED3FC0913801FF
F8020F13FE91393FE07F8091397F001FC0D901FEEB0FE0D903F8EB07F0495A4948EB03F8
495A4948EB01FC137F49C713FE485AA2485A12075B120FA2485A1603123F5BA21607007F
16FC5BA2160F17F8485AA2EE1FF0A2EE3FE0007F16C0167F178016FF1700003F4A5A4B5A
5E6C6C495A4B5A6C6C495A6C6CEB7F806C6C01FEC7FC3901FE07F839007FFFE0011F1380
D907FCC8FC2F3574B33C>111 D<EB0FC0D93FF0EC03C0D97FFCEC07E0D9F07C140F3801
E07E2603C03E141F2607803F15C0120F495A48163F001E1780123E003C5B4A147F007C17
00EA78015CD8F8035D4A5C120013074A13015F130F5C1603011F5D5CA21607013F5D5CA2
160F017F5D91C7FCA2161F5FA3163F4C5AA26D14FF5D6E4890C7FC011F5B6E5A90380FF0
3F903903FFFCFE6D13F09038003FC0EC00015EA215035EA2D803C0495AEA07F0000F4A5A
121F4B5A4B5A4991C8FC5D018013FE391E0001FC001F495A6CEB07E09038801FC03907E0
7F802603FFFEC9FCC613F8EB3F80334C77B339>121 D E /FT 99
128 df<B812FEA300019038C0000F6C6C481301EE007F8383A283A283A5EF0380A594C7
FCB3B33801FFE0B612F8A3313E7DBD39>0 D<B912F8A3D87FE0C71201EE001F6C6CED07
FC6C6C150117006C7E6C6C167C0003173C7F6C7F6C6D151CA26D7E6D6C151E180E6D7E6D
7EA26D6C15006D7EA26D7F6D7F6E7EA26E7E6E7EA2140F6E5AA26E5A5D4A5A4AC9FC141E
5C4A150E14F85C495A4948151E4948151C49C9FC131E013E163C133C5B49167C484816FC
485A48481501000F160748C9EA1FF8001EED01FF003FB8FC5AB9FCA2373E7BBD42>6
D<010FB612E0A3D900030180C7FCDA00FEC8FCA8913807FFC0027F13FC903A03FCFE7F80
D90FE0EB0FE0D93F80EB03F8D9FE00EB00FE4848157F4848ED3F804848ED1FC0000F17E0
4848ED0FF0003F17F8A24848ED07FCA200FF17FEA8007F17FCA26C6CED0FF8A2001F17F0
6C6CED1FE0000717C06C6CED3F806C6CED7F006C6C15FED93F80EB03F8D90FE0EB0FE0D9
03FCEB7F809027007FFFFCC7FC020713C0DA00FEC8FCA8913803FF80010FB612E0A3373E
7BBD42>8 D<0103B612F8A390C701E0C8FCED3F80A8B4EF0FF001C0163FD81FE0EE7F80
000F18006D5E00075FA26D150100035FAB00014C5A7FA3606C6C1507A2017E4B5A137FD9
3F804A5A011F5E02C04AC7FCD90FE0147ED907F05CD901F8EB81F8D900FEEB87E091393F
BFBF8091260FFFFEC8FC020013E0ED3F80A8EDFFE00103B612F8A33C3E7BBD47>I<9138
01FFC0021F13FC9139FF007F80D903F8EB0FE0D90FF0EB07F8D91FC0EB01FCD97F806DB4
FC49C86C7E48486F7E00038348486F7E000F8349150F001F83491507003F83A348486F7E
AA6C6C4B5AA3001F5FA26C6C4B5AA200075F6D151F00035FA26C6C4B5A00005FA2017F4B
C7FC6D157EA26D6C5C010F5DA26D6C495A00E0EF0380010315E0D870019238C007006E13
0301001580A36C0160EC000E003C017049131E263FFFF0ECFFFEA36C5FA339407CBF42>
I<4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383FF03FD907F090397FE07F
80494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01FE6D91C7FC82ADB97EA3
C648C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35>I<4AB4FC021F13C091
387F01F0903901FC0078D907F0131C4948133E494813FF49485A137F1400A213FE6F5A16
3893C7FCAA167FB8FCA33900FE00018182B3AC486CECFF80007FD9FC3F13FEA32F407FBF
33>I<4AB47E021F13F791387F00FFEB01F8903807F001EB0FE0EB1FC0EB3F80137F1400
8101FE80AEB8FCA3C648C77EB3AE486CECFF80007FD9FC3F13FEA32F407FBF33>I<4AB4
ECFF80021FD9C00F13E0913B7F01F03F80F8903C01F80078FE003CD907F0D93FF8130E49
484948131F49484948EB7F804948484913FF137F02005CA201FE92C7FC6FED7F0070141C
96C7FCAAF13F80BBFCA3C648C76CC7FC197F193FB3AC486C4A6CEB7FC0007FD9FC3FD9FE
1FB5FCA348407FBF4C>I<EA01FC127FA3120712031201B3AC487EB512F0A314287DA71A>
16 D<133E133F137F13FFA2EA01FEEA03FCEA07F813F0EA0FE0EA1FC01380EA3E005A5A
1270122010116EBE2D>19 D<B7FCA320037AB52D>22 D<EA03E0EA0FF8EA3C1EEA7007A2
38E00380A638700700A2EA3C1EEA0FF8EA03E0111067C044>I<1660A216C01501168015
031600903801FE0790380FFFC690383F03FCEB7C00D801F0133E48487F4848EB3F80157F
4848EB67C0001FECE7E090C712C348010113F015834890380303F8007E14011406A200FE
010C13FC141C1418143814301470146014E014C0267E018013F8A2EB0300007F1403D83F
0614F0130ED81F0C14E0D80F9CEB07C01398D807F8EB0F806C48EB1F000001143E6C6C5B
9038FF03F09038CFFFC0260181FEC7FCD80380C8FC90C9FC5A1206120E120C5AA226397D
AF2D>28 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AC121EAB120CC7FCA8121EEA7F80A2
EAFFC0A4EA7F80A2EA1E000A4179C019>33 D<001E130F397F803FC000FF137F01C013E0
A201E013F0A3007F133F391E600F3000001300A401E01370491360A3000114E04913C000
03130101001380481303000EEB070048130E0018130C0038131C003013181C1C7DBE2D>
I<EC0FC0EC3FF0ECF878903801F01CEB03E049487E130FEC800F011F7FA2EB3F00A5EC80
0EA25DA25DA25D6D6C5AECC1C0A2ECC38002E7C7387FFFFCEB0FEE14FC4A020713C06D48
913801FE006E5DEF00F06D7E4D5A496C5D010F1503D91DFF4A5A013893C7FC496C6C5B01
E0150E48486C6C131E00036E131C2607801F143C000F6E5B001F6D6C1370263F000714F0
6F485A48D903FE5B913801FF03486D495A0487C8FCED7FCFED3FFE6F4814386D6D5AA200
7F6E6C14786D6D6C14704B6C14F06C6C496C6C13E0001F91393E3FC0016C6C903AFC1FF0
03C03D07FC07F007FC1F800001B5D8C001B512006C6C90C7EA7FFCD90FF8EC0FF03E437C
C047>38 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A31201138012
0313005A120E5A1218123812300B1C79BE19>I<1430147014E0EB01C0EB03801307EB0F
00131E133E133C5B13F85B12015B1203A2485AA2120F5BA2121F90C7FCA25AA3123E127E
A6127C12FCB2127C127EA6123E123FA37EA27F120FA27F1207A26C7EA212017F12007F13
787F133E131E7FEB07801303EB01C0EB00E014701430145A77C323>I<12C07E12707E7E
121E7E6C7E7F12036C7E7F12007F1378137CA27FA2133F7FA21480130FA214C0A3130714
E0A6130314F0B214E01307A614C0130FA31480A2131F1400A25B133EA25BA2137813F85B
12015B485A12075B48C7FC121E121C5A5A5A5A145A7BC323>I<1506150FB3A9007FB912
E0BA12F0A26C18E0C8000FC9FCB3A915063C3C7BB447>43 D<121EEA7F8012FF13C0A213
E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C79
8919>I<B512FEA617067F961E>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919
>I<ED0180ED03C01507A21680150FA216005DA2151E153EA2153C157CA2157815F8A25D
1401A25D1403A25D1407A25D140FA24AC7FCA2141E143EA2143C147CA2147814F8A25C13
01A25C1303A25C1307A25C130FA291C8FC5BA2131E133EA25BA2137813F8A25B1201A25B
1203A25B1207A25B120FA290C9FC5AA2121E123EA2123C127CA2127812F8A25A1260225B
7BC32D>I<EB01FE90380FFFC090383F03F090387C00F849137C48487F48487F4848EB0F
80A2000F15C04848EB07E0A3003F15F0A290C712034815F8A64815FCB3A26C15F8A56C6C
EB07F0A3001F15E0A36C6CEB0FC0A26C6CEB1F80000315006C6C133E6C6C5B017C5B9038
3F03F090380FFFC0D901FEC7FC263F7DBC2D>I<EB01C013031307131F137FEA07FFB5FC
139FEAF81F1200B3B3ACEB7FF0B612F8A31D3D78BC2D>I<EB07FC90383FFF8090B512E0
3903F01FF83907C007FC390F0001FE001E6D7E001C1580003CEC7FC05AED3FE01270B4FC
6DEB1FF07FA56C5A6CC7FC120CC813E0153FA216C0157F168015FF16004A5A5D4A5A4A5A
5D4A5A4A5A4AC7FC147E147C5C495A495A495A495A49C71270133E133C5B4914E0485A48
5A485A48C7120148B6FCA25A4815C0B7FCA3243D7CBC2D>I<EB07FC90383FFF809038F8
0FE03901E003F839078001FCD80F007F000E6D7E001E1580D81F80137F486C14C07FA27F
5BA2121F6C5AC8138015FF1600A24A5AA24A5A5DEC07E04A5A023FC7FCEB1FFCECFF8090
38000FE0EC07F86E7E6E7E6E7E1680ED7FC0A216E0153FA216F0A2120C123F487E487EA3
16E0A249137F6CC713C01278EDFF807E6C4913006C495A3907C007FC3903F80FF0C6B55A
013F1380D907F8C7FC243F7CBC2D>I<150E151E153EA2157EA215FE1401A21403EC077E
1406140E141CA214381470A214E0EB01C0A2EB0380EB0700A2130E5BA25B5BA25B5B1201
485A90C7FC5A120E120C121C5AA25A5AB8FCA3C8EAFE00AC4A7E49B6FCA3283E7EBD2D>
I<00061403D80780131F01F813FE90B5FC5D5D5D15C092C7FC14FCEB3FE090C9FCACEB01
FE90380FFF8090383E03E090387001F8496C7E49137E497F90C713800006141FC813C0A2
16E0150FA316F0A3120C127F7F12FFA416E090C7121F12FC007015C012780038EC3F8012
3C6CEC7F00001F14FE6C6C485A6C6C485A3903F80FE0C6B55A013F90C7FCEB07F8243F7C
BC2D>I<EC1FE0ECFFF8903803F03E90380FC00F90391F000780133E017EEB1FC049133F
4848137F12035B12074848EB3F80ED1F00001F91C7FC5BA2123FA3485AA214FE903887FF
8039FF8F07E090389C01F09038B800FC01B0137E13F0497F16804914C0A2ED1FE0A34914
F0A5127FA6123F6D14E0A2121FED3FC0A26C6C1480A20007EC7F006C6C137E6C6C5B6C6C
485A90387E07F06DB45A010F1380D903FCC7FC243F7CBC2D>I<1238123C123F90B612FC
A316F85A16F016E00078C712010070EC03C0ED078016005D48141E151C153C5DC8127015
F04A5A5D14034A5A92C7FC5C141EA25CA2147C147814F8A213015C1303A31307A3130F5C
A2131FA6133FAA6D5A0107C8FC26407BBD2D>I<EB03FC90381FFF8090387C07E09038F0
01F83901E0007C48487F48487F48C7FCED0F80121E16C0003E1407A4123FA26DEB0F807F
6C6C131F6D140001FC133E6C6C5B9038FF80786C6D5A6CEBF3E06CEBFF806C91C7FC133F
6D13C06D7F013F13F801787F48486C7E3903E01FFF48486C1380260F800313C048487E48
9038007FE0003E143F007E141F007CEC0FF01507481403A31501A46C15E0007C1403A200
7E15C06C14076CEC0F806DEB1F006C6C133ED807F05B3901FC03F86CB512E0011F1380D9
03FCC7FC243F7CBC2D>I<EB03FCEB1FFF90387E07C09038FC03F048486C7E48486C7E48
48137C000F147E4848137F81003F15805B007F15C0A2151F12FF16E0A516F0A5127F153F
A36C7EA2001F147F120F6C6C13FF6D13DF000313013900F8039F90387E0F1FD91FFE13E0
EB07F090C7FCA2ED3FC0A41680157FD80F801400487E486C13FEA24A5A5D49485AEB8007
391E000FE0001F495A260FC07FC7FC3803FFFE6C13F838003FC0243F7CBC2D>I<121EEA
7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A27
79A619>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121E127FEAFF80A213C0A4
127F121E1200A412011380A3120313005A1206120E120C121C5A1230A20A3979A619>I<
007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E03C167BA147>61
D<EB1FF890B5FC3903E01FC0390F0007F0001EEB03F848EB01FC4814FE140000FE14FF7E
7FA46CC7FC123EC7EA01FEA2EC03FCEC07F815F0EC0FC0EC1F80EC3F00143E5C147814F8
5C13015CA2495AA25CAB91C7FC90C8FCA8EB0780EB1FE0A2497EA46D5AA2EB078020407B
BF2B>63 D<15074B7EA34B7EA34B7EA34B7EA34B7E15E7A2913801C7FC15C3A291380381
FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D7EA34A6D7EA349486D7E91B6FCA249
819138800001A249C87EA24982010E157FA2011E82011C153FA2013C820138151FA20178
82170F13FC00034C7ED80FFF4B7EB500F0010FB512F8A33D417DC044>65
D<B712FCEEFF8017F00001903980000FF86C6CC7EA03FE707E701380EF7FC0EF3FE0A2EF
1FF0A218F8A3170F171FA318F0A2EF3FE0177F18C0EFFF804C1300EE03FCEE0FF8EE7FE0
91B6C7FC17E091C7EA07FCEE01FE933800FF80EF7FC0EF3FE0EF1FF018F8170F18FC1707
A218FEA718FC170FA2EF1FF818F0173FEF7FE0EFFFC00403138048486C90380FFE00B85A
17E094C7FC373E7DBD40>I<DB3FF01306912603FFFE130E020F9038FF801E913A3FF007
E03E9139FF8000F8D903FEC7EA7C7ED907F8EC1EFE4948140FD93FE0140749481403495A
91C812014848150012034848167E5B000F173EA24848161EA2123F5B180E127FA3491600
12FFAC127F7F180EA2123FA27F001F171E181C6C7EA20007173C6D16386C6C1678000117
706C6C16F06EEC01E06D6C15C06D6C1403D90FF0EC07806D6CEC1F00D903FE143E902600
FF8013F891393FF007F0020FB512C0020391C7FC9138003FF037427BBF42>I<B712FCEE
FF8017E000019039C0001FF86C6C48EB03FEEE00FF717E717EEF0FE084717E717E170184
717EA21980187F19C0A3F03FE0A519F0AB19E0A5F07FC0A21980A218FF19004D5AA24D5A
6017074D5A4D5AEF7FC04DC7FCEE03FE48486CEB1FF8B85A178004FCC8FC3C3E7DBD45>
I<B912E0A300019038C000016C6C48EB001FEF0FF01703A217011700A31870A41838161C
A41800A2163CA2167C16FC150391B5FCA3EC80031500167C163CA2161CA21807A3180E93
C7FCA4181E181CA2183CA2187CA218F8170117031707171F48486CEB01FFB912F0A3383E
7DBD3E>I<B91280A300019038C000036C6C48EB007FEF1FC0170F1707A21703A31701A4
EF00E0A21638A31800A31678A216F81501150791B5FCA3EC8007150115001678A21638A6
93C8FCAF3801FFE0B612F0A3333E7DBD3B>I<DB3FE0130C912603FFFE131C021F9038FF
803C913A7FF00FC07C9139FF0001F0D903FC90380078FC4948143DD91FE0141F4948140F
4948140701FF15034890C8FC491501485A000716005B000F177C5B001F173CA2485AA218
1C127FA25B95C7FC12FFAB041FB512F0127FA26D9139000FFE00EF03FC123FA27F121FA2
6C7EA212077F12036C7E7F6C7F6D6C14076D7E6D6C140FD907F8141ED903FEEC3C7C9026
00FF80EBF83C913A7FF007F01C021FB5EAC00C020391C8FC9138003FF03C427BBF47>I<
B6D8C01FB512F8A3000101E0C7383FFC0026007F80EC0FF0B3A691B7FCA30280C7120FB3
A92601FFE0EC3FFCB6D8C01FB512F8A33D3E7DBD44>I<B612F0A3C6EBF000EB3FC0B3B3
B2EBFFF0B612F0A31C3E7EBD21>I<011FB512FCA3D9000713006E5A1401B3B3A6123FEA
7F80EAFFC0A44A5A1380D87F005B007C130700385C003C495A6C495A6C495A2603E07EC7
FC3800FFF8EB3FC026407CBD2F>I<B600C090387FFFFCA3000101E0C7000F138026007F
80913807FE0018F818E0604D5A4DC7FC173E5F5F4C5A4C5A4C5A4C5A4CC8FC163E5E5E4B
5A4B5AED07804B7E151F4B7E4B7E15FF913881EFF8913883C7FCEC878791388F03FE9138
9E01FF14BCDAF8007F4A6D7E5C4A6D7E4A6D7EA2707E707EA2707E707EA2707F717E8417
3F717E717EA2717E848419802601FFE04A13C0B600C090B6FCA3403E7DBD47>I<B612F8
A3000101E0C9FC38007F80B3B0EF0380A517071800A45FA35FA25F5F5F4C5A160748486C
133FB8FCA3313E7DBD39>I<B500C093383FFFF0A300016D93387FF800D8007F18E0D977
F016EFA3D973F8ED01CFA2D971FCED038FA3D970FEED070FA26E150E80A26E6C141CA36E
6C1438A26E6C1470A36E6C14E0A26E6CEB01C0A36E6CEB0380A36E6CEB0700A2037F130E
A36F6C5AA26F6C5AA36F6C5AA25FED07F0A2923803F9C0A36FB45AA26F90C7FCA213F848
6C147ED807FFEF3FF8B500F8013C011FB512F0A34C3E7DBD53>I<B56C91B512F88080D8
007F030713006EEC01FC6E6E5A1870EB77FCEB73FEA2EB71FF01707FA26E7E6E7EA26E7E
6E7EA26E7E6E7EA26E7E6E7FA26F7E6F7EA26F7E6F7EA26F7E6F7EA26F7E6F1380A2EE7F
C0EE3FE0A2EE1FF0EE0FF8A2EE07FCEE03FEA2EE01FF7013F0A2177F173FA2171F170FA2
170701F81503487ED807FF1501B500F81400A218703D3E7DBD44>I<ED7FE0913807FFFE
91391FC03F8091397E0007E04948EB03F8D907F0EB00FE4948147F49486E7E49486E7E49
C86C7E01FE6F7E00018349150300038348486F7EA248486F7EA2001F188049167F003F18
C0A3007F18E049163FA300FF18F0AC007F18E06D167FA4003F18C0A26C6CEEFF80A36C6C
4B1300A26C6C4B5A00035F6D150700015F6C6C4B5A6D5E6D6C4A5A6D6C4A5A6D6C4AC7FC
6D6C14FED901FCEB03F8D9007FEB0FE091391FC03F80912607FFFEC8FC9138007FE03C42
7BBF47>I<B712F8EEFF8017E000019039C0003FF86C6C48EB07FCEE01FE707EEF7F80EF
3FC018E0A2EF1FF0A218F8A818F0A2EF3FE0A218C0EF7F80EFFF004C5AEE07FCEE3FF091
B612C04CC7FC0280C9FCB3A73801FFE0B612C0A3353E7DBD3E>I<ED7FE0913807FFFE91
391FC03F8091397F000FE0D901FCEB03F8D907F0EB00FE4948147F49486E7E49486E7E49
C86C7E498248486F7E49150300038348486F7EA2000F834981001F1880A24848EE7FC0A3
007F18E0A249163FA200FF18F0AC007F18E0A26D167FA3003F18C0A26C6CEEFF80A3000F
18006D5D0007DA0F805B6C6C90393FE003FCED70706C6C496C485A6C6C48486C485A017F
D9800E5BD93F819038061FC0D91FC19038073F80D90FE14AC7FCD907F1EB03FE902601FD
C013F8903A007EE007E091271FF03FC013180207B5FC9139007FE1E0DB00011438837113
78A2706C13F0EFFF0318FFA27113E0A37113C0711380711300715AEF01F83D527BBF47>
I<B712C016FCEEFF800001D9C00013E06C6C48EB1FF0EE07FCEE01FE707E84717EA2717E
A284A760177F606017FF95C7FCEE01FCEE07F8EE1FE0EEFF8091B500FCC8FC16F0913880
01FCED003FEE1FC0707E707E83160383160183A383A484A4F0C004190EA28218E0057F13
1E2601FFE0161CB600C0EB3FF094381FF83805071370CA3801FFE09438003F803F407DBD
43>I<D907FC130C90391FFF801C017FEBF03C3901FC03F83A03F0007E7CD807C0EB1FFC
4848130F001F140748C71203003E1401007E1400A2007C157C12FCA2163CA36C151CA27E
A26C6C14007F7FEA3FF8EBFF806C13F86CEBFF806C14F06C14FC6C14FF6C15C0013F14E0
010714F0EB007F020713F89138007FFC150FED07FE15031501ED00FFA200E0157FA3163F
A27EA3163E7E167E6C157C6C15FC6C15F86D13016DEB03F06DEB07E0D8F9FCEB0FC03AF0
7F803F8090391FFFFE00D8E00713F839C0007FC028427BBF33>I<003FB91280A3903AF0
007FE001018090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00
E0A5C81600B3B14B7E4B7E0107B612FEA33B3D7DBC42>I<B600C090B512F8A3000101E0
C70007130026007F80EC01FC715A1870B3B3A4013F16F06E5DA21701011F5E80010F1503
6E4A5A010793C7FC6D6C5C6D6C141E6D6C5C027F14F86E6C485A91390FF00FE00203B512
80020049C8FCED1FF03D407DBD44>I<B691380FFFFEA3000301E0020113E06C01809138
007F806CEF3F00017F163E181C6E153C013F1638A26E1578011F1670A26D6C5DA26E1401
01075EA26E140301035EA26D6C4AC7FCA2806D150EA26F131E027F141CA26F133C023F14
38A26E6C5BA26F13F0020F5CA2EDF80102075CA26E6C485AA2EDFE07020191C8FCA26F5A
6E130EA2ED7F9CA216DCED3FF8A36F5AA36F5AA26F5AA36F5A3F407EBD44>I<B500FE01
7FB5D88007B5FCA3000301C0010101E0C713F86C90C849EC3FE07148EC0F807E7215006E
143F017F190E84A26D6C60A24D7E6D6C60A2EFE7F86D6C60A2933801C3FC6E18F0010761
04037F6E0281140101036104077F17006D6C4D5AA2040EEB7F806D6C4DC7FCA24CEB3FC0
DA7F80160EA24CEB1FE003C0161E023F171C047814F0DBE070010F133C021F173804F014
F84C1307DA0FF05EA2DBF1C0EB03FCDA07F95EA2DBFB80EB01FEDA03FF6F5AA293C8FCA2
6E5FA24B157F020094C8FCA24B81037C153EA20378151E0338151C58407EBD5D>I<007F
B5D8C003B512E0A3C649C7EBFC00D93FF8EC3FE06D48EC1F806D6C92C7FC171E6D6C141C
6D6C143C5F6D6C14706D6D13F04C5ADA7FC05B023F13036F485ADA1FF090C8FC020F5BED
F81E913807FC1C163C6E6C5A913801FF7016F06E5B6F5AA26F7E6F7EA28282153FED3BFE
ED71FF15F103E07F913801C07F0203804B6C7EEC07004A6D7E020E6D7E5C023C6D7E0238
6D7E14784A6D7E4A6D7F130149486E7E4A6E7E130749C86C7E496F7E497ED9FFC04A7E00
076DEC7FFFB500FC0103B512FEA33F3E7EBD44>I<B66C0103B51280A3000101F0C8EBF8
006C6C48ED3FC0725A013F041EC7FC6D7E606D6C15386D6C1578606D6C5D6E14016D5E6D
6D1303606E6C49C8FC6E6C5B170E6E6C131E171C6E6C5B6E6C137817706E6C13F06F5B6E
13016EEB83C05FED7FC7DB3FE7C9FC16EFED1FFE5E150F6F5AB3A4ED1FFC020FB512FCA3
413E7FBD44>I<003FB712F8A391C7EA1FF013F801E0EC3FE00180EC7FC090C8FC003EED
FF80A2003C4A1300007C4A5A12784B5A4B5AA200704A5AA24B5A4B5AA2C8485A4A90C7FC
A24A5A4A5AA24A5AA24A5A4A5AA24A5A4A5AA24990C8FCA2495A4948141CA2495A495AA2
495A495A173C495AA24890C8FC485A1778485A484815F8A24848140116034848140F4848
143FED01FFB8FCA32E3E7BBD38>I<EAFFFCA4EAF000B3B3B3B3ABEAFFFCA40E5B77C319>
I<486C13C00003130101001380481303000EEB070048130E0018130C0038131C00301318
0070133800601330A300E01370481360A400CFEB678039FFC07FE001E013F0A3007F133F
A2003F131F01C013E0390F0007801C1C73BE2D>I<EAFFFCA4EA003CB3B3B3B3ABEAFFFC
A40E5B7FC319>I<1318133C137E13FF3801E7803803C3C0380781E0380F00F0001E1378
48133C48131E48130F00601306180D76BD2D>I<EA0180120313005A120E5A1218123812
3012701260A312E05AA412CFEAFFC013E0A3127FA2123F13C0EA0F000B1C7ABE19>96
D<EB0FF8EBFFFE3903F01F8039078007E0000F6D7E9038E001F8D81FF07F6E7EA3157F6C
5AEA0380C8FCA4EC1FFF0103B5FC90381FF87FEB7F803801FC00EA07F8EA0FE0485A485A
A248C7FCEE038012FEA315FFA3007F5BEC03BF3B3F80071F8700261FC00E13CF3A07F03C
0FFE3A01FFF807FC3A003FC001F0292A7DA82D>I<EA01FC12FFA3120712031201B1EC03
FC91381FFF8091387C07E09039FDE001F09039FFC000FC4A137E91C77E49158049141F17
C0EE0FE0A217F0A2160717F8AA17F0A2160FA217E0161F17C06D1580EE3F006D5C6E13FE
9039F3C001F89039F1E003F09039E0780FC09026C03FFFC7FCC7EA07F82D407EBE33>I<
49B4FC010F13E090383F00F8017C131E4848131F4848137F0007ECFF80485A5B121FA248
48EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3003FEC01C07F001F140316806C6C1307
6C6C14000003140E6C6C131E6C6C137890383F01F090380FFFC0D901FEC7FC222A7DA828
>I<ED01FC15FFA3150715031501B114FF010713E190381F80F990387E003D49131FD803
F81307485A49130348481301121F123F5B127FA290C7FCA25AAA7E7FA2123FA26C7E000F
14037F000714076C6C497E6C6C497ED8007C017913F890383F01F190380FFFC1903A01FE
01FC002D407DBE33>I<EB01FE90380FFFC090383F03F09038FC01F848486C7E4848137E
48487F000F158049131F001F15C04848130FA2127F16E090C7FCA25AA290B6FCA290C9FC
A67EA27F123F16E06C7E1501000F15C06C6C13036DEB07806C6C1400C66C131E017E5B90
381F80F8903807FFE0010090C7FC232A7EA828>I<EC1FC0EC7FF8903801F83C903807E0
7E90380FC0FFEB1FC1EB3F811401137FEC00FE01FE137C1500AEB6FCA3C648C7FCB3AE48
7E007F13FFA320407EBF1C>I<167C903903F801FF903A1FFF078F8090397E0FDE1F9038
F803F83803F001A23B07E000FC0600000F6EC7FC49137E001F147FA8000F147E6D13FE00
075C6C6C485AA23901F803E03903FE0FC026071FFFC8FCEB03F80006CAFC120EA3120FA2
7F7F6CB512E015FE6C6E7E6C15E06C810003813A0FC0001FFC48C7EA01FE003E14004815
7E825A82A46C5D007C153E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017F
C7FC90383FFFFC010313C0293D7EA82D>I<EA01FC12FFA3120712031201B1EC01FE9138
07FFC091381E07E091387803F09138E001F8D9FDC07F148001FF6D7E91C7FCA25BA25BB3
A6486C497EB5D8F87F13FCA32E3F7DBE33>I<EA01E0EA07F8A2487EA46C5AA2EA01E0C8
FCACEA01FC127FA3120712031201B3AC487EB512F0A3143E7DBD1A>I<1478EB01FEA2EB
03FFA4EB01FEA2EB00781400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14
FEA214FCEB81F8EA7F01387C03F0381E07C0380FFF803801FC00185185BD1C>I<EA01FC
12FFA3120712031201B292B51280A392383FFC0016E0168093C7FC153C5D5D4A5AEC07C0
4A5A4AC8FC143E147F4A7E13FD9038FFDFC0EC9FE0140F496C7E01FC7F496C7E1401816E
7E81826F7E151F826F7EA282486C14FEB539F07FFFE0A32B3F7EBE30>I<EA01FC12FFA3
120712031201B3B3B1487EB512F8A3153F7DBE1A>I<2701F801FE14FF00FF902707FFC0
0313E0913B1E07E00F03F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F
000101805C01FBD900FF147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F8
7FD9FC3F13FEA347287DA74C>I<3901F801FE00FF903807FFC091381E07E091387803F0
00079038E001F82603F9C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8
F87F13FCA32E287DA733>I<14FF010713E090381F81F890387E007E01F8131F4848EB0F
804848EB07C04848EB03E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C
15FEA36C6CEB01FCA3001F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D800
7EEB7E0090383F81FC90380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381F
FF8091387C0FE09039FDE003F03A03FFC001FC6C496C7E91C7127F49EC3F805BEE1FC017
E0A2EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F8
9039FDE007F09039FC780FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>
I<02FF131C0107EBC03C90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC15
0748481303121F485A1501485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F
6C7E6C6C133DD8007E137990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8
A32D3A7DA730>I<3901F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB14
01EC00FC01FF1330491300A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FF
F8E03807C03F381F000F003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA
7FF8EBFFC06C13F86C13FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800
E01303A214017E1400A27E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FF
FC38C01FE01D2A7DA824>I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6
FCA2D801FCC7FCB3A215E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FEEB
01F81B397EB723>I<D801FC14FE00FF147FA3000714030003140100011400B3A51501A3
1503120015076DEB06FF017E010E13806D4913FC90381FC078903807FFE00100903880FE
002E297DA733>I<B539E00FFFE0A32707FE000313006C48EB00FC5E00015D7F00005DA2
6D13016D5CA26D6C485AA2ECC007011F91C7FCA290380FE00EA2ECF01E0107131CA26D6C
5AA2ECFC7801011370A2ECFEF001005BA2EC7FC0A36E5AA26EC8FCA3140E2B287EA630>
I<B53BC3FFFE03FFF8A3290FFE003FE00013C06C486D48EB3F806C4817006D010F141E00
016F131C15076D163C00004A6C1338A2017F5E4B7E151DD93F805DED3DFC1538D91FC04A
5AED78FE9238707E03D90FE0017F5BEDE03F02F0140701070387C7FC9138F1C01F02F914
8F010315CE9138FB800F02FF14DE6D15FCED00076D5DA24A1303027E5CA2027C1301023C
5C023813003D287EA642>I<B539F01FFFE0A30003D9C00F1300C690388007F8D97F0013
E002805BD93FC05B011F49C7FC90380FE00EECF01E6D6C5A01035B6D6C5A6E5AEB00FF6E
5A6E5A81141F814A7E81147BECF1FC903801E1FEECC0FF01037F49486C7ED90F007F011E
6D7E013E130F496D7E01FC80486C80000F4A7EB539803FFFF8A32D277FA630>I<B539E0
0FFFE0A32707FE000313006C48EB01FC6F5A00015D7F00005DA2017F495AA2EC8003013F
5CA26D6C48C7FCA26E5A010F130EA26D6C5AA2ECF83C01031338A26D6C5AA2ECFEF00100
5BA2EC7FC0A36E5AA36EC8FCA2140EA2141E141C143C1438A2147800181370127EB45BA2
495AA248485AD87E07C9FCEA780EEA3C3CEA1FF8EA07E02B3A7EA630>I<001FB61280A2
EBE0000180140049485A001E495A121C4A5A003C495A141F00385C4A5A147F5D4AC7FCC6
485AA2495A495A130F5C495A90393FC00380A2EB7F80EBFF005A5B484813071207491400
485A48485BA248485B4848137F00FF495A90B6FCA221277EA628>I<B812F0A22C028098
2D>I<BE12C0A25A0280985B>I<001C130E007FEB3F8039FF807FC0A5397F003F80001CEB
0E001A0977BD2D>127 D E /FU 26 122 df<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00
0A0A77891D>46 D<DB1FF8EB01804AB5EA8003020FECE007913A3FF803F80F9139FF8000
7C4948C7EA1E1FD907FCEC0FBFD90FF0EC07FF49488049488049488049C9127F4848163F
120349161F12074848160FA2485A1807123F5B1803127FA34993C7FC12FFAC127F7FF003
80A2123FA27F001F170719006C7EA26C6C5E0003170E6D161E0001171C6C6C163C6D6C5D
6D6C5D6D6C4A5A6D6C4A5AD907FC4A5AD901FE021FC7FC902600FFC0137E91393FF803F8
020FB512E0020114809126001FFCC8FC39427ABF47>67 D<B600C0011FB5FCA3000101E0
C7000313E026007F806E90C7FC18FC6060EF03C04D5A4DC8FC171E5F5F5FEE03E04C5A4C
5A4CC9FC163E5E5EED01E04B5A15074B7E151F4B7EEDFFF81481913883E7FC913887C3FE
EC8F8391389F01FFDABE007F02FC8002F0137F4A6D7E4A6D7E5C707E707EA2707E707E84
82717E717EA2717E717EA2717E717E84A219C02601FFE04A7FB600C0017FEBFF80A3413E
7BBD4D>75 D<003FB912E0A3903BF0003FF0007F01806D48130F48C7ED07F0007E170300
7C170100781700A300701870A5481838A5C81600B3B14B7E4B7E0103B7FCA33D3D7CBC47
>84 D<B600C090387FFFFCA3000101E0C70003138026007F80913800FE00187C1838B3B3
A4013F16786E1570A218F0011F5E6E1401010F5E6D6C1403606D6C14076D6C4AC7FC6D6C
141E027F147C91393FC001F891390FF00FE00203B55A020049C8FCED1FF03E407BBD4A>
I<EC01E0A24A7EA34A7EA34A7EA24A7E141CA2EC3CFFEC387FA24A6C7EA34A6C7EA20101
80ECC00FA249486C7EA349486C7EA24980010E1301010FB5FC4980A2011CC7FC49147FA2
0178810170143FA201F08149141F1201486C811207486CEC3FF8D8FFFE49B512C0A33231
7DB038>97 D<B612FEEDFFC016F03A03FC0007F86C48EB01FE1500167F1780163F17C0A6
1780167F170016FE4B5AED07F0ED7FE090B6128016F09039F80001FC6F7EEE7F80163FEE
1FC017E0160F17F0A617E0161FA2EE3FC0EE7F80923801FF00486CEB07FEB712F85E93C7
FC2C2F7CAE35>I<DA0FF81330DA7FFF13700103B5EAC0F090390FFC03F190391FE000F9
D97F80133F01FEC7121F4848140F48481407485A000F1503491401121F491400123F5B12
7F1770A248C9FC1700AA6C6C1570A3123F6D15F0121F6D15E0000F15016D15C000071503
6C6C15806C6C14076C6CEC0F00D97F80133ED91FE05B90390FFC03F00103B55AD9007F13
80DA0FF8C7FC2C317BAF36>I<B612FEEDFFE016F83A03FE0007FC6C48EB00FFEE3F8070
7E707E707E707E160183160083A2177FA41880AA1800A317FEA34C5A5F16034C5A5FEE1F
C04C5A04FFC7FC486CEB07FEB712F816E093C8FC312F7DAE39>I<B81280A3D803FEC7FC
6C48EC1FC0160F16071603A21601A317E0ED0E00A31700A2151E153E157E90B512FEA390
38FC007E153E151E150EA21738A392C71270A417F0A2EE01E0A216031607161F486C14FF
B812C0A32D2F7DAE33>I<B8FCA33903FE00016C489038003F80161F160F1607A21603A3
17C01601150EA293C7FCA3151E153E157E90B512FEA39038FC007E153E151E150EA592C8
FCAA487EB512FCA32A2F7DAE31>I<DA0FF81360DAFFFE13E00103EBFF8190390FF807E3
90393FC000F34948137F01FEC7123F4848141F4848140F48481407120F491403485A003F
1501A25B007F1500A348C9FC1700A8031FB5FCA26C7E9238001FF0EE0FE0123F7FA26C7E
120F7F12076C7E6C7E6C6C141FD97F80133FD93FE0137B90390FFC03F10103B512E00100
EC8060DA0FFCC7FC30317BAF3A>I<B5D8F807B512C0A3D803FEC7381FF0006C486E5AB1
90B7FCA301FCC7120FB3486C4A7EB5D8F807B512C0A3322F7DAE38>I<B512F8A33803FE
006C5AB3B3A3487EB512F8A3152F7DAE1B>I<90383FFFFCA39038007FC0EC3F80B3AD12
18127EB4FCA3EC7F005A007C137E007813FE383C01F8381F03F03807FFC0C648C7FC1E30
7CAE27>I<B512FCA3D803FEC8FC6C5AB3A71607A4160EA4161EA2163E167E16FEED03FC
486C130FB7FCA3282F7DAE2F>108 D<D8FFFE923807FFF0A3D803FF92380FFC006C5FD9
DF80141DA3D9CFC01439A2D9C7E01471A3D9C3F014E1A2D9C1F8EB01C1A3D9C0FCEB0381
A2027EEB0701A36E130EA291381F801CA391380FC038A2913807E070A3913803F0E0A391
3801F9C0A2913800FF80A3486CEB7F00487E486C013E497EB5008091B512F0A2151C3C2F
7CAE44>I<D8FFFC91387FFFC07F7F0001923807FC006E6D5A6E6D5AD9DFE06D5AA2EBCF
F0EBC7F8EBC3FCA2EBC1FEEBC0FF6E7EA26E7E6E7EA26E7E6E7E6E7EA26E7E6E7EED7F80
A2ED3FC0ED1FE0ED0FF0A2ED07F8ED03FCA2ED01FEED00FF167FA2163F161F160F487E48
6C1407486C1403B56C1301A21600322F7DAE38>I<EC1FF891B5FC903907F00FE090390F
C003F0013FC712FC017E147E49804848EC1F804848EC0FC04848EC07E0000F16F0491403
001F16F8491401003F16FCA2007F16FE90C9FCA34816FFAA6C6CEC01FEA3003F16FCA26D
1403001F16F86C6CEC07F0A26C6CEC0FE0000316C06C6CEC1F806C6CEC3F00017E147E6D
5C90390FC003F0903907F00FE00100B5C7FCEC1FF830317BAF3A>I<B612FEEDFFC016F0
3A03FE0007FC6C48EB01FEED007FEE3F80A2EE1FC0A217E0A617C0A2EE3F80A2EE7F00ED
01FCED07F890B612E0168001FCC9FCB2487EB512F8A32B2F7DAE33>I<B612F015FF16C0
3A03FE001FF06C48EB03FCED00FE167FA283163F83A55F167F94C7FC16FE4B5A4B5AED1F
E090B6C8FC5D9039FC003F80ED0FC06F7E826F7EA26F7EA582A418E082A281486CED01C0
B500F8EB7F8193381FC38093380FFF00C9EA01FC33307DAE37>114
D<90383FC00C9038FFF81C0003EBFE3C390FE03FFC381F8007EB0003003E130148130015
7C5A153CA36C141CA27E6C14006C7E13E013FE383FFFE06C13FE6CEBFF806C14E0000114
F06C6C13F8010F13FC1300EC07FE14011400157F153F12E0151FA37EA2151E6C143E6C14
3C6C147C6C14F89038C001F039FBF807E000F1B512C0D8E07F130038C007FC20317BAF2A
>I<007FB712F8A39039801FF0073A7E000FE00000781678A20070163800F0163CA34816
1CA5C71500B3A8EC3FF8011FB512F0A32E2E7CAD36>I<B500F890387FFFC0A3D803FEC7
3807FC006C486E5A705A705AB3AB000015016D5D1603017E5D017F14076D6C49C7FC131F
6D6C133ED907F05B903903FC03F00100B55A023F1380DA07FCC8FC32307DAE38>I<B500
E0903807FFF0A3000790C7000113806C48913800FE000001167C0000167817706D15F06D
5DA26D6C495AA26E1303011F5DA26D6C49C7FCA26E5B0107140EA26D6C5BA26E133C0101
14388001005CA26E13F06E5B1581023F5BA215C3021F5B15E7020F90C8FCA2EC07FEA36E
5AA26E5AA36E5AA234307EAE38>I<B500E0903807FFF0A3000790C70001138000019238
00FE006C16F86E5C017F4A5A6D7E6E495A011F5D6D6C13076E49C7FC0107140E6D6C131E
6E5B010114386D6C13786F5A027F5BEC3FC191381FE3C05EEC0FF76EB4C8FC5D14036E5A
B04A7E91B512F0A3342F7EAE38>121 D E /FV 18 90 df<EE03804C7EA34C7EA34C7EA3
4C7EA34C7EA24C7E16E7A203017F16C3A20303801681A2DB07017F82A2030E80177FA24B
80173FA24B6D7EA20378800370130FA203F0804B1307A20201814B7FA24A488183A24AC7
8083A2020E82187FA24A6F7E021FB7FC4A82A30278C8EA1FFC0270150FA202F0824A1507
A249488284A249488384A249CA7FA24984010E177FA2011E84193F133E017F8448486C4C
7E000F01E04B487EB500FE037FEBFFFEA44F557CD458>65 D<B812FEEFFFC018F818FE26
007FF8C73807FF806D4802017F011F6F6C7E727E727E727E727EA2727EA2721380A21AC0
A384A360A21A80A2601A00A24E5A180F614E5A4E5AF0FFE04D1380050790C7FCEF7FFC91
B712E08418FE02F0C73801FF809438007FE0F01FF8727EF003FE857213807213C0F17FE0
A2F13FF0A21AF8191FA21AFCA81AF8193FA21AF0197FF1FFE0A24E13C04E1380604E1300
F03FFC013FEEFFF8496C02075BBA12C096C7FC18FC18C046527AD153>I<DC1FFC14034B
B500C01307030F02F0130F037F14FC912801FFF800FF131F02070180EB1FC04A48C73807
E03FDA3FF8913801F07FDA7FE0EC00F8902601FF80ED3CFF4990C97E494882494882495A
4948824948825C01FF834849177F91CBFC48193F485AA24848181FA2121F49180FA2123F
A2491807127FA31A005B12FFAE127F7FA31A07123F7FA2121FA26D180F000F190EA26C7E
1A1E6C6C181C6C193C806C6D1778137F6E17F06D6CEE01E06D7E6D6CEE03C06D6CEE0780
6D6CEE0F006D6D151E9026007FE0157CDA3FF85DDA0FFEEC03F06E6C6CEB0FE0020101F8
EBFF806E6CB548C7FC030F14F8030114E09226001FFEC8FC48567AD355>I<B812FEEFFF
E018FC18FF26007FFCC7000F13C06D4802017F011F9238003FF8F00FFC727EF001FF727F
737E737E737E190F86737EA2737E737EA21B8085A21BC01A7F1BE0A4F23FF0A51BF8AE1B
F0A4F27FE0A41BC01AFF1B80A24F1300A24F5AA24F5AA24F5A4F5A4F5A4F5A4F5A4E90C7
FC4E5AF00FFCF03FF8013FEEFFE0496C020F5BBAC8FC18FC18E04DC9FC4D527BD159>I<
BB1280A426007FFCC8123F6D48030313C0011F1600193F191F190F1907A21903A2F101E0
A31900A6050E1470A41A00A3171EA3173E177E17FE160791B6FCA49138F800071600177E
173E171EA3170EA31A0EA31A1C94C8FCA41A3C1A38A31A78A31AF0A21901A21903A21907
F10FE0193F19FF013F1603496C153FBBFC1AC0A347527BD150>I<BA12FCA426007FFCC7
12016D489138001FFE011F160318011800197E193EA2191EA285A385A6F10380A2171CA2
96C7FCA5173CA3177C17FC1601160F91B6FCA49138F8000F16011600177C173CA3171CA7
94C9FCB3497E49B4FCB712E0A441527BD14C>I<B70107B612F8A4C66C48C80003EBF000
6D486F5B6D486F5BB3AD91B9FCA402F8C9FCB3B1496C4B7F496C4B7FB70107B612F8A44D
527BD158>72 D<B7FCA439007FFE006D5A6D5AB3B3B3B0497E497EB7FCA420527BD12A>
I<B74AB512FEA4C66C48C9003F13C06D487048C7FC6D4817F0735A1A804FC8FC193E6161
4E5A4E5A4E5A4E5A4EC9FC183E60604D5A4D5A4D5A4D5A4DCAFC173E5F5F4C5A4C5A4C5A
160F4C7E4C7E167F4C7E4B7F5D923807CFFE92380F8FFFED1F07DB3E037F037C804B7E91
26F9E0007FDAFBC080DAFF80137F92C76C7E4A814A141F4A6E7E8483717F8583717F8518
7F727EA2727E727EA2727E727FA2727F727FA2737E86193F8686496C83496C93B512E0B7
020FECFF80A451527BD15B>75 D<B712E0A4C66C90CAFCEB3FFC6D5AB3B3A9191CA51938
A61978A319F819F0A218011803A21807180F181F183F18FF013F1503496C023F13E0BAFC
A43E527BD149>I<B500FC95B512FCA36E5FD8007FF2F800D93DFF943803BFF0011D62A2
011C6DEE073FA36E6C160EA36E6C161CA26E6C1638A36E6C1670A36E6C16E0A36E6CED01
C0A26E6CED0380A36E6DEC0700A36F6C140EA36F6C5CA26F6C5CA36F6C5CA36F6C5CA26F
6C495AA36F6C495AA36F6D48C7FCA393387FC00EA2706C5AA3706C5AA3706C5AA3706C5A
A270B45AA3705BA3013E6E90C8FC137F496C4E7E000701F0027E4B7EB66C4BB612FC173C
A35E527AD16B>I<B500FC030FB512F8A28080D8003FDC007F13006D6DED0FF8735A011D
6D6F5A011C6D6F5AA26E7E81143F6E7E81140F6E7E82806E7F82806F7E82153F6F7EA26F
7E6F7EA26F7F83816F7F83167F707E83161F707E838270138018C0827013E0A2EF7FF0EF
3FF8A2EF1FFC18FE170FEF07FF1981837113C119E183F07FF119F9183FF01FFD19FF8484
A28484A284A2197F193F133E017F171F496C160F000713F0B66C15071903A219014D527B
D158>I<EE7FF80307B57E033F14F09239FFC00FFC913A03FE0001FFDA0FF89038007FC0
DA3FE0EC1FF0DA7F80EC07F84AC86C7E49486F7E49486F7E4948707E4948707E4948707E
4948707E017F844948707E91CA120348854848717EA24848711380A2000F1AC049187F00
1F1AE0A34848F03FF0A3007F1AF8A249181FA300FF1AFCAE6C6CF03FF8A5003F1AF06D18
7FA2001F1AE0A26D18FF000F1AC0A26C6C4D1380A200031A006D5F6C616C6D4C5A6E160F
017F606D6C4C5A6D6C4C5AA26D6C4C5A6D6C4C5A6D6C4B90C7FCD900FFED03FCDA7FC0EC
0FF86E6C4A5ADA0FF8EC7FC0DA03FE4948C8FC913A00FFC00FFC033FB512F00307148092
26007FF8C9FC4E567AD35B>I<B812FCEFFFC018F818FE26007FFCC7381FFF806D480201
7F011F9238007FF0F01FF8727E727E727E841A807213C0A21AE0197FA21AF0A91AE0A219
FF1AC0A24E13801A00604E5A4E5A4E5AF07FE04D485A051F90C7FC91B712FC18F0188002
F8CBFCB3AE497E497EB7FCA444527BD150>I<B812C017FEEFFFC018F026007FFCC713FC
6D48EC0FFF011F03017F9438007FE0727E727E727E180785727EA28684A286A762A26097
C7FCA24E5A614E5A4E5A4E5AF0FFC04D90C8FCEF0FFEEFFFF891B712C04DC9FC839126F8
000113C09338003FF0EF0FFC717EEF01FF85717F727EA2727EA2727EA985A81B0785180F
A21B0F0607140E496C82496C6F141EB76EEB801C72EBC03C96387FE0F896381FFFF0CC00
0713E09638007F8050547BD156>82 D<DA3FF0130349B55B010714C0011FECF00F903A7F
E00FF81F49C712FED801FCEC3F3F4848EC1FBF48486EB4FC48481403485A4980003F8190
C97E5A83127E8312FEA283A37E837FA27F007F93C7FC7F7FEA3FFC7F6C6C7E14F86CEBFF
806C14F8EDFF806C15F06C15FE6C6C806D15C0010F81010315F8D9007F80140F02008003
0F7F03001480161F040713C0160182EF7FE0A2173FEF1FF0A200E0160FA31707A37EA318
E07E170F7E18C06C161F6C17806D153F6D16006D157E6D15FED8FCFC4A5A017F4A5A26F8
3FC0EB0FF0D90FFEEB7FC0D8F003B65A48C64AC7FC023F13F848010113C034567AD341>
I<003FBB12C0A449C79038F0000701F06E48130001C0183F48C8EE0FE0007E1907007C19
03A200781901A400701900A500F01AF0481A70A6C91700B3B3AC4C7E030313FC027FB712
E0A44C517CD055>I<B600FE0303B512F8A4C60280DB007F1300013F90CAEA1FF86DF00F
E06D616D616D6D94C7FC6F161E6D181C6F163C6D18386E6C16786F5E023F5F6F1501021F
5F6E6C15036F4B5A6E94C8FC6E6D5C70140E6E161E705C6E16386F6C1478701470033F15
F070495A6F6C5C030F1403705C6F14076F6D48C9FCEFC00E6F141EEFE01C6F143C706C5A
EFF870043F13F0715AEE1FFF705B60827090CAFCB3AA5E4C13C0031FB612E0A455527FD1
58>89 D E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: a4
%%EndSetup
%%Page: 1 1
1 0 bop 624 1109 a FV(ON)39 b(THE)f(READ)m(ABILITY)g(OF)h(MA)m(CHINE)f
(CHECKABLE)1581 1222 y(F)m(ORMAL)i(PR)m(OOFS)1602 2462
y FU(a)34 b(thesis)f(submitted)g(to)1179 2575 y(The)h(University)f(of)h
(Kent)f(a)-6 b(t)34 b(Canterbur)-6 b(y)1279 2687 y(in)35
b(the)e(subject)g(of)g(computer)g(science)1748 2800 y(f)n(or)g(the)h
(degree)1513 2913 y(of)g(doctor)f(of)h(philosophy.)2047
5080 y FT(By)1777 5193 y(Vincen)m(t)d(Zammit)1871 5306
y(Marc)m(h)g(1999)p eop
%%Page: 2 2
2 1 bop 2977 1257 a FS(T)-9 b(o)41 b(my)h(family)2078
5954 y FT(ii)p eop
%%Page: 3 3
3 2 bop 378 1061 a FR(Con)-6 b(ten)g(ts)3770 1597 y FQ(ii)378
1800 y(List)35 b(of)g(T)-9 b(ables)2718 b(vii)378 2004
y(List)35 b(of)g(Figures)2643 b(viii)378 2208 y(Abstract)2966
b(ix)378 2412 y(Ac)m(kno)m(wledgemen)m(ts)2527 b(x)378
2616 y(1)84 b(In)m(tro)s(duction)2686 b(1)514 2729 y
FT(1.1)94 b(Mac)m(hine)31 b(Chec)m(k)-5 b(able)30 b(Pro)s(ofs)g(and)g
(their)f(Readabilit)m(y)82 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)175 b FT(1)723 2841 y(1.1.1)106 b(F)-8
b(ormalised)30 b(and)g(Mec)m(hanised)g(Mathematics)79
b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175
b FT(1)723 2954 y(1.1.2)106 b(Pro)s(of)30 b(Chec)m(king)g(and)g
(Theorem)g(Pro)m(ving)g(En)m(vironmen)m(ts)49 b FP(:)d(:)g(:)f(:)h(:)g
(:)f(:)175 b FT(2)723 3067 y(1.1.3)106 b(The)30 b(Readabilit)m(y)f(of)i
(Mac)m(hine-Chec)m(k)-5 b(able)31 b(Pro)s(ofs)36 b FP(:)46
b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b FT(2)514
3180 y(1.2)94 b(Preliminaries)71 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b FT(3)723 3293 y(1.2.1)106
b(First-Order)29 b(Logic)j FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175
b FT(3)723 3406 y(1.2.2)106 b(Higher-Order)29 b(Logic)g
FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b FT(4)514
3519 y(1.3)94 b(Outline)29 b(of)h(the)h(Thesis)36 b FP(:)45
b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b FT(5)378
3723 y FQ(2)84 b(On)35 b(the)f(Mec)m(hanisation)i(of)f(Mathematical)f
(Pro)s(ofs)1136 b(7)514 3836 y FT(2.1)94 b(The)30 b(Lev)m(el)h(of)f
(Rigour)g(in)f(Mathematics)69 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b
FT(7)514 3949 y(2.2)94 b(The)30 b(F)-8 b(ormalisation)30
b(of)h(Mathematics)69 b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)175 b FT(8)514
4061 y(2.3)94 b(The)30 b(Mec)m(hanisation)h(of)f(Mathematics)41
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)130 b FT(10)723 4174 y(2.3.1)106 b(Automated)32
b(Deduction)56 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b
FT(10)723 4287 y(2.3.2)106 b(Pro)s(of)30 b(Chec)m(king)g(and)g(Pro)s
(of)g(Dev)m(elopmen)m(t)i(Systems)65 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)
g(:)f(:)130 b FT(12)514 4400 y(2.4)94 b(A)31 b(Brief)f(Ov)m(erview)f
(of)i(the)g(HOL)f(System)90 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(14)723
4513 y(2.4.1)106 b(On)30 b(the)g(LCF)h(Approac)m(h)f(of)g(Theorem)h
(Pro)m(ving)58 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)
130 b FT(14)723 4626 y(2.4.2)106 b(The)30 b(Implemen)m(tation)g(of)g
(HOL)82 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)130 b FT(15)723 4739 y(2.4.3)106
b(A)31 b(Num)m(b)s(er)e(of)i(Mec)m(hanised)f(Pro)s(ofs)g(in)f(HOL)89
b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(17)514 4852 y(2.5)94 b(On)30 b(Readable)g(Mec)m(hanical)h(Pro)s
(ofs)c FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(19)723 4965
y(2.5.1)106 b(The)30 b(Unreadabilit)m(y)f(of)h(Mec)m(hanised)h(Pro)s
(ofs)39 b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)
130 b FT(20)723 5078 y(2.5.2)106 b(Extracting)31 b(Natural)f(Language)h
(Pro)s(ofs)f(from)g(Mec)m(hanised)g(Ones)39 b FP(:)46
b(:)f(:)130 b FT(23)723 5191 y(2.5.3)106 b(Impro)m(ving)29
b(the)i(Readabilit)m(y)e(of)i(Mec)m(hanised)f(Pro)s(ofs)74
b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(24)378
5394 y FQ(3)84 b(Case)35 b(Studies)g(on)g(T)-9 b(actic-Based)35
b(Theorem)f(Pro)m(v)m(ers)1035 b(27)514 5507 y FT(3.1)94
b(In)m(tro)s(duction)29 b(and)h(Motiv)-5 b(ation)24 b
FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(27)514 5620
y(3.2)94 b(A)31 b(F)-8 b(ormalisation)30 b(of)g(URM)h(Computabilit)m(y)
d(in)h(HOL)79 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)130 b FT(28)2065 5954 y(iii)p eop
%%Page: 4 4
4 3 bop 723 396 a FT(3.2.1)106 b(The)30 b(URM)h(Mo)s(del)f(of)g
(Computation)g(in)f(HOL)45 b FP(:)h(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)130 b FT(28)723 509 y(3.2.2)106 b(Building)28
b(URM)i(Programs)87 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(31)723
622 y(3.2.3)106 b(P)m(artial)30 b(Recursiv)m(e)g(F)-8
b(unctions)30 b(are)h(URM)g(Computable)58 b FP(:)45 b(:)h(:)g(:)f(:)h
(:)g(:)f(:)130 b FT(32)723 735 y(3.2.4)106 b(De\014ning)30
b(Computable)f(F)-8 b(unctions)32 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b
FT(33)723 848 y(3.2.5)106 b(Concluding)28 b(Remarks)i(on)h(the)f(HOL)g
(F)-8 b(ormalisation)93 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(33)514 961 y(3.3)94 b(A)31 b(Pro)s(of)f(of)g(the)h
FP(S)1392 928 y FO(m)1387 983 y(n)1489 961 y FT(Theorem)f(in)f(Co)s(q)e
FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)130 b FT(34)723 1074 y(3.3.1)106 b(On)30
b(the)g(Co)s(q)g(Theorem)g(Pro)m(ving)g(En)m(vironmen)m(t)52
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(34)723 1187 y(3.3.2)106 b(The)30 b FN(P)7 b(RF)41
b FT(Programming)29 b(Language)46 b FP(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(35)723 1300
y(3.3.3)106 b FN(P)7 b(RF)41 b FT(Computabilit)m(y)g
FP(:)k(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(39)723 1413 y(3.3.4)106
b(The)30 b FP(S)1262 1380 y FO(m)1257 1435 y(n)1359 1413
y FT(Theorem)h FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)
f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(40)723 1526 y(3.3.5)106 b(Concluding)28 b(Remarks)i(on)h(the)f(Co)
s(q)g(F)-8 b(ormalisation)56 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)130 b FT(42)514 1638 y(3.4)94 b(A)31 b(Comparativ)m(e)g(Study)e(of)
h(HOL)g(and)g(Co)s(q)84 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(42)723 1751 y(3.4.1)106
b(De\014nitions)88 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)130 b FT(43)723 1864 y(3.4.2)106 b(Theorem)30 b(Pro)m(ving)36
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(45)723
1977 y(3.4.3)106 b(Miscellaneous)44 b FP(:)h(:)h(:)g(:)g(:)f(:)h(:)g(:)
f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)130 b FT(48)723 2090 y(3.4.4)106 b(Concluding)28
b(Remarks)44 b FP(:)h(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b
FT(50)514 2203 y(3.5)94 b(On)30 b(T)-8 b(actic)31 b(Pro)s(ofs)56
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(51)378 2407 y FQ(4)84 b(The)35 b(Implemen)m(tation)d(of)j(a)g
(Declarativ)m(e)g(Pro)s(of)h(Language)f(in)g(HOL)376
b(54)514 2520 y FT(4.1)94 b(In)m(tro)s(duction)27 b FP(:)46
b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(54)514 2633 y(4.2)94 b(The)30 b(Structure)g(of)g(SPL)g(Scripts)63
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(55)723 2746 y(4.2.1)106
b(An)30 b(Example)92 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)130 b FT(55)723 2858 y(4.2.2)106 b(Sectioning)30 b(Pro)s(of)g
(Scripts)49 b FP(:)d(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)
g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(58)723 2971
y(4.2.3)106 b(Reasoning)30 b(Items)74 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)130 b FT(59)723 3084 y(4.2.4)106 b(Pro)s(ofs)30
b(and)g(Justi\014cations)d FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(60)723 3197 y(4.2.5)106 b(SPL)30 b(Sen)m(tences)78
b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b
FT(65)514 3310 y(4.3)94 b(Pro)s(of)30 b(Chec)m(king)g(SPL)g(Scripts)f
(in)g(HOL)67 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)
f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(68)723 3423 y(4.3.1)106
b(The)30 b(En)m(vironmen)m(t)g(of)g(SPL)79 b FP(:)45
b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)130 b FT(69)723 3536 y(4.3.2)106 b(The)30
b(Represen)m(tation)h(of)f(SPL)g(F)-8 b(acts)32 b(in)d(HOL)63
b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(70)723 3649 y(4.3.3)106 b(P)m(arsing)30 b(Pro)s(of)g(Scripts)90
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(71)723 3762 y(4.3.4)106
b(Pro)s(cessing)30 b(SPL)f(Constructs)74 b FP(:)46 b(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(71)723 3875 y(4.3.5)106 b(Expanding)29 b(SPL)g(F)-8
b(acts)77 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(73)514
3988 y(4.4)94 b(Pro)s(of)30 b(Supp)s(ort)25 b FP(:)46
b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(74)723 4100 y(4.4.1)106 b(A)31 b(Database)h(of)f(T)-8
b(rivial)28 b(Kno)m(wledge)40 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(74)514
4213 y(4.5)94 b(Conclusions)53 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(77)378 4417
y FQ(5)84 b(A)35 b(T)-9 b(ableau)35 b(Pro)m(v)m(er)h(as)f(a)f(HOL)g
(Deriv)m(ed)i(Rule)1282 b(79)514 4530 y FT(5.1)94 b(In)m(tro)s(duction)
27 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)130 b FT(79)514 4643 y(5.2)94 b(A)31 b(Clausal)e(T)-8
b(ableau)30 b(with)f(Rigid)f(Basic)j(Sup)s(erp)s(osition)71
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(80)723 4756 y(5.2.1)106 b(On)30 b(Clausal)f(T)-8
b(ableaux)30 b(and)f(Rigid)g(Basic)i(Sup)s(erp)s(osition)62
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)130 b FT(80)723 4869
y(5.2.2)106 b(The)30 b FN(C)5 b(B)s(S)i(E)38 b FT(Calculus)80
b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(81)723 4982
y(5.2.3)106 b(Some)31 b(Examples)h FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)130 b FT(84)514 5095 y(5.3)94 b(The)30 b(T)-8
b(ableau)30 b(Calculus)e(in)h(HOL)g FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(86)723 5208 y(5.3.1)106 b(Reasoning)30 b(with)g(P)m(olymorphic)e
(F)-8 b(orm)m(ulae)37 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)130 b FT(87)723 5321 y(5.3.2)106 b(Prepro)s(cessing)
29 b(F)-8 b(orm)m(ulae)70 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(90)723 5433 y(5.3.3)106 b(Pro)s(of)30 b(Searc)m(h)80
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(91)723 5546 y(5.3.4)106 b(Deriving)30 b(a)g(HOL)g(Theorem)58
b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)130 b FT(94)514 5659 y(5.4)94
b(F)-8 b(rom)31 b(Higher-Order)e(to)i(First-Order)e(Logic)50
b FP(:)c(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)130 b FT(95)2066 5954 y(iv)p eop
%%Page: 5 5
5 4 bop 514 396 a FT(5.5)94 b(Conclusions)28 b(and)i(F)-8
b(uture)31 b(W)-8 b(ork)46 b FP(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(96)378 600 y FQ(6)84 b(Structured)35 b(Straigh)m(tforw)m(ard)g
(Justi\014cations)1324 b(98)514 713 y FT(6.1)94 b(Motiv)-5
b(ation)92 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)130 b FT(98)514 826 y(6.2)94 b(On)30 b(Explicitly)d(Stated)
k(Inferences)f(and)g(Implicitly)d(Applied)h(Manipulations)76
b FP(:)85 b FT(100)723 939 y(6.2.1)106 b(Generalising)29
b(Inferences)c FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)
f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(100)723
1052 y(6.2.2)106 b(Straigh)m(tforw)m(ard)30 b(Justi\014cations)f(with)g
(Explicitly)e(Stated)k(Inferences)55 b FP(:)85 b FT(102)514
1165 y(6.3)94 b(The)30 b(Syn)m(tax)h(of)f(Structured)f
(Justi\014cations)82 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)
f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(103)514 1278 y(6.4)94
b(The)30 b(Seman)m(tics)h(of)f(Structured)f(Justi\014cations)i
FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)85 b FT(104)723 1391 y(6.4.1)106 b(Implicit)28 b(First-Order)h
(Inferences)75 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(104)723 1504 y(6.4.2)106
b(Explicitly)28 b(Stated)j(Inferences)55 b FP(:)46 b(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(109)514 1616 y(6.5)94 b(Results)30 b(on)g(Structured)f
(Justi\014cations)90 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)
g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(110)514 1729
y(6.6)94 b(Discussion)39 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(114)378 1933 y
FQ(7)f(A)35 b(Coloured)g(First-Order)f(Logic)1790 b(121)514
2046 y FT(7.1)94 b(In)m(tro)s(duction)27 b FP(:)46 b(:)g(:)f(:)h(:)g(:)
f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(121)514
2159 y(7.2)94 b(A)31 b(First-Order)e(Logic)h(with)g(Coloured)f(F)-8
b(orm)m(ulae)79 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)85 b FT(123)723 2272 y(7.2.1)106 b(Basic)31
b(De\014nitions)58 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(123)723 2385 y(7.2.2)106 b(The)30 b(Consistency)g(of)g(Sets)h(of)f
(Coloured)f(F)-8 b(orm)m(ulae)86 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)
g(:)f(:)85 b FT(126)514 2498 y(7.3)94 b(F)-8 b(rom)31
b(Coloured)e(F)-8 b(orm)m(ulae)31 b(to)h(Uncoloured)d(Ones)52
b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(127)723 2611 y(7.3.1)106 b(The)30 b(De\014nition)f(of)i(a)f
(Decolourisation)i FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)
g(:)f(:)h(:)g(:)f(:)85 b FT(127)723 2724 y(7.3.2)106
b(Correctness)31 b(of)f(the)h(Decolourisation)f(Mapping)41
b FP(:)k(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(131)723 2836 y(7.3.3)106 b(Applications)89 b FP(:)45
b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(133)514
2949 y(7.4)94 b(Changing)30 b(the)g(Colour)f(of)i(F)-8
b(orm)m(ulae)40 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(134)723
3062 y(7.4.1)106 b(The)30 b(De\014nition)f(of)i(Recolouring)e(Mappings)
46 b FP(:)g(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(135)723 3175 y(7.4.2)106 b(Consistency)30 b(Results)f(on)i
(Recoloured)f(Sets)65 b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)85 b FT(136)514 3288 y(7.5)94 b(Coloured)30
b(In)m(terp)s(olan)m(ts)79 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)85 b FT(138)514 3401 y(7.6)94 b(An)30 b(Undecidabilit)m(y)e(Result)
79 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(147)514
3514 y(7.7)94 b(Summary)72 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(148)378 3718
y FQ(8)f(Pro)s(of)36 b(Chec)m(king)g(Structured)f(Straigh)m(tforw)m
(ard)f(Justi\014cations)529 b(149)514 3831 y FT(8.1)94
b(In)m(tro)s(duction)27 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(149)514 3944 y(8.2)94
b(Pro)s(of)30 b(Chec)m(king)g(Implicit)e(First-Order)h(Inferences)76
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(151)723 4056 y(8.2.1)106 b(A)31 b(Restricted)f(Pro)s(of)g(Searc)m
(h)h(for)f(Chec)m(king)g(Implicit)e(Inferences)83 b FP(:)46
b(:)f(:)85 b FT(151)723 4169 y(8.2.2)106 b(Soundness)29
b(of)h(the)h(Restriction)87 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(152)723
4282 y(8.2.3)106 b(Completeness)30 b(of)h(the)f(Restriction)d
FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)85 b FT(155)723 4395 y(8.2.4)106 b(The)27
b(Undecidabilit)m(y)d(of)k(First-Order)d(Implicit)g(and)i(Explicit)e
(Deriv)-5 b(ations)5 b(157)514 4508 y(8.3)94 b(F)-8 b(rom)31
b(Structured)e(Justi\014cations)g(to)i(Coloured)e(Problems)k
FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b
FT(158)723 4621 y(8.3.1)106 b(A)31 b(Restricted)f(Pro)s(of)g(Searc)m(h)
h(for)f(Chec)m(king)g(Structured)f(Justi\014cations)91
b(158)723 4734 y(8.3.2)106 b(A)31 b(Con\015uence)e(Prop)s(ert)m(y)d
FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(163)514 4847 y(8.4)94
b(Soundness)29 b(and)g(Completeness)h(of)g(the)g(Restricted)h(Pro)s(of)
e(Chec)m(king)h(of)g(Struc-)723 4960 y(tured)g(Justi\014cations)42
b FP(:)k(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(167)723 5073 y(8.4.1)106 b(Soundness)29 b(and)g(Completeness)h
(for)g(P)m(articular)g(Cases)94 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)
85 b FT(167)723 5186 y(8.4.2)106 b(On)30 b(W)-8 b(ell-Coloured)29
b(P)m(artitions)83 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(172)723 5299
y(8.4.3)106 b(Soundness)29 b(and)g(Completeness)h(for)g(the)h(General)f
(Case)63 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(178)514
5411 y(8.5)94 b(Mo)s(difying)29 b(the)h FN(C)5 b(B)s(S)i(E)38
b FT(Deriv)m(ed)31 b(Rule)e(to)i(Chec)m(k)g(Structured)e
(Justi\014cations)90 b FP(:)85 b FT(187)514 5524 y(8.6)94
b(Summary)72 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)85 b FT(190)2079 5954 y(v)p eop
%%Page: 6 6
6 5 bop 378 396 a FQ(9)84 b(A)35 b(Mec)m(hanisation)h(of)f(Group)g
(Theory)1573 b(192)514 509 y FT(9.1)94 b(In)m(tro)s(duction)27
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)
85 b FT(192)514 622 y(9.2)94 b(Group)30 b(Theory)g(in)f(SPL)74
b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(193)723
735 y(9.2.1)106 b(The)30 b(De\014nition)f(of)i(Groups)80
b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)85 b FT(193)723 848 y(9.2.2)106
b(Preliminary)28 b(Results)81 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(194)723 961 y(9.2.3)106 b(Preliminary)28 b(Simpli\014ers)e(and)k
(Database)i(Query)d(F)-8 b(unctions)34 b FP(:)46 b(:)f(:)h(:)g(:)f(:)85
b FT(197)723 1074 y(9.2.4)106 b(Subgroups)30 b FP(:)46
b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(199)514 1187 y(9.3)94 b(Congruences,)31 b(Cosets)g(and)e(Subgroup)
g(Pro)s(ducts)79 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)85 b FT(201)514 1300 y(9.4)94 b(F)-8 b(urther)30
b(Results)47 b FP(:)e(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
f(:)85 b FT(204)723 1413 y(9.4.1)106 b(Normal)30 b(Subgroups)f(and)g
(Quotien)m(t)i(Groups)e FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)85 b FT(204)723 1526 y(9.4.2)106 b(Homomorphisms)29
b(and)h(Isomorphisms)88 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)85 b FT(206)514 1638 y(9.5)94
b(Discussion)39 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)
g(:)f(:)h(:)g(:)f(:)85 b FT(209)378 1842 y FQ(10)32 b(Conclusions)2617
b(214)514 1955 y FT(10.1)49 b(Summary)29 b(of)i(the)f(Main)g(Con)m
(tributions)58 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(214)514 2068 y(10.2)49
b(F)-8 b(uture)31 b(W)-8 b(ork)88 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g
(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)
g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(219)378 2272
y FQ(A)57 b(The)35 b(Syn)m(tax)g(of)g(SPL)2282 b(222)514
2385 y FT(A.1)71 b(Reasoning)31 b(Items)81 b FP(:)46
b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(222)514 2498 y(A.2)71 b(Justi\014cations)i FP(:)46
b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(224)514 2611 y(A.3)71 b(Sen)m(tences)j FP(:)46 b(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)
g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b
FT(226)378 2814 y FQ(B)62 b(Seman)m(tic)31 b(T)-9 b(ableaux)32
b(for)g(First-Order)f(Logic)h(With)g(and)g(Without)f(Equalit)m(y)5
b(228)514 2927 y FT(B.1)75 b(The)30 b(Structure)g(of)g(T)-8
b(ableaux)33 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(228)514 3040 y(B.2)75 b(T)-8 b(ableaux-Based)31
b(Pro)s(of)f(Pro)s(cedures)25 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(229)723 3153 y(B.2.1)i(F)-8 b(ree)32 b(V)-8 b(ariable)30
b(T)-8 b(ableaux)91 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(231)723
3266 y(B.2.2)i(Connection)30 b(T)-8 b(ableaux)30 b(Calculus)81
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)85 b FT(233)723 3379 y(B.2.3)i(T)-8 b(ableaux)30
b(Calculi)e(for)i(First-Order)f(Logic)i(with)e(Equalit)m(y)g
FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)85 b FT(234)378 3583
y FQ(C)60 b(A)35 b(Long)h(Pro)s(of)2529 b(239)514 3696
y FT(C.1)73 b FN(K)q FT(-Consistency)31 b(Implies)d FN(K)q
FT(-Satis\014abilit)m(y)68 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(239)378
3900 y FQ(Bibliograph)m(y)2708 b(246)2066 5954 y FT(vi)p
eop
%%Page: 7 7
7 6 bop 378 1061 a FR(List)77 b(of)g(T)-19 b(ables)514
1506 y FT(1)164 b(On)30 b(the)g(Source)h(Co)s(de)f(of)g(the)h(Mec)m
(hanisation)f(in)f(HOL.)57 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)130 b FT(34)514 1619 y(2)164 b(On)30 b(the)g(Source)h(Co)s
(de)f(of)g(the)h(Mec)m(hanisation)f(in)f(Co)s(q.)91 b
FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(43)514 1731 y(3)164 b(On)30 b(the)g(Source)h(Co)s(de)f(of)g(the)h
(Mec)m(hanisation)f(of)h(Group)e(Theory)-8 b(.)118 b
FP(:)46 b(:)f(:)h(:)g(:)f(:)85 b FT(212)514 1844 y(4)164
b(A)31 b(Uniform)e(Notation)i(for)f(First-Order)f(F)-8
b(orm)m(ulae.)85 b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)
g(:)f(:)85 b FT(231)2054 5954 y(vii)p eop
%%Page: 8 8
8 7 bop 378 1061 a FR(List)77 b(of)g(Figures)514 1506
y FT(1)164 b(The)30 b(Primitiv)m(e)f(Inference)h(Rules)f(of)i(the)f
(HOL)g(System.)59 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)130 b FT(16)514 1619 y(2)164 b(The)30 b(Beha)m(viour)h(of)f
FM(Sub)g FP(f)39 b(g)s FT(.)110 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(37)514 1731 y(3)164 b(The)30 b(De\014nition)f(of)i
FM(Subl)e FP(m)h(n)g(f)40 b FT([)p FP(g)2006 1745 y FL(0)2046
1731 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(g)2305 1746 y
FO(k)r FK(\000)p FL(1)2439 1731 y FT(].)104 b FP(:)45
b(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(38)514 1844 y(4)164 b(An)30 b(Example)g(of)g(a)h(T)-8
b(actic)31 b(Pro)s(of.)79 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(52)514 1957 y(5)164 b(An)30 b(Example)g(SPL)f(Pro)s(of)h(Script.)
65 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g
(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(56)514 2070 y(6)164
b(Declaring)30 b(Relev)-5 b(an)m(t)31 b(Pro)s(of)f(Step)h(Results)e(in)
g(SPL)h(Pro)s(ofs.)77 b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130
b FT(62)514 2183 y(7)164 b(The)30 b(Syn)m(tax)h(of)f(SPL)g(Sen)m
(tences.)102 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(65)514
2296 y(8)164 b(The)30 b(Use)h(of)g(Abstractions.)97 b
FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(67)514
2409 y(9)164 b(Pro)s(cessing)30 b(Lo)s(cal)g(Declarations.)89
b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(72)514 2522 y(10)119
b(The)30 b(Implemen)m(tation)g(of)g(a)h(Query)f(F)-8
b(unction.)125 b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f
(:)h(:)g(:)f(:)130 b FT(76)514 2635 y(11)119 b(The)30
b(Inference)g(Rules)g(of)g(the)h FN(C)5 b(B)s(S)i(E)38
b FT(T)-8 b(ableau)30 b(Calculus.)90 b FP(:)46 b(:)g(:)g(:)f(:)h(:)g(:)
f(:)h(:)g(:)f(:)130 b FT(83)514 2748 y(12)119 b(A)31
b(Closed)e FN(C)5 b(B)s(S)i(E)38 b FT(T)-8 b(ableau.)95
b FP(:)46 b(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)130 b FT(85)514
2861 y(13)119 b(Examples)30 b(of)g(Structured)f(Justi\014cations.)66
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)85 b FT(104)514 2973 y(14)119 b(An)30 b(SPL)g(Pro)s(of)g
(Script)f(using)g(Structured)g(Justi\014cations.)115
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(115)514
3086 y(15)119 b(An)30 b(SPL)g(Pro)s(of)g(of)g FM(nonobv)f
FT(using)g(Unstructured)g(Justi\014cations.)71 b FP(:)46
b(:)g(:)f(:)h(:)g(:)f(:)85 b FT(117)514 3199 y(16)119
b(An)30 b(SPL)g(Pro)s(of)g(of)g FM(nonobv)f FT(using)g(Structured)g
(Justi\014cations.)105 b FP(:)45 b(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(118)514 3312 y(17)119 b(An)30 b(SPL)g(Pro)s(of)g(of)g
FM(nonobv)f FT(using)g(Structured)g(Justi\014cations)g(without)g
FM(!)p FT(.)114 b FP(:)45 b(:)85 b FT(119)514 3425 y(23)119
b(Pro)s(ofs)30 b(of)h(the)f(Uniqueness)f(Results.)108
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)85 b FT(196)514 3538 y(24)119 b(The)30
b(Rules)g(for)g(Normalising)e(Group)i(T)-8 b(erms.)89
b FP(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f
(:)85 b FT(197)514 3651 y(25)119 b(The)30 b(Need)h(for)f(the)h(Group)e
(Elemen)m(t)i(Normaliser)e(in)g(Normalising)f(Subsets.)103
b FP(:)85 b FT(204)514 3764 y(26)119 b(A)31 b(SPL)e(Pro)s(of)h(of)h(a)g
(Theorem)f(on)g(Homomorphisms.)82 b FP(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h
(:)g(:)f(:)h(:)g(:)f(:)85 b FT(211)514 3877 y(27)119
b(An)30 b(Example)g(of)g(a)h(T)-8 b(ableau.)116 b FP(:)45
b(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(229)514 3990 y(28)119
b(The)30 b(F)-8 b(ree)32 b(V)-8 b(ariable)30 b(T)-8 b(ableau)30
b(Calculus.)130 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f
(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(232)514 4103 y(29)119
b(The)30 b(Connection)g(T)-8 b(ableau)30 b(Calculus.)66
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g
(:)f(:)h(:)g(:)f(:)85 b FT(234)514 4215 y(30)119 b(An)30
b(Example)g(of)g(a)h(Closed)f(Connection)f(T)-8 b(ableau.)120
b FP(:)46 b(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(235)514 4328 y(31)119 b(Fitting's)30 b(Additional)e(Expansion)h
(and)h(Closure)f(Rules.)67 b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)f(:)85 b FT(235)514 4441 y(32)119 b(T)-8 b(ableau)30
b(Branc)m(hes)h(with)e(Di\013eren)m(t)i(Rigid)e(Equations.)92
b FP(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85
b FT(237)514 4554 y(33)119 b(Additional)28 b(T)-8 b(ableau)30
b(Rules)g(for)g(Rigid)e(Basic)j(Sup)s(erp)s(osition.)47
b FP(:)f(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)85 b FT(238)2041
5954 y(viii)p eop
%%Page: 9 9
9 8 bop 378 1061 a FR(Abstract)378 1506 y FT(It)37 b(is)f(p)s(ossible)f
(to)i(implemen)m(t)f(mathematical)h(pro)s(ofs)g(in)e(a)j(mac)m
(hine-readable)e(language.)61 b(In-)378 1619 y(deed,)34
b(certain)f(pro)s(ofs,)h(esp)s(ecially)e(those)h(deriving)f(prop)s
(erties)g(of)h(safet)m(y-critical)h(systems,)g(are)378
1731 y(often)42 b(required)d(to)j(b)s(e)e(c)m(hec)m(k)m(ed)k(b)m(y)d
(mac)m(hine)f(in)g(order)h(to)h(a)m(v)m(oid)g(h)m(uman)e(errors.)73
b(Ho)m(w)m(ev)m(er,)378 1844 y(mac)m(hine)32 b(c)m(hec)m(k)-5
b(able)33 b(pro)s(ofs)e(are)h(v)m(ery)h(hard)e(to)i(follo)m(w)e(b)m(y)h
(a)g(h)m(uman)g(reader.)45 b(Because)34 b(of)e(their)378
1957 y(unreadabilit)m(y)-8 b(,)36 b(suc)m(h)g(pro)s(ofs)f(are)i(hard)e
(to)i(implemen)m(t,)g(and)e(more)i(di\016cult)d(still)g(to)j(main)m
(tain)378 2070 y(and)31 b(mo)s(dify)-8 b(.)45 b(In)31
b(this)g(thesis)g(w)m(e)h(study)f(the)i(p)s(ossibilit)m(y)28
b(of)k(implemen)m(ting)e(mac)m(hine)i(c)m(hec)m(k)-5
b(able)378 2183 y(pro)s(ofs)30 b(in)f(a)i(more)g(readable)f(format.)43
b(W)-8 b(e)32 b(design)d(a)i(declarativ)m(e)h(pro)s(of)e(language,)h
(SPL,)f(whic)m(h)378 2296 y(is)f(based)h(on)h(the)f(Mizar)h(language.)
519 2409 y(W)-8 b(e)42 b(also)g(implemen)m(t)d(a)j(pro)s(of)e(c)m(hec)m
(k)m(er)k(for)d(SPL)f(whic)m(h)g(deriv)m(es)h(theorems)g(in)f(the)h
(HOL)378 2522 y(system)24 b(from)f(SPL)g(pro)s(of)h(scripts.)37
b(The)23 b(language)i(and)e(its)g(pro)s(of)h(c)m(hec)m(k)m(er)h(are)g
(extensible,)f(in)f(the)378 2635 y(sense)i(that)i(the)e(user)g(can)h
(mo)s(dify)e(and)h(extend)g(the)h(syn)m(tax)g(of)g(the)g(language)f
(and)g(the)h(deductiv)m(e)378 2748 y(p)s(o)m(w)m(er)j(of)g(the)g(pro)s
(of)f(c)m(hec)m(k)m(er)j(during)26 b(the)j(mec)m(hanisation)g(of)g(a)g
(theory)-8 b(.)41 b(A)28 b(deductiv)m(e)h(database)378
2861 y(of)36 b(trivial)e(kno)m(wledge)i(is)f(used)g(b)m(y)h(the)g(pro)s
(of)f(c)m(hec)m(k)m(er)j(to)f(deriv)m(e)e(facts)i(whic)m(h)e(are)h
(considered)378 2973 y(trivial)31 b(b)m(y)i(the)h(dev)m(elop)s(er)e(of)
i(mec)m(hanised)e(theories)h(so)h(that)f(the)h(pro)s(ofs)e(of)h(suc)m
(h)g(facts)h(can)g(b)s(e)378 3086 y(omitted.)59 b(W)-8
b(e)38 b(also)e(in)m(tro)s(duce)f(the)i(notion)f(of)h(structured)e
(straigh)m(tforw)m(ard)h(justi\014cations,)h(in)378 3199
y(whic)m(h)24 b(simple)g(facts,)k(or)e(conclusions,)f(are)h
(justi\014ed)e(b)m(y)i(a)g(n)m(um)m(b)s(er)e(of)i(premises)f(together)i
(with)d(a)378 3312 y(n)m(um)m(b)s(er)h(of)i(inferences)f(whic)m(h)f
(are)i(used)f(in)f(deriving)g(the)h(conclusion)g(from)g(the)g(giv)m(en)
h(premises.)378 3425 y(A)37 b(tableau)g(pro)m(v)m(er)h(for)e
(\014rst-order)h(logic)f(with)g(equalit)m(y)h(is)f(implemen)m(ted)f(as)
j(a)f(HOL)g(deriv)m(ed)378 3538 y(rule)29 b(and)h(used)f(during)g(the)h
(pro)s(of)g(c)m(hec)m(king)h(of)f(SPL)g(scripts.)519
3651 y(The)g(w)m(ork)h(presen)m(ted)g(in)e(this)h(thesis)f(also)i
(includes)d(a)j(case)h(study)e(in)m(v)m(olving)f(the)i(mec)m(hani-)378
3764 y(sation)h(of)g(a)h(n)m(um)m(b)s(er)d(of)j(results)e(in)f(group)i
(theory)g(in)f(SPL,)h(in)f(whic)m(h)f(the)j(deductiv)m(e)f(p)s(o)m(w)m
(er)g(of)378 3877 y(the)f(SPL)e(pro)s(of)h(c)m(hec)m(k)m(er)i(is)d
(extended)i(throughout)f(the)g(dev)m(elopmen)m(t)h(of)g(the)f(theory)-8
b(.)2066 5954 y(ix)p eop
%%Page: 10 10
10 9 bop 378 1061 a FR(Ac)-6 b(kno)g(wledgemen)g(ts)378
1506 y FT(I)38 b(thank)f(m)m(y)h(sup)s(ervisor,)g(Simon)e(Thompson,)j
(for)f(his)e(con)m(tin)m(uous)i(supp)s(ort)e(and)h(encourage-)378
1619 y(men)m(t.)70 b(I)39 b(greatly)i(appreciate)f(the)g(guidance)f(he)
h(has)f(giv)m(en)h(me)g(throughout)g(the)g(three)g(y)m(ear)378
1731 y(p)s(erio)s(d)28 b(of)j(m)m(y)f(study)-8 b(.)519
1844 y(I)30 b(also)g(thank)g(all)f(the)i(academic)f(and)g(non-academic)
h(sta\013)f(of)h(the)f(Computing)f(Lab)s(oratory)378
1957 y(at)d(the)g(Univ)m(ersit)m(y)e(of)i(Ken)m(t)f(for)g(pro)m(viding)
f(an)h(excellen)m(t)g(w)m(orking)g(en)m(vironmen)m(t.)39
b(In)24 b(particular,)378 2070 y(I)e(thank)h(all)e(the)i(sta\013)h(mem)
m(b)s(ers)e(and)g(researc)m(h)h(studen)m(ts)f(of)h(the)g(TCS)f(group)g
(for)g(their)g(commen)m(ts)378 2183 y(on)31 b(parts)f(of)h(the)g(w)m
(ork)g(presen)m(ted)f(in)g(this)f(thesis.)41 b(I)31 b(also)f(thank)h(m)
m(y)g(examiners,)f(Keith)g(Hanna)378 2296 y(and)g(T)-8
b(om)30 b(Melham,)h(for)f(their)f(helpful)f(commen)m(ts)j(on)f(this)f
(thesis.)519 2409 y(I)i(thank)g(the)g(organisers,)h(sp)s(onsors,)e(sp)s
(eak)m(ers)h(and)g(participan)m(ts)f(of)h(the)h(1996)h(BRICS)d(Au-)378
2522 y(tumn)23 b(Sc)m(ho)s(ol)g(on)g(V)-8 b(eri\014cation,)25
b(the)f(1997)h(Marktob)s(erdorf)e(Summer)f(Sc)m(ho)s(ol)i(on)f
(Computational)378 2635 y(Logic,)i(the)f(1996)h(and)d(1997)j(TPHOLs)d
(Conferences)i(and)e(the)i(1997)h(PTP)d(W)-8 b(orkshop)24
b(for)f(making)378 2748 y(suc)m(h)30 b(ev)m(en)m(ts)i(v)m(ery)f
(researc)m(h-stim)m(ulating.)519 2861 y(I)22 b(w)m(armly)g(thank)h
(Geraldina,)g(Helena)f(and)g(Jason)h(for)f(b)s(eing)f(w)m(onderful)g
(o\016ce)j(mates)f(and)f(for)378 2973 y(all)j(the)h(great)h(time)e(w)m
(e)h(sp)s(en)m(t)g(together.)41 b(During)24 b(m)m(y)i(sta)m(y)h(in)d
(Can)m(terbury)h(I)h(met,)h(made)f(friends)378 3086 y(with,)39
b(and)e(shared)g(houses)h(with)e(man)m(y)i(in)m(teresting)g
(individuals)33 b(from)k(all)g(the)h(con)m(tinen)m(ts)h(of)378
3199 y(the)34 b(w)m(orld.)49 b(I)33 b(thank)g(them)h(all)e(for)i(their)
e(friendship)e(and)j(for)g(the)h(time)f(w)m(e)h(sp)s(en)m(t)f
(together.)52 b(I)378 3312 y(esp)s(ecially)26 b(thank)i(Julie)e(for)i
(her)g(companionship.)37 b(I)28 b(thank)g(Mik)m(e,)h(Kevin,)f(Rob)s
(erta)g(and)g(Ingrid)378 3425 y(for)i(making)g(me)g(feel)g(closer)h(to)
g(Malta.)519 3538 y(I)f(also)g(thank)g(all)f(m)m(y)i(friends)d(in)h
(Malta)i(for)f(alw)m(a)m(ys)h(b)s(eing)e(v)m(ery)h(encouraging.)41
b(I)30 b(thank)g(m)m(y)378 3651 y(family)f(for)h(their)f(care,)j(supp)s
(ort)c(and)i(all)f(the)i(things)e(they)i(ha)m(v)m(e)g(done)f(to)i(me.)
519 3764 y(Last,)39 b(but)e(not)g(least,)i(I)e(thank)f(the)h(Computing)
f(Lab)s(oratory)h(for)f(pro)m(viding)f(the)i(funding)378
3877 y(for)28 b(m)m(y)f(studies.)39 b(The)27 b(w)m(ork)h(presen)m(ted)g
(in)e(this)h(thesis)g(w)m(ould)f(not)i(ha)m(v)m(e)h(b)s(een)f(p)s
(ossible)d(without)378 3990 y(this)k(funding.)2079 5954
y(x)p eop
%%Page: 1 11
1 10 bop 378 1019 a FJ(Chapter)65 b(1)378 1434 y FR(In)-6
b(tro)6 b(duction)378 1879 y FT(In)41 b(this)g(thesis)g(w)m(e)h(study)f
(the)h(implemen)m(tation)e(of)i FI(machine-che)-5 b(ckable)44
b(pr)-5 b(o)g(ofs)52 b FT(in)40 b(a)j(format)378 1992
y(that)37 b(can)f(b)s(e)g FI(e)-5 b(asily)39 b(fol)5
b(lowe)-5 b(d)40 b(by)d(a)i(human)g(r)-5 b(e)g(ader)p
FT(.)60 b(The)35 b(implemen)m(tation)g(of)i(mathematical)378
2105 y(pro)s(ofs)g(in)g(a)h(mac)m(hine-c)m(hec)m(k)-5
b(able)40 b(format)e(is)f(usually)f(required)g(when)h(the)i
(correctness)f(of)h(the)378 2218 y(pro)s(ofs)25 b(is)g(a)h(ma)5
b(jor)26 b(concern.)40 b(F)-8 b(or)26 b(example,)h(one)f(requires)f
(that)h(the)h(pro)s(ofs)e(deriving)e(prop)s(erties)378
2331 y(of)43 b(safet)m(y-critical)h(systems)f(are)h(error-free,)i(and)d
(the)g(use)g(of)g(a)h(computer)f(system)g(to)h(c)m(hec)m(k)378
2444 y(suc)m(h)31 b(pro)s(ofs)g(can)g(greatly)h(minimise)d(the)i(n)m
(um)m(b)s(er)f(of)i(errors)f(in)f(comparison)g(with)h(an)g(informal)378
2557 y(pro)s(of.)75 b(Ho)m(w)m(ev)m(er,)47 b(the)42 b(pro)s(ofs)f(whic)
m(h)g(can)h(b)s(e)g(c)m(hec)m(k)m(ed)i(b)m(y)e(curren)m(t)g(computer)f
(systems)h(are)378 2670 y(unreadable)24 b(and)h(hard)f(to)i(follo)m(w,)
g(and)f(it)g(is)f(therefore)i(desirable)e(that)i(more)f(e\013ort)h(is)f
(put)g(in)f(the)378 2783 y(in)m(v)m(estigation)k(of)g(p)s(ossible)e(w)m
(a)m(ys)j(of)f(impro)m(ving)e(the)j(readabilit)m(y)d(of)i(mac)m(hine-c)
m(hec)m(k)-5 b(able)30 b(pro)s(ofs.)378 2895 y(In)37
b(this)h(in)m(tro)s(ductory)f(c)m(hapter)i(w)m(e)f(\014rst)g(brie\015y)
e(discuss)h(the)h(problems)e(concerned)j(with)e(the)378
3008 y(implemen)m(tation)25 b(of)h(readable)f(mec)m(hanised)h(pro)s
(ofs)f(in)f(section)j(1.1.)40 b(Section)26 b(1.2)h(in)m(tro)s(duces)e
(the)378 3121 y(notation)33 b(and)e(de\014nitions)f(whic)m(h)h(are)i
(used)f(in)f(this)g(thesis.)46 b(Section)32 b(1.3)i(giv)m(es)f(a)f
(brief)f(outline)378 3234 y(of)f(the)h(remaining)e(c)m(hapters)h(of)h
(this)e(thesis.)378 3521 y FH(1.1)135 b(Mac)l(hine)45
b(Chec)l(k)-7 b(able)46 b(Pro)t(ofs)f(and)f(their)i(Readabilit)l(y)378
3724 y FT(In)21 b(this)f(section)i(w)m(e)g(illustrate)e(the)h(problems)
f(concerned)i(with)e(the)i(implemen)m(tation)e(of)i(mac)m(hine-)378
3836 y(c)m(hec)m(k)-5 b(able)34 b(pro)s(ofs)f(in)e(a)j(readable)f
(format,)h(and)f(motiv)-5 b(ate)34 b(the)f(w)m(ork)g(presen)m(ted)h(in)
d(this)h(thesis.)378 3949 y(The)e(material)g(giv)m(en)g(here)g(is)g
(discussed)e(in)h(more)i(detail)e(in)g(c)m(hapter)i(2.)378
4193 y FG(1.1.1)112 b(F)-9 b(ormalised)36 b(and)j(Mec)m(hanised)f
(Mathematics)378 4364 y FT(The)33 b(implemen)m(tation)g(of)h
(mathematics)h(in)d(a)i(language)h(whose)f(syn)m(tax)g(and)f(seman)m
(tics)i(is)e(un-)378 4477 y(am)m(biguously)h(de\014ned)h(is)g(referred)
g(to)h(as)g(the)g FI(formalisation)41 b(of)d(mathematics)p
FT(.)59 b(Mathematics)378 4590 y(is)30 b(formalised)f(in)h(order)g(to)i
(ac)m(hiev)m(e)g(a)g(higher)d(degree)j(of)f(precision)e(and)i
(correctness)h(than)e(that)378 4703 y(found)j(in)g(the)i(usual,)g(or)f
FI(informal)p FT(,)k(mathematical)d(texts.)54 b(By)35
b(the)f FI(me)-5 b(chanisation)39 b(of)e(mathe-)378 4816
y(matics)h FT(one)30 b(usually)e(refers)h(to)i(the)f(use)g(of)g(a)g
(mac)m(hine,)g(and)f(esp)s(ecially)f(the)i(use)g(of)g(a)g(computer)378
4929 y(system,)37 b(to)g(p)s(erform)d(mathematical)i(tasks,)i(whic)m(h)
d(include)e(n)m(umerical)i(calculations,)h(manip-)378
5042 y(ulations)c(of)h(mathematical)g(terms)g(and)f(the)i(logical)e
(dev)m(elopmen)m(t)h(of)h(mathematical)f(theories.)378
5155 y(In)d(this)f(thesis)h(w)m(e)h(use)f(the)h(term)f(`mec)m
(hanisation)h(of)f(a)h(mathematical)g(theory')g(to)g(refer)f(to)i(the)
378 5268 y(formalisation)j(of)h(the)h(mathematical)g(theory)g(in)e
(order)h(for)g(pro)s(ofs)g(in)f(it)h(to)h(b)s(e)f(c)m(hec)m(k)m(ed)i(b)
m(y)e(a)378 5381 y(computer)h(system.)59 b(The)37 b(adv)-5
b(an)m(tages)38 b(of)f(using)e(a)i(computer)g(system)g(in)e
(formalising)f(mathe-)378 5494 y(matics)e(include)d(the)j(minimisation)
d(of)j(errors)f(in)g(the)h(de\014nitions)d(and)j(pro)s(ofs,)f(and)g
(the)h(abilit)m(y)378 5606 y(to)f(use)f(sp)s(ecialised)e(to)s(ols)i(to)
h(\014nd)e(formal)h(pro)s(ofs.)2080 5954 y(1)p eop
%%Page: 2 12
2 11 bop 378 5 a FF(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)2031
b FT(2)378 396 y FG(1.1.2)112 b(Pro)s(of)38 b(Chec)m(king)f(and)h
(Theorem)f(Pro)m(ving)f(En)m(vironmen)m(ts)378 568 y
FT(A)31 b FI(pr)-5 b(o)g(of)35 b(che)-5 b(cker)40 b FT(is)30
b(a)h(computer)f(system)h(dev)m(elop)s(ed)f(to)h(c)m(hec)m(k)h(the)f(v)
-5 b(alidit)m(y)29 b(of)i(formal)f(pro)s(ofs.)378 681
y(Examples)20 b(of)i(early)f(pro)s(of)f(c)m(hec)m(k)m(ers)k(include)19
b(A)m(UTOMA)-8 b(TH)22 b(\(de)g(Bruijn)d(1970\))24 b(and)c(Mizar)i(\(T)
-8 b(ry-)378 794 y(bulec)40 b(1978\).)76 b(Mo)s(dern)41
b(computer)h(systems,)i(suc)m(h)d(as)h(HOL)f(\(Gordon)h(and)e(Melham)i
(1993\),)378 907 y(Isab)s(elle)g(\(P)m(aulson)h(1994\),)49
b(Co)s(q)43 b(\(Barras)h(et)g(al.)80 b(1996\),)49 b(LEGO)43
b(\(Luo)h(and)f(P)m(ollac)m(k)h(1992\),)378 1020 y(Nuprl)25
b(\(Constable)i(et)h(al.)39 b(1986\),)30 b(and)d(PVS)g(\(Shank)-5
b(ar,)27 b(Owre,)g(and)g(Rush)m(b)m(y)f(1993\))k(are)d(usually)378
1133 y(called)34 b FI(the)-5 b(or)g(em)40 b(pr)-5 b(oving)38
b(envir)-5 b(onments)44 b FT(since)34 b(they)i(pro)m(vide)e(sev)m(eral)
i(other)f(facilities)f(for)h(the)378 1246 y(mec)m(hanisation)g(of)h
(mathematics)g(apart)g(from)f(pro)s(of)g(c)m(hec)m(king.)57
b(In)35 b(particular,)g(they)h(pro)m(vide)378 1358 y(an)41
b(in)m(teractiv)m(e)i(pro)s(of-disco)m(v)m(ery)e(en)m(vironmen)m(t)g
(based)h(on)f FI(tactics)p FT(.)75 b(In)40 b(a)i(tactic-based)h(en)m
(vi-)378 1471 y(ronmen)m(t,)37 b(theorems)f(are)h(pro)m(v)m(ed)f(b)m(y)
f(sp)s(ecifying)f(them)i(as)g(goals,)i(and)d(then)g(applying)f(tactics)
378 1584 y(in)m(teractiv)m(ely)-8 b(,)27 b(whic)m(h)c(either)h(solv)m
(e)i(the)e(goal)i(automatically)e(or)h(break)g(it)f(in)m(to)h(simpler)d
(subgoals.)378 1697 y(A)27 b(theorem)g(is)f(pro)m(v)m(ed)i(when)e(all)g
(the)h(subgoals)f(of)h(the)g(original)f(goal)h(are)h(solv)m(ed.)39
b(The)26 b(sequence)378 1810 y(of)35 b(tactics)h(required)d(to)j(pro)m
(v)m(e)g(a)f(theorem)g(represen)m(ts)g(a)h(tactic-based)g(pro)s(of)e
(of)h(the)g(theorem.)378 1923 y(The)c(application)g(of)h(a)g(single)f
(tactic)i(can)g(in)m(v)m(olv)m(e)f(v)m(ery)g(p)s(o)m(w)m(erful)f
(inferences.)44 b(F)-8 b(or)33 b(example,)f(a)378 2036
y(commonly)26 b(used)g(class)h(of)g(tactics)h(uses)f(arbitrary)e
(term-rewriting)h(systems)h(to)g(simplify)d(a)j(goal,)378
2149 y(and)j(an)g(application)f(of)h(suc)m(h)g(tactics)i(often)e
(corresp)s(onds)f(to)j(sev)m(eral)e(h)m(undreds)e(of)j(inferences.)378
2392 y FG(1.1.3)112 b(The)38 b(Readabilit)m(y)e(of)h(Mac)m(hine-Chec)m
(k)-6 b(able)38 b(Pro)s(ofs)378 2564 y FT(The)33 b(readabilit)m(y)f(of)
h(a)h(pro)s(of)f(dep)s(ends)e(on)j(the)f(e\013ort)i(required)c(b)m(y)j
(the)f(reader)h(to)g(understand)378 2677 y(it.)39 b(Therefore,)29
b(in)e(order)h(to)h(b)s(e)e(readable,)i(a)f(pro)s(of)g(should)e(con)m
(tain)j(the)f(necessary)h(information)378 2790 y(to)j(b)s(e)g(follo)m
(w)m(ed)f(without)g(undue)f(e\013ort.)45 b(It)32 b(should)e(also)i
(omit)f(irrelev)-5 b(an)m(t)31 b(information,)g(or)h(an)m(y)378
2903 y(information)d(whic)m(h)h(can)h(b)s(e)f(easily)g(deduced)g(b)m(y)
h(the)g(in)m(tended)f(reader)g(of)h(the)g(pro)s(of.)42
b(F)-8 b(urther-)378 3016 y(more,)33 b(in)f(order)f(to)j(facilitate)e
(its)g(readabilit)m(y)-8 b(,)32 b(the)h(information)d(con)m(tained)j
(in)e(a)i(pro)s(of)f(should)378 3128 y(b)s(e)e(organised)f(in)h(a)g(w)m
(a)m(y)i(whic)m(h)d(highligh)m(ts)f(its)i(structure.)519
3241 y(The)i(mec)m(hanised)g(pro)s(ofs)g(that)h(can)g(b)s(e)e(c)m(hec)m
(k)m(ed)k(b)m(y)d(curren)m(t)h(pro)s(of)e(c)m(hec)m(k)m(ers)k(are)e
(not)g(v)m(ery)378 3354 y(readable.)38 b(One)22 b(reason)i(for)f(this)f
(is)g(the)h(fact)h(that)g(the)f(pro)s(of)f(languages)i(accepted)g(b)m
(y)f(most)h(pro)s(of)378 3467 y(c)m(hec)m(k)m(ers)29
b(are)f(not)g(designed)e(for)h(the)g(implemen)m(tation)f(of)i(readable)
f(pro)s(ofs,)g(but)g(for)g(some)h(other)378 3580 y(purp)s(oses.)37
b(F)-8 b(or)27 b(instance,)f(a)g(pro)s(of)f(language)h(based)g(on)f
(tactics)i(is)e(usually)e(designed)i(in)f(order)h(to)378
3693 y(facilitate)30 b(the)g(in)m(teractiv)m(e)h(disco)m(v)m(ery)f(of)g
(pro)s(ofs.)39 b(As)30 b(a)h(result,)e(tactic)i(pro)s(ofs)e(are)h(not)g
(in)m(tended)378 3806 y(to)h(b)s(e)f(easily)g(understo)s(o)s(d)e(b)m(y)
i(a)h(h)m(uman)f(reader)g(and)g(can)h(only)e(b)s(e)h(follo)m(w)m(ed)g
(b)m(y)h(examining)e(the)378 3919 y(e\013ect)37 b(of)f(eac)m(h)i(pro)s
(of)d(step)h(using)e(the)i(in)m(teractiv)m(e)h(theorem)f(pro)m(ving)g
(en)m(vironmen)m(t.)57 b(Because)378 4032 y(of)42 b(their)f
(unreadabilit)m(y)-8 b(,)43 b(it)e(is)g(hard)g(to)i(debug,)h(main)m
(tain)d(and)h(mo)s(dify)e(tactic-based)j(pro)s(ofs)378
4145 y(in)37 b(order)i(to)g(use)f(them)h(to)h(deriv)m(e)e(sligh)m(tly)f
(di\013eren)m(t)h(theorems)h(without)f(feedbac)m(k)h(from)g(the)378
4258 y(theorem)31 b(pro)m(ving)e(en)m(vironmen)m(t.)519
4370 y(The)g(design)g(of)h(a)g(pro)s(of)g(language)g(whose)f(pro)s(ofs)
g(are)i(easy)f(to)h(follo)m(w)e(is)g(not)h(a)g(trivial)e(task.)378
4483 y(F)-8 b(or)31 b(instance,)e(the)h(information)f(con)m(tained)h
(in)e(readable)i(pro)s(ofs)e(should)g(b)s(e)i(at)g(an)g(appropriate)378
4596 y(lev)m(el)42 b(for)f(the)i(in)m(tended)d(reader.)76
b(Ov)m(er-detailed)41 b(pro)s(ofs)h(are)g(tedious)f(to)i(read)f(and)f
(hard)g(to)378 4709 y(understand,)g(while)d(a)j(considerable)e(amoun)m
(t)h(of)h(e\013ort)g(is)e(required)f(to)j(follo)m(w)f(pro)s(ofs)f(whic)
m(h)378 4822 y(con)m(tain)30 b(to)s(o)g(little)e(information.)38
b(It)30 b(is)e(not)h(straigh)m(tforw)m(ard)g(to)h(\014nd)e(this)g(righ)
m(t)h(lev)m(el)g(of)g(detail,)378 4935 y(to)35 b(de\014ne)e(the)i
(appropriate)e(language)i(constructs)f(and)g(inferences)f(to)i(express)
f(pro)s(of)f(steps)h(at)378 5048 y(the)i(required)d(lev)m(el)j(of)f
(detail,)h(and)f(to)i(design)d(and)h(implemen)m(t)f(the)i(algorithms)e
(necessary)i(to)378 5161 y(pro)s(of)30 b(c)m(hec)m(k)i(suc)m(h)e
(inferences)f(e\016cien)m(tly)-8 b(.)519 5274 y(Da)m(vis)31
b(\(1981\))h(and)e(Rudnic)m(ki)e(\(1987\))k(study)d(the)i(notion)e(of)i
FI(obvious)h(infer)-5 b(enc)g(es)p FT(.)41 b(An)30 b(infer-)378
5387 y(ence)d(is)f(ob)m(vious)g(if)f(it)i(can)f(b)s(e)g(easily)g
(deduced)g(b)m(y)g(a)h(h)m(uman)f(reader,)i(and)e(if)f(it)h(can)h(b)s
(e)f(e\016cien)m(tly)378 5500 y(c)m(hec)m(k)m(ed)33 b(b)m(y)f(mac)m
(hine.)43 b(An)31 b(imp)s(ortan)m(t)g(issue)f(discussed)f(in)h(this)h
(thesis)f(is)h(the)g(realisation)f(that)p eop
%%Page: 3 13
3 12 bop 378 5 a FF(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)2031
b FT(3)378 396 y(the)32 b(notion)e(of)i(ob)m(viousness)e(cannot)i(b)s
(e)f(static.)45 b(F)-8 b(or)32 b(instance,)f(the)h(inferences)e(whic)m
(h)g(are)i(con-)378 509 y(sidered)26 b(to)j(b)s(e)f(essen)m(tial)g(to)g
(the)h(readabilit)m(y)d(of)i(the)h(pro)s(ofs)e(of)h(the)g(results)f
(deriv)m(ed)g(in)g(the)h(early)378 622 y(stages)f(of)f(a)g(theory)g
(are)g(v)m(ery)g(often)g(omitted)g(from)f(the)h(pro)s(ofs)f(of)h(the)f
(results)g(giv)m(en)h(later)f(in)g(the)378 735 y(same)30
b(theory)-8 b(.)41 b(What)30 b(is)f(considered)f(to)i(b)s(e)f(ob)m
(vious)g(b)m(y)g(the)h(reader)f(of)h(a)g(pro)s(of)e(dep)s(ends)g(on)h
(his)378 848 y(kno)m(wledge)24 b(of)g(the)h(sub)5 b(ject.)38
b(This)23 b(kno)m(wledge)h(increases)g(as)g(the)g(reader)g(reads)g(and)
g(understands)378 961 y(the)31 b(pro)s(ofs)f(of)h(the)h(results)d(giv)m
(en)i(in)f(the)h(theory)-8 b(.)43 b(This)29 b(suggests)j(that)g(one)f
(cannot)h(use)e(a)i(\014xed)378 1074 y(pro)s(of)c(c)m(hec)m(king)i
(algorithm)d(to)j(c)m(hec)m(k)g(all)e(the)h(mec)m(hanised)f(pro)s(ofs)g
(of)h(a)g(theory)-8 b(.)41 b(The)28 b(dev)m(elop)s(er)378
1187 y(of)f(a)g(mec)m(hanised)g(theory)g(is)f(therefore)h(required)f
(to)h(extend,)h(or)f(impro)m(v)m(e,)h(the)f(deductiv)m(e)g(p)s(o)m(w)m
(er)378 1300 y(of)j(the)h(pro)s(of)f(c)m(hec)m(k)m(er)i(during)c(mec)m
(hanisation.)378 1586 y FH(1.2)135 b(Preliminaries)378
1789 y FT(In)32 b(this)h(section)g(w)m(e)h(giv)m(e)g(a)f(n)m(um)m(b)s
(er)f(of)i(preliminary)29 b(de\014nitions)i(concerning)i(\014rst-order)
f(logic)378 1902 y(and)e(higher-order)f(logic)h(whic)m(h)f(are)i(used)e
(throughout)h(this)f(thesis.)378 2145 y FG(1.2.1)112
b(First-Order)37 b(Logic)378 2317 y FT(The)c(follo)m(wing)f(notation)i
(and)f(de\014nitions)e(of)i(a)h(n)m(um)m(b)s(er)e(of)i(standard)f
(concepts)h(of)g(\014rst-order)378 2430 y(languages)h(and)f
(\014rst-order)g(logic)h(are)g(used)g(in)e(this)h(thesis.)53
b(More)36 b(elab)s(orate)f(treatmen)m(ts)i(can)378 2543
y(b)s(e)30 b(found,)f(for)h(instance,)g(in)f(\(Chang)i(and)e(Keisler)g
(1990\))k(and)d(\(Fitting)g(1996\).)519 2656 y(Let)d
FP(X)34 b FT(b)s(e)26 b(a)g(coun)m(table)h(set)g(of)f(v)-5
b(ariables,)26 b(and)g(let)h(\006)2389 2670 y FO(F)2473
2656 y FT(b)s(e)f(a)h FI(signatur)-5 b(e)p FT(,)28 b(that)f(is,)f(a)h
(collection)378 2769 y(of)k(function)f(sym)m(b)s(ols)g(eac)m(h)i(of)f
(whic)m(h)f(has)h(a)g(\014xed)g(natural)f(n)m(um)m(b)s(er)g(asso)s
(ciated)h(with)f(it)h(called)378 2882 y(the)38 b FI(arity)p
FT(.)65 b(F)-8 b(unction)38 b(sym)m(b)s(ols)f(with)g(zero)i(arit)m(y)f
(are)g(called)g FI(c)-5 b(onstants)p FT(.)66 b(A)38 b
FI(term)46 b FT(is)37 b(either)g(a)378 2995 y(v)-5 b(ariable)38
b(or)h(some)g FP(f)10 b FT(\()p FP(t)1207 3009 y FL(1)1246
2995 y FP(;)15 b(:)g(:)g(:)i(;)e(t)1481 3009 y FO(n)1528
2995 y FT(\))39 b(where)g FP(f)48 b FT(is)38 b(a)i(function)d(sym)m(b)s
(ol,)k FP(n)d FT(is)g(the)h(arit)m(y)h(of)f FP(f)10 b
FT(,)40 b(and)378 3107 y FP(t)411 3121 y FL(1)450 3107
y FT(,)e FP(:)15 b(:)g(:)31 b FT(,)40 b FP(t)747 3121
y FO(n)831 3107 y FT(are)e(terms.)62 b(Constan)m(t)38
b(terms)g FP(c)p FT(\(\))g(are)g(simply)d(denoted)j(b)m(y)f
FP(c)p FT(.)63 b(The)37 b(language)h(of)378 3220 y(\014rst-order)e
(terms)g FN(T)23 b FT(\(\006)1243 3234 y FO(F)1301 3220
y FP(;)15 b(X)7 b FT(\),)40 b(or)c(simply)f FN(T)22 b
FT(,)39 b(is)c(the)i(set)g(of)g(all)e(terms)i(constructed)g(from)f(the)
378 3333 y(function)29 b(sym)m(b)s(ols)g(in)g(\006)1253
3347 y FO(F)1342 3333 y FT(and)g(the)i(v)-5 b(ariables)29
b(in)g FP(X)7 b FT(.)519 3446 y(Let)29 b(\006)746 3460
y FO(R)831 3446 y FT(b)s(e)e(a)i(collection)e(of)h(relation)g(sym)m(b)s
(ols)e(\(also)i(called)g(predicates\))g(with)e(\014xed)i(arities.)378
3559 y(W)-8 b(e)42 b(iden)m(tify)d(t)m(w)m(o)j(predicates)f
FN(>)f FT(and)h FN(?)f FT(with)g(zero)h(arit)m(y)g(in)f(\006)2750
3573 y FO(R)2807 3559 y FT(.)72 b(An)40 b FI(atomic)k(formula)p
FT(,)h(or)378 3672 y FI(atom)p FT(,)29 b(is)d(of)h(the)g(form)f
FP(P)13 b FT(\()p FP(t)1319 3686 y FL(1)1359 3672 y FP(;)i(:)g(:)g(:)h
(;)f(t)1593 3686 y FO(n)1640 3672 y FT(\))27 b(where)g
FP(P)39 b FT(is)26 b(a)h(predicate,)h FP(n)e FT(is)g(the)h(arit)m(y)f
(of)h FP(P)40 b FT(and)26 b FP(t)3575 3686 y FL(1)3614
3672 y FT(,)h FP(:)15 b(:)g(:)32 b FT(,)378 3785 y FP(t)411
3799 y FO(n)491 3785 y FT(are)h(terms.)48 b(First-order)32
b FI(formulae)41 b FT(are)33 b(constructed)g(from)g(atoms,)h(the)f
(unary)f(op)s(erator)h FN(:)p FT(,)378 3898 y(the)k(in\014x)d(binary)h
(op)s(erators)h FN(^)p FT(,)i FN(_)p FT(,)g FN(\))e FT(and)g
FN(,)g FT(whic)m(h)f(are)i(also)f(called)g FI(c)-5 b(onne)g(ctives)p
FT(,)39 b(and)d(the)378 4011 y FI(quanti\014ers)k FN(8)31
b FT(and)g FN(9)p FT(.)45 b(A)32 b FI(liter)-5 b(al)42
b FT(is)31 b(either)g(an)h(atom)h(in)d(whic)m(h)h(case)i(it)e(is)g(a)h
(p)s(ositiv)m(e)f(literal,)g(or)378 4124 y(a)j(negated)g(atom)h(of)e
(the)h(form)f FN(:)p FP(A)p FT(,)h(where)f FP(A)g FT(is)g(atomic,)i(in)
d(whic)m(h)g(case)i(it)f(is)g(negativ)m(e.)51 b(Tw)m(o)378
4237 y(literals)36 b(are)j FI(c)-5 b(omplementary)49
b FT(if)37 b(one)h(is)g(the)g(negation)g(of)h(the)f(other.)64
b(The)38 b FI(c)-5 b(omplement)48 b FT(of)38 b(a)378
4349 y(p)s(ositiv)m(e)31 b(literal)g FP(A)h FT(is)f FN(:)p
FP(A)p FT(,)h(and)g(the)g(complemen)m(t)h(of)f(a)g(negativ)m(e)i
(literal)c FN(:)p FP(A)i FT(is)f FP(A)p FT(.)46 b(W)-8
b(e)33 b(denote)378 4462 y(the)25 b(complemen)m(t)h(of)f(a)h(literal)e
FP(B)29 b FT(b)m(y)1700 4439 y(\026)1679 4462 y FP(B)t
FT(.)39 b(The)25 b(language)h(of)f(\014rst-order)f(form)m(ulae)h
FP(L)p FT(\(\006)3422 4476 y FO(R)3480 4462 y FP(;)15
b FT(\006)3586 4476 y FO(F)3644 4462 y FP(;)g(X)7 b FT(\),)378
4575 y(or)27 b(simply)e FP(L)p FT(,)k(is)d(the)i(set)g(of)g(form)m
(ulae)f(constructed)g(from)h(the)f(predicates)g(in)g(\006)3193
4589 y FO(R)3277 4575 y FT(and)g(the)h(terms)378 4688
y(in)33 b FN(T)22 b FT(.)52 b(F)-8 b(ollo)m(wing)33 b(Fitting)h
(\(1996\),)j(w)m(e)e(also)e(use)h(a)h(coun)m(table)f(set)g(of)g
(constan)m(ts)i FI(P)-7 b(AR)37 b FT(disjoin)m(t)378
4801 y(from)31 b(\006)660 4815 y FO(F)718 4801 y FT(,)i(and)e(de\014ne)
g FP(L)1280 4815 y FE(P)-5 b(AR)1434 4801 y FT(\(\006)1535
4815 y FO(R)1592 4801 y FP(;)15 b FT(\006)1698 4815 y
FO(F)1757 4801 y FP(;)g(X)7 b FT(\),)33 b(or)f(simply)d
FP(L)2439 4815 y FE(P)-5 b(AR)2593 4801 y FT(,)32 b(as)g
FP(L)p FT(\(\006)2926 4815 y FO(R)2984 4801 y FP(;)15
b FT(\006)3090 4815 y FO(F)3170 4801 y FN([)20 b FI(P)-7
b(AR)t FP(;)15 b(X)7 b FT(\).)45 b(The)378 4914 y(elemen)m(ts)31
b(in)e FI(P)-7 b(AR)33 b FT(are)e(called)e FI(p)-5 b(ar)g(ameters)41
b FT(and)30 b(stand)g(for)g(unkno)m(wn)f(elemen)m(ts.)519
5027 y(An)k FI(expr)-5 b(ession)41 b FT(is)33 b(either)f(a)h(term)h(or)
f(a)g(form)m(ula.)48 b(An)33 b(expression)e(is)h(said)h(to)g(b)s(e)g
FI(close)-5 b(d)p FT(,)35 b(or)378 5140 y FI(gr)-5 b(ound)p
FT(,)40 b(if)c(all)f(its)h(v)-5 b(ariables)36 b(are)h(b)s(ound.)58
b(A)37 b FI(sentenc)-5 b(e)44 b FT(is)35 b(a)j(closed)e(form)m(ula.)59
b(A)37 b FI(substitution)378 5253 y FT(is)e(a)i(mapping)e(from)h(v)-5
b(ariables)35 b(to)i(terms.)59 b(W)-8 b(e)38 b(use)e
FN(f)p FP(x)2388 5267 y FL(1)2463 5253 y FN(!)f FP(t)2622
5267 y FL(1)2661 5253 y FP(;)15 b(:)g(:)g(:)i(;)e(x)2915
5267 y FO(n)2998 5253 y FN(!)35 b FP(t)3157 5267 y FO(n)3204
5253 y 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 FP(t)3045 636 y FO(i)3108 622 y FT(for)g
FP(i)e FN(2)g(f)p FT(1)p FP(;)15 b(:)g(:)g(:)i(;)e(n)p
FN(g)p FT(,)378 735 y(with)26 b(suitable)h(renaming)f(of)i(v)-5
b(ariables)27 b(to)h(prev)m(en)m(t)h(free)f(v)-5 b(ariables)26
b(in)h FP(t)2906 749 y FO(i)2962 735 y FT(b)s(ecoming)g(b)s(ound)e
(after)378 848 y(replacemen)m(t.)519 961 y(A)40 b FI(p)-5
b(osition)49 b FT(in)38 b(an)i(expression)e(is)h(a)h(list)e(of)i(p)s
(ositiv)m(e)f(in)m(tegers)h(whic)m(h)e(denotes)j(a)f(path)f(to)378
1074 y(some)32 b(no)s(de)e(in)g(the)i(syn)m(tactic)g(tree)g(represen)m
(tation)g(of)f(the)h(expression.)42 b(In)31 b(particular,)f
FP(A)i FT(is)e(at)378 1187 y(p)s(osition)d([])i(in)f
FP(A)p FT(,)h(and)f FP(B)34 b FT(is)27 b(at)j(p)s(osition)d(\()p
FP(n)e FT(:)h FP(l)r FT(\))j(in)e FP(C)7 b FT(\()p FP(A)2389
1201 y FL(1)2429 1187 y FP(;)15 b(:)g(:)g(:)h(;)f(A)2698
1201 y FO(n)2746 1187 y FT(\))29 b(if)f(it)g(is)g(at)h(p)s(osition)e
FP(l)k FT(in)d FP(A)3756 1201 y FO(n)3803 1187 y FT(,)378
1300 y(where)j FP(C)39 b FT(is)30 b(a)j(function)d(sym)m(b)s(ol)h(or)g
(predicate.)45 b(W)-8 b(e)33 b(denote)g(the)f(sub)s(expression)d(at)j
(p)s(osition)e FP(p)378 1413 y FT(in)f FP(A)h FT(b)m(y)h
FP(A)p FN(j)802 1427 y FO(p)842 1413 y FT(.)519 1526
y(A)43 b FI(structur)-5 b(e)50 b FT(for)42 b(a)h(language)f(of)h
(\014rst-order)f(form)m(ulae)g FP(L)g FT(is)f(a)i(pair)e(\()p
FP(D)s(;)15 b(I)7 b FT(\))44 b(where)e FP(D)j FT(is)378
1638 y(some)32 b(non-empt)m(y)f(set)h(called)e(the)i
FI(domain)p FT(,)h(and)d FP(I)7 b FT(,)32 b(whic)m(h)e(is)g(called)h
(the)g FI(interpr)-5 b(etation)p FT(,)34 b(maps)378 1751
y(constan)m(ts)i(to)f(the)g(elemen)m(ts)g(in)f FP(D)s
FT(,)i FP(n)p FT(-ary)e(function)g(sym)m(b)s(ols)f(to)j
FP(n)p FT(-ary)e(functions)g(o)m(v)m(er)i FP(D)h FT(for)378
1864 y FP(n)31 b(>)g FT(0,)36 b(and)e FP(m)p FT(-ary)g(predicates)g(to)
h FP(m)p FT(-ary)f(relations)g(o)m(v)m(er)h FP(D)s FT(.)52
b(An)34 b FI(assignment)44 b FT(is)33 b(a)i(mapping)378
1977 y(from)45 b(the)g(v)-5 b(ariables)44 b(to)i(the)g(domain.)85
b(The)44 b(in)m(terpretation)h(and)g(assignmen)m(t)g(determine)f(a)378
2090 y(mapping)36 b(from)i(form)m(ulae)f(to)i(the)f(set)g(of)g(truth)f
(v)-5 b(alues)38 b FN(f)p FP(T)8 b(;)15 b(F)e FN(g)p
FT(;)43 b(the)38 b(form)m(ulae)f FN(>)h FT(and)f FN(?)g
FT(are)378 2203 y(alw)m(a)m(ys)22 b(mapp)s(ed)f(in)m(to)h
FP(T)34 b FT(and)22 b FP(F)35 b FT(resp)s(ectiv)m(ely)-8
b(.)37 b(The)22 b(truth)f(v)-5 b(alue)21 b(of)h(a)h(sen)m(tence)g(do)s
(es)e(not)i(dep)s(end)378 2316 y(on)34 b(the)g(assignmen)m(t.)51
b(Tw)m(o)35 b(form)m(ulae)e(are)i(said)e(to)h(b)s(e)g
FI(e)-5 b(quivalent)42 b FT(to)35 b(eac)m(h)g(other)g(if)e(they)h(ha)m
(v)m(e)378 2429 y(the)26 b(same)f(truth)g(v)-5 b(alue)25
b(for)g(all)f(structures)h(and)g(assignmen)m(ts.)39 b(A)25
b(form)m(ula)g(is)f(true)h(in)g(a)g(structure)378 2542
y(if)33 b(its)h(truth)g(v)-5 b(alue)34 b(is)f FP(T)47
b FT(regardless)34 b(of)h(the)f(assignmen)m(t.)53 b(W)-8
b(e)36 b(sa)m(y)f(that)g(suc)m(h)f(a)h(structure)f(is)f(a)378
2655 y FI(mo)-5 b(del)47 b FT(for)36 b(the)g(form)m(ula.)58
b(A)36 b(form)m(ula)f(is)h(said)f(to)i(b)s(e)e FI(valid)46
b FT(if)36 b(it)f(is)g(true)h(in)f(all)g(structures.)58
b(A)378 2768 y(set)28 b(of)h(form)m(ulae)e(is)g FI(satis\014able)36
b FT(in)27 b(a)h(structure)g(if)f(there)h(is)f(an)h(assignmen)m(t)g
(whic)m(h)e(allo)m(ws)i(all)f(the)378 2880 y(mem)m(b)s(ers)32
b(of)i(the)f(set)h(to)g(b)s(e)f(giv)m(en)g(the)g(truth)g(v)-5
b(alue)32 b FP(T)13 b FT(.)49 b(A)34 b(set)g(of)f(form)m(ulae)g(is)f
(satis\014able)g(if)g(it)378 2993 y(is)e(satis\014able)f(in)h(some)h
(structure)g(\(i.e.,)15 b(a)32 b(mo)s(del\).)41 b(A)31
b(Herbrand)e(mo)s(del)h(for)h(a)g(language)g FP(L)g FT(is)e(a)378
3106 y(mo)s(del)k(\()p FP(D)s(;)15 b(I)7 b FT(\))35 b(where)e
FP(D)k FT(is)c(the)h(set)h(of)f(all)f(closed)h(terms)g(in)e
FP(L)i FT(and)g FP(I)7 b FT(\()p FP(t)p FT(\))32 b(=)f
FP(t)i FT(for)h(ev)m(ery)h(closed)378 3219 y(term)30
b FP(t)p FT(.)519 3332 y(A)h(form)m(ula)f(is)f(in)h FI(ne)-5
b(gation)34 b(normal)g(form)39 b FT(\(NNF\))32 b(if)d(it)i(is)e
(constructed)i(from)f(literals)f(using)378 3445 y(the)e(connectiv)m(es)
h FN(^)p FT(,)f FN(_)f FT(and)g(the)h(quan)m(ti\014ers)f
FN(8)p FT(,)h FN(9)p FT(.)39 b(A)27 b(form)m(ula)f(is)f(in)h
FI(Skolemise)-5 b(d)31 b(form)j FT(if)26 b(it)g(do)s(es)378
3558 y(not)35 b(con)m(tain)g(the)h FN(9)e FT(quan)m(ti\014er.)53
b(A)35 b(form)m(ula)f(is)g(in)g FI(pr)-5 b(enex)38 b(form)43
b FT(if)34 b(it)g(is)g(quan)m(ti\014er-free,)i(or)f(of)378
3671 y(the)27 b(form)g FN(8)p FP(x:')g FT(or)g FN(9)p
FP(x:')h FT(where)e FP(')i FT(is)e(a)h(form)m(ula)g(in)f(prenex)g
(form.)39 b(A)27 b FI(clause)35 b FT(is)26 b(a)h(disjunction)e(of)378
3784 y(a)31 b(n)m(um)m(b)s(er)f(of)h(literals.)40 b(The)31
b(clause)f FP(A)1749 3798 y FL(1)1809 3784 y FN(_)20
b(\001)15 b(\001)g(\001)22 b(_)e FP(A)2166 3798 y FO(n)2244
3784 y FT(can)31 b(b)s(e)f(represen)m(ted)h(b)m(y)g(the)g(list)f(of)h
(literals)378 3897 y([)p FP(A)471 3911 y FL(1)511 3897
y FP(;)15 b(:)g(:)g(:)h(;)f(A)780 3911 y FO(n)828 3897
y FT(].)42 b(A)31 b(form)m(ula)f(is)g(in)g FI(clausal)k(form)39
b FT(if)30 b(it)g(is)g(a)h(conjunction)f(of)h(a)g(n)m(um)m(b)s(er)f(of)
h(clauses.)378 4010 y(There)22 b(are)i(transformations)e(of)h(form)m
(ulae)g(in)m(to)g(equiv)-5 b(alen)m(t)22 b(form)m(ulae)h(in)f(negation)
h(normal)f(form,)378 4122 y(Sk)m(olemised)29 b(form,)h(prenex)g(form,)g
(and)g(clausal)f(form)h(\(see)i(for)e(instance)g(\(Andrews)f(1981\)\).)
378 4366 y FG(1.2.2)112 b(Higher-Order)37 b(Logic)378
4538 y FT(The)48 b(fundamen)m(tal)f(di\013erence)h(b)s(et)m(w)m(een)h
(higher-order)e(logic)h(and)f(\014rst-order)h(logic)g(is)f(that)378
4650 y(higher-order)42 b(terms)h(can)g(b)s(e)g(quan)m(ti\014ed)f(o)m(v)
m(er)i(function)e(sym)m(b)s(ols)f(and)i(predicates.)79
b(In)42 b(this)378 4763 y(section)d(w)m(e)g(illustrate)e(brie\015y)g
(the)h(syn)m(tax)i(of)e(the)h(simply-t)m(yp)s(ed)e(p)s(olymorphic)e
(higher-order)378 4876 y(terms.)75 b(A)42 b(more)f(elab)s(orate)i
(treatmen)m(t,)j(whic)m(h)40 b(includes)g(the)i(seman)m(tics)g(of)g
(suc)m(h)f(terms,)k(is)378 4989 y(giv)m(en)30 b(in)f(\(Gordon)i(and)f
(Melham)g(1993\).)519 5102 y(Let)e FP(X)35 b FT(b)s(e)27
b(a)h(coun)m(table)g(set)g(of)g FI(typ)-5 b(e)31 b(variables)p
FT(,)f(and)d(\012)g(a)h(collection)f(of)h FI(typ)-5 b(e)31
b(c)-5 b(onstants)37 b FT(with)378 5215 y(\014xed)c(arities.)49
b(A)34 b(t)m(yp)s(e)f(is)g(either)g(a)h(t)m(yp)s(e)f(v)-5
b(ariable,)34 b(an)f FI(atomic)k(typ)-5 b(e)41 b FT(of)34
b(the)f(form)g FP(c)h FT(where)f FP(c)h FT(is)378 5328
y(a)g(t)m(yp)s(e)f(constan)m(t)i(with)d(zero)j(arit)m(y)-8
b(,)35 b(a)f FI(c)-5 b(omp)g(ound)38 b(typ)-5 b(e)41
b FT(of)33 b(the)h(form)f(\()p FP(\033)2955 5342 y FL(1)2995
5328 y FP(;)15 b(:)g(:)g(:)i(;)e(\033)3249 5342 y FO(n)3296
5328 y FT(\))p FI(op)40 b FT(where)33 b FI(op)378 5441
y FT(is)k(a)i(t)m(yp)s(e)f(constan)m(t)h(with)e(arit)m(y)h
FP(n)g(>)g FT(0)g(and)g FP(\033)2109 5455 y FL(1)2148
5441 y FT(,)g FP(:)15 b(:)g(:)32 b FT(,)40 b FP(\033)2465
5455 y FO(n)2550 5441 y FT(are)e(t)m(yp)s(es,)i(or)f(a)f
FI(function)i(typ)-5 b(e)46 b FT(of)378 5554 y(the)31
b(form)f FP(\033)802 5568 y FL(1)868 5554 y FN(!)25 b
FP(\033)1036 5568 y FL(2)1107 5554 y FT(where)30 b FP(\033)1422
5568 y FL(1)1492 5554 y FT(and)g FP(\033)1721 5568 y
FL(2)1791 5554 y FT(are)h(t)m(yp)s(es.)42 b(The)31 b(atomic)g(t)m(yp)s
(es)g FI(b)-5 b(o)g(ol)42 b FT(and)30 b FI(ind)40 b FT(are)32
b(in)d(\012.)378 5667 y(An)h(instance)g(of)h(the)g(t)m(yp)s(e)g
FP(\033)i FT(is)d(some)h(t)m(yp)s(e)g FP(\033)s FN(f)p
FP(\013)2115 5681 y FL(1)2181 5667 y FN(!)25 b FP(\033)2349
5681 y FL(1)2389 5667 y FP(;)15 b(:)g(:)g(:)h(;)f(\013)2648
5681 y FO(n)2721 5667 y FN(!)26 b FP(\033)2890 5681 y
FO(n)2937 5667 y FN(g)31 b FT(whic)m(h)e(represen)m(ts)i(the)p
eop
%%Page: 5 15
5 14 bop 378 5 a FF(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)2031
b FT(5)378 396 y(result)27 b(of)h(substituting,)f(in)f(parallel,)h(the)
i(t)m(yp)s(es)f FP(\033)2155 410 y FL(1)2194 396 y FT(,)g
FP(:)15 b(:)g(:)32 b FT(,)c FP(\033)2489 410 y FO(n)2564
396 y FT(for)g(t)m(yp)s(e)g(v)-5 b(ariables)27 b FP(\013)3336
410 y FL(1)3376 396 y FT(,)h FP(:)15 b(:)g(:)31 b FT(,)e
FP(\013)3677 410 y FO(n)3752 396 y FT(in)378 509 y FP(\033)s
FT(.)43 b(The)31 b(language)g(of)h(t)m(yp)s(es)f(T)m(y)q(\(\012)p
FP(;)15 b(X)7 b FT(\),)33 b(or)e(simply)e(T)m(y)q(,)i(is)g(the)g(set)h
(of)f(t)m(yp)s(es)g(constructed)g(from)378 622 y(the)g(t)m(yp)s(e)f
(constan)m(ts)i(in)d(\012)h(and)f(the)i(t)m(yp)s(e)f(v)-5
b(ariables)30 b(in)f FP(X)7 b FT(.)519 735 y(Let)31 b
FP(V)50 b FT(b)s(e)29 b(a)i(coun)m(table)f(set)h(of)f
FI(variable)j(names)39 b FT(and)29 b(\006)2511 749 y
FL(T)n(y)2632 735 y FT(a)i(collection)f(of)g FI(c)-5
b(onstant)34 b(names)378 848 y FT(eac)m(h)d(of)g(whic)m(h)e(has)h(a)h
(\014xed)f(t)m(yp)s(e)g(in)f(T)m(y)j(asso)s(ciated)f(with)e(it.)40
b(A)31 b FI(term)38 b FT(is)29 b(either)514 1036 y FN(\017)46
b FT(a)31 b FI(variable)38 b FT(of)30 b(the)h(form)f
FP(v)1545 1050 y FO(\033)1622 1036 y FT(where)g FP(v)k
FT(is)29 b(a)i(v)-5 b(ariable)29 b(name)h(and)g FP(\033)k
FT(is)29 b(a)i(t)m(yp)s(e,)514 1223 y FN(\017)46 b FT(a)22
b FI(c)-5 b(onstant)32 b FP(c)1071 1237 y FO(\033)1140
1223 y FT(where)21 b FP(c)h FT(is)f(a)h(constan)m(t)h(name)f(and)f
FP(\033)j FT(is)d(an)h(instance)f(of)h(the)g(t)m(yp)s(e)f(asso)s
(ciated)605 1336 y(with)29 b FP(c)p FT(,)514 1524 y FN(\017)46
b FT(an)30 b FI(applic)-5 b(ation)41 b FT(\()p FP(t)1267
1539 y FO(\033)1309 1520 y FD(0)1332 1539 y FK(!)p FO(\033)1480
1524 y FP(t)1513 1491 y FK(0)1513 1552 y FO(\033)1555
1533 y FD(0)1582 1524 y FT(\))1617 1538 y FO(\033)1694
1524 y FT(where)30 b FP(t)1990 1539 y FO(\033)2032 1520
y FD(0)2055 1539 y FK(!)p FO(\033)2203 1524 y FT(and)f
FP(t)2412 1491 y FK(0)2412 1552 y FO(\033)2454 1533 y
FD(0)2512 1524 y FT(are)i(terms,)f(or)514 1712 y FN(\017)46
b FT(an)30 b FI(abstr)-5 b(action)40 b FT(\()p FP(\025x)1342
1727 y FO(\033)1384 1708 y FD(0)1412 1712 y FP(:t)1470
1726 y FO(\033)1517 1712 y FT(\))1552 1727 y FO(\033)1594
1708 y FD(0)1617 1727 y FK(!)p FO(\033)1765 1712 y FT(where)30
b FP(x)2080 1727 y FO(\033)2122 1708 y FD(0)2179 1712
y FT(is)g(a)g(v)-5 b(ariable)29 b(and)h FP(t)2897 1726
y FO(\033)2974 1712 y FT(is)g(a)g(term.)378 1899 y(A)e(term)g
FP(t)722 1913 y FO(\033)769 1899 y FT(,)h(also)f(written)f
FP(t)e FT(:)h FP(\033)s FT(,)j(is)e(said)g(to)i(ha)m(v)m(e)h(the)e(t)m
(yp)s(e)g FP(\033)s FT(.)40 b(The)28 b(simply-t)m(yp)s(ed)e(p)s
(olymorphic)378 2012 y(higher-order)31 b(language)i FP(H)7
b FT(\(\006)1461 2026 y FL(T)n(y)1553 2012 y FP(;)15
b(V)20 b FT(\),)34 b(or)f(simply)d FP(H)2243 2026 y FL(T)n(y)2334
2012 y FT(,)j(is)f(the)h(set)g(of)g(terms)f(constructed)h(from)378
2125 y(the)e(constan)m(t)g(names)f(in)g(\006)1352 2139
y FL(T)n(y)1473 2125 y FT(and)g(the)g(v)-5 b(ariable)29
b(names)i(in)e FP(V)20 b FT(.)519 2238 y(An)34 b FI(expr)-5
b(ession)43 b FT(is)34 b(either)g(a)g(t)m(yp)s(e)h(or)g(a)f(term.)53
b(An)34 b(expression)g(is)f(said)h(to)h(b)s(e)f FI(p)-5
b(olymorphic)378 2351 y FT(if)42 b(it)g(con)m(tains)h(a)h(t)m(yp)s(e)f
(v)-5 b(ariable,)45 b(otherwise)d(it)h(is)f(said)g(to)h(b)s(e)f
FI(monomorphic)p FT(.)82 b(Logical)43 b(for-)378 2464
y(m)m(ulae)36 b(are)h(terms)g(of)f(t)m(yp)s(e)h FI(b)-5
b(o)g(ol)11 b FT(,)38 b(and)e(the)h(constan)m(ts)h(T)2429
2479 y FE(b)l(o)l(ol)2592 2464 y FT(and)e(F)2834 2479
y FE(b)l(o)l(ol)2998 2464 y FT(represen)m(t)h(the)g(literals)378
2577 y FN(>)j FT(and)g FN(?)g FT(resp)s(ectiv)m(ely)-8
b(.)70 b(The)40 b(negation)g(op)s(erator)h FN(:)f FT(is)f(giv)m(en)i(b)
m(y)f(the)g(constan)m(t)i FN(:)3482 2592 y FE(b)l(o)l(ol)8
b FK(!)p FE(b)l(o)l(ol)3803 2577 y FT(,)378 2690 y(and)38
b(the)g(connectiv)m(es)h(b)m(y)f(the)h(constan)m(ts)g
FN(\))p FT(,)h FN(^)e FT(and)g FN(_)p FT(,)i(eac)m(h)f(ha)m(ving)f(t)m
(yp)s(e)g FI(b)-5 b(o)g(ol)49 b FN(!)39 b FI(b)-5 b(o)g(ol)49
b FN(!)378 2802 y FI(b)-5 b(o)g(ol)11 b FT(.)52 b(The)34
b(quan)m(ti\014ers)e(are)j(giv)m(en)f(b)m(y)g(the)g(p)s(olymorphic)e
(constan)m(ts)j FN(8)e FT(and)h FN(9)g FT(whic)m(h)e(ha)m(v)m(e)k(the)
378 2915 y(t)m(yp)s(e)43 b(\()p FP(\013)j FN(!)g FI(b)-5
b(o)g(ol)11 b FT(\))46 b FN(!)g FI(b)-5 b(o)g(ol)54 b
FT(suc)m(h)42 b(that,)47 b(for)42 b(instance,)k(a)d(form)m(ula)f
FN(8)p FP(x)2986 2929 y FO(\033)3032 2915 y FP(:t)h FT(is)e(represen)m
(ted)i(b)m(y)378 3028 y(the)37 b(term)f FN(8)815 3047
y FL(\()p FO(\033)r FK(!)p FE(b)l(o)l(ol)9 b FL(\))p
FK(!)p FE(b)l(o)l(ol)1341 3028 y FT(\()p FP(\025x)1481
3042 y FO(\033)1528 3028 y FP(:t)1586 3043 y FE(b)l(o)l(ol)1713
3028 y FT(\).)60 b(The)36 b(equalit)m(y)g(predicate)h(is)e(represen)m
(ted)i(b)m(y)f(the)h(con-)378 3141 y(stan)m(t)c(=)681
3156 y FO(\013)p FK(!)p FO(\013)p FK(!)p FE(b)l(o)l(ol)1072
3141 y FT(whose)e(instan)m(tiation)g(=)1947 3156 y FE(b)l(o)l(ol)8
b FK(!)p FE(b)l(o)l(ol)g FK(!)p FE(b)l(o)l(ol)2494 3141
y FT(also)32 b(represen)m(ts)g(the)g(connectiv)m(e)h
FN(,)p FT(.)378 3254 y(The)j(c)m(hoice)h(op)s(erator)g
FP(")1262 3273 y FL(\()p FO(\013)p FK(!)p FE(b)l(o)l(ol)9
b FL(\))p FK(!)p FO(\013)1712 3254 y FT(is)36 b(included)d(in)i(the)i
(language)g FP(H)2915 3268 y FL(T)n(y)3006 3254 y FT(.)58
b(T)-8 b(erms)36 b(of)h(the)f(form)378 3367 y FP(")420
3385 y FL(\()p FO(\033)r FK(!)p FE(b)l(o)l(ol)9 b FL(\))p
FK(!)p FO(\033)859 3367 y FT(\()p FP(\025x)999 3381 y
FO(\033)1046 3367 y FP(:t)1104 3382 y FE(b)l(o)l(ol)1231
3367 y FT(\))30 b(represen)m(t)g(the)g(expression)e FP("x)2375
3381 y FO(\033)2422 3367 y FP(:t)i FT(whic)m(h)e(\(deterministically\))
f(denotes)378 3480 y(some)i FP(x)h FT(for)e(whic)m(h)g
FP(t)h FT(holds)f(if)g(suc)m(h)h(an)g FP(x)g FT(exists.)40
b(No)29 b(conditions)f(are)h(imp)s(osed)f(on)h(the)g(v)-5
b(alue)29 b(of)378 3593 y FP("x)472 3607 y FO(\033)519
3593 y FP(:t)i FT(if)e(no)h(suc)m(h)g FP(x)g FT(exists.)378
3879 y FH(1.3)135 b(Outline)46 b(of)f(the)g(Thesis)378
4082 y FT(The)30 b(rest)g(of)h(this)e(thesis)h(is)f(organised)h(as)h
(follo)m(ws:)378 4295 y FQ(Chapter)j(2)46 b FT(In)c(the)g(next)h(c)m
(hapter)g(w)m(e)h(discuss)c(the)j(problems)e(concerned)i(with)e(the)i
(mec)m(h-)605 4408 y(anisation)38 b(of)g(mathematics,)k(pa)m(ying)c
(particular)e(atten)m(tion)k(to)f(the)g(implemen)m(tation)e(of)605
4521 y(mac)m(hine-c)m(hec)m(k)-5 b(able)32 b(pro)s(ofs)d(in)g(a)i
(readable)f(format.)378 4708 y FQ(Chapter)k(3)46 b FT(One)28
b(of)h(the)f(most)h(common)g(metho)s(ds)f(of)h(dev)m(eloping)f(mac)m
(hine-c)m(hec)m(k)-5 b(able)30 b(pro)s(ofs)605 4821 y(in)m(v)m(olv)m
(es)i(the)f(in)m(teractiv)m(e)h(disco)m(v)m(ery)g(of)f(the)g(pro)s(ofs)
g(b)m(y)g(the)g(application)f(of)h(pro)s(of)g(pro)s(ce-)605
4934 y(dures)19 b(called)h FI(tactics)p FT(.)38 b(This)19
b(c)m(hapter)i(illustrates)d(t)m(w)m(o)k(case)f(studies)f(in)f(the)h
(implemen)m(tation)605 5047 y(of)35 b(tactic-based)g(pro)s(ofs)f(in)f
(the)i(HOL)f(and)f(Co)s(q)h(systems.)53 b(W)-8 b(e)36
b(argue)f(that)g(suc)m(h)f(pro)s(ofs)605 5160 y(are)i(not)g(easily)f
(read)g(and)g(that)h(other)g(st)m(yles)g(of)g(mec)m(hanising)e
(mathematics)i(should)e(b)s(e)605 5273 y(considered)29
b(if)h(the)g(readabilit)m(y)f(of)i(the)f(pro)s(ofs)g(is)f(a)i
(requiremen)m(t.)378 5460 y FQ(Chapter)j(4)46 b FT(W)-8
b(e)34 b(describ)s(e)e(the)h(declarativ)m(e)h(pro)s(of)e(language)i
(SPL,)e(and)h(the)g(implemen)m(tation)605 5573 y(of)c(a)g(pro)s(of)e(c)
m(hec)m(k)m(er)k(whic)m(h)c(deriv)m(es)g(HOL)h(theorems)h(from)f(SPL)f
(pro)s(of)h(scripts.)39 b(The)28 b(SPL)605 5686 y(language)h(is)e
(based)h(on)g(the)h(Mizar)f(language,)h(and)f(b)s(ecause)g(of)h(their)e
(declarativ)m(e)i(nature,)p eop
%%Page: 6 16
6 15 bop 378 5 a FF(CHAPTER)30 b(1.)71 b(INTR)m(ODUCTION)2031
b FT(6)605 396 y(SPL)41 b(pro)s(ofs)f(are)i(m)m(uc)m(h)g(more)f
(readable)g(than)g(tactic-based)i(pro)s(ofs.)73 b(The)41
b(SPL)g(pro)s(of)605 509 y(c)m(hec)m(k)m(er)28 b(is)c(extensible,)i(in)
e(the)h(sense)h(that)g(its)f(deductiv)m(e)g(p)s(o)m(w)m(er)g(can)h(b)s
(e)f(extended)g(during)605 622 y(the)31 b(mec)m(hanisation)f(of)g(a)h
(theory)-8 b(.)378 810 y FQ(Chapter)34 b(5)46 b FT(A)36
b(tableau)g(calculus)e(for)i(\014rst-order)f(logic)h(with)e(equalit)m
(y)i(is)f(implemen)m(ted)f(as)j(a)605 923 y(HOL)23 b(deriv)m(ed)f(rule)
f(whic)m(h)h(is)g(used)g(as)h(one)g(of)g(the)g(comp)s(onen)m(ts)g(of)g
(the)g(SPL)f(pro)s(of)h(c)m(hec)m(k)m(er.)378 1110 y
FQ(Chapter)34 b(6)46 b FT(This)29 b(c)m(hapter)i(in)m(tro)s(duces)f
(the)h(notion)g(of)g(structured)f(straigh)m(tforw)m(ard)h(justi\014ca-)
605 1223 y(tions.)50 b(Unlik)m(e)33 b(the)h(straigh)m(tforw)m(ard)g
(justi\014cations)e(of)i(Mizar)g(whic)m(h)e(consist)h(of)h(the)g(list)
605 1336 y(of)g(premises)e(required)g(to)j(justify)d(some)i(goal,)i(or)
d(conclusion,)h(structured)e(justi\014cations)605 1449
y(also)25 b(con)m(tain)h(a)g(n)m(um)m(b)s(er)e(of)h(inferences)g(whic)m
(h)f(are)h(used)g(in)f(deriving)f(the)j(conclusion)d(from)605
1562 y(the)38 b(premises)e(in)g(the)h(justi\014cation.)60
b(Structured)36 b(justi\014cations)g(are)i(de\014ned)e(in)g(suc)m(h)g
(a)605 1675 y(w)m(a)m(y)i(that)g(pro)s(ofs)e(in)m(v)m(olving)g(them)h
(are)h(not)f(o)m(v)m(er-detailed)h(and)f(therefore)h(not)f(hard)f(to)
605 1788 y(implemen)m(t.)51 b(It)34 b(is)f(argued)h(that)g(pro)s(ofs)f
(in)m(v)m(olving)g(structured)g(justi\014cations)g(are)h(easier)605
1901 y(to)d(follo)m(w)f(than)g(pro)s(ofs)g(in)m(v)m(olving)e
(unstructured)h(justi\014cations.)378 2088 y FQ(Chapter)34
b(7)46 b FT(W)-8 b(e)25 b(in)m(tro)s(duce)d(a)j(\014rst-order)e(logic)g
(whose)h(form)m(ulae)g(are)g(annotated)h(with)d(colours.)605
2201 y(These)27 b(annotations)h(are)g(used)e(to)i(restrict)f(the)h
(searc)m(h)g(space)g(during)d(\014rst-order)h(theorem)605
2314 y(pro)m(ving.)42 b(The)30 b(results)g(giv)m(en)h(in)e(this)h(c)m
(hapter)i(are)f(used)f(in)g(c)m(hapter)h(8)h(to)f(sho)m(w)g(that)h(the)
605 2427 y(searc)m(h)f(space)f(considered)f(during)f(the)i(pro)s(of)f
(c)m(hec)m(king)i(of)f(structured)f(justi\014cations)g(can)605
2540 y(b)s(e)h(restricted.)378 2728 y FQ(Chapter)k(8)46
b FT(This)27 b(c)m(hapter)j(describ)s(es)d(ho)m(w)i(only)f(a)h
(restricted)g(searc)m(h)h(space)f(needs)g(to)h(b)s(e)e(con-)605
2841 y(sidered)h(during)f(the)i(pro)s(of)g(c)m(hec)m(king)h(of)f(pro)s
(ofs)g(in)m(v)m(olving)f(the)h(structured)f(justi\014cations)605
2954 y(giv)m(en)24 b(in)f(c)m(hapter)h(6.)39 b(As)24
b(a)g(result,)g(structured)f(justi\014cations)g(can)h(b)s(e)f(pro)s(of)
g(c)m(hec)m(k)m(ed)j(more)605 3066 y(e\016cien)m(tly)k(than)g
(unstructured)f(ones.)378 3254 y FQ(Chapter)34 b(9)46
b FT(A)25 b(n)m(um)m(b)s(er)g(of)g(results)g(in)f(group)h(theory)h(are)
g(mec)m(hanised)f(in)f(SPL.)h(This)f(mec)m(hani-)605
3367 y(sation)g(follo)m(ws)f(the)i(textb)s(o)s(ok)f(b)m(y)g(Herstein)g
(\(1975\).)41 b(In)23 b(order)h(to)h(minimise)c(the)j(di\013erence)605
3480 y(b)s(et)m(w)m(een)33 b(the)f(lev)m(els)f(of)h(detail)f(of)i(the)f
(mec)m(hanised)f(pro)s(ofs)g(and)g(the)h(pro)s(ofs)f(in)g(\(Herstein)
605 3593 y(1975\),)47 b(the)41 b(deductiv)m(e)g(p)s(o)m(w)m(er)h(of)f
(the)h(SPL)e(pro)s(of)h(c)m(hec)m(k)m(er)i(is)e(extended)g(a)h(n)m(um)m
(b)s(er)e(of)605 3706 y(times)35 b(during)e(the)i(mec)m(hanisation)g
(so)g(that)h(facts)g(whose)f(pro)s(of)f(is)g(omitted)h(from)g(\(Her-)
605 3819 y(stein)27 b(1975\))i(are)e(deduced)g(automatically)g(b)m(y)f
(the)i(SPL)e(pro)s(of)g(c)m(hec)m(k)m(er)j(and)e(are)g(therefore)605
3932 y(omitted)k(from)f(the)g(mec)m(hanised)g(pro)s(ofs)f(as)i(w)m
(ell.)378 4119 y FQ(Chapter)j(10)46 b FT(W)-8 b(e)28
b(summerise)e(the)h(main)f(con)m(tributions)f(of)j(this)e(thesis)g(and)
g(p)s(oin)m(t)h(out)g(a)g(n)m(um-)605 4232 y(b)s(er)j(of)g(directions)f
(for)h(future)g(w)m(ork.)p eop
%%Page: 7 17
7 16 bop 378 1019 a FJ(Chapter)65 b(2)378 1434 y FR(On)77
b(the)h(Mec)-6 b(hanisation)77 b(of)378 1683 y(Mathematical)h(Pro)6
b(ofs)378 2128 y FT(This)36 b(c)m(hapter)j(describ)s(es)d(the)i(mec)m
(hanisms)f(used)g(in)f(the)j(implemen)m(tation)d(of)i(formal)f(mathe-)
378 2241 y(matical)25 b(theories)g(in)f(a)i(mac)m(hine)f(c)m(hec)m(k)-5
b(able)27 b(language.)39 b(The)25 b(\014rst)g(section)g(discusses)f
(the)i(lev)m(el)f(of)378 2354 y(rigour)i(found)g(in)h(the)g
(mathematical)h(literature,)f(and)g(the)h(e\013orts)g(in)e(formalising)
f(mathematics)378 2467 y(and)32 b(the)g(theoretical)h(and)f(practical)g
(problems)f(in)m(v)m(olv)m(ed)h(are)h(men)m(tioned)f(in)f(section)h
(2.2.)48 b(The)378 2580 y(implemen)m(tation)37 b(of)i(formal)e
(theories)i(with)e(the)i(help)e(of)h(computer)h(systems)f(is)g(describ)
s(ed)e(in)378 2693 y(section)27 b(2.3,)h(in)d(whic)m(h)g(b)s(oth)h
(automated)i(deduction)d(and)h(pro)s(of)f(c)m(hec)m(king)i(are)g
(illustrated.)37 b(Sec-)378 2806 y(tion)25 b(2.4)i(giv)m(es)f(a)g
(brief)f(o)m(v)m(erview)h(of)g(the)g(HOL)g(pro)s(of)f(dev)m(elopmen)m
(t)h(system)g(to)g(giv)m(e)h(an)e(example)378 2919 y(of)i(ho)m(w)g(mec)
m(hanised)f(pro)s(ofs)f(are)j(dev)m(elop)s(ed)e(and)g(also)g(b)s
(ecause)h(most)g(of)g(the)g(w)m(ork)g(describ)s(ed)d(in)378
3032 y(this)31 b(thesis)h(is)f(implemen)m(ted)g(in)g(this)g(system.)47
b(W)-8 b(e)33 b(fo)s(cus)f(on)g(the)h(problems)d(in)h(the)i(implemen-)
378 3144 y(tation)e(of)f(h)m(uman-readable)g(mac)m(hine)g(c)m(hec)m(k)
-5 b(able)32 b(mathematical)f(pro)s(ofs)e(in)h(section)g(2.5,)i(whic)m
(h)378 3257 y(also)e(surv)m(eys)g(the)h(curren)m(t)f(e\013orts)h(in)m
(v)m(olv)m(ed)f(in)f(solving)g(these)i(problems.)378
3544 y FH(2.1)135 b(The)45 b(Lev)l(el)h(of)f(Rigour)h(in)e(Mathematics)
378 3747 y FT(The)34 b(w)m(a)m(y)h(mathematics)g(is)e(practiced)i(is)e
(distinguishable)d(from)k(other)h(sciences)f(for)g(its)g(rigour)378
3860 y(and)h(precision.)53 b(Some)35 b(forms)g(of)g(delib)s(erate)f
(imprecision)f(and)h(am)m(biguit)m(y)h(are)g(ho)m(w)m(ev)m(er)i(com-)
378 3973 y(monplace)43 b(in)f(mathematical)i(texts.)81
b(Mathematical)45 b(argumen)m(ts)f(include)d(rather)j(imprecise)378
4086 y(terms)32 b(suc)m(h)g(as)g(\\)p FI(similarly)8
b FT(")35 b(and)d(\\)p FI(obviously)8 b FT(",)34 b(whic)m(h)d(usually)f
(represen)m(t)i(gaps)g(in)f(pro)s(ofs)g(and)378 4198
y(in)36 b(de\014nitions)f(whic)m(h)h(the)i(reader)f(is)g(exp)s(ected)h
(to)g(\014ll.)60 b(Inconsistencies)36 b(and)h(errors)g(are)h(also)378
4311 y(common)31 b(in)e(mathematics,)i(as)f(illustrated)e(for)j
(instance)f(b)m(y)g(Lecat)i(\(1935\).)519 4424 y(W)-8
b(e)27 b(should)d(note)j(that)f(the)h(imprecision)c(and)i
(incorrectness)h(in)e(mathematical)i(texts)h(can)g(b)s(e)378
4537 y(regarded)d(as)g(part)h(of)f(the)g(w)m(a)m(y)h(mathematical)g
(thinking)d(ev)m(olv)m(es.)40 b(Lak)-5 b(atos)25 b(\(1976\))i(and)d
(Putnam)378 4650 y(\(1979\))i(describ)s(e)d(mathematics)h(as)g
FI(quasi-empiric)-5 b(al)p FT(,)27 b(in)22 b(the)i(sense)g(that)h
(similarly)20 b(to)25 b(the)f(empir-)378 4763 y(ical)k(sciences,)h
(mathematical)g(truth)f(dep)s(ends)f(on)h(its)g(success)h(in)f
(practice,)h(and)f(that)h(it)g(ev)m(olv)m(es)378 4876
y(as)f(fallible)d(kno)m(wledge)j(is)f(replaced)h(b)m(y)f(other)i
(fallible)c(kno)m(wledge.)40 b(In)27 b FI(Pr)-5 b(o)g(ofs)32
b(and)f(R)-5 b(efutations)p FT(,)378 4989 y(Lak)g(atos)44
b(\(1976\))h(illustrates)c(ho)m(w)i(Euler's)e(theorem)i(on)g(p)s
(olyhedra)d(has)j(ev)m(olv)m(ed)g(through)f(a)378 5102
y(rep)s(etitiv)m(e)28 b(pro)s(cess)g(of)h(reform)m(ulations,)e
(\(erroneous\))i(pro)s(ofs)f(and)g(refutations.)39 b(He)29
b(uses)f(this)g(as)378 5215 y(an)33 b(analogy)g(to)g(the)g(w)m(a)m(y)h
(the)f(whole)f(of)h(mathematics)g(is)f(ev)m(olving.)h(Kleiner)e(and)h
(Mo)m(vsho)m(vitz-)378 5328 y(Hadar)k(\(1994\))i(sho)m(w)e(ho)m(w)g
(parado)m(xes,)i(whic)m(h)d(include)e(inconsistencies,)j(coun)m
(terexamples)g(to)378 5440 y(widely)f(held)h(notions,)i
(misconceptions,)g(true)e(statemen)m(ts)j(that)f(seem)f(to)h(b)s(e)e
(false,)j(and)d(false)378 5553 y(statemen)m(ts)41 b(that)g(seem)f(to)g
(b)s(e)f(true,)j(k)m(eep)f(reapp)s(earing)d(in)g(mathematics.)69
b(Suc)m(h)39 b(parado)m(xes)378 5666 y(help)27 b(in)g(a)i(b)s(etter)g
(understanding)d(of)i(the)h(basic)f(concepts)i(in)m(v)m(olv)m(ed,)e
(and)g(result)g(in)f(the)i(gradual)2080 5954 y(7)p eop
%%Page: 8 18
8 17 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)210
b FT(8)378 396 y(adv)-5 b(ancemen)m(t)31 b(of)g(mathematics.)519
509 y(Ho)m(w)m(ev)m(er,)f(as)c(argued)g(b)m(y)h(Ko)s(etsier)f
(\(1991\),)j(a)e(considerable)e(n)m(um)m(b)s(er)g(of)i(mathematical)f
(the-)378 622 y(ories)42 b(b)s(ecome)h(established)e(in)h(practice,)k
(in)41 b(the)i(sense)g(that)g(the)g(de\014nitions)d(giv)m(en)j(in)e
(suc)m(h)378 735 y(theories)d(corresp)s(ond)g(to)h(the)g(in)m(tended)e
(concepts)j(and)e(a)h(substan)m(tial)e(amoun)m(t)i(of)g(imp)s(ortan)m
(t)378 848 y(results)29 b(are)i(iden)m(ti\014ed)e(and)g(correctly)i
(pro)m(v)m(ed.)42 b(Suc)m(h)30 b(theories)g(are)h(not)f(sub)5
b(ject)31 b(to)g(m)m(uc)m(h)f(refu-)378 961 y(tation)k(and)g(their)e
(literature)i(is)e(quite)i(rigorous)f(and)g(do)s(es)h(not)g(con)m(tain)
g(errors.)51 b(As)34 b(describ)s(ed)378 1074 y(later)d(in)f(this)g(c)m
(hapter,)i(the)f(de\014nitions)e(and)i(pro)s(ofs)f(in)g(suc)m(h)h
(established)e(theories)i(can)g(b)s(e)g(for-)378 1187
y(m)m(ulated)d(at)h(a)g(high)d(lev)m(el)i(of)h(rigour)e(and)h
(precision)e(in)h(order)h(to)h(b)s(e)f(c)m(hec)m(k)m(ed)i(b)m(y)e(mac)m
(hine.)40 b(This)378 1300 y(minimises)28 b(the)k(presence)g(of)g(h)m
(uman)e(errors)i(in)e(the)i(pro)s(of)f(argumen)m(ts.)44
b(This)30 b(lev)m(el)i(of)f(rigour)g(is)378 1413 y(generally)c(needed)g
(during)e(the)j(v)m(eri\014cation)f(of)h(safet)m(y)h(critical)d
(computer)i(systems.)40 b(The)27 b(pro)s(ofs)378 1526
y(v)m(erifying)32 b(prop)s(erties)f(of)i(suc)m(h)g(systems)f(are)i
(often)f(quite)f(tedious)h(and)f(length)m(y)-8 b(,)34
b(and)e(therefore)378 1638 y(m)m(uc)m(h)j(prone)f(to)h(h)m(uman)f
(error,)i(although)e(they)h(are)g(often)g(describ)s(ed)e(as)i(shallo)m
(w)m(er)f(in)f(nature)378 1751 y(than)f(those)g(found)f(in)f
(mathematical)j(texts.)46 b(The)31 b(implemen)m(tation)g(of)h(suc)m(h)g
(pro)s(ofs,)f(ho)m(w)m(ev)m(er,)378 1864 y(ma)m(y)39
b(dep)s(end)d(on)i(basic)g(results)f(in)g(standard)g(mathematical)i
(theories)f(suc)m(h)g(as)g(n)m(um)m(b)s(er)f(the-)378
1977 y(ory)f(and)f(real)g(analysis.)56 b(Therefore)36
b(one)g(ma)m(y)g(need)g(to)g(dev)m(elop)g(a)g(n)m(um)m(b)s(er)f(of)h
(mathematical)378 2090 y(theories)30 b(during)e(the)j(v)m
(eri\014cation)f(of)g(computer)h(systems.)519 2203 y(The)42
b(implemen)m(tation)f(of)i(mathematics)g(in)e(a)i(mac)m(hine)f
(readable)g(format)h(has)f(b)s(een)g(ad-)378 2316 y(v)m(o)s(cated)j
(for)f(a)g(n)m(um)m(b)s(er)f(of)h(di\013eren)m(t)g(reasons)g
(\(including)c(educational)k(and)f(cultural)f(ones\))378
2429 y(in)37 b(the)h(QED)f(manifesto)h(\(Anon)m(ymous)g(1994\).)65
b(Although)37 b(one)i(ma)m(y)f(ob)5 b(ject)39 b(to)f(the)h(particu-)378
2542 y(lar)e(motiv)-5 b(ations)38 b(discussed)e(in)g(this)h(manifesto,)
j(the)e(implemen)m(tation)f(of)h(a)g(large)g(n)m(um)m(b)s(er)f(of)378
2655 y(mathematical)42 b(theories)g(in)f(a)h(mac)m(hine)g(c)m(hec)m(k)
-5 b(able)43 b(format)f(is)f(b)s(eliev)m(ed)g(to)i(b)s(e)e(p)s(ossible)
e(and)378 2768 y(desirable)d(\(see)i(\(Harrison)e(1996a\)\).)65
b(There)37 b(are)h(curren)m(tly)e(a)i(n)m(um)m(b)s(er)e(of)i(computer)f
(systems)378 2880 y(whic)m(h)27 b(supp)s(ort)g(a)i(formal)f(pro)s(of)f
(language)i(in)f(whic)m(h)f(a)i(considerable)e(amoun)m(t)i(of)f
(mathematics)378 2993 y(is)h(implemen)m(ted.)378 3280
y FH(2.2)135 b(The)45 b(F)-11 b(ormalisation)46 b(of)g(Mathematics)378
3483 y FT(By)30 b(the)g(formalisation)d(of)j(mathematics)g(w)m(e)g
(mean)g(the)f(implemen)m(tation)g(of)g(mathematics)h(in)e(a)378
3596 y FI(formal)44 b FT(language.)49 b(A)33 b(language)g(is)f
FI(formal)44 b FT(if)32 b(its)g(syn)m(tax)i(and)e(seman)m(tics)h(are)g
(unam)m(biguously)378 3709 y(de\014ned.)48 b(Similarly)29
b(w)m(e)34 b(refer)f(to)g(the)h(dev)m(elopmen)m(t)f(of)h(mathematics)f
(in)f(an)h(informal,)f(though)378 3821 y(rigorous,)c(language)g(as)g
FI(informal)39 b FT(mathematics.)i(A)28 b(language)g(for)g(the)g
(formalisation)e(of)j(math-)378 3934 y(ematics)k(m)m(ust)f(b)s(e)g(ric)
m(h)g(enough)g(to)h(express)g(mathematical)f(ob)5 b(jects,)34
b(statemen)m(ts)h(ab)s(out)d(them)378 4047 y(and)f(v)-5
b(alid)31 b(reasoning)g(in)m(v)m(olving)g(these)h(statemen)m(ts.)48
b(Suc)m(h)31 b(v)-5 b(alid)30 b(reasoning)i(can)g(b)s(e)f(expressed)378
4160 y(as)g(a)g(n)m(um)m(b)s(er)f(of)h(logical)f(rules)g(manipulating)e
(the)j(statemen)m(ts)h(concerning)f(the)g(mathematical)378
4273 y(ob)5 b(jects.)519 4386 y(The)28 b(motiv)-5 b(ations)29
b(for)f(formalising)f(mathematics)i(include)e(the)i(abilit)m(y)e(to)j
(ac)m(hiev)m(e)g(a)f(higher)378 4499 y(degree)45 b(of)g(correctness)g
(and)e(precision)g(than)h(that)h(found)e(in)g(informal)f(mathematics.)
83 b(The)378 4612 y(abilit)m(y)26 b(to)j(express)e(v)-5
b(alid)26 b(mathematical)i(reasoning)g(b)m(y)f(sym)m(b)s(olic)f
(manipulations)f(implies)g(that)378 4725 y(the)k(v)-5
b(alidit)m(y)28 b(of)h(an)g(argumen)m(t)g(can)h(b)s(e)e(c)m(hec)m(k)m
(ed)j(in)d(a)h(mec)m(hanical)g(fashion.)39 b(This)27
b(is)h(b)s(eliev)m(ed)g(to)378 4838 y(b)s(e)i(more)g(reliable)f(than)h
(accepting)h(an)f(informal,)f(but)g(con)m(vincing,)h(argumen)m(t.)519
4951 y(A)i(substan)m(tial)f(amoun)m(t)i(of)f(e\013ort)h(w)m(as)g(put)e
(in)g(using)g(sym)m(b)s(olic)f(manipulations)f(to)k(express)378
5064 y(mathematical)38 b(reasoning)f(during)f(the)i(nineteen)m(th)f
(and)g(t)m(w)m(en)m(tieth)i(cen)m(turies.)62 b(Bo)s(ole)38
b(\(1848\))378 5176 y(dev)m(elop)s(ed)30 b(a)g(formal)g(system)g(for)g
(prop)s(ositional)e(logic)i(in)f(whic)m(h)g(reasoning)h(can)h(b)s(e)e
(p)s(erformed)378 5289 y(through)e(mec)m(hanical)g(calculations)g
(rather)h(than)f(through)g(the)h(in)m(terpretation)f(of)h(the)f(sym)m
(b)s(olic)378 5402 y(statemen)m(ts.)42 b(F)-8 b(rege)29
b(\(1879\))i(included)25 b(quan)m(ti\014ers)i(in)g(the)h(formal)f
(logical)h(system)g(he)g(dev)m(elop)s(ed)378 5515 y(whic)m(h)41
b(w)m(as)h(aimed)g(at)h(expressing)d(the)j(whole)e(of)h(mathematics,)k
(and)c(P)m(eano)h(\()f(97\))i(fo)s(cused)378 5628 y(on)34
b(the)h(implemen)m(tation)e(of)i(mathematics)g(of)f(his)f(p)s(erio)s(d)
f(in)i(a)h(formal)e(sym)m(b)s(olic)g(form)h(whose)p eop
%%Page: 9 19
9 18 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)210
b FT(9)378 396 y(notation)27 b(is)e(closer)h(to)i(informal)c
(mathematics)j(than)f(that)h(of)g(F)-8 b(rege.)41 b(Russell)24
b(included)g(t)m(yp)s(es)i(in)378 509 y(his)31 b(logic)g(to)i(a)m(v)m
(oid)g(inconsistencies)d(in)h(F)-8 b(rege's)33 b(deductiv)m(e)f
(system.)2820 476 y FL(1)2906 509 y FT(Whitehead)f(and)h(Russell)378
622 y(\(1910\))40 b(used)d(this)f(t)m(yp)s(ed)i(logic)f(in)f(their)h
FI(Principia)j(Mathematic)-5 b(a)p FT(.)64 b(Although)37
b(the)g(degree)i(of)378 735 y(rigour)34 b(and)h(precision)e(in)h(the)i
(foundational)e(w)m(ork)h(of)h FI(Principia)h(Mathematic)-5
b(a)45 b FT(is)34 b(considered)378 848 y(to)c(b)s(e)e(m)m(uc)m(h)h(w)m
(eak)m(er)h(than)f(that)g(of)g(F)-8 b(rege,)31 b(the)e(w)m(ork)g(of)g
(Whitehead)g(and)f(Russell)f(sho)m(w)m(ed)i(that)378
961 y(a)i(substan)m(tial)e(amoun)m(t)i(of)f(mathematics)h(can)g(indeed)
e(b)s(e)g(written)h(formally)-8 b(.)519 1074 y(A)m(t)37
b(the)f(turn)f(of)i(the)f(cen)m(tury)-8 b(,)38 b(Hilb)s(ert)c(\(see)j
(\(Kreisel)e(1958\)\))k(prop)s(osed)c(a)h(programme)g(in)378
1187 y(whic)m(h)24 b(mathematical)h(theories)g(are)h(formalised)e(in)f
(\014nitary)h(logical)h(systems)g(that)h(are)g(sho)m(wn)e(to)378
1300 y(b)s(e)k(consisten)m(t.)41 b(Statemen)m(ts)30 b(are)f(v)-5
b(alid)28 b(if)g(they)h(ha)m(v)m(e)h(\(\014nite\))e(pro)s(ofs)g(in)g
(suc)m(h)h(systems.)40 b(Hilb)s(ert)378 1413 y(also)35
b(ask)m(ed)h(whether)f(formal)f(statemen)m(ts)j(can)e(b)s(e)g(sho)m(wn)
g(to)g(b)s(e)g(v)-5 b(alid)34 b(b)m(y)h(purely)e(mec)m(hanical)378
1526 y(means,)42 b(that)f(is,)g(whether)e(there)h(is)f(an)h(algorithm)f
(b)m(y)h(whic)m(h)e(one)i(can)h(decide)e(the)h(truth)f(or)378
1638 y(falsit)m(y)26 b(of)h(a)h(statemen)m(t.)41 b(This)25
b(programme,)j(and)f(the)g(e\013orts)h(of)f(other)g(mathematicians)g
(to)g(\014nd)378 1751 y(a)35 b(deductiv)m(e)f(system)g(in)f(whic)m(h)h
(all)f(v)-5 b(alid)33 b(mathematical)h(statemen)m(ts)i(can)f(b)s(e)f
(formalised)e(and)378 1864 y(justi\014ed)38 b(mec)m(hanically)-8
b(,)41 b(w)m(ere)g(ho)m(w)m(ev)m(er)g(sho)m(wn)e(to)h(b)s(e)f(imp)s
(ossible)e(during)g(the)j(1930's.)70 b(The)378 1977 y(basic)30
b(results)f(disco)m(v)m(ered)i(in)e(this)g(p)s(erio)s(d)f(include:)514
2165 y FN(\017)46 b FT(G\177)-45 b(odel's)26 b(Incompleteness)e
(Theorem)i(\(G\177)-45 b(odel)25 b(1931\))j(whic)m(h)c(states)j(the)e
(non-existence)h(of)g(a)605 2278 y(coun)m(table)j(axiomatisation)g(of)f
(all)g(arithmetic)g(whic)m(h)f(is)h(b)s(oth)g(consisten)m(t)h(and)f
(complete.)514 2465 y FN(\017)46 b FT(The)31 b(undecidabilit)m(y)c(of)k
(pure)f(\014rst-order)g(logic,)h(pro)m(v)m(ed)g(b)m(y)g(T)-8
b(uring)29 b(\(1936\))34 b(and)c(Ch)m(urc)m(h)605 2578
y(\(1936\).)514 2766 y FN(\017)46 b FT(The)28 b(unde\014nabilit)m(y)d
(of)j(truth,)g(pro)m(v)m(ed)h(b)m(y)f(T)-8 b(arski)27
b(\(1936\),)32 b(whic)m(h)27 b(also)h(implies)d(that)k(true)605
2879 y(statemen)m(ts)j(are)f(not)g(recursiv)m(ely)e(de\014nable.)519
3066 y(The)d(ma)5 b(jor)26 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 b(mathematicians)e
(\(using)378 3857 y(the)32 b(p)s(en)f(name)i(Bourbaki\))e(formalised)g
(an)h(impressiv)m(e)e(amoun)m(t)j(of)f(mathematics.)47
b(They)32 b(used)378 3970 y(\014rst-order)h(logic)h(as)g(their)f
(deductiv)m(e)h(system)g(together)i(with)c(an)i(axiomatic)h(set)f
(theory)h(simi-)378 4083 y(lar)f(to)i(Zermelo's.)54 b(Ho)m(w)m(ev)m
(er,)39 b(this)34 b(formalisation)f(w)m(as)j(abandoned)e(b)s(ecause)h
(it)g(w)m(as)g(found)f(to)378 4196 y(b)s(e)g(impracticable)f(and)h(b)s
(ecause)g(of)h(the)f FI(c)-5 b(omplexity)44 b FT(and)34
b FI(unr)-5 b(e)g(adability)45 b FT(of)34 b(the)h(formal)f(texts.)378
4308 y(The)e(earlier)f(e\013orts)i(of)g(Whitehead)f(and)g(Russell)e(w)m
(ere)j(faced)g(with)e(the)h(same)h(problems:)43 b(that)378
4421 y(although)e(the)h(reduction)e(of)i(reasoning)f(in)m(to)g(formal)g
(sym)m(b)s(olic)f(manipulations)f(results)h(in)g(a)378
4534 y(more)34 b(rigorous)f(and)g(precise)h(approac)m(h)g(to)g
(mathematics,)i(formalised)c(de\014nitions)f(and)j(pro)s(ofs)378
4647 y(are)k(long)f(and)g(tedious,)i(and)e(that)h(the)g(resulting)e
(texts)i(are)g(unreadable)f(and)g(barely)f(used)h(in)378
4760 y(practice.)j(F)-8 b(urthermore,)27 b(it)f(is)f(lik)m(ely)g(that)i
(one)g(loses)f(the)g(in)m(tuition)e(b)s(ehind)g(an)i(argumen)m(t)h
(when)378 4873 y(it)33 b(is)g(formalised,)g(whic)m(h)f(as)i(Naur)f
(\(1994\))j(has)e(p)s(oin)m(ted)e(out,)j(ma)m(y)f(result)f(in)f(making)
h(the)h(text)378 4986 y(more)d(prone)g(to)h(errors.)43
b(The)31 b(practical)g(di\016cult)m(y)e(of)j(formalised)d(mathematics)j
(can,)g(ho)m(w)m(ev)m(er,)378 5099 y(b)s(e)e(reliev)m(ed)f(b)m(y)i
(using)d(a)j(computer)f(system)h(to)g(c)m(hec)m(k)h(and)e(ev)m(en)h
(\014nd)e(formal)g(pro)s(ofs.)p 378 5261 1380 4 v 482
5315 a FC(1)516 5346 y FB(An)e(inconsistency)h(in)g(F)-6
b(rege's)29 b(system)e(is)h(the)g(w)n(ell)g(kno)n(wn)g(Russell's)h
(parado)n(x)e(whic)n(h)h(is)h(due)e(is)h(the)g(abilit)n(y)378
5438 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
(\(sym)m(b)s(olic)e(mathe-)378 938 y(matics,)g(or)g(computer)g
(algebra\))h(to)g(mimic,)d(for)i(example,)g(the)h(w)m(a)m(y)g(h)m
(umans)e(di\013eren)m(tiate)g(and)378 1051 y(in)m(tegrate)40
b(functions.)63 b(This)37 b(particular)g(use)h(of)g(computers)h(in)e
(mec)m(hanising)g(mathematics)i(is)378 1164 y(usually)33
b(referred)h(to)i(as)g(the)f(sym)m(b)s(olic)f(mec)m(hanisation)g(of)i
(mathematics.)55 b(The)35 b(sym)m(b)s(olic)e(ma-)378
1277 y(nipulations)i(represen)m(ting)i(formalised)g(reasoning)h(can)g
(also)g(b)s(e)g(mec)m(hanised)g(in)e(order)i(to)h(use)378
1390 y(computer)27 b(systems)f(in)g(the)h(formalisation)e(of)i
(mathematics.)40 b(This)24 b(is)i(referred)g(to)i(as)f(the)f(logical)
378 1503 y(mec)m(hanisation)k(of)h(mathematics,)g(and)f(the)h(sev)m
(eral)f(adv)-5 b(an)m(tages)33 b(of)d(using)f(a)i(computer)g(system)378
1616 y(in)e(formalising)f(mathematics)j(include)d(the)j(follo)m(wing:)
514 1800 y FN(\017)46 b FT(the)38 b(syn)m(tactic)g(correctness)g(of)g
(formal)e(statemen)m(ts)k(and)c(the)i(v)-5 b(alidit)m(y)36
b(of)h(formal)g(pro)s(ofs)605 1913 y(can)31 b(b)s(e)f(c)m(hec)m(k)m(ed)
i(b)m(y)e(simple)f(algorithms,)514 2099 y FN(\017)46
b FT(one)31 b(can)g(use)f(algorithms)f(to)i(searc)m(h)g(for)f(pro)s
(ofs)g(of)g(formal)g(statemen)m(ts,)514 2286 y FN(\017)46
b FT(algorithms)40 b(whic)m(h)h(p)s(erform)f(a)h(sp)s(eci\014c)g
(sequence)g(of)h(v)-5 b(alid)40 b(inferences)g(can)i(b)s(e)e(imple-)605
2399 y(men)m(ted)31 b(to)g(a)m(v)m(oid)g(tedious)f(rep)s(etitions.)378
2583 y(The)39 b(history)g(of)h(the)g(mec)m(hanisation)g(of)g(reasoning)
f(is)g(surv)m(ey)m(ed)h(b)m(y)g(MacKenzie)h(\(1995\).)72
b(In)378 2696 y(this)25 b(thesis)g(w)m(e)i(use)e(the)i(term)f(\\mec)m
(hanisation)g(of)g(mathematics")h(to)g(refer)e(to)i(the)f(dev)m
(elopmen)m(t)378 2809 y(of)34 b(mathematical)f(texts)i(whic)m(h)d(can)h
(b)s(e)g(c)m(hec)m(k)m(ed)j(b)m(y)d(mac)m(hine.)49 b(Similarly)-8
b(,)32 b(w)m(e)h(refer)h(to)g(pro)s(ofs)378 2922 y(whic)m(h)40
b(can)i(b)s(e)f(c)m(hec)m(k)m(ed)j(b)m(y)d(mac)m(hine)g(as)h(mec)m
(hanised)f(pro)s(ofs.)73 b(Mec)m(hanised)42 b(pro)s(ofs)f(can)h(b)s(e)
378 3035 y(found)30 b(b)m(y)i(an)g(algorithm,)f(or)h(implemen)m(ted)f
(b)m(y)g(a)i(h)m(uman)e(b)s(eing)f(with)h(or)g(without)g(the)h(help)f
(of)378 3148 y(computerised)h(pro)s(of)h(to)s(ols.)50
b(In)33 b(this)g(section)g(w)m(e)h(\014rst)f(ha)m(v)m(e)i(a)f(lo)s(ok)f
(at)h(automated)h(deduction)378 3261 y(whic)m(h)29 b(in)m(v)m(olv)m(es)
h(the)h(use)f(of)h(algorithms)e(to)i(\014nd)e(pro)s(ofs,)h(and)f(then)h
(at)i(pro)s(of)d(c)m(hec)m(king.)378 3503 y FG(2.3.1)112
b(Automated)37 b(Deduction)378 3675 y FT(Automated)27
b(deduction)e(is)h(the)g(branc)m(h)g(of)h(computer)f(science)g(and)g
(arti\014cial)f(in)m(telligence)g(whic)m(h)378 3788 y(deals)36
b(with)g(the)h(use)f(of)h(computers)g(to)g(decide)f(the)h(v)-5
b(alidit)m(y)35 b(of)i(logical)g(sen)m(tences.)61 b(Although)378
3901 y(this)38 b(decision)g(problem)g(is)g(undecidable)f(in)h(general,)
j(there)f(are)f(sev)m(eral)h(non-trivial)d(theories)378
4014 y(in)g(whic)m(h)h(the)g(v)-5 b(alidit)m(y)38 b(of)g(sen)m(tences)i
(is)e(decidable.)64 b(F)-8 b(or)39 b(instance,)i(prop)s(ositional)36
b(logic,)41 b(the)378 4127 y(theory)f(of)g(linear)f(arithmetic)g(and)g
(the)h FN(8)1901 4094 y FK(\003)1940 4127 y FN(9)1991
4094 y FK(\003)2070 4127 y FT(fragmen)m(t)g(of)h(\014rst-order)e(logic)
3194 4094 y FL(2)3273 4127 y FT(are)h(decidable.)378
4240 y(Also,)46 b(\014rst-order)41 b(logic)i(is)e(semi-decidable)g(and)
h(therefore)h(one)g(can)g(implemen)m(t)e(algorithms)378
4352 y(whic)m(h)30 b(terminate)i(on)g(v)-5 b(alid)30
b(sen)m(tences,)k(though)d(they)h(ma)m(y)g(not)g(halt)g(on)f(in)m(v)-5
b(alid)30 b(ones.)45 b(This)30 b(is)378 4465 y(usually)23
b(done)i(b)m(y)g(searc)m(hing)g(for)f(a)i(pro)s(of)e(since)h(c)m(hec)m
(king)g(whether)g(a)g(pro)s(of)g(deriv)m(es)f(a)i(particular)378
4578 y(theorem)31 b(is)e(decidable.)519 4691 y(The)d(complexit)m(y)g
(of)h(the)g(decidable)e(decision)g(problems)g(men)m(tioned)h(ab)s(o)m
(v)m(e)i(is)e(ho)m(w)m(ev)m(er)i(v)m(ery)378 4804 y(high.)57
b(The)36 b(problem)e(T)-8 b(A)m(UT)37 b(of)f(deciding)f(the)h(v)-5
b(alidit)m(y)35 b(of)h(prop)s(ositional)e(sen)m(tences)j(\(in)e(con-)
378 4917 y(junctiv)m(e)f(normal)f(form\))h(is)f(in)f(co)p
FN(\000)q(N)13 b(P)8 b FT(,)35 b(and)e(therefore)i(considered)e(to)h(b)
s(e)g(un)m(tractable.)52 b(F)-8 b(ur-)378 5030 y(thermore,)32
b(searc)m(hing)g(for)f(evidence)h(of)g(the)g(v)-5 b(alidit)m(y)30
b(of)i(a)g(sen)m(tence)h(in)e(an)g(undecidable)f(theory)378
5143 y(in)m(v)m(olv)m(es)d(searc)m(hing)f(for)h(a)g(pro)s(of)e(in)h(an)
g(in\014nite)f(searc)m(h)i(space.)40 b(This)25 b(normally)g(in)m(v)m
(olv)m(es)h(the)h(use)378 5256 y(of)g(fair)g(strategies,)i(where)d(one)
i(considers)e(a)h(sequence)h(of)f(\014nite)f(searc)m(h)i(spaces,)h(one)
e(larger)g(than)378 5369 y(the)k(other,)f(in)f(order)h(to)h(ensure)f
(that)h(the)g(v)-5 b(alidit)m(y)28 b(of)j(a)g(sen)m(tence)g(is)f(ev)m
(en)m(tually)g(established.)p 378 5528 1380 4 v 482 5582
a FC(2)516 5614 y FB(The)c Fz(8)717 5582 y FD(\003)753
5614 y Fz(9)796 5582 y FD(\003)858 5614 y FB(fragmen)n(t)f(of)i
(\014rst-order)f(logic)h(is)g(the)e(set)h(of)h(all)g(\014rst-order)f
(sen)n(tences)g(whose)h(prenex)e(form)h(is)g(of)378 5705
y(the)f(form)h Fz(8)p FA(x)779 5713 y FC(1)813 5705 y
FA(;)13 b(:)g(:)g(:)26 b(;)13 b(x)1040 5713 y Fy(n)1082
5705 y FA(:)p Fz(9)p FA(y)1184 5713 y FC(1)1218 5705
y FA(;)h(:)f(:)g(:)26 b(;)13 b(y)1440 5713 y Fy(m)1498
5705 y FA(:P)37 b FB(where)26 b FA(n;)13 b(m)21 b Fz(\025)h
FB(0)k(and)f FA(P)36 b FB(is)26 b(a)g(quan)n(ti\014er)f(free)i(form)n
(ula.)p eop
%%Page: 11 21
11 20 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(11)519 396 y(In)22 b(order)f(to)i(b)s(e)f(e\016cien)m(t,)i
(automated)f(deduction)e(systems)i(are)f(based)g(on)g(deductiv)m(e)g
(systems)378 509 y(whose)j(pro)s(ofs)f(can)h(b)s(e)f(`easily')g(found)g
(b)m(y)h(mec)m(hanical)f(means.)39 b(W)-8 b(e)26 b(can)f(refer)g(to)h
(suc)m(h)e(deductiv)m(e)378 622 y(systems)33 b(as)h FI(se)-5
b(ar)g(ch-oriente)g(d)p FT(,)37 b(and)c(usually)f(require)g(the)i
(follo)m(wing)d(t)m(w)m(o)k(prop)s(erties)d(whic)m(h)g(are)378
735 y(illustrated)c(b)m(y)i(some)h(examples)f(later)g(in)f(this)h
(section.)514 923 y FN(\017)46 b FT(The)30 b(lengths)g(of)g(pro)s(ofs)g
(in)f(these)i(systems)f(are)h(short.)514 1110 y FN(\017)46
b FT(Complete)30 b(pro)s(of)g(searc)m(h)h(strategies)g(are)g(not)g
(faced)f(with)g(to)s(o)h(m)m(uc)m(h)f(non-determinism.)378
1298 y(An)39 b(ideal)f(deductiv)m(e)h(system)h(whic)m(h)e(satis\014es)h
(the)g(ab)s(o)m(v)m(e)i(prop)s(erties)d(do)s(es)h(not)g(seem)h(to)g
(ex-)378 1411 y(ist,)f(ho)m(w)m(ev)m(er)g(a)f(n)m(um)m(b)s(er)e(of)i
(systems)f(ha)m(v)m(e)i(b)s(een)e(dev)m(elop)s(ed)g(in)f(whic)m(h)h
(pro)s(ofs)f(of)i(non-trivial)378 1524 y(theorems)k(can)g(b)s(e)f
(found)f(in)g(a)i(relativ)m(ely)f(short)h(time.)74 b(Despite)41
b(the)h(inheren)m(t)f(di\016cult)m(y)f(of)378 1637 y(automated)32
b(deduction,)f(a)g(n)m(um)m(b)s(er)f(of)h(di\016cult)e(problems)h(in)g
(mathematics)h(ha)m(v)m(e)i(b)s(een)d(solv)m(ed)378 1750
y(b)m(y)35 b(suc)m(h)h(pro)s(of)e(searc)m(h)j(systems.)56
b(A)36 b(recen)m(t)g(example)f(is)g(the)h(pro)s(of)f(of)g(the)h
(Robbins)d(problem)378 1863 y(whic)m(h)h(w)m(as)i(op)s(en)f(for)g(more)
h(than)f(\014ft)m(y)g(y)m(ears)h(and)f(a)h(successful)e(pro)s(of)h(for)
g(this)g(problem)f(w)m(as)378 1976 y(found)c(b)m(y)g(the)h(EQP)g
(theorem)g(pro)m(v)m(er)g(in)f(almost)h(8)g(da)m(ys)g(using)e(30)j
(Megab)m(ytes)h(of)e(memory)g(on)378 2088 y(a)g(UNIX)f(w)m(orkstation)h
(with)e(an)h(RS/6000)i(pro)s(cessor)e(\(McCune)h(1997\).)519
2201 y(Examples)37 b(of)g(searc)m(h-orien)m(ted)i(deductiv)m(e)e
(systems)h(for)f(\014rst-order)g(logic)g(include)f(resolu-)378
2314 y(tion)24 b(\(Robinson)f(1965\),)28 b(the)c(connection)h(\(Bib)s
(el)e(1981\))k(\(or)d(matings)g(\(Andrews)g(1981\)\))i(metho)s(d)378
2427 y(and)f(tableaux-based)h(metho)s(ds)1507 2394 y
FL(3)1546 2427 y FT(.)39 b(W)-8 b(e)27 b(discuss)d(resolution)g(and)h
(the)h(connection)g(metho)s(d)g(brie\015y)378 2540 y(in)g(this)g
(section,)i(and)f(App)s(endix)e(B)i(illustrates)f(tableaux-based)h
(metho)s(ds)f(for)h(\014rst-order)g(logic.)519 2653 y(These)44
b(systems)g(are)g(usually)d(refutational;)50 b(that)45
b(is,)h(a)f(sen)m(tence)g(is)e(sho)m(wn)g(to)h(b)s(e)g(v)-5
b(alid)378 2766 y(b)m(y)36 b(sho)m(wing)f(that)i(its)f(negation)g(is)f
(refutable.)58 b(In)35 b(resolution,)i(a)f(sen)m(tence)i(is)d(refuted)h
(b)m(y)g(\014rst)378 2879 y(transforming)d(it)h(in)m(to)g(clausal)g
(form)f(and)h(then)g(applying)f(the)h(resolution)f(rule)g(rep)s(etitiv)
m(ely)g(to)378 2992 y(create)i(new)f(clauses)f(un)m(til)f(the)i(empt)m
(y)g(clause)g(is)e(deriv)m(ed.)50 b(The)34 b(resolution)e(rule)g(is)h
(de\014ned)g(as)378 3105 y(follo)m(ws:)1238 3181 y([)p
FP(A)1331 3195 y FL(1)1371 3181 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(A)1656 3195 y FO(i)1685 3181 y FP(;)g(:)g(:)g(:)32
b(;)15 b(A)1970 3195 y FO(n)2017 3181 y FT(])91 b([)p
FP(B)2227 3195 y FL(1)2267 3181 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(B)2553 3195 y FO(j)2590 3181 y FP(;)g(:)g(:)g(:)31
b(;)15 b(B)2875 3195 y FO(m)2942 3181 y FT(])p 940 3228
2327 4 v 940 3312 a([)p FP(A)1033 3326 y FL(1)1073 3312
y FP(;)g(:)g(:)g(:)31 b(;)15 b(A)1357 3326 y FO(i)p FK(\000)p
FL(1)1476 3312 y FP(;)g(A)1584 3326 y FO(i)p FL(+1)1703
3312 y FP(;)g(:)g(:)g(:)32 b(;)15 b(A)1988 3326 y FO(n)2035
3312 y FP(;)g(B)2144 3326 y FL(1)2184 3312 y FP(;)g(:)g(:)g(:)32
b(;)15 b(B)2470 3326 y FO(j)t FK(\000)p FL(1)2597 3312
y FP(;)g(B)2706 3326 y FO(j)t FL(+1)2833 3312 y FP(;)g(:)g(:)g(:)32
b(;)15 b(B)3119 3326 y FO(m)3186 3312 y FT(])p FP(\033)378
3479 y FT(where)30 b(the)g(literals)f FP(A)1164 3493
y FO(i)1192 3479 y FP(\033)34 b FT(and)c FP(B)1524 3493
y FO(j)1560 3479 y FP(\033)k FT(are)d(complemen)m(tary)-8
b(.)41 b(F)-8 b(or)31 b(example,)f(giv)m(en)h(the)f(sen)m(tence)1401
3683 y(\(\()p FN(8)p FP(x:P)13 b FT(\()p FP(x)p FT(\))26
b FN(\))f FP(Q)p FT(\()p FP(x)p FT(\)\))c FN(^)f FP(P)13
b FT(\()p FP(c)p FT(\)\))27 b FN(\))e FP(Q)p FT(\()p
FP(c)p FT(\))378 3888 y(its)30 b(negation)g(is)g(transformed)f(in)m(to)
i(the)f(clauses)1389 4092 y([)p FN(:)p FP(P)13 b FT(\()p
FP(x)p FT(\))p FP(;)i(Q)p FT(\()p FP(x)p FT(\)])184 b([)p
FP(P)13 b FT(\()p FP(c)p FT(\)])183 b([)p FN(:)p FP(Q)p
FT(\()p FP(c)p FT(\)])378 4296 y(and)30 b(the)g(follo)m(wing)f
(resolution)g(pro)s(of)h(is)f(found.)1480 4482 y([)p
FN(:)p FP(P)13 b FT(\()p FP(x)p FT(\))p FP(;)i(Q)p FT(\()p
FP(x)p FT(\)])93 b([)p FP(P)13 b FT(\()p FP(c)p FT(\)])p
1480 4524 863 4 v 1627 4609 a([)p FP(Q)p FT(\()p FP(x)p
FT(\)])p FN(f)p FP(x)26 b FN(!)g FP(c)p FN(g)238 b FT([)p
FN(:)p FP(Q)p FT(\()p FP(c)p FT(\)])p 1627 4652 1100
4 v 2141 4732 a FN(?)378 4936 y FT(In)43 b(the)g(connection)h(metho)s
(d,)i(the)d(clauses)g(to)h(b)s(e)f(refuted)g(are)h(represen)m(ted)f(b)m
(y)g(columns)f(in)378 5049 y(a)c(t)m(w)m(o)i(dimensional)35
b(matrix.)63 b(Additional)35 b(clauses)j(\(and)g(hence)g(columns\))f
(can)h(b)s(e)g(added)f(b)m(y)378 5162 y(renaming)29 b(the)i(v)-5
b(ariables)30 b(in)f(an)h(existing)g(clause.)41 b(The)31
b(matrix)f(is)f(refuted)h(if)g(all)f(its)i FI(p)-5 b(aths)40
b FT(ha)m(v)m(e)378 5274 y(a)30 b FI(c)-5 b(onne)g(ction)38
b FT(after)31 b(some)f(substitution)d(is)i(applied)f(to)i(all)f(the)h
(literals)e(in)g(the)i(matrix.)40 b(A)30 b(path)378 5387
y(is)d(a)i(list)e(of)h(literals)f([)p FP(L)1176 5401
y FL(1)1216 5387 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(L)1494
5401 y FO(n)1542 5387 y FT(])28 b(where)g FP(L)1918 5401
y FO(i)1974 5387 y FT(is)g(in)f(the)h FP(i)p FT(th)h(column)e(of)h(the)
h(matrix,)f(and)g(it)f(has)i(a)p 378 5555 1380 4 v 482
5608 a FC(3)516 5640 y FB(Resolution,)e(connection,)f(and)f(tableau)h
(based)g(deductiv)n(e)f(systems)g(for)h(other)g(logics)i(exist)d(as)i
(w)n(ell.)p eop
%%Page: 12 22
12 21 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(12)378 396 y(connection)25 b(if)f(it)g(con)m(tains)h(t)m(w)m(o)h
(complemen)m(tary)f(literals.)38 b(The)24 b(follo)m(wing)f(is)h(a)i
(refutable)e(matrix)378 509 y(represen)m(ting)30 b(a)g(pro)s(of)g(of)h
(the)f(v)-5 b(alidit)m(y)29 b(of)h(the)h(sen)m(tence)h(giv)m(en)e
(earlier.)1464 634 y Fx(\024)1512 706 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 30.62582 15.31291
2.7375 } false /N@nPx 16 {InitRnode } NewNode end end
1512 706 a FN(:)p
FP(P)13 b FT(\()p FP(x)p FT(\))1849 706 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 21.80626 10.90312
2.7375 } false /N@Pc 16 {InitRnode } NewNode end end
1849 706 a
FP(P)g FT(\()p FP(c)p FT(\))2113 706 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 29.21196 14.60597
2.7375 } false /N@nQc 16 {InitRnode } NewNode end end
2113 706 a FN(:)p
FP(Q)p FT(\()p FP(c)p FT(\))1541 819 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 23.43149 11.71574
2.7375 } false /N@Qx 16 {InitRnode } NewNode end end
1541 819 a FP(Q)p
FT(\()p FP(x)p FT(\))2356 634 y Fx(\025)2419 762 y FN(f)p
FP(x)25 b FN(!)g FP(c)p FN(g)2741 762 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@nPx /N@Pc 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
2741 762 a 2741
762 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@Qx /N@nQc 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
2741 762 a 519 1020 a FT(Uni\014cation)32 b(\(Robinson)h(1971\))i
(is)e(used)g(to)h(\014nd)e(the)i(required)e(substitutions)f(during)g
(reso-)378 1133 y(lution)j(and)i(connection-based)g(pro)s(of)g(searc)m
(h,)j(as)d(w)m(ell)f(as)i(during)d(the)i(pro)s(of)g(searc)m(h)g(of)h
(man)m(y)378 1246 y(other)31 b(pro)s(of)e(metho)s(ds,)h(suc)m(h)g(as)h
(tableaux)f(calculi.)519 1359 y(It)35 b(can)f(b)s(e)g(seen)g(from)g
(the)h(ab)s(o)m(v)m(e)g(examples)f(that)h(resolution)e(and)h(matrix)f
(pro)s(ofs)h(are)h(not)378 1472 y(mean)m(t)k(to)f(b)s(e)g(understo)s(o)
s(d)e(b)m(y)i(a)g(h)m(uman)f(reader.)63 b(They)38 b(are)g(rather)g
(compact)h(pro)s(ofs)e(whose)378 1585 y(structure)30
b(allo)m(ws)g(them)g(to)h(b)s(e)f(searc)m(hed)h(for)f(e\016cien)m(tly)
-8 b(.)519 1698 y(Although)38 b(automated)i(deduction)e(systems)h(can)g
(b)s(e)g(v)m(ery)g(p)s(o)m(w)m(erful)e(and)i(can)g(ev)m(en)h(solv)m(e)
378 1811 y(op)s(en)25 b(mathematical)i(problems,)e(they)i(ma)m(y)f
(fail)f(to)i(solv)m(e)f(problems)f(whic)m(h)g(are)h(rather)g(in)m
(tuitiv)m(e)378 1924 y(to)40 b(h)m(umans.)67 b(One)39
b(reason)g(for)h(this)e(is)g(that)i(the)g(formal)f(pro)s(ofs)f(of)i
(certain)f(in)m(tuitiv)m(e)f(results)378 2036 y(can)32
b(b)s(e)e(v)m(ery)i(long,)f(or)g(hard)g(to)h(\014nd,)e(when)g
(formalised)f(in)h(ev)m(en)i(the)g(most)f(e\016cien)m(t)h(deductiv)m(e)
378 2149 y(systems.)50 b(A)33 b(famous)g(result)g(in)f(computational)h
(logic,)h(\014rst)f(pro)m(v)m(ed)g(b)m(y)h(Hak)m(en)g(\(1985\),)j
(states)378 2262 y(that)31 b(the)f(lengths)g(of)g(resolution)f(pro)s
(ofs)h(for)g(the)g(prop)s(ositional)e(represen)m(tation)i(of)h(the)f
(pigeon-)378 2375 y(hole)h(principles)d(are)k(exp)s(onen)m(tial)f(with)
f(resp)s(ect)h(to)i(the)f(lengths)e(of)i(the)g(form)m(ulae.)44
b(In)31 b(general,)378 2488 y(pro)s(of)36 b(searc)m(h)h(algorithms)e
(need)h(to)h(b)s(e)f(targeted)i(to)f(particular)e(problem)g(domains)g
(and)h(their)378 2601 y(p)s(erformance)30 b(on)g(problems)f(outside)g
(this)g(domain)h(is)f(greatly)i(diminished.)378 2844
y FG(2.3.2)112 b(Pro)s(of)38 b(Chec)m(king)f(and)h(Pro)s(of)f(Dev)m
(elopmen)m(t)f(Systems)378 3016 y FT(The)e(purp)s(ose)g(of)h(a)g(pro)s
(of)f(c)m(hec)m(king)i(system)f(is)f(to)i(c)m(hec)m(k)h(the)e
(correctness)h(of)f(a)g(formal)f(pro)s(of,)378 3129 y(whic)m(h)h(can)h
(b)s(e)f(found)g(b)m(y)h(a)g(h)m(uman,)h(mac)m(hine,)g(or)f(b)m(y)g(a)h
(com)m(bined)e(e\013ort)h(from)g(b)s(oth.)57 b(Mo)s(d-)378
3242 y(ern)38 b(pro)s(of)f(c)m(hec)m(k)m(ers)k(are)d(usually)e(called)i
(pro)s(of)f(dev)m(elopmen)m(t)i(systems,)i(or)d(theorem)h(pro)m(ving)
378 3355 y(en)m(vironmen)m(ts,)30 b(b)s(ecause)g(they)h(can)f(con)m
(tribute)g(more)h(to)g(the)f(formalisation)f(pro)s(cess)g(than)h(just)
378 3468 y(pro)s(of)37 b(c)m(hec)m(king.)65 b(Mo)s(dern)37
b(systems)h(lik)m(e)g(Isab)s(elle)e(\(P)m(aulson)i(1994\))i(and)e(HOL)f
(\(Gordon)h(and)378 3581 y(Melham)27 b(1993\))j(include)25
b(a)j(n)m(um)m(b)s(er)e(of)i(decision)e(and)h(semi-decision)f(pro)s
(cedures)g(for)i(particular)378 3694 y(theories)j(to)g(pro)m(v)m(e)h
(certain)f(theorems)g(automatically)-8 b(,)32 b(and)e(a)i(n)m(um)m(b)s
(er)d(of)i(pro)s(of)f(pro)s(cedures)g(to)378 3806 y(automate)i(a)f
(sequence)g(of)f(non-trivial)e(inferences.)378 4047 y
FQ(F)-9 b(oundational)35 b(Systems)f(of)h(Pro)s(of)h(Chec)m(k)m(ers)378
4218 y FT(Since)41 b(pro)s(of)h(c)m(hec)m(king)h(systems)f(are)h(in)e
(general)h(not)h(exp)s(ected)f(to)h(\014nd)e(pro)s(ofs)g(themselv)m
(es,)378 4331 y(the)d(deductiv)m(e)f(systems)g(they)h(implemen)m(t)e
(are)i(usually)d(not)j(searc)m(h-orien)m(ted.)63 b(On)37
b(the)h(other)378 4444 y(hand,)31 b(they)h(are)g(exp)s(ected)g(to)h
(formalise)d(a)i(v)-5 b(ariet)m(y)32 b(of)g(mathematical)g(concepts)g
(and)f(therefore)378 4557 y(they)k(are)h(based)f(on)g(rather)g(ric)m(h)
g(and)g(expressiv)m(e)g(foundational)e(systems.)56 b(As)35
b(a)h(result,)g(most)378 4670 y(mo)s(dern)f(systems)h(are)h(based)f(on)
h(some)f(higher-order)f(logic)i(in)e(order)h(to)h(b)s(e)f(able)g(to)h
(quan)m(tify)378 4783 y(o)m(v)m(er)e(functions)c(and)i(predicates)g
(without)f(ha)m(ving)h(to)h(de\014ne)f(them)g(in)f(terms)h(of)h(other)f
(ob)5 b(jects)378 4896 y(\(suc)m(h)35 b(as)f(sets\).)54
b(The)34 b(use)g(of)h(higher-order)e(logic)h(for)h(this)e(purp)s(ose)g
(w)m(as)i(used)f(b)m(y)g(Hanna)g(and)378 5009 y(Daec)m(he)29
b(\(1985\))h(and)d(Gordon)g(\(1985\))j(in)c(the)i(con)m(text)h(of)e
(formalising)e(and)i(v)m(erifying)f(hardw)m(are.)378
5121 y(The)j(HOL)g(system,)h(whic)m(h)e(implemen)m(ts)g(Ch)m(urc)m(h's)
g(simply)f(t)m(yp)s(ed)i(higher-order)f(logic)h(\(Ch)m(urc)m(h)378
5234 y(1940\))h(with)d(p)s(olymorphism,)e(w)m(as)k(originally)c(dev)m
(elop)s(ed)i(for)h(hardw)m(are)g(v)m(eri\014cation)g(but)f(it)h(can)378
5347 y(also)j(b)s(e)g(used)g(to)h(formalise)f(a)h(substan)m(tial)e(n)m
(um)m(b)s(er)g(of)i(mathematical)g(theories)f(including)d(real)378
5460 y(analysis.)519 5573 y(A)22 b(n)m(um)m(b)s(er)f(of)h(pro)s(of)f
(dev)m(elopmen)m(t)h(systems)g(are)h(based)e(on)h(a)g(constructiv)m(e)h
(t)m(yp)s(e)f(theory)g(suc)m(h)378 5686 y(as)33 b(the)h(Calculus)d(of)i
(Constructions)f(\(Co)s(quand)g(and)h(Huet)g(1986\).)52
b(In)32 b(suc)m(h)h(systems,)h(there)g(is)p eop
%%Page: 13 23
13 22 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(13)378 396 y(a)35 b(corresp)s(ondence)f(\(called)g(the)g(Curry-Ho)
m(w)m(ard)g(Isomorphism\))f(b)s(et)m(w)m(een)i(the)g(inference)e(rules)
378 509 y(of)i(the)h(logic)e(and)h(the)g(w)m(a)m(ys)h(v)-5
b(alid)34 b(terms)h(in)e(a)j(t)m(yp)s(ed)f(lam)m(b)s(da)f(calculus)f
(can)j(b)s(e)e(constructed.)378 622 y(As)h(a)h(result,)f(sen)m(tences)h
(can)g(b)s(e)e(represen)m(ted)h(as)h(t)m(yp)s(es,)g(and)f(pro)s(ofs)f
(as)h(terms.)55 b(Therefore,)36 b(a)378 735 y(sen)m(tence)24
b(can)f(b)s(e)f(sho)m(wn)g(to)i(b)s(e)e(v)-5 b(alid)21
b(if)h(the)h(t)m(yp)s(e)g(represen)m(ting)f(it)g(is)g(not)h(empt)m(y)g
(\(i.e.,)16 b(it)23 b(con)m(tains)378 848 y(a)j(pro)s(of)7
b(\).)39 b(An)25 b(in)m(teresting)g(feature)h(of)f(these)h(systems)g
(is)f(that)h(b)s(oth)f(the)g(logical)g(statemen)m(ts)i(and)378
961 y(their)i(pro)s(ofs)h(can)h(b)s(e)e(represen)m(ted)i(in)e(the)h
(same)h(language.)519 1074 y(The)37 b(reliabilit)m(y)d(of)j(the)h(pro)s
(ofs)e(accepted)j(b)m(y)e(pro)s(of)f(c)m(hec)m(k)m(ers)k(is)c(an)h(imp)
s(ortan)m(t)f(issue.)61 b(In)378 1187 y(order)25 b(to)h(maximise)f
(this,)g(some)h(pro)s(of)f(c)m(hec)m(k)m(ers)j(are)e(designed)e(so)i
(that)g(the)g(correctness)g(of)g(their)378 1300 y(pro)s(ofs)32
b(dep)s(ends)e(only)i(on)g(a)h(small)e(fragmen)m(t)i(of)g(their)e(co)s
(de.)48 b(This)30 b(fragmen)m(t)k(is)d(usually)f(small)378
1413 y(and)f(simple)f(enough)h(to)i(b)s(e)e(w)m(ell)f(understo)s(o)s(d)
g(so)i(that)g(the)g(p)s(ossibilit)m(y)c(of)k(programming)e(errors)378
1526 y(is)40 b(minimised.)68 b(W)-8 b(e)42 b(can)g(refer)e(to)i(this)d
(prop)s(ert)m(y)i(as)g(the)g FI(de)h(Bruijn)g(criterion)49
b FT(since)40 b(it)h(w)m(as)378 1638 y(suggested)e(b)m(y)f(de)g
(Bruijn,)h(who)e(headed)h(the)h(A)m(UTOMA)-8 b(TH)39
b(pro)5 b(ject)39 b(\(de)f(Bruijn)f(1970\))j(\(see)378
1751 y(also)28 b(\(de)h(Bruijn)e(1980\)\))k(|)d(undoubtedly)e(one)j(of)
g(the)f(most)h(in\015uen)m(tial)d(pro)5 b(jects)29 b(in)e(the)i(mec)m
(h-)378 1864 y(anisation)24 b(of)g(mathematics.)40 b(In)24
b(systems)g(lik)m(e)g(Co)s(q)g(\(Barras)i(et)f(al.)39
b(1996\))26 b(whic)m(h)e(are)h(based)f(on)g(a)378 1977
y(constructiv)m(e)31 b(t)m(yp)s(e)g(theory)-8 b(,)32
b(the)f(cen)m(tral)g(pro)s(of)f(c)m(hec)m(king)i(mec)m(hanism)e(is)g
(the)h(relativ)m(ely)f(simple)378 2090 y(t)m(yp)s(e)k(c)m(hec)m(king)g
(algorithm.)49 b(The)33 b(design)f(of)i(the)g(HOL)f(system)g(ensures)g
(that)h(in)m(ternal)e(ob)5 b(jects)378 2203 y(represen)m(ting)30
b(theorems)g(and)g(de\014nitions)e(are)i(created)i(only)d(b)m(y)h(a)h
(small)e(n)m(um)m(b)s(er)g(of)h(functions,)378 2316 y(the)c(implemen)m
(tation)f(of)i(whic)m(h)e(is)g(straigh)m(tforw)m(ard.)39
b(These)26 b(functions)f(are)i(an)f(implemen)m(tation)378
2429 y(of)k(the)f(primitiv)m(e)f(inference)g(rules)g(of)i(a)g(sound)e
(deductiv)m(e)h(system)h(for)f(higher-order)f(logic.)40
b(The)378 2542 y(restriction)34 b(of)h(ha)m(ving)f(a)i(simple)d(pro)s
(of)h(c)m(hec)m(king)i(algorithm)e(constitutes)h(a)g(ma)5
b(jor)35 b(limitation)378 2655 y(on)g(the)g(e\016ciency)g(of)g(pro)s
(of)f(dev)m(elopmen)m(t)h(systems.)54 b(An)34 b(in)m(teresting)h(area)g
(of)g(researc)m(h)h(is)e(the)378 2768 y(implemen)m(tation)41
b(of)h(fast)g(pro)s(of)f(pro)s(cedures)g(in)g(suc)m(h)h(systems.)75
b(An)42 b(alternativ)m(e)g(to)h(a)g(\014xed)378 2880
y(pro)s(of)25 b(c)m(hec)m(king)i(algorithm)e(whic)m(h)g(is)g(gaining)g
(the)h(in)m(terest)g(of)h(researc)m(hers)f(is)f(to)i(use)f(some)g(form)
378 2993 y(of)38 b FI(r)-5 b(e\015e)g(ction)47 b FT(so)38
b(that)h(new)e(inference)g(rules)g(can)h(b)s(e)g(safely)f(included)f
(in)h(the)h(pro)s(of)f(c)m(hec)m(king)378 3106 y(mec)m(hanism)30
b(after)h(their)e(correctness)i(is)f(v)m(eri\014ed)f(within)f(the)i
(system.)378 3346 y FQ(The)35 b(Input)f(Language)h(of)g(Pro)s(of)h(Dev)
m(elopmen)m(t)e(Systems)378 3518 y FT(Although)27 b(the)i(pro)s(of)f(c)
m(hec)m(king)h(algorithm)e(of)i(a)g(theorem)f(pro)m(ving)g(en)m
(vironmen)m(t)g(can)g(b)s(e)g(based)378 3631 y(on)38
b(a)h(v)m(ery)f(simple)e(deductiv)m(e)i(system,)j(the)d(input)e
(language)j(whic)m(h)e(is)g(used)g(for)h(the)g(formal-)378
3744 y(isation,)j(and)e(in)f(particular)g(in)h(the)g(implemen)m(tation)
g(of)g(pro)s(ofs,)j(can)e(\(and)f(usually)e(will\))h(b)s(e)378
3857 y(more)30 b(expressiv)m(e.)40 b(Simple)28 b(statemen)m(ts)j(in)e
(the)h(input)e(language)i(can)g(corresp)s(ond)f(to)i(the)f(appli-)378
3970 y(cation)35 b(of)h(sev)m(eral)f(primitiv)m(e)e(inferences)i(in)e
(order)i(to)h(simplify)c(the)j(theorem)h(pro)m(ving)e(task)i(of)378
4083 y(the)31 b(user.)40 b(F)-8 b(or)31 b(instance,)f(the)h(HOL)f
(system)h(includes)d(a)j(n)m(um)m(b)s(er)e(of)i(high-lev)m(el)e
(inference)g(rules)378 4196 y(whic)m(h)36 b(are)i(deriv)m(ed)e(from)h
(the)g(primitiv)m(e)f(ones.)61 b(Examples)37 b(of)g(suc)m(h)g(deriv)m
(ed)g(rules)e(include)g(a)378 4308 y(term)g(rewriting)e(system,)j(pro)s
(cedures)e(for)g(n)m(umeric)g(calculations,)h(and)g(a)g(n)m(um)m(b)s
(er)e(of)i(decision)378 4421 y(pro)s(cedures.)55 b(Similarly)-8
b(,)34 b(constructs)h(for)h(the)f(straigh)m(tforw)m(ard)h(de\014nition)
d(of)j(recursiv)m(e)f(t)m(yp)s(es,)378 4534 y(primitiv)m(e)28
b(recursiv)m(e)i(functions,)f(inductiv)m(e)g(relations,)h(and)f(other)i
(ob)5 b(jects,)31 b(are)g(also)g(pro)m(vided.)519 4647
y(Most)47 b(pro)s(of)d(dev)m(elopmen)m(t)i(systems)g(supp)s(ort)d(an)j
(en)m(vironmen)m(t)f(and)g(a)h(pro)s(of)e(language)378
4760 y(aimed)28 b(at)h(helping)d(the)i(users)g(to)h(\014nd)d(the)j
(formal)e(pro)s(ofs)h(in)m(teractiv)m(ely)-8 b(.)40 b(A)29
b(famous)f(example)g(of)378 4873 y(this)34 b(is)f(the)i(goal-directed)g
(pro)s(of)f(en)m(vironmen)m(t)g(based)h(on)f FI(tactics)p
FT(.)54 b(In)34 b(suc)m(h)g(an)h(en)m(vironmen)m(t,)378
4986 y(users)j(start)i(the)f(theorem)h(pro)m(ving)e(task)i(b)m(y)f(sp)s
(ecifying)e(a)i(goal)h(to)g(b)s(e)e(pro)m(v)m(ed.)68
b(T)-8 b(actics)40 b(can)378 5099 y(then)f(b)s(e)h(applied)d(whic)m(h)i
(either)g(solv)m(e)h(\(pro)m(v)m(e\))i(the)e(goal)g(automatically)-8
b(,)43 b(or)d(break)f(the)h(goal)378 5212 y(in)m(to)31
b(simpler)e(subgoals.)43 b(This)30 b(is)g(rep)s(eated)h(un)m(til)f(all)
g(the)i(subgoals)e(are)i(solv)m(ed.)44 b(A)m(t)32 b(this)e(stage,)378
5325 y(the)f(theorem)f(pro)m(ving)g(system)h(has)f(enough)g
(information)f(to)i(deriv)m(e)f(a)h(theorem)g(corresp)s(onding)378
5438 y(to)42 b(the)f(original)e(goal.)72 b(The)41 b(application)e(of)i
(a)g(tactic)h(can)g(corresp)s(ond)d(to)j(the)f(\(bac)m(kw)m(ards\))378
5550 y(application)32 b(of)i(sev)m(eral)f(primitiv)m(e)f(inference)g
(rules.)49 b(In)33 b(order)g(to)h(increase)g(the)f(p)s(o)m(w)m(er)h(of)
g(eac)m(h)378 5663 y(user)20 b(in)m(teraction,)k(complex)c(tactics)i
(can)g(b)s(e)e(constructed)i(from)e(simpler)f(ones)i(b)m(y)g(the)h
(application)p eop
%%Page: 14 24
14 23 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(14)378 396 y(of)38 b(sp)s(ecial)f(constructs)h(called)g
FI(tactic)-5 b(als)p FT(.)65 b(F)-8 b(urthermore,)40
b(the)e(theorem)h(pro)m(ving)e(en)m(vironmen)m(t)378
509 y(k)m(eeps)43 b(trac)m(k)h(of)f(the)g(unpro)m(v)m(ed)g(subgoals,)i
(and)d(can)i(supp)s(ort)d(a)i(n)m(um)m(b)s(er)e(of)j(useful)d(features)
378 622 y(suc)m(h)35 b(as)h(undoing)d(the)j(application)e(of)i
(tactics,)i(and)c(c)m(ho)s(osing)i(whic)m(h)e(subgoal)h(to)h(pro)m(v)m
(e)g(\014rst.)378 735 y(The)43 b(main)f(adv)-5 b(an)m(tage)45
b(of)f(this)e(approac)m(h)i(is)e(that)i(the)g(theorem)f(pro)m(ving)g
(system)g(p)s(erforms)378 848 y(substan)m(tial)38 b(automation)h(and)g
(b)s(o)s(okk)m(eeping)g(tasks)g(while)e(the)j(user)e(is)g(lo)s(oking)g
(for)h(a)h(formal)378 961 y(pro)s(of.)k(A)32 b(disadv)-5
b(an)m(tage)33 b(of)f(this)f(approac)m(h)h(is)f(the)h(di\016cult)m(y)e
(for)i(a)g(h)m(uman)f(reader)h(to)g(follo)m(w)f(a)378
1074 y(pro)s(of)c(consisting)g(of)i(a)f(list)f(of)h(tactics)i(and)d
(tacticals.)41 b(Tw)m(o)28 b(case)h(studies)e(in)g(the)h(mec)m
(hanisation)378 1187 y(of)41 b(mathematical)h(theories)f(using)f
(tactic-based)i(pro)s(of)f(dev)m(elopmen)m(t)g(are)h(illustrated)d(in)h
(the)378 1300 y(next)30 b(c)m(hapter.)519 1413 y(The)f(input)e
(language)i(for)g(a)g(theorem)g(pro)m(ving)f(system)i(can)f(b)s(e)f
(designed)g(to)i(mak)m(e)f(it)g(easier)378 1526 y(for)37
b(a)h(h)m(uman)f(reader)g(to)i(follo)m(w)d(the)i(mec)m(hanised)f(pro)s
(ofs.)61 b(A)38 b(go)s(o)s(d)f(example)h(of)f(suc)m(h)h(a)f(lan-)378
1638 y(guage)h(is)e(Mizar)h(\(T)-8 b(rybulec)37 b(1978\).)63
b(The)36 b(Mizar)h(system)g(is)g(aimed)f(at)i(the)f(mec)m(hanisation)g
(of)378 1751 y(mathematics)26 b(in)f(general)h(and)f(a)h(substan)m
(tial)e(n)m(um)m(b)s(er)h(of)h(results)e(ha)m(v)m(e)j(b)s(een)e
(formalised)g(in)f(this)378 1864 y(system.)65 b(The)38
b(success)h(of)g(the)f(Mizar)h(pro)5 b(ject)39 b(is)e(mainly)g
(attributed)h(to)h(the)g(e\013ort)g(put)f(in)m(to)378
1977 y(k)m(eeping)33 b(its)f(logical)g(foundations)f(and)h(input)f
(language)i(as)g(similar)e(as)i(p)s(ossible)d(to)j(those)h(used)378
2090 y(b)m(y)i(mathematicians.)59 b(Unlik)m(e)35 b(most)i(other)g
(systems,)h(its)e(logical)g(foundation)f(is)g(set-theoretic)378
2203 y(rather)23 b(than)h(t)m(yp)s(e-theoretic.)40 b(Mizar)23
b(pro)s(of)g(scripts)f(are)j(mean)m(t)f(to)g(b)s(e)f(follo)m(w)m(ed)h
(and)f(understo)s(o)s(d)378 2316 y(b)m(y)35 b(the)h(p)s(erson)e
(implemen)m(ting)f(them,)k(and)d(therefore)i(they)g(state)g(explicitly)
d(whic)m(h)h(steps)h(are)378 2429 y(b)s(eing)24 b(deriv)m(ed)g
(throughout)h(the)g(pro)s(of,)h(rather)f(than)g(merely)g(giving)f(the)h
(instructions)e(to)j(deriv)m(e)378 2542 y(them.)53 b(Also,)35
b(the)f(language)h(constructs)g(are)g(English)d(w)m(ords,)j(suc)m(h)f
(as)g Fw(assume)n FT(,)h Fw(consider)c FT(and)378 2655
y Fw(then)42 b Fv(:)14 b(:)g(:)43 b Fw(by)g Fv(:)14 b(:)g(:)p
FT(,)31 b(whose)g(meaning)f(is)g(similar)e(to)k(the)f(formal)f(seman)m
(tics)i(of)f(the)g(corresp)s(onding)378 2768 y(construct.)72
b(As)41 b(a)g(result,)i(Mizar)e(scripts)f(are)h(more)g(readable)f(when)
g(compared)h(to)g(those)h(of)378 2880 y(other)35 b(systems.)53
b(A)35 b(disadv)-5 b(an)m(tage)35 b(of)g(using)e(the)i(Mizar)f(system)h
(is)f(that)h(no)f(mac)m(hine)h(supp)s(ort)378 2993 y(is)g(giv)m(en)i
(for)f(the)g(in)m(teractiv)m(e)i(disco)m(v)m(ery)e(of)h(pro)s(ofs.)58
b(The)36 b(pro)s(cess)f(of)i(implemen)m(ting)d(a)j(Mizar)378
3106 y(pro)s(of)30 b(script)f(is)g(similar)f(to)j(the)g(pro)s(cess)f
(of)g(implemen)m(ting)f(a)i(\(syn)m(tactically\))g(correct)g(program)
378 3219 y(using)23 b(a)j(text-editor)g(and)e(a)h(compiler.)38
b(Pro)s(of)24 b(scripts)g(are)h(giv)m(en)g(to)h(the)f(Mizar)g(v)m
(eri\014er)f(for)h(pro)s(of)378 3332 y(c)m(hec)m(king)30
b(whic)m(h)f(returns)f(a)i(list)f(of)g(error)h(messages)g(in)f(case)h
(of)g(in)m(v)-5 b(alid)27 b(de\014nitions)g(and)j(pro)s(ofs.)378
3619 y FH(2.4)135 b(A)45 b(Brief)g(Ov)l(erview)h(of)f(the)h(HOL)e
(System)378 3821 y FT(The)d(HOL)h(system)g(w)m(as)g(dev)m(elop)s(ed)g
(b)m(y)f(M.J.C.)i(Gordon)f(\(1988\))i(for)e(the)g(sp)s(eci\014cation)e
(and)378 3934 y(v)m(eri\014cation)d(of)h(hardw)m(are,)i(although)d(it)g
(is)g(also)h(used)f(in)f(soft)m(w)m(are)j(v)m(eri\014cation)f(and)f
(the)h(for-)378 4047 y(malisation)e(of)i(mathematics)g(in)f(general.)62
b(The)37 b(system)h(is)f(based)g(on)h(the)g(higher-order)e(logic)378
4160 y(describ)s(ed)28 b(brie\015y)h(in)g(section)h(1.2.2,)j(and)d(in)f
(detail)g(in)g(\(Gordon)i(and)f(Melham)g(1993\).)378
4404 y FG(2.4.1)112 b(On)38 b(the)g(LCF)g(Approac)m(h)f(of)h(Theorem)f
(Pro)m(ving)378 4575 y FT(The)j(HOL)h(theorem)g(pro)m(v)m(er)g(is)f(a)i
(descendan)m(t)f(of)g(the)g(LCF)g(system)g(\(Gordon,)i(Milner,)f(and)
378 4688 y(W)-8 b(adsw)m(orth)31 b(1979\),)i(with)c(whic)m(h)g(it)h
(shares)g(a)h(n)m(um)m(b)s(er)e(of)h(signi\014can)m(t)f(features,)i(in)
e(particular:)514 4876 y FN(\017)46 b FT(The)28 b(mec)m(hanisation)f
(of)i(the)f(logic)g(is)f(implemen)m(ted)g(in)f(ML)j(and)e(includes)f
(ML)i(t)m(yp)s(es)g(rep-)605 4989 y(resen)m(ting)k(the)g(logic's)g
(theorems,)h(terms)f(and)g(t)m(yp)s(es.)45 b(The)32 b(t)m(yp)s(e)g
(represen)m(ting)g(theorems)605 5102 y(is)d(an)g(abstract)i(data)f(t)m
(yp)s(e)f(and)g(the)h(functions)e(in)g(its)h(signature)g(whic)m(h)f
(return)h(theorems)605 5215 y(are)d(an)g(implemen)m(tation)e(of)i(the)f
(primitiv)m(e)f(inference)g(rules)g(of)i(the)g(logic)f(\(and)h(other)f
(rules)605 5327 y(for)30 b(in)m(tro)s(ducing)d(axioms)i(and)h
(de\014nitions\).)38 b(As)30 b(a)g(result)e(theorems)i(in)f(the)g(HOL)h
(system)605 5440 y(can)24 b(only)g(b)s(e)f(constructed)h(through)f(the)
h(application)f(of)h(one)g(or)g(more)g(primitiv)m(e)e(inference)605
5553 y(rules.)60 b(This)36 b(ensures)g(that)i(only)f(v)-5
b(alid)36 b(sen)m(tences)i(can)g(b)s(e)e(deriv)m(ed)h(as)g(HOL)g
(theorems.)p eop
%%Page: 15 25
15 24 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(15)605 396 y(The)35 b(implemen)m(tation)f(of)h(this)f(abstract)i
(data)g(t)m(yp)s(e)f(is)g(usually)e(referred)h(to)i(as)f(the)h
FI(c)-5 b(or)g(e)605 509 y(infer)g(enc)g(e)33 b(engine)p
FT(.)514 697 y FN(\017)46 b FT(HOL)34 b(users)f(can)h(extend)g(the)g
(system)g(through)f(the)h(implemen)m(tation)e(of)i(ML)g(functions.)605
810 y(F)-8 b(or)41 b(instance,)h(one)e(can)f(implemen)m(t)g(b)s(oth)g
(functions)f(whic)m(h)g(represen)m(t)i(new)f(\(deriv)m(ed\))605
923 y(inference)f(rules)f(and)h(also)g(decision)f(pro)s(cedures)h(that)
h(mak)m(e)g(use)f(of)h(theorems)g(deriv)m(ed)605 1036
y(during)28 b(the)j(mec)m(hanisation)f(of)g(some)h(particular)e
(mathematical)i(theory)-8 b(.)514 1223 y FN(\017)46 b
FT(The)28 b(HOL)g(system)h(supp)s(orts)d(a)j(tactic-based)h
(goal-directed)e(pro)s(of)g(searc)m(h)h(en)m(vironmen)m(t.)378
1411 y(In)e(general,)i(pro)s(of)e(dev)m(elopmen)m(t)i(systems)e(in)g
(whic)m(h)g(theorems)h(can)g(only)f(b)s(e)h(deriv)m(ed)f(b)m(y)h(a)g
(core)378 1524 y(inference)f(engine,)g(whic)m(h)f(can)i(b)s(e)f
(extended)g(b)m(y)g(the)h(users,)f(and)g(whic)m(h)f(supp)s(ort)g(a)i
(tactic-based)378 1637 y(pro)s(of)i(en)m(vironmen)m(t)g(are)g(called)g
(LCF-st)m(yle)h(theorem)f(pro)m(v)m(ers.)378 1880 y FG(2.4.2)112
b(The)38 b(Implemen)m(tation)d(of)j(HOL)378 2052 y FT(The)26
b(latest)h(v)m(ersions)f(of)g(the)h(HOL)f(system)g(are)h(the)g(HOL90)g
(system)f(implemen)m(ted)f(in)g(Standard)378 2165 y(ML)40
b(of)g(New)h(Jersey)-8 b(,)43 b(and)c(the)h(recen)m(tly)h(released)f
(Hol98)g(implemen)m(ted)f(in)g(Mosco)m(w)i(ML.)g(In)378
2278 y(these)g(systems)f(the)g(ML)g(data)h(t)m(yp)s(es)f(for)g(HOL)g(t)
m(yp)s(es,)j(terms)d(and)f(theorems)i(are)f Fw(hol_type)m
FT(,)378 2391 y Fw(term)c FT(and)h Fw(thm)f FT(resp)s(ectiv)m(ely)-8
b(.)63 b(The)37 b(ob)5 b(ject)38 b(language)h(em)m(b)s(edding)c(system)
j(of)g(Slind)d(\(1991\))40 b(is)378 2504 y(used)34 b(for)h(em)m(b)s
(edding)e(a)i(language)g(with)f(a)h(user-friendly)d(syn)m(tax)j(for)g
(HOL)g(terms)f(and)h(t)m(yp)s(es.)378 2616 y(One)26 b(can)h(sp)s(ecify)
e(HOL)h(t)m(yp)s(es)g(and)g(terms)h(b)m(y)f(enclosing)f(expressions)g
(in)h(bac)m(kquotes)h(whic)m(h)e(are)378 2729 y(then)30
b(parsed)g(b)m(y)g(the)g(t)m(yp)s(e)h(and)f(term)g(parsers)g(in)m(to)g
(their)g(in)m(ternal)f(ML)h(represen)m(tation.)519 2842
y(As)f(men)m(tioned)f(earlier,)g(ob)5 b(jects)30 b(of)f(the)g(abstract)
h(data)f(t)m(yp)s(e)g(of)g(theorems)g Fw(thm)f FT(can)h(only)f(b)s(e)
378 2955 y(created)c(using)e(an)h(implemen)m(tation)e(of)i(a)h(simple)d
(deductiv)m(e)i(system,)i(and)d(b)m(y)h(a)g(small)f(n)m(um)m(b)s(er)g
(of)378 3068 y(other)29 b(ML)g(functions)e(whic)m(h)h(allo)m(w)g(one)h
(to)g(in)m(tro)s(duce)f(axioms)h(and)f(de\014nitions)e(in)h(a)j
(particular)378 3181 y(HOL)24 b(theory)-8 b(.)40 b(F)-8
b(or)26 b(completeness,)g(w)m(e)f(giv)m(e)g(the)g(inference)f(rules)g
(of)h(the)g(HOL)f(deductiv)m(e)h(system)378 3294 y(in)38
b(\014gure)g(1.)68 b(Since)38 b(the)i(implemen)m(tation)e(of)h(this)f
(abstract)i(data)g(t)m(yp)s(e)f(is)g(rather)g(small)e(and)378
3407 y(straigh)m(tforw)m(ard,)g(the)e(HOL)g(system)h(satis\014es)f(the)
g(de)g(Bruijn)f(criterion.)54 b(All)34 b(other)i(inference)378
3520 y(rules,)k(decision)d(pro)s(cedures,)j(and)e(a)h(n)m(um)m(b)s(er)e
(of)i(functions)f(whic)m(h)f(allo)m(w)h(the)h(user)f(to)i(de\014ne)378
3633 y(constan)m(ts)i(are)g(implemen)m(ted)e(using)g(only)g(the)h
(functions)f(in)g(the)i(signature)e(of)i(the)f(abstract)378
3746 y(t)m(yp)s(e)30 b Fw(thm)g FT(to)h(construct)f(ob)5
b(jects)32 b(of)e(that)h(t)m(yp)s(e.)519 3858 y(The)i(pro)s(of)g
(language)h(of)g(the)g(HOL)f(system)h(is)f(basically)f(the)i(ML)g
(language)3285 3825 y FL(4)3325 3858 y FT(.)50 b(HOL)34
b(users)378 3971 y(usually)39 b(formalise)g(their)h(theories)h(using)e
(the)j(facilities)d(of)i(the)g(ML)g(standard)f(en)m(vironmen)m(t.)378
4084 y(The)25 b(functions)g(represen)m(ting)g(the)h(primitiv)m(e)e(and)
h(deriv)m(ed)g(inference)g(rules)f(are)j(used)e(directly)f(to)378
4197 y(pro)m(v)m(e)h(theorems.)39 b(De\014nitions,)24
b(theorems)h(and)f(axioms)g(are)h(referred)e(to)i(b)m(y)g(their)e(ML)h
(iden)m(ti\014er.)378 4310 y(The)32 b(HOL)g(system)g(includes)e(a)j(n)m
(um)m(b)s(er)e(of)i(functions)e(whic)m(h)g(create)j(and)d(manipulate)g
(ob)5 b(jects)378 4423 y(of)36 b(t)m(yp)s(es)f Fw(hol_type)d
FT(and)j Fw(term)n FT(.)56 b(These)35 b(are)h(used)e(b)m(y)i(the)f
(users)g(to)h(implemen)m(t)e(new)h(inference)378 4536
y(rules,)29 b(de\014nition)f(mec)m(hanisms,)i(and)g(also)g(complete)h
(pro)s(of)f(en)m(vironmen)m(ts.)519 4649 y(As)f(stated)h(ab)s(o)m(v)m
(e,)h(the)f(HOL)f(system)g(supp)s(orts)f(a)h(tactic-based)i(pro)s(of)d
(en)m(vironmen)m(t.)40 b(HOL)378 4762 y(tactics)33 b(are)f(implemen)m
(ted)e(as)i(sp)s(ecial)f(ML)g(functions)g(whic)m(h)f(tak)m(e)k(a)e
(goal)g(and)f(return)g(a)h(list)f(of)378 4875 y(subgoals)24
b(together)h(with)e(a)i(v)-5 b(alidation)23 b(function.)37
b(A)24 b(goal)h(is)f(a)g(sequen)m(t)h(\(whic)m(h)e(consists)h(of)h(a)f
(list)378 4988 y(of)33 b(assumptions)e(and)h(a)i(conclusion\))d
(represen)m(ting)h(an)h(unpro)m(v)m(ed)f(statemen)m(t.)50
b(The)33 b(v)-5 b(alidation)378 5100 y(function)20 b(deriv)m(es)g(the)h
(goal)h(as)f(a)g(HOL)g(theorem)h(when)e(all)f(the)j(subgoals)e(are)h
(themselv)m(es)h(deriv)m(ed.)378 5213 y(T)-8 b(acticals)37
b(are)h(implemen)m(ted)d(as)i(ML)g(functions)e(whic)m(h)h(tak)m(e)j
(and)d(return)g(tactics.)61 b(Unpro)m(v)m(ed)378 5326
y(goals)30 b(are)g(organised)f(in)g(a)h FI(go)-5 b(alstack)41
b FT(data)31 b(structure,)e(and)h(a)g(n)m(um)m(b)s(er)e(of)i(ML)g
(functions)e(whic)m(h)p 378 5488 1380 4 v 482 5542 a
FC(4)516 5574 y FB(The)c(ML)h(language)g(w)n(as)g(actually)f(dev)n
(elop)r(ed)g(as)g(the)g(meta-language)g(for)h(the)e(LCF)i(system;)f(ML)
g(stands)g(for)378 5665 y(meta-language.)p eop
%%Page: 16 26
16 25 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(16)p 378 848 3453 4 v 376 5170 4 4322 v 1828 1088
185 4 v 1828 1162 a Fv(t)37 b Fu(`)g Fv(t)2054 1109 y
Ft(\()p Fw(ASSUME)n Ft(\))p 1835 1378 259 4 v 1835 1452
a Fu(`)g Fv(t)23 b Ft(=)f Fv(t)2135 1399 y Ft(\()p Fw(REFL)n
Ft(\))p 1378 1648 954 4 v 1378 1726 a Fu(`)37 b Ft(\()p
Fv(\025)q(x:)14 b(t)1661 1738 y Fs(1)1700 1726 y Ft(\))p
Fv(t)1762 1738 y Fs(2)1822 1726 y Ft(=)23 b Fv(t)1940
1738 y Fs(1)1977 1726 y Fu(f)p Fv(x)37 b Fu(!)g Fv(t)2253
1738 y Fs(2)2290 1726 y Fu(g)2373 1689 y Ft(\()p Fw(BETA_CONV)m
Ft(\))1002 1984 y(\000)1054 1996 y Fs(1)1128 1984 y Fu(`)g
Fv(t)1246 1996 y Fs(1)1306 1984 y Ft(=)22 b Fv(t)1423
1954 y Fr(0)1423 2005 y Fs(1)1544 1984 y Fu(\001)14 b(\001)g(\001)96
b Ft(\000)1789 1996 y Fq(n)1871 1984 y Fu(`)37 b Fv(t)1989
1996 y Fq(n)2057 1984 y Ft(=)23 b Fv(t)2175 1954 y Fr(0)2175
2005 y Fq(n)2303 1984 y Ft(\000)37 b Fu(`)g Fv(t)p Ft([)p
Fv(t)2563 1996 y Fs(1)2600 1984 y Fv(;)14 b(:)g(:)g(:)f(;)h(t)2814
1996 y Fq(n)2859 1984 y Ft(])p 1002 2025 1881 4 v 1373
2104 a(\000)1425 2116 y Fs(1)1480 2104 y Fu([)19 b(\001)14
b(\001)g(\001)k([)h Ft(\000)1795 2116 y Fq(n)1859 2104
y Fu([)f Ft(\000)37 b Fu(`)g Fv(t)p Ft([)p Fv(t)2192
2074 y Fr(0)2192 2124 y Fs(1)2229 2104 y Fv(;)14 b(:)g(:)g(:)g(;)g(t)
2444 2074 y Fr(0)2444 2124 y Fq(n)2489 2104 y Ft(])2924
2066 y(\()p Fw(SUBST)n Ft(\))1775 2357 y(\000)37 b Fu(`)f
Fv(t)1981 2369 y Fs(1)2042 2357 y Ft(=)22 b Fv(t)2159
2369 y Fs(2)p 1576 2389 820 4 v 1576 2468 a Ft(\000)37
b Fu(`)g Ft(\()p Fv(\025)q(x:)14 b(t)1948 2480 y Fs(1)1986
2468 y Ft(\))24 b(=)e(\()p Fv(\025)q(x:)14 b(t)2324 2480
y Fs(2)2363 2468 y Ft(\))2437 2430 y(\()p Fw(ABS)o Ft(\))1752
2720 y(\000)37 b Fu(`)f Fv(t)p 1278 2740 1154 4 v 1278
2819 a Ft(\000)h Fu(`)g Fv(t)p Fu(f)p Fv(\013)1580 2831
y Fs(1)1653 2819 y Fu(!)g Fv(\033)1820 2831 y Fs(1)1858
2819 y Fv(;)14 b(:)g(:)g(:)g(;)g(\013)2096 2831 y Fq(n)2178
2819 y Fu(!)37 b Fv(\033)2345 2831 y Fq(n)2390 2819 y
Fu(g)2473 2782 y Ft(\()p Fw(INST_TYPE)m Ft(\))1820 3072
y(\000)g Fu(`)g Fv(t)2027 3084 y Fs(2)p 1573 3104 738
4 v 1573 3183 a Ft(\000)19 b Fu(\000)f(f)p Fv(t)1799
3195 y Fs(1)1835 3183 y Fu(g)37 b(`)f Fv(t)2031 3195
y Fs(1)2115 3183 y Fu(\))46 b Fv(t)2274 3195 y Fs(2)2353
3146 y Ft(\()p Fw(DISCH)n Ft(\))1564 3436 y(\000)1616
3448 y Fs(1)1690 3436 y Fu(`)37 b Fv(t)1808 3448 y Fs(1)1891
3436 y Fu(\))46 b Fv(t)2050 3448 y Fs(2)2170 3436 y Ft(\000)2222
3448 y Fs(2)2296 3436 y Fu(`)37 b Fv(t)2414 3448 y Fs(1)p
1564 3468 888 4 v 1776 3543 a Ft(\000)1828 3555 y Fs(1)1884
3543 y Fu([)19 b Ft(\000)2010 3555 y Fs(2)2084 3543 y
Fu(`)37 b Fv(t)2202 3555 y Fs(2)2493 3502 y Ft(\()p Fw(MP)o
Ft(\))568 3738 y Fu(\017)45 b Ft(Expressions)21 b(of)i(the)h(form)f
(\000)37 b Fu(`)g Fv(t)23 b Ft(are)f(HOL)h(theorems)g(with)h
(conclusion)e Fv(t)i Ft(and)f(assumption)g(list)655 3837
y(\000.)568 3987 y Fu(\017)45 b Ft(The)28 b(rules)f(can)g(b)r(e)h
(applied)g(only)f(if)h(the)g(follo)n(wing)e(conditions)h(hold:)745
4149 y(1.)45 b(In)28 b(the)g Fw(ABS)e Ft(rule,)i(the)g(v)-5
b(ariable)26 b Fv(x)i Ft(is)g(not)f(free)h(in)g(\000.)745
4282 y(2.)45 b(In)29 b(the)h Fw(INST_TYPE)c Ft(rule,)j(the)h(term)f
Fv(t)p Fu(f)p Fv(\013)2188 4294 y Fs(1)2262 4282 y Fu(!)36
b Fv(\033)2428 4294 y Fs(1)2466 4282 y Fv(;)14 b(:)g(:)g(:)g(;)g(\013)
2704 4294 y Fq(n)2786 4282 y Fu(!)37 b Fv(\033)2953 4294
y Fq(n)2998 4282 y Fu(g)29 b Ft(is)g(the)h(result)f(of)g(sub-)855
4381 y(stituting,)d(in)g(parallel,)e(the)i(t)n(yp)r(es)f
Fv(\033)2029 4393 y Fs(1)2067 4381 y Fv(;)14 b(:)g(:)g(:)27
b(;)14 b(\033)2312 4393 y Fq(n)2383 4381 y Ft(for)25
b(t)n(yp)r(e)g(v)-5 b(ariables)24 b Fv(\013)3087 4393
y Fs(1)3124 4381 y Fv(;)14 b(:)g(:)g(:)28 b(;)14 b(\013)3376
4393 y Fq(n)3446 4381 y Ft(in)25 b Fv(t)p Ft(,)h(with)855
4481 y(the)i(t)n(w)n(o)f(restrictions)874 4614 y(\(a\))45
b(none)27 b(of)h(the)g(t)n(yp)r(e)g(v)-5 b(ariables)26
b Fv(\013)2045 4626 y Fs(1)2082 4614 y Fv(;)14 b(:)g(:)g(:)28
b(;)14 b(\013)2334 4626 y Fq(n)2407 4614 y Ft(o)r(ccur)27
b(in)g(\000,)h(and)869 4730 y(\(b\))46 b(no)27 b(distinct)i(v)-5
b(ariables)26 b(in)i Fv(t)f Ft(b)r(ecome)h(iden)n(ti\014ed)g(after)f
(the)h(instan)n(tiation.)894 5001 y FT(Figure)i(1:)41
b(The)30 b(Primitiv)m(e)e(Inference)i(Rules)f(of)i(the)g(HOL)f(System.)
p 3829 5170 4 4322 v 378 5173 3453 4 v eop
%%Page: 17 27
17 26 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(17)378 396 y(for)27 b(instance,)g(allo)m(w)f(the)i(user)e(to)h
(apply)f(tactics)i(to)f(the)g(curren)m(t)g(goal,)h(c)m(ho)s(ose)g(the)f
(curren)m(t)g(goal,)378 509 y(and)40 b(undo)g(the)h(application)e(of)i
(a)g(n)m(um)m(b)s(er)f(of)h(tactics,)k(are)c(included)d(in)i(the)h
(system.)72 b(Since)378 622 y(tactics)40 b(and)e(tacticals)i(are)f
(simply)e(sp)s(ecial)h(kinds)f(of)i(ML)g(functions,)h(HOL)e(users)g
(can)i(easily)378 735 y(implemen)m(t)29 b(new)h(ones)g(during)e(the)j
(mec)m(hanisation)f(of)g(a)h(theory)-8 b(.)519 848 y(As)28
b(discussed)f(in)g(the)h(next)g(c)m(hapter,)i(the)e(fact)h(that)g(the)g
(pro)s(of)e(language)i(of)f(HOL)g(is)f(a)i(p)s(o)m(w-)378
961 y(erful)24 b(general-purp)s(ose)g(programming)g(language)i(is)e
(one)i(of)f(the)h(strongest)g(features)f(of)h(the)f(HOL)378
1074 y(system.)58 b(This)35 b(particular)f(approac)m(h)i(to)h(theorem)g
(pro)m(ving,)g(ho)m(w)m(ev)m(er,)i(has)d(the)g(disadv)-5
b(an)m(tage)378 1187 y(that)37 b(it)g(is)f(v)m(ery)h(hard)f(to)h(dev)m
(elop)g(e\013ectiv)m(e)h(user)f(in)m(terfaces)g(and)f(other)h(pro)s(of)
f(to)s(ols)h(without)378 1300 y(compromising)29 b(the)h(\015exibilit)m
(y)e(of)i(the)h(system.)378 1543 y FG(2.4.3)112 b(A)37
b(Num)m(b)s(er)g(of)h(Mec)m(hanised)g(Pro)s(ofs)g(in)e(HOL)378
1715 y FT(In)26 b(this)g(section)h(w)m(e)h(illustrate)d(some)i
(examples)g(of)g(mec)m(hanical)g(pro)s(ofs)f(using)g(the)h(HOL)f
(system.)378 1828 y(In)k(eac)m(h)h(case)h(w)m(e)e(deriv)m(e)g(the)h
(follo)m(wing)e(simple)f(statemen)m(t:)1418 2032 y(\()p
FP(A)d FN(\))h FP(B)5 b FT(\))25 b FN(\))g FT(\()p FP(B)30
b FN(\))25 b FP(C)7 b FT(\))25 b FN(\))g FT(\()p FP(A)h
FN(\))f FP(C)7 b FT(\))p FP(:)378 2272 y FQ(The)35 b(Pro)s(of)g(in)g
(Sequen)m(t)g(Calculus)378 2444 y FT(The)25 b(ab)s(o)m(v)m(e)h
(statemen)m(t)h(can)f(b)s(e)f(deriv)m(ed)f(in)g(the)i(deductiv)m(e)f
(system)g(giv)m(en)h(in)e(\014gure)g(1)i(as)g(follo)m(ws.)p
655 2706 679 4 v 655 2786 a FP(B)j FN(\))d FP(C)31 b
FN(`)25 b FP(B)30 b FN(\))25 b FP(C)1375 2729 y FT(\()p
Fw(ASSUME)n FT(\))p 1798 2606 673 4 v 1798 2686 a FP(A)h
FN(\))f FP(B)30 b FN(`)24 b FP(A)i FN(\))f FP(B)2512
2629 y FT(\()p Fw(ASSUME)n FT(\))p 2935 2606 243 4 v
2935 2686 a FP(A)h FN(`)e FP(A)3219 2629 y FT(\()p Fw(ASSUME)n
FT(\))p 1798 2706 1380 4 v 2202 2786 a FP(A;)15 b(A)26
b FN(\))f FP(B)30 b FN(`)25 b FP(B)3219 2729 y FT(\()p
Fw(MP)p FT(\))p 655 2823 2119 4 v 1266 2903 a FP(A;)15
b(A)26 b FN(\))f FP(B)5 b(;)15 b(B)30 b FN(\))25 b FP(C)32
b FN(`)25 b FP(C)2815 2846 y FT(\()p Fw(MP)p FT(\))p
1216 2941 998 4 v 1216 3021 a FP(A)g FN(\))g FP(B)5 b(;)15
b(B)30 b FN(\))25 b FP(C)32 b FN(`)25 b FP(A)g FN(\))g
FP(C)2254 2964 y FT(\()p Fw(DISCH)n FT(\))p 1094 3058
1240 4 v 1094 3143 a FP(A)h FN(\))f FP(B)30 b FN(`)24
b FT(\()p FP(B)31 b FN(\))25 b FP(C)7 b FT(\))25 b FN(\))g
FT(\()p FP(A)h FN(\))f FP(C)7 b FT(\))2375 3081 y(\()p
Fw(DISCH)o FT(\))p 1001 3186 1427 4 v 1001 3270 a FN(`)25
b FT(\()p FP(A)g FN(\))h FP(B)5 b FT(\))25 b FN(\))g
FT(\()p FP(B)30 b FN(\))25 b FP(C)7 b FT(\))25 b FN(\))g
FT(\()p FP(A)h FN(\))f FP(C)7 b FT(\))2469 3208 y(\()p
Fw(DISCH)n FT(\))378 3510 y FQ(A)35 b(F)-9 b(orw)m(ard)35
b(Pro)s(of)g(in)g(HOL)378 3682 y FT(The)30 b(ab)s(o)m(v)m(e)h(pro)s(of)
f(can)h(b)s(e)f(mec)m(hanised)f(in)g(HOL)h(using)f(the)i(implemen)m
(tation)e(of)i(the)f(primitiv)m(e)378 3795 y(inference)g(rules)e
Fw(ASSUME)n FT(,)j Fw(DISCH)d FT(and)i Fw(MP)o FT(:)378
3983 y Fw(ASSUME:)41 b(term)h Fu(!)h Fw(thm)h FT(whic)m(h)29
b(tak)m(es)j(a)f(term)g Fv(t)p Fw(:bool)d FT(and)h(returns)h(the)g
(theorem)h Fv(t)37 b Fu(`)f Fv(t)p FT(;)378 4170 y Fw(DISCH:)41
b(term)h Fu(!)i Fw(thm)e Fu(!)i Fw(thm)g FT(whic)m(h)25
b(tak)m(es)k(a)e(term)g Fv(t)p Fw(:bool)d FT(and)j(a)g(theorem)g(of)g
(the)g(form)f Ft(\000)37 b Fu(`)g Fv(q)605 4283 y FT(and)30
b(returns)f(the)i(theorem)f Ft(\000)19 b Fu(\000)f(f)p
Fv(t)p Fu(g)36 b(`)g Fv(t)23 b Fu(\))h Fv(q)s FT(;)378
4471 y Fw(MP:)42 b(thm)h Fu(!)g Fw(thm)g Fu(!)g Fw(thm)h
FT(whic)m(h)38 b(tak)m(es)i(t)m(w)m(o)g(theorems)e Ft(\000)2469
4483 y Fs(1)2543 4471 y Fu(`)f Fv(p)23 b Fu(\))g Fv(q)41
b FT(and)d Ft(\000)3117 4483 y Fs(2)3191 4471 y Fu(`)f
Fv(p)h FT(and)g(returns)605 4584 y(the)31 b(theorem)g
Ft(\000)1168 4596 y Fs(1)1237 4584 y Fu([)i Ft(\000)1377
4596 y Fs(2)1451 4584 y Fu(`)j Fv(q)s FT(;)378 4771 y(and)30
b(the)g(deriv)m(ed)g(rule)378 4959 y Fw(DISCH_ALL:)39
b(thm)k Fu(!)g Fw(thm)i FT(whic)m(h)29 b(disc)m(harges)h(all)f(the)i(h)
m(yp)s(otheses)f(of)g(a)h(giv)m(en)f(theorem.)378 5147
y(HOL)36 b(terms)g(can)h(b)s(e)e(constructed)i(b)m(y)f(enclosing)f
(them)h(b)s(et)m(w)m(een)h Fw(--`)e FT(and)h Fw(`--)n
FT(,)i(so)f(that)g(they)378 5260 y(can)31 b(b)s(e)e(parsed)h(in)m(to)g
(ob)5 b(jects)31 b(of)g(t)m(yp)s(e)g Fw(term)n FT(.)41
b(The)29 b(follo)m(wing)g(is)h(the)g(required)f(pro)s(of)g(in)g(HOL.)
473 5447 y FM(DISCH_ALL)46 b(\(DISCH)g(\(--`A:bool`--\))521
5560 y(\(MP)h(\(ASSUME)f(\(--`B)g FN(\))i FM(C`--\)\))712
5673 y(\(MP)f(\(ASSUME)f(\(--`A)g FN(\))i FM(B`--\)\))e(\(ASSUME)g
(\(--`A:bool`--\)\)\)\)\);)p eop
%%Page: 18 28
18 27 bop 378 5 a FF(CHAPTER)30 b(2.)61 b(ON)30 b(THE)g(MECHANISA)-8
b(TION)30 b(OF)g(MA)-8 b(THEMA)g(TICAL)32 b(PR)m(OOFS)175
b FT(18)378 396 y FQ(Deriving)36 b(an)e(Inference)h(Rule)378
568 y FT(The)25 b(mec)m(hanism)g(of)h(the)h(forw)m(ard)e(pro)s(of)g
(giv)m(en)h(ab)s(o)m(v)m(e)h(can)f(b)s(e)f(used)g(to)i(deriv)m(e)e(an)h
(inference)f(rule)378 681 y Fw(IMP_TRANS)l FT(.)1318
760 y(\000)1375 774 y FL(1)1440 760 y FN(`)f FP(A)i FN(\))f
FP(B)96 b FT(\000)1952 774 y FL(2)2016 760 y FN(`)25
b FP(B)30 b FN(\))25 b FP(C)p 1318 794 1066 4 v 1510
874 a FT(\000)1567 888 y FL(1)1627 874 y FN([)20 b FT(\000)1765
888 y FL(2)1829 874 y FN(`)25 b FP(A)g FN(\))h FP(C)2425
817 y FT(\()p Fw(IMP_TRANS)l FT(\))378 1040 y(This)39
b(can)j(b)s(e)e(implemen)m(ted)g(as)h(an)g(ML)h(function)d(whic)m(h)h
(tak)m(es)j(t)m(w)m(o)f(theorems)g(of)f(the)g(form)378
1153 y Ft(\000)430 1165 y Fs(1)481 1153 y Fu(`)14 b Fv(X)7
b Fu(\))p Fv(Y)47 b FT(and)28 b Ft(\000)1027 1165 y Fs(2)1078
1153 y Fu(`)14 b Fv(Y)k Fu(\))p Fv(Z)35 b FT(and)29 b(returns)f(the)h
(theorem)h Ft(\000)2433 1165 y Fs(1)2484 1153 y Fu([)15
b Ft(\000)2606 1165 y Fs(2)2657 1153 y Fu(`)f Fv(X)7
b Fu(\))p Fv(Z)e FT(.)40 b(W)-8 b(e)30 b(use)f(the)g(follo)m(wing)378
1266 y(functions)g(on)h(HOL)g(terms)g(and)g(theorems:)378
1448 y Fw(concl:)85 b(thm)42 b Fu(!)i Fw(term)f FT(tak)m(es)32
b(a)f(theorem)g(and)f(returns)f(its)g(conclusion.)378
1633 y Fw(dest_imp:)83 b(term)42 b Fu(!)i Fw(\(term)d(*)j(term\))f
FT(tak)m(es)24 b(a)f(term)g(of)g(the)g(form)g Fv(X)d
Fu(\))15 b Fv(Y)41 b FT(and)23 b(returns)e(the)i(pair)605
1746 y Fw(\()p Fv(X)7 b Fw(,)p Fv(Y)17 b Fw(\))p FT(.)378
1928 y(The)30 b(deriv)m(ed)f(rule)g(can)i(then)f(b)s(e)g(implemen)m
(ted)f(in)g(SML)h(as)g(follo)m(ws:)521 2109 y FM(fun)47
b(IMP_TRANS)e(\(AB_thm:)h(thm\))g(\(BC_thm:)g(thm\))h(:)g(thm)g(=)617
2222 y(let)f(val)h(AB_term)f(=)i(concl)e(AB_thm)807 2335
y(val)h(A_term)94 b(=)48 b(fst)f(\(dest_imp)e(AB_term\))664
2448 y(in)i(DISCH)g(A_term)f(\(MP)h(BC_thm)f(\(MP)h(AB_thm)f(\(ASSUME)g
(A_term\)\)\))617 2561 y(end;)378 2742 y FT(Alternativ)m(ely)-8
b(,)42 b(one)e(can)g(implemen)m(t)f(this)g(deriv)m(ed)f(rule)h(using)f
(the)i(theorem)h(pro)m(v)m(ed)f(earlier.)378 2855 y(The)g(rule)g(can)h
(simply)e(instan)m(tiate)i(the)g(v)-5 b(ariables)39 b(in)h(the)h
(theorem)g(according)g(to)h(the)f(giv)m(en)378 2968 y(argumen)m(ts.)g
(This)29 b(approac)m(h)h(can)h(often)g(b)s(e)e(used)h(to)h(implemen)m
(t)e(e\016cien)m(t)i(deriv)m(ed)e(rules.)378 3207 y FQ(A)35
b(Bac)m(kw)m(ard)g(Pro)s(of)h(in)f(HOL)378 3379 y FT(The)30
b(same)h(theorem)f(can)h(b)s(e)f(deriv)m(ed)f(in)m(teractiv)m(ely)i
(using)e(the)h(follo)m(wing)f(t)m(w)m(o)j(tactics:)378
3560 y Fw(DISCH_TAC:)39 b(tactic)k FT(whic)m(h)24 b(simpli\014es)d(a)k
(goal)h(with)d(conclusion)h(of)h(the)g(form)f Fv(t)p
Fu(\))p Fv(q)k FT(in)m(to)c(the)h(goal)605 3673 y(with)k(conclusion)g
Fv(q)34 b FT(and)29 b(with)g(the)i(extra)g(assumption)e
Fv(t)p FT(.)378 3859 y Fw(RES_TAC:)40 b(tactic)j FT(whic)m(h,)h
(amongst)f(other)f(things,)i(adds)d(an)h(assumption)e
Fv(q)45 b FT(to)e(the)f(curren)m(t)605 3971 y(goal)d(if)f(it)g(con)m
(tains)h(t)m(w)m(o)g(assumptions)e(of)i(the)g(form)f
Fv(t)14 b Fu(\))h Fv(q)41 b FT(and)d Fv(t)p FT(.)65 b(The)38
b(goal)h(is)f(solv)m(ed)605 4084 y(automatically)30 b(if)g(its)g
(conclusion)e(is)i Fv(q)s FT(.)378 4266 y(The)g(goal)h(represen)m(ting)
e(the)i(required)e(theorem)h(can)h(b)s(e)f(deriv)m(ed)f(b)m(y)489
4448 y(1.)46 b(Applying)33 b Fw(DISCH_TAC)f FT(three)j(times)g(whic)m
(h)f(results)g(in)g(the)i(goal)g(with)e(conclusion)g
Fw(C)g FT(and)605 4560 y(the)d(assumptions)d Fw(A)44
b Fu(\))f Fw(B)p FT(,)30 b Fw(B)44 b Fu(\))f Fw(C)30
b FT(and)g Fw(A)o FT(;)489 4746 y(2.)46 b(Applying)28
b Fw(RES_TAC)g FT(to)j(add)e(the)i(extra)g(assumption)e
Fw(B)p FT(;)489 4931 y(3.)46 b(Applying)28 b Fw(RES_TAC)g
FT(again)i(to)h(add)f(the)g(assumption)f Fw(C)h FT(and)g(th)m(us)g
(solving)f(the)i(goal.)378 5113 y(This)e(pro)s(of)g(can)i(b)s(e)f(giv)m
(en)g(as)h(a)f(single)f(tactic)j(b)m(y)e(using)f(the)i(tacticals)378
5294 y Fw(REPEAT:)41 b(tactic)g Fu(!)i Fw(tactic)g FT(whic)m(h)32
b(applies)f(a)i(giv)m(en)f(tactic)i(rep)s(eatedly)e(un)m(til)f(it)h(is)
g(no)g(longer)605 5407 y(v)-5 b(alid;)378 5592 y Fw(THEN:)41
b(tactic)h(*)h(tactic)e Fu(!)i Fw(tactic)g FT(whic)m(h)37
b(is)g(an)g(in\014x)f(tactical)j(and)e(applies)f(the)i(tactic)h(on)605
5705 y(its)30 b(left)g(and)g(then)g(the)g(tactic)i(on)e(its)g(righ)m
(t.)p eop
%%Page: 19 29
19 28 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(19)378 396 y(The)30 b(required)e(tactic)k(pro)s(of)e(is:)473
584 y FM(REPEAT)46 b(DISCH_TAC)g(THEN)473 697 y(REPEAT)g(RES_TAC;)519
885 y FT(This)30 b(pro)s(of)h(is)g(m)m(uc)m(h)h(shorter)g(than)g(the)g
(forw)m(ard)f(pro)s(of)h(giv)m(en)f(earlier.)45 b(In)31
b(particular,)g(this)378 998 y(pro)s(of)f(do)s(es)h(not)g(con)m(tain)g
(an)m(y)h(subterms)e(from)g(the)h(goal,)h(and)f(in)m(v)m(olv)m(es)g
(pro)s(of)f(steps)h(whic)m(h)e(are)378 1110 y(rep)s(eated)h(un)m(til)f
(they)i(fail.)39 b(As)30 b(a)h(result)e(it)h(can)h(b)s(e)f(used)f(to)i
(deriv)m(e)f(similar)e(theorems,)j(suc)m(h)f(as:)1131
1315 y(\()p FP(W)39 b FN(\))25 b FP(X)7 b FT(\))26 b
FN(\))f FT(\()p FP(X)33 b FN(\))25 b FP(Y)20 b FT(\))26
b FN(\))f FT(\()p FP(Y)46 b FN(\))25 b FP(Z)7 b FT(\))25
b FN(\))g FT(\()p FP(W)38 b FN(\))25 b FP(Z)7 b FT(\))p
FP(:)378 1519 y FT(Ho)m(w)m(ev)m(er,)31 b(it)c(is)g(v)m(ery)h(hard)f
(to)i(\014gure)e(out)h(what)g(the)g(ab)s(o)m(v)m(e)h(pro)s(of)e(is)g
(actually)g(deriving)f(without)378 1632 y(kno)m(wing)g(b)s(eforehand)f
(the)i(statemen)m(t)h(of)f(the)f(theorem.)40 b(In)26
b(general,)i(the)e(only)g(practical)g(w)m(a)m(y)i(of)378
1745 y(follo)m(wing)g(a)h(tactic-based)i(pro)s(of)d(is)h(to)h(use)f
(the)g(theorem)h(pro)m(v)m(er)g(to)g(see)g(the)f(result)g(of)g
(applying)378 1858 y(the)e(individual)c(tactics)28 b(in)e(the)i(pro)s
(of)e(one)i(b)m(y)f(one.)40 b(This)25 b(is)i(reasonable,)h(since)e(the)
i(tactic-based)378 1971 y(en)m(vironmen)m(t)22 b(is)g(dev)m(elop)s(ed)g
(to)h(facilitate)g(in)m(teractiv)m(e)h(pro)s(of)d(disco)m(v)m(ery)-8
b(,)26 b(rather)c(than)h(to)g(pro)s(duce)378 2083 y(h)m(uman)29
b(readable)h(scripts.)378 2324 y FQ(A)35 b(Pro)s(of)h(using)f(a)g
(Decision)h(Pro)s(cedure)378 2495 y FT(Simple)23 b(statemen)m(ts)k(suc)
m(h)e(as)g(the)h(one)f(w)m(e)h(are)f(pro)m(ving)g(in)f(this)g(section)h
(can)h(b)s(e)f(easily)f(deriv)m(ed)g(in)378 2608 y(HOL)g(using)g
(appropriate)g(decision)f(pro)s(cedures.)38 b(In)24 b(this)g(case,)j(w)
m(e)e(can)h(use)e(the)h(HOL)g(tautology)378 2721 y(c)m(hec)m(k)m(er)32
b(to)f(deriv)m(e)f(the)h(ab)s(o)m(v)m(e)h(theorem)e(automatically)-8
b(.)41 b(The)30 b(required)f(ML)h(function)f(is)378 2909
y Fw(TAUT_PROVE:)39 b(term)j Fu(!)i Fw(thm)g FT(whic)m(h)36
b(tak)m(es)j(a)e(term)h Fv(t)p Fw(:bool)d FT(and)i(returns)f(the)h
(theorem)h Fu(`)f Fv(t)g FT(if)f FP(t)605 3022 y FT(is)d(a)g(tautology)
-8 b(.)51 b(This)32 b(function)g(is)g(a)i(sligh)m(tly)d(optimised)h
(implemen)m(tation)g(of)h(the)h(truth)605 3134 y(tables)c(metho)s(d)g
(of)h(tautology)g(c)m(hec)m(king.)378 3322 y(The)f(theorem)h(can)f
(therefore)h(b)s(e)f(deriv)m(ed)f(b)m(y)i(the)f(ML)h(expression)473
3510 y FM(TAUT_PROVE)45 b(\(--`\(A)15 b FN(\))g FM(B\))48
b FN(\))f FM(\(B)16 b FN(\))f FM(C\))48 b FN(\))f FM(\(A)16
b FN(\))g FM(C\)`--\))378 3697 y FT(The)29 b(use)g(of)h(decision)e(pro)
s(cedures)g(can)i(greatly)g(facilitate)f(the)h(implemen)m(tation)e(of)h
(mec)m(hanised)378 3810 y(pro)s(ofs.)44 b(The)32 b(readabilit)m(y)e(of)
i(pro)s(ofs)f(can)h(also)g(b)s(e)f(impro)m(v)m(ed)h(if)e(one)j
(implemen)m(ts)d(and)h(uses)h(the)378 3923 y(required)27
b(decision)g(pro)s(cedures)g(to)i(deriv)m(e)f(automatically)g(statemen)
m(ts)j(whic)m(h)c(readers)h(consider)378 4036 y(trivial.)60
b(Ho)m(w)m(ev)m(er,)41 b(b)s(ecause)c(of)h(the)f(di\013erence)g(b)s(et)
m(w)m(een)h(the)f(nature)g(of)h(the)f(inferences)g(used)378
4149 y(in)g(informal)f(and)i(formal)f(texts,)k(and)d(b)s(ecause)g(of)h
(the)f(di\016culties)e(in)h(automating)h(reasoning)378
4262 y(e\016cien)m(tly)-8 b(,)31 b(suc)m(h)f(a)h(task)f(is)g(not)h
(trivial.)378 4548 y FH(2.5)135 b(On)45 b(Readable)h(Mec)l(hanical)g
(Pro)t(ofs)378 4751 y FT(The)27 b(presen)m(tation)h(of)g(clear)g
(mathematical)g(concepts,)h(whether)f(it)f(is)g(in)f(an)i(informal)e
(or)i(formal)378 4864 y(language,)34 b(is)d(in)h(itself)f(not)i(a)g
(trivial)d(task.)48 b(Th)m(urston)32 b(\(1994\))j(explains)30
b(that)k(one)e(of)h(the)g(main)378 4977 y(aims)f(of)h(mathematicians)f
(is)g(to)h(adv)-5 b(ance)33 b FI(human)j(understanding)g(of)f
(mathematics)p FT(.)50 b(This)31 b(un-)378 5090 y(derstanding)j(is)g
(often)i(a)f(v)m(ery)h(p)s(ersonal)e(and)g(individual)d(matter.)56
b(Di\013eren)m(t)36 b(p)s(eople)e(visualise)378 5203
y(mathematical)k(concepts)g(in)e(di\013eren)m(t)h(w)m(a)m(ys,)j(whic)m
(h)c(often)i(dep)s(end)e(on)h(the)h(particular)e(bac)m(k-)378
5316 y(ground)i(of)i(the)g(individuals.)63 b(Suc)m(h)39
b(ideas)g(are)h(therefore)g(hard)f(to)h(comm)m(unicate,)j(esp)s
(ecially)378 5429 y(in)34 b(writing,)g(where)h(the)g(author)g(is)g
(required)e(to)j(translate)f(her)g(concepts)h(in)m(to)f(sym)m(b)s(ols,)
g(logic,)378 5542 y(and)28 b(statemen)m(ts)j(in)c(a)i(natural)f(\(or)h
(formal\))g(language.)40 b(The)28 b(readers)h(are)g(then)g(required)e
(to)i(use)378 5655 y(these)35 b(texts)h(to)f(build)d(their)i(o)m(wn)g
(in)m(tuition)f(of)i(the)g(sub)5 b(ject.)53 b(The)34
b(clarit)m(y)h(of)f(a)i(mathematical)p eop
%%Page: 20 30
20 29 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(20)378 396 y(exp)s(osition)37 b(is)g(therefore)i(extremely)g(imp)s
(ortan)m(t)f(in)f(order)h(to)h(facilitate)f(the)h(reader's)g(task)g(of)
378 509 y(understanding)23 b(it.)39 b(Halmos)25 b(\(1983\))k(argues)d
(that)g(a)g(go)s(o)s(d)g(exp)s(osition)e(is)h(based)g(on)h(its)f(\\con)
m(ten)m(t,)378 622 y(aim)j(and)f(organisation,)i(plus)d(the)i(vitally)f
(imp)s(ortan)m(t)g(details)g(of)i(grammar,)g(diction,)e(and)h(nota-)378
735 y(tion",)e(and)e(giv)m(es)h(a)g(n)m(um)m(b)s(er)e(of)i(suggestions)
g(to)g(ac)m(hiev)m(e)h(this.)38 b(v)-5 b(an)24 b(Gasteren)i(\(1990\))h
(fo)s(cuses)d(on)378 848 y(the)h(problems)f(of)h(presen)m(ting)g
(mathematical)g(pro)s(ofs)g(clearly)-8 b(.)38 b(Both)26
b(Halmos)f(and)g(v)-5 b(an)25 b(Gasteren)378 961 y(stress)i(the)h(imp)s
(ortance)e(of)h(reducing)f(the)i(e\013ort)g(needed)f(b)m(y)g(the)g
(reader)h(to)g(follo)m(w)e(an)h(argumen)m(t)378 1074
y(in)33 b(a)j(pro)s(of.)53 b(This)33 b(can)i(b)s(e)f(obtained)g(b)m(y)h
(b)s(eing)e(explicit)h(ab)s(out)g(what)h(is)f(needed)g(in)g(the)h(pro)s
(of)378 1187 y(and)30 b(through)h(the)g(omission)e(of)j(trivial)d(and)h
(sup)s(er\015uous)f(information.)41 b(Their)29 b(opinion)g(di\013ers,)
378 1300 y(ho)m(w)m(ev)m(er,)g(on)e(the)g(use)g(of)g(formalism.)37
b(Halmos)27 b(suggests)h(a)f(minimal)d(use)j(of)g(sym)m(b)s(ols,)f
(while)f(v)-5 b(an)378 1413 y(Gasteren)31 b(encourages)h(the)e
(practice)h(of)f(sym)m(b)s(olic)f(manipulation)f(without)h(in)m
(terpretation.)519 1526 y(In)c(general,)i(mathematical)g(pro)s(ofs)e
(implemen)m(ted)f(in)h(a)h(formal)g(language)g(are)g(harder)f(to)i
(fol-)378 1638 y(lo)m(w)33 b(than)f(those)h(written)f(in)g(an)g
(informal)f(language.)49 b(Although)32 b(formal)g(mathematical)h
(texts,)378 1751 y(and)38 b(formal)g(pro)s(ofs)g(in)f(particular,)i
(are)g(unam)m(biguous)e(and)h(quite)g(straigh)m(tforw)m(ard)g(to)i(pro)
s(of)378 1864 y(c)m(hec)m(k)46 b(in)d(a)h(mec)m(hanical)g(fashion,)j
(they)d(are)h(v)m(ery)g(distan)m(t)f(from)f(the)i(original)d(ideas)i
(in)f(the)378 1977 y(mathematician's)32 b(mind.)42 b(F)-8
b(ormalisation)31 b(is)g(often)h(accused)g(of)g(remo)m(ving)f(all)g(in)
m(tuition)e(from)i(a)378 2090 y(mathematical)f(exp)s(osition.)39
b(Ho)m(w)m(ev)m(er,)32 b(w)m(e)e(stress)g(that,)g(in)f(general,)h(the)g
(main)e(aim)h(of)h(formali-)378 2203 y(sation)d(is)g(not)g(to)h(comm)m
(unicate)g(suc)m(h)g(in)m(tuitiv)m(e)e(concepts)i(stored)f(in)g(a)g
(mathematician's)h(mind,)378 2316 y(but)35 b(to)h(pro)s(duce)f(precise)
g(and)f(rigorous)h(mathematics)h(whic)m(h)f(usually)e(has)i(to)i(b)s(e)
d(c)m(hec)m(k)m(ed)k(b)m(y)378 2429 y(mac)m(hine.)h(This)24
b(is)i(required)e(when)h(the)i(correctness)g(of)f(a)h(particular)e(pro)
s(of)g(is)g(a)i(ma)5 b(jor)26 b(concern.)378 2542 y(An)41
b(example)h(of)g(suc)m(h)f(pro)s(ofs)g(is)g(those)i(whic)m(h)d(deriv)m
(e)h(certain)h(prop)s(erties)e(of)i(safet)m(y-critical)378
2655 y(computer)30 b(systems.)519 2768 y(The)36 b(implemen)m(tation)e
(of)j(mec)m(hanised)e(pro)s(ofs)g(in)g(a)h(format)h(that)g(is)e(easily)
g(follo)m(w)m(ed)h(b)m(y)f(a)378 2880 y(h)m(uman)42 b(reader)h(is,)i
(ho)m(w)m(ev)m(er,)i(v)m(ery)c(desirable.)76 b(Apart)43
b(from)f(b)s(eing)g(able)g(to)h(follo)m(w)f(a)i(pro)s(of)378
2993 y(for)32 b(its)g(o)m(wn)g(sak)m(e,)i(the)e(abilit)m(y)f(to)i
(understand)d(pro)s(ofs)h(easily)h(is)f(v)m(ery)i(imp)s(ortan)m(t)e
(during)f(their)378 3106 y(implemen)m(tation.)46 b(It)33
b(is)f(m)m(uc)m(h)h(easier)g(to)g(correct)h(errors)e(in)g(readable)g
(pro)s(ofs,)h(for)f(instance.)48 b(It)378 3219 y(is)30
b(also)i(easier)f(to)h(mo)s(dify)e(a)h(pro)s(of)g(that)h(can)g(b)s(e)e
(follo)m(w)m(ed)h(easily)g(in)f(order)h(to)h(deriv)m(e)f(a)h(sligh)m
(tly)378 3332 y(di\013eren)m(t)f(theorem.)47 b(This)30
b(is)h(often)h(the)g(case)i(during)29 b(mec)m(hanisation.)46
b(The)31 b(formal)g(de\014nitions)378 3445 y(and)25 b(the)g(statemen)m
(ts)i(of)f(certain)f(prop)s(erties)f(ma)m(y)i(c)m(hange)g(sligh)m(tly)e
(during)f(the)i(implemen)m(tation)378 3558 y(due)30 b(to)h(o)m(v)m
(ersigh)m(ts)h(from)e(the)h(pro)s(of)e(dev)m(elop)s(er.)41
b(Understanding)29 b(someone)i(else's)g(pro)s(of)e(is)h(also)378
3671 y(imp)s(ortan)m(t)42 b(when)g(a)i(team)g(of)f(p)s(eople)f(are)i
(engaged)g(in)e(the)h(mec)m(hanisation)g(of)g(a)h(particular)378
3784 y(theory)-8 b(.)519 3897 y(It)29 b(is)g(our)f(aim)h(to)h(in)m(v)m
(estigate)g(w)m(a)m(ys)g(of)f(pro)s(ducing)e(pro)s(ofs)h(whic)m(h)g
(can)i(b)s(e)e(mac)m(hine)h(c)m(hec)m(k)m(ed)378 4010
y(as)j(w)m(ell)f(as)i(easily)e(follo)m(w)m(ed)g(b)m(y)h(a)h(h)m(uman)e
(reader.)46 b(W)-8 b(e)33 b(remark)f(that)g(this)f(aim)h(is)f(only)g(a)
h(small)378 4122 y(requiremen)m(t)e(for)g(the)h(implemen)m(tation)f(of)
h(h)m(uman-readable)f(formalised)f(mathematical)i(texts,)378
4235 y(whic)m(h)f(apart)h(from)g(the)h(form)m(ulation)e(and)g(pro)s(of)
h(of)g(theorems,)h(also)f(include)e(the)j(in)m(tro)s(duction)378
4348 y(of)38 b(formal)f(de\014nitions)f(and)i(the)g(implemen)m(tation)e
(of)j(pro)s(of)e(pro)s(cedures.)63 b(F)-8 b(or)38 b(instance,)i(it)e
(is)378 4461 y(imp)s(ortan)m(t)27 b(that)i(formal)f(statemen)m(ts)h
(and)f(de\014nitions)e(are)i(easily)g(understo)s(o)s(d)e(so)i(that)h
(one)g(can)378 4574 y(b)s(e)h(sure)f(that)i(they)g(corresp)s(ond)e(to)i
(the)g(in)m(tended)e(mathematical)i(concepts.)378 4818
y FG(2.5.1)112 b(The)38 b(Unreadabilit)m(y)e(of)i(Mec)m(hanised)g(Pro)s
(ofs)378 4989 y FT(There)24 b(are)h(t)m(w)m(o)g(imp)s(ortan)m(t)f
(kinds)e(of)j(limitations)d(on)i(the)h(readabilit)m(y)-8
b(,)24 b(as)h(w)m(ell)e(as)i(the)g FI(writability)378
5102 y FT(\(ease)32 b(of)e(implemen)m(tation\),)g(of)g(mec)m(hanised)g
(pro)s(ofs:)514 5290 y FN(\017)46 b FT(limitations)30
b(due)h(to)i FI(formalisation)42 b FT(whic)m(h)30 b(dictates)j(that)g
(ev)m(ery)f(construct)h(in)d(the)j(pro)s(of)605 5403
y(language)e(has)f(a)h(precise)f(meaning,)514 5590 y
FN(\017)46 b FT(and)41 b(the)g(limitations)e(due)i(to)h(the)f(fact)i
(that)f(the)f(pro)s(ofs)f(are)i(required)e(to)i(b)s(e)e
FI(che)-5 b(cke)g(d)605 5703 y(by)34 b(machine)p FT(,)e(and)f
(therefore)h(the)g(pro)s(of)f(language)g(dep)s(ends)f(on)h(what)h(can)f
(b)s(e)g(e\016cien)m(tly)p eop
%%Page: 21 31
21 30 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(21)605 396 y(parsed)30 b(and)g(pro)s(of)f(c)m(hec)m(k)m(ed.)519
584 y(In)j(this)g(section,)i(w)m(e)f(ha)m(v)m(e)h(a)f(lo)s(ok)g(at)g
(these)h(t)m(w)m(o)g(limitations,)d(and)h(what)h(is)f(required)f(for)h
(a)378 697 y(mec)m(hanised)f(pro)s(of)f(to)i(b)s(e)e(easily)g(understo)
s(o)s(d)g(b)m(y)h(a)g(h)m(uman)f(reader.)43 b(T)-8 b(o)m(w)m(ards)32
b(the)f(end)g(of)g(this)378 810 y(section,)g(w)m(e)g(men)m(tion)f(the)g
(issue)f(of)i(the)f(in)m(tro)s(duction)f(of)h(notation)h(b)m(y)f
(mathematicians.)378 1050 y FQ(Unreadabilit)m(y)35 b(due)g(to)g(F)-9
b(ormalisation)378 1222 y FT(As)39 b(explained)e(earlier)h(this)g
(section,)k(it)d(is)f(hard)g(to)i(comm)m(unicate)g(mathematical)f
(ideas)g(in)e(a)378 1335 y(formal)28 b(language)g(b)s(ecause)h(of)f
(the)h(di\013erence)f(b)s(et)m(w)m(een)h(the)g(w)m(a)m(ys)g(that)g(a)g
(concept)g(is)f(visualised)378 1447 y(b)m(y)40 b(mathematicians)f(and)h
(the)g(w)m(a)m(ys)h(that)f(it)g(can)g(b)s(e)g(represen)m(ted)f
(formally)-8 b(.)69 b(Giv)m(en)40 b(an)g(un-)378 1560
y(derstanding)34 b(of)i(a)f(mathematical)h(concept,)i(a)e(h)m(uman)f
(reader)g(can)h(easily)f(infer)f(certain)h(basic)378
1673 y(statemen)m(ts)g(without)d(considering)f(a)j(formal)e(deductiv)m
(e)h(pro)s(of.)48 b(F)-8 b(or)34 b(example,)g(one)f(can)h(easily)378
1786 y(accept)29 b(that)g(the)f(union)e(of)j(t)m(w)m(o)g(\014nite)e
(sets)h(is)f(\014nite)g(giv)m(en)h(a)h(reasonable)e(visualisation)f(of)
i(\014nite)378 1899 y(sets)36 b(and)e(of)i(the)g(notion)e(of)i(union.)
54 b(On)34 b(the)i(other)f(hand,)h(a)g(formal)f(pro)s(of)f(of)i(this)e
(statemen)m(t)378 2012 y(w)m(ould)25 b(in)m(v)m(olv)m(e)h(a)h(rigorous)
e(argumen)m(t)i(in)m(v)m(olving)e(the)h(precise)g(de\014nition)e(of)i
(sets,)i(\014niteness)c(and)378 2125 y(union.)37 b(F)-8
b(urthermore,)27 b(h)m(uman)f(b)s(eings)e(are)i(capable)g(of)h
(understanding)c(the)j(precise)f(meaning)h(of)378 2238
y(an)34 b(informal)d(argumen)m(t)j(despite)f(it)g(b)s(eing)g(p)s(oten)m
(tially)f(am)m(biguous.)50 b(They)33 b(mak)m(e)i(use)e(of)h(their)378
2351 y(abilities)c(to)k(generalise)e(a)h(statemen)m(t)i(correctly)e
(giv)m(en)g(enough)f(evidence,)i(to)f(sp)s(ot)f(similarities)378
2464 y(b)s(et)m(w)m(een)h(concepts,)h(to)f(infer)e(what)h(is)f(in)m
(tended)h(\(rather)g(than)g(what)g(is)g(actually)g(said\))f(and)h(to)
378 2577 y(use)e(their)f(kno)m(wledge)i(and)f(exp)s(erience)f
(e\013ectiv)m(ely)-8 b(.)519 2689 y(During)24 b(the)i(writing)d(of)j(a)
f(pro)s(of,)h(authors)f(of)h(informal)d(mathematics)j(can)g(therefore)f
(rely)g(on)378 2802 y(their)h(reader's)i(abilit)m(y)e(to)i(infer)e(kno)
m(wledge)h(from)g(her)g(understanding)e(of)j(a)g(mathematical)f(con-)
378 2915 y(cept,)k(and)f(the)h(ab)s(o)m(v)m(e)h(men)m(tioned)e
(abilities)e(to)j(gain)f(understanding)e(through)i(`non-deductiv)m(e')
378 3028 y(means.)51 b(They)33 b(can)h(also)g(fo)s(cus)f(on)h(these)g
(abilities)d(in)h(order)i(to)g(mak)m(e)h(their)e(exp)s(osition)f
(easier)378 3141 y(to)k(follo)m(w.)54 b(On)34 b(the)h(other)h(hand,)f
(authors)g(of)g(formal)f(pro)s(ofs)g(can)i(only)e(rely)g(on)h(the)g
(precisely)378 3254 y(de\014ned)30 b(constructs)i(of)g(the)g(formal)f
(language.)46 b(In)31 b(this)f(case,)k(all)c(concepts)j(are)f(represen)
m(ted)g(as)378 3367 y(sym)m(b)s(olic)24 b(expressions)g(and)h(all)f
(inferences)g(are)i(reduced)f(to)h(the)g(sym)m(b)s(olic)d
(manipulations)g(giv)m(en)378 3480 y(b)m(y)k(a)h(sound)e(deductiv)m(e)h
(system.)40 b(Because)29 b(of)e(this,)h(argumen)m(ts)f(whic)m(h)f(can)i
(b)s(e)f(expressed)g(easily)378 3593 y(in)i(informal)g(mathematics)i
(and)f(whic)m(h)f(are)i(easily)f(follo)m(w)m(ed)g(b)m(y)h(a)g(h)m(uman)
e(reader)i(can)g(b)s(e)f(hard)378 3706 y(to)k(express)g(formally)-8
b(.)50 b(As)33 b(a)h(result,)g(formal)f(pro)s(ofs)g(are)h(generally)f
(to)s(o)i(detailed,)f(in)e(the)i(sense)378 3819 y(that)k(they)f(con)m
(tain)h(details)f(whic)m(h)f(h)m(uman)g(readers)h(can)h(easily)f(infer)
e(without)i(di\016cult)m(y)f(but)378 3931 y(whose)30
b(deriv)-5 b(ation)29 b(in)g(the)i(formal)e(language)i(is)f(not)g
(trivially)e(expressible.)519 4044 y(One)i(can)h(argue)h(that)f(the)g
(c)m(haracteristics)g(of)g(informal)e(pro)s(ofs)h(whic)m(h)f(mak)m(e)j
(them)f(easy)g(to)378 4157 y(follo)m(w)h(and)h(to)h(accept)h(are)e
(those)h(whic)m(h)e(can)i(p)s(oten)m(tially)e(in)m(tro)s(duce)g
(errors.)49 b(Mathematics)34 b(is)378 4270 y(k)m(ept)d(aliv)m(e)f(b)m
(y)h(the)f(p)s(eople)g(who)g(practice)h(it)f(and)f(k)m(eep)j(on)e
(re\014ning)f(de\014nitions,)f(\014lling)g(in)h(gaps)378
4383 y(in)35 b(argumen)m(ts,)i(and)f(correcting)g(errors.)56
b(The)36 b(formalisation)e(of)i(a)g(mathematical)g(theory)h(can)378
4496 y(b)s(e)c(seen)h(as)g(a)g(test)h(of)e(the)h(lev)m(el)g(of)g
(rigour)e(and)h(of)h(the)g(correctness)h(the)e(theory)h(has)g(ac)m
(hiev)m(ed,)378 4609 y(and)25 b(as)h(a)g(means)f(of)h(impro)m(ving)e
(this)g(lev)m(el)i(if)e(needed.)39 b(F)-8 b(urthermore,)27
b(the)f(abilit)m(y)e(to)i(formalise)e(a)378 4722 y(theory)h(requires)e
(the)i(clari\014cation)e(of)i(its)f(fundamen)m(tal)g(concepts,)i(and)e
(formalisation)f(therefore)378 4835 y(results)30 b(in)f(a)i(b)s(etter)g
(understanding)d(of)j(suc)m(h)g(concepts.)43 b(This)29
b(giv)m(es)i(another)g(reason)g(wh)m(y)f(it)h(is)378
4948 y(desirable)e(to)i(implemen)m(t)e(formal)g(pro)s(ofs)h(in)f(an)h
(easily)g(understo)s(o)s(d)e(format.)519 5061 y(The)g(implemen)m
(tation)e(of)j(formal)e(pro)s(ofs)g(in)g(a)i(h)m(uman)e(readable)g
(format)i(therefore)f(requires)378 5174 y(the)d(de\014nition)d(and)i
(use)g(of)g(inferences)g(whic)m(h)f(more)i(or)f(less)g(corresp)s(ond)f
(to)i(the)g(argumen)m(ts)g(used)378 5286 y(in)33 b(writing)f(clear)i
(informal)f(pro)s(ofs.)51 b(This)33 b(in)m(v)m(olv)m(es)h
(understanding)e(what)i(a)g(h)m(uman)g(reader)g(is)378
5399 y(able)39 b(to)h(infer)e(without)g(di\016cult)m(y)g(and)h
(deriving)e(theorems)j(and)f(rules)e(whic)m(h)i(represen)m(t)g(this)378
5512 y(abilit)m(y)-8 b(.)59 b(A)37 b(n)m(um)m(b)s(er)e(of)i(suc)m(h)f
(inference)g(rules)g(ma)m(y)h(b)s(e)f(used)g(in)g(sev)m(eral)h
(mathematical)g(theo-)378 5625 y(ries,)d(while)e(others)i(ma)m(y)g
(only)f(b)s(e)g(used)h(in)e(a)i(small)f(part)g(of)h(a)h(particular)d
(theory)-8 b(.)52 b(Iden)m(tifying)p eop
%%Page: 22 32
22 31 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(22)378 396 y(inferences)40 b(whic)m(h)g(are)i(commonly)f(used)f
(in)g(a)i(mathematical)f(theory)h(and)f(mimic)m(king)e(them)378
509 y(e\013ectiv)m(ely)32 b(in)e(a)h(formal)f(framew)m(ork)h(o\013ers)h
(an)f(extremely)g(e\013ectiv)m(e)i(to)s(ol)e(in)e(the)j(formalisation)
378 622 y(of)k(mathematics.)58 b(F)-8 b(urthermore,)37
b(and)f(more)g(imp)s(ortan)m(tly)-8 b(,)36 b(the)g(iden)m(ti\014cation)
f(of)h(these)g(rules)378 735 y(ma)m(y)42 b(o\013er)f(a)h(deep)s(er)e
(understanding)e(of)j(the)h(mathematical)f(theory)g(concerned)h(whic)m
(h)d(can-)378 848 y(not)h(b)s(e)f(ac)m(hiev)m(ed)h(through)f(informal)e
(argumen)m(ts,)43 b(or)c(naiv)m(e)h(formalisation)e(whic)m(h)g(results)
g(in)378 961 y(unreadable)29 b(pro)s(ofs.)378 1195 y
FQ(Unreadabilit)m(y)35 b(due)g(to)g(Mac)m(hine)g(Chec)m(king)378
1367 y FT(Apart)i(from)g(b)s(eing)f(unam)m(biguously)f(de\014ned,)j
(the)g(inferences)f(whic)m(h)f(can)i(b)s(e)e(used)h(in)f(mec)m(h-)378
1479 y(anised)g(pro)s(ofs)h(are)h(also)f(required)f(to)i(b)s(e)e
(e\016cien)m(tly)i(c)m(hec)m(k)m(ed)h(b)m(y)e(mac)m(hine.)62
b(In)36 b(other)i(w)m(ords,)378 1592 y(ev)m(en)e(though)f(one)g(can)g
(de\014ne)g(a)g(formal)f(inference)h(rule)f(whic)m(h)f(corresp)s(onds)h
(to)i(a)f(commonly)378 1705 y(used)26 b(informal)f(one,)j(its)f(use)f
(in)g(the)h(mec)m(hanisation)g(of)g(mathematics)g(dep)s(ends)e(on)i
(whether)g(the)378 1818 y(problem)32 b(of)h(c)m(hec)m(king)h(the)f(v)-5
b(alidit)m(y)32 b(of)i(instances)e(of)i(this)e(inference)g(is)h
(tractable.)50 b(T)-8 b(ec)m(hniques)378 1931 y(used)21
b(in)g(automated)j(deduction)d(for)g(the)i(implemen)m(tation)e(of)h
(e\016cien)m(t)h(decision)d(pro)s(cedures)h(ma)m(y)378
2044 y(therefore)i(need)g(to)g(b)s(e)f(used)g(in)g(pro)s(ducing)e(h)m
(uman)i(readable)h(mec)m(hanised)f(pro)s(ofs.)37 b(The)22
b(problem)378 2157 y(domains)31 b(usually)f(considered)i(in)f
(automated)j(deduction,)e(ho)m(w)m(ev)m(er,)i(are)f(di\013eren)m(t)f
(from)g(those)378 2270 y(in)m(v)m(olv)m(ed)d(in)f(this)g(case.)41
b(Instead)29 b(of)h(lo)s(oking)e(for)h(pro)s(ofs)f(of)h(p)s(ossibly)e
(non-trivial)g(theorems,)j(the)378 2383 y(required)c(algorithms)i(ha)m
(v)m(e)h(to)g(b)s(e)f(designed)f(to)i(\014ll)d(in)h(the)i(gaps)f(b)s
(et)m(w)m(een)h(pro)s(ofs)e(of)i(a)g(rigorous,)378 2496
y(y)m(et)i(easy)g(to)h(follo)m(w,)d(argumen)m(ts.)519
2609 y(Ho)m(w)m(ev)m(er,)46 b(most)41 b(of)h(the)f(curren)m(t)g(pro)s
(of)f(languages)h(and)g(inference)f(systems)h(used)f(in)g(the)378
2721 y(mec)m(hanisation)23 b(of)g(mathematics)h(are)g(not)f(orien)m
(ted)h(to)m(w)m(ards)g(the)f(dev)m(elopmen)m(t)h(of)f(h)m(uman)g(read-)
378 2834 y(able)30 b(pro)s(ofs.)40 b(They)30 b(are)g(instead)g
(designed)f(for)h(other)h(purp)s(oses,)e(whic)m(h)g(include:)514
2989 y FN(\017)46 b FT(E\016cien)m(t)25 b(pro)s(of)f(searc)m(h:)39
b(The)24 b(deductiv)m(e)g(systems)h(of)g(automated)h(deduction)e(pro)s
(cedures,)605 3102 y(suc)m(h)36 b(as)h(those)g(based)f(on)g(the)g
(resolution)f(principle)e(and)j(the)g(connection)h(metho)s(d,)g(are)605
3215 y(searc)m(h-orien)m(ted.)42 b(The)30 b(pro)s(ofs)f(found)g(b)m(y)i
(suc)m(h)f(systems)g(are)h(v)m(ery)f(di\013eren)m(t)g(in)f(structure)
605 3328 y(to)i(those)g(found)e(in)g(mathematical)i(texts.)514
3502 y FN(\017)46 b FT(In)m(teractiv)m(e)30 b(pro)s(of)d(disco)m(v)m
(ery:)40 b(The)28 b(pro)s(ofs)f(implemen)m(ted)g(in)g(suc)m(h)h(a)g
(pro)s(of)g(language)g(are)605 3615 y(made)21 b(up)e(of)i(the)g(user)f
(in)m(teractions)g(required)f(to)j(deriv)m(e)e(the)h(result.)36
b(The)20 b(user)g(in)m(teractions)605 3728 y(c)m(hange)34
b(the)e(state)h(of)g(the)f(pro)s(of)f(dev)m(elopmen)m(t)i(en)m
(vironmen)m(t)f(un)m(til)e(a)j(complete)f(pro)s(of)g(is)605
3841 y(found.)37 b(In)22 b(general,)i(it)e(is)g(not)h(p)s(ossible)d(to)
j(follo)m(w)f(suc)m(h)h(a)g(list)e(of)i(user)e(in)m(teractions)i
(without)605 3954 y(seeing)30 b(their)g(e\013ect)i(on)e(the)h(state)g
(of)g(the)f(system.)514 4129 y FN(\017)46 b FT(Chec)m(k)-5
b(able)26 b(b)m(y)g(a)g(simple)e(algorithm:)38 b(An)25
b(example)h(of)g(suc)m(h)f(pro)s(ofs)h(are)g(the)g(pro)s(of)f(ob)5
b(jects)605 4241 y(in)33 b(the)g(theorem)h(pro)m(ving)f(systems)h(Co)s
(q)f(and)g(LEGO.)h(Suc)m(h)f(pro)s(ofs)f(can)i(b)s(e)f(c)m(hec)m(k)m
(ed)j(b)m(y)605 4354 y(a)28 b(t)m(yp)s(e-c)m(hec)m(king)h(algorithm)d
(whose)h(implemen)m(tation)f(is)g(simple)f(and)i(easily)f(understo)s(o)
s(d.)605 4467 y(Pro)s(ofs)k(of)h(this)e(kind)f(can)j(b)s(e)f(to)s(o)h
(detailed)e(to)i(b)s(e)f(follo)m(w)m(ed)g(easily)g(b)m(y)g(a)h(h)m
(uman)e(reader.)378 4622 y(W)-8 b(e)40 b(shall)d(see)j(in)e(section)h
(2.5.2)i(b)s(elo)m(w)d(that)i(there)f(is)f(ongoing)h(researc)m(h)h(in)d
(automating)j(the)378 4735 y(transformation)34 b(of)i(pro)s(ofs)e(in)g
(suc)m(h)h(inference)f(systems)h(in)m(to)g(h)m(uman)g(readable)f(pro)s
(of)h(scripts.)378 4848 y(An)27 b(adv)-5 b(an)m(tage)29
b(of)f(suc)m(h)f(an)g(approac)m(h)h(is)f(to)h(use)f(pro)s(of)g
(languages)g(orien)m(ted)h(to)m(w)m(ards)g(the)g(ab)s(o)m(v)m(e)378
4961 y(men)m(tioned)j(purp)s(oses,)e(and)i(still)d(b)s(e)j(able)f(to)i
(obtain)f(pro)s(ofs)f(whic)m(h)g(a)h(h)m(uman)f(can)h(follo)m(w.)42
b(The)378 5074 y(aim)28 b(of)g(our)g(researc)m(h,)i(though,)f(is)e(to)j
(study)d(the)i(p)s(ossibilit)m(y)c(of)j(dev)m(eloping)g(mec)m(hanised)g
(pro)s(ofs)378 5187 y(whic)m(h)h(can)i(b)s(e)e(easily)h(follo)m(w)m(ed)
g(b)m(y)g(h)m(umans.)378 5421 y FQ(On)k(the)h(In)m(tro)s(duction)g(of)g
(Notation)378 5592 y FT(W)-8 b(e)27 b(conclude)f(this)f(section)h(b)m
(y)g(p)s(oin)m(ting)f(out)h(that)h(one)f(factor)h(whic)m(h)e(impro)m(v)
m(es)h(the)g(readabilit)m(y)378 5705 y(of)39 b(informal)f(pro)s(ofs)g
(is)h(the)g(abilit)m(y)f(of)h(mathematicians)g(to)h(in)m(tro)s(duce)f
(new)g(notation)g(as)h(the)p eop
%%Page: 23 33
23 32 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(23)378 396 y(theory)40 b(dev)m(elops.)69 b(Appropriate)38
b(notation)j(is)d(c)m(hosen)j(to)f(represen)m(t)g(expressions)f
(compactly)-8 b(,)378 509 y(sometimes)30 b(through)e(the)i(omission)e
(of)i(information)e(whic)m(h)g(can)i(b)s(e)f(induced)e(from)i(the)h
(con)m(text)378 622 y(in)i(whic)m(h)h(the)h(expressions)e(are)i(used.)
51 b(An)33 b(example)h(of)f(this,)h(is)f(the)h(omission)e(of)i(the)g
(pro)s(duct)378 735 y(sym)m(b)s(ol)25 b(from)g(expressions)g(represen)m
(ting)g(the)h(pro)s(duct)f(of)h(t)m(w)m(o)h(group)e(elemen)m(ts.)40
b(Since,)26 b(expres-)378 848 y(sions)37 b(in)h(a)g(formal)g(language)h
(m)m(ust)g(ha)m(v)m(e)g(an)g(unam)m(biguous)d(meaning,)k(suc)m(h)f
(omissions)d(ma)m(y)378 961 y(not)c(b)s(e)e(p)s(ossible)f(b)s(ecause)i
(they)h(can)f(in)m(tro)s(duce)g(am)m(biguit)m(y)-8 b(.)43
b(The)31 b(juxtap)s(osition)e(of)j(t)m(w)m(o)g(group)378
1074 y(elemen)m(ts)f(is)e(am)m(biguous)h(if)f(there)i(are)f(t)m(w)m(o)i
(p)s(ossible)c(pro)s(ducts)h(whic)m(h)g(can)i(b)s(e)e(used.)519
1187 y(Appropriate)k(notation)h(is)f(also)h(in)m(tro)s(duced)e(in)h
(informal)e(theories)j(to)h(facilitate)e(reasoning)378
1300 y(on)24 b(certain)g(ob)5 b(jects.)39 b(By)24 b(omitting)f(the)h
(paren)m(theses)g(in)f(represen)m(ting)g(the)h(pro)s(duct)f(of)h(a)g(n)
m(um)m(b)s(er)378 1413 y(of)h(group)f(elemen)m(ts)h(one)g(can)g(infer)e
(the)i(equalit)m(y)g(of)g(t)m(w)m(o)h(suc)m(h)e(expressions)g(syn)m
(tactically)-8 b(,)26 b(rather)378 1526 y(than)k(through)g(the)g(rep)s
(etitiv)m(e)g(application)f(of)h(the)h(asso)s(ciativ)m(e)g(la)m(w.)519
1638 y(The)38 b(abilit)m(y)e(to)j(omit)f(information)f(without)g
(danger)h(of)g(am)m(biguit)m(y)g(and)f(to)i(enhance)g(the)378
1751 y(grammar)30 b(of)f(a)h(formal)f(language)h(through)f(the)h(in)m
(tro)s(duction)d(of)j(theory-sp)s(eci\014c)f(notation)h(is)e(a)378
1864 y(desirable)20 b(feature)j(in)e(the)h(mec)m(hanisation)g(of)g
(mathematics.)39 b(Issues)21 b(regarding)h(whether)f(one)i(can)378
1977 y(safely)f(extend)g(the)h(term)f(language)h(of)f(a)h(pro)s(of)e
(dev)m(elopmen)m(t)i(system)f(in)f(order)h(to)h(in)m(tro)s(duce)e(new)
378 2090 y(notation)32 b(are)g(not)f(considered)g(in)f(this)g(thesis,)h
(although)g(w)m(e)h(p)s(oin)m(t)f(out)g(that)i(this)d(is)g(necessary)
378 2203 y(for)g(the)h(minimisation)c(of)j(the)h(di\013erence)f(b)s(et)
m(w)m(een)h(formal)e(and)h(informal)e(texts.)378 2446
y FG(2.5.2)112 b(Extracting)36 b(Natural)h(Language)i(Pro)s(ofs)f(from)
f(Mec)m(hanised)h(Ones)378 2618 y FT(In)28 b(the)g(previous)f(section)i
(w)m(e)g(stated)g(that)h(the)e(mec)m(hanisation)g(of)h(pro)s(ofs)f(is)f
(usually)f(p)s(erformed)378 2731 y(using)35 b(inference)g(systems)h
(and)g(pro)s(of)g(languages)g(designed)f(for)h(e\016cien)m(t)h(pro)s
(of)e(searc)m(h,)k(in)m(ter-)378 2844 y(activ)m(e)e(pro)s(of)e(disco)m
(v)m(ery)-8 b(,)38 b(or)e(to)h(b)s(e)e(capable)h(of)g(b)s(eing)e(c)m
(hec)m(k)m(ed)k(b)m(y)e(a)g(simple)e(algorithm.)56 b(The)378
2957 y(pro)s(ofs)24 b(dev)m(elop)s(ed)g(in)g(suc)m(h)h(framew)m(orks)f
(are)i(not)f(easily)f(follo)m(w)m(ed)h(b)m(y)f(h)m(umans,)i(ho)m(w)m
(ev)m(er)g(certain)378 3070 y(systems)33 b(o\013er)h(the)f(p)s
(ossibilit)m(y)d(of)j(extracting)h(a)g(natural)e(language)i(pro)s(of)e
(from)h(their)f(in)m(ternal)378 3183 y(pro)s(of)e(represen)m(tation.)
519 3296 y(Cosco)m(y)-8 b(,)33 b(Hahn,)e(and)f(Th)m(\023)-43
b(ery)31 b(\(1997\))i(ha)m(v)m(e)g(dev)m(elop)s(ed)d(an)h(algorithm,)f
(whic)m(h)f(w)m(as)j(later)f(im-)378 3408 y(pro)m(v)m(ed)c(b)m(y)g
(Cosco)m(y)i(\(1997\),)h(to)e(translate)f(Co)s(q)g(pro)s(ofs)f(in)m
(ternally)f(represen)m(ted)i(in)f(the)h(Calculus)378
3521 y(of)39 b(Inductiv)m(e)f(Constructions)g(in)m(to)h(English)d
(text.)67 b(In)39 b(order)f(to)i(impro)m(v)m(e)e(the)i(qualit)m(y)e(of)
h(the)378 3634 y(resulting)29 b(texts,)i(certain)f FI(wel)5
b(l-known)39 b FT(inferences)29 b(are)i(omitted.)41 b(These)30
b(include)e(the)j(unfolding)378 3747 y(of)i(w)m(ell-kno)m(wn)f(constan)
m(ts)i(and)f(the)g(in)m(tro)s(duction)e(and)i(elimination)d(of)j(w)m
(ell-kno)m(wn)f(inductiv)m(e)378 3860 y(de\014nitions.)64
b(The)38 b(user)g(can)i(declare)e(whic)m(h)g(constan)m(ts)i(and)e
(inductiv)m(e)g(de\014nitions)e(are)k(w)m(ell-)378 3973
y(kno)m(wn.)519 4086 y(Another)h(system)h(dev)m(elop)s(ed)f(for)g(the)h
(v)m(erbalisation)e(of)h(pro)s(ofs)g(is)f(PR)m(O)m(VERB)j(whic)m(h)d
(is)378 4199 y(em)m(b)s(edded)32 b(in)g(the)i(\012mega)g(pro)s(of)e
(dev)m(elopmen)m(t)i(en)m(vironmen)m(t)f(\(Benzm)s(\177)-48
b(uller)32 b(et)i(al.)49 b(1997\).)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
(hine)g(c)m(hec)m(k)-5 b(able)30 b(pro)s(ofs)f(can)h(b)s(e)e(seen)i(as)
f(the)h(in)m(v)m(erse)378 5215 y(pro)s(cess)39 b(of)g(pro)s(of)f(v)m
(erbalisation:)57 b(pro)s(ofs)38 b(are)h(implemen)m(ted)f(at)i(a)f
(high)f(lev)m(el)h(of)g(abstraction)378 5328 y(and)e(then)g
(transformed)g(in)m(to)h(lo)m(w-lev)m(el)f(inferences)g(for)g(pro)s(of)
g(c)m(hec)m(king.)63 b(An)37 b(imp)s(ortan)m(t)g(dif-)378
5441 y(ference)30 b(b)s(et)m(w)m(een)g(these)g(t)m(w)m(o)h(pro)s
(cesses)e(is)f(that)i(the)g(high-lev)m(el)e(mac)m(hine)h(c)m(hec)m(k)-5
b(able)31 b(pro)s(ofs)d(are)378 5554 y(necessarily)h(formal,)h(while)e
(high-lev)m(el)i(`extracted')i(pro)s(ofs)d(ma)m(y)i(b)s(e)f(informal.)p
eop
%%Page: 24 34
24 33 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(24)378 396 y FG(2.5.3)112 b(Impro)m(ving)36 b(the)i(Readabilit)m
(y)d(of)j(Mec)m(hanised)g(Pro)s(ofs)378 568 y FT(In)45
b(this)f(section)i(w)m(e)g(ha)m(v)m(e)h(a)f(lo)s(ok)f(at)h(e\013orts)g
(at)h(impro)m(ving)d(the)h(readabilit)m(y)f(of)i(the)g(input)378
681 y(language)39 b(of)g(mec)m(hanised)f(pro)s(ofs.)65
b(Suc)m(h)38 b(e\013orts)h(range)g(from)g(the)g(inclusion)c(of)k
(explanatory)378 794 y(information)27 b(to)i(help)f(h)m(uman)f(readers)
i(understand)e(ho)m(w)h(pro)s(ofs)g(w)m(ork,)h(to)h(the)f(dev)m
(elopmen)m(t)g(of)378 907 y(pro)s(of)h(languages)g(and)g(en)m(vironmen)
m(ts)g(in)f(whic)m(h)g(pro)s(ofs)g(are)i(easier)g(to)g(follo)m(w.)378
1147 y FQ(Presen)m(ting)36 b(Pro)s(ofs)g(in)f(a)f(Hierarc)m(hical)i
(Structure)378 1319 y FT(Lamp)s(ort)h(\(1995\))j(notes)e(that)g
(expressing)e(form)m(ulae)i(and)f(pro)s(ofs)f(in)g(a)i(format)g(whic)m
(h)f(rev)m(eals)378 1431 y(their)c(structure)h(usually)e(mak)m(es)j
(them)f(easier)g(to)h(understand)d(and)i(less)f(am)m(biguous.)51
b(He)35 b(pro-)378 1544 y(p)s(oses)f(a)h(st)m(yle)f(for)h(writing)d
(\(informal\))h(pro)s(ofs)h(in)f(whic)m(h)g(their)h(hierarc)m(hical)f
(structure)h(is)f(pre-)378 1657 y(sen)m(ted)k(explicitly)-8
b(.)57 b(A)37 b(pro)s(of)e(is)h(presen)m(ted)g(as)h(an)f(en)m(umerated)
h(sequence)g(of)g(steps)f(whic)m(h)f(are)378 1770 y(themselv)m(es)k
(justi\014ed)f(b)m(y)h(more)g(detailed)f(pro)s(ofs.)66
b(A)40 b(similar)c(format)k(is)e(prop)s(osed)g(b)m(y)h(Bac)m(k,)378
1883 y(Grundy)-8 b(,)39 b(and)e(v)m(on)h(W)-8 b(righ)m(t)39
b(\(1996\))h(where)e FI(c)-5 b(alculational)42 b(pr)-5
b(o)g(ofs)47 b FT(\(see)39 b(\(Gries)f(and)f(Sc)m(hneider)378
1996 y(1995\)\))c(are)e(presen)m(ted)f(in)f(a)i(nested)f(hierarc)m
(hical)f(structure.)519 2109 y(Hierarc)m(hical)e(and)f(calculational)g
(pro)s(of)h(formats)g(can)h(also)f(b)s(e)f(used)h(in)f(the)h(implemen)m
(tation)378 2222 y(and)k(represen)m(tation)h(of)g(formal)f(pro)s(ofs.)
45 b(Praset)m(y)m(a)33 b(\(1993\))i(implemen)m(ted)30
b(t)m(w)m(o)j(pac)m(k)-5 b(ages)34 b(based)378 2335 y(on)k(the)g
(tactic-based)g(pro)s(of)f(en)m(vironmen)m(t)h(of)g(HOL.)f(One)h(pac)m
(k)-5 b(age)39 b(allo)m(ws)e(the)h(deriv)-5 b(ation)37
b(of)378 2448 y(calculational)30 b(st)m(yle)h(pro)s(ofs)f(through)g
(iterativ)m(e)i(equalities)e(and)g(inequalities)e(justi\014ed)h(b)m(y)i
(HOL)378 2561 y(tactics.)42 b(The)30 b(other)g(pac)m(k)-5
b(age)33 b(allo)m(ws)c(the)i(deriv)-5 b(ation)29 b(of)h(pro)s(ofs)g(as)
h(a)f(sequence)h(of)g(lemmas.)519 2673 y(Grundy)24 b(and)g(L)-11
b(\027)-57 b(angbac)m(k)-5 b(a)27 b(\(1997\))h(dev)m(elop)s(ed)d(to)s
(ols)g(for)g(recording)f(HOL)h(pro)s(ofs)f(in)g(a)i(bro)m(ws-)378
2786 y(able)e(hierarc)m(hical)f(format)i(similar)d(to)j(the)g(hierarc)m
(hical)e(calculational)h(pro)s(ofs)f(of)i(Bac)m(k,)j(Grundy)-8
b(,)378 2899 y(and)25 b(v)m(on)i(W)-8 b(righ)m(t)26 b(\(1996\).)42
b(Theorems)25 b(are)i(deriv)m(ed)e(in)m(teractiv)m(ely)h(using)e(the)i
(windo)m(ws)f(inference)378 3012 y(st)m(yle)j(of)g(reasoning)g
(\(Robinson)f(and)g(Staples)g(1993;)k(Grundy)c(1996\).)42
b(The)27 b(resulting)f(pro)s(ofs)i(can)378 3125 y(then)36
b(b)s(e)g(presen)m(ted)h(in)f(a)h(bro)m(wsable)f(format)h(whic)m(h)e
(allo)m(ws)h(the)h(user)f(to)i(c)m(ho)s(ose)f(the)g(lev)m(el)g(of)378
3238 y(detail)29 b(at)j(whic)m(h)d(particular)f(pro)s(of)i(step)g
(justi\014cations)f(are)i(sho)m(wn.)378 3478 y FQ(Explaining)k(Pro)s
(of)h(Scripts)378 3650 y FT(Kalv)-5 b(ala)40 b(\(1994\))j(illustrates)
38 b(the)j(use)f(of)h(annotations)f(on)h(HOL)f(terms)g(and)g(pro)s(ofs)
g(to)h(carry)378 3763 y(information)j(of)i(an)f(informal)f(nature.)86
b(Suc)m(h)44 b(information)g(can)i(consist)f(of)h(hin)m(ts)f(to)h
(guide)378 3876 y(the)39 b(user)f(during)e(in)m(teractiv)m(e)k(pro)s
(of)e(disco)m(v)m(ery)h(and)f(as)h(an)g(explanatory)f(aid.)65
b(F)-8 b(or)40 b(example,)378 3988 y(HOL)f(constan)m(ts)h(can)g(b)s(e)e
(annotated)j(with)d(a)h(text)i(giving)d(an)h(informal)e(description)g
(of)j(their)378 4101 y(b)s(eha)m(viour.)56 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 b(follo)m(w)m(ed)h(and)f(also)h(easily)f
(mac)m(hine)h(c)m(hec)m(k)m(ed.)62 b(The)37 b(actual)g(de\014nition)e
(of)i(ob)m(viousness)378 509 y(in)f(Mizar)i(is)f(giv)m(en)h(through)f
(the)h(pro)s(of)f(c)m(hec)m(king)i(algorithm)d(implemen)m(ted)h(in)f
(its)h(v)-5 b(alidator.)378 622 y(Exp)s(erience)29 b(in)f(mec)m
(hanising)h(mathematics)h(in)f(Mizar)h(suggests)g(that)h(pro)s(of)e(c)m
(hec)m(king)i(sp)s(eed)e(is)378 735 y(giv)m(en)h(more)h(imp)s(ortance)e
(than)i(p)s(o)m(w)m(er)f(\(Rudnic)m(ki)e(1992\).)519
848 y(The)41 b(deductiv)m(e)g(p)s(o)m(w)m(er)g(of)h(the)f(pro)s(of)g(c)
m(hec)m(k)m(er)i(of)f(Mizar)f(do)s(es)g(not)g(increase)h(during)d(the)
378 961 y(dev)m(elopmen)m(t)24 b(of)g(a)g(particular)e(mathematical)i
(theory)-8 b(,)26 b(and)d(therefore)h(the)g(de\014nition)e(of)h(ob)m
(vious)378 1074 y(inferences)34 b(is)h(\014xed.)55 b(This)33
b(is)i(not)g(consisten)m(t)h(with)e(the)h(notion)g(of)h(what)f(is)f
(considered)g(to)j(b)s(e)378 1187 y(ob)m(vious)20 b(during)e(the)j(dev)
m(elopmen)m(t)g(of)g(informal)e(texts.)38 b(As)21 b(a)g(h)m(uman)e
(reader)i(progresses)g(through)378 1300 y(a)35 b(mathematical)f(text)i
(and)e(gains)f(understanding)f(on)i(the)h(sub)5 b(ject,)35
b(his)e(abilit)m(y)g(to)i(infer)e(facts)378 1413 y(ab)s(out)h(the)g
(concepts)h(concerned)f(increases.)52 b(Therefore,)35
b(the)g(notion)e(of)i(ob)m(viousness)e(c)m(hanges)378
1526 y(throughout)40 b(the)h(dev)m(elopmen)m(t)g(of)f(a)h(theory)-8
b(.)72 b(It)40 b(is)g(th)m(us)g(desirable)f(that)i(the)f(implemen)m
(tors)378 1638 y(of)c(mec)m(hanised)f(pro)s(ofs)g(are)h(giv)m(en)f(the)
h(p)s(ossibilit)m(y)c(to)37 b(extend)e(the)h(automation)g(p)s(o)m(w)m
(er)g(of)g(the)378 1751 y(pro)s(of)28 b(c)m(hec)m(k)m(er)j(usually)26
b(to)k(mak)m(e)f(use)g(of)g(particular)e(result)g(automatically)-8
b(.)41 b(The)28 b(Mizar)h(system)378 1864 y(lac)m(ks)d(suc)m(h)g
FI(extensibility)p FT(,)i(and)d(the)i(need)f(for)g(suc)m(h)f(a)i(prop)s
(ert)m(y)f(is)f(men)m(tioned)g(in)g(the)i(concluding)378
1977 y(remarks)d(of)h(\(Rudnic)m(ki)d(and)i(T)-8 b(rybulec)24
b(1997\).)41 b(The)24 b(future)f(w)m(ork)i(section)g(of)f(\(Syme)h
(1997a\))i(also)378 2090 y(men)m(tions)g(the)h(p)s(ossibilit)m(y)c(of)k
(making)g(DECLARE)f(extensible.)39 b(The)27 b(Mizar)h(mo)s(de)f(of)h
(Harrison)378 2203 y(\(1996b\))23 b(allo)m(ws)e(the)g(use)g(of)h
(arbitrary)e(HOL)h(tactics)h(for)f(justifying)e(pro)s(of)i(steps,)i
(and)e(is)f(therefore)378 2316 y(extensible.)60 b(The)36
b(SPL)g(language)i(describ)s(ed)d(in)h(c)m(hapter)h(4)h(is)e(implemen)m
(ted)g(on)h(top)g(of)g(HOL)378 2429 y(and)30 b(is)f(also)h(extensible)g
(though)g(it)g(do)s(es)g(not)g(rely)g(on)g(HOL)g(tactics.)p
eop
%%Page: 27 37
27 36 bop 378 1019 a FJ(Chapter)65 b(3)378 1434 y FR(Case)77
b(Studies)g(on)h(T)-19 b(actic-Based)378 1683 y(Theorem)77
b(Pro)-6 b(v)g(ers)378 2165 y FH(3.1)135 b(In)l(tro)t(duction)45
b(and)g(Motiv)-7 b(ation)378 2368 y FT(In)28 b(this)g(c)m(hapter)h(w)m
(e)h(describ)s(e)d(the)i(mec)m(hanisation)g(of)g(t)m(w)m(o)h(results)d
(from)i(the)g(theory)g(of)g(compu-)378 2481 y(tation)c(in)f(t)m(w)m(o)i
(LCF-st)m(yle)g(theorem)f(pro)m(v)m(ers:)38 b(the)26
b(HOL)e(system)h(\(see)h(section)f(2.4\))i(and)d(the)h(Co)s(q)378
2594 y(system)35 b(\(Barras)h(et)g(al.)56 b(1996\).)i(The)34
b(theory)i(of)f(computation)h(has)f(b)s(een)f(widely)g(explored)g(in)
378 2706 y(mathematical)c(and)f(computer)g(science)h(literature)e
(\(see)j(\(T)-8 b(ourlakis)28 b(1984;)k(Sommerhalder)27
b(and)378 2819 y(v)-5 b(an)37 b(W)-8 b(estrhenen)37 b(1988;)k(Cutland)
35 b(1980\)\).)63 b(The)36 b(mec)m(hanisation)g(in)g(HOL)g(includes)e
(the)j(de\014-)378 2932 y(nition)c(of)i(a)h(computable)e(function)g
(according)g(to)i(the)f(Unlimited)e(Register)i(Mac)m(hine)g(\(URM\))378
3045 y(mo)s(del)43 b(of)i(computation.)83 b(It)45 b(includes)d(a)j(pro)
s(of)f(that)h(the)g(set)g(of)f(URM)h(computable)f(func-)378
3158 y(tions)31 b(con)m(tains)h(the)g(set)h(of)f(partial)f(recursiv)m
(e)g(functions.)44 b(The)31 b(mec)m(hanisation)h(in)e(Co)s(q)i
(de\014nes)378 3271 y(computable)f(functions)f(according)i(to)g(a)h(mo)
s(del)d(based)h(on)h(the)g(de\014nition)d(of)j(partial)e(recursiv)m(e)
378 3384 y(functions,)f(and)h(includes)e(a)j(pro)s(of)e(of)i(the)f
FP(S)1955 3351 y FO(m)1950 3406 y(n)2052 3384 y FT(theorem.)519
3497 y(One)c(of)g(the)h(aims)e(of)i(these)g(mec)m(hanisations)e(is)h
(to)h(giv)m(e)g(an)f(illustration)d(of)k(ho)m(w)f(a)h(particular)378
3610 y(mathematical)33 b(theory)g(is)f(mec)m(hanised)g(using)f
(existing)h(pro)s(of)g(dev)m(elopmen)m(t)i(systems.)48
b(W)-8 b(e)33 b(are)378 3723 y(mostly)d(in)m(terested)h(in)e(the)i(pro)
s(cess)f(of)h(\014nding)d(pro)s(ofs)i(using)f(a)i(tactic-based)h(in)m
(teractiv)m(e)g(pro)s(of)378 3836 y(en)m(vironmen)m(t,)24
b(and)e(the)h(t)m(w)m(o)h(mec)m(hanisations)f(presen)m(ted)f(here)h
(mak)m(e)h(substan)m(tial)d(use)i(of)g(tactics.)378 3949
y(The)j(mec)m(hanisation)f(in)g(HOL)h(is)f(based)h(on)g(the)g(textb)s
(o)s(ok)h(of)f(Cutland)f(\(1980\),)k(and)d(therefore)g(it)378
4061 y(o\013ers)31 b(us)f(a)i(p)s(ossibilit)m(y)27 b(of)k(comparing)f
(mec)m(hanised)h(pro)s(ofs)f(with)f(their)h(informal)f(coun)m(terpart.)
378 4174 y(On)c(the)g(other)h(hand,)g(the)f(mec)m(hanisation)g(in)f(Co)
s(q)h(do)s(es)h(not)f(follo)m(w)g(an)g(existing)g(textb)s(o)s(ok.)39
b(The)378 4287 y(particular)34 b(pro)s(ofs)i(implemen)m(ted)e(in)h(Co)s
(q)h(w)m(ere)h(found)e(b)m(y)h(the)g(user)g(during)e(mec)m(hanisation)
3764 4254 y FL(1)3803 4287 y FT(.)378 4400 y(This)25
b(exercise)j(in)d(Co)s(q)i(serv)m(es)g(as)h(a)f(study)f(in)g(the)h(pro)
s(cess)g(of)g(\014nding)e(mec)m(hanical)i(pro)s(ofs)f(in)g(the)378
4513 y(absence)31 b(of)f(informal)f(ones.)519 4626 y(Another)d(aim)f
(of)h(the)f(w)m(ork)h(presen)m(ted)g(in)e(this)h(c)m(hapter)h(is)f(to)h
(compare)g(the)g(di\013eren)m(t)f(w)m(a)m(ys)h(a)378
4739 y(theory)k(is)f(mec)m(hanised)g(in)f(HOL)h(and)h(in)e(Co)s(q.)40
b(Although)29 b(b)s(oth)g(HOL)g(and)g(Co)s(q)g(are)i(LCF-st)m(yle)378
4852 y(theorem)36 b(pro)m(ving)f(systems,)i(they)f(are)g(di\013eren)m
(t)f(in)g(some)h(imp)s(ortan)m(t)f(resp)s(ects.)56 b(HOL)35
b(imple-)378 4965 y(men)m(ts)30 b(a)g(classical)e(simply-t)m(yp)s(ed)f
(higher-order)h(logic,)i(while)e(Co)s(q)h(implemen)m(ts)f(a)h
(constructiv)m(e)378 5078 y(logic)k(based)g(on)g(a)g(m)m(uc)m(h)g(ric)m
(her)f(t)m(yp)s(e)i(system.)49 b(The)32 b(di\013erence)h(in)f(the)h
(foundational)f(logic)g(af-)378 5191 y(fects)c(b)s(oth)e(the)h(w)m(a)m
(y)h(ob)5 b(jects)28 b(are)f(de\014ned)f(as)h(w)m(ell)f(as)h(the)h(w)m
(a)m(y)g(pro)s(ofs)e(are)h(dev)m(elop)s(ed.)39 b(Another)378
5303 y(di\013erence)34 b(b)s(et)m(w)m(een)h(the)g(t)m(w)m(o)h(systems)e
(is)g(that)h(HOL)f(users)g(usually)e(apply)h(ML)i(functions)e(di-)378
5416 y(rectly)25 b(during)d(the)j(dev)m(elopmen)m(t)g(of)g(a)g(theory)
-8 b(,)27 b(while)c(Co)s(q)h(users)g(dev)m(elop)h(a)g(theory)g(through)
f(the)p 378 5578 1380 4 v 482 5632 a FC(1)516 5664 y
FB(Or)i(rather,)g(re-disco)n(v)n(ered)f(b)n(y)g(the)g(user)h(since)g
(suc)n(h)f(pro)r(ofs)i(did)e(exist)h(b)r(eforehand.)2057
5954 y FT(27)p eop
%%Page: 28 38
28 37 bop 378 5 a FF(CHAPTER)30 b(3.)61 b(CASE)29 b(STUDIES)h(ON)g(T)-8
b(A)m(CTIC-BASED)30 b(THEOREM)g(PR)m(O)m(VERS)185 b FT(28)378
396 y(sp)s(eci\014cation)35 b(and)g(pro)s(of)g(language)i
FM(Gallina)p FT(.)56 b(A)36 b(comparativ)m(e)h(study)e(whic)m(h)g
(illustrates)f(the)378 509 y(e\013ect)i(of)f(the)f(di\013erences)g(of)h
(the)f(t)m(w)m(o)i(systems)f(can)f(b)s(e)g(useful)f(b)s(oth)h(to)h
(users)f(of)g(the)h(systems)378 622 y(and)30 b(to)h(dev)m(elop)s(ers)f
(of)g(theorem)h(pro)m(v)m(ers.)519 735 y(The)24 b(mec)m(hanisation)h
(in)e(HOL)i(is)f(giv)m(en)h(in)e(the)i(next)g(section)g(and)g(section)g
(3.3)h(illustrates)d(the)378 848 y(mec)m(hanisation)33
b(in)e(Co)s(q.)49 b(These)33 b(mec)m(hanisations)f(are)i(describ)s(ed)c
(in)i(more)h(detail)g(in)e(\(Zammit)378 961 y(1996\))39
b(and)d(in)g(\(Zammit)g(1997\).)62 b(The)36 b(theorem)h(pro)m(ving)f
(approac)m(hes)i(of)f(the)g(HOL)f(and)g(Co)s(q)378 1074
y(systems)41 b(are)g(compared)g(in)f(section)h(3.4,)k(and)40
b(some)h(remarks)g(on)f(the)h(tactic-based)i(st)m(yle)e(of)378
1187 y(theorem)31 b(pro)m(ving)e(are)i(giv)m(en)f(in)f(section)i(3.5.)
378 1473 y FH(3.2)135 b(A)45 b(F)-11 b(ormalisation)46
b(of)f(URM)h(Computabilit)l(y)g(in)f(HOL)378 1676 y FT(In)34
b(this)f(section)i(w)m(e)g(illustrate)e(the)h(mec)m(hanisation)g(of)h
(a)g(n)m(um)m(b)s(er)e(of)i(results)e(in)g(the)i(theory)g(of)378
1789 y(computation.)40 b(W)-8 b(e)31 b(use)e(the)g(Unlimited)e
(Register)j(Mac)m(hine)f(mo)s(del)g(of)g(computation,)h(and)f(base)378
1901 y(the)i(mec)m(hanisation)e(on)i(the)f(textb)s(o)s(ok)h(b)m(y)f
(Cutland)f(\(1980\).)378 2144 y FG(3.2.1)112 b(The)38
b(URM)g(Mo)s(del)f(of)h(Computation)e(in)h(HOL)378 2316
y FQ(The)e(Unlimited)f(Register)h(Mac)m(hine)378 2488
y FT(An)24 b(Unlimited)e(Register)j(Mac)m(hine,)i(or)d(URM,)i(consists)
e(of)h(a)g(coun)m(tably)g(in\014nite)d(set)j(of)g(registers)378
2601 y(eac)m(h)34 b(con)m(taining)e(a)i(natural)e(n)m(um)m(b)s(er.)47
b(This)31 b(set)i(of)g(registers)g(is)f(called)g(the)h
FI(memory)42 b FT(or)33 b FI(stor)-5 b(e)p FT(.)378 2713
y(The)27 b(registers)g(are)h(n)m(um)m(b)s(ered)d FP(R)1555
2727 y FL(0)1595 2713 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(R)1881 2727 y FO(n)1928 2713 y FP(;)g(:)g(:)g(:)i FT(,)28
b(and)f(the)g(v)-5 b(alue)27 b(stored)g(in)f(the)i(register)f
FP(R)3619 2727 y FO(n)3666 2713 y FT(,)h(for)378 2826
y FP(n)d FN(\025)g FT(0,)31 b(is)f(giv)m(en)g(b)m(y)h
FP(r)1152 2840 y FO(n)1199 2826 y FT(.)41 b(The)30 b(register)g
FP(R)1845 2840 y FO(n)1922 2826 y FT(is)g(said)f(to)j(b)s(e)d(cleared)i
(if)e FP(r)2869 2840 y FO(n)2942 2826 y FT(=)c(0.)41
b(A)31 b(URM)g(executes)378 2939 y(a)g FI(pr)-5 b(o)g(gr)g(am)p
FT(,)34 b(whic)m(h)29 b(is)g(a)i(\014nite)e(list)g(of)i(the)f(follo)m
(wing)f(kinds)g(of)h(instructions:)378 3124 y FQ(Zero:)46
b Fw(ZR)c Fv(n)30 b FT(sets)h FP(r)1089 3138 y FO(n)1167
3124 y FT(to)g(0;)378 3310 y FQ(Successor:)47 b Fw(SC)c
Fv(n)30 b FT(incremen)m(ts)g FP(r)1597 3324 y FO(n)1674
3310 y FT(b)m(y)h(1;)378 3497 y FQ(T)-9 b(ransfer:)46
b Fw(TF)c Fv(n)i(m)30 b FT(sets)h FP(r)1379 3511 y FO(m)1476
3497 y FT(to)g FP(r)1628 3511 y FO(n)1675 3497 y FT(;)378
3683 y FQ(Jump:)45 b Fw(JP)e Fv(n)g(m)g(p)31 b FT(jumps)d(to)j(the)g
FP(p)p FT(th)f(instruction)e(of)j(the)f(program)g(if)g
FP(r)3032 3697 y FO(n)3104 3683 y FT(=)25 b FP(r)3241
3697 y FO(m)3307 3683 y FT(.)378 3868 y(The)k(p)s(osition)f(of)h(the)h
(next)g(instruction)d(to)j(b)s(e)f(executed)i(is)d(stored)i(in)e(a)i
FI(pr)-5 b(o)g(gr)g(am)35 b(c)-5 b(ounter)p FT(,)30 b(and)378
3981 y(the)d FI(c)-5 b(on\014gur)g(ation)35 b FT(of)27
b(a)g(URM)g(is)e(giv)m(en)i(b)m(y)f(a)h(pair)e(consisting)g(of)i(the)g
(curren)m(t)f(program)g(coun)m(ter)378 4094 y(and)40
b(the)g(store.)71 b(A)40 b(con\014guration)g(is)f(said)g(to)i(b)s(e)f
FI(initial)50 b FT(if)39 b(the)h(program)g(coun)m(ter)h(is)e(set)i(to)
378 4207 y(the)c(index)e(of)i(the)g(\014rst)f(instruction)e(\(i.e.,)16
b(to)38 b(0\),)h(and)d(it)g(is)g(said)g(to)h(b)s(e)f
FI(\014nal)47 b FT(with)35 b(resp)s(ect)i(to)378 4320
y(some)g(program)f(if)g(the)h(program)f(coun)m(ter)h(is)f(greater)i
(than)e(the)h(index)e(of)i(the)g(program's)f(last)378
4432 y(instruction.)519 4545 y(Since)k(the)g(instruction)f(set)i(of)g
(the)g(URM)g(can)f(b)s(e)g(regarded)h(as)g(a)g(v)m(ery)g(simple)d(mac)m
(hine)378 4658 y(language,)53 b(w)m(e)48 b(follo)m(w)f(some)h(of)g(the)
g(tec)m(hniques)f(used)g(in)f(sp)s(ecifying)g(real)h(w)m(orld)g(arc)m
(hitec-)378 4771 y(tures)29 b(\(Windley)e(1994\).)42
b(A)29 b(URM)h(store)f(is)f(represen)m(ted)h(as)g(a)g(function)f(from)g
(natural)g(n)m(um)m(b)s(ers)378 4884 y(to)43 b(natural)f(n)m(um)m(b)s
(ers)f(and)h(con\014gurations)f(as)i(pairs)e(consisting)g(of)i(a)g
(natural)e(n)m(um)m(b)s(er)g(\(the)378 4997 y(program)30
b(coun)m(ter\))h(and)f(a)h(store.)473 5182 y FM(store)94
b(==)48 b(:num)e FN(!)i FM(num)473 5295 y(config)e(==)i(:num)e
FN(\002)i FM(store)378 5479 y FT(The)20 b(syn)m(tax)i(of)f(the)g(URM)g
(instruction)e(set)i(is)f(then)h(sp)s(eci\014ed)e(through)h(the)h
(de\014nition)e(of)i(the)g(t)m(yp)s(e)378 5592 y Fw(:instruction)29
b FT(using)k(the)i(t)m(yp)s(e)g(de\014nition)d(pac)m(k)-5
b(age)36 b(of)f(HOL)f(\(Melham)h(1988\))h(and)e(programs)378
5705 y(are)d(de\014ned)e(as)h(lists)f(of)i(instructions.)p
eop
%%Page: 29 39
29 38 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(29)473
396 y FM(instruction)45 b(::=)i(ZR)g(num)1142 509 y(|)g(SC)g(num)1142
622 y(|)g(TF)g(num)g FN(!)h FM(num)1142 735 y(|)f(JP)g(num)g
FN(!)h FM(num)f FN(!)g FM(num)473 961 y(program)f(==)h(:instruction)e
(list)378 1149 y FT(The)24 b(seman)m(tics)g(of)g(the)h(instruction)d
(set)j(is)e(then)h(sp)s(eci\014ed)e(through)i(the)g(de\014nition)e(of)j
(a)f(function)378 1262 y Fw(exec_instruction)o(:)81 b(instruction)39
b Fu(!)44 b Fw(config)d Fu(!)i Fw(config)33 b FT(whic)m(h)g(tak)m(es)k
(an)e(instruction)d(and)378 1374 y(a)f(con\014guration)f(and)h(returns)
e(the)i(con\014guration)f(resulting)f(from)i(the)g(execution)g(of)g
(the)g(giv)m(en)378 1487 y(instruction.)38 b(The)26 b(predicate)h
Fw(Initial:)84 b(config)41 b Fu(!)i Fw(bool)26 b FT(to)h(represen)m(t)g
(initial)e(con\014gurations)378 1600 y(and)30 b(the)g(predicate)g
Fw(Final:)85 b(program)41 b Fu(!)i Fw(config)f Fu(!)h
Fw(bool)29 b FT(for)h(\014nal)f(ones)i(are)f(also)h(de\014ned.)378
1840 y FQ(Computations)378 2012 y FT(The)h(instructions)e(in)h(a)i
(program)f(are)h(executed)g(one)g(b)m(y)f(one)g(starting)h(from)e(an)i
(initial)c(con\014g-)378 2125 y(uration)h(to)h(giv)m(e)h(a)f
FI(c)-5 b(omputation)p FT(.)45 b(The)30 b(execution)h(of)g(a)g(URM)h
(instruction)c(on)j(a)g(\014nal)f(con\014gu-)378 2238
y(ration)g(has)g(no)h(e\013ect.)43 b(A)30 b(computation)h(is)e
(de\014ned)h(as)g(an)h(in\014nite)d(sequence)j(of)g(con\014gurations)
378 2351 y FN(h)p FP(c)452 2365 y FL(0)492 2351 y FP(;)15
b(c)571 2365 y FL(1)611 2351 y FP(;)g(:)g(:)g(:)i FN(i)31
b FT(where)f FP(c)1141 2365 y FL(0)1212 2351 y FT(is)g(initial,)e(and)i
(giv)m(en)h(also)g(a)g(program)f FP(P)13 b FT(,)32 b(its)e(computation)
g(can)i(b)s(e)e(de-)378 2464 y(noted)h(b)m(y)h FP(P)13
b FN(h)p FP(c)904 2478 y FL(0)944 2464 y FN(i)p FT(,)32
b(or)f(simply)e(b)m(y)i FP(P)13 b FT(\()p FP(r)s FT(\))32
b(where)f FP(c)2087 2478 y FL(0)2153 2464 y FT(=)c(\(0)p
FP(;)15 b(r)s FT(\).)45 b(A)31 b(store)h(is)f(usually)e(represen)m(ted)
i(b)m(y)378 2577 y(the)d(sequence)g(\(in)e(paren)m(thesis\))i(of)f(the)
h(v)-5 b(alues)27 b(in)f(its)h(registers)h(\()p FP(r)2734
2591 y FL(0)2774 2577 y FP(;)15 b(r)2855 2591 y FL(1)2894
2577 y FP(;)g(:)g(:)g(:)i FT(\).)40 b(A)28 b(\014nite)e(sequence)378
2689 y(\()p FP(r)454 2703 y FL(0)494 2689 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(r)752 2703 y FO(n)799 2689 y FT(\))26 b(is)e(used)h(to)i
(represen)m(t)e(the)h(store)g(\()p FP(r)2089 2703 y FL(0)2129
2689 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(r)2387 2703 y
FO(n)2434 2689 y FP(;)g FT(0)p FP(;)g FT(0)p FP(;)g(:)g(:)g(:)k
FT(\))26 b(where)f FP(r)3128 2703 y FO(m)3220 2689 y
FT(=)g(0,)i(for)e FP(m)g(>)g(n)p FT(.)378 2802 y(W)-8
b(e)31 b(also)g(use)f(the)g(notation)h FP(P)13 b FN(h)p
FP(c)1539 2816 y FL(0)1579 2802 y FN(i)26 b(!)1731 2816
y FO(n)1803 2802 y FP(c)1842 2769 y FK(0)1896 2802 y
FT(to)31 b(express)f(that)g FP(c)2559 2769 y FK(0)2613
2802 y FT(is)g(the)g FP(n)p FT(th)g(elemen)m(t)h(in)e
FP(P)13 b FT(\()p FP(c)3619 2816 y FL(0)3659 2802 y FT(\).)41
b(A)378 2915 y(computation)32 b(is)g(said)f(to)i FI(c)-5
b(onver)g(ge)40 b FT(if)31 b(it)h(con)m(tains)h(a)f(\014nal)f
(con\014guration,)i(otherwise)f(it)f(is)h(said)378 3028
y(to)g FI(diver)-5 b(ge)p FT(.)45 b(The)31 b FI(value)38
b FT(of)32 b(a)g(con)m(v)m(ergen)m(t)h(computation)f(is)e(giv)m(en)i(b)
m(y)f(the)h(con)m(ten)m(ts)h(of)f(the)g(\014rst)378 3141
y(register)25 b(in)e(an)m(y)i(of)g(its)f(\014nal)g(con\014gurations.)38
b(The)24 b(v)-5 b(alue)24 b(is)g(w)m(ell-de\014ned)f(as)i(program)g
(execution)378 3254 y(do)s(es)30 b(not)h(a\013ect)h(a)e(\014nal)f
(con\014guration.)519 3367 y(The)g(function)f Fw(EXEC_STEPS:)39
b(num)k Fu(!)g Fw(program)e Fu(!)j Fw(config)d Fu(!)i
Fw(config)27 b FT(is)i(de\014ned)f(b)m(y)i(prim-)378
3480 y(itiv)m(e)39 b(recursion)e(in)h(HOL)h(to)g(represen)m(t)g
(computations;)44 b(The)38 b(term)i Fw(EXEC_STEPS)f Fv(n)k(P)56
b(c)3601 3492 y Fs(0)3682 3480 y Fw(=)43 b Fv(c)3805
3450 y Fr(0)378 3593 y FT(holds)29 b(if)g(and)h(only)f(if)h
FP(P)13 b FN(h)p FP(c)1304 3607 y FL(0)1344 3593 y FN(i)25
b(!)1495 3607 y FO(n)1568 3593 y FP(c)1607 3560 y FK(0)473
3780 y FN(`)529 3795 y FE(def)734 3780 y FM(\()p FN(8)p
FP(P)60 b(c)p FM(.)48 b(EXEC_STEPS)d FT(0)j FP(P)61 b(c)48
b FN(\021)f FP(c)p FM(\))96 b FN(^)760 3893 y FM(\()p
FN(8)p FP(n)46 b(P)61 b(c)p FM(.)48 b(EXEC_STEPS)d(\(SUC)h
FP(n)p FM(\))h FP(P)61 b(c)1237 4006 y FN(\021)47 b FM(EXEC_STEPS)e
FP(n)j(P)60 b FM(\(EXEC)p 2347 4006 29 4 v 33 w(STEP)47
b FP(P)61 b(c)p FM(\)\))378 4194 y FT(where)30 b Fw(EXEC_STEP:)39
b(program)i Fu(\))j Fw(config)d Fu(\))i Fw(config)28
b FT(represen)m(ts)i(the)h(execution)g(of)f(one)h(step.)473
4381 y FN(`)529 4396 y FE(def)686 4381 y FN(8)p FP(P)60
b(c)p FM(.)48 b(EXEC_STEP)d FP(P)61 b(c)855 4494 y FN(\021)48
b FM(\(\(Final)d FP(P)61 b(c)p FM(\))48 b FN(!)g FP(c)g
FM(|)998 4607 y(\(exec_instruction)43 b(\(EL)k(\(FST)g
FP(c)p FM(\))h FP(P)13 b FM(\))47 b FP(c)p FM(\)\))519
4795 y FT(A)29 b(n)m(um)m(b)s(er)f(of)h(ML)h(functions)d(called)i(con)m
(v)m(ersions)g(are)h(implemen)m(ted)d(to)j(sim)m(ulate)e(formally)378
4908 y(the)37 b(b)s(eha)m(viour)f(of)i(the)f(ab)s(o)m(v)m(e)i
(de\014ned)d(functions.)60 b(A)37 b(con)m(v)m(ersion)h(tak)m(es)h(a)e
(HOL)g(term)g FP(t)g FT(and)378 5021 y(if)k(successful)f(it)h(returns)f
(a)i(theorem)g Fu(`)14 b Fv(t)43 b Fw(=)h Fv(t)2032 4991
y Fr(0)2055 5021 y FT(.)74 b(A)42 b(con)m(v)m(ersion)g(can)g(sim)m
(ulate)f(the)g(b)s(eha)m(viour)378 5134 y(of)g(a)g(function)e
Fv(f)50 b FT(b)m(y)40 b(taking)h(terms)g(of)g(the)g(form)f
Fv(f)52 b(x)41 b FT(and)f(returns)g(the)h(theorem)g Fu(`)14
b Fv(f)52 b(x)44 b Fw(=)f Fv(c)378 5247 y FT(where)30
b Fv(c)g FT(is)g(the)g(v)-5 b(alue)30 b(of)h(the)f(application)f
Fv(f)52 b(x)q FT(.)41 b(One)30 b(of)g(the)h(con)m(v)m(ersions)g
(implemen)m(ted)e(in)g(the)378 5359 y(mec)m(hanisation)36
b(tak)m(es)h(a)g(term)f(of)g(the)g(form)g Fw(EXEC_STEPS)j
Fv(n)44 b(P)55 b(c)36 b FT(and)f(uses)h(the)g(de\014nitions)e(of)378
5472 y(the)j(ab)s(o)m(v)m(e)h(functions)d(systematically)h(to)h(deriv)m
(e)f(a)h(theorem)g Fu(`)14 b Fw(EXEC_STEPS)39 b Fv(n)44
b(P)55 b(c)44 b Fw(=)f(\()p Fv(p)p Fw(,)f Fv(r)r Fw(\))q
FT(,)378 5585 y(where)30 b Fw(\()p Fv(p)p Fw(,)p Fv(r)r
Fw(\))g FT(is)g(the)h(result)f(of)h(executing)g(the)g(giv)m(en)g
(program)f(\(i.e.,)16 b Fv(P)c FT(\))31 b Fv(n)f FT(times)h(starting)f
(from)378 5698 y Fv(c)p FT(.)45 b(Suc)m(h)31 b(con)m(v)m(ersions)h(are)
h(useful)d(in)g(c)m(hec)m(king)j(that)g(the)f(de\014nitions)d(represen)
m(t)j(their)f(in)m(tended)p eop
%%Page: 30 40
30 39 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(30)378
396 y(concepts,)27 b(and)d(can)h(also)f(b)s(e)g(used)g(to)i(aid)e(the)g
(theorem)h(pro)m(ving)f(pro)s(cess)h(b)m(y)f(calculating)g(certain)378
509 y(results)29 b(automatically)-8 b(.)519 622 y(The)46
b(predicate)h Fw(CONVERGES:)39 b(program)i Fu(!)j Fw(\(num)e(list\))f
Fu(!)j Fw(num)e Fu(!)i Fw(bool)h FT(is)h(de\014ned)f(suc)m(h)378
735 y(that)c Fw(CONVERGES)f Fv(P)56 b(r)46 b(v)e FT(holds)39
b(if)h(there)h(is)e(a)i(\014nal)f(con\014guration)g FP(c)2882
750 y FO(f)2968 735 y FT(in)g FP(P)13 b FT(\()p FP(r)s
FT(\))40 b(with)g(store)h FP(r)3805 702 y FK(0)378 848
y FT(suc)m(h)31 b(that)g FP(r)825 815 y FK(0)822 872
y FL(0)888 848 y FT(=)25 b FP(v)s FT(,)32 b(i.e.,)15
b(it)31 b(con)m(v)m(erges)i(with)c(v)-5 b(alue)31 b FP(v)s
FT(.)42 b(\(Note)33 b(that)e FP(r)j FT(is)c(a)h(\014nite)f(list.\))41
b(Similarly)-8 b(,)378 961 y Fw(DIVERGES:)40 b(program)h
Fu(!)i Fw(\(num)f(list\))g Fu(!)h Fw(bool)37 b FT(is)h(de\014ned)g(suc)
m(h)h(that)g Fw(DIVERGES)h Fv(P)56 b(r)41 b FT(holds)d(if)378
1074 y FP(P)13 b FT(\()p FP(r)s FT(\))30 b(div)m(erges.)519
1187 y(A)40 b(n)m(um)m(b)s(er)e(of)h(theorems)h(represen)m(ting)f(in)m
(tuitiv)m(e)f(prop)s(erties)g(concerning)h(con\014gurations)378
1300 y(and)26 b(computations)h(are)h(then)f(deriv)m(ed)f(so)h(that)h
(they)f(can)h(b)s(e)e(used)h(later)g(in)f(the)h(mec)m(hanisation.)378
1413 y(These)d(include)d(the)j(fact)h(that)f(ev)m(ery)h(program)f(con)m
(v)m(erges)h(to)g(a)f(unique)e(v)-5 b(alue)23 b(unless)f(it)i(div)m
(erges.)473 1600 y FN(`)48 b(8)p FP(P)60 b(r)s FM(.)47
b(\()p FN(9)p FT(!)p FP(v)s FM(.)g(CONVERGES)e FP(P)61
b(r)50 b(v)s FM(\))e FN(_)g FM(\(DIVERGES)d FP(P)61 b(r)s
FM(\))378 1840 y FQ(URM)36 b(Computable)d(F)-9 b(unctions)378
2012 y FT(A)31 b(URM)h(program)e(can)i(b)s(e)e(used)h(to)g(de\014ne)g
(an)g FP(n)p FT(-ary)g(partial)f(function)f(for)i(an)m(y)h(giv)m(en)f
(natural)378 2125 y(n)m(um)m(b)s(er)40 b FP(n)p FT(.)71
b(The)41 b FP(n)f FT(argumen)m(ts)h(of)g(the)g(function)f(are)h(placed)
g(in)e(the)i(\014rst)f FP(n)h FT(registers)f(of)i(a)378
2238 y(cleared)31 b(URM)h(store)h(and)e(then)g(the)h(program)f(is)g
(executed.)45 b(The)31 b(result)f(of)i(the)g(application)e(of)378
2351 y(the)g(function)f(is)g(the)h(v)-5 b(alue)29 b(of)h(the)g
(computation)g(if)f(it)g(is)g(con)m(v)m(ergen)m(t,)k(or)d(unde\014ned)d
(otherwise.)378 2464 y(W)-8 b(e)27 b(sa)m(y)g(that)f(a)g(program)g
FP(P)39 b FI(c)-5 b(omputes)35 b FT(an)26 b FP(n)p FT(-ary)f(function)g
FP(f)35 b FT(if,)26 b(for)g(ev)m(ery)g FP(a)3115 2478
y FL(0)3155 2464 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(a)3420
2478 y FO(n)p FK(\000)p FL(1)3583 2464 y FT(and)25 b
FP(v)s FT(,)378 2577 y(the)32 b(computation)g FP(P)13
b FT(\()p FP(a)1222 2591 y FL(0)1261 2577 y FP(;)i(:)g(:)g(:)32
b(;)15 b(a)1526 2591 y FO(n)p FK(\000)p FL(1)1664 2577
y FT(\))32 b(con)m(v)m(erges)h(to)g FP(v)i FT(if)30 b(and)i(only)e(if)h
FP(f)10 b FT(\()p FP(a)3017 2591 y FL(0)3056 2577 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(a)3321 2591 y FO(n)p FK(\000)p
FL(1)3459 2577 y FT(\))32 b(is)e(equal)378 2689 y(to)f
FP(v)s FT(.)41 b(This)27 b(de\014nition)f(implies)g(that)k(the)e
(computation)h FP(P)13 b FT(\()p FP(a)2548 2703 y FL(0)2588
2689 y FP(;)i(:)g(:)g(:)32 b(;)15 b(a)2853 2703 y FO(n)p
FK(\000)p FL(1)2990 2689 y FT(\))29 b(div)m(erges)g(if)f(and)g(only)378
2802 y(if)k FP(f)10 b FT(\()p FP(a)602 2816 y FL(0)641
2802 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(a)906 2816 y FO(n)p
FK(\000)p FL(1)1043 2802 y FT(\))33 b(is)f(unde\014ned.)46
b(A)32 b(function)g(is)g(said)f(to)j(b)s(e)e FI(URM-c)-5
b(omputable)40 b FT(if)32 b(there)h(is)e(a)378 2915 y(program)f(whic)m
(h)f(computes)i(it.)519 3028 y(Since)g(functions)g(in)g(HOL)h(are)h
(total,)g(w)m(e)g(in)m(tro)s(duce)e(a)i(p)s(olymorphic)c(t)m(yp)s(e)k
(of)f(p)s(ossibly)e(un-)378 3141 y(de\014ned)f(v)-5 b(alues)473
3329 y FP(\013)49 b FM(PP)e(::=)94 b(Undef)285 b(\(*)48
b(Undefined)d(*\))855 3442 y(|)j(Value)e FP(\013)191
b FM(\(*)48 b(Defined)d(with)i(this)g(particular)e(value)h(*\))378
3629 y FT(and)40 b(de\014ne)g(the)g(t)m(yp)s(e)h(of)f
FP(n)p FT(-ary)h(partial)e(functions)g(as)i(mappings)e(from)h(lists)e
(of)j(n)m(um)m(b)s(ers)e(to)378 3742 y(p)s(ossibly)27
b(unde\014ned)i(n)m(um)m(b)s(ers.)473 3930 y FM(pfunc)47
b(==)g(:num)g(list)f FN(!)i FM(num)f(PP)378 4117 y FT(The)36
b(arit)m(y)g(of)g(partial)f(functions)f(is)h(not)i(represen)m(ted)f(in)
e(their)i(t)m(yp)s(es)g(and)f(m)m(ust)h(therefore)h(b)s(e)378
4230 y(explicitly)28 b(men)m(tioned)i(in)f(HOL)h(statemen)m(ts.)43
b(F)-8 b(or)32 b(example,)e(the)h(computabilit)m(y)e(of)h(a)h(function)
378 4343 y(is)f(giv)m(en)h(b)m(y)g(the)g(predicate)g
Fw(COMPUTES:)40 b(num)i Fu(!)i Fw(program)c Fu(!)k Fw(pfunc)d
Fu(!)j Fw(bool)29 b FT(whic)m(h)h(is)g(de\014ned)378
4456 y(as)h(follo)m(ws)473 4644 y FN(`)529 4659 y FE(def)686
4644 y FN(8)p FP(n)47 b(P)61 b(f)10 b FM(.)46 b(COMPUTES)g
FP(n)h(P)61 b(f)760 4757 y FN(\021)47 b FM(\()p FN(8)p
FP(l)i(v)s FM(.)f(\(LENGTH)e FP(l)j FM(=)f FP(n)p FM(\))f
FN(\))951 4870 y FM(\(CONVERGES)e FP(P)60 b(l)50 b(v)h
FM(=)d(\()p FP(f)56 b(l)50 b FM(=)d(Value)g FP(v)s FM(\)\)\))378
5057 y FT(A)35 b(n)m(um)m(b)s(er)e(of)i(prop)s(erties,)f(including)d
(the)k(uniqueness)d(of)j(the)g(function)e(computed)i(b)m(y)f(a)h(pro-)
378 5170 y(gram,)c(are)g(then)f(deriv)m(ed.)39 b(Finally)-8
b(,)29 b(the)i(de\014nition)d(of)j(a)f(computable)g(function)f(is)h
(giv)m(en)g(b)m(y)473 5358 y FN(`)529 5373 y FE(def)686
5358 y FN(8)p FP(n)47 b(f)10 b FM(.)47 b(COMPUTABLE)e
FP(n)i(f)57 b FN(\021)47 b FM(\()p FN(9)p FP(P)13 b FM(.)47
b(COMPUTES)f FP(n)h(P)61 b(f)10 b FM(\))p eop
%%Page: 31 41
31 40 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(31)378
396 y FG(3.2.2)112 b(Building)36 b(URM)h(Programs)378
568 y FT(Pro)m(ving)30 b(that)h(a)g(particular)e(function)g(is)g
(computable)h(usually)f(in)m(v)m(olv)m(es)h(the)h(construction)f(of)g
(a)378 681 y(URM)35 b(program)g(whic)m(h)f(is)g(sho)m(wn)h(to)g
(compute)h(it.)54 b(In)35 b(order)f(to)i(simplify)c(this)i(task,)j
(Cutland)378 794 y(\(1980\))e(giv)m(es)e(the)g(de\014nition)d(of)j(a)f
(concatenation)i(op)s(erator)f(on)g(programs.)47 b(In)m(tuitiv)m(ely)-8
b(,)32 b(giv)m(en)378 907 y(t)m(w)m(o)f(programs)e FP(P)42
b FT(and)29 b FP(Q)p FT(,)h(the)g(computation)f(of)h(their)e
(concatenation)j FP(P)13 b(Q)30 b FT(should)d(corresp)s(ond)378
1020 y(\(in)g(some)h(sense\))h(to)f(that)h(of)f FP(P)41
b FT(follo)m(w)m(ed)28 b(b)m(y)f(that)i(of)f FP(Q)p FT(.)40
b(In)27 b(order)h(to)g(ac)m(hiev)m(e)i(this)d(w)m(e)h(need)g(the)378
1133 y(follo)m(wing:)514 1282 y FN(\017)46 b FT(The)32
b(jumps)e(in)h FP(P)45 b FT(are)32 b(not)g(to)s(o)h(far)f(a)m(w)m(a)m
(y)-8 b(,)35 b(that)d(is,)g(the)g(destination)f(of)h(all)f(the)h(jumps)
e(in)605 1395 y FP(P)44 b FT(should)28 b(b)s(e)i(less)g(or)g(equal)g
(to)h(the)g(length)f(of)g FP(P)13 b FT(.)41 b(A)31 b(program)f(whic)m
(h)f(has)h(this)g(prop)s(ert)m(y)605 1508 y(is)c(said)f(to)i(b)s(e)f
(in)f FI(standar)-5 b(d)32 b(form)p FT(,)c(and)e(an)m(y)g(program)h
(can)f(b)s(e)g(transformed)f(in)m(to)i(standard)605 1620
y(form)j(without)g(a\013ecting)h(the)f(store)h(of)g(its)f(\014nal)f
(con\014gurations.)514 1793 y FN(\017)46 b FT(Since)31
b(URM)h(jumps)e(are)i(absolute,)g(the)g(jumps)e(in)h
FP(Q)g FT(need)h(to)g(b)s(e)g(shifted)e(b)m(y)i(the)f(length)605
1906 y(of)g FP(P)13 b FT(.)519 2055 y(This)19 b(concatenation)k(is)d
(de\014ned)g(b)m(y)h(the)g(function)e Fw(SAPP:)42 b(program)f
Fu(!)i Fw(program)e Fu(!)j Fw(program)m FT(,)378 2168
y(and)j(since)h(it)f(is)g(often)h(required)e(to)j(concatenate)i(more)d
(than)f(t)m(w)m(o)j(programs,)i(a)c(function)378 2280
y Fw(SAPPL:)41 b(program)g(list)h Fu(!)h Fw(program)d
FT(whic)m(h)h(concatenates)k(a)d(giv)m(en)h(list)e(of)h(programs)g(is)g
(also)378 2393 y(de\014ned.)519 2506 y(The)30 b(follo)m(wing)f(three)i
(program)f(mo)s(dules)f(\(functions)h(whic)m(h)f(return)h(programs\))g
(whic)m(h)f(are)378 2619 y(used)h(quite)f(often)i(in)e(the)i
(construction)f(of)g(URM)h(programs)f(are)h(also)f(de\014ned:)378
2768 y Fw(SET_FST_ZERO)39 b Fv(n)45 b FT(clears)30 b(the)h(registers)f
FP(R)1879 2782 y FL(0)1918 2768 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(R)2204 2782 y FO(n)2251 2768 y FT(.)378 2940
y Fw(TRANSFER_TO)39 b Fv(p)k(n)j FT(stores)25 b(the)g(v)-5
b(alues)24 b(of)h(the)g(\014rst)f FP(n)g FT(registers)h(of)g(the)g(URM)
g(in)m(to)g(those)g(starting)605 3053 y(from)30 b FP(R)889
3067 y FO(p)929 3053 y FT(.)378 3226 y Fw(TRANSFER_FROM)38
b Fv(p)44 b(n)h FT(stores)30 b(the)g(v)-5 b(alues)29
b(of)h(the)g FP(n)f FT(registers)h(starting)f(from)g
FP(R)3177 3240 y FO(p)3247 3226 y FT(in)m(to)g(the)h(\014rst)f
FP(n)605 3338 y FT(registers)h(of)h(the)f(URM.)378 3488
y(Registers)37 b(need)f(to)h(b)s(e)g(cleared)f(since)g(programs)g
(computing)g(functions)f(assume)i(that)g(all)f(the)378
3600 y(registers)e(not)g(con)m(taining)g(the)g(argumen)m(ts)g(are)h
(set)f(to)h(0.)53 b(The)33 b(last)h(t)m(w)m(o)h(mo)s(dules)e(are)h
(needed)378 3713 y(to)h(mo)m(v)m(e)h(the)e(argumen)m(ts)h(to)f(and)g
(from)g(di\013eren)m(t)g(memory)g(lo)s(cations.)51 b(Similarly)31
b(to)k(Cutland)378 3826 y(\(1980\))c(w)m(e)e(de\014ne)f(a)h(program)f
(mo)s(dule)f(whic)m(h)h(tak)m(es)i(its)e(argumen)m(ts)h(from)f(a)h
(di\013eren)m(t)f(memory)378 3939 y(lo)s(cation)23 b(rather)h(than)g
(from)f(the)i(\014rst)e(registers.)38 b(This)22 b(is)h(giv)m(en)h(b)m
(y)g(the)g(function)f Fw(PSHIFT)e FT(de\014ned)378 4052
y(b)s(elo)m(w)29 b(suc)m(h)g(that)h(the)f(program)h Fw(PSHIFT)41
b Fv(P)55 b(s)44 b(n)f(d)30 b FT(clears)f(all)f(the)i(registers)f(it)g
(uses,)h(tak)m(es)h(its)d FP(n)378 4165 y FT(argumen)m(ts)d(from)f(the)
h(memory)g(segmen)m(t)h(at)f(o\013set)h FP(s)e FT(and)h(stores)g(the)g
(v)-5 b(alue)24 b(of)h(the)g(computation)378 4278 y(of)30
b FP(P)44 b FT(in)29 b(the)h(register)h FP(R)1239 4293
y FO(d)1279 4278 y FT(:)473 4427 y FN(`)529 4442 y FE(def)686
4427 y FN(8)p FP(P)60 b(s)48 b(n)f(d)p FM(.)h(PSHIFT)e
FP(P)61 b(s)47 b(n)g(d)h FN(\021)903 4540 y FM(SAPPL)e([SET_FST_ZERO)e
(\(MAXREG)i FP(P)13 b FM(\);)1237 4653 y(TRANSFER_FROM)44
b FP(s)k(n)p FM(;)1237 4766 y FP(P)13 b FM(;)1237 4879
y([TF)47 b(0)g FP(d)p FM(]])378 5028 y FT(where)30 b
Fw(MAXREG)41 b Fv(P)h FT(returns)29 b(the)i(maxim)m(um)e(register)h
(used)g(b)m(y)g FP(P)13 b FT(.)519 5141 y(Because)37
b(of)e(their)f(tec)m(hnical)h(nature,)h(deriving)d(the)j(necessarily)e
(prop)s(erties)f(to)j(sho)m(w)f(that)378 5253 y(the)23
b(functions)e(men)m(tioned)i(in)e(this)h(section)h(con)m(v)m(ey)h
(their)e(exp)s(ected)h(b)s(eha)m(viour)f(to)s(ok)h(substan)m(tial)378
5366 y(e\013ort.)43 b(T)-8 b(able)30 b(1)h(on)g(page)g(34)h(sho)m(ws)e
(that)h(the)g(implemen)m(tation)f(of)g(the)h(de\014nitions)e(and)h(pro)
s(ofs)378 5479 y(in)c(this)h(part)h(of)g(the)g(mec)m(hanisation)f
(consists)g(of)h(2800)h(lines)d(of)i(ML)g(co)s(de.)40
b(Most)29 b(of)f(the)g(deriv)m(ed)378 5592 y(prop)s(erties)d(are)h
(considered)f(to)i(b)s(e)f(ob)m(vious)f(in)g(\(Cutland)g(1980\),)k
(whic)m(h)c(dedicates)h(only)g(3)g(pages)378 5705 y(on)k(building)d
(URM)j(programs.)p eop
%%Page: 32 42
32 41 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(32)378
396 y FG(3.2.3)112 b(P)m(artial)36 b(Recursiv)m(e)g(F)-9
b(unctions)37 b(are)h(URM)g(Computable)378 568 y FT(The)29
b(mec)m(hanisation)h(includes)e(a)i(pro)s(of)f(that)i(the)f(partial)f
(recursiv)m(e)h(functions)e(are)j(URM)f(com-)378 681
y(putable.)47 b(The)32 b(set)h(of)g(partial)e(recursiv)m(e)i(functions)
e(\(as)i(de\014ned)f(in)f(\(Cutland)g(1980\)\))k(includes)378
794 y(the)c(follo)m(wing)d(three)j(basic)f(t)m(yp)s(es)g(of)h
(functions:)378 974 y FQ(Zero)46 b FT(The)29 b(zero)j(functions)d
FN(Z)1468 988 y FO(n)1545 974 y FT(of)h(arit)m(y)h FP(n)24
b FN(\025)h FT(0,)31 b(whic)m(h)f(return)f(the)h(v)-5
b(alue)30 b(0)h(for)f(an)m(y)h(input,)378 1159 y FQ(Successor)47
b FT(The)30 b(unary)f(successor)i(function)e FN(S)37
b FT(whic)m(h)29 b(incremen)m(ts)h(its)g(input)e(b)m(y)j(one,)378
1344 y FQ(Pro)6 b(jection)46 b FT(The)30 b(pro)5 b(jection)29
b(functions)f FN(U)1977 1311 y FO(i)1968 1367 y(n)2045
1344 y FT(\(where)h FP(i)d(<)f(n)p FT(\))k(of)h(arit)m(y)g
FP(n)f FT(whic)m(h)f(return)h(the)h FP(i)p FT(th)605
1457 y(comp)s(onen)m(t)h(of)f(their)g(argumen)m(ts,)378
1638 y(and)g(is)f(closed)h(under)f(the)i(follo)m(wing)e(op)s(erations)g
(on)h(functions:)378 1818 y FQ(Substitution)45 b FT(The)26
b(substitution)d(of)j FP(k)j FT(functions)24 b(with)h(arit)m(y)g
FP(n)p FT(,)i(sa)m(y)f FP(g)j FT(=)c(\()p FP(g)3153 1832
y FL(0)3193 1818 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(g)3453
1833 y FO(k)r FK(\000)p FL(1)3587 1818 y FT(\),)27 b(in)m(to)605
1931 y(a)j FP(k)s FT(-ary)f(function)f FP(f)38 b FT(giv)m(es)30
b(the)f FP(n)p FT(-ary)g(function)f(pro)s(duced)f(b)m(y)i(applying)e
FP(f)38 b FT(to)30 b(the)f(results)605 2044 y(of)i(the)f(applications)f
(of)h FP(g)s FT(.)42 b(That)30 b(is,)g(the)g(substitution)e
FP(f)2648 2043 y FT(^)2648 2044 y FN(\016)p FP(g)34 b
FT(is)29 b(de\014ned)g(b)m(y)917 2248 y FP(f)972 2247
y FT(^)972 2248 y FN(\016)p FP(g)s FT(\()p FP(x)1150
2262 y FL(0)1190 2248 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(x)1459 2262 y FO(n)p FK(\000)p FL(1)1596 2248 y FT(\))26
b(=)f FP(f)10 b FT(\()p FP(g)1886 2262 y FL(0)1925 2248
y FT(\()p FP(x)2012 2262 y FL(0)2052 2248 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)2321 2262 y FO(n)p FK(\000)p FL(1)2458 2248
y FT(\))p FP(;)g(:)g(:)g(:)33 b(;)15 b(g)2754 2263 y
FO(k)r FK(\000)p FL(1)2887 2248 y FT(\()p FP(x)2974 2262
y FL(0)3014 2248 y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)3283
2262 y FO(n)p FK(\000)p FL(1)3420 2248 y FT(\)\))p FP(:)378
2488 y FQ(Primitiv)m(e)34 b(Recursion)47 b FT(Giv)m(en)29
b(an)h FP(n)p FT(-ary)f(base)g(case)h(function)e FP(f)10
b FT(,)29 b(and)g(an)g(\()p FP(n)17 b FT(+)h(2\)-ary)30
b(recur-)605 2601 y(sion)c(step)g(function)g FP(g)s FT(,)i(the)e(\()p
FP(n)13 b FT(+)g(1\)-ary)27 b(primitiv)m(e)d(recursiv)m(e)i(function)f
FN(R)p FT(\()p FP(f)5 b(;)15 b(g)s FT(\))28 b(is)d(de\014ned)605
2714 y(as)31 b(follo)m(ws:)910 2918 y FN(R)p FT(\()p
FP(f)5 b(;)15 b(g)s FT(\)\(0)p FP(;)g(x)1365 2932 y FL(0)1406
2918 y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)1675 2932 y FO(n)p
FK(\000)p FL(1)1812 2918 y FT(\))26 b(=)f FP(f)10 b FT(\()p
FP(x)2111 2932 y FL(0)2150 2918 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)2419 2932 y FO(n)p FK(\000)p FL(1)2556 2918
y FT(\))746 3056 y FN(R)p FT(\()p FP(f)5 b(;)15 b(g)s
FT(\)\()p FP(x)22 b FT(+)e(1)p FP(;)15 b(x)1366 3070
y FL(0)1406 3056 y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)1675
3070 y FO(n)p FK(\000)p FL(1)1812 3056 y FT(\))26 b(=)f
FP(g)s FT(\()p FP(x;)15 b FN(R)p FT(\()p FP(f)5 b(;)15
b(g)s FT(\)\()p FP(x;)g(x)2604 3070 y FL(0)2646 3056
y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)2915 3070 y FO(n)p FK(\000)p
FL(1)3053 3056 y FT(\))p FP(;)g(x)3180 3070 y FL(0)3220
3056 y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)3489 3070 y FO(n)p
FK(\000)p FL(1)3626 3056 y FT(\))p FP(:)605 3260 y FT(The)25
b(\014rst)g(argumen)m(t)h(of)g FN(R)p FT(\()p FP(f)5
b(;)15 b(g)s FT(\))26 b(is)e(the)i(depth)f(of)g(the)h(recursion,)f(or)h
(the)g(n)m(um)m(b)s(er)e(of)h(times)605 3373 y(the)35
b(function)e FP(g)38 b FT(is)33 b(applied)g(after)i FP(f)43
b FT(is.)52 b(Note)36 b(that)f(the)g(depth)e(of)i(the)g(recursion)d(is)
i(also)605 3486 y(giv)m(en)d(as)f(an)g(argumen)m(t)h(to)g(the)g(step)f
(function)f FP(g)s FT(.)378 3671 y FQ(Un)m(b)s(ounded)36
b(Minimalisation)45 b FT(The)c(un)m(b)s(ounded)e(minimalisation)f
FP(\026)3033 3686 y FO(f)3120 3671 y FT(of)j(an)h(\()p
FP(n)27 b FT(+)h(1\)-ary)605 3784 y(function)43 b FP(f)53
b FT(is)43 b(the)h FP(n)p FT(-ary)g(function)f(whic)m(h)f(giv)m(en)i
(the)g(argumen)m(ts)g(\()p FP(x)3213 3798 y FL(0)3253
3784 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)3522 3798 y
FO(n)p FK(\000)p FL(1)3659 3784 y FT(\),)48 b(it)605
3897 y(returns)29 b(the)i(least)g FP(x)f FT(suc)m(h)g(that)689
4082 y(1.)46 b(the)31 b(v)-5 b(alue)30 b(of)g FP(f)10
b FT(\()p FP(x;)15 b(x)1534 4096 y FL(0)1574 4082 y FP(;)g(:)g(:)g(:)31
b(;)15 b(x)1842 4096 y FO(n)p FK(\000)p FL(1)1980 4082
y FT(\))25 b(=)g(0,)31 b(and)689 4225 y(2.)46 b(for)30
b(all)g FP(y)e(<)d(x)p FT(,)30 b(the)h(application)d
FP(f)10 b FT(\()p FP(y)s(;)15 b(x)2203 4239 y FL(0)2243
4225 y FP(;)g(:)g(:)g(:)31 b(;)15 b(x)2511 4239 y FO(n)p
FK(\000)p FL(1)2649 4225 y FT(\))31 b(is)e(de\014ned.)605
4410 y(The)h(v)-5 b(alue)30 b(of)g FP(\026)1185 4425
y FO(f)1231 4410 y FT(\()p FP(x)1318 4424 y FL(0)1357
4410 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)1626 4424 y
FO(n)p FK(\000)p FL(1)1764 4410 y FT(\))30 b(is)g(unde\014ned)e(if)h
(no)h(suc)m(h)h FP(x)f FT(exists.)519 4590 y(The)40 b(mec)m(hanisation)
g(includes)e(de\014nitions)f(of)k(the)f(ab)s(o)m(v)m(e)i(basic)e
(functions)f(and)g(function)378 4703 y(op)s(erations,)28
b(and)g(pro)s(ofs)f(that)i(the)f(three)g(basic)g(kinds)e(of)i
(functions)f(are)h(computable,)h(and)e(that)378 4816
y(functions)e(de\014ned)g(b)m(y)i(substitution,)e(recursion,)h(or)g
(minimalisation)d(on)j(computable)g(functions)378 4929
y(are)h(themselv)m(es)f(computable.)39 b(In)25 b(eac)m(h)j(case,)g(the)
f(pro)s(of)e(that)i(these)g(functions)e(are)h(computable)378
5042 y(is)j(as)i(follo)m(ws:)489 5223 y(1.)46 b(The)30
b(criteria)g(under)f(whic)m(h)g(the)h(function)f(is)h(de\014ned)f(are)i
(iden)m(ti\014ed,)489 5407 y(2.)46 b(A)31 b(URM)f(program)h(is)e
(de\014ned)g(and)h(is)f(sho)m(wn)h(to)h(compute)g(the)f(function)g(as)g
(follo)m(ws:)644 5592 y(\(a\))46 b(the)28 b(criteria)f(under)e(whic)m
(h)i(the)g(computation)g(of)h(the)g(program)f(con)m(v)m(erges)i(are)f
(iden-)805 5705 y(ti\014ed,)p eop
%%Page: 33 43
33 42 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(33)639
396 y(\(b\))45 b(sho)m(wing)40 b(that)h(whenev)m(er)g(the)g(program)f
(div)m(erges,)k(the)d(v)-5 b(alue)40 b(of)h(the)f(function)g(is)805
509 y(unde\014ned,)649 655 y(\(c\))46 b(sho)m(wing)35
b(that)h(whenev)m(er)f(the)h(program)f(con)m(v)m(erges,)k(the)d(v)-5
b(alue)35 b(of)h(the)g(function)e(is)805 768 y(de\014ned)29
b(and)h(iden)m(tical)f(to)i(the)g(v)-5 b(alue)30 b(of)g(the)h
(computation.)378 956 y(Sho)m(wing)41 b(that)i(the)g(basic)e(functions)
g(are)i(computable)f(is)f(rather)i(straigh)m(tforw)m(ard.)76
b(On)42 b(the)378 1069 y(other)i(hand,)i(the)d(pro)s(ofs)g(that)h
(substitution,)g(recursion)e(and)h(minimalisation)d(preserv)m(e)k(the)
378 1182 y(computabilit)m(y)36 b(of)h(functions)f(con)m(tain)h(sev)m
(eral)h(cases,)i(eac)m(h)e(of)g(whic)m(h)e(is)g(not)i(trivial.)59
b(F)-8 b(or)38 b(in-)378 1295 y(stance,)32 b(the)e(programs)g(whic)m(h)
g(compute)h(primitiv)m(e)d(recursiv)m(e)i(functions)f(and)h
(minimalisations)378 1408 y(con)m(tain)i(lo)s(ops)e(and)h(therefore)h
(in)m(v)-5 b(arian)m(ts)31 b(had)f(to)j(b)s(e)d(found.)43
b(On)31 b(the)g(other)h(hand,)f(the)h(pro)s(ofs)378 1521
y(in)d(\(Cutland)g(1980\))j(are)f(based)f(on)g(informal)f(\015o)m(w)h
(diagrams.)378 1764 y FG(3.2.4)112 b(De\014ning)38 b(Computable)f(F)-9
b(unctions)378 1936 y FT(The)33 b(language)h(of)f(partial)f(recursiv)m
(e)h(functions)f(can)h(b)s(e)g(considered)f(as)i(a)f(high-lev)m(el)f
(language)378 2049 y(for)27 b(expressing)e(computable)i(functions.)38
b(F)-8 b(or)28 b(instance,)g(addition)d(can)i(b)s(e)f(de\014ned)g(b)m
(y)h(primitiv)m(e)378 2161 y(recursion)j(on)h(the)g(iden)m(tit)m(y)g
(and)g(the)g(successor)h(functions,)e(or)i(formally)d(b)m(y)j
FN(R)p FT(\()p FN(U)3313 2128 y FL(0)3304 2186 y(1)3352
2161 y FP(;)15 b FN(S)3454 2160 y FT(^)3454 2161 y FN(\016)q
FT([)p FN(U)3591 2128 y FL(1)3582 2186 y(3)3631 2161
y FT(]\).)44 b(A)378 2274 y(n)m(um)m(b)s(er)24 b(of)i(functions)e(w)m
(ere)i(de\014ned)f(in)f(this)h(manner,)h(and)f(the)h(deriv)-5
b(ation)24 b(that)i(suc)m(h)f(functions)378 2387 y(are)e(computable)f
(w)m(as)h(automated)h(through)e(the)g(systematic)i(application)d(of)h
(the)h(theorems)g(men-)378 2500 y(tioned)i(in)g(the)h(previous)f
(section.)39 b(Sho)m(wing)25 b(that)i(the)f(function)f(de\014ned)g(in)g
(terms)h(of)g(the)g(partial)378 2613 y(recursiv)m(e)39
b(op)s(erators)g(corresp)s(onds)f(to)i(the)f(in)m(tended)f(one)i(needs)
f(some)g(w)m(ork.)68 b(F)-8 b(or)40 b(example,)378 2726
y(sho)m(wing)26 b(that)h(the)g(ab)s(o)m(v)m(e)h(de\014nition)c(of)j
(addition)e(actually)h(corresp)s(onds)f(to)j(the)f(addition)d(func-)378
2839 y(tion)35 b(requires)f(mathematical)i(induction.)53
b(A)36 b(con)m(v)m(ersion)f(whic)m(h)g(sim)m(ulates)f(the)i(b)s(eha)m
(viour)e(of)378 2952 y(partial)29 b(recursiv)m(e)h(functions)f(is)g
(implemen)m(ted)g(to)i(help)e(this)h(pro)s(cess.)378
3195 y FG(3.2.5)112 b(Concluding)37 b(Remarks)g(on)h(the)f(HOL)h(F)-9
b(ormalisation)378 3367 y FT(W)h(e)32 b(ha)m(v)m(e)g(illustrated)c(the)
j(HOL)g(mec)m(hanisation)f(of)h(URM)g(computabilit)m(y)e(whic)m(h)g
(includes)g(the)378 3480 y(result)f(that)h(partial)f(recursiv)m(e)g
(functions)f(are)j(URM)f(computable.)39 b(T)-8 b(able)29
b(1)g(sho)m(ws)g(the)g(lengths)378 3593 y(of)h(di\013eren)m(t)e(parts)i
(of)f(the)h(source)f(co)s(de)h(of)g(the)f(mec)m(hanisation)g(with)f
(commen)m(ts)j(remo)m(v)m(ed.)41 b(F)-8 b(or)378 3706
y(eac)m(h)31 b(part,)g(the)g(lengths)e(listed)g(in)g(the)i(table)f(are)
h(divided)c(as)k(follo)m(ws:)514 3893 y FN(\017)46 b
FT(ML)35 b(declarations:)48 b(ML)34 b(de\014nitions)e(of)i(pro)s(of)g
(pro)s(cedures)f(and)h(tactics)h(whic)m(h)e(are)i(used)605
4006 y(in)29 b(the)i(pro)s(of)e(of)i(more)g(than)f(one)g(theorem.)514
4194 y FN(\017)46 b FT(HOL)25 b(de\014nitions:)35 b(the)25
b(application)e(of)i(ML)g(functions)f(whic)m(h)f(in)m(tro)s(duce)h(new)
g(HOL)h(t)m(yp)s(es)605 4307 y(and)30 b(constan)m(ts.)514
4494 y FN(\017)46 b FT(HOL)32 b(pro)s(ofs:)43 b(the)32
b(application)f(of)h(ML)g(functions)f(whic)m(h)f(deriv)m(e)i
(particular)e(HOL)i(theo-)605 4607 y(rems.)519 4795 y(It)45
b(can)f(b)s(e)g(seen)h(that)g(a)g(substan)m(tial)e(part)h(of)h(the)g
(mec)m(hanisation)f(is)f(dedicated)h(to)i(the)378 4908
y(deriv)-5 b(ation)41 b(of)i(theorems,)j(most)d(of)f(whic)m(h)g(w)m
(ere)h(pro)m(v)m(ed)g(b)m(y)f(applying)e(tactics)k(in)m(teractiv)m(ely)
378 5021 y(in)35 b(a)h(goal)h(directed)e(fashion.)56
b(A)36 b(small)f(n)m(um)m(b)s(er)g(of)h(tactics)h(and)e(other)i(pro)s
(of)e(pro)s(cedures)f(are)378 5134 y(implemen)m(ted)29
b(to)j(automate)h(inferences)d(sp)s(eci\014c)f(to)j(this)e(mec)m
(hanisation.)42 b(Ev)m(en)31 b(though)f(these)378 5247
y(pro)s(of)37 b(pro)s(cedures)f(w)m(ere)i(used)e(in)g(sev)m(eral)i(o)s
(ccasions)g(during)d(theorem)j(pro)m(ving,)g(most)g(of)g(the)378
5359 y(pro)s(of)43 b(steps)i(in)m(v)m(olv)m(e)f(the)g(standard)g(HOL)g
(tactics)h(and)f(tacticals.)83 b(There)43 b(is)h(a)g(substan)m(tial)378
5472 y(di\013erence)30 b(b)s(et)m(w)m(een)h(the)g(lev)m(el)f(of)h
(detail)f(\(and)g(therefore)h(the)g(length\))f(of)h(the)g(HOL)f(pro)s
(ofs)f(and)378 5585 y(the)43 b(pro)s(ofs)f(found)g(in)g(the)h
(literature.)78 b(The)42 b(mec)m(hanisation)h(includes)d(the)j(pro)s
(of)g(of)g(dozens)378 5698 y(of)37 b(theorems)h(whic)m(h)e(w)m(ould)g
(b)s(e)h(considered)f(to)i(b)s(e)f(trivial)e(in)h(an)h(informal)f(exp)s
(osition.)60 b(Suc)m(h)p eop
%%Page: 34 44
34 43 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(34)428
384 y FQ(In)m(tro)s(ductory)35 b(Mec)m(hanisation)646
497 y FT(ML)30 b(declarations:)205 b(170)31 b(lines)646
610 y(HOL)f(de\014nitions:)258 b(10)31 b(lines)646 723
y(HOL)f(pro)s(ofs:)381 b(440)31 b(lines)736 836 y FQ(T)-9
b(otal:)467 b(620)35 b(lines)428 1062 y(De\014nition)g(of)g(URM)h
(Computabilit)m(y)646 1175 y FT(ML)30 b(declarations:)205
b(380)31 b(lines)646 1287 y(HOL)f(de\014nitions:)213
b(130)31 b(lines)646 1400 y(HOL)f(pro)s(ofs:)381 b(370)31
b(lines)736 1513 y FQ(T)-9 b(otal:)467 b(880)35 b(lines)428
1739 y(Building)h(URM)g(programs)646 1852 y FT(ML)30
b(declarations:)250 b(70)31 b(lines)646 1965 y(HOL)f(de\014nitions:)258
b(90)31 b(lines)646 2078 y(HOL)f(pro)s(ofs:)335 b(2660)32
b(lines)736 2191 y FQ(T)-9 b(otal:)414 b(2820)36 b(lines)428
2417 y(P)m(artial)f(Recursiv)m(e)h(F)-9 b(unctions)36
b(are)e(URM)i(Computable)646 2529 y FT(ML)30 b(declarations:)205
b(180)31 b(lines)646 2642 y(HOL)f(de\014nitions:)258
b(60)31 b(lines)646 2755 y(HOL)f(pro)s(ofs:)335 b(3290)32
b(lines)736 2868 y FQ(T)-9 b(otal:)414 b(3530)36 b(lines)910
3057 y FT(T)-8 b(able)30 b(1:)41 b(On)30 b(the)g(Source)g(Co)s(de)g(of)
h(the)f(Mec)m(hanisation)h(in)e(HOL.)378 3314 y(`shallo)m(w)22
b(theorems')h(are)g(used)f(throughout)g(the)h(mec)m(hanisation,)h(ev)m
(en)f(in)e(the)i(pro)s(of)f(of)h(theorems)378 3427 y(whic)m(h)43
b(state)i(m)m(uc)m(h)g(deep)s(er)e(results.)81 b(On)43
b(the)i(other)f(hand,)j(the)e(simple)d(results)h(pro)m(v)m(ed)h(in)378
3540 y(informal)30 b(texts)i(are)g(usually)e(tak)m(en)i(for)g(gran)m
(ted)g(once)h(they)f(ha)m(v)m(e)g(b)s(een)f(stated,)j(illustrated)29
b(b)m(y)378 3653 y(a)i(n)m(um)m(b)s(er)e(of)h(examples,)g(and)g(deriv)m
(ed.)378 3939 y FH(3.3)135 b(A)45 b(Pro)t(of)g(of)g(the)g
Fp(S)1667 3896 y Fv(m)1660 3969 y(n)1789 3939 y FH(Theorem)g(in)g(Co)t
(q)378 4145 y FG(3.3.1)112 b(On)38 b(the)g(Co)s(q)f(Theorem)g(Pro)m
(ving)f(En)m(vironmen)m(t)378 4317 y FT(The)27 b(Co)s(q)h(system)g(is)f
(an)h(implemen)m(tation)e(in)h(CAML)h(of)g(the)g(Calculus)e(of)i
(Inductiv)m(e)f(Construc-)378 4430 y(tions)g(\(CIC\))h(\(Co)s(quand)e
(and)i(Huet)g(1986\),)j(a)d(v)-5 b(arian)m(t)28 b(of)g(t)m(yp)s(e)g
(theory)g(related)f(to)i(Martin-L\177)-45 b(of)7 b('s)378
4543 y(In)m(tuitionistic)32 b(T)m(yp)s(e)i(Theory)g(\(Martin-L\177)-45
b(of)34 b(1984;)39 b(Nordstr\177)-45 b(om,)35 b(P)m(etersson,)h(and)e
(Smith)f(1990\))378 4656 y(and)j(Girard's)f(p)s(olymorphic)e
FP(\025)p FT(-calculus)j FP(F)1945 4670 y FO(!)2032 4656
y FT(\(Girard)f(1972\).)61 b(T)-8 b(erms)36 b(in)f(CIC)h(are)g(t)m(yp)s
(ed)g(and)378 4769 y(t)m(yp)s(es)c(are)h(also)f(terms.)45
b(Suc)m(h)32 b(a)g(t)m(yp)s(e)g(theory)h(can)f(b)s(e)f(treated)j(as)e
(a)g(logic)g(through)g(the)g FI(Curry-)378 4881 y(Howar)-5
b(d)43 b(isomorphism)50 b FT(\(see)40 b(\(Thompson)f(1991;)46
b(Nordstr\177)-45 b(om,)42 b(P)m(etersson,)h(and)c(Smith)f(1990\))378
4994 y(for)32 b(in)m(tro)s(ductions)f(of)i(the)f(Curry-Ho)m(w)m(ard)g
(isomorphism\))e(where)i(prop)s(ositions)e(are)j(expressed)378
5107 y(as)i(t)m(yp)s(es.)54 b(F)-8 b(or)35 b(instance,)h(a)f
(conjunction)f FP(A)23 b FN(^)g FP(B)39 b FT(is)34 b(represen)m(ted)h
(b)m(y)f(a)i(pro)s(duct)d(t)m(yp)s(e)i FP(A)23 b FN(\002)g
FP(B)5 b FT(,)378 5220 y(and)30 b(an)h(implication)d
FP(A)f FN(\))f FP(B)35 b FT(is)30 b(represen)m(ted)h(b)m(y)g(a)g
(function)f(t)m(yp)s(e)h FP(A)26 b FN(!)g FP(B)5 b FT(.)42
b(Also,)31 b(a)h(term)f(can)378 5333 y(b)s(e)f(seen)g(as)h(a)g(pro)s
(of)e(of)i(the)g(prop)s(osition)d(represen)m(ted)i(b)m(y)g(its)g(t)m
(yp)s(e,)h(and)f(th)m(us)g(theorems)h(in)e(the)378 5446
y(logic)h(are)h(nonempt)m(y)f(t)m(yp)s(es.)41 b(F)-8
b(or)31 b(example,)f(the)h(function)1598 5650 y FM(curry)24
b FT(=)h FP(\025f)5 b(:\025x:\025y)s(:f)10 b FT(\()p
FP(x;)15 b(y)s FT(\))p eop
%%Page: 35 45
35 44 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(35)378
396 y(whic)m(h)31 b(has)i(t)m(yp)s(e)g(\(\()p FP(A)22
b FN(\002)g FP(B)5 b FT(\))29 b FN(!)g FP(C)7 b FT(\))29
b FN(!)g FP(A)g FN(!)g FP(B)34 b FN(!)29 b FP(C)39 b
FT(is)32 b(a)h(pro)s(of)f(of)h(the)g(theorem)g(\(\()p
FP(A)23 b FN(^)e FP(B)5 b FT(\))29 b FN(\))378 509 y
FP(C)7 b FT(\))34 b FN(\))g FT(\()p FP(A)g FN(\))g FP(B)k
FN(\))c FP(C)7 b FT(\).)56 b(Ob)5 b(jects)36 b(whic)m(h)e(ha)m(v)m(e)j
(the)e(same)h(normal)f(form)g(according)h(to)g FP(\013\014)5
b(\016)s(\023)p FT(-)378 622 y(con)m(v)m(ersion)26 b(\(simply)f(called)
g(con)m(v)m(ertible)h(ob)5 b(jects\))28 b(are)e(treated)h(as)g(the)f
(same)h(term)f(b)m(y)g(the)h(logic.)378 735 y FP(\016)s
FT(-con)m(v)m(ersion)e(in)m(v)m(olv)m(es)e(the)h(substitution)d(of)j(a)
g(constan)m(t)h(b)m(y)e(its)g(de\014ning)f(term)h(and)g
FP(\023)p FT(-con)m(v)m(ersion)378 848 y(is)29 b(automation)i(of)g
(inductiv)m(e)e(de\014nitions.)519 961 y(Under)41 b(the)g(Curry-Ho)m(w)
m(ard)g(isomorphism,)h(theorem)g(pro)m(ving)e(corresp)s(onds)g(to)j
(the)e(con-)378 1074 y(struction)35 b(of)g(w)m(ell-t)m(yp)s(ed)g(terms)
g(and)g(the)g(core)i(inference)d(engine)h(of)h(Co)s(q)f(is)f(basically)
g(a)i(t)m(yp)s(e)378 1187 y(c)m(hec)m(king)h(algorithm)e(for)h(CIC)f
(terms.)59 b(T)-8 b(erms)35 b(whose)h(t)m(yp)s(e)h(is)e(a)i(theorem)f
(are)h(usually)d(called)378 1300 y(pro)s(of)g(ob)5 b(jects)36
b(and)e(are)i(stored)f(in)e(Co)s(q)i(theories.)54 b(The)35
b(Co)s(q)f(system)h(pro)m(vides)f(the)h(sp)s(eci\014ca-)378
1413 y(tion)27 b(and)g(pro)s(of)g(language)h FM(Gallina)e
FT(in)g(whic)m(h)g(users)h(p)s(erform)f(the)i(actual)g(in)m(teractiv)m
(e)g(theorem)378 1526 y(pro)m(ving.)38 b FM(Gallina)24
b FT(constructs)h(include)e(commands)i(for)g(sp)s(ecifying)f
(de\014nitions)f(and)h(for)h(tactic-)378 1638 y(based)c(theorem)h(pro)m
(ving)e(and)h(Co)s(q)g(users)f(can)i(extend)f(the)h FM(Gallina)d
FT(language)j(b)m(y)f(implemen)m(ting)378 1751 y(new)29
b(constructs)g(in)f(CAML.)i(The)f(\014les)f(whic)m(h)g
FM(Gallina)f FT(accepts)k(during)c(theorem)j(pro)m(ving)e(are)378
1864 y(called)i(pro)s(of)f(scripts,)h(or)g(simply)e(scripts.)378
2108 y FG(3.3.2)112 b(The)38 b Fo(P)8 b(RF)48 b FG(Programming)35
b(Language)378 2279 y FT(In)k(this)g(section)h(w)m(e)g(giv)m(e)h(the)f
(syn)m(tax)g(and)g(seman)m(tics)g(of)g(the)g FN(P)7 b(RF)50
b FT(language)40 b(of)g(programs)378 2392 y(whic)m(h)33
b(w)m(e)i(em)m(b)s(ed)f(in)f(Co)s(q.)53 b(The)34 b FN(P)7
b(RF)44 b FT(language)35 b(is)f(v)m(ery)h(close)f(to)i(the)e
(de\014nition)e(of)j(partial)378 2505 y(recursiv)m(e)30
b(functions.)378 2745 y FQ(The)35 b(Syn)m(tax)f(of)i
FN(P)7 b(RF)378 2917 y FT(The)32 b(syn)m(tax)g(of)g(the)h
FN(P)7 b(RF)42 b FT(language)32 b(is)f(de\014ned)g(in)g(Co)s(q)h(in)f
(terms)h(of)g(a)g(data)h(t)m(yp)s(e)f Fw(prf)f FT(whose)378
3030 y(constructors)44 b(corresp)s(ond)f(to)i(the)g(three)f(basic)g
(functions)e(and)i(the)g(three)h(op)s(erators)f(whic)m(h)378
3143 y(de\014ne)30 b(partial)f(recursiv)m(e)h(functions.)473
3330 y FM(prf)47 b(::=)g(Zero:)g(prf)760 3443 y(|)g(Succ:)g(prf)760
3556 y(|)g(Proj:)g(nat)f FN(!)i FM(prf)760 3669 y(|)f(Sub:)95
b(prf)46 b FN(!)i FM(prf)f FN(!)h FM(nat)f FN(!)g FM(nat)g
FN(!)h FM(prf)760 3782 y(|)f(Rec:)95 b(prf)46 b FN(!)i
FM(prf)f FN(!)h FM(prf)760 3895 y(|)f(Min:)95 b(prf)46
b FN(!)i FM(prf)378 4083 y FT(It)43 b(should)f(b)s(e)h(noted)g(that)h
(an)m(y)g(particular)e FN(P)7 b(RF)53 b FT(program)43
b(represen)m(ts)g(a)h(di\013eren)m(t)f(partial)378 4195
y(recursiv)m(e)e(function)f(for)h(eac)m(h)i(arit)m(y)-8
b(.)74 b(F)-8 b(or)42 b(example,)i(although)d Fw(Succ)f
FT(is)g(de\014ned)g(in)g(order)h(to)378 4308 y(represen)m(t)21
b(the)g(successor)g(function)f FN(S)7 b FT(,)22 b(it)f(also)g(represen)
m(ts)f(the)h FP(n)p FT(-ary)g(function)f(whic)m(h)f(returns)h(the)378
4421 y(successor)k(of)g(the)f(\014rst)g(n)m(um)m(b)s(er)f(of)i(its)f
(input:)35 b FP(\025)p FT(\()p FP(x)2144 4435 y FL(0)2184
4421 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)2453 4435 y
FO(n)p FK(\000)p FL(1)2591 4421 y FT(\))p FP(:)p FN(S)7
b FT(\()p FP(x)2800 4435 y FL(0)2840 4421 y FT(\))24
b(for)f(eac)m(h)i(v)-5 b(alue)23 b(of)g FP(n)p FT(.)38
b(The)378 4534 y(t)m(yp)s(e)29 b(of)h(the)f(constructor)h
Fw(Sub)e FT(in)g(the)i(ab)s(o)m(v)m(e)g(de\014nition)d(of)j
Fw(prf)e FT(is)g(di\013eren)m(t)h(from)g(the)g(exp)s(ected)378
4647 y Fw(:)14 b(prf)42 b Fu(!)h Fw(\(list)f(prf\))g
Fu(!)h Fw(prf)d FT(corresp)s(onding)f(to)j(the)f(substitution)e(of)i(a)
h(function)e(on)h(a)g(list)378 4760 y(of)34 b(functions.)49
b(A)33 b(t)m(yp)s(e)h(de\014nition)d(with)i(suc)m(h)g(a)h(constructor)g
(has)f(a)h(non-p)s(ositiv)m(e)e(o)s(ccurrence,)378 4873
y(and)e(is)f(not)h(accepted)i(b)m(y)e(the)g(v)m(ersion)g(of)g(Co)s(q)g
(used)f(in)g(the)h(mec)m(hanisation.)41 b(The)29 b(substitution)378
4986 y(construct)f Fw(Sub)f FT(in)g FN(P)7 b(RF)38 b
FT(tak)m(es)29 b(t)m(w)m(o)g(programs)f FP(f)37 b FT(and)28
b FP(g)s FT(,)h(and)e(t)m(w)m(o)i(natural)f(n)m(um)m(b)s(ers)e
FP(n)i FT(and)f FP(m)p FT(,)378 5099 y(and)h(corresp)s(onds)f(to)j(the)
f(application)d(of)j FP(f)38 b FT(on)29 b(the)g(output)f(of)h
FP(g)j FT(and)c(part)g(of)h(its)f(input)f(\(the)i FP(m)378
5212 y FT(argumen)m(ts)e(of)g FP(g)k FT(starting)c(from)f(the)i
FP(n)p FT(th\).)39 b(The)26 b(b)s(eha)m(viour)g(of)h
Fw(Sub)f FT(is)g(describ)s(ed)f(in)h(more)h(detail)378
5325 y(b)s(elo)m(w)32 b(where)h(the)g(seman)m(tics)g(of)g
FN(P)7 b(RF)43 b FT(programs)33 b(is)f(de\014ned.)47
b(A)33 b(program)g(corresp)s(onding)e(to)378 5437 y(the)g(substitution)
d(on)i(a)h(list)e(of)h(functions)f(is)g(then)i(de\014ned)e(in)g(terms)h
(of)h Fw(Sub)n FT(.)p eop
%%Page: 36 46
36 45 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(36)378
396 y FQ(The)35 b(Seman)m(tics)f(of)h FN(P)7 b(RF)378
568 y(P)g(RF)32 b FT(programs)21 b(tak)m(e)j(a)e(list)f(of)h(natural)f
(n)m(um)m(b)s(ers)f(and)i(return)e(a)j(natural)e(n)m(um)m(b)s(er)f(if)h
(they)h(termi-)378 681 y(nate.)39 b(A)23 b(program)g(assumes)f(the)h
(input)e(list)h(to)h(b)s(e)g(of)g(a)g(particular)e(length.)38
b(When)23 b(a)g(program)g(re-)378 794 y(quires)c(an)i(elemen)m(t)g(at)h
(a)f(p)s(osition)e(greater)j(than)e(the)h(length)g(of)g(the)g(list,)g
(the)g(v)-5 b(alue)20 b(of)h(the)g(elemen)m(t)378 907
y(is)k(assumed)g(to)i(b)s(e)e(0.)40 b(Lists)25 b(are)h(indexed)e(using)
h(the)h(function)f Fw(zel:)d(nat)43 b Fu(!)g Fw(\(list)f(nat\))g
Fu(!)h Fw(nat)378 1020 y FT(whic)m(h)29 b(is)g(de\014ned)f(suc)m(h)i
(that)g Fw(zel)43 b Fv(i)g(l)31 b FT(returns)e(the)h(\()p
FP(i)20 b FT(+)f(1\)th)31 b(elemen)m(t)f(in)f FP(l)j
FT(if)d FP(i)h FT(is)f(less)g(than)h(the)378 1133 y(length)k(of)g
FP(l)r FT(,)i(or)e(0)h(otherwise.)52 b(In)34 b(the)h(follo)m(wing,)f(w)
m(e)h(use)f(the)g(notation)h FP(x)d FN(\030)3179 1147
y FO(R)3268 1133 y FP(y)38 b FT(to)d(represen)m(t)378
1246 y(the)h(prop)s(osition)e Fv(R)44 b(x)g(y)s FT(,)38
b(where)e Fv(R)q Fw(:)86 b Fv(A)44 b Fu(!)f Fv(B)48 b
Fu(!)c Fw(Prop)34 b FT(is)h(a)i(binary)d(relation)i(on)g(the)g(sets)h
FP(A)378 1358 y FT(and)30 b FP(B)5 b FT(.)519 1471 y(The)38
b(seman)m(tics)h(of)f FN(P)7 b(RF)49 b FT(programs)38
b(is)f(giv)m(en)i(through)e(the)i(inductiv)m(e)e(de\014nition)f(of)j
(the)378 1584 y(predicate)26 b Fw(converges_to:)82 b(prf)42
b Fu(!)i Fw(\(list)d(nat\))h Fu(!)i Fw(nat)e Fu(!)i Fw(Prop)25
b FT(giv)m(en)h(b)s(elo)m(w.)39 b(W)-8 b(e)27 b(sa)m(y)g(that)378
1697 y FP(p)j FT(con)m(v)m(erges)i(to)f FP(v)j FT(on)c(input)f
FP(l)r FT(,)h(and)g(write)g FP(p)p FN(h)p FP(l)r FN(i)25
b(#)h FP(v)s FT(,)31 b(if)e Fw(converges_to)39 b Fv(p)k(l)i(v)34
b FT(holds.)378 1869 y FQ(Zero)46 b FT(F)-8 b(or)31 b(an)m(y)f(list)f
FP(l)r FT(,)i(the)f(program)h Fw(Zero)d FT(con)m(v)m(erges)33
b(to)e(0.)p 2009 2007 416 4 v 2009 2092 a Fw(Zero)n FN(h)p
FP(l)r FN(i)26 b(#)g FT(0)378 2330 y FQ(Successor)47
b FT(Giv)m(en)32 b(a)g(non-empt)m(y)g(list,)f Fw(Succ)f
FT(con)m(v)m(erges)k(to)f(the)f(successor)g(of)g(its)f(head,)h(other-)
605 2443 y(wise)e(it)g(con)m(v)m(erges)i(to)f Fw(\(S)43
b(0\))29 b FT(\(i.e.,)16 b(1\).)p 1461 2585 610 4 v 1461
2670 a Fw(Succ)o FN(h)p FT([])p FN(i)26 b(#)g Fw(\(S)43
b(0\))p 2253 2585 720 4 v 181 w(Succ)n FN(h)p FP(x)25
b FT(:)h FP(l)r FN(i)f(#)h Fw(\(S)43 b Fv(x)p Fw(\))378
2909 y FQ(Pro)6 b(jections)47 b FT(Giv)m(en)29 b(a)g(list)f
FP(l)r FT(,)i(the)f(pro)5 b(jection)29 b Fw(Proj)42 b
Fv(i)29 b FT(con)m(v)m(erges)i(to)f(the)f(\()p FP(i)19
b FT(+)f(1\)th)29 b(elemen)m(t)h(in)605 3021 y FP(l)r
FT(,)h(or)f(to)h(0)g(if)e FP(i)i FT(is)e(greater)j(than)e(the)h(length)
e(of)i FP(l)r FT(.)p 1815 3159 803 4 v 1815 3243 a Fw(Proj)42
b Fv(i)p FN(h)p FP(l)r FN(i)26 b(#)g Fw(\(zel)42 b Fv(i)h(l)r
Fw(\))378 3482 y FQ(Substitution)i FT(Giv)m(en)30 b(an)f(input)f(list)g
FP(l)r FT(,)i(the)g(program)g Fw(Sub)42 b Fv(f)52 b(g)46
b(n)e(m)29 b FT(\014rst)g(applies)f FP(g)33 b FT(to)e
FP(l)r FT(,)e(and)605 3595 y(then)g(applies)d FP(f)38
b FT(to)30 b(the)f FP(m)f FT(elemen)m(ts)h(in)e FP(l)k
FT(starting)d(from)g(the)h FP(n)p FT(th)f(one)h(together)h(with)e(the)
605 3708 y(output)i(of)h FP(g)j FT(\(see)d(\014gure)f(2\).)41
b(W)-8 b(e)32 b(de\014ne)860 3912 y Fw(pcombine)40 b
Fv(n)k(m)f(l)i(x)26 b FT(=)f([)p Fw(zel)42 b Fv(n)i(l)q
FP(;)15 b Fw(zel)43 b Ft(\()p Fv(n)19 b Ft(+)f(1\))43
b Fv(l)q FP(;)15 b(:)g(:)g(:)32 b(;)15 b Fw(zel)43 b
Ft(\()p Fv(n)18 b Ft(+)g Fv(m)h Fu(\000)f Ft(1\))43 b
Fv(l)r FP(;)15 b(x)p FT(])605 4116 y(and)30 b(the)h(seman)m(tics)f(of)h
Fw(Sub)e FT(is)g(giv)m(en)i(b)m(y)1530 4302 y FP(g)s
FN(h)p FP(l)r FN(i)26 b(#)g FP(x)91 b(f)10 b FN(h)p Fw(pcombine)40
b Fv(n)j(m)h(l)h(x)p FN(i)26 b(#)f FP(y)p 1530 4344 1374
4 v 1791 4429 a Fw(\(Sub)42 b Fv(f)52 b(g)47 b(n)c(m)p
Fw(\))o FN(h)p FP(l)r FN(i)26 b(#)g FP(y)378 4702 y FQ(Recursion)47
b FT(The)22 b(primitiv)m(e)f(recursiv)m(e)i(program)g
Fw(Rec)42 b Fv(f)52 b(g)26 b FT(has)c(base)i(case)g FP(f)32
b FT(and)22 b(recursion)g(step)605 4815 y FP(g)s FT(.)42
b(The)29 b(depth)h(of)g(the)h(recursion)e(is)g(giv)m(en)i(b)m(y)f(the)h
(\014rst)e(elemen)m(t)i(of)g(the)f(input)f(list.)804
5000 y FP(f)10 b FN(h)p FT([])p FN(i)25 b(#)h FP(x)p
640 5043 650 4 v 640 5128 a FT(\()p Fw(Rec)43 b Fv(f)52
b(g)s FT(\))p FN(h)p FT([])q FN(i)26 b(#)f FP(x)1696
5000 y(f)10 b FN(h)p FP(l)r FN(i)25 b(#)h FP(x)p 1472
5043 750 4 v 1472 5128 a FT(\()p Fw(Rec)43 b Fv(f)52
b(g)s FT(\))p FN(h)p FT(0)26 b(:)g FP(l)q FN(i)g(#)g
FP(x)2404 5000 y FT(\()p Fw(Rec)42 b Fv(f)52 b(g)s FT(\))p
FN(h)p FP(h)26 b FT(:)g FP(l)r FN(i)f(#)h FP(y)94 b(g)s
FN(h)p FP(h)26 b FT(:)g FP(y)i FT(:)e FP(l)q FN(i)g(#)g
FP(x)p 2404 5043 1390 4 v 2643 5128 a FT(\()p Fw(Rec)43
b Fv(f)52 b(g)s FT(\))p FN(h)p FT(\()p Fw(S)44 b Fv(h)p
FT(\))25 b(:)h FP(l)r FN(i)f(#)h FP(x)378 5366 y FQ(Minimalisation)45
b FT(The)f(program)h Fw(Min)d Fv(f)53 b FT(denotes)45
b(the)g(un)m(b)s(ounded)d(minimalisation)f(of)k(the)605
5479 y(function)33 b FP(f)10 b FT(.)52 b(Giv)m(en)34
b(the)g(input)f(list)f FP(l)r FT(,)k(it)e(returns)f(the)h(smallest)f(n)
m(um)m(b)s(er)g FP(h)i FT(suc)m(h)f(that)h FP(f)605 5592
y FT(returns)28 b(0)h(on)g(input)d FP(h)g FT(:)f FP(l)31
b FT(and)d(terminates)h(for)f(all)g(input)f FP(y)h FT(:)d
FP(l)31 b FT(where)d FP(y)g(<)d(h)p FT(.)41 b(In)28 b(order)g(to)605
5705 y(de\014ne)j(the)i(b)s(eha)m(viour)d(of)i Fw(Min)f
FT(w)m(e)h(\014rst)g(in)m(tro)s(duce)f(the)h(predicates)f
Fw(all_successors)26 b FT(and)p eop
%%Page: 37 47
37 46 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(37)922
1775 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 56.90549 113.81097
113.81097 113.81097 113.81097 56.90549 56.90549 56.90549 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial 1649 1091
a Fn(g)1402 991 y
tx@Dict begin tx@NodeDict begin {4.71457 0.0 6.57257 3.28629 2.35728
} false /N@N 16 {InitRnode } NewNode end end
1402 991 a FP(n)1430 1161 y
tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 0.0 } false /N@O
16 {InitRnode } NewNode end end
1430 1161
a 922 1775 a
tx@Dict begin gsave STV newpath 0.8 SLW 0. setgray /ArrowA { BeginArrow
1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow 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@N /N@O
InitNC { NCLine } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
922 1775 a 1451 1086 a FP(m)922 1775 y @beginspecial
@setspecial
tx@Dict begin STP 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 [ 56.90549 108.1205 5.69046 108.1205 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 56.90549 96.73915 5.69046 96.73915 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 56.90549 73.9773 5.69046 73.9773 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 56.90549 62.59595 5.69046 62.59595 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial 1169 897 a FT(.)1169 931 y(.)1169
964 y(.)1169 1039 y(.)1169 1072 y(.)1169 1105 y(.)1169
1181 y(.)1169 1214 y(.)1169 1247 y(.)922 1775 y @beginspecial
@setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 199.1692 113.81097
256.07469 113.81097 256.07469 56.90549 199.1692 56.90549 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial 2751 1118 a Fn(f)922 1775 y
@beginspecial @setspecial
tx@Dict begin STP 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 [ 278.83696 85.35823 256.07469 85.35823 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 19.91682 96.73915 Dot end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 199.1692 102.43004 176.40692 102.43004 176.40692 150.79962
19.91682 150.79962 19.91682 96.73915 /Lineto /lineto load def false
Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial
2468 992 a FT(.)2468 1025 y(.)2468 1058 y(.)1724 590
y(.)1724 623 y(.)1724 657 y(.)922 1775 y @beginspecial
@setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 42.67911 73.9773 Dot end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 199.1692 79.66776 153.64464 79.66776 153.64464 128.03734 42.67911
128.03734 42.67911 73.9773 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 199.1692 68.2864 142.26372 68.2864 142.26372 85.35823 113.81097
85.35823 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial 2600 954 a
tx@Dict begin tx@NodeDict begin {7.05666 0.0 5.475 2.7375 3.52832
} false /N@Z 16 {InitRnode } NewNode end end
2600
954 a FT(0)2595 1227 y FP(m)1357 1645 y FT(Figure)30
b(2:)41 b(The)30 b(Beha)m(viour)g(of)h FM(Sub)e FP(f)40
b(g)s FT(.)605 2038 y Fw(minl)o FT(.)64 b(The)38 b(prop)s(osition)e
Fw(all_successors)i Fv(R)44 b(n)39 b FT(holds)e(if)g(for)h(all)g
FP(m)g FN(\024)g FP(n)p FT(,)j(there)d(exists)605 2151
y(some)31 b FP(k)i FT(suc)m(h)e(that)g FP(m)25 b FN(\030)1492
2165 y FO(R)1574 2151 y FT(\()p Fw(S)44 b Fv(k)s FT(\).)1142
2347 y(0)26 b FN(\030)1284 2361 y FO(R)1367 2347 y FT(\()p
Fw(S)43 b Fv(k)s FT(\))p 955 2390 803 4 v 955 2463 a
Fw(all_successors)38 b Fv(R)44 b Ft(0)1940 2342 y FT(\()p
Fw(S)f Fv(m)p FT(\))26 b FN(\030)2267 2356 y FO(R)2349
2342 y FT(\()p Fw(S)44 b Fv(k)s FT(\))91 b Fw(all_successors)38
b Fv(R)44 b(m)p 1940 2384 1539 4 v 2216 2463 a Fw(all_successors)38
b Fv(R)44 b Ft(\()p Fw(S)f Fv(m)p Ft(\))605 2668 y FT(The)27
b(prop)s(osition)e Fw(minl)42 b Fv(R)i(n)27 b FT(holds)f(if)g
FP(n)h FT(is)f(the)i(smallest)e(n)m(um)m(b)s(er)g(suc)m(h)h(that)h
FP(n)d FN(\030)3526 2682 y FO(R)3609 2668 y FT(0)i(and)605
2780 y(for)j(all)g FP(m)25 b FN(\024)g FP(n)p FT(,)30
b(there)g(is)g(some)h FP(k)i FT(suc)m(h)d(that)h FP(n)25
b FN(\030)2367 2794 y FO(R)2450 2780 y FP(k)s FT(.)1324
2980 y(0)h FN(\030)1466 2994 y FO(R)1548 2980 y FT(0)p
1275 3014 367 4 v 1275 3087 a Fw(minl)42 b Fv(R)j Ft(0)1824
2966 y FT(\()p Fw(S)e Fv(n)p FT(\))26 b FN(\030)2128
2980 y FO(R)2211 2966 y FT(0)91 b Fw(all_successors)38
b Fv(R)44 b(n)p 1824 3009 1334 4 v 2227 3087 a Fw(minl)e
Fv(R)j Ft(\()p Fw(S)e Fv(n)p Ft(\))605 3292 y FT(The)30
b(b)s(eha)m(viour)f(of)i Fw(Min)42 b Fv(f)d FT(is)29
b(then)h(giv)m(en)h(b)m(y)f(the)h(rule)1569 3471 y Fw(minl)41
b Ft(\()p Fv(\025h:)p Fw(converges_to)f Fv(f)52 b(h)23
b Ft(:)g Fv(l)q Ft(\))44 b Fv(x)p 1569 3512 1297 4 v
1946 3597 a FT(\()p Fw(Min)e Fv(f)9 b FT(\))p FN(h)p
FP(l)r FN(i)26 b(#)g FP(x)378 3838 y FT(The)35 b(mec)m(hanisation)h(in)
e(Co)s(q)i(con)m(tains)g(a)g(pro)s(of)f(that)i(the)f(relation)f
Fw(converges_to)c FT(as)36 b(de\014ned)378 3951 y(ab)s(o)m(v)m(e)c(is)d
(\(at)i(most\))g(single-v)-5 b(alued,)29 b(that)i(is)f
FN(P)7 b(RF)40 b FT(programs)30 b(are)h(deterministic.)519
4064 y(A)k(URM)g(program)g(uses)f(a)h(sp)s(eci\014c)f(n)m(um)m(b)s(er)f
(of)i(elemen)m(ts)h(from)e(the)h(list.)53 b(The)34 b(maxim)m(um)378
4177 y(v)-5 b(alue)33 b(of)h(the)f(p)s(ositions)f(of)i(the)f(elemen)m
(ts)h(used)f(b)m(y)g(a)h(program)g(is)e(called)h(the)h
FI(natur)-5 b(al)37 b(arity)43 b FT(of)378 4290 y(the)31
b(program,)f(and)g(is)f(de\014ned)g(as)i(follo)m(ws:)473
4478 y FN(`)529 4493 y FE(def)686 4478 y FM(natarity)46
b(Zero)g(=)i FT(0)998 4591 y FM(|)g(Succ)e(=)i(\(S)f
FT(0)p FM(\))998 4703 y(|)h(Proj)e FP(i)j FM(=)e(\(S)g
FP(i)p FM(\))998 4816 y(|)h(Sub)f FP(f)57 b(g)51 b(n)c(m)h
FM(=)f(max)g(\(natarity)f FP(g)s FM(\))i(\()p FP(n)67
b FT(+)h FP(m)p FM(\))998 4929 y(|)48 b(Rec)f FP(b)h(s)f
FM(=)g(max)g(\(S)g(\(natarity)f FP(b)p FM(\))h(\(pred)f(\(natarity)g
FP(s)p FM(\)\))998 5042 y(|)i(Min)f FP(f)57 b FM(=)47
b(pred)g(\(natarity)e FP(f)10 b FM(\))378 5230 y FT(The)49
b(maxim)m(um)f(natural)g(arit)m(y)h(of)g(a)h(list)e(of)h(programs)g(is)
f(then)h(de\014ned)e(as)j(the)f(function)378 5343 y Fw(maxarity:)83
b(\(list)42 b(prf\))g Fu(!)h Fw(nat)o FT(.)67 b(It)40
b(is)e(then)h(sho)m(wn)g(that)h(the)f(b)s(eha)m(viour)f(of)h(a)h
(program)f(is)378 5456 y(not)28 b(a\013ected)i(b)m(y)e(the)h(elemen)m
(ts)g(in)e(its)g(input)g(list)f(at)k(a)e(p)s(osition)f(greater)i(than)f
(its)g(natural)f(arit)m(y)-8 b(.)p eop
%%Page: 38 48
38 47 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(38)473
1539 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 34.14365 68.2873 68.2873
68.2873 68.2873 34.14365 34.14365 34.14365 /Lineto /lineto load def
false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial 885 1124
a FP(g)928 1138 y FL(0)473 1539 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 102.43094 68.2873
136.57458 68.2873 136.57458 34.14365 102.43094 34.14365 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial 1452 1124 a FP(g)1495 1138 y FL(1)473 1539
y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 170.71823 68.2873
204.86188 68.2873 204.86188 34.14365 170.71823 34.14365 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial 2019 1124 a
FP(g)2062 1138 y FL(2)473 1539 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 273.14917 68.2873
307.29282 68.2873 307.29282 34.14365 273.14917 34.14365 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial 2823 1122 a FP(g)2866 1137 y FO(k)r FK(\000)p
FL(1)473 1539 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray [ 341.43646 68.2873
375.58011 68.2873 375.58011 34.14365 341.43646 34.14365 /Lineto /lineto
load def false Polygon gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore end
@endspecial
3451 1108 a FP(f)2319 1134 y FN(\001)15 b(\001)g(\001)473
1539 y @beginspecial @setspecial
tx@Dict begin STP 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 [ 34.14365 64.87297 3.4143 64.87297 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial 772 1028
a FL(0)473 1539 y @beginspecial @setspecial
tx@Dict begin STP 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 [ 34.14365 51.21547 3.4143 51.21547 /Lineto /lineto load def
false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
759 1134 a FO(m)473 1539 y @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 9.38954 64.87297 Dot end
@endspecial @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 23.04706 51.21547 Dot end
@endspecial 595
1032 a FT(.)595 1065 y(.)595 1098 y(.)473 1539 y @beginspecial
@setspecial
tx@Dict begin STP 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 [ 102.43094 64.87297 93.04138 64.87297 93.04138 95.60231 9.38954
95.60231 9.38954 64.87297 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 102.43094 51.21547 84.50546 51.21547 84.50546 81.9448 23.04706
81.9448 23.04706 51.21547 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 102.43094 42.67955 75.96956 42.67955 75.96956 51.21547 68.2873
51.21547 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 93.04138 95.60231 Dot end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 170.71823 64.87297 161.32867 64.87297 161.32867 95.60231 93.04138
95.60231 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 84.50546 81.9448 Dot end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 170.71823 51.21547 152.79276 51.21547 152.79276 81.9448 84.50546
81.9448 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 88.77342 42.67955 Dot end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 170.71823 42.67955 147.67116 42.67955 147.67116 78.53049 88.77342
78.53049 88.77342 42.67955 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 170.71823 39.26524 143.4032 39.26524 143.4032 51.21547 136.57458
51.21547 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial @beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 161.32867 95.60231 Dot end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 216.81209 95.60231 161.32867 95.60231 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 152.79276 81.9448 Dot end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 216.81209 81.9448 152.79276 81.9448 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 147.67116 78.53049 Dot end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 216.81209 78.53049 147.67116 78.53049 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 216.81209 51.21547 204.86188 51.21547 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 157.06071 39.26524 Dot end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 216.81209 75.1159 157.06071 75.1159 157.06071 39.26524 /Lineto
/lineto load def false Line gsave 0.8 SLW 0. setgray 0 setlinecap
stroke grestore end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 273.14917 64.87297 244.12712 64.87297 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP 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 [ 273.14917 51.21547 244.12712 51.21547 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 273.14917 42.67955 244.12712 42.67955 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 273.14917 37.55795 244.12712 37.55795 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
@beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 248.39507 42.67955 Dot end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 341.43646 61.45866 332.0469 61.45866 332.0469 95.60231 248.39507
95.60231 248.39507 42.67955 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial 3323 1056 a FL(0)473 1539 y
@beginspecial @setspecial
tx@Dict begin STP newpath 0.8 SLW 0. setgray /DS 1.8 1.8 CLW mul
add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot
/Dot { moveto (b) show } bind def 262.05258 37.55795 Dot end
@endspecial @beginspecial
@setspecial
tx@Dict begin STP 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 [ 341.43646 47.80115 323.51099 47.80115 323.51099 81.9448 262.05258
81.9448 262.05258 37.55795 /Lineto /lineto load def false Line gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore end
@endspecial @beginspecial @setspecial
tx@Dict begin STP 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 [ 341.43646 40.97226 314.97508 40.97226 314.97508 51.21547 307.29282
51.21547 /Lineto /lineto load def false Line gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore end
@endspecial
3319 1224 a FO(k)r FK(\000)p FL(1)2870 777 y FT(.)2870
810 y(.)2870 843 y(.)999 777 y(.)999 810 y(.)999 843
y(.)473 1539 y @beginspecial @setspecial
tx@Dict begin STP 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 [ 389.23761 51.21547 375.58011 51.21547 /Lineto /lineto load
def false Line gsave 0.8 SLW 0. setgray 0 setlinecap stroke grestore
end
@endspecial
742 1496 a FP(m)25 b FT(=)g Fw(maxarity)40 b([)p Fv(g)1419
1508 y Fs(0)1455 1496 y Fw(,)p Fv(:)14 b(:)g(:)p Fw(,)p
Fv(g)1680 1508 y Fq(k)q Fr(\000)p Fs(1)1805 1496 y Fw(])20
b FN(\000)f FT(1)1001 1692 y(Figure)30 b(3:)41 b(The)30
b(De\014nition)f(of)h FM(Subl)f FP(m)h(n)g(f)40 b FT([)p
FP(g)2681 1706 y FL(0)2721 1692 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(g)2981 1707 y FO(k)r FK(\000)p FL(1)3114 1692
y FT(].)378 2090 y FQ(Substitution)35 b(of)g(a)f(List)h(of)g(F)-9
b(unctions)378 2262 y FT(W)h(e)37 b(no)m(w)e(de\014ne)g(the)h(function)
e Fw(Subl:)85 b(prf)43 b Fu(!)g Fw(\(list)f(prf\))g Fu(!)h
Fw(prf)35 b FT(suc)m(h)g(that,)j(giv)m(en)d(a)h(pro-)378
2375 y(gram)31 b FP(f)40 b FT(and)30 b(a)h(list)e(of)i(programs)f
FP(g)f FT(=)c([)p FP(g)1841 2389 y FL(0)1881 2375 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(g)2141 2390 y FO(k)r FK(\000)p
FL(1)2275 2375 y FT(],)31 b(the)g(program)f Fw(Subl)42
b Fv(f)52 b(g)33 b FT(con)m(v)m(erges)g(to)e FP(y)378
2488 y FT(on)f(input)f FP(l)r FT(,)h(if)514 2675 y FN(\017)46
b FT(for)30 b(all)g FP(i)25 b(<)g(k)s FT(,)31 b(there)f(is)g(some)h
FP(x)1733 2689 y FO(i)1791 2675 y FT(suc)m(h)f(that)h
FP(g)2236 2689 y FO(i)2265 2675 y FN(h)p FP(l)r FN(i)26
b(#)f FP(x)2512 2689 y FO(i)2540 2675 y FT(,)31 b(and)514
2863 y FN(\017)46 b FT(the)31 b(program)f FP(f)10 b FN(h)p
FT([)p FP(x)1293 2877 y FL(0)1332 2863 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)1601 2878 y FO(k)r FK(\000)p FL(1)1734 2863
y FT(])p FN(i)26 b(#)g FP(y)s FT(.)1036 3128 y FP(g)1079
3142 y FL(0)1119 3128 y FN(h)p FP(l)r FN(i)g(#)g FP(x)1367
3142 y FL(0)1497 3128 y FN(\001)15 b(\001)g(\001)107
b FP(g)1752 3143 y FO(k)r FK(\000)p FL(1)1885 3128 y
FN(h)p FP(l)r FN(i)26 b(#)g FP(x)2133 3143 y FO(k)r FK(\000)p
FL(1)2357 3128 y FP(f)10 b FN(h)p FT([)p FP(x)2524 3142
y FL(0)2563 3128 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)2832
3143 y FO(k)r FK(\000)p FL(1)2965 3128 y FT(])p FN(i)26
b(#)g FP(y)p 1036 3171 2133 4 v 1547 3256 a FT(\()p Fw(Subl)42
b Fv(f)52 b Ft([)p Fv(g)1956 3268 y Fs(0)1993 3256 y
Fv(;)14 b(:)g(:)g(:)28 b(;)14 b(g)2232 3268 y Fq(k)q
Fr(\000)p Fs(1)2357 3256 y Ft(])p FT(\))q FN(h)p FP(l)r
FN(i)25 b(#)h FP(y)378 3460 y FT(This)21 b(is)i(ac)m(hiev)m(ed)h(b)m(y)
f(using)f(the)h(op)s(erator)h Fw(Sub)e FT(to)i(pass)e(the)i(input)d
(list)h(together)j(with)d(the)h(output)378 3573 y(v)-5
b(alues)26 b FP(x)697 3587 y FL(0)736 3573 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(x)1005 3587 y FO(i)p FK(\000)p
FL(1)1150 3573 y FT(to)27 b(the)g(program)f FP(g)1813
3587 y FO(i)1868 3573 y FT(where)g FP(i)g(<)f(k)s FT(.)39
b(The)26 b FP(k)k FT(outputs)c FP(x)3036 3587 y FL(0)3075
3573 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)3344 3588 y
FO(k)r FK(\000)p FL(1)3503 3573 y FT(are)27 b(then)378
3686 y(giv)m(en)41 b(to)h(the)f(program)f FP(f)51 b FT(\(see)41
b(\014gure)g(3\).)73 b(F)-8 b(or)42 b FP(i)h(<)f(k)s
FT(,)i(the)d(output)g(v)-5 b(alues)40 b FP(x)3308 3700
y FO(i)3377 3686 y FT(are)i(k)m(ept)f(in)378 3799 y(the)32
b(input)e(list)h(of)h(the)h(program)e FP(g)1606 3813
y FO(i)p FL(+1)1757 3799 y FT(at)i(a)f(p)s(osition)f(whic)m(h)f(is)h
(greater)j(than)e(its)f(natural)g(arit)m(y)-8 b(,)378
3911 y(and)36 b(therefore)i(do)s(es)f(not)g(a\013ect)h(the)g(output)e
(v)-5 b(alue)37 b FP(x)2343 3925 y FO(i)p FL(+1)2461
3911 y FT(.)61 b(The)36 b(function)g Fw(Subl)f FT(is)h(de\014ned)g(b)m
(y)378 4024 y(structural)29 b(recursion)g(o)m(v)m(er)j(the)e(list)f(of)
i(programs)f([)p FP(g)2263 4038 y FL(0)2303 4024 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(g)2563 4039 y FO(k)r FK(\000)p
FL(1)2696 4024 y FT(])31 b(as)f(follo)m(ws:)473 4212
y FN(`)529 4227 y FE(def)686 4212 y FM(Subl)p 884 4212
29 4 v 33 w(in)48 b FP(f)57 b(m)47 b FT([])h FP(n)143
b FN(\021)48 b FM(Sub)f FP(f)57 b FM(Zero)46 b FT(0)i(0)951
4325 y FM(|)f FP(f)57 b(m)48 b FT([)p FP(g)1344 4339
y FL(0)1384 4325 y FT(])g FP(n)63 b FN(\021)47 b FM(Sub)g
FP(f)57 b(g)2029 4339 y FL(0)2117 4325 y FP(m)47 b(n)951
4438 y FM(|)g FP(f)57 b(m)48 b FT(\()p FP(g)1354 4452
y FL(0)1419 4438 y FT(:)26 b FP(g)1513 4452 y FL(1)1578
4438 y FT(:)f FP(g)s FT(\))49 b FP(n)1142 4551 y FN(\021)e
FM(Sub)g(\(Subl)p 1697 4551 V 33 w(in)g FP(f)57 b(m)48
b FM(\()p FP(g)2188 4565 y FL(1)2252 4551 y FT(:)26 b
FP(g)s FM(\))48 b(\(S)f FP(n)p FM(\)\))g FP(g)2829 4565
y FL(0)2916 4551 y FT(0)i(\()p FP(m)20 b FT(+)g FP(n)p
FT(\))473 4777 y FN(`)529 4792 y FE(def)686 4777 y FM(Subl)47
b FP(f)57 b(g)51 b FN(\021)d FM(\(Subl)p 1486 4777 V
33 w(in)f FP(f)57 b FM(\(maxarity)45 b FP(g)s FM(\))j
FP(g)k FT(0)p FM(\))519 4964 y FT(The)28 b(follo)m(wing)f(theorem)i(is)
f(then)g(deriv)m(ed)g(to)i(sho)m(w)e(that)h(programs)g(constructed)g
(using)e(the)378 5077 y(function)i Fw(Subl)g FT(ha)m(v)m(e)i(the)g(exp)
s(ected)g(b)s(eha)m(viour.)473 5265 y FN(`)48 b(8)p FP(f)56
b(g)s(l)50 b(l)g(x)p FM(.)569 5378 y(\(converges)p 1055
5378 V 31 w(to)e(\(Subl)e FP(f)57 b(g)s(l)r FM(\))48
b FP(l)i(x)p FM(\))d FN(,)569 5491 y FM(\()p FN(9)15
b FP(xl)r FM(.)47 b(\(mapR)g(prf)g(nat)g(\()p FP(\025g)s
FM(.)g(converges)p 2208 5491 V 32 w(to)g FP(g)52 b(l)r
FM(\))47 b FP(g)s(l)j(xl)r FM(\))e FN(^)855 5603 y FM(\(converges)p
1341 5603 V 32 w(to)f FP(f)57 b(xl)50 b(x)p FM(\)\))p
eop
%%Page: 39 49
39 48 bop 378 5 a FF(CHAPTER)30 b(3.)61 b(CASE)29 b(STUDIES)h(ON)g(T)-8
b(A)m(CTIC-BASED)30 b(THEOREM)g(PR)m(O)m(VERS)185 b FT(39)378
396 y(The)30 b(relation)g Fw(mapR:)42 b(\()p Fv(A)h Fu(!)h
Fv(B)j Fu(!)d Fw(Prop\))d Fu(!)j Fw(\(list)e Fv(A)p Fw(\))h
Fu(!)h Fw(\(list)d Fv(B)t Fw(\))i Fu(!)h Fw(Prop)29 b
FT(is)h(de\014ned)f(in)378 509 y(Co)s(q)22 b(suc)m(h)f(that)i
FP(l)974 523 y FL(1)1039 509 y FN(\030)1110 535 y FL(\()p
Fw(mapR)42 b Fv(R)p FL(\))1475 509 y FP(l)1502 523 y
FL(2)1564 509 y FT(holds)20 b(if)h FP(l)1895 523 y FL(1)1957
509 y FT(and)g FP(l)2152 523 y FL(2)2214 509 y FT(ha)m(v)m(e)i(the)f
(same)h(length)e(and)g(all)g(the)i(elemen)m(ts)378 622
y(in)29 b FP(l)511 636 y FL(1)581 622 y FT(relate)i(\(with)e(resp)s
(ect)h(to)i FP(R)q FT(\))e(with)f(the)i(corresp)s(onding)d(elemen)m(ts)
j(in)e FP(l)3087 636 y FL(2)3126 622 y FT(.)p 1270 872
563 4 v 1270 956 a([])d FN(\030)1417 982 y FL(\()p Fw(mapR)42
b Fv(R)q FL(\))1782 956 y FT([])2026 808 y FP(a)26 b
FN(\030)2171 822 y FO(R)2253 808 y FP(b)91 b(k)29 b FN(\030)2530
833 y FL(\()p Fw(mapR)42 b Fv(R)p FL(\))2895 808 y FP(l)p
2015 872 921 4 v 2015 956 a FT(\()p FP(a)25 b FT(:)h
FP(k)s FT(\))g FN(\030)2356 982 y FL(\()p Fw(mapR)42
b Fv(R)q FL(\))2721 956 y FT(\()p FP(b)26 b FT(:)g FP(l)r
FT(\))378 1200 y FG(3.3.3)112 b Fo(P)8 b(RF)48 b FG(Computabilit)m(y)
378 1371 y FT(The)24 b FN(P)7 b(RF)34 b FT(language)25
b(is)e(used)g(as)i(a)f(mo)s(del)g(of)g(computation)g(b)m(y)g
(de\014ning)f(computable)g(functions)378 1484 y(as)31
b(those)g(whic)m(h)e(can)i(b)s(e)f(computed)h(b)m(y)f(a)h
FN(P)7 b(RF)41 b FT(program.)g(Similarly)27 b(to)k(the)g(implemen)m
(tation)378 1597 y(in)40 b(HOL)i(w)m(e)g(\014rst)f(de\014ne)g(the)h(t)m
(yp)s(e)g(of)g FP(n)p FT(-ary)f(partial)g(functions)f(from)h(natural)g
(n)m(um)m(b)s(ers)f(to)378 1710 y(natural)29 b(n)m(um)m(b)s(ers,)h(and)
f(then)h(de\014ne)g(the)h(notion)e(of)i FN(P)7 b(RF)j
FT(-computable)30 b(functions.)378 1950 y FQ(V)-9 b(ectors)36
b(and)f(P)m(artial)f(F)-9 b(unctions)378 2122 y FT(The)30
b(set)h(of)f(v)m(ectors)i(o)m(v)m(er)g(a)f(set)f FP(A)h
FT(is)e(de\014ned)h(inductiv)m(ely)e(b)m(y)473 2310 y
FM(vector)46 b FP(A)i FM(nat)f(::=)g(Vnil:)g(\(vector)e
FP(A)j FT(0)p FM(\))1110 2422 y(|)95 b(Vcons:)46 b(\()p
FP(n)p FM(:)h(nat\))f FN(!)i FP(A)g FN(!)g FM(\(vector)d
FP(A)j(n)p FM(\))1810 2535 y FN(!)f FM(\(vector)f FP(A)i
FM(\(S)f FP(n)p FM(\)\))378 2723 y FT(The)25 b(t)m(yp)s(e)h
Fw(vector)42 b Fv(A)h(n)26 b FT(of)g(v)m(ectors)i(with)c
FP(n)i FT(elemen)m(ts)g(of)g(t)m(yp)s(e)g FP(A)g FT(is)f
FI(dep)-5 b(endent)36 b FT(on)26 b(the)g FI(values)34
b FT(of)378 2836 y FP(A)25 b FT(and)g FP(n)p FT(.)39
b(Suc)m(h)25 b(a)g(t)m(yp)s(e)h(cannot)g(b)s(e)f(de\014ned)f(in)g(HOL)h
(b)s(ecause)g(of)h(its)f(w)m(eak)m(er)i(t)m(yp)s(e)e(system.)39
b(The)378 2949 y(head,)34 b(tail)e(and)g(the)h(elemen)m(ts)g(in)f(a)h
(particular)f(p)s(osition)f(in)g(a)j(v)m(ector)g(are)f(de\014ned)f
(inductiv)m(ely)378 3062 y(b)m(y)e(the)h(relations:)p
771 3192 1203 4 v 771 3266 a Fw(Vhd)43 b Fv(A)h Fw(\(S)e
Fv(n)p Fw(\))h(\(Vcons)e Fv(n)j(h)f(t)p Fw(\))g Fv(h)p
2156 3192 1279 4 v 182 w Fw(Vtl)f Fv(A)i Fw(\(S)f Fv(n)p
Fw(\))g Fv(n)g Fw(\(Vcons)f Fv(n)h(h)g(t)p Fw(\))h Fv(t)p
644 3449 1288 4 v 644 3523 a Fw(Vel)e Fv(A)i Ft(0)f Fw(\(S)g
Fv(n)p Fw(\))g(\(Vcons)e Fv(n)i(h)h(t)p Fw(\))f Fv(h)2554
3429 y Fw(Vel)g Fv(A)h(i)f(n)g(t)h(x)p 2113 3449 1450
4 v 2113 3523 a Fw(Vel)f Fv(A)h Fw(\(S)e Fv(i)p Fw(\))h(\(S)g
Fv(n)p Fw(\))g(\(Vcons)e Fv(n)i(h)h(t)p Fw(\))f Fv(x)378
3690 y FT(The)f(head,)j(tail,)f(and)e(the)g FP(i)p FT(th)g(elemen)m(t)h
(in)e(a)h(v)m(ector)i(for)e(some)h FP(i)p FT(,)i(are)e(also)f
(de\014ned)f(b)m(y)h(the)378 3803 y(functions:)426 3991
y FM(vhd:)k(\()p FP(A)p FM(:)i(Set\))e FN(!)i FM(\()p
FP(n)p FM(:)f(nat\))f FN(!)i FM(\(vector)e FP(A)i FM(\(S)f
FP(n)p FM(\)\))g FN(!)g FP(A)426 4104 y FM(vtl:)f(\()p
FP(A)p FM(:)i(Set\))e FN(!)i FM(\()p FP(n)p FM(:)f(nat\))f
FN(!)i FM(\(vector)e FP(A)i FM(\(S)f FP(n)p FM(\)\))g
FN(!)g FM(\(vector)f FP(A)i(n)p FM(\))426 4217 y(vel:)e(\()p
FP(A)p FM(:)i(Set\))e FN(!)i FM(\()p FP(i)p FM(:)f(nat\))g
FN(!)h FM(\()p FP(n)p FM(:)e(nat\))h FN(!)h FM(\(Hl:)e
FP(i)i(<)g(n)p FM(\))1332 4330 y FN(!)g FM(\(vector)e
FP(A)i(n)p FM(\))f FN(!)h FP(A)378 4517 y FT(Note)23
b(that)e(the)h(t)m(yp)s(e)f(of)h(the)f(fourth)g(argumen)m(t)g(of)h
Fw(vel)e FT(is)g(the)h(prop)s(osition)e Fw(\()p Fv(i)43
b(<)g(n)p Fw(\))21 b FT(and)f(therefore)378 4630 y(terms)30
b(in)m(v)m(olving)e Fw(vel)g FT(need)i(a)g(pro)s(of)f(that)h(the)g
(second)g(argumen)m(t)g(is)f(smaller)f(than)h(the)h(third)e(in)378
4743 y(order)i(to)h(b)s(e)f(correctly)g(t)m(yp)s(ed.)519
4856 y(In)24 b(general,)i(theorems)f(in)m(v)m(olving)f(the)h
Fw(Vhd)n FT(,)h Fw(Vtl)e FT(and)g Fw(Vel)f FT(relations)h(are)h(easier)
g(to)h(pro)m(v)m(e)f(than)378 4969 y(those)30 b(in)m(v)m(olving)f(the)h
(functional)e(coun)m(terparts)i(if)f(rule)f(induction)g(can)i(b)s(e)f
(used.)40 b(On)29 b(the)h(other)378 5082 y(hand,)44 b(theorems)e(and)f
(assumptions)g(in)m(v)m(olving)f(equalities)h(on)g(terms)h(con)m
(taining)g(the)g(ab)s(o)m(v)m(e)378 5195 y(functions)35
b(can)i(b)s(e)g(used)f(as)h(rewriting)e(rules.)58 b(In)36
b(order)h(to)g(obtain)f(the)h(b)s(est)g(of)g(b)s(oth)f(w)m(orlds,)378
5308 y(the)f(t)m(w)m(o)i(kinds)c(of)i(de\014nitions)e(are)i(in)m(tro)s
(duced)f(in)f(the)j(mec)m(hanisation)e(and)h(are)g(sho)m(wn)g(to)h(b)s
(e)378 5420 y(equiv)-5 b(alen)m(t.)519 5533 y(The)41
b(follo)m(wing)f(t)m(w)m(o)j(functions)d(whic)m(h)g(map)h(v)m(ectors)i
(in)m(to)f(lists)e(and)h(vice-v)m(ersa)h(are)g(also)378
5646 y(de\014ned:)p eop
%%Page: 40 50
40 49 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(40)426
396 y FM(listify:)93 b(\()p FP(A)p FM(:)47 b(Set\))g
FN(!)h FM(\()p FP(n)p FM(:)e(nat\))h FN(!)h FM(\(vector)d
FP(A)j(n)p FM(\))f FN(!)h FM(\(list)e FP(A)p FM(\))426
509 y(vectrify:)f(\()p FP(A)p FM(:)i(Set\))g FN(!)h FM(\()p
FP(l)r FM(:)f(list)f FP(A)p FM(\))i FN(!)g FM(\(vector)d
FP(A)j FM(\(length)e FP(A)i(l)r FM(\)\))378 697 y FT(where)30
b Fw(length)41 b Fv(A)j(l)31 b FT(returns)f(the)g(length)g(of)h(list)e
FP(l)j FT(whose)e(elemen)m(ts)h(are)f(in)f(the)i(set)g
FP(A)p FT(.)519 810 y(The)e(t)m(yp)s(e)g(of)g FP(n)p
FT(-ary)g(partial)f(recursiv)m(e)g(functions)g(o)m(v)m(er)i(the)g
(natural)e(n)m(um)m(b)s(ers)f(is)h(de\014ned)g(to)378
923 y(b)s(e)i(that)h(of)f(the)h(single-v)-5 b(alued)28
b(relations)i(b)s(et)m(w)m(een)h Fw(vector)41 b(nat)h
Fv(n)30 b FT(and)g Fw(nat)o FT(:)473 1110 y FM(pfunc)47
b FI(arity)57 b FN(\021)47 b FM(mk)p 1223 1110 29 4 v
34 w(pfunc)760 1223 y FN(f)h FM(reln)285 b(:)48 b(\(Rel)e(\(vector)g
(nat)h FI(arity)9 b FM(\))48 b(nat\);)855 1336 y(One)p
1005 1336 V 34 w(valued:)e(\(one)p 1613 1336 V 33 w(valued)g(\(vector)g
(nat)h FI(arity)9 b FM(\))47 b(nat)g(reln\))p FN(g)378
1524 y FT(where)21 b Fw(Rel)42 b Fv(A)i(B)25 b FT(is)c(the)g(t)m(yp)s
(e)h(of)f(the)h(relations)e(b)s(et)m(w)m(een)i(the)f(sets)h
FP(A)g FT(and)e FP(B)5 b FT(,)23 b(and)e(the)g(prop)s(osition)378
1637 y Fw(one_valued)39 b Fv(A)44 b(B)k(R)31 b FT(holds)e(if)g(the)i
(relation)e FP(R)i FT(is)f(single-v)-5 b(alued.)473 1824
y FN(`)529 1839 y FE(def)686 1824 y FM(one_valued)45
b FP(A)j(B)k(R)d FN(\021)e(8)p FP(a)h(b)f(c)p FM(.)h
FT(\()p FP(R)h(a)f(b)p FT(\))g FN(\))f FT(\()p FP(R)i(a)f(c)p
FT(\))g FN(\))g FT(\()p FP(b)73 b FT(=)g FP(c)p FT(\))519
2012 y(The)36 b(t)m(yp)s(e)h Fw(pfunc)e FT(is)h(a)h(record)f(where)h
(the)g(\014eld)e Fw(reln)g FT(is)h(a)h(relation)f(b)s(et)m(w)m(een)h(v)
m(ectors)h(and)378 2125 y(natural)21 b(n)m(um)m(b)s(ers,)h(and)g(the)g
(\014eld)e Fw(One_valued)e FT(is)j(a)h(theorem)h(stating)f(that)g
Fw(reln)f FT(is)g(single-v)-5 b(alued.)378 2238 y(It)34
b(can)g(b)s(e)f(seen)h(that)h(this)e(is)g(a)h(dep)s(enden)m(t)f(record)
h(as)g(the)g(t)m(yp)s(e)g(of)g(the)g(second)g(\014eld)f(dep)s(ends)378
2351 y(on)g(the)g(v)-5 b(alue)32 b(of)h(the)g(\014rst)f(\014eld.)46
b(The)32 b(t)m(yp)s(e)h Fw(pfunc)e FT(can)i(b)s(e)f(considered)g(as)h
(a)g(subt)m(yp)s(e)e(of)i Fw(reln)o FT(,)378 2464 y(since)d(ob)5
b(jects)31 b(of)f(t)m(yp)s(e)h Fw(pfunc)d FT(are)i(the)h(ob)5
b(jects)31 b(of)f(t)m(yp)s(e)h Fw(reln)e FT(whic)m(h)g(are)h(pro)m(v)m
(ed)h(to)g(satisfy)f(the)378 2577 y(prop)s(ert)m(y)g(giv)m(en)g(b)m(y)g
Fw(One_valued)l FT(.)519 2690 y(Giv)m(en)21 b(a)h(function)e
Fw(g:\(vector)39 b(nat)k Fv(n)p Fw(\))g Fu(!)g Fw(nat)20
b FT(one)i(can)f(construct)h(a)f(total)h(single-v)-5
b(alued)20 b(re-)378 2802 y(lation)h FP(G)g FT(suc)m(h)g(that)i
FP(v)28 b FN(\030)1248 2816 y FO(G)1332 2802 y FP(c)22
b FT(if)f(and)g(only)g(if)f FP(g)s FT(\()p FP(v)s FT(\))27
b(=)e FP(c)p FT(,)f(and)d(therefore)h(an)g(ob)5 b(ject)23
b(of)e(t)m(yp)s(e)h Fw(pfunc)42 b Fv(n)p FT(.)378 2915
y(The)30 b(function)e Fw(pfuncize:)84 b(\()p Fv(n)p Fw(:)i(nat\))42
b Fu(!)i Fw(\(\(vector)c(nat)i Fv(n)p Fw(\))h Fu(!)h
Fw(nat\)\))d Fu(!)j Fw(\(pfunc)d Fv(n)p Fw(\))30 b FT(is)f(de-)378
3028 y(\014ned)g(in)g(order)h(to)h(pro)s(duce)e(this)h(particular)e
(construction.)378 3268 y FN(P)7 b(RF)45 b FQ(Computable)33
b(F)-9 b(unctions)378 3440 y FT(A)41 b FN(P)7 b(RF)51
b FT(program)40 b Fv(p)p Fw(:prf)f FT(is)h(said)g(to)h(compute)h(an)e
FP(n)p FT(-ary)h(partial)f(function)f Fv(f)9 b Fw(:pfunc)41
b Fv(n)f FT(if)g FP(p)378 3553 y FT(con)m(v)m(erges)32
b(to)f(the)g(same)g(v)-5 b(alues)29 b(the)i(relation)f(in)f
FP(f)39 b FT(\(giv)m(en)31 b(b)m(y)f Fw(reln)42 b Fv(n)i(f)8
b FT(\))31 b(holds.)473 3741 y FN(`)529 3756 y FE(def)686
3741 y FN(8)p FP(p)47 b(n)g(f)10 b FM(.)47 b(computes)f
FP(p)h(n)h(f)57 b FN(\021)712 3853 y(8)p FP(v)51 b(x)p
FM(.)c(\(reln)f FP(n)i(f)57 b(v)51 b(x)p FM(\))c FN(,)1094
3966 y FM(\(converges)p 1580 3966 V 31 w(to)h FP(p)f
FM(\(listify)f(nat)h FP(n)g(v)s FM(\))h FP(x)p FM(\))378
4154 y FT(A)38 b(partial)f(function)g(is)g(de\014ned)f(to)j(b)s(e)e
FN(P)7 b(RF)k FT(-computable)37 b(if)g(there)i(is)e(some)h
FN(P)7 b(RF)48 b FT(program)378 4267 y(whic)m(h)29 b(computes)i(it.)473
4455 y FN(`)529 4470 y FE(def)686 4455 y FN(8)p FP(n)47
b(f)10 b FM(.)47 b(computable)e FP(n)i(f)57 b FN(\021)47
b(9)p FP(p)p FM(.)g(computes)f FP(p)h(n)g(f)378 4642
y FT(The)33 b(mec)m(hanisation)f(includes)f(the)i(de\014nition)e(of)i
(sev)m(eral)h(partial)e(functions)g(\(mostly)h(through)378
4755 y(the)28 b(use)g(of)g Fw(pfuncize)m FT(\))g(whic)m(h)f(are)i(sho)m
(wn)e(to)i(b)s(e)f FN(P)7 b(RF)j FT(-computable.)40 b(A)28
b(list)e(of)j(these)f(functions)378 4868 y(together)k(with)d(the)h
FN(P)7 b(RF)41 b FT(programs)30 b(that)h(compute)g(them)f(is)f(giv)m
(en)i(in)e(\(Zammit)g(1997\).)378 5111 y FG(3.3.4)112
b(The)38 b Fm(S)1015 5075 y FO(m)1009 5136 y(n)1119 5111
y FG(Theorem)378 5283 y FQ(En)m(umerating)c FN(P)7 b(RF)45
b FQ(programs)378 5455 y FT(A)30 b(set)g FP(A)g FT(is)f(said)f(to)j(b)s
(e)e FI(e\013e)-5 b(ctively)31 b(denumer)-5 b(able)38
b FT(if)29 b(there)h(is)e(a)i(bijection)f FP(f)35 b FT(:)25
b FP(A)g FN(!)h Fl(N)42 b FT(suc)m(h)29 b(that)378 5568
y(b)s(oth)20 b FP(f)30 b FT(and)21 b FP(f)881 5535 y
FK(\000)p FL(1)995 5568 y FT(are)h FI(e\013e)-5 b(ctively)23
b(c)-5 b(omputable)30 b FT(functions.)36 b(A)21 b(function)f(is)g
(e\013ectiv)m(ely)h(computable)378 5680 y(if)33 b(it)g(is)f(computable)
h(in)g(some)h(informal)d(sense)j(\(unless)e(the)i(notion)f(of)h
(computabilit)m(y)e(on)h(that)p eop
%%Page: 41 51
41 50 bop 378 5 a FF(CHAPTER)30 b(3.)61 b(CASE)29 b(STUDIES)h(ON)g(T)-8
b(A)m(CTIC-BASED)30 b(THEOREM)g(PR)m(O)m(VERS)185 b FT(41)378
396 y(t)m(yp)s(e)36 b(of)g(function)f(is)g(formalised,)g(e.g.,)17
b(if)35 b(its)g(range)h(and)g(domain)e(are)j(the)f(natural)f(n)m(um)m
(b)s(ers\),)378 509 y(and)f(therefore)h(this)e(notion)h(is)g(not)h
(de\014ned)e(in)g(Co)s(q.)53 b(Ho)m(w)m(ev)m(er,)38 b(b)s(ecause)c(of)h
(the)f(constructiv)m(e)378 622 y(nature)41 b(of)g(the)g(Calculus)e(of)i
(Constructions,)h(ev)m(ery)g(function)d(de\014ned)h(in)g(Co)s(q)g(is)g
(e\013ectiv)m(ely)378 735 y(computable,)i(hence)d(one)h(can)g(sho)m(w)g
(that)g(a)g(set)g FP(A)g FT(is)f(e\013ectiv)m(ely)h(den)m(umerable)e(b)
m(y)i(de\014ning)378 848 y(t)m(w)m(o)d(Co)s(q)e(functions)g
Fv(f)9 b Fw(:)86 b Fv(A)44 b Fu(!)g Fw(nat)34 b FT(and)h
Fv(g)s Fw(:)87 b(nat)42 b Fu(!)i Fv(A)36 b FT(and)f(b)m(y)h(sho)m(wing)
e(that)j FP(f)45 b FT(and)35 b FP(g)k FT(are)378 961
y(bijections)29 b(and)h(in)m(v)m(erses)g(of)g(eac)m(h)i(other.)519
1074 y(The)27 b(mec)m(hanisation)g(in)f(Co)s(q)g(includes)f(the)j
(de\014nitions)d(of)i(a)h(function)e Fw(Godel:)84 b(prf)43
b Fu(!)g Fw(nat)378 1187 y FT(whic)m(h)32 b(asso)s(ciates)j(a)f(n)m(um)
m(b)s(er)e(\(the)i FI(G\177)-46 b(odel)37 b(numb)-5 b(er)10
b FT(\))34 b(with)e(a)i FN(P)7 b(RF)44 b FT(program,)34
b(and)f(a)h(function)378 1300 y Fw(Prog:)85 b(nat)42
b Fu(!)i Fw(prf)30 b FT(whic)m(h)h(en)m(umerates)h FN(P)7
b(RF)41 b FT(programs.)j(These)32 b(t)m(w)m(o)g(functions)f(are)h(pro)m
(v)m(ed)378 1413 y(to)38 b(b)s(e)f(the)h(in)m(v)m(erses)f(of)h(eac)m(h)
h(other)f(and)f(bijectiv)m(e,)i(and)e(so)h(w)m(e)g(pro)m(v)m(e)g(that)g
(the)g(set)g(of)g FN(P)7 b(RF)378 1526 y FT(programs)45
b(is)g(e\013ectiv)m(ely)h(den)m(umerable.)85 b(The)45
b(function)f Fw(Prog)g FT(is)h(then)g(used)g(to)h(de\014ne)f(the)378
1638 y(function)d Fw(pf_compute_Prog)o(:)81 b(\()p Fv(n)p
Fw(:nat\))41 b Fu(!)j Fw(\()p Fv(e)p Fw(:nat\))c Fu(!)k
Fw(pfunc)e Fv(n)g FT(whic)m(h)g(tak)m(es)i(t)m(w)m(o)h(natural)378
1751 y(n)m(um)m(b)s(ers)c FP(n)h FT(and)f FP(e)i FT(and)f(returns)f
(the)h(partial)g(function)f(of)h(arit)m(y)h FP(n)e FT(whic)m(h)g(is)h
(computed)g(b)m(y)378 1864 y(the)32 b(program)g(with)e(G\177)-45
b(odel)32 b(n)m(um)m(b)s(er)e FP(e)p FT(.)46 b(\(Note)33
b(that)g(ev)m(ery)f(computable)f(function)g(can)h(th)m(us)g(b)s(e)378
1977 y(e\013ectiv)m(ely)d(represen)m(ted)e(b)m(y)h(its)f(arit)m(y)h
(and)f(the)h(G\177)-45 b(odel)28 b(n)m(um)m(b)s(er)e(of)i(a)h(program)e
(whic)m(h)g(computes)378 2090 y(it.\))473 2278 y FN(`)529
2293 y FE(def)686 2278 y FN(8)p FP(n)47 b(e)p FM(.)h(fcompute_Prog)c
FP(n)j(e)617 2391 y FN(\021)g FP(\025v)s FM(.)h(\(converges_to)c
(\(Prog)i FP(e)p FM(\))i(listify)e(nat)h FP(n)g(v)s FM(\))473
2616 y FN(`)529 2631 y FE(def)686 2616 y FN(8)p FP(n)g(e)p
FM(.)h(pf_compute_Prog)43 b FP(n)48 b(e)617 2729 y FN(\021)f
FM(mk_pfunc)f FP(n)h FM(\(fcompute_Prog)d FP(n)j(e)p
FM(\))1237 2842 y(\(fcompute_Prog_one_value)o(d)42 b
FP(n)47 b(e)p FM(\)\))378 3030 y FT(where)c Fw(fcompute_Prog_one)o(_va)
o(lu)o(ed)38 b FT(is)43 b(the)h(theorem)g(whic)m(h)f(states)i(that)g
(the)f(relation)f(con-)378 3143 y(structed)c(b)m(y)f
Fw(fcompute_Prog)c FT(is)j(single-v)-5 b(alued.)64 b(The)38
b(function)g Fw(pf_compute_Prog)f Fv(n)44 b(e)38 b FT(is)g(de-)378
3267 y(noted)30 b(b)m(y)h FP(\036)811 3219 y FL(\()p
FO(n)p FL(\))811 3279 y FO(e)913 3267 y FT(.)378 3507
y FQ(The)k FP(S)653 3474 y FO(m)648 3530 y(n)754 3507
y FQ(Theorem)378 3679 y FT(Giv)m(en)e(an)h(\()p FP(m)23
b FT(+)f FP(n)p FT(\)-ary)33 b(function)g FP(f)10 b FT(,)34
b(and)f FP(m)g FT(n)m(um)m(b)s(ers)f FP(x)f FT(=)f(\()p
FP(x)2695 3693 y FL(0)2735 3679 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)3004 3693 y FO(m)p FK(\000)p FL(1)3161 3679
y FT(\),)35 b(one)f(can)g(de\014ne)378 3792 y(an)c FP(n)p
FT(-ary)g(function)f FP(g)34 b FT(b)m(y)d(\014xing)e(the)h(\014rst)g
FP(m)g FT(argumen)m(ts)h(of)f FP(f)40 b FT(to)31 b FP(x)p
FT(.)1159 3996 y FP(g)s FT(\()p FP(y)1285 4010 y FL(0)1325
3996 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(y)1587 4010 y
FO(n)p FK(\000)p FL(1)1724 3996 y FT(\))26 b(=)f FP(f)10
b FT(\()p FP(x)2023 4010 y FL(0)2062 3996 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)2331 4010 y FO(m)p FK(\000)p FL(1)2488 3996
y FP(;)g(y)2573 4010 y FL(0)2612 3996 y FP(;)g(:)g(:)g(:)32
b(;)15 b(y)2874 4010 y FO(n)p FK(\000)p FL(1)3011 3996
y FT(\))378 4200 y(The)31 b FP(S)627 4167 y FO(m)622
4223 y(n)726 4200 y FT(theorem,)i(also)f(called)f(the)h
FI(p)-5 b(ar)g(ametrisation)44 b FT(theorem,)33 b(states)g(that)g(for)e
(\014xed)h FP(m)f FT(and)378 4313 y FP(n)d FT(if)f FP(f)37
b FT(is)27 b(computed)h(b)m(y)g(some)h(program)f(with)e(G\177)-45
b(odel)28 b(n)m(um)m(b)s(er)f FP(e)p FT(,)i(then)f(the)g(G\177)-45
b(odel)28 b(n)m(um)m(b)s(er)f(of)h(a)378 4426 y(program)e(computing)g
FP(g)k FT(can)d(b)s(e)f(computed)g(from)g FP(m)p FT(,)h
FP(n)p FT(,)g FP(e)g FT(and)f FP(x)2699 4440 y FL(0)2738
4426 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)3007 4440 y
FO(m)p FK(\000)p FL(1)3164 4426 y FT(.)40 b(In)25 b(other)i(w)m(ords,)
378 4539 y(for)j(all)f FP(m)h FT(and)g FP(n)p FT(,)g(there)h(is)e(a)i
(total)g(computable)f(\()p FP(m)21 b FT(+)e(1\)-ary)32
b(function)d FP(s)3043 4506 y FO(m)3043 4562 y(n)3139
4539 y FT(suc)m(h)i(that)1398 4759 y FN(8)p FP(e;)15
b(x;)g(y)s(:)31 b(\036)1781 4711 y FL(\()p FO(n)p FL(\))1781
4792 y FO(s)1814 4773 y Fy(m)1814 4809 y(n)1872 4792
y FL(\()p FO(e;x)p FL(\))2023 4759 y FT(\()p FP(y)s FT(\))57
b(=)e FP(\036)2378 4722 y FL(\()p FO(m)p FL(+)p FO(n)q
FL(\))2378 4782 y FO(e)2597 4759 y FT(\()p FP(x;)15 b(y)s
FT(\))378 4977 y(where)30 b FP(x)25 b FT(=)g(\()p FP(x)901
4991 y FL(0)941 4977 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(x)1210 4991 y FO(m)p FK(\000)p FL(1)1367 4977 y FT(\))30
b(and)g FP(y)e FT(=)d(\()p FP(y)1858 4991 y FL(0)1897
4977 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(y)2159 4991 y
FO(n)p FK(\000)p FL(1)2296 4977 y FT(\).)519 5105 y(The)38
b(function)g FP(\036)1133 5058 y FL(\()p FO(n)p FL(\))1133
5139 y FO(s)1166 5120 y Fy(m)1166 5155 y(n)1224 5139
y FL(\()p FO(e;x)p FL(\))1414 5105 y FT(can)h(b)s(e)g(computed)f(b)m(y)
h(a)g(program)g(whic)m(h)e(tak)m(es)k(the)e(argumen)m(ts)378
5261 y(\()p FP(y)458 5275 y FL(0)497 5261 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(y)759 5275 y FO(n)p FK(\000)p FL(1)896 5261
y FT(\))31 b(and)f(applies)e(the)j(program)f(whic)m(h)f(computes)i
FP(\036)2682 5213 y FL(\()p FO(m)p FL(+)p FO(n)p FL(\))2682
5273 y FO(e)2901 5261 y FT(\()p FP(x;)15 b(y)s FT(\))31
b(to)g(the)g(list)1545 5465 y([)p FP(x)1622 5479 y FL(0)1661
5465 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)1930 5479 y
FO(m)p FK(\000)p FL(1)2087 5465 y FP(;)g(y)2172 5479
y FL(0)2212 5465 y FP(;)g(:)g(:)g(:)31 b(;)15 b(y)2473
5479 y FO(n)p FK(\000)p FL(1)2611 5465 y FT(])p FP(:)378
5670 y FT(This)29 b(program)h(is)f(de\014ned)g(in)g(Co)s(q)h(as)h(the)f
(function)f Fw(smnprf)n FT(:)p eop
%%Page: 42 52
42 51 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(42)473
509 y FN(`)529 524 y FE(def)686 509 y FN(8)p FP(m)47
b(n)h(e)g(x)p FM(.)f(smnprf)f FP(m)i(n)f(e)h(x)569 622
y FN(\021)f FM(Subl)g(\(Prog)f FP(e)p FM(\))i(\(\(constants)d
FT([)p FP(x)2000 636 y FL(0)2039 622 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(x)2308 636 y FO(m)2375 622 y FT(])p FM(\))48
b(++)f(\(projections)d FP(n)p FM(\)\))378 804 y FT(where)30
b(the)g(functions)f Fw(constants)e FT(and)j Fw(projections)25
b FT(are)31 b(de\014ned)e(suc)m(h)h(that)473 985 y FM(constants)46
b FT([)p FP(c)1015 999 y FL(0)1055 985 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(c)1310 1000 y FO(l)1337 985 y FT(])48 b FM(=)g
FT([)p FM(Constant)d FP(c)1999 999 y FL(0)2039 985 y
FP(;)j(:)15 b(:)g(:)h(;)48 b FM(Constant)e FP(c)2775
1000 y FO(l)2801 985 y FT(])473 1098 y FM(projections)f
FP(n)i FM(=)h FT([)p FM(Proj)f FT(0)p FP(;)h(:)15 b(:)g(:)h(;)48
b FM(Proj)f FT(\()p FP(n)20 b FN(\000)g FT(1\)])378 1279
y(and)30 b(where)g(the)g(program)g Fw(Constant)41 b Fv(c)30
b FT(alw)m(a)m(ys)h(con)m(v)m(erges)h(with)d(v)-5 b(alue)30
b FP(c)p FT(.)519 1392 y(The)g(function)f FP(s)1105 1359
y FO(m)1105 1414 y(n)1202 1392 y FT(is)g(then)h(giv)m(en)g(b)m(y)h(the)
f(ob)5 b(ject)31 b Fw(\(pf_smnprf)40 b Fv(m)j(n)p Fw(\):)24
b(pfunc)41 b Fv(m)473 1573 y FN(`)529 1588 y FE(def)686
1573 y FN(8)p FP(m)47 b(n)h(v)s FM(.)f(vf_smnprf)f FP(m)h(n)g(v)760
1686 y FN(\021)g FM(\(Godel)f(\(smnprf)g FP(m)i(n)f FM(\(vhd)g(nat)f
FP(m)i(v)s FM(\))1094 1799 y(\(listify)d(nat)i FP(m)h
FM(\(vtl)e(nat)h FP(m)h(v)s FM(\)\)\)\))473 2025 y FN(`)529
2040 y FE(def)686 2025 y FM(pf_smnprf)d FN(\021)j FP(\025m)f(n)p
FM(.)g(\(pfuncize)f(\(S)h FP(m)p FM(\))g(\(vf_smnprf)e
FP(m)j(n)p FM(\)\))519 2206 y FT(Sho)m(wing)27 b(that)h(the)h(function)
d Fw(pf_smnprf)40 b Fv(m)k(n)28 b FT(is)f(computable)g(for)h(all)f
FP(m)h FT(and)f FP(n)h FT(w)m(as)g(rather)378 2319 y(lab)s(orious)22
b(and)i(needed)h(sev)m(eral)g(lemmas,)g(most)g(of)g(whic)m(h)e(w)m(ere)
i(pro)m(v)m(ed)g(b)m(y)g(\(rule,)g(or)g(structural\))378
2432 y(induction.)63 b(In)38 b(particular)f(the)i(pro)s(of)f(needed)g
(the)h(fact)h(that)f(all)e(the)i(functions)e(used)h(in)g(the)378
2545 y(de\014nition)30 b(of)i Fw(Prog)f FT(are)h FN(P)7
b(RF)43 b FT(computable.)i(Unfortunately)-8 b(,)33 b(the)f(statemen)m
(ts)i(of)f(some)f(of)h(the)378 2658 y(lemmas)i(needed)h(for)g(the)g
(pro)s(of)f(of)h(the)g FP(S)1903 2625 y FO(m)1898 2680
y(n)2006 2658 y FT(theorem)g(in)m(v)m(olv)m(ed)g(constan)m(t)h(names)f
(whic)m(h)e(w)m(ere)378 2771 y(de\014ned)j(only)g(to)i(b)s(e)f(used)f
(in)g(the)i(de\014nition)d(of)i(other)h(constan)m(ts.)65
b(F)-8 b(or)39 b(example,)i(a)d(n)m(um)m(b)s(er)378 2884
y(of)h(lemmas)f(in)m(v)m(olv)m(ed)h(the)g(function)e
Fw(Subl_in)f FT(whic)m(h)h(w)m(as)j(de\014ned)d(only)h(to)i(b)s(e)e
(used)g(in)f Fw(Subl)o FT(.)378 2997 y(Actually)-8 b(,)41
b(most)e(of)g(the)h(lemmas)e(concerning)g(prop)s(erties)g(of)h(the)g
(function)e Fw(smnprf)g FT(\(whic)m(h)h(is)378 3109 y(de\014ned)29
b(in)h(terms)g(of)h Fw(Subl)n FT(\))g(are)g(pro)m(v)m(ed)g(b)m(y)f
(induction)f(on)h(more)h(general)f(prop)s(erties)f(in)m(v)m(olving)378
3222 y Fw(Subl_in)m FT(.)58 b(This)34 b(is)i(probably)e(due)h(to)i(a)g
(bad)e(theory)i(structure.)57 b(F)-8 b(or)37 b(instance,)h(more)e
(general)378 3335 y(results)d(on)h Fw(Subl)e FT(can)i(probably)e(b)s(e)
i(deriv)m(ed)f(in)g(the)h(mo)s(dule)e(deriving)g(it,)j(so)f(that)h(no)e
(lemmas)378 3448 y(on)c Fw(Subl_in)d FT(are)k(required)e(outside)g
(this)g(mo)s(dule.)39 b(W)-8 b(e)30 b(p)s(oin)m(t)f(out)g(that)h(there)
f(is)g(a)g(lac)m(k)h(of)g(pro)s(of)378 3561 y(dev)m(elopmen)m(t)35
b(to)s(ols)g(aimed)g(at)h(the)f(structuring)e(and)i(re-structuring)e
(of)j(mec)m(hanised)e(theories)378 3674 y(in)m(teractiv)m(ely)-8
b(.)378 3916 y FG(3.3.5)112 b(Concluding)37 b(Remarks)g(on)h(the)f(Co)s
(q)h(F)-9 b(ormalisation)378 4088 y FT(This)30 b(section)i(describ)s
(es)e(the)i(mec)m(hanisation)f(of)h(the)g FP(S)2358 4055
y FO(m)2353 4110 y(n)2456 4088 y FT(theorem)g(in)f(the)h(Co)s(q)f
(system.)45 b(Com-)378 4201 y(putabilit)m(y)29 b(w)m(as)i(formalised)f
(according)h(to)h(a)f(mo)s(del)f(of)h(computation)g(based)g(on)g(the)g
(de\014nition)378 4314 y(of)k(partial)f(recursiv)m(e)h(functions,)g
(and)g(all)f(the)h(results)f(in)g(this)g(mec)m(hanisation)h(are)h
(deriv)m(ed)e(b)m(y)378 4427 y(constructiv)m(e)g(pro)s(ofs.)51
b(T)-8 b(able)33 b(2)i(sho)m(ws)e(the)h(lengths)f(of)h(di\013eren)m(t)g
(parts)f(of)h(the)h(source)f(co)s(de)g(of)378 4540 y(the)k(mec)m
(hanisation)f(with)f(commen)m(ts)j(remo)m(v)m(ed.)64
b(The)37 b(part)g(with)g(title)g(\\)p FN(P)7 b(RF)48
b FT(Computabil-)378 4652 y(it)m(y")35 b(is)f(rather)g(length)m(y)g
(since)g(it)h(includes)d(the)j(pro)s(ofs)e(of)i(the)g(computabilit)m(y)
e(of)h(a)h(n)m(um)m(b)s(er)f(of)378 4765 y(functions.)44
b(This)30 b(in)m(v)m(olv)m(es)h(the)h(de\014nition)e(of)i
FN(P)7 b(RF)42 b FT(programs)31 b(whic)m(h)g(are)h(sho)m(wn)f(to)i
(compute)378 4878 y(the)g(particular)e(functions.)46
b(Similarly)29 b(to)34 b(the)e(mec)m(hanisation)h(of)g(computabilit)m
(y)e(in)g(HOL,)i(the)378 4991 y(pro)s(ofs)c(in)f(Co)s(q)h(are)h(v)m
(ery)g(detailed)f(and)g(a)h(large)g(n)m(um)m(b)s(er)f(of)g(them)h
(deriv)m(e)f(results)g(whic)m(h)f(w)m(ould)378 5104 y(b)s(e)i
(considered)f(trivial)f(in)h(the)i(informal)d(mathematical)j
(literature.)378 5389 y FH(3.4)135 b(A)45 b(Comparativ)l(e)i(Study)d
(of)h(HOL)g(and)g(Co)t(q)378 5592 y FT(It)38 b(can)h(b)s(e)f(noted)g
(from)g(the)h(t)m(w)m(o)h(case)f(studies)e(describ)s(ed)f(in)i
(sections)g(3.2)h(and)f(3.3)i(that)f(the)378 5705 y(strongest)33
b(p)s(oin)m(t)e(of)h(the)g(Co)s(q)g(system)g(is)f(the)h(expressiv)m(e)g
(p)s(o)m(w)m(er)g(of)g(the)h(Calculus)d(of)i(Inductiv)m(e)p
eop
%%Page: 43 53
43 52 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(43)428
384 y FQ(In)m(tro)s(ductory)35 b(Mec)m(hanisation)646
497 y FT(de\014nitions:)248 b(40)31 b(lines)646 610 y(pro)s(ofs:)370
b(890)32 b(lines)736 723 y FQ(T)-9 b(otal:)230 b(930)36
b(lines)428 949 y(The)e FN(P)7 b(RF)45 b FQ(Language)646
1062 y FT(de\014nitions:)248 b(50)31 b(lines)646 1175
y(pro)s(ofs:)325 b(1210)32 b(lines)736 1287 y FQ(T)-9
b(otal:)178 b(1260)36 b(lines)428 1513 y FN(P)7 b(RF)45
b FQ(Computabilit)m(y)646 1626 y FT(de\014nitions:)202
b(490)32 b(lines)646 1739 y(pro)s(ofs:)325 b(6450)32
b(lines)736 1852 y FQ(T)-9 b(otal:)178 b(6940)36 b(lines)428
2078 y(En)m(umerating)e(Programs)h(and)f(the)h FP(S)1974
2045 y FO(m)1969 2100 y(n)2075 2078 y FQ(Theorem)646
2191 y FT(de\014nitions:)202 b(100)32 b(lines)646 2304
y(pro)s(ofs:)325 b(2170)32 b(lines)736 2417 y FQ(T)-9
b(otal:)178 b(2270)36 b(lines)927 2605 y FT(T)-8 b(able)30
b(2:)41 b(On)30 b(the)g(Source)g(Co)s(de)g(of)h(the)f(Mec)m(hanisation)
h(in)e(Co)s(q.)378 2880 y(Constructions.)64 b(The)39
b(HOL)f(logic)g(is)g(m)m(uc)m(h)h(simpler)d(but)i(users)g(can)h(rely)f
(on)h(a)g(greater)h(\015ex-)378 2993 y(ibilit)m(y)34
b(o\013ered)j(b)m(y)g(the)f(metalanguage.)62 b(As)36
b(a)h(result)f(HOL)g(theorem)h(pro)m(ving)f(is)g(m)m(uc)m(h)h(more)378
3106 y(implemen)m(tation-orien)m(ted,)25 b(while)e(in)h(Co)s(q)g(the)h
(implemen)m(tation)f(of)h(simple)d(tactics)k(\(whic)m(h)e(ma)m(y)378
3219 y(not)30 b(b)s(e)e(used)h(often)h(during)d(a)j(mec)m(hanisation\))
f(is)g(discouraged)f(b)m(y)i(ha)m(ving)e(a)i(sp)s(eci\014cation)e(and)
378 3332 y(pro)s(of)e(language)h(\(on)f(top)h(of)g(the)f
(metalanguage\))j(in)c(whic)m(h)g(all)g(user)h(in)m(teractions)g(are)h
(made.)40 b(In)378 3444 y(this)32 b(section)i(w)m(e)f(compare)h(the)g
(w)m(a)m(y)g(ob)5 b(jects)34 b(are)f(de\014ned)f(\(section)i(3.4.1\))i
(and)c(ho)m(w)i(theorems)378 3557 y(are)g(pro)m(v)m(ed)h(\(section)f
(3.4.2\))i(in)d(these)h(t)m(w)m(o)h(systems.)52 b(Other)33
b(considerations)g(are)h(discussed)e(in)378 3670 y(section)e(3.4.3,)j
(and)d(some)g(concluding)f(remarks)h(are)h(giv)m(en)f(in)f(section)i
(3.4.4.)378 3914 y FG(3.4.1)112 b(De\014nitions)378 4085
y FT(A)28 b(de\014nition)e(can)i(b)s(e)f(considered)g(as)h(a)g(name)g
(giv)m(en)g(to)h(an)f(ob)5 b(ject)28 b(b)m(y)g(whic)m(h)f(it)g(can)h(b)
s(e)g(referred)378 4198 y(to)f(in)f(a)h(theory)-8 b(.)40
b(A)26 b(concept)i(can)f(b)s(e)f(formalised)f(b)m(y)h(de\014ning)f(it)h
(in)f(terms)i(of)g(previously)d(de\014ned)378 4311 y(concepts,)42
b(or)d(b)m(y)g(deriving)e(its)h(existence)i(and)e(asso)s(ciating)h(a)g
(name)g(with)f(it.)66 b(Concepts)39 b(can)378 4424 y(also)34
b(b)s(e)g(formalised)f(through)h(the)g(declaration)h(of)f(axioms)g(and)
g(b)s(oth)g(systems)h(allo)m(w)e(users)h(to)378 4537
y(in)m(tro)s(duce)e(axioms)i(in)e(theories.)50 b(Ho)m(w)m(ev)m(er,)37
b(an)d(axiomatic)f(theory)h(can)g(b)s(e)f(inconsisten)m(t)g(while)378
4650 y(the)e(de\014nition)e(mec)m(hanisms)i(of)g(Co)s(q)g(and)g(HOL)f
(guaran)m(tee)j(that)f(purely)d(de\014nitional)g(theories)378
4763 y(are)i(alw)m(a)m(ys)g(consisten)m(t.)519 4876 y(The)26
b(de\014nition)e(mec)m(hanism)h(in)g(Co)s(q)h(in)m(tro)s(duces)f(new)h
(constan)m(t)i(names)e(in)f(an)h(en)m(vironmen)m(t)378
4989 y(and)41 b(allo)m(ws)g(these)h(terms)g(to)h(b)s(e)e(con)m(v)m
(ertible)g(with)g(their)g(de\014ning)e(terms.)75 b(This)40
b(applies)g(to)378 5102 y(b)s(oth)26 b(simple)e(abbreviations)h(\()p
FP(\016)s FT(-con)m(v)m(ersion\))k(and)d(inductiv)m(e)f(de\014nitions)f
(\()p FP(\023)p FT(-con)m(v)m(ersion\).)41 b(Since)378
5214 y(pro)s(ofs)35 b(and)g(theorems)h(are)g(\014rst)f(class)h(ob)5
b(jects)37 b(in)d(CIC,)h(the)h(name)g(of)g(a)g(theorem)g(is)f(actually)
378 5327 y(a)j(constan)m(t)h(de\014nition)d(giv)m(en)i(to)g(its)g(pro)s
(of)f(term.)63 b(In)37 b(fact,)k(although)d(the)g(sp)s(eci\014cation)e
(lan-)378 5440 y(guage)j FM(Gallina)d FT(giv)m(es)i(di\013eren)m(t)f
(constructs)h(for)g(de\014ning)e(terms)h(and)h(for)f(theorem)h(pro)m
(ving,)378 5553 y(one)28 b(can,)h(for)f(instance,)h(use)f(tactics)h(to)
g(de\014ne)e(terms)h(and)g(the)g(de\014nition)e(mec)m(hanism)h(to)i
(pro)m(v)m(e)378 5666 y(theorems.)46 b(The)31 b(system)h(di\013eren)m
(tiates)g(b)s(et)m(w)m(een)h(de\014nitions)c(and)j(theorems)g(b)m(y)g
(lab)s(elling)d(the)p eop
%%Page: 44 54
44 53 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(44)378
396 y(former)36 b(ob)5 b(jects)37 b(as)g FI(tr)-5 b(ansp)g(ar)g(ent)49
b FT(and)36 b(the)h(latter)g(as)g FI(op)-5 b(aque)p FT(.)61
b(T)-8 b(ransparen)m(t)36 b(ob)5 b(jects)37 b(are)g(con-)378
509 y(v)m(ertible)29 b(with)f(their)h(de\014ning)f(terms)h(while)f
(opaque)i(ob)5 b(jects)31 b(are)f(not.)41 b(The)29 b
FM(Gallina)e FT(language)378 622 y(pro)m(vides)i(commands)h(for)g(lab)s
(elling)e(ob)5 b(jects)31 b(as)f(opaque)h(or)f(transparen)m(t)h(man)m
(ually)-8 b(.)519 735 y(The)29 b(HOL)g(logic)h(treats)g(t)m(yp)s(e)g
(and)f(constan)m(t)i(de\014nitions)c(di\013eren)m(tly)-8
b(,)29 b(and)g(the)h(core)h(system)378 848 y(pro)m(vides)21
b(one)h(primitiv)m(e)e(inference)h(rule)f(for)i(t)m(yp)s(e)g
(de\014nitions)e(and)h(t)m(w)m(o)i(for)f(constan)m(t)h(de\014nitions.)
378 961 y(Other)e(inference)g(rules)g(are)h(giv)m(en)g(for)g(deriving)d
(theorems.)39 b(The)21 b(function)f(of)j(the)f(HOL)f(primitiv)m(e)378
1074 y(rules)38 b(for)h(de\014nitions)e(is)h(illustrated)f(b)s(elo)m
(w,)k(where)e(the)g(di\013erences)g(b)s(et)m(w)m(een)h(the)f
(de\014nition)378 1187 y(mec)m(hanism)30 b(for)g(constan)m(ts)h(in)e
(HOL)h(and)g(in)f(Co)s(q)h(are)h(discussed.)378 1427
y FQ(T)m(yp)s(e)k(De\014nitions)378 1599 y FT(The)i(HOL)g(system)h(has)
f(one)h(primitiv)m(e)d(rule)h(for)h(t)m(yp)s(e)h(de\014nitions,)f(whic)
m(h)f(in)m(tro)s(duces)h(a)g(new)378 1711 y(t)m(yp)s(e)31
b(expression)e(as)i(a)g(nonempt)m(y)f(subset)g(of)h(an)f(existing)g(t)m
(yp)s(e)g FP(\033)s FT(,)h(giv)m(en)g(a)g(term)f FP(P)39
b FT(:)26 b FP(\033)i FN(!)e FI(b)-5 b(o)g(ol)378 1824
y FT(whic)m(h)39 b(denotes)h(its)f(c)m(haracteristic)i(predicate.)69
b(Ho)m(w)m(ev)m(er,)45 b(in)39 b(practice,)k(the)d(user)f(in)m(tro)s
(duces)378 1937 y(new)28 b(t)m(yp)s(es)g(through)g(the)h(t)m(yp)s(e)f
(de\014nition)e(pac)m(k)-5 b(age)31 b(\(Melham)d(1988\))j(whic)m(h)c
(sp)s(eci\014es)g(ML)i(st)m(yle)378 2050 y(p)s(olymorphic)37
b(recursiv)m(e)i(t)m(yp)s(es)h(as)g(w)m(ell)e(as)i(automatically)g
(deriving)d(a)j(n)m(um)m(b)s(er)f(of)h(theorems)378 2163
y(sp)s(ecifying)24 b(certain)i(prop)s(erties)e(ab)s(out)i(the)g(t)m(yp)
s(e)g(\(suc)m(h)g(as)h(the)f(fact)h(that)f(the)g(t)m(yp)s(e)h
(constructors)378 2276 y(are)k(injectiv)m(e\).)519 2389
y(Suc)m(h)g(t)m(yp)s(es)h(are)g(sp)s(eci\014ed)e(in)h(Co)s(q)g(b)m(y)h
(inductiv)m(ely)e(de\014ned)g(sets)i(and)g(t)m(yp)s(es,)g(and)f(the)h
(cor-)378 2502 y(resp)s(onding)h(theorems)j(deriv)m(ed)f(b)m(y)g(HOL's)
h(t)m(yp)s(e)g(de\014nition)d(pac)m(k)-5 b(age)38 b(are)e(either)f
(returned)g(as)378 2615 y(theorems)i(b)m(y)f(the)g(de\014nition)e(mec)m
(hanism)i(of)g FM(Gallina)f FT(or)h(follo)m(w)f(from)h(the)h
(elimination)c(and)378 2728 y(in)m(tro)s(duction)28 b(rules)h(of)i(the)
f(set)h(or)g(t)m(yp)s(e.)519 2841 y(The)25 b(ob)m(vious)h(adv)-5
b(an)m(tage)27 b(of)f(ha)m(ving)f(t)m(yp)s(es)h(as)g(terms)g(in)e(CIC)h
(o)m(v)m(er)i(HOL's)f(simple)d(t)m(yp)s(e)j(the-)378
2953 y(ory)33 b(is)f(a)h(m)m(uc)m(h)g(more)f(expressiv)m(e)h(t)m(yp)s
(e)g(system)g(whic)m(h)e(allo)m(ws)h(quan)m(ti\014cation)h(o)m(v)m(er)h
(t)m(yp)s(es)e(and)378 3066 y(dep)s(enden)m(t)k(t)m(yp)s(es.)60
b(F)-8 b(or)38 b(instance,)g(the)f(dep)s(enden)m(t)f(record)h(t)m(yp)s
(e)g(of)g FP(n)p FT(-ary)f(partial)g(functions,)378 3179
y Fw(pfunc)41 b Fv(n)p FT(,)g(w)m(as)d(in)m(tro)s(duced)f(in)g(the)h
(mec)m(hanisation)g(in)f(Co)s(q)g(so)i(that)g(the)f(arit)m(y)g(of)h(a)f
(function)378 3292 y(can)j(b)s(e)f(declared)g(in)f(its)h(t)m(yp)s(e.)72
b(Suc)m(h)40 b(information)f(cannot)i(b)s(e)f(stored)h(in)e(the)i
(simple)e(t)m(yp)s(es)378 3405 y(of)c(HOL)f(and)g(therefore)h(w)m(as)g
(declared)f(in)g(all)f(the)i(statemen)m(ts)h(in)m(v)m(olving)e
FP(n)p FT(-ary)g(partial)f(func-)378 3518 y(tions.)73
b(\(Compare)42 b(the)f(de\014nition)e(of)j Fw(COMPUTES)c
FT(in)i(section)i(3.2.1)h(and)e(that)h(of)f Fw(computes)d
FT(in)378 3631 y(section)30 b(3.3.3.\))519 3744 y(A)d(mec)m(hanism)f
(whic)m(h)f(translates)h(ob)5 b(jects)28 b(in)d(a)i(dep)s(enden)m(t)e
(t)m(yp)s(e)i(theory)g(in)m(to)f(HOL)g(ob)5 b(jects)378
3857 y(is)30 b(describ)s(ed)e(b)m(y)j(Jacobs)g(and)f(Melham)h(\(1993\))
i(and)d(an)h(extension)f(of)h(the)g(HOL)f(logic)h(to)g(co)m(v)m(er)378
3970 y(quan)m(ti\014cation)f(o)m(v)m(er)h(t)m(yp)s(es)g(is)e(prop)s
(osed)g(b)m(y)i(Melham)f(\(1992\).)378 4210 y FQ(Constan)m(t)k
(De\014nitions)378 4381 y FT(Here)g(w)m(e)g(list)e(the)i(di\013eren)m
(t)f(mec)m(hanism)f(b)m(y)i(whic)m(h)e(constan)m(t)j(de\014nitions)c
(can)j(b)s(e)e(sp)s(eci\014ed)g(in)378 4494 y(Co)s(q)e(and)g(in)f(HOL.)
378 4707 y FQ(Simple)34 b(De\014nitions)46 b FT(In)40
b(HOL)55 b(giv)m(en)41 b(a)g(closed)g(term)f FP(e)12
b FT(:)28 b FP(\034)10 b FT(,)44 b(a)d(new)f(constan)m(t)i
FP(c)12 b FT(:)28 b FP(\034)51 b FT(can)41 b(b)s(e)605
4820 y(in)m(tro)s(duced)24 b(in)h(the)h(curren)m(t)g(theory)g(b)m(y)f
(the)h(primitiv)m(e)e(rule)h(of)h(constan)m(t)h(de\014nition)c(whic)m
(h)605 4933 y(also)39 b(yields)d(the)j(theorem)f FN(`)g
FP(c)h FT(=)f FP(e)p FT(.)65 b(Th)m(us,)40 b(while)c(in)h(the)i
(Calculus)d(of)i(Constructions)605 5046 y(constan)m(ts)j(are)f(con)m(v)
m(ertible)f(\()p FP(\016)s FT(-con)m(v)m(ertible\))j(with)c(their)h
(de\014ning)e(terms,)42 b(in)d(HOL)g(the)605 5159 y(in)m(terc)m
(hangeabilit)m(y)29 b(of)h FP(c)g FT(and)g FP(e)g FT(is)f(justi\014ed)f
(b)m(y)h(the)h(ab)s(o)m(v)m(e)h(theorem,)g(whic)m(h)e(needs)g(to)i(b)s
(e)605 5271 y(used)f(whenev)m(er)g FP(c)h FT(and)e FP(e)i
FT(ha)m(v)m(e)h(to)f(b)s(e)f(substituted)e(for)i(eac)m(h)i(other)f(in)e
(other)h(theorems.)378 5459 y FQ(Sp)s(eci\014cations)47
b FT(The)28 b(second)i(primitiv)m(e)d(rule)h(whic)m(h)g(in)m(tro)s
(duces)g(constan)m(ts)i(in)e(HOL)h(theories)605 5572
y(is)34 b(called)g(the)g(rule)g(of)g(constan)m(t)i(sp)s(eci\014cation.)
52 b(It)35 b(in)m(tro)s(duces)e(a)i(constan)m(t)h FP(c)r
FT(:)17 b FP(\034)45 b FT(ob)s(eying)605 5685 y(some)35
b(prop)s(ert)m(y)f FP(P)13 b FT(\()p FP(c)p FT(\))35
b(if)f(its)g(existence)h(can)f(b)s(e)g(sho)m(wn)g(b)m(y)h(a)f(theorem)h
FN(`)d(9)p FP(x:P)13 b FT(\()p FP(x)p FT(\).)53 b(The)p
eop
%%Page: 45 55
45 54 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(45)605
396 y(theorem)30 b FN(`)25 b FP(P)13 b FT(\()p FP(c)p
FT(\))30 b(is)e(returned)g(b)m(y)h(the)g(rule.)39 b(Note)30
b(that)g(only)e(the)i(existence)f(of)g(some)h FP(x)f
FT(is)605 509 y(required,)d(rather)h(than)f(the)h(existence)h(of)f(a)g
(unique)e FP(x)p FT(,)i(and)f(nothing)g(else)h(can)g(b)s(e)f(inferred)
605 622 y(ab)s(out)f FP(c)g FT(apart)g(from)f FP(P)13
b FT(\()p FP(c)p FT(\))26 b(\(and)f(an)m(ything)f(whic)m(h)g(can)h(b)s
(e)f(inferred)f(from)h FP(P)13 b FT(\()p FP(c)p FT(\)\).)41
b(Because)605 735 y(of)32 b(its)g(in)m(tuitionistic)d(nature,)j(there)g
(is)f(no)h(suc)m(h)g(rule)e(in)h(the)h(Calculus)e(of)i(Constructions)
605 848 y(although)h(an)m(y)h(constructiv)m(e)g(pro)s(of)f(of)g
FN(9)p FP(x)p FT(:)16 b FP(\034)5 b(:P)13 b FT(\()p FP(x)p
FT(\))34 b(is)f(actually)g(a)h(pair)e(\()p FP(w)h FT(:)e
FP(\034)5 b(;)15 b(p)31 b FT(:)f FP(P)13 b FT(\()p FP(w)r
FT(\)\))605 961 y(con)m(taining)35 b(a)h(term)f(of)h(t)m(yp)s(e)g
FP(\034)45 b FT(and)35 b(a)h(pro)s(of)e(stating)i(that)g(this)e(term)i
(satis\014es)f FP(P)13 b FT(.)55 b(The)605 1074 y(HOL)26
b(man)m(ual)g(\(Gordon)h(and)f(Melham)g(1993\))j(in)m(tro)s(duces)d(a)h
(primitiv)m(e)d(inference)i(rule)f(for)605 1187 y(t)m(yp)s(e)31
b(sp)s(eci\014cation)e(as)i(w)m(ell,)e(but)h(there)g(is)g(no)g
(implemen)m(tation)f(of)i(this)e(rule)g(y)m(et.)378 1374
y FQ(Recursiv)m(e)36 b(De\014nitions)46 b FT(The)25 b(de\014nition)d
(of)j(primitiv)m(e)e(recursiv)m(e)h(functions)f(o)m(v)m(er)j(a)f
(recursiv)m(e)605 1487 y(t)m(yp)s(e)31 b(is)f(justi\014ed)f(in)h(HOL)g
(b)m(y)h(a)g(theorem)g(stating)g(the)g(principle)d(of)j(primitiv)m(e)d
(recursion)605 1600 y(whic)m(h)i(can)i(b)s(e)e(automatically)h(deriv)m
(ed)g(b)m(y)g(the)g(t)m(yp)s(e)h(de\014nition)d(pac)m(k)-5
b(age.)45 b(A)31 b(library)e(for)605 1713 y(de\014ning)e(w)m
(ell-founded)g(recursiv)m(e)h(functions,)g(whic)m(h)g(in)f(general)i
(requires)f(user)g(in)m(terv)m(en-)605 1826 y(tion)f(for)g(pro)m(ving)g
(that)h(a)f(relation)g(is)f(w)m(ell-formed,)h(is)f(also)i(included)c
(in)i(the)i(HOL)f(system)605 1939 y(\(Slind)h(1996\).)45
b(In)30 b(Co)s(q,)h(recursiv)m(e)f(functions)f(are)j(de\014ned)d(b)m(y)
i(a)g(\014xp)s(oin)m(t)f(op)s(erator.)42 b(The)605 2052
y(syn)m(tax)h(of)g(actually)f(de\014ning)f(suc)m(h)h(functions)g
(implicitly)c(in)k(the)g(Co)s(q)h(is)e(v)m(ery)i(crude.)605
2165 y(Ho)m(w)m(ev)m(er,)k(a)c(mec)m(hanism)e(whic)m(h)g(allo)m(ws)g
(function)g(de\014nitions)e(in)i(an)h(ML)g(lik)m(e)f(syn)m(tax)605
2278 y(with)27 b(pattern)h(matc)m(hing)h(is)e(pro)m(vided)g(in)g(the)h
FM(Gallina)e FT(language.)41 b(This)26 b(mec)m(hanism)i(can)605
2391 y(also)j(b)s(e)e(used)h(on)g(the)h(de\014nition)d(of)i(functions)f
(o)m(v)m(er)j(dep)s(enden)m(t)d(t)m(yp)s(es.)378 2578
y FQ(Inductiv)m(e)35 b(De\014nitions)46 b FT(The)33 b(CIC)f(includes)f
(rules)h(for)h(inductiv)m(e)e(de\014nitions)g(and)i(are)g(th)m(us)605
2691 y(in)m(built)27 b(in)i(Co)s(q.)40 b(The)30 b FM(Gallina)e
FT(sp)s(eci\014cation)h(language)h(pro)m(vides)f(constructs)h(for)g(in)
m(tro-)605 2804 y(ducing)37 b(\(p)s(ossibly)f(m)m(utually\))i(inductiv)
m(e)f(de\014nitions)f(as)j(w)m(ell)f(as)g(tactics)i(for)e(reasoning)605
2917 y(ab)s(out)26 b(them.)39 b(Inductiv)m(e)24 b(de\014nitions)f(can)j
(b)s(e)f(used)g(for)g(in)m(tro)s(ducing)f(inductiv)m(e)g(t)m(yp)s(es)h
(and)605 3030 y(sets)39 b(as)f(recursiv)m(e)g(data)h(t)m(yp)s(es)f(and)
g(also)g(for)g(inductiv)m(ely)e(de\014ned)h(relations.)63
b(Supp)s(ort)605 3143 y(for)38 b(coinductiv)m(e)g(and)g(corecursiv)m(e)
h(de\014nitions)d(and)h(reasoning)h(b)m(y)h(coinduction)d(is)i(also)605
3256 y(pro)m(vided)29 b(b)m(y)h(the)h(Co)s(q)f(implemen)m(tation)f(of)h
(CIC.)605 3406 y(The)i(HOL)g(system)h(pro)m(vides)f(a)g(n)m(um)m(b)s
(er)g(of)g(pac)m(k)-5 b(ages)35 b(for)d(de\014ning)e(inductiv)m(e)i
(relations,)605 3519 y(whic)m(h)39 b(include)e(the)j(pac)m(k)-5
b(age)42 b(b)m(y)d(Melham)h(\(1991\))i(\(see)f(also)e(\(Camilleri)e
(and)i(Melham)605 3632 y(1992\)\),)50 b(supp)s(ort)42
b(for)i(m)m(utually)e(inductiv)m(e)h(de\014nitions)e(\(Ro)m(xas)k
(1993\))h(and)e(the)g(more)605 3745 y(recen)m(t)35 b(implemen)m(tation)
e(due)g(to)i(Harrison)e(\(1995b\).)53 b(Besides)33 b(pro)m(viding)f(a)j
(mec)m(hanism)605 3858 y(for)24 b(sp)s(ecifying)e(de\014nitions,)i
(these)g(pac)m(k)-5 b(ages)26 b(include)c(ML)j(functions)d(for)i
(reasoning)g(ab)s(out)605 3971 y(them)e(and)g(for)f(automating)i(them.)
38 b(It)22 b(is)f(argued)h(\(for)g(instance)f(in)g(\(Harrison)g
(1995a\)\))k(that)605 4083 y(inductiv)m(e)31 b(de\014nitions)f(can)i(b)
s(e)g(in)m(tro)s(duced)e(earlier)i(in)e(the)j(HOL)f(system)g(and)g(a)g
(n)m(um)m(b)s(er)605 4196 y(of)25 b(frequen)m(tly)f(used)f(relations)h
(in)f(existing)h(theories)g(\(suc)m(h)h(as)g(the)f(inequalities)e(on)j
(natural)605 4309 y(n)m(um)m(b)s(ers\))f(can)i(b)s(e)f(rede\014ned)f
(inductiv)m(ely)f(so)i(that)h(users)f(can)g(apply)f(the)i(principle)c
(of)j(rule)605 4422 y(induction)j(on)j(them,)f(m)m(uc)m(h)h(in)e(the)h
(same)h(fashion)e(that)i(it)f(is)f(done)i(b)m(y)f(Co)s(q)g(users.)378
4666 y FG(3.4.2)112 b(Theorem)37 b(Pro)m(ving)378 4837
y FT(This)23 b(section)i(illustrates)e(the)h(di\013eren)m(t)h(pro)s(of)
f(strategies)h(b)m(y)g(whic)m(h)e(users)h(of)h(the)g(Co)s(q)f(and)g
(HOL)378 4950 y(systems)30 b(p)s(erform)f(the)i(actual)g(theorem)f(pro)
m(ving.)378 5190 y FQ(F)-9 b(orw)m(ard)35 b(Pro)m(ving)378
5362 y FT(F)-8 b(orw)m(ard)41 b(theorem)g(pro)m(ving)f(is)g(p)s
(erformed)g(in)f(HOL)i(b)m(y)f(applying)f(ML)i(functions)e(whic)m(h)h
(re-)378 5475 y(turn)33 b(theorems.)52 b(This)32 b(is)h(done)h(in)f(Co)
s(q)h(b)m(y)g(constructing)f(terms)h(whose)g(t)m(yp)s(e)g(corresp)s
(onds)f(to)378 5588 y(theorems.)56 b(Ho)m(w)m(ev)m(er)38
b(since)d(HOL)g(users)g(ha)m(v)m(e)h(direct)f(access)i(to)g(the)e
(metalanguage,)k(one)d(can)p eop
%%Page: 46 56
46 55 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(46)378
396 y(implemen)m(t)35 b(more)h(elab)s(orate)h(mec)m(hanisms)e(for)h
(forw)m(ard)f(theorem)i(pro)m(ving)e(than)h(simple)e(con-)378
509 y(structions)c(of)g(terms)h(in)e(Co)s(q.)41 b(In)30
b(general,)h(theorem)g(pro)m(ving)f(in)f(Co)s(q)h(is)g(done)g(in)g(a)h
(bac)m(kw)m(ards)378 622 y(manner)e(b)m(y)i(applying)d(tactics.)378
862 y FQ(Bac)m(kw)m(ard)35 b(Pro)m(ving)378 1034 y FT(Both)44
b(theorem)f(pro)m(v)m(ers)g(supp)s(ort)f(in)m(teractiv)m(e)i
(tactic-based)g(goal-directed)f(reasoning.)78 b(The)378
1147 y(required)20 b(theorem)i(is)f(stated)i(as)f(a)h(goal)f(and)f(the)
i(user)e(applies)f(tactics)j(whic)m(h)e(break)g(the)h(goal)h(in)m(to)
378 1260 y(simpler)31 b(subgoals)i(un)m(til)f(they)i(can)g(b)s(e)f(pro)
m(v)m(ed)h(directly)-8 b(.)50 b(T)-8 b(actics)35 b(also)f(pro)m(vide)f
(a)h FI(justi\014c)-5 b(ation)378 1373 y FT(for)31 b(the)g
(simpli\014cation)d(of)j(a)g(goal)h(in)m(to)f(subgoals,)f(whic)m(h)g
(deriv)m(es)h(the)g(goal)g(as)h(a)f(theorem)g(from)378
1486 y(deriv)-5 b(ations)31 b(of)h(the)h(subgoals.)46
b(A)33 b(goal)g(usually)d(consists)i(of)g(the)h(statemen)m(t)h(whic)m
(h)d(is)h(required)378 1599 y(to)e(b)s(e)f(pro)m(v)m(ed)g(together)i
(with)d(a)i(n)m(um)m(b)s(er)e(of)h(assumptions)e(whic)m(h)h(a)i(pro)s
(of)f(of)g(the)h(goal)f(can)h(use.)519 1711 y(As)35 b(men)m(tioned)f
(in)g(section)h(2.4.2,)j(bac)m(kw)m(ard)d(pro)m(ving)f(is)g(supp)s
(orted)f(in)g(HOL)i(through)f(an)378 1824 y(implemen)m(tation)j(of)i(a)
g FI(go)-5 b(alstack)50 b FT(data)40 b(structure)e(whic)m(h)g(pro)m
(vides)f(a)i(n)m(um)m(b)s(er)f(of)h(op)s(erations)378
1937 y(\(including)h(sp)s(ecifying)h(goals,)46 b(applying)41
b(tactics,)47 b(mo)m(ving)c(around)f(subgoals,)k(etc.)16
b(\))79 b(as)43 b(ML)378 2050 y(functions.)61 b(T)-8
b(actics)39 b(and)e(tacticals)i(are)f(also)g(ML)f(functions)g(and)g
(users)g(can)h(implemen)m(t)e(new)378 2163 y(tactics)f(during)e(theory)
h(dev)m(elopmen)m(t.)54 b(On)33 b(the)i(other)f(hand,)h(Co)s(q)f
(tactics,)j(tacticals)e(and)f(the)378 2276 y(op)s(erations)23
b(on)h(the)g(in)m(ternal)f(goalstac)m(k)j(are)e(pro)m(vided)f(as)h
(constructs)g(of)g(the)g FM(Gallina)e FT(language.)378
2389 y(As)27 b(a)g(result,)f(implemen)m(ting)f(a)i(new)f(tactic)i(in)e
(Co)s(q)g(in)m(v)m(olv)m(es)h(the)g(non-trivial)d(task)k(of)f
(extending)378 2502 y(the)35 b FM(Gallina)d FT(language)j(and)f(in)g
(general)g(Co)s(q)g(users)g(tend)g(to)i(implemen)m(t)d(less)h(tactics)h
(during)378 2615 y(theory)43 b(dev)m(elopmen)m(t)g(than)f(HOL)h(users)f
(do.)77 b(Moreo)m(v)m(er,)48 b(HOL)43 b(users)f(can)h(also)f(implemen)m
(t)378 2728 y(tactics)36 b(`on)f(the)g(\015y')g(b)m(y)g(com)m(bining)e
(di\013eren)m(t)i(tactics,)i(tacticals,)g(and)d(general)h(ML)g
(functions)378 2841 y(during)28 b(a)j(single)e(in)m(teraction.)519
2953 y(W)-8 b(e)29 b(also)f(remark)f(that)i(HOL)e(tactics)i(are)f(m)m
(uc)m(h)g(more)g(elab)s(orate)g(and)f(n)m(umerous)g(than)g(Co)s(q)378
3066 y(ones.)50 b(One)33 b(reason)h(for)f(this)f(arises)h(from)g(the)h
(di\013eren)m(t)e(nature)i(of)f(the)h(Calculus)d(of)j(Inductiv)m(e)378
3179 y(Constructions)k(and)h(the)h(HOL)f(logic.)69 b(Since)38
b(theorems)i(in)e(Co)s(q)i(are)g(essen)m(tially)e(t)m(yp)s(es,)k(tac-)
378 3292 y(tics)35 b(corresp)s(ond)f(to)i(the)f(di\013eren)m(t)g(w)m(a)
m(ys)h(terms)f(can)g(b)s(e)g(constructed)g(and)g(brok)m(en)g(do)m(wn)f
(\(the)378 3405 y(in)m(tro)s(duction)k(and)h(elimination)e(rules)h(of)h
(the)h(constructs\).)69 b(On)39 b(the)h(other)g(hand,)h(tactics)f(in)
378 3518 y(HOL)28 b(ha)m(v)m(e)j(to)e(b)s(e)g(implemen)m(ted)e(using)g
(the)i(m)m(uc)m(h)g(less)g(p)s(o)m(w)m(erful)e(\(and)i(less)f
(general\))h(primitiv)m(e)378 3631 y(inference)36 b(rules.)57
b(Moreo)m(v)m(er,)40 b(the)d(p)s(o)m(w)m(erful)e(notion)g(of)i(con)m(v)
m(ertible)f(terms)h(of)f(CIC)g(mak)m(es)h(in-)378 3744
y(ference)32 b(rules)e(suc)m(h)h(as)h(rewriting)d(with)h(the)i
(de\014nitions)d(and)i(b)s(eta)h(con)m(v)m(ersion)f(unnecessary)g(in)
378 3857 y(Co)s(q.)40 b(Ho)m(w)m(ev)m(er,)32 b(tactics)e(for)f
(unfolding)e(de\014nitions)g(and)h(c)m(hanging)i(a)f(goal)h(or)g
(assumption)d(to)j(a)378 3970 y(con)m(v)m(ertible)i(one)h(are)f(also)g
(pro)m(vided,)g(b)s(oth)f(b)s(ecause)i(it)e(facilitates)h(theorem)h
(pro)m(ving)e(and)h(also)378 4083 y(b)s(ecause)25 b(higher-order)f
(uni\014cation)g(is)h(undecidable)e(and)i(user)f(in)m(terv)m(en)m(tion)
i(ma)m(y)g(sometimes)g(b)s(e)378 4196 y(essen)m(tial.)519
4308 y(The)42 b(considerable)f(di\013erence)h(b)s(et)m(w)m(een)g(the)h
(n)m(um)m(b)s(er)e(\(and)h(nature\))g(of)h(tactics)g(in)e(HOL)378
4421 y(and)e(in)f(Co)s(q)h(and)f(the)i(a)m(v)-5 b(ailabilit)m(y)37
b(of)j(a)g(sp)s(eci\014cation)e(and)h(pro)s(of)f(language)i(mak)m(es)g
(Co)s(q)f(an)378 4534 y(easier)32 b(system)g(to)h(learn.)45
b(New)32 b(HOL)g(users)f(are)h(faced)h(with)e(h)m(undreds)e(of)k
(inference)e(rules)f(and)378 4647 y(tactics)39 b(to)g(learn,)g(and)f(p)
s(ossibly)d(a)k(new)e(programming)g(language)i(to)g(master)f(in)f
(order)h(to)h(b)s(e)378 4760 y(used)g(e\013ectiv)m(ely)i(as)g(a)f
(metalanguage.)72 b(New)40 b(Co)s(q)g(users)f(need)h(to)h(learn)e(ho)m
(w)h(to)h(use)f(ab)s(out)378 4873 y(\014ft)m(y)34 b(language)h
(constructs)g(and)f(most)h(theory)f(dev)m(elopmen)m(t)h(can)g(b)s(e)f
(done)g(without)f(the)i(need)378 4986 y(of)30 b(extending)g
FM(Gallina)p FT(.)519 5099 y(Finally)37 b(w)m(e)j(note)g(that)g
(assumptions)d(in)h(Co)s(q)h(are)h(lab)s(elled)d(with)h(names)h(while)e
(they)i(are)378 5212 y(unnamed)e(in)h(HOL.)h(This)e(a\013ects)k(the)e
(w)m(a)m(y)h(users)e(of)h(the)g(systems)g(use)g(assumptions)e(during)
378 5325 y(the)44 b(construction)f(of)h(a)g(pro)s(of.)80
b(Basically)-8 b(,)46 b(Co)s(q)e(users)e(select)j(the)e(assumptions)f
(they)i(need)378 5438 y(b)m(y)e(their)f(name)h(while)e(HOL)h(users)g
(apply)g(tactics)i(whic)m(h)d(try)i(to)h(use)e(all)g(the)h
(assumptions.)378 5550 y(Nev)m(ertheless,)36 b(HOL)e(users)g(can)g
(implemen)m(t)f(tactics)j(\(on)f(the)f(\015y)-8 b(,)36
b(or)e(otherwise\))g(whic)m(h)f(select)378 5663 y(a)46
b(subset)f(of,)k(or)d(a)f(particular)f(elemen)m(t)i(from,)j(the)d(list)
e(of)h(assumptions)f(through)h(\014ltering)p eop
%%Page: 47 57
47 56 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(47)378
396 y(functions)38 b(and)h(other)g(tec)m(hniques)g(discussed)f(in)g
(\(Blac)m(k)j(and)d(Windley)g(1995\).)70 b(Ho)m(w)m(ev)m(er)41
b(w)m(e)378 509 y(stress)23 b(that)h(selecting)f(an)h(assumption)d
(simply)g(b)m(y)j(its)e(name)i(is)e(de\014nitely)g(more)h(straigh)m
(tforw)m(ard)378 622 y(than)44 b(an)m(y)h(suc)m(h)f(tec)m(hniques.)83
b(During)43 b(the)i(implemen)m(tation)e(describ)s(ed)f(in)h(section)i
(3.2)h(the)378 735 y(need)c(of)g(writing)e(sev)m(eral)j(\014ltering)d
(functions)h(w)m(as)h(sometimes)h(tedious)e(and)h(o)m(v)m(erwhelming.)
378 848 y(T)-8 b(actics)46 b(whic)m(h)e(mak)m(e)j(use)e(of)h(all)e(the)
i(assumptions)e(can)i(ho)m(w)m(ev)m(er)h(b)s(e)d(quite)h(p)s(o)m(w)m
(erful)f(and)378 961 y(ma)m(y)d(sa)m(v)m(e)h(sev)m(eral)f(rep)s(etitiv)
m(e)e(pro)s(of)h(steps.)71 b(One)40 b(can)g(for)g(instance)g(consider)g
(the)g(p)s(o)m(w)m(er)h(of)378 1074 y FM(ASM_REWRITE_TAC)19
b FT(in)i(HOL)i(whic)m(h)e(rep)s(etitiv)m(ely)h(rewrites)g(with)g(all)f
(the)i(assumptions,)g(together)378 1187 y(with)j(a)j(n)m(um)m(b)s(er)d
(of)i(theorems)g(supplied)c(b)m(y)k(the)g(user)f(and)g(a)h(list)f(of)h
(basic)f(pre-pro)m(v)m(ed)h(theorems)378 1300 y(\(suc)m(h)i(as)h
Fu(`)14 b(8)f Fv(A)p Fw(.)h Ft(\()p Fu(>)43 b(_)h Fv(A)p
Ft(\))g Fw(=)g Fu(>)o FT(.\))378 1540 y FQ(Automation)378
1711 y FT(The)26 b(HOL)g(system)h(is)f(equipp)s(ed)e(with)h(more)i
(decision)e(pro)s(cedures)h(and)g(automation)h(to)s(ols)f(than)378
1824 y(Co)s(q.)47 b(HOL)33 b(\(HOL90)g(v)m(ersion)f(9)p
FP(:)p FT(1)p FP(\013)j FT(and)d(Hol98\))i(includes)c(automation)j(for)
g(rewriting,)e(a)i(tau-)378 1937 y(tology)j(c)m(hec)m(k)m(er,)j
(semidecision)33 b(pro)s(cedures)h(for)g(\014rst-order)h(reasoning,)h
(a)f(decision)f(pro)s(cedure)378 2050 y(for)i(Presburger)f(arithmetic,)
j(as)e(w)m(ell)g(as)g(an)h(implemen)m(tation)e(of)h(Nelson)g(and)g(Opp)
s(en's)f(tec)m(h-)378 2163 y(nique)k(for)h(com)m(bining)f(decision)f
(pro)s(cedures.)69 b(Since)40 b(most)g(pro)s(ofs)g(in)e(the)j(mec)m
(hanisation)f(of)378 2276 y(computabilit)m(y)30 b(in)h(HOL)h(are)h(of)f
(a)g(highly)e(tec)m(hnical)i(nature,)h(the)f(use)g(of)g(suc)m(h)g
(decision)f(pro)s(ce-)378 2389 y(dures)i(sa)m(v)m(ed)j(a)f(lot)g(of)f
(time)h(and)e(thinking)g(ab)s(out)h(trivial)f(pro)s(ofs.)52
b(The)34 b(Co)s(q)g(system)h(\(v)m(ersion)378 2502 y(6)p
FP(:)p FT(2\))40 b(pro)m(vides)d(tactics)j(for)e(tautology)i(c)m(hec)m
(king,)i(decision)37 b(pro)s(cedures)g(for)h(in)m(tuitionistic)e(Di-)
378 2614 y(rect)f(Predicate)f(Calculus)f(\(whic)m(h)g(is)g(the)i
(\014rst-order)e(Sequen)m(t)i(Calculus)d(of)i(Gen)m(tzen)i(without)378
2727 y(con)m(traction)i(rules\),)g(for)f(Presburger)f(arithmetic,)j
(and)d(for)h(a)h(n)m(um)m(b)s(er)e(of)h(problems)e(concern-)378
2840 y(ing)i(Ab)s(elian)e(rings.)61 b(The)37 b FM(Gallina)f
FT(language)i(main)m(tains)e(a)i(user)f(de\014nable)f(hin)m(t)h(list,)h
(where)378 2953 y(tactics)32 b(can)g(b)s(e)e(included)f(in)m(to)i(the)h
(list)e(and)g(goals)i(can)g(then)f(b)s(e)f(automatically)h(solv)m(ed)h
(b)m(y)f(the)378 3066 y(application)e(of)h(one)h(or)f(more)h(of)f
(these)h(tactics.)378 3306 y FQ(Reasoning)36 b(with)e(Equalit)m(y)h
(and)g(Equiv)-6 b(alence)378 3478 y FT(HOL's)38 b(notion)f(of)h
(equalit)m(y)g(is)f(extremely)h(p)s(o)m(w)m(erful)f(and)g(since)h
(equiv)-5 b(alence)37 b(of)h(prop)s(ositions)378 3591
y(is)33 b(de\014ned)f(as)i(equalit)m(y)f(on)h(b)s(o)s(olean)e(v)-5
b(alues,)34 b(the)g(same)g(prop)s(erties)e(enjo)m(y)m(ed)j(b)m(y)e
(equalit)m(y)g(hold)378 3703 y(also)j(for)h(equiv)-5
b(alence.)58 b(Equalit)m(y)36 b(is)f(in)m(tro)s(duced)g(in)g(HOL)h(b)m
(y)h(a)f(primitiv)m(e)f(rule,)i FM(REFL)p FT(,)e(whic)m(h)378
3816 y(returns)25 b(the)h(theorem)g Fu(`)43 b Fv(t)h
Fw(=)f Fv(t)26 b FT(for)g(an)m(y)g(term)g FP(t)p FT(;)h(and)f(the)g
(primitiv)m(e)d(rule)i(of)h(substitution)e(allo)m(ws)378
3929 y(an)m(y)k(subterms)f(of)h(a)h(theorem)f(to)h(b)s(e)e(substituted)
g(b)m(y)h(their)f(equals.)39 b(The)28 b(rule)e(of)i(extensionalit)m(y)
378 4042 y(\(whic)m(h)i(can)h(b)s(e)f(deriv)m(ed)g(in)g(HOL\))h(yields)
e(the)i(equalit)m(y)g(of)g(an)m(y)g(t)m(w)m(o)h(functions)e(whic)m(h)f
(giv)m(e)j(the)378 4155 y(same)26 b(results)e(when)g(applied)f(to)j
(the)g(same)f(v)-5 b(alues.)39 b(\(More)26 b(formally)-8
b(,)26 b(the)f(rule)f(of)h(extensionalit)m(y)378 4268
y(is)34 b FN(8)p FP(x:f)10 b FT(\()p FP(x)p FT(\))33
b(=)g FP(g)s FT(\()p FP(x)p FT(\))i FN(`)e FP(f)43 b
FT(=)33 b FP(g)s FT(.\))56 b(As)35 b(a)h(result,)g(equiv)-5
b(alen)m(t)34 b(predicates)h(can)h(b)s(e)e(substituted)g(for)378
4381 y(eac)m(h)28 b(other)f(and)f(assumptions)f(can)i(b)s(e)g
(substituted)e(with)g(the)i(truth)f(v)-5 b(alue)27 b
FN(>)p FT(.)39 b(Hence,)28 b(theorem)378 4494 y(pro)m(ving)j(in)f(HOL)i
(can)g(rely)f(a)h(lot)g(on)f(rewriting,)g(for)g(example,)h(statemen)m
(ts)i(lik)m(e)d FP(a)21 b FN(^)g FP(b)27 b FN(\))h FP(a)21
b FN(_)g FP(c)378 4607 y FT(can)31 b(b)s(e)e(easily)h(pro)m(v)m(ed)h(b)
m(y)f(the)g(tactic:)521 4817 y FM(DISCH_TAC)45 b(THEN)521
4930 y(ASM_REWRITE_TAC)f([])519 5141 y FT(The)39 b(imp)s(ortance)g(of)h
(equalit)m(y)f(in)f(HOL)h(theorem)h(pro)m(ving)f(is)g(emphasised)f(b)m
(y)h(a)h(class)g(of)378 5254 y(inference)32 b(rules)g(called)g
FI(c)-5 b(onversions)42 b FT(\(see)34 b(section)f(3.2.1,)j(page)e(29\))
g(whic)m(h)e(are)h(sp)s(ecialised)e(for)378 5367 y(deriving)d
(equalities.)519 5480 y(Equalit)m(y)i(in)f(CIC)g(is)g(in)m(tro)s(duced)
g(b)m(y)h(the)h(inductiv)m(e)e(de\014nition)p 1680 5632
368 4 v 1680 5705 a Fw(eq)43 b Fv(A)h(a)f(a)2090 5648
y Fw(refl_equal)p eop
%%Page: 48 58
48 57 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(48)378
396 y(and)35 b(results)f(lik)m(e)h(symmetry)-8 b(,)37
b(transitivit)m(y)d(and)h(congruence)h(can)g(then)f(b)s(e)g(deriv)m
(ed.)55 b(Ho)m(w)m(ev)m(er)378 509 y(functions)34 b(are)h(in)m
(tensional)e(and)i(equiv)-5 b(alence)35 b(of)g(prop)s(ositions)d(is)j
(di\013eren)m(t)f(from)h(their)f(equal-)378 622 y(it)m(y)-8
b(.)71 b(Basically)-8 b(,)42 b(t)m(w)m(o)g(prop)s(ositions,)e
FP(a)h FT(and)e FP(b)p FT(,)k(can)e(b)s(e)e(pro)m(v)m(ed)i(to)g(b)s(e)f
(equiv)-5 b(alen)m(t)39 b(in)g(Co)s(q)h(b)m(y)378 735
y(constructing)33 b(a)g(term)h(with)d(t)m(yp)s(e)j(\(\()p
FP(a)c FN(!)g FP(b)p FT(\))23 b FN(\002)f FT(\()p FP(b)30
b FN(!)g FP(a)p FT(\)\))k(and)e(little)g(supp)s(ort)g(is)g(giv)m(en)h
(for)g(tak-)378 848 y(ing)g(adv)-5 b(an)m(tage)37 b(of)d(the)g
(symmetric)g(nature)g(of)g(bi-implication.)49 b(The)34
b(need)g(for)g(more)g(p)s(o)m(w)m(erful)378 961 y(supp)s(ort)40
b(of)i(equalit)m(y)f(is)g(reduced)g(b)m(y)h(ha)m(ving)f(the)h(notion)f
(of)h(con)m(v)m(ertible)g(terms.)75 b(Ho)m(w)m(ev)m(er,)378
1074 y(here)39 b(w)m(e)h(remark)f(on)g(the)g(inabilit)m(y)e(to)j
(construct)g(a)f(term)h FP(t)10 b FT(:)24 b FP(T)2700
1088 y FL(1)2779 1074 y FT(directly)-8 b(,)41 b(where)e
FP(t)g FT(has)g(t)m(yp)s(e)378 1187 y FP(T)431 1201 y
FL(2)510 1187 y FT(whic)m(h)g(is)g(not)h(con)m(v)m(ertible)g(with)f
FP(T)1794 1201 y FL(1)1873 1187 y FT(and)g(it)h(can)g(b)s(e)f(pro)m(v)m
(ed)h(that)h FP(T)3034 1201 y FL(1)3113 1187 y FT(and)e
FP(T)3352 1201 y FL(2)3432 1187 y FT(are)h(equal.)378
1300 y(F)-8 b(or)29 b(example,)g(giv)m(en)g(some)g(term)g
Fv(v)s Fw(:)87 b(\(vector)41 b(nat)h(\()p Fv(n)18 b Ft(+)g
Fv(m)p Fw(\)\))28 b FT(where)g FP(m)g FT(and)h FP(n)f
FT(are)h(v)-5 b(ariables,)378 1413 y(then)29 b(one)h(cannot)g(sp)s
(ecify)e FP(v)k FT(as)e(ha)m(ving)f(t)m(yp)s(e)g Fw(vector)41
b(nat)i Ft(\()p Fv(m)18 b Ft(+)g Fv(n)p Ft(\))30 b FT(ev)m(en)g(though)
f(\()p FP(n)18 b FT(+)g FP(m)p FT(\))29 b(and)378 1526
y(\()p FP(m)c FT(+)g FP(n)p FT(\))37 b(are)h(equal.)61
b(This)36 b(problem)g(is)g(encoun)m(tered)i(in)e(the)i(mec)m
(hanisation)f(in)f(section)i(3.3,)378 1638 y(and)d(for)h(this)e
(particular)g(example)i(it)f(is)g(solv)m(ed)g(b)m(y)h(de\014ning)d(a)k
(function)d Fw(Change_arity)-5 b FT(,)37 b(suc)m(h)378
1751 y(that,)46 b(giv)m(en)c(a)g(v)m(ector)i Fv(v)s Fw(:)87
b(\(vector)41 b Fv(A)j(n)p Fw(\))d FT(and)h(a)g(pro)s(of)g
FP(t)f FT(of)i(\()p FP(n)h FT(=)h FP(m)p FT(\),)g(then)d(the)g(t)m(yp)s
(e)h(of)378 1864 y Fw(Change_arity)c Fv(n)k(m)g(t)h(A)g(v)33
b FT(is)d Fw(\(vector)41 b Fv(A)i(m)p Fw(\))p FT(:)473
2077 y FN(`)529 2092 y FE(def)686 2077 y FM(Change_arity)664
2190 y FN(\021)48 b FP(\025n;)15 b(m)p FM(:nat,)46 b
FP(t)p FM(:\()p FP(n)24 b FT(=)h FP(m)p FM(\),)47 b FP(A)p
FM(:)h(Set,)e FP(v)s FM(:)i(\(vector)e FP(A)i(n)p FM(\).)855
2303 y(eq_rec)e(nat)h FP(n)h FM(\(vector)d FP(A)p FM(\))j
FP(v)j(m)d(t)p FM(\).)378 2515 y FT(and)30 b(it)g(is)f(pro)m(v)m(ed)i
(that)664 2703 y FN(8)p FP(n)p FM(:nat,)46 b FP(t)p FM(:\()p
FP(n)24 b FT(=)h FP(n)p FM(\),)47 b FP(A)p FM(:Set,)f
FP(v)s FM(:\(vector)g FP(A)i(n)p FM(\).)951 2816 y(Change_arity)c
FP(n)j(n)h(t)f(A)h(v)j FM(=)d FP(v)378 3003 y FT(This)29
b(theorem)h(is)g(pro)m(v)m(ed)g(using)f(the)i Fw(eq_rec_eq)26
b FT(axiom.)519 3116 y(No)m(w,)31 b(if)e Fw(plus_sym)e
FT(represen)m(ts)j(the)h(theorem)f FN(8)p FP(n;)15 b(m:n)k
FT(+)h FP(m)25 b FT(=)g FP(m)20 b FT(+)f FP(n)p FT(,)30
b(and)g(the)g(term)h FP(v)i FT(has)378 3229 y(t)m(yp)s(e)d
Fw(vector)42 b(nat)g Ft(\()p Fv(n)19 b Ft(+)f Fv(m)p
Ft(\))30 b FT(then)473 3417 y FM(Change_arity)45 b(\()p
FP(n)67 b FT(+)h FP(m)p FM(\))47 b(\()p FP(m)67 b FT(+)h
FP(n)p FM(\))47 b(\(plus_sym)e FP(n)j(m)p FM(\))f(nat)g
FP(v)378 3604 y FT(has)30 b(the)h(required)d(t)m(yp)s(e)j
Fw(vector)41 b(nat)h Ft(\()p Fv(m)19 b Ft(+)f Fv(n)p
Ft(\))p FT(.)378 3848 y FG(3.4.3)112 b(Miscellaneous)378
4019 y FT(This)33 b(section)h(lists)f(some)i(other)g(considerations)e
(of)i(the)g(di\013erences)e(b)s(et)m(w)m(een)j(the)e(approac)m(hes)378
4132 y(of)c(Co)s(q)g(and)g(HOL)g(to)h(the)g(mec)m(hanisation)f(of)g
(theories.)378 4373 y FQ(Classical)35 b(and)g(Constructiv)m(e)g
(Reasoning)378 4544 y FT(HOL's)26 b(logic)g(is)g(classical,)h(and)e
(the)i(axiom)f(of)h(the)g(excluded)e(middle)f(is)i(in)m(tro)s(duced)e
(in)i(the)g(HOL)378 4657 y(theory)38 b(whic)m(h)e(de\014nes)h(b)s(o)s
(olean)f(v)-5 b(alues.)62 b(One)37 b(can)h(ask)g(ho)m(w)m(ev)m(er)h
(whether)e(an)m(y)h(supp)s(ort)e(can)378 4770 y(b)s(e)c(giv)m(en)h(to)h
(users)e(who)g(ma)m(y)i(w)m(an)m(t)g(to)f(use)g(HOL)g(and)f(still)f
(reason)i(constructiv)m(ely)-8 b(.)48 b(The)33 b(CIC)378
4883 y(is)j(constructiv)m(e)h(and)g(so)g(the)g(la)m(w)g(of)g(the)g
(excluded)f(middle)f(cannot)j(b)s(e)e(deriv)m(ed)g(and)g(all)g(Co)s(q)
378 4996 y(functions)20 b(ha)m(v)m(e)j(to)f(b)s(e)f(computable.)38
b(Ho)m(w)m(ev)m(er,)25 b(one)d(can)g(still)d(reason)j(classically)e(to)
i(some)g(exten)m(t)378 5109 y(in)32 b(Co)s(q)h(b)m(y)h(loading)e(a)i
(classical)f(theory)h(whic)m(h)e(sp)s(eci\014es)g(the)i(la)m(w)f(of)h
(the)g(excluded)e(middle)g(as)378 5222 y(an)e(axiom,)g(although)g(it)g
(should)e(b)s(e)h(stressed)h(that)h(this)e(do)s(es)h(not)g(giv)m(e)h
(Co)s(q)e(the)i(full)d(p)s(o)m(w)m(ers)i(of)378 5335
y(classical)f(reasoning.)519 5447 y(Since)24 b(all)f(functions)g(in)h
(Co)s(q)g(are)h(computable,)h FP(n)p FT(-ary)e(partial)f(functions)h
(are)h(de\014ned)e(in)g(Co)s(q)378 5560 y(as)31 b(single-v)-5
b(alued)28 b(relations)i(rather)g(than)g(as)h(Co)s(q)f(functions,)f(so)
i(that)g(partial)e(functions)g(whic)m(h)378 5673 y(are)41
b(not)g(computable)f(can)h(still)d(b)s(e)i(sp)s(eci\014ed)f(in)g(the)i
(mec)m(hanisation.)71 b(On)39 b(the)i(other)g(hand,)p
eop
%%Page: 49 59
49 58 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(49)378
396 y(functions)26 b(in)g(HOL)h(need)g(not)h(b)s(e)f(computable)f(\(a)j
(classical)d(pro)s(of)h(of)g(their)g(existence)h(is)e(enough)378
509 y(to)34 b(de\014ne)e(them\),)i(and)e FP(n)p FT(-ary)h(partial)e
(functions)h(are)h(de\014ned)f(in)f(HOL)i(as)g(functions)f(mapping)378
622 y(lists)g(of)i(natural)f(n)m(um)m(b)s(ers)g(to)h(the)g(represen)m
(tation)g(of)g(`p)s(ossibly)e(unde\014ned)f(natural)i(n)m(um)m(b)s
(ers')378 735 y(giv)m(en)e(in)f(section)h(3.2.1,)i(page)e(30.)44
b(The)30 b(adv)-5 b(an)m(tage)33 b(of)e(the)g(formalisation)e(of)j
(partial)d(functions)378 848 y(in)g(HOL)h(is)f(that)i(a)g(function)e
(application)g(can)i(b)s(e)e(directly)g(substituted)g(b)m(y)h(its)g(v)
-5 b(alue.)519 961 y(The)43 b(pro)s(of)g(of)h(the)g FP(S)1318
928 y FO(m)1313 983 y(n)1428 961 y FT(theorem)h(in)d(Co)s(q)h(is)g
(constructiv)m(e;)51 b(ho)m(w)m(ev)m(er,)e(the)44 b(literature)f(of)378
1074 y(computabilit)m(y)30 b(con)m(tains)i(a)h(n)m(um)m(b)s(er)d(of)j
(theorems)f(whose)f(pro)s(of)h(requires)e(classical)h(reasoning.)378
1187 y(In)i(particular,)h(w)m(e)g(men)m(tion)g(the)g(theorem)g(whic)m
(h)f(states)i(the)f(existence)g(of)h(an)e(uncomputable)378
1300 y(function,)k(for)g(example,)h(in)e(\(Cutland)f(1980\).)63
b(The)36 b(pro)s(of)g(of)h(this)f(theorem)h(in)f(Co)s(q)g(w)m(as)i(not)
378 1413 y(attempted)31 b(b)m(y)f(the)h(author,)f(and)g(it)g(is)f
(unclear)g(whether)h(this)f(theorem)i(can)f(b)s(e)g(pro)m(v)m(ed)g(in)f
(Co)s(q)378 1526 y(without)k(using)f(the)i(la)m(w)g(of)g(the)f
(excluded)g(middle.)49 b(W)-8 b(e)35 b(also)e(p)s(oin)m(t)g(out)h(that)
h(the)f(mec)m(hanisa-)378 1638 y(tion)c(of)h(the)f(theory)h(of)g
(computation)f(in)g(Co)s(q)g(required)e(the)j(notion)f(of)h(e\013ectiv)
m(ely)g(computable)378 1751 y(functions.)57 b(Suc)m(h)35
b(notion)h(is)f(informal)f(b)m(y)i(nature,)i(and)e(therefore)g(w)m(as)h
(not)f(formalised.)57 b(It)36 b(is)378 1864 y(p)s(oin)m(ted)23
b(out)i(in)e(section)i(3.3.4,)j(ho)m(w)m(ev)m(er,)f(that)e(b)s(ecause)f
(of)h(the)g(constructiv)m(e)g(nature)f(of)h(the)f(Co)s(q)378
1977 y(logic,)j(the)g(formal)e(de\014nition)f(of)i(e\013ectiv)m(ely)i
(computable)d(functions)g(is)h(not)g(required)f(as)h(all)f(Co)s(q)378
2090 y(functions)j(are)j(e\013ectiv)m(e)g(b)m(y)f(nature.)40
b(The)30 b(pro)s(ofs)f(in)f(HOL)i(of)g(theorems)g(whic)m(h)f(use)g(the)
h(notion)378 2203 y(of)g(e\013ectiv)m(ely)i(computable)d(functions)g(w)
m(ere)i(not)f(attempted)i(during)c(the)i(mec)m(hanisation.)40
b(The)378 2316 y(author)30 b(is)g(again)g(not)h(sure)e(whether)h(suc)m
(h)g(results)f(can)i(b)s(e)f(deriv)m(ed)f(in)g(HOL.)378
2556 y FQ(The)35 b(Use)g(of)g(Pro)s(of)h(Ob)6 b(jects)378
2728 y FT(The)30 b(Co)s(q)g(system)g(stores)h(pro)s(of)f(terms)g(in)f
(its)h(theory)g(\014les)g(and)g(uses)f(for)i(these)f(terms)h(include:)
489 2915 y(1.)46 b(Program)40 b(extraction:)60 b(Giv)m(en)40
b(some)g(program)g(sp)s(eci\014cation)e FP(S)5 b FT(,)42
b(a)e(constructiv)m(e)h(pro)s(of)605 3028 y(that)22 b(there)e(is)g
(some)h(program)g(satisfying)e(it)h(con)m(tains)h(an)g(instance)f(of)h
(a)g(program)g(for)f(whic)m(h)605 3141 y FP(S)31 b FT(holds,)26
b(hence)g(one)g(can)g(obtain)g(a)g(certi\014ed)g(program)f(from)h(a)g
(pro)s(of)f(of)h(its)g(sp)s(eci\014cation.)605 3254 y(This)42
b(facilit)m(y)g(is)h(supp)s(orted)e(b)m(y)j(the)f(Co)s(q)g(system)h
(whic)m(h)e(pro)m(vides)g(a)i(pac)m(k)-5 b(age)45 b(whic)m(h)605
3367 y(extracts)h(an)e(ML)h(program)f(from)g(a)h(pro)s(of)f(term,)k(as)
d(w)m(ell)e(as)i(pro)m(viding)d(supp)s(ort)h(for)605
3480 y(pro)m(ving)35 b(the)h(sp)s(eci\014cation)f(of)h(functions)e
(written)h(in)f(an)i(ML)g(syn)m(tax)g(\(P)m(aulin-Mohring)605
3593 y(1989;)d(P)m(aren)m(t)e(1993;)h(P)m(aulin-Mohring)c(and)i(W)-8
b(erner)31 b(1993\).)489 3780 y(2.)46 b(Extracting)30
b(pro)s(of)g(texts)h(written)e(in)f(a)j(natural)e(language:)41
b(A)30 b(pro)s(of)f(term)h(of)g(t)m(yp)s(e)h FP(\034)40
b FT(can)605 3893 y(b)s(e)32 b(seen)g(as)g(an)g(accoun)m(t)i(of)e(the)g
(pro)s(of)g(steps)g(in)m(v)m(olv)m(ed)f(in)g(deriving)f(the)i(theorem)h
FP(\034)10 b FT(,)32 b(and)605 4006 y(Co)s(q)26 b(pro)m(vides)f(to)s
(ols)h(for)h(extracting)g(a)f(pro)s(of)g(written)f(in)g(a)i(natural)e
(language)i(from)f(pro)s(of)605 4119 y(ob)5 b(jects)31
b(\(see)h(section)e(2.5.2\).)489 4307 y(3.)46 b(Indep)s(enden)m(t)25
b(pro)s(of)h(c)m(hec)m(king:)40 b(Pro)s(of)26 b(terms)g(can)h(b)s(e)f
(c)m(hec)m(k)m(ed)i(b)m(y)f(an)f(indep)s(enden)m(t)f(pro)s(of)605
4420 y(c)m(hec)m(k)m(er)34 b(to)e(gain)f(more)g(con\014dence)h(in)e
(their)g(correctness.)45 b(Moreo)m(v)m(er,)34 b(suc)m(h)d(pro)s(of)g
(terms)605 4533 y(can)h(b)s(e)f(easier)g(to)i(translate)e(in)m(to)h
(pro)s(of)f(accoun)m(ts)h(of)g(another)g(theorem)g(pro)m(v)m(er)g(than)
f(an)605 4645 y(actual)k(pro)s(of)f(script)f(or)i(an)f(ML)h(program)f
(\(as)h(HOL)f(pro)s(of)g(scripts)f(actually)h(are\).)54
b(The)605 4758 y(HOL)33 b(system)h(is)e(truth-based)h(rather)g(than)g
(pro)s(of-based)g(and)g(it)g(do)s(es)g(not)h(store)g(pro)s(ofs)605
4871 y(in)29 b(its)h(theories.)378 5111 y FQ(The)35 b(Sectioning)g(Mec)
m(hanism)378 5283 y FT(The)30 b FM(Gallina)f FT(sp)s(eci\014cation)h
(language)h(allo)m(ws)f(Co)s(q)h(pro)s(of)f(scripts)f(to)j(b)s(e)e
(structured)g(in)m(to)h(sec-)378 5396 y(tions,)f(and)g(one)g(can)h(mak)
m(e)h(de\014nitions)27 b(and)j(pro)m(v)m(e)h(theorems)g(whic)m(h)e(are)
i(lo)s(cal)f(to)h(a)g(particular)378 5509 y(section.)66
b(The)38 b(need)g(of)h(lo)s(cal)f(de\014nitions)e(and)i(results)f(is)h
(often)h(encoun)m(tered)g(during)e(theory)378 5622 y(dev)m(elopmen)m
(t,)28 b(where)e(for)g(instance,)h(the)g(de\014nition)d(of)i(some)h
(particular)e(concept)i(can)g(facilitate)p eop
%%Page: 50 60
50 59 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(50)378
396 y(the)21 b(pro)s(of)g(of)g(a)h(n)m(um)m(b)s(er)e(of)h(results)f
(but)h(do)s(es)g(not)g(con)m(tribute)g(m)m(uc)m(h)h(to)g(the)f(o)m(v)m
(erall)h(formalisation)378 509 y(of)30 b(the)h(theory)-8
b(.)378 753 y FG(3.4.4)112 b(Concluding)37 b(Remarks)378
924 y FT(The)c(t)m(w)m(o)i(case)f(studies,)f(and)g(esp)s(ecially)f
(more)h(extensiv)m(e)h(mec)m(hanisations)f(of)h(di\013eren)m(t)f(math-)
378 1037 y(ematical)k(theories,)i(sho)m(w)f(that)f(b)s(oth)g(HOL)g(and)
g(Co)s(q)f(are)i(robust)e(systems)i(and)e(practical)h(in)378
1150 y(mec)m(hanising)23 b(mathematical)h(results.)38
b(The)24 b(strongest)h(p)s(oin)m(t)e(of)h(HOL)g(is)f(the)i
(\015exibilit)m(y)c(giv)m(en)j(to)378 1263 y(the)i(users)f(b)m(y)g
(means)h(of)g(the)g(metalanguage;)j(while)24 b(Co)s(q)h(theorem)h(pro)m
(ving)f(relies)f(on)i(the)g(p)s(o)m(w)m(er)378 1376 y(of)31
b(the)h(Calculus)d(of)i(Inductiv)m(e)g(Constructions.)42
b(Here,)32 b(w)m(e)f(giv)m(e)h(some)g(concluding)d(remarks)i(on)378
1489 y(these)g(features.)378 1729 y FQ(The)k(Flexibilit)m(y)f(of)i(the)
e(Metalanguage)378 1901 y FT(By)h(allo)m(wing)f(a)h(theorem)g(pro)m
(ving)g(session)f(to)h(b)s(e)g(giv)m(en)g(within)d(a)j(general)g(purp)s
(ose)e(metalan-)378 2014 y(guage,)28 b(HOL)d(o\013ers)g(a)h(higher)e
(degree)i(of)f(\015exibilit)m(y)d(than)j(Co)s(q.)39 b(As)25
b(a)h(result,)f(HOL)g(users)f(imple-)378 2127 y(men)m(t)i(a)g(larger)f
(n)m(um)m(b)s(er)f(of)i(new)f(inference)f(rules)g(during)g(theory)h
(dev)m(elopmen)m(t)h(than)f(Co)s(q)g(users.)378 2239
y(F)-8 b(or)32 b(example,)f(the)h(mec)m(hanisation)f(of)g(the)h(theory)
f(of)g(computation)h(in)e(HOL)46 b(includes)29 b(sev)m(eral)378
2352 y(con)m(v)m(ersions)c(for)f(animating)g(the)h(de\014nitions,)e
(simple)g(and)h(more)h(elab)s(orate)g(tactics)h(whic)m(h)d(a)m(v)m(oid)
378 2465 y(rep)s(etitiv)m(e)k(inferences)h(and)f(most)i(bac)m(kw)m(ard)
f(pro)s(ofs)f(include)f(tactics)j(implemen)m(ted)e(`on)h(the)g(\015y')
378 2578 y(using)33 b(tacticals)j(and)e(other)h(ML)g(functions.)52
b(The)34 b(syn)m(tax)h(of)g FM(Gallina)e FT(can)i(b)s(e)f(extended,)i
(sa)m(y)378 2691 y(with)41 b(predicates)h(on)g(terms)g(so)g(that)h(one)
f(can)h(\014lter)e(a)i(sublist)c(of)k(assumptions)d(to)j(b)s(e)f(used)
378 2804 y(b)m(y)33 b(some)h(tactic,)h(but)e(then)g(one)h(asks)f
(whether)g(a)h(sp)s(eci\014cation)e(language)i(as)f(p)s(o)m(w)m(erful)f
(as)i(the)378 2917 y(metalanguage)f(is)e(required)f(to)i(implemen)m(t)e
(the)i(required)e(\014ltering)g(functions)g(during)f(theorem)378
3030 y(pro)m(ving.)39 b(Ha)m(ving)28 b(a)h(sp)s(eci\014cation)e
(language)h(surely)f(has)h(its)f(adv)-5 b(an)m(tages:)41
b(the)29 b(system)f(is)f(easier)378 3143 y(to)36 b(learn)f(b)m(y)h(new)
f(users,)i(and)e(pro)s(of)g(scripts)g(are)h(in)e(general)i(easier)g(to)
g(follo)m(w;)i(also,)f(theorem)378 3256 y(pro)m(ving)24
b(supp)s(ort)f(to)s(ols)i(lik)m(e)f(a)h(debugger)g(or)g(a)g(graphical)f
(user)g(in)m(terface)h(are)g(probably)e(easier)i(to)378
3369 y(dev)m(elop)h(for)h(a)g(sp)s(eci\014cation)e(language)i(with)f(a)
h(limited)d(syn)m(tax)j(rather)g(than)f(for)g(a)h(general)g(pur-)378
3481 y(p)s(ose)35 b(programming)e(language.)55 b(Ho)m(w)m(ev)m(er,)39
b(the)c(p)s(o)m(w)m(er)g(of)g(a)g(T)-8 b(uring-complete)34
b(metalanguage)378 3594 y(is)29 b(not)h(to)g(b)s(e)g(underestimated,)e
(for)i(it)f(can)h(b)s(e)g(used)f(for)g(instance)h(to)g(deriv)m(e)g
(theorems)g(through)378 3707 y(the)h(manipulation)c(of)k(pro)s(of)e
(terms.)378 3947 y FQ(The)35 b(Expressiv)m(eness)g(of)h(the)e(Calculus)
h(of)g(Inductiv)m(e)g(Constructions)378 4119 y FT(The)26
b(restrictions)f(due)g(to)i(the)g(sp)s(eci\014cation)e(language)h(are)h
(reliev)m(ed)f(b)m(y)g(the)g(p)s(o)m(w)m(er)g(of)h(CIC.)e(The)378
4232 y(fact)32 b(that)h(theorems)e(are)h(pro)m(v)m(ed)g(b)m(y)g(simply)
d(constructing)i(and)g(breaking)f(do)m(wn)i(terms)f(mak)m(es)378
4345 y(the)25 b(implemen)m(tation)f(of)h(tactics)i(sp)s(ecialised)22
b(for)j(particular)f(logic)h(constructs)g(unnecessary)f(and)378
4458 y(the)e(p)s(o)m(w)m(erful)e(notion)h(of)h(con)m(v)m(ertibilit)m(y)
e(replaces)i(the)g(implemen)m(tation)e(of)i(con)m(v)m(ersions)g(for)f
(ev)m(ery)378 4571 y(de\014nition.)36 b(No)23 b(new)f(tactics)i(or)e
(inference)g(rules)f(are)i(implemen)m(ted)f(in)f(the)i(mec)m
(hanisation)f(of)h(the)378 4684 y(theory)g(of)g(computation)g(in)e(Co)s
(q,)k(b)s(oth)d(b)s(ecause)h(the)g(inference)f(p)s(o)m(w)m(er)h(of)g
(the)g(simple)e(constructs)378 4796 y(of)g FM(Gallina)e
FT(is)i(enough)f(for)h(most)h(reasoning,)g(and)f(also)g(b)s(ecause)g
(the)g(non-trivial)e(task)j(of)f(actually)378 4909 y(implemen)m(ting)42
b(a)i(new)g(elab)s(orate)g(tactic)h(in)e(Co)s(q)g(discourages)h(the)g
(dev)m(elopmen)m(t)h(of)f(simple)378 5022 y(tactics)23
b(whic)m(h)e(are)i(used)f(only)f(to)i(substitute)f(a)g(small)f(n)m(um)m
(b)s(er)g(of)i(inferences.)37 b(The)22 b(p)s(o)m(w)m(er)g(of)h(CIC)378
5135 y(is)j(also)h(emphasised)f(b)m(y)h(its)g(highly)e(expressiv)m(e)i
(t)m(yp)s(e)g(system)g(whic)m(h)f(allo)m(ws)h(quan)m(ti\014cation)g(o)m
(v)m(er)378 5248 y(t)m(yp)s(es)e(and)g(dep)s(enden)m(t)f(t)m(yp)s(es)i
(and)e(th)m(us)h(giv)m(es)h(a)g(more)f(natural)f(formalisation)g(of)i
(mathematical)378 5361 y(concepts)31 b(than)f(a)g(simple)f(t)m(yp)s(e)h
(theory)-8 b(.)41 b(W)-8 b(e)32 b(ha)m(v)m(e)f(seen)f(ho)m(w)m(ev)m
(er,)i(ho)m(w)e(the)h(stronger)f(notion)g(of)378 5474
y(equalit)m(y)g(and)g(equiv)-5 b(alence)29 b(in)h(HOL)g(simpli\014es)d
(most)j(formalisations.)519 5587 y(The)e(primitiv)m(e)e(inference)h
(rules)g(of)h(HOL)g(are)g(to)s(o)h(simple)d(and)h(are)i(rarely)e(used)h
(in)e(practice;)378 5700 y(most)32 b(reasoning)e(is)g(p)s(erformed)g(b)
m(y)h(higher)f(lev)m(el)h(inferences.)42 b(The)31 b(simplicit)m(y)d(of)
k(the)f(primitiv)m(e)p eop
%%Page: 51 61
51 60 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(51)378
396 y(rules)41 b(giv)m(es)i(a)g(straigh)m(tforw)m(ard)g(implemen)m
(tation)e(of)i(the)g(core)g(inference)f(engine,)k(on)c(whose)378
509 y(correctness)d(the)f(soundness)e(of)i(the)g(HOL)f(system)h
(relies.)62 b(Although)37 b(CIC)g(is)g(more)h(complex)378
622 y(than)27 b(the)g(HOL)g(logic,)h(it)f(is)g(sound)f(and)g(due)h(to)h
(the)f(Curry-Ho)m(w)m(ard)g(isomorphism)e(theorems)i(in)378
735 y(CIC)j(can)h(b)s(e)f(c)m(hec)m(k)m(ed)j(b)m(y)e(a)g(t)m(yp)s(e)g
(c)m(hec)m(king)h(algorithm,)e(on)h(whose)g(correctness)g(the)g
(soundness)378 848 y(of)e(the)g(Co)s(q)f(system)h(relies.)39
b(Th)m(us,)29 b(one)g(can)g(ha)m(v)m(e)h(a)f(v)m(ery)h(p)s(o)m(w)m
(erful)d(logic)i(whose)f(theorems)i(can)378 961 y(still)e(b)s(e)i(c)m
(hec)m(k)m(ed)i(b)m(y)f(a)f(simple)f(algorithm.)519 1074
y(The)35 b(feasibilit)m(y)f(of)i(actually)f(doing)g(so)h(ma)m(y)g(ho)m
(w)m(ev)m(er)h(b)s(e)e(questioned.)56 b(Pro)s(of)36 b(terms)f(ma)m(y)
378 1187 y(b)s(ecome)h(v)m(ery)h(large,)h(and)d FP(\014)5
b(\016)s(\023)p FT(-con)m(v)m(ertibilit)m(y)37 b(ma)m(y)f(b)s(ecome)h
(infeasible)c(for)j(large)g(ob)5 b(jects,)39 b(al-)378
1300 y(though)29 b(these)i(factors)f(do)g(not)g(yield)e(an)m(y)i
(signi\014can)m(t)f(problems)f(for)h(the)h(mec)m(hanisation)g(of)g(the)
378 1413 y(results)f(in)g(section)i(3.3.)378 1699 y FH(3.5)135
b(On)45 b(T)-11 b(actic)45 b(Pro)t(ofs)378 1902 y FT(T)-8
b(actic-based)30 b(in)m(teractiv)m(e)f(pro)s(of)f(disco)m(v)m(ery)i(is)
d(one)j(of)e(the)i(most)f(commonly)f(used)g(metho)s(ds)g(for)378
2015 y(implemen)m(ting)c(mec)m(hanised)i(pro)s(ofs.)38
b(Most)28 b(of)e(the)h(pro)s(ofs)e(implemen)m(ted)g(in)g(the)h(mec)m
(hanisation)378 2128 y(of)31 b(computabilit)m(y)f(in)g(HOL,)i(and)e
(all)h(the)g(pro)s(ofs)g(implemen)m(ted)f(in)g(the)h(mec)m(hanisation)g
(in)f(Co)s(q)378 2241 y(w)m(ere)e(disco)m(v)m(ered)g(in)m(teractiv)m
(ely)g(b)m(y)f(applying)f(tactics.)41 b(This)25 b(mec)m(hanism)i(is)g
(indeed)f(quite)h(e\013ec-)378 2354 y(tiv)m(e)d(for)f(the)g(in)m
(teractiv)m(e)h(disco)m(v)m(ery)g(of)f(pro)s(ofs)g(b)s(ecause)g(users)f
(can)i(use)f(and)f(implemen)m(t)g(p)s(o)m(w)m(erful)378
2467 y(tactics)32 b(to)g(automate)g(sev)m(eral)g(pro)s(of)e(steps,)h
(and)f(usually)f(users)h(do)h(not)g(need)g(to)h(remem)m(b)s(er)e(all)
378 2579 y(the)j(previous)d(steps)j(of)f(in)m(teraction)h(during)d
(theorem)i(pro)m(ving.)46 b(Ho)m(w)m(ev)m(er,)36 b(since)31
b(tactic)j(pro)s(ofs)378 2692 y(are)d(essen)m(tially)e(lists)g(of)h(in)
m(teraction)h(steps)f(they)h(are)f(unreadable)f(and)h(hard)f(to)j
(follo)m(w.)519 2805 y(Figure)25 b(4)h(giv)m(es)f(an)h(example)f(of)g
(a)h(short)f(HOL)g(tactic)i(pro)s(of)d(tak)m(en)j(from)e(the)g(mec)m
(hanisation)378 2918 y(of)33 b(computabilit)m(y)e(theory)-8
b(.)50 b(Tw)m(elv)m(e)33 b(tactics)h(w)m(ere)f(applied)e(b)s(efore)i
(the)g(goal)g(w)m(as)h(pro)m(v)m(ed.)48 b(The)378 3031
y(c)m(hoice)40 b(of)g(whic)m(h)f(tactic)i(to)f(apply)e(during)g(eac)m
(h)j(in)m(teraction)f(step)f(w)m(as)h(determined)f(rapidly)-8
b(,)378 3144 y(and)40 b(the)g(pro)s(of)g(w)m(as)h(found)e(in)g(a)h(few)
h(min)m(utes.)69 b(This)39 b(is)g(mostly)h(due)g(to)h(the)f(fact)i
(that)f(the)378 3257 y(goal)34 b(is)e(rather)i(simple,)e(and)h(b)s
(ecause)h(of)f(the)h(fact)g(that)g(the)g(o)m(v)m(erall)g(strategy)h
(for)e(\014nding)e(this)378 3370 y(particular)42 b(pro)s(of)i(w)m(as)g
(kno)m(wn)g(b)m(y)g(the)g(author.)82 b(It)44 b(should)e(b)s(e)h(noted,)
48 b(ho)m(w)m(ev)m(er,)h(that)c(this)378 3483 y(particular)33
b(theorem)i(is)f(a)h(v)m(ery)h(simple)c(one,)37 b(and)d(sev)m(eral)h
(suc)m(h)g(theorems)g(are)g(pro)m(v)m(ed)g(during)378
3596 y(the)42 b(mec)m(hanisation)f(b)s(efore)g(non-trivial)e(results)h
(can)i(b)s(e)f(deriv)m(ed.)73 b(The)41 b(\014gures)g(in)f(tables)i(1)
378 3709 y(and)34 b(2)g(sho)m(w)h(that)f(successful)g(tactic)h(pro)s
(ofs)e(of)i(imp)s(ortan)m(t)e(results)g(require)g(sev)m(eral)i(h)m
(undreds)378 3821 y(of)d(tactics.)47 b(Finding)30 b(a)i(pro)s(of)f(ma)m
(y)i(require)e(man)m(y)h(more)g(in)m(teraction)g(steps)g(than)f(those)i
(in)e(the)378 3934 y(successful)j(pro)s(of)g(b)s(ecause)i(the)f(user)f
(ma)m(y)i(ha)m(v)m(e)h(to)f(bac)m(ktrac)m(k)h(through)d(the)i
(application)d(of)j(a)378 4047 y(n)m(um)m(b)s(er)29 b(of)i(tactics)g
(whic)m(h)e(resulted)g(in)g(unpro)m(v)-5 b(able)29 b(subgoals.)519
4160 y(Unfortunately)-8 b(,)36 b(b)s(ecause)e(of)h(their)e
(unreadabilit)m(y)-8 b(,)34 b(tactic)i(pro)s(ofs)d(lik)m(e)h(the)h(one)
f(in)g(\014gure)f(4)378 4273 y(do)26 b(not)g(o\013er)g(m)m(uc)m(h)g
(more)h(than)e(a)i(list)d(of)i(in)m(teraction)g(steps)g(whic)m(h)f(pro)
m(v)m(e)i(a)f(particular)e(theorem)378 4386 y(when)34
b(applied)e(to)k(a)f(particular)e(release)i(of)g(a)g(pro)s(of)f(dev)m
(elopmen)m(t)h(system.)54 b(The)35 b(tactic)h(pro)s(of)378
4499 y(is)31 b(en)m(tirely)h(targeted)i(at)f(the)g(pro)s(of)e(dev)m
(elopmen)m(t)i(system,)h(and)d(no)i(additional)d(information)h(is)378
4612 y(giv)m(en)f(to)h(the)g(user)f(to)h(help)e(her)h(understand)e(it.)
519 4725 y(The)45 b(abilit)m(y)e(to)j(follo)m(w)f(a)g(pro)s(of)f(can)i
(b)s(e)e(v)m(ery)i(imp)s(ortan)m(t)e(if)g(one)i(needs)e(to)i(implemen)m
(t)378 4838 y(a)41 b(di\013eren)m(t)f(pro)s(of)g(to)h(deriv)m(e)f(a)h
(similar)d(theorem,)44 b(or)d(to)g(deriv)m(e)f(the)h(same)g(theorem)g
(after)g(a)378 4951 y(de\014nition)27 b(has)h(b)s(een)g(mo)s(di\014ed)f
(sligh)m(tly)-8 b(.)39 b(Because)31 b(of)e(the)g(in)m(teractiv)m(e)h
(nature)e(of)h(tactic)i(pro)s(ofs,)378 5064 y(their)40
b(mo)s(di\014cation)g(often)i(relies)e(on)h(feedbac)m(k)h(from)f(the)h
(pro)s(of)f(dev)m(elopmen)m(t)g(system.)74 b(F)-8 b(or)378
5176 y(example,)41 b(pro)s(ofs)d(in)m(v)m(olving)g(a)h(mo)s(di\014ed)e
(de\014nition)f(are)k(re-run)d(un)m(til)h(one)h(fails.)65
b(The)38 b(failed)378 5289 y(pro)s(of)26 b(is)f(then)h(mo)s(di\014ed)e
(b)m(y)j(disco)m(v)m(ering)f(new)f(pro)s(of)h(steps)g(in)m(teractiv)m
(ely)-8 b(.)40 b(Users)26 b(w)m(ould)g(b)s(e)f(able)378
5402 y(to)k(mak)m(e)h(more)f(mo)s(di\014cations)e(without)h(the)h(need)
f(of)h(feedbac)m(k)h(from)e(the)h(system)g(if)f(the)h(pro)s(ofs)378
5515 y(can)i(b)s(e)e(follo)m(w)m(ed)h(without)g(running)d(them.)519
5628 y(The)g(pro)s(ofs)f(implemen)m(ted)f(in)h(the)h(case)h(studies)e
(often)h(mak)m(e)h(use)f(of)g(de\014nitions)d(in)m(tro)s(duced)p
eop
%%Page: 52 62
52 61 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(52)p
378 416 3453 4 v 376 2910 4 2494 v 602 652 a Fw(val)42
b(EXEC_STEP_MAXREG)37 b(=)44 b(prove)d(\()689 751 y(--`)p
Fu(8)13 b Fw(P)43 b(m)g(p1)g(r1)g(p2)f(r2.)733 851 y(\(EXEC_STEP)d(P)k
(\(p1,)f(r1\))h(=)g(\(p2,)f(r2\)\))g Fu(\))733 951 y
Fw(\(MAXREG)f(P)i(<)g(m\))g Fu(\))733 1050 y Fw(\(r1)f(m)h(=)h(r2)e
(m\)`--,)689 1150 y(REPEAT)f(GEN_TAC)g(THEN)689 1249
y(ASM_CASES_TAC)d(\(--`Final)i(P)k(\(p1,)d(r1\)`--\))g(THENL)689
1349 y([REPEAT)g(STRIP_TAC)f(THEN)733 1449 y(IMP_RES_THEN)820
1548 y(\(fn)i(t)i(=>)e(RULE_ASSUM_TAC)c(\(REWRITE_RULE)g([t]\)\))820
1648 y(Final_EXEC_STEP)f(THEN)733 1748 y(IMP_RES_TAC)i(PAIR_EQ_EQ)g
(THEN)733 1847 y(ASM_REWRITE_TAC)e([],)733 1947 y(ASM_REWRITE_TAC)g
([EXEC_STEP])i(THEN)733 2046 y(IMP_RES_TAC)g(NOT_Final)h(THEN)733
2146 y(IMP_RES_TAC)f(MAXREG_instructi)o(on)o(_MA)o(XR)o(EG)e(THEN)733
2246 y(REPEAT)k(STRIP_TAC)f(THEN)733 2345 y(IMP_RES_TAC)f
(LESS_EQ_LESS_TRA)o(NS)e(THEN)733 2445 y(IMP_RES_TAC)i
(MAXREG_exec_inst)o(ru)o(cti)o(on)o(]\);)1273 2740 y
FT(Figure)29 b(4:)42 b(An)30 b(Example)f(of)i(a)f(T)-8
b(actic)32 b(Pro)s(of.)p 3829 2910 V 378 2913 3453 4
v 378 3270 a(m)m(uc)m(h)21 b(earlier)f(in)g(the)h(mec)m(hanisation)f
(or)h(v)m(ery)h(simple)d(results)g(ab)s(out)i(the)g(de\014ned)f(ob)5
b(jects,)24 b(rather)378 3383 y(than)33 b(theorems)g(stating)g(some)g
(high-lev)m(el)e(prop)s(erties)h(of)h(the)g(de\014ned)e(concepts.)49
b(This)31 b(can)j(b)s(e)378 3496 y(attributed)27 b(to)h(bad)e(theory)i
(design,)f(in)f(the)i(sense)f(that)h(not)g(enough)f(prop)s(erties)f
(concerning)h(the)378 3609 y(de\014ned)g(concepts)h(are)h(deriv)m(ed.)
39 b(It)28 b(is)f(therefore)h(probable)f(that)h(sev)m(eral)h(similar)c
(prop)s(erties)h(are)378 3722 y(deriv)m(ed)32 b(as)i(subgoals)f(of)h
(di\013eren)m(t)f(theorems.)50 b(Ideally)-8 b(,)33 b(suc)m(h)h(prop)s
(erties)d(should)h(b)s(e)h(iden)m(ti\014ed)378 3835 y(to)e(\014nd)d
(out)i(whether)g(some)g(lemma)g(whic)m(h)e(generalises)i(them)g(can)g
(b)s(e)f(deriv)m(ed.)40 b(Ho)m(w)m(ev)m(er,)32 b(it)e(is)378
3948 y(hard)d(to)h(iden)m(tify)e(these)i(prop)s(erties)e(and)h(the)h
(pro)s(of)f(fragmen)m(ts)h(whic)m(h)e(deriv)m(e)i(them)f(b)m(y)h
(reading)378 4061 y(the)42 b(tactic)h(pro)s(of)d(steps.)75
b(Suc)m(h)41 b(prop)s(erties)f(can)i(b)s(e)f(iden)m(ti\014ed)f(during)f
(in)m(teractiv)m(e)j(theorem)378 4174 y(pro)m(ving)29
b(if)h(the)g(user)g(notices)h(that)g(similar)c(subgoals)j(k)m(eep)h
(reapp)s(earing.)519 4287 y(Since)26 b(theorems)i(stating)f(simple)e
(results)h(are)h(also)g(used)g(in)e(the)j(later)f(stages)h(of)g(some)f
(mec)m(h-)378 4400 y(anisation,)36 b(the)g(pro)s(of)f(steps)h(in)e(a)i
(tactic)h(pro)s(of)e(can)h(use)f(theorems)h(represen)m(ting)f(results)g
(of)g(a)378 4512 y(wide)24 b(range)h(of)g(complexit)m(y:)37
b(high-lev)m(el)24 b(results)g(and)g(v)m(ery)h(trivial)e(results)h(are)
h(used)f(in)g(the)h(pro)s(of)378 4625 y(steps)31 b(of)h(the)f(same)h
(pro)s(of.)42 b(This)30 b(inhomogeneit)m(y)h(in)f(the)h(pro)s(of)g
(steps)g(can)h(also)f(b)s(e)f(seen)i(in)e(the)378 4738
y(complexit)m(y)38 b(of)g(the)g(tactics)h(used.)62 b(Sp)s(ecialised)36
b(tactics)j(whic)m(h)d(automate)k(man)m(y)e(pro)s(of)f(steps)378
4851 y(are)31 b(used)e(together)j(with)d(tactics)j(whic)m(h)d(automate)
j(a)f(few.)40 b(Apart)31 b(from)f(making)f(tactic)j(pro)s(ofs)378
4964 y(harder)40 b(to)i(follo)m(w,)h(this)d(inhomogeneit)m(y)g(also)h
(a\013ects)h(the)g(e\013ort)f(required)e(in)h(implemen)m(ting)378
5077 y(tactic)h(pro)s(ofs)d(since)h(the)h(n)m(um)m(b)s(er)f(of)g
(theorems)h(and)f(tactics)i(whic)m(h)d(a)i(user)f(has)h(to)g(consider)
378 5190 y(increases)h(as)h(the)g(theory)g(is)f(mec)m(hanised.)74
b(The)42 b(inhomogeneit)m(y)f(in)f(the)i(complexit)m(y)g(of)g(the)378
5303 y(pro)s(of)28 b(steps)g(can)h(also)g(b)s(e)e(noticed)i(in)e(the)i
(tactic)h(pro)s(ofs)d(of)i(other)g(HOL)f(theories)g(\(for)h(example,)
378 5416 y(those)35 b(supplied)d(with)i(the)h(HOL)g(system\),)h(as)g(w)
m(ell)e(as)h(in)e(pro)s(ofs)h(of)i(other)f(tactic-based)h(theo-)378
5529 y(rem)30 b(pro)m(v)m(ers.)41 b(It)30 b(can)h(also)f(b)s(e)g
(noticed)g(in)f(Mizar)h(pro)s(ofs)f(since)h(theorems)h(deriv)m(ed)e(in)
g(the)h(early)378 5642 y(stages)e(of)e(a)h(mec)m(hanisation,)g(or)f(in)
f(v)m(ery)i(basic)e(theories,)i(are)g(also)f(used)g(in)f(pro)s(ofs)g
(implemen)m(ted)p eop
%%Page: 53 63
53 62 bop 378 5 a FF(CHAPTER)30 b(3.)71 b(CASE)30 b(STUDIES)f(ON)h(T)-8
b(A)m(CTIC-BASED)31 b(THEOREM)e(PR)m(O)m(VERS)175 b FT(53)378
396 y(to)m(w)m(ards)31 b(the)g(end)e(of)i(the)f(mec)m(hanisation.)519
509 y(W)-8 b(e)39 b(therefore)f(argue)g(that)g(although)f(the)h
(tactic-based)h(pro)s(of)d(st)m(yle)i(is)f(quite)g(e\013ectiv)m(e)i(in)
378 622 y(the)28 b(in)m(teractiv)m(e)h(disco)m(v)m(ery)f(of)h(a)f(pro)s
(of,)g(the)g(implemen)m(tation)f(of)h(tactic)h(pro)s(ofs)e(relies)g(to)
s(o)i(m)m(uc)m(h)378 735 y(on)e(feedbac)m(k)h(from)e(the)i(system.)39
b(It)28 b(is)e(not)h(practical)g(to)h(implemen)m(t,)e(follo)m(w,)h(mo)s
(dify)f(or)h(correct)378 848 y(tactic)k(pro)s(ofs)e(without)g(feedbac)m
(k.)42 b(Ho)m(w)m(ev)m(er,)32 b(sev)m(eral)f(activities,)f(whic)m(h)e
(include)g(the)i(structur-)378 961 y(ing)35 b(of)h(a)g(mec)m(hanised)f
(theory)-8 b(,)38 b(and)d(the)h(actual)g(implemen)m(tation)f(of)h(the)g
(pro)s(of,)g(ma)m(y)h(dep)s(end)378 1074 y(on)30 b(the)h(abilit)m(y)d
(of)j(the)f(user)g(to)h(follo)m(w)f(and)f(understand)g(the)h(mec)m
(hanised)g(pro)s(ofs.)40 b(As)30 b(a)h(result,)378 1187
y(systems)39 b(whic)m(h)e(use)h(tactic-based)i(pro)s(of)e(implemen)m
(tation)f(ma)m(y)i(require)f(to)s(ols)g(and)g(e\013ectiv)m(e)378
1300 y(user-in)m(terfaces)43 b(whic)m(h)e(aid)g(the)i(user)f(to)h(p)s
(erform)e(these)i(activities)f(without)g(ha)m(ving)g(to)h(fol-)378
1413 y(lo)m(w)33 b(the)h(pro)s(ofs.)48 b(Alternativ)m(ely)-8
b(,)35 b(pro)s(of)d(st)m(yles)i(whic)m(h)e(do)h(not)h(rely)e(on)i(to)s
(o)g(m)m(uc)m(h)f(\014ne-grained)378 1526 y(in)m(teraction)38
b(with)f(the)i(system)f(to)h(follo)m(w)f(the)g(pro)s(ofs)g(can)g(b)s(e)
g(more)g(suitable)f(for)h(the)h(o)m(v)m(erall)378 1638
y(mec)m(hanisation)29 b(of)h(a)g(theory)g(than)g(one)g(whic)m(h)e
(relies)h(solely)f(on)i(tactic-based)h(implemen)m(tation.)378
1751 y(The)c(abilit)m(y)e(to)j(implemen)m(t)e(mec)m(hanised)g(pro)s
(ofs)g(whic)m(h)g(are)i(easy)f(to)h(follo)m(w)f(can)g(therefore)g
(o\013er)378 1864 y(sev)m(eral)k(adv)-5 b(an)m(tages)32
b(to)f(the)g(mec)m(hanisation)e(of)i(mathematical)g(theories.)p
eop
%%Page: 54 64
54 63 bop 378 1019 a FJ(Chapter)65 b(4)378 1434 y FR(The)77
b(Implemen)-6 b(tation)76 b(of)i(a)378 1683 y(Declarativ)-6
b(e)76 b(Pro)6 b(of)78 b(Language)g(in)378 1932 y(HOL)378
2414 y FH(4.1)135 b(In)l(tro)t(duction)378 2617 y FT(In)26
b(section)g(2.4)i(w)m(e)f(discussed)d(the)j(fact)g(that)g(the)g(HOL)f
(theorem)h(pro)m(v)m(er)g(\(Gordon)f(and)g(Melham)378
2730 y(1993\))32 b(is)e(implemen)m(ted)f(according)h(to)h(the)g(LCF)f
(philosoph)m(y)-8 b(,)29 b(in)g(the)h(sense)h(that:)514
2905 y FN(\017)46 b FT(HOL)e(theorems)g(are)h(represen)m(ted)f(b)m(y)g
(an)g(ML)g(abstract)h(data)f(t)m(yp)s(e)h(whose)f(signature)605
3018 y(functions)38 b(corresp)s(ond)h(to)h(the)g(primitiv)m(e)d(rules)h
(of)i(a)g(sound)e(deductiv)m(e)h(system)h(of)g(the)605
3131 y(HOL)29 b(logic.)40 b(This)28 b(ensures)h(that)h(theorems)f
(deriv)m(ed)g(in)f(the)h(system)h(are)g(v)-5 b(alid)28
b(sen)m(tences.)514 3314 y FN(\017)46 b FT(The)i(user)f(is)g(giv)m(en)h
(the)h(\015exibilit)m(y)c(to)k(implemen)m(t)d(pro)s(of)i(pro)s(cedures)
f(in)f(the)j(meta-)605 3427 y(language)31 b(ML)f(in)g(order)f(to)j
(facilitate)e(the)g(theorem)h(pro)m(ving)f(pro)s(cess.)514
3610 y FN(\017)46 b FT(The)38 b(HOL)g(system)g(includes)e(a)i(n)m(um)m
(b)s(er)f(of)h(ML)g(functions)f(whic)m(h)g(allo)m(w)g(users)h(to)g
(\014nd)605 3723 y(pro)s(ofs)30 b(in)m(teractiv)m(ely)g(b)m(y)g
(applying)f(tactics.)378 3899 y(The)20 b(ma)5 b(jorit)m(y)21
b(of)g(pro)s(ofs)f(implemen)m(ted)f(in)h(HOL,)g(and)g(most)h(other)g
(pro)s(of)f(dev)m(elopmen)m(t)i(systems,)378 4011 y(are)35
b(found)e(in)m(teractiv)m(ely)i(using)e(the)i(tactic-based)h
(goal-orien)m(ted)f(en)m(vironmen)m(t.)54 b(Ho)m(w)m(ev)m(er,)38
b(as)378 4124 y(sho)m(wn)g(in)g(the)h(case)h(studies)e(in)g(Chapter)g
(3,)k(tactic-based)e(pro)s(ofs)e(are)h(not)h(informativ)m(e)e(to)h(a)
378 4237 y(h)m(uman)32 b(reader)i(and)e(it)h(is)g(hard)f(to)i(mo)s
(dify)e(and)h(main)m(tain)f(them)h(without)g(feedbac)m(k)h(from)f(the)
378 4350 y(in)m(teractiv)m(e)e(theorem)f(pro)m(v)m(er.)41
b(On)29 b(the)g(other)h(hand,)f(pro)s(ofs)g(implemen)m(ted)g(in)f(the)i
(Mizar)g(pro)s(of)378 4463 y(language)37 b(\(T)-8 b(rybulec)36
b(1978\))j(are)f(easier)e(to)i(follo)m(w)e(since)h(they)g(o\013er)g
(more)g(v)-5 b(aluable)36 b(informa-)378 4576 y(tion)h(to)i(a)f(h)m
(uman)f(reader)g(than)h(do)f(tactic)i(pro)s(ofs.)62 b(Mizar)38
b(pro)s(ofs)e(are)j(usually)c(describ)s(ed)h(as)378 4689
y(declarativ)m(e,)c(since)f(pro)s(of)f(steps)h(explicitly)e(state)k
(the)e(conclusion)f(and)h(what)g(is)f(used)g(to)i(deriv)m(e)378
4802 y(it,)e(as)f(opp)s(osed)g(to)i(tactic-based)f(pro)s(cedural)e(pro)
s(ofs)h(whic)m(h)f(consist)h(of)h(the)g(list)e(of)i(in)m(teractions)378
4915 y(required)f(to)i(deriv)m(e)f(the)g(pro)s(of.)519
5028 y(In)j(this)f(c)m(hapter)i(w)m(e)g(illustrate)e(the)h(implemen)m
(tation)f(of)i(a)g(declarativ)m(e)g(pro)s(of)e(language)i(in)378
5141 y(HOL.)27 b(The)f(language)h(is)f(called)g(SPL,)g(standing)g(for)h
(Simple)d(Pro)s(of)j(Language,)h(and)f(is)e(based)i(on)378
5253 y(the)36 b(theorem)h(pro)m(ving)e(fragmen)m(t)i(of)g(Mizar.)58
b(The)36 b(motiv)-5 b(ation)36 b(of)g(this)f(implemen)m(tation)g(is)g
(to)378 5366 y(exp)s(erimen)m(t)27 b(with)f(p)s(ossible)f(w)m(a)m(ys)k
(of)e(increasing)g(the)g(theorem)h(pro)m(ving)f(p)s(o)m(w)m(er)h(of)g
(the)f(language)378 5479 y(during)d(the)j(mec)m(hanisation)g(of)g(a)g
(theory)-8 b(.)40 b(The)26 b(SPL)g(language)h(is)f(extensible,)g(in)g
(the)h(sense)f(that)378 5592 y(the)g(user)e(can)i(implemen)m(t)e(new)h
(theorem)h(pro)m(ving)f(constructs)h(and)f(include)e(them)i(in)g(the)g
(syn)m(tax)378 5705 y(of)d(the)g(language.)38 b(Suc)m(h)21
b(extensibilit)m(y)e(is)i(imp)s(ortan)m(t)g(b)s(ecause)g(theory-sp)s
(eci\014c)g(pro)s(of)g(pro)s(cedures)2057 5954 y(54)p
eop
%%Page: 55 65
55 64 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(55)378
396 y(whic)m(h)32 b(use)h(facts)h(deriv)m(ed)e(during)f(the)i(dev)m
(elopmen)m(t)h(of)f(a)g(theory)h(can)f(b)s(e)g(implemen)m(ted.)47
b(The)378 509 y(Mizar)32 b(language)h(is)f(not)g(extensible,)g(and)g
(this)g(feature)h(is)e(often)i(claimed)e(to)j(b)s(e)d(desirable)g
(\(see)378 622 y(the)g(conclusions)d(of)j(\(Rudnic)m(ki)d(and)i(T)-8
b(rybulec)29 b(1997\)\).)519 735 y(Our)42 b(w)m(ork)i(is)e(in)h(some)h
(resp)s(ect)f(similar)e(to)j(that)g(done)f(b)m(y)h(Harrison)e
(\(1996b\))k(who)d(im-)378 848 y(plemen)m(ted)g(a)g(Mizar)g(mo)s(de)g
(in)e(HOL.)i(This)f(mo)s(de)g(is,)k(ho)m(w)m(ev)m(er,)i(v)m(ery)43
b(m)m(uc)m(h)g(based)g(on)g(the)378 961 y(tactic-based)29
b(en)m(vironmen)m(t)e(in)g(HOL)g(since)g(Mizar)h(pro)s(of)f(constructs)
h(are)g(translated)g(in)m(to)f(HOL)378 1074 y(tactics.)55
b(The)34 b(SPL)g(language)h(is)f(ric)m(her)g(than)h(the)g(Mizar)g(mo)s
(de)f(in)f(HOL)i(since,)g(for)g(instance,)378 1187 y(SPL)28
b(scripts)g(can)i(b)s(e)e(structured)h(in)m(to)g(sections)g(to)h(allo)m
(w)f(a)g(more)h(mo)s(dular)d(presen)m(tation.)40 b(The)378
1300 y(pro)s(cessing)e(of)i(SPL)f(scripts)g(is)f(not)i(based)g(on)f
(HOL)h(tactics.)69 b(Recen)m(tly)-8 b(,)41 b(Syme)f(\(1997a\))i(has)378
1413 y(dev)m(elop)s(ed)32 b(a)h(declarativ)m(e)h(pro)s(of)e(language,)i
(DECLARE,)e(for)h(soft)m(w)m(are)h(v)m(eri\014cation)e(and)h(used)378
1526 y(it)i(to)i(v)m(erify)e(the)g(t)m(yp)s(e)h(correctness)h(of)f(Ja)m
(v)-5 b(a)36 b(\(Syme)g(1997b;)k(Syme)35 b(1998\).)59
b(This)34 b(language)i(is,)378 1638 y(ho)m(w)m(ev)m(er,)c(not)e
(extensible,)f(although)h(this)f(is)g(suggested)i(in)e(the)h(future)f
(w)m(ork)h(section)h(of)f(\(Syme)378 1751 y(1997a\).)519
1864 y(In)35 b(the)i(follo)m(wing)d(section)j(w)m(e)f(illustrate)f(the)
h(SPL)f(language)i(with)e(a)h(small)f(example)g(and)378
1977 y(describ)s(e)30 b(the)i(use)f(of)h(the)f(SPL)g(pro)s(of)g
(constructs.)45 b(The)31 b(pro)s(cessing)f(of)i(SPL)e(scripts)h(in)m
(to)g(HOL)378 2090 y(inferences)43 b(is)f(then)i(describ)s(ed)d(in)h
(section)i(4.3.)82 b(The)43 b(di\013eren)m(t)g(t)m(yp)s(es)h(of)f(pro)s
(of)g(pro)s(cedures)378 2203 y(whic)m(h)31 b(can)i(b)s(e)f(implemen)m
(ted)f(to)j(extend)e(the)h(language)g(are)g(listed)e(in)g(section)i
(4.4,)i(whic)m(h)c(also)378 2316 y(describ)s(es)e(the)h(use)g(of)h(a)f
(database)i(of)e(trivial)f(kno)m(wledge)h(whic)m(h)f(can)i(b)s(e)e
(used)h(to)h(deriv)m(e)f(trivial)378 2429 y(facts)h(automatically)-8
b(.)41 b(A)31 b(n)m(um)m(b)s(er)e(of)h(concluding)f(remarks)h(are)g
(then)h(giv)m(en)f(in)f(section)i(4.5.)378 2715 y FH(4.2)135
b(The)45 b(Structure)f(of)i(SPL)e(Scripts)378 2918 y
FT(The)22 b(SPL)f(pro)s(of)g(language)i(is)e(based)h(on)g(the)g
(theorem)h(pro)m(ving)e(fragmen)m(t)i(of)f(the)g(Mizar)g(language)378
3031 y(although)32 b(there)i(are)f(a)h(n)m(um)m(b)s(er)d(of)j
(di\013erences)e(b)s(et)m(w)m(een)i(the)f(t)m(w)m(o)h(languages.)49
b(In)33 b(this)f(section)378 3144 y(w)m(e)24 b(giv)m(e)g(an)f(o)m(v)m
(erview)h(of)g(the)g(structure)f(of)g(SPL)g(scripts)f(b)m(y)i(\014rst)e
(illustrating)f(it)i(with)f(the)i(help)e(of)378 3257
y(a)32 b(simple)d(example,)j(and)f(then)g(discussing)e(the)j
(signi\014cance)e(of)i(the)g(di\013eren)m(t)f(SPL)f(constructs.)378
3369 y(The)g(syn)m(tax)h(of)f(SPL)g(is)f(giv)m(en)i(in)e(App)s(endix)e
(A)2078 3337 y FL(1)2118 3369 y FT(.)378 3613 y FG(4.2.1)112
b(An)38 b(Example)378 3784 y FT(Figure)24 b(5)i(giv)m(es)f(an)g
(example)f(of)h(a)h(small)d(SPL)h(script)g(whic)m(h)g(con)m(tains)h
(one)g(section)g(and)g(in)e(whic)m(h)378 3897 y(the)31
b(follo)m(wing)d(theorems)j(are)g(deriv)m(ed:)473 4083
y FM(R_refl)46 b FT(=)569 4196 y FN(`)h(8)p FP(R)q FM(.)g(Symmetric)e
FP(R)k FN(\))e FM(Transitive)e FP(R)k FN(\))903 4309
y FM(\()p FN(8)p FP(x)p FM(.)e FN(9)p FP(y)s FM(.)f FP(R)j(x)f(y)s
FM(\))f FN(\))g FM(Reflexive)f FP(R)473 4534 y FM(R_equiv)g
FT(=)569 4647 y FN(`)h(8)p FP(R)q FM(.)g(Symmetric)e
FP(R)k FN(\))e FM(Transitive)e FP(R)k FN(\))903 4760
y FM(\()p FN(8)p FP(x)p FM(.)e FN(9)p FP(y)s FM(.)f FP(R)j(x)f(y)s
FM(\))f FN(\))g FM(Equivalence)e FP(R)378 4946 y FT(The)21
b(predicates)g Fw(Reflexive)l FT(,)j Fw(Symmetric)l FT(,)g
Fw(Transitive)17 b FT(and)k Fw(Equivalence)c FT(are)22
b(de\014ned)e(as)i(follo)m(ws:)473 5132 y FN(`)529 5147
y FE(def)686 5132 y FN(8)p FP(R)q FM(.)47 b(Reflexive)e
FP(R)k FN(\021)e FM(\()p FN(8)p FP(x)p FM(.)g FP(R)h(x)g(x)p
FM(\))473 5358 y FN(`)529 5373 y FE(def)686 5358 y FN(8)p
FP(R)q FM(.)f(Symmetric)e FP(R)k FN(\021)e FM(\()p FN(8)p
FP(x)g(y)s FM(.)g FP(R)i(x)e(y)k FM(=)c FP(R)i(y)h(x)p
FM(\))p 378 5528 1380 4 v 482 5582 a FC(1)516 5614 y
FB(F)-6 b(or)46 b(comparison,)k(the)45 b(syn)n(tax)g(of)h(the)f(Mizar)i
(language)f(is)g(a)n(v)l(ailable)h(on)e(the)g(W)-6 b(orld)45
b(Wide)h(W)-6 b(eb)44 b(as)378 5705 y Fk(http://www.mizar.org/language)
q(/synt)q(ax.h)q(tml.)p eop
%%Page: 56 66
56 65 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(56)p
378 416 3453 4 v 376 3608 4 3192 v 515 552 a Fw(section)41
b(on_symm_and_tra)o(ns)602 751 y(given)h(type)g(":'a";)602
851 y(let)g("R:'a)g Fu(!)i Fw('a)e Fu(!)i Fw(bool";)602
1050 y(assume)d(R_symm:)84 b("Symmetric)40 b(R")907 1150
y(R_trans:)g("Transitive)g(R")907 1249 y(R_ex:)172 b(")p
Fu(8)14 b Fw(x.)42 b Fu(9)15 b Fw(y.)43 b(R)g(x)g(y";)602
1449 y(theorem)e(R_refl:)f("Reflexive)g(R")602 1548 y(proof)689
1748 y(simplify)h(with)g(Reflexive,)f(Symmetric)g(and)i(Transitive;)689
1947 y(given)g("x:'a";)689 2046 y(there)g(is)h(some)e("y:'a")h(such)g
(that)907 2146 y(Rxy:)g("R)h(x)g(y")g(by)g(R_ex;)776
2246 y(so)g(Ryx:)f("R)h(y)g(x")g(by)g(R_symm,)d(Rxy;)689
2345 y(hence)i("R)h(x)g(x")f(by)h(R_trans,)e(Rxy,)g(Ryx;)602
2545 y(qed;)602 2744 y(theorem)g(R_equiv:)f("Equivalence)f(R")864
2843 y(<Equivalence>)f(by)43 b(R_refl,)d(R_symm)h(and)i(R_trans;)515
3043 y(end;)1266 3438 y FT(Figure)30 b(5:)41 b(An)30
b(Example)g(SPL)f(Pro)s(of)h(Script.)p 3829 3608 V 378
3611 3453 4 v 473 4081 a FN(`)529 4096 y FE(def)686 4081
y FN(8)p FP(R)q FM(.)47 b(Transitive)e FP(R)j FN(\021)g
FM(\()p FN(8)p FP(x)f(y)s FM(.)g FP(R)h(x)g(y)j FN(\))c(8)p
FP(z)t FM(.)h FP(R)g(y)j(z)h FN(\))47 b FP(R)i(x)e(z)t
FM(\))473 4307 y FN(`)529 4322 y FE(def)686 4307 y FN(8)p
FP(R)q FM(.)g(Equivalence)e FP(R)j FN(\021)f FM(\(Reflexive)e
FP(R)k FN(^)e FM(Symmetric)f FP(R)i FN(^)f FM(Transitive)e
FP(R)q FM(\))378 4494 y FT(These)37 b(de\014nitions)e(are)j(de\014ned)f
(in)f(HOL)h(and)g(are)h(imp)s(orted)e(in)m(to)h(the)h(en)m(vironmen)m
(t)f(of)h(SPL)378 4607 y(using)29 b(a)i(n)m(um)m(b)s(er)e(of)h
(appropriate)g(functions)f(\(as)h(will)e(b)s(e)i(describ)s(ed)e(later)j
(in)e(section)h(4.3\).)519 4720 y(The)36 b(\014rst)f(line)f(of)i(the)g
(script)f(op)s(ens)g(a)i(section)f(with)e(name)i Fw(on_symm_and_trans)
29 b FT(whic)m(h)35 b(is)378 4833 y(closed)g(b)m(y)g(the)g
Fw(end;)e FT(on)i(the)h(last)e(line.)54 b(Sections)34
b(are)i(op)s(ened)e(in)g(order)g(to)i(declare)f FI(r)-5
b(e)g(asoning)378 4946 y(items)p FT(,)50 b(whic)m(h)44
b(include)f(the)j(in)m(tro)s(duction)d(of)j(assumptions,)i(the)d
(declaration)g(and)g(pro)s(of)g(of)378 5059 y(theorems,)31
b(etc.)519 5172 y(The)24 b(\014rst)g(t)m(w)m(o)i(reasoning)d(items)i
(in)e(this)g(section)i(are)g(called)e(generalisations,)i(and)f(in)m
(tro)s(duce)378 5285 y(the)38 b(t)m(yp)s(e)f(v)-5 b(ariable)37
b Fw(:'a)f FT(and)h(the)g(v)-5 b(ariable)37 b Fv(R)h
FT(so)g(that)g(they)f(can)h(b)s(e)f(used)g(in)f(later)h(reasoning)378
5397 y(items.)k(T)m(yp)s(e)30 b(v)-5 b(ariables)29 b(and)i(HOL)f(v)-5
b(ariables)29 b(in)m(tro)s(duced)g(b)m(y)i(generalisations)e
(implicitly)e(bind)p eop
%%Page: 57 67
57 66 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(57)378
396 y(all)37 b(their)g(free)h(o)s(ccurrences)f(in)g(the)h(form)m(ulae)f
(within)f(their)h(scop)s(e.)2829 363 y FL(2)2931 396
y FT(In)g(our)h(case,)j(the)d(scop)s(e)378 509 y(of)e(the)g(v)-5
b(ariables)34 b Fw(:'a)h FT(and)g Fv(R)i FT(starts)f(from)f(their)g
(declaration)h(and)f(ends)g(when)g(the)h(section)g(is)378
622 y(closed.)519 735 y(The)h(t)m(w)m(o)i(generalisations)d(are)i
(follo)m(w)m(ed)f(b)m(y)h(the)f(in)m(tro)s(duction)f(of)i(three)f
(assumptions)f(la-)378 848 y(b)s(elled)e(with)h Fw(R_symm)n
FT(,)j Fw(R_trans)c FT(and)i Fw(R_ex)n FT(.)59 b(Lab)s(els)35
b(are)i(used)f(to)h(denote)g FI(facts)45 b FT(whic)m(h)35
b(include)378 961 y(axioms,)30 b(de\014nitions,)e(assumptions,)h
(theorems)i(and)f(the)g(results)f(in)h(pro)s(of)f(steps.)519
1074 y(The)35 b(\014rst)g(theorem,)j Fw(R_refl)n FT(,)f(is)e(then)g
(declared)h(and)f(pro)m(v)m(ed.)57 b(The)35 b(pro)s(of)g(consists)h(of)
g(the)378 1187 y(list)42 b(of)h(reasoning)g(items)g(b)s(et)m(w)m(een)h
(the)f Fw(proof)e FT(and)i(the)g Fw(qed)f FT(constructs.)80
b(The)43 b(\014rst)f(line)g(of)378 1300 y(the)34 b(pro)s(of)f(declares)
g(a)h(n)m(um)m(b)s(er)e(of)i FI(simpli\014ers)42 b FT(whic)m(h)33
b(are)h(used)f(during)e(the)j(theorem)g(pro)m(ving)378
1413 y(pro)s(cess.)41 b(This)29 b(particular)g(declaration)i(states)h
(that)f(the)g(de\014nitions)d(of)j Fw(Reflexive)m FT(,)g
Fw(Symmetric)378 1526 y FT(and)36 b Fw(Transitive)c FT(will)i(b)s(e)h
(used)h(automatically)g(to)i(simplify)32 b(the)37 b(assumptions)e(and)h
(theorems)378 1638 y(used)h(in)g(the)h(pro)s(of.)63 b(\(In)37
b(the)i(particular)d(implemen)m(tation)g(of)j(the)f(SPL)f(on)h(top)g
(of)g(the)g(HOL)378 1751 y(theorem)i(pro)m(v)m(er)h(describ)s(ed)d(in)h
(this)g(c)m(hapter,)k(the)e(simpli\014ers)36 b(are)41
b(applied)c(during)h(the)j(\014rst)378 1864 y(step)h(of)f(pro)s(of-c)m
(hec)m(king.\))76 b(As)41 b(a)h(result,)i(the)e(user)e(do)s(es)i(not)g
(ha)m(v)m(e)g(to)h(use)e(suc)m(h)g(de\014nitions)378
1977 y(explicitly)g(in)h(later)h(justi\014cations.)79
b(In)42 b(other)i(w)m(ords,)i(the)e(use)f(of)h(the)f(ab)s(o)m(v)m(e)i
(de\014nitions)c(is)378 2090 y(assumed)28 b(to)h(b)s(e)f(trivial)f(in)g
(the)i(con)m(text)h(of)f(this)e(pro)s(of.)40 b(A)28 b(new)g
(generalising)f(v)-5 b(ariable)28 b Fv(x)h FT(is)e(then)378
2203 y(in)m(tro)s(duced,)38 b(the)f(scop)s(e)h(of)f(whic)m(h)f(extends)
i(to)g(the)f(end)g(of)h(this)e(pro)s(of.)61 b(The)37
b(next)g(reasoning)378 2316 y(item)25 b(is)e(an)i(existen)m(tial)f
(result.)38 b(It)25 b(in)m(tro)s(duces)f(a)h(new)f(v)-5
b(ariable)24 b Fv(y)k FT(and)c(the)h(result)f Fv(R)44
b(x)g(y)27 b FT(lab)s(elled)378 2429 y(with)36 b Fw(Rxy)o
FT(.)62 b(The)37 b(v)-5 b(ariable)36 b Fv(y)k FT(existen)m(tially)c
(quan)m(ti\014es)g(all)h(the)g(statemen)m(ts)j(in)c(its)g(scop)s(e)i
(\(that)378 2542 y(is,)c(the)h(pro)s(of)7 b(\).)52 b(The)33
b(result)h Fu(9)p Fv(x)p Fw(.)p Fv(R)44 b(x)g(y)37 b
FT(is)c(justi\014ed)f(b)m(y)j(the)f(fact)h(denoted)f(b)m(y)g(the)h(lab)
s(el)d Fw(R_ex)o FT(,)378 2655 y(i.e.,)16 b(the)30 b(assumption)f
Fu(8)p Fv(x)p Fw(.)p Fu(9)p Fv(y)s Fw(.)p Fv(R)43 b(x)h(y)r
FT(.)d(Justi\014cations)29 b(of)h(the)h(form)521 2842
y FP(:)15 b(:)g(:)49 b FM(by)e FI(pr)-5 b(emise)1128
2856 y FL(1)1168 2842 y FM(,)47 b FI(pr)-5 b(emise)1573
2856 y FL(2)1612 2842 y FM(,)48 b FP(:)15 b(:)g(:)49
b FM(;)378 3030 y FT(are)38 b(called)f(straigh)m(tforw)m(ard)h
(justi\014cations)e(\(see)j(app)s(endix)c(A)j(for)g(the)g(general)g
(form)f(of)h(suc)m(h)378 3143 y(justi\014cations\).)74
b(The)41 b(conclusion)f(of)i(the)g(justi\014cation)e(is)h(deriv)m(ed)g
(automatically)g(from)h(the)378 3256 y(premises)37 b(using)f(an)i(in)m
(built)e(pro)m(v)m(er.)64 b(The)38 b(pro)s(of)f(then)h(follo)m(ws)f(to)
i(deriv)m(e)f(t)m(w)m(o)i(more)e(results,)378 3369 y
Fv(R)44 b(y)i(x)32 b FT(and)e Fv(R)44 b(x)g(x)p FT(,)31
b(b)s(oth)f(of)h(whic)m(h)e(are)j(justi\014ed)c(using)h(straigh)m
(tforw)m(ard)i(justi\014cations.)40 b(Cer-)378 3482 y(tain)27
b(constructs)g(suc)m(h)g(as)g Fw(so)o FT(,)h Fw(hence)n
FT(,)g Fw(then)n FT(,)g(and)f Fw(therefore)c FT(are)28
b(ignored)e(b)m(y)h(the)g(pro)s(of)g(c)m(hec)m(k)m(er,)378
3594 y(and)h(they)h(are)h(only)e(used)g(to)h(mak)m(e)h(the)f(pro)s(of)f
(more)h(readable.)40 b(In)28 b(Mizar,)i(suc)m(h)e(constructs)h(are)378
3707 y(used)35 b(to)h(sho)m(w)g(that)g(the)g(previous)e(result)h(is)g
(used)g(automatically)g(in)g(the)g(justi\014cation)g(of)h(the)378
3820 y(curren)m(t)28 b(statemen)m(t.)42 b(The)27 b(last)h(deriv)m(ed)f
(result)g(corresp)s(onds)f(to)j(the)f(statemen)m(t)i(of)e(the)g
(theorem)378 3933 y(and)i(therefore)h(it)e(completes)i(the)g(pro)s(of.)
519 4046 y(The)38 b(second)g(theorem)h(is)e(deriv)m(ed)g(b)m(y)i(a)f
(straigh)m(tforw)m(ard)g(justi\014cation.)63 b(The)38
b(expression)378 4159 y Fw(<Equivalence>)25 b FT(is)k(a)i(simpli\014er)
c(declaration)j(whic)m(h)f(is)g(lo)s(cal)h(only)f(to)i(the)g
(justi\014cation.)519 4272 y(All)44 b(declarations)g(\(assumptions,)k
(generalising)43 b(v)-5 b(ariables,)48 b(simpli\014ers,)d(etc.)16
b(\))85 b(with)44 b(the)378 4385 y(exception)32 b(of)f(theorems,)h
(exist)f(only)g(within)e(the)i(section)h(or)f(pro)s(of)g(they)g(are)h
(in)m(tro)s(duced.)42 b(The)378 4498 y(scop)s(e)22 b(of)h(theorems)f
(starts)h(from)e(after)i(they)f(are)h(justi\014ed)d(and)i(extends)g(to)
h(the)f(end)g(of)g(the)h(script.)378 4611 y(The)37 b(theorems)g(deriv)m
(ed)g(in)f(the)i(script)e(giv)m(en)h(in)f(\014gure)h(5)h(can)f(still)f
(b)s(e)h(used)f(outside)h(section)378 4724 y Fw(on_symm_and_tran)o(s)-6
b FT(,)40 b(ho)m(w)m(ev)m(er)f(their)e(statemen)m(ts)i(are)f(expanded,)
h(or)f(generalised,)h(according)378 4836 y(to)c(the)g(v)-5
b(ariables)33 b(and)h(assumptions)f(lo)s(cal)h(to)h(this)e(section,)j
(that)f(is)f(to)h(the)g(statemen)m(ts)h(giv)m(en)378
4949 y(in)29 b(page)i(55.)p 378 5112 1380 4 v 482 5165
a FC(2)516 5197 y FB(Note)e(that)g(the)f(represen)n(tation)i(of)f(HOL)f
(terms)h(do)r(es)g(not)g(include)g(quan)n(ti\014cation)f(o)n(v)n(er)h
(t)n(yp)r(es)f(|)h(all)h(t)n(yp)r(e)378 5288 y(v)l(ariables)f(are)g
(implicitly)f(univ)n(ersally)h(quan)n(ti\014ed.)41 b(W)-6
b(e)28 b(use)g(a)h(simple)f(mec)n(hanism)f(for)i(univ)n(ersally)f(quan)
n(tifying)378 5380 y(t)n(yp)r(e)d(v)l(ariables)h(explicitly)g(whic)n(h)
g(is)g(describ)r(ed)g(in)f(section)i(4.3.2.)p eop
%%Page: 58 68
58 67 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(58)378
396 y FG(4.2.2)112 b(Sectioning)36 b(Pro)s(of)i(Scripts)378
568 y FT(SPL)d(scripts)g(are)i(structured)e(in)m(to)h(sections)h(so)f
(that)h(results)e(whose)h(pro)s(ofs)f(mak)m(e)i(use)f(of)h(the)378
681 y(same)d(declarations)g(can)g(b)s(e)f(organised)g(together.)53
b(The)33 b(approac)m(h)h(presen)m(ted)g(here)g(is)f(in)f(some)378
794 y(resp)s(ect)38 b(similar)d(to)k(the)f(sectioning)f(mec)m(hanism)h
(of)g(the)g(Co)s(q)f(system)h(\(Barras)h(et)f(al.)64
b(1996\).)378 907 y(A)34 b(pro)s(of)f(script)f(consists)h(of)h(a)g
(list)e(of)i(sections,)h(and)e(sections)g(can)h(b)s(e)f(nested)h(to)g
(impro)m(v)m(e)g(the)378 1020 y(o)m(v)m(erall)28 b(structure)g(of)g
(scripts.)39 b(The)27 b(adv)-5 b(an)m(tages)30 b(of)e(declaring)f
(information)g(lo)s(cally)f(can)j(also)f(b)s(e)378 1133
y(seen)g(in)e(the)i(simple)e(example)h(giv)m(en)h(earlier)e(in)h
(\014gure)g(5.)40 b(In)27 b(particular,)g(the)g(statemen)m(ts)j(of)e
(the)378 1246 y(theorems)k(declared)g(in)f(the)h(pro)s(of)f(script)g
(are)i(shorter)e(than)h(their)f(fully)f(expanded)h(form)h(giv)m(en)378
1358 y(in)d(page)i(55,)h(and)d(therefore:)514 1546 y
FN(\017)46 b FT(Rep)s(etitiv)m(e)31 b(information)f(in)g(the)i
(statemen)m(ts)h(of)e(theorems)h(is)e(a)m(v)m(oided,)j(for)e(instance)g
(the)605 1659 y(an)m(teceden)m(ts)g(of)e(the)f(t)m(w)m(o)i(theorems)f
(in)e(our)i(example)f(are)h(declared)f(once)h(as)g(the)g(assump-)605
1772 y(tions)h(lo)s(cal)g(to)h(b)s(oth)e(theorems.)514
1960 y FN(\017)46 b FT(The)33 b(unexpanded)d(form)j(of)g(the)g
(statemen)m(t)h(of)f(theorems)g(in)f(the)h(section)g(in)e(whic)m(h)h
(they)605 2072 y(are)27 b(deriv)m(ed)d(is)h(due)h(to)g(the)g(fact)h
(that)g(they)f(are)g(sp)s(ecialised)d(b)m(y)j(the)g(information)e
(declared)605 2185 y(lo)s(cally)-8 b(,)39 b(whic)m(h)e(includes)e(the)j
(generalising)e(v)-5 b(ariables)37 b(and)g(assumptions.)62
b(As)37 b(a)i(result,)605 2298 y(justi\014cations)19
b(using)g(suc)m(h)h(theorems)h(do)f(not)g(ha)m(v)m(e)i(to)f(include)d
(the)j(assumptions)d(whic)m(h)h(are)605 2411 y(used)24
b(in)f(deriving)g(them.)38 b(F)-8 b(or)26 b(example,)f(when)f(the)h
(theorem)g Fw(R_refl)d FT(is)h(used)h(in)f(justifying)605
2524 y(the)30 b(theorem)g Fw(R_equiv)n FT(,)g(there)g(w)m(as)g(no)f
(need)h(to)h(include)c(the)j(three)g(assumptions)e(used)h(in)605
2637 y(deriving)40 b Fw(R_equiv)m FT(.)76 b(As)42 b(a)g(result,)i
(justi\014cations)d(whic)m(h)g(use)g(unexpanded)g(results)f(are)605
2750 y(shorter,)35 b(and)e(also)h(easier)g(to)h(pro)s(of)e(c)m(hec)m
(k,)j(than)e(those)g(whic)m(h)f(use)h(the)g(results)e(in)h(their)605
2863 y(fully)28 b(generalised)i(form.)514 3050 y FN(\017)46
b FT(Since)30 b(pro)s(of)f(statemen)m(ts)j(and)e(pro)s(ofs)g(are)g
(shorter,)h(scripts)e(are)i(easier)f(to)h(read.)378 3238
y(In)26 b(order)f(to)j(maximise)d(the)h(adv)-5 b(an)m(tages)28
b(of)f(readabilit)m(y)e(and)g(pro)s(of-c)m(hec)m(king)i(e\016ciency)-8
b(,)28 b(scripts)378 3351 y(can)45 b(b)s(e)f(organised)f(b)m(y)i
(implemen)m(ting)d(pro)s(ofs)i(whic)m(h)f(share)h(the)h(same)g
(information)d(in)i(one)378 3464 y(section.)73 b(This)39
b(results)h(in)f(a)j(b)s(etter)f(o)m(v)m(erall)g(structuring)e(of)i
(the)h(pro)s(of)e(script,)j(esp)s(ecially)c(if)378 3577
y(nested)29 b(sections)h(are)g(used)f(to)h(presen)m(t)g(the)f(hierarc)m
(hical)f(structure)i(of)f(the)h(mec)m(hanised)f(theory)-8
b(.)519 3690 y(A)38 b(section)g(corresp)s(onds)f(to)h(a)g(lo)s(cal)f
(con)m(text)j(within)35 b(the)j(SPL)f(en)m(vironmen)m(t.)63
b(All)37 b(decla-)378 3803 y(rations,)42 b(with)d(the)h(exception)g(of)
g(theorems,)j(exist)c(and)h(are)g(visible)e(from)h(the)h(line)e(they)i
(are)378 3916 y(declared)28 b(un)m(til)f(the)j(end)e(of)h(their)f(con)m
(text.)42 b(As)29 b(men)m(tioned)f(earlier,)h(theorems)g(exist)f(from)h
(their)378 4028 y(justi\014cation)23 b(to)h(the)g(end)g(of)g(the)g
(script,)g(and)g(are)g(expanded)f(when)g(their)g(con)m(text)j(is)d
(closed.)38 b(The)378 4141 y(expansion)31 b(mec)m(hanism)g(in)m(v)m
(olv)m(es)g(the)h(generalisation)f(of)h(the)g(theorem)g(according)g(to)
g(the)g(v)-5 b(ari-)378 4254 y(ables)29 b(and)g(assumptions)f(lo)s(cal)
h(to)h(the)g(con)m(text)i(the)e(theorem)g(is)f(sp)s(eci\014ed.)38
b(Only)28 b(the)i(v)-5 b(ariables)378 4367 y(free)29
b(in)e(the)i(theorem)h(and)e(the)h(assumptions)e(used)h(in)f(its)h(pro)
s(of)g(are)i(considered)d(for)i(expansion.)378 4480 y(This)g(mec)m
(hanism)g(is)h(describ)s(ed)e(in)h(more)h(detail)g(in)f(section)i
(4.3.5.)519 4593 y(Lo)s(cal)37 b(con)m(texts)h(can)g(also)e(b)s(e)h
(created)g(b)m(y)g(other)g(SPL)f(constructs.)61 b(F)-8
b(or)37 b(instance,)i(pro)s(ofs)378 4706 y(create)i(lo)s(cal)d(con)m
(texts;)46 b(all)38 b(pro)s(of)h(steps)g(deriv)m(ed)g(within)d(a)k
(particular)e(pro)s(of)g(are)i(lo)s(cal)f(only)378 4819
y(to)e(its)e(con)m(text)j(and)d(therefore)i(they)f(cannot)g(b)s(e)g
(used)f(outside)g(it.)57 b(Declarations)36 b(also)g(can)h(b)s(e)378
4932 y(sp)s(eci\014ed)29 b(lo)s(cally)g(to)i(a)f(segmen)m(t)i(of)e(a)h
(script)e(using)g(the)i(follo)m(wing)e(construct.)473
5119 y FM(local)569 5232 y FI(lo)-5 b(c)g(al)59 b(de)-5
b(clar)g(ations)473 5345 y FM(in)569 5458 y FI(script)57
b(se)-5 b(gment)473 5571 y FM(end;)p eop
%%Page: 59 69
59 68 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(59)378
396 y(In)37 b(this)f(construct,)k(the)e(scop)s(e)f(of)h(the)g(lo)s(cal)
e(declarations)h(extends)h(to)g(the)f Fw(end)g FT(of)g(the)h(script)378
509 y(segmen)m(t.)i(The)23 b(scop)s(e)h(of)g(the)g(declarations)f(in)g
(this)f(segmen)m(t)j(extends)f(to)h(the)f(end)f(of)h(the)g(con)m(text)
378 622 y(the)31 b Fw(local)41 b Fv(:)14 b(:)g(:)43 b
Fw(in)g Fv(:)14 b(:)g(:)44 b Fw(end)29 b FT(is)g(sp)s(eci\014ed.)378
866 y FG(4.2.3)112 b(Reasoning)38 b(Items)378 1037 y
FT(Reasoning)32 b(items)g(corresp)s(ond)f(to)i(the)g(individual)28
b(pro)s(of)j(steps)i(and)f(declarations)f(sp)s(eci\014ed)g(in)378
1150 y(SPL)e(scripts.)40 b(The)30 b(di\013eren)m(t)g(kinds)e(of)j
(reasoning)e(items)h(are)h(describ)s(ed)d(b)s(elo)m(w.)378
1390 y FQ(Generalisations)35 b(and)g(Assumptions)378
1562 y FT(Generalisations)30 b(in)m(tro)s(duce)g(v)-5
b(ariables)30 b(and)h(t)m(yp)s(e)h(v)-5 b(ariables)29
b(whic)m(h)i(univ)m(ersally)d(quan)m(tify)j(their)378
1675 y(free)24 b(o)s(ccurrences)g(in)f(the)i(pro)s(of)e(script)g(form)m
(ulae)h(implicitly)-8 b(.)35 b(Assumptions)23 b(represen)m(t)h(h)m(yp)s
(othe-)378 1788 y(ses)36 b(whic)m(h)f(are)i(in)m(tro)s(duced)e(in)g
(order)g(to)i(b)s(e)f(used)g(in)e(justi\014cations.)57
b(The)36 b(free)h(v)-5 b(ariables)34 b(and)378 1901 y(t)m(yp)s(e)e(v)-5
b(ariables)31 b(of)h(an)g(assumption)e(are)i(automatically)g(in)m(tro)s
(duced)e(as)j(generalisations)e(unless)378 2014 y(they)42
b(ha)m(v)m(e)g(already)f(b)s(een)g(in)m(tro)s(duced)f(earlier)g(in)g
(the)i(curren)m(t)f(con)m(text.)76 b(Assumptions)39 b(and)378
2127 y(v)-5 b(ariables)29 b(can)i(also)f(b)s(e)g(in)m(tro)s(duced)f
(together)i(b)m(y)g(declaring)e(quan)m(ti\014ed)g(assumptions,)g(suc)m
(h)h(as)473 2314 y FM(given)47 b(some)f("x:num")g(and)h("y:num")f(such)
h(that)569 2427 y(le_x_y:)f("x)h(<)g(y";)378 2667 y FQ(Theorems)34
b(and)h(Results)378 2839 y FT(Results)26 b(or)h(facts)g(are)h(in)m(tro)
s(duced)d(b)m(y)i(declaring)e(them)i(as)g(lab)s(elled)e(statemen)m(ts)j
(and)e(then)h(justi-)378 2952 y(fying)j(them.)45 b(Results)30
b(whic)m(h)h(are)h(required)d(outside)i(their)g(section)h(are)g(sp)s
(eci\014ed)e(as)h(theorems.)378 3065 y(Most)24 b(results,)f(ho)m(w)m
(ev)m(er,)k(are)c(used)f(only)g(within)f(the)i(pro)s(of)f(or)h(section)
g(they)h(are)f(deriv)m(ed)f(and)h(can)378 3178 y(b)s(e)29
b(called)g(pro)s(of)g(step)h(results,)f(or)h(simply)d(pro)s(of)i
(steps.)41 b(Pro)s(of)29 b(steps)h(can)g(also)g(b)s(e)f(existen)m
(tially)378 3290 y(quan)m(ti\014ed,)g(for)i(example:)473
3478 y FM(there)47 b(is)g(some)g("x:num")e(and)i("y:num")f(such)h(that)
569 3591 y(le_x_y:)f("x)h(<)g(y")473 3704 y FI(justi\014c)-5
b(ation)56 b(of)68 b FN(9)p FP(x)15 b(y)s(:)g(x)47 b(<)h(y)i
FM(;)378 3892 y FT(The)33 b(ab)s(o)m(v)m(e)i(statemen)m(t)g(is)e
(called)f(an)i(existen)m(tial)f(result)f(and)h(in)m(tro)s(duces)f(the)i
(v)-5 b(ariables)32 b FP(x)h FT(and)378 4004 y FP(y)41
b FT(in)d(the)h(curren)m(t)f(con)m(text)j(and)d(the)h(result)f(lab)s
(elled)e(with)h Fw(le_x_y)n FT(.)66 b(The)38 b(v)-5 b(ariables)37
b FP(x)i FT(and)f FP(y)378 4117 y FT(existen)m(tially)19
b(quan)m(tify)h(all)f(the)i(form)m(ulae)f(in)f(their)g(con)m(text.)39
b(The)20 b(di\013eren)m(t)g(kind)f(of)h(justi\014cations)378
4230 y(whic)m(h)29 b(can)i(b)s(e)e(used)h(in)f(deriving)f(results)i
(are)g(discussed)f(in)g(section)h(4.2.4.)378 4470 y FQ(Abbreviations)
378 4642 y FT(Arbitrary)j(terms)h(can)h(b)s(e)e(represen)m(ted)i(b)m(y)
f(an)g(abbreviation)f(whic)m(h)g(can)i(b)s(e)f(declared)g(lo)s(cally)-8
b(.)378 4755 y(F)g(or)31 b(example,)f(the)h(abbreviation)e(declaration)
473 4943 y FM(define)46 b(y_def:)h("y)g(=)g(\(x)g(*)h(2)f(+)h(1\)";)378
5130 y FT(in)m(tro)s(duces)35 b(the)h(v)-5 b(ariable)34
b FP(y)39 b FT(as)d(an)g(abbreviation)e(for)i Fv(x)44
b Fw(*)f Ft(2)g Fw(+)g Ft(1)p FT(.)57 b(It)36 b(also)g(in)m(tro)s
(duces)e(the)i(as-)378 5243 y(sumption)g Fv(y)47 b Fw(=)c
Fv(x)h Fw(*)f Ft(2)g Fw(+)g Ft(1)38 b FT(lab)s(elled)e(with)h
Fw(y_def)f FT(so)i(that)h(it)f(can)g(b)s(e)g(used)g(to)h(substitute)e
(the)378 5356 y(abbreviating)c(v)-5 b(ariable)34 b(with)f(the)i(term)f
(it)g(represen)m(ts.)54 b(An)34 b(abbreviating)f(v)-5
b(ariable)34 b(implicitly)378 5469 y(binds)e(all)h(its)h(free)g(o)s
(ccurrences)g(in)f(the)i(form)m(ulae)e(in)g(its)h(con)m(text.)54
b(The)34 b(role)g(of)g(abbreviations)378 5582 y(is)28
b(to)i(reduce)g(the)f(size)g(of)h(sen)m(tences,)h(whic)m(h)d(results)g
(in)g(b)s(etter)i(readabilit)m(y)d(of)j(SPL)e(scripts)g(and)378
5695 y(also)i(in)f(faster)i(pro)s(of-c)m(hec)m(king.)p
eop
%%Page: 60 70
60 69 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(60)378
396 y FQ(Declaring)36 b(Simpli\014ers)378 568 y FT(Simpli\014ers)k(are)
45 b(pro)s(of)f(pro)s(cedures)f(whic)m(h)h(mo)s(dify)f(sen)m(tences,)49
b(usually)42 b(in)m(to)j(an)f(equiv)-5 b(alen)m(t)378
681 y(simpler)18 b(form)i(\(hence)h(the)g(term)f(simpli\014ers\).)34
b(Simpli\014ers)16 b(are)21 b(denoted)g(in)e(SPL)h(b)m(y)g(an)g(iden)m
(ti\014er.)378 794 y(F)-8 b(or)38 b(example,)i(the)e(iden)m(ti\014er)e
Fw(lambda)f FT(denotes)j(a)g(pro)s(of)f(pro)s(cedure)g(whic)m(h)f
(normalises)g(terms)378 907 y(in)c(the)i(lam)m(b)s(da)f(calculus)f(in)m
(to)h FP(\014)5 b(\021)s FT(-long)34 b(normal)f(form.)50
b(The)33 b(lab)s(els)f(of)i(facts)g(whic)m(h)e(consist)i(of)378
1020 y(equalities)29 b(denote)i(a)g(simpli\014er)c(whic)m(h)i(uses)i
(the)f(fact)i(as)f(a)g(rewriting)d(rule.)40 b(The)30
b(user)g(can)h(also)378 1133 y(implemen)m(t)k(simpli\014ers)d(as)37
b(HOL)f(pro)s(of)g(pro)s(cedures)f(during)f(the)i(mec)m(hanisation)g
(of)h(a)g(theory)378 1246 y(and)30 b(asso)s(ciate)h(SPL)f(iden)m
(ti\014ers)e(with)h(them.)519 1358 y(Simpli\014ers)23
b(can)k(b)s(e)f(declared)h(so)g(that)g(sen)m(tences)i(are)e
(automatically)g(simpli\014ed)c(when)j(they)378 1471
y(are)d(sp)s(eci\014ed.)36 b(F)-8 b(or)24 b(example,)g(the)f
(conclusion)e(and)h(premises)f(of)i(a)g(straigh)m(tforw)m(ard)f
(justi\014cation)378 1584 y(are)36 b(simpli\014ed)c(according)k(to)g
(the)g(declared)f(simpli\014ers)d(during)i(pro)s(of)h(searc)m(h.)57
b(The)35 b(declared)378 1697 y(simpli\014ers)24 b(are)k(applied)e(one)i
(b)m(y)g(one)g(\(no)g(particular)f(order)g(should)f(b)s(e)h(assumed\))h
(un)m(til)e(none)i(is)378 1810 y(applicable.)36 b(A)23
b(term)g(rewriting)d(system)j(can)g(therefore)g(b)s(e)f(used)g(to)i
(simplify)19 b(terms)k(b)m(y)f(declaring)378 1923 y(the)31
b(equalities)e(represen)m(ting)g(the)i(rewrite)e(rules)g(of)i(the)f
(system)h(as)f(simpli\014ers.)519 2036 y(A)44 b(n)m(um)m(b)s(er)e(of)h
(mathematical)h(theories)f(are)h FI(c)-5 b(anonisable)p
FT(,)49 b(that)44 b(is,)i(their)c(terms)i(can)g(b)s(e)378
2149 y(uniquely)24 b(represen)m(ted)j(b)m(y)g(a)g(canonical,)h(or)f
(normal)f(form.)39 b(Theories)26 b(whose)h(terms)g(can)g(b)s(e)f(nor-)
378 2262 y(malised)g(e\013ectiv)m(ely)j(ha)m(v)m(e)g(a)f(decidable)f(w)
m(ord)g(problem)g(since)g(t)m(w)m(o)i(terms)f(are)g(equal)g(if)e(and)i
(only)378 2375 y(if)h(their)f(resp)s(ectiv)m(e)i(normal)e(forms)h(are)h
(syn)m(tactically)g(iden)m(tical.)39 b(The)29 b(main)g(role)g(of)h
(simpli\014ers)378 2488 y(is)40 b(to)i(allo)m(w)e(the)h(user)g(to)g
(implemen)m(t)f(theory-sp)s(eci\014c)g(normalisers)f(so)i(that)h(the)f
(equalit)m(y)g(of)378 2600 y(terms)30 b(do)s(es)g(not)h(ha)m(v)m(e)h
(to)f(b)s(e)e(pro)m(v)m(ed)i(explicitly)-8 b(.)519 2713
y(The)40 b(disco)m(v)m(ery)g(of)h(normal)e(forms)h(is)f(a)i(v)m(ery)f
(imp)s(ortan)m(t)g(task)g(in)f(mathematics)i(and)f(the)378
2826 y(mathematical)27 b(literature)e(often)i(includes)c(metho)s(ds)j
(of)g(transforming)f(terms)h(in)m(to)h(their)e(normal)378
2939 y(form.)41 b(The)30 b(implemen)m(tation)g(of)g(normalisers)f(is)h
(actually)g(a)h(formal)f(w)m(a)m(y)h(of)g(represen)m(ting)f(suc)m(h)378
3052 y(metho)s(ds.)69 b(W)-8 b(e)41 b(therefore)f(argue)h(that)f(the)h
(implemen)m(tation)d(of)i(normalisers)e(is)h(an)h(essen)m(tial)378
3165 y(part)i(of)g(a)g(formal)f(mathematical)i(text.)76
b(The)42 b(use)f(of)i(simpli\014ers)38 b(for)j(the)i(normalisation)d
(of)378 3278 y(terms)26 b(has)g(b)s(een)f(used)h(in)e(our)i(case)h
(study)e(in)g(c)m(hapter)i(9)g(to)f(reduce)g(the)h(length)e(of)h
(formal)g(pro)s(ofs)378 3391 y(considerably)-8 b(.)39
b(W)-8 b(e)31 b(also)e(b)s(eliev)m(e)g(that)h(this)e(has)h(impro)m(v)m
(ed)g(the)h(readabilit)m(y)e(of)h(the)h(pro)s(ofs)f(since)378
3504 y(normalisations)d(are)i(often)g(considered)e(to)i(b)s(e)f
(trivial)f(in)g(informal)g(pro)s(ofs)h(once)h(they)g(ha)m(v)m(e)h(b)s
(een)378 3617 y(disco)m(v)m(ered)i(and)f(do)s(cumen)m(ted.)41
b(This)29 b(underlines)e(our)j(argumen)m(t)h(that)h(the)e(implemen)m
(tation)g(of)378 3730 y(normalisers,)24 b(and)g(pro)s(of)g(pro)s
(cedures)g(in)f(general,)j(should)d(b)s(e)h(considered)g(as)h(an)f(imp)
s(ortan)m(t)g(part)378 3842 y(of)30 b(the)h(mec)m(hanisation)f(of)h
(mathematics.)378 4081 y FQ(Declaring)36 b(T)-9 b(rivial)35
b(F)-9 b(acts)378 4253 y FT(F)h(acts)32 b(whic)m(h)d(are)i(considered)e
(trivial)f(can)j(b)s(e)e(stored)i(in)e(a)h(kno)m(wledge)h(database)g
(whic)m(h)e(can)i(b)s(e)378 4366 y(used)26 b(b)m(y)i(SPL)e(pro)s(of)h
(pro)s(cedures)f(during)f(pro)s(of-c)m(hec)m(king.)40
b(The)27 b(database)h(organises)f(facts)h(in)m(to)378
4479 y(categories,)38 b(and)c(the)h(SPL)e(language)j(includes)c(the)j
(kno)m(wledge)f(declaration)h(construct)g(of)g(the)378
4592 y(form)473 4772 y FM(consider)46 b FI(Cate)-5 b(gory)57
b(F)-7 b(act)1488 4786 y FL(1)1527 4772 y FM(,)48 b FI(F)-7
b(act)1802 4786 y FL(2)1842 4772 y FM(,)47 b FP(:)15
b(:)g(:)49 b FM(;)378 4953 y FT(to)c(store)h(the)e(facts)i
FI(F)-7 b(act)1320 4967 y FL(1)1360 4953 y FP(;)15 b
FI(F)-7 b(act)1580 4967 y FL(2)1619 4953 y FP(;)15 b(:)g(:)g(:)61
b FT(in)44 b(the)h(category)h FI(Cate)-5 b(gory)10 b
FT(.)83 b(These)45 b(facts)g(can)g(then)378 5066 y(b)s(e)32
b(used)g(automatically)h(b)m(y)g(the)g(pro)s(of)f(pro)s(cedures)g(whic)
m(h)g(are)h(able)g(to)g(query)g(the)g(kno)m(wledge)378
5179 y(database.)39 b(The)24 b(use)f(of)h(the)g(kno)m(wledge)f
(database)i(is)e(describ)s(ed)e(in)i(more)h(detail)f(in)f(section)i
(4.4.1.)378 5421 y FG(4.2.4)112 b(Pro)s(ofs)38 b(and)g
(Justi\014cations)378 5592 y FT(The)20 b(statemen)m(ts)i(of)f(theorems)
g(and)f(pro)s(of)g(step)h(results)e(are)i(follo)m(w)m(ed)f(b)m(y)h
(their)e(justi\014cation.)36 b(The)378 5705 y(length)21
b(and)h(complexit)m(y)f(of)i(justi\014cations)d(ranges)i(from)g(one)g
(line)e(in)h(the)h(case)h(of)f(straigh)m(tforw)m(ard)p
eop
%%Page: 61 71
61 70 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(61)378
396 y(justi\014cations,)30 b(to)j(sev)m(eral)e(p)s(ossible)e(nested)j
(argumen)m(ts.)44 b(W)-8 b(e)33 b(refer)e(to)h(the)g(statemen)m(t)h
(whic)m(h)d(a)378 509 y(particular)f(justi\014cation)g(is)g(deriving)f
(as)j(the)f(conclusion)f(of)i(the)f(justi\014cation.)378
749 y FQ(Straigh)m(tforw)m(ard)k(Justi\014cations)378
921 y FT(Straigh)m(tforw)m(ard)g(justi\014cations)g(are)h(the)g
(simplest)f(kind)f(of)i(justi\014cations)e(and)i(consist)g(of)g(the)378
1034 y Fw(by)f FT(construct,)j(an)e(optional)g FI(pr)-5
b(over)46 b FT(name,)37 b(and)d(the)h(argumen)m(ts)h(of)f(the)h(pro)m
(v)m(er.)55 b(A)35 b(pro)m(v)m(er)h(is)378 1147 y(a)f(\(HOL\))g
(decision)e(pro)s(cedure)h(whic)m(h)f(deriv)m(es)h(the)h(conclusion)e
(of)i(the)g(justi\014cation)e(from)i(the)378 1260 y(giv)m(en)f(argumen)
m(ts.)52 b(F)-8 b(or)34 b(example,)h(a)g(decision)d(pro)s(cedure)h(for)
h(prop)s(osition)d(logic)j(can)g(b)s(e)g(used)378 1373
y(to)41 b(justify)e(the)i(conclusion)e(\()p FP(A)k FN(\))f
FP(B)5 b FT(\))41 b(from)f(the)h(argumen)m(ts)g FP(A)h
FN(\))g FT(\()p FP(C)34 b FN(_)27 b FP(B)5 b FT(\))40
b(and)g FP(C)49 b FN(\))42 b FP(B)5 b FT(.)378 1486 y(If)39
b(no)g(pro)m(v)m(er)h(name)f(is)g(giv)m(en,)i(a)f(default)f(one)g(is)g
(assumed.)67 b(In)38 b(the)i(examples)f(giv)m(en)g(in)f(this)378
1599 y(c)m(hapter,)30 b(the)g(default)e(pro)m(v)m(er)i(is)e(assumed)h
(to)h(b)s(e)e(a)i(tableau-based)f(pro)m(v)m(er)h(for)f(\014rst-order)f
(logic)378 1711 y(with)43 b(equalit)m(y)-8 b(.)81 b(The)43
b(calculus)g(this)f(pro)m(v)m(er)j(implemen)m(ts)d(is)h(complete)h(for)
g(\014rst-order)f(logic)378 1824 y(with)35 b(equalit)m(y)-8
b(.)59 b(Ho)m(w)m(ev)m(er,)40 b(b)s(ecause)c(of)h(the)f(simplicit)m(y)e
(of)i(the)h(justi\014cations)e(of)h(SPL)g(scripts,)378
1937 y(v)m(ery)e(restrictiv)m(e)f(resource)h(b)s(ounds)e(are)i(used)e
(during)g(the)h(pro)s(of)g(searc)m(h)i(pro)s(cess)e(so)h(that)g(only)
378 2050 y(a)k(small)f(\014nite)g(searc)m(h)i(space)f(is)f(considered.)
63 b(The)37 b(iden)m(ti\014er)g(of)h(this)f(pro)m(v)m(er)h(is)f
Fw(fol)o FT(,)j(and)d(its)378 2163 y(implemen)m(tation)e(as)h(a)h(HOL)f
(pro)s(of)f(pro)s(cedure)g(is)h(describ)s(ed)e(in)h(the)h(next)h(c)m
(hapter.)58 b(The)36 b Fw(fol)378 2276 y FT(pro)m(v)m(er)c(tak)m(es)h
(a)f(p)s(ossibly)d(empt)m(y)j(list)f(of)h(sen)m(tences)h(as)f(an)f
(argumen)m(t.)45 b(A)32 b(n)m(um)m(b)s(er)f(of)h(\015ags)g(can)378
2389 y(also)d(b)s(e)g(sp)s(eci\014ed)e(b)s(efore)i(or)g(after)h(the)f
(pro)m(v)m(er)h(name.)41 b(F)-8 b(or)30 b(example,)f(the)g(follo)m
(wing)f(statemen)m(t)378 2502 y(uses)g(the)g(\015ag)g
Fw(pure)f FT(whic)m(h)f(instructs)h(the)h(\014rst-order)g(pro)m(v)m(er)
g(not)g(to)h(giv)m(e)g(sp)s(ecial)d(treatmen)m(t)k(to)378
2615 y(equalities.)473 2802 y FM(")p FN(8)16 b FM(x)47
b(y.)g(\(x)g(=)h(y\))f FN(_)g(:)p FM(\(x)g(=)h(y\)")e(by)i(pure)e(fol;)
378 2990 y FT(A)26 b(list)f(of)h(simpli\014ers)d(can)j(b)s(e)g(sp)s
(eci\014ed)e(b)s(efore)i(the)g Fw(by)g FT(tok)m(en)h(as)f(illustrated)e
(in)h(the)h(justi\014cation)378 3103 y(of)k(the)h(last)f(theorem)h(in)e
(\014gure)h(5.)519 3216 y(The)41 b(default)f(pro)m(v)m(er)i(used)e(in)g
(the)h(case)h(study)f(in)e(c)m(hapter)j(9)g(tak)m(es)g(an)f(expression)
f(con-)378 3329 y(structed)j(b)m(y)f(a)h(n)m(um)m(b)s(er)f(of)g(sen)m
(tences)i(and)e(the)h(op)s(erators)g Fw(on)o FT(,)j Fw(then)41
b FT(and)h Fw(and)o FT(,)k(in)41 b(order)h(to)378 3442
y(increase)28 b(the)g(readabilit)m(y)f(of)h(the)g(scripts)f(and)h(for)f
(pro)s(of-c)m(hec)m(king)i(e\016ciency)-8 b(.)40 b(Suc)m(h)28
b(structured)378 3555 y(justi\014cations)h(are)i(in)m(tro)s(duced)d(in)
h(c)m(hapter)i(6.)378 3795 y FQ(Pro)s(of)36 b(Justi\014cations)378
3966 y FT(The)d(pro)s(ofs)f(of)i(theorems)f(usually)e(consist)i(of)g
(sev)m(eral)h(argumen)m(ts)g(rather)f(than)g(a)g(straigh)m(tfor-)378
4079 y(w)m(ard)28 b(justi\014cation.)39 b(Suc)m(h)28
b(argumen)m(ts)h(are)g(giv)m(en)g(in)e(a)i(pro)s(of)f(justi\014cation)f
(whic)m(h)h(consists)g(of)g(a)378 4192 y(sequence)j(of)f(reasoning)g
(items)g(enclosed)g(b)s(et)m(w)m(een)h(a)g Fw(proof)d
FT(and)i(a)h Fw(qed)e FT(or)h Fw(end)o FT(.)519 4305
y(A)39 b(pro)s(of)f(justi\014cation)f(creates)j(a)f(new)f(con)m(text)j
(in)c(the)i(SPL)f(en)m(vironmen)m(t)g(in)f(whic)m(h)h(the)378
4418 y(necessarily)i(pro)s(of)h(results)f(are)i(deriv)m(ed.)73
b(A)42 b(n)m(um)m(b)s(er)e(of)i(results)e(can)i(b)s(e)f(declared)g(as)g
(b)s(eing)378 4531 y(relev)-5 b(an)m(t)29 b(for)g(the)g
(justi\014cation)e(of)i(the)g(pro)m(v)m(er)h(using)d(the)i
Fw(case)e FT(directiv)m(e,)i(as)g(illustrated)e(b)m(y)i(the)378
4644 y(example)h(in)f(\014gure)h(6.)519 4757 y(The)j(conjunction)g(of)g
(the)h(relev)-5 b(an)m(t)34 b(results)e(is)h(expanded)g(according)g(to)
i(the)e(v)-5 b(ariables)32 b(and)378 4870 y(the)d(assumptions)f(in)m
(tro)s(duced)f(in)h(the)h(pro)s(of.)40 b(If)28 b(no)h(results)f(are)h
(sp)s(eci\014ed)f(as)h(relev)-5 b(an)m(t,)30 b(the)f(last)378
4983 y(result)35 b(deriv)m(ed)g(in)g(the)h(pro)s(of)g(is)f(instead)g
(expanded)h(and)f(used)g(for)h(justifying)e(the)i(conclusion)378
5095 y(of)43 b(the)g(pro)s(of.)76 b(The)43 b(expanded)e(result)h(\(or)h
(conjunction)f(of)g(the)h(relev)-5 b(an)m(t)43 b(results\))f(is)g
(called)378 5208 y(the)36 b(justifying)d(fact,)38 b(and)d(the)g(aim)g
(of)h(a)g(pro)s(of)f(justi\014cation)f(is)g(to)j(construct)e(an)h
(appropriate)378 5321 y(justifying)31 b(fact.)49 b(An)32
b(optional)h(straigh)m(tforw)m(ard)f(justi\014cation)g(can)h(b)s(e)f
(sp)s(eci\014ed)g(after)h(the)g Fw(qed)378 5434 y FT(statemen)m(t)38
b(in)e(order)g(to)h(b)s(e)g(used)e(with)h(the)h(justifying)d(fact)k(to)
f(deriv)m(e)f(the)h(pro)s(of)f(conclusion.)378 5547 y(Suc)m(h)29
b(a)g(straigh)m(tforw)m(ard)h(justi\014cation)e(can)h(also)h(b)s(e)e
(sp)s(eci\014ed)g(at)i(the)g(start)g(of)f(the)h(pro)s(of)e(using)378
5660 y(the)j Fw(proceed)c FT(construct,)k(as)g(sho)m(wn)e(b)s(elo)m(w.)
p eop
%%Page: 62 72
62 71 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(62)p
378 988 3453 4 v 376 5031 4 4043 v 515 1124 a Fw(theorem)41
b(Rel_equiv:)e("Equivalence)g(Rel")515 1223 y(proof)602
1422 y(case)j("Reflexive)d(Rel")689 1522 y(proof)776
1622 y(let)k("x:'a";)907 1702 y(.)907 1735 y(.)907 1768
y(.)776 1868 y("Rel)f(x)i(x")e(by)h Fv(:)14 b(:)g(:)44
b Fw(;)776 1967 y(simplify)d(with)h(Reflexive;)689 2067
y(end;)602 2266 y(case)g("Symmetric)d(Rel")689 2366 y(proof)776
2465 y(given)j("x:'a")f(and)h("y:'a")g(such)f(that)864
2565 y(xRy:)h("Rel)f(x)j(y";)907 2652 y(.)907 2685 y(.)907
2718 y(.)776 2818 y("Rel)e(y)i(x")e(by)h Fv(:)14 b(:)g(:)44
b Fw(;)776 2918 y(simplify)d(with)h(Symmetric;)689 3017
y(end;)602 3217 y(case)g("Transitive)d(Rel")689 3316
y(proof)776 3416 y(given)j("x:'a",)f("y:'a")g(and)h("z:'a")f(such)h
(that)864 3515 y(xRy:)g("Rel)f(x)j(y")e(and)864 3615
y(yRz:)g("Rel)f(y)j(z";)907 3702 y(.)907 3735 y(.)907
3768 y(.)776 3868 y("Rel)e(x)i(z")e(by)h Fv(:)14 b(:)g(:)44
b Fw(;)776 3968 y(simplify)d(with)h(Transitive;)689 4067
y(end;)602 4267 y(simplify)e(with)i(Equivalence;)515
4466 y(qed;)833 4861 y FT(Figure)30 b(6:)41 b(Declaring)30
b(Relev)-5 b(an)m(t)31 b(Pro)s(of)f(Step)g(Results)f(in)g(SPL)h(Pro)s
(ofs.)p 3829 5031 V 378 5034 3453 4 v eop
%%Page: 63 73
63 72 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(63)473
396 y FM(theorem)46 b(")p FN(8)15 b FM(n.)48 b(n)f FN(\024)g
FM(Factorial)f(n")473 509 y(proof)569 622 y(proceed)g(by)h(induction)e
(on)i("n";)569 848 y(case)f(base:)h("0)g FN(\024)g FM(Factorial)f(0")
664 961 y(proof)760 1051 y(.)760 1084 y(.)760 1117 y(.)664
1230 y(end;)569 1456 y(case)g(ind:)h("\(n)g FN(\024)g
FM(Factorial\))e FN(\))j FM(\(SUC)f(n)g FN(\024)h FM(Factorial)d(\(SUC)
i(n\)\)")664 1569 y(proof)760 1658 y(.)760 1692 y(.)760
1725 y(.)664 1838 y(end;)473 2064 y(qed;)378 2251 y FT(where)31
b Fw(induction)d FT(is)j(assumed)f(to)j(b)s(e)e(the)h(iden)m(ti\014er)d
(of)j(a)g(pro)m(v)m(er)g(whic)m(h)f(uses)g(the)g(principle)e(of)378
2364 y(mathematical)e(induction)d(on)j(the)g(conjunction)f(of)h(the)g
(base)g(case)g(and)g(the)g(induction)d(step)j(case)378
2477 y(to)k(justify)e(its)h(conclusion.)519 2590 y(If)36
b(no)h(straigh)m(tforw)m(ard)f(justi\014cation)g(is)f(sp)s(eci\014ed,)i
(a)g(default)f(pro)m(v)m(er)h(\()p Fw(fol)f FT(in)f(the)i(case)h(of)378
2703 y(the)31 b(examples)e(giv)m(en)i(in)e(this)g(c)m(hapter\))j(is)d
(used.)519 2816 y(The)g(ab)s(o)m(v)m(e)h(treatmen)m(t)h(of)e(pro)s(of)f
(justi\014cations)g(is)g(di\013eren)m(t)h(from)f(that)i(used)f(b)m(y)g
(other)g(sys-)378 2929 y(tems)i(whic)m(h)e(include)f(the)j(Mizar)g(mo)s
(de)f(in)f(HOL)h(of)h(Harrison)e(\(1996b\).)43 b(In)30
b(Harrison's)g(system)378 3042 y(reasoning)e(items)g(are)g(used)g(in)f
(a)h(pro)s(of)g(to)h(break)f(do)m(wn)g(the)h(conclusion)d(whic)m(h)h
(can)i(b)s(e)f(referred)378 3154 y(to)f(b)m(y)f(a)g Fw(thesis)d
FT(directiv)m(e.)39 b(F)-8 b(or)27 b(example,)g(the)f(in)m(tro)s
(duction)e(of)i(an)g(assumption)f(within)e(a)k(pro)s(of)378
3267 y(corresp)s(onds)32 b(to)j(the)e(application)f(of)i(the)g(HOL)f
(tactic)i Fw(DISCH_TAC)30 b FT(whic)m(h)i(simpli\014es)e(a)k(conclu-)
378 3380 y(sion)i(\(thesis\))g(of)h(the)g(form)f Fv(A)44
b Fu(\))f Fv(C)g FT(in)m(to)37 b Fv(C)42 b FT(and)36
b(includes)f(the)h(assumption)g Fv(A)p FT(.)59 b(As)37
b(a)g(result,)378 3493 y(the)29 b(structure)f(of)h(the)g(pro)s(ofs)f
(in)g(this)f(system)i(are)g(v)m(ery)h(m)m(uc)m(h)e(based)h(on)g(the)g
(structure)f(of)h(their)378 3606 y(conclusions.)40 b(The)30
b(structural)f(dep)s(endency)g(of)i(a)g(pro)s(of)f(on)g(its)g
(conclusion)f(is)h(also)g(observ)m(ed)h(in)378 3719 y(Mizar)37
b(pro)s(ofs.)61 b(On)37 b(the)g(other)h(hand,)g(SPL)f(pro)s(ofs)f
(construct)i(a)g(justifying)d(fact)j(irresp)s(ectiv)m(e)378
3832 y(of)g(the)g(structure)g(of)g(their)f(conclusion.)62
b(The)37 b(deriv)-5 b(ation)37 b(of)h(the)g(conclusion)e(from)i(the)g
(justi-)378 3945 y(fying)32 b(fact)j(is)d(then)h(done)h(automatically)
-8 b(,)34 b(or)g(as)f(instructed)g(b)m(y)g(the)g(optional)g(straigh)m
(tforw)m(ard)378 4058 y(justi\014cation.)64 b(This)36
b(particular)h(approac)m(h)i(o\013ers)f(greater)i(\015exibilit)m(y)c
(in)h(the)h(w)m(a)m(y)i(pro)s(ofs)d(are)378 4171 y(implemen)m(ted.)50
b(F)-8 b(or)35 b(instance,)f(the)g(user)g(can)g(form)m(ulate)g(a)g
(theorem)g(in)f(a)h(statemen)m(t)i(whic)m(h)d(is)378
4284 y(adequate)26 b(for)e(its)g(later)g(use,)i(and)e(pro)s(ceed)g(to)i
(pro)m(v)m(e)f(an)g(equiv)-5 b(alen)m(t)24 b(statemen)m(t)i(whose)e
(structure)378 4397 y(ma)m(y)32 b(mak)m(e)g(it)f(easier)h(to)g(pro)m(v)
m(e.)44 b(T)-8 b(o)32 b(illustrate)e(this,)h(v)-5 b(an)31
b(Gasteren)h(\(1990\))i(giv)m(es)e(the)f(example)378
4509 y(that)i(results)f(stating)h(the)g(symmetry)f(of)h(some)g
(relation)f FN(\030)h FT(are)g(more)g(useable)f(if)g(they)h(are)g(for-)
378 4622 y(m)m(ulated)25 b(b)m(y)g(an)g(equalit)m(y)g
FP(x)g FN(\030)g FP(y)j FT(=)d FP(y)j FN(\030)d FP(x)p
FT(,)h(although)f(it)f(ma)m(y)i(b)s(e)f(easier)g(to)h(pro)m(v)m(e)g
(the)f(statemen)m(t)378 4735 y FP(x)31 b FN(\030)g FP(y)j
FN(\))d FP(y)j FN(\030)d FP(x)p FT(;)k(an)f(equalit)m(y)g(is)f(used)g
(in)f(the)i(de\014nition)e(of)i(symmetry)f(in)g(page)i(56,)h(but)d(the)
378 4848 y(justifying)28 b(statemen)m(t)k(of)f(the)g(relev)-5
b(an)m(t)31 b(subpro)s(of)d(in)h(\014gure)h(6)h(is)f(an)g(implication)e
(\()p Fw("Rel)42 b(x)h(y")30 b FT(is)378 4961 y(assumed)g(and)f
Fw("Rel)42 b(y)h(x")30 b FT(is)f(deriv)m(ed\).)519 5074
y(W)-8 b(e)28 b(b)s(eliev)m(e)d(that)i(this)f(approac)m(h)g(is)g(more)g
(true)g(to)i(the)e(declarativ)m(e)h(st)m(yle)g(of)f(reasoning)g(than)
378 5187 y(one)40 b(in)f(whic)m(h)g(the)h(structure)f(of)h(pro)s(ofs)f
(is)g(greatly)i(in\015uenced)d(b)m(y)h(their)g(conclusion.)68
b(With)378 5300 y(hindsigh)m(t,)35 b(ho)m(w)m(ev)m(er,)k(most)e(pro)s
(ofs)e(in)g(the)h(case)h(study)e(illustrated)f(in)h(c)m(hapter)i(9)f
(pro)s(ceed)g(b)m(y)378 5413 y(generalising)23 b(on)h(the)h(univ)m
(ersal)d(v)-5 b(ariables)23 b(of)i(the)g(conclusion,)f(and)g(in)m(tro)s
(ducing)e(its)i(an)m(teceden)m(ts)378 5526 y(as)35 b(assumptions)f
(\(though)h(not)h(necessarily)e(in)g(the)h(same)h(order)f(as)g(they)h
(are)f(sp)s(eci\014ed)f(in)g(the)378 5639 y(conclusion\).)52
b(As)35 b(a)g(result,)g(the)f(pro)m(v)m(ers)h(whic)m(h)f(automate)i
(the)f(deriv)-5 b(ation)33 b(of)i(the)f(conclusion)p
eop
%%Page: 64 74
64 73 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(64)378
396 y(from)20 b(a)g(justifying)e(statemen)m(t)23 b(ma)m(y)e(assume)f
(that)g(these)h(probably)e(ha)m(v)m(e)i(a)g(v)m(ery)g(similar)c
(structure)378 509 y(in)29 b(order)h(to)h(increase)f(the)h(pro)s(of-c)m
(hec)m(king)g(e\016ciency)-8 b(.)378 746 y FQ(Iterativ)m(e)34
b(Equalities)378 918 y FT(Similarly)27 b(to)k(Mizar,)f(results)f(can)i
(b)s(e)f(justi\014ed)e(b)m(y)j(iterativ)m(e)f(equalities)f(suc)m(h)h
(as:)473 1090 y FM(abc:)47 b("a)g(+)h(\(b)f(+)g(c\))95
b(=)g(a)48 b(+)f(\(c)g(+)h(b\)")f(by)g(commutativity)1237
1203 y(.")g(=)95 b(\(a)48 b(+)f(c\))g(+)h(b")f(by)g(associativity)1237
1316 y(.")g(=)95 b(\(c)48 b(+)f(a\))g(+)h(b")f(by)g(commutativity;)378
1487 y FT(This)d(justi\014cation)h(deriv)m(es)g(the)h(result)f
Fw("a)e(+)g(\(b)g(+)g(c\))g(=)g(\(c)g(+)g(a\))g(+)g(b")i
FT(lab)s(elled)e(with)i Fw(abc)o FT(.)378 1600 y(The)35
b(structure)h(of)g(suc)m(h)f(calculational)g(justi\014cations)f
(greatly)j(impro)m(v)m(es)e(the)h(readabilit)m(y)e(and)378
1713 y(writabilit)m(y)28 b(of)i(pro)s(of)g(scripts.)39
b(In)30 b(SPL,)g(one)g(can)h(also)f(lab)s(el)f(the)i(individual)25
b(lines,)k(as)i(in)473 1885 y FM(abc:)47 b("a)g(+)h(\(b)f(+)g(c\))95
b(=)g(a)48 b(+)f(\(c)g(+)h(b\)")f(\(1\))g(by)g(commutativity)1237
1997 y(.")g(=)95 b(\(a)48 b(+)f(c\))g(+)h(b")f(\(2\))g(by)g
(associativity)1237 2110 y(.")g(=)95 b(\(c)48 b(+)f(a\))g(+)h(b")238
b(by)47 b(commutativity;)378 2282 y FT(suc)m(h)40 b(that)h(fragmen)m
(ts)g(of)g(the)f(ab)s(o)m(v)m(e)i(sequence)f(can)f(also)h(b)s(e)f
(referred)f(to)i(later.)71 b(Giv)m(en)41 b(t)m(w)m(o)378
2395 y(lines)d(lab)s(elled)f(with)h Fw(l)1200 2407 y
Fs(1)1276 2395 y FT(and)h Fw(l)1506 2407 y Fs(2)1543
2395 y FT(,)i(one)f(can)g(use)f(the)h(lab)s(el)d Fw(abc)p
Fu(f)p Fw(l)2739 2407 y Fs(1)2775 2395 y Fw(-l)2863 2407
y Fs(2)2899 2395 y Fu(g)i FT(to)h(refer)f(to)h(the)g(result)378
2508 y Fw(")p Fj(R)486 2520 y Fs(1)566 2508 y Fw(=)j
Fj(R)717 2520 y Fs(2)754 2508 y Fw(")28 b FT(where)g
Fj(R)1151 2520 y Fq(i)1206 2508 y FT(refers)g(to)h(the)f(term)g(on)g
(the)g(righ)m(t)g(hand)f(side)g(of)h(the)h(equalit)m(y)e(in)g(the)i
(line)378 2621 y(with)24 b(lab)s(el)g Fw(l)839 2633 y
Fq(i)866 2621 y FT(.)39 b(Similarly)-8 b(,)23 b(the)i(lab)s(el)f
Fw(abc)p Fu(f)p Fw(-l)1958 2633 y Fq(i)1983 2621 y Fu(g)h
FT(refers)g(the)g(result)g Fw(")p Fj(L)43 b Fw(=)g Fj(R)2981
2633 y Fs(2)3018 2621 y Fw(")25 b FT(and)f Fw(abc)p Fu(f)p
Fw(l)3476 2633 y Fq(i)3502 2621 y Fw(-)p Fu(g)g FT(refers)378
2734 y(to)j Fw(")p Fj(R)593 2746 y Fq(i)663 2734 y Fw(=)43
b Fj(R)s Fw(")25 b FT(where)h Fj(L)g FT(is)e(the)i(left)g(hand)f(side)g
(term)g(of)h(\014rst)f(line,)h(and)f Fj(R)k FT(is)24
b(the)i(one)h(on)e(the)h(righ)m(t)378 2847 y(hand)36
b(side)g(in)g(the)i(last)f(line.)59 b(In)37 b(our)f(example,)j(the)f
(follo)m(wing)d(lab)s(elled)g(results)h(are)i(deriv)m(ed:)428
3064 y Fw(abc)p Fu(f)p Fw(-1)p Fu(g)p Fw(:)83 b("a)43
b(+)g(\(b)g(+)g(c\))g(=)g(a)g(+)h(\(c)e(+)i(b\)")98 b(abc)p
Fu(f)p Fw(-2)p Fu(g)p Fw(:)84 b("a)42 b(+)i(\(b)e(+)i(c\))e(=)h(\(a)g
(+)g(c\))g(+)g(b")428 3177 y(abc)p Fu(f)p Fw(1-2)p Fu(g)p
Fw(:)35 b("a)42 b(+)i(\(c)e(+)h(b\))g(=)g(\(a)g(+)g(c\))g(+)g(b")104
b(abc)p Fu(f)p Fw(1-)p Fu(g)p Fw(:)84 b("a)42 b(+)i(\(c)e(+)i(b\))e(=)h
(\(c)g(+)g(a\))g(+)g(b")428 3290 y(abc)p Fu(f)p Fw(2-)p
Fu(g)p Fw(:)83 b("\(a)43 b(+)g(c\))g(+)g(b)g(=)g(\(c)g(+)g(a\))g(+)g
(b")268 b(abc:)86 b("a)43 b(+)g(\(b)g(+)g(c\))g(=)g(\(c)f(+)i(a\))e(+)i
(b")519 3508 y FT(The)33 b(syn)m(tax)i(for)e(iterativ)m(e)h(equalities)
f(can)h(b)s(e)f(extended)g(to)i(consider)e(other)g(transitiv)m(e)h(re-)
378 3621 y(lations)43 b(apart)i(from)e(equalit)m(y)-8
b(,)48 b(and)c(the)g(SPL)f(kno)m(wledge)h(database)h(can)g(b)s(e)e
(used)g(to)i(store)378 3734 y(the)39 b(required)f(transitivit)m(y)g
(results)g(required)f(b)m(y)i(the)h(pro)s(of)e(c)m(hec)m(k)m(er.)70
b(This)37 b(feature)j(w)m(as)g(not)378 3847 y(implemen)m(ted)f(in)h
(the)h(pro)s(of)f(c)m(hec)m(k)m(er)j(in)d(HOL)g(since)h(its)f(use)g(w)m
(as)i(not)f(required)e(during)g(the)378 3960 y(dev)m(elopmen)m(t)31
b(of)f(the)h(case)g(study)-8 b(.)378 4197 y FQ(Case)34
b(Splitting)378 4369 y FT(A)f(case)i(splitting)c(justi\014cation)h
(corresp)s(onds)g(to)i(the)f(natural)f(deduction)g(rule)g(for)i
(eliminating)378 4482 y(disjunctions,)28 b(and)i(has)g(the)g(follo)m
(wing)f(structure:)473 4653 y FM(")p FP(C)7 b FM(")473
4766 y(consider)46 b(cases)g FT([)i FI(str)-5 b(aightforwar)g(d)62
b(justi\014c)-5 b(ation)56 b(of)67 b FP(A)2639 4780 y
FL(1)2699 4766 y FN(_)20 b(\001)15 b(\001)g(\001)21 b(_)h
FP(A)3057 4780 y FO(n)3151 4766 y FM(;)48 b FT(])569
4992 y FM(suppose)e(")p FP(A)1067 5006 y FL(1)1106 4992
y FM(":)h FI(justi\014c)-5 b(ation)56 b(of)68 b FP(C)569
5174 y FM(.)569 5208 y(.)569 5241 y(.)569 5467 y(suppose)46
b(")p FP(A)1067 5481 y FO(n)1114 5467 y FM(":)h FI(justi\014c)-5
b(ation)56 b(of)67 b FP(C)473 5693 y FM(end)47 b(cases;)p
eop
%%Page: 65 75
65 74 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(65)p
378 416 3453 4 v 376 1814 4 1398 v 515 552 a Fj(Sentenc)l(e)49
b Ft(=)43 b([)h Fw(<)f Fj(Simpli\014ers)51 b Fw(>)43
b Ft(])h Fj(Unsimpli\014e)l(d)p 2191 552 27 4 v 41 w(Sentenc)l(e)515
751 y(Unsimpli\014e)l(d)p 976 751 V 41 w(Sentenc)l(e)49
b Ft(=)689 851 y([)44 b Fw([)f Fj(A)n(bstr)l(actions)50
b Fw(])43 b Ft(])h(\()g Fj(L)l(ab)l(el)p 1767 851 V 41
w(Identi\014er)52 b Fu(j)44 b Fj(F)-6 b(ormula)50 b Ft(\))44
b([)g Fw([)f Fj(Applic)l(ations)52 b Fw(])43 b Ft(])602
951 y Fu(j)h Fj(Comp)l(ound)p 1066 951 V 41 w(Sentenc)l(e)515
1150 y(Comp)l(ound)p 912 1150 V 41 w(Sentenc)l(e)49 b
Ft(=)689 1249 y Fw(\()43 b Fj(Comp)l(ound)p 1173 1249
V 41 w(Sentenc)l(e)49 b Fw(\))602 1349 y Fu(j)44 b Fj(R)n(ule)p
843 1349 V 37 w(Identi\014er)53 b(R)n(ule)p 1428 1349
V 37 w(Par)l(ams)1734 1361 y Fi(R)n(ule)p 1878 1361 V
35 w(Identi\014er)1284 1645 y FT(Figure)30 b(7:)41 b(The)30
b(Syn)m(tax)g(of)h(SPL)e(Sen)m(tences.)p 3829 1814 4
1398 v 378 1818 3453 4 v 378 2175 a FG(4.2.5)112 b(SPL)38
b(Sen)m(tences)378 2346 y FT(SPL)g(sen)m(tences)j(are)e(the)h
(expressions)e(in)f(the)j(syn)m(tax)g(of)f(the)g(language)h(whic)m(h)e
(denote)i(facts.)378 2459 y(In)31 b(their)g(simplest)e(form,)j(sen)m
(tences)h(consist)e(of)h(the)g(lab)s(el)e(denoting)h(some)h(fact)h(in)d
(the)i(curren)m(t)378 2572 y(en)m(vironmen)m(t,)39 b(suc)m(h)e(as)g(a)h
(deriv)m(ed)f(result)f(or)h(an)g(assumption.)60 b(A)38
b(sen)m(tence)g(can)g(also)f(consist)378 2685 y(of)g(a)h(HOL)f(form)m
(ula,)h(in)e(whic)m(h)g(case)j(the)e(form)m(ula)g(is)f(in)m(tro)s
(duced)g(as)h(an)g(assumption)f(\(unless)378 2798 y(it)43
b(already)g(o)s(ccurs)g(as)h(a)g(fact)g(in)f(the)g(en)m(vironmen)m(t\))
h(so)g(that)g(the)f(sen)m(tence)i(can)f(denote)g(it.)378
2911 y(Ho)m(w)m(ev)m(er,)38 b(as)d(sho)m(wn)f(b)m(y)h(their)f(syn)m
(tax)h(giv)m(en)g(in)f(\014gure)g(7,)j(sen)m(tences)f(can)f(b)s(e)f
(constructed)h(b)m(y)378 3024 y(applying)27 b(a)i(n)m(um)m(b)s(er)f(of)
h(inferences)g(whic)m(h)e(include)g(simpli\014cations,)g(abstractions)i
(\(generalisa-)378 3137 y(tions\),)34 b(applications)e(\(sp)s
(ecialisations\))g(and)h(other)h(inference)e(rules)g(implemen)m(ted)g
(during)g(the)378 3249 y(mec)m(hanisation)e(of)g(a)h(theory)-8
b(.)378 3490 y FQ(Simpli\014cations)378 3661 y FT(Simpli\014ers)31
b(can)k(b)s(e)g(applied)e(to)j(individual)30 b(sen)m(tences)37
b(so)e(that)h(the)f(facts)h(they)f(represen)m(t)g(are)378
3774 y(automatically)h(simpli\014ed)c(with)j(the)h(applied)e
(simpli\014ers)e(as)k(w)m(ell)f(as)h(with)f(those)i(declared)e(in)378
3887 y(the)f(en)m(vironmen)m(t.)52 b(Since)33 b(a)i(fact)g(consisting)e
(of)h(an)g(equalit)m(y)g(can)h(b)s(e)e(denoted)i(as)f(a)h(simpli\014er)
378 4000 y(to)c(represen)m(t)f(a)h(rewriting)e(rule,)g(one)i(can)f(use)
g(expressions)f(of)i(the)f(form)664 4188 y FP(:)15 b(:)g(:)49
b FM(<)p FI(R)n(ule)6 b FM(>)p FI(Sentenc)-5 b(e)55 b
FP(:)15 b(:)g(:)378 4375 y FT(to)30 b(use)f(the)g(fact)i
Fj(R)n(ule)j FT(to)c(rewrite)f(the)g(fact)i(denoted)e(b)m(y)g
Fj(Sentenc)l(e)35 b FT(during)27 b(pro)s(of-c)m(hec)m(king.)40
b(The)378 4488 y(use)31 b(of)h(suc)m(h)f(explicit)e(rewrites)i(for)g
(equalit)m(y)g(reasoning)g(can)g(reduce)h(the)f(pro)s(of-c)m(hec)m
(king)h(time,)378 4601 y(although)e(if)f(o)m(v)m(erused)j(it)e(results)
f(in)g(a)i(pro)s(cedural)e(st)m(yle)i(of)f(pro)s(of)g(implemen)m
(tation)f(whic)m(h)h(can)378 4714 y(b)s(e)g(hard)f(to)i(follo)m(w.)378
4954 y FQ(Abstractions)378 5126 y FT(The)e(facts)h(in)m(tro)s(duced)d
(in)h(a)i(con)m(text)h(are)f(sp)s(ecialised)c(according)k(to)g(the)f
(lo)s(cally)f(declared)g(v)-5 b(ari-)378 5239 y(ables)21
b(and)g(assumptions.)37 b(As)22 b(a)g(result)f(unnecessary)g
(inferences)g(suc)m(h)g(as)h(v)-5 b(ariable)21 b(instan)m(tiations)378
5351 y(are)28 b(a)m(v)m(oided)h(during)c(pro)s(of-c)m(hec)m(king.)41
b(Ho)m(w)m(ev)m(er,)31 b(during)25 b(the)k(implemen)m(tation)d(of)i(a)h
(pro)s(of,)f(one)378 5464 y(ma)m(y)h(need)f(a)h(more)g(general)f(form)g
(of)h(a)g(result)e(than)h(the)h(one)g(whic)m(h)e(is)g(a)m(v)-5
b(ailable)28 b(in)f(the)i(curren)m(t)378 5577 y(con)m(text.)50
b(The)32 b(role)h(of)g(abstractions)g(is)f(to)h(generalise)g(a)g(fact)h
(according)e(to)i(the)f(v)-5 b(ariables)31 b(and)p eop
%%Page: 66 76
66 75 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(66)378
396 y(assumptions)28 b(in)m(tro)s(duced)g(in)g(its)h(con)m(text)i(whic)
m(h)d(implicitly)e(sp)s(ecialise)i(it.)40 b(This)27 b(inference)i(cor-)
378 509 y(resp)s(onds)h(to)j(the)f(w)m(a)m(y)h(functions)e(can)h(b)s(e)
f(constructed)i(b)m(y)f(lam)m(b)s(da)f(abstraction)h(in)e(functional)
378 622 y(programs)37 b(and)f(the)i(lam)m(b)s(da)e(calculus.)60
b(SPL)36 b(facts)i(can)f(b)s(e)g(generalised)f(using)g(the)h(follo)m
(wing)378 735 y(three)31 b(kinds)d(of)i(abstractions:)514
923 y FN(\017)46 b FT(Generalising)29 b(t)m(yp)s(e)h(v)-5
b(ariables)29 b(so)i(that)g(they)g(can)f(b)s(e)g(instan)m(tiated,)514
1110 y FN(\017)46 b FT(Generalising)29 b(v)-5 b(ariables)29
b(o)s(ccurring)g(freely)h(in)f(a)h(fact)i(so)e(that)h(it)f(can)h(sp)s
(ecialised,)514 1298 y FN(\017)46 b FT(Disc)m(harging)27
b(assumptions)f(deriving)f(a)j(fact,)h(so)f(that)g(free)g(v)-5
b(ariables)26 b(in)g(the)i(assumptions)605 1411 y(can)h(b)s(e)e
(generalised)h(b)m(y)g(the)g(ab)s(o)m(v)m(e)i(kind)c(of)i(abstraction,)
h(and)f(the)g(resulting)f(fact)i(can)g(b)s(e)605 1524
y(applied)f(to)k(di\013eren)m(t)d(facts.)378 1712 y(T)m(yp)s(e)34
b(v)-5 b(ariable)33 b(and)g(free)i(v)-5 b(ariable)33
b(abstractions)h(are)h(denoted)f(b)m(y)g(the)h(abstracted)g(HOL)f(t)m
(yp)s(e)378 1824 y(or)29 b(term.)41 b(Assumption)27 b(abstractions)j
(are)g(denoted)f(b)m(y)g(the)h(lab)s(el)e(of)h(the)h(assumption.)39
b(W)-8 b(e)30 b(\014nd)378 1937 y(the)d(inferences)g(giv)m(en)g(b)m(y)g
(abstractions)g(to)h(b)s(e)e(v)m(ery)i(useful)d(when)h(the)i
(sectioning)e(mec)m(hanism)h(is)378 2050 y(used)g(to)i(structure)f(pro)
s(of)f(scripts.)39 b(Figure)27 b(8)i(illustrates)d(the)i(use)g(of)g
(abstractions)h(to)f(generalise)378 2163 y(the)j(lo)s(cal)e(statemen)m
(t)j(of)f(the)f(fact)i Fw(Q_P)d FT(from)h Fv(P)25 b(x)31
b FT(in)m(to)g Fu(8)p Fv(x)p Fw(.)p Fv(Q)14 b(x)42 b
Fu(\))i Fv(P)25 b(x)q FT(.)378 2403 y FQ(Applications)378
2575 y FT(Applications)30 b(are)i(the)h(in)m(v)m(erse)e(of)i
(abstractions,)f(in)f(the)h(sense)g(that)h(they)f(in)m(v)m(olv)m(e)g
(the)h(explicit)378 2688 y(sp)s(ecialisation)28 b(of)j(facts.)41
b(The)30 b(p)s(ossible)e(kinds)g(of)j(applications)d(include:)514
2875 y FN(\017)46 b FT(Instan)m(tiating)30 b(t)m(yp)s(e)h(v)-5
b(ariables.)514 3063 y FN(\017)46 b FT(Sp)s(ecialising)27
b(univ)m(ersally)h(quan)m(ti\014ed)h(facts.)514 3251
y FN(\017)46 b FT(Eliminating)27 b(implications)h(through)i(the)g(rule)
f(of)i(Mo)s(dus)e(P)m(onens.)378 3438 y(Although)j(in)g(a)i(declarativ)
m(e)g(language)f(abstractions)g(can)h(b)s(e)f(unam)m(biguously)d
(determined)i(b)m(y)378 3551 y(the)41 b(name)g(of)g(the)h(free)f(v)-5
b(ariable)40 b(or)h(b)m(y)g(the)g(lab)s(el)e(of)i(the)h(assumption,)g
(in)e(general)h(applica-)378 3664 y(tions)f(cannot.)73
b(F)-8 b(or)42 b(example,)h(it)e(is)f(not)h(clear)g(whether)f(the)i
(application)d(of)i(the)g(statemen)m(t)378 3777 y FN(8)p
FP(x;)15 b(y)s(:P)e FT(\()p FP(x;)i(y)s FT(\))34 b(to)f(some)g(constan)
m(t)h(term)f FP(c)g FT(should)e(result)h(in)f FN(8)p
FP(y)s(:P)13 b FT(\()p FP(c;)i(y)s FT(\),)35 b FN(8)p
FP(x:P)13 b FT(\()p FP(x;)i(c)p FT(\))33 b(or)g FP(P)13
b FT(\()p FP(c;)i(c)p FT(\))378 3890 y(unless)22 b(applications)g(are)i
(de\014ned)f(pro)s(cedurally)e(according)i(to)i(a)f(w)m(ell)f(sp)s
(eci\014ed)f(algorithm.)37 b(The)378 4003 y(role)22 b(of)g
(applications)e(is)h(to)h(reduce)g(the)g(searc)m(h)h(space)f(through)g
(explicit)e(instan)m(tiations)h(and)g(elim-)378 4116
y(ination)j(of)h(implications.)37 b(It)25 b(should)e(b)s(e)i(noted)h
(that)f(an)h(instan)m(tiation)e(of)h(a)h(v)-5 b(ariable)24
b(ma)m(y)i(result)378 4229 y(in)32 b(sp)s(ecialising)f(a)j
(higher-order)f(theorem)h(in)m(to)g(a)g(\014rst-order)f(one,)i(and)e
(can)h(therefore)g(greatly)378 4342 y(reduce)d(the)h(pro)s(of-c)m(hec)m
(king)g(time.)43 b(Since)31 b(suc)m(h)g(inferences)f(can)i(b)s(e)f(of)g
(a)h(great)h(adv)-5 b(an)m(tage,)34 b(the)378 4455 y(follo)m(wing)29
b(applications)f(are)j(supp)s(orted)e(b)m(y)h(SPL:)514
4642 y FN(\017)46 b FT(t)m(yp)s(e)34 b(v)-5 b(ariables)33
b(can)h(b)s(e)f(instan)m(tiated)h(sim)m(ultaneously)d(with)i(eac)m(h)i
(other)f(b)m(y)g(an)g(explicit)605 4755 y(substitution,)514
4943 y FN(\017)46 b FT(\(term\))23 b(v)-5 b(ariables)20
b(can)i(b)s(e)f(instan)m(tiated)g(individually)-8 b(,)19
b(either)i(b)m(y)g(an)h(explicit)e(substitution,)605
5056 y(or)37 b(b)m(y)g(giving)f(a)i(term)f(in)f(whic)m(h)g(case)i(the)f
(\014rst)g(v)-5 b(ariable)35 b(\(reading)i(the)g(term)h(from)e(left)605
5169 y(to)d(righ)m(t\))f(in)f(the)h(sen)m(tence)h(matc)m(hing)f(the)g
(t)m(yp)s(e)g(of)h(the)f(term)g(is)f(instan)m(tiated.)45
b(In)31 b(either)605 5281 y(case)i(the)g(v)-5 b(ariable)30
b(to)j(b)s(e)f(instan)m(tiated)f(is)h(mo)m(v)m(ed)h(to)f(the)h(b)s
(eginning)c(of)j(the)h(theorem)f(b)m(y)605 5394 y(the)j(usual)e(rules)g
(whic)m(h)g(transform)g(\(classical\))i(form)m(ulae)f(in)m(to)g(prenex)
g(form)g(b)s(efore)g(the)605 5507 y(theorem)d(is)e(sp)s(ecialised.)p
eop
%%Page: 67 77
67 76 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(67)p
378 1364 3453 4 v 376 4655 4 3291 v 515 1500 a Fw(let)42
b("x:'a")f(and)i("y:'a";)515 1699 y(section)e(on_P)602
1898 y(assume)g(Px:)i("P)f(x")733 1998 y(and)g(Py:)h("P)f(y";)602
2197 y(theorem)f(P_unique:)f("x)i(=)i(y")e(by)h(Px,)f(Py,)h
Fv(:)14 b(:)g(:)43 b Fw(;)515 2396 y(end)f(on_P;)515
2695 y(section)f(on_Q)602 2894 y(assume)g(Qx:)i("Q)f(x";)602
3094 y(theorem)f(Q_P:)h("P)g(x")h(by)g(Qx,)f Fv(:)14
b(:)g(:)44 b Fw(;)602 3293 y(assume)d(Qy:)i("Q)f(y";)602
3492 y(theorem)f(Q_unique:)f("x)i(=)i(y")602 3592 y(proof)689
3691 y(Py:)f("P)f(y")h(by)g(["x",Qx])14 b(Q_P,)38 b(Qy;)689
3791 y("x)43 b(=)g(y")g(by)g(P_unique,)d(Q_P,)i(Py;)602
3891 y(end;)515 4090 y(end)g(on_Q;)1388 4485 y FT(Figure)30
b(8:)41 b(The)30 b(Use)h(of)f(Abstractions.)p 3829 4655
V 378 4658 3453 4 v eop
%%Page: 68 78
68 77 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(68)378
396 y(The)26 b(applications)f(whic)m(h)h(corresp)s(ond)f(to)j(the)f
(elimination)d(of)j(implication)d(are)j(not)g(considered.)378
509 y(W)-8 b(e)47 b(remark)e(that)h(the)g(e\013ect)h(of)e(this)g(kind)e
(of)j(applications)d(\(i.e.,)16 b(Mo)s(dus)45 b(P)m(onens\))h(can)g(b)s
(e)378 622 y(ac)m(hiev)m(ed)31 b(b)m(y)f(using)f(structured)h
(justi\014cations)f(as)h(describ)s(ed)e(in)h(section)i(6.2.2,)h(page)g
(102.)519 735 y(Alternativ)m(es)e(to)h(the)f(approac)m(h)h(describ)s
(ed)d(ab)s(o)m(v)m(e)j(include)d(the)j(represen)m(tation)f(of)g(v)-5
b(ariable)378 848 y(applications)19 b(simply)g(b)m(y)h(terms)h
(\(rather)g(than)g(explicit)e(substitutions\))g(and)i(pro)s(of)f(searc)
m(h)i(heuris-)378 961 y(tics)32 b(can)g(b)s(e)g(dev)m(elop)s(ed)f(for)h
(fo)s(cusing)f(the)h(instan)m(tiation)f(of)i(v)-5 b(ariables)31
b(according)h(to)h(the)f(giv)m(en)378 1074 y(applications)855
1041 y FL(3)893 1074 y FT(.)48 b(It)33 b(is)e(not)j(clear,)f(ho)m(w)m
(ev)m(er,)i(whether)e(suc)m(h)f(an)h(approac)m(h)g(w)m(ould)f(result)f
(in)h(sub-)378 1187 y(stan)m(tial)f(e\016ciency)f(gains.)42
b(One)30 b(can)h(also)g(mo)s(dify)e(the)i(represen)m(tation)f(of)h(HOL)
g(terms)f(so)h(that)378 1300 y(subterms)24 b(can)h(b)s(e)g(lab)s
(elled.)36 b(This)24 b(w)m(ould)g(allo)m(w)g(the)i(user)e(to)i(state)h
(explicitly)22 b(whic)m(h)i(h)m(yp)s(othesis)378 1413
y(is)29 b(b)s(eing)g(eliminated)g(so)h(that)h(implication-elimination)
26 b(application)j(can)h(b)s(e)g(implemen)m(ted.)378
1653 y FQ(Inference)35 b(Rules)378 1824 y FT(A)45 b(sen)m(tence)g(can)g
(also)f(b)s(e)g(constructed)h(b)m(y)f(applying)f(some)h(inference)g
(rule)f(explicitly)-8 b(.)81 b(An)378 1937 y(inference)36
b(rule)g(is)g(denoted)h(b)m(y)g(an)g(iden)m(ti\014er,)g(and)f(the)h
(user)g(can)g(implemen)m(t)e(theory-sp)s(eci\014c)378
2050 y(HOL)44 b(inference)g(rules)f(and)g(include)g(them)h(in)f(the)i
(syn)m(tax)g(of)f(SPL)g(during)e(mec)m(hanisation.)378
2163 y(Ho)m(w)m(ev)m(er,)c(the)d(use)g(of)g(inference)f(rules)f(in)h
(the)h(construction)f(of)h(sen)m(tences)h(is)e(not)h(encouraged)378
2276 y(b)s(ecause)25 b(of)h(its)e(pro)s(cedural)g(nature.)38
b(The)25 b(only)f(SPL)h(inference)f(rule)g(whic)m(h)g(is)g(implemen)m
(ted)g(has)378 2389 y(the)h(iden)m(ti\014er)e Fw(select)f
FT(and)i(is)g(used)g(to)i(construct)f(facts)h(in)m(v)m(olving)d(the)i
(Hilb)s(ert)e(c)m(hoice)i(op)s(erator.)378 2502 y(It)30
b(tak)m(es)i(a)f(v)-5 b(ariable)29 b Fv(v)34 b FT(and)29
b(a)i(sen)m(tence)h(denoting)e(some)g(fact)i Fv(P)12
b Ft([)p Fv(v)s Ft(])30 b FT(and)g(deriv)m(es)g Fv(P)12
b Ft([)p Fv(")q(v)s(:P)g Ft([)p Fv(v)s Ft(]])p FT(.)378
2788 y FH(4.3)135 b(Pro)t(of)45 b(Chec)l(king)h(SPL)e(Scripts)h(in)g
(HOL)378 2991 y FT(The)30 b(pro)s(of)g(c)m(hec)m(k)m(er)j(of)d(the)h
(SPL)f(language)h(implemen)m(ted)e(in)g(HOL)i(pro)s(cesses)f(pro)s(of)g
(scripts)f(in)378 3104 y(t)m(w)m(o)j(steps:)514 3292
y FN(\017)46 b FT(P)m(arsing)32 b(the)g(input)e(text)j(in)m(to)f(an)f
(in)m(ternal)g(\(ML\))i(represen)m(tation)f(of)g(the)g(language)g(con-)
605 3405 y(struct;)514 3592 y FN(\017)46 b FT(Pro)s(cessing)30
b(the)g(constructs)h(to)g(mo)s(dify)e(the)h(en)m(vironmen)m(t)g(of)h
(the)f(pro)s(of)g(c)m(hec)m(k)m(er.)378 3780 y(The)d(SPL)h(state)h(is)e
(represen)m(ted)h(b)m(y)g(an)g(ML)g(ob)5 b(ject)29 b(of)f(t)m(yp)s(e)h
Fw(reason_state)23 b FT(and)k(consists)g(of)i(the)378
3893 y(input)i(string)g(and)h(the)h(en)m(vironmen)m(t)f(of)g(t)m(yp)s
(e)h Fw(reason_environme)o(nt)-6 b FT(.)47 b(The)32 b(implemen)m
(tation)f(of)378 4006 y(the)k(pro)s(of)f(c)m(hec)m(k)m(er)k(consists)c
(of)h(a)h(n)m(um)m(b)s(er)d(of)j(ML)f(functions)e(whic)m(h)h(parse)h
(and)f(pro)s(cess)h(SPL)378 4119 y(constructs.)42 b(Suc)m(h)30
b(functions)f(tak)m(e)j(and)e(return)g(ob)5 b(jects)31
b(of)g(t)m(yp)s(e)g Fw(reason_state)-5 b FT(.)42 b(A)30
b(n)m(um)m(b)s(er)g(of)378 4232 y(other)e(functions)e(whic)m(h)h(act)h
(on)g(ob)5 b(jects)29 b(of)e(t)m(yp)s(e)h Fw(reason_state)23
b FT(are)28 b(also)g(implemen)m(ted.)38 b(These)378 4344
y(include)21 b(functions)g(whic)m(h)h(extract)j(pro)m(v)m(ed)e
(theorems)g(from)g(the)g(SPL)f(en)m(vironmen)m(t)h(so)g(that)h(they)378
4457 y(can)36 b(b)s(e)f(used)f(in)h(HOL,)g(add)g(HOL)g(axioms,)i
(de\014nitions)c(and)i(theorems)h(to)g(the)g(en)m(vironmen)m(t,)378
4570 y(and)30 b(add)f(new)h(input)f(text)i(in)e(order)h(to)h(b)s(e)f
(parsed)g(and)f(pro)s(cessed.)519 4683 y(The)45 b(pro)s(cessing)e(of)i
(SPL)g(scripts)e(can)j(therefore)f(b)s(e)f(in)m(v)m(ok)m(ed)i(during)c
(a)k(HOL)e(theorem)378 4796 y(pro)m(ving)e(session)g(b)m(y)g(calling)g
(the)h(appropriate)e(ML)i(functions.)77 b(As)42 b(a)i(result,)h(the)e
(user)f(can)378 4909 y(implemen)m(t)32 b(an)h(SPL)f(script,)h(pro)s
(cess)g(it)f(within)f(a)i(HOL)g(session)f(and)h(use)g(the)g(deriv)m(ed)
f(results)378 5022 y(in)41 b(HOL)h(inference)f(rules)g(and)h(tactics)h
(or)g(in)e(the)h(implemen)m(tation)f(of)i(pro)s(of)e(pro)s(cedures)g
(in)378 5135 y(ML.)h(Moreo)m(v)m(er,)47 b(the)42 b(SPL)e(language)i(is)
f(extensible:)62 b(the)42 b(user)f(can)h(implemen)m(t)e(HOL)i(pro)s(of)
378 5248 y(pro)s(cedures)e(and)h(include)e(them)i(in)g(the)g(language)h
(syn)m(tax.)75 b(Therefore,)44 b(one)d(can)h(dev)m(elop)g(a)378
5361 y(theory)31 b(b)m(y)f(rep)s(eating)g(the)g(follo)m(wing)f(steps:)p
378 5523 1380 4 v 482 5577 a FC(3)516 5608 y FB(F)-6
b(or)29 b(example,)g(one)g(can)g(giv)n(e)g(priorit)n(y)g(to)g(instan)n
(tiations)h(suggested)g(b)n(y)e(the)g(applications)i(o)n(v)n(er)f
(those)g(sug-)378 5700 y(gested)d(b)n(y)f(uni\014cation)g(during)h(pro)
r(of)h(searc)n(h.)p eop
%%Page: 69 79
69 78 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(69)464
396 y(\(i\))45 b(deriving)28 b(a)j(n)m(um)m(b)s(er)e(of)i(theorems)f
(using)f(SPL)h(pro)s(ofs,)439 579 y(\(ii\))44 b(using)29
b(the)i(deriv)m(ed)e(theorems)i(in)e(the)h(implemen)m(tation)f(of)i
(HOL)f(pro)s(of)g(pro)s(cedures,)413 762 y(\(iii\))44
b(extending)30 b(the)g(SPL)g(language)h(to)g(mak)m(e)g(use)f(of)h(the)f
(new)g(pro)s(of)g(pro)s(cedures.)378 939 y(This)24 b(approac)m(h)i(com)
m(bines)f(the)h(readabilit)m(y)e(of)i(SPL)e(pro)s(ofs)h(with)f(the)i
(extensibilit)m(y)e(of)h(the)h(HOL)378 1051 y(system.)53
b(The)33 b(mec)m(hanisation)h(of)h(group)e(theory)i(describ)s(ed)d(in)h
(c)m(hapter)i(9)f(is)g(dev)m(elop)s(ed)f(using)378 1164
y(this)d(approac)m(h.)42 b(In)31 b(this)f(case,)i(new)e(pro)s(of)h(pro)
s(cedures)e(w)m(ere)j(implemen)m(ted)d(as)i(the)g(theory)h(w)m(as)378
1277 y(mec)m(hanised)g(in)f(order)g(to)j(automate)f(the)g(pro)s(of)f
(steps)g(whic)m(h)f(w)m(ould)g(b)s(e)g(considered)h(trivial)e(b)m(y)378
1390 y(the)h(reader.)519 1503 y(ML)43 b(references)g(are)g(used)f(to)i
(store)f(the)g(functions)e(whic)m(h)h(parse)g(and)g(pro)s(cess)h(the)g
(SPL)378 1616 y(language)35 b(constructs)h(\(including)c(the)j(pro)s
(cessors)f(of)i(reasoning)e(items\))h(so)g(that)h(they)f(can)h(b)s(e)
378 1729 y(up)s(dated)c(b)m(y)i(the)g(user)f(during)f(the)i(dev)m
(elopmen)m(t)g(of)g(a)h(theory)-8 b(.)51 b(This)33 b(implemen)m(tation)
f(design)378 1842 y(w)m(as)37 b(originally)e(used)h(to)h(allo)m(w)g
(the)g(author)g(to)g(alter)g(the)g(syn)m(tax)h(and)e(seman)m(tics)h(of)
g(the)g(lan-)378 1955 y(guage)d(easily)e(during)f(the)i(dev)m(elopmen)m
(t)g(of)g(a)g(theory)g(when)f(the)h(implemen)m(tation)f(of)h(the)g(SPL)
378 2068 y(language)j(w)m(as)f(still)f(in)g(its)g(exp)s(erimen)m(tal)h
(stages.)57 b(Ho)m(w)m(ev)m(er,)38 b(w)m(e)e(no)m(w)g(b)s(eliev)m(e)e
(that)i(the)g(\015exi-)378 2181 y(bilit)m(y)30 b(o\013ered)i(b)m(y)g
(this)f(design)g(can)i(indeed)d(b)s(e)i(a)g(desirable)e(feature)j(of)f
(pro)s(of)g(languages.)46 b(This)378 2293 y(allo)m(ws)27
b(the)g(pro)s(of)g(implemen)m(tor,)f(for)h(instance,)h(to)g(include)d
(new)i(reasoning)g(items)g(\(rather)g(than)378 2406 y(just)k(pro)s(of)f
(pro)s(cedures\))g(whic)m(h)g(mak)m(e)j(use)e(of)g(deriv)m(ed)f
(theorems)i(during)d(the)i(implemen)m(tation)378 2519
y(of)e(a)h(theory)-8 b(.)41 b(One)29 b(can)g(also)g(c)m(hange)i
(substan)m(tial)d(parts)h(of)g(the)g(syn)m(tax)h(of)g(the)f(language)h
(to)g(one)378 2632 y(whic)m(h)37 b(is)h(b)s(eliev)m(ed)f(to)j(b)s(e)e
(more)g(appropriate)g(to)h(the)g(particular)e(theory)i(b)s(eing)e(mec)m
(hanised.)378 2745 y(Ideally)-8 b(,)26 b(an)m(y)h(alterations)f(made)h
(to)g(the)g(syn)m(tax)g(of)f(the)h(language)g(should)d(b)s(e)i(lo)s
(cal)f(to)j(particular)378 2858 y(sections.)41 b(In)30
b(order)g(to)h(ac)m(hiev)m(e)g(this,)f(one)h(needs)f(a)h(n)m(um)m(b)s
(er)e(of)h(design)g(c)m(hanges)h(to)g(the)g(curren)m(t)378
2971 y(implemen)m(tation)24 b(of)h(the)h(language)f(since)g(the)g(use)g
(of)g(ML)h(references)f(allo)m(ws)g(the)g(user)g(to)h(up)s(date)378
3084 y(the)31 b(syn)m(tax)f(globally)f(rather)i(than)f(lo)s(cally)-8
b(.)519 3197 y(In)37 b(the)h(follo)m(wing)e(sections)i(w)m(e)h(\014rst)
e(lo)s(ok)g(at)i(ho)m(w)e(the)h(SPL)f(en)m(vironmen)m(t)h(and)f(facts)h
(are)378 3310 y(represen)m(ted)30 b(and)g(then)g(describ)s(e)f(the)h
(parsing)f(and)h(pro)s(cessing)f(mec)m(hanisms.)378 3551
y FG(4.3.1)112 b(The)38 b(En)m(vironmen)m(t)e(of)i(SPL)378
3723 y FT(The)d(SPL)f(en)m(vironmen)m(t)h(consists)f(of)i(the)f
(information)f(whic)m(h)g(has)h(b)s(een)f(declared)h(or)g(deriv)m(ed)
378 3835 y(b)m(y)e(the)h(SPL)e(constructs.)50 b(Because)35
b(of)e(the)h(hierarc)m(hical)e(structure)g(of)i(SPL)e(scripts,)i(the)f
(en)m(vi-)378 3948 y(ronmen)m(t)h(is)e(structured)h(as)h(a)g(stac)m(k)h
(of)f FI(layers)42 b FT(con)m(taining)33 b(the)h(information)e
(declared)h(lo)s(cally)-8 b(.)378 4061 y(An)29 b(empt)m(y)i(la)m(y)m
(er)f(is)f(created)i(and)e(pushed)f(on)i(top)g(of)g(the)g(stac)m(k)h
(at)f(the)g(b)s(eginning)e(of)i(a)g(section)378 4174
y(or)k(pro)s(of.)50 b(Pro)s(cessing)33 b(reasoning)g(items)h(a\013ects)
h(only)e(the)h(information)e(in)g(the)i(top)g(la)m(y)m(er.)52
b(A)m(t)378 4287 y(the)32 b(end)g(of)g(a)h(section)f(or)g(pro)s(of,)g
(the)h(top)f(la)m(y)m(er)h(is)e(p)s(opp)s(ed)f(from)i(the)g(stac)m(k)i
(and)e(all)f(the)h(infor-)378 4400 y(mation)c(stored)g(in)f(this)h(la)m
(y)m(er,)h(with)e(the)i(exception)f(of)h(theorems,)g(is)f(destro)m(y)m
(ed.)41 b(Theorems)27 b(are)378 4513 y(expanded)36 b(and)h(inserted)f
(in)m(to)h(the)g(new)g(top)g(la)m(y)m(er.)61 b(W)-8 b(e)39
b(sa)m(y)f(that)f(a)h(la)m(y)m(er)f(has)g(b)s(een)g(op)s(ened)378
4626 y(when)32 b(it)g(is)g(pushed)f(on)i(top)g(of)g(the)g(en)m
(vironmen)m(t)g(stac)m(k.)49 b(W)-8 b(e)34 b(also)f(sa)m(y)h(that)f(a)h
(la)m(y)m(er)f(has)g(b)s(een)378 4739 y(closed)d(when)g(it)f(is)h(p)s
(opp)s(ed)e(from)i(the)h(stac)m(k.)519 4852 y(Eac)m(h)c(la)m(y)m(er)g
(con)m(tains)f(a)h(list)e(of)h(lo)s(cally)f(deriv)m(ed)h(or)g(assumed)g
(facts)h(lab)s(elled)c(b)m(y)k(their)e(iden)m(ti-)378
4965 y(\014er,)f(a)f(list)e(of)i(v)-5 b(ariables)21 b(and)h(t)m(yp)s(e)
h(v)-5 b(ariables)22 b(in)m(tro)s(duced)f(b)m(y)h(reasoning)g(items,)i
(a)f(list)f(of)g(declared)378 5077 y(simpli\014ers,)33
b(a)j(list)e(of)i(facts)h(stored)e(in)g(the)g(trivial)f(kno)m(wledge)i
(database)g(and)f(some)h(other)g(in-)378 5190 y(formation)29
b(\(e.g.,)17 b(the)30 b(name)g(of)g(the)g(section,)g(the)g(curren)m(t)f
(conclusion)f(in)h(case)h(of)g(a)g(pro)s(of)f(la)m(y)m(er,)378
5303 y(etc.)17 b(\).)519 5416 y(There)30 b(are)h(three)f(kinds)f(of)h
(v)-5 b(ariables)29 b(whic)m(h)g(can)i(b)s(e)f(in)m(tro)s(duced:)378
5592 y FQ(Univ)m(ersal)46 b FT(v)-5 b(ariables)37 b(whic)m(h)h(are)h
(in)m(tro)s(duced)e(b)m(y)h(generalisations)f(and)h(quan)m(ti\014ed)g
(assump-)605 5705 y(tions,)p eop
%%Page: 70 80
70 79 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(70)378
396 y FQ(Existen)m(tial)45 b FT(v)-5 b(ariables)29 b(whic)m(h)g(are)i
(in)m(tro)s(duced)e(b)m(y)h(existen)m(tial)g(results,)378
584 y FQ(Abbreviating)46 b FT(v)-5 b(ariables)29 b(whic)m(h)g(are)i(in)
m(tro)s(duced)d(b)m(y)j(abbreviations.)378 772 y(These)20
b(kinds)f(of)i(v)-5 b(ariables)19 b(implicitly)e(bind)h(all)h(their)h
(free)h(o)s(ccurrences)f(in)g(the)g(form)m(ulae)h(sp)s(eci\014ed)378
885 y(in)29 b(their)g(con)m(text,)k(and)d(can)g(b)s(e)g(called)g
(binding)d(v)-5 b(ariables.)378 1128 y FG(4.3.2)112 b(The)38
b(Represen)m(tation)f(of)h(SPL)f(F)-9 b(acts)38 b(in)f(HOL)378
1300 y FT(SPL)d(facts)h(are)g(represen)m(ted)f(b)m(y)g(pairs)g
Fw(\()p Fj(vl)8 b Fw(,)43 b Ft(\000)23 b Fu(`)g Fv(t)p
Fw(\))34 b FT(where)g Fj(vl)43 b FT(is)34 b(a)h(list)e(of)h(t)m(yp)s(e)
h(v)-5 b(ariables,)34 b(and)378 1413 y Ft(\000)23 b Fu(`)g
Fv(t)30 b FT(is)f(a)h(HOL)g(theorem.)41 b(The)30 b(conclusion)f
Fv(t)h FT(represen)m(ts)g(the)g(statemen)m(t)i(of)e(the)h(fact,)g(and)f
(the)378 1526 y(h)m(yp)s(othesis)g(list)f Ft(\000)i FT(is)f(the)h(list)
f(of)h(SPL)f(assumptions)g(used)g(in)g(deriving)f(it.)42
b(An)m(y)31 b(t)m(yp)s(e)g(v)-5 b(ariables)378 1638 y(in)37
b Fj(vl)46 b FT(univ)m(ersally)36 b(quan)m(tify)h(the)h(statemen)m(t)i
Fv(t)o FT(.)64 b(The)37 b(t)m(yp)s(e)h(v)-5 b(ariables)37
b(o)s(ccurring)f(in)h Fv(t)g FT(but)h(not)378 1751 y(in)29
b Fj(vl)39 b FT(do)30 b(not)h(univ)m(ersally)d(quan)m(tify)h(the)h
(fact)i Fj(t)37 b FT(and)30 b(therefore)h(cannot)g(b)s(e)e(instan)m
(tiated)h(during)378 1864 y(pro)s(of)g(searc)m(h.)519
1977 y(The)35 b(list)e(of)i(t)m(yp)s(e)g(v)-5 b(ariables)34
b(quan)m(tifying)f(SPL)h(facts)i(is)e(required)f(in)h(their)g(represen)
m(tation)378 2090 y(b)s(ecause)28 b(the)g(HOL)f(term)h(syn)m(tax)h(do)s
(es)e(not)h(include)e(explicit)g(quan)m(ti\014cation)h(o)m(v)m(er)i(t)m
(yp)s(es.)40 b(T)m(yp)s(e)378 2203 y(v)-5 b(ariables)36
b(are)i(included)d(in)h(the)h(simple)f(t)m(yp)s(es)h(of)h(HOL)f(in)f
(order)h(to)h(construct)g(p)s(olymorphic)378 2316 y(theorems.)55
b(The)35 b(scop)s(e)g(of)h(t)m(yp)s(e)f(v)-5 b(ariables)34
b(includes)f(b)s(oth)h(the)i(theorem)f(h)m(yp)s(otheses)g(and)g(the)378
2429 y(conclusion,)26 b(and)g(therefore)h(p)s(olymorphic)d(HOL)j
(theorems)g(can)g(b)s(e)f(assumed)g(to)h(b)s(e)f(`templates')378
2542 y(whic)m(h)32 b(can)i(generate)h(new)e(theorems)h(through)f(t)m
(yp)s(e)h(instan)m(tiation.)49 b(A)33 b(theorem)h Ft(\000)23
b Fu(`)g Fv(t)34 b FT(can)g(b)s(e)378 2655 y(seen)c(as)h(b)s(eing)e
(univ)m(ersally)f(quan)m(ti\014ed)h(b)m(y)h(all)g(the)g(t)m(yp)s(e)h(v)
-5 b(ariables)29 b(whic)m(h)g(o)s(ccur)h(in)f(it,)h(that)h(is:)1701
2859 y FN(8)15 b Fw(TyVars)2028 2881 y FL(\000)d FK(`)g
FO(t)2168 2859 y FP(:)j FT(\(\000)41 b FN(`)f FP(t)p
FT(\))378 3063 y(where)i Fw(TyVars)917 3075 y Fq(')1005
3063 y FT(represen)m(ts)g(the)h(t)m(yp)s(e)g(v)-5 b(ariables)41
b(in)g Fv(')p FT(.)77 b(Ho)m(w)m(ev)m(er,)48 b(the)43
b(HOL)f(rule)f(for)i(t)m(yp)s(e)378 3176 y(instan)m(tiation,)34
b Fw(INST_TYPE)29 b FT(restricts)34 b(the)f(instan)m(tiation)g(to)h
(the)g(t)m(yp)s(e)g(v)-5 b(ariables)32 b(whic)m(h)h(are)h(not)378
3289 y(free)j(in)e(the)i(h)m(yp)s(otheses)f Ft(\000)p
FT(,)j(although)d(the)h(more)g(general)f(rule)g(of)h(instan)m(tiating)e
(all)h(the)h(t)m(yp)s(e)378 3402 y(v)-5 b(ariables)34
b(o)s(ccurring)g(in)f(a)j(theorem)f(can)h(b)s(e)e(easily)h(deriv)m(ed)f
(b)m(y)h(disc)m(harging)f(the)h(h)m(yp)s(otheses,)378
3515 y(instan)m(tiating,)26 b(and)f(undisc)m(harging)e(the)j(h)m(yp)s
(otheses)g(bac)m(k.)40 b(This)24 b(suggests)i(that)g(t)m(yp)s(e)g(v)-5
b(ariables)378 3628 y(are)25 b(seen)f(as)g(quan)m(tifying)f(only)g(the)
i(conclusion)e(of)h(a)h(theorem,)h(that)f(is,)f(a)h(p)s(olymorphic)c
(theorem)378 3741 y Ft(\000)i Fu(`)g Fv(t)30 b FT(is)f(visualised)f(as)
1553 3853 y(\000)40 b FN(`)g(8)p FT(\()p Fw(TyVars)2094
3876 y FO(t)2144 3853 y FN(\000)19 b Fw(TyVars)2496 3876
y FL(\000)2544 3853 y FT(\))p FP(:)c(t)378 4020 y FT(This)26
b(particular)h(visualisation)f(someho)m(w)i(corresp)s(onds)f(to)i(the)g
(HOL)f(approac)m(h)g(of)h(considering)378 4133 y(the)34
b(list)f(of)h(h)m(yp)s(otheses)f(more)h(of)g(a)h FI(working)h(sp)-5
b(ac)g(e)42 b FT(during)32 b(theorem)i(pro)m(ving)f(rather)h(than)g(as)
378 4246 y(part)c(of)h(the)f(theorem)h(statemen)m(t.)519
4359 y(W)-8 b(e)31 b(cannot)g(assume)e(that)i(SPL)e(form)m(ulae)g(are)i
(implicitly)26 b(quan)m(ti\014ed)j(b)m(y)g(all)g(the)h(t)m(yp)s(e)g(v)
-5 b(ari-)378 4472 y(ables)40 b(o)s(ccurring)g(in)g(them)h(since)g(one)
g(cannot)h(instan)m(tiate)f(the)g(t)m(yp)s(e)g(v)-5 b(ariables)40
b(whic)m(h)g(o)s(ccur)378 4585 y(in)d(the)h(assumptions)e(of)j(a)f(pro)
s(of.)63 b(Suc)m(h)38 b(an)g(instan)m(tiation)f(w)m(ould)g(require)g
(the)h(instan)m(tiation)378 4698 y(of)e(the)g(t)m(yp)s(e)g(v)-5
b(ariables)34 b(in)g(the)i(assumptions)e(as)i(w)m(ell)f(in)f(order)i
(to)g(b)s(e)f(sound.)56 b(Therefore,)37 b(the)378 4811
y(t)m(yp)s(e)h(v)-5 b(ariables)36 b(o)s(ccurring)g(in)g(the)i
(assumptions)e(are)h(in)m(tro)s(duced)f(as)i(generalisations)f(so)g
(that)378 4924 y(they)27 b(bind)d(the)j(t)m(yp)s(e)g(v)-5
b(ariables)25 b(of)i(the)f(form)m(ulae)g(sp)s(eci\014ed)f(in)h(the)g
(pro)s(of,)h(and)f(therefore)h(cannot)378 5037 y(b)s(e)34
b(instan)m(tiated)h(during)d(theorem)k(pro)m(ving.)53
b(On)34 b(the)h(other)h(hand,)f(one)g(cannot)h(eliminate)d(all)378
5149 y(t)m(yp)s(e)f(instan)m(tiations)e(as)i(otherwise)f(p)s
(olymorphic)d(theorems)k(could)f(not)g(b)s(e)g(used.)43
b(As)32 b(a)g(result,)378 5262 y(in)g(order)h(to)h(use)g(t)m(yp)s(e)f
(v)-5 b(ariables)32 b(soundly)g(and)h(e\013ectiv)m(ely)-8
b(,)36 b(one)e(is)e(required)g(to)i(sp)s(ecify)e(whic)m(h)378
5375 y(t)m(yp)s(e)h(v)-5 b(ariables)31 b(o)s(ccurring)h(in)f(SPL)h
(facts)h(can)g(b)s(e)f(instan)m(tiated.)48 b(This)31
b(explains)g(wh)m(y)h(the)h(t)m(yp)s(e)378 5488 y(v)-5
b(ariables)29 b(quan)m(tifying)g(facts)i(are)g(included)c(in)i(their)h
(represen)m(tation.)p eop
%%Page: 71 81
71 80 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(71)378
396 y FG(4.3.3)112 b(P)m(arsing)37 b(Pro)s(of)g(Scripts)378
568 y FT(The)g(ob)5 b(ject)38 b(em)m(b)s(edding)e(system)i(of)f(Slind)e
(\(1991\))40 b(is)c(used)h(to)h(em)m(b)s(ed)f(the)h(SPL)e(language)i
(in)378 681 y(SML.)21 b(Basically)-8 b(,)23 b(using)d(this)g(system)i
(the)f(text)i(of)e(SPL)g(scripts)f(and)h(script)f(fragmen)m(ts)i(is)e
(enclosed)378 794 y(in)33 b(bac)m(kquotes)i(\()p Fw(`)p
FT(\))f(so)h(that)f(they)g(can)h(b)s(e)e(easily)g(written)h(and)f
(read.)52 b(The)33 b(texts)i(are)g(ho)m(w)m(ev)m(er)378
907 y(in)m(ternally)f(represen)m(ted)j(as)g(ML)f(ob)5
b(jects)37 b(from)f(whic)m(h)g(ML)g(strings)g(represen)m(ting)f(the)i
(lines)e(of)378 1020 y(the)c(pro)s(of)f(texts)i(can)f(b)s(e)f
(extracted.)43 b(Once)31 b(extracted)h(the)f(strings)f(are)h(then)f
(parsed)g(using)g(the)378 1133 y(SPL)f(language)i(parser.)519
1246 y(The)43 b(SPL)g(language)i(uses)e(the)h(HOL)g(syn)m(tax)g(for)g
(terms)g(and)f(t)m(yp)s(es.)81 b(SPL)43 b(expressions)378
1358 y(represen)m(ting)33 b(terms)g(and)g(t)m(yp)s(es)g(are)h(giv)m(en)
f(to)i(the)e(in)m(ternal)f(HOL)h(parser)g(after)h(a)g(simple)d(pre-)378
1471 y(pro)s(cessing)36 b(stage)i(whic)m(h,)f(for)g(instance,)h(giv)m
(es)g(the)f(t)m(yp)s(e)g Fw(:bool)e FT(to)i(expressions)f(represen)m
(ting)378 1584 y(form)m(ulae,)29 b(and)f(inserts)f(t)m(yp)s(es)i(for)g
(an)m(y)g(free)g(v)-5 b(ariables)27 b(in)g(terms)i(according)g(to)g
(the)g(t)m(yp)s(es)g(of)g(the)378 1697 y(curren)m(t)h(list)f(of)i
(binding)c(v)-5 b(ariables.)519 1810 y(The)21 b(implemen)m(tation)f(of)
i(the)g(parser)f(is)g(quite)g(straigh)m(tforw)m(ard,)i(and)e(is)g
(based)g(on)h(the)f(syn)m(tax)378 1923 y(giv)m(en)30
b(in)f(app)s(endix)f(A.)378 2165 y FG(4.3.4)112 b(Pro)s(cessing)37
b(SPL)h(Constructs)378 2337 y FT(This)29 b(section)h(lists)f(the)i
(e\013ect)g(of)g(pro)s(cessing)e(the)i(individual)25
b(SPL)30 b(constructs.)378 2576 y FQ(Sections)378 2748
y FT(As)h(describ)s(ed)e(in)h(section)h(4.3.1,)i(a)f(section)f(op)s
(ens)f(a)h(new)g(la)m(y)m(er)h(whic)m(h)d(is)h(closed)h(at)h(the)f(end)
g(of)378 2861 y(the)g(section.)378 3100 y FQ(Lo)s(cal)k(Declarations)
378 3271 y FT(Lo)s(cal)30 b(declarations)g(of)h(the)f(form)473
3453 y FM(local)569 3566 y FI(lo)-5 b(c)g(al)59 b(de)-5
b(clar)g(ations)473 3679 y FM(in)569 3792 y FI(script)57
b(se)-5 b(gment)473 3905 y FM(end;)378 4087 y FT(are)33
b(pro)s(cessed)f(b)m(y)h(\014rst)f(op)s(ening)g(a)h(new)f(la)m(y)m(er)h
(to)h(store)f(the)g(lo)s(cal)f(declarations.)48 b(When)32
b(these)378 4200 y(are)24 b(pro)s(cessed,)g(another)g(la)m(y)m(er)g(is)
f(op)s(ened)f(to)j(store)f(the)f(declarations)g(in)f(the)i(script)e
(segmen)m(t.)40 b(A)m(t)378 4313 y(the)24 b(end)f(of)h(the)g(script)f
(segmen)m(t,)j(the)e(t)m(w)m(o)h(la)m(y)m(ers)f(are)h(closed)e(and)g
(the)h(information)e(stored)i(in)f(the)378 4425 y(segmen)m(t)k(la)m(y)m
(er)f(is)e(transferred)g(to)j(the)e(original)f(la)m(y)m(er)i(whic)m(h)e
(is)h(no)m(w)g(on)g(top)h(of)g(the)f(en)m(vironmen)m(t)378
4538 y(stac)m(k)34 b(\(see)f(\014gure)f(9\).)47 b(An)m(y)32
b(results)f(deriv)m(ed)h(in)f(the)h(script)f(segmen)m(t)j(are)e
(expanded)g(according)378 4651 y(to)41 b(the)f(v)-5 b(ariables)38
b(and)i(assumptions)e(in)m(tro)s(duced)g(in)h(the)h(lo)s(cal)f
(declarations.)69 b(F)-8 b(or)41 b(example,)378 4764
y(after)h(the)g(script)e(fragmen)m(t)i(giv)m(en)g(b)s(elo)m(w)e(is)h
(pro)s(cessed,)j(the)d(lab)s(el)f Fw(Qx)h FT(will)d(denote)k(the)g
(fact)378 4877 y Fu(8)p Fv(x)p Fw(:num.)p Fv(P)53 b(x)44
b Fu(\))g Fv(Q)f(x)p FT(.)473 5059 y FM(local)569 5172
y(let)k("x:)g(num";)569 5285 y(assume)f(Px:)h("P)g(x";)473
5398 y(in)569 5511 y(Qx:)g("Q)g(x")g(by)g(Px;)473 5624
y(end;)p eop
%%Page: 72 82
72 81 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(72)p
378 416 3453 4 v 376 1728 4 1312 v 888 570 a
19208274 6249267 10525081 22892052 29733355 29141319 startTexFig
/showpage{}def /erasepage{}def /copypage{}def
888 570
a
%%BeginDocument: local_layers.ps
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/l {lineto} bind def
/m {moveto} bind def
/s {stroke} bind def
/n {newpath} bind def
/gs {gsave} bind def
/gr {grestore} bind def
/clp {closepath} bind def
/graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
4 -2 roll mul setrgbcolor} bind def
/col-1 {} def
/col0 {0 0 0 setrgbcolor} bind def
/col1 {0 0 1 setrgbcolor} bind def
/col2 {0 1 0 setrgbcolor} bind def
/col3 {0 1 1 setrgbcolor} bind def
/col4 {1 0 0 setrgbcolor} bind def
/col5 {1 0 1 setrgbcolor} bind def
/col6 {1 1 0 setrgbcolor} bind def
/col7 {1 1 1 setrgbcolor} bind def
end
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
$F2psBegin
0 setlinecap 0 setlinejoin
-26.0 603.5 translate 0.810 -0.810 scale
0.500 setlinewidth
% Polyline
n 376 199 m 369 199 369 232 7 arcto 4 {pop} repeat 369 239 442 239 7 arcto 4 {pop} repeat 449 239 449 206 7 arcto 4 {pop} repeat 449 199 376 199 7 arcto 4 {pop} repeat clp gs col-1 s gr
% Polyline
n 376 249 m 369 249 369 282 7 arcto 4 {pop} repeat 369 289 442 289 7 arcto 4 {pop} repeat 449 289 449 256 7 arcto 4 {pop} repeat 449 249 376 249 7 arcto 4 {pop} repeat clp gs col-1 s gr
% Polyline
n 369 314 m 369 299 l 449 299 l 449 314 l gs col-1 s gr
% Polyline
n 509 314 m 509 299 l 589 299 l 589 314 l gs col-1 s gr
% Polyline
n 236 249 m 229 249 229 282 7 arcto 4 {pop} repeat 229 289 302 289 7 arcto 4 {pop} repeat 309 289 309 256 7 arcto 4 {pop} repeat 309 249 236 249 7 arcto 4 {pop} repeat clp gs col-1 s gr
% Polyline
n 229 314 m 229 299 l 309 299 l 309 314 l gs col-1 s gr
% Polyline
n 529 254 m 529 279 l gs col-1 s gr
n 531.000 271.000 m 529.000 279.000 l 527.000 271.000 l gs 2 setlinejoin col-1 s gr
% Polyline
n 549 254 m 549 279 l gs col-1 s gr
n 551.000 271.000 m 549.000 279.000 l 547.000 271.000 l gs 2 setlinejoin col-1 s gr
% Polyline
n 569 254 m 569 279 l gs col-1 s gr
n 571.000 271.000 m 569.000 279.000 l 567.000 271.000 l gs 2 setlinejoin col-1 s gr
/Times-Roman findfont 12.00 scalefont setfont
384 214 m
gs 1 -1 scale (Segment) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
384 229 m
gs 1 -1 scale (Layer) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
239 264 m
gs 1 -1 scale (Local) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
239 282 m
gs 1 -1 scale (Declarations) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
379 264 m
gs 1 -1 scale (Local) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
379 282 m
gs 1 -1 scale (Declarations) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
524 209 m
gs 1 -1 scale (Segment) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
524 227 m
gs 1 -1 scale (Layer) col-1 show gr
/Times-Roman findfont 12.00 scalefont setfont
524 244 m
gs 1 -1 scale (Information) col-1 show gr
showpage
$F2psEnd
%%EndDocument
endTexFig
1277 1558 a FT(Figure)30 b(9:)41 b(Pro)s(cessing)30
b(Lo)s(cal)g(Declarations.)p 3829 1728 4 1312 v 378 1731
3453 4 v 378 2088 a FQ(Generalisations)378 2260 y FT(Generalisations)f
(in)m(tro)s(duce)h(v)-5 b(ariables)29 b(and)g(t)m(yp)s(e)i(v)-5
b(ariables)29 b(as)h(univ)m(ersal)f(v)-5 b(ariables.)378
2500 y FQ(Assumptions)378 2671 y FT(An)30 b(assumption)f(of)h(a)h(lab)s
(elled)d(form)m(ula)h Fv(L)p Fw(:)14 b Fv(A)31 b FT(in)m(tro)s(duces)e
(the)h(fact)i Fw(\()p Ft([])p Fw(,)p Fv(A)22 b Fu(`)h
Fv(A)p Fw(\))30 b FT(with)f(lab)s(el)g Fv(L)p FT(.)378
2912 y FQ(Results)378 3083 y FT(A)g(theorem)h(or)f(pro)s(of)g(step)g
(result)f(op)s(ens)h(a)h(new)e(la)m(y)m(er)i(to)g(store)g(the)g
(declarations)e(of)i(its)e(justi\014-)378 3196 y(cation.)41
b(The)29 b(justi\014cation)f(pro)s(ceeds)h(b)m(y)g(constructing)g(a)h
(justifying)e(fact)i(whic)m(h)e(is)h(used)f(in)h(the)378
3309 y(deriv)-5 b(ation)34 b(of)h(the)g(conclusion)f(of)h(the)g(result)
f(\(see)i(section)f(4.2.4\).)57 b(The)35 b(justi\014cation)e(la)m(y)m
(er)j(is)378 3422 y(closed)30 b(when)g(the)g(conclusion)f(is)g(deriv)m
(ed)h(whic)m(h)f(is)g(then)h(included)e(as)j(a)f(fact.)378
3662 y FQ(Existen)m(tial)k(Results)378 3834 y FT(Existen)m(tial)29
b(results)g(of)i(the)g(form)473 4021 y FM(there)47 b(is)g(some)g
FP(x)g FM(such)g(that)569 4134 y FP(L)p FM(:)g(")p FP(P)28
b(x)p FM(")473 4247 y FI(justi\014c)-5 b(ation)56 b(of)68
b FN(9)p FP(x:P)28 b(x)47 b FM(;)378 4435 y FT(are)36
b(justi\014ed)d(in)h(the)h(same)h(w)m(a)m(y)g(as)f(non-existen)m(tial)g
(results)e(are.)56 b(When)35 b(the)g(existen)m(tial)g(fact)378
4548 y Fw(\()p Ft([])p Fw(,)p Ft(\000)h Fu(`)h(9)p Fv(x:P)26
b(x)p Fw(\))k FT(is)f(deriv)m(ed)g(using)f(some)j(assumptions)d
Ft(\000)p FT(,)i(the)g(v)-5 b(ariable)29 b Fv(x)h FT(is)f(in)m(tro)s
(duced)g(as)h(an)378 4661 y(existen)m(tial)g(v)-5 b(ariable,)29
b(and)h(the)g(fact)h Fw(\()p Ft([])p Fw(,)p Fv(P)25 b(x)38
b Fu(`)e Fv(P)26 b(x)p Fw(\))k FT(is)f(in)m(tro)s(duced)g(with)g(lab)s
(el)g Fv(L)o FT(.)41 b(As)30 b(a)h(result,)378 4773 y(the)23
b(lab)s(el)f Fv(L)h FT(denotes)h(the)f(exp)s(ected)h(statemen)m(t,)j
(but)c(all)f(results)g(deriv)m(ed)g(in)g(the)i(curren)m(t)f(con)m(text)
378 4886 y(using)i(it)g(will)f(ha)m(v)m(e)j(the)f(assumption)f
Fv(P)g(x)i FT(rather)f(than)g Ft(\000)p FT(.)39 b(The)26
b(justi\014ed)e(fact)j Fw(\()p Ft([])p Fw(,)p Ft(\000)36
b Fu(`)g(9)p Fv(x:P)27 b(x)p Fw(\))f FT(is)378 4999 y(then)32
b(used)f(to)i(replace)g(the)f(assumption)f Fv(P)25 b(x)33
b FT(with)e Ft(\000)h FT(when)f(suc)m(h)h(results)f(are)i(expanded.)46
b(This)378 5112 y(is)29 b(explained)g(in)g(detail)h(in)f(section)h
(4.3.5)i(b)s(elo)m(w.)378 5352 y FQ(Abbreviations)378
5524 y FT(An)g(abbreviation)f Fv(L)p Fw(:)13 b(")p Fv(a)44
b Fw(=)f Fv(t)p Fw(")32 b FT(in)m(tro)s(duces)f(the)h(v)-5
b(ariable)31 b Fv(a)h FT(as)h(an)f(abbreviating)f(v)-5
b(ariable.)45 b(The)378 5637 y(statemen)m(t)32 b(of)f(the)f
(abbreviation)f(is)h(in)m(tro)s(duced)f(as)h(an)g(assumption.)p
eop
%%Page: 73 83
73 82 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(73)378
396 y FQ(Declarations)35 b(of)g(Simpli\014ers)g(and)g(T)-9
b(rivial)35 b(F)-9 b(acts)378 568 y FT(The)32 b(declarations)h(of)g
(simpli\014ers)c(and)k(trivial)e(facts)j(are)f(simply)e(included)f(in)i
(the)h(top)g(la)m(y)m(er)h(of)378 681 y(the)d(en)m(vironmen)m(t)f(stac)
m(k.)378 924 y FG(4.3.5)112 b(Expanding)38 b(SPL)g(F)-9
b(acts)378 1096 y FT(When)28 b(an)g(en)m(vironmen)m(t)g(la)m(y)m(er)g
(is)f(closed,)i(theorems)f(are)h(expanded,)f(or)g(generalised,)g
(according)378 1209 y(to)f(the)g(assumptions)d(and)i(binding)e(v)-5
b(ariables)25 b(used)g(in)g(deriving)g(them.)39 b(The)26
b(expansion)f(pro)s(cess)378 1322 y(is)k(p)s(erformed)g(as)i(follo)m
(ws:)489 1510 y(1.)46 b(The)35 b(result)g(is)f(\014rst)h(expanded)g
(according)h(to)g(the)g(in)m(tro)s(duced)e(binding)e(v)-5
b(ariables.)55 b(The)605 1622 y(v)-5 b(ariables)26 b(are)i(considered)e
(in)g(the)i(rev)m(erse)g(order)f(they)g(are)h(in)m(tro)s(duced,)f(and)f
(the)i(e\013ect)h(of)605 1735 y(the)i(expansion)e(is)g(as)i(follo)m
(ws:)714 1923 y FN(\017)46 b FT(Expanding)30 b(according)i(to)g(a)g
(univ)m(ersal)e(v)-5 b(ariable,)31 b FP(u)h FT(sa)m(y)-8
b(,)34 b(in)m(v)m(olv)m(es)d(the)h(disc)m(harging)805
2036 y(of)42 b(all)f(the)h(assumptions)e(in)g(whic)m(h)h
FP(u)g FT(o)s(ccurs)h(freely)f(and)g(then)h(generalising)e(the)805
2149 y(result)30 b(if)f FP(u)h FT(is)g(free)g(in)f(its)h(conclusion.)
714 2295 y FN(\017)46 b FT(An)39 b(existen)m(tial)g(v)-5
b(ariable,)41 b FP(x)p FT(,)h(is)d(in)m(tro)s(duced)f(only)g(when)h
(some)h(existen)m(tial)f(result)805 2408 y Fw(\()p Ft([])p
Fw(,)p Ft(\000)d Fu(`)h(9)p Fv(x:P)26 b(x)p Fw(\))35
b FT(is)f(deriv)m(ed.)53 b(The)34 b(theorems)h(using)e(this)h(result)f
(will)f(ha)m(v)m(e)k(the)f(as-)805 2521 y(sumption)22
b Fv(P)k(x)e FT(rather)g(than)g Ft(\000)f FT(and)h(the)g(role)f(of)h
(the)g(expansion)f(pro)s(cess)g(is)g(to)i(replace)805
2634 y(the)31 b(assumption)e Fv(P)c(x)31 b FT(with)e
Ft(\000)p FT(.)805 2763 y(Expanding)23 b(with)h Fv(x)i
FT(pro)s(ceeds)e(b)m(y)h(disc)m(harging)f(all)g(h)m(yp)s(otheses,)j
(with)c(the)j(exception)805 2876 y(of)33 b Fv(P)25 b(x)q
FT(,)33 b(whic)m(h)e(con)m(tain)i(a)f(free)h(o)s(ccurrence)f(of)h
Fv(x)p FT(,)g(and)f(in)m(tro)s(ducing)e(the)i(existen)m(tial)805
2989 y(quan)m(ti\014er)26 b(if)f Fv(x)i FT(is)f(free)h(in)e(the)i
(conclusion)e(of)h(the)h(theorem.)40 b(The)26 b(assumption)f
Fv(P)h(x)h FT(is)805 3102 y(then)32 b(remo)m(v)m(ed)g(b)m(y)g
(eliminating)d(the)j(existen)m(tial)f(quan)m(ti\014er)g(from)g(the)h
(deriv)m(ed)f(fact)805 3215 y Fw(\()p Ft([])p Fw(,)p
Ft(\000)36 b Fu(`)h(9)p Fv(x:P)26 b(x)p Fw(\))p FT(.)57
b(F)-8 b(or)36 b(example,)h(if)d(after)j(disc)m(harging)d(the)i(relev)
-5 b(an)m(t)36 b(assumptions)805 3328 y(and)e(in)m(tro)s(ducing)d(the)j
(existen)m(tial)g(quan)m(ti\014er)f(the)h(statemen)m(t)i(of)e(the)g
(theorem)g(has)805 3441 y(b)s(een)c(expanded)f(to)2039
3554 y Ft(\001)q FP(;)15 b Fv(P)26 b(x)41 b FN(`)e Fu(9)p
Fv(x:Q)805 3720 y FT(the)31 b(h)m(yp)s(othesis)e Fv(P)c(x)31
b FT(is)f(then)g(replaced)g(with)f Ft(\000)h FT(b)m(y)1320
3912 y Ft(\000)37 b Fu(`)f(9)p Fv(x:P)27 b(x)91 b Ft(\001)p
Fv(;)14 b(P)26 b(x)37 b Fu(`)g(9)p Fv(x:Q)p 1320 3948
1050 4 v 1612 4022 a Ft(\000)p Fv(;)14 b Ft(\001)37 b
Fu(`)g(9)p Fv(x:Q)2411 3971 y Fw(CHOOSE)k(\()p Fv(x)p
Fw(,)p Ft(\000)36 b Fu(`)h(9)p Fv(x:P)26 b(x)p Fw(\))805
4227 y FT(where)49 b Fw(CHOOSE:)41 b(term)h Fu(\002)h
Fw(thm)f Fu(!)i Fw(thm)e Fu(!)i Fw(thm)k FT(is)g(the)h(follo)m(wing)f
(HOL)h(inference)805 4339 y(rule:)1295 4410 y Ft(\000)1347
4422 y Fs(1)1421 4410 y Fu(`)37 b(9)p Fv(x:s)91 b Ft(\000)1807
4422 y Fs(2)1844 4410 y Fv(;)14 b(s)p Fu(f)p Fv(x)37
b Fu(!)g Fv(v)s Fu(g)f(`)h Fv(t)p 1295 4451 1111 4 v
1665 4525 a Ft(\000)1717 4537 y Fs(1)1754 4525 y Fv(;)14
b Ft(\000)1843 4537 y Fs(2)1917 4525 y Fu(`)37 b Fv(t)2446
4473 y Fw(CHOOSE)42 b(\()p Fv(v)s Fw(,)h Ft(\000)2978
4485 y Fs(1)3052 4473 y Fu(`)36 b(9)p Fv(x:s)p Fw(\))714
4708 y FN(\017)46 b FT(If)24 b(a)h(lo)s(cal)f(abbreviation)f
Fv(a)43 b Fw(=)g Fv(t)25 b FT(is)e(in)m(tro)s(duced,)h(the)h(v)-5
b(ariable)23 b Fv(a)h FT(needs)g(to)i(b)s(e)d(replaced)805
4821 y(with)32 b(the)i(term)f(it)g(abbreviates)f(if)h(it)f(o)s(ccurs)h
(in)f(the)i(statemen)m(t)h(of)e(some)h(theorems)805 4934
y(when)20 b(the)h(curren)m(t)g(la)m(y)m(er)g(is)f(closed.)38
b(Basically)-8 b(,)22 b(this)e(substitution)f(\(in)h(b)s(oth)g(h)m(yp)s
(othe-)805 5047 y(ses)26 b(and)f(conclusion)f(of)i(the)g(theorems\))g
(is)f(done)h(using)e(the)i(assumption)e Fv(a)43 b Fw(=)h
Fv(t)p FT(.)39 b(This)805 5160 y(assumption)28 b(is)g(then)h(disc)m
(harged)f(from)h(the)g(h)m(yp)s(otheses)g(of)g(the)g(theorem,)h(the)f
(v)-5 b(ari-)805 5273 y(able)31 b Fv(a)h FT(is)e(generalised)h(and)f
(then)i(sp)s(ecialised)d(to)j(the)g(term)f Fv(t)p FT(,)h(whic)m(h)e
(results)h(in)f(the)805 5386 y(tautological)h(an)m(teceden)m(t)i
Fv(t)43 b Fw(=)g Fv(t)31 b FT(whic)m(h)e(can)i(b)s(e)e(easily)h
(eliminated.)489 5573 y(2.)46 b(An)m(y)28 b(lo)s(cal)f(assumptions)e
(whic)m(h)i(are)h(not)f(disc)m(harged)g(during)f(the)h(previous)f(step)
i(are)g(no)m(w)605 5686 y(disc)m(harged.)p eop
%%Page: 74 84
74 83 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(74)489
396 y(3.)46 b(The)22 b(result)f(is)h(then)g(univ)m(ersally)e(quan)m
(ti\014ed)h(with)g(an)m(y)h(t)m(yp)s(e)h(v)-5 b(ariables)21
b(in)m(tro)s(duced)g(lo)s(cally)-8 b(.)378 681 y FH(4.4)135
b(Pro)t(of)45 b(Supp)t(ort)378 884 y FT(The)31 b(follo)m(wing)f(kinds)f
(of)j(pro)s(of)e(pro)s(cedures)g(are)i(supp)s(orted)d(b)m(y)j(the)f
(SPL)g(language.)44 b(The)31 b(user)378 997 y(can)36
b(implemen)m(t)e(an)m(y)i(of)g(these)h(kinds)c(of)j(pro)s(of)f(pro)s
(cedures)g(in)f(ML)i(during)e(the)h(dev)m(elopmen)m(t)378
1110 y(of)30 b(a)h(theory)-8 b(,)31 b(asso)s(ciate)g(SPL)f(iden)m
(ti\014ers)e(with)h(them,)h(and)g(include)e(them)i(in)f(the)i(syn)m
(tax)f(of)h(the)378 1223 y(language.)378 1402 y FQ(Inference)k(Rules)46
b FT(whic)m(h)37 b(allo)m(w)g(the)i(user)e(to)i(deriv)m(e)f(facts)h(in)
d(a)j(pro)s(cedural)d(manner)h(using)605 1514 y(an)m(y)g(forw)m(ard)f
(inference)g(rule.)59 b(The)36 b(use)g(of)h(these)g(rules)f(is)f(not)i
(encouraged)h(b)s(ecause)e(it)605 1627 y(ma)m(y)31 b(reduce)f(the)h
(readabilit)m(y)e(of)h(pro)s(of)g(scripts.)378 1811 y
FQ(Simpli\014ers)45 b FT(whic)m(h)30 b(can)i(b)s(e)e(used)g(to)i
(normalise)e(terms,)i(and)e(to)i(p)s(erform)e(calculations)g(whic)m(h)
605 1924 y(w)m(ould)36 b(b)s(e)g(considered)g(trivial)f(in)h(an)h
(informal)e(pro)s(of.)59 b(An)m(y)38 b(HOL)e(con)m(v)m(ersions)h(can)h
(b)s(e)605 2037 y(included)28 b(b)m(y)i(the)h(user)e(as)i(SPL)e
(simpli\014ers.)378 2221 y FQ(Pro)s(of)36 b(Searc)m(h)f(Pro)s(cedures)
47 b FT(whic)m(h)19 b(are)j(used)d(to)j(deriv)m(e)e(the)h(conclusions)e
(of)i(straigh)m(tforw)m(ard)605 2334 y(justi\014cations.)37
b(The)23 b(follo)m(wing)e(pro)m(v)m(ers)j(are)g(used)e(to)i(supp)s(ort)
d(the)j(pro)s(of-c)m(hec)m(king)f(of)h(SPL)605 2447 y(scripts:)605
2654 y Fw(fol)44 b FT(the)36 b(tableau)f(calculus)f(for)h
(\014rst-order)f(logic)h(with)f(equalit)m(y)h(describ)s(ed)e(in)h(the)i
(next)805 2767 y(c)m(hapter.)605 2910 y Fw(cfol)44 b
FT(the)30 b Fw(fol)e FT(pro)m(v)m(er)i(mo)s(di\014ed)d(for)i(coloured)g
(\014rst-order)g(logic.)40 b(\(see)31 b(c)m(hapters)f(7)f(and)g(8,)805
3023 y(and)h(in)f(particular)g(section)h(8.5.\))605 3165
y Fw(taut)44 b FT(a)31 b(tautology)g(c)m(hec)m(k)m(er.)519
3372 y(The)c(SPL)g(implemen)m(tation)f(includes)g(a)i(kno)m(wledge)g
(database)g(whic)m(h)f(can)h(b)s(e)f(used)g(to)h(store)378
3485 y(facts)38 b(whic)m(h)d(are)j(considered)d(to)j(b)s(e)e(trivial.)
59 b(This)35 b(database)j(can)f(b)s(e)f(queried)g(b)m(y)g(an)m(y)i(of)f
(the)378 3598 y(ab)s(o)m(v)m(e)c(kinds)c(of)i(pro)s(of)g(pro)s(cedures)
f(in)g(order)h(to)h(obtain)f(trivial)e(facts)j(automatically)-8
b(.)44 b(The)31 b(use)378 3711 y(of)f(this)g(database)h(is)e(describ)s
(ed)g(in)g(the)h(next)h(section.)378 3953 y FG(4.4.1)112
b(A)37 b(Database)i(of)f(T)-9 b(rivial)35 b(Kno)m(wledge)378
4124 y FT(One)30 b(ma)5 b(jor)31 b(di\013erence)f(b)s(et)m(w)m(een)i
(formal)e(and)g(informal)f(pro)s(ofs)h(is)g(the)h(lev)m(el)f(of)h
(detail)f(b)s(et)m(w)m(een)378 4237 y(the)36 b(t)m(w)m(o.)57
b(Informal)33 b(pro)s(ofs)i(con)m(tain)h(gaps)f(in)f(their)h(reasoning)
f(whic)m(h)g(the)i(reader)f(is)g(required)378 4350 y(to)i(\014ll)d(in)g
(order)i(to)h(understand)d(the)i(pro)s(of.)57 b(The)35
b(author)h(of)g(an)g(informal)e(pro)s(of)h(usually)f(has)378
4463 y(a)42 b(sp)s(eci\014c)e(t)m(yp)s(e)i(of)g(reader)g(in)e(mind,)j
(one)e(who)h(has)f(a)h(certain)g(amoun)m(t)g(of)g(kno)m(wledge)f(in)f
(a)378 4576 y(n)m(um)m(b)s(er)d(of)i(mathematical)f(\014elds,)i(and)e
(one)g(who)g(has)g(read)h(and)f(understo)s(o)s(d)e(the)j(preceding)378
4689 y(sections)f(of)f(the)h(literature)f(con)m(taining)g(the)h(pro)s
(of.)61 b(The)37 b(author)h(can)g(therefore)g(rely)f(on)g(his,)378
4802 y(usually)i(justi\014ed,)i(assumptions)e(ab)s(out)i(what)g(the)g
(in)m(tended)e(reader)i(is)f(able)g(to)i(understand)378
4915 y(when)29 b(deciding)f(what)i(to)h(include)c(in)i(an)h(informal)e
(pro)s(of)h(and)g(what)h(can)h(b)s(e)e(easily)g(inferred)f(b)m(y)378
5028 y(the)e(reader,)h(and)e(can)h(\(or)g(m)m(ust\))g(therefore)g(b)s
(e)f(unjusti\014ed.)36 b(F)-8 b(or)27 b(example,)f(if)f(one)h(assumes)f
(that)378 5141 y(some)d(set)g FP(A)g FT(is)f(a)h(subset)f(of)h
FP(B)5 b FT(,)23 b(and)e(that)h(some)g(elemen)m(t)g FP(a)g
FT(is)f(a)h(mem)m(b)s(er)f(of)h FP(A)p FT(,)h(then)f(the)g(inference)
378 5254 y(whic)m(h)j(deriv)m(es)i(the)g(mem)m(b)s(ership)d(of)j
FP(a)g FT(in)e FP(B)32 b FT(can)27 b(usually)d(b)s(e)i(omitted)h(if)f
(the)h(reader)g(is)f(assumed)378 5366 y(to)h(b)s(e)e(familiar)e(with)i
(the)h(notions)f(of)h(set)h(mem)m(b)s(ership)c(and)i(con)m(tainmen)m
(t.)40 b(On)26 b(the)g(other)g(hand,)378 5479 y(the)h(case)h(studies)e
(describ)s(ed)e(in)i(c)m(hapter)i(3)f(sho)m(w)g(that)g(ev)m(en)h(when)e
(a)h(substan)m(tial)f(fragmen)m(t)h(of)h(a)378 5592 y(theory)33
b(has)g(b)s(een)f(dev)m(elop)s(ed,)h(formal)f(tactic)i(pro)s(ofs)e(ma)m
(y)h(still)e(con)m(tain)i(inferences)f(whic)m(h)f(use)378
5705 y(trivial)d(results)i(whic)m(h)f(ha)m(v)m(e)i(b)s(een)f(deriv)m
(ed)f(m)m(uc)m(h)i(earlier)e(in)g(the)i(mec)m(hanisation.)p
eop
%%Page: 75 85
75 84 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(75)519
396 y(Since)44 b(the)g(need)h(to)g(include)d(explicitly)g(suc)m(h)j
(trivial)d(inferences)i(in)f(most)i(formal)f(pro)s(of)378
509 y(systems)39 b(results)g(in)f(the)i(observ)m(ed)f(di\013erence)g(b)
s(et)m(w)m(een)h(the)g(size)f(and)g(readabilit)m(y)f(of)i(formal)378
622 y(and)33 b(informal)e(pro)s(ofs,)j(w)m(e)g(ha)m(v)m(e)h(exp)s
(erimen)m(ted)d(with)g(the)i(implemen)m(tation)e(of)i(a)g(simple)d
(user-)378 735 y(extensible)e(kno)m(wledge)h(database)i(whic)m(h)d(pro)
s(of)g(pro)s(cedures)g(can)i(query)f(to)h(deriv)m(e)f(trivial)e(facts)
378 848 y(automatically)-8 b(.)519 961 y(The)29 b(kno)m(wledge)h(in)f
(the)h(database)g(is)f(organised)h(in)m(to)f(categories)j(eac)m(h)f
(con)m(taining)e(a)h(list)f(of)378 1074 y(facts.)40 b(New)25
b(categories)h(can)f(b)s(e)g(added)f(during)e(the)j(dev)m(elopmen)m(t)h
(of)f(a)g(theory)-8 b(.)40 b(F)-8 b(or)25 b(example,)h(in)378
1187 y(order)i(to)i(deriv)m(e)e(the)h(trivial)e(inference)h
(illustrated)e(in)i(the)h(example)f(giv)m(en)h(earlier)f(this)f
(section,)378 1300 y(one)32 b(can)g(include)d(a)j(mem)m(b)s(ership)e
(category)j(with)e(iden)m(ti\014er)f Fw(in_set)f FT(in)h(order)h(to)i
(include)c(facts)378 1413 y(of)38 b(the)g(form)g Fv(x)44
b Fj(is)51 b(a)f(memb)l(er)j(of)62 b Fv(X)6 b FT(,)40
b(and)e(a)g(con)m(tainmen)m(t)h(category)h Fw(subset)c
FT(whic)m(h)g(includes)378 1526 y(facts)j(of)f(the)g(form)f
Fv(X)50 b Fj(is)h(a)f(subset)h(of)62 b Fv(Y)19 b FT(.)63
b(SPL)37 b(facts)h(can)h(then)e(b)s(e)g(stored)h(in)f(the)h(database)
378 1638 y(during)28 b(pro)s(of)i(implemen)m(tation)f(using)g(the)h
(construct:)473 1826 y FM(consider)46 b(in_set)g FP(a)i
FI(is)55 b(a)g(memb)-5 b(er)59 b(of)67 b FP(A)903 1939
y FM(subset)46 b FP(A)i FI(is)55 b(a)g(subset)i(of)67
b FP(B)52 b FM(;)378 2127 y FT(In)33 b(order)f(that)i(these)g(facts)g
(can)g(b)s(e)e(used)h(b)m(y)g(pro)s(of)g(pro)s(cedures,)g(the)g(user)g
(is)f(also)i(required)d(to)378 2240 y(implemen)m(t)c(ML)i(functions)e
(whic)m(h)h(query)g(the)g(database.)42 b(Suc)m(h)28 b(functions)f(tak)m
(e)j(the)f(kno)m(wledge)378 2352 y(database)h(as)f(an)g(argumen)m(t)g
(together)h(with)e(a)h(n)m(um)m(b)s(er)e(of)i(other)g(argumen)m(ts)h
(dep)s(ending)c(on)j(the)378 2465 y(category)38 b(they)d(query)-8
b(.)57 b(F)-8 b(or)36 b(example,)h(a)f(function)e(to)j(query)e(the)g
Fw(in_set)e FT(category)38 b(ma)m(y)e(tak)m(e)378 2578
y(a)g(pair)e(of)h(terms)h(represen)m(ting)e(an)h(elemen)m(t)h(and)f(a)h
(set.)56 b(Query)35 b(functions)e(return)i(a)g(theorem)378
2691 y(when)i(they)i(succeed.)65 b(ML)38 b(references)h(can)f(b)s(e)g
(used)f(to)i(store)g(the)g(searc)m(hing)f(routine)f(of)i(the)378
2804 y(query)29 b(function)f(so)h(that)h(it)f(can)g(b)s(e)g(up)s(dated)
f(during)f(the)i(dev)m(elopmen)m(t)h(of)f(a)h(theory)-8
b(,)30 b(as)f(sho)m(wn)378 2917 y(in)g(the)i(SML)f(fragmen)m(t)h(in)e
(\014gure)g(10.)519 3030 y(The)j(user)f(can)i(then)e(implemen)m(t)g
(pro)s(of)h(pro)s(cedures)e(\(suc)m(h)j(as)f(simpli\014ers\))d(whic)m
(h)i(call)g(this)378 3143 y(query)f(function.)519 3256
y(Query)c(functions)f(can)i(also)g(b)s(e)f(implemen)m(ted)f(to)i
(handle)e(existen)m(tial)h(queries.)39 b(F)-8 b(or)27
b(example)378 3369 y(an)32 b(existen)m(tial)f(query)h(function)f(for)g
(the)i Fw(subset)c FT(category)34 b(can)e(tak)m(e)i(a)e(set)h
Fv(X)38 b FT(as)33 b(an)e(argumen)m(t)378 3482 y(and)i(lo)s(oks)g(for)g
(a)h(fact)h(of)f(the)g(form)f Fv(X)50 b Fj(subset)h(of)62
b Fv(Y)52 b FT(for)33 b(some)h(set)g Fv(Y)19 b FT(.)51
b(A)33 b(di\013eren)m(t)g(existen)m(tial)378 3594 y(query)k(function)f
(on)i(the)f(same)h(category)i(w)m(ould)c(lo)s(ok)h(for)h(some)g(fact)g
Fv(Y)62 b Fj(subset)51 b(of)62 b Fv(X)7 b FT(.)62 b(Since)378
3707 y(man)m(y)28 b(suc)m(h)f(facts)h(ma)m(y)g(b)s(e)f(deriv)m(ed)g(b)m
(y)h(the)f(kno)m(wledge)h(database,)h(existen)m(tial)e(query)g
(functions)378 3820 y(are)k(implemen)m(ted)e(to)i(return)e(a)i(lazy)f
(sequence)h(of)f(facts)i(satisfying)d(the)h(query)-8
b(.)519 3933 y(Query)28 b(functions)f(can)i(b)s(e)g(up)s(dated)e(when)h
(new)g(results)g(are)h(deriv)m(ed)f(whic)m(h)f(can)i(b)s(e)f(used)g(in)
378 4046 y(the)j(automatic)g(deduction)e(of)i(trivial)d(facts.)42
b(F)-8 b(or)31 b(example,)f(giv)m(en)g(the)h(deriv)m(ed)e(fact)1012
4250 y FN(8)p FP(x;)15 b(X)r(;)g(Y)5 b(:)p FT(\()p FP(x)31
b FI(is)38 b(in)f FP(X)7 b FT(\))26 b FN(\))f FT(\()p
FP(X)39 b FI(subset)f(of)50 b FP(Y)20 b FT(\))26 b FN(\))f
FT(\()p FP(x)31 b FI(is)38 b(in)f FP(Y)20 b FT(\))378
4455 y(one)31 b(can)f(then)g(up)s(date)g(the)g Fw(in_set)e
FT(query)i(function)f(so)i(that)g(giv)m(en)f(some)h(query)f
Fv(a)43 b Fj(is)51 b(in)f Fv(B)34 b FT(it)489 4642 y(1.)46
b(calls)33 b(the)h(appropriate)f(existen)m(tial)h Fw(subset)d
FT(query)j(function)e(to)j(c)m(hec)m(k)g(whether)f(there)g(is)605
4755 y(some)d(set)g Fv(A)f FT(suc)m(h)h(that)g Fv(A)43
b Fj(subset)51 b(of)62 b Fv(B)35 b FT(can)30 b(b)s(e)g(deriv)m(ed)g
(from)f(the)i(database,)h(and)489 4943 y(2.)46 b(queries)29
b Fw(in_set)e FT(\(recursiv)m(ely\))i(to)i(c)m(hec)m(k)g(whether)e
Fv(a)44 b Fj(is)51 b(in)f Fv(A)30 b FT(for)f(some)h Fv(A)g
FT(satisfying)f(the)605 5056 y(previous)g(query)-8 b(.)519
5243 y(Giv)m(en)29 b(the)g(required)e(facts,)j(the)g(new)e
Fw(in_set)e FT(query)j(function)e(can)j(then)e(deriv)m(e)h(and)f
(return)378 5356 y(the)36 b(fact)h Fv(a)44 b Fj(is)50
b(in)h Fv(B)40 b FT(using)34 b(the)i(ab)s(o)m(v)m(e)i(result.)56
b(As)36 b(the)h(searc)m(h)f(function)f(is)g(stored)h(in)f(an)h(ML)378
5469 y(reference,)h(up)s(dating)d(a)h(query)g(function)f(a\013ects)j
(the)e(b)s(eha)m(viour)f(of)i(all)e(the)i(pro)s(of)e(pro)s(cedures)378
5582 y(whic)m(h)29 b(use)h(it.)p eop
%%Page: 76 86
76 85 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(76)p
378 1513 3453 4 v 376 4505 4 2992 v 515 1649 a Fw(fun)42
b(in_set_search)c(kdbs)k(\(e,)h(s\))g(=)646 1749 y Fj(lo)l(ok)53
b(for)h(the)c(fact)i Ft(``)p Fj(e)e(is)h(in)f(s)7 b Ft(")43
b Fj(in)50 b Fw(kdbs)646 1848 y Fj(and)i(r)l(eturn)d(it)j(if)62
b(found)9 b Fv(;)646 1948 y Fj(otherwise)51 b(r)l(aise)g(an)f(exc)l
(eption)515 2147 y Fw(local)602 2247 y(\(*)43 b Fj(stor)l(e)50
b(the)g(se)l(ar)l(ch)g(function)g(in)g(a)h(r)l(efer)l(enc)l(e)f
Fw(*\))602 2346 y(val)42 b(in_set_ref)e(=)j(ref)f(in_set_search)515
2446 y(in)602 2645 y(\(*)h Fj(the)50 b(query)h(c)l(al)t(ls)g(the)g
(stor)l(e)l(d)h(se)l(ar)l(ch)e(function)6 b Fv(:)45 b
Fw(*\))602 2745 y(fun)d(in_set)g(kdbs)f(query)h(=)h(\(!in_set_ref\))38
b(kdbs)k(query)602 2944 y(\(*)h Fj(up)l(dating)51 b(the)f(query)i
(function)e Fw(*\))602 3044 y(fun)42 b(update_in_set)d(new_qf)i(=)689
3143 y(let)i(val)f(old_in_set)d(=)44 b(!in_set_ref)864
3243 y(fun)e(new_in_set)d(kdbs)j(query)g(=)1125 3343
y(old_in_set)e(kdbs)i(query)f(\(*)i Fj(try)51 b(the)f(old)j(query)7
b Fv(:)44 b Fw(*\))1125 3442 y(handle)d(_)i(=>)479 b(\(*)43
b Fj(if)62 b(it)51 b(fails)h Fw(*\))1212 3542 y(new_qf)41
b(kdbs)h(query)129 b(\(*)h Fj(try)51 b(the)f(new)i(one)6
b Fv(:)45 b Fw(*\))733 3642 y(in)e(in_set_ref)c(:=)k(new_in_set)170
b(\(*)43 b Fj(up)l(date)50 b(the)g(stor)l(e)g(function)6
b Fv(:)44 b Fw(*\))689 3741 y(end)515 3940 y(end;)1026
4336 y FT(Figure)30 b(10:)41 b(The)30 b(Implemen)m(tation)g(of)g(a)h
(Query)e(F)-8 b(unction.)p 3829 4505 V 378 4509 3453
4 v eop
%%Page: 77 87
77 86 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(77)519
396 y(Since)38 b(some)i(searc)m(h)g(is)e(needed)h(in)f(the)h(handling)e
(of)i(most)g(queries,)i(and)e(since)f(the)h(same)378
509 y(query)26 b(ma)m(y)h(b)s(e)e(made)i(sev)m(eral)g(times)f(during)e
(theorem)i(pro)m(ving,)h(the)g(output)f(of)g(successful)f(non-)378
622 y(existen)m(tial)36 b(queries)g(is)g(cac)m(hed)i(to)f(a)m(v)m(oid)g
(rep)s(eated)g(searc)m(h.)61 b(In)36 b(the)h(curren)m(t)f(implemen)m
(tation)378 735 y(cac)m(hes)f(are)g(stored)f(globally)e(and)h(are)i
(reset)f(when)f(a)h(la)m(y)m(er)h(con)m(taining)e(kno)m(wledge)h(whic)m
(h)f(can)378 848 y(a\013ect)28 b(the)g(query)e(concerned)h(is)f
(closed.)39 b(A)27 b(b)s(etter)g(approac)m(h)h(w)m(ould)d(b)s(e)i(to)g
(store)h(cac)m(hes)g(lo)s(cally)378 961 y(in)h(eac)m(h)j(la)m(y)m(er.)
519 1074 y(Case)44 b(studies)e(in)m(v)m(olving)g(the)i(implemen)m
(tation)e(of)h(formal)g(pro)s(ofs)g(in)f(SPL)h(sho)m(w)m(ed)g(that)378
1187 y(the)36 b(length)e(of)i(the)g(pro)s(ofs)e(can)i(b)s(e)f(substan)m
(tially)e(reduced)i(through)g(the)g(use)g(of)h(a)g(kno)m(wledge)378
1300 y(database.)k(This)24 b(reduction)h(of)h(pro)s(of)f(length)h(is)f
(due)g(to)i(the)f(implemen)m(tation)e(of)i(theory-sp)s(eci\014c)378
1413 y(query)32 b(functions)f(whic)m(h)g(mak)m(e)i(use)g(of)f(deriv)m
(ed)g(theorems,)h(as)g(w)m(ell)e(as)i(the)g(implemen)m(tation)e(of)378
1526 y(pro)s(of)g(pro)s(cedures)g(whic)m(h)f(are)j(able)e(to)i(query)e
(the)h(database.)47 b(W)-8 b(e)33 b(notice)f(that)h(the)f(implemen-)378
1638 y(tation)e(of)g(suc)m(h)g(functions)f(with)g(the)h(in)m(ten)m
(tion)f(of)h(minimising)c(the)k(di\013erence)g(b)s(et)m(w)m(een)h
(formal)378 1751 y(and)d(informal)f(pro)s(ofs)g(in)m(v)m(olv)m(es)i
(the)g(understanding)d(of)i(what)h(authors)f(of)h(informal)e(pro)s(ofs)
g(con-)378 1864 y(sider)32 b(to)j(b)s(e)e(trivial)g(b)m(y)g(the)h(in)m
(tended)f(reader.)52 b(Therefore,)35 b(the)f(implemen)m(tation)e(of)i
(functions)378 1977 y(capable)26 b(of)h(deriving)e(facts)i(whic)m(h)e
(are)i(considered)f(to)h(b)s(e)f(trivial)f(b)m(y)h(a)h(kno)m
(wledgeable)f(reader)h(is)378 2090 y(a)j(formal)f(means)g(of)h
(illustrating)c(what)k(can)f(b)s(e)g(considered)g(ob)m(vious)g(in)f
(some)i(particular)e(pro)s(of)378 2203 y(and)36 b(ho)m(w)g(suc)m(h)g
(ob)m(vious)g(facts)h(can)g(b)s(e)e(deriv)m(ed.)58 b(W)-8
b(e)38 b(argue)e(that)h(this)f(is)f(a)i(formal)e(means)i(of)378
2316 y(represen)m(ting)30 b(a)g(particular)f(kind)g(of)h(kno)m(wledge)h
(and)f(understanding)e(in)h(a)i(mathematical)f(\014eld)378
2429 y(other)k(than)g(giving)f(a)h(list)f(of)h(detailed)f(formal)h(pro)
s(ofs.)50 b(W)-8 b(e)36 b(b)s(eliev)m(e)d(that)h(the)h(presen)m(tation)
f(of)378 2542 y(suc)m(h)c(information)f(should)f(b)s(e)i(included)d(in)
j(a)g(formal)g(dev)m(elopmen)m(t)h(of)f(a)h(mathematical)g(\014eld.)519
2655 y(In)f(our)h(case)h(study)-8 b(,)30 b(the)i(only)e(pro)s(of)g(pro)
s(cedures)f(whic)m(h)h(use)h(the)g(kno)m(wledge)g(database)g(are)378
2768 y(the)j(simplifying)c(pro)s(cedures.)50 b(The)33
b(main)g(reason)h(for)g(this)f(is)g(the)h(fact)g(that)h(the)f(pro)s(of)
f(searc)m(h)378 2880 y(pro)s(cedures)23 b(w)m(ere)i(implemen)m(ted)e(b)
s(efore)h(the)h(exp)s(erimen)m(tal)f(database)h(w)m(as)g(designed.)38
b(Ho)m(w)m(ev)m(er,)378 2993 y(in)22 b(principle)f(the)j(pro)s(of)f
(pro)s(cedures)f(can)i(b)s(e)f(redesigned)g(and)g(implemen)m(ted)f(to)j
(b)s(e)e(able)g(to)h(query)378 3106 y(the)31 b(database.)43
b(W)-8 b(e)32 b(will)c(consider)i(this)g(area)h(for)g(future)f(w)m(ork)
h(and)f(b)s(eliev)m(e)g(that)h(the)g(length)g(of)378
3219 y(formal)f(pro)s(ofs)f(can)i(b)s(e)e(greatly)i(reduced)f(with)f
(suc)m(h)h(a)h(feature.)378 3506 y FH(4.5)135 b(Conclusions)378
3709 y FT(In)22 b(this)g(c)m(hapter)i(w)m(e)f(ha)m(v)m(e)i(illustrated)
20 b(the)k(implemen)m(tation)d(of)j(an)f(extensible)e(pro)s(of)i
(language)g(in)378 3821 y(the)f(HOL)g(system.)38 b(The)22
b(language)h(supp)s(orts)d(a)j(declarativ)m(e)f(st)m(yle)h(of)f(pro)s
(of)g(implemen)m(tation)e(and)378 3934 y(is)k(v)m(ery)i(similar)d(to)j
(the)g(Mizar)f(language)h(although)f(the)h(t)m(w)m(o)h(languages)e
(di\013er)g(in)f(man)m(y)h(asp)s(ects.)378 4047 y(In)20
b(particular)g(the)h(pro)s(of-c)m(hec)m(king)h(p)s(o)m(w)m(er)f(of)g
(the)h(SPL)e(pro)s(of)g(language)i(can)f(b)s(e)g(extended)g(during)378
4160 y(the)h(dev)m(elopmen)m(t)h(of)f(a)h(theory)f(b)m(y)g(implemen)m
(ting)e(pro)s(of)i(pro)s(cedures)f(whic)m(h)g(mak)m(e)i(use)f(of)g
(results)378 4273 y(deriv)m(ed)27 b(in)h(earlier)f(sections)h(of)h(the)
g(theory)-8 b(.)40 b(W)-8 b(e)30 b(ha)m(v)m(e)g(argued)e(in)f(section)i
(2.5.3)h(\(page)g(25\))f(that)378 4386 y(suc)m(h)36 b(extensibilit)m(y)
d(of)k(a)f(pro)s(of)f(language)i(is)e(necessary)h(for)g(the)g(implemen)
m(tation)f(of)h(mac)m(hine)378 4499 y(c)m(hec)m(k)-5
b(able)24 b(pro)s(ofs)f(whic)m(h)f(can)i(also)f(b)s(e)g(follo)m(w)m(ed)
g(b)m(y)g(a)h(h)m(uman)e(reader.)39 b(During)22 b(the)h(dev)m(elopmen)m
(t)378 4612 y(of)30 b(a)h(particular)e(theory)-8 b(,)31
b(the)g(user)e(can)i(extend:)514 4799 y FN(\017)46 b
FT(the)31 b(pro)s(of)e(pro)s(cedures)h(used)f(to)i(justify)e(the)i(pro)
s(of)e(statemen)m(ts,)514 4987 y FN(\017)46 b FT(the)31
b(simpli\014ers)26 b(whic)m(h)j(normalise)g(terms)i(in)m(to)f
(canonical)g(forms;)514 5175 y FN(\017)46 b FT(the)35
b(inference)f(rules)g(used)g(to)h(deriv)m(e)g(facts)g(in)f(a)h(forw)m
(ard)f(manner)g(\(although)h(it)f(is)g(sug-)605 5288
y(gested)45 b(that)g(the)g(frequen)m(t)f(use)g(of)h(suc)m(h)f(rules)e
(should)h(b)s(e)g(a)m(v)m(oided)i(b)s(ecause)f(of)h(their)605
5401 y(pro)s(cedural)29 b(nature\);)h(and,)514 5588 y
FN(\017)46 b FT(the)22 b(kno)m(wledge)f(database)h(b)m(y)f(adding)f
(new)h(kno)m(wledge)g(categories,)k(and)c(b)m(y)g(implemen)m(ting)605
5701 y(and)30 b(up)s(dating)e(appropriate)i(query)g(functions.)p
eop
%%Page: 78 88
78 87 bop 378 5 a FF(CHAPTER)30 b(4.)122 b(A)30 b(DECLARA)-8
b(TIVE)30 b(PR)m(OOF)h(LANGUA)m(GE)h(IN)e(HOL)625 b FT(78)519
396 y(The)40 b(user)f(can)i(also)f(extend)g(the)g(syn)m(tax)h(and)e
(seman)m(tics)i(of)f(the)g(language)h(b)m(y)f(up)s(dating)378
509 y(or)45 b(mo)s(difying)d(the)j(language)g(parser)g(and)f(pro)s
(cessor.)84 b(Ho)m(w)m(ev)m(er,)50 b(the)45 b(author)g(has)g(not)g(y)m
(et)378 622 y(exp)s(erimen)m(ted)35 b(with)g(extensiv)m(e)i(case)g
(studies)e(on)h(using)e(suc)m(h)i(a)h(feature,)h(although)e(its)f(use)h
(in)378 735 y(the)31 b(mec)m(hanisation)e(of)i(mathematics)g(seems)f
(to)i(b)s(e)d(adv)-5 b(an)m(tageous.)519 848 y(ML)38
b(references)g(are)g(used)f(in)f(order)h(to)i(store)f(the)g(functions)e
(whic)m(h)g(ma)m(y)j(b)s(e)e(up)s(dated)f(b)m(y)378 961
y(the)h(theory)f(dev)m(elop)s(er.)59 b(It)37 b(is)e(desirable)g(that)i
(the)g(ab)s(o)m(v)m(e-men)m(tioned)g(extensions)f(b)s(e)g(lo)s(cal)g
(to)378 1074 y(particular)29 b(theories,)h(or)h(to)g(theory)g
(sections,)g(and)f(this)f(requires)h(a)g(n)m(um)m(b)s(er)g(of)g(design)
g(c)m(hanges)378 1187 y(to)h(the)g(curren)m(t)f(implemen)m(tation.)519
1300 y(A)42 b(sectioning)e(mec)m(hanism)h(is)g(used)f(to)i(structure)f
(theories)h(in)e(a)i(mo)s(dular)d(fashion.)73 b(As-)378
1413 y(sumptions)25 b(and)h(other)h(information)e(can)i(b)s(e)f
(declared)h(lo)s(cal)f(to)h(certain)g(sections)g(and,)g(with)f(the)378
1526 y(exception)31 b(of)f(pro)m(v)m(ed)h(theorems,)g(lo)s(cal)e
(information)g(is)g(not)i(visible)d(in)h(di\013eren)m(t)h(con)m(texts.)
519 1638 y(W)-8 b(e)34 b(strongly)e(b)s(eliev)m(e)g(in)f(the)i
(necessit)m(y)g(of)g(the)g(extensibilit)m(y)d(of)j(the)g(language)g
(since,)g(sim-)378 1751 y(ilarly)c(to)j(informal)d(mathematics,)k
(formal)d(mathematical)i(texts)g(should)e(not)h(include)e(only)i(the)
378 1864 y(implemen)m(tation)36 b(of)i(pro)s(ofs.)62
b(Informal)36 b(mathematics)i(also)g(includes,)g(amongst)g(other)g
(things)378 1977 y(suc)m(h)21 b(as)h(examples)f(and)g(coun)m
(terexamples,)j(tec)m(hniques)e(for)f(\014nding)e(the)j(normal)f(forms)
g(of)h(terms,)378 2090 y(algorithms)31 b(for)g(sp)s(eci\014c)g
(calculations,)h(rules)e(of)i(th)m(um)m(bs)f(for)h(\014nding)d(the)j
(pro)s(ofs)f(of)h(theorems,)378 2203 y(etc.)17 b(A)33
b(formal)g(w)m(a)m(y)h(of)f(presen)m(ting)g(these)h(is)e(b)m(y)h
(implemen)m(ting)e(the)j(appropriate)e(pro)s(of)h(pro)s(ce-)378
2316 y(dures,)39 b(whic)m(h)e(also)h(results)f(in)g(reducing)g(the)h
(length)g(of)g(formal)f(pro)s(ofs.)64 b(If)37 b(suc)m(h)h(pro)s
(cedures)378 2429 y(are)30 b(used)f(to)i(minimise)c(the)j(di\013erence)
g(b)s(et)m(w)m(een)g(formal)f(and)h(informal)e(pro)s(ofs,)h(then)h
(they)g(also)378 2542 y(con)m(tribute)g(to)h(the)g(comprehensibilit)m
(y)c(of)j(formal)g(mathematical)h(texts.)p eop
%%Page: 79 89
79 88 bop 378 1019 a FJ(Chapter)65 b(5)378 1434 y FR(A)77
b(T)-19 b(ableau)76 b(Pro)-6 b(v)g(er)76 b(as)h(a)h(HOL)378
1683 y(Deriv)-6 b(ed)76 b(Rule)378 2165 y FH(5.1)135
b(In)l(tro)t(duction)378 2368 y FT(In)23 b(the)i(previous)d(c)m(hapter)
j(w)m(e)g(illustrated)d(the)i(simple)e(pro)s(of)i(language)h(SPL)e(and)
g(the)i(implemen-)378 2481 y(tation)k(of)h(a)f(pro)s(of)g(c)m(hec)m(k)m
(er)i(for)e(this)f(language)h(in)f(the)i(HOL)e(pro)s(of)h(dev)m
(elopmen)m(t)g(system.)41 b(This)378 2594 y(pro)s(of)26
b(c)m(hec)m(k)m(er)k(deriv)m(es)c(HOL)h(theorems)g(from)g(SPL)f(facts)i
(and)f(it)g(is)f(supp)s(orted)f(b)m(y)i(a)h(n)m(um)m(b)s(er)e(of)378
2706 y(user-de\014ned)g(and)h(in)m(built)e(pro)s(of)i(pro)s(cedures.)39
b(In)27 b(particular,)f(a)i(tableau)g(pro)m(v)m(er)g(for)g
(\014rst-order)378 2819 y(logic)38 b(with)f(equalit)m(y)g(is)g(used)h
(to)g(c)m(hec)m(k)i(most)f(of)f(the)g(straigh)m(tforw)m(ard)g
(justi\014cations)f(of)h(SPL)378 2932 y(results.)52 b(This)33
b(pro)m(v)m(er)i(is)f(implemen)m(ted)f(as)i(a)g(deriv)m(ed)f(rule)f(in)
h(HOL,)g(and)g(in)g(this)f(c)m(hapter)i(w)m(e)378 3045
y(illustrate)29 b(the)h(pro)s(of)g(calculus)f(used)g(and)h(its)g
(implemen)m(tation.)519 3158 y(The)35 b(design)f(of)h(pro)s(of)f
(calculi)f(for)i(the)h(automated)g(deduction)e(of)h(theorems)g(in)f
(\014rst-order)378 3271 y(logic)g(with)f(equalit)m(y)-8
b(,)36 b(and)d(the)i(implemen)m(tation)e(of)h(pro)s(of)g(pro)s(cedures)
f(based)h(on)g(suc)m(h)g(calculi)378 3384 y(is)28 b(in)g(general)h(not)
g(a)g(trivial)e(task)j(b)s(ecause)f(of)g(the)g(man)m(y)g(w)m(a)m(ys)h
(equations)f(can)g(b)s(e)f(used)h(to)g(infer)378 3497
y(results.)64 b(In)38 b(particular,)h(the)g(handling)d(of)i(equalit)m
(y)g(in)g(tableau-based)g(calculi)f(needs)h(sp)s(ecial)378
3610 y(atten)m(tion)30 b(since)f(the)g(problem)f(of)h(deciding)f
(whether)h(a)g(tableau)g(can)h(b)s(e)f(closed)g(b)m(y)g(considering)378
3723 y(only)h(its)g(literals)f(is)h(undecidable)e(\(V)-8
b(o)s(da)32 b(and)e(Komara)h(1995\).)44 b(The)30 b(calculus)f(implemen)
m(ted)h(as)378 3836 y(a)i(HOL)g(deriv)m(ed)f(rule)g(is)g(based)g(on)h
(the)g FN(T)23 b(B)s(S)7 b(E)39 b FT(calculus)31 b(of)h(Degt)m(y)m
(arev)j(and)d(V)-8 b(oronk)m(o)m(v)33 b(\(1998\))378
3949 y(whic)m(h)22 b(giv)m(es)i(a)g(complete)g(semi-decision)e(pro)s
(cedure)h(for)g(\014rst-order)g(logic)g(with)f(equalit)m(y)i(despite)
378 4061 y(this)29 b(problem.)519 4174 y(In)24 b(order)h(to)h(guaran)m
(tee)g(the)g(correctness)g(of)f(the)g(theorems)h(deriv)m(ed)e(in)f(the)
j(HOL)e(system,)j(all)378 4287 y(HOL)33 b(inferences)f(are)i(p)s
(erformed)d(b)m(y)j(a)f(simple)e(core)j(inference)f(engine.)49
b(The)32 b(implemen)m(tation)378 4400 y(of)26 b(the)h(tableau)f
(calculus)f(as)i(a)f(HOL)g(deriv)m(ed)g(rule)f(therefore)i(requires)d
(the)j(use)f(of)g(this)g(inference)378 4513 y(engine)34
b(in)g(deriving)e(the)j(required)e(theorem.)54 b(F)-8
b(or)36 b(e\016ciency)e(reasons)h(the)g(pro)s(of)f(searc)m(h)h(stage)
378 4626 y(of)41 b(the)h(algorithm)e(do)s(es)h(not)h(use)e(the)i(HOL)f
(represen)m(tation)g(for)g(terms)g(and)g(theorems,)j(and)378
4739 y(only)32 b(when)f(a)i(closed)g(tableau)g(is)e(found)h(is)f(the)i
(core)h(inference)e(engine)g(used)g(to)h(deriv)m(e)f(a)h(HOL)378
4852 y(theorem.)519 4965 y(The)27 b(de\014nition)f(of)i(the)f(calculus)
g(is)g(giv)m(en)g(in)g(the)h(next)f(section,)i(and)e(its)g(implemen)m
(tation)g(of)378 5078 y(the)e(HOL)g(deriv)m(ed)f(rule)g(is)g(describ)s
(ed)f(in)h(section)i(5.3.)40 b(Since)24 b(the)h(deriv)m(ed)g(rule)e
(can)j(only)e(b)s(e)h(used)378 5191 y(to)33 b(reason)g(with)e
(\014rst-order)g(form)m(ulae,)i(a)g(mec)m(hanism)f(for)g(translating)f
(higher-order)g(form)m(ulae)378 5303 y(in)m(to)42 b(equiv)-5
b(alen)m(t)41 b(\014rst-order)h(ones)g(is)f(describ)s(ed)f(in)h
(section)h(5.4.)77 b(A)42 b(n)m(um)m(b)s(er)f(of)h(concluding)378
5416 y(remarks)30 b(and)g(directions)f(for)h(future)f(w)m(ork)i(are)f
(giv)m(en)h(in)e(section)h(5.5.)519 5529 y(In)36 b(this)f(c)m(hapter)i
(w)m(e)g(use)f(the)g(notation)h FP(s)e FN(\031)f FP(t)j
FT(to)g(am)m(biguously)d(represen)m(t)j(the)f(equations)378
5642 y FP(x)25 b FT(=)g FP(y)33 b FT(and)d FP(y)e FT(=)d
FP(x)p FT(.)41 b(Similarly)-8 b(,)27 b(w)m(e)k(use)f
FP(x)25 b FN(6\031)g FP(y)33 b FT(for)d(b)s(oth)g FN(:)p
FT(\()p FP(x)25 b FT(=)g FP(y)s FT(\))31 b(and)e FN(:)p
FT(\()p FP(y)f FT(=)d FP(x)p FT(\).)2057 5954 y(79)p
eop
%%Page: 80 90
80 89 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(80)378 396 y FH(5.2)135 b(A)45 b(Clausal)h(T)-11
b(ableau)45 b(with)g(Rigid)h(Basic)f(Sup)t(erp)t(osition)378
599 y FT(The)40 b(calculus)f(describ)s(ed)f(here)i(refutes)g(a)h(list)e
(of)h(clauses)g(\(sk)m(olemised)g(\014rst-order)g(sen)m(tences)378
712 y(in)g(conjunctiv)m(e)h(normal)f(form\))h(b)m(y)g(lo)s(oking)f(for)
h(a)h(closed)e(tableau)h(\(i.e.,)16 b(a)42 b(tableau)f(whic)m(h)f(is)
378 825 y(sho)m(wn)26 b(to)i(represen)m(t)f(an)g(unsatis\014able)e
(form)m(ula\).)39 b(The)26 b(reader)h(unfamiliar)d(with)h(the)i
(notions)g(of)378 938 y(seman)m(tic)j(tableaux)f(and)f(tableau-based)i
(calculi)d(is)i(referred)f(to)i(app)s(endix)d(B)j(whic)m(h)e
(illustrates)378 1051 y(the)j(use)f(of)g(tableaux)g(in)f(refuting)h
(sen)m(tences)h(in)e(\014rst-order)h(logic)g(with)f(or)h(without)g
(equalit)m(y)-8 b(.)519 1164 y(In)37 b(this)g(section)h(w)m(e)g
(\014rst)f(giv)m(e)i(a)f(brief)e(discussion)g(on)h(clausal)g(tableaux)h
(and)f(on)h(the)g(use)378 1277 y(of)45 b(tableaux)f(in)g(reasoning)g
(in)f(\014rst-order)h(logic)h(with)e(equalit)m(y)-8 b(.)84
b(In)44 b(section)h(5.2.2)h(w)m(e)f(giv)m(e)378 1390
y(the)g(de\014nition)d(of)j(the)g(calculus)e(whic)m(h)h(is)f(implemen)m
(ted)h(as)h(a)g(HOL)f(deriv)m(ed)g(rule,)j(and)d(in)378
1503 y(section)30 b(5.2.3)j(w)m(e)d(illustrate)f(it)h(with)f(the)i
(help)e(of)h(some)h(examples.)378 1746 y FG(5.2.1)112
b(On)38 b(Clausal)f(T)-9 b(ableaux)38 b(and)h(Rigid)c(Basic)i(Sup)s
(erp)s(osition)378 1918 y FT(W)-8 b(e)29 b(use)f(the)h(m)m(ultiset)e
(notation)i(for)f(represen)m(ting)g(tableaux:)39 b(A)28
b(tableau)h(is)e(a)i(m)m(ultiset)e(of)i(op)s(en)378 2031
y(branc)m(hes,)h(and)g(a)h(branc)m(h)f(is)f(a)i(m)m(ultiset)e(of)i
(form)m(ulae.)40 b(The)30 b(tableau)1271 2235 y FN(f)15
b(f)p FP(L)1438 2249 y FL(11)1514 2235 y FP(;)g(:)g(:)g(:)32
b(;)15 b(L)1793 2249 y FL(1)p FO(n)1871 2258 y FC(1)1910
2235 y FN(g)p FP(;)g(:)g(:)g(:)32 b(;)15 b FN(f)p FP(L)2279
2249 y FO(m)p FL(1)2382 2235 y FP(;)g(:)g(:)g(:)31 b(;)15
b(L)2660 2249 y FO(mn)2765 2257 y Fy(m)2829 2235 y FN(g)g(g)378
2439 y FT(is)29 b(denoted)i(b)m(y)1395 2552 y FP(L)1457
2566 y FL(11)1532 2552 y FP(;)15 b(:)g(:)g(:)31 b(;)15
b(L)1810 2566 y FL(1)p FO(n)1888 2575 y FC(1)1943 2552
y FN(j)30 b(\001)15 b(\001)g(\001)32 b(j)15 b FP(L)2237
2566 y FO(m)p FL(1)2339 2552 y FP(;)g(:)g(:)g(:)31 b(;)15
b(L)2617 2566 y FO(mn)2722 2574 y Fy(m)2786 2552 y FP(:)519
2719 y FT(A)31 b(branc)m(h)f FP(B)g FT(=)c FN(f)p FP(L)1222
2733 y FL(1)1262 2719 y FP(;)15 b(:)g(:)g(:)31 b(;)15
b(L)1540 2733 y FO(n)1588 2719 y FN(g)31 b FT(is)f(refutable)f(if)h
(the)h(sen)m(tence)h FN(8)o FP(~)-44 b(x)o(:)p FT(\()p
FP(L)2964 2733 y FL(1)3025 2719 y FN(^)20 b(\001)15 b(\001)g(\001)21
b(^)f FP(L)3375 2733 y FO(n)3422 2719 y FT(\))31 b(is)f(unsat-)378
2832 y(is\014able,)f(where)f FP(~)-43 b(x)30 b FT(represen)m(ts)g(the)h
(list)e(of)h(v)-5 b(ariables)29 b(free)i(in)e FP(B)5
b FT(.)40 b(A)31 b(tableau)1314 3036 y FP(T)38 b FT(=)25
b FP(L)1563 3050 y FL(11)1638 3036 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(L)1916 3050 y FL(1)p FO(n)1994 3059 y FC(1)2049
3036 y FN(j)30 b(\001)15 b(\001)g(\001)32 b(j)15 b FP(L)2343
3050 y FO(m)p FL(1)2445 3036 y FP(;)g(:)g(:)g(:)32 b(;)15
b(L)2724 3050 y FO(mn)2829 3058 y Fy(m)378 3240 y FT(is)29
b(refutable)h(if)1154 3353 y FN(8)o FP(~)-44 b(y)r(:)p
FT(\()p FP(L)1374 3367 y FL(11)1470 3353 y FN(^)19 b(\001)c(\001)g
(\001)22 b(^)d FP(L)1819 3367 y FL(1)p FO(n)1897 3376
y FC(1)1936 3353 y FT(\))i FN(_)f(\001)15 b(\001)g(\001)21
b(_)f FT(\()p FP(L)2377 3367 y FO(m)p FL(1)2499 3353
y FN(^)g(\001)15 b(\001)g(\001)21 b(^)f FP(L)2849 3367
y FO(mn)2954 3375 y Fy(m)3017 3353 y FT(\))378 3520 y(is)29
b(unsatis\014able,)g(where)g FP(~)-44 b(y)33 b FT(is)c(the)i(list)e(of)
h(v)-5 b(ariables)29 b(free)i(in)e FP(T)13 b FT(.)519
3633 y(An)43 b(adv)-5 b(an)m(tage)46 b(of)e(refuting)f(a)h(set)g(of)g
(clauses)f(o)m(v)m(er)i(general)f(form)m(ulae)f(is)g(that)i(one)f(can)
378 3746 y(restrict)h(the)g(application)f(of)h(the)g(tableau)g
(expansion)f(rules)g(to)h(those)h(whic)m(h)e(result)g(in)g(the)378
3859 y(immediate)21 b(closure)h(of)g(a)h(branc)m(h)e(without)h
(a\013ecting)h(the)f(completeness)g(of)h(the)f(calculus)f(for)h(pure)
378 3972 y(\014rst-order)32 b(logic.)46 b(Because)34
b(of)f(this)e(restriction,)i(suc)m(h)f FI(c)-5 b(onne)g(ction)41
b FT(tableau)32 b(calculi)f(\(see)j(\(Letz)378 4085 y(1993\)\),)50
b(whic)m(h)43 b(include)f(mo)s(del)h(elimination)e(based)j(metho)s(ds)g
(\(Lo)m(v)m(eland)g(1968\),)50 b(are)45 b(m)m(uc)m(h)378
4198 y(more)32 b(e\016cien)m(t)g(than)g(non-clausal)f(tableau)h
(calculi.)43 b(Unfortunately)-8 b(,)33 b(tableau)e(calculi)g(for)g
(\014rst-)378 4310 y(order)26 b(logic)h(with)e(equalit)m(y)i(cannot)g
(b)s(e)f(restricted)h(to)h(tableaux)e(with)g(this)g(connection)h(prop)s
(ert)m(y)378 4423 y(without)i(losing)g(their)h(completeness.)519
4536 y(Reasoning)41 b(in)e(\014rst-order)i(logic)f(with)g(equalit)m(y)h
(is)f(not)h(straigh)m(tforw)m(ard)g(b)s(ecause)g(of)g(the)378
4649 y(man)m(y)34 b(w)m(a)m(ys)g(an)g(equation)f(can)h(b)s(e)f(used)g
(\(e.g.,)17 b(an)33 b(equation)h FP(a)c FT(=)h FP(b)i
FT(can)h(b)s(e)f(used)g(to)h(infer)e FP(P)13 b FT([)p
FP(b)p FT(])378 4762 y(from)36 b FP(P)13 b FT([)p FP(a)p
FT(])38 b(and)e FP(Q)p FT([)p FP(a)p FT(])h(from)f FP(Q)p
FT([)p FP(b)p FT(],)j(and)d(it)h(is)f(tautological)h(if)f
FP(a)g FT(and)h FP(b)f FT(are)i(the)f(same\).)60 b(If)37
b(one)378 4875 y(do)s(es)44 b(not)h(tak)m(e)h(sp)s(ecial)d(care,)49
b(the)c(pro)s(of)e(searc)m(h)i(can)g(easily)f(b)s(ecome)h(in)m
(tractable)f(ev)m(en)h(for)378 4988 y(trivial)28 b(problems.)38
b(In)29 b(the)h(case)g(of)g(tableau)f(calculi,)g(the)g(problem)f(of)i
(whether)f(the)g(literals)f(in)g(a)378 5101 y(branc)m(h)d(can)g(b)s(e)g
(refuted)g(is)g FN(N)13 b(P)7 b FT(-complete)27 b(\(Gallier,)e
(Narendran,)h(Plaisted,)f(and)g(Sn)m(yder)g(1990\),)378
5214 y(and)31 b(the)h(problem)f(of)h(whether)f(a)h(tableau)g(can)g(b)s
(e)f(refuted)h(b)m(y)f(considering)f(only)h(its)h(literals)e(is)378
5327 y(undecidable)e(\(V)-8 b(o)s(da)31 b(and)f(Komara)h(1995\).)519
5440 y(Recen)m(tly)-8 b(,)25 b(Degt)m(y)m(arev)h(and)c(V)-8
b(oronk)m(o)m(v)24 b(\(1998\))h(prop)s(osed)c(a)i(tableau)f(calculus,)h
FN(T)g(B)s(S)7 b(E)g FT(,)24 b(whic)m(h)378 5552 y(is)31
b(complete)i(for)f(\014rst-order)g(logic)g(with)f(equalit)m(y)h(and)g
(is)f(based)h(on)g(rigid)f(basic)g(sup)s(erp)s(osition)378
5665 y(\()p FN(B)s(S)7 b(E)h FT(\).)55 b(Although)34
b(the)h(inference)g(rules)e(of)i FN(T)23 b(B)s(S)7 b(E)43
b FT(do)35 b(not)g(\(and)g(cannot\))h(close)f(all)f(tableaux)p
eop
%%Page: 81 91
81 90 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(81)378 396 y(whose)41 b(literals)e(represen)m(t)i(in)m(v)-5
b(alid)38 b(sen)m(tences,)45 b(all)40 b(refutable)g(tableaux)h(can)g(b)
s(e)f(expanded)g(to)378 509 y(ones)32 b(whic)m(h)f(can)h(b)s(e)g
(closed)f(b)m(y)h(this)f(calculus)2029 476 y FL(1)2067
509 y FT(.)46 b(The)31 b(basic)h(restriction,)f(whic)m(h)g(w)m(as)h
(originally)378 622 y(used)39 b(in)f(narro)m(wing)h(\(F)-8
b(a)m(y)41 b(1979;)47 b(Hullot)38 b(1980\))k(and)d(in)m(v)m(olv)m(es)h
(the)f(application)f(of)i(equalities)378 735 y(on)e(non-v)-5
b(ariable)37 b(subterms,)i(is)f(used)f(to)j(reduce)e(the)g(searc)m(h)h
(space.)65 b(The)38 b(inference)g(rules)f(of)378 848
y(the)27 b(calculus)f(are)h(also)g(restricted)g(b)m(y)g
FI(or)-5 b(dering)31 b(e)-5 b(quality)30 b(c)-5 b(onstr)g(aints)37
b FT(whic)m(h)25 b(are)j(quan)m(ti\014er)e(free)378 961
y(\014rst-order)j(form)m(ulae)h(on)h(literals)d(of)j(the)g(form:)514
1149 y FN(\017)46 b FP(s)25 b FN(')g FP(t)30 b FT(represen)m(ting)g
(the)g(equalit)m(y)g(of)h FP(s)f FT(and)g FP(t)p FT(,)514
1336 y FN(\017)46 b FP(s)25 b FN(\037)g FP(t)30 b FT(where)g
FN(\037)g FT(is)f(a)i(reduction)f(ordering)f(\(see)i(\(Klop)f(1992\)\))
i(total)g(on)e(ground)f(terms.)378 1524 y(A)35 b(solution)e(of)i(a)g
(constrain)m(t)g FN(C)40 b FT(is)34 b(a)h(substitution)e
FP(\033)38 b FT(suc)m(h)c(that)i FN(C)5 b FP(\033)38
b FT(is)c(v)-5 b(alid.)52 b(A)35 b(constrain)m(t)g(is)378
1637 y(said)g(to)h(b)s(e)f(satis\014able)g(if)g(it)g(has)h(a)g
(solution.)55 b(A)36 b(commonly)g(used)f(reduction)f(ordering)h(is)g
(the)378 1750 y(lexicographical)29 b(path)i(ordering)e(\(Kamin)h(and)g
(L)m(\023)-43 b(evy)32 b(1980\))h(whic)m(h)c(is)h(de\014ned)g(as)h(an)f
(extension)393 1863 y FP(>)464 1878 y FL(lp)r(o)606 1863
y FT(of)h(an)m(y)f(total)i(ordering)d FP(>)h FT(on)g(function)f(sym)m
(b)s(ols)g(as)i(follo)m(ws:)519 1976 y(Giv)m(en)f FP(s)25
b FT(=)g FP(f)10 b FT(\()p FP(s)1079 1990 y FL(1)1118
1976 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(s)1378 1990 y
FO(m)1444 1976 y FT(\))31 b(and)f FP(t)25 b FT(=)g FP(g)s
FT(\()p FP(t)1955 1990 y FL(1)2010 1976 y FP(:)15 b(:)g(:)31
b(;)15 b(t)2219 1990 y FO(n)2267 1976 y FT(\),)31 b(then)f
FP(s)15 b(>)2693 1991 y FL(lp)r(o)2805 1976 y FP(t)30
b FT(if)g(and)f(only)h(if:)514 2163 y FN(\017)46 b FP(s)648
2177 y FO(i)691 2163 y FN(\025)762 2180 y FL(lp)r(o)874
2163 y FP(t)30 b FT(for)g(some)h FP(i)25 b FN(2)g(f)p
FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15 b(m)p FN(g)p FT(,)31
b(or)514 2351 y FN(\017)46 b FP(f)35 b(>)25 b(g)s FT(,)31
b(and)f FP(s)15 b(>)1188 2366 y FL(lp)r(o)1300 2351 y
FP(t)1333 2365 y FO(j)1400 2351 y FT(for)30 b(all)f FP(j)i
FN(2)25 b(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p
FN(g)p FT(,)31 b(or)514 2538 y FN(\017)46 b FP(f)j FT(=)39
b FP(g)s FT(,)j FN(h)p FP(s)1000 2552 y FL(1)1039 2538
y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(s)1299 2552 y FO(m)1365
2538 y FN(i)g FP(>)1487 2505 y FL(lex)1487 2566 y(lp)r(o)1617
2538 y FN(h)p FP(t)1685 2552 y FL(1)1725 2538 y FP(;)g(:)g(:)g(:)32
b(;)15 b(t)1975 2552 y FO(n)2022 2538 y FN(i)p FT(,)41
b(and)d FP(s)15 b(>)2437 2553 y FL(lp)r(o)2549 2538 y
FP(t)2582 2552 y FO(j)2657 2538 y FT(for)39 b(all)e FP(j)45
b FN(2)39 b(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15
b(n)p FN(g)p FT(,)41 b(where)605 2666 y FN(h)p FP(x)692
2680 y FL(1)732 2666 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(x)1001 2681 y FO(l)1027 2666 y FN(i)g Fl(m)1148 2633
y FL(lex)1252 2666 y FN(h)p FP(y)1332 2680 y FL(1)1372
2666 y FP(;)g(:)g(:)g(:)32 b(;)15 b(y)1634 2681 y FO(l)1660
2666 y FN(i)27 b FT(for)g(a)g(giv)m(en)g(ordering)f Fl(m)h
FT(if)f(there)h(is)f(some)i FP(j)j FN(\024)25 b FP(l)k
FT(suc)m(h)d(that)605 2779 y FP(x)657 2793 y FO(i)711
2779 y FT(=)f FP(y)852 2793 y FO(i)910 2779 y FT(for)30
b(all)f FP(i)d(<)f(j)36 b FT(and)29 b FP(x)1629 2793
y FO(j)1686 2779 y Fl(m)20 b FP(y)1822 2793 y FO(j)1858
2779 y FT(.)378 2967 y(Algorithms)33 b(for)i(solving)e(suc)m(h)h
(constrain)m(ts)h(are)g(giv)m(en)g(in)e(\(Comon)i(1990;)k(Nieu)m(w)m
(enh)m(uis)33 b(1993;)378 3080 y(Nieu)m(w)m(enh)m(uis)c(and)g(Rubio)g
(1995\).)519 3193 y(The)24 b(calculus)g FN(C)5 b(B)s(S)i(E)32
b FT(describ)s(ed)23 b(in)h(this)g(c)m(hapter)h(refutes)g(a)g(list)f
(of)h(clauses)f(using)g(rigid)e(basic)378 3306 y(sup)s(erp)s(osition.)
48 b(It)35 b(is)e(basically)f(the)j FN(T)22 b(B)s(S)7
b(E)42 b FT(calculus)33 b(mo)s(di\014ed)f(sligh)m(tly)g(to)j(lo)s(ok)f
(for)g(a)g(closed)378 3419 y(connected)d(tableau)e(if)g(p)s(ossible,)f
(and)h(relies)f(on)i FN(B)s(S)7 b(E)37 b FT(if)29 b(this)g(fails.)39
b(T)-8 b(ableau)29 b(branc)m(hes)h(are)g(also)378 3532
y(closed)35 b(when)e(they)i(can)g(b)s(e)f(refuted)h(without)e(instan)m
(tiating)h(their)g(free)h(v)-5 b(ariables.)52 b(Reasoning)378
3644 y(with)42 b(ground)h(equations)g(is)g(m)m(uc)m(h)g(simpler)e(than)
j(reasoning)e(with)h(non-ground)f(ones.)80 b(The)378
3757 y(ground)28 b(literals)f(in)g(a)i(tableau)g(branc)m(h)f(can)h(b)s
(e)f(sho)m(wn)g(to)i(b)s(e)e(refutable)g(in)f(p)s(olynomial)g(time)h(b)
m(y)378 3870 y(using,)i(for)h(instance,)g(algorithms)f(based)h(on)g
(congruence)g(closure)g(\(Shostak)g(1978;)j(Nelson)c(and)378
3983 y(Opp)s(en)e(1980\).)378 4227 y FG(5.2.2)112 b(The)38
b Fo(C)6 b(B)s(S)h(E)48 b FG(Calculus)378 4398 y FT(The)29
b(inference)g(rules)f(of)i(the)f FN(C)5 b(B)s(S)i(E)37
b FT(calculus)28 b(are)i(applied)e(to)i FI(c)-5 b(onstr)g(aint)34
b(table)-5 b(aux)42 b FT(of)30 b(the)f(form)378 4511
y FP(T)48 b FN(\001)36 b(C)f FT(where)30 b FP(T)43 b
FT(is)29 b(a)i(tableau)f(and)g FN(C)36 b FT(is)29 b(an)h(ordering)f
(equalit)m(y)h(constrain)m(t.)519 4624 y(Giv)m(en)d(a)h(set)f(of)g
(clauses)g(\000,)h(a)f(tableau)g(is)f(expanded)h(b)m(y)g(c)m(ho)s
(osing)g(a)g(branc)m(h)g FP(B)k FT(and)c(a)g(clause)378
4737 y(in)c(\000)i(whose)f(free)h(v)-5 b(ariables)23
b(are)i(instan)m(tiated)f(to)i(new)e(ones)h(whic)m(h)e(do)h(not)h(o)s
(ccur)g(in)e(the)i(tableau.)378 4850 y(The)34 b(leaf)g(no)s(de)f(in)g
FP(B)38 b FT(is)33 b(then)h(branc)m(hed)g(b)m(y)g(all)f(the)h(literals)
e(in)h(the)i(instan)m(tiated)e(clause,)i(and)378 4963
y(some)j(inequalities)d(are)j(added)f(in)g(the)h(resulting)d(branc)m
(hes)j(in)e(order)h(to)i(b)s(e)e(used)g(in)f(equalit)m(y)378
5076 y(reasoning.)46 b(More)33 b(precisely)-8 b(,)32
b(giv)m(en)g(a)h(literal)e FP(L)h FT(and)g(a)g(branc)m(h)g
FP(B)5 b FT(,)32 b(w)m(e)h(de\014ne)f(the)g(insertion)f(of)p
378 5238 1380 4 v 482 5292 a FC(1)516 5323 y FB(W)-6
b(e)23 b(stress)h(that)e(a)i(tableau)f(is)h Fh(r)l(efutable)29
b FB(if)24 b(it)f(represen)n(ts)g(an)g(in)n(v)l(alid)g(form)n(ula,)h
(and)f(it)g(is)g Fh(close)l(d)32 b FB(if)23 b(it)h(is)f
Fh(shown)378 5415 y FB(to)j(b)r(e)f(refutable.)p eop
%%Page: 82 92
82 91 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(82)378 396 y FP(L)30 b FT(in)f FP(B)5 b FT(,)30
b(denoted)h(b)m(y)f FP(B)25 b FN(\016)20 b FP(L)p FT(,)31
b(b)m(y:)668 601 y FP(B)24 b FN(\016)d FP(P)13 b FT(\()p
FP(s)976 615 y FL(1)1015 601 y FP(;)i(:)g(:)g(:)32 b(;)15
b(s)1275 615 y FO(n)1322 601 y FT(\))26 b(=)849 739 y
FP(B)5 b(;)15 b(P)e FT(\()p FP(s)1112 753 y FL(1)1152
739 y FP(;)i(:)g(:)g(:)31 b(;)15 b(s)1411 753 y FO(n)1458
739 y FT(\))21 b FN([)f(fh)p FP(s)1718 753 y FL(1)1758
739 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(s)2017 753 y FO(n)2064
739 y FN(i)26 b(6)p FT(=)f FN(h)p FP(t)2289 753 y FL(1)2329
739 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(t)2578 753 y FO(n)2626
739 y FN(i)25 b(j)h(:)p FP(P)13 b FT(\()p FP(t)2937 753
y FL(1)2976 739 y FP(;)i(:)g(:)g(:)32 b(;)15 b(t)3226
753 y FO(n)3273 739 y FT(\))25 b FN(2)g FP(B)5 b FN(g)668
876 y FP(B)24 b FN(\016)d(:)p FP(P)13 b FT(\()p FP(s)1037
890 y FL(1)1076 876 y FP(;)i(:)g(:)g(:)32 b(;)15 b(s)1336
890 y FO(n)1383 876 y FT(\))25 b(=)849 1014 y FP(B)5
b(;)15 b FN(:)p FP(P)e FT(\()p FP(s)1173 1028 y FL(1)1212
1014 y FP(;)i(:)g(:)g(:)32 b(;)15 b(s)1472 1028 y FO(n)1519
1014 y FT(\))21 b FN([)e(fh)p FP(s)1778 1028 y FL(1)1818
1014 y FP(;)c(:)g(:)g(:)32 b(;)15 b(s)2078 1028 y FO(n)2125
1014 y FN(i)26 b(6)p FT(=)f FN(h)p FP(t)2350 1028 y FL(1)2389
1014 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(t)2639 1028 y
FO(n)2686 1014 y FN(i)26 b(j)f FP(P)13 b FT(\()p FP(t)2936
1028 y FL(1)2976 1014 y FP(;)i(:)g(:)g(:)32 b(;)15 b(t)3226
1028 y FO(n)3273 1014 y FT(\))25 b FN(2)g FP(B)5 b FN(g)668
1152 y FP(B)24 b FN(\016)d FT(\()p FP(s)k FT(=)g FP(t)p
FT(\))h(=)f FP(B)5 b(;)15 b FT(\()p FP(s)25 b FT(=)g
FP(t)p FT(\))668 1290 y FP(B)f FN(\016)d FT(\()p FP(s)k
FN(6)p FT(=)g FP(t)p FT(\))h(=)f FP(B)5 b(;)15 b FT(\()p
FP(s)25 b FN(6)p FT(=)g FP(t)p FT(\))378 1494 y(where)46
b FP(P)60 b FT(is)46 b(a)i(predicate)e(sym)m(b)s(ol)g(other)h(than)g
(equalit)m(y)f(and)h(an)f(expression)g(of)h(the)g(form)378
1607 y FN(h)p FP(t)446 1621 y FL(1)486 1607 y FP(;)15
b(:)g(:)g(:)31 b(;)15 b(t)735 1621 y FO(n)783 1607 y
FN(i)35 b FT(denotes)h(the)f(term)h FN(hi)1641 1621 y
FO(n)1688 1607 y FT(\()p FP(t)1756 1621 y FL(1)1796 1607
y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(t)2046 1621 y FO(n)2093
1607 y FT(\))36 b(where)e FN(hi)2501 1621 y FL(0)2542
1607 y FT(,)i FN(hi)2673 1621 y FL(1)2714 1607 y FT(,)g(etc.)17
b(,)37 b(are)e(function)f(sym)m(b)s(ols)378 1720 y(whic)m(h)29
b(do)h(not)h(o)s(ccur)f(in)f(\000.)519 1833 y(The)39
b(ab)s(o)m(v)m(e)j(metho)s(d)d(of)h(inserting)e(literals)g(in)m(to)i(a)
h(branc)m(h)e(allo)m(ws)g(one)h(to)h(consider)e(only)378
1946 y(the)j(equations)f(and)h(inequations)e(in)g(the)i(branc)m(hes)g
(in)e(closing)h(the)h(tableau)f(without)g(losing)378
2059 y(refutational)33 b(completeness)g(\(see)h(\(Gallier,)f
(Narendran,)h(Plaisted,)f(Raatz,)j(and)d(Sn)m(yder)f(1993\))378
2171 y(and)e(\(Bec)m(k)m(ert)j(1997\)\).)519 2284 y(T)-8
b(ableau)37 b(branc)m(hes)g(can)h(b)s(e)e(simpli\014ed)e(or)j(ev)m(en)h
(refuted)f(b)m(y)g(using)f(tec)m(hniques)h(to)h(reason)378
2397 y(with)e(ground)g(equations)h(in)f(order)h(to)h(a)m(v)m(oid)g
(redundan)m(t)e(instan)m(tiations.)60 b(Let)38 b FP(E)43
b FT(b)s(e)36 b(a)i(set)g(of)378 2510 y(equations,)c(and)f(let)g(the)g
(relation)g FN($)1715 2524 y FO(E)1807 2510 y FT(b)s(e)g(de\014ned)f
(suc)m(h)h(that)h FP(s)29 b FN($)2821 2524 y FO(E)2911
2510 y FP(t)k FT(if)f(and)h(only)f(if)g(there)i(is)378
2623 y(some)j(term)f FP(p)g FT(and)f(some)i FP(a)e FN(\031)f
FP(b)i FT(in)f FP(E)42 b FT(suc)m(h)35 b(that)i FP(s)e
FT(=)f FP(p)p FT([)p FP(a)p FT(])j(and)e FP(t)g FT(=)g
FP(p)p FT([)p FP(b)p FT(].)58 b(Therefore,)38 b(if)d(the)378
2736 y(equations)29 b(in)g FP(E)24 b FN([)19 b(f)p FP(s)25
b FT(=)g FP(t)p FN(g)30 b FT(are)g(ground,)g(then)f FP(E)i
FN(`)24 b FP(s)h FT(=)g FP(t)30 b FT(if)f(and)g(only)g(if)f
FP(s)d FN($)3173 2703 y FK(\003)3173 2763 y FO(E)3258
2736 y FP(t)p FT(.)40 b(A)30 b(branc)m(h)f FP(B)378 2849
y FT(whic)m(h)i(is)h(in)f(a)i(constrain)m(t)g(tableau)f
FP(T)50 b FN(\001)37 b(C)h FT(and)32 b(con)m(tains)h(an)f(inequalit)m
(y)f FP(s)e FN(6)p FT(=)f FP(t)33 b FT(can)f(b)s(e)g(refuted)378
2962 y(if)h FP(s\033)h FN($)685 2929 y FK(\003)685 2993
y FL(Eq\()p FO(B)s FL(\))p FO(\033)960 2962 y FP(t\033)s
FT(,)h(where)f(Eq)o(\()p FP(B)5 b FT(\))35 b(is)e(the)h(set)h(of)f
(equalities)e(in)h FP(B)39 b FT(and)33 b FP(\033)k FT(is)c(the)h(most)h
(general)378 3075 y(substitution)28 b(satisfying)h(the)i(constrain)m(t)
f FN(C)5 b FT(.)41 b(Similarly)-8 b(,)27 b(an)k(equalit)m(y)f(in)f
FP(B)34 b FT(of)d(the)f(form)1174 3279 y FN(h)p FP(:)15
b(:)g(:)32 b(;)15 b(s)1429 3293 y FO(i)p FK(\000)p FL(1)1547
3279 y FP(;)g(s)1630 3293 y FO(i)1659 3279 y FP(;)g(s)1742
3293 y FO(i)p FL(+1)1860 3279 y FP(;)g(:)g(:)g(:)i FN(i)25
b(6)p FT(=)g FN(h)p FP(:)15 b(:)g(:)32 b(;)15 b(t)2423
3293 y FO(i)p FK(\000)p FL(1)2542 3279 y FP(;)g(t)2615
3293 y FO(i)2643 3279 y FP(;)g(t)2716 3293 y FO(i)p FL(+1)2835
3279 y FP(;)g(:)g(:)g(:)h FN(i)378 3483 y FT(can)31 b(b)s(e)e
(simpli\014ed)e(to)1281 3687 y FN(h)p FP(:)15 b(:)g(:)31
b(;)15 b(s)1535 3701 y FO(i)p FK(\000)p FL(1)1654 3687
y FP(;)g(s)1737 3701 y FO(i)p FL(+1)1855 3687 y FP(;)g(:)g(:)g(:)i
FN(i)25 b(6)p FT(=)g FN(h)p FP(:)15 b(:)g(:)32 b(;)15
b(t)2418 3701 y FO(i)p FK(\000)p FL(1)2537 3687 y FP(;)g(t)2610
3701 y FO(i)p FL(+1)2728 3687 y FP(;)g(:)g(:)g(:)i FN(i)378
3892 y FT(if)k FP(s)496 3906 y FO(i)524 3892 y FP(\033)28
b FN($)695 3859 y FK(\003)695 3923 y FL(Eq\()p FO(B)s
FL(\))p FO(\033)964 3892 y FP(t)997 3906 y FO(i)1025
3892 y FP(\033)s FT(.)38 b(Congruence)22 b(closure)f(algorithms)g(can)h
(b)s(e)f(used)g(to)i(decide)e(whether)h FP(s)i FN($)3710
3859 y FK(\003)3710 3919 y FO(E)3795 3892 y FP(t)378
4005 y FT(for)30 b(terms)g FP(s)g FT(and)g FP(t)g FT(and)g(a)h(set)g
(of)f(equations)g FP(E)5 b FT(.)519 4118 y(The)38 b(inference)g(rules)g
(of)g(the)h FN(B)s(S)7 b(E)47 b FT(calculus)37 b(\(i.e.,)16
b(rigid)36 b(basic)j(sup)s(erp)s(osition)c(with)i(equa-)378
4230 y(tional)30 b(re\015exivit)m(y\))g(are)h(used)f(on)g(tableau)h
(branc)m(hes)f(that)h(ma)m(y)g(need)g(the)f(instan)m(tiation)g(of)h
(free)378 4343 y(v)-5 b(ariables)21 b(to)i(b)s(e)f(closed.)38
b(Because)24 b(of)f(the)f(fact)i(that)f(the)f(tableau)h(expansion)e
(rules)g(together)j(with)378 4456 y(the)31 b(rules)e(of)i(the)f
FN(B)s(S)7 b(E)38 b FT(calculus)30 b(for)g(closing)g(branc)m(hes)g(giv)
m(e)h(a)g(complete)g(semi-decision)e(pro)s(ce-)378 4569
y(dure)j(for)h(\014rst-order)g(logic)g(with)e(equalit)m(y)-8
b(,)35 b(the)e FN(C)5 b(B)s(S)i(E)41 b FT(calculus,)33
b(whic)m(h)f(is)g(giv)m(en)h(in)f(\014gure)h(11,)378
4682 y(is)d(refutationally)g(complete)i(for)f(\014rst-order)f(logic)h
(with)f(equalit)m(y)-8 b(.)43 b(In)31 b(the)g(implemen)m(tation)f(de-)
378 4795 y(scrib)s(ed)37 b(in)i(section)g(5.3,)k(the)d(expansion)f
(rule)f(tries)h(to)h(select)g(a)g(clause)f(whic)m(h)g(results)f(in)g
(an)378 4908 y(immediate)j(closure)g(of)i(a)f(branc)m(h)f(in)g(order)g
(to)i(gain)f(some)g(of)g(the)h(e\016ciency)f(of)g(connection)378
5021 y(tableau)30 b(calculi.)p eop
%%Page: 83 93
83 92 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(83)p 378 938 3453 4 v 376 5080 4 4143 v 1661 1138
610 4 v 1661 1214 a Fv(L)1718 1226 y Fs(1)1768 1214 y
Fu(j)28 b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(L)2038
1226 y Fq(m)2132 1214 y Fu(\001)33 b(fg)2308 1157 y Ft(\(Start\))1633
1388 y Fv(B)1696 1400 y Fs(1)1747 1388 y Fu(j)28 b(\001)14
b(\001)g(\001)27 b(j)14 b Fv(B)2022 1400 y Fq(n)2099
1388 y Fu(\001)33 b(C)p 1285 1425 1267 4 v 1285 1501
a Fv(B)1348 1513 y Fs(1)1403 1501 y Fu(\016)18 b Fv(L)1520
1513 y Fs(1)1571 1501 y Fu(j)27 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)1846 1513 y Fs(1)1902 1501 y Fu(\016)k
Fv(L)2019 1513 y Fq(m)2095 1501 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)2370 1513 y Fq(n)2448 1501 y Fu(\001)32
b(C)2589 1444 y Ft(\(Expand\))746 1642 y Fv(B)809 1654
y Fs(1)847 1642 y Fv(;)14 b Fu(h)p Fv(:)g(:)g(:)27 b(;)14
b(s)1116 1654 y Fq(i)p Fr(\000)p Fs(1)1229 1642 y Fv(;)g(s)1305
1654 y Fq(i)1332 1642 y Fv(;)g(s)1408 1654 y Fq(i)p Fs(+1)1520
1642 y Fv(;)g(:)g(:)g(:)f Fu(i)24 b(6\031)e(h)p Fv(:)14
b(:)g(:)28 b(;)14 b(t)2034 1654 y Fq(i)p Fr(\000)p Fs(1)2147
1642 y Fv(;)g(t)2214 1654 y Fq(i)2241 1642 y Fv(;)g(t)2308
1654 y Fq(i)p Fs(+1)2420 1642 y Fv(;)g(:)g(:)g(:)f Fu(i)h(j)28
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)2889 1654 y
Fq(n)2966 1642 y Fu(\001)32 b(C)p 746 1679 2324 4 v 845
1756 a Fv(B)908 1768 y Fs(1)946 1756 y Fv(;)14 b Fu(h)p
Fv(:)g(:)g(:)27 b(;)14 b(s)1215 1768 y Fq(i)p Fr(\000)p
Fs(1)1328 1756 y Fv(;)g(s)1404 1768 y Fq(i)p Fs(+1)1515
1756 y Fv(;)g(:)g(:)g(:)g Fu(i)23 b(6\031)g(h)p Fv(:)14
b(:)g(:)28 b(;)14 b(t)2030 1768 y Fq(i)p Fr(\000)p Fs(1)2142
1756 y Fv(;)g(t)2209 1768 y Fq(i)p Fs(+1)2321 1756 y
Fv(;)g(:)g(:)g(:)f Fu(i)h(j)28 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)2790 1768 y Fq(n)2867 1756 y Fu(\001)32
b(C)3108 1699 y Ft(\(Simplify\))1358 1897 y Fv(B)1421
1909 y Fs(1)1458 1897 y Fv(;)14 b(s)23 b Fu(6\031)g Fv(t)14
b Fu(j)g Fv(B)1789 1909 y Fs(2)1840 1897 y Fu(j)27 b(\001)14
b(\001)g(\001)28 b(j)14 b Fv(B)2115 1909 y Fq(n)2192
1897 y Fu(\001)33 b(C)p 1358 1934 939 4 v 1542 2010 a
Fv(B)1605 2022 y Fs(2)1656 2010 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)1931 2022 y Fq(n)2009 2010 y Fu(\001)32
b(C)2334 1953 y Ft(\(T)-7 b(rivial)27 b(Close\))1449
2151 y Fv(B)1512 2163 y Fs(1)1549 2151 y Fv(;)14 b(l)24
b Fu(\031)f Fv(r)n(;)14 b(s)p Ft([)p Fv(p)p Ft(])23 b
Fu(\031)g Fv(t)14 b Fu(j)27 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)2352 2163 y Fq(n)2429 2151 y Fu(\001)32
b(C)p 1016 2188 1950 4 v 1016 2264 a Fv(B)1079 2276 y
Fs(1)1116 2264 y Fv(;)14 b(l)25 b Fu(\031)d Fv(r)n(;)14
b(s)p Ft([)p Fv(r)r Ft(])24 b Fu(\031)f Fv(t)14 b Fu(j)27
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1917 2276 y
Fq(n)1994 2264 y Fu(\001)33 b(C)23 b([)18 b(f)p Fv(l)24
b Fu(\037)f Fv(r)n(;)14 b(s)p Ft([)p Fv(p)p Ft(])23 b
Fu(\037)g Fv(t;)14 b(l)24 b Fu(')e Fv(p)p Fu(g)3003 2207
y Ft(\(lrbs\))1444 2405 y Fv(B)1507 2417 y Fs(1)1544
2405 y Fv(;)14 b(l)25 b Fu(\031)d Fv(r)n(;)14 b(s)p Ft([)p
Fv(p)p Ft(])23 b Fu(6\031)g Fv(t)14 b Fu(j)27 b(\001)14
b(\001)g(\001)28 b(j)14 b Fv(B)2347 2417 y Fq(n)2424
2405 y Fu(\001)33 b(C)p 1011 2442 V 1011 2518 a Fv(B)1074
2530 y Fs(1)1111 2518 y Fv(;)14 b(l)25 b Fu(\031)d Fv(r)n(;)14
b(s)p Ft([)p Fv(r)r Ft(])24 b Fu(6\031)f Fv(t)14 b Fu(j)28
b(\001)14 b(\001)g(\001)27 b(j)14 b Fv(B)1912 2530 y
Fq(n)1989 2518 y Fu(\001)33 b(C)23 b([)c(f)p Fv(l)24
b Fu(\037)e Fv(r)n(;)14 b(s)p Ft([)p Fv(p)p Ft(])23 b
Fu(\037)g Fv(t;)14 b(l)24 b Fu(')f Fv(p)p Fu(g)2999 2461
y Ft(\(rrbs\))1554 2659 y Fv(B)1617 2671 y Fs(1)1655
2659 y Fv(;)14 b(s)22 b Fu(6\031)h Fv(t)14 b Fu(j)g Fv(B)1985
2671 y Fs(2)2036 2659 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)2311 2671 y Fq(n)2389 2659 y Fu(\001)32
b(C)p 1554 2696 939 4 v 1560 2773 a Fv(B)1623 2785 y
Fs(2)1675 2773 y Fu(j)27 b(\001)14 b(\001)g(\001)28 b(j)14
b Fv(B)1950 2785 y Fq(n)2027 2773 y Fu(\001)33 b(C)23
b([)c(f)p Fv(s)j Fu(')h Fv(t)p Fu(g)2530 2716 y Ft(\(er\))568
2934 y Fu(\017)45 b Ft(The)24 b(terms)f(lrbs,)g(rrbs)g(and)g(er)g
(stand)g(for)g(Left)h(Rigid)f(Basic)g(Sup)r(erp)r(osition,)h(Righ)n(t)f
(Rigid)h(Basic)655 3034 y(Sup)r(erp)r(osition)k(and)f(Equational)f
(Re\015exivit)n(y)h(resp)r(ectiv)n(ely)-7 b(.)568 3183
y Fu(\017)45 b Ft(The)28 b(rules)f(are)f(only)i(applicable)f(if)h(the)g
(follo)n(wing)e(conditions)h(hold:)745 3345 y(1.)45 b(The)28
b(constrain)n(t)e(at)i(the)g(conclusion)e(of)i(eac)n(h)f(rule)g(is)g
(satis\014able.)745 3478 y(2.)45 b(In)31 b(the)f(start)g(and)g(expand)g
(rules,)h Fv(L)2047 3490 y Fs(1)2104 3478 y Fu(_)21 b(\001)14
b(\001)g(\001)20 b(_)g Fv(L)2429 3490 y Fq(m)2522 3478
y Ft(is)31 b(an)f(instance)g Fv(C)6 b(\033)34 b Ft(of)c(a)g(clause)g
Fv(C)37 b Ft(in)855 3577 y(the)25 b(giv)n(en)f(set)g(of)h(clauses)e
(where)h Fv(\033)k Ft(maps)c(all)h(the)g(free)f(v)-5
b(ariables)23 b(in)i Fv(C)31 b Ft(to)24 b(some)g(v)-5
b(ariables)855 3677 y(whic)n(h)28 b(do)f(not)g(o)r(ccur)g(in)h(the)g
(constrain)n(t)f(tableau)g(in)h(the)g(premise.)745 3810
y(3.)45 b(In)26 b(the)f(simplify)h(rule,)g Fv(s)1636
3822 y Fq(i)1663 3810 y Fv(\034)33 b Fu($)1815 3780 y
Fr(\003)1815 3837 y Fs(Eq)o(\()p Fq(B)1970 3845 y Fg(1)2003
3837 y Fq(\034)6 b Fs(\))2093 3810 y Fv(t)2123 3822 y
Fq(i)2151 3810 y Fv(\034)j Ft(,)27 b(where)d Fv(\034)35
b Ft(is)26 b(the)f(most)g(general)f(solution)h(of)g(the)855
3909 y(constrain)n(t)h Fu(C)5 b Ft(.)745 4042 y(4.)45
b(In)30 b(the)g(trivial)g(close)f(rule,)h Fv(s\034)36
b Fu($)1943 4012 y Fr(\003)1943 4069 y Fs(Eq\()p Fq(B)2099
4077 y Fg(1)2131 4069 y Fq(\034)6 b Fs(\))2225 4042 y
Fv(t\034)j Ft(,)32 b(where)d Fv(\034)39 b Ft(is)30 b(the)g(most)g
(general)e(solution)i(of)855 4142 y(the)e(constrain)n(t)e
Fu(C)5 b Ft(.)745 4275 y(5.)45 b(In)28 b(the)g(basic)f(sup)r(erp)r
(osition)g(rules,)g(the)h(term)g Fv(p)f Ft(is)g(not)h(a)f(v)-5
b(ariable.)745 4408 y(6.)45 b(the)28 b(righ)n(t-hand)e(side)i(of)f(the)
h(rigid)f(equation)g(at)g(the)h(premise)f(of)h(eac)n(h)f(rule)g(is)g
(not)h(of)f(the)855 4507 y(form)g Fv(q)f Fu(\031)d Fv(q)s
Ft(.)745 4640 y(7.)45 b(In)28 b(the)g(left)g(basic)f(sup)r(erp)r
(osition)g(rule,)h Fv(s)p Ft([)p Fv(r)r Ft(])23 b Fu(6)p
Ft(=)g Fv(t)p Ft(.)853 4911 y FT(Figure)30 b(11:)41 b(The)30
b(Inference)g(Rules)g(of)g(the)h FN(C)5 b(B)s(S)i(E)38
b FT(T)-8 b(ableau)30 b(Calculus.)p 3829 5080 4 4143
v 378 5084 3453 4 v eop
%%Page: 84 94
84 93 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(84)378 396 y FG(5.2.3)112 b(Some)38 b(Examples)378
568 y FT(In)f(this)f(section)i(w)m(e)g(giv)m(e)g(a)g(n)m(um)m(b)s(er)e
(of)i(simple)d(examples)j(to)g(illustrate)e(the)h(ab)s(o)m(v)m(e)i
(calculus.)378 681 y(The)30 b(\014rst)f(example)i(\014nds)d(a)j(pro)s
(of)f(for)g(the)g(sen)m(tence)1363 898 y(\()p FP(G)p
FT(\()p FP(e)p FT(\))21 b FN(^)f FT(\()p FN(8)p FP(x:G)p
FT(\()p FP(x)p FT(\))25 b FN(\))h FP(p)p FT(\()p FP(e;)15
b(x)p FT(\))26 b(=)f FP(x)p FT(\)\))h FN(\))1363 1053
y FT(\()p FP(G)p FT(\()p FP(f)10 b FT(\))20 b FN(^)g
FT(\()p FN(8)p FP(x:G)p FT(\()p FP(x)p FT(\))25 b FN(\))g
FP(p)p FT(\()p FP(x;)15 b(f)10 b FT(\))26 b(=)f FP(x)p
FT(\)\))h FN(\))1515 1209 y FT(\()p FP(e)f FT(=)g FP(f)10
b FT(\))378 1413 y(b)m(y)30 b(refuting)f(the)i(set)g(of)f(clauses:)1707
1610 y FN(:)p FP(G)p FT(\()p FP(x)p FT(\))20 b FN(_)g
FP(p)p FT(\()p FP(e;)15 b(x)p FT(\))26 b(=)f FP(x)1707
1723 y FN(:)p FP(G)p FT(\()p FP(x)p FT(\))20 b FN(_)g
FP(p)p FT(\()p FP(x;)15 b(f)10 b FT(\))25 b(=)g FP(x)1707
1835 y(G)p FT(\()p FP(e)p FT(\))1707 1948 y FP(G)p FT(\()p
FP(f)10 b FT(\))1707 2061 y FP(e)26 b FN(6)p FT(=)e FP(f)378
2263 y FT(where)31 b FP(e)h FT(and)f FP(f)41 b FT(are)32
b(constan)m(ts.)46 b(F)-8 b(orm)m(ulae)32 b(of)g(the)g(form)f
FP(G)p FT(\()p FP(x)p FT(\))c FN(\))h FP(f)10 b FT([)p
FP(x)p FT(])27 b(=)g FP(g)s FT([)p FP(x)p FT(])32 b(o)s(ccurred)f
(quite)378 2376 y(often)j(in)e(the)h(mec)m(hanisation)g(of)g(group)g
(theory)g(describ)s(ed)e(in)h(c)m(hapter)i(9,)h(where)d(prop)s
(ositions)378 2489 y(of)d(the)f(form)g FP(G)p FT(\()p
FP(x)p FT(\))h(are)g(used)f(to)h(denote)g(the)g(fact)g(that)g
FP(x)g FT(is)e(a)i(mem)m(b)s(er)f(of)h(some)g(set)g FP(G)f
FT(\(usually)378 2602 y(assumed)35 b(to)h(b)s(e)e(a)i(group\).)55
b(The)35 b(ab)s(o)m(v)m(e)h(sen)m(tence)h(states)f(that)g(if)f(a)g
(left)g(iden)m(tit)m(y)g(and)g(a)g(righ)m(t)378 2715
y(iden)m(tit)m(y)30 b(exist)g(in)f(a)i(set,)g(then)f(they)h(are)f
(equal.)519 2827 y(The)d(pro)s(of)f(searc)m(h)i(is)e(initialised)d(b)m
(y)k(starting)g(with)e(the)j(\014rst)e(clause.)39 b(This)25
b(is)i(then)f(follo)m(w)m(ed)378 2940 y(b)m(y)33 b(an)f(expansion)g
(step)h(with)e(the)i(fourth)f(clause)g(since)g(equalit)m(y)h
(re\015exivit)m(y)f(can)h(immediately)378 3053 y(b)s(e)d(used)f(to)i
(close)g(one)g(of)f(the)h(branc)m(hes:)p 1396 3198 1017
4 v 1396 3283 a FN(:)p FP(G)p FT(\()p FP(v)1608 3297
y FL(1)1647 3283 y FT(\))15 b FN(j)g FP(p)p FT(\()p FP(e;)g(v)1944
3297 y FL(1)1985 3283 y FT(\))26 b(=)f FP(v)2186 3297
y FL(1)2261 3283 y FN(\001)35 b(fg)2454 3221 y FT(\(start\))p
1057 3326 1694 4 v 1057 3411 a FN(:)p FP(G)p FT(\()p
FP(v)1269 3425 y FL(1)1308 3411 y FT(\))p FP(;)15 b(G)p
FT(\()p FP(f)10 b FT(\))p FP(;)15 b FN(h)p FP(v)1699
3425 y FL(1)1740 3411 y FN(i)25 b(6)p FT(=)g FN(h)p FP(f)10
b FN(i)15 b(j)g FP(p)p FT(\()p FP(e;)g(v)2283 3425 y
FL(1)2324 3411 y FT(\))26 b(=)f FP(v)2525 3425 y FL(1)2600
3411 y FN(\001)35 b(fg)2793 3349 y FT(\(expand\))p 1057
3453 V 1438 3538 a FP(p)p FT(\()p FP(e;)15 b(v)1645 3552
y FL(1)1685 3538 y FT(\))25 b(=)g FP(v)1885 3552 y FL(1)1960
3538 y FN(\001)36 b(f)p FP(v)2110 3552 y FL(1)2175 3538
y FN(')25 b FP(f)10 b FN(g)2793 3476 y FT(\(er\))378
3742 y(Note)35 b(that)g(the)f(inequalit)m(y)e FN(h)p
FP(v)1466 3756 y FL(1)1506 3742 y FN(i)f(6)p FT(=)g FN(h)p
FP(f)10 b FN(i)34 b FT(is)f(included)e(in)i(the)h(branc)m(h)f(when)g
(the)h(literal)f FP(G)p FT(\()p FP(f)10 b FT(\))33 b(is)378
3855 y(inserted)27 b(in)f(the)i(branc)m(h)g FN(f)p FP(G)p
FT(\()p FP(v)1472 3869 y FL(1)1512 3855 y FT(\))p FN(g)p
FT(.)40 b(The)28 b(constrain)m(t)g FN(f)p FP(v)2356 3869
y FL(1)2421 3855 y FN(')d FP(f)10 b FN(g)28 b FT(is)e(a)j(simpli\014ed)
24 b(equiv)-5 b(alen)m(t)27 b(form)378 3968 y(of)j FN(fh)p
FP(v)605 3982 y FL(1)646 3968 y FN(i)25 b(')g(h)p FP(f)10
b FN(ig)p FT(.)519 4081 y(A)m(t)43 b(this)e(p)s(oin)m(t)g(there)i(is)e
(no)h(clause)g(whic)m(h)f(can)h(b)s(e)g(used)f(for)h(an)g(expansion)f
(step)h(whic)m(h)378 4194 y(can)31 b(b)s(e)e(immediately)g(follo)m(w)m
(ed)h(b)m(y)g(the)h(closure)e(of)i(a)g(branc)m(h.)40
b(Unlik)m(e)29 b(the)i(connection)f(tableau)378 4307
y(calculus)g(for)i(pure)f(\014rst-order)g(logic)g(this)g(do)s(es)g(not)
i(imply)c(the)j(failure)e(of)i(the)g(curren)m(t)g(path)f(in)378
4420 y(the)h(pro)s(of)e(searc)m(h.)45 b(The)31 b(second)g(clause)h(is)e
(used)h(for)g(expansion,)g(and)g(this)f(can)i(b)s(e)f(follo)m(w)m(ed)g
(b)m(y)378 4533 y(an)f(expansion)f(with)h(the)g(third)f(clause)h(and)g
(an)g(equational)g(re\015exivit)m(y)f(step.)1438 4718
y FP(p)p FT(\()p FP(e;)15 b(v)1645 4732 y FL(1)1685 4718
y FT(\))25 b(=)g FP(v)1885 4732 y FL(1)1960 4718 y FN(\001)36
b(f)p FP(v)2110 4732 y FL(1)2175 4718 y FN(')25 b FP(f)10
b FN(g)p 733 4761 2343 4 v 733 4846 a FP(p)p FT(\()p
FP(e;)15 b(v)940 4860 y FL(1)980 4846 y FT(\))26 b(=)f
FP(v)1181 4860 y FL(1)1220 4846 y FP(;)15 b FN(:)p FP(G)p
FT(\()p FP(v)1472 4860 y FL(2)1512 4846 y FT(\))g FN(j)g
FP(p)p FT(\()p FP(e;)g(v)1809 4860 y FL(1)1850 4846 y
FT(\))25 b(=)g FP(v)2050 4860 y FL(1)2090 4846 y FP(;)15
b(p)p FT(\()p FP(v)2255 4860 y FL(2)2295 4846 y FP(;)g(f)10
b FT(\))25 b(=)g FP(v)2590 4860 y FL(2)2665 4846 y FN(\001)36
b(f)p FP(v)2815 4860 y FL(1)2880 4846 y FN(')25 b FP(f)10
b FN(g)3117 4784 y FT(\(expand\))p 406 4888 2997 4 v
406 4973 a FP(p)p FT(\()p FP(e;)15 b(v)613 4987 y FL(1)653
4973 y FT(\))26 b(=)f FP(v)854 4987 y FL(1)893 4973 y
FP(;)15 b FN(:)p FP(G)p FT(\()p FP(v)1145 4987 y FL(2)1185
4973 y FT(\))p FP(;)g(G)p FT(\()p FP(e)p FT(\))p FP(;)g
FN(h)p FP(v)1563 4987 y FL(2)1604 4973 y FN(i)26 b(6)p
FT(=)f FN(h)p FP(e)p FN(i)15 b(j)g FP(p)p FT(\()p FP(e;)g(v)2135
4987 y FL(1)2177 4973 y FT(\))25 b(=)g FP(v)2377 4987
y FL(1)2417 4973 y FP(;)15 b(p)p FT(\()p FP(v)2582 4987
y FL(2)2622 4973 y FP(;)g(f)10 b FT(\))25 b(=)g FP(v)2917
4987 y FL(2)2992 4973 y FN(\001)35 b(f)p FP(v)3141 4987
y FL(1)3207 4973 y FN(')25 b FP(f)10 b FN(g)3444 4911
y FT(\(expand\))p 406 5016 V 1026 5100 a FP(p)p FT(\()p
FP(e;)15 b(v)1233 5114 y FL(1)1274 5100 y FT(\))25 b(=)g
FP(v)1474 5114 y FL(1)1514 5100 y FP(;)15 b(p)p FT(\()p
FP(v)1679 5114 y FL(2)1719 5100 y FP(;)g(f)10 b FT(\))25
b(=)g FP(v)2014 5114 y FL(2)2089 5100 y FN(\001)36 b(f)p
FP(v)2239 5114 y FL(1)2304 5100 y FN(')25 b FP(f)5 b(;)15
b(v)2534 5114 y FL(2)2598 5100 y FN(')25 b FP(e)p FN(g)3444
5038 y FT(\(er\))378 5305 y(Finally)-8 b(,)38 b(the)g(last)f(clause)g
(is)g(used)f(for)i(expansion.)61 b(This)35 b(results)i(in)f(a)i
(tableau)f(with)f(a)i(single)378 5418 y(branc)m(h.)g(Since)23
b(the)i(substitution)d(in)h(the)h(constrain)m(t)g(maps)g(all)f(the)i
(free)f(v)-5 b(ariables)23 b(in)g(the)h(branc)m(h)378
5531 y(to)34 b(constan)m(ts,)h(the)f(trivial)d(closure)i(rule)f(whic)m
(h)g(uses)h(reasoning)g(on)g(ground)f(equations)h(can)h(b)s(e)p
eop
%%Page: 85 95
85 94 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(85)1936 527 y
tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 3.30017 } false /N@T-0
16 {InitRnode } NewNode end end
1936 527 a 1415 776 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 34.48392 17.24196
3.30017 } false /N@T-0-0 16 {InitRnode } NewNode end end
1415 776 a FN(:)p
FP(G)p FT(\()p FP(v)1627 790 y FL(1)1666 776 y FT(\))1558
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-0-0 InitNC { NCLine } if end gsave 0.8 SLW 0.
setgray 0 setlinecap stroke grestore grestore end
1558 748 a 1460 1025 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 23.66573 11.83286
3.30017 } false /N@T-0-0-0 16 {InitRnode } NewNode end end
1460 1025 a FP(G)p FT(\()p
FP(f)10 b FT(\))1558 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-0-0 /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 0 0 /N@T-0-1-1-0 /N@T-0-1-1-0-0 InitNC { NCLine } if end grestore
end
2598 1379 a 1363
1917 a FT(Figure)30 b(12:)41 b(A)31 b(Closed)e FN(C)5
b(B)s(S)i(E)38 b FT(T)-8 b(ableau.)378 2311 y(used)30
b(to)h(close)f(the)h(tableau.)940 2478 y FP(p)p FT(\()p
FP(e;)15 b(v)1147 2492 y FL(1)1188 2478 y FT(\))25 b(=)g
FP(v)1388 2492 y FL(1)1428 2478 y FP(;)15 b(p)p FT(\()p
FP(v)1593 2492 y FL(2)1633 2478 y FP(;)g(f)10 b FT(\))25
b(=)g FP(v)1928 2492 y FL(2)2003 2478 y FN(\001)35 b(f)p
FP(v)2152 2492 y FL(1)2218 2478 y FN(')24 b FP(f)5 b(;)15
b(v)2447 2492 y FL(2)2512 2478 y FN(')25 b FP(e)p FN(g)p
811 2521 2014 4 v 811 2606 a FP(p)p FT(\()p FP(e;)15
b(v)1018 2620 y FL(1)1058 2606 y FT(\))26 b(=)f FP(v)1259
2620 y FL(1)1298 2606 y FP(;)15 b(p)p FT(\()p FP(v)1463
2620 y FL(2)1503 2606 y FP(;)g(f)10 b FT(\))26 b(=)f
FP(v)1799 2620 y FL(2)1838 2606 y FP(;)15 b(e)26 b FN(6)p
FT(=)f FP(f)45 b FN(\001)35 b(f)p FP(v)2281 2620 y FL(1)2347
2606 y FN(')25 b FP(f)5 b(;)15 b(v)2577 2620 y FL(2)2641
2606 y FN(')25 b FP(e)p FN(g)2866 2544 y FT(\(expand\))p
811 2648 V 1408 2733 a FN(fg)37 b(\001)e(f)p FP(v)1684
2747 y FL(1)1749 2733 y FN(')25 b FP(f)5 b(;)15 b(v)1979
2747 y FL(2)2044 2733 y FN(')25 b FP(e)p FN(g)2866 2671
y FT(\(trivial)k(close\))378 2937 y(The)h(closed)g(tableau)g(found)f(b)
m(y)i(this)e(pro)s(of)g(is)h(illustrated)e(in)h(\014gure)h(12.)519
3050 y(In)g(the)i(second)f(example,)g(w)m(e)g(illustrate)f(the)h(use)g
(of)g(the)g(simplify)d(rule)h(b)m(y)i(refuting)f(the)h(set)378
3163 y(of)f(clauses)h(b)s(elo)m(w:)1622 3251 y FN(:)p
FP(I)7 b FT(\()p FP(x)p FT(\))21 b FN(_)f FP(x)25 b FT(=)g
FP(e)163 b(I)7 b FT(\()p FP(f)j FT(\))1760 3364 y FP(P)j
FT(\()p FP(e;)i(y)s FT(\))221 b FN(:)p FP(P)13 b FT(\()p
FP(f)5 b(;)15 b(c)p FT(\))378 3523 y(where)30 b FP(e)p
FT(,)h FP(f)39 b FT(and)30 b FP(c)h FT(are)f(constan)m(ts.)519
3636 y(The)g(follo)m(wing)f(is)g(a)i FN(C)5 b(B)s(S)i(E)38
b FT(refutation)30 b(of)g(these)h(clauses:)p 1514 3798
752 4 v 1514 3883 a FN(:)p FP(I)7 b FT(\()p FP(v)1701
3897 y FL(1)1741 3883 y FT(\))15 b FN(j)g FP(v)1875 3897
y FL(1)1941 3883 y FT(=)25 b FP(e)35 b FN(\001)h(fg)2307
3821 y FT(\(start\))p 1188 3926 1405 4 v 1188 4011 a
FN(:)p FP(I)7 b FT(\()p FP(v)1375 4025 y FL(1)1414 4011
y FT(\))p FP(;)15 b(I)7 b FT(\()p FP(f)j FT(\))p FP(;)15
b FN(h)p FP(v)1780 4025 y FL(1)1821 4011 y FN(i)26 b(6)p
FT(=)f FN(h)p FP(f)10 b FN(i)15 b(j)g FP(v)2202 4025
y FL(1)2267 4011 y FT(=)25 b FP(e)36 b FN(\001)f(fg)2634
3949 y FT(\(expand\))p 1188 4053 V 1544 4138 a FP(v)1588
4152 y FL(1)1652 4138 y FT(=)25 b FP(e)36 b FN(\001)g(f)p
FP(v)1976 4152 y FL(1)2041 4138 y FN(')25 b FP(f)10 b
FN(g)2634 4076 y FT(\(er\))p 1369 4181 1042 4 v 1369
4266 a FP(v)1413 4280 y FL(1)1478 4266 y FT(=)25 b FP(e;)15
b(P)e FT(\()p FP(e;)i(v)1888 4280 y FL(2)1929 4266 y
FT(\))36 b FN(\001)g(f)p FP(v)2150 4280 y FL(1)2215 4266
y FN(')25 b FP(f)10 b FN(g)2452 4204 y FT(\(expand\))p
884 4308 2012 4 v 884 4393 a FP(v)928 4407 y FL(1)993
4393 y FT(=)25 b FP(e;)15 b(P)e FT(\()p FP(e;)i(v)1403
4407 y FL(2)1444 4393 y FT(\))p FP(;)g FN(:)p FP(P)e
FT(\()p FP(f)5 b(;)15 b(c)p FT(\))p FP(;)g FN(h)p FP(e;)g(v)2051
4407 y FL(2)2093 4393 y FN(i)26 b(6)p FT(=)f FN(h)p FP(f)5
b(;)15 b(c)p FN(i)36 b(\001)g(f)p FP(v)2635 4407 y FL(1)2700
4393 y FN(')25 b FP(f)10 b FN(g)2937 4331 y FT(\(expand\))p
884 4436 V 971 4520 a FP(v)1015 4534 y FL(1)1079 4520
y FT(=)25 b FP(e;)15 b(P)e FT(\()p FP(e;)i(v)1489 4534
y FL(2)1531 4520 y FT(\))p FP(;)g FN(:)p FP(P)e FT(\()p
FP(f)5 b(;)15 b(c)p FT(\))p FP(;)g FN(h)p FP(v)2056 4534
y FL(2)2097 4520 y FN(i)26 b(6)p FT(=)f FN(h)p FP(c)p
FN(i)36 b(\001)g(f)p FP(v)2549 4534 y FL(1)2614 4520
y FN(')25 b FP(f)10 b FN(g)2937 4458 y FT(\(simplify\))p
971 4563 1839 4 v 1482 4648 a FN(fg)36 b(\001)g(f)p FP(v)1758
4662 y FL(1)1823 4648 y FN(')25 b FP(f)5 b(;)15 b(v)2053
4662 y FL(2)2117 4648 y FN(')25 b FP(c)p FN(g)2851 4586
y FT(\(er\))378 4852 y(The)36 b(inequalit)m(y)f FN(h)p
FP(e;)15 b(v)1160 4866 y FL(2)1201 4852 y FN(i)35 b(6)p
FT(=)h FN(h)p FP(f)5 b(;)15 b(c)p FN(i)37 b FT(is)f(simpli\014ed)c(in)m
(to)37 b FN(h)p FP(v)2397 4866 y FL(2)2437 4852 y FN(i)f(6)p
FT(=)f FN(h)p FP(c)p FN(i)j FT(so)e(that)i(equalit)m(y)e(re\015exivit)m
(y)378 4965 y(can)31 b(b)s(e)e(used)h(to)h(close)g(the)f(tableau.)519
5078 y(In)37 b(the)i(follo)m(wing)d(example)i(w)m(e)h(sho)m(w)f(ho)m(w)
g(a)g(branc)m(h)g(is)f(closed)h(using)f(the)h(rules)f(of)h(rigid)378
5191 y(basic)30 b(sup)s(erp)s(osition.)37 b(The)30 b(branc)m(h)g(w)m(e)
g(consider)g(is)f(the)i(follo)m(wing:)712 5395 y FP(f)10
b FT(\()p FP(a)p FT(\))25 b(=)g FP(a;)46 b(g)s FT(\()p
FP(f)10 b FT(\()p FP(x)p FT(\)\))27 b(=)e FP(f)10 b FT(\()p
FP(g)s FT(\()p FP(x)p FT(\)\))p FP(;)47 b(h)p FT(\()p
FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s FT(\))27
b(=)e FP(f)10 b FT(\()p FP(y)s FT(\))p FP(;)46 b(h)p
FT(\()p FP(g)s FT(\()p FP(z)t FT(\))p FP(;)15 b(z)t FT(\))29
b FN(6)p FT(=)24 b FP(g)s FT(\()p FP(a)p FT(\))37 b FN(\001)f(fg)378
5599 y FT(where)j FP(x)p FT(,)i FP(y)h FT(and)d FP(z)k
FT(are)d(free)f(v)-5 b(ariables)38 b(and)h FP(a)g FT(is)f(a)i(constan)m
(t.)69 b(W)-8 b(e)40 b(use)f(the)g(lexicographical)p
eop
%%Page: 86 96
86 95 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(86)378 396 y(path)36 b(ordering)g(with)f FP(f)45
b FN(\037)36 b FP(g)j FN(\037)d FP(h)g FN(\037)f FP(a)i
FT(as)g(the)g(required)e(reduction)g(ordering)h(in)f(the)i(tableau)378
509 y(constrain)m(ts.)54 b(If)34 b(the)h(equations)g(in)e(this)h(branc)
m(h)g(are)h(treated)h(naiv)m(ely)-8 b(,)36 b(then)e(the)h(searc)m(h)g
(space)378 622 y(whic)m(h)29 b(needs)h(to)h(b)s(e)f(considered)f(is)g
(v)m(ery)i(large)g(since)e(there)i(are)g(n)m(umerous)e(p)s(ossible)f
(inferences)378 735 y(whic)m(h)35 b(can)i(b)s(e)e(considered.)57
b(Ho)m(w)m(ev)m(er,)41 b(the)36 b(searc)m(h)h(space)g(whic)m(h)e(is)g
(considered)g(b)m(y)h(the)h(rigid)378 848 y(basic)d(sup)s(erp)s
(osition)d(rules)h(is)i(m)m(uc)m(h)g(restricted.)53 b(F)-8
b(or)35 b(example,)g(there)f(are)h(only)f(t)m(w)m(o)h(p)s(ossible)378
961 y(inferences)21 b(whic)m(h)h(can)g(b)s(e)g(applied)e(on)i(the)h(ab)
s(o)m(v)m(e)h(branc)m(h.)37 b(These)22 b(are)h(a)g(left)f(rigid)e(sup)s
(erp)s(osition)378 1074 y(of)27 b(the)h(\014rst)f(literal)e(on)j(the)f
(third,)g(and)f(a)i(left)f(rigid)e(sup)s(erp)s(osition)f(of)k(the)f
(second)h(literal)d(on)j(the)378 1187 y(third.)60 b(W)-8
b(e)38 b(denote)g(the)g(application)d(of)j(a)g(sup)s(erp)s(osition)33
b(rule)j(b)m(y)i(orien)m(ting)e(the)i(equalit)m(y)f(of)378
1300 y(the)c(sup)s(erp)s(ositioning)c(literal)j(using)g
FP(l)g FN(!)e FP(r)36 b FT(or)d FP(l)f FN( )e FP(r)s
FT(,)j(and)g(underlining)c(the)k(sup)s(erp)s(ositioned)378
1413 y(subterm.)519 1526 y(The)40 b(\014rst)f(p)s(ossible)f(inference)h
(is)h(a)g(sup)s(erp)s(osition)d(of)j FP(f)10 b FT(\()p
FP(a)p FT(\))42 b FN(!)f FP(a)f FT(on)g FP(h)p FT(\()p
FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s FT(\))45
b(=)c FP(f)10 b FT(\()p FP(y)s FT(\))p 3630 1563 173
4 v(,)378 1638 y(whic)m(h)29 b(giv)m(es)i(the)f(branc)m(h)666
1843 y FP(f)10 b FT(\()p FP(a)p FT(\))25 b(=)g FP(a;)46
b(g)s FT(\()p FP(f)10 b FT(\()p FP(x)p FT(\)\))27 b(=)e
FP(f)10 b FT(\()p FP(g)s FT(\()p FP(x)p FT(\)\))p FP(;)47
b(h)p FT(\()p FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s
FT(\))27 b(=)e FP(a;)46 b(h)p FT(\()p FP(g)s FT(\()p
FP(z)t FT(\))p FP(;)15 b(z)t FT(\))29 b FN(6)p FT(=)c
FP(g)s FT(\()p FP(a)p FT(\))36 b FN(\001)g(f)p FP(y)28
b FN(')d FP(a)p FN(g)378 2047 y FT(W)-8 b(e)30 b(do)f(not)g(include)d
(the)j(ordered)g(constrain)m(ts)f FP(f)10 b FT(\()p FP(y)s
FT(\))25 b FN(\037)g FP(h)p FT(\()p FP(g)s FT(\()p FP(y)s
FT(\))p FP(;)15 b(y)s FT(\))32 b(and)c FP(f)10 b FT(\()p
FP(a)p FT(\))25 b FN(\037)g FP(a)k FT(b)s(ecause)g(they)378
2160 y(are)k(trivially)c(satis\014able.)46 b(The)32 b(only)f(p)s
(ossible)f(inferences)i(at)h(this)e(stage)j(is)d(a)i(sup)s(erp)s
(osition)c(of)378 2273 y FP(h)p FT(\()p FP(g)s FT(\()p
FP(y)s FT(\))p FP(;)15 b(y)s FT(\))32 b FN(!)c FP(a)33
b FT(on)g FP(h)p FT(\()p FP(g)s FT(\()p FP(z)t FT(\))p
FP(;)15 b(z)t FT(\))p 1113 2310 374 4 v 32 w FN(6)p FT(=)29
b FP(g)s FT(\()p FP(a)p FT(\),)35 b(resulting)30 b(in)i(the)h(follo)m
(wing)e(branc)m(h)h(whic)m(h)f(cannot)j(b)s(e)378 2386
y(closed,)c(and)g(no)g(other)h(rigid)d(basic)i(sup)s(erp)s(osition)d
(rule)i(is)h(applicable)e(to)j(it:)701 2590 y FP(f)10
b FT(\()p FP(a)p FT(\))25 b(=)g FP(a;)46 b(g)s FT(\()p
FP(f)10 b FT(\()p FP(x)p FT(\)\))26 b(=)f FP(f)10 b FT(\()p
FP(g)s FT(\()p FP(x)p FT(\)\))p FP(;)47 b(h)p FT(\()p
FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s FT(\))28
b(=)d FP(a;)46 b(a)25 b FN(6)p FT(=)g FP(g)s FT(\()p
FP(a)p FT(\))37 b FN(\001)e(f)p FP(y)29 b FN(')c FP(a;)15
b(z)30 b FN(')25 b FP(a)p FN(g)519 2794 y FT(W)-8 b(e)31
b(no)m(w)e(consider)f(the)h(second)h(p)s(ossible)d(inference)h(whic)m
(h)g(can)h(b)s(e)g(applied)e(on)i(the)h(original)378
2907 y(branc)m(h.)39 b(This)27 b(is)h(a)h(sup)s(erp)s(osition)c(of)j
FP(g)s FT(\()p FP(f)10 b FT(\()p FP(x)p FT(\)\))27 b
FN( )e FP(f)10 b FT(\()p FP(g)s FT(\()p FP(x)p FT(\)\))30
b(on)e FP(h)p FT(\()p FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15
b(y)s FT(\))28 b(=)d FP(f)10 b FT(\()p FP(y)s FT(\))p
3125 2944 173 4 v(,)29 b(whic)m(h)e(giv)m(es:)482 3111
y FP(f)10 b FT(\()p FP(a)p FT(\))26 b(=)f FP(a;)45 b(g)s
FT(\()p FP(f)10 b FT(\()p FP(x)p FT(\)\))27 b(=)e FP(f)10
b FT(\()p FP(g)s FT(\()p FP(x)p FT(\)\))p FP(;)47 b(h)p
FT(\()p FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s FT(\))28
b(=)c FP(g)s FT(\()p FP(f)10 b FT(\()p FP(x)p FT(\)\))p
FP(;)47 b(h)p FT(\()p FP(g)s FT(\()p FP(z)t FT(\))p FP(;)15
b(z)t FT(\))29 b FN(6)p FT(=)c FP(g)s FT(\()p FP(a)p
FT(\))37 b FN(\001)e(f)p FP(y)29 b FN(')c FP(g)s FT(\()p
FP(x)p FT(\))p FN(g)378 3316 y FT(This)k(can)h(only)g(b)s(e)f(follo)m
(w)m(ed)h(b)m(y)h(a)f(sup)s(erp)s(osition)d(of)k FP(f)10
b FT(\()p FP(a)p FT(\))25 b FN(!)g FP(a)31 b FT(on)f
FP(h)p FT(\()p FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15
b(y)s FT(\))28 b(=)d FP(g)s FT(\()p FP(f)10 b FT(\()p
FP(x)p FT(\))p 3392 3353 177 4 v 1 w(\):)416 3520 y FP(f)g
FT(\()p FP(a)p FT(\))25 b(=)g FP(a;)46 b(g)s FT(\()p
FP(f)10 b FT(\()p FP(x)p FT(\)\))26 b(=)f FP(f)10 b FT(\()p
FP(g)s FT(\()p FP(x)p FT(\)\))p FP(;)47 b(h)p FT(\()p
FP(g)s FT(\()p FP(y)s FT(\))p FP(;)15 b(y)s FT(\))28
b(=)d FP(g)s FT(\()p FP(a)p FT(\))p FP(;)47 b(h)p FT(\()p
FP(g)s FT(\()p FP(z)t FT(\))p FP(;)15 b(z)t FT(\))29
b FN(6)p FT(=)c FP(g)s FT(\()p FP(a)p FT(\))37 b FN(\001)e(f)p
FP(x)26 b FN(')f FP(a;)15 b(y)28 b FN(')d FP(g)s FT(\()p
FP(x)p FT(\))p FN(g)378 3724 y FT(and)e(then)g(only)f(b)m(y)i(a)g(sup)s
(erp)s(osition)19 b(of)24 b FP(h)p FT(\()p FP(g)s FT(\()p
FP(y)s FT(\))p FP(;)15 b(y)s FT(\))28 b FN(!)d FP(g)s
FT(\()p FP(a)p FT(\))g(on)e FP(h)p FT(\()p FP(g)s FT(\()p
FP(z)t FT(\))p FP(;)15 b(z)t FT(\))p 2606 3761 374 4
v 29 w FN(6)p FT(=)25 b FP(g)s FT(\()p FP(a)p FT(\),)h(whic)m(h)c
(results)378 3837 y(in)29 b(the)i(follo)m(wing)d(trivially)g(refutable)
i(branc)m(h:)379 4041 y FP(f)10 b FT(\()p FP(a)p FT(\))26
b(=)f FP(a;)45 b(g)s FT(\()p FP(f)10 b FT(\()p FP(x)p
FT(\)\))27 b(=)e FP(f)10 b FT(\()p FP(g)s FT(\()p FP(x)p
FT(\)\))p FP(;)47 b(h)p FT(\()p FP(g)s FT(\()p FP(y)s
FT(\))p FP(;)15 b(y)s FT(\))28 b(=)d FP(g)s FT(\()p FP(a)p
FT(\))p FP(;)47 b(g)s FT(\()p FP(a)p FT(\))26 b FN(6)p
FT(=)f FP(g)s FT(\()p FP(a)p FT(\))37 b FN(\001)f(f)p
FP(x)25 b FN(')g FP(a;)15 b(y)29 b FN(')c FP(g)s FT(\()p
FP(x)p FT(\))p FP(;)15 b(z)31 b FN(')25 b FP(y)s FN(g)p
FP(:)378 4328 y FH(5.3)135 b(The)45 b(T)-11 b(ableau)45
b(Calculus)g(in)g(HOL)378 4531 y FT(In)29 b(this)g(section)h(w)m(e)h
(describ)s(e)d(the)i(implemen)m(tation)f(of)h(the)g FN(C)5
b(B)s(S)i(E)38 b FT(calculus)29 b(as)h(a)g(HOL)g(deriv)m(ed)378
4643 y(rule.)55 b(This)33 b(rule)h(tak)m(es)j(a)f(list)e(of)i(theorems)
f(\000)2046 4657 y FL(1)2119 4643 y FN(`)e FP(t)2241
4657 y FL(1)2281 4643 y FP(;)15 b(:)g(:)g(:)31 b(;)15
b FT(\000)2554 4657 y FO(n)2635 4643 y FN(`)33 b FP(t)2757
4657 y FO(n)2840 4643 y FT(and)h(refutes)i(the)f(form)m(ulae)378
4756 y FP(t)411 4770 y FL(1)450 4756 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(t)700 4770 y FO(n)777 4756 y FT(to)32 b(return)d(a)i(theorem)f
(with)g(the)g(conclusion)f FN(?)p FT(,)h(that)h(is:)1532
4937 y(\000)1589 4951 y FL(1)1653 4937 y FN(`)25 b FP(t)1767
4951 y FL(1)1897 4937 y FN(\001)15 b(\001)g(\001)107
b FT(\000)2166 4951 y FO(n)2238 4937 y FN(`)25 b FP(t)2352
4951 y FO(n)p 1532 4970 868 4 v 1623 5050 a FT(\000)1680
5064 y FL(1)1739 5050 y FN([)20 b(\001)15 b(\001)g(\001)21
b([)f FT(\000)2084 5064 y FO(n)2156 5050 y FN(`)25 b(?)2441
5001 y(C)5 b(B)s(S)i(E)p eop
%%Page: 87 97
87 96 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(87)378 396 y(This)20 b(rule)h(can)h(b)s(e)f(used,)i(for)f
(instance)f(to)i(pro)m(v)m(e)g(a)f(goal)h FP(p)e FT(b)m(y)h(refuting)f
(the)h(conclusion)e(of)i FN(:)p FP(p)j FN(`)g(:)p FP(p)378
509 y FT(to)31 b(return)e FN(:)p FP(p)c FN(`)g(?)p FT(,)30
b(whic)m(h)f(can)i(b)s(e)f(used)f(to)i(infer)e FP(p)p
FT(:)p 1553 661 319 4 v 1553 741 a FN(:)p FP(p)c FN(`)g(:)p
FP(p)1914 677 y Fw(ASSUME)41 b(o)i(mk_neg)e Fv(p)p 1553
779 V 1571 858 a FN(:)p FP(p)25 b FN(`)f(?)1914 810 y(C)5
b(B)s(S)i(E)p 1571 896 284 4 v 1649 976 a(`)25 b FP(p)1896
913 y Fw(CCONTR)41 b Fv(p)378 1180 y FT(The)32 b(form)m(ulae)h
FP(t)977 1194 y FL(1)1016 1180 y FT(,)g FP(:)15 b(:)g(:)32
b FT(,)p FP(t)1269 1194 y FO(n)1348 1180 y FT(are)i(assumed)e(to)i(b)s
(e)e(\014rst-order.)47 b(As)33 b(one)g(often)h(requires)d(to)j(reason)
378 1293 y(with)21 b(higher-order)g(form)m(ulae,)j(a)f(mec)m(hanism)f
(for)g(translating)g(higher-order)f(form)m(ulae)h(in)m(to)g(\014rst-)
378 1406 y(order)30 b(ones)g(is)g(describ)s(ed)e(in)h(section)i(5.4.)
519 1519 y(F)-8 b(or)38 b(e\016ciency)g(reasons,)h(the)f(implemen)m
(tation)e(of)i(the)f(pro)s(of)g(searc)m(h)h(algorithm)f(do)s(es)g(not)
378 1632 y(use)g(the)h(HOL)f(term)h(represen)m(tation,)h(but)e(a)h
(simple)d(represen)m(tation)j(b)s(etter)g(suited)e(for)h(\014rst-)378
1744 y(order)j(form)m(ulae.)69 b(The)40 b FN(C)5 b(B)s(S)i(E)48
b FT(rule)39 b(transforms)g(the)h(giv)m(en)g(HOL)g(theorems)h(in)m(to)f
(\014rst-order)378 1857 y(clauses)33 b(in)f(this)g(represen)m(tation)i
(and)f(then)g(uses)g(them)g(to)h(\014nd)e(a)i(closed)f(tableau.)50
b(The)33 b(list)f(of)378 1970 y(inferences)g(required)g(to)h(\014nd)f
(the)h(closed)g(tableau)g(are)h(then)f(used)f(to)i(deriv)m(e)f(a)g(HOL)
g(theorem.)378 2083 y(The)h(refutation)g(pro)s(cess)g(of)g(the)h(deriv)
m(ed)e(rule)g(can)i(therefore)g(b)s(e)f(seen)g(as)h(consisting)e(of)i
(three)378 2196 y(distinct)29 b(stages:)489 2384 y(1.)46
b(the)41 b(prepro)s(cessing)e(stage,)45 b(in)39 b(whic)m(h)h(HOL)g
(theorems)h(are)g(transformed)f(in)m(to)g(a)i(set)f(of)605
2497 y(clauses;)489 2684 y(2.)46 b(the)30 b(actual)h(pro)s(of)e(searc)m
(h,)i(in)d(whic)m(h)h(the)h FN(C)5 b(B)s(S)i(E)37 b FT(rules)29
b(are)h(applied)e(to)j(the)f(set)g(of)g(clauses)605 2797
y(to)h(\014nd)e(a)i(closed)f(tableau;)489 2985 y(3.)46
b(the)29 b(pro)s(of)e(transformation)h(stage,)i(where)e(a)g(successful)
f(sequence)i(of)f FN(C)5 b(B)s(S)i(E)36 b FT(inferences)27
b(is)605 3098 y(used)j(to)h(deriv)m(e)f(the)g(required)f(HOL)h
(theorem.)519 3285 y(W)-8 b(e)24 b(remark)e(that)h(the)f(main)f(motiv)
-5 b(ation)22 b(of)h(this)e(implemen)m(tation)g(is)g(to)i(use)f(the)h
(deriv)m(ed)e(rule)378 3398 y(as)33 b(a)h(pro)s(of)e(c)m(hec)m(king)i
(supp)s(ort)d(for)i(the)g(SPL)f(language.)49 b(Since)32
b(the)i(straigh)m(tforw)m(ard)f(justi\014ca-)378 3511
y(tions)20 b(in)g(SPL)f(scripts)h(in)g(general)g(corresp)s(ond)g(to)h
(rather)g(simple)e(problems,)i(our)f(implemen)m(tation)378
3624 y(is)29 b(not)h(mean)m(t)h(to)g(b)s(e)e(used)g(as)i(an)f
(e\016cien)m(t)g(to)s(ol)g(for)g(\014nding)d(non-trivial)h(pro)s(ofs.)
40 b(In)29 b(particular,)378 3737 y(w)m(e)g(ha)m(v)m(e)i(not)e(exp)s
(erimen)m(ted)f(with)g(a)h(wide)f(range)i(of)f(searc)m(h)g(strategies)h
(and)f(heuristics)e(to)j(cop)s(e)378 3850 y(with)c(large)i(searc)m(h)g
(spaces,)g(and)f(w)m(e)h(ha)m(v)m(e)g(not)g(put)f(substan)m(tial)f
(e\013ort)i(in)e(remo)m(ving)h(an)m(y)h(redun-)378 3963
y(dan)m(t)k(inferences)e(from)h(the)h(pro)s(of)f(found)f(b)m(y)i(the)f
(searc)m(h)i(stage)g(of)e(the)h(implemen)m(tation)e(b)s(efore)378
4076 y(a)h(HOL)f(theorem)h(is)e(deriv)m(ed.)519 4189
y(The)48 b(three)g(stages)h(of)f(the)g(refutation)g(pro)s(cess)f(are)i
(describ)s(ed)c(in)i(more)h(detail)f(in)g(sec-)378 4302
y(tions)42 b(5.3.2)i(\(prepro)s(cessing)c(theorems\),)46
b(5.3.3)e(\(pro)s(of)e(searc)m(h\),)47 b(and)41 b(5.3.4)j(\(pro)s(of)e
(transfor-)378 4414 y(mation\).)j(Since)31 b(terms)g(in)g(the)g(HOL)h
(logic)f(can)h(b)s(e)f(p)s(olymorphic,)f(w)m(e)i(\014rst)f(illustrate)f
(the)i(w)m(a)m(y)378 4527 y(p)s(olymorphic)27 b(theorems)k(are)g
(handled)d(in)h(section)i(5.3.1.)378 4771 y FG(5.3.1)112
b(Reasoning)38 b(with)e(P)m(olymorphic)f(F)-9 b(orm)m(ulae)378
4942 y FT(As)33 b(describ)s(ed)e(in)g(section)j(1.2.2,)h(HOL)e(terms)g
(are)g(t)m(yp)s(ed)g(b)m(y)g(simple)e(\(i.e.,)15 b(\014rst-order\))33
b(expres-)378 5055 y(sions)k(whic)m(h)g(ma)m(y)h(con)m(tain)h(t)m(yp)s
(e)f(v)-5 b(ariables.)63 b(T)m(yp)s(e)37 b(v)-5 b(ariables)37
b(can)i(b)s(e)e(instan)m(tiated)h(to)h(other)378 5168
y(t)m(yp)s(es,)32 b(and)f(this)f(pro)m(vides)g(a)i(means)f(of)h
(de\014ning)d(p)s(olymorphic)f(constan)m(ts)33 b(and)e(deriving)e(p)s
(oly-)378 5281 y(morphic)36 b(theorems.)64 b(In)37 b(order)h(to)h(use)e
(suc)m(h)h(p)s(olymorphic)d(form)m(ulae)j(e\013ectiv)m(ely)-8
b(,)41 b(the)d(imple-)378 5394 y(men)m(tation)30 b(of)g(\014rst-order)f
(pro)s(of)h(calculi)e(as)i(deriv)m(ed)f(rules)g(in)f(a)j(theorem)f(pro)
m(v)m(er)g(m)m(ust)g(b)s(e)f(able)378 5507 y(to)36 b(instan)m(tiate)g
(t)m(yp)s(e)g(v)-5 b(ariables)34 b(during)g(the)i(refutation)f(pro)s
(cess.)56 b(Ho)m(w)m(ev)m(er,)39 b(most)d(commonly)378
5620 y(used)26 b(implemen)m(tations)g(do)h(not)h(instan)m(tiate)f(t)m
(yp)s(e)g(v)-5 b(ariables)26 b(during)f(the)j(pro)s(of)e(searc)m(h)i
(pro)s(cess,)p eop
%%Page: 88 98
88 97 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(88)378 396 y(but)24 b(treat)i(p)s(olymorphism)21
b(in)j(a)h(rather)f(indirect)f(w)m(a)m(y)-8 b(.)41 b(F)-8
b(or)25 b(instance,)h(t)m(yp)s(e)f(instan)m(tiation)f(is)g(p)s(er-)378
509 y(formed)34 b(in)e(the)j(prepro)s(cessing)d(stage)k(of)e(the)h
Fw(MESON)d FT(pro)m(v)m(er)i(supplied)d(with)i(the)i(HOL)e(system.)378
622 y(Giv)m(en)24 b(a)h(list)e(of)i(theorems)f(to)i(b)s(e)d(used)h(for)
g(refutation,)i(a)e(p)s(olymorphic)e(theorem)j(is)e(instan)m(tiated)378
735 y(to)36 b(a)g(n)m(um)m(b)s(er)f(of)h(less)f(general)g(theorems)h
(according)g(to)h(the)f(ground)e(t)m(yp)s(es)i(in)e(the)i(input)e
(list.)378 848 y(This)26 b(metho)s(d)h(is)g(incomplete)g(and)g(often)h
(generates)h(sev)m(eral)f(redundan)m(t)f(clauses.)40
b(The)27 b(classical)378 961 y(pro)m(v)m(er)k(of)f(Isab)s(elle)f
(considers)g(terms)h(to)h(b)s(e)f(un)m(t)m(yp)s(ed)f(during)f(the)j
(pro)s(of)e(searc)m(h,)i(and)f(an)m(y)h(t)m(yp)s(e)378
1074 y(instan)m(tiations)h(are)i(p)s(erformed)e(during)f(the)j(pro)s
(of)e(transformation)h(stage.)51 b(If)33 b(an)h(in)m(v)-5
b(alid)31 b(t)m(yp)s(e)378 1187 y(instan)m(tiation)i(is)g(encoun)m
(tered)i(during)d(the)i(transformation)f(pro)s(cess,)i(the)f(pro)s(of)g
(searc)m(h)g(stage)378 1300 y(is)h(used)h(again)g(to)h(\014nd)e
(another)h(\(p)s(ossibly)e(in)m(v)-5 b(alid\))34 b(pro)s(of.)58
b(Ho)m(w)m(ev)m(er)39 b(t)m(yp)s(e)d(instan)m(tiation)f(\(of)378
1413 y(simple)24 b(t)m(yp)s(es\))i(can)g(b)s(e)f(easily)g(incorp)s
(orated)g(in)g(the)h(pro)s(of)f(searc)m(h)h(pro)s(cess)g(of)g(a)g
(\014rst-order)f(logic)378 1526 y(calculus.)38 b(In)27
b(this)f(section,)i(w)m(e)g(illustrate)e(ho)m(w)h(this)f(can)i(b)s(e)e
(done)h(after)h(remarking)f(on)g(a)g(couple)378 1638
y(of)j(p)s(oin)m(ts)g(on)g(the)h(v)-5 b(alidit)m(y)28
b(of)j(t)m(yp)s(e)f(instan)m(tiations.)378 1878 y FQ(The)35
b(V)-9 b(alidit)m(y)35 b(of)g(T)m(yp)s(e)f(Instan)m(tiations)378
2050 y FT(It)27 b(should)e(b)s(e)h(noted)h(that)h(not)f(all)f(t)m(yp)s
(e)h(instan)m(tiations)f(are)h(v)-5 b(alid.)38 b(Giv)m(en)27
b(a)g(theorem)g(\000)e FN(`)g FP(t)p FT(,)j(the)378 2163
y(instan)m(tiation)h(of)i(the)f(t)m(yp)s(es)h(in)e FP(t)h
FT(\(without)g(the)g(instan)m(tiation)g(of)g(the)h(t)m(yp)s(es)f(in)f
(\000\))i(is)e(v)-5 b(alid)29 b(if)489 2347 y(1.)46 b(no)30
b(t)m(yp)s(e)h(v)-5 b(ariables)29 b(o)s(ccurring)g(in)g(\000)h(are)h
(instan)m(tiated,)489 2534 y(2.)46 b(no)33 b(distinct)f(v)-5
b(ariables)31 b(b)s(ecome)j(iden)m(ti\014ed)d(after)j(the)f(instan)m
(tiation.)48 b(This)31 b(o)s(ccurs)i(when)605 2647 y(t)m(w)m(o)24
b(v)-5 b(ariables)21 b(with)h(the)g(same)h(name)g(but)f(with)f
(di\013eren)m(t)h(t)m(yp)s(es)h(\(suc)m(h)g(as)f Fv(x)p
Fw(:'a)g FT(and)g Fv(x)p Fw(:'b)o FT(\),)605 2760 y(whic)m(h)i(are)i
(considered)e(as)h(distinct)f(in)g(HOL,)h(are)g(instan)m(tiated)g(to)h
(the)f(same)h(v)-5 b(ariable)24 b(\(for)605 2872 y(example)30
b(with)f FN(f)p Fw('a)c FN(!)g Fw('b)p FN(g)p FT(\).)378
3057 y(The)f(\014rst)g(restriction)g(implies)e(that)j(giv)m(en)g(the)g
(input)e(list)g(of)i(theorems)g(\000)2970 3071 y FL(1)3035
3057 y FN(`)f FP(t)3148 3071 y FL(1)3188 3057 y FP(;)15
b(:)g(:)g(:)31 b(;)15 b FT(\000)3461 3071 y FO(n)3534
3057 y FN(`)25 b FP(t)3648 3071 y FO(n)3719 3057 y FT(for)378
3170 y(refutation,)j(the)f(t)m(yp)s(e)g(v)-5 b(ariables)26
b(in)g FP(t)1687 3184 y FO(i)1742 3170 y FT(whic)m(h)f(are)j(also)f(in)
f(\000)2486 3184 y FO(i)2514 3170 y FT(,)i(for)f(1)e
FN(\024)g FP(i)h FN(\024)f FP(n)p FT(,)i(should)e(b)s(e)i(mark)m(ed)378
3283 y(as)g FI(uninstantiatable)35 b FT(and)25 b(the)i(rest)g(as)f
FI(instantiatable)p FT(,)k(suc)m(h)c(that)h(only)e(the)i(instan)m
(tiatable)f(t)m(yp)s(e)378 3396 y(v)-5 b(ariables)42
b(can)h(b)s(e)f(considered)g(for)h(instan)m(tiation)f(during)f(pro)s
(of)h(searc)m(h.)80 b(W)-8 b(e)44 b(remark,)i(that)378
3509 y(the)39 b(instan)m(tiation)e(of)i(an)f(uninstan)m(tiatable)f(t)m
(yp)s(e)h(v)-5 b(ariable)38 b(ma)m(y)h(result)e(in)g(the)i(deriv)-5
b(ation)37 b(of)378 3622 y(an)i(unexp)s(ected)f(theorem,)k(or)d
(otherwise)g(in)f(a)h(failed)f(pro)s(of)g(transformation.)66
b(F)-8 b(or)40 b(example,)378 3735 y(supp)s(ose)29 b(the)h
FN(C)5 b(B)s(S)i(E)38 b FT(calculus)29 b(is)h(used)f(to)i(deriv)m(e)f
(the)h(\(in)m(v)-5 b(alid\))29 b(form)m(ula)473 3919
y FP(P)61 b FM(\()p FP(c)p FM(:num)47 b(list\))f FN(\))i(9)15
b FP(x)p FM(:'a)47 b(list.)15 b FP(P)60 b(x)378 4104
y FT(where)41 b FP(P)54 b FT(is)40 b(some)i(p)s(olymorphic)c(predicate)
j(of)g(t)m(yp)s(e)h Fw(:'a)g(list)g Fu(!)h Fw(bool)d
FT(and)h Fv(c)p Fw(:num)g(list)f FT(is)378 4217 y(some)31
b(constan)m(t.)42 b(This)28 b(is)i(done)g(b)m(y)g(refuting)f(the)i
(conclusion)e(of)473 4402 y FP(P)61 b FM(\()p FP(c)p
FM(:num)47 b(list\))f FN(\))i(9)15 b FP(x)p FM(:'a)47
b(list.)15 b FP(P)60 b(x)760 4515 y FN(`)47 b FP(P)61
b FM(\()p FP(c)p FM(:num)46 b(list\))h FN(\))g(9)16 b
FP(x)p FM(:'a)47 b(list.)14 b FP(P)61 b(x)378 4699 y
FT(whic)m(h)29 b(is)g(transformed)h(in)m(to)g(the)h(clauses)569
4884 y FN(:)p FP(P)60 b FM(\()p FP(x)p FM(:'a)47 b(list\))569
4997 y FP(P)61 b FM(\()p FP(c)p FM(:num)46 b(list\))378
5182 y FT(where)20 b(the)g(t)m(yp)s(e)h(v)-5 b(ariable)19
b(\()p Fw(:'a)o FT(\))h(is)g(mark)m(ed)g(as)h(uninstan)m(tiatable,)g
(and)e(as)i(a)g(result)e(the)h(refutation)378 5295 y(fails.)47
b(Please)33 b(note)h(that)f(the)g(ab)s(o)m(v)m(e)h(sen)m(tence)h(is)d
(in)f(general)i(not)g(v)-5 b(alid.)47 b(This)31 b(can)j(b)s(e)e(seen)h
(b)m(y)378 5408 y(substituting)25 b Fv(P)h(x)i FT(with)f
Fw(LENGTH)41 b(\(SETIFY)f Fv(x)p Fw(\))k(>)f(1)p FT(,)28
b(and)f FP(c)h FT(with)f Fw([1,2])n FT(.)40 b(The)27
b(resulting)f(sen)m(tence)378 5520 y(is)j(not)i(v)-5
b(alid)29 b(b)s(ecause)h(it)g(could)f(b)s(e)h(used)g(to)h(infer)473
5705 y FM(LENGTH)46 b(\(SETIFY)g([1,2]\))g(>)i(1)f FN(\))h(9)15
b FP(x)p FM(:'a)47 b(list.)g(LENGTH)f(\(SETIFY)f FP(x)p
FM(\))j(>)f(1)p eop
%%Page: 89 99
89 98 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(89)378 396 y(whic)m(h)29 b(yields)473 584 y FN(9)16
b FP(x)p FM(:'a)47 b(list.)f(LENGTH)g(\(SETIFY)g FP(x)p
FM(\))h(>)h(1)378 772 y FT(The)36 b(t)m(yp)s(e)h(instan)m(tiation)f
FN(f)p Fw(:'a)f FN(!)h Fw(:one)n FN(g)h FT(will)d(then)j(result)f(in)f
(an)i(in)m(v)-5 b(alid)34 b(result)i(as)h(the)g(t)m(yp)s(e)378
885 y Fw(:one)29 b FT(con)m(tains)h(only)g(one)g(distinct)f(elemen)m
(t.)519 998 y(On)f(the)h(other)g(hand)e(the)i(deriv)-5
b(ation)28 b(of)h Fu(:)p Fv(P)12 b Fw(\()p Fv(c)p Fw(:num)41
b(list\))27 b FT(from)h Fu(`)43 b(8)14 b Fw(\()p Fv(x)p
Fw(:'a)42 b(list\).)p Fu(:)p Fv(P)54 b(x)29 b FT(is)378
1110 y(equiv)-5 b(alen)m(t)35 b(to)g(the)h(refutation)e(of)i(the)f
(same)h(t)m(w)m(o)g(clauses)f(with)f(the)h(di\013erence)g(that)g(the)h
(t)m(yp)s(e)378 1223 y(v)-5 b(ariable)36 b(\()p Fw(:'a)o
FT(\))h(is)f(mark)m(ed)h(as)g(instan)m(tiatable,)h(and)e(therefore)i
(the)f(refutation)f(succeeds)h(with)378 1336 y(the)31
b(t)m(yp)s(e)f(substitution)e FN(f)p Fw('a)d FN(!)g Fw(num)o
FN(g)31 b FT(and)e(the)i(substitution)d FN(f)p FP(x)e
FN(!)f FP(c)p FN(g)p FT(.)519 1449 y(The)35 b(second)h(restriction)f
(giv)m(en)g(earlier)g(suggests)h(that)h(distinct)d(v)-5
b(ariables)34 b(with)h(the)g(same)378 1562 y(v)-5 b(ariable)27
b(name)h(\(but)g(di\013eren)m(t)g(t)m(yp)s(e\))h(should)d(b)s(e)i
(renamed)g(b)s(efore)f(pro)s(of)h(searc)m(h.)41 b(This)26
b(restric-)378 1675 y(tion)41 b(a)m(v)m(oids,)46 b(for)c(instance,)i
(the)e(in)m(v)-5 b(alid)40 b(instan)m(tiation)h(of)h
Fu(9)15 b Fw(\()p Fv(x)p Fw(:'a\),\()p Fv(x)p Fw(:'b\).)p
Fv(P)21 b Fw(\()p Fv(x)p Fw(:'a,)13 b Fv(x)p Fw(:'b\))378
1788 y FT(in)m(to)30 b Fu(9)15 b Fw(\()p Fv(x)p Fw(:'a\),\()p
Fv(x)p Fw(:'a\).)p Fv(P)22 b Fw(\()p Fv(x)p Fw(:'a,)12
b Fv(x)p Fw(:'a\))30 b FT(whic)m(h)f(is)g(equiv)-5 b(alen)m(t)30
b(to)h Fu(9)15 b Fw(\()p Fv(x)p Fw(:'a\).)p Fv(P)23 b
Fw(\()p Fv(x)p Fw(:'a,)13 b Fv(x)p Fw(:'a\))o FT(.)378
2028 y FQ(F)-9 b(rom)34 b(P)m(olymorphic)i(First-Order)e(F)-9
b(orm)m(ulae)35 b(to)g(Un)m(t)m(yp)s(ed)g(Ones)378 2200
y FT(Giv)m(en)h(t)m(w)m(o)i(\014rst-order)e(term)g(languages)h
FP(L)f FT(=)f FP(L)p FT(\(\006)2268 2214 y FO(L)2320
2200 y FP(;)15 b(X)2435 2214 y FO(L)2488 2200 y FT(\))37
b(and)e FP(T)49 b FT(=)35 b FP(T)13 b FT(\(\006)3117
2214 y FO(T)3172 2200 y FP(;)i(X)3287 2214 y FO(T)3343
2200 y FT(\),)38 b(where)e(\006)3776 2214 y FO(L)378
2313 y FT(and)c(\006)623 2327 y FO(T)710 2313 y FT(are)h(disjoin)m(t)f
(collections)g(of)h(function)e(sym)m(b)s(ols)g(with)h(\014xed)g
(arities,)g(and)g FP(X)3433 2327 y FO(L)3519 2313 y FT(and)g
FP(X)3773 2327 y FO(T)378 2425 y FT(are)e(disjoin)m(t)d(sets)j(of)f(v)
-5 b(ariables,)28 b(one)i(can)f(de\014ne)g(the)g(t)m(yp)s(ed)g
(language)h FP(L)2973 2444 y Fw(typ)n FL(\()p FO(T)10
b FL(\))3242 2425 y FT(of)30 b(the)f(terms)g(in)378 2538
y FP(L)h FT(t)m(yp)s(ed)g(with)f(the)i(terms)f(in)f FP(T)13
b FT(,)30 b(as)h(the)g(set)g(consisting)e(of:)2498 2505
y FL(2)489 2726 y FT(1.)46 b(t)m(yp)s(ed)30 b(v)-5 b(ariables)29
b FP(x)d FT(:)f FP(\033)s FT(,)31 b(where)f FP(x)g FT(is)f(in)g
FP(X)2093 2740 y FO(L)2176 2726 y FT(and)h FP(\033)j
FT(\(called)d(the)h(t)m(yp)s(e)f(of)h FP(x)25 b FT(:)h
FP(\033)s FT(\))31 b(is)e(in)g FP(T)13 b FT(,)489 2914
y(2.)46 b(t)m(yp)s(ed)28 b(constan)m(ts)i FP(c)25 b FT(:)h
FP(\033)s FT(,)j(where)f FP(c)g FT(is)g(a)g(constan)m(t)i(in)d(\006)
2509 2928 y FO(L)2561 2914 y FT(,)i(and)e FP(\033)32
b FT(\(called)c(the)g(t)m(yp)s(e)h(of)f FP(c)e FT(:)f
FP(\033)s FT(\))605 3027 y(is)30 b(in)f FP(T)13 b FT(,)489
3214 y(3.)46 b(t)m(yp)s(ed)30 b(comp)s(ound)f(terms)h(of)h(the)g(form:)
1509 3418 y(\()p FP(f)k FT(:)25 b FP(\014)1725 3432 y
FL(1)1790 3418 y FN(!)h(\001)15 b(\001)g(\001)26 b(!)f
FP(\014)2205 3432 y FO(n)2278 3418 y FN(!)g FP(\033)s
FT(\)\()p FP(t)2552 3432 y FL(1)2592 3418 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(t)2842 3432 y FO(n)2889 3418 y FT(\))605 3623
y(where)23 b FP(f)32 b FT(is)21 b(a)j(non-constan)m(t)f(function)f(in)f
(\006)2138 3637 y FO(L)2190 3623 y FT(,)k(and)d FP(\014)2460
3637 y FO(i)2511 3623 y FT(is)g(the)h(t)m(yp)s(e)g(of)g
FP(t)3070 3637 y FO(i)3121 3623 y FT(for)f FP(i)k FN(2)f(f)p
FT(1)p FP(;)15 b(:)g(:)g(:)32 b(;)15 b(n)p FN(g)p FT(,)605
3736 y(and)30 b(w)m(e)h(call)f FP(\033)j FT(the)e(t)m(yp)s(e)f(of)h
(the)f(ab)s(o)m(v)m(e)i(comp)s(ound)d(term.)519 3923
y(As)k(a)f(consequence)i(of)e(the)h(ab)s(o)m(v)m(e)g(discussion)d(on)j
(the)f(v)-5 b(alidit)m(y)31 b(of)i(t)m(yp)s(e)f(instan)m(tiations,)g(w)
m(e)378 4036 y(partition)24 b(the)h(set)g(of)g(t)m(yp)s(e)g(v)-5
b(ariables)24 b FP(X)1783 4050 y FO(T)1863 4036 y FT(in)m(to)h(t)m(w)m
(o)h(coun)m(table)f(sets:)39 b(a)25 b(set)g FP(X)3113
4003 y FO(i)3106 4063 y(T)3186 4036 y FT(of)h(instan)m(tiatable)378
4149 y(v)-5 b(ariables,)29 b(and)h(a)h(set)g FP(X)1257
4116 y FO(u)1250 4176 y(T)1335 4149 y FT(of)g(uninstan)m(tiatable)e(v)
-5 b(ariables.)519 4262 y(W)d(e)45 b(de\014ne)d(the)i(\(un)m(t)m(yp)s
(ed\))f(\014rst-order)f(language)i FN(f)p FP(T)13 b FN(g)p
FP(L)44 b FT(as)f(the)h(set)f(of)h(terms)f(o)m(v)m(er)i(the)378
4375 y(signature)30 b(\006)839 4393 y FK(f)p FO(T)10
b FK(g)p FO(L)1042 4375 y FT(and)30 b(the)h(set)f(of)h(v)-5
b(ariables)29 b FP(X)2073 4393 y FK(f)p FO(T)10 b FK(g)p
FO(L)2247 4375 y FT(,)30 b(where)514 4562 y FN(\017)46
b FP(X)680 4581 y FK(f)p FO(T)10 b FK(g)p FO(L)879 4562
y FT(=)25 b FP(X)1050 4576 y FO(L)1123 4562 y FN([)20
b FP(X)1286 4529 y FO(i)1279 4589 y(T)1334 4562 y FT(,)31
b(i.e.,)15 b(the)31 b(v)-5 b(ariables)29 b(in)g FP(L)h
FT(and)g(the)g(instan)m(tiatable)g(v)-5 b(ariables)29
b(in)g FP(T)13 b FT(,)514 4750 y FN(\017)46 b FT(\006)671
4769 y FK(f)p FO(T)10 b FK(g)p FO(L)870 4750 y FT(=)25
b(\006)1032 4764 y FO(L)1096 4750 y FN([)13 b FT(\006)1236
4764 y FO(T)1303 4750 y FN([)g FP(X)1459 4717 y FO(u)1452
4777 y(T)1520 4750 y FN([)g(f)p Fw(pair)n FN(g)p FT(,)28
b(where)e Fw(pair)f FT(is)h(a)h(new)f(binary)f(function)g(sym)m(b)s
(ol,)h(and)605 4863 y(w)m(e)32 b(write)f(\()p FP(s;)15
b(t)p FT(\))31 b(to)h(denote)g Fw(pair)o FT(\()p FP(s;)15
b(t)p FT(\).)44 b(The)30 b(set)i(of)g(function)e(sym)m(b)s(ols)g(in)g
FN(f)p FP(T)13 b FN(g)p FP(L)31 b FT(consists)605 4976
y(of)j(the)f(set)h(of)f(functions)f(in)g FP(L)p FT(,)i(the)g(functions)
d(in)h FP(T)13 b FT(,)34 b(the)g(uninstan)m(tiatable)d(v)-5
b(ariables)32 b(in)605 5089 y FP(T)13 b FT(,)31 b(and)e(the)i(new)f
(sym)m(b)s(ol)f Fw(pair)n FT(.)p 378 5252 1380 4 v 482
5305 a FC(2)516 5337 y FB(F)-6 b(or)24 b(the)f(purp)r(oses)g(of)h(this)
g(section)g(w)n(e)g(do)f(not)g(imp)r(ose)g(the)g(restriction)i(that)e
(constan)n(ts)h(and)f(functions)g(m)n(ust)378 5429 y(b)r(e)j(of)g(a)g
(sp)r(eci\014c)g(t)n(yp)r(e,)f(e.g.,)15 b(0)26 b(m)n(ust)e(b)r(e)h(of)i
(t)n(yp)r(e)e Fk(num.)p eop
%%Page: 90 100
90 99 bop 378 5 a FF(CHAPTER)30 b(5.)61 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)560
b FT(90)519 396 y(W)-8 b(e)32 b(no)m(w)e(de\014ne)g(the)g
(transformation)g FN(U)k FT(:)26 b FP(L)2108 415 y Fw(typ)n
FL(\()p FO(T)10 b FL(\))2373 396 y FN(!)26 b(f)p FP(T)13
b FN(g)p FP(L)30 b FT(as)h(follo)m(ws:)792 612 y FN(U)9
b FT(\()p FP(x)25 b FT(:)h FP(\033)s FT(\))g FN(7!)f
FT(\()p FP(\033)n(;)15 b(x)p FT(\),)32 b(where)d FP(x)i
FT(is)e(a)i(v)-5 b(ariable)792 750 y FN(U)9 b FT(\()p
FP(c)26 b FT(:)f FP(\033)s FT(\))h FN(7!)f FT(\()p FP(\033)n(;)15
b(c)p FT(\),)32 b(where)e FP(c)h FT(is)e(a)i(constan)m(t)792
888 y FN(U)9 b FT(\(\()p FP(f)35 b FT(:)26 b FP(\014)1110
902 y FL(1)1175 888 y FN(!)f(\001)15 b(\001)g(\001)26
b(!)f FP(\014)1589 902 y FO(n)1662 888 y FN(!)g FP(\033)s
FT(\)\()p FP(t)1936 902 y FL(1)1976 888 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(t)2226 902 y FO(n)2273 888 y FT(\)\))26 b FN(7!)f
FT(\()p FP(\033)n(;)15 b(f)10 b FT(\()p FN(U)f FT(\()p
FP(t)2834 902 y FL(1)2874 888 y FT(\))p FP(;)15 b(:)g(:)g(:)33
b(;)15 b FN(U)9 b FT(\()p FP(t)3261 902 y FO(n)3308 888
y FT(\)\)\))519 1092 y(In)23 b(other)h(w)m(ords,)h(w)m(e)f(transform)f
(a)h(t)m(yp)s(ed)f(term)h FP(t)h FT(:)g FP(\033)i FT(in)m(to)c(a)i
(pair)d(\()p FP(\033)n(;)15 b(t)2976 1059 y FK(0)3000
1092 y FT(\))24 b(\(where)f FP(t)3383 1059 y FK(0)3430
1092 y FT(represen)m(ts)378 1205 y(the)30 b(term)g FP(t)f
FT(whose)h(subterms)e(are)i(all)f(transformed)g(recursiv)m(ely)f(in)m
(to)i(pairs)e(as)i(describ)s(ed)e(here\))378 1318 y(and)23
b(treat)i(them)e(as)h(un)m(t)m(yp)s(ed)f(\014rst-order)g(terms.)38
b(Uninstan)m(tiatable)23 b(t)m(yp)s(e)h(v)-5 b(ariables)22
b(are)i(treated)378 1431 y(as)h(constan)m(ts,)j(and)c(the)h(use)g(of)g
(the)g Fw(pair)f FT(function)g(sym)m(b)s(ol)f(ensures)h(that)i(the)f
(uni\014cation)e(of)j(t)m(w)m(o)378 1544 y(paired)e(terms)h(results)f
(in)g(the)i(instan)m(tiation)e(of)i(t)m(yp)s(e)f(v)-5
b(ariables)24 b(to)i(t)m(yp)s(es,)h(and)d(the)i(instan)m(tiation)378
1656 y(of)h(\(term\))g(v)-5 b(ariables)25 b(to)i(un)m(t)m(yp)s(ed)e
(terms.)40 b(It)26 b(should)f(b)s(e)g(noted,)j(though,)f(that)g(in)e
(order)h(to)h(a)m(v)m(oid)378 1769 y(in)m(v)-5 b(alid)31
b(instan)m(tiations,)j(no)g(distinct)e(t)m(yp)s(ed)i(v)-5
b(ariables)32 b(should)g(ha)m(v)m(e)j(the)f(same)g(name)g(\(see)h(the)
378 1882 y(discussion)28 b(earlier)h(this)g(section\).)519
1995 y(W)-8 b(e)38 b(illustrate)d(this)h(transformation)g(pro)s(cess)g
(with)g(the)h(follo)m(wing)e(t)m(w)m(o)j(simple)d(examples.)378
2108 y(The)30 b(literal)569 2267 y FP(P)13 b FT(\()p
FP(x)25 b FT(:)h Fw('a)43 b(list)n FP(;)63 b Fw(LENGTH)45
b FP(x)p FT(\))378 2425 y(is)29 b(transformed)h(in)m(to)569
2584 y FP(P)13 b FT(\(\()p Fw(list)o FT(\()p FP(w)985
2598 y FL(1)1025 2584 y FT(\))p FP(;)63 b(v)1192 2598
y FL(1)1232 2584 y FT(\))p FP(;)g FT(\()p Fw(num)o FP(;)g
Fw(LENGTH)n FT(\()p Fw(list)o FT(\()p FP(w)2181 2598
y FL(1)2221 2584 y FP(;)g(v)2353 2598 y FL(1)2392 2584
y FT(\)\)\)\))378 2742 y(and)30 b(the)g(literal)569 2901
y FP(Q)p FT(\()p FP(x)25 b FT(:)h Fw('a)o FP(;)63 b FT(\()p
FP(x)26 b FT(:)f Fw(num)o FT(\))48 b Fw(+)f FT(1\))378
3059 y(in)m(to)569 3218 y FP(Q)p FT(\(\()p FP(w)776 3232
y FL(1)816 3218 y FP(;)63 b(v)948 3232 y FL(1)988 3218
y FT(\))p FP(;)g FT(\()p Fw(num)o FP(;)g Fw(+)p FT(\(\()p
Fw(num)o FP(;)g(v)1742 3232 y FL(2)1782 3218 y FT(\))p
FP(;)g FT(\()p Fw(num)p FP(;)g FT(1\)\)\)\))378 3376
y(where)30 b FP(w)706 3390 y FL(1)745 3376 y FT(,)h FP(v)845
3390 y FL(1)915 3376 y FT(and)e FP(v)1135 3390 y FL(2)1205
3376 y FT(are)i(new)f(distinct)f(v)-5 b(ariables.)378
3614 y FG(5.3.2)112 b(Prepro)s(cessing)37 b(F)-9 b(orm)m(ulae)378
3786 y FT(The)42 b(role)g(of)h(the)g(prepro)s(cessing)e(stage)j(is)e
(to)h(transform)f(the)h(giv)m(en)f(list)g(of)h(theorems)f(\000)3687
3800 y FL(1)3772 3786 y FN(`)378 3899 y FP(t)411 3913
y FL(1)450 3899 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b FT(\000)724
3913 y FO(n)806 3899 y FN(`)34 b FP(t)929 3913 y FO(n)1011
3899 y FT(in)m(to)i(a)g(list)f(of)h(\014rst-order)f(clauses)g(represen)
m(ted)h(in)f(the)h(format)g(accepted)h(b)m(y)378 4011
y(the)k(pro)s(of)g(searc)m(h)g(stage.)75 b(First-order)40
b(clauses)h(are)g(represen)m(ted)g(as)h(lists)d(of)j(literals,)g(and)e
(a)378 4124 y(literal)31 b(is)g(either)h(an)h(equation,)g(an)f
(inequation,)g(a)h(p)s(ositiv)m(e)e(non-equation,)i(or)g(a)g(negated)g
(non-)378 4237 y(equation.)39 b(Equations)23 b(and)h(inequations)e(con)
m(tain)j(a)g(pair)e(of)h(terms,)i(and)e(non-equations)g(con)m(tain)378
4350 y(a)33 b(predicate)f(sym)m(b)s(ol)f(and)h(a)h(list)e(of)h(terms.)
47 b(A)32 b(term)h(is)e(represen)m(ted)h(as)h(a)g(pair)e(consisting)g
(of)h(a)378 4463 y(t)m(yp)s(e)e(and)g(an)h(un)m(t)m(yp)s(ed)e(term)h
(\(as)h(illustrated)e(in)g(the)h(previous)f(section\).)519
4576 y(The)g(giv)m(en)h(theorem)g(is)f(\014rst)f(con)m(v)m(erted)k(in)m
(to)d(sk)m(olemised)g(conjunctiv)m(e)h(normal)e(form)h(using)378
4689 y(a)e(n)m(um)m(b)s(er)e(of)i(deriv)m(ed)f(rules)f(supplied)f(with)
h(the)i(HOL)f(system.)40 b(The)26 b(univ)m(ersal)f(quan)m(ti\014ers)g
(and)378 4802 y(the)j(conjunctions)f(in)f(the)i(conclusions)f(of)h(eac)
m(h)h(theorem)f(are)g(then)g(eliminated)e(to)j(giv)m(e)f(a)g(list)f(of)
378 4915 y(disjunctiv)m(e)i(theorems.)42 b(Finally)-8
b(,)30 b(the)h(conclusions)e(of)i(the)g(resulting)d(disjunctiv)m(e)h
(theorems)i(are)378 5028 y(translated)h(in)m(to)g(the)g(pro)s(of)f
(searc)m(h)i(represen)m(tation)f(marking)f(the)h(appropriate)f(t)m(yp)s
(e)h(v)-5 b(ariables)378 5141 y(as)32 b(uninstan)m(tiatable,)f(and)g(b)
s(eing)g(sure)g(that)h(distinct)f(v)-5 b(ariables)30
b(are)i(giv)m(en)g(distinct)f(names.)45 b(It)378 5253
y(should)35 b(b)s(e)i(noted)g(that)h(care)g(m)m(ust)f(b)s(e)f(tak)m(en)
i(to)g(mark)f(the)h(t)m(yp)s(e)f(v)-5 b(ariables)36 b(in)g(the)h(h)m
(yp)s(othe-)378 5366 y(ses)d(of)g(the)g FI(original)45
b FT(theorems)34 b(as)g(uninstan)m(tiatable,)g(rather)f(than)h(the)g(t)
m(yp)s(e)g(v)-5 b(ariables)33 b(in)g(the)378 5479 y(h)m(yp)s(otheses)h
(of)g(the)g(\014nal)f(disjunctiv)m(e)g(theorems)h(whic)m(h)f(ma)m(y)i
(con)m(tain)f(additional)e(h)m(yp)s(otheses)378 5592
y(included)39 b(during)g(prepro)s(cessing.)72 b(F)-8
b(or)42 b(instance,)i(the)e(sk)m(olemisation)f(of)g(a)h(theorem)g(adds)
e(a)378 5705 y(h)m(yp)s(othesis)29 b(represen)m(ting)h(the)g
(de\014nition)e(of)j(the)f(Sk)m(olem)g(function,)g(e.g.,)16
b(sk)m(olemising)p eop
%%Page: 91 101
91 100 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(91)473 396 y(\000)48 b FN(`)f(8)p FP(x)p FM(.)p
FN(9)p FP(y)s FM(.)14 b FP(P)39 b(x)25 b(y)378 584 y
FT(results)k(in)473 772 y(\000)p FM(,)16 b FT(\()p FP(s)g
FM(=)f FP(\025x)p FM(.)p FP("y)s FM(.)h FP(P)38 b(x)25
b FT(\()p FP(y)k(x)p FT(\)\))48 b FN(`)g FP(P)38 b(x)25
b FT(\()p FP(s)g(x)p FT(\))378 959 y(and)31 b(an)m(y)i(t)m(yp)s(e)f(v)
-5 b(ariables)30 b(whic)m(h)h(o)s(ccur)h(in)e Fv(P)c(x)14
b(y)35 b FT(but)c(not)h(in)f(\000,)h(o)s(ccur)g(also)f(in)g(the)h(h)m
(yp)s(otheses)378 1072 y(of)f(the)h(ab)s(o)m(v)m(e)g(sk)m(olemised)e
(theorem.)44 b(Instan)m(tiations)31 b(on)g(these)g(t)m(yp)s(e)h(v)-5
b(ariables)29 b(are)j(v)-5 b(alid)30 b(since)378 1185
y(they)g(do)h(not)f(instan)m(tiate)h(the)f(t)m(yp)s(es)h(in)e(\000.)378
1429 y FG(5.3.3)112 b(Pro)s(of)38 b(Searc)m(h)378 1600
y FT(The)43 b(pro)s(of)g(searc)m(h)i(stage)g(tak)m(es)h(a)e(list)f(of)h
(clauses)f(and)g(lo)s(oks)h(for)f(a)i(closed)e(tableau)h(whic)m(h)378
1713 y(can)39 b(b)s(e)f(constructed)h(using)e(an)i(implemen)m(tation)e
(of)i(the)g(inference)e(rules)h(in)f(\014gure)h(11.)66
b(The)378 1826 y(searc)m(h)34 b(strategy)i(used)d(is)f(suitable)h(for)g
(pro)s(of)g(c)m(hec)m(king)i(the)f(straigh)m(tforw)m(ard)f
(justi\014cations)g(of)378 1939 y(SPL)f(pro)s(ofs,)g(but)g(is)g(rather)
g(ine\016cien)m(t)g(in)f(solving)g(complex)i(problems.)45
b(Shostak's)33 b(algorithm)378 2052 y(for)40 b(congruence)h(closure)f
(\(Shostak)h(1978\))i(is)d(used)f(to)j(reason)f(with)e(ground)g
(equations,)k(and)378 2165 y(constrain)m(ts)33 b(are)h(solv)m(ed)e
(using)g(a)h(simple,)f(but)h(incomplete,)g(algorithm.)48
b(W)-8 b(e)34 b(\014rst)f(ha)m(v)m(e)h(a)f(lo)s(ok)378
2278 y(at)g(the)f(congruence)h(closure)e(algorithm,)h(the)g(w)m(a)m(y)h
(constrain)m(ts)f(are)h(handled,)e(and)g(then)h(at)h(the)378
2391 y(searc)m(h)e(strategy)h(used.)378 2631 y FQ(Congruence)j(Closure)
378 2802 y FT(Congruence)h(closure)g(algorithms)f(construct)i(the)f
(congruence)h(classes)f(of)h(a)f(set)h(of)g(\014rst-order)378
2915 y(terms)30 b(according)f(to)i(a)f(\014nite)f(set)h(of)g(ground)f
(equations.)40 b(More)31 b(formally)-8 b(,)29 b(let)h
FP(T)38 b FT(=)25 b FP(T)13 b FT(\(\006)p FP(;)i(X)7
b FT(\))31 b(b)s(e)378 3028 y(the)25 b(set)h(of)g(terms)f(o)m(v)m(er)h
(a)g(signature)e(\006)h(and)g(a)g(set)h(of)g(v)-5 b(ariables)23
b FP(X)7 b FT(,)27 b(then)e(the)h(congruence)f(closure)378
3141 y(of)30 b(a)h(binary)e(relation)g FP(R)i FT(o)m(v)m(er)h(the)f
(terms)f(in)f FP(T)43 b FT(is)30 b(the)g(least)h(binary)d(relation)3183
3118 y(^)3163 3141 y FP(R)j FT(satisfying:)726 3344 y
FP(aR)q(b)p 726 3385 157 4 v 726 3482 a(a)794 3459 y
FT(^)774 3482 y FP(R)q(b)p 1085 3385 166 4 v 202 w(a)1153
3459 y FT(^)1133 3482 y FP(R)q(a)1453 3344 y(a)1520 3322
y FT(^)1501 3344 y FP(Rb)p 1453 3385 157 4 v 1453 3482
a(b)1511 3459 y FT(^)1492 3482 y FP(Ra)1811 3344 y(a)1879
3322 y FT(^)1859 3344 y FP(R)q(b)91 b(b)2118 3322 y FT(^)2098
3344 y FP(R)q(c)p 1811 3385 396 4 v 1930 3482 a(a)1998
3459 y FT(^)1978 3482 y FP(R)q(c)2572 3344 y(a)2620 3358
y FL(1)2679 3322 y FT(^)2660 3344 y FP(Rb)2768 3358 y
FL(1)2883 3344 y FN(\001)15 b(\001)g(\001)77 b FP(a)3113
3358 y FO(n)3180 3322 y FT(^)3160 3344 y FP(R)q(b)3269
3358 y FO(n)p 2409 3385 1071 4 v 2409 3482 a FP(f)10
b FT(\()p FP(a)2547 3496 y FL(1)2586 3482 y FP(;)15 b(:)g(:)g(:)h(;)f
(a)2835 3496 y FO(n)2883 3482 y FT(\))2938 3459 y(^)2918
3482 y FP(R)q(f)10 b FT(\()p FP(b)3117 3496 y FL(1)3156
3482 y FP(;)15 b(:)g(:)g(:)i(;)e(b)3397 3496 y FO(n)3444
3482 y FT(\))378 3668 y(for)30 b(ev)m(ery)h(terms)g FP(a;)15
b(a)1147 3682 y FL(1)1186 3668 y FP(;)g(:)g(:)g(:)32
b(;)15 b(a)1451 3682 y FO(n)1499 3668 y FP(;)g(b;)g(b)1657
3682 y FL(1)1697 3668 y FP(;)g(:)g(:)g(:)32 b(;)15 b(b)1953
3682 y FO(n)2030 3668 y FT(in)29 b FP(T)44 b FT(and)29
b(function)g FP(f)40 b FT(in)29 b(\006.)519 3780 y(Giv)m(en)i(a)h
(\014nite)e(set)i(of)f(ground)f(equations)h FP(E)5 b
FT(,)32 b(a)g(congruence)g(closure)e(algorithm)h(computes)398
3870 y(^)378 3893 y FP(R)447 3907 y FO(E)537 3893 y FT(for)f(the)h
(relation)e FP(R)1235 3907 y FO(E)1325 3893 y FT(de\014ned)g(as)i
(follo)m(ws:)1455 4098 y FP(aR)1572 4112 y FO(E)1631
4098 y FP(b)91 b FT(if)29 b(and)h(only)g(if)59 b FP(a)26
b FT(=)f FP(b)g FN(2)g FP(E)5 b(:)378 4302 y FT(It)30
b(can)g(b)s(e)g(sho)m(wn)f(b)m(y)h(Birkho\013)7 b('s)29
b(theorem)i(\(Birkho\013)e(1935\))j(that)f(for)e(arbitrary)g(ground)g
(terms)378 4415 y FP(a)e FT(and)f FP(b)h FT(the)g(statemen)m(t)i
FP(a)1332 4392 y FT(^)1312 4415 y FP(R)1381 4429 y FO(E)1440
4415 y FP(b)e FT(is)f(equiv)-5 b(alen)m(t)27 b(to)g FP(a)f
FN($)2295 4382 y FK(\003)2295 4442 y FO(E)2379 4415 y
FP(b)h FT(and)f(equiv)-5 b(alen)m(t)27 b(to)g(deciding)f(whether)378
4528 y(the)g(equalit)m(y)g FP(x)f FT(=)g FP(y)k FT(can)e(b)s(e)e
(deduced)h(from)f(the)i(equations)f(in)e FP(E)32 b FT(using)25
b(the)h(rules)f(of)h(re\015exivit)m(y)-8 b(,)378 4641
y(symmetry)g(,)31 b(transitivit)m(y)e(and)g(substitution)f(of)j(equals)
f(for)g(equals.)519 4754 y(Congruence)21 b(closure)g(algorithms)f
(\(Shostak)i(1978;)k(Nelson)21 b(and)f(Opp)s(en)g(1980\))j(can)e
(therefore)378 4866 y(b)s(e)40 b(used)g(to)i(decide)e(a)i(ground)d
(equalit)m(y)i(giv)m(en)g(a)g(\014nite)f(list)g(of)h(ground)e
(equational)i(axioms.)378 4979 y(Equiv)-5 b(alen)m(tly)d(,)46
b(they)d(can)h(b)s(e)f(used)g(to)h(decide)f(the)h(equalit)m(y)f(of)h(t)
m(w)m(o)h(\(p)s(ossibly)c(non-ground\))378 5092 y(terms)f(from)f(an)h
(equational)g(theory)g(when)f(the)h(instan)m(tiation)f(of)h(v)-5
b(ariables)39 b(is)g(not)h(required.)378 5205 y(Suc)m(h)32
b(algorithms)g(are)h(usually)d(quite)i(e\016cien)m(t,)i(deciding)d(the)
i(required)e(equalit)m(y)h(in)f(quadratic)378 5318 y(time)f(with)f
(resp)s(ect)h(to)i(the)e(n)m(um)m(b)s(er)f(of)i(equations)f(in)f
FP(E)5 b FT(.)519 5431 y(W)-8 b(e)25 b(ha)m(v)m(e)g(used)e(Shostak's)g
(algorithm)g(for)h(congruence)g(closure)f(\(Shostak)h(1978\))i(rather)d
(than)378 5544 y(other)33 b(algorithms)e(since)h(the)h(congruence)g
(classes)g(are)g(computed)f(incremen)m(tally)f(without)h(the)378
5657 y(need)37 b(of)h(an)m(y)g(precomputation.)62 b(This)36
b(is)h(relev)-5 b(an)m(t)38 b(in)e(our)h(case)i(b)s(ecause)f
(congruence)g(classes)p eop
%%Page: 92 102
92 101 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(92)378 396 y(are)34 b(built)e(as)i(the)g(tableau)g(branc)m(hes)g
(are)g(expanded)f(and)h(are)g(required)e(to)j(close)f(and)f(simplify)
378 509 y(branc)m(hes)d(at)h(ev)m(ery)g(stage)h(during)c(the)j(pro)s
(of)e(searc)m(h.)519 622 y(Similarly)g(to)k(Nelson)f(and)g(Opp)s(en's)f
(algorithm,)h(Shostak's)h(algorithm)f(uses)g(the)g(follo)m(wing)378
735 y(data)f(structures:)489 923 y(1.)46 b Fw(use)o FT(:)41
b(storing)30 b(ho)m(w)g(terms)g(are)h(con)m(tained)g(within)d(eac)m(h)j
(other;)g Fw(use)p Ft(\()p Fv(a)p Ft(\))f FT(returns)f(the)i(list)e(of)
605 1036 y(terms)i(of)f(the)h(form)f FP(f)10 b FT(\()p
FP(:)15 b(:)g(:)g(;)g(a;)g(:)g(:)g(:)r FT(\))31 b(in)e(the)i(set)f(of)h
(terms)f(b)s(eing)f(considered;)489 1223 y(2.)46 b Fw(find)o
FT(:)38 b(storing)25 b(the)g(actual)h(congruence)g(classes;)i
Fw(find)p Ft(\()p Fv(a)p Ft(\))c FT(returns)g(a)i(represen)m(tativ)m(e)
h(mem-)605 1336 y(b)s(er)j(of)g(the)h(congruence)g(class)f(of)g
FP(a)p FT(.)378 1524 y(Shostak's)g(algorithm)g(also)g(uses)g(the)h
(follo)m(wing)e(data)i(structure)f(for)g(e\016ciency:)489
1712 y(3.)46 b Fw(sig)o FT(:)41 b(ha)m(ving)30 b(the)g(in)m(v)-5
b(arian)m(t)30 b Fw(sig)n FT(\()p FP(f)10 b FT(\()p FP(u)1940
1726 y FL(1)1980 1712 y FP(;)15 b(:)g(:)g(:)i(;)e(u)2234
1726 y FO(n)2281 1712 y FT(\)\))26 b(=)f FP(f)10 b FT(\()p
Fw(find)n FT(\()p FP(u)2824 1726 y FL(1)2864 1712 y FT(\))p
FP(;)15 b(:)g(:)g(:)i(;)e Fw(find)n FT(\()p FP(u)3362
1726 y FO(n)3410 1712 y FT(\)\).)378 1899 y(The)30 b(follo)m(wing)f
(pro)s(cedures)g(are)i(used)e(b)m(y)h(the)h(algorithm:)489
2087 y(1.)46 b Fw(merge)n FT(:)39 b(where)25 b Fw(merge)n
FT(\()p FP(a;)15 b(b)p FT(\))27 b(merges)f(the)g(congruence)g(classes)g
(of)g FP(a)g FT(and)f FP(b)h FT(b)m(y)g(up)s(dating)e(the)605
2200 y Fw(use)o FT(,)31 b Fw(find)d FT(and)i Fw(sig)f
FT(data)i(structures.)489 2387 y(2.)46 b Fw(canon)n FT(:)72
b(where)46 b Fw(canon)n FT(\()p FP(a)p FT(\))h(up)s(dates)e(the)i
Fw(use)d FT(and)i Fw(sig)f FT(data)i(structures,)j(and)45
b(returns)605 2500 y Fw(find)o FT(\()p FP(a)p FT(\).)378
2688 y(The)25 b(main)g(lo)s(op)g(of)h(the)g(algorithm)f(applies)f
Fw(merge)m FT(\()p Fw(canon)o FT(\()p FP(a)p FT(\))p
FP(;)15 b Fw(canon)o FT(\()p FP(b)p FT(\)\))27 b(on)e(eac)m(h)i
(equalit)m(y)f FP(a)f FT(=)g FP(b)378 2801 y FT(in)30
b(the)h(giv)m(en)g(equational)f(theory)-8 b(.)43 b(An)31
b(equalit)m(y)g FP(x)26 b FT(=)g FP(y)33 b FT(is)d(then)h(decided)f(b)m
(y)h(c)m(hec)m(king)h(whether)378 2914 y(the)44 b(represen)m(tativ)m(e)
g(mem)m(b)s(ers)f(of)h(the)g(congruence)g(classes)g(of)g
FP(x)f FT(and)g FP(y)k FT(are)d(equal,)j(that)d(is)378
3027 y(whether)36 b Fw(canon)n FT(\()p FP(x)p FT(\))g(=)g
Fw(canon)m FT(\()p FP(y)s FT(\).)60 b(Cyrluk,)37 b(Lincoln,)f(and)g
(Shank)-5 b(ar)36 b(\(1996\))j(giv)m(e)e(a)g(v)m(ery)g(clear)378
3140 y(presen)m(tation)44 b(of)h(Shostak's)f(algorithm)f(for)h
(congruence)h(closure)e(\(as)i(w)m(ell)e(as)i(Shostak's)f(al-)378
3252 y(gorithm)39 b(for)h(com)m(bining)e(decision)h(pro)s(cedures\).)68
b(Kapur's)39 b(treatmen)m(t)j(of)e(this)f(algorithm)g(as)378
3365 y(completion)30 b(is)f(also)h(v)m(ery)h(illuminating)26
b(\(Kapur)k(1997\).)378 3605 y FQ(Solving)36 b(Constrain)m(ts)378
3777 y FT(Ordering)26 b(equalit)m(y)h(constrain)m(ts)h(are)g(used)f(to)
i(restrict)e(the)h(pro)s(of)f(searc)m(h)i(space,)g(and)e(a)h(solution)
378 3890 y(to)34 b(the)f(constrain)m(t)g(in)f(a)h(closed)g(tableau)g
(giv)m(es)g(a)h(global)e(substitution)f(whic)m(h)h(instan)m(tiates)h
(the)378 4003 y(tableau)c(in)m(to)h(a)g(trivially)c(refutable)j(one)h
(\(that)g(is,)f(one)h(whic)m(h)e(is)h(refutable)g(when)f(its)h(terms)g
(are)378 4116 y(considered)21 b(to)i(b)s(e)e(ground\).)38
b(As)22 b(explained)e(in)h(section)h(5.2.1,)k(ordering)21
b(equalit)m(y)g(constrain)m(ts)h(are)378 4229 y(quan)m(ti\014er)34
b(free)h(\014rst-order)f(form)m(ulae)g(on)h(literals)e(with)h(the)h
(predicate)g(sym)m(b)s(ols)e FN(')i FT(\(equalit)m(y\))378
4342 y(and)k FN(\037)h FT(whic)m(h)e(is)h(a)h(reduction)f(ordering)g
(total)h(on)g(ground)f(terms.)69 b(A)40 b(lexicographical)e(path)378
4455 y(ordering)i(is)g(used)g(as)h(the)g(required)f(reduction)f
(ordering.)72 b(In)40 b(the)h(curren)m(t)g(implemen)m(tation,)378
4567 y(w)m(e)j(ha)m(v)m(e)g(not)g(solv)m(ed)f(the)h(constrain)m(ts)f
(using)f(the)i(complete)f(algorithms)g(illustrated)e(in)h(the)378
4680 y(literature)36 b(\(Comon)h(1990;)43 b(Nieu)m(w)m(enh)m(uis)35
b(1993;)42 b(Nieu)m(w)m(enh)m(uis)36 b(and)g(Rubio)g(1995\))j(b)s
(ecause)e(of)378 4793 y(their)43 b(exp)s(onen)m(tial)h(nature,)k(and)c
(mostly)g(b)s(ecause)g(of)h(the)f(simplicit)m(y)e(of)i(the)h(problems)e
(the)378 4906 y(deriv)m(ed)35 b(rule)f(is)g(used)h(to)h(solv)m(e.)57
b(Although)35 b(w)m(e)h(use)f(complete)h(metho)s(ds)f(for)g(solving)f
(equalit)m(y)378 5019 y(constrain)m(ts)26 b(\(i.e.,)16
b(a)26 b(syn)m(tactic)h(uni\014cation)d(algorithm\),)i(ordering)f
(constrain)m(ts)g(are)i(sho)m(wn)e(to)i(b)s(e)378 5132
y(unsatis\014able)d(only)h(if)g(their)g(transitiv)m(e)h(closure)f(can)h
(b)s(e)g(easily)f(rejected)i(when)e(substituted)f(with)378
5245 y(the)29 b(solution)f(of)h(the)g(equalit)m(y)f(constrain)m(t.)41
b(An)28 b(ordering)g(constrain)m(t)h FP(s)c FN(\037)g
FP(t)j FT(is)g(easily)g(rejected)i(if)378 5358 y FP(s)g
FT(and)g FP(t)g FT(are)h(ground)e(and)h FP(t)25 b FN(\027)g
FP(s)p FT(,)30 b(or)g(if)g FP(s)f FT(is)h(a)h(subterm)e(of)h
FP(t)p FT(.)519 5471 y(The)43 b(tableau)g(constrain)m(ts)g(are)g
(represen)m(ted)g(as)h(a)f(pair)f(\()p FN(C)2687 5485
y FK(')2746 5471 y FP(;)15 b FN(C)2834 5485 y FK(\037)2893
5471 y FT(\))44 b(where)e FN(C)3295 5485 y FK(')3397
5471 y FT(is)h(a)g(list)f(of)378 5584 y(equalit)m(y)f(constrain)m(ts)g
(in)g(solv)m(ed)g(form)g(\(i.e.,)16 b(a)42 b(substitution\),)g(and)f
FN(C)2932 5598 y FK(\037)3032 5584 y FT(is)g(a)g(list)g(of)g(ordering)
378 5697 y(constrain)m(ts.)g(Including)27 b(an)j(equalit)m(y)f
(constrain)m(t)h FP(s)25 b FN(')g FP(t)30 b FT(in)f FN(C)2557
5711 y FK(')2646 5697 y FT(also)h(in)m(v)m(olv)m(es)g(the)g(instan)m
(tiation)p eop
%%Page: 93 103
93 102 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(93)378 396 y(of)28 b(the)h(constrain)m(ts)f(in)f
FN(C)1247 410 y FK(\037)1333 396 y FT(with)g(the)i(solution)d(of)j
FN(C)2184 410 y FK(')2258 396 y FN([)16 b(f)p FP(s)25
b FN(')g FP(t)p FN(g)p FT(.)40 b(When)28 b(an)g(ordering)f(constrain)m
(t)378 509 y FP(s)39 b FN(\037)g FP(t)g FT(is)f(inserted)g(in)g
FN(C)1258 523 y FK(\037)1316 509 y FT(,)k(the)d(constrain)m(t)g
FP(s)g FN(\037)g FP(u)g FT(is)f(also)h(inserted)f(for)h(ev)m(ery)h
FP(t)f FN(\037)g FP(u)g FT(in)f FN(C)3744 523 y FK(\037)3803
509 y FT(.)378 622 y(Because)e(of)e(the)h(incompleteness)e(of)h(this)f
(metho)s(d,)i(the)g(searc)m(h)g(space)g(considered)e(during)f(the)378
735 y(refutation)j(pro)s(cess)g(is)f(larger)h(than)g(the)g(ideal)f
(one.)56 b(More)35 b(e\016cien)m(t)h(incomplete)f(metho)s(ds)f(for)378
848 y(solving)29 b(constrain)m(ts)h(are)h(giv)m(en)g(b)m(y)f(Plaisted)f
(\(1993b\))378 1081 y FQ(The)35 b(Searc)m(h)g(Strategy)378
1252 y FT(A)h(tableau)h(branc)m(h)e(is)h(represen)m(ted)g(as)g(a)h
(pair)e(consisting)g(of)i(a)f(list)f(of)i(literals)d(together)k(with)
378 1365 y(the)30 b(data)h(structures)f(represen)m(ting)g(the)g
(congruence)h(closure)f(of)g(the)h(equational)f(theory)g(of)h(the)378
1478 y(branc)m(h.)54 b(A)36 b(constrain)m(t)f(tableau)g(is)f(represen)m
(ted)h(as)h(a)f(pair)f(consisting)g(of)i(a)f(lazy)g(list)f(of)i(op)s
(en)378 1591 y(branc)m(hes)k(and)f(the)i(constrain)m(ts.)70
b(The)40 b(head)g(of)g(the)h(lazy)f(list)f(corresp)s(onds)f(to)j(the)g
(leftmost)378 1704 y(branc)m(h)e(of)i(the)f(tableau,)j(and)c(the)h
(last)g(elemen)m(t)h(of)f(the)g(list)f(corresp)s(onds)g(to)i(the)f
(righ)m(tmost)378 1817 y(one.)71 b(The)40 b(strategy)h(giv)m(en)g(b)s
(elo)m(w)e(is)h(used)f(for)h(lo)s(oking)f(for)h(a)h(closed)f(tableau.)
71 b(W)-8 b(e)41 b(remark)378 1930 y(that)32 b(although)g(this)f
(strategy)i(suits)e(our)g(purp)s(oses,)g(it)g(is)g(not)h(recommended)g
(for)f(solving)g(hard)378 2043 y(problems.)514 2208 y
FN(\017)46 b FT(The)36 b(inference)e(rules)h(are)h(applied)e(to)i(the)g
(leftmost)g(op)s(en)f(branc)m(h)g(of)h(the)g(tableau,)h(and)605
2321 y(therefore)k(only)e(the)h(head)g(of)h(the)f(lazy)g(list)f(is)g
(considered)g(at)i(an)m(y)f(stage)i(of)e(the)h(pro)s(of)605
2434 y(searc)m(h.)f(The)26 b(\014rst)g(elemen)m(t)h(of)f(the)h(tail)f
(of)g(the)h(list)e(is)g(computed)i(only)e(when)h(it)g(is)f(needed,)605
2547 y(that)30 b(is)e(when)g(the)i(branc)m(h)e(represen)m(ted)h(b)m(y)g
(the)h(head)f(elemen)m(t)g(is)g(closed)g(and)f(discarded.)514
2719 y FN(\017)46 b FT(A)21 b(b)s(ound)e(is)h(giv)m(en)h(on)g(the)h(n)m
(um)m(b)s(er)d(of)i(times)g(that)h(a)f(clause)g(can)g(b)s(e)g(used)f(b)
m(y)h(the)g(expansion)605 2832 y(rule.)51 b(The)33 b(least-used)h
(clauses)g(are)h(giv)m(en)f(higher)e(priorit)m(y)-8 b(,)34
b(and)g(expansions)f(whic)m(h)g(can)605 2945 y(b)s(e)h(immediately)f
(follo)m(w)m(ed)h(b)m(y)g(the)h(closure)f(of)h(a)g(branc)m(h)e
(\(through)i(a)f(trivial)f(closure)h(or)605 3058 y(an)j(equalit)m(y)f
(re\015exivit)m(y)g(rule\))f(are)i(applied)e(\014rst.)58
b(This)35 b(giv)m(es)i(a)g(certain)g(degree)g(of)g(the)605
3171 y(goal-directedness)g(of)g(the)f(connection)h(tableau.)60
b(Clauses)35 b(whic)m(h)h(con)m(tain)h(an)f(equation)605
3284 y(are)31 b(then)f(giv)m(en)g(a)h(higher)e(priorit)m(y)g(to)i
(those)g(whic)m(h)e(do)h(not.)514 3456 y FN(\017)46 b
FT(The)35 b(simplify)d(and)j(trivial)e(closure)i(rules)f(are)i(applied)
d(eagerly)j(on)f(an)m(y)h(inequalities)d(in-)605 3569
y(serted)43 b(in)f(the)h(branc)m(h.)78 b(The)43 b(congruence)h(closure)
e(of)h(the)h(equational)e(theory)h(of)h(the)605 3681
y(branc)m(h)d(is)g(computed)g(incremen)m(tally)f(as)i(the)g(tableau)g
(expands.)73 b(When)42 b(an)f(equation)605 3794 y FP(a)26
b FT(=)e FP(b)31 b FT(is)e(inserted)g(in)g(the)i(branc)m(h,)f(the)g
(congruence)h(classes)g(of)f FP(a)g FT(and)g FP(b)g FT(are)h(merged)f
(and)605 3907 y(the)h(inequations)e(of)h(the)h(branc)m(h)e(are)i
(simpli\014ed)c(and)j(p)s(ossibly)d(refuted.)605 4050
y(The)46 b(congruence)h(closure)f(is)f(also)i(up)s(dated)e(whenev)m(er)
h(new)g(free)g(v)-5 b(ariables)45 b(are)i(con-)605 4163
y(strained)40 b(\(substituted\).)70 b(If)40 b(a)h(previously)e
(unconstrained)g(v)-5 b(ariable)39 b FP(v)44 b FT(is)39
b(constrained,)605 4276 y(to)29 b FP(t)g FT(sa)m(y)-8
b(,)30 b(the)f(congruence)g(classes)f(of)h FP(v)j FT(and)c
FP(t)g FT(are)h(merged.)40 b(As)28 b(a)h(result)f(the)g(congruence)605
4388 y(closure)35 b(of)h(the)f(branc)m(h)g(can)h(b)s(e)e(seen)i(as)f(b)
s(eing)f(instan)m(tiated)h(b)m(y)h(a)g(global)e(substitution)605
4501 y(\(i.e.,)16 b(the)31 b(most)f(general)h(solution)e(of)h(the)h
(constrain)m(t\))g(applied)d(to)j(the)g(tableau.)514
4673 y FN(\017)46 b FT(The)37 b(equational)g(re\015exivit)m(y)f(rule)g
(is)h(tried)f(on)h(inequalities)e(after)j(they)f(are)h(inserted)e(in)
605 4786 y(the)31 b(branc)m(h)f(and)f(simpli\014ed.)514
4958 y FN(\017)46 b FT(Since)33 b(Shostak's)i(algorithm)e(for)h
(congruence)h(closure)f(refutes)g(an)g(inequation)f FP(a)e
FN(6)p FT(=)h FP(b)i FT(b)m(y)605 5071 y(computing)22
b(the)h(canonical)f(form)h(of)f FP(a)p FT(,)j Fw(canon)n
FT(\()p FP(a)p FT(\),)g(and)d(the)h(canonical)f(form)h(of)g
FP(b)p FT(,)h Fw(canon)n FT(\()p FP(b)p FT(\),)605 5184
y(and)30 b(c)m(hec)m(ks)i(whether)e Fw(canon)n FT(\()p
FP(a)p FT(\))c(=)f Fw(canon)n FT(\()p FP(b)p FT(\),)31
b(the)g(computed)f(canonical)g(forms)g(ma)m(y)h(also)605
5297 y(b)s(e)23 b(used)f(to)i(refute)f(the)h(tableau)f(if)f(they)h(are)
h(uni\014able.)35 b(This)22 b(pro)s(cedure)g(can)h(b)s(e)g(describ)s
(ed)605 5410 y(as)31 b(a)g(new)e(rule)1523 5578 y FP(B)1592
5592 y FL(1)1631 5578 y FP(;)15 b(s)25 b FN(6\031)g FP(t)15
b FN(j)g FP(B)1992 5592 y FL(2)2047 5578 y FN(j)31 b(\001)15
b(\001)g(\001)31 b(j)15 b FP(B)2348 5592 y FO(n)2431
5578 y FN(\001)36 b(C)p 1240 5620 1587 4 v 1240 5705
a FP(B)1309 5719 y FL(2)1364 5705 y FN(j)30 b(\001)15
b(\001)g(\001)32 b(j)15 b FP(B)1665 5719 y FO(n)1747
5705 y FN(\001)36 b(C)25 b([)20 b(f)p Fw(canon)o FT(\()p
FP(s)p FT(\))25 b FN(')g Fw(canon)n FT(\()p FP(t)p FT(\))p
FN(g)2869 5640 y FT(er-)p Fw(canon)p eop
%%Page: 94 104
94 103 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(94)605 396 y(whic)m(h)28 b(dep)s(ends)g(on)h(the)g(actual)h
(implemen)m(tation)e(of)h(the)g(congruence)h(closure)f(algorithm)605
509 y(used.)40 b(This)29 b(rule)g(is)g(applied)f(after)j(equational)f
(re\015exivit)m(y)g(fails)f(to)i(close)f(the)h(branc)m(h.)514
697 y FN(\017)46 b FT(When)31 b(all)e(the)i(clauses)f(ha)m(v)m(e)i(b)s
(een)d(used)h(the)h(same)g(n)m(um)m(b)s(er)e(of)i(times)f(b)m(y)g(the)h
(expansion)605 810 y(rule,)42 b(the)e(rigid)e(basic)i(sup)s(erp)s
(osition)d(rules)i(with)f(equalit)m(y)i(re\015exivit)m(y)g(are)g
(applied)e(to)605 923 y(close)e(the)f(branc)m(h.)55 b(If)34
b(the)i(branc)m(h)e(cannot)i(b)s(e)e(closed,)j(the)e(clauses)g(are)h
(used)e(again)h(for)605 1036 y(expansion.)60 b(This)35
b(is)h(rep)s(eated)h(un)m(til)e(the)j(clauses)e(are)i(used)e(a)h(giv)m
(en)g(n)m(um)m(b)s(er)f(of)h(times)605 1149 y(\(the)31
b(b)s(ound)d(men)m(tioned)i(earlier\).)514 1336 y FN(\017)46
b FT(If)29 b(a)g(closed)g(tableau)f(is)g(not)h(found,)g(the)g(curren)m
(t)f(b)s(ound)f(is)h(incremen)m(ted)g(b)m(y)h(one)g(and)g(the)605
1449 y(pro)s(of)c(searc)m(h)h(is)e(applied)g(again.)39
b(This)23 b(is)i(rep)s(eated)g(un)m(til)f(a)i(maxim)m(um)e(b)s(ound)f
(is)i(reac)m(hed.)605 1562 y(Since)33 b(the)g(problems)f(the)h(pro)s
(of)g(pro)s(cedure)f(is)h(exp)s(ected)g(to)h(solv)m(e)g(are)g(rather)f
(simple,)g(a)605 1675 y(v)m(ery)28 b(lo)m(w)f(maxim)m(um)f(b)s(ound)g
(is)g(c)m(hosen)i(\(only)f(3\).)40 b(The)27 b(n)m(um)m(b)s(er)f(of)i
(times)f(the)g(rigid)e(basic)605 1788 y(sup)s(erp)s(osition)i(rules)i
(are)i(applied)d(to)j(close)g(a)g(branc)m(h)e(is)h(also)g(b)s(ounded)e
(\(b)m(y)j(5\).)519 2000 y(When)23 b(a)g(closed)g(constrain)m(t)g
(tableau)g(is)f(found,)h(the)g(pro)s(of)g(searc)m(h)g(stage)i(returns)c
(a)j(simpli\014ed)378 2113 y(list)h(of)i(the)g(inferences)f(used)g
(together)i(with)d(a)j(substitution)c(solving)h(the)i(constrain)m(t.)40
b(These)26 b(are)378 2226 y(used)i(b)m(y)g(the)h(pro)s(of)f
(transformation)g(stage)j(to)e(deriv)m(e)f(a)h(HOL)g(theorem.)40
b(The)28 b(simpli\014ed)d(list)j(of)378 2339 y(inferences)h(consists)h
(of:)514 2527 y FN(\017)46 b FT(Expansions,)29 b(whic)m(h)g(also)h(con)
m(tain)h(the)g(instance)f(of)g(the)h(clause)f(used.)514
2714 y FN(\017)46 b FT(Closures:)38 b(in)27 b(this)g(case,)j(no)f
(distinction)c(is)j(made)g(b)s(et)m(w)m(een)h(the)f(di\013eren)m(t)g
(inference)f(rules)605 2827 y(\(trivial)k(close,)j(equational)e
(re\015exivit)m(y)-8 b(,)34 b(or)e(equational)h(re\015exivit)m(y)e(on)i
FM(canon)p FT(ical)e(terms\))605 2940 y(whic)m(h)e(are)i(used.)378
3184 y FG(5.3.4)112 b(Deriving)36 b(a)i(HOL)g(Theorem)378
3355 y FT(The)30 b(role)h(of)g(this)f(\014nal)g(stage)i(of)g(the)f
(deriv)m(ed)f(rule)f(is)i(to)g(construct)h(a)f(HOL)g(theorem)g(from)g
(the)378 3468 y(closed)g(tableau)h(found)e(b)m(y)h(the)h(previous)e
(stage.)45 b(The)31 b(substitution)f(and)h(the)g(list)f(of)i(expansion)
378 3581 y(and)23 b(closure)h(rules)e(giv)m(en)j(b)m(y)f(the)g(pro)s
(of)f(searc)m(h)i(stage)g(are)g(translated)e(in)m(to)i(HOL)e(natural)g
(deduc-)378 3694 y(tion)j(inferences.)38 b(The)26 b(closure)g(of)h(a)g
(branc)m(h)e FP(B)31 b FT(is)26 b(translated)g(in)m(to)g(the)h(deriv)-5
b(ation)25 b(of)i(a)f(theorem)378 3807 y FP(B)k FN(`)25
b(?)30 b FT(and)f(an)i(expansion)e(rule)g(is)g(translated)h(in)m(to)g
(an)h(elimination)c(of)k(the)g(disjunctiv)m(e)d(clause)378
3920 y(used)38 b(for)h(expansion)e(instan)m(tiated)i(with)e(the)i(giv)m
(en)g(substitution.)64 b(The)39 b(translating)e(pro)s(cess)378
4033 y(pro)m(v)m(es)d(a)f(HOL)g(theorem)h(stating)f(the)h
(inconsistency)e(of)h(instan)m(tiations)f(of)i(the)f(list)f(of)h
(clauses)378 4146 y(deriv)m(ed)i(during)e(the)k(prepro)s(cessing)d
(stage.)58 b(This)34 b(theorem)j(can)f(then)f(b)s(e)h(used)f(to)h
(deriv)m(e)g(the)378 4259 y(required)29 b(inconsistency)g(of)h(the)h
(list)e(of)h(theorems)h(giv)m(en)f(as)h(argumen)m(ts)g(to)g(the)f
(deriv)m(ed)g(rule.)519 4372 y(Congruence)36 b(closure)g(is)f(used)g
(to)i(deriv)m(e)f(the)g(inconsistency)f(of)i(a)f(giv)m(en)g(branc)m(h)g
FP(B)5 b FT(.)58 b(This)378 4484 y(is)44 b(done)g(b)m(y)h(computing)f
(the)h(congruence)g(classes)g(according)f(to)i(the)f(equational)f
(theory)h(of)378 4597 y(the)37 b(branc)m(h)f(and)g(b)m(y)h(lo)s(oking)f
(for)g(an)h(inequalit)m(y)e FP(s)h FN(6)p FT(=)f FP(t)i
FT(suc)m(h)f(that)i FP(s)e FT(and)g FP(t)h FT(are)g(in)e(the)i(same)378
4710 y(congruence)c(class,)h(or)f(for)g(t)m(w)m(o)h(literals)d
FP(P)13 b FT(\()p FP(s)1972 4724 y FL(1)2012 4710 y FP(;)i(:)g(:)g(:)32
b(;)15 b(s)2272 4724 y FO(n)2319 4710 y FT(\))33 b(and)f
FN(:)p FP(P)13 b FT(\()p FP(t)2766 4724 y FL(1)2805 4710
y FP(;)i(:)g(:)g(:)32 b(;)15 b(t)3055 4724 y FO(n)3102
4710 y FT(\))34 b(suc)m(h)e(that)i FP(s)3621 4724 y FO(i)3681
4710 y FT(and)378 4823 y FP(t)411 4837 y FO(i)476 4823
y FT(are)j(in)f(the)h(same)h(class)f(for)f FP(i)h FN(2)f(f)p
FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p FN(g)p FT(.)61
b(Ho)m(w)m(ev)m(er,)41 b(since)36 b(w)m(e)i(need)e(to)i(deriv)m(e)f(a)g
(HOL)378 4936 y(theorem,)48 b(the)c(congruence)h(closure)e(algorithm)g
(describ)s(ed)f(in)h(section)h(5.3.3)i(is)d(mo)s(di\014ed)f(to)378
5049 y(b)s(e)d(used)g(as)g(a)h(HOL)f(deriv)m(ed)g(rule.)67
b(The)39 b(data)h(structures)f(and)g(the)g(functions)g(in)f(Shostak's)
378 5162 y(algorithm)i(are)i(mo)s(di\014ed)d(to)j(store)f(and)g(return)
f(HOL)h(theorems.)73 b(F)-8 b(or)42 b(example,)i(the)d
Fw(canon)378 5275 y FT(function)33 b(whic)m(h)h(computes)g(the)h
(canonical)g(form)f(of)h(a)g(giv)m(en)f(term)h FP(t)f
FT(is)g(mo)s(di\014ed)f(to)i(return)e(a)378 5388 y(theorem)473
5575 y(\000)48 b FN(`)f FP(t)h FM(=)f FP(t)890 5542 y
FK(0)p eop
%%Page: 95 105
95 104 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(95)378 396 y(where)33 b FP(t)677 363 y FK(0)733
396 y FT(is)g(the)g(canonical)g(form)g(of)g FP(t)g FT(and)g(\000)g(is)g
(the)g(list)f(of)i(equations)f(used)f(in)g(computing)h
FP(t)3780 363 y FK(0)3803 396 y FT(.)378 509 y(F)-8 b(or)33
b(e\016ciency)f(purp)s(oses,)e(lazy)i(theorems)h(\(Boulton)f(1993\))i
(are)e(used)f(in)g(the)h(implemen)m(tation.)378 622 y(These)e(are)g(ML)
g(functions)e(whic)m(h)h(deriv)m(e)g(a)h(theorem)g(only)f(when)g(it)g
(is)g(needed,)h(and)f(can)h(there-)378 735 y(fore)37
b(b)s(e)e(used)h(to)h(a)m(v)m(oid)g(the)g(computation)g(of)f
(unnecessarily)f(HOL)h(inferences.)58 b(W)-8 b(e)38 b(use)e(lazy)378
848 y(theorems)27 b(of)h(t)m(yp)s(e)f Fw(converters)39
b Fu(!)44 b Fw(thm)26 b FT(where)g Fw(converters)d FT(is)j(the)i(t)m
(yp)s(e)f(of)g(the)h(SML)e(functions)378 961 y(whic)m(h)f(translate)h
(the)h(terms)f(from)g(the)g(in)m(ternal)f(term)i(represen)m(tation)f
(used)g(b)m(y)g(the)g(congruence)378 1074 y(closure)38
b(algorithm)g(in)m(to)h(HOL)g(terms)g(and)f(vice-v)m(ersa.)68
b(By)39 b(using)e(suc)m(h)i(lazy)g(theorems,)j(the)378
1187 y(implemen)m(tation)28 b(of)h(the)h(congruence)g(closure)f
(algorithm)f(is)h(indep)s(enden)m(t)e(of)i(the)h(w)m(a)m(y)g(its)f
(term)378 1300 y(represen)m(tation)h(is)g(translated)g(in)m(to)g(HOL)g
(terms.)378 1586 y FH(5.4)135 b(F)-11 b(rom)45 b(Higher-Order)g(to)h
(First-Order)e(Logic)378 1789 y FT(The)33 b FN(C)5 b(B)s(S)i(E)41
b FT(deriv)m(ed)33 b(rule)f(and)h(other)h(semi-decision)d(pro)s
(cedures)h(for)i(\014rst-order)e(logic)i(can)g(b)s(e)378
1902 y(used)g(to)i(reason)f(with)e(higher-order)h(form)m(ulae)g(b)m(y)h
(transforming)e(them)i(in)m(to)g(equiv)-5 b(alen)m(t)34
b(\014rst-)378 2015 y(order)c(ones.)41 b(Suc)m(h)29 b(a)i
(transformation)f(can)h(b)s(e)e(done)h(in)g(three)g(steps:)489
2203 y(1.)46 b(Normalising)29 b(the)h(terms)g(in)m(to)h
FP(\021)s FT(-long)f FP(\014)36 b FT(normal)29 b(form;)489
2390 y(2.)46 b(Eliminating)27 b(quan)m(ti\014cation)j(o)m(v)m(er)i
(functions)d(and)h(predicates;)489 2578 y(3.)46 b(Eliminating)27
b(lam)m(b)s(da)j(abstractions.)519 2765 y(The)44 b(\014rst)g(step)g(is)
g(quite)f(straigh)m(tforw)m(ard)i(and)e(can)i(b)s(e)f(p)s(erformed)f
(in)g(HOL)h(using)f(the)378 2878 y(appropriate)26 b(inference)g(rules)g
(supplied)e(with)h(the)j(system.)39 b(Quan)m(ti\014cation)27
b(o)m(v)m(er)h(functions)d(and)378 2991 y(predicates)35
b(is)f(usually)f(eliminated)h(b)m(y)h(in)m(tro)s(ducing)e(a)j(new)f
(constan)m(t)h FP(\013)e FT(:)g(\()p FP(\015)39 b FN(!)34
b FP(\016)s FT(\))h FN(!)e FP(\015)39 b FN(!)33 b FP(\016)378
3104 y FT(\()p FP(\013)38 b FT(for)f(\\apply"\))h(and)f(then)g
(transforming)f(terms)h(of)h(the)g(form)f(\()p FP(f)46
b(x)p FT(\))38 b(in)m(to)g(\()p FP(\013)g(f)46 b(x)p
FT(\))38 b(\(see)g(for)378 3217 y(example)d(\(Kerb)s(er)g(1990\)\).)60
b(As)35 b(a)h(result,)h(higher-order)d(form)m(ulae,)j(suc)m(h)e(as)h
FN(8)p FP(P)s(:)15 b(P)47 b(x)35 b FN(\))f FP(P)47 b(y)s
FT(,)378 3330 y(are)f(transformed)e(in)m(to)i(\014rst-order)e(ones,)50
b FN(8)p FP(P)s(:)15 b(\013)50 b(P)64 b(x)50 b FN(\))g
FP(\013)h(P)63 b(y)s FT(.)86 b(The)45 b(third)e(step)j(giv)m(en)378
3443 y(ab)s(o)m(v)m(e)37 b(in)m(v)m(olv)m(es)g(the)f(transformation)g
(of)g(lam)m(b)s(da)f(abstractions)i(in)m(to)f(equiv)-5
b(alen)m(t)36 b(lam)m(b)s(da-free)378 3556 y(terms,)j(usually)c
(through)i(the)g(in)m(tro)s(duction)f(of)h(new)g(constan)m(ts.)63
b(It)37 b(should)e(b)s(e)i(noted)h(that)f(a)378 3669
y(straigh)m(tforw)m(ard)h(renaming)f(of)i(abstractions)f(in)m(to)g(new)
g(constan)m(ts)h(is)f(often)g(not)h(appropriate)378 3782
y(since)j(ev)m(en)h(trivial)e(sen)m(tences)i(are)g(not)g(transformed)f
(in)m(to)h(v)-5 b(alid)40 b(\014rst-order)i(form)m(ulae.)77
b(F)-8 b(or)378 3895 y(instance,)30 b(the)h(sen)m(tence)1309
4099 y(\()p FP(a)26 b FT(=)f FP(b)p FT(\))h FN(\))f FP(P)38
b FT(\()p FP(\025x:)15 b(f)35 b(x)26 b(a)p FT(\))f FN(\))g
FP(P)39 b FT(\()p FP(\025x:)15 b(f)35 b(x)25 b(b)p FT(\))378
4303 y(is)47 b(not)g(transformed)g(in)m(to)g(a)h(v)-5
b(alid)46 b(one)i(if)f(the)g(t)m(w)m(o)i(terms)f(\()p
FP(\025x:)15 b(f)63 b(x)54 b(a)p FT(\))48 b(and)f(\()p
FP(\025x:)15 b(f)64 b(x)54 b(b)p FT(\))378 4416 y(are)42
b(renamed)g(in)m(to)f(di\013eren)m(t)h(constan)m(ts.)76
b(Ho)m(w)m(ev)m(er,)47 b(one)42 b(can)g(con)m(v)m(ert)i(these)e(t)m(w)m
(o)h(terms)f(to)378 4529 y(\(\()p FP(\025y)s(;)15 b(z)t(;)g(x:)g(y)30
b(x)25 b(z)t FT(\))h FP(f)35 b(a)p FT(\))30 b(and)g(\(\()p
FP(\025y)s(;)15 b(z)t(;)g(x:)g(y)30 b(x)25 b(z)t FT(\))h
FP(f)35 b(b)p FT(\))c(and)e(in)m(tro)s(duce)h(a)g(constan)m(t)1725
4733 y FP(g)f FT(=)c(\()p FP(\025y)s(;)15 b(z)t(;)g(x:)g(y)30
b(x)25 b(z)t FT(\))378 4937 y(suc)m(h)30 b(that)h(the)g(ab)s(o)m(v)m(e)
g(sen)m(tence)h(is)d(transformed)h(in)m(to)g(the)h(v)-5
b(alid)28 b(\014rst-order)i(form)m(ula:)1460 5142 y(\()p
FP(a)c FT(=)f FP(b)p FT(\))g FN(\))h FP(P)38 b FT(\()p
FP(g)29 b(f)35 b(a)p FT(\))25 b FN(\))g FP(P)39 b FT(\()p
FP(g)29 b(f)34 b(b)p FT(\))519 5346 y(In)c(general,)g(giv)m(en)h(a)f
(term)h(of)f(the)h(form)1691 5550 y FP(\025v)1788 5564
y FL(1)1828 5550 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(v)2089
5564 y FO(m)2156 5550 y FP(:)g(t)2229 5564 y FL(1)2299
5550 y FN(\001)g(\001)g(\001)31 b FP(t)2468 5564 y FO(n)p
eop
%%Page: 96 106
96 105 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(96)378 396 y(w)m(e)29 b(\014rst)f(abstract)i(all)e(the)h(o)s
(ccurrences)g(of)g(the)g(v)-5 b(ariables)28 b FP(v)2493
410 y FL(1)2532 396 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(v)2793
410 y FO(m)2889 396 y FT(from)28 b(the)h(terms)g FP(t)3542
410 y FL(1)3612 396 y FN(\001)15 b(\001)g(\001)31 b FP(t)3781
410 y FO(n)378 509 y FT(and)24 b(eliminating)e(the)j(abstractions)g
(recursiv)m(ely)e(from)h(them.)39 b(By)25 b(abstracting)g(an)m(y)g(o)s
(ccurrences)378 622 y(of)f(the)h(term)f FP(s)f FT(from)h(another)g
(term)g FP(t)g FT(w)m(e)h(mean)f(the)g(transformation)g(of)g
FP(t)g FT(in)m(to)g(the)g FP(\021)s FT(-con)m(v)m(ertible)378
735 y(\(\()p FP(\025x:)15 b(t)626 702 y FK(0)650 735
y FT(\))50 b FP(s)p FT(\))45 b(where)g FP(t)1169 702
y FK(0)1237 735 y FT(is)f(the)i(term)f FP(t)g FT(with)f(all)g(its)g(o)s
(ccurrences)h(of)h FP(s)e FT(substituted)g(with)g FP(x)p
FT(.)378 848 y(The)30 b(resulting)f(terms)i(with)f(the)h(exception)g
(of)g(the)g(v)-5 b(ariables)30 b FP(v)2649 862 y FL(1)2689
848 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(v)2949 862 y FO(n)2997
848 y FT(,)31 b(are)g(then)g(abstracted)378 961 y(from)g(the)h(main)e
(term.)44 b(The)31 b(resulting)f(abstraction)i(is)e(\014nally)g
(renamed)h(in)m(to)g(a)h(new)f(constan)m(t.)378 1074
y(Abstractions)37 b(whic)m(h)f(are)i FP(\013)p FT(-con)m(v)m(ertible)g
(are)f(giv)m(en)g(the)h(same)f(constan)m(t.)63 b(W)-8
b(e)39 b(illustrate)c(this)378 1187 y(pro)s(cedure)29
b(with)g(an)i(example.)40 b(Giv)m(en)30 b(the)h(term)1515
1391 y FP(\025v)1612 1405 y FL(1)1651 1391 y FP(;)15
b(v)1735 1405 y FL(2)1775 1391 y FP(:)g(f)35 b FT(\()p
FP(\025v)2027 1405 y FL(3)2067 1391 y FP(:)15 b(v)2151
1405 y FL(1)2191 1391 y FT(\))25 b(\()p FP(\025v)2383
1405 y FL(3)2423 1391 y FP(:)15 b(v)2507 1405 y FL(2)2572
1391 y FP(v)2616 1405 y FL(3)2656 1391 y FT(\))378 1595
y(w)m(e)27 b(\014rst)g(abstract)h(the)f(b)s(ound)e(v)-5
b(ariables)26 b FP(v)1892 1609 y FL(1)1958 1595 y FT(and)g
FP(v)2175 1609 y FL(2)2242 1595 y FT(from)g(the)i(terms)f(\()p
FP(\025v)2989 1609 y FL(3)3028 1595 y FP(:)15 b(v)3112
1609 y FL(1)3152 1595 y FT(\))28 b(and)e(\()p FP(\025v)3520
1609 y FL(3)3560 1595 y FP(:)15 b(v)3644 1609 y FL(2)3709
1595 y FP(v)3753 1609 y FL(3)3793 1595 y FT(\))378 1708
y(to)31 b(giv)m(e)1193 1821 y FP(\025v)1290 1835 y FL(1)1329
1821 y FP(;)15 b(v)1413 1835 y FL(2)1453 1821 y FP(:)g(f)35
b FT(\(\()p FP(\025x)1748 1835 y FL(1)1788 1821 y FP(;)15
b(v)1872 1835 y FL(3)1912 1821 y FP(:)g(x)2004 1835 y
FL(1)2044 1821 y FT(\))26 b FP(v)2149 1835 y FL(1)2188
1821 y FT(\))31 b(\(\()p FP(\025x)2429 1835 y FL(1)2469
1821 y FP(;)15 b(v)2553 1835 y FL(3)2593 1821 y FP(:)g(x)2685
1835 y FL(1)2750 1821 y FP(v)2794 1835 y FL(3)2834 1821
y FT(\))25 b FP(v)2938 1835 y FL(2)2978 1821 y FT(\))378
1988 y(and)i(then)g(eliminate)g(the)h(abstractions)f(recursiv)m(ely)g
(from)g(the)h(b)s(o)s(dy)e(terms,)i(whic)m(h)f(in)f(this)h(case)378
2101 y(in)m(v)m(olv)m(es)j(the)h(in)m(tro)s(duction)d(of)j(the)f(new)g
(constan)m(ts)1652 2305 y FP(c)1691 2319 y FL(1)1814
2305 y FT(=)83 b FP(\025v)2065 2319 y FL(1)2104 2305
y FP(;)15 b(v)2188 2319 y FL(2)2228 2305 y FP(:)g(v)2312
2319 y FL(1)2352 2305 y FT(,)30 b(and)1652 2443 y FP(c)1691
2457 y FL(2)1814 2443 y FT(=)83 b FP(\025v)2065 2457
y FL(1)2104 2443 y FP(;)15 b(v)2188 2457 y FL(2)2228
2443 y FP(:)g(v)2312 2457 y FL(1)2377 2443 y FP(v)2421
2457 y FL(2)2461 2443 y FP(;)378 2647 y FT(to)31 b(giv)m(e)g(the)g
(term:)1629 2760 y FP(\025v)1726 2774 y FL(1)1766 2760
y FP(;)15 b(v)1850 2774 y FL(2)1890 2760 y FP(:)g(f)35
b FT(\()p FP(c)2084 2774 y FL(1)2149 2760 y FP(v)2193
2774 y FL(1)2233 2760 y FT(\))25 b(\()p FP(c)2367 2774
y FL(2)2433 2760 y FP(v)2477 2774 y FL(2)2516 2760 y
FT(\))p FP(:)378 2927 y FT(W)-8 b(e)32 b(no)m(w)e(abstract)h(the)g
(terms)f(in)f(the)i(b)s(o)s(dy)e(with)g(the)i(exception)f(of)h
FP(v)2899 2941 y FL(1)2969 2927 y FT(and)f FP(v)3190
2941 y FL(2)3259 2927 y FT(from)g(the)h(main)378 3040
y(term:)1239 3153 y(\()p FP(\025x)1379 3167 y FL(1)1419
3153 y FP(;)15 b(x)1511 3167 y FL(2)1551 3153 y FP(;)g(x)1643
3167 y FL(3)1682 3153 y FP(;)g(v)1766 3167 y FL(1)1806
3153 y FP(;)g(v)1890 3167 y FL(2)1930 3153 y FP(:)g(x)2022
3167 y FL(1)2087 3153 y FT(\()p FP(x)2174 3167 y FL(2)2239
3153 y FP(v)2283 3167 y FL(1)2323 3153 y FT(\))g(\()p
FP(x)2460 3167 y FL(3)2525 3153 y FP(v)2569 3167 y FL(2)2609
3153 y FT(\)\))26 b FP(f)34 b(c)2823 3167 y FL(1)2888
3153 y FP(c)2927 3167 y FL(2)378 3320 y FT(and)c(\014nally)e(w)m(e)j
(rename)f(the)h(abstraction:)1920 3524 y FP(c)1959 3538
y FL(3)2024 3524 y FP(f)j(c)2142 3538 y FL(1)2207 3524
y FP(c)2246 3538 y FL(2)1135 3819 y FT(where)c FP(c)1437
3833 y FL(3)1560 3819 y FT(=)82 b FP(\025v)1810 3833
y FL(1)1850 3819 y FP(;)15 b(v)1934 3833 y FL(2)1974
3819 y FP(;)g(v)2058 3833 y FL(3)2098 3819 y FP(;)g(v)2182
3833 y FL(4)2222 3819 y FP(;)g(v)2306 3833 y FL(5)2345
3819 y FP(:)g(v)2429 3833 y FL(1)2495 3819 y FT(\()p
FP(v)2574 3833 y FL(2)2639 3819 y FP(v)2683 3833 y FL(4)2722
3819 y FT(\))26 b(\()p FP(v)2862 3833 y FL(3)2927 3819
y FP(v)2971 3833 y FL(5)3010 3819 y FT(\))p FP(:)378
4024 y FT(W)-8 b(e)26 b(remark)e(that)i(although)e(this)g(translation)f
(from)i(higher-order)e(logic)i(to)g(\014rst-order)f(logic)h(w)m(as)378
4137 y(e\013ectiv)m(e)39 b(in)d(transforming)g(higher-order)f(form)m
(ulae)i(in)m(to)h(equiv)-5 b(alen)m(t)36 b(\014rst-order)h(ones)g
(during)378 4250 y(our)c(case)i(study)-8 b(,)35 b(the)f(t)m(w)m(o)h
(logics)e(are)i(v)m(ery)f(di\013eren)m(t)f(in)g(nature)g(and)g(no)h
(suc)m(h)g(transformation)378 4362 y(can)d(b)s(e)e(complete.)378
4649 y FH(5.5)135 b(Conclusions)46 b(and)e(F)-11 b(uture)44
b(W)-11 b(ork)378 4852 y FT(In)35 b(this)h(c)m(hapter)g(w)m(e)h(ha)m(v)
m(e)g(illustrated)d(the)j(implemen)m(tation)e(of)h(a)h(tableau)f
(calculus)e(for)i(\014rst-)378 4965 y(order)k(logic)h(as)g(a)g(deriv)m
(ed)f(rule)f(in)h(the)h(HOL)f(theorem)h(pro)m(v)m(er.)73
b(This)39 b(deriv)m(ed)g(rule)h(is)g(used)378 5078 y(as)h(the)f(main)f
(pro)m(v)m(er)i(for)f(c)m(hec)m(king)h(the)g(straigh)m(tforw)m(ard)f
(justi\014cations)f(of)h(the)h(SPL)e(scripts)378 5191
y(implemen)m(ted)f(in)f(the)j(mec)m(hanisation)e(of)i(group)e(theory)h
(describ)s(ed)e(in)h(Chapter)h(9.)67 b(Since)38 b(in)378
5303 y(general)g(suc)m(h)f(justi\014cations)f(do)i(not)g(represen)m(t)f
(hard)g(problems,)h(there)g(w)m(as)g(no)g(need)f(to)i(put)378
5416 y(a)j(considerable)e(amoun)m(t)i(of)f(e\013ort)h(in)e(handling)f
(v)m(ery)j(large)f(searc)m(h)h(spaces,)j(and)c(in)f(\014nding)378
5529 y(long)31 b(pro)s(ofs.)42 b(Although)31 b(the)g(pro)s(of)g
(calculus)e(is)i(complete)g(for)g(\014rst-order)g(logic)g(with)f
(equalit)m(y)-8 b(,)378 5642 y(w)m(e)35 b(imp)s(ose)e(v)m(ery)i(strict)
f(resource)h(b)s(ounds)d(during)h(pro)s(of)g(searc)m(h.)54
b(F)-8 b(urthermore,)36 b(the)e(metho)s(d)p eop
%%Page: 97 107
97 106 bop 378 5 a FF(CHAPTER)30 b(5.)71 b(A)30 b(T)-8
b(ABLEA)m(U)32 b(PR)m(O)m(VER)f(AS)f(A)g(HOL)g(DERIVED)h(R)m(ULE)550
b FT(97)378 396 y(used)38 b(for)h(solving)f(equalit)m(y)h(constrain)m
(ts)g(is)f(v)m(ery)i(simple)d(and)i(incomplete)3139 363
y FL(3)3179 396 y FT(.)67 b(Although)38 b(this)378 509
y(implemen)m(tation)30 b(is)h(suitable)f(for)i(its)f(purp)s(ose,)f
(more)i(e\016cien)m(t)h(searc)m(h)f(strategies)h(are)f(required)378
622 y(if)d(one)i(needs)f(to)h(use)f(it)g(in)f(deriving)f(less)i
(trivial)e(statemen)m(ts.)519 735 y(An)33 b(in)m(teresting)g(direction)
f(for)h(future)g(researc)m(h)h(is)e(the)i(in)m(v)m(estigation)f(of)h(w)
m(a)m(ys)g(of)g(incorp)s(o-)378 848 y(rating)f(theory)h(sp)s(eci\014c)f
(decision)f(pro)s(cedures)h(with)f(suc)m(h)h(a)i(calculus.)49
b(The)33 b(pro)s(of)g(c)m(hec)m(king)i(of)378 961 y(SPL)e(scripts)f(in)
m(v)m(olv)m(es)i(the)g(application)e(of)i(theory)g(sp)s(eci\014c)e
(simpli\014ers)e(b)s(efore)j(the)h(refutation)378 1074
y(pro)s(cess.)39 b(Although)25 b(this)f(metho)s(d)h(pro)m(v)m(ed)h(to)h
(b)s(e)e(quite)g(e\013ectiv)m(e,)k(simpli\014ers)22 b(and)j(also)g
(decision)378 1187 y(pro)s(cedures)34 b(can)i(b)s(e)f(used)f(b)m(y)i(a)
f(\014rst-order)g(pro)s(of)g(calculus)f(during)f(the)j(refutational)e
(pro)s(cess)378 1300 y(in)h(order)i(to)g(enhance)g(its)f(deductiv)m(e)g
(p)s(o)m(w)m(er.)60 b(Suc)m(h)36 b(tec)m(hniques)h(ha)m(v)m(e)h(b)s
(een)e(studied)f(recen)m(tly)378 1413 y(in)c(\(Bj\034rner,)h(Stic)m(k)m
(el,)h(and)e(Urib)s(e)g(1997\))j(where,)e(for)g(example,)g(decision)f
(pro)s(cedures)f(are)j(used)378 1526 y(b)m(y)c(a)h(\014rst-order)f(pro)
m(v)m(er)h(to)g(suggest)g(a)g(substitution)d(whic)m(h)i(p)s(oten)m
(tially)f(refutes)h(a)h(giv)m(en)g(set)g(of)378 1638
y(clauses.)519 1751 y(A)j(database)g(of)g(trivial)d(kno)m(wledge)j(is)f
(used)f(in)h(the)g(automatic)i(deriv)-5 b(ation)31 b(of)i(simple)d
(facts)378 1864 y(during)24 b(the)i(pro)s(of)g(c)m(hec)m(king)h(of)f
(SPL)f(scripts.)39 b(Suc)m(h)25 b(database)i(can)g(b)s(e)e(queried)g(b)
m(y)h(other)h(theory)378 1977 y(sp)s(eci\014c)j(\(or)j(more)e
(general\))i(pro)s(of)d(pro)s(cedures.)44 b(It)32 b(is)f(sho)m(wn)g(in)
f(our)h(case)i(study)e(that)h(the)g(use)378 2090 y(of)k(simpli\014ers)c
(whic)m(h)j(are)i(able)e(to)i(query)f(this)f(database)i(can)f(greatly)g
(increase)g(the)h(p)s(o)m(w)m(er)f(of)378 2203 y(the)30
b(SPL)f(pro)s(of)g(c)m(hec)m(k)m(er)j(during)c(the)i(mec)m(hanisation)g
(of)g(a)g(theory)-8 b(.)42 b(This)28 b(results)h(in)f(the)i(abilit)m(y)
378 2316 y(to)37 b(write)e(formal)g(pro)s(ofs)g(whic)m(h)g(are)h(quite)
f(similar)e(to)k(those)g(found)d(in)h(informal)f(texts)i(where)378
2429 y(trivial)29 b(facts)k(are)f(often)f(omitted.)44
b(W)-8 b(e)33 b(ha)m(v)m(e)g(not)e(y)m(et)i(tried)d(to)i(mo)s(dify)e
(the)i(implemen)m(tation)e(of)378 2542 y(the)f FN(C)5
b(B)s(S)i(E)35 b FT(calculus)27 b(presen)m(ted)i(here)f(to)h(b)s(e)f
(able)g(to)h(query)e(suc)m(h)i(a)f(database.)41 b(W)-8
b(e)30 b(b)s(eliev)m(e)d(that)378 2655 y(suc)m(h)37 b(a)g(mo)s
(di\014cation)e(will)f(result)i(in)g(the)h(abilit)m(y)e(to)j(implemen)m
(t)e(shorter)g(and)h(p)s(ossibly)d(more)378 2768 y(readable)c(formal)f
(pro)s(ofs.)p 378 5254 1380 4 v 482 5308 a FC(3)516 5340
y FB(Note)19 b(that)f(the)g(use)g(of)h(an)g(incomplete)f(constrain)n(t)
h(solving)g(metho)r(d)e(do)r(es)i(not)g(con\015ict)f(with)h(the)f
(completeness)378 5431 y(of)32 b(the)e(calculus)i(for)f(\014rst-order)g
(logic)h(with)f(equalit)n(y)-6 b(.)50 b(The)31 b(consequence)f(of)i
(using)f(an)g(incomplete)f(constrain)n(t)378 5522 y(solving)18
b(algorithm)g(is)h(that)e(inferences)i(whic)n(h)e(in)h(principle)g(w)n
(ould)g(fail)h(due)e(to)h(the)f(inconsistency)h(of)g(the)f(constrain)n
(t)378 5614 y(in)26 b(their)h(conclusion)g(can)g(still)g(b)r(e)g
(considered)g(during)f(pro)r(of)h(searc)n(h.)38 b(As)26
b(a)h(result)f(the)g(searc)n(h)h(space)g(considered)378
5705 y(during)e(pro)r(of)i(searc)n(h)f(is)g(larger)h(than)f(the)f
(ideal)h(one.)p eop
%%Page: 98 108
98 107 bop 378 1019 a FJ(Chapter)65 b(6)378 1434 y FR(Structured)79
b(Straigh)-6 b(tforw)g(ard)378 1683 y(Justi\014cations)378
2165 y FH(6.1)135 b(Motiv)-7 b(ation)378 2368 y FT(The)41
b(Mizar)g(pro)s(of)f(language,)45 b(and)c(similar)d(languages)k(suc)m
(h)f(as)g(SPL)g(\(c)m(hapter)h(4\))g(and)e(DE-)378 2481
y(CLARE)j(\(Syme)h(1997a;)54 b(Syme)43 b(1998\),)50 b(are)45
b(often)f(describ)s(ed)e(as)i(supp)s(orting)e(a)i(declarativ)m(e)378
2594 y(pro)s(of)32 b(st)m(yle)i(as)f(opp)s(osed)f(to)i(the)f(more)g
(pro)s(cedural)e(st)m(yle)j(of)f(tactic-based)h(pro)s(of)e(dev)m
(elopmen)m(t)378 2706 y(\(see,)e(for)f(instance,)g(\(Harrison)f(1997\))
j(for)d(a)i(comparison)e(of)h(di\013eren)m(t)f(pro)s(of)g(st)m(yles\).)
41 b(Although)378 2819 y(the)28 b(distinction)e(b)s(et)m(w)m(een)j(a)g
(declarativ)m(e)f(and)g(a)h(pro)s(cedural)d(st)m(yle)j(is)e(somewhat)h
(v)-5 b(ague,)30 b(declar-)378 2932 y(ativ)m(e)25 b(pro)s(ofs)f(do)g
(not)h(explicitly)d(state)k(all)d(the)i(details)f(on)g
FI(how)36 b FT(a)24 b(theorem)h(is)f(pro)m(v)m(ed,)i(but)e(rather)378
3045 y(state)36 b FI(what)46 b FT(is)34 b(needed.)55
b(F)-8 b(or)36 b(instance,)g(simple)d(results)h(in)g(a)h(pro)s(of)g
(script)f(can)h(b)s(e)g(deriv)m(ed)f(b)m(y)378 3158 y(straigh)m(tforw)m
(ard)c(justi\014cations)f(whic)m(h)g(are)i(usually)d(of)i(the)h(form)
473 3336 y FP(C)102 b FM(by)48 b FP(P)842 3350 y FL(1)881
3336 y FM(,)g FP(:)15 b(:)g(:)q FM(,)47 b FP(P)1236 3350
y FO(n)378 3514 y FT(where)39 b FP(P)708 3528 y FL(1)748
3514 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(P)1023 3528 y
FO(n)1110 3514 y FT(are)40 b(the)g(premises)e(of)i(the)g
(justi\014cation)e(and)h FP(C)46 b FT(is)39 b(its)g(conclusion.)67
b(Suc)m(h)378 3627 y(statemen)m(ts)514 3805 y FN(\017)46
b FT(state)32 b(explicitly)c(whic)m(h)h(conclusion)g(is)g(b)s(eing)g
(justi\014ed,)514 3989 y FN(\017)46 b FT(list)29 b(the)i(premises)e
(whic)m(h)g(are)i(required)d(to)j(deriv)m(e)f(the)h(conclusion,)514
4172 y FN(\017)46 b FT(do)30 b(not)h(explain)e(ho)m(w)h(the)h(premises)
e(are)h(used)g(in)f(deriving)f(the)j(conclusion.)519
4350 y(Straigh)m(tforw)m(ard)48 b(justi\014cations)e(are)j(c)m(hec)m(k)
m(ed)h(b)m(y)d(using)g(a)h(simple)e(automatic)j(theorem)378
4463 y(pro)m(v)m(er)23 b(whic)m(h)d(lo)s(oks)i(for)g(a)h(pro)s(of)e(of)
h(the)h(conclusion)e(from)g(the)i(giv)m(en)f(premises.)36
b(The)22 b(complexit)m(y)378 4576 y(of)30 b(the)g(pro)s(ofs)g(that)g
(can)h(b)s(e)e(found)g(automatically)h(b)m(y)g(the)g(pro)s(of)g(c)m
(hec)m(k)m(er)i(is)d(a)h(v)m(ery)h(imp)s(ortan)m(t)378
4689 y(factor)39 b(in)d(determining)g(the)i(readabilit)m(y)e(of)i(the)g
(scripts)f(whic)m(h)g(can)h(b)s(e)f(implemen)m(ted)f(in)h(the)378
4802 y(system.)k(If)29 b(the)i(pro)s(of)e(c)m(hec)m(k)m(er)k(can)d
(automate)i(complex)e(pro)s(ofs)f(whic)m(h)g(are)i(v)m(ery)f(hard)f(to)
i(\014nd,)378 4915 y(then)36 b(quite)f(uninformativ)m(e)f(pro)s(ofs)h
(can)i(b)s(e)e(implemen)m(ted)g(in)f(the)j(system,)g(and)f
(furthermore,)378 5028 y(suc)m(h)27 b(pro)s(ofs)g(w)m(ould)f(require)g
(substan)m(tial)h(resources)g(in)g(order)g(to)h(b)s(e)f(mac)m(hine)g(c)
m(hec)m(k)m(ed.)42 b(On)27 b(the)378 5141 y(other)i(hand,)f(if)f(only)h
(v)m(ery)h(simple)d(inferences)i(can)h(b)s(e)f(implemen)m(ted,)f(the)i
(resulting)e(pro)s(ofs)h(will)378 5253 y(b)s(e)i(to)s(o)h(detailed)e
(to)i(follo)m(w)f(and)g(hard)f(to)i(implemen)m(t.)519
5366 y(The)45 b(inferences)f(whic)m(h)g(are)i(allo)m(w)m(ed)f(to)g(b)s
(e)g(mac)m(hine)g(c)m(hec)m(k)m(ed)i(are)f(often)f(restricted)g(to)378
5479 y(those)35 b(whic)m(h)e(are)h FI(obvious)42 b FT(according)34
b(to)h(some)g(sp)s(eci\014c)e(de\014nition)f(of)i(ob)m(viousness.)51
b(Ob)m(vious)378 5592 y(inferences)23 b(are)h(those)h(whic)m(h)d(are)j
(considered)d(to)j(b)s(e)e(easily)g(follo)m(w)m(ed)h(b)m(y)g(a)g(h)m
(uman)f(reader)h(as)g(w)m(ell)378 5705 y(as)33 b(e\016cien)m(tly)f(c)m
(hec)m(k)m(ed)i(b)m(y)e(mac)m(hine.)46 b(Sp)s(eci\014c)31
b(de\014nitions)f(of)i(ob)m(vious)g(inferences)f(are)i(usually)2057
5954 y(98)p eop
%%Page: 99 109
99 108 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)288
b FT(99)378 396 y(based)32 b(on)g(the)g(e\013ort)h(required)e(to)i(c)m
(hec)m(k)g(the)g(inference.)45 b(F)-8 b(or)33 b(instance,)g(Da)m(vis)f
(\(1981\))j(de\014ned)378 509 y(ob)m(vious)23 b(inferences)g(as)h
(those)h(that)f(ha)m(v)m(e)h(a)f(pro)s(of)f(in)m(v)m(olving)g(at)h
(most)h(one)f(substitution)d(instance)378 622 y(of)h(eac)m(h)g
(premise.)37 b(Rudnic)m(ki)19 b(\(1987\))24 b(observ)m(ed)e(that)g(suc)
m(h)f(inferences)f(ma)m(y)j(still)c(b)s(e)i(hard)f(to)j(pro)s(of)378
735 y(c)m(hec)m(k)33 b(and)e(in)f(general,)i(one)g(can)g(justify)e(an)m
(y)i(conclusion)e(with)g(a)i(Da)m(vis)g(ob)m(vious)f(inference)g(b)m(y)
378 848 y(rep)s(eating)f(the)g(premises)g(of)g(the)h(justi\014cation.)
40 b(Rudnic)m(ki)28 b(prop)s(osed)h(an)h(alternativ)m(e)h(de\014nition)
378 961 y(of)36 b(ob)m(vious)g(inferences,)h(according)g(to)g(whic)m(h)
e(an)h(inference)g(is)f(ob)m(vious)h(if)f(there)i(is)e(not)i(m)m(uc)m
(h)378 1074 y(non-determinism)j(in)m(v)m(olv)m(ed)i(in)f(\014nding)g
(its)h(pro)s(of)f(when)h(using)f(a)i(sp)s(eci\014c)e(algorithm)h(giv)m
(en)378 1187 y(in)29 b(\(Rudnic)m(ki)f(1987\).)519 1300
y(In)35 b(practice)g(it)g(is)g(quite)g(hard)f(to)i(formalise)e(ob)m
(viousness)h(b)m(y)g(a)h(rigid)d(de\014nition)g(based)i(on)378
1413 y(a)44 b(general)g(deductiv)m(e)g(mec)m(hanism.)81
b(The)44 b(actual)g(de\014nition)e(of)i(the)g(notion)g(of)g(ob)m
(viousness)378 1526 y(in)38 b(a)j(particular)d(system)i(is)f(simply)e
(determined)i(b)m(y)g(the)h(implemen)m(tation)f(of)h(the)g(algorithm)
378 1638 y(used)34 b(in)g(the)h(pro)s(of)g(c)m(hec)m(king)g(pro)s
(cess,)h(and)f(suc)m(h)g(an)g(algorithm)f(is)g(impro)m(v)m(ed)g(and)h
(optimised)378 1751 y(as)k(new)g(v)m(ersions)f(of)h(the)h(system)f(are)
g(released.)67 b(As)39 b(w)m(e)g(argued)g(in)f(section)h(2.5.1,)k(a)d
(h)m(uman)378 1864 y(reader)27 b(often)g(relies)f(on)h(his)e
FI(understanding)37 b FT(to)27 b(infer)f(facts)i(rather)e(than)h(on)g
(mec)m(hanical)g(means,)378 1977 y(and)j(therefore)i(the)f(notions)f
(of)h(h)m(uman)f(ob)m(viousness)g(and)g(mac)m(hine)h(ob)m(viousness)f
(can)h(b)s(e)f(quite)378 2090 y(di\013eren)m(t.)80 b(Giv)m(en)43
b(the)h(di\016cult)m(y)e(of)h(de\014ning)f(a)i(practical)f(notion)g(of)
h(ob)m(viousness,)i(w)m(e)e(call)378 2203 y(the)d(inferences)f(whic)m
(h)g(can)h(b)s(e)f(pro)s(of)g(c)m(hec)m(k)m(ed)j(b)m(y)e(a)g
(particular)e(system)i(as)g FI(str)-5 b(aightforwar)g(d)378
2316 y FT(inferences,)41 b(taking)e(the)g(adjectiv)m(e)h(`straigh)m
(tforw)m(ard')g(from)f(`straigh)m(tforw)m(ard)g(justi\014cations'.)378
2429 y(W)-8 b(e)37 b(can)g(also)f(denote)g(the)h(inference)e(of)h(a)h
(conclusion)d(from)i(a)g(n)m(um)m(b)s(er)f(of)h(premises)f(giv)m(en)h
(in)378 2542 y(straigh)m(tforw)m(ard)30 b(justi\014cations)f(b)m(y)h
(an)g(inference)g(rule)1482 2721 y FP(P)1540 2735 y FL(1)1670
2721 y FN(\001)15 b(\001)g(\001)107 b FP(P)1940 2735
y FO(n)p 1482 2755 507 4 v 1699 2833 a FP(C)2029 2777
y FT(\(Straigh)m(tforw)m(ard\))378 3038 y(whic)m(h)32
b(w)m(e)i(call)f(the)h(straigh)m(tforw)m(ard)f(inference)g(rule.)49
b(This)32 b(rule)g(dep)s(ends)g(on)h(the)h(\(particular)378
3151 y(v)m(ersion)24 b(of)h(the\))h(particular)d(system)i(considered.)
37 b(In)25 b(SPL)e(\(as)j(w)m(ell)e(as)h(in)e(other)i(systems)g(suc)m
(h)g(as)378 3264 y(the)h(Mizar)g(mo)s(de)g(in)f(HOL)g(of)i(Harrison)d
(\(1996b\)\),)30 b(the)c(user)f(can)i(use)f(di\013eren)m(t)f(straigh)m
(tforw)m(ard)378 3376 y(rules)k(b)m(y)h(explicitly)e(stating)j(whic)m
(h)e(pro)m(v)m(er)i(is)e(used)h(during)e(the)i(pro)s(of)g(c)m(hec)m
(king)h(pro)s(cess.)519 3489 y(Although)h(straigh)m(tforw)m(ard)g
(justi\014cations)f(do)h(not)h(men)m(tion)f(explicitly)e(the)i
(particular)f(in-)378 3602 y(ferences)24 b(whic)m(h)f(are)h(used)g(in)e
(deriving)g(the)i(conclusion)f(from)g(the)i(premises,)f(it)f(is)g
(often)i(observ)m(ed)378 3715 y(\(b)m(y)38 b(v)-5 b(an)38
b(Gasteren)h(\(1990\))h(for)e(example\))g(that)g(men)m(tioning)f
(certain)h(inferences)f(used)g(in)g(the)378 3828 y(justi\014cation)e
(can)h(impro)m(v)m(e)g(the)h(readabilit)m(y)d(of)j(the)f(pro)s(of.)57
b(The)36 b(reason)h(for)f(this)f(is)g(that)i(the)378
3941 y(readabilit)m(y)29 b(of)h(a)h(pro)s(of)e(dep)s(ends)g(on)h(the)g
(e\013ort)h(required)e(b)m(y)h(the)g(reader)g(to)h(\014ll)d(in)h(the)i
(gaps)f(in)378 4054 y(the)d(pro)s(of,)g(and)f(therefore)i(men)m
(tioning)e(a)h(n)m(um)m(b)s(er)e(of)i(the)h(inferences)d(used)i(can)g
(reduce)f(suc)m(h)h(an)378 4167 y(e\013ort.)41 b(The)28
b(use)g(of)g(`inference-less')g(\(general)h(or)f(sp)s(eci\014c\))g
(straigh)m(tforw)m(ard)g(rules)f(in)g(justifying)378
4280 y(pro)s(of)36 b(results)g(ma)m(y)i(not)f(b)s(e)g(ideal)f(for)h
(the)g(dev)m(elopmen)m(t)h(of)f(readable)g(pro)s(ofs.)60
b(On)36 b(the)i(other)378 4393 y(hand,)27 b(a)g(pro)s(of)f(whic)m(h)g
(explicitly)e(states)k(all)e(the)h(inferences)f(used)g(is)g(to)s(o)i
(detailed)e(and)g(lo)m(w-lev)m(el)378 4506 y(to)31 b(b)s(e)f(follo)m(w)
m(ed)g(easily)-8 b(.)519 4618 y(In)35 b(this)g(c)m(hapter)h(w)m(e)g(in)
m(tro)s(duce)f(the)g(notion)h(of)f(straigh)m(tforw)m(ard)h
(justi\014cations)e(whic)m(h)g(ex-)378 4731 y(plicitly)k(state)k
FI(some)49 b FT(of)41 b(the)g(inferences)f(used)g(in)f(the)i(deriv)-5
b(ation)40 b(of)h(their)f(conclusion.)70 b(The)378 4844
y(motiv)-5 b(ations)30 b(for)g(the)h(use)f(of)g(suc)m(h)g
(justi\014cations)f(include:)514 5009 y FN(\017)46 b
FT(impro)m(ving)24 b(the)i(readabilit)m(y)e(of)i(the)g(pro)s(ofs)f(b)m
(y)h(giving)f(more)h FI(r)-5 b(elevant)35 b FT(information)24
b(to)j(the)605 5122 y(reader;)514 5301 y FN(\017)46 b
FT(giving)27 b(more)h(relev)-5 b(an)m(t)28 b(information)e(to)j(the)f
(pro)s(of)f(c)m(hec)m(k)m(er)j(so)e(that)h(pro)s(ofs)e(can)h(b)s(e)f
(found)605 5414 y(more)k(e\016cien)m(tly;)514 5592 y
FN(\017)46 b FT(exploring)31 b(whether)h(some)h(inferences)e(can)i(b)s
(e)f(stated)h(in)e(straigh)m(tforw)m(ard)h(justi\014cations)605
5705 y(without)e(making)f(the)i(resulting)d(pro)s(ofs)i(to)s(o)h
(detailed)e(or)i(pro)s(cedural;)p eop
%%Page: 100 110
100 109 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(100)514 396 y FN(\017)46 b FT(exploring)34 b(whether)i(simple)d
(results)i(can)h(b)s(e)f(deriv)m(ed)g(b)m(y)h(a)g(less)f(implemen)m
(tation-based)605 509 y(mec)m(hanism)26 b(than)g(that)h(of)g(using)e
(straigh)m(tforw)m(ard)i(rules)e(in)m(tended)g(to)i(automate)h(ob)m
(vious)605 622 y(inferences.)378 810 y(The)i(mec)m(hanism)g(w)m(e)h
(use)f(in)m(v)m(olv)m(es)h(the)g(distinction)d(b)s(et)m(w)m(een)j
(trivial)e(inferences)g(and)h(relev)-5 b(an)m(t,)378
923 y(or)44 b FI(substantial)p FT(,)50 b(inferences,)e(and)c(using)f
(these)i(notions)e(in)h(de\014ning)e FI(gener)-5 b(alise)g(d)56
b FT(inferences)378 1036 y(whic)m(h)42 b(in)m(v)m(olv)m(e)h(the)g
(application)e(of)i(a)g(relev)-5 b(an)m(t)43 b(inference)g(and)f(sev)m
(eral)h(trivial)e(ones.)79 b(Only)378 1149 y(suc)m(h)31
b(generalised)g(inferences)g(can)h(b)s(e)g(used)f(in)f(straigh)m(tforw)
m(ard)i(justi\014cations.)43 b(The)32 b(resulting)378
1262 y(justi\014cations)39 b(are)h(called)f(structured)h(straigh)m
(tforw)m(ard)f(justi\014cations)g(since)h(the)g(generalised)378
1374 y(inferences)30 b(used)f(are)i(represen)m(ted)g(b)m(y)f(binary)f
(op)s(erators)i(on)g(premises)e(whic)m(h)g(giv)m(e)i(them)g(more)378
1487 y(structure)f(than)g(inference-less)f(justi\014cations.)519
1600 y(In)41 b(the)i(next)f(section,)j(w)m(e)e(discuss)d(ho)m(w)i
(inference)f(rules)g(can)h(b)s(e)g(generalised)f(according)378
1713 y(to)c(a)f(n)m(um)m(b)s(er)f(of)h(trivial)e(inferences,)i(or)g
(manipulations)d(on)j(form)m(ulae)g(whic)m(h)f(can)h(b)s(e)f(applied)
378 1826 y(implicitly)g(to)k(the)f(premises)f(and)h(conclusion)f(of)i
(the)f(rules.)64 b(W)-8 b(e)40 b(in)m(tro)s(duce)d(the)i(syn)m(tax)f
(and)378 1939 y(seman)m(tics)28 b(of)f(structured)g(straigh)m(tforw)m
(ard)g(justi\014cations)f(in)g(sections)h(6.3)i(and)e(6.4.)41
b(A)27 b(n)m(um)m(b)s(er)378 2052 y(of)32 b(results)g(on)g(suc)m(h)g
(justi\014cations)f(are)h(giv)m(en)h(in)e(section)h(6.5,)i(and)e(a)h
(concluding)d(discussion)g(is)378 2165 y(giv)m(en)e(in)f(section)h
(6.6.)41 b(A)28 b(mec)m(hanism)f(for)h(restricting)f(the)h(pro)s(of)g
(searc)m(h)g(required)e(for)i(v)m(erifying)378 2278 y(structured)g
(justi\014cations)f(is)i(then)f(illustrated)f(in)h(c)m(hapter)h(8,)h
(after)g(the)f(relev)-5 b(an)m(t)29 b(notation)g(and)378
2391 y(results)22 b(required)g(for)i(de\014ning)d(this)i(mec)m(hanism)g
(and)g(pro)m(ving)g(its)g(soundness)f(and)h(completeness)378
2504 y(are)31 b(dev)m(elop)s(ed)e(in)h(c)m(hapter)h(7.)378
2790 y FH(6.2)135 b(On)30 b(Explicitly)i(Stated)f(Inferences)g(and)f
(Implicitly)i(Applied)684 2939 y(Manipulations)378 3142
y FT(It)i(is)f(men)m(tioned)h(in)f(section)h(3.5)i(that)e(tactic-based)
i(pro)s(ofs)d(often)i(con)m(tain)f(v)m(ery)h(basic)e(results)378
3255 y(and)f(inferences,)h(ev)m(en)h(when)e(the)i(pro)s(ofs)e(are)h
(implemen)m(ted)f(at)i(a)f(mature)g(stage)i(of)e(the)g(mec)m(h-)378
3368 y(anisation)41 b(where)h(sev)m(eral)h(high-lev)m(el)f(results)f
(ha)m(v)m(e)j(b)s(een)d(deriv)m(ed.)76 b(Suc)m(h)42 b(trivial)f
(inferences)378 3481 y(rarely)36 b(con)m(tribute)h(to)g(the)h
(comprehensibilit)m(y)33 b(of)k(the)g(pro)s(ofs,)h(and)e(it)h(is)f
(often)h(the)g(case)h(that)378 3594 y(o)m(v)m(er-detailed)j(pro)s(ofs)f
(are)h(hard)f(to)h(follo)m(w)f(as)h(w)m(ell)e(as)i(tedious)f(to)h
(implemen)m(t.)70 b(It)41 b(is)e(there-)378 3707 y(fore)29
b(desirable)e(that)j(suc)m(h)e(inferences)g(are)i(omitted)f(from)f(pro)
s(ofs)g(b)m(y)h(pro)m(viding)e(the)i(necessarily)378
3820 y(automation)39 b(to)g(deriv)m(e)f(them)h(`implicitly'.)62
b(Of)38 b(course,)j(not)d(all)g(the)g(steps)h(of)g(a)f(mec)m(hanised)
378 3933 y(pro)s(of)h(are)g(trivial.)66 b(A)39 b(considerable)f(n)m(um)
m(b)s(er)g(of)i(steps)f(use)g(high-lev)m(el)f(theorems)i(and)e(apply)
378 4046 y(theory-sp)s(eci\014c)28 b(pro)s(of)f(pro)s(cedures.)39
b(Suc)m(h)27 b(pro)s(of)h(steps)g(can)h(giv)m(e)g(a)f(go)s(o)s(d)g
(idea)g(of)h(ho)m(w)f(the)g(con-)378 4159 y(clusion)23
b(of)h(the)h(pro)s(of)f(is)f(deriv)m(ed.)38 b(A)25 b(mec)m(hanised)f
(pro)s(of)f(can)i(therefore)g(b)s(e)f(seen)h(as)f(con)m(taining)g(a)378
4271 y(n)m(um)m(b)s(er)g(of)i(substan)m(tial)f(inferences)g(whic)m(h)f
(con)m(tribute)i(to)g(the)g(comprehensibilit)m(y)d(of)i(the)h(pro)s
(of,)378 4384 y(together)f(with)d(a)j(n)m(um)m(b)s(er)d(of)i(trivial)e
(ones)h(whic)m(h)g(p)s(oten)m(tially)f(hinder)g(it.)38
b(In)22 b(this)h(section)h(w)m(e)g(dis-)378 4497 y(cuss)31
b(the)g(p)s(ossibilit)m(y)d(of)j(implemen)m(ting)f(pro)s(ofs)g(whic)m
(h)g(consist)h(only)f(of)i(substan)m(tial)e(inferences)378
4610 y(and)h(an)m(y)i(trivial)d(inferences)h(can)h(b)s(e)f(applied)f
(implicitly)-8 b(.)42 b(In)31 b(section)h(6.2.1)i(b)s(elo)m(w)d(w)m(e)h
(describ)s(e)378 4723 y(the)c(notion)g(of)g FI(gener)-5
b(alising)37 b FT(an)28 b(inference)f(whic)m(h)g(in)m(v)m(olv)m(es)h
(the)g(de\014nition)e(of)i(an)g(inference)f(rule)378
4836 y(whose)d(premises)f(and)h(conclusion)f(can)i(b)s(e)f(implicitly)d
(manipulated)h(according)j(to)g(a)g(giv)m(en)g(set)g(of)378
4949 y(inferences.)48 b(Structured)32 b(straigh)m(tforw)m(ard)h
(justi\014cations,)f(in)g(whic)m(h)g(a)i(n)m(um)m(b)s(er)e(of)h
(generalised)378 5062 y(inferences)c(are)i(stated)g(explicitly)-8
b(,)29 b(are)i(in)m(tro)s(duced)e(in)g(section)h(6.2.2.)378
5305 y FG(6.2.1)112 b(Generalising)37 b(Inferences)378
5477 y FT(Ideally)-8 b(,)43 b(the)f(inference)e(rules)g(whic)m(h)g(are)
h(used)g(in)e(the)j(mec)m(hanisation)f(of)g(pro)s(ofs)f(should)g(b)s(e)
378 5590 y(de\014ned)e(in)h(suc)m(h)g(a)h(manner)e(that)j(no)e(trivial)
f(inferences)g(are)i(needed)f(in)g(pro)s(of)g(implemen)m(ta-)378
5703 y(tion.)58 b(If)36 b(a)h(n)m(um)m(b)s(er)e(of)i(inferences)e(are)i
(iden)m(ti\014ed)e(as)h(trivial,)g(one)h(can)g(usually)d(generalise)i
(an)p eop
%%Page: 101 111
101 110 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(101)378 396 y(arbitrary)35 b(substan)m(tial)f(inference)h(rule)g
(b)m(y)g(applying)f(the)i(trivial)e(inferences)h(b)s(efore)g(and)g
(after)378 509 y(the)h(substan)m(tial)e(inference)h(is)g(applied.)55
b(More)36 b(formally)-8 b(,)36 b(let)g(us)f(consider)g(a)h(set)g(of)g
(inferences)378 622 y FN(I)42 b FT(=)36 b FN(f)p FP(I)662
636 y FL(1)702 622 y FP(;)15 b(I)782 636 y FL(2)822 622
y FP(;)g(:)g(:)g(:)h FN(g)p FT(.)61 b(Eac)m(h)38 b(rule)d(tak)m(es)j
(one)g(premise)d(from)i(whic)m(h)e(it)i(infers)e(a)i(conclusion,)g(and)
378 735 y(this)28 b(inference)h(is)g(assumed)f(to)j(b)s(e)e(trivial,)f
(in)g(the)h(sense)h(that)g(it)f(can)h(\(and)f(should\))f(b)s(e)h
(omitted)378 848 y(from)39 b(the)h(implemen)m(tation)e(of)h(pro)s(ofs.)
67 b(Note)41 b(that)f(in)e(this)g(thesis)h(w)m(e)h(consider)e(only)h
(trivial)378 961 y(inferences)30 b(whic)m(h)f(tak)m(e)j(a)f(single)e
(premise)h FP(A)g FT(and)g(return)g(a)h(conclusion)e
FP(B)5 b FT(,)30 b(or)h(in)e(other)i(w)m(ords,)378 1074
y(whic)m(h)36 b(implicitly)d(manipulate)i(the)i(form)m(ula)g
FP(A)g FT(in)m(to)g FP(B)5 b FT(.)60 b(T)-8 b(rivial)34
b(inferences)i(whic)m(h)g(can)h(tak)m(e)378 1187 y(more)f(than)f(one)h
(premise)e(ma)m(y)i(b)s(e)f(considered)f(in)g(future.)55
b(W)-8 b(e)37 b(can)e(de\014ne)g(a)h(binary)e(relation)378
1300 y FN(!)469 1314 y FK(I)547 1300 y FT(o)m(v)m(er)d(form)m(ulae)f
(suc)m(h)g(that)1092 1540 y FP(A)25 b FN(!)1276 1554
y FK(I)1349 1540 y FP(B)95 b FT(if)29 b(and)h(only)g(if)2119
1476 y FP(A)p 2117 1495 74 4 v 2117 1574 a(B)2232 1518
y FT(\()p FP(I)2307 1532 y FO(i)2335 1518 y FT(\))91
b(for)31 b(some)f FP(I)2868 1532 y FO(i)2927 1518 y FT(in)f
FN(I)7 b FT(.)378 1778 y(W)-8 b(e)32 b(can)e(also)h(denote)f(the)h
(expression)e FP(A)d FN(!)1956 1792 y FK(I)2028 1778
y FP(B)35 b FT(b)m(y)30 b(an)h(instance)f(of)g(an)g(inference)g(rule)f
(\()p FN(I)7 b FT(\):)1457 1963 y FP(A)p 1454 1983 V
1454 2062 a(B)1569 2006 y FT(\()p FN(I)g FT(\))91 b(if)29
b(and)h(only)g(if)120 b FP(A)25 b FN(!)2605 2020 y FK(I)2678
2006 y FP(B)378 2266 y FT(although)j(suc)m(h)g(a)h(rule)e(is)g
(non-deterministic)f(as)j(sev)m(eral)g(inferences)e(in)h
FN(I)34 b FT(can)29 b(b)s(e)f(applicable)e(to)378 2379
y(the)31 b(premise)f FP(A)p FT(,)i(and)f(therefore)g(sev)m(eral)h(p)s
(ossible)d(conclusions)g(can)j(b)s(e)e(inferred)g(b)m(y)h(\()p
FN(I)7 b FT(\).)43 b(No)m(w,)378 2492 y(giv)m(en)33 b(an)f(inference)g
(rule,)h(denoted)f(b)m(y)h FP(R)g FT(sa)m(y)-8 b(,)35
b(whic)m(h)c(infers)h(a)h(conclusion)e(from)h(a)h(n)m(um)m(b)s(er)f(of)
378 2605 y(premises)1759 2693 y FP(P)1817 2707 y FL(1)1948
2693 y FN(\001)15 b(\001)g(\001)107 b FP(P)2218 2707
y FO(n)p 1759 2726 507 4 v 1976 2805 a FP(C)2307 2749
y FT(\()p FP(R)q FT(\))378 2972 y(it)27 b(can)h(b)s(e)e
FI(gener)-5 b(alise)g(d)39 b FT(in)m(to)27 b(a)h(rule)e
FP(R)1718 2986 y FK(I)1793 2972 y FT(in)g(whic)m(h)g(a)i(n)m(um)m(b)s
(er)e(of)i(inferences)e(in)g FN(I)34 b FT(can)27 b(b)s(e)g(applied)378
3085 y(implicitly)g(to)k(its)e(premises)g(and)h(conclusion.)39
b(If)30 b(w)m(e)h(de\014ne)f(the)g(rule)f FN(I)2927 3052
y FK(\003)2996 3085 y FT(suc)m(h)h(that)1452 3264 y FP(A)p
1450 3284 74 4 v 1450 3363 a(B)1565 3305 y FT(\()p FN(I)1657
3272 y FK(\003)1696 3305 y FT(\))91 b(if)29 b(and)h(only)g(if)90
b FP(A)25 b FN(!)2610 3272 y FK(\003)2610 3332 y(I)2683
3305 y FP(B)378 3567 y FT(where)f FN(!)726 3534 y FK(\003)726
3594 y(I)798 3567 y FT(is)f(the)i(re\015exiv)m(e)f(transitiv)m(e)g
(closure)g(of)g FN(!)2269 3581 y FK(I)2317 3567 y FT(,)i(then)e(the)g
(rule)f FP(R)2964 3581 y FK(I)3036 3567 y FT(is)h(de\014ned)f(as)i
(follo)m(ws:)784 3878 y FP(P)842 3892 y FL(1)973 3878
y FN(\001)15 b(\001)g(\001)107 b FP(P)1243 3892 y FO(n)p
784 3912 507 4 v 1002 3991 a FP(C)1332 3935 y FT(\(R)1434
3949 y FK(I)1482 3935 y FT(\))121 b(if)30 b(and)g(only)f(if)2333
3756 y FP(P)2391 3770 y FL(1)p 2333 3789 98 4 v 2333
3876 a FP(P)2404 3843 y FK(0)2391 3900 y FL(1)2472 3812
y FT(\()p FN(I)2564 3779 y FK(\003)2603 3812 y FT(\))2820
3876 y FN(\001)15 b(\001)g(\001)3108 3756 y FP(P)3166
3770 y FO(n)p 3108 3789 106 4 v 3108 3876 a FP(P)3179
3843 y FK(0)3166 3898 y FO(n)3255 3812 y FT(\()p FN(I)3347
3779 y FK(\003)3386 3812 y FT(\))p 2333 3920 881 4 v
2726 4007 a FP(C)2798 3974 y FK(0)3255 3943 y FT(\(R\))p
2726 4026 95 4 v 2738 4105 a FP(C)2862 4049 y FT(\()p
FN(I)2954 4016 y FK(\003)2993 4049 y FT(\))378 4309 y(for)30
b(some)h(form)m(ulae)f FP(P)1190 4276 y FK(0)1177 4334
y FL(1)1217 4309 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(P)1504
4276 y FK(0)1491 4332 y FO(n)1569 4309 y FT(and)30 b
FP(C)1818 4276 y FK(0)1841 4309 y FT(.)519 4422 y(W)-8
b(e)36 b(sa)m(y)g(that)f(the)g(conclusion)f FP(C)41 b
FT(is)34 b(deriv)m(ed)g(from)h(the)g(premises)e FP(P)2987
4436 y FL(1)3027 4422 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(P)3302 4436 y FO(n)3384 4422 y FT(b)m(y)35 b(the)g(rule)378
4535 y(\()p FP(R)q FT(\))41 b(and)g(the)g(implicit)d(application)h(of)i
(the)g(inferences)f(in)g FN(I)7 b FT(.)71 b(W)-8 b(e)42
b(also)f(sa)m(y)h(that)f(\()p FP(R)3556 4549 y FK(I)3604
4535 y FT(\))h(is)d(a)378 4648 y(generalisation)29 b(of)i(\()p
FP(R)q FT(\))g(according)f(to)h(the)g(implicit)c(inferences)j(in)f
FN(I)7 b FT(.)519 4761 y(F)-8 b(or)29 b(example,)f(let)g(us)f(consider)
g(the)h(inferences)f(giv)m(en)h(b)m(y)g(the)g(follo)m(wing)e(rules)h
(to)h(b)s(e)g(trivial:)791 4947 y FP(P)13 b FT([)p FP(x)21
b FT(+)e(0])p 791 4989 331 4 v 869 5074 a FP(P)13 b FT([)p
FP(x)p FT(])1163 5012 y(\(+0\))1531 4947 y FP(P)g FT([)p
FP(x)21 b FT(+)f FP(y)s FT(])p 1531 4989 333 4 v 1531
5074 a FP(P)13 b FT([)p FP(y)24 b FT(+)c FP(x)p FT(])1905
5012 y(\(+comm\))2466 4947 y FP(P)13 b FT([\()p FP(x)21
b FT(+)e FP(y)s FT(\))i(+)f FP(z)t FT(])p 2466 4989 561
4 v 2466 5074 a FP(P)13 b FT([)p FP(x)20 b FT(+)g(\()p
FP(y)k FT(+)19 b FP(z)t FT(\)])3068 5012 y(\(+asso)s(c\))1271
5265 y FP(P)13 b FT([)p FP(n)20 b FT(+)g FP(m)p FT(])p
1271 5307 367 4 v 1379 5392 a FP(P)13 b FT([)p FP(l)r
FT(])1680 5330 y(\(+calc)1937 5344 y FL(1)1977 5330 y
FT(\))2302 5265 y FP(P)g FT([)p FP(l)r FT(])p 2194 5307
V 2194 5392 a FP(P)g FT([)p FP(n)20 b FT(+)g FP(m)p FT(])2602
5330 y(\(+calc)2859 5344 y FL(2)2899 5330 y FT(\))378
5559 y(where)26 b(in)f(the)i(\(+calc)1149 5573 y FL(1)1189
5559 y FT(\))f(and)g(\(+calc)1680 5573 y FL(2)1720 5559
y FT(\))h(rules,)f(the)g(n)m(um)m(b)s(er)g FP(l)i FT(is)d(the)i(sum)e
(of)i(the)g(n)m(um)m(b)s(ers)e FP(n)g FT(and)378 5672
y FP(m)p eop
%%Page: 102 112
102 111 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(102)519 396 y(W)-8 b(e)24 b(de\014ne)f(the)g(set)h
FN(A)h FT(=)g FN(f)p FT(+0)p FP(;)15 b FT(+comm)q FP(;)g
FT(+asso)s(c)p FP(;)g FT(+calc)2490 410 y FL(1)2530 396
y FP(;)g FT(+calc)2792 410 y FL(2)2832 396 y FN(g)p FT(,)26
b(and)c(giv)m(en)h(the)h(inference)378 509 y(rule)1646
567 y FP(x)i(>)f(y)93 b(y)28 b(>)d(z)p 1646 604 528 4
v 1800 670 a(x)h(>)f(z)2215 627 y FT(\(+trans\))378 837
y(w)m(e)31 b(can)f(de\014ne)g(the)h(generalised)e(rule)g(\(+trans)2054
851 y FK(A)2115 837 y FT(\),)h(whic)m(h)g(for)g(instance)g(can)g(b)s(e)
g(used)g(to)h(deriv)m(e)1018 1027 y(1)21 b(+)f(\(2)p
FP(a)h FT(+)f(3\))26 b FP(>)f FT(4)p FP(b)182 b FT(\(3)21
b(+)f(1\))p FP(b)26 b(>)f(b)20 b FT(+)g(\(0)h(+)f FP(a)p
FT(\))p 1018 1070 1724 4 v 1595 1150 a(2)p FP(a)h FT(+)f(4)25
b FP(>)g(a)c FT(+)f FP(b)2783 1093 y FT(\(+trans)3092
1107 y FK(A)3152 1093 y FT(\))378 1354 y(A)34 b(mec)m(hanism)f(for)g(c)
m(hec)m(king)i(instances)e(of)h(\(+trans)2272 1368 y
FK(A)2332 1354 y FT(\))g(can)g(b)s(e)f(implemen)m(ted)f(b)m(y)i
(\014rst)f(simpli-)378 1467 y(fying)28 b(the)h(terms)g(in)f(the)h
(premises)f(and)g(the)h(conclusion)f(in)m(to)h(some)h(normal)e(form)g
(according)h(to)378 1580 y(the)e(inferences)e(in)h FN(A)g
FT(and)g(then)g(c)m(hec)m(king)i(whether)e(the)g(resulting)f(form)m
(ulae)h(are)h(as)g(required)e(b)m(y)378 1693 y(the)31
b(inference)e(rule)g(\(+trans\).)519 1806 y(Theory-sp)s(eci\014c)i
(simpli\014ers)d(can)k(b)s(e)g(declared)f(in)g(SPL)g(scripts)g(so)h
(that)h(they)f(can)g(b)s(e)g(used)378 1918 y(automatically)25
b(to)i(normalise)d(the)i(terms)f(in)f(the)i(premises)e(and)h
(conclusions)f(of)i(straigh)m(tforw)m(ard)378 2031 y(justi\014cations)
31 b(during)f(pro)s(of)h(c)m(hec)m(king.)47 b(The)32
b(calculations)f(p)s(erformed)g(b)m(y)h(the)g(simpli\014ers)d(can)378
2144 y(therefore)f(b)s(e)f(seen)g(as)h(the)g(implicit)c(inferences)j
(generalising)e(the)j(straigh)m(tforw)m(ard)f(rule)f(used)h(to)378
2257 y(c)m(hec)m(k)g(SPL)d(justi\014cations)g(\(i.e.,)16
b(the)25 b FN(C)5 b(B)s(S)i(E)33 b FT(deriv)m(ed)24 b(rule)g
(illustrated)f(in)h(the)h(previous)f(c)m(hapter\).)378
2370 y(Note)30 b(that)g(the)f(straigh)m(tforw)m(ard)g(rule)f
(generalised)g(with)g(the)h(implicit)d(inferences)j(giv)m(en)g(b)m(y)g
(the)378 2483 y(simpli\014ers)e(do)s(es)k(not)g(corresp)s(ond)e(to)j
(the)f(straigh)m(tforw)m(ard)g(rule)e(augmen)m(ted)j(with)e(the)h
(simpli-)378 2596 y(\014ers)g(\(whic)m(h)g(in)m(v)m(olv)m(es)i(the)f
(use)g(of)g(the)g(simpli\014ers)c FI(during)41 b FT(the)32
b(pro)s(of)f(c)m(hec)m(king)i(mec)m(hanism)f(of)378 2709
y(the)j(straigh)m(tforw)m(ard)g(rule,)h(rather)f(than)g(just)f
FI(b)-5 b(efor)g(e)43 b FT(or)36 b FI(after)10 b FT(\).)56
b(F)-8 b(or)36 b(example,)g(the)f(straigh)m(t-)378 2822
y(forw)m(ard)c(rule)f(generalised)g(with)g(the)i(simpli\014er)27
b(giv)m(en)32 b(b)m(y)f(the)g(rule)f FP(x)21 b FT(+)g(0)27
b FN(!)g FP(x)k FT(do)s(es)g(not)h(solv)m(e)378 2935
y(the)f(goal)f FN(9)p FP(a:b)20 b FT(+)g FP(a)26 b FT(=)f
FP(b)p FT(,)30 b(though)g(an)h(augmen)m(ted)g(rule)e(w)m(ould.)378
3178 y FG(6.2.2)112 b(Straigh)m(tforw)m(ard)37 b(Justi\014cations)f
(with)h(Explicitly)c(Stated)38 b(Inferences)378 3350
y FT(W)-8 b(e)38 b(no)m(w)f(consider)f(the)h(de\014nition)d(of)j
(straigh)m(tforw)m(ard)g(justi\014cations)e(whic)m(h)h(explicitly)f
(state)378 3463 y(a)46 b(n)m(um)m(b)s(er)e(of)h(the)g(\014rst-order)g
(inferences)f(whic)m(h)g(are)i(used)e(in)g(deriving)f(the)j(conclusion)
e(of)378 3576 y(the)d(justi\014cation)f(from)g(the)h(giv)m(en)g
(premises.)71 b(Ho)m(w)m(ev)m(er,)46 b(these)41 b(rules)e(are)j
(generalised)e(b)m(y)g(a)378 3688 y(n)m(um)m(b)s(er)f(of)h(trivial)e
(inferences)h(whic)m(h)g(manipulate)f(\014rst-order)h(form)m(ulae)h(in)
m(to)f(equiv)-5 b(alen)m(t)40 b(or)378 3801 y(w)m(eak)m(er)46
b(ones.)83 b(As)45 b(a)f(result,)k(although)c(suc)m(h)g
(justi\014cations)f(con)m(tain)i(a)f(certain)h(amoun)m(t)g(of)378
3914 y(information)28 b(on)i(what)g(inferences)f(are)h(used)f(in)g(the)
h(deriv)-5 b(ation,)29 b(this)g(information)f(is)h(not)h(o)m(v)m(er-)
378 4027 y(detailed)38 b(since)g(a)i(n)m(um)m(b)s(er)d(of)j(inferences)
e(are)h(applied)e(implicitly)e(in)j(the)h(deriv)-5 b(ation)37
b(pro)s(cess)378 4140 y(and)c(therefore)g(not)h(men)m(tioned)f(in)f
(the)h(justi\014cation.)48 b(This)32 b(is)g(an)h(alternativ)m(e)h
(metho)s(d)f(to)h(the)378 4253 y(use)40 b(of)h(a)g(straigh)m(tforw)m
(ard)g(justi\014cation)e(con)m(tains)i(a)g(list)f(of)h(premises,)h(and)
e(no)h(information)378 4366 y(ab)s(out)34 b(whic)m(h)f(\014rst-order)g
(inferences)h(are)g(used)g(in)f(justifying)e(the)k(conclusion)e(is)g
(giv)m(en)h(\(apart)378 4479 y(from)24 b(the)i(fact)f(that)h(the)f(o)m
(v)m(erall)g(inference)f(is)g(ob)m(vious)h(according)g(to)g(an)g
(implemen)m(tation-based)378 4592 y(de\014nition)j(of)j(ob)m
(viousness\).)519 4705 y(The)38 b(\014rst-order)g(inferences)g(used)f
(implicitly)e(in)i(deriving)g(the)i(conclusion)e(of)i(a)g(justi\014ca-)
378 4818 y(tion)31 b(are)h(describ)s(ed)d(in)h(section)i(6.4.1)h(and)e
(corresp)s(ond)f(to)i(simple)d(manipulations)g(suc)m(h)i(as)h(the)378
4930 y(instan)m(tiation)39 b(of)h(univ)m(ersally)d(quan)m(ti\014ed)i(v)
-5 b(ariables)38 b(and)h(the)h(application)f(of)g(the)h(comm)m(uta-)378
5043 y(tivit)m(y)32 b(of)h(the)g(conjunction)f(and)g(disjunction)e(op)s
(erators.)48 b(Inferences)32 b(are)h(stated)h(explicitly)c(b)m(y)378
5156 y(constructing)g(expressions)f(using)g(the)h(follo)m(wing)f
(binary)g(op)s(erators:)378 5344 y Fw(on)45 b FT(whic)m(h)29
b(corresp)s(onds)g(to)i(the)f(rule)f(of)i(Mo)s(dus)e(P)m(onens:)41
b(\()p Ft(\()p Fv(A)24 b Fu(\))f Fv(B)t Ft(\))44 b Fw(on)f
Fv(A)p FT(\))31 b(deriv)m(es)f Fv(B)t FT(.)378 5532 y
Fw(and)44 b FT(whic)m(h)29 b(corresp)s(onds)g(to)i(the)g(in)m(tro)s
(duction)d(of)j(conjunction:)40 b(\()p Fv(A)k Fw(and)e
Fv(B)t FT(\))31 b(deriv)m(es)f Fv(A)p Fu(^)p Fv(B)5 b
FT(.)p eop
%%Page: 103 113
103 112 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(103)378 396 y Fw(then)44 b FT(whic)m(h)25 b(is)g(used)h(to)h
(abbreviate)f(certain)g(expressions)f(in)m(v)m(olving)g(the)h
Fw(on)g FT(op)s(erator,)h(and)f(cor-)605 509 y(resp)s(onds)e(to)i(the)g
(transitivit)m(y)e(of)i(implication:)36 b Ft(\()p Fv(A)23
b Fu(\))g Fv(B)t Ft(\))44 b Fw(then)e Ft(\()p Fv(B)28
b Fu(\))23 b Fv(C)6 b Ft(\))26 b FT(deriv)m(es)f Fv(A)e
Fu(\))g Fv(C)7 b FT(.)605 622 y(An)30 b(expression)f(of)i(the)f(form)g
Ft(\()p Fv(X)51 b Fw(then)41 b Fv(Y)19 b Ft(\))44 b Fw(on)f
Fv(Z)36 b FT(is)29 b(equiv)-5 b(alen)m(t)30 b(to)h Fv(Y)62
b Fw(on)43 b Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Z)6 b Ft(\))p
FT(.)378 799 y(Straigh)m(tforw)m(ard)39 b(justi\014cations)e
(constructed)j(using)d(the)i(ab)s(o)m(v)m(e)i(op)s(erators)e(are)h
(called)e(struc-)378 912 y(tured)e(straigh)m(tforw)m(ard)h
(justi\014cations,)g(or)g(simply)e(structured)h(justi\014cations,)h(as)
g(opp)s(osed)f(to)378 1025 y(the)i(unstructured)e(ones)i(whic)m(h)f
(simply)f(list)g(the)i(required)e(premises.)63 b(It)38
b(is)f(not)h(hard)f(to)i(im-)378 1138 y(plemen)m(t)29
b(pro)s(ofs)g(in)m(v)m(olving)f(structured)h(justi\014cations)g(since)g
(only)g(three)h(op)s(erators)f(need)h(to)h(b)s(e)378
1251 y(remem)m(b)s(ered)38 b(and)g(understo)s(o)s(d.)65
b(F)-8 b(urthermore,)41 b(since)d(these)h(op)s(erators)g(corresp)s(ond)
f(to)h(gen-)378 1364 y(eralised)k(inferences,)48 b(structured)c
(justi\014cations)f(omit)i(sev)m(eral)g(tedious)f(details)g(suc)m(h)h
(as)g(the)378 1477 y(instan)m(tiation)25 b(of)h(v)-5
b(ariables)25 b(and)g(structural)g(manipulations)e(on)i(form)m(ulae.)39
b(The)26 b(follo)m(wing)e(is)h(an)378 1589 y(example)30
b(of)h(a)f(v)-5 b(alid)29 b(structured)g(justi\014cation.)473
1766 y FM(")p FN(9)p FM(c.)p FN(8)15 b FM(x.)g(x)47 b(>)h(c)f
FN(\))h FM(x)f(>)h(d")f(by)664 1879 y(")p FN(8)p FM(x)26
b(y)g(z.)15 b(\(x)48 b(>)f(y\))g FN(^)h FM(\(y)f(>)g(z\))g
FN(\))h FM(x)f(>)h(z")f(on)g(")p FN(9)p FM(c.)15 b(c)47
b(>)h(d";)378 2056 y FT(It)31 b(should)d(b)s(e)i(noted)h(that)h(a)f
(structured)e(justi\014cation)h(can)h(b)s(e)f(used)g(to)h(justify)e
(sev)m(eral)i(conclu-)378 2169 y(sions.)44 b(F)-8 b(or)33
b(instance,)f(the)g(justi\014cation)f(of)h(the)g(ab)s(o)m(v)m(e)h
(statemen)m(t)h(can)e(also)g(b)s(e)g(used)f(to)h(deriv)m(e)378
2282 y(the)f(follo)m(wing)d(conclusion:)473 2459 y FM(")p
FN(9)p FM(c.)p FN(8)15 b FM(z.)g(d)47 b(>)h(z)f FN(\))h
FM(c)f(>)h(z")378 2636 y FT(Because)36 b(of)f(their)f
(non-deterministic)e(nature,)k(the)f(generalised)f(inferences)f
(corresp)s(onding)g(to)378 2749 y(the)41 b Fw(on)p FT(,)j
Fw(and)c FT(and)g Fw(then)g FT(op)s(erators)h(cannot)h(b)s(e)f
(implemen)m(ted)e(as)j(functions)e(whic)m(h)g(tak)m(e)j(t)m(w)m(o)378
2861 y(premises)d(and)g(infer)f(a)j(conclusion,)g(but)f(rather)g(as)g
(pro)s(of)f(c)m(hec)m(king)i(functions)e(whic)m(h)f(c)m(hec)m(k)378
2974 y(whether)d(a)h(giv)m(en)f(conclusion)f(follo)m(ws)h(from)g(the)h
(giv)m(en)f(premises.)58 b(The)36 b(formal)g(de\014nition)e(of)378
3087 y(the)d(syn)m(tax)f(and)g(seman)m(tics)h(of)f(structured)g
(justi\014cations)f(is)g(giv)m(en)h(in)f(the)i(next)g(t)m(w)m(o)g
(sections.)378 3372 y FH(6.3)135 b(The)45 b(Syn)l(tax)g(of)g
(Structured)f(Justi\014cations)378 3575 y FT(F)-8 b(or)30
b(the)f(purp)s(oses)e(of)i(this)f(c)m(hapter,)j(the)e(syn)m(tax)g(of)h
(structured)e(straigh)m(tforw)m(ard)g(justi\014cations)378
3687 y(is)h(de\014ned)g(as)i(follo)m(ws:)473 3864 y FI(Structur)-5
b(e)g(d)p 893 3864 29 4 v 46 w(Justi\014c)g(ation)56
b FT(=)95 b FM(by)g FI(Structur)-5 b(e)g(d)p 2238 3864
V 45 w(Expr)g(ession)473 4064 y(Structur)g(e)g(d)p 893
4064 V 46 w(Expr)g(ession)56 b FT(=)47 b FI(Sentenc)-5
b(e)1428 4177 y FN(j)48 b FI(Then)p 1718 4177 V 42 w(Expr)-5
b(ession)56 b FM(on)47 b FI(Structur)-5 b(e)g(d)p 2790
4177 V 46 w(Expr)g(ession)1428 4290 y FN(j)48 b FI(Structur)-5
b(e)g(d)p 1921 4290 V 45 w(Expr)g(ession)57 b FM(and)46
b FI(Structur)-5 b(e)g(d)p 3042 4290 V 46 w(Expr)g(ession)473
4489 y(Then)p 690 4489 V 42 w(Expr)g(ession)56 b FT(=)48
b FI(Structur)-5 b(e)g(d)p 1738 4489 V 45 w(Expr)g(ession)1237
4602 y FN(j)48 b FI(Then)p 1527 4602 V 42 w(Expr)-5 b(ession)56
b FM(then)47 b FI(Then)p 2492 4602 V 42 w(Expr)-5 b(ession)378
4802 y FT(Suc)m(h)31 b(justi\014cations)e(are)j(preceded)f(b)m(y)g
(their)g(conclusion)e(in)h(pro)s(of)h(scripts,)g(and)f(a)i
FI(Sentenc)-5 b(e)38 b FT(in)378 4915 y(the)e(ab)s(o)m(v)m(e)g(syn)m
(tax)g(represen)m(ts)f(a)h(premise)e(in)g(the)i(justi\014cation.)54
b(Expressions)33 b(whic)m(h)i(con)m(tain)378 5028 y(the)d
Fw(then)e FT(op)s(erator)j(at)g(the)f(top)g(lev)m(el)g(\(denoted)g(b)m
(y)g FI(Then)p 2478 5028 28 4 v 40 w(Expr)-5 b(ession)9
b FT(s)32 b(in)f(the)h(ab)s(o)m(v)m(e)h(syn)m(tax\))378
5141 y(can)d(only)f(o)s(ccur)h(on)g(the)g(left-hand)f(side)g(of)h(an)g
Fw(on)f FT(op)s(erator.)41 b(The)29 b Fw(on)o FT(,)i
Fw(and)d FT(and)i Fw(then)e FT(op)s(erators)378 5253
y(asso)s(ciate)39 b(to)f(the)g(left,)i(and)d Fw(and)f
FT(has)i(a)g(higher)e(precedence)i(than)g Fw(then)n FT(,)i(whic)m(h)d
(has)g(a)h(higher)378 5366 y(precedence)e(than)g Fw(on)o
FT(.)56 b(Examples)35 b(of)h(conclusions)e(justi\014ed)g(b)m(y)i
(structured)e(justi\014cations)g(are)378 5479 y(giv)m(en)c(in)f
(\014gure)h(13.)519 5592 y(W)-8 b(e)39 b(recall)d(that)i(the)f
(premises)f(of)h(an)h(SPL)e(justi\014cation)g(can)h(b)s(e)g(giv)m(en)g
(as)h(argumen)m(ts)f(to)378 5705 y(sp)s(eci\014c)28 b(pro)s(of)g(c)m
(hec)m(k)m(ers)j(\(simply)c(called)h(pro)m(v)m(ers\).)41
b(F)-8 b(or)30 b(the)f(case)h(of)f(structured)f(justi\014cations,)p
eop
%%Page: 104 114
104 113 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(104)p 378 416 3453 4 v 376 1914 4 1498 v 515 652
a Fw(")p Fu(9)p Fw(x.)13 b(C\(x\)")42 b(by)h(")p Fu(8)p
Fw(x.)12 b(A\(x\))42 b Fu(\))i Fw(C\(x)e(+)h(1\)")f(and)h(")p
Fu(8)p Fw(x.)12 b(B\(x\))42 b Fu(\))i Fw(C\(x)e(+)h(2\)")1125
751 y(on)g(")p Fu(8)p Fw(x.)12 b(A\(x\))42 b Fu(_)i Fw(B\(x\)";)515
951 y("R\(c,e\)")c(by)j(")p Fu(8)p Fw(x,y,z.)11 b(R\(x,y\))41
b Fu(\))i Fw(R\(y,z\))f Fu(\))h Fw(R\(x,z\)")e(on)733
1050 y("R\(c,d\)")f(and)i(\(")p Fu(8)p Fw(x,y.)12 b(R\(x,y\))41
b Fu(\))i Fw(R\(y,x\)")e(on)i("R\(e,d\)"\);)515 1249
y(")p Fu(9)p Fw(c.)13 b(D\(c\)")42 b(by)h(")p Fu(8)p
Fw(x,y.)11 b(\(A\(x,y\))41 b Fu(^)j Fw(B\(x\)\))d Fu(,)j
Fw(C\(y,x\)")1125 1349 y(then)e(")p Fu(8)p Fw(x,y.)12
b(C\(x,y\))41 b Fu(\))i Fw(D\(x\)")1212 1449 y(on)g(")p
Fu(8)p Fw(x.)12 b Fu(9)p Fw(c.)i(A\(x,c\)")41 b(and)h("B\(d\)";)1102
1744 y FT(Figure)30 b(13:)42 b(Examples)29 b(of)i(Structured)e
(Justi\014cations.)p 3829 1914 V 378 1917 3453 4 v 378
2274 a(one)j(can)f(use)h(pro)m(v)m(ers)f(whic)m(h)f(accept)j
(structured)e(expressions)f(as)i(argumen)m(ts.)44 b(The)31
b(\014rst-order)378 2387 y(tableau)42 b(pro)m(v)m(er)g(describ)s(ed)e
(in)h(c)m(hapter)h(5)h(used)e(to)i(c)m(hec)m(k)g(unstructured)d(SPL)h
(justi\014cations)378 2500 y(is)h(mo)s(di\014ed)e(\(see)j(c)m(hapter)h
(8\))f(so)g(that)g(it)f(can)h(b)s(e)f(used)f(to)j(c)m(hec)m(k)g
(structured)d(justi\014cations)378 2613 y(e\016cien)m(tly)-8
b(.)41 b(This)27 b(mo)s(di\014ed)h(pro)m(v)m(er)i(is)f(used)f(as)i(the)
g(default)f(pro)m(v)m(er)h(during)d(the)j(mec)m(hanisation)378
2726 y(of)g(group)g(theory)h(describ)s(ed)d(in)h(c)m(hapter)i(9.)378
3012 y FH(6.4)135 b(The)45 b(Seman)l(tics)h(of)f(Structured)f
(Justi\014cations)378 3215 y FT(The)32 b(seman)m(tics)g(of)h
(structured)e(justi\014cations)g(is)g(giv)m(en)i(in)e(terms)h(of)h(the)
f(inferences)f(whic)m(h)g(are)378 3328 y(assumed)k(implicitly)d(during)
h(the)j(implemen)m(tation)e(of)i(pro)s(ofs,)g(and)f(the)h(seman)m(tics)
g(of)g(the)g Fw(on)o FT(,)378 3441 y Fw(and)o FT(,)h(and)f
Fw(then)e FT(op)s(erators)i(whic)m(h)f(corresp)s(ond)f(to)j(the)f
(inferences)f(that)i(are)f(stated)h(explicitly)-8 b(.)378
3554 y(W)g(e)34 b(\014rst)e(de\014ne)h(the)g(set)g(of)h(implicit)c
(inferences)i(and)g(then)h(giv)m(e)g(the)g(seman)m(tics)h(of)f
(structured)378 3667 y(expressions.)378 3910 y FG(6.4.1)112
b(Implicit)34 b(First-Order)j(Inferences)378 4082 y FT(The)d
(inferences)f(whic)m(h)g(are)h(assumed)g(implicitly)c(in)j(structured)g
(justi\014cations)g(are)h(de\014ned)f(in)378 4195 y(terms)23
b(of)h(a)g(binary)e(relation)h Ff(\032)h FT(o)m(v)m(er)h(\014rst-order)
d(form)m(ulae.)38 b(In)23 b(other)h(w)m(ords,)h(the)f(manipulation)378
4308 y(of)29 b(a)h(form)m(ula)e FP(A)i FT(in)m(to)f FP(B)34
b FT(using)27 b(a)j(n)m(um)m(b)s(er)e(of)h(these)h(inferences,)f(that)g
(is)g FP(A)c Ff(\032)3168 4275 y FK(\003)3233 4308 y
FP(B)34 b FT(where)28 b Ff(\032)3698 4275 y FK(\003)3767
4308 y FT(is)378 4421 y(the)j(re\015exiv)m(e)f(transitiv)m(e)g(closure)
g(of)g Ff(\032)p FT(,)h(is)e(omitted)i(from)f(structured)f
(justi\014cations.)40 b(W)-8 b(e)31 b(giv)m(e)378 4534
y(the)g(follo)m(wing)d(de\014nitions.)378 4746 y FQ(De\014nition)35
b(6.1)h(\(Single)f(Step)f(Implicit)g(Deriv)-6 b(ation\))46
b FT(The)30 b(relation)g Ff(\032)h FT(o)m(v)m(er)h(\014rst-order)378
4859 y(form)m(ulae)i(is)f(the)h(smallest)f(binary)f(relation)h(whic)m
(h)g(satis\014es)g(the)i(follo)m(wing)d(rules,)i(categorised)378
4972 y(in)m(to)c(9)h(groups:)489 5160 y(1.)46 b(F)-8
b(or)39 b(all)e(form)m(ulae)g FP(A)h FT(and)f FP(B)43
b FT(whic)m(h)36 b(ha)m(v)m(e)j(the)f(same)h(negation)f(normal)f(form)g
(\(see)i(sec-)605 5272 y(tion)30 b(1.2.1\),)j(it)d(is)f(the)i(case)g
(that)g FP(A)26 b Ff(\032)f FP(B)5 b FT(.)489 5460 y(2.)46
b(F)-8 b(or)31 b(ev)m(ery)h(form)m(ula)d FP(A)p FT(,)i(w)m(e)g(ha)m(v)m
(e)1835 5664 y FP(A)26 b Ff(\032)f FN(>)182 b(?)24 b
Ff(\032)i FP(A)p eop
%%Page: 105 115
105 114 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(105)489 396 y(3.)46 b(F)-8 b(or)31 b(ev)m(ery)h(form)m(ula)d
FP(A)p FT(,)1669 509 y FP(A)c Ff(\032)g FP(A)c FN(^)e
FP(A)183 b(A)20 b FN(_)g FP(A)25 b Ff(\032)g FP(A)489
713 y FT(4.)46 b(F)-8 b(or)31 b(all)f(form)m(ulae)g FP(A)g
FT(and)g FP(B)5 b FT(,)1658 955 y FP(A)20 b FN(^)g FP(B)30
b Ff(\032)25 b FP(A)193 b(A)25 b Ff(\032)h FP(A)20 b
FN(_)g FP(B)1658 1092 y(A)g FN(^)g FP(B)30 b Ff(\032)25
b FP(B)187 b(B)29 b Ff(\032)d FP(A)20 b FN(_)g FP(B)489
1333 y FT(5.)46 b(F)-8 b(or)31 b(all)f(form)m(ulae)g
FP(A)p FT(,)g FP(B)35 b FT(and)30 b FP(C)7 b FT(,)1655
1550 y FP(A)20 b FN(^)g FT(\()p FP(B)25 b FN(_)20 b FP(C)7
b FT(\))25 b Ff(\032)g FT(\()p FP(A)c FN(^)f FP(B)5 b
FT(\))20 b FN(_)g FT(\()p FP(A)h FN(^)e FP(C)7 b FT(\))1415
1705 y(\()p FP(A)21 b FN(_)f FP(B)5 b FT(\))20 b FN(^)f
FT(\()p FP(A)i FN(_)f FP(C)7 b FT(\))25 b Ff(\032)g FP(A)c
FN(_)f FT(\()p FP(B)25 b FN(^)19 b FP(C)7 b FT(\))489
1946 y(6.)46 b(F)-8 b(or)31 b(ev)m(ery)h(v)-5 b(ariable)29
b FP(x)p FT(,)h(and)g(form)m(ula)g FP(A)p FT(,)g(if)g
FP(x)g FT(is)f(not)i(free)f(in)f FP(A)p FT(,)i(then)1639
2188 y(\()p FN(8)p FP(x:A)p FT(\))26 b Ff(\032)f FP(A)183
b(A)25 b Ff(\032)g FT(\()p FN(9)p FP(x:A)p FT(\))1639
2344 y(\()p FN(9)p FP(x:A)p FT(\))h Ff(\032)f FP(A)183
b(A)25 b Ff(\032)g FT(\()p FN(8)p FP(x:A)p FT(\))489
2584 y(7.)46 b(F)-8 b(or)31 b(ev)m(ery)h(v)-5 b(ariable)29
b FP(x)p FT(,)h(and)g(all)f(form)m(ulae)h FP(B)35 b FT(and)30
b FP(C)7 b FT(,)30 b(if)f FP(x)h FT(is)g(not)g(free)h(in)e
FP(C)7 b FT(,)30 b(then)1085 2826 y(\()p FN(8)p FP(x:B)5
b FT(\))20 b FN(^)g FP(C)32 b Ff(\032)25 b FN(8)p FP(x:)p
FT(\()p FP(B)g FN(^)20 b FP(C)7 b FT(\))182 b FN(9)p
FP(x:)p FT(\()p FP(B)24 b FN(_)c FP(C)7 b FT(\))25 b
Ff(\032)g FT(\()p FN(9)p FP(x:B)5 b FT(\))21 b FN(_)e
FP(C)1085 2982 y FT(\()p FN(9)p FP(x:B)5 b FT(\))20 b
FN(^)g FP(C)32 b Ff(\032)25 b FN(9)p FP(x:)p FT(\()p
FP(B)g FN(^)20 b FP(C)7 b FT(\))182 b FN(8)p FP(x:)p
FT(\()p FP(B)24 b FN(_)c FP(C)7 b FT(\))25 b Ff(\032)g
FT(\()p FN(8)p FP(x:B)5 b FT(\))21 b FN(_)e FP(C)489
3222 y FT(8.)46 b(F)-8 b(or)28 b(ev)m(ery)g(v)-5 b(ariable)26
b FP(x)p FT(,)i(form)m(ula)e FP(A)i FT(and)e(term)h FP(t)p
FT(,)h(if)e(no)h(free)h(v)-5 b(ariable)26 b(in)g FP(t)h
FT(b)s(ecomes)g(b)s(ound)605 3335 y(in)i FP(A)p FN(f)p
FP(x)d FN(!)f FP(t)p FN(g)p FT(,)31 b(then)1265 3539
y FN(8)p FP(x:A)25 b Ff(\032)h FN(8)p FP(x:A)p FN(f)p
FP(x)f FN(!)g FP(t)p FN(g)183 b(9)p FP(x:A)p FN(f)p FP(x)25
b FN(!)g FP(t)p FN(g)h Ff(\032)f FN(9)p FP(x:A)489 3780
y FT(9.)46 b(F)-8 b(or)31 b(all)f(form)m(ulae)g FP(A)g
FT(and)g FP(B)5 b FT(,)30 b(if)f FP(A)d Ff(\032)f FP(B)35
b FT(then)30 b(for)g(ev)m(ery)h(form)m(ula)f FP(C)7 b
FT(,)1487 4022 y FP(A)21 b FN(^)f FP(C)31 b Ff(\032)26
b FP(B)e FN(^)c FP(C)189 b(A)20 b FN(_)g FP(C)31 b Ff(\032)26
b FP(B)e FN(_)c FP(C)1532 4160 y FN(8)p FP(x:A)25 b Ff(\032)h
FN(8)p FP(x:B)275 b FN(9)p FP(x:A)25 b Ff(\032)h FN(9)p
FP(x:B)378 4400 y FT(W)-8 b(e)32 b(sa)m(y)f(that)g FP(A)f
FT(implicitly)d(deriv)m(es)i FP(B)35 b FT(in)29 b(a)i(single)e(step)h
(if)g FP(A)25 b Ff(\032)h FP(B)34 b FT(holds.)688 b Ff(\003)378
4607 y FQ(De\014nition)35 b(6.2)h(\(Implicit)d(Deriv)-6
b(ations\))46 b FT(F)-8 b(or)31 b(all)e(\014rst-order)g(form)m(ulae)g
FP(A)i FT(and)e FP(B)5 b FT(,)30 b(w)m(e)g(sa)m(y)378
4720 y(that)35 b FP(A)g FT(deriv)m(es)f FP(B)39 b FT(implicitly)31
b(if)j FP(A)e Ff(\032)1798 4687 y FK(\003)1870 4720 y
FP(B)39 b FT(where)34 b Ff(\032)2346 4687 y FK(\003)2420
4720 y FT(is)g(the)h(re\015exiv)m(e)g(transitiv)m(e)f(closure)g(of)378
4833 y Ff(\032)p FT(.)41 b(W)-8 b(e)31 b(also)g(de\014ne)e(the)i(follo)
m(wing)e(inference)g(rule)g(denoted)i(b)m(y)f Ff(\032)2821
4800 y FK(\003)2860 4833 y FT(:)1447 5012 y FP(A)p 1444
5032 74 4 v 1444 5111 a(B)1559 5055 y FT(\()p Ff(\032)1695
5022 y FK(\003)1735 5055 y FT(\))91 b(if)29 b(and)h(only)g(if)f
FP(A)c Ff(\032)2598 5022 y FK(\003)2663 5055 y FP(B)5
b FT(.)378 5315 y Ff(\003)378 5522 y FT(The)30 b(follo)m(wing)f(t)m(w)m
(o)i(results)f(follo)m(w)f(immediately)g(from)h(the)g(ab)s(o)m(v)m(e)i
(de\014nition.)378 5705 y FQ(Prop)s(osition)k(6.1)g(\(Correctness)f(of)
g(Implicit)f(Inferences\))45 b FI(F)-7 b(or)30 b(al)5
b(l)30 b(formulae)h FP(A)e FI(and)i FP(B)5 b FI(,)p eop
%%Page: 106 116
106 115 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(106)485 396 y FI(1.)46 b(if)33 b FP(A)25 b Ff(\032)g
FP(B)37 b FI(then)c FN(8)o FP(~)-44 b(x:)p FT(\()p FP(A)26
b FN(\))f FP(B)5 b FT(\))p FI(;)485 584 y(2.)46 b(if)33
b FP(A)25 b Ff(\032)888 551 y FK(\003)953 584 y FP(B)37
b FI(then)c FN(8)o FP(~)-44 b(x)o(:)p FT(\()p FP(A)26
b FN(\))f FP(B)5 b FT(\))p FI(;)378 772 y(wher)-5 b(e)32
b FP(~)-43 b(x)32 b FI(denotes)i(the)f(list)g(of)g(variables)g(fr)-5
b(e)g(e)33 b(in)g FP(A)25 b FN(\))g FP(B)5 b FI(.)378
959 y FQ(Pro)s(of)p FT(:)28 b(The)e(\014rst)g(statemen)m(t)j(can)e(b)s
(e)f(easily)g(c)m(hec)m(k)m(ed)j(for)d(eac)m(h)i(rule)e(in)f(the)i(ab)s
(o)m(v)m(e)h(de\014nition)d(of)378 1072 y Ff(\032)32
b FT(using)g(the)g(standard)g(results)f(on)i(the)g(v)-5
b(alidit)m(y)31 b(of)h(classical)g(implication)e(\(giv)m(en)j(in)e
(\(Fitting)378 1185 y(1996\))e(for)e(instance\).)39 b(The)27
b(second)g(statemen)m(t)i(follo)m(ws)d(from)h(the)g(\014rst)f(one,)i
(and)f(the)g(re\015exivit)m(y)378 1298 y(and)j(transitivit)m(y)f(of)h
(implication.)2154 b Ff(\004)378 1511 y FQ(Prop)s(osition)36
b(6.2)g(\(Con)m(trap)s(ositiv)m(eness)e(of)i Ff(\032)e
FQ(and)h Ff(\032)2604 1478 y FK(\003)2644 1511 y FQ(\))45
b FI(F)-7 b(or)33 b(al)5 b(l)34 b(formulae)f FP(A)g FI(and)h
FP(B)5 b FI(,)485 1698 y(1.)46 b(if)33 b FP(A)25 b Ff(\032)g
FP(B)37 b FI(then)c FN(:)p FP(B)d Ff(\032)1482 1665 y
FK(\003)1546 1698 y FN(:)p FP(A)p FI(;)485 1886 y(2.)46
b(if)33 b FP(A)25 b Ff(\032)888 1853 y FK(\003)953 1886
y FP(B)37 b FI(then)c FN(:)p FP(B)c Ff(\032)1521 1853
y FK(\003)1586 1886 y FN(:)p FP(A)p FI(.)378 2098 y FQ(Pro)s(of)p
FT(:)39 b(F)-8 b(or)38 b(eac)m(h)h(rule)e FP(X)44 b Ff(\032)38
b FP(Y)58 b FT(in)36 b(de\014nition)f(6.1,)41 b(it)d(follo)m(ws)e(that)
j FN(:)p FP(Y)57 b Ff(\032)3222 2065 y FK(\003)3299 2098
y FN(:)p FP(X)44 b FT(from)38 b(the)378 2211 y(rule)29
b(adjacen)m(t)i(to)g FP(X)h Ff(\032)26 b FP(Y)50 b FT(in)28
b(the)j(de\014nition)c(\(or)k(from)e(the)h(other)h(rule)d(in)h(the)h
(same)h(group)e(for)378 2324 y(the)35 b(case)h(of)e(the)h(rules)e(in)h
(group)g(5\),)j(and)d(from)g(the)h(fact)g(that)h(t)m(w)m(o)g(form)m
(ulae)e(can)h(b)s(e)f(deriv)m(ed)378 2437 y(from)c(eac)m(h)i(other)f
(implicitly)c(if)j(they)h(ha)m(v)m(e)h(the)f(same)g(negation)h(normal)d
(form)i(\(i.e.,)16 b(the)31 b(rule)e(in)378 2550 y(group)i(1\).)45
b(F)-8 b(or)32 b(example,)g(it)f(can)h(b)s(e)e(sho)m(wn)h(that)h
FN(:)p FP(A)27 b Ff(\032)2450 2517 y FK(\003)2517 2550
y FN(:)p FT(\()p FP(A)21 b FN(^)f FP(B)5 b FT(\))32 b(giv)m(en)f(that)h
FP(A)21 b FN(^)g FP(B)32 b Ff(\032)27 b FP(A)378 2663
y FT(as)k(follo)m(ws:)1046 2867 y FN(:)p FP(A)25 b Ff(\032)g
FN(:)p FP(A)20 b FN(_)g(:)p FP(B)65 b FT(b)m(y)30 b(the)g(top)h(righ)m
(t)f(rule)f(in)g(defn.)h(6.1\(3\))1200 3023 y Ff(\032)25
b FN(:)p FT(\()p FP(A)20 b FN(^)g FP(B)5 b FT(\))61 b(b)m(y)30
b(the)g(rule)g(in)f(defn.)g(6.1\(1\).)378 3227 y(The)36
b(second)g(statemen)m(t)j(of)d(this)g(prop)s(osition)e(follo)m(ws)h
(from)h(the)h(\014rst)f(one)g(and)g(the)h(fact)g(that)378
3340 y Ff(\032)479 3307 y FK(\003)549 3340 y FT(is)29
b(the)i(re\015exiv)m(e)f(transitiv)m(e)g(closure)g(of)g
Ff(\032)p FT(.)1666 b Ff(\004)519 3566 y FT(It)29 b(should)e(b)s(e)h
(noted)h(that)h(the)f(inference)f(giv)m(en)h(b)m(y)g
Ff(\032)2469 3533 y FK(\003)2537 3566 y FT(is)f(w)m(eak)m(er)i(than)f
(the)g(classical)f(\014rst-)378 3679 y(order)k(implication.)45
b(F)-8 b(or)33 b(instance,)g(for)g(an)m(y)g(form)m(ula)e
FP(A)i FT(whose)f(negation)h(normal)f(form)g(is)g(not)378
3792 y FN(>)e FT(or)g FN(?)p FT(,)h(and)e(for)i(an)m(y)f(form)m(ula)g
FP(B)35 b FT(whose)30 b(negation)g(normal)g(form)g(is)f(not)i
FN(?)p FT(,)955 3996 y FP(A)21 b FN(^)e(:)p FP(A)25 b
FN(6)p Ff(\032)1379 3958 y FK(\003)1444 3996 y FN(?)182
b(>)25 b(6)p Ff(\032)1894 3958 y FK(\003)1958 3996 y
FP(A)c FN(_)e(:)p FP(A)182 b FT(\()p FP(A)26 b FN(\))f
FP(B)5 b FT(\))20 b FN(^)g FP(A)25 b FN(6)p Ff(\032)3087
3958 y FK(\003)3152 3996 y FP(B)5 b(:)378 4200 y FT(Suc)m(h)25
b(inferences)f(are)i(therefore)g(required)d(to)k(b)s(e)d(stated)j
(explicitly)c(in)h(structured)g(justi\014cations.)519
4313 y(The)31 b(implicit)e(inferences)h(are)i(ho)m(w)m(ev)m(er)h
(strong)f(enough)f(to)i(deriv)m(e)e(a)h(large)f(n)m(um)m(b)s(er)g(of)g
(ma-)378 4426 y(nipulations)c(on)k(form)m(ulae,)f(for)g(example)h(it)f
(can)g(b)s(e)g(sho)m(wn)g(b)m(y)h(the)f(rules)f(in)h(groups)f(3,)j(4)f
(and)f(9)378 4539 y(that)1000 4743 y FP(A)21 b FN(^)e
FP(B)30 b Ff(\032)25 b FT(\()p FP(A)c FN(^)f FP(B)5 b
FT(\))20 b FN(^)g FT(\()p FP(A)h FN(^)e FP(B)5 b FT(\))61
b(b)m(y)30 b(defn.)g(6.1\(3\))1268 4899 y Ff(\032)25
b FP(B)g FN(^)20 b FT(\()p FP(A)h FN(^)f FP(B)5 b FT(\))60
b(b)m(y)30 b(defn.)g(6.1\(4\))j(and)c(6.1\(9\))1268 5054
y Ff(\032)c FP(B)g FN(^)20 b FP(A)61 b FT(b)m(y)30 b(defn.)g(6.1\(4\))i
(and)e(6.1\(9\),)1000 5347 y FP(A)21 b FN(_)e FP(B)30
b Ff(\032)25 b FT(\()p FP(B)h FN(_)19 b FP(A)p FT(\))i
FN(_)f FP(B)65 b FT(b)m(y)30 b(defn.)g(6.1\(4\))j(and)c(6.1\(9\))1268
5503 y Ff(\032)c FT(\()p FP(B)h FN(_)19 b FP(A)p FT(\))i
FN(_)f FT(\()p FP(B)25 b FN(_)20 b FP(A)p FT(\))61 b(b)m(y)30
b(defn.)g(6.1\(4\))j(and)c(6.1\(9\))1268 5659 y Ff(\032)c
FT(\()p FP(B)h FN(_)19 b FP(A)p FT(\))61 b(b)m(y)31 b(defn.)f
(6.1\(3\),)p eop
%%Page: 107 117
107 116 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(107)378 396 y(and)30 b(similarly)849 638 y FP(A)21
b FN(^)e FT(\()p FP(B)26 b FN(^)19 b FP(C)7 b FT(\))25
b Ff(\032)1461 601 y FK(\003)1526 638 y FT(\()p FP(A)c
FN(^)f FP(B)5 b FT(\))20 b FN(^)g FP(C)188 b FT(\()p
FP(A)21 b FN(^)e FP(B)5 b FT(\))20 b FN(^)g FP(C)32 b
Ff(\032)2806 601 y FK(\003)2871 638 y FP(A)20 b FN(^)g
FT(\()p FP(B)25 b FN(^)20 b FP(C)7 b FT(\))849 794 y
FP(A)21 b FN(_)e FT(\()p FP(B)26 b FN(_)19 b FP(C)7 b
FT(\))25 b Ff(\032)1461 756 y FK(\003)1526 794 y FT(\()p
FP(A)c FN(_)f FP(B)5 b FT(\))20 b FN(_)g FP(C)188 b FT(\()p
FP(A)21 b FN(_)e FP(B)5 b FT(\))20 b FN(_)g FP(C)32 b
Ff(\032)2806 756 y FK(\003)2871 794 y FP(A)20 b FN(_)g
FT(\()p FP(B)25 b FN(_)20 b FP(C)7 b FT(\))1267 949 y
FP(A)25 b Ff(\032)1461 912 y FK(\003)1526 949 y FP(A)c
FN(_)e FT(\()p FP(A)i FN(^)f FP(B)5 b FT(\))188 b FP(A)21
b FN(_)e FT(\()p FP(A)i FN(^)f FP(B)5 b FT(\))25 b Ff(\032)2806
912 y FK(\003)2871 949 y FP(A)1267 1105 y(A)g Ff(\032)1461
1067 y FK(\003)1526 1105 y FP(A)c FN(^)e FT(\()p FP(A)i
FN(_)f FP(B)5 b FT(\))188 b FP(A)21 b FN(^)e FT(\()p
FP(A)i FN(_)f FP(B)5 b FT(\))25 b Ff(\032)2806 1067 y
FK(\003)2871 1105 y FP(A:)378 1309 y FT(The)20 b(\014fth)g(group)g(of)h
(rules)f(allo)m(ws)g(form)m(ulae)g(to)i(b)s(e)e(manipulated)f(in)m(to)i
(eac)m(h)h(other)f(b)m(y)g(distributing)378 1422 y(the)31
b(conjunctions)e(o)m(v)m(er)j(the)e(disjunctions,)e(and)i(vice-v)m
(ersa.)41 b(The)30 b(manipulations)609 1626 y(\()p FP(A)21
b FN(^)f FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)h FN(^)e
FP(C)7 b FT(\))25 b Ff(\032)1461 1589 y FK(\003)1526
1626 y FP(A)c FN(^)e FT(\()p FP(B)25 b FN(_)20 b FP(C)7
b FT(\))182 b FP(A)20 b FN(_)g FT(\()p FP(B)25 b FN(^)20
b FP(C)7 b FT(\))25 b Ff(\032)2806 1589 y FK(\003)2871
1626 y FT(\()p FP(A)20 b FN(_)g FP(B)5 b FT(\))20 b FN(^)g
FT(\()p FP(A)h FN(_)f FP(C)7 b FT(\))378 1831 y(can)31
b(b)s(e)e(deriv)m(ed)h(as)g(follo)m(ws:)761 2035 y(\()p
FP(A)21 b FN(^)f FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)g
FN(^)g FP(C)7 b FT(\))786 2190 y Ff(\032)26 b FT(\(\()p
FP(A)21 b FN(^)f FP(B)5 b FT(\))20 b FN(_)f FT(\()p FP(A)i
FN(^)f FP(C)7 b FT(\)\))20 b FN(^)g FT(\(\()p FP(A)h
FN(^)f FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)h FN(^)e
FP(C)7 b FT(\)\))61 b(b)m(y)31 b(defn.)e(6.1\(3\))786
2346 y Ff(\032)d FT(\()p FP(A)20 b FN(_)g FT(\()p FP(A)h
FN(^)f FP(C)7 b FT(\)\))20 b FN(^)g FT(\(\()p FP(A)h
FN(^)f FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)g FN(^)g
FP(C)7 b FT(\)\))61 b(b)m(y)30 b(defn.)g(6.1\(4\))j(and)c(6.1\(9\))786
2502 y Ff(\032)d FT(\()p FP(A)20 b FN(_)g FP(A)p FT(\))h
FN(^)f FT(\(\()p FP(A)h FN(^)f FP(B)5 b FT(\))20 b FN(_)g
FT(\()p FP(A)g FN(^)g FP(C)7 b FT(\)\))61 b(b)m(y)30
b(defn.)g(6.1\(4\))i(and)e(6.1\(9\))786 2657 y Ff(\032)c
FP(A)20 b FN(^)g FT(\(\()p FP(A)h FN(^)f FP(B)5 b FT(\))20
b FN(_)g FT(\()p FP(A)g FN(^)g FP(C)7 b FT(\)\))61 b(b)m(y)30
b(defn.)g(6.1\(3\))j(and)c(6.1\(9\))786 2813 y Ff(\032)d
FP(A)20 b FN(^)g FT(\()p FP(B)25 b FN(_)20 b FT(\()p
FP(A)g FN(^)g FP(C)7 b FT(\)\))61 b(b)m(y)30 b(defn.)g(6.1\(4\))j(and)c
(6.1\(9\))786 2968 y Ff(\032)d FP(A)20 b FN(^)g FT(\()p
FP(B)25 b FN(_)20 b FP(C)7 b FT(\))60 b(b)m(y)30 b(defn.)g(6.1\(4\))j
(and)d(6.1\(9\))r FP(:)639 3282 y(A)20 b FN(_)g FT(\()p
FP(B)25 b FN(^)20 b FP(C)7 b FT(\))664 3437 y Ff(\032)25
b FT(\()p FP(A)c FN(^)f FP(A)p FT(\))g FN(_)g FT(\()p
FP(B)25 b FN(^)20 b FP(C)7 b FT(\))60 b(b)m(y)31 b(defn.)e(6.1\(3\))k
(and)d(6.1\(9\))664 3593 y Ff(\032)25 b FT(\(\()p FP(A)c
FN(_)f FP(B)5 b FT(\))20 b FN(^)g FP(A)p FT(\))h FN(_)e
FT(\()p FP(B)25 b FN(^)20 b FP(C)7 b FT(\))61 b(b)m(y)30
b(defn.)g(6.1\(4\))i(and)e(6.1\(9\))664 3748 y Ff(\032)25
b FT(\(\()p FP(A)c FN(_)f FP(B)5 b FT(\))20 b FN(^)g
FT(\()p FP(A)h FN(_)e FP(C)7 b FT(\)\))21 b FN(_)f FT(\()p
FP(B)25 b FN(^)19 b FP(C)7 b FT(\))61 b(b)m(y)30 b(defn.)g(6.1\(4\))j
(and)c(6.1\(9\))664 3904 y Ff(\032)c FT(\(\()p FP(A)c
FN(_)f FP(B)5 b FT(\))20 b FN(^)g FT(\()p FP(A)h FN(_)e
FP(C)7 b FT(\)\))21 b FN(_)f FT(\(\()p FP(A)h FN(_)e
FP(B)5 b FT(\))21 b FN(^)e FP(C)7 b FT(\))61 b(b)m(y)30
b(defn.)g(6.1\(4\))i(and)e(6.1\(9\))664 4059 y Ff(\032)25
b FT(\(\()p FP(A)c FN(_)f FP(B)5 b FT(\))20 b FN(^)g
FT(\()p FP(A)h FN(_)e FP(C)7 b FT(\)\))21 b FN(_)f FT(\(\()p
FP(A)h FN(_)e FP(B)5 b FT(\))21 b FN(^)e FT(\()p FP(A)i
FN(_)f FP(C)7 b FT(\)\))61 b(b)m(y)30 b(defn.)g(6.1\(4\))i(and)e
(6.1\(9\))664 4215 y Ff(\032)25 b FT(\()p FP(A)c FN(_)f
FP(B)5 b FT(\))20 b FN(^)f FT(\()p FP(A)i FN(_)f FP(C)7
b FT(\))60 b(b)m(y)31 b(defn.)e(6.1\(3\))s FP(:)378 4419
y FT(The)h(sixth)g(group)g(of)i(rules)d(in)h(the)h(de\014nition)d(remo)
m(v)m(es)k(and)f(adds)f(an)m(y)h(redundan)m(t)f(quan)m(ti\014ers,)378
4532 y(and)40 b(the)h(sev)m(en)m(th)h(group)e(allo)m(ws)h(t)m(w)m(o)h
(form)m(ulae)e(whic)m(h)g(ha)m(v)m(e)i(the)f(same)g(prenex)g(form)f(to)
i(b)s(e)378 4645 y(implicitly)27 b(deriv)-5 b(able)28
b(from)i(eac)m(h)i(other.)41 b(Note)31 b(that)g(the)g(rule)1563
4849 y FN(8)p FP(x:)p FT(\()p FP(B)25 b FN(^)20 b FP(C)7
b FT(\))25 b Ff(\032)2134 4812 y FK(\003)2198 4849 y
FT(\()p FN(8)p FP(x:B)5 b FT(\))21 b FN(^)e FP(C)p eop
%%Page: 108 118
108 117 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(108)378 396 y(where)30 b(the)g(v)-5 b(ariable)30
b FP(x)g FT(is)f(not)i(free)f(in)f FP(C)7 b FT(,)30 b(can)h(b)s(e)f
(deriv)m(ed)f(as)i(follo)m(ws:)819 613 y FN(8)p FP(x:)p
FT(\()p FP(B)25 b FN(^)20 b FP(C)7 b FT(\))25 b Ff(\032)g
FT(\()p FN(8)p FP(x:)p FT(\()p FP(B)g FN(^)20 b FP(C)7
b FT(\)\))21 b FN(^)e FT(\()p FN(8)p FP(x:)p FT(\()p
FP(B)25 b FN(^)20 b FP(C)7 b FT(\)\))61 b(b)m(y)30 b(defn.)g(6.1\(3\))
1289 769 y Ff(\032)25 b FT(\()p FN(8)p FP(x:B)5 b FT(\))20
b FN(^)g FT(\()p FN(8)p FP(x:)p FT(\()p FP(B)25 b FN(^)20
b FP(C)7 b FT(\)\))61 b(b)m(y)30 b(defn.)g(6.1\(4\))j(and)c(6.1\(9\))
1289 925 y Ff(\032)c FT(\()p FN(8)p FP(x:B)5 b FT(\))20
b FN(^)g FT(\()p FN(8)p FP(x:C)7 b FT(\))61 b(b)m(y)30
b(defn.)g(6.1\(4\))i(and)e(6.1\(9\))1289 1080 y Ff(\032)25
b FT(\()p FN(8)p FP(x:B)5 b FT(\))20 b FN(^)g FP(C)67
b FT(b)m(y)31 b(defn.)e(6.1\(6\))k(and)d(6.1\(9\))378
1284 y(and)g(the)g(rules)1563 1489 y FN(9)p FP(x:)p FT(\()p
FP(B)25 b FN(^)20 b FP(C)7 b FT(\))25 b Ff(\032)2134
1451 y FK(\003)2198 1489 y FT(\()p FN(9)p FP(x:B)5 b
FT(\))21 b FN(^)e FP(C)1563 1644 y FT(\()p FN(9)p FP(x:B)5
b FT(\))20 b FN(_)g FP(C)32 b Ff(\032)2134 1607 y FK(\003)2198
1644 y FN(9)p FP(x:)p FT(\()p FP(B)25 b FN(_)20 b FP(C)7
b FT(\))1563 1800 y(\()p FN(8)p FP(x:B)e FT(\))20 b FN(_)g
FP(C)32 b Ff(\032)2134 1762 y FK(\003)2198 1800 y FN(8)p
FP(x:)p FT(\()p FP(B)25 b FN(_)20 b FP(C)7 b FT(\))378
2004 y(where)37 b FP(x)h FT(is)f(not)h(free)g(in)f FP(C)7
b FT(,)39 b(can)f(b)s(e)f(deriv)m(ed)g(similarly)-8 b(.)61
b(The)37 b(t)m(w)m(o)i(rules)e(in)f(the)i(eigh)m(t)g(group)378
2117 y(allo)m(w)k(the)h(sp)s(ecialisation)e(of)i(univ)m(ersally)e(quan)
m(ti\014ed)g(v)-5 b(ariables,)45 b(and)e(the)g(generalisation)f(of)378
2230 y(existen)m(tially)29 b(quan)m(ti\014ed)g(ones,)i(and)f(can)g(b)s
(e)g(used)g(for)g(instance)g(to)h(deriv)m(e)562 2472
y FN(8)p FP(x:P)13 b FT(\()p FP(x)p FT(\))26 b Ff(\032)f
FN(8)p FP(x:P)13 b FT(\()p FP(f)d FT(\()p FP(x)p FT(\)\))615
b FN(9)p FP(x:P)13 b FT(\()p FP(f)d FT(\()p FP(x)p FT(\)\))26
b Ff(\032)f FN(9)p FP(x:P)13 b FT(\()p FP(x)p FT(\))p
FP(:)562 2765 y FN(8)p FP(x:P)g FT(\()p FP(x)p FT(\))26
b Ff(\032)f FN(8)p FP(x:P)13 b FT(\()p FP(c)p FT(\))1018
b FP(P)13 b FT(\()p FP(c)p FT(\))27 b Ff(\032)e FN(9)p
FP(x:P)13 b FT(\()p FP(c)p FT(\))31 b(\(b)m(y)g(gp.)40
b(7\))909 2921 y Ff(\032)25 b FP(P)13 b FT(\()p FP(c)p
FT(\))31 b(\(b)m(y)g(gp.)40 b(7\))919 b Ff(\032)25 b
FN(9)p FP(x:P)13 b FT(\()p FP(x)p FT(\))p FP(:)562 3214
y FN(8)p FP(x:P)g FT(\()p FP(x)p FT(\))26 b Ff(\032)f
FN(8)p FP(y)s(:)p FN(8)p FP(x:P)13 b FT(\()p FP(x)p FT(\))30
b(\(b)m(y)h(gp.)40 b(6\))308 b FN(9)p FP(x:P)13 b FT(\()p
FP(x)p FT(\))26 b Ff(\032)f FN(9)p FP(x:)p FN(9)p FP(y)s(:P)13
b FT(\()p FP(x)p FT(\))30 b(\(b)m(y)h(gp.)40 b(6,9\))909
3369 y Ff(\032)25 b FN(8)p FP(y)s(:)p FN(8)p FP(x:P)13
b FT(\()p FP(y)s FT(\))1092 b Ff(\032)25 b FN(9)p FP(x:)p
FN(9)p FP(y)s(:P)13 b FT(\()p FP(y)s FT(\))909 3525 y
Ff(\032)25 b FN(8)p FP(y)s(:P)13 b FT(\()p FP(y)s FT(\))30
b(\(b)m(y)h(gp.)41 b(6,9\))716 b Ff(\032)25 b FN(9)p
FP(y)s(:P)13 b FT(\()p FP(y)s FT(\))30 b(\(b)m(y)h(gp.)41
b(6\))p FP(:)378 3729 y FT(where)c FP(y)j FT(is)c(not)h(free)g(in)f
FP(P)13 b FT(\()p FP(x)p FT(\).)62 b(The)37 b(last)g(group)f(of)i
(rules)d(in)h(the)i(de\014nition)d(states)j(that)g(the)378
3842 y(relation)32 b Ff(\032)h FT(and)g(hence)g Ff(\032)1384
3809 y FK(\003)1456 3842 y FT(are)h(monotonic)f(with)f(resp)s(ect)h(to)
g FN(^)p FT(,)h FN(_)p FT(,)f FN(8)g FT(and)f FN(9)p
FT(.)48 b(It)34 b(is)e(also)h(the)378 3955 y(case)e(that)g(the)g(follo)
m(wing)e(manipulations)e(hold:)974 4172 y FN(8)p FP(x:)p
FT(\()p FP(P)13 b FT(\()p FP(x)p FT(\))21 b FN(^)e FP(Q)p
FT(\()p FP(x)p FT(\)\))27 b Ff(\032)e FT(\()p FN(8)p
FP(x:)p FT(\()p FP(P)13 b FT(\()p FP(x)p FT(\))21 b FN(^)f
FP(Q)p FT(\()p FP(x)p FT(\)\)\))h FN(^)f FT(\()p FN(8)p
FP(x:)p FT(\()p FP(P)13 b FT(\()p FP(x)p FT(\))22 b FN(^)d
FP(Q)p FT(\()p FP(x)p FT(\)\)\))1687 4328 y Ff(\032)1788
4290 y FK(\003)1852 4328 y FT(\()p FN(8)p FP(x:P)13 b
FT(\()p FP(x)p FT(\)\))22 b FN(^)d FT(\()p FN(8)p FP(x:Q)p
FT(\()p FP(x)p FT(\)\))775 4621 y(\()p FN(8)p FP(x:P)13
b FT(\()p FP(x)p FT(\)\))22 b FN(^)d FT(\()p FN(8)p FP(x:Q)p
FT(\()p FP(x)p FT(\)\))27 b Ff(\032)e FN(8)p FP(y)s(:)p
FT(\(\()p FN(8)p FP(x:P)13 b FT(\()p FP(x)p FT(\)\))21
b FN(^)f FT(\()p FN(8)p FP(x:Q)p FT(\()p FP(x)p FT(\)\)\))1687
4776 y Ff(\032)1788 4739 y FK(\003)1852 4776 y FN(8)p
FP(y)s(:)p FT(\(\()p FN(8)p FP(x:P)13 b FT(\()p FP(y)s
FT(\)\))21 b FN(^)f FT(\()p FN(8)p FP(x:Q)p FT(\()p FP(y)s
FT(\)\)\))1687 4932 y Ff(\032)1788 4895 y FK(\003)1852
4932 y FN(8)p FP(y)s(:)p FT(\()p FP(P)13 b FT(\()p FP(y)s
FT(\))21 b FN(^)f FP(Q)p FT(\()p FP(y)s FT(\)\))1687
5088 y Ff(\032)1788 5050 y FK(\003)1852 5088 y FN(8)p
FP(x:)p FT(\()p FP(P)13 b FT(\()p FP(x)p FT(\))21 b FN(^)f
FP(Q)p FT(\()p FP(x)p FT(\)\))p FP(:)p eop
%%Page: 109 119
109 118 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(109)378 396 y(where)34 b(the)h(v)-5 b(ariable)34
b FP(y)k FT(is)c(not)h(free)g(in)e FP(P)13 b FT(\()p
FP(x)p FT(\))36 b(and)e FP(Q)p FT(\()p FP(x)p FT(\).)55
b(The)34 b(follo)m(wing)g(can)h(b)s(e)f(deriv)m(ed)g(in)g(a)378
509 y(similar)28 b(fashion:)1109 714 y(\()p FN(9)p FP(x:P)13
b FT(\()p FP(x)p FT(\)\))21 b FN(_)f FT(\()p FN(9)p FP(x:Q)p
FT(\()p FP(x)p FT(\)\))26 b Ff(\032)2121 676 y FK(\003)2186
714 y FN(9)p FP(x:)p FT(\()p FP(P)13 b FT(\()p FP(x)p
FT(\))21 b FN(_)f FP(Q)p FT(\()p FP(x)p FT(\)\))1307
869 y FN(9)p FP(x:)p FT(\()p FP(P)13 b FT(\()p FP(x)p
FT(\))21 b FN(_)f FP(Q)p FT(\()p FP(x)p FT(\)\))26 b
Ff(\032)2121 832 y FK(\003)2186 869 y FT(\()p FN(9)p
FP(x:P)13 b FT(\()p FP(x)p FT(\)\))21 b FN(_)f FT(\()p
FN(9)p FP(x:Q)p FT(\()p FP(x)p FT(\)\))p FP(:)378 1112
y FG(6.4.2)112 b(Explicitly)34 b(Stated)k(Inferences)378
1284 y FT(Inferences)c(are)h(stated)g(explicitly)d(in)h(structured)g
(justi\014cations)g(b)m(y)i(using)d(structured)i(expres-)378
1397 y(sions)25 b(whic)m(h)h(in)m(v)m(olv)m(e)g(the)h(op)s(erators)f
Fw(on)p FT(,)h Fw(and)e FT(and)h Fw(then)n FT(.)40 b(W)-8
b(e)28 b(giv)m(e)e(the)h(follo)m(wing)e(de\014nitions)f(for)378
1510 y(the)31 b(seman)m(tics)f(of)h(structured)e(expressions)g(and)h
(structured)f(justi\014cations.)378 1722 y FQ(De\014nition)35
b(6.3)h(\(Explicit)e(Deriv)-6 b(ation\))46 b FT(W)-8
b(e)25 b(sa)m(y)g(that)g(a)g(structured)e(expression)g
FP(X)32 b FT(explic-)378 1835 y(itly)k(deriv)m(es)h(a)h(form)m(ula)e
FP(C)44 b FT(if)36 b FP(X)45 b Ff( )36 b FP(C)7 b FT(,)39
b(where)e(the)h(binary)d(relation)i Ff( )g FT(b)s(et)m(w)m(een)h
(structured)378 1947 y(expressions)26 b(and)h(form)m(ulae)g(is)f
(de\014ned)h(as)g(the)h(smallest)e(relation)h(satisfying)f(the)i(follo)
m(wing)e(four)378 2060 y(rules:)1938 2133 y FP(A)f Ff(\032)2132
2100 y FK(\003)2197 2133 y FP(C)p 1938 2173 331 4 v 1962
2256 a(A)h Ff( )f FP(C)1627 2478 y(X)33 b Ff( )25 b FT(\()p
FP(A)h FN(\))f FP(B)5 b FT(\))91 b FP(Y)45 b Ff( )25
b FP(A)p 1627 2519 952 4 v 1804 2602 a FT(\()p FP(X)38
b FM(on)30 b FP(Y)20 b FT(\))26 b Ff( )f FP(B)1434 2884
y(X)33 b Ff( )25 b FP(A)91 b(Y)45 b Ff( )25 b FP(B)96
b FT(\()p FP(A)21 b FN(^)e FP(B)5 b FT(\))26 b Ff(\032)2636
2851 y FK(\003)2700 2884 y FP(C)p 1434 2925 1338 4 v
1782 3008 a FT(\()p FP(X)38 b FM(and)29 b FP(Y)20 b FT(\))26
b Ff( )f FP(C)1483 3290 y(X)33 b Ff( )25 b FT(\()p FP(A)h
FN(\))f FP(B)5 b FT(\))90 b FP(Y)46 b Ff( )25 b FT(\()p
FP(B)30 b FN(\))25 b FP(C)7 b FT(\))p 1483 3330 1241
4 v 1618 3413 a(\()p FP(X)38 b FM(then)29 b FP(Y)20 b
FT(\))25 b Ff( )h FT(\()p FP(A)f FN(\))h FP(C)7 b FT(\))378
3599 y(where)30 b FP(A)p FT(,)h FP(B)j FT(and)c FP(C)37
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 Fw(.)14
b Fv(B)t Ft(\()p Fv(x;)g(c)p Ft(\)\))45 b Fw(and)d Ft(\()p
Fv(A)p Ft(\()p Fv(a;)14 b(b)p Ft(\)\))26 b Ff( )f Ft(\()p
Fu(9)p Fv(c)p Fw(.)15 b Fv(A)p Ft(\()p Fv(a;)f(b)p Ft(\))43
b Fu(^)h Fv(B)t Ft(\()p Fv(b;)14 b(c)p Ft(\)\))564 b
FT(\(1\))378 5143 y(b)m(y)30 b(using)f(the)i(third)d(rule)h(in)g
(de\014nition)f(6.3,)k(and)e(the)g(follo)m(wing:)514
5330 y FN(\017)46 b Ft(\()p Fu(8)p Fv(x)p Fw(.)p Fu(9)p
Fv(c)p Fw(.)14 b Fv(B)t Ft(\()p Fv(x;)g(c)p Ft(\)\))26
b Ff(\032)1325 5297 y FK(\003)1390 5330 y Fu(9)p Fv(c)p
Fw(.)14 b Fv(B)t Ft(\()p Fv(b;)g(c)p Ft(\))p FT(,)31
b(and)f(so)g Ft(\()p Fu(8)p Fv(x)p Fw(.)p Fu(9)p Fv(c)p
Fw(.)14 b Fv(B)t Ft(\()p Fv(x;)g(c)p Ft(\)\))26 b Ff( )g
Fu(9)p Fv(c)p Fw(.)14 b Fv(B)t Ft(\()p Fv(b;)g(c)p Ft(\))p
FT(;)514 5518 y FN(\017)46 b Fv(A)p Ft(\()p Fv(a;)14
b(b)p Ft(\))26 b Ff(\032)975 5485 y FK(\003)1039 5518
y Fv(A)p Ft(\()p Fv(a;)14 b(b)p Ft(\))q FT(,)30 b(and)g(so)h
Fv(A)p Ft(\()p Fv(a;)14 b(b)p Ft(\))25 b Ff( )g Fv(A)p
Ft(\()p Fv(a;)14 b(b)p Ft(\))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 y(as)1071 1147 y Fu(8)p Fv(x;)14 b(y)s(;)g(z)t
Fw(.)e Fv(A)p Ft(\()p Fv(x;)i(y)s Ft(\))44 b Fu(\))g
Fv(B)t Ft(\()p Fv(y)s(;)14 b(z)t Ft(\))43 b Fu(\))h Fv(C)6
b Ft(\()p Fv(x;)14 b(z)t Ft(\))1237 1285 y Ff(\032)1338
1247 y FK(\003)1402 1285 y Fu(8)p Fv(x;)g(y)s Fw(.)g
Ft(\()p Fu(9)p Fv(z)t Fw(.)f Fv(A)p Ft(\()p Fv(x;)h(y)s
Ft(\))45 b Fu(^)f Fv(B)t Ft(\()p Fv(y)s(;)14 b(z)t Ft(\)\))43
b Fu(\))h Ft(\()p Fu(8)p Fv(z)t Fw(.)13 b Fv(C)6 b Ft(\()p
Fv(x;)14 b(z)t Ft(\)\))1237 1423 y Ff(\032)1338 1385
y FK(\003)1402 1423 y Fu(8)p Fv(x;)g(y)s Fw(.)g Ft(\()p
Fu(9)p Fv(z)t Fw(.)f Fv(A)p Ft(\()p Fv(x;)h(y)s Ft(\))45
b Fu(^)f Fv(B)t Ft(\()p Fv(y)s(;)14 b(z)t Ft(\)\))43
b Fu(\))h Ft(\()p Fu(9)p Fv(z)t Fw(.)14 b Fv(C)6 b Ft(\()p
Fv(x;)14 b(z)t Ft(\)\))1237 1560 y Ff(\032)1338 1523
y FK(\003)1402 1560 y Ft(\()p Fu(9)p Fv(c)p Fw(.)h Fv(A)p
Ft(\()p Fv(a;)f(b)p Ft(\))44 b Fu(^)g Fv(B)t Ft(\()p
Fv(b;)14 b(c)p Ft(\)\))44 b Fu(\))f Ft(\()p Fu(9)p Fv(c)p
Fw(.)15 b Fv(C)6 b Ft(\()p Fv(a;)14 b(c)p Ft(\)\))p FP(:)378
1765 y FT(Therefore,)28 b(b)m(y)f(the)h(second)g(rule)e(of)i
(de\014nition)d(6.3)j(and)f(equations)g(\(1\))i(and)e(\(2\))i(ab)s(o)m
(v)m(e,)g(it)e(is)g(the)378 1878 y(case)k(that)461 2082
y Ft(\(\()p 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(\))h Fv(C)6 b Ft(\()p
Fv(x;)14 b(z)t Ft(\)\))44 b Fw(on)f Ft(\()p Fu(8)p Fv(x)p
Fw(.)p Fu(9)p Fv(c)p Fw(.)13 b Fv(B)t Ft(\()p Fv(x;)h(c)p
Ft(\)\))45 b Fw(and)e Ft(\()p Fv(A)p Ft(\()p Fv(a;)14
b(b)p Ft(\)\)\))2025 2220 y Ff( )55 b Fu(9)p Fv(c)p Fw(.)15
b Fv(C)6 b Ft(\()p Fv(a;)14 b(c)p Ft(\))p FP(:)1174 b
Ff(\003)378 2506 y FH(6.5)135 b(Results)46 b(on)f(Structured)f
(Justi\014cations)378 2709 y FT(In)30 b(this)g(section)h(w)m(e)h(giv)m
(e)f(a)g(n)m(um)m(b)s(er)f(of)h(results)f(on)h(the)g(structured)f
(justi\014cations)f(giv)m(en)i(in)f(the)378 2822 y(previous)39
b(section.)71 b(W)-8 b(e)41 b(start)g(b)m(y)f(sho)m(wing)g(that)h(the)f
Fw(on)g FT(and)f Fw(and)g FT(op)s(erators)i(generalise)f(the)378
2935 y(inference)32 b(rules)f(of)i(Mo)s(dus)f(P)m(onens)h(and)f(the)h
(in)m(tro)s(duction)d(of)j(conjunction)f(resp)s(ectiv)m(ely)-8
b(,)33 b(and)378 3048 y(that)e(an)f(expression)f(of)i(the)f(form)g
Ft(\()p Fv(X)50 b Fw(then)42 b Fv(Y)19 b Ft(\))44 b Fw(on)f
Fv(Z)36 b FT(is)29 b(equiv)-5 b(alen)m(t)30 b(to)h Fv(Y)62
b Fw(on)43 b Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Z)6 b Ft(\))p
FT(.)378 3260 y FQ(Prop)s(osition)36 b(6.3)g(\()p FM(on)e
FQ(Generalises)h(Mo)s(dus)h(P)m(onens\))46 b FI(F)-7
b(or)40 b(al)5 b(l)40 b(formulae)h FP(A)p FI(,)g FP(B)j
FI(and)d FP(C)7 b FI(,)378 3373 y(the)33 b(expr)-5 b(ession)34
b Fv(A)44 b Fw(on)f Fv(B)29 b Ff( )c FP(C)39 b FI(holds)c(if)d(and)i
(only)f(if)f(ther)-5 b(e)34 b(ar)-5 b(e)33 b(some)h FP(P)13
b FI(,)32 b FP(Q)g FI(and)i FP(R)f FI(such)g(that)485
3561 y(1.)46 b FP(A)26 b Ff(\032)800 3528 y FK(\003)864
3561 y FT(\()p FP(P)39 b FN(\))25 b FP(Q)p FT(\))p FI(,)485
3748 y(2.)46 b FP(B)30 b Ff(\032)805 3715 y FK(\003)870
3748 y FP(P)13 b FI(,)32 b(and)485 3936 y(3.)46 b FP(Q)25
b Ff(\032)803 3903 y FK(\003)868 3936 y FP(C)7 b FI(,)378
4124 y(so)33 b(that)h FP(C)39 b FI(c)-5 b(an)33 b(b)-5
b(e)33 b(derive)-5 b(d)33 b(fr)-5 b(om)34 b FP(A)f FI(and)g
FP(B)k FI(by:)1731 4303 y FP(A)p 1623 4323 285 4 v 1623
4402 a(P)h FN(\))26 b FP(Q)1949 4346 y FT(\()p Ff(\032)2085
4313 y FK(\003)2125 4346 y FT(\))2253 4303 y FP(B)p 2253
4323 74 4 v 2254 4402 a(P)2368 4346 y FT(\()p Ff(\032)2504
4313 y FK(\003)2544 4346 y FT(\))p 1623 4439 703 4 v
1938 4518 a FP(Q)2367 4462 y FT(\(MP\))p 1938 4556 72
4 v 1939 4634 a FP(C)2052 4578 y FT(\()p Ff(\032)2188
4545 y FK(\003)2227 4578 y FT(\))378 4847 y FQ(Pro)s(of)p
FT(:)i(Giv)m(en)e(that)h Fv(A)44 b Fw(on)f Fv(B)29 b
Ff( )c FP(C)7 b FT(,)27 b(then)g(from)f(de\014nition)e(6.3)k(it)e(m)m
(ust)g(b)s(e)g(the)h(case)h(that)f(there)378 4960 y(is)34
b(some)i FP(D)h FT(suc)m(h)e(that)h FP(A)d Ff( )g FT(\()p
FP(D)j FN(\))d FP(C)7 b FT(\))35 b(and)g FP(B)i Ff( )d
FP(D)s FT(,)i(and)e(therefore)i(from)e(de\014nition)f(6.3)j(it)378
5073 y(follo)m(ws)29 b(that)h FP(A)c Ff(\032)1067 5040
y FK(\003)1132 5073 y FT(\()p FP(D)i FN(\))d FP(C)7 b
FT(\))30 b(and)f FP(B)h Ff(\032)1899 5040 y FK(\003)1964
5073 y FP(D)s FT(.)40 b(The)29 b(ab)s(o)m(v)m(e)i(three)g(results)d(in)
h(the)h(statemen)m(t)i(of)378 5186 y(the)25 b(prop)s(osition)e(can)i(b)
s(e)g(satis\014ed)f(b)m(y)h(c)m(ho)s(osing)g FP(P)38
b FT(to)26 b(b)s(e)e FP(D)k FT(and)c FP(Q)h FT(to)h(b)s(e)f
FP(C)7 b FT(.)38 b(F)-8 b(or)26 b(the)f(con)m(v)m(erse,)378
5299 y(giv)m(en)30 b(the)h(ab)s(o)m(v)m(e)g(three)g(h)m(yp)s(otheses,)f
(it)g(follo)m(ws)g(that)1322 5503 y FP(A)25 b Ff(\032)1516
5465 y FK(\003)1581 5503 y FT(\()p FP(P)38 b FN(\))25
b FP(Q)p FT(\))61 b(b)m(y)31 b(the)f(\014rst)g(h)m(yp)s(othesis)1415
5641 y Ff(\032)1516 5603 y FK(\003)1581 5641 y FT(\()p
FP(P)38 b FN(\))25 b FP(C)7 b FT(\))61 b(b)m(y)30 b(the)h(third,)p
eop
%%Page: 111 121
111 120 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(111)378 396 y(and)33 b(therefore)h FP(A)c Ff( )h
FT(\()p FP(P)44 b FN(\))30 b FP(C)7 b FT(\).)50 b(F)-8
b(rom)34 b(the)g(second)f(h)m(yp)s(othesis,)g(w)m(e)h(get)h
FP(B)g Ff( )30 b FP(P)13 b FT(,)35 b(and)e(hence)378
509 y Fv(A)44 b Fw(on)f Fv(B)29 b Ff( )c FP(C)37 b FT(as)31
b(required.)2367 b Ff(\004)378 722 y FQ(Prop)s(osition)36
b(6.4)g(\()p FM(and)d FQ(Generalises)j FN(^)p FQ(-In)m(tro)s(duction\))
45 b FI(F)-7 b(or)33 b(al)5 b(l)32 b(formulae)h FP(A)p
FI(,)f FP(B)37 b FI(and)c FP(C)7 b FI(,)378 835 y(the)33
b(expr)-5 b(ession)34 b Fv(A)44 b Fw(and)e Fv(B)30 b
Ff( )25 b FP(C)39 b FI(holds)34 b(if)f(and)g(only)h(if)e(ther)-5
b(e)33 b(ar)-5 b(e)34 b(some)f FP(P)46 b FI(and)33 b
FP(Q)g FI(such)f(that)485 1022 y(1.)46 b FP(A)26 b Ff(\032)800
989 y FK(\003)864 1022 y FP(P)13 b FI(,)485 1210 y(2.)46
b FP(B)30 b Ff(\032)805 1177 y FK(\003)870 1210 y FP(Q)p
FI(,)i(and)485 1398 y(3.)46 b FP(P)33 b FN(^)20 b FP(Q)25
b Ff(\032)975 1365 y FK(\003)1040 1398 y FP(C)7 b FI(,)378
1585 y(so)33 b(that)h FP(C)39 b FI(c)-5 b(an)33 b(b)-5
b(e)33 b(derive)-5 b(d)33 b(fr)-5 b(om)34 b FP(A)f FI(and)g
FP(B)k FI(by:)1660 1765 y FP(A)p 1658 1785 71 4 v 1658
1863 a(P)1771 1807 y FT(\()p Ff(\032)1907 1774 y FK(\003)1946
1807 y FT(\))2075 1765 y FP(B)p 2075 1785 74 4 v 2076
1863 a(Q)2190 1807 y FT(\()p Ff(\032)2326 1774 y FK(\003)2366
1807 y FT(\))p 1658 1901 490 4 v 1781 1980 a FP(P)c FN(^)20
b FP(Q)2189 1924 y FT(\()p FN(^)p FT(-In)m(tro\))p 1781
2017 244 4 v 1867 2096 a FP(C)2066 2040 y FT(\()p Ff(\032)2202
2007 y FK(\003)2242 2040 y FT(\))378 2309 y FQ(Pro)s(of)p
FT(:)31 b(Similarly)c(to)k(prop)s(osition)d(6.3,)k(this)d(result)g
(follo)m(ws)h(from)g(de\014nition)e(6.3.)434 b Ff(\004)519
2534 y FT(Before)37 b(sho)m(wing)e(that)i(structured)e(expressions)f
(in)m(v)m(olving)h(the)h Fw(then)e FT(op)s(erator)i(are)h(equiv-)378
2647 y(alen)m(t)i(to)g(certain)f(expressions)e(in)m(v)m(olving)h(the)i
Fw(on)e FT(op)s(erator,)k(w)m(e)d(\014rst)g(giv)m(e)h(the)f
(de\014nitions)e(of)378 2760 y(equiv)-5 b(alence)30 b(on)g(structured)f
(expressions.)378 2973 y FQ(De\014nition)35 b(6.5)h(\(Equiv)-6
b(alen)m(t)34 b(Structured)h(Expressions\))46 b FT(Tw)m(o)28
b(structured)e(expressions)378 3086 y FP(X)40 b FT(and)32
b FP(Y)52 b FT(are)33 b(equiv)-5 b(alen)m(t)32 b(if)f
FP(X)36 b Ff( )29 b FP(C)39 b FT(holds)31 b(if)g(and)h(only)g(if)f
FP(Y)49 b Ff( )28 b FP(C)39 b FT(holds)31 b(for)h(ev)m(ery)i(form)m
(ula)378 3199 y FP(C)7 b FT(.)3282 b Ff(\003)378 3411
y FQ(Prop)s(osition)36 b(6.5)g(\(Elimination)d(of)i FM(then)p
FQ(\))44 b FI(F)-7 b(or)32 b(al)5 b(l)32 b(structur)-5
b(e)g(d)33 b(expr)-5 b(essions)33 b FP(X)7 b FI(,)32
b FP(Y)51 b FI(and)33 b FP(Z)7 b FI(,)378 3524 y(the)33
b(expr)-5 b(ession)1750 3637 y Ft(\()p Fv(X)51 b Fw(then)42
b Fv(Y)18 b Ft(\))44 b Fw(on)f Fv(Z)378 3804 y FI(is)33
b(e)-5 b(quivalent)32 b(to)1781 3917 y Fv(Y)63 b Fw(on)42
b Ft(\()p Fv(X)51 b Fw(on)42 b Fv(Z)6 b Ft(\))p FP(:)378
4129 y FQ(Pro)s(of)p FT(:)23 b(F)-8 b(or)24 b(all)d(form)m(ulae)h
FP(C)7 b FT(,)23 b(giv)m(en)g(that)g Ft(\()p Fv(X)50
b Fw(then)42 b Fv(Y)18 b Ft(\))44 b Fw(on)f Fv(Z)31 b
Ff( )25 b FP(C)k FT(then)22 b(there)h(is)e(some)i(form)m(ula)378
4242 y FP(A)34 b FT(suc)m(h)f(that)h Ft(\()p Fv(X)51
b Fw(then)42 b Fv(Y)18 b Ft(\))31 b Ff( )g FT(\()p FP(A)g
FN(\))g FP(C)7 b FT(\))33 b(and)g Fv(Z)k Ff( )31 b FP(A)p
FT(,)j(and)g(so)f(there)h(m)m(ust)g(b)s(e)f(some)h FP(B)k
FT(suc)m(h)378 4355 y(that)k FP(X)52 b Ff( )43 b FT(\()p
FP(A)i FN(\))f FP(B)5 b FT(\))41 b(and)g FP(Y)64 b Ff( )44
b FT(\()p FP(B)49 b FN(\))44 b FP(C)7 b FT(\).)74 b(Hence,)45
b(w)m(e)d(can)g(deriv)m(e)f FP(C)48 b FT(explicitly)39
b(from)378 4468 y Fv(Y)62 b Fw(on)43 b Ft(\()p Fv(X)50
b Fw(on)43 b Fv(Z)6 b Ft(\))30 b FT(as)h(follo)m(ws:)1308
4778 y Fv(Y)44 b Ff( )25 b FT(\()p FP(B)30 b FN(\))25
b FP(C)7 b FT(\))1964 4657 y Fv(X)32 b Ff( )25 b FT(\()p
FP(A)h FN(\))f FP(B)5 b FT(\))91 b Fv(Z)30 b Ff( )c FP(A)p
1964 4699 934 4 v 2135 4778 a Ft(\()p Fv(X)50 b Fw(on)43
b Fv(Z)6 b Ft(\))25 b Ff( )g FP(B)p 1308 4821 1419 4
v 1602 4900 a Fv(Y)62 b Fw(on)43 b Ft(\()p Fv(X)50 b
Fw(on)43 b Fv(Z)6 b Ft(\))25 b Ff( )g FP(C)378 5104 y
FT(F)-8 b(or)32 b(the)f(con)m(v)m(erse,)i(if)d Fv(Y)62
b Fw(on)43 b Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Z)6 b Ft(\))26
b Ff( )g FP(C)7 b FT(,)31 b(then)g(there)g(is)f(some)h
FP(B)36 b FT(suc)m(h)31 b(that)g Fv(Y)45 b Ff( )26 b
FP(B)31 b FN(\))26 b FP(C)378 5217 y FT(and)k Fv(X)49
b Fw(on)43 b Fv(Z)31 b Ff( )25 b FP(B)5 b FT(,)30 b(and)g(therefore)h
(there)f(is)f(some)i(form)m(ula)f FP(A)g FT(suc)m(h)g(that)h
Fv(X)g Ff( )25 b FT(\()p FP(A)h FN(\))f FP(B)5 b FT(\))30
b(and)378 5330 y Fv(Z)h Ff( )25 b FP(A)p FT(.)41 b(Hence,)1308
5424 y Fv(X)32 b Ff( )25 b FT(\()p FP(A)h FN(\))f FP(B)5
b FT(\))91 b Fv(Y)43 b Ff( )26 b FT(\()p FP(B)k FN(\))25
b FP(C)7 b FT(\))p 1308 5466 1227 4 v 1441 5551 a Ft(\()p
Fv(X)50 b Fw(then)42 b Fv(Y)19 b Ft(\))25 b Ff( )g FT(\()p
FP(A)h FN(\))f FP(C)7 b FT(\))224 b Fv(Z)30 b Ff( )c
FP(A)p 1441 5594 1458 4 v 1710 5673 a Ft(\()p Fv(X)51
b Fw(then)41 b Fv(Y)19 b Ft(\))44 b Fw(on)f Fv(Z)31 b
Ff( )25 b FP(C)p eop
%%Page: 112 122
112 121 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(112)378 396 y(Therefore,)29 b(the)g(structured)f(expressions)f
Ft(\()p Fv(X)50 b Fw(then)42 b Fv(Y)19 b Ft(\))44 b Fw(on)e
Fv(Z)35 b FT(and)28 b Fv(Y)62 b Fw(on)43 b Ft(\()p Fv(X)50
b Fw(on)43 b Fv(Z)6 b Ft(\))28 b FT(are)i(equiv-)378
509 y(alen)m(t.)3160 b Ff(\004)519 735 y FT(Since)27
b(the)h(syn)m(tax)h(of)f(structured)f(justi\014cations)g(restricts)h
(the)g(use)g(of)g(the)g Fw(then)f FT(op)s(erator)h(to)378
848 y(the)j(left)f(hand)f(side)g(of)i(an)f Fw(on)g FT(op)s(erator,)h
(one)f(can)h(in)e(general)h(rewrite)g(a)h(structured)e(expression)378
961 y(in)m(v)m(olving)g(the)h Fw(then)f FT(op)s(erator)i(in)m(to)f
(equiv)-5 b(alen)m(t)30 b(ones)g(whic)m(h)g(do)g(not.)519
1074 y(A)35 b(n)m(um)m(b)s(er)e(of)i(other)g(results)e(on)i(the)g(prop)
s(erties)e(of)i(structured)e(expressions)h(are)h(giv)m(en)f(in)378
1187 y(the)d(next)f(prop)s(osition.)378 1379 y FQ(Prop)s(osition)36
b(6.6)46 b FI(F)-7 b(or)35 b(al)5 b(l)35 b(structur)-5
b(e)g(d)35 b(expr)-5 b(essions)36 b FP(X)7 b FI(,)34
b FP(Y)54 b FI(and)35 b FP(Z)7 b FI(,)34 b(and)h(formulae)g
FP(A)f FI(and)h FP(B)5 b FI(,)378 1492 y(the)33 b(fol)5
b(lowing)34 b(r)-5 b(esults)33 b(hold:)485 1662 y(1.)46
b(If)33 b FP(X)40 b FI(is)32 b(not)i(a)f Fw(then)d FI(expr)-5
b(ession)35 b(and)e FP(X)g Ff( )25 b FP(A)33 b FI(and)g
FP(X)g Ff( )25 b FP(B)37 b FI(then)c FP(X)g Ff( )25 b
FT(\()p FP(A)c FN(^)f FP(B)5 b FT(\))p FI(.)485 1843
y(2.)46 b(The)33 b(expr)-5 b(ession)34 b Ft(\()p Fv(X)51
b Fw(on)42 b Fv(Y)19 b Ft(\))44 b Fw(on)f Fv(Z)38 b FI(is)32
b(e)-5 b(quivalent)33 b(to)g Ft(\()p Fv(X)50 b Fw(on)43
b Fv(Z)6 b Ft(\))44 b Fw(on)e Fv(Y)19 b FI(.)485 2024
y(3.)46 b(The)33 b(expr)-5 b(ession)34 b Fv(X)50 b Fw(and)43
b Fv(Y)51 b FI(is)32 b(e)-5 b(quivalent)33 b(to)g Fv(Y)63
b Fw(and)42 b Fv(X)6 b FI(.)485 2204 y(4.)46 b(The)33
b(expr)-5 b(ession)34 b Fv(X)50 b Fw(on)43 b Ft(\()p
Fv(Y)63 b Fw(and)42 b Fv(Z)6 b Ft(\))32 b FI(is)h(e)-5
b(quivalent)33 b(to)g Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Y)18
b Ft(\))44 b Fw(on)f Fv(Z)6 b FI(.)485 2385 y(5.)46 b(The)33
b(expr)-5 b(ession)34 b Ft(\()p Fv(X)51 b Fw(and)42 b
Fv(Y)19 b Ft(\))43 b Fw(and)g Fv(Z)38 b FI(is)32 b(e)-5
b(quivalent)33 b(to)g Fv(X)50 b Fw(and)43 b Ft(\()p Fv(Y)62
b Fw(and)43 b Fv(Z)6 b Ft(\))o FI(.)485 2566 y(6.)46
b Ft(\()p Fv(X)k Fw(then)42 b Fv(Y)19 b Ft(\))44 b Fw(then)e
Fv(Z)31 b Ff( )25 b FT(\()p FP(A)h FN(\))f FP(B)5 b FT(\))32
b FI(if)g(and)i(only)f(if)g Fv(X)50 b Fw(then)42 b Ft(\()p
Fv(Y)62 b Fw(then)42 b Fv(Z)6 b Ft(\))25 b Ff( )g FT(\()p
FP(A)h FN(\))f FP(B)5 b FT(\))p FI(.)485 2747 y(7.)46
b(If)33 b Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Z)6 b Ft(\))43
b Fw(and)f Ft(\()p Fv(Y)63 b Fw(on)43 b Fv(Z)6 b Ft(\))25
b Ff( )g FP(A)33 b FI(then)g Ft(\()p Fv(X)50 b Fw(and)42
b Fv(Y)19 b Ft(\))44 b Fw(on)f Fv(Z)31 b Ff( )25 b FP(A)p
FI(.)378 2939 y FQ(Pro)s(of)p FT(:)34 b(The)f(\014rst)f(statemen)m(t)j
(follo)m(ws)d(b)m(y)h(induction)d(on)j(the)g(structure)g(of)g
Fv(X)6 b FT(.)49 b(In)32 b(the)h(ligh)m(t)g(of)378 3052
y(prop)s(osition)23 b(6.5)k(w)m(e)g(can)f(assume)f(without)g(loss)g(of)
h(generalit)m(y)g(that)g(the)g(expression)f FP(X)33 b
FT(do)s(es)26 b(not)378 3165 y(con)m(tain)31 b(the)f
Fw(then)f FT(op)s(erator.)41 b(W)-8 b(e)31 b(need)g(to)g(consider)e
(the)i(follo)m(wing)d(three)j(cases:)514 3335 y FN(\017)46
b FT(The)25 b(expression)g FP(X)33 b FT(is)25 b(a)h(form)m(ula:)37
b(Therefore)26 b(w)m(e)g(are)g(required)e(to)i(sho)m(w)g(that)g(if)e
FP(X)33 b Ff(\032)3695 3302 y FK(\003)3760 3335 y FP(A)605
3448 y FT(and)d FP(X)j Ff(\032)991 3415 y FK(\003)1055
3448 y FP(B)i FT(then)30 b FP(X)j Ff(\032)1575 3415 y
FK(\003)1640 3448 y FT(\()p FP(A)20 b FN(^)g FP(B)5 b
FT(\))30 b(whic)m(h)g(follo)m(ws)f(b)m(y)1410 3652 y
FP(X)k Ff(\032)25 b FT(\()p FP(X)j FN(^)20 b FP(X)7 b
FT(\))26 b Ff(\032)2107 3615 y FK(\003)2171 3652 y FT(\()p
FP(A)21 b FN(^)f FP(X)7 b FT(\))26 b Ff(\032)2620 3615
y FK(\003)2685 3652 y FT(\()p FP(A)20 b FN(^)g FP(B)5
b FT(\))p FP(:)514 3890 y FN(\017)46 b FT(The)35 b(expression)g
FP(X)43 b FT(is)35 b(some)h Fw(on)e FT(expression)h Fv(Y)62
b Fw(on)43 b Fv(Z)e FT(where)35 b(if)g FP(Y)54 b Ff( )34
b FP(P)3241 3904 y FL(1)3316 3890 y FT(and)h FP(Y)54
b Ff( )34 b FP(P)3788 3904 y FL(2)605 4003 y FT(then)39
b FP(Y)58 b Ff( )39 b FT(\()p FP(P)1155 4017 y FL(1)1221
4003 y FN(^)25 b FP(P)1365 4017 y FL(2)1405 4003 y FT(\),)41
b(and)d(if)g FP(Z)45 b Ff( )39 b FP(P)2078 4017 y FL(1)2157
4003 y FT(and)f FP(Z)45 b Ff( )39 b FP(P)2637 4017 y
FL(2)2715 4003 y FT(then)g FP(Z)45 b Ff( )39 b FT(\()p
FP(P)3261 4017 y FL(1)3327 4003 y FN(^)25 b FP(P)3471
4017 y FL(2)3511 4003 y FT(\))39 b(for)f(all)605 4116
y(form)m(ulae)25 b FP(P)1032 4130 y FL(1)1098 4116 y
FT(and)g FP(P)1328 4130 y FL(2)1368 4116 y FT(.)39 b(No)m(w,)27
b(since)e(\()p Fv(Y)63 b Fw(on)42 b Fv(Z)6 b FT(\))26
b Ff( )f FP(A)h FT(then)f(there)h(is)e(some)i(form)m(ula)f
FP(C)32 b FT(suc)m(h)605 4229 y(that)1637 4433 y FP(Y)45
b Ff( )25 b FT(\()p FP(C)32 b FN(\))25 b FP(A)p FT(\))84
b(and)e FP(Z)32 b Ff( )25 b FP(C)605 4637 y FT(and)30
b(since)g(\()p Fv(Y)62 b Fw(on)43 b Fv(Z)6 b FT(\))25
b Ff( )g FP(B)35 b FT(then)30 b(there)h(is)e(some)i(form)m(ula)f
FP(D)j FT(suc)m(h)d(that)1615 4842 y FP(Y)45 b Ff( )25
b FT(\()p FP(D)k FN(\))c FP(B)5 b FT(\))83 b(and)f FP(Z)32
b Ff( )25 b FP(D)s(:)605 5046 y FT(As)31 b(a)f(result,)1697
5250 y FP(Y)45 b Ff( )25 b FT(\()p FP(C)32 b FN(\))25
b FP(A)p FT(\))c FN(^)f FT(\()p FP(D)28 b FN(\))d FP(B)5
b FT(\))1795 5388 y Ff(\032)1896 5350 y FK(\003)1961
5388 y FT(\()p FP(C)27 b FN(^)19 b FP(D)s FT(\))26 b
FN(\))f FT(\()p FP(A)c FN(^)f FP(B)5 b FT(\))605 5592
y(and)1951 5705 y FP(Z)32 b Ff( )25 b FT(\()p FP(C)i
FN(^)20 b FP(D)s FT(\))p eop
%%Page: 113 123
113 122 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(113)605 396 y(and)30 b(therefore)1825 509 y Fv(Y)62
b Fw(on)43 b Fv(Z)31 b Ff( )25 b FT(\()p FP(A)c FN(^)e
FP(B)5 b FT(\))p FP(:)514 709 y FN(\017)46 b FT(The)31
b(expression)e FP(X)39 b FT(is)30 b(some)h Fw(and)f FT(expression)g
Fv(Y)62 b Fw(and)42 b Fv(Z)37 b FT(where)30 b(if)g FP(Y)46
b Ff( )27 b FP(P)3266 723 y FL(1)3336 709 y FT(and)k
FP(Y)46 b Ff( )26 b FP(P)3788 723 y FL(2)605 822 y FT(then)39
b FP(Y)58 b Ff( )39 b FT(\()p FP(P)1155 836 y FL(1)1221
822 y FN(^)25 b FP(P)1365 836 y FL(2)1405 822 y FT(\),)41
b(and)d(if)g FP(Z)45 b Ff( )39 b FP(P)2078 836 y FL(1)2157
822 y FT(and)f FP(Z)45 b Ff( )39 b FP(P)2637 836 y FL(2)2715
822 y FT(then)g FP(Z)45 b Ff( )39 b FT(\()p FP(P)3261
836 y FL(1)3327 822 y FN(^)25 b FP(P)3471 836 y FL(2)3511
822 y FT(\))39 b(for)f(all)605 935 y(form)m(ulae)f FP(P)1044
949 y FL(1)1120 935 y FT(and)f FP(P)1361 949 y FL(2)1401
935 y FT(.)59 b(No)m(w,)40 b(since)c(\()p Fv(Y)62 b Fw(and)43
b Fv(Z)5 b FT(\))36 b Ff( )g FP(A)h FT(then)f(there)h(are)g(some)g
(form)m(ulae)605 1048 y FP(A)673 1062 y FO(Y)764 1048
y FT(and)30 b FP(A)1009 1062 y FO(Z)1096 1048 y FT(suc)m(h)g(that)1240
1252 y FP(Y)45 b Ff( )25 b FP(A)1522 1266 y FO(Y)1583
1252 y FP(;)197 b(Z)32 b Ff( )25 b FP(A)2083 1266 y FO(Z)2140
1252 y FP(;)106 b FT(and)91 b(\()p FP(A)2612 1266 y FO(Y)2693
1252 y FN(^)20 b FP(A)2842 1266 y FO(Z)2899 1252 y FT(\))26
b Ff(\032)3061 1214 y FK(\003)3125 1252 y FP(A)605 1456
y FT(and)k(since)g(\()p Fv(Y)62 b Fw(and)43 b Fv(Z)5
b FT(\))26 b Ff( )f FP(B)35 b FT(there)30 b(are)h(some)g(form)m(ulae)f
FP(B)2723 1470 y FO(Y)2814 1456 y FT(and)g FP(B)3060
1470 y FO(Z)3147 1456 y FT(suc)m(h)g(that)1223 1660 y
FP(Y)45 b Ff( )25 b FP(B)1506 1674 y FO(Y)1567 1660 y
FP(;)197 b(Z)32 b Ff( )25 b FP(B)2068 1674 y FO(Z)2125
1660 y FP(;)106 b FT(and)91 b(\()p FP(B)2598 1674 y FO(Y)2679
1660 y FN(^)20 b FP(B)2829 1674 y FO(Z)2885 1660 y FT(\))26
b Ff(\032)3047 1623 y FK(\003)3111 1660 y FP(B)5 b(:)605
1864 y FT(As)31 b(a)f(result,)1422 2069 y FP(Y)45 b Ff( )25
b FT(\()p FP(A)1739 2083 y FO(Y)1821 2069 y FN(^)20 b
FP(B)1971 2083 y FO(Y)2031 2069 y FT(\))84 b(and)e FP(Z)32
b Ff( )25 b FT(\()p FP(A)2692 2083 y FO(Z)2769 2069 y
FN(^)20 b FP(B)2919 2083 y FO(Z)2976 2069 y FT(\))605
2273 y(and)30 b(it)g(is)f(the)i(case)g(that)1168 2477
y(\()p FP(A)1271 2491 y FO(Y)1352 2477 y FN(^)20 b FP(B)1502
2491 y FO(Y)1563 2477 y FT(\))g FN(^)g FT(\()p FP(A)1802
2491 y FO(Z)1879 2477 y FN(^)g FP(B)2029 2491 y FO(Z)2086
2477 y FT(\))25 b Ff(\032)2247 2440 y FK(\003)2312 2477
y FT(\()p FP(A)2415 2491 y FO(Y)2497 2477 y FN(^)19 b
FP(A)2645 2491 y FO(Z)2702 2477 y FT(\))i FN(^)f FT(\()p
FP(B)2943 2491 y FO(Y)3024 2477 y FN(^)g FP(B)3174 2491
y FO(Z)3230 2477 y FT(\))2146 2615 y Ff(\032)2247 2577
y FK(\003)2312 2615 y FP(A)g FN(^)g FP(B)605 2819 y FT(and)30
b(therefore)1803 2932 y Fv(Y)62 b Fw(and)43 b Fv(Z)30
b Ff( )c FT(\()p FP(A)20 b FN(^)g FP(B)5 b FT(\))p FP(:)519
3132 y FT(The)34 b(next)g(\014v)m(e)g(statemen)m(ts)i(in)d(the)i
(curren)m(t)e(prop)s(osition)f(are)j(quite)e(straigh)m(tforw)m(ard,)i
(and)378 3244 y(their)g(pro)s(ofs)g(are)i(similar)c(to)k(that)g(of)f
(prop)s(osition)e(6.5.)59 b(The)35 b(pro)s(of)g(of)i(the)f(last)g
(statemen)m(t)i(is)378 3357 y(giv)m(en)30 b(b)s(elo)m(w.)519
3470 y(If)h Ft(\()p Fv(X)51 b Fw(on)42 b Fv(Z)6 b Ft(\))44
b Fw(and)e Ft(\()p Fv(Y)63 b Fw(on)42 b Fv(Z)6 b Ft(\))28
b Ff( )f FP(A)32 b FT(then)f(it)h(follo)m(ws)e(from)i(the)g
(de\014nition)d(of)j Ff( )f FT(that)i(there)378 3583
y(m)m(ust)j(b)s(e)g(some)h(form)m(ulae)f FP(H)44 b FT(and)35
b FP(I)44 b FT(suc)m(h)36 b(that)h Ft(\()p Fv(X)50 b
Fw(on)43 b Fv(Z)6 b Ft(\))36 b Ff( )f FP(H)7 b FT(,)38
b Ft(\()p Fv(Y)63 b Fw(on)42 b Fv(Z)6 b Ft(\))36 b Ff( )f
FP(I)7 b FT(,)38 b(and)e(that)378 3696 y FP(H)25 b FN(^)19
b FP(I)32 b Ff(\032)732 3663 y FK(\003)797 3696 y FP(A)p
FT(.)40 b(No)m(w,)31 b(from)e Ft(\()p Fv(X)50 b Fw(on)43
b Fv(Z)6 b Ft(\))25 b Ff( )g FP(H)37 b FT(w)m(e)30 b(get)g(that)g
(there)g(is)f(some)h(form)m(ula)e FP(J)39 b FT(suc)m(h)29
b(that)378 3809 y FP(X)42 b Ff( )34 b FT(\()p FP(J)44
b FN(\))34 b FP(H)7 b FT(\))36 b(and)f FP(Z)41 b Ff( )35
b FP(J)9 b FT(,)37 b(and)e(from)h Ft(\()p Fv(Y)62 b Fw(on)43
b Fv(Z)6 b Ft(\))34 b Ff( )g FP(I)43 b FT(it)36 b(follo)m(ws)f(that)h
(there)g(is)f(some)h FP(K)378 3922 y FT(suc)m(h)30 b(that)h
Fv(Y)44 b Ff( )25 b FT(\()p FP(K)32 b FN(\))26 b FP(I)7
b FT(\))30 b(and)g Fv(Z)h Ff( )25 b FP(K)7 b FT(.)519
4035 y(In)33 b(order)h(that)h Ft(\()p Fv(X)50 b Fw(and)42
b Fv(Y)19 b Ft(\))44 b Fw(on)e Fv(Z)37 b Ff( )32 b FP(A)p
FT(,)j(it)f(is)f(su\016cien)m(t)g(that)i(there)f(exist)g(some)g(form)m
(ulae)378 4148 y FP(U)10 b FT(,)30 b FP(V)51 b FT(and)30
b FP(W)43 b FT(suc)m(h)30 b(that)514 4311 y FN(\017)46
b FP(X)33 b Ff( )25 b FP(U)10 b FT(,)514 4489 y FN(\017)46
b FP(Y)f Ff( )26 b FP(V)20 b FT(,)514 4667 y FN(\017)46
b FP(Z)32 b Ff( )25 b FP(W)13 b FT(,)30 b(and)514 4845
y FN(\017)46 b FT(\()p FP(U)31 b FN(^)20 b FP(V)g FT(\))25
b Ff(\032)1048 4812 y FK(\003)1113 4845 y FT(\()p FP(W)38
b FN(\))25 b FP(A)p FT(\),)378 5009 y(so)31 b(that:)1084
5085 y Fv(X)h Ff( )25 b FP(U)101 b Fv(Y)44 b Ff( )25
b FP(V)111 b FT(\()p FP(U)30 b FN(^)20 b FP(V)g FT(\))26
b Ff(\032)2279 5052 y FK(\003)2344 5085 y FT(\()p FP(W)38
b FN(\))25 b FP(A)p FT(\))p 1084 5128 1639 4 v 1431 5213
a Ft(\()p Fv(X)50 b Fw(and)42 b Fv(Y)19 b Ft(\))26 b
Ff( )f FT(\()p FP(W)38 b FN(\))25 b FP(A)p FT(\))438
b FP(Z)32 b Ff( )25 b FP(W)p 1431 5255 1691 4 v 1841
5334 a Ft(\()p Fv(X)50 b Fw(and)42 b Fv(Y)19 b Ft(\))44
b Fw(on)f Fv(Z)31 b Ff( )25 b FP(A)378 5501 y FT(W)-8
b(e)32 b(c)m(ho)s(ose)f(the)f(form)m(ulae)g FP(U)10 b
FT(,)31 b FP(V)51 b FT(and)29 b FP(W)43 b FT(to)31 b(b)s(e)1147
5705 y FP(U)k FT(=)25 b(\()p FP(J)35 b FN(\))25 b FP(H)7
b FT(\))182 b FP(V)46 b FT(=)25 b(\()p FP(K)32 b FN(\))25
b FP(I)7 b FT(\))182 b FP(W)38 b FT(=)25 b FP(J)k FN(^)20
b FP(K)p eop
%%Page: 114 124
114 123 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(114)378 396 y(and)30 b(c)m(hec)m(k)i(that)f(they)f(satisfy)g(the)h
(ab)s(o)m(v)m(e)g(four)f(requiremen)m(ts:)514 584 y FN(\017)46
b FT(It)31 b(is)e(the)i(case)g(that)g Fv(X)h Ff( )25
b FT(\()p FP(J)35 b FN(\))25 b FP(H)7 b FT(\),)31 b(and)e(that)514
772 y FN(\017)46 b Fv(Y)e Ff( )25 b FT(\()p FP(K)33 b
FN(\))25 b FP(I)7 b FT(\).)514 959 y FN(\017)46 b FT(Since)c
Fv(Z)51 b Ff( )45 b FP(J)51 b FT(and)42 b Fv(Z)51 b Ff( )46
b FP(K)7 b FT(,)45 b(then)d Fv(Z)51 b Ff( )45 b FT(\()p
FP(J)38 b FN(^)28 b FP(K)7 b FT(\))42 b(b)m(y)h(the)f(\014rst)g
(statemen)m(t)i(of)f(this)605 1072 y(prop)s(osition.)514
1260 y FN(\017)j FT(It)31 b(also)f(follo)m(ws)g(that)g(\(\()p
FP(J)36 b FN(\))25 b FP(H)7 b FT(\))20 b FN(^)g FT(\()p
FP(K)32 b FN(\))26 b FP(I)7 b FT(\)\))26 b Ff(\032)2377
1227 y FK(\003)2441 1260 y FT(\(\()p FP(J)k FN(^)20 b
FP(K)7 b FT(\))26 b FN(\))f FP(A)p FT(\),)31 b(as)f(sho)m(wn)g(b)s(elo)
m(w:)793 1464 y(\()p FP(J)35 b FN(\))25 b FP(H)7 b FT(\))20
b FN(^)g FT(\()p FP(K)32 b FN(\))25 b FP(I)7 b FT(\))959
1602 y Ff(\032)55 b FT(\()p FN(:)p FP(J)30 b FN(_)19
b FP(H)7 b FT(\))21 b FN(^)f FT(\()p FN(:)p FP(K)27 b
FN(_)19 b FP(I)7 b FT(\))62 b(\(same)31 b(NNF\))959 1740
y Ff(\032)1060 1702 y FK(\003)1155 1740 y FT(\()p FN(:)p
FP(J)e FN(^)20 b(:)p FP(K)7 b FT(\))20 b FN(_)g FT(\()p
FN(:)p FP(J)29 b FN(^)20 b FP(I)7 b FT(\))20 b FN(_)g
FT(\()p FP(H)28 b FN(^)20 b(:)p FP(K)7 b FT(\))19 b FN(_)h
FT(\()p FP(H)28 b FN(^)20 b FP(I)7 b FT(\))61 b(\(distributivit)m(y\))
959 1878 y Ff(\032)1060 1840 y FK(\003)1155 1878 y FN(:)p
FP(J)29 b FN(_)20 b(:)p FP(J)29 b FN(_)19 b(:)p FP(K)27
b FN(_)20 b FT(\()p FP(H)27 b FN(^)20 b FP(I)7 b FT(\))61
b(\(w)m(eak)m(ening)31 b(the)g(\014rst)e(three)i(disjuncts\))959
2015 y Ff(\032)1060 1978 y FK(\003)1155 2015 y FT(\()p
FN(:)p FP(J)e FN(_)20 b(:)p FP(K)7 b FT(\))20 b FN(_)g
FT(\()p FP(H)27 b FN(^)20 b FP(I)7 b FT(\))61 b(\(re-brac)m(k)m
(eting\))959 2153 y Ff(\032)55 b FT(\()p FP(J)30 b FN(^)20
b FP(K)7 b FT(\))25 b FN(\))g FT(\()p FP(H)j FN(^)20
b FP(I)7 b FT(\))61 b(\(same)31 b(NNF\))q FP(:)378 2395
y FT(Therefore,)f(if)g Ft(\()p Fv(X)50 b Fw(on)43 b Fv(Z)6
b Ft(\))43 b Fw(and)f Ft(\()p Fv(Y)63 b Fw(on)43 b Fv(Z)6
b Ft(\))25 b Ff( )g FP(A)30 b FT(then)h Ft(\()p Fv(X)50
b Fw(and)42 b Fv(Y)19 b Ft(\))44 b Fw(on)e Fv(Z)31 b
Ff( )25 b FP(A)p FT(.)553 b Ff(\004)519 2621 y FT(It)31
b(should)d(b)s(e)i(noted)h(that)g(the)g(con)m(v)m(erse)h(of)f(prop)s
(osition)d(6.6\(7\))33 b(do)s(es)d(not)h(hold)e(in)h(general,)378
2734 y(as)h(seen)f(b)m(y)g(the)h(follo)m(wing)e(coun)m(terexample.)378
2946 y FQ(Example)34 b(6.2)h(\(Coun)m(terexample)e(to)i(the)g(Con)m(v)m
(erse)g(of)g(Prop.)h(6.6\(7\)\))45 b FT(It)29 b(is)f(the)i(case)378
3059 y(that)h Ft(\(\()p Fv(A)44 b Fw(and)f Ft(\()p Fv(B)48
b Fu(\))43 b Fv(C)6 b Ft(\)\))45 b Fw(on)e Ft(\()p Fv(A)h
Fu(\))f Fv(B)t Ft(\)\))26 b Ff( )f FT(\()p FP(A)c FN(^)f
FP(C)7 b FT(\))30 b(holds,)f(as)1348 3251 y FP(A)20 b
FN(^)g FT(\()p FP(B)30 b FN(\))c FP(C)7 b FT(\))25 b
Ff(\032)g FP(A)20 b FN(^)g FT(\()p FN(:)p FP(B)25 b FN(_)20
b FP(C)7 b FT(\))1900 3389 y Ff(\032)25 b FT(\()p FP(A)c
FN(^)e(:)p FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)h
FN(^)f FP(C)7 b FT(\))1900 3527 y Ff(\032)25 b FT(\()p
FP(A)h FN(\))f FP(B)5 b FT(\))25 b FN(\))g FT(\()p FP(A)c
FN(^)f FP(C)7 b FT(\))p FP(:)378 3724 y FT(Ho)m(w)m(ev)m(er,)32
b Ft(\()p Fv(A)44 b Fw(on)f Ft(\()p Fv(A)h Fu(\))g Fv(B)t
Ft(\)\))g Fw(and)e Ft(\(\()p Fv(B)49 b Fu(\))43 b Fv(C)6
b Ft(\))44 b Fw(on)f Ft(\()p Fv(A)h Fu(\))g Fv(B)t Ft(\)\))26
b Ff( )f FT(\()p FP(A)19 b FN(^)g FP(C)7 b FT(\))29 b(do)s(es)g(not)h
(hold.)39 b(Al-)378 3837 y(though)44 b(this)f(statemen)m(t)i(seems)g
(implausible,)e(w)m(e)i(do)f(not)g(ha)m(v)m(e)h(the)f(necessarily)f
(results)g(to)378 3950 y(sho)m(w)31 b(in)f(a)h(more)g(formal)f(manner)h
(that)g(it)g(do)s(es)g(not)g(hold.)41 b(The)31 b(required)e(results)h
(are)h(giv)m(en)g(in)378 4063 y(c)m(hapters)g(7)g(and)e(8,)i(and)f(the)
h(ab)s(o)m(v)m(e)g(statemen)m(t)h(is)e(sho)m(wn)g(to)h(b)s(e)f(false)g
(in)f(example)h(8.5.)241 b Ff(\003)378 4350 y FH(6.6)135
b(Discussion)378 4553 y FT(This)29 b(c)m(hapter)i(giv)m(es)g(the)g
(de\014nition)d(of)j(the)g(syn)m(tax)g(and)f(seman)m(tics)h(of)g
(structured)e(straigh)m(tfor-)378 4665 y(w)m(ard)d(justi\014cations)g
(whic)m(h)g(state)i(some)f(of)g(the)g(\014rst-order)f(logic)h
(inferences)f(used)g(in)g(deriving)e(a)378 4778 y(conclusion)29
b(from)i(a)g(n)m(um)m(b)s(er)e(of)i(premises.)41 b(These)30
b(justi\014cations,)g(ho)m(w)m(ev)m(er,)i(omit)f(sev)m(eral)g(sim-)378
4891 y(ple)36 b(inferences)f(suc)m(h)h(as)h(the)g(instan)m(tiation)e
(of)i(univ)m(ersally)d(quan)m(ti\014ed)h(v)-5 b(ariables)35
b(and)h(certain)378 5004 y(manipulations)g(on)k(the)f(structure)g(of)h
(form)m(ulae.)67 b(In)39 b(c)m(hapter)h(8)f(w)m(e)h(illustrate)e(a)i
(mec)m(hanism)378 5117 y(for)33 b(c)m(hec)m(king)g(structured)f
(justi\014cations)f(b)m(y)i(lo)s(oking)f(for)g(a)i(pro)s(of)e(of)h(the)
g(conclusion)e(from)i(the)378 5230 y(premises)28 b(in)h(a)h(v)m(ery)h
(restricted)e(searc)m(h)i(space.)41 b(The)29 b(restriction)g(on)h(the)g
(searc)m(h)g(space)h(dep)s(ends)378 5343 y(on)24 b(the)h(inferences)e
(whic)m(h)g(are)i(explicitly)d(stated)j(in)e(the)h(justi\014cation.)37
b(In)24 b(the)g(follo)m(wing)f(c)m(hapter)378 5456 y(w)m(e)k(in)m(tro)s
(duce)f(a)h(n)m(um)m(b)s(er)e(of)i(de\014nitions)d(and)i(results)f
(whic)m(h)h(are)h(used)e(in)h(sho)m(wing)f(that)j(a)f(pro)s(of)378
5569 y(searc)m(h)39 b(based)f(on)g(the)h(restrictions)e(on)h(the)g
(searc)m(h)h(space)g(giv)m(en)f(in)f(c)m(hapter)i(8)g(is)e(sound)g(and)
378 5682 y(complete)f(according)g(to)g(the)g(seman)m(tics)f(of)h
(structured)f(justi\014cations)f(giv)m(en)h(in)g(this)f(c)m(hapter.)p
eop
%%Page: 115 125
115 124 bop 378 5 a FF(CHAPTER)30 b(6.)61 b(STR)m(UCTURED)30
b(STRAIGHTF)m(OR)-10 b(W)g(ARD)30 b(JUSTIFICA)-8 b(TIONS)254
b FT(115)p 378 416 3453 4 v 376 3707 4 3291 v 515 652
a Fw(section)41 b(on_symm_and_tra)o(ns)602 851 y(given)h(type)g(":'a";)
602 951 y(let)g("R:'a)g Fu(!)i Fw('a)e Fu(!)i Fw(bool";)602
1150 y(assume)d(R_symm:)84 b("Symmetric)40 b(R")907 1249
y(R_trans:)g("Transitive)g(R")907 1349 y(R_ex:)172 b(")p
Fu(8)14 b Fw(x.)42 b Fu(9)15 b Fw(y.)43 b(R)g(x)g(y";)602
1548 y(theorem)e(R_refl:)f("Reflexive)g(R")602 1648 y(proof)689
1847 y(simplify)h(with)g(Reflexive,)f(Symmetric)g(and)i(Transitive;)689
2046 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(<Equivalence>)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 y(3:)g("Q\(x,a\)")d(by)j(sq)g(on)f(auxstep;)602
3990 y(hence)g("Q\(x,y\)")e(by)j(tq)f(on)h(auxstep)e(and)h(3;)515
4189 y(qed;)525 4585 y FT(Figure)30 b(17:)41 b(An)30
b(SPL)g(Pro)s(of)g(of)g FM(nonobv)f FT(using)g(Structured)g
(Justi\014cations)g(without)g FM(!)p FT(.)p 3829 4754
V 378 4758 3453 4 v eop
%%Page: 120 130
120 129 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(120)378 396 y(implemen)m(tation-based)32 b(mec)m(hanism)g(than)h
(that)g(giv)m(en)g(b)m(y)g(the)g(use)g(of)g(a)g(theorem)h(pro)m(ving)e
(al-)378 509 y(gorithm)j(whic)m(h)f(de\014nes)g(a)i(notion)f(of)g(ob)m
(vious)g(inferences)f(\(see)j(section)e(6.1,)j(page)e(100\).)57
b(The)378 622 y(curren)m(t)36 b(de\014nition)f(of)h(the)h(seman)m(tics)
g(of)g(structured)e(justi\014cations)g(do)s(es)i(not)g(dep)s(end)e(on)h
(an)378 735 y(algorithm)23 b(for)h(c)m(hec)m(king)h(the)f
(justi\014cations.)37 b(Instead,)25 b(the)f(seman)m(tics)h(is)e(giv)m
(en)h(in)e(terms)i(of)g(triv-)378 848 y(ial)33 b(manipulations)e(on)j
(\014rst-order)f(form)m(ulae,)i(and)e(in)g(terms)h(of)g(three)g(quite)g
(simple)e(inference)378 961 y(rules.)66 b(F)-8 b(urthermore,)42
b(the)d(mec)m(hanism)g(for)g(restricting)f(the)h(searc)m(h)h(space)g
(during)d(the)j(pro)s(of)378 1074 y(c)m(hec)m(king)g(of)f(structured)f
(justi\014cations)f(do)s(es)i(not)g(dep)s(end)e(on)i(the)g(pro)s(of)f
(calculus)f(or)i(searc)m(h)378 1187 y(strategy)30 b(used.)40
b(These)29 b(remarks)g(therefore)g(suggest)h(that)g(the)f(de\014nition)
e(of)i(structured)f(justi\014-)378 1300 y(cations)33
b(is)g(indep)s(enden)m(t)e(of)i(the)h(algorithm)e(used)g(in)g(c)m(hec)m
(king)i(them.)50 b(Ho)m(w)m(ev)m(er,)36 b(the)d(problem)378
1413 y(of)j(c)m(hec)m(king)h(the)f(v)-5 b(alidit)m(y)35
b(of)h(structured)f(justi\014cations)f(is)h(undecidable)f(and)h(th)m
(us)h(one)g(needs)378 1526 y(to)i(imp)s(ose)f(implemen)m(tation-based)f
(b)s(ounds)f(on)j(the)f(searc)m(h)i(space)f(considered)e(during)g(pro)s
(of)378 1638 y(c)m(hec)m(king.)k(Because)25 b(of)g(this,)g(the)g(seman)
m(tics)f(of)h(structured)e(justi\014cations)g(that)j(can)e(b)s(e)g(mac)
m(hine)378 1751 y(c)m(hec)m(k)m(ed)32 b(in)d(practice)i(is)e(not)i(en)m
(tirely)f(implemen)m(tation)f(indep)s(enden)m(t.)p eop
%%Page: 121 131
121 130 bop 378 1019 a FJ(Chapter)65 b(7)378 1434 y FR(A)77
b(Coloured)g(First-Order)g(Logic)378 1916 y FH(7.1)135
b(In)l(tro)t(duction)378 2119 y FT(This)35 b(c)m(hapter)i(giv)m(es)h
(the)f(de\014nition)d(of)j(a)g(pure)f(\014rst-order)g(logic)h(in)e
(whic)m(h)h(form)m(ulae)g(are)i(an-)378 2232 y(notated)43
b(with)e(colours.)76 b(The)41 b(annotations)h(are)h(used)e(to)i
(restrict)f(the)h(searc)m(h)f(space)h(during)378 2345
y(automated)28 b(theorem)f(pro)m(ving.)39 b(The)26 b(de\014nitions)e
(and)i(results)f(giv)m(en)i(here)g(are)g(used)e(in)h(the)h(next)378
2457 y(c)m(hapter)h(to)g(sho)m(w)f(ho)m(w)g(the)h(inferences)e(stated)i
(explicitly)d(in)h(structured)g(straigh)m(tforw)m(ard)h(justi-)378
2570 y(\014cations)f(\(c)m(hapter)i(6\))g(can)f(b)s(e)f(used)g(to)h
(reduce)g(the)g(e\013ort)h(required)d(during)f(the)j(pro)s(of)f(c)m
(hec)m(king)378 2683 y(pro)s(cess)k(of)g(suc)m(h)g(justi\014cations.)
519 2796 y(The)i(pro)s(cess)g(of)h(automating)f(the)h(disco)m(v)m(ery)g
(of)f(a)h(pro)s(of)f(of)g(a)h(\014rst-order)f(sen)m(tence,)i(whic)m(h)
378 2909 y(can)41 b(b)s(e)f(called)g(the)h(conclusion)e(or)i(goal,)j
(from)d(a)g(n)m(um)m(b)s(er)e(of)i(assumptions,)h(or)f(h)m(yp)s
(otheses,)378 3022 y(usually)26 b(in)m(v)m(olv)m(es)i(the)h(refutation)
f(of)g(the)g(set)h(of)g(sen)m(tences)g(consisting)e(of)h(the)h
(assumptions)d(and)378 3135 y(the)i(negation)g(of)g(the)h(goal.)40
b(The)27 b(refutation)h(is)f(done)h(b)m(y)f(sho)m(wing)g(the)i
(inconsistency)d(of)i(the)g(set)378 3248 y(of)k(sen)m(tences,)i(that)f
(is,)f(sho)m(wing)f(that)h(one)h(can)f(deriv)m(e)g(falsit)m(y)f(\()p
FN(?)p FT(\))i(or)f(an)g(inconsisten)m(t)f(pair)g(of)378
3361 y(sen)m(tences)k FP(X)41 b FT(and)33 b FN(:)p FP(X)7
b FT(.)52 b(In)33 b(general,)i(one)f(can)g(restrict)g(the)g
(refutational)f(pro)s(cess)g(to)i(consider)378 3474 y(only)k(the)h
(literals)e(of)i(a)g(giv)m(en)g(set)h(of)f(sen)m(tences.)70
b(This)38 b(can)i(b)s(e)f(seen)h(for)g(instance)f(from)h(the)378
3587 y(de\014nition)28 b(of)j(a)f(consistency)h(prop)s(ert)m(y)e(giv)m
(en)i(in)e(\(Fitting)h(1996\))i(and)e(sho)m(wn)g(here:)378
3799 y FQ(De\014nition)35 b(7.1)h(\(First-Order)d(Consistency)j(Prop)s
(ert)m(y\))46 b FT(Let)33 b FN(C)k FT(b)s(e)32 b(a)h(collection)g(of)f
(sets)378 3912 y(of)f(\014rst-order)g(sen)m(tences.)45
b(It)31 b(is)f(called)h(a)h(consistency)f(prop)s(ert)m(y)g(with)f(resp)
s(ect)h(to)h(a)g(\014rst-order)378 4025 y(language)f
FP(L)p FT(,)f(if)g(for)g(ev)m(ery)h(set)g FP(S)f FN(2)25
b(C)5 b FT(:)489 4213 y(1.)46 b(F)-8 b(or)31 b(ev)m(ery)h(literal)c
FP(A)j FT(in)e FP(L)p FT(,)h(not)h(b)s(oth)f FP(A)g FT(and)g
FN(:)p FP(A)g FT(are)h(in)e FP(S)5 b FT(.)489 4400 y(2.)46
b(The)30 b(literal)f FN(?)35 b FP(=)-55 b FN(2)25 b FP(S)5
b FT(.)489 4588 y(3.)46 b(If)30 b FP(')21 b FN(^)f FP( )28
b FN(2)d FP(S)35 b FT(then)c FP(S)25 b FN([)20 b(f)p
FP(';)15 b( )s FN(g)27 b(2)e(C)5 b FT(.)489 4775 y(4.)46
b(If)30 b FP(')21 b FN(_)f FP( )28 b FN(2)d FP(S)35 b
FT(then)c FP(S)25 b FN([)20 b(f)p FP(')p FN(g)26 b(2)f(C)36
b FT(or)30 b FP(S)25 b FN([)20 b(f)p FP( )s FN(g)27 b(2)d(C)5
b FT(.)489 4963 y(5.)46 b(If)30 b FN(8)p FP(x:')c FN(2)e
FP(S)36 b FT(then)30 b FP(S)25 b FN([)20 b(f)p FP(')p
FN(f)p FP(x)26 b FN(!)g FP(t)p FN(gg)g(2)e(C)36 b FT(for)30
b(ev)m(ery)h(closed)f(term)h FP(t)f FT(of)g FP(L)p FT(.)489
5151 y(6.)46 b(If)24 b FN(9)p FP(x:')h FN(2)g FP(S)k
FT(then)24 b FP(S)13 b FN([)8 b(f)p FP(')p FN(f)p FP(x)25
b FN(!)g FP(p)p FN(gg)h(2)f(C)k FT(for)24 b(some)g(parameter)h
FP(p)f FT(of)g FP(L)3091 5165 y FE(P)-5 b(AR)3269 5151
y FT(\(the)24 b(de\014nition)605 5264 y(of)31 b FP(L)771
5278 y FE(P)-5 b(AR)955 5264 y FT(and)30 b(parameters)g(is)g(giv)m(en)g
(in)f(section)i(1.2.1\).)1171 b Ff(\003)378 5476 y FT(Note)41
b(that)g(in)e(the)h(\014rst)f(condition)g(in)g(the)h(ab)s(o)m(v)m(e)i
(de\014nition,)e(the)g(form)m(ulae)g FP(A)g FT(and)g
FN(:)p FP(A)f FT(are)378 5589 y(literals.)82 b(It)45
b(is)f(also)g(sho)m(wn)g(that)i(a)f(set)g(of)g(sen)m(tences)g(is)f
(satis\014ed)g(in)f(some)i(mo)s(del)f(if)g(it)g(is)378
5702 y(consisten)m(t.)d(This)29 b(result)g(is)g(giv)m(en)i(b)m(y)f(the)
h(mo)s(del)e(existence)i(theorem:)2035 5954 y(121)p eop
%%Page: 122 132
122 131 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(122)378 396 y FQ(Theorem)34
b(7.1)h(\(Mo)s(del)g(Existence)g(Theorem\))45 b FI(If)33
b FN(C)40 b FI(is)34 b(a)g(c)-5 b(onsistency)35 b(pr)-5
b(op)g(erty)37 b(with)e(r)-5 b(e-)378 509 y(sp)g(e)g(ct)41
b(to)f(a)g(\014rst-or)-5 b(der)41 b(language)f FP(L)p
FI(,)h(and)g FP(S)i FN(2)37 b(C)45 b FI(then)40 b FP(S)k
FI(is)c(satis\014able)h(\(in)e(some)i(Herbr)-5 b(and)378
622 y(mo)g(del)34 b(for)f FP(L)845 636 y FE(P)-5 b(AR)999
622 y FI(\).)378 835 y FQ(Pro)s(of)p FT(:)31 b(see)g(for)g(instance)f
(\(Fitting)g(1996\).)1847 b Ff(\004)519 1061 y FT(In)m(tuitiv)m(ely)-8
b(,)28 b(a)g(set)h(of)g(sen)m(tences)g(can)f(b)s(e)g(sho)m(wn)f(to)i(b)
s(e)f(satis\014able)f(b)m(y)h(c)m(hec)m(king)h(that)g(all)e(the)378
1174 y(sets)35 b(of)g(literals)f(whic)m(h)f(can)i(b)s(e)g(deriv)m(ed)f
(from)g(it)h(are)g(consisten)m(t.)55 b(Con)m(v)m(ersely)-8
b(,)36 b(a)g(refutational)378 1286 y(pro)s(cess)21 b(c)m(hec)m(ks)i
(that)f(an)g(inconsisten)m(t)e(set)i(of)g(literals)e(can)i(b)s(e)f
(deriv)m(ed)f(from)h(the)h(giv)m(en)g(sen)m(tences.)519
1399 y(In)h(this)g(c)m(hapter)i(w)m(e)f(giv)m(e)g(a)h(mec)m(hanism)e
(for)h(restricting)e(the)j(refutational)e(pro)s(cess)g(b)m(y)h(c)m(hec)
m(k-)378 1512 y(ing)31 b(the)h(inconsistency)e(of)i(certain)g(literals)
e(only)-8 b(.)44 b(This)30 b(is)h(done)h(b)m(y)f(annotating)h(the)g
(literals)e(in)378 1625 y(a)40 b(giv)m(en)f(set)h(of)g(sen)m(tences)h
(with)d(colours)h(and)g(allo)m(wing)f(only)g(pairs)g(of)i(literals)e
(of)i(particular)378 1738 y(colours)e(to)h(b)s(e)e(considered)g
(inconsisten)m(t.)64 b(The)37 b(restriction)h(is)f(giv)m(en)h(through)g
(the)g(de\014nition)378 1851 y(of)d(a)h(connectabilit)m(y)e(relation)g
(b)s(et)m(w)m(een)i(colours:)50 b(t)m(w)m(o)36 b(literals)d(are)j(allo)
m(w)m(ed)f(to)h(b)s(e)e(considered)378 1964 y(inconsisten)m(t)h(if)h
(and)g(only)f(if)h(they)g(are)h(complemen)m(tary)g(and)e(their)h
(colours)g(relate)h(with)e(eac)m(h)378 2077 y(other)c(according)f(to)h
(the)g(connectabilit)m(y)e(relation.)519 2190 y(This)37
b(mec)m(hanism)g(can)i(b)s(e)e(used)h(to)h(restrict)f(the)g(w)m(a)m(y)h
(the)g(giv)m(en)f(sen)m(tences)i(can)e(b)s(e)g(used)378
2303 y(during)29 b(theorem)j(pro)m(ving.)42 b(This)30
b(results)g(in)g(a)h(more)h(e\016cien)m(t)f(pro)s(of)g(c)m(hec)m(king)h
(pro)s(cess)f(since)f(a)378 2416 y(smaller)22 b(searc)m(h)j(space)f(is)
f(considered.)38 b(F)-8 b(or)24 b(instance,)h(let)f(us)f(consider)g
(the)h(pro)s(of)f(of)i(the)f(sen)m(tence)378 2528 y FP(X)40
b FT(from)33 b(the)h(assumptions)d FP(Y)50 b FN(\))30
b FP(X)40 b FT(and)33 b FP(Y)53 b FT(using)31 b(the)j(connection)f
(metho)s(d)g(\(Andrews)f(1981;)378 2641 y(Bib)s(el)d(1981\))j(\(see)f
(also)g(section)f(2.3.1\).)43 b(This)29 b(in)m(v)m(olv)m(es)h(the)h
(refutation)f(of)g(the)h(three)f(clauses)1655 2846 y
FP(Y)201 b FN(:)p FP(Y)40 b FN(_)20 b FP(X)189 b FN(:)p
FP(X)378 3050 y FT(b)m(y)30 b(the)h(follo)m(wing)e(matrix:)1797
3083 y Fx(\024)1845 3155 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 8.79042 4.3952 3.74124
} false /N@a1 16 {InitRnode } NewNode end end
1845 3155 a FP(Y)2001 3155
y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 16.09044 8.04521 3.74124
} false /N@a2 16 {InitRnode } 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 grestore end
2409 3211 a 378 3427 a
FT(The)j(literals)e(in)h(the)h(clause)g(corresp)s(onding)e(to)j(the)f
(implication)e FP(Y)48 b FN(\))28 b FP(X)40 b FT(are)32
b(connected)h(with)378 3540 y(the)24 b(literals)d(in)h(the)i(other)g
(clauses,)g FN(:)p FP(X)31 b FT(and)23 b FP(Y)d FT(,)25
b(suc)m(h)e(that)h(ev)m(ery)g(path)f(in)f(the)i(ab)s(o)m(v)m(e)g
(matrix)f(has)378 3653 y(a)31 b(connection.)42 b(In)30
b(general,)h(the)g(matrix)f(pro)s(of)g(of)h(some)g(goal)g(from)f(t)m(w)
m(o)i(h)m(yp)s(otheses)f(using)e(the)378 3766 y(elimination)e(of)j
(implication)e(has)h(the)h(ab)s(o)m(v)m(e)h(form:)40
b(the)30 b(literals)f(in)f(the)i(clauses)g(corresp)s(onding)378
3879 y(to)39 b(the)f(implication)e(connect)j(with)e(the)h(literals)e
(of)j(the)f(other)g(clauses.)64 b(Therefore)38 b(if)f(w)m(e)h(are)378
3992 y(giv)m(en)33 b(the)h(information)d(that)j(a)g(conclusion)d
FP(C)40 b FT(can)33 b(b)s(e)g(deriv)m(ed)f(from)h(t)m(w)m(o)i(sen)m
(tences)f FP(I)40 b FT(and)33 b FP(J)378 4105 y FT(b)m(y)i(the)h
(elimination)d(of)j(implication)d(in)h FP(I)43 b FT(b)m(y)35
b FP(J)9 b FT(,)37 b(then)e(one)h(can)g(restrict)f(the)h(pro)s(of)f
(searc)m(h)h(to)378 4217 y(only)i(lo)s(ok)h(for)g(connections)g(b)s(et)
m(w)m(een)h(the)f(literals)f(in)g(the)h(clauses)g(of)g
FP(I)46 b FT(with)38 b(the)i(literals)d(in)378 4330 y(the)d(clauses)g
(of)g FN(:)p FP(C)40 b FT(and)33 b FP(J)9 b FT(.)52 b(Note)35
b(that)f(in)f(general,)i(there)f(ma)m(y)h(b)s(e)e(literals)f(in)h(the)h
(clauses)g(of)378 4443 y FN(:)p FP(C)43 b FT(whic)m(h)35
b(can)i(b)s(e)f(connected)h(with)e(the)i(literals)e(in)g(the)i(clauses)
f(if)g FP(J)9 b FT(.)59 b(By)37 b(using)e(the)i(ab)s(o)m(v)m(e)378
4556 y(men)m(tioned)27 b(restriction,)h(suc)m(h)f(connections)h(are)g
(ignored)f(during)e(pro)s(of)i(searc)m(h)h(and)f(therefore)h(a)378
4669 y(smaller)h(searc)m(h)i(space)g(is)e(considered.)519
4782 y(The)h(particular)f(restriction)g(on)h(the)h(pro)s(of)e(searc)m
(h)i(men)m(tioned)f(in)f(the)i(previous)d(paragraph)378
4895 y(can)35 b(b)s(e)e(done)i(b)m(y)f(annotating)h(the)f(literals)f
(in)g FP(I)7 b FT(,)36 b FP(C)41 b FT(and)34 b FP(J)43
b FT(with)33 b(the)i(colours)f Fw(red)o FT(,)h Fw(green)d
FT(and)378 5008 y Fw(blue)g FT(resp)s(ectiv)m(ely)-8
b(,)34 b(sa)m(y)-8 b(,)36 b(and)d(allo)m(wing)f(the)i(connection)g(of)g
Fw(red)e FT(literals)g(with)g Fw(green)g FT(and)h Fw(blue)378
5121 y FT(ones)44 b(only)-8 b(.)82 b(In)44 b(general,)k(the)c
(inferences)g(stated)h(explicitly)d(in)h(structured)g(straigh)m(tforw)m
(ard)378 5234 y(justi\014cations)30 b(can)h(b)s(e)g(used)f(to)i
(restrict)f(the)h(searc)m(h)g(space)f(considered)g(during)e(pro)s(of)h
(c)m(hec)m(king)378 5347 y(using)f(the)j(colouring)d(mec)m(hanism)i
(describ)s(ed)d(in)i(this)g(c)m(hapter.)43 b(This)29
b(restriction)h(is)g(illustrated)378 5459 y(in)f(c)m(hapter)i(8.)519
5572 y(In)37 b(the)h(next)f(section)h(w)m(e)g(in)m(tro)s(duce)f(the)h
(basic)f(de\014nitions)e(of)i(the)h(\014rst-order)f(logic)g(with)378
5685 y(coloured)44 b(form)m(ulae.)82 b(In)43 b(section)i(7.3)g(w)m(e)g
(sho)m(w)f(ho)m(w)g(a)h(set)f(of)h(coloured)f(sen)m(tences)h(can)g(b)s
(e)p eop
%%Page: 123 133
123 132 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(123)378 396 y(mapp)s(ed)42
b(in)m(to)g(an)h(equiv)-5 b(alen)m(t)43 b(set)g(of)g(uncoloured)f(sen)m
(tences,)47 b(and)c(in)e(section)j(7.4)g(w)m(e)f(sho)m(w)378
509 y(ho)m(w)38 b(certain)h(recolourings)e(of)i(the)f(form)m(ulae)g
(preserv)m(e)h(the)g(consistency)f(or)h(inconsistency)e(of)378
622 y(coloured)31 b(sen)m(tences.)45 b(An)31 b(in)m(terp)s(olation)f
(theorem)i(for)f(the)h(coloured)f(\014rst-order)f(logic)h(is)g(giv)m
(en)378 735 y(in)f(section)i(7.5,)i(and)d(an)g(undecidabilit)m(y)d
(result)j(is)g(giv)m(en)g(in)g(section)h(7.6.)45 b(A)32
b(brief)e(summary)h(of)378 848 y(this)e(c)m(hapter)i(is)f(giv)m(en)g
(in)f(section)i(7.7.)378 1133 y FH(7.2)135 b(A)45 b(First-Order)g
(Logic)g(with)g(Coloured)h(F)-11 b(orm)l(ulae)378 1339
y FG(7.2.1)112 b(Basic)37 b(De\014nitions)378 1511 y
FT(In)22 b(this)g(section)i(w)m(e)f(in)m(tro)s(duce)f(a)i(set)f(of)h
(colours)e FN(P)7 b FT(,)26 b(and)c(a)i(\014rst-order)e(language)h
(whose)g(form)m(ulae)378 1624 y(are)33 b(coloured)f(with)g
FN(P)7 b FT(.)48 b(A)m(tomic)34 b(form)m(ulae)e(can)h(b)s(e)f(asso)s
(ciated)h(with)f(only)f(one)i(colour,)h(so)f(it)f(is)378
1737 y(enough)h(to)h(annotate)h(only)d(the)i(predicate)f(sym)m(b)s(ols)
e(with)h(colours)h(since)g(there)g(is)g(exactly)h(one)378
1849 y(predicate)c(sym)m(b)s(ol)f(in)g(ev)m(ery)i(atom.)519
1962 y(An)g(atomic)i(form)m(ula)e(in)f(this)g(language)j(is)d(a)i(pair)
f(consisting)f(of)i(an)g(uncoloured)e(atom)i(and)378
2075 y(a)f(colour.)378 2277 y FQ(De\014nition)k(7.2)h(\(P)m(alette\))44
b FT(A)30 b(palette)h(is)f(a)h(coun)m(table)f(set)h(of)g(colours.)740
b Ff(\003)378 2479 y FT(Note)42 b(that)e(in)f(general,)k(the)e(role)f
(of)g(the)g(colours)g(is)f(to)i(restrict)f(the)h(searc)m(h)g(space)g
(during)c(a)378 2592 y(refutational)c(pro)s(of,)h(and)f(therefore)h
(only)e(a)i(\014nite)f(set)h(of)f(colours)g(is)g(considered)f(during)g
(pro)s(of)378 2705 y(searc)m(h)h(\(since)f(pro)s(ofs)g(are)h
(\014nite\).)46 b(Ho)m(w)m(ev)m(er,)35 b(w)m(e)e(need)f(a)h(palette)g
(to)g(b)s(e)f(coun)m(tably)g(in\014nite)e(in)378 2818
y(certain)j(cases)g(where,)g(for)g(instance,)g(w)m(e)h(need)e(the)h
(existence)g(of)g(some)g(new)g(colour)f FP(j)38 b FT(whic)m(h)32
b(is)378 2930 y(not)39 b(in)f(some)i(giv)m(en)f(palette)h
FN(P)47 b FT(\(for)40 b(example,)h(in)d(de\014nition)f(8.4)j(in)e(the)i
(next)f(c)m(hapter\).)69 b(In)378 3043 y(this)30 b(c)m(hapter)h(and)g
(in)e(c)m(hapter)j(8,)g(all)e(sets)h(of)g(colours)g(are)g(in\014nite)e
(unless)g(otherwise)h(stated.)44 b(A)378 3156 y(coloured)30
b(\014rst-order)f(language)i(is)f(no)m(w)g(de\014ned)f(as)i(follo)m
(ws.)378 3358 y FQ(De\014nition)k(7.3)h(\(Coloured)e(Language\))46
b FT(Let)27 b FN(P)35 b FT(b)s(e)26 b(a)i(palette,)g(a)g(coloured)e
(\014rst-order)g(lan-)378 3471 y(guage)e(is)e(a)i(\014rst-order)e
(language)h FP(L)p FT(\(\006)1733 3438 y FK(P)1733 3498
y FO(R)1792 3471 y FP(;)15 b FT(\006)1898 3485 y FO(F)1957
3471 y FP(;)g(X)7 b FT(\))24 b(where)e(\006)2459 3485
y FO(R)2540 3471 y FT(is)g(a)h(collection)g(of)g(relation)f(sym)m(b)s
(ols)378 3584 y(with)34 b(\014xed)g(arities,)h(\006)1184
3598 y FO(F)1277 3584 y FT(is)f(a)i(collection)e(of)h(function)f(sym)m
(b)s(ols)f(with)h(\014xed)g(arities,)i FP(X)42 b FT(is)34
b(a)h(set)378 3697 y(of)g(v)-5 b(ariables,)36 b(and)e(\006)1142
3664 y FK(P)1142 3724 y FO(R)1236 3697 y FT(is)g(the)i(collection)e(of)
i(relation)e(sym)m(b)s(ols)g(of)h(the)h(form)e(\()p FP(P)s(;)15
b(i)p FT(\))37 b(with)d(arit)m(y)378 3810 y FP(n)p FT(,)j(where)f
FP(P)49 b FT(is)35 b(in)g(\006)1146 3824 y FO(R)1203
3810 y FT(,)j(the)e(colour)g FP(i)g FT(is)f(in)g FN(P)7
b FT(,)38 b(and)e FP(n)g FT(is)f(the)h(arit)m(y)g(of)g
FP(P)13 b FT(.)58 b(F)-8 b(or)37 b(simplicit)m(y)c(w)m(e)378
3923 y(will)23 b(refer)j(to)h(the)f(language)h FP(L)p
FT(\(\006)1548 3890 y FK(P)1548 3950 y FO(R)1607 3923
y FP(;)15 b FT(\006)1713 3937 y FO(F)1772 3923 y FP(;)g(X)7
b FT(\))27 b(b)m(y)f FP(L)2140 3890 y FK(P)2199 3923
y FT(.)39 b(W)-8 b(e)27 b(represen)m(t)g(a)f(coloured)g(predicate)g(\()
p FP(P)s(;)15 b(i)p FT(\))378 4036 y(with)29 b FP(P)656
4003 y FO(i)684 4036 y FT(.)3048 b Ff(\003)519 4237 y
FT(F)-8 b(or)29 b(simplicit)m(y)-8 b(,)27 b(w)m(e)i(assume)f(that)h
(all)f(form)m(ulae)g(are)g(in)g(negation)g(normal)g(form)g(\(NNF\))h
(and)378 4350 y(that)39 b(expressions)e(suc)m(h)g(as)i
FP(A)f FN(\))h FP(B)j FT(and)c FN(:)p FT(\()p FP(A)g
FN(\))g FP(B)5 b FT(\))38 b(are)h(syn)m(tactic)g(sugarings)e(for)h
FN(:)p FP(A)25 b FN(_)g FP(B)378 4463 y FT(and)31 b FP(A)21
b FN(_)g(:)p FP(B)5 b FT(.)43 b(The)31 b(set)h(of)g(relation)f(sym)m(b)
s(ols,)f FP(R)q FT(,)i(con)m(tains)g(the)g(n)m(ullary)d(predicate)j
FN(>)f FT(and)g(the)378 4576 y(literals)26 b FN(>)p FT(\(\))i(and)f
FN(:>)p FT(\(\))h(are)g(denoted)g(b)m(y)g FN(>)f FT(and)g
FN(?)h FT(resp)s(ectiv)m(ely)-8 b(.)40 b(It)27 b(should)f(b)s(e)h
(noted)h(that)h(the)378 4689 y(language)34 b FP(L)822
4656 y FK(P)915 4689 y FT(do)s(es)g(not)h(con)m(tain)f(the)g(literals)f
FN(>)h FT(and)f FN(?)p FT(,)i(but)e(rather)h(literals)f(of)h(the)g
(form)g FN(>)3800 4656 y FO(i)378 4802 y FT(and)c FN(?)626
4769 y FO(i)684 4802 y FT(for)g FP(i)c FN(2)e(P)7 b FT(.)519
4915 y(W)-8 b(e)24 b(also)f(giv)m(e)h(the)f(de\014nition)e(of)i(a)h
(connectabilit)m(y)f(relation)f(b)s(et)m(w)m(een)i(colours.)37
b(This)22 b(relation)378 5028 y(determines)42 b(whic)m(h)g(complemen)m
(tary)i(literals)e(are)i(allo)m(w)m(ed)f(to)h(b)s(e)f(regarded)g(as)h
(inconsisten)m(t)378 5141 y(during)31 b(a)i(refutational)f(pro)s(of)g
(searc)m(h:)46 b(a)34 b(complemen)m(tary)f(pair)e(of)i(literals)f(will)
e(b)s(e)i(considered)378 5253 y(to)d(b)s(e)f(inconsisten)m(t)f(if)g
(their)g(colours)h(relate)h(with)e(eac)m(h)i(other)g(according)f(to)h
(the)f(connectabilit)m(y)378 5366 y(relation)37 b(considered.)62
b(Since)37 b(the)h(complemen)m(tary)h(relation)e(o)m(v)m(er)i(literals)
d(is)h(symmetric,)j(the)378 5479 y(connectabilit)m(y)31
b(relation)f(is)g(required)f(to)j(b)s(e)f(symmetric)f(as)h(w)m(ell.)42
b(W)-8 b(e)32 b(can)g(also)f(assume)g(that)h(a)378 5592
y(connectabilit)m(y)24 b(relation)f(is)g(\014nite)g(since)g(only)h(a)g
(\014nite)f(n)m(um)m(b)s(er)g(of)h(sen)m(tences)h(\(and)f(th)m(us)g
(colours\))378 5705 y(are)31 b(used)e(in)g(an)m(y)i(particular)e(pro)s
(of.)p eop
%%Page: 124 134
124 133 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(124)378 396 y FQ(De\014nition)35
b(7.4)h(\(Connectabilit)m(y)e(Relation\))45 b FT(A)26
b(connectabilit)m(y)f(relation)f(is)h(a)h(\014nite)e(sym-)378
509 y(metric)33 b(relation)g(o)m(v)m(er)i(a)g(set)f(of)g(colours.)50
b(If)33 b FN(K)j FT(is)c(a)j(connectabilit)m(y)e(relation)g(and)g
FP(i)p FT(,)i FP(j)k FT(are)c(t)m(w)m(o)378 622 y(colours,)27
b(then)f(w)m(e)h(use)f(the)g(notation)h FP(i)e FN(\030)1834
636 y FK(K)1917 622 y FP(j)32 b FT(to)27 b(denote)g(the)g(fact)g(that)g
FP(i)f FT(and)g FP(j)32 b FT(relate)27 b(with)e(eac)m(h)378
735 y(other)31 b(in)e FN(K)j FT(\(i.e.,)15 b(\()p FP(i;)g(j)5
b FT(\))28 b FN(2)d(K)q FT(\).)2316 b Ff(\003)519 943
y FT(A)34 b(connectabilit)m(y)f(relation)g(is)g(usually)e(sp)s
(eci\014ed)h(using)g(the)i(follo)m(wing)e(de\014nition)g(and)h(no-)378
1056 y(tation.)378 1264 y FQ(De\014nition)i(7.5)h(\(F)-9
b(ull)34 b(Connection\))46 b FT(Giv)m(en)f(t)m(w)m(o)i(\014nite)e(sets)
h(of)f(colours)g FN(P)3299 1278 y FL(1)3384 1264 y FT(and)g
FN(P)3639 1278 y FL(2)3724 1264 y FT(w)m(e)378 1377 y(de\014ne)22
b(the)h(full)d(connection)j(b)s(et)m(w)m(een)g FN(P)1786
1391 y FL(1)1848 1377 y FT(and)f FN(P)2080 1391 y FL(2)2120
1377 y FT(,)i(denoted)f(b)m(y)g FN(P)2687 1391 y FL(1)2752
1377 y FN($)i(P)2931 1391 y FL(2)2970 1377 y FT(,)g(as)e(the)f
(connectabilit)m(y)378 1490 y(relation)30 b(in)f(whic)m(h)g(all)g(the)i
(colours)e(in)g FN(P)1838 1504 y FL(1)1908 1490 y FT(relate)i(with)e
(all)h(the)g(colours)g(in)f FN(P)3129 1504 y FL(2)3199
1490 y FT(and)h(vice-v)m(ersa:)1419 1694 y FN(P)1482
1708 y FL(1)1547 1694 y FN($)25 b(P)1726 1708 y FL(2)1791
1694 y FT(=)g(\()p FN(P)1985 1708 y FL(1)2045 1694 y
FN(\002)20 b(P)2199 1708 y FL(2)2238 1694 y FT(\))h FN([)f
FT(\()p FN(P)2473 1708 y FL(2)2533 1694 y FN(\002)g(P)2687
1708 y FL(1)2727 1694 y FT(\))p FP(:)378 1898 y FT(F)-8
b(or)26 b(simplicit)m(y)d(w)m(e)j(denote)g FN(f)p FP(i)p
FN(g)g($)f(P)7 b FT(,)28 b FN(f)p FP(i)p FN(g)e($)f(f)p
FP(j)5 b FN(g)p FT(,)29 b(and)c FN(P)32 b($)26 b(f)p
FP(i)p FN(g)g FT(b)m(y)g FP(i)f FN($)g(P)7 b FT(,)28
b FP(i)d FN($)g FP(j)5 b FT(,)28 b(and)d FN(P)32 b($)26
b FP(i)378 2011 y FT(resp)s(ectiv)m(ely)-8 b(,)29 b(where)g
FN(P)37 b FT(is)28 b(some)h(palette)h(and)e FP(i)i FT(and)e
FP(j)35 b FT(are)29 b(colours.)40 b(W)-8 b(e)31 b(also)e(use)f(the)i
(notation)378 2124 y FN(P)441 2138 y FL(1)506 2124 y
FN($)25 b(P)685 2138 y FL(2)750 2124 y FN($)g(P)929 2138
y FL(3)994 2124 y FN($)g(\001)15 b(\001)g(\001)27 b($)e(P)1421
2138 y FO(n)p FK(\000)p FL(1)1583 2124 y FN($)g(P)1762
2138 y FO(n)1840 2124 y FT(to)31 b(represen)m(t)g(the)f(relation)1313
2344 y FN(P)1376 2358 y FL(1)1441 2344 y FN($)25 b(P)1620
2358 y FL(2)1680 2344 y FN([)20 b(P)1824 2358 y FL(2)1889
2344 y FN($)25 b(P)2068 2358 y FL(3)2128 2344 y FN([)20
b(\001)15 b(\001)g(\001)21 b([)f(P)2479 2358 y FO(n)p
FK(\000)p FL(1)2641 2344 y FN($)25 b(P)2820 2358 y FO(n)2868
2344 y FP(:)864 b Ff(\003)519 2552 y FT(The)30 b(atoms)i(of)e(an)h
(uncoloured)e(\014rst-order)h(form)m(ula)g(can)g(b)s(e)g(annotated)i
(with)d(some)i(colour)378 2665 y(using)e(the)h(follo)m(wing)f(mapping.)
378 2873 y FQ(De\014nition)35 b(7.6)h(\(Colouring)f(F)-9
b(orm)m(ulae\))44 b FT(Giv)m(en)32 b(a)g(colour)f FP(i)d
FN(2)e(P)40 b FT(and)31 b(a)h(form)m(ula)e(\011)i(in)e(an)378
2986 y(uncoloured)f(\014rst-order)g(language)i FP(L)p
FT(,)g(w)m(e)f(de\014ne)g(the)h(form)m(ula)e(\011)2722
2953 y FO(i)2780 2986 y FT(in)g FP(L)2948 2953 y FK(P)3038
2986 y FT(as)h(follo)m(ws:)1410 3190 y(\()p FP(P)13 b
FT(\()p FP(t)1584 3204 y FL(1)1624 3190 y FP(;)i(:)g(:)g(:)32
b(;)15 b(t)1874 3204 y FO(n)1921 3190 y FT(\)\))1991
3152 y FO(i)2103 3190 y FT(=)83 b FP(P)2328 3152 y FO(i)2356
3190 y FT(\()p FP(t)2424 3204 y FL(1)2463 3190 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(t)2713 3204 y FO(n)2760 3190
y FT(\))1801 3328 y(\()p FN(:)p FP(')p FT(\))1991 3290
y FO(i)2103 3328 y FT(=)83 b FN(:)p FT(\()p FP(')2412
3290 y FO(i)2440 3328 y FT(\))1698 3465 y(\()p FP(')21
b FN(^)f FP( )s FT(\))1991 3428 y FO(i)2103 3465 y FT(=)83
b(\()p FP(')2351 3428 y FO(i)2380 3465 y FT(\))20 b FN(^)g
FT(\()p FP( )2613 3428 y FO(i)2642 3465 y FT(\))1698
3603 y(\()p FP(')h FN(_)f FP( )s FT(\))1991 3566 y FO(i)2103
3603 y FT(=)83 b(\()p FP(')2351 3566 y FO(i)2380 3603
y FT(\))20 b FN(_)g FT(\()p FP( )2613 3566 y FO(i)2642
3603 y FT(\))1734 3741 y(\()p FN(8)p FP(x:')p FT(\))1991
3703 y FO(i)2103 3741 y FT(=)83 b FN(8)p FP(x:)p FT(\()p
FP(')2479 3703 y FO(i)2507 3741 y FT(\))1731 3879 y(\()p
FN(9)p FP(x: )s FT(\))1991 3841 y FO(i)2103 3879 y FT(=)g
FN(9)p FP(x:)p FT(\()p FP( )2482 3841 y FO(i)2510 3879
y FT(\))378 4083 y(where)32 b FP(P)44 b FT(is)32 b(a)g(predicate,)h
FP(')f FT(and)f FP( )36 b FT(are)c(form)m(ulae)g(and)f
FP(x)h FT(is)f(a)i(v)-5 b(ariable.)44 b(W)-8 b(e)34 b(will)29
b(refer)j(to)h(the)378 4196 y(set)e FN(f)p FT(\010)631
4163 y FO(i)684 4196 y FN(j)26 b FT(\010)f FN(2)g FP(S)5
b FN(g)30 b FT(b)m(y)h FP(S)1236 4163 y FO(i)1264 4196
y FT(.)2468 b Ff(\003)519 4404 y FT(W)-8 b(e)23 b(also)f(giv)m(e)g(the)
g(follo)m(wing)f(de\014nitions)e(on)j(coloured)f(form)m(ulae,)i(sets)g
(of)f(coloured)f(form)m(ulae,)378 4517 y(and)30 b(connectabilit)m(y)g
(relations.)378 4725 y FQ(De\014nition)35 b(7.7)h(\(Ha)m(ving)e(some)h
(Colour,)g(Homogeneously)g(Coloured\))46 b FT(W)-8 b(e)43
b(sa)m(y)f(that)378 4838 y(a)33 b(form)m(ula)f(\011)c
FN(2)h FP(L)1043 4805 y FK(P)1134 4838 y FT(has)j(colour)g
FP(i)h FT(if)f(there)g(is)g(some)h(\010)28 b FN(2)h FP(L)j
FT(suc)m(h)g(that)h(\011)c(=)g(\010)3231 4805 y FO(i)3258
4838 y FT(.)47 b(A)33 b(form)m(ula)f(is)378 4951 y(homogeneously)e
(coloured)g(if)f(all)h(its)g(literals)e(ha)m(v)m(e)k(the)e(same)h
(colour.)889 b Ff(\003)378 5158 y FT(Note)46 b(that)f(a)f(form)m(ula)g
(\011)g(is)f(homogeneously)i(coloured)f(if)f(and)h(only)f(if)h(\011)k
(=)g(\010)3405 5125 y FO(i)3477 5158 y FT(for)c(some)378
5271 y(uncoloured)29 b(form)m(ula)g(\010)h(and)g(colour)g
FP(i)p FT(.)378 5479 y FQ(De\014nition)35 b(7.8)h(\(Colours)f(in)g(a)f
(F)-9 b(orm)m(ula,)35 b(new)f(to)h(a)f(F)-9 b(orm)m(ula\))45
b FT(A)38 b(colour)f FP(i)h FT(is)f(said)g(to)378 5592
y(b)s(e)29 b(in)g(the)h(coloured)g(form)m(ula)f(\010)h(if)f(there)h(is)
f(some)i(atom)g FP(A)2490 5559 y FO(i)2548 5592 y FT(in)e(\010;)h(it)g
(is)f(in)g(the)h(set)h FP(S)k FT(if)29 b(there)h(is)378
5705 y(some)d(\010)f(in)g FP(S)31 b FT(suc)m(h)c(that)g
FP(i)g FT(is)f(in)f(\010,)i(and)f(it)g(is)g(in)g(the)g(connectabilit)m
(y)h(relation)f FN(K)i FT(o)m(v)m(er)g FN(P)34 b FT(if)26
b(there)p eop
%%Page: 125 135
125 134 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(125)378 396 y(is)32 b(some)h
FP(j)i FN(2)29 b(P)40 b FT(suc)m(h)33 b(that)g FP(i)d
FN(\030)1506 410 y FK(K)1593 396 y FP(j)5 b FT(.)49 b(A)33
b(colour)f(is)g(new)g(to)i(\010)e(if)g(it)g(is)g(not)h(in)f(\010,)h
(and)f(w)m(e)h(denote)378 509 y(the)f(set)g(of)f(colours)g(in)f(\010)i
(b)m(y)f Fe(C)p FT(\(\010\).)44 b(W)-8 b(e)33 b(giv)m(e)f(similar)c
(de\014nitions)h(for)j(`new)f(to)h FP(S)5 b FT(',)32
b(`new)g(to)g FN(K)q FT(',)378 622 y Fe(C)p FT(\()p FP(S)5
b FT(\))31 b(and)e Fe(C)p FT(\()p FN(K)q FT(\).)2764
b Ff(\003)378 806 y FQ(De\014nition)35 b(7.9)h(\(Connectabilit)m(y)e
(Relation)h(on)g(Sets\))45 b FT(Giv)m(en)i(t)m(w)m(o)g(sets)g
FP(S)3335 820 y FL(1)3420 806 y FT(and)f FP(S)3669 820
y FL(2)3755 806 y FT(of)378 919 y(coloured)e(sen)m(tences)h(and)f(a)h
(connectabilit)m(y)f(relation)g FN(K)q FT(,)k(w)m(e)d(sa)m(y)g(that)g
FP(S)3125 933 y FL(1)3209 919 y FT(relates)g(with)e FP(S)3789
933 y FL(2)378 1032 y FT(in)32 b(the)h(extension)g(of)h
FN(K)g FT(to)g(sets,)h(and)e(write)f FP(S)2053 1046 y
FL(1)2122 1032 y FN(\031)2193 1046 y FK(K)2281 1032 y
FP(S)2337 1046 y FL(2)2377 1032 y FT(,)i(if)e(there)h(are)h(some)g
(colours)e FP(i)f FN(2)e Fe(C)p FT(\()p FP(S)3753 1046
y FL(1)3793 1032 y FT(\))378 1145 y(and)h FP(j)h FN(2)24
b Fe(C)p FT(\()p FP(S)855 1159 y FL(2)895 1145 y FT(\))30
b(suc)m(h)g(that)h FP(i)26 b FN(\030)1490 1159 y FK(K)1573
1145 y FP(j)5 b FT(.)2117 b Ff(\003)378 1328 y FQ(De\014nition)35
b(7.10)h(\(Uncoloured)g(Pro)6 b(jection\))46 b FT(Let)37
b FP(B)k FT(=)35 b FP(A)2706 1295 y FO(i)2771 1328 y
FT(b)s(e)i(a)g(coloured)f(literal.)58 b(The)378 1441
y(uncoloured)26 b(pro)5 b(jection)28 b(of)g FP(B)33 b
FT(\(denoted)28 b(b)m(y)g FP(B)2047 1408 y FK(U)2101
1441 y FT(\))h(is)e(the)h(literal)e FP(A)p FT(.)40 b(W)-8
b(e)29 b(also)f(de\014ne)g FP(')3455 1408 y FK(U)3538
1441 y FT(and)f FP(S)3773 1408 y FK(U)378 1554 y FT(as)32
b(the)f(uncoloured)f(coun)m(terparts)i(of)f(a)h(coloured)f(form)m(ula)f
FP( )35 b FT(and)30 b(a)i(set)g(of)f(coloured)g(form)m(ulae)378
1667 y FP(S)k FT(resp)s(ectiv)m(ely)-8 b(.)2805 b Ff(\003)378
1850 y FQ(De\014nition)35 b(7.11)h(\(Range\))45 b FT(Giv)m(en)29
b(a)h(colour)f FP(i)g FT(and)g(a)g(connectabilit)m(y)g(relation)g
FN(K)q FT(,)h(w)m(e)g(de\014ne)378 1963 y(the)h(range)f(of)h
FP(i)f FT(in)f FN(K)j FT(\(and)e(denote)h(it)f(b)m(y)h
FN(K)q FT(\()p FP(i)p FT(\)\))h(as)e(the)h(set)g(of)f(colours)g(in)f
FN(K)j FT(whic)m(h)d(relate)i(to)g FP(i)p FT(:)1712 2185
y FN(K)q FT(\()p FP(i)p FT(\))c(=)e FN(f)p FP(j)31 b
FN(j)26 b FP(i)f FN(\030)2297 2199 y FK(K)2380 2185 y
FP(j)5 b FN(g)p FP(:)1265 b Ff(\003)378 2369 y FQ(De\014nition)35
b(7.12)h(\(Restriction)f(of)g FN(K)i FQ(to)d FP(S)5 b
FQ(\))46 b FT(Giv)m(en)e(the)h(connectabilit)m(y)g(relation)f
FN(K)i FT(and)378 2482 y(set)36 b FP(S)k FT(of)c(coloured)f(form)m
(ulae,)i(the)f(restriction)e(of)i FN(K)h FT(to)f FP(S)5
b FT(,)37 b(also)f(called)e(the)i(subrelation)e(of)h
FN(K)378 2594 y FT(relev)-5 b(an)m(t)31 b(to)g FP(S)5
b FT(,)30 b(is)g(the)g(connectabilit)m(y)g(relation)g
FN(K)q(d)p FP(S)5 b FN(e)31 b FT(de\014ned)f(as)g(follo)m(ws:)1065
2816 y FN(K)q(d)p FP(S)5 b FN(e)57 b FT(=)e FN(f)p FT(\()p
FP(i;)15 b(j)5 b FT(\))28 b FN(j)d FP(i)h FN(\030)1893
2830 y FK(K)1976 2816 y FP(j)36 b FT(and)30 b(b)s(oth)f
FP(i)i FT(and)f FP(j)36 b FT(are)30 b(in)f FP(S)5 b FN(g)p
FP(:)617 b Ff(\003)519 3000 y FT(The)30 b(follo)m(wing)f(example)h
(illustrates)e(the)j(de\014nitions)d(giv)m(en)i(in)f(this)g(section.)
378 3183 y FQ(Example)34 b(7.1)46 b FT(Let)36 b(the)g(palette)g
FN(P)42 b FT(=)34 b FN(f)p FP(i;)15 b(j;)g(k)s(;)g(l)r
FN(g)39 b FT(where)c FP(i)p FT(,)j FP(j)5 b FT(,)38 b
FP(k)h FT(and)c FP(l)j FT(are)e(distinct)e(colours,)378
3296 y(and)c(let)g(us)g(denote)h(the)f(connectabilit)m(y)g(relation)g
FN(f)p FP(i;)15 b(j)5 b FN(g)27 b($)e(f)p FP(k)s(;)15
b(l)r FN(g)32 b FT(b)m(y)e FN(K)2954 3310 y FL(1)2994
3296 y FT(.)41 b(Then)1237 3501 y FP(i)26 b FN(\030)1365
3515 y FK(K)1419 3524 y FC(1)1482 3501 y FP(k)185 b(i)26
b FN(\030)1842 3515 y FK(K)1896 3524 y FC(1)1959 3501
y FP(l)184 b(j)31 b FN(\030)2309 3515 y FK(K)2363 3524
y FC(1)2426 3501 y FP(k)185 b(j)31 b FN(\030)2797 3515
y FK(K)2851 3524 y FC(1)2914 3501 y FP(l)r(:)378 3705
y FT(Also,)f(let)h FN(K)809 3719 y FL(2)874 3705 y FT(=)24
b FP(i)i FN($)f FP(j)31 b FN($)25 b FP(k)s FT(,)31 b(then)1702
3909 y FP(i)26 b FN(\030)1830 3923 y FK(K)1884 3932 y
FC(2)1947 3909 y FP(j)188 b(j)31 b FN(\030)2311 3923
y FK(K)2365 3932 y FC(2)2428 3909 y FP(k)s(:)378 4113
y FT(Then)e FN(K)684 4127 y FL(1)724 4113 y FT(\()p FP(i)p
FT(\))d(=)f FN(f)p FP(k)s(;)15 b(l)r FN(g)32 b FT(and)e
FN(K)1434 4127 y FL(2)1473 4113 y FT(\()p FP(j)5 b FT(\))27
b(=)e FN(f)p FP(i;)15 b(k)s FN(g)p FT(.)42 b(No)m(w)32
b(let)1603 4335 y FP(S)1659 4349 y FL(1)1724 4335 y FT(=)25
b FN(f)p FP(A)1933 4298 y FO(i)1962 4335 y FP(;)15 b
FT(\()p FP(B)25 b FN(^)20 b FP(C)7 b FT(\))2319 4298
y FO(j)2355 4335 y FN(g)p FT(,)31 b(and)1603 4491 y FP(S)1659
4505 y FL(2)1724 4491 y FT(=)25 b FN(f)p FP(A)1933 4453
y FO(i)1962 4491 y FP(;)15 b(B)2076 4453 y FO(j)2132
4491 y FN(^)20 b FP(C)2285 4453 y FO(k)2327 4491 y FN(g)p
FP(;)378 4695 y FT(then)30 b FP(l)i FT(is)e(new)g(to)h
FP(S)1090 4709 y FL(1)1129 4695 y FT(,)f(to)h FP(S)1351
4709 y FL(2)1421 4695 y FT(and)f(to)h FN(K)1778 4709
y FL(2)1848 4695 y FT(but)f(it)f(is)h(in)f FN(K)2372
4709 y FL(1)2412 4695 y FT(.)40 b(W)-8 b(e)32 b(also)e(ha)m(v)m(e)i
(that)1315 4912 y FP(S)1371 4926 y FL(1)1436 4912 y FN(6\031)1507
4926 y FK(K)1561 4935 y FC(1)1624 4912 y FP(S)1680 4926
y FL(1)1901 4912 y FP(S)1957 4926 y FL(1)2021 4912 y
FN(\031)2092 4926 y FK(K)2146 4935 y FC(1)2210 4912 y
FP(S)2266 4926 y FL(2)2487 4912 y FP(S)2543 4926 y FL(2)2607
4912 y FN(\031)2678 4926 y FK(K)2732 4935 y FC(1)2796
4912 y FP(S)2852 4926 y FL(2)1315 5068 y FP(S)1371 5082
y FL(1)1436 5068 y FN(\031)1507 5082 y FK(K)1561 5091
y FC(2)1624 5068 y FP(S)1680 5082 y FL(1)1901 5068 y
FP(S)1957 5082 y FL(1)2021 5068 y FN(\031)2092 5082 y
FK(K)2146 5091 y FC(2)2210 5068 y FP(S)2266 5082 y FL(2)2487
5068 y FP(S)2543 5082 y FL(2)2607 5068 y FN(\031)2678
5082 y FK(K)2732 5091 y FC(2)2796 5068 y FP(S)2852 5082
y FL(2)378 5272 y FT(and)e(that)600 5477 y FN(K)669 5491
y FL(1)709 5477 y FN(d)p FP(S)805 5491 y FL(1)844 5477
y FN(e)c FT(=)f FN(fg)182 b(K)1347 5491 y FL(2)1387 5477
y FN(d)p FP(S)1483 5491 y FL(1)1523 5477 y FN(e)25 b
FT(=)g FP(i)h FN($)f FP(j)188 b FN(K)2151 5491 y FL(1)2190
5477 y FN(d)p FP(S)2286 5491 y FL(2)2326 5477 y FN(e)26
b FT(=)e FN(f)p FP(i;)15 b(j)5 b FN(g)28 b($)d FP(k)185
b FN(K)3135 5491 y FL(2)3175 5477 y FN(d)p FP(S)3271
5491 y FL(2)3310 5477 y FN(e)26 b FT(=)f FN(K)3541 5491
y FL(2)3581 5477 y FP(:)378 5681 y FT(Finally)-8 b(,)29
b(the)i(uncoloured)d(pro)5 b(jection)31 b FP(S)1820 5648
y FK(U)1815 5705 y FL(1)1900 5681 y FT(=)25 b FP(S)2057
5648 y FK(U)2052 5705 y FL(2)2137 5681 y FT(=)g FN(f)p
FP(A;)15 b(B)25 b FN(^)20 b FP(C)7 b FN(g)p FT(.)1054
b Ff(\003)p eop
%%Page: 126 136
126 135 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(126)378 396 y FG(7.2.2)112
b(The)38 b(Consistency)f(of)h(Sets)f(of)h(Coloured)f(F)-9
b(orm)m(ulae)378 568 y FT(It)41 b(should)e(b)s(e)h(noted)i(that)f
(since)g(the)g(coloured)f(language)i FP(L)2618 535 y
FK(P)2718 568 y FT(is)e(de\014ned)f(as)j(the)f(\014rst-order)378
681 y(language)f FP(L)p FT(\(\006)929 648 y FK(P)929
708 y FO(R)988 681 y FP(;)15 b FT(\006)1094 695 y FO(F)1153
681 y FP(;)g(X)7 b FT(\),)43 b(the)d(same)h(notions)e(of)h(v)-5
b(alidit)m(y)38 b(\(for)i(form)m(ulae\))g(and)f(satis\014abilit)m(y)378
794 y(\(for)32 b(sets)g(of)g(form)m(ulae\))g(that)h(apply)e(in)f(the)i
(standard)f(\(i.e.,)16 b(uncoloured\))31 b(\014rst-order)g(logic)h
(still)378 907 y(apply)38 b(for)g FP(L)846 874 y FK(P)905
907 y FT(.)66 b(F)-8 b(or)40 b(example,)h(the)e(set)h
FP(S)k FT(=)39 b FN(f)p FP(A)2199 874 y FO(i)2228 907
y FP(;)15 b FN(:)p FP(A)2397 874 y FO(j)2434 907 y FN(g)39
b FT(is)f(satis\014able)g(if)g FP(i)i FN(6)p FT(=)f FP(j)44
b FT(as)39 b(the)h(t)m(w)m(o)378 1020 y(prop)s(ositions)28
b FP(A)960 987 y FO(i)1019 1020 y FT(and)i FP(A)1264
987 y FO(j)1331 1020 y FT(are)h(di\013eren)m(t.)41 b(Ho)m(w)m(ev)m(er,)
33 b(w)m(e)e(require)e(a)i(notion)f(of)h(consistency)f(with)378
1133 y(resp)s(ect)g(to)i(some)f(connectabilit)m(y)f(relation)f
FN(K)q FT(.)42 b(In)30 b(particular)f(w)m(e)i(w)m(an)m(t)g(the)g(set)g
FP(S)36 b FT(ab)s(o)m(v)m(e)31 b(to)h(b)s(e)378 1246
y(inconsisten)m(t)24 b(with)h(resp)s(ect)g(to)h FN(K)h
FT(if)d(and)h(only)g(if)f FP(i)i FN(\030)2233 1260 y
FK(K)2316 1246 y FP(j)5 b FT(.)39 b(Basically)-8 b(,)27
b(w)m(e)f(de\014ne)e(a)i FN(K)q FT(-consistency)378 1358
y(prop)s(ert)m(y)h(whic)m(h)f(is)g(equiv)-5 b(alen)m(t)27
b(to)h(the)f(uncoloured)f(de\014nition)f(of)j(consistency)f
(\(De\014nition)f(7.1\))378 1471 y(with)34 b(the)h(exception)g(that)g
(a)g(complemen)m(tary)g(pair)f(of)h(literals)e(mak)m(e)j(a)f(set)h
(inconsisten)m(t)e(only)378 1584 y(if)29 b(their)h(colours)g(relate)g
(in)f FN(K)q FT(.)42 b(Similarly)-8 b(,)27 b(w)m(e)k(deem)f(a)h(set)g
(of)f(coloured)g(sen)m(tences)i(con)m(taining)e(a)378
1697 y(literal)f FN(?)712 1664 y FO(i)770 1697 y FT(to)i(b)s(e)f
(inconsisten)m(t)f(if)h FP(i)g FT(is)g(in)f(the)h(connectabilit)m(y)g
(relation)g(considered.)378 1896 y FQ(De\014nition)35
b(7.13)h(\(Coloured)f(Consistency)g(Prop)s(ert)m(y\))46
b FT(Let)26 b FN(C)31 b FT(b)s(e)25 b(a)h(collection)f(of)h(sets)g(of)
378 2009 y(coloured)c(sen)m(tences,)j(and)d FN(K)i FT(a)f
(connectabilit)m(y)f(relation.)37 b(Then)22 b FN(C)27
b FT(is)22 b(said)f(to)i(b)s(e)f(a)h FN(K)q FT(-consistency)378
2122 y(prop)s(ert)m(y)36 b(with)f(resp)s(ect)i(to)g(a)g(coloured)g
(language)g FP(L)2304 2089 y FK(P)2399 2122 y FT(if)f(for)g(ev)m(ery)i
(set)f FP(S)j FN(2)c(C)42 b FT(the)36 b(follo)m(wing)378
2235 y(conditions)29 b(hold:)489 2411 y(1.)46 b(F)-8
b(or)31 b(ev)m(ery)h(pair)d(of)h(colours)g FP(i)p FT(,)h
FP(j)5 b FT(,)32 b(suc)m(h)e(that)h FP(i)25 b FN(\030)2323
2425 y FK(K)2407 2411 y FP(j)5 b FT(,)31 b(and)f(ev)m(ery)h(literal)e
FP(A)c FN(2)g FP(L)p FT(,)31 b(not)g(b)s(oth)605 2524
y FP(A)673 2491 y FO(i)732 2524 y FT(and)f FN(:)p FP(A)1038
2491 y FO(j)1104 2524 y FT(are)h(in)e FP(S)5 b FT(.)489
2707 y(2.)46 b(F)-8 b(or)31 b(ev)m(ery)h(colour)d FP(i)i
FT(in)e FN(K)q FT(,)i(the)g(literal)e FN(?)2066 2675
y FO(i)2129 2707 y FP(=)-55 b FN(2)25 b FP(S)5 b FT(.)489
2891 y(3.)46 b(If)30 b FP(')21 b FN(^)f FP( )28 b FN(2)d
FP(S)35 b FT(then)c FP(S)25 b FN([)20 b(f)p FP(';)15
b( )s FN(g)27 b(2)e(C)5 b FT(.)489 3074 y(4.)46 b(If)30
b FP(')21 b FN(_)f FP( )28 b FN(2)d FP(S)35 b FT(then)c
FP(S)25 b FN([)20 b(f)p FP(')p FN(g)26 b(2)f(C)36 b FT(or)30
b FP(S)25 b FN([)20 b(f)p FP( )s FN(g)27 b(2)d(C)5 b
FT(.)489 3257 y(5.)46 b(If)30 b FN(8)p FP(x:')c FN(2)e
FP(S)36 b FT(then)30 b FP(S)25 b FN([)20 b(f)p FP(')p
FN(f)p FP(x)26 b FN(!)g FP(t)p FN(gg)g(2)e(C)36 b FT(for)30
b(ev)m(ery)h(closed)f(term)h FP(t)f FT(of)g FP(L)p FT(.)489
3440 y(6.)46 b(If)30 b FN(9)p FP(x:')c FN(2)e FP(S)36
b FT(then)30 b FP(S)25 b FN([)20 b(f)p FP(')p FN(f)p
FP(x)26 b FN(!)g FP(p)p FN(gg)f(2)g(C)36 b FT(for)30
b(some)h(parameter)g FP(p)f FT(of)g FP(L)3173 3454 y
FE(P)-5 b(AR)3327 3440 y FT(.)405 b Ff(\003)519 3639
y FT(Note)23 b(that)e(conditions)f(3{6)i(of)g(the)f(de\014nition)e(of)i
(a)h FN(K)q FT(-consistency)g(prop)s(ert)m(y)e(giv)m(en)h(ab)s(o)m(v)m
(e)h(are)378 3752 y(iden)m(tical)h(to)i(those)f(of)g(de\014nition)e
(7.1)j(of)f(a)h(consistency)f(prop)s(ert)m(y)-8 b(.)38
b(W)-8 b(e)25 b(no)m(w)f(de\014ne)g FN(K)q FT(-consisten)m(t)378
3865 y(and)30 b FN(K)q FT(-inconsisten)m(t)g(sets)h(of)f(sen)m(tences,)
i(and)e(giv)m(e)h(a)g(n)m(um)m(b)s(er)e(of)h(examples.)378
4064 y FQ(De\014nition)35 b(7.14)h(\(Consisten)m(t)e(Sets)h(of)g
(Coloured)g(F)-9 b(orm)m(ulae\))45 b FT(A)34 b(set)i(of)f(coloured)f
(sen-)378 4177 y(tences)f FP(S)k FT(is)31 b(said)g(to)i(b)s(e)e
(consisten)m(t)i(with)e(resp)s(ect)h(to)h(a)f(connectabilit)m(y)g
(relation)f FN(K)q FT(,)i(or)f(simply)378 4290 y FN(K)q
FT(-consisten)m(t,)37 b(if)c(it)h(is)f(a)h(mem)m(b)s(er)g(of)g(some)h
FN(K)q FT(-consistency)g(prop)s(ert)m(y)-8 b(,)35 b(otherwise)f(it)g
(is)f(said)g(to)378 4403 y(b)s(e)d(inconsisten)m(t)f(with)g(resp)s(ect)
i(to)g FN(K)h FT(\(or)e FN(K)q FT(-inconsisten)m(t\).)1251
b Ff(\003)378 4602 y FQ(Example)34 b(7.2)h(\(Consisten)m(t)f(and)h
(Inconsisten)m(t)g(Sets)g(of)g(Coloured)g(F)-9 b(orm)m(ulae\))514
4801 y FN(\017)46 b FT(The)32 b(set)h FN(f)p FP(X)1065
4768 y FO(i)1094 4801 y FP(;)15 b FN(:)p FP(X)1277 4768
y FO(j)1314 4801 y FP(;)g(X)1436 4768 y FO(k)1480 4801
y FP(;)g FN(:)p FP(X)1663 4768 y FO(l)1689 4801 y FN(g)33
b FT(is)e(consisten)m(t)i(with)e(resp)s(ect)h(to)h FP(i)c
FN($)g FP(k)24 b FN([)d FP(j)35 b FN($)28 b FP(l)34 b
FT(but)e(it)g(is)605 4914 y(inconsisten)m(t)e(with)f(resp)s(ect)h(to)h
FP(i)26 b FN($)f FP(l)r FT(.)514 5097 y FN(\017)46 b
FT(The)30 b(set)h FN(f)p FP(X)1061 5064 y FO(i)1110 5097
y FN(^)20 b(:)p FP(X)1334 5064 y FO(j)1371 5097 y FN(g)30
b FT(is)g FP(i)25 b FN($)h FP(j)5 b FT(-inconsisten)m(t.)514
5280 y FN(\017)46 b FT(The)35 b(set)h FN(f)p FP(X)1071
5247 y FO(i)1100 5280 y FP(;)15 b FT(\()p FP(X)42 b FN(\))33
b FP(Y)20 b FT(\))1524 5247 y FO(j)1561 5280 y FP(;)15
b FN(:)p FP(Y)1735 5247 y FO(k)1778 5280 y FN(g)36 b
FT(is)e FN(f)p FP(i;)15 b(j)5 b FN(g)36 b($)d FP(k)s
FT(-consisten)m(t)k(but)d(it)h(is)g(not)g FP(i)f FN($)g
FP(j)39 b FN($)33 b FP(k)s FT(-)605 5393 y(consisten)m(t.)2735
b Ff(\003)519 5592 y FT(The)26 b(follo)m(wing)f(prop)s(osition)f(follo)
m(ws)h(immediately)g(from)h(the)h(de\014nition)d(of)i(the)h
(consistency)378 5705 y(of)j(a)h(coloured)f(set)h(of)g(sen)m(tences.)p
eop
%%Page: 127 137
127 136 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(127)378 396 y FQ(Prop)s(osition)36
b(7.1)46 b FI(L)-5 b(et)34 b FP(S)k FI(b)-5 b(e)33 b(a)g(set)g(of)h(c)
-5 b(olour)g(e)g(d)35 b(sentenc)-5 b(es)33 b(and)h FN(K)h
FI(a)f(c)-5 b(onne)g(ctability)34 b(r)-5 b(elation.)378
509 y(If)34 b(al)5 b(l)34 b(the)h(c)-5 b(olours)36 b(in)e
FP(S)39 b FI(r)-5 b(elate)35 b(with)g(e)-5 b(ach)35 b(other,)g(that)h
(is)e FP(i)28 b FN(\030)2617 523 y FK(K)2703 509 y FP(j)40
b FI(for)34 b(al)5 b(l)35 b FP(i)p FI(,)f FP(j)40 b FI(in)34
b FP(S)5 b FI(,)34 b(then)h FP(S)k FI(is)378 622 y FN(K)q
FI(-c)-5 b(onsistent)34 b(if)e(and)i(only)f(if)f FP(S)1515
589 y FK(U)1603 622 y FI(is)g(c)-5 b(onsistent.)378 835
y FQ(Pro)s(of)p FT(:)31 b(T)-8 b(rivial;)29 b(b)m(y)h(de\014nitions)e
(7.1)j(and)f(7.13.)1677 b Ff(\004)519 1061 y FT(It)36
b(is)f(often)h(con)m(v)m(enien)m(t)h(to)f(represen)m(t)g(a)g(set)g(of)g
(coloured)f(form)m(ulae)g(and)g(a)h(connectabilit)m(y)378
1174 y(relation)d(as)h(a)h(single)d(en)m(tit)m(y)-8 b(.)53
b(This)32 b(is)h(giv)m(en)h(b)m(y)g(the)g(follo)m(wing)e(de\014nition)g
(of)i(a)g FI(c)-5 b(olour)g(e)g(d)39 b(\014rst-)378 1286
y(or)-5 b(der)34 b(pr)-5 b(oblem)p FT(.)378 1499 y FQ(De\014nition)35
b(7.15)h(\(Coloured)f(First-Order)f(Problem\))45 b FT(A)22
b(pair)f(\()p FP(S;)15 b FN(K)q FT(\))23 b(consisting)e(of)h(a)g(set)
378 1612 y(of)33 b(coloured)g(sen)m(tences)h FP(S)39
b FT(and)32 b(a)i(connectabilit)m(y)f(relation)f FN(K)j
FT(is)d(called)g(a)i(coloured)f(\014rst-order)378 1725
y(problem,)42 b(or)f(simply)d(a)j(coloured)g(problem.)70
b(W)-8 b(e)42 b(sa)m(y)g(that)f(\()p FP(S;)15 b FN(K)q
FT(\))43 b(is)c(consisten)m(t)j(if)d FP(S)46 b FT(is)40
b FN(K)q FT(-)378 1838 y(consisten)m(t,)31 b(otherwise)f(\()p
FP(S;)15 b FN(K)q FT(\))32 b(is)d(said)g(to)j(b)s(e)d(inconsisten)m(t.)
1251 b Ff(\003)378 2124 y FH(7.3)135 b(F)-11 b(rom)45
b(Coloured)g(F)-11 b(orm)l(ulae)46 b(to)f(Uncoloured)g(Ones)378
2327 y FT(In)38 b(this)h(section)g(w)m(e)h(de\014ne)e(a)i(mapping)e
(from)g(sets)i(of)g(coloured)e(sen)m(tences)j(in)m(to)e(`equiv)-5
b(alen)m(t')378 2440 y(sets)32 b(of)g(uncoloured)e(ones.)45
b(More)33 b(precisely)-8 b(,)31 b(giv)m(en)h(a)g(connectabilit)m(y)f
(relation)g FN(K)i FT(o)m(v)m(er)g(a)f(set)h(of)378 2553
y(colours)d FN(P)7 b FT(,)31 b(w)m(e)g(de\014ne)f(a)g(mapping)1386
2757 y FN(D)1456 2771 y FK(K)1540 2757 y FT(:)25 b FP(L)p
FT(\(\006)1753 2720 y FK(P)1753 2780 y FO(R)1812 2757
y FP(;)15 b FT(\006)1918 2771 y FO(F)1977 2757 y FP(;)g(X)7
b FT(\))26 b FN(!)f FP(L)p FT(\(\006)2439 2720 y FK(0)2439
2780 y FO(R)2497 2757 y FP(;)15 b FT(\006)2603 2771 y
FO(F)2662 2757 y FP(;)g(X)7 b FT(\))378 2961 y(where)27
b(\006)704 2928 y FK(0)704 2988 y FO(R)789 2961 y FT(is)g(some)h
(collection)f(of)g(predicate)h(sym)m(b)s(ols)e(with)g(\014xed)h
(arities,)h(suc)m(h)f(that)h(a)g(set)g FP(S)33 b FT(of)378
3074 y(coloured)24 b(sen)m(tences)h(is)f FN(K)q FT(-consisten)m(t)i(if)
d(and)h(only)g(if)f FN(D)2320 3088 y FK(K)2378 3074 y
FT(\()p FP(S)5 b FT(\))26 b(=)f FN(fD)2746 3088 y FK(K)2805
3074 y FT(\(\010\))h FN(j)f FT(\010)g FN(2)g FP(S)5 b
FN(g)25 b FT(is)e(consisten)m(t,)378 3187 y(or)37 b(equiv)-5
b(alen)m(tly)35 b(satis\014able.)59 b(W)-8 b(e)38 b(call)e(the)h
(mapping)e FN(D)2431 3201 y FK(K)2526 3187 y FT(a)i FI(de)-5
b(c)g(olourisation)47 b FT(mapping.)58 b(The)378 3300
y(main)21 b(application)g(of)i(this)e(mapping)g(is)g(to)i(b)s(e)f(able)
g(to)h(extend)g(a)g(n)m(um)m(b)s(er)e(of)h(results)g(in)f
(\014rst-order)378 3413 y(logic)30 b(to)h(the)g(coloured)f(logic)g(b)m
(y)g(means)g(of)h(their)e(represen)m(tation)i(in)e(\014rst-order)g
(logic.)378 3656 y FG(7.3.1)112 b(The)38 b(De\014nition)e(of)i(a)g
(Decolourisation)378 3828 y FT(The)29 b(required)e(mapping)h(is)h(giv)m
(en)g(in)f(de\014nition)f(7.17)k(and)e(maps)g(a)h(literal)e(in)g
FP(L)p FT(\(\006)3336 3795 y FK(P)3336 3855 y FO(R)3395
3828 y FP(;)15 b FT(\006)3501 3842 y FO(F)3560 3828 y
FP(;)g(X)7 b FT(\))30 b(to)378 3941 y(a)k(literal)f(in)g
FP(L)p FT(\(\006)997 3903 y FK(P)6 b(\002P)997 3970 y
FO(R)1188 3941 y FN([)23 b(f>)p FP(;)15 b FN(?g)p FP(;)g
FT(\006)1650 3955 y FO(F)1709 3941 y FP(;)g(X)7 b FT(\).)53
b(The)34 b(atoms)h(in)e(the)h(form)m(ulae)g(of)g(the)g(range)h(of)f
(this)378 4054 y(mapping)27 b(are)i(annotated)h(with)d(a)i(pair)f(of)g
(colours.)40 b(F)-8 b(or)30 b(simplicit)m(y)-8 b(,)27
b(w)m(e)i(will)d(refer)i(to)i(a)f(form)m(ula)378 4167
y FP(')437 4134 y FL(\()p FO(i;j)t FL(\))603 4167 y FT(b)m(y)h
FP(')788 4134 y FO(ij)849 4167 y FT(.)41 b(Please)30
b(note)h(that)g(if)f FP(i)25 b FN(6)p FT(=)g FP(j)5 b
FT(,)31 b(then)f FP(')2193 4134 y FO(ij)2280 4167 y FN(6)p
FT(=)25 b FP(')2435 4134 y FO(j)t(i)2496 4167 y FT(.)519
4280 y(Before)40 b(w)m(e)g(giv)m(e)f(the)h(de\014nition)d(of)i(the)g
(mapping)f FN(D)2459 4294 y FK(K)2556 4280 y FT(w)m(e)i(\014rst)e
(consider)g(the)h(conditions)378 4393 y(it)d(needs)f(to)i(satisfy)f
(for)g(the)g(simple)e(case)j(when)e FN(K)i FT(=)d(\()p
FP(i)i FN($)f FP(j)5 b FT(\),)38 b(and)e FP(i)f FN(6)p
FT(=)g FP(j)5 b FT(.)58 b(One)36 b(imp)s(ortan)m(t)378
4506 y(condition)29 b(is)g(that)i(for)f(ev)m(ery)i(literal)d
FP(A)1670 4710 y FN(D)1740 4724 y FK(K)1798 4710 y FT(\()p
FP(A)1901 4672 y FO(i)1930 4710 y FT(\))c(=)g FN(:D)2217
4724 y FK(K)2275 4710 y FT(\()p FN(:)p FP(A)2439 4672
y FO(j)2476 4710 y FT(\))p FP(;)378 4914 y FT(suc)m(h)i(that)h(when)f
(it)g(is)f(applied)g(to)i(the)g(elemen)m(ts)g(of)f(the)h
FN(K)q FT(-inconsisten)m(t)g(set)g FP(S)i FT(=)25 b FN(f)p
FP(A)3417 4881 y FO(i)3446 4914 y FP(;)15 b FN(:)p FP(A)3615
4881 y FO(j)3651 4914 y FN(g)28 b FT(w)m(e)378 5027 y(get)996
5140 y FN(D)1066 5154 y FK(K)1124 5140 y FT(\()p FP(S)5
b FT(\))26 b(=)f FN(fD)1492 5154 y FK(K)1551 5140 y FT(\()p
FP(A)1654 5102 y FO(i)1683 5140 y FT(\))p FP(;)15 b FN(D)1828
5154 y FK(K)1887 5140 y FT(\()p FN(:)p FP(A)2051 5102
y FO(j)2087 5140 y FT(\))p FN(g)26 b FT(=)f FN(f:D)2465
5154 y FK(K)2524 5140 y FT(\()p FN(:)p FP(A)2688 5102
y FO(j)2724 5140 y FT(\))p FP(;)15 b FN(D)2869 5154 y
FK(K)2929 5140 y FT(\()p FN(:)p FP(A)3093 5102 y FO(j)3129
5140 y FT(\))p FN(g)378 5307 y FT(whic)m(h)35 b(is)h(unsatis\014able.)
58 b(Similarly)-8 b(,)35 b(w)m(e)i(also)g(need)g(that)g
FN(D)2551 5321 y FK(K)2609 5307 y FT(\()p FN(:)p FP(A)2773
5274 y FO(i)2802 5307 y FT(\))f(=)f FN(:D)3110 5321 y
FK(K)3168 5307 y FT(\()p FP(A)3271 5274 y FO(j)3308 5307
y FT(\))i(so)g(that)h(the)378 5420 y(set)e FN(f:)p FP(A)699
5387 y FO(i)728 5420 y FP(;)15 b(A)836 5387 y FO(j)873
5420 y FN(g)36 b FT(is)f(mapp)s(ed)g(in)m(to)h(an)f(unsatis\014able)f
(set.)58 b(Other)35 b(prop)s(erties)g(of)h(this)f(particular)p
eop
%%Page: 128 138
128 137 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(128)378 396 y(mapping)29
b(should)f(include:)1532 601 y FN(D)1602 615 y FK(K)1660
601 y FT(\()p FP(A)1763 563 y FO(i)1792 601 y FT(\))83
b FN(6)p FT(=)g FN(:D)2195 615 y FK(K)2253 601 y FT(\()p
FN(:)p FP(A)2417 563 y FO(i)2445 601 y FT(\),)31 b(and)1523
739 y FN(D)1593 753 y FK(K)1652 739 y FT(\()p FP(A)1755
701 y FO(j)1792 739 y FT(\))83 b FN(6)p FT(=)g FN(:D)2195
753 y FK(K)2253 739 y FT(\()p FN(:)p FP(A)2417 701 y
FO(j)2454 739 y FT(\))p FP(;)378 943 y FT(so)31 b(that)g(the)g(sets)g
FN(f)p FP(A)1135 910 y FO(i)1164 943 y FP(;)15 b FN(:)p
FP(A)1333 910 y FO(i)1361 943 y FN(g)31 b FT(and)f FN(f)p
FP(A)1727 910 y FO(j)1764 943 y FP(;)15 b FN(:)p FP(A)1933
910 y FO(j)1970 943 y FN(g)31 b FT(are)g(mapp)s(ed)e(in)m(to)h
(satis\014able)g(sets)h(\(since)f FP(i)c FN(6\030)3713
957 y FK(K)3797 943 y FP(i)378 1056 y FT(and)k FP(j)h
FN(6\030)694 1070 y FK(K)777 1056 y FP(j)5 b FT(\).)519
1169 y(No)m(w,)31 b(giv)m(en)e(a)h(coloured)f(literal)f
FP(B)1761 1136 y FO(i)1788 1169 y FT(,)i(w)m(e)g(de\014ne)f(the)g
(literal)f(\()p FP(B)2764 1136 y FO(i)2792 1169 y FT(\))2827
1136 y Fd(x)p FO(j)2964 1169 y FT(in)g FP(L)3131 1136
y FK(P)6 b(\002P)3329 1169 y FT(as)30 b(the)f(literal)378
1281 y FP(B)41 b FT(coloured)c(with)e(a)j(pair)d(con)m(taining)i
FP(i)g FT(and)f FP(j)43 b FT(\(that)38 b(is,)f(either)g(\()p
FP(i;)15 b(j)5 b FT(\))39 b(or)e(\()p FP(j;)15 b(i)p
FT(\)\))39 b(so)e(that)g(it)g(is)378 1394 y(complemen)m(tary)31
b(to)g(the)f(literal)f(\()p FN(:)p FP(B)1715 1361 y FO(j)1751
1394 y FT(\))1786 1361 y Fd(x)p FO(i)1885 1394 y FT(.)378
1607 y FQ(De\014nition)35 b(7.16)h(\(Decolourisation)g(according)g(to)f
(a)g(Single)g(Colour\))45 b FT(Giv)m(en)27 b(a)g(literal)378
1720 y FP(B)452 1687 y FO(i)506 1720 y FT(in)e(the)i(coloured)e
(language)i FP(L)p FT(\(\006)1658 1687 y FK(P)1658 1747
y FO(R)1717 1720 y FP(;)15 b FT(\006)1823 1734 y FO(F)1882
1720 y FP(;)g(X)7 b FT(\),)28 b(and)e(a)h(colour)f FP(j)31
b FN(2)24 b(P)7 b FT(,)28 b(the)f(literal)e(\()p FP(B)3403
1687 y FO(i)3431 1720 y FT(\))3466 1687 y Fd(x)p FO(j)3600
1720 y FT(in)g(the)378 1833 y(language)31 b FP(L)p FT(\(\006)920
1795 y FK(P)6 b(\002P)920 1861 y FO(R)1109 1833 y FN([)19
b(f>)p FP(;)c FN(?g)p FP(;)g FT(\006)1567 1847 y FO(F)1627
1833 y FP(;)g(X)7 b FT(\))31 b(is)f(de\014ned)f(as)i(follo)m(ws:)1200
2061 y(\()p FP(B)1309 2023 y FO(i)1337 2061 y FT(\))1372
2023 y Fd(x)p FO(j)1505 2061 y FT(=)25 b FN(>)p FT(,)30
b(if)g FP(B)f FT(=)c FN(>)1505 2216 y FT(=)g FN(?)p FT(,)30
b(if)g FP(B)f FT(=)c FN(?)1505 2367 y FT(=)g FP(A)1669
2329 y FO(ij)1730 2367 y FT(,)30 b(if)g FP(B)f FT(=)c
FP(A)p FT(,)31 b(for)f(atomic)h FP(A)26 b FN(6)p FT(=)f
FN(>)1505 2517 y FT(=)g FN(:)p FP(A)1730 2480 y FO(j)t(i)1790
2517 y FT(,)31 b(if)e FP(B)h FT(=)25 b FN(:)p FP(A)p
FT(,)30 b(for)g(atomic)h FP(A)26 b FN(6)p FT(=)f FN(>)751
b Ff(\003)378 2730 y FT(In)30 b(other)g(w)m(ords,)g(w)m(e)h(ha)m(v)m(e)
1353 2972 y(\()p FN(>)1459 2934 y FO(i)1487 2972 y FT(\))1522
2934 y Fd(x)p FO(j)1685 2972 y FT(=)55 b FN(>)298 b FT(\()p
FN(?)2286 2934 y FO(i)2314 2972 y FT(\))2349 2934 y Fd(x)p
FO(j)2512 2972 y FT(=)55 b FN(?)1355 3127 y FT(\()p FP(A)1458
3090 y FO(i)1487 3127 y FT(\))1522 3090 y Fd(x)p FO(j)1685
3127 y FT(=)g FP(A)1879 3090 y FO(ij)2122 3127 y FT(\()p
FN(:)p FP(A)2286 3090 y FO(i)2314 3127 y FT(\))2349 3090
y Fd(x)p FO(j)2512 3127 y FT(=)g FN(:)p FP(A)2767 3090
y FO(j)t(i)2828 3127 y FP(:)378 3331 y FT(where)30 b
FP(A)g FT(is)g(atomic)h(and)e FP(A)d FN(6)p FT(=)f FN(>)p
FT(.)519 3444 y(It)30 b(is)f(easy)h(to)g(c)m(hec)m(k)h(that)g(for)e
(the)h(case)h(of)e FN(K)e FT(=)e FN(f)p FP(i)h FN($)f
FP(j)5 b FN(g)31 b FT(if)e(the)h(mapping)e FN(D)3256
3458 y FK(K)3344 3444 y FT(is)g(de\014ned)h(as)1704 3649
y FN(D)1774 3663 y FK(K)1832 3649 y FT(\()p FP(A)1935
3611 y FO(i)1964 3649 y FT(\))83 b(=)g(\()p FP(A)2339
3611 y FO(i)2368 3649 y FT(\))2403 3611 y Fd(x)p FO(j)1696
3786 y FN(D)1766 3800 y FK(K)1824 3786 y FT(\()p FP(A)1927
3749 y FO(j)1964 3786 y FT(\))g(=)g(\()p FP(A)2339 3749
y FO(j)2376 3786 y FT(\))2411 3749 y Fd(x)p FO(i)378
3991 y FT(for)30 b(an)m(y)h(literal)e FP(A)h FT(then)g(it)g
(satis\014es)g(the)h(conditions)d(discussed)h(earlier)g(this)g
(section.)378 4203 y FQ(Example)34 b(7.3)46 b FT(The)29
b(follo)m(wing)g(are)h(some)g(examples)g(of)g(the)g(use)g(of)g(the)g
(mapping)f FP(X)3460 4170 y Fd(x)p FO(c)3595 4203 y FT(where)378
4316 y FP(X)38 b FT(is)29 b(a)i(literal)e(and)h FP(c)g
FT(a)h(colour.)929 4545 y FN(f)p FT(\()p FP(A)1077 4508
y FO(i)1107 4545 y FT(\))1142 4508 y Fd(x)p FO(j)1249
4545 y FP(;)15 b FT(\()p FN(:)p FP(A)1453 4508 y FO(i)1482
4545 y FT(\))1517 4508 y Fd(x)p FO(j)1624 4545 y FN(g)26
b FT(=)f FN(f)p FP(A)1904 4508 y FO(ij)1965 4545 y FP(;)15
b FN(:)p FP(A)2134 4508 y FO(j)t(i)2195 4545 y FN(g)91
b FT(and)30 b(is)f(th)m(us)h(satis\014able.)929 4701
y FN(f)p FT(\()p FP(A)1077 4663 y FO(i)1107 4701 y FT(\))1142
4663 y Fd(x)p FO(j)1249 4701 y FP(;)15 b FT(\()p FN(:)p
FP(A)1453 4663 y FO(j)1490 4701 y FT(\))1525 4663 y Fd(x)p
FO(i)1624 4701 y FN(g)26 b FT(=)f FN(f)p FP(A)1904 4663
y FO(ij)1965 4701 y FP(;)15 b FN(:)p FP(A)2134 4663 y
FO(ij)2195 4701 y FN(g)91 b FT(and)30 b(hence)g(unsatis\014able.)929
4856 y FN(f)p FT(\()p FP(A)1077 4819 y FO(j)1115 4856
y FT(\))1150 4819 y Fd(x)p FO(i)1249 4856 y FP(;)15 b
FT(\()p FN(:)p FP(A)1453 4819 y FO(i)1482 4856 y FT(\))1517
4819 y Fd(x)p FO(j)1624 4856 y FN(g)26 b FT(=)f FN(f)p
FP(A)1904 4819 y FO(j)t(i)1965 4856 y FP(;)15 b FN(:)p
FP(A)2134 4819 y FO(j)t(i)2195 4856 y FN(g)91 b FT(and)30
b(hence)g(unsatis\014able.)929 5012 y FN(f)p FT(\()p
FP(A)1077 4974 y FO(j)1115 5012 y FT(\))1150 4974 y Fd(x)p
FO(i)1249 5012 y FP(;)15 b FT(\()p FN(:)p FP(A)1453 4974
y FO(j)1490 5012 y FT(\))1525 4974 y Fd(x)p FO(i)1624
5012 y FN(g)26 b FT(=)f FN(f)p FP(A)1904 4974 y FO(ij)1965
5012 y FP(;)15 b FN(:)p FP(A)2134 4974 y FO(ij)2195 5012
y FN(g)91 b FT(and)30 b(hence)g(satis\014able.)581 b
Ff(\003)519 5224 y FT(W)-8 b(e)32 b(usually)27 b(write)j
FP(B)1293 5191 y FO(i)p Fd(x)p FO(j)1454 5224 y FT(instead)f(of)i(\()p
FP(B)1980 5191 y FO(i)2008 5224 y FT(\))2043 5191 y Fd(x)p
FO(j)2150 5224 y FT(.)41 b(W)-8 b(e)31 b(will)d(no)m(w)i(see)h(that)f
FP(B)3150 5191 y FO(i)p Fd(x)p FO(j)3311 5224 y FT(and)g(\()p
FN(:)p FP(B)3658 5191 y FO(j)3694 5224 y FT(\))3729 5191
y Fd(x)p FO(i)378 5337 y FT(are)g(indeed)e(complemen)m(tary)h
(literals,)g(and)f(that)j(the)e(mapping)f(\()2710 5304
y Fd(x)p FO(j)2817 5337 y FT(\))i(is)f(injectiv)m(e)g(on)g(the)h(set)g
(of)378 5450 y(coloured)g(literals)f FP(A)1109 5417 y
FO(c)1174 5450 y FT(where)h FP(c)g FT(is)g(a)g(colour)g(and)g
FP(A)h FT(is)e(a)i(literal)e(other)h(than)h FN(>)f FT(and)f
FN(?)p FT(.)378 5663 y FQ(Prop)s(osition)36 b(7.2)46
b FI(F)-7 b(or)34 b(every)e(liter)-5 b(al)34 b FP(B)1865
5630 y FO(i)1893 5663 y FI(,)f(it)f(is)h(the)g(c)-5 b(ase)33
b(that)h FN(:)p FT(\()p FP(B)2844 5630 y FO(i)p Fd(x)p
FO(j)2974 5663 y FT(\))26 b(=)f(\()p FN(:)p FP(B)3301
5630 y FO(j)3337 5663 y FT(\))3372 5630 y Fd(x)p FO(i)3471
5663 y FI(.)p eop
%%Page: 129 139
129 138 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(129)378 396 y FQ(Pro)s(of)p
FT(:)31 b(W)-8 b(e)32 b(consider)d(the)i(follo)m(wing)e(four)g(cases:)
514 584 y FN(\017)46 b FT(Let)31 b FP(B)f FT(=)25 b FN(>)p
FT(,)30 b(then)1755 689 y FN(:)p FT(\()p FN(>)1922 656
y FO(i)p Fd(x)p FO(j)2053 689 y FT(\))83 b(=)g FN(:>)153
b FT(=)82 b FN(?)1574 802 y FT(and)30 b(\()p FN(:>)1918
769 y FO(i)1946 802 y FT(\))1981 769 y Fd(x)p FO(j)2171
802 y FT(=)83 b FN(?)2396 769 y FO(i)p Fd(x)p FO(j)2610
802 y FT(=)f FN(?)p FP(:)514 999 y FN(\017)46 b FT(Let)31
b FP(B)f FT(=)25 b FN(?)p FT(,)30 b(then)1755 1104 y
FN(:)p FT(\()p FN(?)1922 1071 y FO(i)p Fd(x)p FO(j)2053
1104 y FT(\))83 b(=)g FN(:?)153 b FT(=)82 b FN(>)1574
1217 y FT(and)30 b(\()p FN(:?)1918 1184 y FO(i)1946 1217
y FT(\))1981 1184 y Fd(x)p FO(j)2171 1217 y FT(=)83 b
FN(>)2396 1184 y FO(i)p Fd(x)p FO(j)2610 1217 y FT(=)f
FN(>)p FP(:)514 1413 y FN(\017)46 b FT(Let)31 b FP(B)f
FT(=)25 b FP(A)30 b FT(for)g(some)h(atom)h FP(A)25 b
FN(6)p FT(=)g FN(>)p FT(.)40 b(Then)1873 1617 y FN(:)p
FT(\()p FP(A)2037 1580 y FO(i)p Fd(x)p FO(j)2168 1617
y FT(\))84 b(=)f FN(:)p FT(\()p FP(A)2605 1580 y FO(ij)2665
1617 y FT(\))p FP(;)1692 1755 y FT(and)30 b(\()p FN(:)p
FP(A)2033 1718 y FO(j)2070 1755 y FT(\))2105 1718 y Fd(x)p
FO(i)2287 1755 y FT(=)83 b FN(:)p FT(\()p FP(A)2605 1718
y FO(ij)2665 1755 y FT(\))p FP(:)514 1997 y FN(\017)46
b FT(Let)31 b FP(B)f FT(=)25 b FN(:)p FP(A)30 b FT(for)g(some)h(atom)g
FP(A)26 b FN(6)p FT(=)f FN(>)p FT(.)40 b(Then)1563 2193
y FN(:)p FT(\(\()p FN(:)p FP(A)1823 2160 y FO(i)1851
2193 y FT(\))1886 2160 y Fd(x)p FO(j)1993 2193 y FT(\))84
b(=)e FN(:)p FT(\()p FN(:)p FT(\()p FP(A)2525 2160 y
FO(j)t(i)2586 2193 y FT(\)\))i(=)e FP(A)2961 2160 y FO(j)t(i)3022
2193 y FP(;)1386 2306 y FT(and)30 b(\()p FN(:)p FT(\()p
FN(:)p FP(A)1823 2273 y FO(j)1859 2306 y FT(\)\))1929
2273 y Fd(x)p FO(i)2112 2306 y FT(=)82 b(\()p FP(A)2368
2273 y FO(j)2405 2306 y FT(\))2440 2273 y Fd(x)p FO(i)2740
2306 y FT(=)g FP(A)2961 2273 y FO(j)t(i)3022 2306 y FP(:)519
2554 y FT(Therefore,)30 b FN(:)p FT(\()p FP(B)1126 2521
y FO(i)p Fd(x)p FO(j)1257 2554 y FT(\))c(=)f(\()p FN(:)p
FP(B)1584 2521 y FO(j)1620 2554 y FT(\))1655 2521 y Fd(x)p
FO(i)1784 2554 y FT(for)30 b(ev)m(ery)h(literal)e FP(B)2500
2521 y FO(i)2528 2554 y FT(.)1204 b Ff(\004)378 2766
y FQ(Prop)s(osition)36 b(7.3)46 b FI(F)-7 b(or)35 b(al)5
b(l)33 b(liter)-5 b(als)35 b FP(B)1789 2780 y FL(1)1829
2766 y FI(,)e FP(B)1959 2780 y FL(2)1998 2766 y FI(,)h(and)g(c)-5
b(olours)35 b FP(i)p FI(,)e FP(j)5 b FI(,)34 b FP(m)f
FI(and)i FP(n)p FI(,)e(if)g(neither)g FP(B)3619 2780
y FL(1)3692 2766 y FI(nor)378 2879 y FP(B)447 2893 y
FL(2)519 2879 y FI(ar)-5 b(e)33 b FN(>)g FI(or)g FN(?)p
FI(,)f(and)i(if)e FP(B)1364 2835 y FO(i)p Fd(x)p FO(j)1359
2905 y FL(1)1520 2879 y FT(=)25 b FP(B)1690 2846 y FO(m)p
Fd(x)p FO(n)1685 2904 y FL(2)1902 2879 y FI(then)33 b
FP(B)2173 2893 y FL(1)2237 2879 y FT(=)25 b FP(B)2402
2893 y FL(2)2442 2879 y FI(,)32 b FP(i)26 b FT(=)f FP(m)32
b FI(and)i FP(j)d FT(=)25 b FP(n)p FI(.)378 3101 y FQ(Pro)s(of)p
FT(:)36 b(F)-8 b(rom)35 b(the)f(de\014nition)f(of)h FP(B)1685
3057 y FO(i)p Fd(x)p FO(j)1680 3127 y FL(1)1850 3101
y FT(and)g FP(B)2105 3068 y FO(m)p Fd(x)p FO(n)2100 3126
y FL(2)2319 3101 y FT(w)m(e)h(can)g(assume)f(that)h(if)f
FP(B)3309 3057 y FO(i)p Fd(x)p FO(j)3304 3127 y FL(1)3472
3101 y FT(=)d FP(B)3648 3068 y FO(m)p Fd(x)p FO(n)3643
3126 y FL(2)378 3214 y FT(either)f(b)s(oth)f FP(B)919
3228 y FL(1)989 3214 y FT(and)h FP(B)1235 3228 y FL(2)1305
3214 y FT(are)g(p)s(ositiv)m(e)g(literals)e(or)j(else)f(they)h(are)f(b)
s(oth)g(negativ)m(e:)514 3402 y FN(\017)46 b FT(If)36
b(b)s(oth)f FP(B)991 3416 y FL(1)1066 3402 y FT(and)g
FP(B)1317 3416 y FL(2)1392 3402 y FT(are)i(p)s(ositiv)m(e)d(then)i
FP(B)2173 3416 y FL(1)2247 3402 y FT(=)e FP(A)2420 3416
y FL(1)2495 3402 y FT(and)i FP(B)2747 3416 y FL(2)2820
3402 y FT(=)e FP(A)2993 3416 y FL(2)3069 3402 y FT(for)h(some)i(atoms)f
FP(A)3788 3416 y FL(1)605 3515 y FT(and)30 b FP(A)850
3529 y FL(2)890 3515 y FT(.)41 b(So)30 b FP(B)1156 3471
y FO(i)p Fd(x)p FO(j)1151 3541 y FL(1)1312 3515 y FT(=)25
b FP(A)1476 3471 y FO(i)p Fd(x)p FO(j)1476 3541 y FL(1)1633
3515 y FT(=)g FP(A)1797 3471 y FO(ij)1797 3541 y FL(1)1888
3515 y FT(and)30 b FP(B)2139 3482 y FO(m)p Fd(x)p FO(n)2134
3539 y FL(2)2344 3515 y FT(=)25 b FP(A)2508 3482 y FO(m)p
Fd(x)p FO(n)2508 3539 y FL(2)2714 3515 y FT(=)g FP(A)2878
3482 y FO(mn)2878 3539 y FL(2)2988 3515 y FT(.)41 b(Therefore)30
b FP(A)3534 3529 y FL(1)3599 3515 y FT(=)25 b FP(A)3763
3529 y FL(2)3803 3515 y FT(,)605 3628 y FP(i)h FT(=)f
FP(m)30 b FT(and)g FP(j)h FT(=)25 b FP(n)p FT(.)514 3815
y FN(\017)46 b FT(No)m(w,)39 b(if)d(b)s(oth)g FP(B)1225
3829 y FL(1)1301 3815 y FT(and)g FP(B)1553 3829 y FL(2)1629
3815 y FT(are)h(negativ)m(e,)i(then)e FP(B)2461 3829
y FL(1)2536 3815 y FT(=)e FN(:)p FP(A)2771 3829 y FL(1)2847
3815 y FT(and)h FP(B)3099 3829 y FL(2)3174 3815 y FT(=)f
FN(:)p FP(A)3409 3829 y FL(2)3485 3815 y FT(for)h(some)605
3928 y(atoms)j FP(A)949 3942 y FL(1)1026 3928 y FT(and)e
FP(A)1278 3942 y FL(2)1317 3928 y FT(,)j(and)d(so)h FP(B)1759
3884 y FO(i)p Fd(x)p FO(j)1754 3954 y FL(1)1927 3928
y FT(=)f(\()p FN(:)p FP(A)2199 3895 y FO(i)2199 3953
y FL(1)2239 3928 y FT(\))2274 3895 y Fd(x)p FO(j)2418
3928 y FT(=)g FN(:)p FP(A)2655 3884 y FO(j)t(i)2655 3954
y FL(1)2753 3928 y FT(and)g FP(B)3011 3895 y FO(m)p Fd(x)p
FO(n)3006 3953 y FL(2)3228 3928 y FT(=)g(\()p FN(:)p
FP(A)3500 3895 y FO(n)3500 3953 y FL(2)3547 3928 y FT(\))3582
3895 y Fd(x)p FO(m)3757 3928 y FT(=)605 4041 y FN(:)p
FP(A)734 4008 y FO(nm)734 4066 y FL(2)843 4041 y FT(.)k(And)30
b(again)g FP(A)1419 4055 y FL(1)1484 4041 y FT(=)25 b
FP(A)1648 4055 y FL(2)1687 4041 y FT(,)31 b FP(j)g FT(=)25
b FP(n)30 b FT(and)f FP(i)d FT(=)f FP(m)p FT(.)378 4229
y(In)30 b(either)f(case)j FP(B)1011 4243 y FL(1)1075
4229 y FT(=)25 b FP(B)1240 4243 y FL(2)1280 4229 y FT(,)30
b FP(i)c FT(=)f FP(m)30 b FT(and)g FP(j)h FT(=)25 b FP(n)p
FT(.)1738 b Ff(\004)519 4455 y FT(W)-8 b(e)32 b(no)m(w)f(consider)e
(the)i(conditions)f(whic)m(h)f FN(D)2143 4469 y FK(K)2232
4455 y FT(needs)i(to)g(satisfy)f(if)g FN(K)i FT(con)m(tains)f(more)g
(than)378 4568 y(one)42 b(pair)f(of)h(colours.)75 b(Basically)42
b(if)f FP(i)k FN(\030)1889 4582 y FK(K)1991 4568 y FP(j)2028
4582 y FL(1)2068 4568 y FT(,)g FP(i)g FN(\030)2285 4582
y FK(K)2388 4568 y FP(j)2425 4582 y FL(2)2465 4568 y
FT(,)d FP(:)15 b(:)g(:)32 b FT(,)45 b FP(i)g FN(\030)2886
4582 y FK(K)2988 4568 y FP(j)3025 4582 y FO(n)3073 4568
y FT(,)g(w)m(e)d(need)g(the)g(sets)378 4680 y FN(fD)493
4694 y FK(K)552 4680 y FT(\()p FP(A)655 4647 y FO(i)683
4680 y FT(\))p FP(;)15 b FN(D)828 4694 y FK(K)888 4680
y FT(\()p FN(:)p FP(A)1052 4647 y FO(j)1081 4655 y Fy(x)1123
4680 y FT(\))p FN(g)34 b FT(to)f(b)s(e)f(unsatis\014able)e(for)i(all)g
FP(x)d FN(2)f(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15
b(n)p FN(g)p FT(.)47 b(F)-8 b(urthermore,)33 b(if)f FP(i)d
FN(6\030)3691 4694 y FK(K)3778 4680 y FP(k)378 4793 y
FT(then)d(the)h(set)h FN(fD)988 4807 y FK(K)1047 4793
y FT(\()p FP(A)1150 4760 y FO(i)1178 4793 y FT(\))p FP(;)15
b FN(D)1323 4807 y FK(K)1382 4793 y FT(\()p FN(:)p FP(A)1546
4760 y FO(k)1589 4793 y FT(\))p FN(g)28 b FT(has)e(to)i(b)s(e)e
(satis\014able)f(\(ev)m(en)j(if)e FP(i)g FT(=)f FP(k)s
FT(\).)40 b(W)-8 b(e)28 b(de\014ne)e FN(D)3603 4807 y
FK(K)3661 4793 y FT(\()p FP(A)3764 4760 y FO(i)3793 4793
y FT(\))378 4906 y(to)h(b)s(e)f(the)h(conjunction)e(of)i(all)e(the)i
(literals)d(in)h FN(f)p FP(A)2130 4873 y FO(i)p Fd(x)p
FO(k)2294 4906 y FN(j)g FP(i)h FN(\030)2472 4920 y FK(K)2555
4906 y FP(k)s FN(g)p FT(,)i(so)f(that)g(if)e FP(i)h FN(\030)3211
4920 y FK(K)3294 4906 y FP(j)5 b FT(,)28 b(then)e(one)h(of)378
5019 y(the)33 b(conjuncts)f(in)f FN(D)1124 5033 y FK(K)1183
5019 y FT(\()p FP(A)1286 4986 y FO(i)1315 5019 y FT(\))h(\(whic)m(h)g
(is)g FP(A)1842 4986 y FO(i)p Fd(x)p FO(j)1973 5019 y
FT(\))h(is)f(the)h(complemen)m(t)f(of)h(one)g(of)g(the)g(conjuncts)f
(in)378 5132 y FN(D)448 5146 y FK(K)506 5132 y FT(\()p
FN(:)p FP(A)670 5099 y FO(j)707 5132 y FT(\))i(\(\()p
FN(:)p FP(A)975 5099 y FO(j)1012 5132 y FT(\))1047 5099
y Fd(x)p FO(i)1177 5132 y FT(=)c FN(:)p FT(\()p FP(A)1442
5099 y FO(i)p Fd(x)p FO(j)1574 5132 y FT(\)\),)35 b(and)e(th)m(us)g
(the)h(set)g FN(fD)2507 5146 y FK(K)2566 5132 y FT(\()p
FP(A)2669 5099 y FO(i)2698 5132 y FT(\))p FP(;)15 b FN(D)2843
5146 y FK(K)2902 5132 y FT(\()p FN(:)p FP(A)3066 5099
y FO(j)3103 5132 y FT(\))p FN(g)34 b FT(is)f(unsatis\014able.)378
5245 y(Giv)m(en)e(a)h(connectabilit)m(y)f(relation)f
FN(K)j FT(and)e(a)g(form)m(ula)g(\010,)g(the)h(result)e(of)i(the)f
(required)f(mapping)378 5358 y FN(D)448 5372 y FK(K)541
5358 y FT(is)j(de\014ned)g(b)s(elo)m(w)g(as)i(\010)1396
5325 y Fd(x)p FK(K)1521 5337 y FD(\024)1610 5358 y FT(where)f
FN(\024)g FT(is)f(some)i(total)g(ordering)e(on)h(the)h(palette)f
FN(P)7 b FT(.)54 b(Note)378 5471 y(that)30 b(since)f
FN(P)37 b FT(is)29 b(a)g(coun)m(table)h(set)g(of)g(colours,)f(then)g
(there)h(is)f(at)h(least)g(one)f(suc)m(h)h(ordering.)39
b(An)m(y)378 5584 y(total)31 b(ordering)e FN(\024)i FT(on)f
FN(P)38 b FT(can)31 b(b)s(e)f(used)f(in)h(the)g(follo)m(wing)f
(de\014nition.)39 b(As)30 b(usual,)g(w)m(e)h(write)e
FP(i)d(<)f(j)378 5697 y FT(if)k FP(i)d FN(\024)f FP(j)36
b FT(and)30 b FP(i)25 b FN(6)p FT(=)g FP(j)5 b FT(,)31
b(and)f FP(i)c FN(\025)f FP(j)35 b FT(and)30 b FP(i)c(>)f(j)36
b FT(if)29 b FP(j)i FN(\024)25 b FP(i)31 b FT(and)e FP(j)i(<)25
b(i)31 b FT(resp)s(ectiv)m(ely)-8 b(.)p eop
%%Page: 130 140
130 139 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(130)378 396 y FQ(De\014nition)35
b(7.17)h(\(Decolourisation\))46 b FT(Giv)m(en)29 b(a)h(coloured)f(form)
m(ula)g(\010)g(in)g(the)g(coloured)g(lan-)378 509 y(guage)43
b FP(L)p FT(\(\006)810 476 y FK(P)810 536 y FO(R)869
509 y FP(;)15 b FT(\006)975 523 y FO(F)1034 509 y FP(;)g(X)7
b FT(\),)46 b(a)d(connectabilit)m(y)f(relation)f FN(K)q
FT(,)46 b(and)41 b(a)i(total)g(ordering)e FN(\024)h FT(on)g
FN(P)7 b FT(,)46 b(the)378 622 y(form)m(ula)30 b(\010)778
589 y Fd(x)p FK(K)903 601 y FD(\024)958 622 y FT(,)g(or)h(simply)c
(\010)1481 589 y Fd(x)p FK(K)1610 622 y FT(,)k(in)e FP(L)p
FT(\(\006)1935 584 y FK(P)6 b(\002P)1935 651 y FO(R)2123
622 y FN([)20 b(f>)p FP(;)15 b FN(?g)p FP(;)g FT(\006)2582
636 y FO(F)2642 622 y FP(;)g(X)7 b FT(\))31 b(is)f(de\014ned)f(as)h
(follo)m(ws:)1384 827 y(\()p FP(A)1487 789 y FO(i)1516
827 y FT(\))1551 789 y Fd(x)p FK(K)1676 801 y FD(\024)1815
827 y FT(=)82 b FN(>)p FT(,)31 b(if)e FP(i)d FN(62)e
Fe(C)p FT(\()p FN(K)q FT(\))1815 978 y(=)2081 892 y Fx(^)1968
1093 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])2244
1105 y FD(\024)2311 978 y FP(A)2379 940 y FO(i)p Fd(x)p
FO(j)2510 978 y FT(,)31 b(otherwise)1266 1240 y(\()p
FP(\036)21 b FN(^)e FP(')p FT(\))1550 1203 y Fd(x)p FK(K)1675
1215 y FD(\024)1815 1240 y FT(=)82 b(\()p FP(\036)2057
1203 y Fd(x)p FK(K)2182 1215 y FD(\024)2238 1240 y FT(\))21
b FN(^)f FT(\()p FP(')2469 1203 y Fd(x)p FK(K)2594 1215
y FD(\024)2650 1240 y FT(\))1266 1378 y(\()p FP(\036)h
FN(_)e FP(')p FT(\))1550 1341 y Fd(x)p FK(K)1675 1353
y FD(\024)1815 1378 y FT(=)82 b(\()p FP(\036)2057 1341
y Fd(x)p FK(K)2182 1353 y FD(\024)2238 1378 y FT(\))21
b FN(_)f FT(\()p FP(')2469 1341 y Fd(x)p FK(K)2594 1353
y FD(\024)2650 1378 y FT(\))1293 1516 y(\()p FN(8)p FP(x:')p
FT(\))1550 1478 y Fd(x)p FK(K)1675 1490 y FD(\024)1815
1516 y FT(=)82 b FN(8)p FP(x:)p FT(\()p FP(')2190 1478
y Fd(x)p FK(K)2315 1490 y FD(\024)2371 1516 y FT(\))1293
1654 y(\()p FN(9)p FP(x:')p FT(\))1550 1616 y Fd(x)p
FK(K)1675 1628 y FD(\024)1815 1654 y FT(=)g FN(9)p FP(x:)p
FT(\()p FP(')2190 1616 y Fd(x)p FK(K)2315 1628 y FD(\024)2371
1654 y FT(\))378 1858 y(where)1501 1884 y Fx(^)1347 2086
y FO(j)t FK( )p FL([)p FO(x)1511 2095 y FC(1)1544 2086
y FO(;:::)11 b(;x)1695 2094 y Fy(n)1736 2086 y FL(])1771
1971 y FP(P)i FT(\()p FP(j)5 b FT(\))27 b(=)e FP(P)13
b FT(\()p FP(x)2235 1985 y FL(1)2274 1971 y FT(\))21
b FN(^)f(\001)15 b(\001)g(\001)21 b(^)f FP(P)13 b FT(\()p
FP(x)2776 1985 y FO(n)2823 1971 y FT(\))378 2242 y(and)24
b([)p FN(K)q FT(\()p FP(i)p FT(\)]\))805 2256 y FK(\024)890
2242 y FT(is)g(the)g(\014nite)f(list)g(con)m(taining)h(the)h(colours)e
(in)g(the)i(range)f FN(K)q FT(\()p FP(i)p FT(\))i(sorted)f(in)e
(ascending)378 2355 y(order)34 b(according)h(to)h(the)f(ordering)e
FN(\024)p FT(.)54 b(If)34 b FP(S)40 b FT(is)34 b(a)h(set)g(of)g
(coloured)f(form)m(ulae,)i(w)m(e)f(will)e(refer)h(to)378
2468 y(the)d(set)f FN(f)p FT(\010)787 2435 y Fd(x)p FK(K)912
2447 y FD(\024)993 2468 y FN(j)c FT(\010)f FN(2)g FP(S)5
b FN(g)30 b FT(b)m(y)h FP(S)1545 2435 y Fd(x)p FK(K)1670
2447 y FD(\024)1725 2468 y FT(.)2007 b Ff(\003)519 2635
y FT(In)26 b(the)g(follo)m(wing,)g(w)m(e)g(write)g FP(X)1627
2602 y Fd(x)p FK(K)1782 2635 y FT(instead)f(of)i FP(X)2273
2602 y Fd(x)p FK(K)2398 2614 y FD(\024)2480 2635 y FT(whenev)m(er)f
(the)g(total)h(ordering)e FN(\024)h FT(can)378 2748 y(b)s(e)j(understo)
s(o)s(d)e(from)i(the)g(con)m(text.)42 b(W)-8 b(e)30 b(will)d(also)i
(write)g FP(X)2523 2715 y FO(i)p Fd(x)p FK(K)2705 2748
y FT(instead)f(of)i(\()p FP(X)3237 2715 y FO(i)3266 2748
y FT(\))3301 2715 y Fd(x)p FK(K)3459 2748 y FT(whenev)m(er)378
2861 y(there)h(is)e(no)h(danger)g(of)h(am)m(biguit)m(y)-8
b(.)378 3029 y FQ(Example)34 b(7.4)46 b FT(Let)29 b(the)g(palette)h
FN(P)j FT(=)25 b FN(f)p FP(i;)15 b(j;)g(k)s(;)g(l)r FN(g)32
b FT(where)d FP(i)c(<)g(j)31 b(<)25 b(k)k(<)c(l)r FT(,)k(and)f(let)h
FP(X)37 b FT(b)s(e)28 b(atomic)378 3141 y(and)i(not)g(equal)g(to)h
FN(>)p FT(.)489 3292 y(1.)46 b(If)30 b FP(S)752 3306
y FL(1)817 3292 y FT(=)25 b FN(f)p FP(X)1040 3259 y FO(i)1089
3292 y FN(^)20 b(:)p FP(X)1313 3259 y FO(j)1349 3292
y FN(g)31 b FT(and)f FN(K)1671 3306 y FL(1)1736 3292
y FT(=)25 b FP(i)g FN($)g FP(j)36 b FT(then)1762 3496
y FP(S)1823 3455 y Fd(x)p FK(K)1948 3464 y FC(1)1818
3522 y FL(1)2011 3496 y FT(=)25 b FN(f)p FP(X)2234 3458
y FO(ij)2316 3496 y FN(^)20 b(:)p FP(X)2540 3458 y FO(ij)2600
3496 y FN(g)p FP(:)489 3730 y FT(2.)46 b(If)30 b FP(S)752
3744 y FL(2)817 3730 y FT(=)25 b FN(f)p FP(X)1040 3697
y FO(i)1069 3730 y FP(;)15 b FT(\()p FN(:)p FP(X)28 b
FN(_)19 b FP(Y)h FT(\))1496 3697 y FO(j)1533 3730 y FP(;)15
b FN(:)p FP(Y)1707 3697 y FO(k)1750 3730 y FN(g)31 b
FT(and)e FN(K)2071 3744 y FL(2)2136 3730 y FT(=)c FP(i)h
FN($)f FP(j)31 b FN($)25 b FP(k)34 b FT(then)1196 3934
y FP(S)1257 3894 y Fd(x)p FK(K)1382 3903 y FC(2)1252
3960 y FL(2)1445 3934 y FT(=)25 b FN(f)p FP(X)1668 3897
y FO(ij)1729 3934 y FP(;)15 b FT(\()p FN(:)p FP(X)1947
3897 y FO(ij)2029 3934 y FN(^)20 b(:)p FP(X)2253 3897
y FO(k)r(j)2328 3934 y FT(\))g FN(_)g FT(\()p FP(Y)2573
3897 y FO(j)t(i)2653 3934 y FN(^)g FP(Y)2807 3897 y FO(j)t(k)2882
3934 y FT(\))p FP(;)15 b FN(:)p FP(Y)3092 3897 y FO(j)t(k)3167
3934 y FN(g)p FP(:)489 4168 y FT(3.)46 b(If)30 b FP(S)752
4182 y FL(3)817 4168 y FT(=)25 b FN(f)p FP(X)1040 4135
y FO(i)1069 4168 y FP(;)15 b FN(:)p FP(X)1252 4135 y
FO(j)1289 4168 y FP(;)g(X)1411 4135 y FO(k)1454 4168
y FP(;)g FN(:)p FP(X)1637 4135 y FO(l)1664 4168 y FN(g)30
b FT(and)g FN(K)1985 4182 y FL(3)2050 4168 y FT(=)25
b FP(i)h FN($)f FP(k)e FN([)d FP(j)31 b FN($)25 b FP(l)32
b FT(then)1575 4372 y FP(S)1636 4332 y Fd(x)p FK(K)1761
4341 y FC(3)1631 4398 y FL(3)1824 4372 y FT(=)25 b FN(f)p
FP(X)2047 4335 y FO(ik)2115 4372 y FP(;)15 b FN(:)p FP(X)2298
4335 y FO(l)q(j)2356 4372 y FP(;)g(X)2478 4335 y FO(k)r(i)2546
4372 y FP(;)g FN(:)p FP(X)2729 4335 y FO(j)t(l)2788 4372
y FN(g)p FP(:)489 4607 y FT(4.)46 b(If)30 b FP(S)752
4621 y FL(4)817 4607 y FT(=)25 b FN(f)p FP(X)1040 4574
y FO(i)1089 4607 y FN(^)20 b FP(X)1252 4574 y FO(j)1289
4607 y FP(;)15 b FN(:)p FP(X)1472 4574 y FO(k)1535 4607
y FN(_)20 b FP(Y)1689 4574 y FO(l)1715 4607 y FN(g)30
b FT(and)g FN(K)2036 4621 y FL(4)2101 4607 y FT(=)25
b FN(f)p FP(i;)15 b(j)5 b FN(g)28 b($)d FP(k)33 b FT(then)1376
4811 y FP(S)1437 4770 y Fd(x)p FK(K)1562 4779 y FC(4)1432
4837 y FL(4)1625 4811 y FT(=)25 b FN(f)p FP(X)1848 4773
y FO(ik)1936 4811 y FN(^)20 b FP(X)2099 4773 y FO(j)t(k)2175
4811 y FP(;)15 b FT(\()p FN(:)p FP(X)2393 4773 y FO(ik)2480
4811 y FN(^)20 b(:)p FP(X)2704 4773 y FO(j)t(k)2779 4811
y FT(\))h FN(_)e(>g)p FP(:)489 5045 y FT(5.)46 b(If)30
b FP(S)752 5059 y FL(5)817 5045 y FT(=)25 b FN(f)p FP(X)1040
5012 y FO(i)1069 5045 y FP(;)15 b FN(:)p FP(X)1252 5012
y FO(i)1280 5045 y FN(g)31 b FT(and)f FN(K)1602 5059
y FL(5)1667 5045 y FT(=)25 b FP(i)g FN($)g FP(i)31 b
FT(then)1770 5249 y FP(S)1831 5209 y Fd(x)p FK(K)1956
5218 y FC(5)1826 5275 y FL(5)2020 5249 y FT(=)25 b FN(f)p
FP(X)2243 5212 y FO(ii)2316 5249 y FN(^)20 b(:)p FP(X)2540
5212 y FO(ii)2592 5249 y FN(g)p FP(:)489 5483 y FT(6.)46
b(If)30 b FP(S)752 5497 y FL(6)817 5483 y FT(=)25 b FN(f)p
FP(X)1040 5450 y FO(i)1069 5483 y FP(;)15 b FN(:)p FP(X)1252
5450 y FO(i)1280 5483 y FN(g)31 b FT(and)f FN(K)1602
5497 y FL(6)1667 5483 y FT(=)25 b FP(i)g FN($)g FP(j)36
b FT(then)1762 5705 y FP(S)1823 5665 y Fd(x)p FK(K)1948
5674 y FC(6)1818 5731 y FL(6)2011 5705 y FT(=)25 b FN(f)p
FP(X)2234 5668 y FO(ij)2316 5705 y FN(^)20 b(:)p FP(X)2540
5668 y FO(j)t(i)2600 5705 y FN(g)p FP(:)1087 b Ff(\003)p
eop
%%Page: 131 141
131 140 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(131)519 396 y(W)-8 b(e)32
b(no)m(w)e(giv)m(e)h(the)f(follo)m(wing)f(de\014nition)f(of)j
(satis\014abilit)m(y)d(b)m(y)i(decolourisation.)378 609
y FQ(De\014nition)35 b(7.18)h(\(Satis\014able)e(b)m(y)h
(Decolourisation\))47 b FT(A)c(set)i FP(S)j FT(of)c(coloured)f
(\014rst-order)378 722 y(form)m(ulae)30 b(is)g(said)f(to)i(b)s(e)f
(satis\014able)f(with)h(resp)s(ect)g(to)h(the)g(decolourisation)e
(according)h(to)i FN(K)q FT(,)f(or)378 835 y(simply)f
FN(K)q FT(-satis\014able,)i(if)f FP(S)1365 802 y Fd(x)p
FK(K)1526 835 y FT(is)g(satis\014able.)45 b(Similarly)-8
b(,)30 b FP(S)37 b FT(is)31 b FN(K)q FT(-unsatis\014able)g(if)g
FP(S)3443 802 y Fd(x)p FK(K)3603 835 y FT(is)h(not)378
948 y(satis\014able.)2963 b Ff(\003)378 1160 y FQ(Example)34
b(7.5)h(\(Satis\014able)f(and)h(Unsatis\014able)g(sets)g(b)m(y)h
(Decolourisation\))46 b FT(The)36 b(sets)378 1273 y(in)21
b(Example)g(7.4)j(ab)s(o)m(v)m(e)f(are)g(as)f(follo)m(ws:)36
b(the)23 b(set)f FP(S)2148 1287 y FL(1)2210 1273 y FT(is)f
FN(K)2362 1287 y FL(1)2402 1273 y FT(-unsatis\014able,)h
FP(S)3028 1287 y FL(2)3089 1273 y FT(is)g FN(K)3242 1287
y FL(2)3281 1273 y FT(-unsatis\014able,)378 1386 y FP(S)434
1400 y FL(3)511 1386 y FT(is)37 b FN(K)679 1400 y FL(3)719
1386 y FT(-satis\014able,)i FP(S)1260 1400 y FL(4)1338
1386 y FT(is)e FN(K)1506 1400 y FL(4)1545 1386 y FT(-satis\014able,)j
FP(S)2087 1400 y FL(5)2164 1386 y FT(is)d FN(K)2332 1400
y FL(5)2372 1386 y FT(-unsatis\014able,)h(and)g(the)g(set)h
FP(S)3513 1400 y FL(6)3590 1386 y FT(is)e FN(K)3758 1400
y FL(6)3798 1386 y FT(-)378 1499 y(satis\014able.)2963
b Ff(\003)378 1742 y FG(7.3.2)112 b(Correctness)38 b(of)f(the)h
(Decolourisation)d(Mapping)378 1914 y FT(In)f(this)g(section)i(w)m(e)f
(will)e(sho)m(w)i(that)g(the)h(decolourisation)d(mapping)h(giv)m(en)h
(in)f(de\014nition)e(7.17)378 2027 y(ab)s(o)m(v)m(e)c(is)d(correct.)41
b(In)25 b(other)i(w)m(ords,)g(w)m(e)g(will)d(sho)m(w)i(that)h(a)g(set)g
(is)f(satis\014able)f(b)m(y)h(decolourisation)378 2140
y(if)j(and)h(only)g(if)f(it)h(is)f(consisten)m(t)i(according)f(to)h
(the)g(connectabilit)m(y)f(relation)f(considered.)519
2253 y(First)h(of)g(all,)g(it)g(is)f(straigh)m(tforw)m(ard)h(to)h(sho)m
(w)g(that)g(the)f(follo)m(wing)f(results)g(hold.)378
2465 y FQ(Prop)s(osition)36 b(7.4)46 b FI(L)-5 b(et)36
b FP(S)k FI(b)-5 b(e)36 b(a)f(set)h(of)g(c)-5 b(olour)g(e)g(d)37
b(sentenc)-5 b(es)36 b(and)g FN(K)h FI(a)f(c)-5 b(onne)g(ctability)37
b(r)-5 b(elation)378 2578 y(then:)485 2766 y(1.)46 b(If)33
b FT(\()p FP(')21 b FN(^)f FP( )s FT(\))994 2733 y FO(i)1048
2766 y FN(2)25 b FP(S)37 b FI(and)d FP(S)j FI(is)c FN(K)q
FI(-satis\014able)h(then)f FP(S)25 b FN([)20 b(f)p FP(')2582
2733 y FO(i)2611 2766 y FP(;)15 b( )2713 2733 y FO(i)2742
2766 y FN(g)33 b FI(is)g FN(K)q FI(-satis\014able.)485
2953 y(2.)46 b(If)33 b FT(\()p FP(')21 b FN(_)f FP( )s
FT(\))994 2921 y FO(i)1048 2953 y FN(2)25 b FP(S)37 b
FI(and)d FP(S)j FI(is)c FN(K)q FI(-satis\014able)h(then)f
FP(S)25 b FN([)20 b(f)p FP(')2582 2921 y FO(i)2611 2953
y FN(g)33 b FI(or)g FP(S)25 b FN([)20 b(f)p FP( )3075
2921 y FO(i)3104 2953 y FN(g)33 b FI(is)g FN(K)q FI(-satis\014able.)485
3141 y(3.)46 b(If)40 b FT(\()p FN(8)p FP(x:')p FT(\))965
3108 y FO(i)1031 3141 y FN(2)e FP(S)44 b FI(and)c FP(S)45
b FI(is)39 b FN(K)q FI(-satis\014able)i(then)f FP(S)30
b FN([)25 b(f)p FP(')p FN(f)p FP(x)39 b FN(!)f FP(t)p
FN(g)2972 3108 y FO(i)3000 3141 y FN(g)i FI(is)f FN(K)q
FI(-satis\014able)i(for)605 3254 y(every)33 b(close)-5
b(d)34 b(term)f FP(t)p FI(.)485 3442 y(4.)46 b(If)40
b FT(\()p FN(9)p FP(x:')p FT(\))965 3409 y FO(i)1031
3442 y FN(2)e FP(S)44 b FI(and)c FP(S)45 b FI(is)39 b
FN(K)q FI(-satis\014able)i(then)f FP(S)30 b FN([)25 b(f)p
FP(')p FN(f)p FP(x)39 b FN(!)f FP(t)p FN(g)2972 3409
y FO(i)3000 3442 y FN(g)i FI(is)f FN(K)q FI(-satis\014able)i(for)605
3555 y(some)34 b(close)-5 b(d)33 b(term)h FP(t)p FI(.)485
3742 y(5.)46 b(If)40 b FT(\()p FN(9)p FP(x:')p FT(\))965
3709 y FO(i)1031 3742 y FN(2)e FP(S)44 b FI(and)c FP(S)45
b FI(is)39 b FN(K)q FI(-satis\014able)i(then)f FP(S)30
b FN([)25 b(f)p FP(')p FN(f)p FP(x)39 b FN(!)f FP(t)p
FN(g)2972 3709 y FO(i)3000 3742 y FN(g)i FI(is)f FN(K)q
FI(-satis\014able)i(for)605 3855 y(every)33 b(close)-5
b(d)34 b(term)f FP(t)f FI(whose)i(r)-5 b(o)g(ot)35 b(is)d(new)h(to)g
FP(S)5 b FI(.)485 4043 y(6.)46 b(The)31 b(set)f FP(S)20
b FN([)15 b(f8)p FP(x:')1312 4010 y FO(i)1340 4043 y
FN(g)31 b FI(is)f FN(K)q FI(-satis\014able)i(if)e(and)h(only)g(if)f
FP(S)20 b FN([)15 b(f)p FP(')p FN(f)p FP(x)26 b FN(!)f
FP(t)p FN(g)3142 4010 y FO(i)3171 4043 y FN(g)30 b FI(is)h
FN(K)q FI(-satis\014able)605 4156 y(for)i(al)5 b(l)34
b(close)-5 b(d)33 b(term)h FP(t)p FI(.)485 4343 y(7.)46
b(The)31 b(set)f FP(S)20 b FN([)15 b(f9)p FP(x:')1312
4310 y FO(i)1340 4343 y FN(g)31 b FI(is)f FN(K)q FI(-satis\014able)i
(if)e(and)h(only)g(if)f FP(S)20 b FN([)15 b(f)p FP(')p
FN(f)p FP(x)26 b FN(!)f FP(t)p FN(g)3142 4310 y FO(i)3171
4343 y FN(g)30 b FI(is)h FN(K)q FI(-satis\014able)605
4456 y(for)i(every)g(close)-5 b(d)34 b(term)f FP(t)f
FI(whose)i(r)-5 b(o)g(ot)35 b(is)d(new)h(to)g FP(S)26
b FN([)19 b(f9)p FP(x:')2774 4423 y FO(i)2803 4456 y
FN(g)p FI(.)485 4644 y(8.)46 b(L)-5 b(et)33 b FP(i)p
FI(,)g FP(j)38 b FI(b)-5 b(e)33 b(c)-5 b(olours)34 b(such)f(that)g
FP(i)26 b FN(\030)1871 4658 y FK(K)1954 4644 y FP(j)5
b FI(.)43 b(If)32 b(ther)-5 b(e)34 b(is)e(some)i(sentenc)-5
b(e)32 b FP(')h FI(such)g(that)h FP(')3627 4611 y FO(i)3681
4644 y FN(2)25 b FP(S)605 4757 y FI(and)34 b FN(:)p FP(')902
4724 y FO(j)964 4757 y FN(2)24 b FP(S)5 b FI(,)33 b(then)g
FP(S)38 b FI(is)32 b FN(K)q FI(-unsatis\014able.)485
4944 y(9.)46 b(L)-5 b(et)33 b FP(i)26 b FN(2)f Fe(C)p
FT(\()p FN(K)q FT(\))p FI(,)33 b(if)f FN(?)1321 4911
y FO(i)1374 4944 y FN(2)25 b FP(S)38 b FI(then)33 b FP(S)k
FI(is)c FN(K)q FI(-unsatis\014able.)378 5157 y FQ(Pro)s(of)p
FT(:)k(F)-8 b(or)36 b(eac)m(h)h(case,)h(the)d(pro)s(of)g(follo)m(ws)g
(from)g(the)h(de\014nition)d(of)j FN(K)q FT(-satis\014abilit)m(y)e(and)
h(the)378 5270 y(coun)m(terpart)42 b(of)g(the)g(prop)s(osition)e(for)i
(an)f(uncoloured)g(language.)75 b(W)-8 b(e)44 b(illustrate)c(b)s(elo)m
(w)h(the)378 5383 y(pro)s(of)30 b(for)g(the)g(\014rst)g(case.)p
eop
%%Page: 132 142
132 141 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(132)519 396 y(Let)31 b(\()p
FP(')21 b FN(^)f FP( )s FT(\))975 363 y FO(i)1029 396
y FN(2)25 b FP(S)5 b FT(,)30 b(and)g(let)h FP(S)1601
363 y Fd(x)p FK(K)1760 396 y FT(b)s(e)e(satis\014able)h(\(i.e.,)15
b FP(S)36 b FT(is)29 b FN(K)q FT(-satis\014able\).)768
618 y(No)m(w,)j(\()p FP(')21 b FN(^)f FP( )s FT(\))1294
581 y FO(i)1348 618 y FN(2)25 b FP(S)30 b FN(\))25 b
FT(\()p FP(')c FN(^)f FP( )s FT(\))1929 581 y FO(i)p
Fd(x)p FK(K)2108 618 y FN(2)25 b FP(S)2255 581 y Fd(x)p
FK(K)1520 774 y FN(\))g FT(\()p FP(')1730 736 y FO(i)p
Fd(x)p FK(K)1904 774 y FN(^)20 b FP( )2047 736 y FO(j)t
Fd(x)p FK(K)2208 774 y FT(\))26 b FN(2)f FP(S)2416 736
y Fd(x)p FK(K)1520 930 y FN(\))g FP(S)1697 892 y Fd(x)p
FK(K)1846 930 y FN([)20 b(f)p FP(\036)2026 892 y FO(i)p
Fd(x)p FK(K)2179 930 y FP(;)15 b(')2278 892 y FO(i)p
Fd(x)p FK(K)2432 930 y FN(g)31 b FT(is)e(satis\014able)g(as)i
FP(S)3192 897 y Fd(x)p FK(K)3351 930 y FT(is.)1520 1085
y FN(\))25 b FP(S)g FN([)20 b(f)p FP(\036)1897 1048 y
FO(i)1926 1085 y FP(;)15 b(')2025 1048 y FO(i)2054 1085
y FN(g)31 b FT(is)e FN(K)q FT(-satis\014able.)378 1289
y(The)h(pro)s(ofs)f(of)i(the)f(other)h(cases)g(pro)s(ceed)f(similarly)
-8 b(.)1495 b Ff(\004)519 1515 y FT(Giv)m(en)34 b(this)f(prop)s
(osition,)g(it)g(follo)m(ws)g(that)i(ev)m(ery)g FN(K)q
FT(-satis\014able)f(set)g(of)g(coloured)g(sen)m(tences)378
1628 y(is)29 b FN(K)q FT(-consisten)m(t.)378 1841 y FQ(Theorem)34
b(7.2)46 b(\()p FN(K)q FQ(-Satis\014abilit)m(y)e(Implies)f
FN(K)q FQ(-Consistency\))e FI(F)-7 b(or)41 b(every)f(c)-5
b(onne)g(ctability)378 1954 y(r)g(elation)34 b FN(K)q
FI(,)f(the)g(c)-5 b(ol)5 b(le)-5 b(ction)34 b(of)f(al)5
b(l)33 b FN(K)q FI(-satis\014able)h(sets)f(is)g(a)g FN(K)q
FI(-c)-5 b(onsistency)33 b(pr)-5 b(op)g(erty.)378 2166
y FQ(Pro)s(of)p FT(:)33 b(Giv)m(en)e FN(K)j FT(then)d(for)h(an)m(y)g
FN(K)q FT(-satis\014able)f(set)h(of)g(coloured)f(sen)m(tences,)j(all)c
(the)i(conditions)378 2279 y(in)d(De\014nition)g(7.13)j(hold)d(b)m(y)h
(Prop)s(osition)f(7.4.)1700 b Ff(\004)519 2505 y FT(T)-8
b(o)29 b(deduce)g(the)g(con)m(v)m(erse)h(of)f(this)f(theorem)h(w)m(e)g
(need)g(to)g(sho)m(w)g(that)g(giv)m(en)g(a)g FN(K)q FT(-consistency)378
2618 y(prop)s(ert)m(y)35 b FN(C)5 b FT(,)36 b(all)f(the)g(sets)h(in)e
FN(C)1509 2585 y Fd(x)p FK(K)1671 2618 y FT(=)f FN(f)p
FP(S)1881 2585 y Fd(x)p FK(K)2044 2618 y FN(j)h FP(S)k
FN(2)33 b(C)5 b(g)36 b FT(are)g(satis\014able.)54 b(This)34
b(task)h(w)m(ould)g(b)s(e)378 2731 y(quite)22 b(straigh)m(tforw)m(ard)h
(if)e(w)m(e)j(could)e(sho)m(w)g(that)i FN(C)2130 2698
y Fd(x)p FK(K)2281 2731 y FT(is)e(a)h(consistency)g(prop)s(ert)m(y)-8
b(,)24 b(but)e(in)g(general)378 2844 y(this)k(is)f(not)i(the)g(case.)41
b(The)26 b(reason)h(for)f(this)g(is)f(that)j(some)f(of)g(the)f
(literals)f(in)h FN(C)32 b FT(are)27 b(mapp)s(ed)e(in)m(to)378
2956 y(conjunctions)k(in)f FN(C)1064 2923 y Fd(x)p FK(K)1223
2956 y FT(and)h(as)h(a)g(result)f(the)h(third)e(condition)g(in)g
(de\014nition)g(7.1)j(ma)m(y)f(not)g(hold.)378 3069 y(That)d(is,)f(if)g
FP(S)18 b FN([)13 b(f)p FP(')g FN(^)g FP( )s FN(g)27
b(2)d(C)1406 3036 y Fd(x)p FK(K)1562 3069 y FT(then)j(it)f(ma)m(y)h
(not)g(b)s(e)g(the)f(case)i(that)g FP(S)18 b FN([)13
b(f)p FP(')g FN(^)g FP( )s(;)i(';)g( )s FN(g)28 b(2)d(C)3674
3036 y Fd(x)p FK(K)3803 3069 y FT(.)378 3182 y(An)30
b(example)g(of)h(this)e(is)g(giv)m(en)i(here.)378 3395
y FQ(Example)j(7.6)h(\()p FN(C)1072 3362 y Fd(x)p FK(K)1236
3395 y FQ(is)g(not)g(a)f(Consistency)i(Prop)s(ert)m(y\))46
b FT(Let)30 b(the)h(set)1619 3599 y FP(S)f FT(=)25 b
FN(f)p FP(A)1914 3562 y FO(i)1943 3599 y FP(;)15 b(B)2057
3562 y FO(j)2114 3599 y FN(_)k(:)p FP(A)2323 3562 y FO(k)2366
3599 y FP(;)c(B)2480 3562 y FO(j)2516 3599 y FN(g)p FP(;)378
3803 y FT(and)30 b(the)g(connectabilit)m(y)g(relation)g
FN(K)d FT(=)e FP(i)g FN($)g FP(j)31 b FN($)25 b FP(k)34
b FT(with)29 b FP(i)c(<)g(j)31 b(<)25 b(k)s FT(,)31 b(so)g(that)1213
4008 y FP(S)1274 3970 y Fd(x)p FK(K)1428 4008 y FT(=)25
b FN(f)p FP(A)1637 3970 y FO(ij)1698 4008 y FP(;)15 b
FT(\()p FP(B)1847 3970 y FO(j)t(i)1928 4008 y FN(^)20
b FP(B)2083 3970 y FO(j)t(k)2158 4008 y FT(\))g FN(_)g(:)p
FP(A)2423 3970 y FO(j)t(k)2498 4008 y FP(;)15 b(B)2612
3970 y FO(j)t(i)2693 4008 y FN(^)k FP(B)2847 3970 y FO(j)t(k)2922
4008 y FN(g)p FP(:)378 4212 y FT(Note)35 b(that)f(although)f(the)g
(singleton)g(set)h FN(f)p FP(S)5 b FN(g)34 b FT(is)f(a)h
FN(K)q FT(-consistency)g(prop)s(ert)m(y)-8 b(,)34 b FN(f)p
FP(S)3315 4179 y Fd(x)p FK(K)3444 4212 y FN(g)g FT(is)f(not)g(a)378
4325 y(\014rst-order)c(consistency)i(prop)s(ert)m(y)-8
b(,)30 b(as)h(it)f(do)s(es)g(not)g(con)m(tain)h(the)g(set)g
FP(S)2921 4292 y Fd(x)p FK(K)3070 4325 y FN([)19 b(f)p
FP(B)3269 4292 y FO(j)t(i)3330 4325 y FP(;)c(B)3444 4292
y FO(j)t(k)3519 4325 y FN(g)p FT(.)168 b Ff(\003)519
4537 y FT(Ho)m(w)m(ev)m(er,)30 b(w)m(e)d(can)g(extend)g(the)h(set)f
FN(f)p FP(S)1891 4504 y Fd(x)p FK(K)2020 4537 y FN(g)g
FT(in)f(example)h(7.6)h(ab)s(o)m(v)m(e)g(b)m(y)e(the)i(set)f(of)g(form)
m(ulae:)787 4741 y FP(S)848 4704 y Fd(x)p FK(K)997 4741
y FN([)19 b(f)p FP(B)1196 4704 y FO(j)t(i)1257 4741 y
FP(;)c(B)1371 4704 y FO(j)t(k)1446 4741 y FN(g)26 b FT(=)f
FN(f)p FP(A)1726 4704 y FO(ij)1787 4741 y FP(;)15 b FT(\()p
FP(B)1936 4704 y FO(j)t(i)2017 4741 y FN(^)20 b FP(B)2172
4704 y FO(j)t(k)2246 4741 y FT(\))h FN(_)f(:)p FP(A)2512
4704 y FO(j)t(k)2586 4741 y FP(;)15 b(B)2700 4704 y FO(j)t(i)2781
4741 y FN(^)20 b FP(B)2936 4704 y FO(j)t(k)3010 4741
y FP(;)15 b(B)3124 4704 y FO(j)t(i)3185 4741 y FP(;)g(B)3299
4704 y FO(j)t(k)3374 4741 y FN(g)378 4946 y FT(suc)m(h)36
b(that)g FN(f)p FP(S)897 4913 y Fd(x)p FK(K)1026 4946
y FP(;)15 b(S)1127 4913 y Fd(x)p FK(K)1280 4946 y FN([)24
b(f)p FP(B)1484 4913 y FO(j)t(i)1544 4946 y FP(;)15 b(B)1658
4913 y FO(j)t(k)1733 4946 y FN(gg)37 b FT(is)e(a)h(consistency)g(prop)s
(ert)m(y)-8 b(.)57 b(In)35 b(general,)j(giv)m(en)e(a)g
FN(K)q FT(-)378 5059 y(consistency)29 b(prop)s(ert)m(y)f
FN(C)5 b FT(,)29 b(w)m(e)g(can)g(alw)m(a)m(ys)h(construct)f(a)g
(\014rst-order)f(consistency)g(prop)s(ert)m(y)g(con-)378
5171 y(taining)35 b FN(C)745 5138 y Fd(x)p FK(K)874 5171
y FT(.)59 b(Unfortunately)-8 b(,)38 b(a)f(precise)f(de\014nition)e(of)j
(the)f(required)f(construction)h(is)f(quite)378 5284
y(elab)s(orate)g(and)g(one)g(needs)f(a)i(length)m(y)e(pro)s(of)h(to)g
(c)m(hec)m(k)i(its)d(correctness.)55 b(The)35 b(follo)m(wing)e(theo-)
378 5397 y(rem)j(has)h(its)f(pro)s(of)g(sk)m(etc)m(hed)i(b)s(elo)m(w,)g
(and)e(the)h(sceptical)f(reader)h(is)e(directed)h(to)i(the)f(detailed)
378 5510 y(presen)m(tation)30 b(in)f(App)s(endix)f(C.1.)p
eop
%%Page: 133 143
133 142 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(133)378 396 y FQ(Theorem)34
b(7.3)h(\()p FN(K)q FQ(-Consistency)h(Implies)e FN(K)q
FQ(-Satis\014abilit)m(y\))45 b FI(If)28 b(a)h(c)-5 b(ol)5
b(le)-5 b(ction)30 b(of)f(c)-5 b(olour)g(e)g(d)378 509
y(sentenc)g(es)34 b FN(C)40 b FI(is)34 b(c)-5 b(onsistent)35
b(with)h(r)-5 b(esp)g(e)g(ct)35 b(to)g(some)g(c)-5 b(onne)g(ctability)
36 b(r)-5 b(elation)36 b FN(K)q FI(,)f(then)g(every)f(set)378
622 y(in)e FN(C)38 b FI(is)33 b FN(K)q FI(-satis\014able.)378
835 y FQ(Pro)s(of)40 b(\(Sk)m(etc)m(h\))p FT(:)34 b(Let)g
FN(C)1331 802 y Fd(x)p FK(K)1494 835 y FT(b)s(e)f FN(f)p
FP(S)1727 802 y Fd(x)p FK(K)1888 835 y FN(j)e FP(S)36
b FN(2)31 b(C)5 b(g)p FT(.)52 b(No)m(w)35 b(let)f FN(C)2702
802 y FK(0)2759 835 y FT(b)s(e)f(some)i(set)f(whic)m(h)f(extends)378
948 y FN(C)431 915 y Fd(x)p FK(K)596 948 y FT(suc)m(h)i(that)h(for)g
(all)e FP(S)29 b FN([)23 b(f)p FP(')i FN(^)e FP( )s FN(g)36
b(2)d(C)1956 915 y FK(0)1980 948 y FT(,)k(the)f(set)g
FP(S)29 b FN([)23 b(f)p FP(')i FN(^)e FP( )s(;)15 b(';)g( )s
FN(g)37 b(2)d(C)3226 915 y FK(0)3249 948 y FT(.)57 b(Then)35
b FN(C)3627 915 y FK(0)3686 948 y FT(is)f(a)378 1061
y(consistency)h(prop)s(ert)m(y)f(as)i(it)e(satis\014es)h(all)f(the)h
(conditions)f(in)f(de\014nition)g(7.1.)56 b(No)m(w,)37
b(for)e(ev)m(ery)378 1174 y(set)j FP(S)j FN(2)36 b(C)5
b FT(,)39 b(it)e(follo)m(ws)f(that)i FP(S)1506 1141 y
Fd(x)p FK(K)1671 1174 y FN(2)e(C)1821 1141 y Fd(x)p FK(K)1986
1174 y FN(\022)h(C)2147 1141 y FK(0)2170 1174 y FT(.)61
b(So)37 b FP(S)2450 1141 y Fd(x)p FK(K)2616 1174 y FT(is)f
(satis\014able)g(and)g(th)m(us)h FP(S)42 b FT(is)36 b
FN(K)q FT(-)378 1286 y(satis\014able.)49 b(A)34 b(more)g(detailed)e
(pro)s(of)h(is)g(giv)m(en)h(in)e(app)s(endix)f(C.1)j(where)f(it)g(is)g
(sho)m(wn)g(ho)m(w)h(the)378 1399 y(required)29 b(set)h
FN(C)929 1366 y FK(0)983 1399 y FT(can)h(b)s(e)f(constructed)g(from)g
FN(C)5 b FT(.)1700 b Ff(\004)519 1625 y FT(Due)35 b(to)h(theorems)f
(7.2)h(and)e(7.3,)j(the)e(notions)f(of)h FN(K)q FT(-satis\014abilit)m
(y)f(and)g FN(K)q FT(-consistency)h(are)378 1738 y(equiv)-5
b(alen)m(t.)45 b(Th)m(us)31 b(all)f(the)j(results)d(whic)m(h)h(hold)g
(for)g FN(K)q FT(-satis\014abilit)m(y)g(\(and)g(in)g(particular)f
(those)378 1851 y(giv)m(en)g(in)f(prop)s(osition)f(7.4\))j(also)f(hold)
f(for)h FN(K)q FT(-consistency)-8 b(.)42 b(The)29 b(equiv)-5
b(alence)30 b(of)g FN(K)q FT(-consistency)378 1964 y(and)39
b FN(K)q FT(-satis\014abilit)m(y)f(is)g(stated)i(in)f(the)g(follo)m
(wing)f(theorem,)k(and)d(some)h(applications)d(of)j(this)378
2077 y(result)29 b(are)i(giv)m(en)f(in)f(the)i(next)g(section.)378
2289 y FQ(Theorem)j(7.4)h(\()p FN(K)q FQ(-Satis\014abilit)m(y)g(is)g
(Equiv)-6 b(alen)m(t)35 b(to)g FN(K)q FQ(-Consistency\))46
b FI(F)-7 b(or)25 b(every)f(set)h(of)378 2402 y(c)-5
b(olour)g(e)g(d)33 b(sentenc)-5 b(es)31 b FP(S)k FI(and)d(a)f(c)-5
b(onne)g(ctability)32 b(r)-5 b(elation)32 b FN(K)q FI(,)f(then)g
FP(S)36 b FI(is)30 b FN(K)q FI(-satis\014able)i(if)f(and)g(only)378
2515 y(if)h(it)h(is)f(a)h(memb)-5 b(er)34 b(of)f(some)g
FN(K)q FI(-c)-5 b(onsistency)34 b(pr)-5 b(op)g(erty.)378
2728 y FQ(Pro)s(of)p FT(:)31 b(follo)m(ws)f(from)g(theorems)g(7.2)i
(and)e(7.3.)1703 b Ff(\004)378 2971 y FG(7.3.3)112 b(Applications)378
3143 y FT(The)35 b(\014rst)g(result)f(deriv)m(ed)h(b)s(elo)m(w)g(in)f
(this)h(section)h(allo)m(ws)e(us)h(to)i(sho)m(w)e(that)h(a)g(set)h(of)e
(coloured)378 3256 y(sen)m(tences)f(is)e(consisten)m(t)h(according)g
(to)h(some)g(connectabilit)m(y)e(relation)g(giv)m(en)h(the)g
(assumption)378 3369 y(that)e(it)e(is)g(kno)m(wn)h(to)g(b)s(e)g
(consisten)m(t)g(according)g(to)h(some)f(other)g(connectabilit)m(y)g
(relation.)40 b(Note)378 3482 y(that)23 b(giv)m(en)f(t)m(w)m(o)i
(connectabilit)m(y)d(relations)h FN(K)1968 3496 y FL(1)2007
3482 y FT(,)i FN(K)2125 3496 y FL(2)2187 3482 y FT(and)e(a)g
FN(K)2492 3496 y FL(1)2532 3482 y FT(-consisten)m(t)h(set)g
FP(S)5 b FT(,)24 b(then)e(in)e(order)i(to)378 3594 y(sho)m(w)k(that)g
FP(S)31 b FT(is)25 b(also)g FN(K)1211 3608 y FL(2)1251
3594 y FT(-consisten)m(t)i(one)f(needs)f(only)g(to)i(sho)m(w)e(that)i
(the)f(\014rst)f(t)m(w)m(o)i(conditions)d(in)378 3707
y(de\014nition)h(7.13)k(hold)d(since)h(conditions)f(3{6)j(do)f(not)f
(dep)s(end)f(on)i(the)g(particular)d(connectabilit)m(y)378
3820 y(relation)30 b(b)s(eing)f(considered.)40 b(This)28
b(is)i(giv)m(en)g(b)m(y)h(the)f(follo)m(wing)f(prop)s(osition)f(and)i
(is)g(used)g(in)f(the)378 3933 y(pro)s(of)h(of)g(prop)s(ositions)e(7.6)
j(and)f(7.7.)378 4146 y FQ(Prop)s(osition)36 b(7.5)46
b FI(L)-5 b(et)32 b FN(K)1347 4160 y FL(1)1418 4146 y
FI(and)h FN(K)1663 4160 y FL(2)1734 4146 y FI(b)-5 b(e)31
b(two)i(c)-5 b(onne)g(ctability)33 b(r)-5 b(elations.)44
b(In)31 b(or)-5 b(der)33 b(to)g(show)g(that)378 4259
y(every)f FN(K)687 4273 y FL(1)727 4259 y FI(-c)-5 b(onsistent)33
b(set)g(is)g(also)h FN(K)1677 4273 y FL(2)1716 4259 y
FI(-c)-5 b(onsistent)34 b(it)e(is)h(su\016cient)f(to)h(show)i(that:)485
4446 y(1.)46 b(F)-7 b(or)35 b(every)e FN(K)1086 4460
y FL(1)1125 4446 y FI(-c)-5 b(onsistent)34 b(set)g FP(S)5
b FI(,)33 b(c)-5 b(olours)35 b FP(i)p FI(,)e FP(j)39
b FI(and)c(liter)-5 b(al)34 b FP(A)p FI(,)g(if)f FP(i)26
b FN(\030)3109 4460 y FK(K)3163 4469 y FC(2)3228 4446
y FP(j)39 b FI(then)34 b(not)g(b)-5 b(oth)605 4559 y
FP(A)673 4526 y FO(i)734 4559 y FI(and)34 b FN(:)p FP(A)1040
4526 y FO(j)1109 4559 y FI(ar)-5 b(e)33 b(in)g FP(S)5
b FI(.)485 4747 y(2.)46 b(F)-7 b(or)34 b(every)e FN(K)1084
4761 y FL(1)1124 4747 y FI(-c)-5 b(onsistent)33 b(set)g
FP(S)38 b FI(and)33 b(c)-5 b(olour)34 b FP(i)26 b FN(2)f
Fe(C)p FT(\()p FN(K)2567 4761 y FL(2)2606 4747 y FT(\))p
FI(,)33 b FN(?)2773 4714 y FO(i)2836 4747 y FP(=)-55
b FN(2)25 b FP(S)5 b FI(.)378 4959 y FQ(Pro)s(of)p FT(:)35
b(Let)g FN(C)k FT(b)s(e)33 b(the)i(set)f(of)g(all)f FN(K)1684
4973 y FL(1)1724 4959 y FT(-satis\014able)g(sets)i(of)f(coloured)f(sen)
m(tences.)53 b(Then)33 b FN(C)40 b FT(is)33 b(the)378
5072 y(set)38 b(of)g(all)e FN(K)840 5086 y FL(1)880 5072
y FT(-consisten)m(t)i(sets)g(b)m(y)g(theorem)g(7.4)g(and)f(is)g(also)h
(a)g FN(K)2798 5086 y FL(1)2837 5072 y FT(-consistency)g(prop)s(ert)m
(y)f(b)m(y)378 5185 y(prop)s(osition)d(7.4.)59 b(If)36
b(w)m(e)g(assume)g(further)f(that)i(the)f(ab)s(o)m(v)m(e)h(t)m(w)m(o)h
(conditions)c(hold)h(then)h FN(C)41 b FT(is)35 b(a)378
5298 y FN(K)447 5312 y FL(2)487 5298 y FT(-consistency)27
b(prop)s(ert)m(y)f(as)h(w)m(ell.)38 b(This)25 b(follo)m(ws)h(from)g
(the)h(fact)h(that)f(the)g(\014rst)f(t)m(w)m(o)i(conditions)378
5411 y(of)i(de\014nition)e(7.13)j(corresp)s(ond)e(to)h(the)h(ab)s(o)m
(v)m(e)g(assumptions,)d(and)i(conditions)e(3{6)j(follo)m(w)e(from)378
5524 y(the)i(fact)g(that)g FN(C)k FT(is)30 b(a)g FN(K)1230
5538 y FL(1)1270 5524 y FT(-consistency)h(prop)s(ert)m(y)-8
b(.)1619 b Ff(\004)p eop
%%Page: 134 144
134 143 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(134)519 396 y(W)-8 b(e)42
b(can)g(no)m(w)f(use)g(the)g(prop)s(osition)e(ab)s(o)m(v)m(e)j(c)m
(haracterising)f FN(K)2815 410 y FL(2)2855 396 y FT(-consistency)g(in)f
(terms)h(of)378 509 y FN(K)447 523 y FL(1)487 509 y FT(-consistency)32
b(to)h(sho)m(w)f(that)h(a)g(set)g(consisten)m(t)g(according)f(to)h
(some)g(connectabilit)m(y)f(relation)378 622 y FN(K)447
636 y FL(1)517 622 y FT(is)e(also)g(consisten)m(t)h(according)g(to)g
(an)m(y)g(subrelation)d(of)j FN(K)2550 636 y FL(1)2590
622 y FT(.)41 b(In)m(tuitiv)m(ely)-8 b(,)30 b(the)g(connectabilit)m(y)
378 735 y(relation)22 b(is)g(a)h(restriction)f(on)g(whic)m(h)g
(literals)f(can)i(b)s(e)f(used)g(in)g(sho)m(wing)g(that)h(a)g(set)h(is)
e(inconsisten)m(t.)378 848 y(If)28 b(a)g(set)h(cannot)f(b)s(e)g(sho)m
(wn)f(to)i(b)s(e)f(inconsisten)m(t)f(according)h(to)h(a)f(particular)f
(restriction,)h(then)f(it)378 961 y(cannot)k(b)s(e)f(sho)m(wn)f(to)j(b)
s(e)d(inconsisten)m(t)h(according)g(to)h(a)g(stronger)g(restriction.)
378 1174 y FQ(Prop)s(osition)36 b(7.6)46 b FI(Given)37
b(a)g(set)f FP(S)42 b FI(of)37 b(c)-5 b(olour)g(e)g(d)39
b(sentenc)-5 b(es)37 b(and)g(c)-5 b(onne)g(ctability)39
b(r)-5 b(elations)38 b FN(K)3788 1188 y FL(1)378 1286
y FI(and)c FN(K)624 1300 y FL(2)696 1286 y FI(such)f(that)g
FN(K)1157 1300 y FL(2)1222 1286 y FN(\022)25 b(K)1387
1300 y FL(1)1427 1286 y FI(,)32 b(if)h FP(S)k FI(is)c
FN(K)1836 1300 y FL(1)1875 1286 y FI(-c)-5 b(onsistent)34
b(then)f(it)g(is)f(also)i FN(K)2976 1300 y FL(2)3016
1286 y FI(-c)-5 b(onsistent.)378 1499 y FQ(Pro)s(of)p
FT(:)31 b(Let)g FP(S)36 b FT(b)s(e)29 b FN(K)1134 1513
y FL(1)1174 1499 y FT(-consisten)m(t,)i(and)f(let)g FN(K)2029
1513 y FL(2)2094 1499 y FN(\022)25 b(K)2259 1513 y FL(1)2299
1499 y FT(.)489 1687 y(1.)46 b(W)-8 b(e)38 b(sho)m(w)e(that)h(for)f(ev)
m(ery)h(colours)e FP(i)i FT(and)e FP(j)42 b FT(suc)m(h)36
b(that)h FP(i)e FN(\030)2787 1701 y FK(K)2841 1710 y
FC(2)2914 1687 y FP(j)5 b FT(,)39 b(if)c(the)h(literal)f
FP(A)3608 1654 y FO(i)3671 1687 y FN(2)g FP(S)605 1800
y FT(then)g FN(:)p FP(A)946 1767 y FO(j)1026 1800 y FP(=)-55
b FN(2)33 b FP(S)5 b FT(.)55 b(No)m(w,)38 b(since)c FN(K)1786
1814 y FL(2)1859 1800 y FN(\022)f(K)2032 1814 y FL(1)2107
1800 y FT(it)i(follo)m(ws)g(that)g FP(i)f FN(\030)2844
1814 y FK(K)2898 1823 y FC(1)2970 1800 y FP(j)40 b FT(and)35
b(so)h(not)f(b)s(oth)g FP(A)3800 1767 y FO(i)605 1912
y FT(and)30 b FN(:)p FP(A)911 1879 y FO(j)978 1912 y
FT(are)g(in)f FP(S)5 b FT(.)489 2100 y(2.)46 b(F)-8 b(or)41
b(the)f(second)g(case)h(w)m(e)f(need)g(to)g(sho)m(w)g(that)h(if)d
FP(i)k FN(2)e Fe(C)p FT(\()p FN(K)2802 2114 y FL(2)2842
2100 y FT(\))g(then)g FN(?)3205 2067 y FO(i)3284 2100
y FP(=)-55 b FN(2)41 b FP(S)5 b FT(.)69 b(No)m(w,)43
b(if)605 2213 y FP(i)26 b FN(2)f Fe(C)p FT(\()p FN(K)908
2227 y FL(2)947 2213 y FT(\))31 b(then)f FP(i)c FN(2)f
Fe(C)p FT(\()p FN(K)1523 2227 y FL(1)1562 2213 y FT(\))31
b(and)f(therefore)h FN(?)2258 2180 y FO(i)2321 2213 y
FP(=)-55 b FN(2)25 b FP(S)35 b FT(as)30 b FP(S)36 b FT(is)29
b FN(K)2851 2227 y FL(1)2891 2213 y FT(-consisten)m(t.)378
2401 y(Hence,)i(it)f(follo)m(ws)g(that)h FP(S)k FT(is)29
b FN(K)1511 2415 y FL(2)1551 2401 y FT(-consisten)m(t)i(b)m(y)f(prop)s
(osition)f(7.5.)1008 b Ff(\004)519 2626 y FT(The)36 b(role)g(of)g(the)h
(next)f(prop)s(osition)e(is)h(to)i(allo)m(w)f(us)g(to)h(simplify)c(a)j
(giv)m(en)h(problem)d(\()p FP(S;)15 b FN(K)q FT(\))378
2739 y(in)m(to)30 b(one)g(whic)m(h)f(considers)g(only)g(the)h(subset)f
(of)h FN(K)i FT(whic)m(h)d(is)g(relev)-5 b(an)m(t)30
b(to)g FP(S)5 b FT(,)31 b(that)f(is)f(\()p FP(S;)15 b
FN(K)q(d)p FP(S)5 b FN(e)p FT(\).)378 2952 y FQ(Prop)s(osition)36
b(7.7)46 b FI(A)25 b(set)h FP(S)31 b FI(of)26 b(c)-5
b(olour)g(e)g(d)28 b(sentenc)-5 b(es)25 b(is)h FN(K)q
FI(-c)-5 b(onsistent)27 b(if)e(and)i(only)f(if)g(it)f(is)h
FN(K)q(d)p FP(S)5 b FN(e)p FI(-)378 3065 y(c)-5 b(onsistent.)378
3277 y FQ(Pro)s(of)p FT(:)25 b(If)e FP(S)29 b FT(is)23
b FN(K)q FT(-consisten)m(t,)k(then)c(it)h(is)f(also)g
FN(K)q(d)p FP(S)5 b FN(e)p FT(-consisten)m(t)26 b(b)m(y)e(prop)s
(osition)e(7.6)j(as)f FN(K)q(d)p FP(S)5 b FN(e)26 b(\022)378
3390 y(K)q FT(.)41 b(No)m(w,)32 b(let)e FP(S)35 b FT(b)s(e)30
b FN(K)q(d)p FP(S)5 b FN(e)p FT(-consisten)m(t.)489 3578
y(1.)46 b(W)-8 b(e)28 b(need)f(to)g(sho)m(w)g(that)g(for)g(an)m(y)g
(literal)e FP(A)i FT(and)f(colours)h FP(i)p FT(,)h FP(j)k
FT(suc)m(h)26 b(that)i FP(i)d FN(\030)3301 3592 y FK(K)3384
3578 y FP(j)5 b FT(,)29 b(if)d FP(A)3628 3545 y FO(i)3681
3578 y FN(2)f FP(S)605 3691 y FT(then)34 b FN(:)p FP(A)945
3658 y FO(j)1023 3691 y FP(=)-55 b FN(2)30 b FP(S)5 b
FT(.)52 b(If)34 b(w)m(e)g(assume)g(that)g(there)h(is)e(some)h(literal)f
FP(A)h FT(suc)m(h)f(that)i(b)s(oth)e FP(A)3619 3658 y
FO(i)3681 3691 y FT(and)605 3804 y FN(:)p FP(A)734 3771
y FO(j)797 3804 y FT(are)28 b(in)d FP(S)32 b FT(then)26
b(the)h(colours)g FP(i)g FT(and)f FP(j)32 b FT(are)c(in)d
FP(S)32 b FT(and)26 b(so)h FP(i)f FN(\030)2846 3822 y
FK(Kd)p FO(S)t FK(e)3038 3804 y FP(j)5 b FT(,)29 b(whic)m(h)c(con)m
(tradicts)605 3917 y(the)31 b(assumptions)d(that)j(b)s(oth)f
FP(A)1758 3884 y FO(i)1812 3917 y FN(2)25 b FP(S)35 b
FT(and)30 b FN(:)p FP(A)2295 3884 y FO(j)2356 3917 y
FN(2)25 b FP(S)35 b FT(and)30 b(that)h FP(S)k FT(is)30
b FN(K)q(d)p FP(S)5 b FN(e)p FT(-consisten)m(t.)489 4104
y(2.)46 b(W)-8 b(e)34 b(sho)m(w)e(that)i(if)d FP(i)e
FN(2)g Fe(C)p FT(\()p FN(K)q FT(\))k(then)f FN(?)1935
4071 y FO(i)2002 4104 y FP(=)-55 b FN(2)29 b FP(S)37
b FT(b)m(y)32 b(con)m(tradiction.)48 b(If)32 b FN(?)3063
4071 y FO(i)3119 4104 y FN(2)d FP(S)37 b FT(and)32 b
FP(i)e FN(2)e Fe(C)p FT(\()p FN(K)q FT(\))605 4217 y(then)22
b FP(i)h FT(is)e(also)h(in)f Fe(C)p FT(\()p FP(S)5 b
FT(\))23 b(and)e(hence)i(in)e Fe(C)p FT(\()p FN(K)q(d)p
FP(S)5 b FN(e)p FT(\))24 b(as)e(w)m(ell.)37 b(But)23
b(once)g(again)f(this)f(con)m(tradicts)605 4330 y(the)31
b(assumption)e(that)i FP(S)k FT(is)29 b FN(K)q(d)p FP(S)5
b FN(e)p FT(-consisten)m(t.)378 4518 y(Hence,)31 b(it)f(follo)m(ws)g
(that)h FP(S)k FT(is)29 b FN(K)q(d)p FP(S)5 b FN(e)p
FT(-consisten)m(t)33 b(b)m(y)d(prop)s(osition)e(7.5.)905
b Ff(\004)378 4804 y FH(7.4)135 b(Changing)46 b(the)f(Colour)g(of)h(F)
-11 b(orm)l(ulae)378 5007 y FT(The)28 b(colours)h(in)e(a)i(coloured)g
(problem)e(\()p FP(S;)15 b FN(K)q FT(\))30 b(are)f(simply)e(a)i(mec)m
(hanism)f(for)h(iden)m(tifying)d(whic)m(h)378 5120 y(complemen)m(tary)g
(literals)f(in)g FP(S)31 b FT(are)c(allo)m(w)m(ed)f(to)h(con)m(tribute)
f(to)h(the)f(refutation)g(of)h FP(S)5 b FT(.)39 b(The)26
b(actual)378 5233 y(names)44 b(of)h(the)f(colours)g(in)f(the)i(problem)
e(\()p FP(S;)15 b FN(K)q FT(\))46 b(is)d(irrelev)-5 b(an)m(t)44
b(and)g(one)g(can)h(rename)f(some)378 5346 y(colour)39
b FP(i)h FT(in)e(\()p FP(S;)15 b FN(K)q FT(\))42 b(to)e(some)g(new)f
(colour)g(whic)m(h)g(is)f(not)i(in)f(the)g(problem)f(without)h
(a\013ecting)378 5459 y(the)33 b(consistency)g(of)g(\()p
FP(S;)15 b FN(K)q FT(\).)51 b(In)32 b(this)g(section)h(w)m(e)g(giv)m(e)
h(a)f(n)m(um)m(b)s(er)f(of)h(de\014nitions)e(whic)m(h)h(allo)m(ws)378
5572 y(us)g(to)i(recolour)e(literals,)g(and)g(sho)m(w)h(ho)m(w)g(the)g
(consistency)g(of)g(a)g(problem)e(ma)m(y)i(b)s(e)g(a\013ected)h(b)m(y)
378 5685 y(recolouring)29 b(certain)h(literals)f(in)g(it.)p
eop
%%Page: 135 145
135 144 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(135)378 396 y FG(7.4.1)112
b(The)38 b(De\014nition)e(of)i(Recolouring)d(Mappings)378
568 y FT(A)26 b(recolouring)g(mapping)e(is)i(de\014ned)f(b)s(elo)m(w)h
(as)g(a)h(mapping)e(whic)m(h)g(c)m(hanges)j(all,)e(or)g(some)h(of,)h
(the)378 681 y(colours)i(in)f(a)i(form)m(ula,)e(set)i(or)g(problem)e
(to)i(some)f(single)f(colour.)378 884 y FQ(De\014nition)35
b(7.19)h(\(Recolouring)g(Mapping\))46 b FT(Giv)m(en)29
b(some)g(coloured)g(literal)e FP(A)i FT(and)f(colour)378
997 y FP(j)5 b FT(,)36 b(w)m(e)e(de\014ne)f(the)h FP(j)5
b FT(-recolouring)34 b(of)g FP(A)g FT(as)h(\()p FP(A)2019
964 y FK(U)2074 997 y FT(\))2109 964 y FO(j)2180 997
y FT(and)e(denote)i(it)e(b)m(y)h FP(A)2949 964 y FK(!)p
FO(j)3056 997 y FT(.)52 b(Similarly)-8 b(,)32 b(giv)m(en)h(a)378
1110 y(form)m(ula)26 b FP(')h FT(and)f(a)h(set)h(of)f(coloured)f(form)m
(ulae)g FP(S)5 b FT(,)28 b(w)m(e)f(de\014ne)f FP( )2573
1077 y FK(!)p FO(j)2708 1110 y FT(and)g FP(S)2942 1077
y FK(!)p FO(j)3076 1110 y FT(as)h(\()p FP(')3278 1077
y FK(U)3333 1110 y FT(\))3368 1077 y FO(j)3432 1110 y
FT(and)f(\()p FP(S)3701 1077 y FK(U)3756 1110 y FT(\))3791
1077 y FO(j)378 1223 y FT(resp)s(ectiv)m(ely)-8 b(.)42
b(Giv)m(en)30 b(a)h(set)h(of)f(colours)f FN(P)7 b FT(,)32
b(w)m(e)f(de\014ne)f(the)h FN(P)38 b FT(to)32 b FP(j)k
FT(recolouring)30 b(of)h(the)g(literal)e FP(A)378 1336
y FT(\(denoted)i(b)m(y)f FP(A)951 1303 y FL(\()p FK(P)6
b(!)p FO(j)t FL(\))1168 1336 y FT(\))31 b(as)f(follo)m(ws:)980
1540 y FP(A)1048 1503 y FL(\()p FK(P)6 b(!)p FO(j)t FL(\))1348
1540 y FT(=)83 b FP(B)1576 1503 y FO(j)1612 1540 y FT(,)61
b(if)29 b FP(A)c FT(=)g FP(B)2044 1507 y FO(i)2102 1540
y FT(for)31 b(some)f(literal)f FP(B)35 b FT(and)30 b
FP(i)25 b FN(2)g(P)1348 1678 y FT(=)83 b FP(A)p FT(,)61
b(otherwise.)378 1882 y(The)27 b(form)m(ula)g FP(')952
1849 y FL(\()p FK(P)6 b(!)p FO(j)t FL(\))1197 1882 y
FT(is)27 b(de\014ned)g(similarly)d(as)k(the)g(form)m(ula)f
FP(')h FT(with)f(all)g(its)g FN(P)35 b FT(coloured)28
b(literals)378 1995 y(recoloured)44 b(with)g FP(j)5 b
FT(,)49 b(and)44 b FP(S)1422 1962 y FL(\()p FK(P)6 b(!)p
FO(j)t FL(\))1684 1995 y FT(denotes)45 b(the)g(set)g
FN(f)p FP( )2461 1962 y FL(\()p FK(P)6 b(!)p FO(j)t FL(\))2728
1995 y FN(j)50 b FP( )j FN(2)c FP(S)5 b FN(g)p FT(.)84
b(W)-8 b(e)46 b(abbreviate)378 2108 y FP(A)446 2075 y
FL(\()p FK(f)p FO(i)p FK(g!)p FO(j)t FL(\))703 2108 y
FT(,)30 b FP(')817 2075 y FL(\()p FK(f)p FO(i)p FK(g!)p
FO(j)t FL(\))1105 2108 y FT(and)f FP(S)1342 2075 y FL(\()p
FK(f)p FO(i)p FK(g!)p FO(j)t FL(\))1599 2108 y FT(,)h(etc.)17
b(b)m(y)30 b FP(A)2005 2075 y FL(\()p FO(i)p FK(!)p FO(j)t
FL(\))2192 2108 y FT(,)g FP(')2306 2075 y FL(\()p FO(i)p
FK(!)p FO(j)t FL(\))2523 2108 y FT(and)g FP(S)2761 2075
y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))2947 2108 y FT(,)g(etc.)615
b Ff(\003)519 2311 y FT(W)-8 b(e)27 b(no)m(w)e(de\014ne)f(the)i
FI(r)-5 b(enaming)34 b FT(of)26 b(a)f(colour)g(whic)m(h)f(in)m(v)m(olv)
m(es)h(the)h(recolouring)e(of)h(the)h(literals)378 2424
y(of)i(some)h(particular)d(colour)i(in)f(a)h(form)m(ula)f(\(set)i(of)g
(form)m(ulae,)f(connectabilit)m(y)g(relation,)g(etc.)16
b(\))40 b(to)378 2537 y(a)31 b(colour)f(whic)m(h)f(is)g(new)h(to)h(the)
g(form)m(ula)e(\(set,)j(relation,)e(etc.)16 b(\).)378
2741 y FQ(De\014nition)35 b(7.20)h(\(Renaming)e(Colours\))46
b FT(Giv)m(en)c(t)m(w)m(o)j(coloured)d(form)m(ulae)h(\011)f(and)h
(\010,)i(w)m(e)378 2854 y(sa)m(y)31 b(that)g(\011)f(is)g(obtained)g
(from)g(\010)g(b)m(y)g(renaming)g(one)h(colour,)f(and)g(write)g(\010)25
b FN(!)3193 2868 y FL(rc)3281 2854 y FT(\011,)31 b(if)e(for)h(some)378
2966 y(colour)g FP(i)c FN(2)e Fe(C)p FT(\(\010\))31 b(and)e
FP(j)41 b(=)-55 b FN(2)25 b Fe(C)p FT(\(\010\),)31 b(then)1869
3171 y(\011)24 b(=)h(\010)2126 3133 y FL(\()p FO(i)p
FK(!)p FO(j)t FL(\))2312 3171 y FP(:)378 3375 y FT(W)-8
b(e)25 b(denote)f(the)g(re\015exiv)m(e)f(transitiv)m(e)g(closure)g(of)h
(the)f(relation)g FN(!)2672 3389 y FL(rc)2758 3375 y
FT(b)m(y)h(the)g(relation)e Fl(u)3425 3389 y FL(rc)3488
3375 y FT(.)38 b(W)-8 b(e)25 b(sa)m(y)378 3488 y(that)k(t)m(w)m(o)h
(form)m(ulae,)f(\010)f(and)g(\011)g(are)h(isomorphic)e(b)m(y)h
(renaming)f(colours)h(if)g(and)g(only)g(if)f(\010)e Fl(u)3644
3502 y FL(rc)3732 3488 y FT(\011.)378 3601 y(The)g(de\014nition)d(and)j
(notation)g(of)g(colour)g(renaming)f(can)h(b)s(e)g(extended)g(to)h
(sets)f(of)g(form)m(ulae,)h(sets)378 3714 y(of)k(colours,)h
(connectabilit)m(y)e(relations,)h(coloured)g(problems,)f(etc.)1042
b Ff(\003)519 3917 y FT(The)27 b(follo)m(wing)e(prop)s(osition)g(sho)m
(ws)h(that)i(the)f(relation)f Fl(u)2563 3931 y FL(rc)2653
3917 y FT(is)g(symmetric)g(and)h(therefore)g(an)378 4030
y(equiv)-5 b(alence)30 b(relation.)378 4233 y FQ(Prop)s(osition)36
b(7.8)g(\(Symmetry)c(of)j FN(!)1871 4247 y FL(rc)1934
4233 y FQ(,)g(Equiv)-6 b(alence)36 b(of)f Fl(u)2764 4247
y FL(rc)2827 4233 y FQ(\))45 b FI(The)d(r)-5 b(elations)44
b FN(!)3579 4247 y FL(rc)3684 4233 y FI(and)378 4346
y Fl(u)449 4360 y FL(rc)544 4346 y FI(ar)-5 b(e)34 b(symmetric,)f(and)h
(the)f(r)-5 b(elation)34 b Fl(u)1901 4360 y FL(rc)1996
4346 y FI(is)f(an)g(e)-5 b(quivalenc)g(e)33 b(r)-5 b(elation.)378
4549 y FQ(Pro)s(of)p FT(:)31 b(W)-8 b(e)32 b(\014rst)d(sho)m(w)h(that)h
FN(!)1544 4563 y FL(rc)1637 4549 y FT(is)e(symmetric.)40
b(Giv)m(en)30 b(the)g(t)m(w)m(o)i(coloured)d(form)m(ulae)h(\(or)g(sets)
378 4662 y(of)g(form)m(ulae,)h(coloured)f(problems,)f(etc.)16
b(\))41 b(\010)30 b(and)g(\011,)g(if)f(\010)c FN(!)2556
4676 y FL(rc)2644 4662 y FT(\011)30 b(then)1881 4867
y(\011)25 b(=)g(\010)2139 4829 y FL(\()p FO(i)p FK(!)p
FO(j)t FL(\))378 5071 y FT(for)30 b(some)h FP(i)25 b
FN(2)g Fe(C)p FT(\(\010\))31 b(and)e FP(j)41 b(=)-55
b FN(2)25 b Fe(C)p FT(\(\010\).)41 b(Therefore)30 b FP(j)h
FN(2)25 b Fe(C)p FT(\(\011\))30 b(and)g FP(i)36 b(=)-56
b FN(2)25 b Fe(C)p FT(\(\011\),)31 b(and)f(also)1881
5275 y(\010)25 b(=)g(\011)2139 5238 y FL(\()p FO(j)t
FK(!)p FO(i)p FL(\))378 5479 y FT(and)37 b(hence)h(\011)g
FN(!)1022 5493 y FL(rc)1123 5479 y FT(\010.)63 b(Consequen)m(tly)37
b(the)h(relation)f Fl(u)2424 5493 y FL(rc)2525 5479 y
FT(is)g(symmetric)g(as)h(w)m(ell)f(as)h(it)g(is)f(the)378
5592 y(re\015exiv)m(e)h(transitiv)m(e)g(closure)f(of)h
FN(!)1668 5606 y FL(rc)1731 5592 y FT(.)64 b(As)38 b
Fl(u)2033 5606 y FL(rc)2134 5592 y FT(is)f(re\015exiv)m(e)h(and)g
(transitiv)m(e)f(b)m(y)h(de\014nition,)g(it)378 5705
y(follo)m(ws)29 b(that)i(it)f(is)g(an)g(equiv)-5 b(alence)30
b(relation.)1766 b Ff(\004)p eop
%%Page: 136 146
136 145 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(136)519 509 y(An)49 b(alternativ)m(e)h
(w)m(a)m(y)g(of)g(c)m(haracterising)f(the)h(colour)f(renaming)f
(relation)h Fl(u)3395 523 y FL(rc)3507 509 y FT(b)s(et)m(w)m(een)378
622 y(coloured)38 b(ob)5 b(jects)40 b(\(suc)m(h)f(as)g(form)m(ulae,)h
(sets)g(of)f(form)m(ulae,)h(etc.)17 b(\))66 b(is)38 b(b)m(y)g(a)i
(bijectiv)m(e)e(function)378 735 y(mapping)32 b(the)j(colours)e(in)g
(one)h(ob)5 b(ject)35 b(to)g(another.)52 b(This)32 b(is)h(giv)m(en)h(b)
m(y)g(the)g(follo)m(wing)e(prop)s(osi-)378 848 y(tion.)378
1016 y FQ(Prop)s(osition)k(7.9)g(\(Recolouring)g(Bijection\))46
b FI(Given)26 b(two)h(c)-5 b(olour)g(e)g(d)28 b(formulae)f(\(or)g
(alterna-)378 1128 y(tively)35 b(sets)h(of)f(c)-5 b(olour)g(e)g(d)38
b(formulae,)f(sets)f(of)f(c)-5 b(olours,)38 b(etc.)15
b(\))50 b FP(A)35 b FI(and)h FP(B)5 b FI(,)36 b(wher)-5
b(e)36 b(the)g(set)f Fe(C)p FT(\()p FP(A)p FT(\))h FI(is)378
1241 y(\014nite,)29 b(then)g FP(A)d Fl(u)1000 1255 y
FL(rc)1088 1241 y FP(B)33 b FI(if)28 b(and)i(only)f(if)f(ther)-5
b(e)30 b(is)e(a)h(r)-5 b(e)g(c)g(olouring)31 b(bije)-5
b(ction)29 b Fe(R)f FI(mapping)i(the)f(c)-5 b(olours)378
1354 y(in)32 b FP(A)h FI(to)g(the)g(c)-5 b(olours)35
b(in)d FP(B)37 b FI(such)c(that)h Fe(R)p FT(\()p FP(A)p
FT(\))26 b(=)f FP(B)5 b FI(.)378 1522 y FQ(Pro)s(of)p
FT(:)30 b(W)-8 b(e)30 b(\014rst)e(sho)m(w)h(the)g(`only)g(if)7
b(')28 b(direction.)39 b(Giv)m(en)29 b(that)g(there)g(is)f(a)i
(bijection)d Fe(R)j FT(mapping)378 1635 y(the)g(colours)g(in)f
FP(A)h FT(to)h FP(B)k FT(then)30 b(the)g(sets)g Fe(C)p
FT(\()p FP(A)p FT(\))h(and)f Fe(C)p FT(\()p FP(B)5 b
FT(\))30 b(ha)m(v)m(e)h(the)g(same)f(n)m(um)m(b)s(er)f(of)h(elemen)m
(ts,)378 1747 y FP(n)f FT(sa)m(y)-8 b(.)42 b(No)m(w,)31
b(let)e Fe(C)p FT(\()p FP(A)p FT(\))d(=)f FN(f)p FP(i)1401
1761 y FL(1)1441 1747 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(i)1689 1761 y FO(n)1737 1747 y FN(g)30 b FT(and)f Fe(C)p
FT(\()p FP(B)5 b FT(\))25 b(=)g FN(f)p FP(j)2391 1761
y FL(1)2432 1747 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(j)2685
1761 y FO(n)2733 1747 y FN(g)p FT(,)31 b(and)e(let)h
FN(f)p FP(k)3233 1761 y FL(1)3273 1747 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(k)3537 1761 y FO(n)3584 1747 y FN(g)30 b FT(b)s(e)f(a)378
1860 y(set)34 b(of)g FP(n)g FT(colours)f(suc)m(h)h(that)g(for)g(all)f
FP(x)e FN(2)g(f)p FT(1)p FP(;)15 b(:)g(:)g(:)32 b(;)15
b(n)p FN(g)34 b FT(the)h(colour)e FP(k)2813 1874 y FO(x)2898
1860 y FP(=)-55 b FN(2)31 b Fe(C)p FT(\()p FP(A)p FT(\))j(and)g
FP(k)3436 1874 y FO(x)3521 1860 y FP(=)-55 b FN(2)31
b Fe(C)p FT(\()p FP(B)5 b FT(\).)378 1973 y(Then)1096
2178 y FP(A)83 b FN(!)1338 2192 y FL(rc)1484 2178 y FP(A)1552
2140 y FL(\()p FO(i)1603 2149 y FC(1)1638 2140 y FK(!)p
FO(k)1746 2149 y FC(1)1780 2140 y FL(\))1247 2315 y FN(!)1338
2282 y FK(\003)1338 2338 y FL(rc)1484 2315 y FT(\(\()p
FP(A)1622 2278 y FL(\()p FO(i)1673 2287 y FC(1)1709 2278
y FK(!)p FO(k)1817 2287 y FC(1)1851 2278 y FL(\))1882
2315 y FT(\))1917 2278 y FK(\001\001\001)1981 2315 y
FT(\))2016 2278 y FL(\()p FO(i)2067 2286 y Fy(n)2110
2278 y FK(!)p FO(k)2218 2286 y Fy(n)2260 2278 y FL(\))1247
2453 y FN(!)1338 2420 y FK(\003)1338 2476 y FL(rc)1484
2453 y FT(\(\(\()p FP(A)1657 2416 y FL(\()p FO(i)1708
2425 y FC(1)1744 2416 y FK(!)p FO(k)1852 2425 y FC(1)1886
2416 y FL(\))1918 2453 y FT(\))1953 2416 y FK(\001\001\001)2016
2453 y FT(\))2051 2416 y FL(\()p FO(i)2102 2424 y Fy(n)2145
2416 y FK(!)p FO(k)2253 2424 y Fy(n)2295 2416 y FL(\))2327
2453 y FT(\))2362 2416 y FL(\()p FO(k)2426 2425 y FC(1)2461
2416 y FK(!)p FO(j)2561 2425 y FC(1)2594 2416 y FL(\))1247
2591 y FN(!)1338 2558 y FK(\003)1338 2613 y FL(rc)1484
2591 y FT(\(\(\(\(\()p FP(A)1727 2553 y FL(\()p FO(i)1778
2562 y FC(1)1815 2553 y FK(!)p FO(k)1923 2562 y FC(1)1957
2553 y FL(\))1988 2591 y FT(\))2023 2553 y FK(\001\001\001)2087
2591 y FT(\))2122 2553 y FL(\()p FO(i)2173 2561 y Fy(n)2216
2553 y FK(!)p FO(k)2324 2561 y Fy(n)2366 2553 y FL(\))2397
2591 y FT(\))2432 2553 y FL(\()p FO(k)2496 2562 y FC(1)2531
2553 y FK(!)p FO(j)2631 2562 y FC(1)2665 2553 y FL(\))2697
2591 y FT(\))2732 2553 y FK(\001\001\001)2795 2591 y
FT(\))2830 2553 y FL(\()p FO(k)2894 2561 y Fy(n)2937
2553 y FK(!)p FO(j)3037 2561 y Fy(n)3079 2553 y FL(\))1288
2729 y FT(=)125 b FP(B)5 b(:)378 2933 y FT(Hence)31 b
FP(A)26 b Fl(u)813 2947 y FL(rc)901 2933 y FP(B)5 b FT(.)519
3046 y(F)-8 b(or)41 b(the)g(`if)7 b(')40 b(direction,)i(w)m(e)f
(\014rst)e(sho)m(w)i(that)g(if)e FP(A)j FN(!)2505 3060
y FL(rc)2610 3046 y FP(B)j FT(then)40 b(there)h(is)f(a)g(bijection)g
Fe(R)378 3159 y FT(mapping)24 b(the)i(colours)g(in)f
FP(A)h FT(to)g(the)h(colours)e(in)g FP(B)30 b FT(suc)m(h)c(that)g
Fe(R)p FT(\()p FP(A)p FT(\))h(=)e FP(B)5 b FT(.)38 b(Giv)m(en)26
b(that)h FP(A)e FN(!)3666 3173 y FL(rc)3754 3159 y FP(B)378
3272 y FT(then)36 b(there)h(is)e(a)i(colour)f FP(i)g
FN(2)f Fe(C)p FT(\()p FP(A)p FT(\))i(and)f FP(j)41 b
FN(2)35 b Fe(C)p FT(\()p FP(B)5 b FT(\))36 b(suc)m(h)g(that)h
FP(B)j FT(=)35 b FP(A)2972 3239 y FL(\()p FO(i)p FK(!)p
FO(j)t FL(\))3158 3272 y FT(,)j(and)e(so)h Fe(C)p FT(\()p
FP(B)5 b FT(\))35 b(=)378 3385 y(\()p Fe(C)p FT(\()p
FP(A)p FT(\))21 b FN(\000)f(f)p FP(i)p FN(g)p FT(\))i
FN([)e(f)p FP(j)5 b FN(g)p FT(.)42 b(Therefore,)30 b(w)m(e)h(de\014ne)f
Fe(R)g FT(as)h(follo)m(ws:)1649 3589 y Fe(R)p FT(\()p
FP(x)p FT(\))84 b(=)f FP(x)p FT(,)30 b(if)g FP(x)25 b
FN(6)p FT(=)g FP(i)1930 3727 y FT(=)83 b FP(j)36 b FT(otherwise.)378
3931 y(No)m(w)23 b(to)g(sho)m(w)g(that)g(suc)m(h)f(a)h(mapping)d
Fe(R)j FT(exists)f(giv)m(en)h(that)g FP(A)i Fl(u)2641
3945 y FL(rc)2729 3931 y FP(B)5 b FT(,)24 b(w)m(e)f(notice)g(that)g(if)
e FP(A)k Fl(u)3666 3945 y FL(rc)3754 3931 y FP(B)378
4044 y FT(then)30 b(there)h(is)e(a)i(\014nite)e(sequence)i(of)f(form)m
(ulae)g FN(h)p FP(X)2181 4058 y FL(1)2221 4044 y FP(;)15
b(X)2336 4058 y FL(2)2377 4044 y FP(;)g(:)g(:)g(:)31
b(;)15 b(X)2668 4058 y FO(n)2716 4044 y FN(i)31 b FT(suc)m(h)f(that)
1363 4248 y FP(A)25 b FT(=)g FP(X)1627 4262 y FL(1)1692
4248 y FN(!)1783 4262 y FL(rc)1871 4248 y FP(X)1946 4262
y FL(2)2011 4248 y FN(!)2102 4262 y FL(rc)2190 4248 y
FN(\001)15 b(\001)g(\001)27 b(!)2413 4262 y FL(rc)2501
4248 y FP(X)2576 4262 y FO(n)2648 4248 y FT(=)e FP(B)5
b(:)378 4452 y FT(Therefore)27 b(there)g(are)g(bijectiv)m(e)g(mappings)
e Fe(R)2003 4466 y FL(1)2042 4452 y FP(;)15 b Fe(R)2157
4466 y FL(2)2198 4452 y FP(;)g(:)g(:)g(:)31 b(;)15 b
Fe(R)2489 4466 y FO(n)p FK(\000)p FL(1)2654 4452 y FT(where)26
b Fe(R)2988 4466 y FO(x)3059 4452 y FT(maps)h(the)g(colours)f(in)378
4565 y FP(X)453 4579 y FO(x)525 4565 y FT(to)j(the)g(colours)f(in)f
FP(X)1275 4579 y FO(x)p FL(+1)1437 4565 y FT(for)h FP(x)e
FN(2)e(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)h
FN(\000)g FT(1)p FN(g)p FT(.)41 b(Hence)29 b(w)m(e)g(de\014ne)e
Fe(R)i FT(to)g(b)s(e)e(\(\()p Fe(R)3498 4579 y FL(1)3555
4565 y FN(\016)16 b Fe(R)3691 4579 y FL(2)3731 4565 y
FT(\))g FN(\016)378 4678 y(\001)f(\001)g(\001)21 b(\016)g
Fe(R)645 4692 y FO(n)p FK(\000)p FL(1)782 4678 y FT(\).)2915
b Ff(\004)378 4914 y FG(7.4.2)112 b(Consistency)38 b(Results)e(on)i
(Recoloured)e(Sets)378 5086 y FT(It)28 b(is)f(straigh)m(tforw)m(ard)h
(to)h(sho)m(w)f(that)h(if)e(t)m(w)m(o)i(coloured)f(problems)e(are)j
(isomorphic)d(b)m(y)i(renaming)378 5199 y(colours)36
b(then)h(they)g(are)h(equiv)-5 b(alen)m(t,)38 b(in)e(the)h(sense)g
(that)h(they)f(are)h(either)e(b)s(oth)g(consisten)m(t)i(or)378
5312 y(b)s(oth)30 b(inconsisten)m(t.)378 5479 y FQ(Prop)s(osition)36
b(7.10)g(\(Renaming)e(Colours)h(Preserv)m(es)h FN(K)q
FQ(-Consistency\))47 b FI(Given)39 b(sets)h(of)378 5592
y(sentenc)-5 b(es)40 b FP(S)843 5606 y FL(1)882 5592
y FI(,)i FP(S)1008 5606 y FL(2)1087 5592 y FI(and)f(c)-5
b(onne)g(ctability)41 b(r)-5 b(elations)42 b FN(K)2294
5606 y FL(1)2334 5592 y FI(,)f FN(K)2472 5606 y FL(2)2552
5592 y FI(such)f(that)h FT(\()p FP(S)3050 5606 y FL(1)3090
5592 y FP(;)15 b FN(K)3199 5606 y FL(1)3239 5592 y FT(\))39
b Fl(u)3384 5606 y FL(rc)3485 5592 y FT(\()p FP(S)3576
5606 y FL(2)3616 5592 y FP(;)15 b FN(K)3725 5606 y FL(2)3765
5592 y FT(\))p FI(,)378 5705 y(then)33 b FP(S)636 5719
y FL(1)708 5705 y FI(is)f FN(K)874 5719 y FL(1)914 5705
y FI(-c)-5 b(onsistent)33 b(if)g(and)g(only)h(if)e FP(S)1976
5719 y FL(2)2048 5705 y FI(is)g FN(K)2214 5719 y FL(2)2254
5705 y FI(-c)-5 b(onsistent.)p eop
%%Page: 137 147
137 146 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(137)378 396 y FQ(Pro)s(of)p
FT(:)31 b(Straigh)m(tforw)m(ard;)f(b)m(y)h(sho)m(wing)e(that)i
FP(S)2098 356 y Fd(x)p FK(K)2223 365 y FC(1)2093 422
y FL(1)2291 396 y FT(is)f(satis\014able)f(if)g(and)h(only)g(if)f
FP(S)3407 356 y Fd(x)p FK(K)3532 365 y FC(2)3402 422
y FL(2)3600 396 y FT(is.)70 b Ff(\004)519 622 y FT(One)35
b(can)h(also)g(recolour)f(literals)f(of)i(more)f(than)h(one)g(colour)f
(in)m(to)h(a)g(single)e(one)i(in)e(certain)378 735 y(coloured)29
b(problems)g(without)g(a\013ecting)h(their)f(consistency)-8
b(.)41 b(F)-8 b(or)31 b(example,)f(let)g(us)f(consider)g(the)378
848 y(connectabilit)m(y)37 b(relation)g FN(K)i FT(=)e
FP(i)h FN($)f FP(j)44 b FN($)37 b FP(k)s FT(.)63 b(A)38
b(pair)e(of)i(coloured)f(literals)f(is)h FN(K)q FT(-inconsisten)m(t)378
961 y(if)f(and)h(only)f(their)g(uncoloured)g(pro)5 b(jections)37
b(are)g(complemen)m(tary)h(and)e(one)i(of)f(the)g(literals)f(is)378
1074 y(coloured)28 b(with)f FP(j)34 b FT(and)28 b(the)g(other)h(one)g
(with)e FP(i)i FT(or)f FP(k)s FT(.)40 b(One)28 b(can)h(th)m(us)f
(recolour)g(all)f(the)i FP(k)s FT(-coloured)378 1187
y(literals)42 b(\(if)g(an)m(y\))i(in)e(the)h(pair)f(with)g
FP(i)i FT(without)e(a\013ecting)i(its)e(consistency)-8
b(.)80 b(In)42 b(general,)47 b(for)378 1300 y FN(K)27
b FT(=)e FP(i)g FN($)g FP(j)31 b FN($)26 b FP(k)s FT(,)j(a)h(set)f
FP(S)34 b FT(is)28 b FN(K)q FT(-consisten)m(t)i(if)e(and)h(only)f
FP(S)2465 1267 y FL(\()p FO(k)r FK(!)p FO(i)p FL(\))2686
1300 y FT(is.)39 b(The)29 b(follo)m(wing)e(prop)s(osition)378
1413 y(giv)m(es)32 b(a)g(more)f(general)h(statemen)m(t)h(on)f(the)g
(recolouring)e(of)i(a)g(n)m(um)m(b)s(er)e(of)h(literals)f(in)h(a)g
(problem)378 1526 y(without)e(a\013ecting)i(its)e(consistency)-8
b(.)41 b(Basically)29 b(if)g(t)m(w)m(o)i(disjoin)m(t)d(sets)j(of)f
(colours)f FN(P)3328 1540 y FL(1)3398 1526 y FT(and)g
FN(P)3637 1540 y FL(2)3706 1526 y FT(are)378 1638 y(iden)m(ti\014ed)21
b(in)i(the)g(colours)g(of)h(a)f(problem)f(\()p FP(S;)15
b FN(K)q FT(\))25 b(suc)m(h)e(that)h FN(P)2542 1652 y
FL(1)2605 1638 y FT(is)f(the)g(range)h(under)e FN(K)j
FT(for)e(eac)m(h)i(of)378 1751 y(the)k(colours)f(in)f
FN(P)1007 1765 y FL(2)1047 1751 y FT(,)i(then)f(all)g(the)g(literals)f
(coloured)i(with)e FN(P)2513 1765 y FL(2)2581 1751 y
FT(can)i(b)s(e)f(recoloured)g(to)h(an)m(y)g(colour)378
1864 y(in)f FN(P)546 1878 y FL(2)614 1864 y FT(without)g(a\013ecting)i
(the)f(consistency)g(of)g(the)g(problem.)39 b(F)-8 b(or)30
b(the)f(case)h(of)f FN(K)e FT(=)e FP(i)g FN($)g FP(j)31
b FN($)26 b FP(k)s FT(,)378 1977 y(w)m(e)31 b(can)f(see)h(that)g(the)g
(set)g FN(P)1384 1991 y FL(1)1449 1977 y FT(=)25 b FN(f)p
FP(j)5 b FN(g)32 b FT(and)d FN(P)1948 1991 y FL(2)2013
1977 y FT(=)c FN(f)p FP(i;)15 b(k)s FN(g)p FT(.)43 b(This)28
b(result)i(is)f(deriv)m(ed)g(here.)378 2190 y FQ(Theorem)34
b(7.5)46 b FI(Given)39 b(a)h(c)-5 b(onne)g(ctability)41
b(r)-5 b(elation)41 b FN(K)q FI(,)h(and)e(two)h(disjoint)f(sets)g(of)f
(c)-5 b(olours)41 b FN(P)3788 2204 y FL(1)378 2303 y
FI(and)34 b FN(P)618 2317 y FL(2)691 2303 y FI(such)g(that)h
FN(K)q FT(\()p FP(i)p FT(\))28 b(=)e FN(P)1445 2317 y
FL(1)1518 2303 y FI(for)34 b(every)g FP(i)27 b FN(2)f(P)2114
2317 y FL(2)2154 2303 y FI(,)33 b(then)h(for)g(every)f(c)-5
b(olour)35 b FP(m)27 b FN(2)f(P)3336 2317 y FL(2)3376
2303 y FI(,)33 b(a)h(set)f FP(S)39 b FI(of)378 2416 y(c)-5
b(olour)g(e)g(d)38 b(\014rst-or)-5 b(der)38 b(sentenc)-5
b(es)37 b(is)f FN(K)q FI(-c)-5 b(onsistent)37 b(if)f(and)h(only)g(if)e
(the)i(r)-5 b(e)g(c)g(olour)g(e)g(d)39 b(set)d FP(S)3552
2383 y FL(\()p FK(P)3628 2392 y FC(2)3663 2383 y FK(!)p
FO(m)p FL(\))378 2528 y FI(is)d FN(K)q FI(-c)-5 b(onsistent.)378
2741 y FQ(Pro)s(of)p FT(:)28 b(First)d(of)i(all)e(w)m(e)i(notice)g
(that)g(if)f FN(P)1853 2755 y FL(1)1919 2741 y FT(and)g
FN(P)2155 2755 y FL(2)2221 2741 y FT(are)h(giv)m(en)f(as)h(required,)f
(then)g(for)g(all)f(colours)378 2854 y FP(i)31 b FT(and)e
FP(j)36 b FT(in)29 b FN(P)858 2868 y FL(2)898 2854 y
FT(,)i(the)f(colours)g FP(i)c FN(6\030)1547 2868 y FK(K)1630
2854 y FP(j)36 b FT(since)29 b(the)i(set)g FN(K)q FT(\()p
FP(i)p FT(\))c(=)e FN(P)2581 2868 y FL(1)2651 2854 y
FT(and)k FN(P)2890 2868 y FL(1)2950 2854 y FN(\\)20 b(P)3094
2868 y FL(2)3159 2854 y FT(=)25 b FN(fg)p FT(.)519 2967
y(No)m(w)44 b(giv)m(en)f(a)g(connectabilit)m(y)f(relation)g
FN(K)q FT(,)47 b(the)c(sets)h(of)f(colours)f FN(P)3023
2981 y FL(1)3063 2967 y FT(,)k FN(P)3197 2981 y FL(2)3236
2967 y FT(,)h(and)42 b(a)h(colour)378 3080 y FP(m)25
b FN(2)g(P)632 3094 y FL(2)672 3080 y FT(,)30 b(w)m(e)h(pro)m(v)m(e)g
(the)g(`only-if)7 b(')29 b(direction)g(b)m(y)h(de\014ning)f(the)h
(collection)g(of)h(sets)g FN(C)3342 3094 y FL(1)3411
3080 y FT(as)g(follo)m(ws:)1042 3284 y FN(C)1090 3298
y FL(1)1212 3284 y FT(=)83 b FN(f)p FP(S)1472 3246 y
FL(\()p FK(P)1548 3255 y FC(2)1584 3246 y FK(!)p FO(m)p
FL(\))1774 3284 y FN(j)25 b FP(S)35 b FT(is)30 b FN(K)q
FT(-consisten)m(t)h(and)f FN(P)2770 3298 y FL(2)2835
3284 y FN(\022)25 b Fe(C)p FT(\()p FP(S)5 b FT(\))q FN(g)378
3488 y FT(and)38 b(w)m(e)h(deduce)f(that)h FN(C)1269
3502 y FL(1)1347 3488 y FT(is)f(a)h FN(K)q FT(-consistency)g(prop)s
(ert)m(y)-8 b(.)65 b(Sho)m(wing)38 b(that)h FN(C)3165
3502 y FL(1)3242 3488 y FT(satis\014es)g(condi-)378 3601
y(tions)32 b(2{6)i(of)e(De\014nition)f(7.13)j(is)e(routine)f(and)h(w)m
(e)h(illustrate)e(here)h(only)g(the)h(pro)s(of)e(of)i(the)g(\014rst)378
3714 y(condition.)514 3902 y FN(\017)46 b FT(Let)41 b(some)f(set)g
FP(S)1227 3869 y FL(\()p FK(P)1303 3878 y FC(2)1339 3869
y FK(!)p FO(m)p FL(\))1545 3902 y FN(2)g(C)1694 3916
y FL(1)1773 3902 y FT(and)g(supp)s(ose)e(that)j(some)f(literals)e
FP(A)3131 3869 y FO(i)3199 3902 y FT(and)i FN(:)p FP(A)3515
3869 y FO(j)3591 3902 y FT(are)g(in)605 4015 y FP(S)666
3982 y FL(\()p FK(P)742 3991 y FC(2)778 3982 y FK(!)p
FO(m)p FL(\))942 4015 y FT(,)32 b(and)f(that)i FP(i)27
b FN(\030)1505 4029 y FK(K)1590 4015 y FP(j)5 b FT(.)46
b(If)31 b FP(i)c FT(=)g FP(m)32 b FT(and)f FP(j)i FT(=)27
b FP(m)k FT(then)g FP(i)d FN(6\030)2858 4029 y FK(K)2943
4015 y FP(j)38 b FT(as)31 b FP(m)d FN(6\030)3309 4029
y FK(K)3394 4015 y FP(m)p FT(.)44 b(Also,)32 b(if)605
4127 y FP(i)26 b FN(6)p FT(=)f FP(m)j FT(and)g FP(j)j
FN(6)p FT(=)25 b FP(m)j FT(then)g FP(A)1586 4094 y FO(i)1643
4127 y FT(and)g FN(:)p FP(A)1947 4094 y FO(j)2012 4127
y FT(are)h(b)s(oth)f(in)f FP(S)5 b FT(,)29 b(whic)m(h)f(is)f(a)i(con)m
(tradiction)g(as)f FP(S)34 b FT(is)605 4240 y FN(K)q
FT(-consisten)m(t.)42 b(Therefore,)29 b(one)g(of)g(the)g(colours,)g(sa)
m(y)g FP(j)5 b FT(,)30 b(is)e(equal)h(to)g FP(m)p FT(,)g(and)g(the)g
(other,)g FP(i)p FT(,)605 4353 y(is)i(not.)46 b(So,)32
b FP(A)1121 4320 y FO(i)1177 4353 y FN(2)27 b FP(S)37
b FT(and)31 b FN(:)p FP(A)1665 4320 y FO(k)1736 4353
y FN(2)c FP(S)37 b FT(for)31 b(some)i FP(k)e FN(2)c(P)2516
4367 y FL(2)2556 4353 y FT(,)32 b(and)f(th)m(us)h FP(i)c
FN(\030)3123 4367 y FK(K)3208 4353 y FP(k)s FT(.)46 b(But)32
b(this)f(is)f(a)605 4466 y(con)m(tradiction,)h(as)f FP(S)36
b FT(is)29 b FN(K)q FT(-consisten)m(t.)378 4654 y(Th)m(us)g
FN(C)656 4668 y FL(1)726 4654 y FT(is)g(a)i FN(K)q FT(-consistency)g
(prop)s(ert)m(y)f(and)f(so)i(whenev)m(er)f FP(S)36 b
FT(is)29 b FN(K)q FT(-consisten)m(t,)j(so)e(is)g FP(S)3527
4621 y FL(\()p FK(P)3603 4630 y FC(2)3638 4621 y FK(!)p
FO(m)p FL(\))3803 4654 y FT(.)519 4767 y(The)g(pro)s(of)g(of)g(the)h
(`if)7 b(')30 b(case)h(pro)s(ceeds)f(similarly)-8 b(.)38
b(First)29 b(w)m(e)i(de\014ne)f FN(C)2975 4781 y FL(2)3045
4767 y FT(as)g(follo)m(ws:)1042 4971 y FN(C)1090 4985
y FL(2)1212 4971 y FT(=)83 b FN(f)p FP(S)31 b FN(j)25
b FP(S)1609 4938 y FL(\()p FK(P)1685 4947 y FC(2)1721
4938 y FK(!)p FO(m)p FL(\))1915 4971 y FT(is)30 b FN(K)q
FT(-consisten)m(t)h(and)f FN(P)2770 4985 y FL(2)2835
4971 y FN(\022)25 b Fe(C)p FT(\()p FP(S)5 b FT(\))q FN(g)378
5175 y FT(and)26 b(sho)m(w)g(that)i FN(C)1014 5189 y
FL(2)1080 5175 y FT(is)d(a)i FN(K)q FT(-consistency)h(prop)s(ert)m(y)-8
b(.)39 b(Once)27 b(again,)g(w)m(e)g(only)f(giv)m(e)h(the)g(pro)s(of)f
(of)h(the)378 5288 y(\014rst)j(condition.)514 5476 y
FN(\017)46 b FT(Let)38 b(some)g(set)h FP(S)j FN(2)37
b(C)1404 5490 y FL(2)1481 5476 y FT(con)m(tain)h(the)g(literals)e
FP(A)2347 5443 y FO(i)2413 5476 y FT(and)h FN(:)p FP(A)2726
5443 y FO(j)2800 5476 y FT(and)g(let)g FP(i)h FN(\030)3262
5490 y FK(K)3357 5476 y FP(j)5 b FT(.)63 b(No)m(w)39
b(the)605 5589 y(colours)27 b FP(i)g FT(and)f FP(j)33
b FT(cannot)27 b(b)s(e)g(b)s(oth)f(outside)g FN(P)2211
5603 y FL(2)2278 5589 y FT(as)h FP(S)2447 5556 y FL(\()p
FK(P)2523 5565 y FC(2)2558 5556 y FK(!)p FO(m)p FL(\))2750
5589 y FT(is)f FN(K)q FT(-consisten)m(t.)41 b(Also,)27
b FP(i)g FT(and)605 5702 y FP(j)41 b FT(cannot)35 b(b)s(e)f(b)s(oth)g
(in)f FN(P)1505 5716 y FL(2)1580 5702 y FT(as)i FP(i)e
FN(\030)1831 5716 y FK(K)1921 5702 y FP(j)5 b FT(,)37
b(and)d(therefore)h(one,)i(sa)m(y)e FP(i)p FT(,)h(is)e(outside)g
FN(P)3520 5716 y FL(2)3560 5702 y FT(,)i(while)p eop
%%Page: 138 148
138 147 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(138)605 396 y(the)33 b(other,)h
FP(j)5 b FT(,)34 b(is)d(in)h FN(P)1396 410 y FL(2)1435
396 y FT(.)48 b(As)32 b FP(i)d FN(\030)1775 410 y FK(K)1862
396 y FP(j)5 b FT(,)34 b(then)f FP(i)c FN(2)g(K)q FT(\()p
FP(i)p FT(\))h(=)f FN(P)2687 410 y FL(1)2726 396 y FT(.)48
b(Hence,)34 b FP(A)3165 363 y FO(i)3222 396 y FN(2)29
b FP(S)3373 363 y FL(\()p FK(P)3449 372 y FC(2)3484 363
y FK(!)p FO(m)p FL(\))3681 396 y FT(and)605 509 y FN(:)p
FP(A)734 476 y FO(m)837 509 y FN(2)35 b FP(S)994 476
y FL(\()p FK(P)1070 485 y FC(2)1106 476 y FK(!)p FO(m)p
FL(\))1270 509 y FT(,)k(but)d(this)g(is)g(a)h(con)m(tradiction)g(as)g
FP(i)g FN(\030)2688 523 y FK(K)2782 509 y FP(m)f FT(and)h(the)g(set)g
FP(S)3454 476 y FL(\()p FK(P)3530 485 y FC(2)3565 476
y FK(!)p FO(m)p FL(\))3767 509 y FT(is)605 622 y FN(K)q
FT(-consisten)m(t.)378 801 y(Hence,)28 b FN(C)718 815
y FL(2)783 801 y FT(is)d(a)h FN(K)q FT(-consistency)h(prop)s(ert)m(y)e
(and)h(consequen)m(tly)-8 b(,)27 b(the)f(set)h FP(S)j
FT(is)25 b FN(K)q FT(-consisten)m(t)j(when-)378 914 y(ev)m(er)j
FP(S)631 881 y FL(\()p FK(P)707 890 y FC(2)743 881 y
FK(!)p FO(m)p FL(\))937 914 y FT(is.)2733 b Ff(\004)378
1199 y FH(7.5)135 b(Coloured)46 b(In)l(terp)t(olan)l(ts)378
1402 y FT(In)25 b(this)g(section)h(w)m(e)g(deriv)m(e)f(an)h(in)m(terp)s
(olation)e(theorem)i(for)g(the)g(coloured)f(\014rst-order)g(logic.)39
b(Our)378 1515 y(notion)31 b(of)h(an)f(in)m(terp)s(olan)m(t)g(is)f(a)i
(generalisation)f(of)h(the)g(standard)e(de\014nition)g(of)h(an)h(in)m
(terp)s(olan)m(t)378 1628 y(for)e(\014rst-order)g(sen)m(tences.)41
b(An)30 b(\(uncoloured\))g(in)m(terp)s(olan)m(t)f(is)h(de\014ned)f(as)i
(follo)m(ws:)378 1806 y FQ(De\014nition)k(7.21)h(\(In)m(terp)s(olan)m
(t)e(for)h(Sen)m(tences\))45 b FT(The)24 b(\014rst-order)f(sen)m(tence)
j FP(I)31 b FT(is)23 b(called)g(an)378 1919 y(in)m(terp)s(olan)m(t)k
(for)h(the)h(sen)m(tence)g FP(X)k FN(\))25 b FP(Y)48
b FT(if)27 b(ev)m(ery)i(function)e(sym)m(b)s(ol)g(and)g(relation)h(sym)
m(b)s(ol)f(\(with)378 2032 y(the)k(exception)g(of)g FN(>)g
FT(and)f FN(?)p FT(\))h(in)f FP(I)38 b FT(o)s(ccurs)31
b(in)e(b)s(oth)i FP(X)38 b FT(and)30 b FP(Y)51 b FT(and)30
b(the)i(sen)m(tences)g FP(X)h FN(\))26 b FP(I)38 b FT(and)378
2145 y FP(I)32 b FN(\))25 b FP(Y)51 b FT(are)31 b(v)-5
b(alid.)2719 b Ff(\003)378 2324 y FT(Or)32 b(equiv)-5
b(alen)m(tly)d(,)32 b(in)m(terp)s(olan)m(ts)g(can)h(b)s(e)f(de\014ned)f
(on)i(\014nite)e(sets)i(of)g(sen)m(tences.)49 b(W)-8
b(e)34 b(\014rst)e(de\014ne)378 2437 y(the)45 b(notion)f(of)g(a)h(set)g
(of)g(sen)m(tences)h(partitioned)d(b)m(y)h(a)h(pair)e(of)i(sen)m
(tences)h(and)d(then)i(de\014ne)378 2550 y(in)m(terp)s(olan)m(ts)29
b(for)h(partitions.)378 2752 y FQ(De\014nition)35 b(7.22)h(\(P)m
(artitions\))45 b FT(Let)25 b FP(S)5 b FT(,)26 b FP(S)1992
2766 y FL(1)2055 2752 y FT(and)e FP(S)2282 2766 y FL(2)2346
2752 y FT(b)s(e)g(sets)h(of)f(sen)m(tences.)40 b(The)24
b(pair)f(\()p FP(S)3617 2766 y FL(1)3657 2752 y FP(;)15
b(S)3753 2766 y FL(2)3793 2752 y FT(\))378 2865 y(is)29
b(a)i(partition)e(of)i(the)f(set)h FP(S)k FT(if)514 3043
y FN(\017)46 b FP(S)661 3057 y FL(1)721 3043 y FN([)19
b FP(S)857 3057 y FL(2)922 3043 y FT(=)25 b FP(S)5 b
FT(,)30 b(and)514 3228 y FN(\017)46 b FP(S)661 3242 y
FL(1)721 3228 y FN(\\)19 b FP(S)857 3242 y FL(2)922 3228
y FT(=)25 b FN(fg)p FT(.)2624 b Ff(\003)378 3429 y FT(It)30
b(is)g(clear)g(that)h(if)f(\()p FP(S)1157 3443 y FL(1)1196
3429 y FP(;)15 b(S)1292 3443 y FL(2)1332 3429 y FT(\))30
b(is)g(a)h(partition)e(of)h(some)h(set)g FP(S)k FT(then)30
b(so)h(is)e(\()p FP(S)3009 3443 y FL(2)3049 3429 y FP(;)15
b(S)3145 3443 y FL(1)3184 3429 y FT(\).)378 3631 y FQ(De\014nition)35
b(7.23)h(\(In)m(terp)s(olan)m(t)e(for)h(Sets\))45 b FT(Giv)m(en)27
b(that)h FP(S)33 b FT(is)26 b(a)i(\014nite)e(set)i(of)g(sen)m(tences)g
(and)378 3744 y(that)40 b(\()p FP(S)675 3758 y FL(1)715
3744 y FP(;)15 b(S)811 3758 y FL(2)851 3744 y FT(\))40
b(is)f(a)h(partition)e(of)i FP(S)45 b FT(then)40 b(the)g(sen)m(tence)h
FP(I)47 b FT(is)39 b(said)g(to)h(b)s(e)f(an)h(in)m(terp)s(olan)m(t)f
(for)378 3857 y(\()p FP(S)469 3871 y FL(1)508 3857 y
FP(;)15 b(S)604 3871 y FL(2)644 3857 y FT(\))32 b(if)e(ev)m(ery)i
(function)e(sym)m(b)s(ol)g(and)h(relation)f(sym)m(b)s(ol)g(\(with)h
(the)g(exception)h(of)f FN(>)g FT(and)g FN(?)p FT(\))378
3970 y(in)24 b FP(I)32 b FT(o)s(ccurs)25 b(in)f(b)s(oth)h
FP(S)1194 3984 y FL(1)1258 3970 y FT(and)g FP(S)1486
3984 y FL(2)1550 3970 y FT(and)g(the)g(sets)h FP(S)2102
3984 y FL(1)2151 3970 y FN([)10 b(f)p FP(I)d FN(g)26
b FT(and)e FP(S)2612 3984 y FL(2)2661 3970 y FN([)10
b(f:)p FP(I)d FN(g)26 b FT(are)f(b)s(oth)g(unsatis\014able.)378
4083 y Ff(\003)519 4285 y FT(Note)42 b(that)e FP(I)47
b FT(is)40 b(an)g(in)m(terp)s(olan)m(t)f(for)h FP(X)49
b FN(\))41 b FP(Y)60 b FT(if)39 b(and)h(only)f(if)g FN(:)p
FP(I)47 b FT(is)39 b(an)h(in)m(terp)s(olan)m(t)f(for)378
4398 y(\()p FN(f)p FP(X)7 b FN(g)p FP(;)15 b FN(f:)p
FP(Y)22 b FN(g)p FT(\).)519 4511 y(An)30 b(in)m(teresting)f(and)g
(quite)g(imp)s(ortan)m(t)g(result)g(in)g(\014rst-order)g(logic)g
(states)i(that)g(ev)m(ery)g(v)-5 b(alid)378 4624 y(implication)29
b(has)i(an)g(in)m(terp)s(olan)m(t.)42 b(This)30 b(result)g(is)g(due)h
(to)h(Craig)e(\(1957\))k(and)d(is)f(called)g(Craig's)378
4737 y(in)m(terp)s(olation)f(theorem.)378 4939 y FQ(Theorem)34
b(7.6)h(\(Craig\))45 b FI(If)35 b(a)h(\014rst-or)-5 b(der)37
b(sentenc)-5 b(e)35 b FP(X)j FN(\))30 b FP(Y)55 b FI(is)36
b(valid)g(then)g(it)f(has)h(an)g(inter-)378 5052 y(p)-5
b(olant.)63 b(Or)38 b(e)-5 b(quivalently,)41 b(every)e(p)-5
b(air)40 b(of)f(sets)h(which)f(p)-5 b(artition)42 b(a)d(\014nite)g
(unsatis\014able)h(set)f(of)378 5164 y(sentenc)-5 b(es)33
b(has)g(an)h(interp)-5 b(olant.)378 5366 y FQ(Pro)s(of)p
FT(:)31 b(see)g(for)g(instance)f(\(Fitting)g(1996\))1872
b Ff(\004)519 5592 y FT(W)-8 b(e)43 b(are)g(in)m(terested)f(in)e
(generalising)h(this)g(result)g(to)h(the)h(coloured)e(\014rst-order)g
(logic.)76 b(In)378 5705 y(particular)37 b(w)m(e)j(w)m(ould)e(lik)m(e)g
(to)i(sho)m(w)f(that)h(giv)m(en)f(a)g(\014nite)g FN(K)q
FT(-inconsisten)m(t)g(set)g FP(S)44 b FT(partitioned)p
eop
%%Page: 139 149
139 148 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(139)378 396 y(b)m(y)34
b(\()p FP(S)599 410 y FL(1)639 396 y FP(;)15 b(S)735
410 y FL(2)774 396 y FT(\),)36 b(then)e(there)g(is)g(some)g(uncoloured)
f(`in)m(terp)s(olan)m(t')h FP(I)41 b FT(suc)m(h)34 b(that)h(the)f(sets)
h FP(S)3529 363 y FK(U)3524 421 y FL(1)3607 396 y FN([)22
b(f)p FP(I)7 b FN(g)378 509 y FT(and)36 b FP(S)622 476
y FK(U)617 534 y FL(2)702 509 y FN([)24 b(f:)p FP(I)7
b FN(g)38 b FT(are)f(unsatis\014able.)59 b(F)-8 b(urthermore,)38
b(w)m(e)g(w)m(an)m(t)g(these)f(sets)h(to)f(b)s(e)g(inconsisten)m(t)378
622 y(according)30 b(to)h(the)f(restrictions)g(giv)m(en)g(b)m(y)g(the)g
(connectabilit)m(y)g(relation)f FN(K)q FT(.)42 b(Therefore)29
b(w)m(e)i(need)378 735 y(the)j(set)g FP(S)739 749 y FL(1)800
735 y FN([)22 b(f)p FP(X)1003 749 y FL(1)1044 735 y FN(g)34
b FT(to)g(b)s(e)f FN(K)q FT(-inconsisten)m(t)h(for)f(some)h(coloured)f
(sen)m(tence)i FP(X)3147 749 y FL(1)3220 735 y FT(where)e
FP(X)3568 702 y FK(U)3561 760 y FL(1)3654 735 y FT(=)e
FP(I)7 b FT(.)378 848 y(Similarly)-8 b(,)42 b FP(S)853
862 y FL(2)920 848 y FN([)28 b(f)p FP(X)1129 862 y FL(2)1169
848 y FN(g)42 b FT(has)g(to)h(b)s(e)e FN(K)q FT(-inconsisten)m(t)h(for)
g(some)h(coloured)e(sen)m(tence)j FP(X)3514 862 y FL(2)3595
848 y FT(where)378 961 y FP(X)460 928 y FK(U)453 985
y FL(2)542 961 y FT(=)26 b FN(:)p FP(I)7 b FT(.)42 b(In)31
b(general)g FP(X)1318 975 y FL(1)1388 961 y FT(and)g
FP(X)1641 975 y FL(2)1712 961 y FT(ma)m(y)g(b)s(e)g(of)g(di\013eren)m
(t)f(colours,)h(although)g(w)m(e)g(restrict)g(that)378
1074 y(all)d(the)i(coloured)e(predicates)h(in)f FP(X)1630
1088 y FL(1)1699 1074 y FT(o)s(ccur)h(in)f FP(S)2104
1088 y FL(2)2143 1074 y FT(,)i(and)f(similarly)c(that)30
b(all)e(coloured)h(predicates)378 1187 y(in)g FP(X)559
1201 y FL(2)629 1187 y FT(o)s(ccur)h(in)f FP(S)1036 1201
y FL(1)1076 1187 y FT(.)40 b(W)-8 b(e)32 b(de\014ne)d(coloured)h(in)m
(terp)s(olan)m(ts)g(as)g(follo)m(ws.)378 1393 y FQ(De\014nition)35
b(7.24)h(\(Coloured)f(In)m(terp)s(olan)m(t\))44 b FT(The)26
b(pair)f(of)i(coloured)f(sen)m(tences)i(\()p FP(X)3510
1407 y FL(1)3550 1393 y FP(;)15 b(X)3665 1407 y FL(2)3705
1393 y FT(\))27 b(is)378 1506 y(said)i(to)i(b)s(e)f(a)h
FN(K)q FT(-in)m(terp)s(olan)m(t)f(for)g(the)h(partition)e(\()p
FP(S)2210 1520 y FL(1)2250 1506 y FP(;)15 b(S)2346 1520
y FL(2)2385 1506 y FT(\))31 b(of)f(some)h(\014nite)e(set,)j(if:)489
1688 y(1.)46 b(All)29 b(the)i(function)e(sym)m(b)s(ols)g(in)g
FP(X)1795 1702 y FL(1)1865 1688 y FT(and)h FP(X)2117
1702 y FL(2)2187 1688 y FT(o)s(ccur)g(in)f(b)s(oth)h
FP(S)2809 1702 y FL(1)2878 1688 y FT(and)g FP(S)3111
1702 y FL(2)3150 1688 y FT(.)489 1874 y(2.)46 b(The)30
b(sets)h FP(S)1026 1888 y FL(1)1085 1874 y FN([)20 b(f)p
FP(X)1286 1888 y FL(1)1326 1874 y FN(g)31 b FT(and)f
FP(S)1635 1888 y FL(2)1694 1874 y FN([)20 b(f)p FP(X)1895
1888 y FL(2)1935 1874 y FN(g)31 b FT(are)g FN(K)q FT(-inconsisten)m(t.)
489 2059 y(3.)46 b(Let)33 b FP(X)852 2026 y FK(0)845
2084 y FL(1)917 2059 y FT(b)s(e)f(the)h(negation)f(normal)g(form)f(of)i
FP(X)2280 2073 y FL(1)2352 2059 y FT(and)f FP(X)2613
2026 y FK(0)2606 2084 y FL(2)2678 2059 y FT(b)s(e)g(the)g(negation)h
(normal)e(form)605 2172 y(of)g FN(:)p FP(X)845 2186 y
FL(2)884 2172 y FT(,)g(then)644 2358 y(\(a\))46 b FP(X)887
2325 y FK(0)880 2382 y FL(1)920 2325 y FK(U)1000 2358
y FT(=)25 b FP(X)1178 2325 y FK(0)1171 2382 y FL(2)1211
2325 y FK(U)1266 2358 y FT(;)639 2502 y(\(b\))45 b(for)38
b(ev)m(ery)h(p)s(osition)d FP(p)p FT(,)k(if)d FP(X)1838
2469 y FK(0)1831 2526 y FL(1)1870 2502 y FN(j)1895 2516
y FO(p)1973 2502 y FT(=)h FP(P)2153 2469 y FO(i)2181
2502 y FT(\()2210 2485 y FP(~)2216 2502 y(t)p FT(\))g(and)g
FP(X)2589 2469 y FK(0)2582 2526 y FL(2)2622 2502 y FN(j)2647
2516 y FO(p)2724 2502 y FT(=)g FP(P)2904 2469 y FO(j)2940
2502 y FT(\()2969 2485 y FP(~)2975 2502 y(t)q FT(\))g(for)g(some)g
(predicate)805 2615 y(sym)m(b)s(ol)29 b FP(P)42 b FT(and)29
b(list)f(of)i(terms)1893 2598 y FP(~)1899 2615 y(t)p
FT(,)g(then)f FP(i)c FN(\030)2320 2629 y FK(K)2404 2615
y FP(j)35 b FT(and)29 b(if)f FP(P)39 b FN(6)p FT(=)25
b FN(>)k FT(and)g FP(P)38 b FN(6)p FT(=)25 b FN(?)k FT(then)h(the)805
2728 y(coloured)g(predicate)g(sym)m(b)s(ol)f FP(P)1945
2695 y FO(i)2004 2728 y FT(o)s(ccurs)h(in)f FP(S)2447
2742 y FL(2)2516 2728 y FT(and)h FP(P)2764 2695 y FO(j)2831
2728 y FT(o)s(ccurs)g(in)f FP(S)3274 2742 y FL(1)3313
2728 y FT(.)419 b Ff(\003)378 2934 y FQ(Example)34 b(7.7)46
b FT(Let)31 b(some)f(set)h FP(S)36 b FT(b)s(e)29 b(partitioned)g(b)m(y)
h(the)h(pair)1042 3138 y(\()p FP(S)1133 3152 y FL(1)1173
3138 y FP(;)15 b(S)1269 3152 y FL(2)1308 3138 y FT(\))26
b(=)f(\()p FN(f)p FP(C)1617 3101 y FO(i)1645 3138 y FP(;)15
b FN(8)p FP(x:A)1881 3101 y FO(i)1910 3138 y FT(\()p
FP(x)p FT(\))21 b FN(_)f(:)p FP(B)2269 3101 y FO(k)2310
3138 y FN(g)p FP(;)15 b FN(f:)p FP(A)2569 3101 y FO(j)2607
3138 y FT(\()p FP(c)p FT(\))21 b FN(^)f FP(D)2896 3101
y FO(j)2932 3138 y FP(;)15 b(B)3046 3101 y FO(j)3083
3138 y FN(g)p FT(\))378 3343 y(and)30 b(let)g(the)h(connectabilit)m(y)e
(relation)h FN(K)i FT(b)s(e)e FP(i)25 b FN($)g FP(j)31
b FN($)25 b FP(k)s FT(.)41 b(Then)1422 3547 y(\()p FN(9)p
FP(x:)p FN(:)p FP(A)1714 3509 y FO(j)1750 3547 y FT(\()p
FP(x)p FT(\))21 b FN(^)f FP(B)2048 3509 y FO(j)2084 3547
y FP(;)15 b FN(8)p FP(x:A)2320 3509 y FO(i)2348 3547
y FT(\()p FP(x)p FT(\))21 b FN(_)f(:)p FP(B)2707 3509
y FO(k)2749 3547 y FT(\))378 3751 y(is)29 b(a)i FN(K)q
FT(-in)m(terp)s(olan)m(t)f(for)h(\()p FP(S)1344 3765
y FL(1)1383 3751 y FP(;)15 b(S)1479 3765 y FL(2)1519
3751 y FT(\).)2178 b Ff(\003)519 3957 y FT(Note)37 b(that)g(this)e
(notion)g(of)h(a)g(coloured)g(in)m(terp)s(olan)m(t)f(generalises)g(the)
h(standard)f(de\014nition)378 4070 y(of)f(uncoloured)d(in)m(terp)s
(olan)m(ts,)j(in)e(the)h(sense)h(that)g(if)e(\()p FP(X)2386
4084 y FL(1)2426 4070 y FP(;)15 b(X)2541 4084 y FL(2)2581
4070 y FT(\))34 b(is)f(a)g FN(K)q FT(-in)m(terp)s(olan)m(t)h(of)f(\()p
FP(S)3592 4084 y FL(1)3632 4070 y FP(;)15 b(S)3728 4084
y FL(2)3767 4070 y FT(\),)378 4183 y(then)32 b FP(X)669
4150 y FK(U)662 4207 y FL(1)756 4183 y FT(is)g(an)g(in)m(terp)s(olan)m
(t)f(of)i(\()p FP(S)1649 4150 y FK(U)1644 4207 y FL(1)1704
4183 y FP(;)15 b(S)1805 4150 y FK(U)1800 4207 y FL(2)1860
4183 y FT(\).)47 b(In)31 b(particular)g(\()p FP(I)2586
4150 y FO(i)2614 4183 y FP(;)15 b FN(:)p FP(I)2762 4150
y FO(i)2791 4183 y FT(\))32 b(is)f(an)i(\()p FP(i)c FN($)f
FP(i)p FT(\)-in)m(terp)s(olan)m(t)378 4296 y(for)i(\()p
FP(R)622 4263 y FO(i)621 4320 y FL(1)661 4296 y FP(;)15
b(R)771 4263 y FO(i)770 4320 y FL(2)810 4296 y FT(\))30
b(if)g(and)g(only)f(if)g FP(I)38 b FT(is)29 b(an)h(in)m(terp)s(olan)m
(t)g(for)g(\()p FP(R)2424 4310 y FL(1)2464 4296 y FP(;)15
b(R)2573 4310 y FL(2)2613 4296 y FT(\).)519 4409 y(W)-8
b(e)27 b(no)m(w)e(sho)m(w)g(that)h(ev)m(ery)g(partition)e(of)h(a)h
(\014nite)e FN(K)q FT(-inconsisten)m(t)i(set)g(of)f(coloured)g(sen)m
(tences)378 4522 y(has)37 b(a)h FN(K)q FT(-in)m(terp)s(olan)m(t.)62
b(W)-8 b(e)38 b(\014rst)f(in)m(tro)s(duce)g(some)g(notion)g(of)h
(consistency)f(whic)m(h)f(w)m(e)i(call)f FN(K)q FT(-)378
4635 y(in)m(terp)s(olation)20 b(consistency)h(and)g(sho)m(w)g(that)h
FN(K)q FT(-in)m(terp)s(olation)f(consisten)m(t)h(sets)g(are)f
FN(K)q FT(-consisten)m(t.)378 4841 y FQ(De\014nition)35
b(7.25)h(\(Coloured)f(In)m(terp)s(olation)f(Consistency\))46
b FT(A)29 b(set)h(of)f(sen)m(tences)h(is)e(said)378 4954
y(to)j(b)s(e)f FN(K)q FT(-in)m(terp)s(olation)f(consisten)m(t)i(if)e
(it)h(has)g(some)h(partition)e(without)h(a)g FN(K)q FT(-in)m(terp)s
(olan)m(t.)160 b Ff(\003)378 5160 y FQ(Lemma)33 b(7.1)46
b FI(The)37 b(c)-5 b(ol)5 b(le)-5 b(ction)38 b(of)f(al)5
b(l)37 b FN(K)q FI(-interp)-5 b(olation)40 b(c)-5 b(onsistent)38
b(sets)g(of)f(sentenc)-5 b(es)37 b(is)g(a)g FN(K)q FI(-)378
5273 y(c)-5 b(onsistency)34 b(pr)-5 b(op)g(erty.)378
5479 y FQ(Pro)s(of)p FT(:)30 b(The)e(pro)s(of)h(of)g(this)f(lemma)g
(generalises)g(the)i(pro)s(of)e(of)h(Craig's)f(In)m(terp)s(olation)g
(Theorem)378 5592 y(giv)m(en)k(in)f(\(Fitting)g(1996\).)48
b(Giv)m(en)32 b(a)g(connectabilit)m(y)f(relation)g FN(K)q
FT(,)j(w)m(e)e(sho)m(w)g(that)g(if)f(some)h(set)h FP(S)378
5705 y FT(is)c FN(K)q FT(-in)m(terp)s(olation)h(consisten)m(t)h(then)f
(it)g(satis\014es)f(all)h(the)g(conditions)f(in)g(De\014nition)g(7.13;)
p eop
%%Page: 140 150
140 149 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(140)489 396 y(1.)46 b(Supp)s(ose)33
b(that)i(for)f(some)g(literal)f FP(A)h FT(and)g(colours)g
FP(i)p FT(,)i FP(j)5 b FT(,)36 b(b)s(oth)d FP(A)2885
363 y FO(i)2945 396 y FN(2)f FP(S)39 b FT(and)34 b FN(:)p
FP(A)3443 363 y FO(j)3511 396 y FN(2)d FP(S)5 b FT(,)35
b(w)m(e)605 509 y(sho)m(w)d(that)g(if)f FP(i)c FN(\030)1244
523 y FK(K)1330 509 y FP(j)37 b FT(then)31 b FP(S)37
b FT(is)31 b(not)g FN(K)q FT(-in)m(terp)s(olation)g(consisten)m(t.)45
b(Let)32 b(the)g(pair)f(of)g(sets)605 622 y(\()p FP(S)696
636 y FL(1)736 622 y FP(;)15 b(S)832 636 y FL(2)871 622
y FT(\))33 b(partition)e FP(S)5 b FT(,)33 b(then)e(either)h(b)s(oth)g
FP(A)2192 589 y FO(i)2252 622 y FT(and)g FN(:)p FP(A)2560
589 y FO(j)2628 622 y FT(are)h(in)e(the)h(same)h(set)g(\()p
FP(S)3513 636 y FL(1)3584 622 y FT(or)f FP(S)3753 636
y FL(2)3793 622 y FT(\))605 735 y(or)f(else)g(they)h(are)f(in)f
(di\013eren)m(t)g(sets.)44 b(If)30 b(b)s(oth)h(literals)e(are)j(in)d
(the)j(same)f(set)h FP(S)3380 749 y FL(1)3419 735 y FT(,)g(sa)m(y)-8
b(,)32 b(then)605 848 y(let)c FP(X)809 862 y FL(1)874
848 y FT(=)d FN(>)1041 815 y FO(i)1097 848 y FT(and)j(let)g
FP(X)1476 862 y FL(2)1541 848 y FT(=)d FN(?)1708 815
y FO(j)1744 848 y FT(.)40 b(It)28 b(is)f(easy)i(to)g(see)g(that)f(\()p
FP(X)2750 862 y FL(1)2791 848 y FP(;)15 b(X)2906 862
y FL(2)2946 848 y FT(\))28 b(satis\014es)g(the)g(\014rst)f(and)605
961 y(last)g(conditions)f(of)h(De\014nition)e(7.24;)30
b(and)d(since)f FP(i)g FN(\030)2461 975 y FK(K)2544 961
y FP(j)5 b FT(,)28 b(b)s(oth)f FP(S)2907 975 y FL(1)2959
961 y FN([)13 b(f)p FP(X)3153 975 y FL(1)3194 961 y FN(g)27
b FT(and)g FP(S)3496 975 y FL(2)3548 961 y FN([)13 b(f)p
FP(X)3742 975 y FL(2)3782 961 y FN(g)605 1074 y FT(are)29
b FN(K)q FT(-inconsisten)m(t.)40 b(Th)m(us,)28 b(\()p
FP(X)1753 1088 y FL(1)1793 1074 y FP(;)15 b(X)1908 1088
y FL(2)1948 1074 y FT(\))29 b(is)e(a)i FN(K)q FT(-in)m(terp)s(olan)m(t)
f(for)g(\()p FP(S)2969 1088 y FL(1)3009 1074 y FP(;)15
b(S)3105 1088 y FL(2)3144 1074 y FT(\).)41 b(No)m(w,)30
b(if)d FP(A)3625 1041 y FO(i)3681 1074 y FT(and)605 1187
y FN(:)p FP(A)734 1154 y FO(j)810 1187 y FT(are)39 b(in)f(di\013eren)m
(t)h(sets,)j(sa)m(y)d FP(A)1900 1154 y FO(i)1969 1187
y FN(2)g FP(S)2125 1201 y FL(1)2203 1187 y FT(and)g FN(:)p
FP(A)2518 1154 y FO(j)2594 1187 y FN(2)g FP(S)2750 1201
y FL(2)2790 1187 y FT(,)i(then)e(let)g FP(X)3287 1201
y FL(1)3367 1187 y FT(=)g FN(:)p FP(A)3606 1154 y FO(j)3681
1187 y FT(and)605 1300 y FP(X)680 1314 y FL(2)748 1300
y FT(=)27 b FP(A)914 1267 y FO(i)942 1300 y FT(.)45 b(Once)32
b(more,)h(\()p FP(X)1611 1314 y FL(1)1651 1300 y FP(;)15
b(X)1766 1314 y FL(2)1806 1300 y FT(\))32 b(is)f(a)h
FN(K)q FT(-in)m(terp)s(olan)m(t)g(for)g(\()p FP(S)2845
1314 y FL(1)2884 1300 y FP(;)15 b(S)2980 1314 y FL(2)3020
1300 y FT(\),)33 b(and)e(therefore)h FP(S)37 b FT(is)605
1413 y(not)31 b FN(K)q FT(-in)m(terp)s(olation)e(consisten)m(t.)489
1590 y(2.)46 b(Let)33 b FN(?)841 1557 y FO(i)898 1590
y FN(2)c FP(S)38 b FT(and)32 b FP(i)d FN(\030)1392 1604
y FK(K)1479 1590 y FP(j)39 b FT(for)32 b(some)h(colour)g
FP(j)5 b FT(.)48 b(W)-8 b(e)34 b(sho)m(w)e(that)i(ev)m(ery)f(partition)
e(\()p FP(S)3617 1604 y FL(1)3657 1590 y FP(;)15 b(S)3753
1604 y FL(2)3793 1590 y FT(\))605 1703 y(of)29 b FP(S)34
b FT(has)29 b(a)h FN(K)q FT(-in)m(terp)s(olan)m(t.)40
b(Basically)-8 b(,)29 b(if)f FN(?)2191 1670 y FO(i)2244
1703 y FN(2)d FP(S)2386 1717 y FL(1)2454 1703 y FT(then)k(let)g
FP(X)2865 1717 y FL(1)2930 1703 y FT(=)c FN(>)3097 1670
y FO(i)3154 1703 y FT(and)k(let)g FP(X)3535 1717 y FL(2)3600
1703 y FT(=)24 b FN(?)3766 1670 y FO(j)3803 1703 y FT(.)605
1816 y(The)34 b(pair)g(\()p FP(X)1098 1830 y FL(1)1138
1816 y FP(;)15 b(X)1253 1830 y FL(2)1293 1816 y FT(\))35
b(satis\014es)f(all)f(the)i(conditions)e(in)h(De\014nition)f(7.24)j
(and)e(is)g(th)m(us)g(a)h FN(K)q FT(-)605 1929 y(in)m(terp)s(olan)m(t)
30 b(for)g(\()p FP(S)1303 1943 y FL(1)1342 1929 y FP(;)15
b(S)1438 1943 y FL(2)1478 1929 y FT(\).)41 b(The)30 b(argumen)m(t)h(is)
e(similar)f(if)h FN(?)2717 1896 y FO(i)2770 1929 y FN(2)c
FP(S)2912 1943 y FL(2)2951 1929 y FT(.)489 2107 y(3.)46
b(Supp)s(ose)41 b(that)h FP(')29 b FN(^)e FP( )49 b FN(2)44
b FP(S)5 b FT(,)45 b(w)m(e)e(need)f(to)h(sho)m(w)f(that)h
FP(S)33 b FN([)27 b(f)p FP(';)15 b( )s FN(g)45 b FT(is)c
FN(K)q FT(-in)m(terp)s(olation)605 2220 y(consisten)m(t.)e(Let)24
b(us)f(assume)g(that)h FP(S)12 b FN([)6 b(f)p FP(';)15
b( )s FN(g)25 b FT(is)e(not)g FN(K)q FT(-in)m(terp)s(olation)g
(consisten)m(t,)j(that)e(is,)605 2333 y(ev)m(ery)h(partition)e(of)h
FP(S)13 b FN([)8 b(f)p FP(';)15 b( )s FN(g)26 b FT(has)d(a)i
FN(K)q FT(-in)m(terp)s(olan)m(t,)g(and)f(w)m(e)h(sho)m(w)f(that)g(ev)m
(ery)h(partition)605 2446 y(of)43 b FP(S)k FT(has)42
b(a)g FN(K)q FT(-in)m(terp)s(olan)m(t)g(as)h(w)m(ell.)75
b(Let)42 b(\()p FP(S)2309 2460 y FL(1)2349 2446 y FP(;)15
b(S)2445 2460 y FL(2)2484 2446 y FT(\))43 b(partition)e
FP(S)5 b FT(,)45 b(and)d(let)g(us)f(assume)605 2558 y(without)31
b(loss)f(of)i(generalit)m(y)f(that)h FP(')21 b FN(^)g
FP( )30 b FN(2)c FP(S)2235 2572 y FL(1)2275 2558 y FT(.)43
b(Then)30 b(\()p FP(S)2672 2572 y FL(1)2733 2558 y FN([)20
b(f)p FP(';)15 b( )s FN(g)p FP(;)g(S)3161 2572 y FL(2)3203
2558 y FT(\))32 b(partitions)e FP(S)25 b FN([)605 2671
y(f)p FP(';)15 b( )s FN(g)33 b FT(and)c(therefore)i(has)f(some)h
FN(K)q FT(-in)m(terp)s(olan)m(t)f(\()p FP(X)2515 2685
y FL(1)2555 2671 y FP(;)15 b(X)2670 2685 y FL(2)2711
2671 y FT(\).)41 b(Therefore)30 b FP(S)3280 2685 y FL(1)3339
2671 y FN([)20 b(f)p FP(';)15 b( )s(;)g(X)3741 2685 y
FL(1)3782 2671 y FN(g)605 2784 y FT(and)20 b FP(S)828
2798 y FL(2)869 2784 y FN([)q(f)p FP(X)1051 2798 y FL(2)1091
2784 y FN(g)h FT(are)g FN(K)q FT(-inconsisten)m(t.)38
b(No)m(w,)24 b(\()p FP(X)2266 2798 y FL(1)2306 2784 y
FP(;)15 b(X)2421 2798 y FL(2)2461 2784 y FT(\))21 b(is)f(also)h(a)g
FN(K)q FT(-in)m(terp)s(olan)m(t)g(for)f(\()p FP(S)3617
2798 y FL(1)3657 2784 y FP(;)15 b(S)3753 2798 y FL(2)3793
2784 y FT(\))605 2897 y(as)34 b(all)e(the)i(function)e(sym)m(b)s(ols,)g
(and)h(coloured)g(predicates)g(in)f FP(S)2888 2911 y
FL(1)2949 2897 y FN([)22 b(f)p FP(';)15 b( )s FN(g)36
b FT(o)s(ccur)d(also)g(in)605 3010 y FP(S)661 3024 y
FL(1)700 3010 y FT(,)e(and)f FP(S)989 3024 y FL(1)1048
3010 y FN([)20 b(f)p FP(X)1249 3024 y FL(1)1289 3010
y FN(g)31 b FT(is)e FN(K)q FT(-inconsisten)m(t)i(since)e
FP(S)2334 3024 y FL(1)2394 3010 y FN([)20 b(f)p FP(';)15
b( )s(;)g(X)2796 3024 y FL(1)2838 3010 y FN(g)30 b FT(is.)489
3188 y(4.)46 b(Let)30 b FP(')18 b FN(_)f FP( )29 b FN(2)c
FP(S)5 b FT(,)29 b(w)m(e)h(need)f(to)h(sho)m(w)f(that)g(either)g
FP(S)22 b FN([)c(f)p FP(')p FN(g)30 b FT(or)f FP(S)23
b FN([)17 b(f)p FP( )s FN(g)31 b FT(is)d FN(K)q FT(-in)m(terp)s
(olation)605 3301 y(consisten)m(t.)40 b(W)-8 b(e)27 b(pro)m(v)m(e)f
(the)g(con)m(trap)s(ositiv)m(e,)i(that)e(is,)g(w)m(e)g(assume)g(that)g
(b)s(oth)g FP(S)16 b FN([)11 b(f)p FP(')p FN(g)26 b FT(and)605
3414 y FP(S)18 b FN([)13 b(f)p FP( )s FN(g)28 b FT(are)f(not)g
FN(K)q FT(-in)m(terp)s(olation)e(consisten)m(t)i(and)g(sho)m(w)f(that)h
FP(S)32 b FT(is)26 b(not)h FN(K)q FT(-in)m(terp)s(olation)605
3527 y(consisten)m(t.)54 b(Let)35 b(\()p FP(S)1334 3541
y FL(1)1374 3527 y FP(;)15 b(S)1470 3541 y FL(2)1509
3527 y FT(\))35 b(partition)f FP(S)5 b FT(,)35 b(and)f(let)h
FP(')24 b FN(_)e FP( )36 b FN(2)c FP(S)2811 3541 y FL(1)2850
3527 y FT(.)54 b(The)34 b(pro)s(of)g(for)g(the)h(case)605
3639 y(where)41 b FP(')28 b FN(_)f FP( )46 b FN(2)d FP(S)1319
3653 y FL(2)1399 3639 y FT(pro)s(ceeds)e(similarly)-8
b(.)70 b(Then)40 b(\()p FP(S)2553 3653 y FL(1)2620 3639
y FN([)27 b(f)p FP(')p FN(g)p FP(;)15 b(S)2953 3653 y
FL(2)2994 3639 y FT(\))41 b(and)g(\()p FP(S)3349 3653
y FL(1)3416 3639 y FN([)27 b(f)p FP( )s FN(g)p FP(;)15
b(S)3752 3653 y FL(2)3793 3639 y FT(\))605 3752 y(partition)40
b(the)g(sets)h FP(S)32 b FN([)27 b(f)p FP(')p FN(g)42
b FT(and)e FP(S)32 b FN([)27 b(f)p FP( )s FN(g)41 b FT(resp)s(ectiv)m
(ely)-8 b(,)44 b(and)c(th)m(us)g(they)h(ha)m(v)m(e)g(some)605
3865 y(in)m(terp)s(olan)m(ts)32 b(\()p FP(X)1221 3879
y FL(1)1261 3865 y FP(;)15 b(X)1376 3879 y FL(2)1416
3865 y FT(\))33 b(and)f(\()p FP(Y)1751 3879 y FL(1)1791
3865 y FP(;)15 b(Y)1884 3879 y FL(2)1923 3865 y FT(\).)49
b(Therefore,)33 b FP(S)2528 3879 y FL(1)2589 3865 y FN([)21
b(f)p FP(';)15 b(X)2890 3879 y FL(1)2931 3865 y FN(g)33
b FT(and)f FP(S)3244 3879 y FL(1)3305 3865 y FN([)22
b(f)p FP( )s(;)15 b(Y)3588 3879 y FL(1)3628 3865 y FN(g)33
b FT(are)605 3978 y FN(K)q FT(-inconsisten)m(t)e(and)e(hence)i
FP(S)1690 3992 y FL(1)1749 3978 y FN([)20 b(f)p FP(X)1950
3992 y FL(1)2010 3978 y FN(^)g FP(Y)2144 3992 y FL(1)2184
3978 y FN(g)30 b FT(is)g(also)g FN(K)q FT(-inconsisten)m(t,)h
(otherwise)1131 4182 y FP(S)1187 4196 y FL(1)1246 4182
y FN([)20 b(f)p FP(X)1447 4196 y FL(1)1507 4182 y FN(^)g
FP(Y)1641 4196 y FL(1)1681 4182 y FN(g)61 b FT(is)29
b FN(K)q FT(-consisten)m(t)1214 4320 y FN(\))83 b FP(S)1444
4334 y FL(1)1503 4320 y FN([)20 b(f)p FP(X)1704 4334
y FL(1)1744 4320 y FP(;)15 b(Y)1837 4334 y FL(1)1877
4320 y FN(g)61 b FT(is)29 b FN(K)q FT(-consisten)m(t)1214
4458 y FN(\))83 b FP(S)1444 4472 y FL(1)1503 4458 y FN([)20
b(f)p FP(';)15 b(X)1803 4472 y FL(1)1844 4458 y FP(;)g(Y)1937
4472 y FL(1)1977 4458 y FN(g)61 b FT(is)29 b FN(K)q FT(-consisten)m(t,)
j(or)1448 4596 y FP(S)1504 4610 y FL(1)1564 4596 y FN([)20
b(f)p FP( )s(;)15 b(X)1867 4610 y FL(1)1908 4596 y FP(;)g(Y)2001
4610 y FL(1)2040 4596 y FN(g)61 b FT(is)30 b FN(K)q FT(-consisten)m(t)h
(as)g FP(')21 b FN(_)f FP( )28 b FN(2)d FP(S)3263 4610
y FL(1)1214 4734 y FN(\))83 b FP(S)1444 4748 y FL(1)1503
4734 y FN([)20 b(f)p FP(';)15 b(X)1803 4748 y FL(1)1844
4734 y FN(g)61 b FT(is)30 b FN(K)q FT(-consisten)m(t,)h(or)1448
4872 y FP(S)1504 4886 y FL(1)1564 4872 y FN([)20 b(f)p
FP( )s(;)15 b(Y)1845 4886 y FL(1)1885 4872 y FN(g)61
b FT(is)30 b FN(K)q FT(-consisten)m(t)q FP(:)605 5076
y FT(Also,)40 b FP(S)901 5090 y FL(2)965 5076 y FN([)25
b(f)p FP(X)1171 5090 y FL(2)1211 5076 y FN(g)39 b FT(and)e
FP(S)1535 5090 y FL(2)1599 5076 y FN([)25 b(f)p FP(Y)1783
5090 y FL(2)1823 5076 y FN(g)38 b FT(are)g FN(K)q FT(-inconsisten)m(t)g
(and)f(so)h FP(S)3031 5090 y FL(2)3096 5076 y FN([)25
b(f)p FP(X)3302 5090 y FL(2)3367 5076 y FN(_)g FP(Y)3506
5090 y FL(2)3545 5076 y FN(g)38 b FT(is)f FN(K)q FT(-)605
5189 y(inconsisten)m(t.)46 b(Hence)33 b(\()p FP(X)1527
5203 y FL(1)1588 5189 y FN(^)21 b FP(Y)1723 5203 y FL(1)1762
5189 y FP(;)15 b(X)1877 5203 y FL(2)1939 5189 y FN(_)21
b FP(Y)2074 5203 y FL(2)2113 5189 y FT(\))33 b(is)e(a)h
FN(K)q FT(-in)m(terp)s(olan)m(t)h(for)e(\()p FP(S)3153
5203 y FL(1)3193 5189 y FP(;)15 b(S)3289 5203 y FL(2)3329
5189 y FT(\),)33 b(as)f FN(:)p FT(\()p FP(X)3706 5203
y FL(1)3767 5189 y FN(^)605 5302 y FP(Y)658 5316 y FL(1)697
5302 y FT(\))732 5269 y FK(U)813 5302 y FT(=)25 b FN(:)p
FP(X)1052 5269 y FK(U)1045 5326 y FL(1)1123 5302 y FN(_)16
b(:)p FP(Y)1334 5269 y FK(U)1314 5326 y FL(1)1414 5302
y FT(=)25 b(\()p FP(X)1620 5316 y FL(2)1676 5302 y FN(_)17
b FP(Y)1807 5316 y FL(2)1846 5302 y FT(\))1881 5269 y
FK(U)1964 5302 y FT(and)28 b(the)h(sets)g FP(S)2526 5316
y FL(1)2582 5302 y FN([)16 b(f)p FP(')p FN(g)29 b FT(and)f
FP(S)3068 5316 y FL(1)3124 5302 y FN([)16 b(f)p FP( )s
FN(g)30 b FT(con)m(tain)f(the)605 5415 y(same)i(predicates)f(and)g
(function)f(sym)m(b)s(ols)g(as)h(the)h(set)g FP(S)2609
5429 y FL(1)2648 5415 y FT(.)489 5592 y(5.)46 b(Let)39
b FN(8)p FP(x:')e FN(2)h FP(S)5 b FT(.)63 b(Supp)s(ose)36
b(that)i FP(S)31 b FN([)24 b(f)p FP(')p FN(f)p FP(x)39
b FN(!)f FP(t)p FN(gg)h FT(is)d(not)j FN(K)q FT(-in)m(terp)s(olation)e
(consisten)m(t)605 5705 y(for)h(some)h(closed)f(term)g
FP(t)p FT(,)i(w)m(e)e(sho)m(w)g(that)h(ev)m(ery)g(partition)e(of)h
FP(S)43 b FT(has)38 b(a)h FN(K)q FT(-in)m(terp)s(olan)m(t.)p
eop
%%Page: 141 151
141 150 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(141)605 396 y(Supp)s(ose)26
b(that)h(\()p FP(S)1243 410 y FL(1)1283 396 y FP(;)15
b(S)1379 410 y FL(2)1418 396 y FT(\))28 b(partitions)d
FP(S)32 b FT(and)27 b(let)g FN(8)p FP(x:')g FT(b)s(e)g(in)f(one)h(of)g
(\()p FP(S)3074 410 y FL(1)3114 396 y FP(;)15 b(S)3210
410 y FL(2)3249 396 y FT(\),)29 b(sa)m(y)e FP(S)3547
410 y FL(1)3587 396 y FT(.)39 b(No)m(w)605 509 y(\()p
FP(S)696 523 y FL(1)743 509 y FN([)7 b(f)p FP(')p FN(f)p
FP(x)26 b FN(!)f FP(t)p FN(gg)p FP(;)15 b(S)1373 523
y FL(2)1413 509 y FT(\))24 b(is)f(a)h(partition)e(of)i
FP(S)12 b FN([)7 b(f)p FP(')p FN(f)p FP(x)26 b FN(!)f
FP(t)p FN(gg)f FT(and)f(therefore)h(it)g(has)f(some)h
FN(K)q FT(-)605 622 y(in)m(terp)s(olan)m(t)29 b(\()p
FP(X)1182 636 y FL(1)1223 622 y FP(;)15 b(X)1338 636
y FL(2)1378 622 y FT(\).)41 b(Therefore)30 b FP(S)1947
636 y FL(1)2006 622 y FN([)19 b(f)p FP(')p FN(f)p FP(x)27
b FN(!)e FP(t)p FN(g)p FP(;)15 b(X)2623 636 y FL(1)2663
622 y FN(g)31 b FT(is)e FN(K)q FT(-inconsisten)m(t)h(and)g(hence)605
735 y(so)37 b(is)f FP(S)877 749 y FL(1)941 735 y FN([)24
b(f)p FP(X)1146 749 y FL(1)1186 735 y FN(g)38 b FT(b)m(y)e(Prop)s
(osition)f(7.4\(3\).)63 b(Also,)38 b FP(S)2509 749 y
FL(2)2573 735 y FN([)24 b(f)p FP(X)2778 749 y FL(2)2818
735 y FN(g)37 b FT(is)f FN(K)q FT(-inconsisten)m(t.)60
b(But,)605 848 y(w)m(e)37 b(cannot)g(assume)f(that)h(\()p
FP(X)1682 862 y FL(1)1723 848 y FP(;)15 b(X)1838 862
y FL(2)1878 848 y FT(\))36 b(is)g(a)h FN(K)q FT(-in)m(terp)s(olan)m(t)f
(of)h(\()p FP(S)2904 862 y FL(1)2943 848 y FP(;)15 b(S)3039
862 y FL(2)3079 848 y FT(\))37 b(as)f(some)h(function)605
961 y(sym)m(b)s(ols)32 b(in)f FP(X)1138 975 y FL(1)1211
961 y FT(\(and)h FP(X)1500 975 y FL(2)1540 961 y FT(\))h(ma)m(y)h(b)s
(e)e(found)g(in)f FP(t)p FT(,)j(and)e(so)h(in)f FP(S)2850
975 y FL(1)2911 961 y FN([)21 b(f)p FP(')p FN(f)p FP(x)31
b FN(!)e FP(t)p FN(gg)p FT(,)34 b(but)f(not)605 1074
y(in)f FP(S)770 1088 y FL(1)809 1074 y FT(.)49 b(Ho)m(w)m(ev)m(er,)36
b(if)d(this)f(is)g(the)h(case)h(then)f(there)h(m)m(ust)f(b)s(e)f(some)i
(term)f FP(t)3277 1041 y FK(0)3333 1074 y FT(in)f FP(X)3517
1088 y FL(1)3590 1074 y FT(whose)605 1187 y(ro)s(ot)c(is)e(not)i(found)
e(in)g FP(S)1455 1201 y FL(1)1495 1187 y FT(.)39 b(No)m(w,)29
b(let)f FP(X)2000 1154 y FK(0)1993 1211 y FL(1)2058 1187
y FT(=)d FP(X)2229 1201 y FL(1)2268 1187 y FN(f)p FP(t)2346
1154 y FK(0)2395 1187 y FN(!)g FP(y)s FN(g)j FT(and)f
FP(X)2888 1154 y FK(0)2881 1211 y FL(2)2946 1187 y FT(=)e
FP(X)3117 1201 y FL(2)3157 1187 y FN(f)p FP(t)3235 1154
y FK(0)3283 1187 y FN(!)h FP(y)s FN(g)h FT(where)g FP(y)605
1300 y FT(is)i(some)h(v)-5 b(ariable)29 b(whic)m(h)f(do)s(es)i(not)g(o)
s(ccur)f(in)g FP(X)2314 1314 y FL(1)2354 1300 y FT(,)h(then)f
FP(S)2671 1314 y FL(1)2729 1300 y FN([)19 b(f9)p FP(y)s(:X)3060
1267 y FK(0)3053 1324 y FL(1)3093 1300 y FN(g)30 b FT(is)f
FN(K)q FT(-inconsisten)m(t)605 1413 y(b)m(y)h(Prop)s(osition)e
(7.4\(5\),)33 b(and)c(so)i(is)e FP(S)1943 1427 y FL(2)2001
1413 y FN([)20 b(f8)p FP(y)s(:X)2333 1380 y FK(0)2326
1437 y FL(2)2365 1413 y FN(g)31 b FT(b)m(y)f(Prop)s(osition)e
(7.4\(3\).)43 b(Hence,)31 b(if)e(all)605 1526 y(the)d(function)f(sym)m
(b)s(ols)g(in)f FP(X)1635 1493 y FK(0)1628 1550 y FL(1)1694
1526 y FT(are)j(found)d(in)h FP(S)2250 1540 y FL(1)2315
1526 y FT(then)h(\()p FN(9)p FP(y)s(:X)2759 1493 y FK(0)2752
1550 y FL(1)2791 1526 y FP(;)15 b FN(8)p FP(y)s(:X)3037
1493 y FK(0)3030 1550 y FL(2)3070 1526 y FT(\))26 b(is)g(a)g
FN(K)q FT(-in)m(terp)s(olan)m(t)605 1638 y(for)40 b(\()p
FP(S)845 1652 y FL(1)885 1638 y FP(;)15 b(S)981 1652
y FL(2)1020 1638 y FT(\).)71 b(If)40 b(not,)k(w)m(e)c(can)h(rep)s(eat)f
(the)h(same)g(pro)s(cess)f(on)g(\()p FN(9)p FP(y)s(:X)3171
1605 y FK(0)3164 1663 y FL(1)3204 1638 y FP(;)15 b FN(8)p
FP(y)s(:X)3450 1605 y FK(0)3443 1663 y FL(2)3482 1638
y FT(\))41 b(un)m(til)d(a)605 1751 y FN(K)q FT(-in)m(terp)s(olan)m(t)31
b(is)e(constructed.)489 1939 y(6.)46 b(Let)40 b FN(9)p
FP(x:')h FN(2)f FP(S)5 b FT(,)43 b(and)c(that)h(for)f(ev)m(ery)i
(parameter)f FP(p)f FT(the)h(set)g FP(S)31 b FN([)26
b(f)p FP(')p FN(f)p FP(x)42 b FN(!)f FP(p)p FN(gg)f FT(is)f(not)605
2052 y FN(K)q FT(-in)m(terp)s(olation)e(consisten)m(t,)k(w)m(e)d(sho)m
(w)g(that)g(ev)m(ery)h(partition)e(\()p FP(S)3016 2066
y FL(1)3055 2052 y FP(;)15 b(S)3151 2066 y FL(2)3191
2052 y FT(\))38 b(of)g FP(S)43 b FT(has)38 b(a)g FN(K)q
FT(-)605 2164 y(in)m(terp)s(olan)m(t.)47 b(Let)33 b(us)f(assume)g(that)
i FN(9)p FP(x:')29 b FN(2)f FP(S)2276 2178 y FL(1)2316
2164 y FT(,)33 b(and)f(let)h FP(p)f FT(b)s(e)g(some)h(parameter)g(new)f
(to)605 2277 y FP(S)661 2291 y FL(1)730 2277 y FT(and)e
FP(S)963 2291 y FL(2)1002 2277 y FT(.)41 b(No)m(w)30
b(\()p FP(S)1365 2291 y FL(1)1425 2277 y FN([)19 b(f)p
FP(')p FN(f)p FP(x)26 b FN(!)g FP(p)p FN(gg)p FP(;)15
b(S)2081 2291 y FL(2)2121 2277 y FT(\))30 b(partitions)f(the)h(set)h
FP(S)24 b FN([)c(f)p FP(')p FN(f)p FP(x)26 b FN(!)f FP(p)p
FN(gg)31 b FT(and)f(so)605 2390 y(it)g(has)g(some)h(in)m(terp)s(olan)m
(t)e(\()p FP(X)1663 2404 y FL(1)1704 2390 y FP(;)15 b(X)1819
2404 y FL(2)1859 2390 y FT(\).)41 b(So)1087 2594 y FP(S)1143
2608 y FL(1)1202 2594 y FN([)20 b(f)p FP(')p FN(f)p FP(x)27
b FN(!)e FP(p)p FN(g)p FP(;)15 b(X)1833 2608 y FL(1)1873
2594 y FN(g)61 b FT(is)30 b FN(K)q FT(-inconsisten)m(t)1170
2732 y FN(\))83 b FP(S)1400 2746 y FL(1)1459 2732 y FN([)20
b(f)p FP(')p FN(f)p FP(x)27 b FN(!)e FP(p)p FN(g)p FP(;)15
b FN(9)p FP(y)s(:X)2214 2746 y FL(1)2254 2732 y FN(f)p
FP(p)25 b FN(!)g FP(y)s FN(gg)62 b FT(is)29 b FN(K)q
FT(-inconsisten)m(t)1170 2870 y FN(\))83 b FP(S)1400
2884 y FL(1)1459 2870 y FN([)20 b(f9)p FP(y)s(:X)1784
2884 y FL(1)1824 2870 y FN(f)p FP(p)25 b FN(!)h FP(y)s
FN(gg)61 b FT(is)29 b FN(K)q FT(-inconsisten)m(t)p FP(:)605
3074 y FT(Also)f(since)f FP(S)1084 3088 y FL(2)1139 3074
y FN([)14 b(f)p FP(X)1334 3088 y FL(2)1375 3074 y FN(g)28
b FT(is)f FN(K)q FT(-inconsisten)m(t,)h(then)g(so)g(is)f
FP(S)2618 3088 y FL(2)2673 3074 y FN([)14 b(f8)p FP(y)s(:X)2992
3088 y FL(2)3032 3074 y FN(f)p FP(p)26 b FN(!)f FP(y)s
FN(gg)j FT(and)g(hence)605 3187 y(\()p FN(9)p FP(y)s(:X)839
3201 y FL(1)879 3187 y FN(f)p FP(p)f FN(!)h FP(y)s FN(g)p
FP(;)15 b FN(8)p FP(y)s(:X)1448 3201 y FL(2)1488 3187
y FN(f)p FP(p)27 b FN(!)g FP(y)s FN(g)p FT(\))32 b(is)f(a)h
FN(K)q FT(-in)m(terp)s(olan)m(t)f(for)h(\()p FP(S)2855
3201 y FL(1)2894 3187 y FP(;)15 b(S)2990 3201 y FL(2)3030
3187 y FT(\).)44 b(The)32 b(pro)s(of)e(for)i(the)605
3300 y(case)g(where)e FN(9)p FP(x:')25 b FN(2)g FP(S)1415
3314 y FL(2)1484 3300 y FT(pro)s(ceeds)30 b(similarly)-8
b(.)519 3487 y(Th)m(us)33 b(the)h(collection)g(of)g(all)f
FN(K)q FT(-in)m(terp)s(olation)h(consisten)m(t)g(sets)g(of)h(coloured)e
(sen)m(tences)i(is)e(a)378 3600 y FN(K)q FT(-consistency)e(prop)s(ert)m
(y)-8 b(.)2441 b Ff(\004)378 3924 y FQ(Theorem)34 b(7.7)46
b FI(Given)28 b(a)g(c)-5 b(onne)g(ctability)29 b(r)-5
b(elation)30 b FN(K)f FI(and)g(a)g(\014nite)f FN(K)q
FI(-inc)-5 b(onsistent)29 b(set)f FP(S)5 b FI(,)28 b(then)378
4037 y(every)k(p)-5 b(artition)35 b(of)e FP(S)38 b FI(has)33
b(some)h FN(K)q FI(-interp)-5 b(olant.)378 4248 y FQ(Pro)s(of)p
FT(:)34 b(Supp)s(ose)d(that)i FP(S)38 b FT(is)32 b(partitioned)f(b)m(y)
i(\()p FP(S)2128 4262 y FL(1)2167 4248 y FP(;)15 b(S)2263
4262 y FL(2)2303 4248 y FT(\))33 b(and)f(let)h(us)f(assume)h(that)g(\()
p FP(S)3410 4262 y FL(1)3450 4248 y FP(;)15 b(S)3546
4262 y FL(2)3585 4248 y FT(\))33 b(do)s(es)378 4361 y(not)23
b(ha)m(v)m(e)i(a)e FN(K)q FT(-in)m(terp)s(olan)m(t.)38
b(Then)22 b FP(S)28 b FT(is)22 b FN(K)q FT(-in)m(terp)s(olation)h
(consisten)m(t)g(and)g(b)m(y)g(the)g(ab)s(o)m(v)m(e)h(lemma,)378
4474 y FP(S)32 b FT(is)27 b FN(K)q FT(-consisten)m(t.)41
b(Consequen)m(tly)-8 b(,)28 b(giv)m(en)f(that)h FP(S)33
b FT(is)26 b FN(K)q FT(-inconsisten)m(t)h(then)h(\()p
FP(S)3189 4488 y FL(1)3228 4474 y FP(;)15 b(S)3324 4488
y FL(2)3364 4474 y FT(\))27 b(m)m(ust)h(ha)m(v)m(e)378
4587 y(some)j FN(K)q FT(-in)m(terp)s(olan)m(t.)2588 b
Ff(\004)519 4813 y FT(Unfortunately)-8 b(,)33 b(the)f(con)m(v)m(erse)i
(of)e(this)f(theorem)h(do)s(es)g(not)h(hold.)44 b(In)32
b(other)g(w)m(ords,)g(if)f(some)378 4926 y(partition)k(of)i(a)g
(\014nite)f(set)h FP(S)k FT(of)c(coloured)f(sen)m(tences)i(has)e(a)h
FN(K)q FT(-in)m(terp)s(olan)m(t,)i(then)d(it)g(do)s(es)h(not)378
5039 y(follo)m(w)30 b(that)h FP(S)k FT(is)29 b FN(K)q
FT(-inconsisten)m(t.)41 b(This)29 b(is)g(illustrated)f(in)h(the)i
(follo)m(wing)e(coun)m(terexample.)378 5225 y FQ(Example)34
b(7.8)46 b FT(Let)31 b FN(K)g FT(b)s(e)f(the)h(connectabilit)m(y)f
(relation)f(\()p FP(i)d FN($)f FP(j)31 b FN($)25 b FP(k)k
FN($)c FP(l)r FT(\),)31 b(and)f(let)1342 5430 y(\()p
FP(S)1433 5444 y FL(1)1473 5430 y FP(;)15 b(S)1569 5444
y FL(2)1608 5430 y FT(\))84 b(=)f(\()p FN(f:)p FP(A)2090
5392 y FO(i)2138 5430 y FN(_)20 b(:)p FP(A)2348 5392
y FO(k)2391 5430 y FN(g)p FP(;)15 b FN(f)p FP(A)2589
5392 y FO(j)2647 5430 y FN(_)20 b FP(A)2796 5392 y FO(l)2822
5430 y FN(g)p FT(\))1303 5567 y(\()p FP(X)1413 5581 y
FL(1)1453 5567 y FP(;)15 b(X)1568 5581 y FL(2)1608 5567
y FT(\))84 b(=)f(\()p FP(A)1984 5530 y FO(j)2021 5567
y FP(;)15 b FN(:)p FP(A)2190 5530 y FO(k)2232 5567 y
FT(\))p eop
%%Page: 142 152
142 151 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(142)378 396 y(then)22
b FP(S)633 410 y FL(1)676 396 y FN([)t(f)p FP(X)861 410
y FL(1)901 396 y FN(g)g FT(and)g FP(S)1193 410 y FL(2)1236
396 y FN([)t(f)p FP(X)1421 410 y FL(2)1461 396 y FN(g)g
FT(are)h(b)s(oth)e FN(K)q FT(-inconsisten)m(t)i(as)f(it)g(can)g(b)s(e)g
(seen)g(from)g(the)g(follo)m(wing)378 509 y(matrix)30
b(represen)m(tations)1549 543 y Fx(\024)1605 614 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 18.90648 9.45323 4.63118
} false /N@x 16 {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 FT(-inconsisten)m(t)h(whenev)
m(er)f(the)h(sets)h FP(S)3284 1809 y FL(1)3349 1795 y
FN([)26 b(f)p FP(X)3556 1809 y FL(1)3596 1795 y FN(g)40
b FT(and)378 1908 y FP(S)434 1922 y FL(2)493 1908 y FN([)20
b(f)p FP(X)694 1922 y FL(2)734 1908 y FN(g)30 b FT(are,)h(one)g
(requires)e(that)h(there)h(is)e(some)i(subset)e FN(P)2584
1922 y FL(1)2654 1908 y FT(of)i(the)f(colours)g(in)f
FP(S)3385 1922 y FL(1)3454 1908 y FT(and)g(some)378 2021
y(subset)h FN(P)720 2035 y FL(2)790 2021 y FT(of)g(the)h(colours)f(in)f
FP(S)1521 2035 y FL(2)1590 2021 y FT(suc)m(h)h(that:)514
2176 y FN(\017)46 b FT(The)30 b(colours)g(in)f FP(S)1263
2190 y FL(1)1332 2176 y FT(are)i(disjoin)m(t)e(from)h(the)g(colours)g
(in)f FP(S)2650 2190 y FL(2)2690 2176 y FT(,)h(i.e.,)16
b Fe(C)p FT(\()p FP(S)3049 2190 y FL(1)3088 2176 y FT(\))k
FN(\\)g Fe(C)p FT(\()p FP(S)3371 2190 y FL(2)3410 2176
y FT(\))26 b(=)f FN(fg)p FT(.)514 2351 y FN(\017)46 b
FT(All)29 b(the)i(colours)f(in)f FN(P)1389 2365 y FL(1)1459
2351 y FT(relate)h(with)g(all)f(the)h(colours)g(in)f
FN(P)2679 2365 y FL(2)2719 2351 y FT(,)h(i.e.,)16 b(\()p
FN(P)3029 2365 y FL(1)3094 2351 y FN($)25 b(P)3273 2365
y FL(2)3313 2351 y FT(\))h FN(\022)f(K)q FT(.)514 2526
y FN(\017)46 b FT(The)34 b(only)f(colours)g(in)f FP(S)1476
2540 y FL(1)1549 2526 y FT(that)j(relate)f(with)f(some)h(colour)f(in)g
FP(S)2890 2540 y FL(2)2963 2526 y FT(are)h(the)g(colours)f(in)g
FN(P)3763 2540 y FL(1)3803 2526 y FT(.)605 2639 y(Similarly)-8
b(,)25 b(the)i(only)e(colours)i(in)e FP(S)1819 2653 y
FL(2)1885 2639 y FT(that)i(relate)g(with)f(some)h(colour)f(in)g
FP(S)3183 2653 y FL(1)3248 2639 y FT(are)i(the)e(colours)605
2751 y(in)j FN(P)774 2765 y FL(2)814 2751 y FT(,)i(that)f(is)1487
2968 y FN(P)1550 2982 y FL(1)1615 2968 y FT(=)25 b FN(f)p
FP(i)h FN(2)f Fe(C)p FT(\()p FP(S)2046 2982 y FL(1)2085
2968 y FT(\))h FN(j)f FP(i)h FN(\030)2324 2982 y FK(K)2407
2968 y FP(j;)15 b(j)32 b FN(2)25 b Fe(C)p FT(\()p FP(S)2786
2982 y FL(2)2825 2968 y FT(\))p FN(g)p FP(;)1487 3124
y FN(P)1550 3138 y FL(2)1615 3124 y FT(=)g FN(f)p FP(i)h
FN(2)f Fe(C)p FT(\()p FP(S)2046 3138 y FL(2)2085 3124
y FT(\))h FN(j)f FP(i)h FN(\030)2324 3138 y FK(K)2407
3124 y FP(j;)15 b(j)32 b FN(2)25 b Fe(C)p FT(\()p FP(S)2786
3138 y FL(1)2825 3124 y FT(\))p FN(g)p FP(:)514 3359
y FN(\017)46 b FT(All)33 b(the)h(colours)g(in)f FN(P)1404
3373 y FL(1)1477 3359 y FT(relate)i(with)e(all)g(the)h(colours)f(in)g
FP(X)2732 3373 y FL(1)2772 3359 y FT(,)i(and)f(all)f(the)h(colours)f
(in)g FN(P)3788 3373 y FL(2)605 3472 y FT(relate)e(with)e(all)g(the)i
(colours)f(in)f FP(X)1838 3486 y FL(2)1878 3472 y FT(,)h(that)h(is)1466
3676 y(\(\()p FN(P)1599 3690 y FL(1)1665 3676 y FN($)25
b Fe(C)p FT(\()p FP(X)1947 3690 y FL(1)1987 3676 y FT(\)\))c
FN([)f FT(\()p FN(P)2257 3690 y FL(2)2322 3676 y FN($)25
b Fe(C)p FT(\()p FP(X)2604 3690 y FL(2)2644 3676 y FT(\)\)\))h
FN(\022)f(K)q FP(:)514 3911 y FN(\017)46 b FT(The)33
b(colours)f(in)g FP(X)1290 3925 y FL(1)1362 3911 y FT(relate)i(with)d
(no)i(other)g(colour)g(in)e FP(S)2636 3925 y FL(1)2708
3911 y FT(apart)j(from)e(those)h(in)f FN(P)3583 3925
y FL(1)3623 3911 y FT(,)h(and)605 4024 y(the)f(colours)f(in)g
FP(X)1256 4038 y FL(2)1327 4024 y FT(relate)h(with)e(no)i(other)g
(colour)f(in)g FP(S)2594 4038 y FL(2)2664 4024 y FT(apart)h(from)g(the)
g(colours)f(in)f FN(P)3763 4038 y FL(2)3803 4024 y FT(,)605
4137 y(that)h(is)1477 4341 y FN(f)p FP(i)26 b FN(2)f
Fe(C)p FT(\()p FP(X)1831 4355 y FL(1)1871 4341 y FT(\))h
FN(j)f FP(i)h FN(\030)2110 4355 y FK(K)2193 4341 y FP(j;)15
b(j)32 b FN(2)25 b Fe(C)p FT(\()p FP(S)2572 4355 y FL(1)2611
4341 y FT(\))p FN(g)h(\022)f(P)2876 4355 y FL(1)2915
4341 y FP(;)1477 4497 y FN(f)p FP(i)h FN(2)f Fe(C)p FT(\()p
FP(X)1831 4511 y FL(1)1871 4497 y FT(\))h FN(j)f FP(i)h
FN(\030)2110 4511 y FK(K)2193 4497 y FP(j;)15 b(j)32
b FN(2)25 b Fe(C)p FT(\()p FP(S)2572 4511 y FL(1)2611
4497 y FT(\))p FN(g)h(\022)f(P)2876 4511 y FL(1)2915
4497 y FP(:)378 4732 y FT(It)41 b(can)g(b)s(e)g(c)m(hec)m(k)m(ed)i
(that)e(in)f(example)h(7.8,)k(the)c(partition)e(\()p
FP(S)2666 4746 y FL(1)2706 4732 y FP(;)15 b(S)2802 4746
y FL(2)2842 4732 y FT(\))41 b(and)f(the)h(connectabilit)m(y)378
4845 y(relation)c FN(K)h FT(do)g(not)f(satisfy)g(the)h(ab)s(o)m(v)m(e)g
(conditions.)60 b(In)37 b(particular)f(there)h(is)g(no)g(sets)h
FN(P)3605 4859 y FL(1)3681 4845 y FT(and)378 4958 y FN(P)441
4972 y FL(2)517 4958 y FT(whic)m(h)d(satisfy)h(the)h(second)f(and)g
(third)f(conditions.)57 b(Ho)m(w)m(ev)m(er,)40 b(the)d(ab)s(o)m(v)m(e)h
(conditions)c(are)378 5071 y(satis\014ed)29 b(for)i(\()p
FP(S)958 5085 y FL(1)997 5071 y FP(;)15 b(S)1093 5085
y FL(2)1133 5071 y FT(\))30 b(and)g(the)h(connectabilit)m(y)f(relation)
f FN(K)22 b([)e FP(i)25 b FN($)g FP(l)33 b FT(with)1570
5275 y FN(P)1633 5289 y FL(1)1698 5275 y FT(=)25 b FN(f)p
FP(i;)15 b(k)s FN(g)184 b(P)2252 5289 y FL(2)2317 5275
y FT(=)25 b FN(f)p FP(j;)15 b(l)r FN(g)p FP(:)378 5479
y FT(It)30 b(can)h(also)f(b)s(e)g(c)m(hec)m(k)m(ed)i(that)f(the)g(set)g
FP(S)1834 5493 y FL(1)1893 5479 y FN([)20 b FP(S)2030
5493 y FL(2)2100 5479 y FT(is)29 b(\()p FN(K)22 b([)e
FP(i)26 b FN($)f FP(l)r FT(\)-inconsisten)m(t.)519 5592
y(W)-8 b(e)28 b(call)f(a)g(partition)f FI(wel)5 b(l-c)-5
b(olour)g(e)g(d)38 b FT(if)26 b(it)h(satis\014es)f(the)h(\014rst)f
(three)i(conditions)d(of)i(the)g(ab)s(o)m(v)m(e.)378
5705 y(This)i(notion)g(is)h(de\014ned)f(b)s(elo)m(w)g(in)h
(de\014nition)e(7.27)k(whic)m(h)d(requires)g(the)h(follo)m(wing)f
(de\014nition.)p eop
%%Page: 143 153
143 152 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(143)378 396 y FQ(De\014nition)35
b(7.26)h(\(Outside)e(Connecting)h(Colours\))46 b FT(Giv)m(en)34
b(the)h(connectabilit)m(y)f(relation)378 509 y FN(K)27
b FT(and)e(sets)g FP(S)874 523 y FL(1)939 509 y FT(and)g
FP(S)1167 523 y FL(2)1231 509 y FT(of)h(coloured)f(form)m(ulae,)h(w)m
(e)g(denote)g(the)f(set)h(of)g(colours)f(in)f FP(S)3349
523 y FL(1)3413 509 y FT(that)i(relate)378 622 y(with)j(some)i(colours)
f(in)f FP(S)1284 636 y FL(2)1353 622 y FT(b)m(y)1070
846 y FP(S)1126 860 y FL(1)1209 794 y FK(K)1191 846 y
FN(!)c FP(S)1363 860 y FL(2)1427 846 y FT(=)g FN(f)p
FP(i)h FN(2)f Fe(C)p FT(\()p FP(S)1858 860 y FL(1)1897
846 y FT(\))h FN(j)g FP(i)f FN(\030)2136 860 y FK(K)2219
846 y FP(j)67 b FT(for)30 b(some)h FP(j)f FN(2)25 b Fe(C)p
FT(\()p FP(S)2990 860 y FL(2)3029 846 y FT(\))q FN(g)p
FP(:)378 1050 y FT(Similarly)-8 b(,)27 b(w)m(e)k(de\014ne)f(the)g
(follo)m(wing:)1070 1273 y FP(S)1126 1287 y FL(1)1209
1222 y FK(K)1191 1273 y FN( )25 b FP(S)1363 1287 y FL(2)1427
1273 y FT(=)g FN(f)p FP(j)32 b FN(2)25 b Fe(C)p FT(\()p
FP(S)1870 1287 y FL(2)1909 1273 y FT(\))g FN(j)h FP(i)f
FN(\030)2147 1287 y FK(K)2231 1273 y FP(j)66 b FT(for)30
b(some)h FP(i)25 b FN(2)g Fe(C)p FT(\()p FP(S)2990 1287
y FL(1)3029 1273 y FT(\))q FN(g)p FP(:)378 1477 y FT(W)-8
b(e)45 b(also)e(de\014ne)g(the)h(out)m(w)m(ard)g(connecting)g(colours)f
(in)g(some)h(set)g(of)g(coloured)f(sen)m(tences)i FP(S)378
1590 y FT(according)c(to)g FN(K)q FT(,)j(and)c(denote)h(it)g(b)m(y)f
FN(K)k(")f FP(S)5 b FT(,)44 b(as)d(the)g(colours)f(in)f
FP(S)46 b FT(that)41 b(relate)g(with)e(some)378 1703
y(colours)30 b(not)g(in)f FP(S)5 b FT(:)1157 1907 y FN(K)27
b(")e FP(S)30 b FT(=)25 b FN(f)p FP(i)h FN(2)f Fe(C)p
FT(\()p FP(S)5 b FT(\))26 b FN(j)f FP(i)h FN(\030)2084
1921 y FK(K)2167 1907 y FP(j)66 b FT(for)31 b(some)f
FP(j)h FN(62)25 b Fe(C)p FT(\()p FP(S)5 b FT(\))q FN(g)p
FP(:)708 b Ff(\003)378 2120 y FT(The)30 b(follo)m(wing)f(result)g
(follo)m(ws)g(from)h(the)h(ab)s(o)m(v)m(e)g(de\014nition.)378
2308 y FQ(Prop)s(osition)36 b(7.11)46 b FI(Given)30 b(the)g(sets)g
FP(S)1817 2322 y FL(1)1885 2308 y FI(and)h FP(S)2115
2322 y FL(2)2184 2308 y FI(of)f(c)-5 b(olour)g(e)g(d)31
b(formulae,)h(and)e(a)g(c)-5 b(onne)g(ctability)378 2440
y(r)g(elation)34 b FN(K)q FI(,)f(then)g FT(\()p FP(S)1135
2454 y FL(1)1219 2388 y FK(K)1200 2440 y FN(!)25 b FP(S)1372
2454 y FL(2)1412 2440 y FT(\))g(=)g(\()p FP(S)1659 2454
y FL(2)1742 2388 y FK(K)1724 2440 y FN( )g FP(S)1896
2454 y FL(1)1935 2440 y FT(\))p FI(.)378 2651 y FQ(Pro)s(of)p
FT(:)31 b(follo)m(ws)f(from)g(the)g(de\014nitions)e(of)j
FP(S)1957 2665 y FL(1)2040 2600 y FK(K)2021 2651 y FN(!)25
b FP(S)2193 2665 y FL(2)2263 2651 y FT(and)30 b FP(S)2496
2665 y FL(2)2579 2600 y FK(K)2560 2651 y FN( )25 b FP(S)2732
2665 y FL(1)2771 2651 y FT(.)961 b Ff(\004)519 2877 y
FT(W)-8 b(ell-coloured)30 b(partitions)f(are)i(no)m(w)f(de\014ned)f(as)
i(follo)m(ws.)378 3090 y FQ(De\014nition)k(7.27)h(\(W)-9
b(ell-Coloured)35 b(P)m(artition\))45 b FT(A)k(pair)e(of)i(sets)g(of)g
(coloured)f(sen)m(tences)378 3203 y(\()p FP(S)469 3217
y FL(1)508 3203 y FP(;)15 b(S)604 3217 y FL(2)644 3203
y FT(\))45 b(is)e(said)h(to)h(b)s(e)e(a)i(w)m(ell-coloured)e(partition)
g(of)i(some)g(set)g FP(S)k FT(with)43 b(resp)s(ect)h(to)h(some)378
3316 y(connectabilit)m(y)30 b(relation)f FN(K)j FT(if)489
3503 y(1.)46 b FP(S)661 3517 y FL(1)721 3503 y FN([)19
b FP(S)857 3517 y FL(2)922 3503 y FT(=)25 b FP(S)5 b
FT(,)489 3691 y(2.)46 b Fe(C)p FT(\()p FP(S)752 3705
y FL(1)791 3691 y FT(\))21 b FN(\\)f Fe(C)p FT(\()p FP(S)1075
3705 y FL(2)1114 3691 y FT(\))26 b(=)f FN(fg)p FT(,)489
3903 y(3.)46 b(for)30 b(ev)m(ery)i(colour)e FP(i)25 b
FN(2)g FT(\()p FP(S)1491 3917 y FL(1)1574 3851 y FK(K)1556
3903 y FN(!)g FP(S)1728 3917 y FL(2)1767 3903 y FT(\))31
b(and)e FP(j)i FN(2)25 b FT(\()p FP(S)2254 3917 y FL(1)2337
3851 y FK(K)2319 3903 y FN( )g FP(S)2491 3917 y FL(2)2530
3903 y FT(\))31 b(it)f(is)f(the)i(case)g(that)g FP(i)26
b FN(\030)3452 3917 y FK(K)3535 3903 y FP(j)5 b FT(.)155
b Ff(\003)378 4115 y FT(Note)41 b(that)f(the)g(third)d(condition)i(in)f
(the)i(ab)s(o)m(v)m(e)g(de\014nition)e(corresp)s(onds)g(to)i(the)g
(second)f(and)378 4228 y(third)34 b(conditions)h(giv)m(en)h(on)g(page)h
(142)g(for)f(the)h(set)f FP(S)2328 4242 y FL(1)2392 4228
y FN([)23 b FP(S)2532 4242 y FL(2)2607 4228 y FT(to)37
b(b)s(e)f FN(K)q FT(-inconsisten)m(t)g(whenev)m(er)378
4341 y FP(S)434 4355 y FL(1)497 4341 y FN([)23 b(f)p
FP(X)701 4355 y FL(1)741 4341 y FN(g)36 b FT(and)f FP(S)1060
4355 y FL(2)1122 4341 y FN([)24 b(f)p FP(X)1327 4355
y FL(2)1367 4341 y FN(g)36 b FT(are)f FN(K)q FT(-inconsisten)m(t)h
(where)f FP(X)2559 4308 y FK(U)2552 4365 y FL(1)2648
4341 y FT(=)e FN(:)p FP(X)2895 4308 y FK(U)2888 4365
y FL(2)2950 4341 y FT(.)56 b(The)35 b(sets)g FN(P)3468
4355 y FL(1)3543 4341 y FT(and)g FN(P)3788 4355 y FL(2)378
4454 y FT(giv)m(en)30 b(in)f(the)i(conditions)e(on)h(page)h(142)h(can)f
(b)s(e)e(de\014ned)g(b)m(y)1373 4677 y FN(P)1436 4691
y FL(1)1501 4677 y FT(=)c(\()p FP(S)1688 4691 y FL(1)1771
4626 y FK(K)1753 4677 y FN(!)g FP(S)1925 4691 y FL(2)1964
4677 y FT(\))182 b FN(P)2244 4691 y FL(2)2309 4677 y
FT(=)25 b(\()p FP(S)2496 4691 y FL(1)2579 4626 y FK(K)2561
4677 y FN( )g FP(S)2733 4691 y FL(2)2772 4677 y FT(\))p
FP(:)378 4881 y FT(It)35 b(is)e(clear)i(that)g(if)f(\()p
FP(S)1178 4895 y FL(1)1217 4881 y FP(;)15 b(S)1313 4895
y FL(2)1353 4881 y FT(\))35 b(is)e(a)i(w)m(ell-coloured)f(partition)f
(of)i(some)g(set)g FP(S)k FT(with)33 b(resp)s(ect)i(to)g
FN(K)378 4994 y FT(then)28 b(so)h(is)f(\()p FP(S)874
5008 y FL(2)913 4994 y FP(;)15 b(S)1009 5008 y FL(1)1049
4994 y FT(\);)29 b(it)f(also)h(the)g(case)g(that)g(\()p
FP(S)2039 5008 y FL(1)2079 4994 y FP(;)15 b(S)2175 5008
y FL(2)2214 4994 y FT(\))29 b(is)f(a)h(partition)e(\(as)i
FP(S)3020 5008 y FL(1)3076 4994 y FN(\\)16 b FP(S)3209
5008 y FL(2)3273 4994 y FT(=)25 b FN(fg)30 b FT(from)e(the)378
5107 y(second)i(condition)f(in)g(de\014nition)f(7.27\).)519
5220 y(The)i(third)e(condition)h(in)h(the)g(ab)s(o)m(v)m(e)i
(de\014nition)c(can)i(b)s(e)g(substituted)f(with)g(the)i(equation)1115
5443 y FN(K)q(d)p FP(S)5 b FN(e)27 b FT(=)e FN(K)q(d)p
FP(S)1615 5457 y FL(1)1655 5443 y FN(e)c([)f(K)q(d)p
FP(S)1963 5457 y FL(2)2003 5443 y FN(e)g([)g FT(\()p
FP(S)2235 5457 y FL(1)2318 5392 y FK(K)2300 5443 y FN(!)25
b FP(S)2472 5457 y FL(2)2511 5443 y FT(\))h FN($)f FT(\()p
FP(S)2779 5457 y FL(1)2862 5392 y FK(K)2844 5443 y FN( )g
FP(S)3016 5457 y FL(2)3055 5443 y FT(\))378 5648 y(as)31
b(sho)m(wn)e(in)g(the)i(follo)m(wing)e(prop)s(osition.)p
eop
%%Page: 144 154
144 153 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(144)378 396 y FQ(Prop)s(osition)36
b(7.12)46 b FI(Given)30 b(a)f(p)-5 b(artition)32 b FT(\()p
FP(S)1973 410 y FL(1)2012 396 y FP(;)15 b(S)2108 410
y FL(2)2148 396 y FT(\))30 b FI(of)f(a)h(set)f FP(S)34
b FI(of)c(c)-5 b(olour)g(e)g(d)32 b(sentenc)-5 b(es)29
b(such)h(that)378 509 y Fe(C)p FT(\()p FP(S)525 523 y
FL(1)564 509 y FT(\))21 b FN(\\)f Fe(C)p FT(\()p FP(S)848
523 y FL(2)887 509 y FT(\))27 b(=)e FN(fg)34 b FI(then)g
FT(\()p FP(S)1463 523 y FL(1)1502 509 y FP(;)15 b(S)1598
523 y FL(2)1638 509 y FT(\))33 b FI(is)g(a)h(wel)5 b(l-c)-5
b(olour)g(e)g(d)35 b(p)-5 b(aritition)35 b(of)e FP(S)38
b FI(with)c(r)-5 b(esp)g(e)g(ct)35 b(to)e(some)378 646
y(c)-5 b(onne)g(ctability)27 b(r)-5 b(elation)28 b FN(K)f
FI(if)e(and)i(only)f(if)f FN(K)q(d)p FP(S)5 b FN(e)27
b FT(=)e FN(K)q(d)p FP(S)2385 660 y FL(1)2425 646 y FN(e)5
b([)g(K)q(d)p FP(S)2702 660 y FL(2)2743 646 y FN(e)g([)g
FT(\()p FP(S)2945 660 y FL(1)3028 595 y FK(K)3009 646
y FN(!)26 b FP(S)3182 660 y FL(2)3221 646 y FT(\))f FN($)h
FT(\()p FP(S)3489 660 y FL(1)3572 595 y FK(K)3553 646
y FN( )g FP(S)3726 660 y FL(2)3765 646 y FT(\))p FI(.)378
859 y FQ(Pro)s(of)p FT(:)32 b(Let)f(us)f(assume)g(that)h(\()p
FP(S)1570 873 y FL(1)1610 859 y FP(;)15 b(S)1706 873
y FL(2)1745 859 y FT(\))31 b(is)f(a)h(w)m(ell-coloured)e(paritition)g
(of)i FP(S)k FT(with)29 b(resp)s(ect)i(to)g FN(K)378
972 y FT(and)f(that)h Fe(C)p FT(\()p FP(S)899 986 y FL(1)938
972 y FT(\))20 b FN(\\)g Fe(C)p FT(\()p FP(S)1221 986
y FL(2)1260 972 y FT(\))26 b(=)f FN(fg)p FT(.)42 b(The)29
b(third)g(condition)g(in)g(de\014nition)f(7.27)k(is)d(equiv)-5
b(alen)m(t)30 b(to)1521 1200 y(\()p FP(S)1612 1214 y
FL(1)1695 1149 y FK(K)1677 1200 y FN(!)25 b FP(S)1849
1214 y FL(2)1888 1200 y FT(\))h FN($)f FT(\()p FP(S)2156
1214 y FL(1)2239 1149 y FK(K)2221 1200 y FN( )g FP(S)2393
1214 y FL(2)2432 1200 y FT(\))h FN(\022)f(K)q FP(:)378
1404 y FT(No)m(w,)31 b(since)1501 1628 y FP(i)25 b FN(2)g
FT(\()p FP(S)1734 1642 y FL(1)1817 1576 y FK(K)1799 1628
y FN(!)g FP(S)1971 1642 y FL(2)2010 1628 y FT(\))h FN(\))f
FP(i)g FN(2)g Fe(C)p FT(\()p FP(S)2476 1642 y FL(1)2516
1628 y FT(\))g FN(\022)g Fe(C)p FT(\()p FP(S)5 b FT(\))1324
1790 y(and)30 b FP(j)g FN(2)25 b FT(\()p FP(S)1745 1804
y FL(1)1829 1738 y FK(K)1810 1790 y FN( )g FP(S)1982
1804 y FL(2)2021 1790 y FT(\))h FN(\))f FP(j)31 b FN(2)25
b Fe(C)p FT(\()p FP(S)2499 1804 y FL(2)2538 1790 y FT(\))h
FN(\022)f Fe(C)p FT(\()p FP(S)5 b FT(\))378 2018 y(then)30
b(all)f(the)i(colours)f(in)f(\()p FP(S)1374 2032 y FL(1)1457
1967 y FK(K)1439 2018 y FN(!)c FP(S)1611 2032 y FL(2)1650
2018 y FT(\))g FN($)h FT(\()p FP(S)1918 2032 y FL(1)2001
1967 y FK(K)1982 2018 y FN( )g FP(S)2155 2032 y FL(2)2194
2018 y FT(\))k(are)h(in)e FP(S)36 b FT(and)29 b(therefore)793
2246 y(\()p FP(S)884 2260 y FL(1)967 2195 y FK(K)949
2246 y FN(!)c FP(S)1121 2260 y FL(2)1160 2246 y FT(\))h
FN($)f FT(\()p FP(S)1428 2260 y FL(1)1511 2195 y FK(K)1493
2246 y FN( )g FP(S)1665 2260 y FL(2)1704 2246 y FT(\))h
FN(\022)e(K)58 b(\))d FT(\()p FP(S)2224 2260 y FL(1)2307
2195 y FK(K)2289 2246 y FN(!)25 b FP(S)2461 2260 y FL(2)2500
2246 y FT(\))h FN($)f FT(\()p FP(S)2768 2260 y FL(1)2851
2195 y FK(K)2833 2246 y FN( )g FP(S)3005 2260 y FL(2)3044
2246 y FT(\))h FN(\022)f(K)q(d)p FP(S)5 b FN(e)378 2451
y FT(and)30 b(so)g(since)g FN(K)q(d)p FP(S)5 b FN(e)27
b(\022)e(K)32 b FT(our)d(goal)i(is)f(equiv)-5 b(alen)m(t)30
b(to)1020 2679 y(\()p FP(S)1111 2693 y FL(1)1194 2628
y FK(K)1176 2679 y FN(!)25 b FP(S)1348 2693 y FL(2)1387
2679 y FT(\))g FN($)h FT(\()p FP(S)1655 2693 y FL(1)1738
2628 y FK(K)1719 2679 y FN( )g FP(S)1892 2693 y FL(2)1931
2679 y FT(\))f FN(\022)g(K)q(d)p FP(S)5 b FN(e)93 b FT(if)29
b(and)h(only)f(if)1186 2841 y FN(K)q(d)p FP(S)5 b FN(e)26
b FT(=)f FN(K)q(d)p FP(S)1685 2855 y FL(1)1726 2841 y
FN(e)20 b([)g(K)q(d)p FP(S)2033 2855 y FL(2)2073 2841
y FN(e)h([)f FT(\()p FP(S)2306 2855 y FL(1)2389 2789
y FK(K)2370 2841 y FN(!)25 b FP(S)2542 2855 y FL(2)2582
2841 y FT(\))g FN($)g FT(\()p FP(S)2849 2855 y FL(1)2933
2789 y FK(K)2914 2841 y FN( )g FP(S)3086 2855 y FL(2)3126
2841 y FT(\))p FP(:)378 3045 y FT(The)f(`if)7 b(')24
b(direction)g(of)h(the)g(ab)s(o)m(v)m(e)h(is)d(straigh)m(tforw)m(ard)i
(and)f(w)m(e)h(sho)m(w)g(that)g(the)g(`only)f(if)7 b(')24
b(direction)378 3158 y(holds)29 b(b)m(y)h(assuming)f(its)h(left-hand)f
(side)h(and)f(considering)g(the)h(follo)m(wing)f(t)m(w)m(o)j(cases:)514
3365 y FN(\017)46 b(K)q(d)p FP(S)5 b FN(e)35 b(\022)f(K)q(d)p
FP(S)1122 3379 y FL(1)1162 3365 y FN(e)24 b([)f(K)q(d)p
FP(S)1476 3379 y FL(2)1517 3365 y FN(e)h([)f FT(\()p
FP(S)1756 3379 y FL(1)1848 3313 y FK(K)1829 3365 y FN(!)34
b FP(S)2010 3379 y FL(2)2049 3365 y FT(\))g FN($)g FT(\()p
FP(S)2334 3379 y FL(1)2426 3313 y FK(K)2407 3365 y FN( )g
FP(S)2588 3379 y FL(2)2627 3365 y FT(\):)52 b(if)34 b(\()p
FP(i;)15 b(j)5 b FT(\))36 b FN(2)d(K)q(d)p FP(S)5 b FN(e)37
b FT(then)e(either)605 3478 y(b)s(oth)44 b FP(i)h FT(and)f
FP(j)50 b FT(are)44 b(in)g(the)g(same)h(set)g(in)e(the)i(partition)e
(\(i.e.,)16 b(in)43 b FP(S)3094 3492 y FL(1)3178 3478
y FT(or)h FP(S)3359 3492 y FL(2)3398 3478 y FT(\))h(in)e(whic)m(h)605
3591 y(case)c(\()p FP(i;)15 b(j)5 b FT(\))39 b(is)e(in)f
FN(K)q(d)p FP(S)1405 3605 y FL(1)1445 3591 y FN(e)i FT(or)g
FN(K)q(d)p FP(S)1808 3605 y FL(2)1848 3591 y FN(e)p FT(,)i(or)d(else)h
(they)f(are)h(in)e(di\013eren)m(t)h(sets)h(in)f(whic)m(h)f(case)605
3728 y(\()p FP(i;)15 b(j)5 b FT(\))28 b FN(2)c FT(\()p
FP(S)992 3742 y FL(1)1076 3676 y FK(K)1057 3728 y FN(!)h
FP(S)1229 3742 y FL(2)1269 3728 y FT(\))g FN($)g FT(\()p
FP(S)1536 3742 y FL(1)1620 3676 y FK(K)1601 3728 y FN( )g
FP(S)1773 3742 y FL(2)1812 3728 y FT(\).)514 3939 y FN(\017)46
b(K)q(d)p FP(S)771 3953 y FL(1)811 3939 y FN(e)q([)q(K)q(d)p
FP(S)1080 3953 y FL(2)1121 3939 y FN(e)q([)q FT(\()p
FP(S)1315 3953 y FL(1)1400 3888 y FK(K)1381 3939 y FN(!)25
b FP(S)1553 3953 y FL(2)1593 3939 y FT(\))g FN($)g FT(\()p
FP(S)1860 3953 y FL(1)1944 3888 y FK(K)1925 3939 y FN( )g
FP(S)2097 3953 y FL(2)2136 3939 y FT(\))h FN(\022)f(K)q(d)p
FP(S)5 b FN(e)p FT(:)37 b(it)21 b(is)f(the)h(case)h(that)g
FN(K)q(d)p FP(S)3414 3953 y FL(1)3454 3939 y FN(e)k(\022)f(K)q(d)p
FP(S)5 b FN(e)605 4076 y FT(and)31 b(that)h FN(K)q(d)p
FP(S)1147 4090 y FL(2)1187 4076 y FN(e)c(\022)f(K)q(d)p
FP(S)5 b FN(e)p FT(,)33 b(and)e(it)g(is)f(already)i(assumed)e(that)i
(\()p FP(S)2961 4090 y FL(1)3046 4025 y FK(K)3028 4076
y FN(!)27 b FP(S)3202 4090 y FL(2)3241 4076 y FT(\))h
FN($)f FT(\()p FP(S)3513 4090 y FL(1)3598 4025 y FK(K)3579
4076 y FN( )g FP(S)3753 4090 y FL(2)3793 4076 y FT(\))605
4189 y(is)j(a)g(subset)g(of)h FN(K)q(d)p FP(S)5 b FN(e)p
FT(.)2366 b Ff(\004)519 4377 y FT(Theorem)29 b(7.8)i(b)s(elo)m(w)d(giv)
m(es)i(a)f(n)m(um)m(b)s(er)f(of)i(su\016cien)m(t)f(conditions)e(for)i
(whic)m(h)f(the)i(set)g FP(S)3597 4391 y FL(1)3654 4377
y FN([)18 b FP(S)3789 4391 y FL(2)378 4490 y FT(is)35
b FN(K)q FT(-unsatis\014able)g(whenev)m(er)g(the)i(sets)f
FP(S)1909 4504 y FL(1)1972 4490 y FN([)23 b(f)p FP(X)2176
4504 y FL(1)2217 4490 y FN(g)36 b FT(and)f FP(S)2536
4504 y FL(2)2599 4490 y FN([)24 b(f)p FP(X)2804 4504
y FL(2)2844 4490 y FN(g)36 b FT(are)h(for)e(some)i(sen)m(tences)378
4603 y FP(X)453 4617 y FL(1)524 4603 y FT(and)31 b FP(X)777
4617 y FL(2)848 4603 y FT(suc)m(h)g(that)h FN(:)p FP(X)1395
4570 y FK(U)1388 4627 y FL(1)1477 4603 y FT(=)27 b FP(X)1657
4570 y FK(U)1650 4627 y FL(2)1712 4603 y FT(.)44 b(The)30
b(conditions)g(giv)m(en)i(in)e(this)g(theorem)i(corresp)s(ond)e(to)378
4716 y(those)35 b(giv)m(en)f(on)g(page)g(142.)53 b(The)34
b(\014rst)f(three)i(conditions)d(on)i(page)h(142)g(are)g(giv)m(en)f(b)m
(y)g(the)g(fact)378 4829 y(that)28 b(the)h(partition)d(\()p
FP(S)1194 4843 y FL(1)1234 4829 y FP(;)15 b(S)1330 4843
y FL(2)1369 4829 y FT(\))29 b(is)e(required)f(to)i(b)s(e)g(w)m
(ell-coloured)f(with)f(resp)s(ect)i(to)h FN(K)q FT(,)g(so)f(that)h(the)
378 4942 y(sets)i FN(P)619 4956 y FL(1)689 4942 y FT(and)e
FN(P)928 4956 y FL(2)998 4942 y FT(men)m(tioned)h(on)h(page)g(142)g
(are)g(giv)m(en)g(b)m(y:)1373 5165 y FN(P)1436 5179 y
FL(1)1501 5165 y FT(=)25 b(\()p FP(S)1688 5179 y FL(1)1771
5113 y FK(K)1753 5165 y FN(!)g FP(S)1925 5179 y FL(2)1964
5165 y FT(\))182 b FN(P)2244 5179 y FL(2)2309 5165 y
FT(=)25 b(\()p FP(S)2496 5179 y FL(2)2579 5113 y FK(K)2561
5165 y FN(!)g FP(S)2733 5179 y FL(1)2772 5165 y FT(\))p
FP(:)378 5369 y FT(The)35 b(last)g(t)m(w)m(o)i(conditions)d(on)h(page)h
(142)h(are)f(satis\014ed)e(b)m(y)i(restricting)e(the)i(sen)m(tences)g
FP(X)3606 5383 y FL(1)3681 5369 y FT(and)378 5482 y FP(X)453
5496 y FL(2)521 5482 y FT(to)30 b(b)s(e)e(homogeneously)g(coloured,)h
(b)m(y)f FP(m)h FT(and)f FP(n)g FT(resp)s(ectiv)m(ely)-8
b(,)29 b(sa)m(y)-8 b(,)30 b(and)e(b)m(y)g(the)h(conditions:)489
5689 y(1.)46 b FN(K)q FT(\()p FP(m)p FT(\))27 b(=)e(\()p
FP(S)1039 5703 y FL(1)1122 5637 y FK(K)1103 5689 y FN(!)g
FP(S)1275 5703 y FL(2)1315 5689 y FT(\))30 b(and)g FN(K)q
FT(\()p FP(n)p FT(\))c(=)f(\()p FP(S)1965 5703 y FL(2)2048
5637 y FK(K)2030 5689 y FN(!)g FP(S)2202 5703 y FL(1)2241
5689 y FT(\),)p eop
%%Page: 145 155
145 154 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(145)489 396 y(2.)46 b
FP(m)35 b(=)-55 b FN(2)25 b Fe(C)p FT(\()p FP(S)943 410
y FL(1)982 396 y FT(\))31 b(and)f FP(n)35 b(=)-55 b FN(2)24
b Fe(C)p FT(\()p FP(S)1537 410 y FL(2)1577 396 y FT(\).)378
560 y(The)30 b(results)f(on)h(recolouring)f(literals)g(giv)m(en)h(in)f
(section)i(7.4)g(can)g(b)s(e)e(used)h(with)f(theorem)i(7.8)g(to)378
673 y(sho)m(w)g(that)h(the)f(conditions)f(giv)m(en)h(on)g(page)h(142)h
(are)e(also)g(su\016cien)m(t)g(for)g(the)g(set)h FP(S)3355
687 y FL(1)3415 673 y FN([)20 b FP(S)3552 687 y FL(2)3623
673 y FT(to)32 b(b)s(e)378 786 y FN(K)q FT(-unsatis\014able)d(whenev)m
(er)h(the)h(sets)g FP(S)1790 800 y FL(1)1849 786 y FN([)20
b(f)p FP(X)2050 800 y FL(1)2090 786 y FN(g)31 b FT(and)f
FP(S)2399 800 y FL(2)2458 786 y FN([)20 b(f)p FP(X)2659
800 y FL(2)2699 786 y FN(g)31 b FT(are.)378 970 y FQ(Theorem)j(7.8)46
b FI(Given)29 b(a)g(c)-5 b(onne)g(ctability)31 b(r)-5
b(elation)30 b FN(K)q FI(,)g(two)g(sets)f(of)h(c)-5 b(olour)g(e)g(d)31
b(sentenc)-5 b(es)29 b FP(S)3608 984 y FL(1)3647 970
y FI(,)h FP(S)3761 984 y FL(2)3800 970 y FI(,)378 1083
y(such)38 b(that)h FT(\()p FP(S)872 1097 y FL(1)912 1083
y FP(;)15 b(S)1008 1097 y FL(2)1047 1083 y FT(\))39 b
FI(is)e(a)i(wel)5 b(l-c)-5 b(olour)g(e)g(d)40 b(p)-5
b(artition)40 b(\(of)e FP(S)2434 1097 y FL(1)2498 1083
y FN([)23 b FP(S)2638 1097 y FL(2)2678 1083 y FI(\))38
b(with)g(r)-5 b(esp)g(e)g(ct)40 b(to)e FN(K)q FI(,)i(and)f(two)378
1195 y(c)-5 b(olours)34 b FP(m)e FI(and)i FP(n)e FI(such)h(that)485
1359 y(1.)46 b FN(K)q FT(\()p FP(m)p FT(\))27 b(=)e(\()p
FP(S)1039 1373 y FL(1)1122 1307 y FK(K)1103 1359 y FN(!)g
FP(S)1275 1373 y FL(2)1315 1359 y FT(\))33 b FI(and)g
FN(K)q FT(\()p FP(n)p FT(\))26 b(=)f(\()p FP(S)1967 1373
y FL(2)2050 1307 y FK(K)2032 1359 y FN(!)g FP(S)2204
1373 y FL(1)2243 1359 y FT(\))p FI(,)485 1537 y(2.)46
b FP(m)35 b(=)-55 b FN(2)25 b Fe(C)p FT(\()p FP(S)943
1551 y FL(1)982 1537 y FT(\))33 b FI(and)h FP(n)h(=)-55
b FN(2)24 b Fe(C)p FT(\()p FP(S)1539 1551 y FL(2)1579
1537 y FT(\))p FI(,)378 1701 y(then,)32 b(the)f(set)h
FP(S)953 1715 y FL(1)1009 1701 y FN([)17 b FP(S)1143
1715 y FL(2)1213 1701 y FI(is)32 b FN(K)q FI(-c)-5 b(onsistent)32
b(if)f(and)h(only)g(if)f(ther)-5 b(e)32 b(is)g(some)g(unc)-5
b(olour)g(e)g(d)33 b(sentenc)-5 b(e)32 b FP(X)378 1814
y FI(such)h(that)h(the)f(sets)f FP(S)1156 1828 y FL(1)1216
1814 y FN([)20 b(f)p FP(X)1424 1781 y FO(m)1491 1814
y FN(g)33 b FI(and)h FP(S)1802 1828 y FL(2)1861 1814
y FN([)20 b(f:)p FP(X)2130 1781 y FO(n)2177 1814 y FN(g)33
b FI(ar)-5 b(e)34 b FN(K)q FI(-c)-5 b(onsistent.)378
1997 y FQ(Pro)s(of)p FT(:)36 b(W)-8 b(e)37 b(\014rst)d(sho)m(w)h(the)h
(`only)e(if)7 b(')35 b(direction.)54 b(Giv)m(en)35 b(the)h
(connectabilit)m(y)e(relation)h FN(K)h FT(and)378 2110
y(t)m(w)m(o)c(colours)e FP(m)g FT(and)f FP(n)p FT(,)i(w)m(e)f(de\014ne)
g(the)h(palettes)f FN(P)2210 2124 y FL(1)2280 2110 y
FT(and)g FN(P)2520 2124 y FL(2)2590 2110 y FT(as)h(follo)m(ws:)1471
2314 y FN(P)1534 2328 y FL(1)1599 2314 y FT(=)25 b FN(K)q
FT(\()p FP(m)p FT(\))115 b(and)e FN(P)2353 2328 y FL(2)2417
2314 y FT(=)25 b FN(K)q FT(\()p FP(n)p FT(\))p FP(:)378
2519 y FT(W)-8 b(e)32 b(further)f(assume)g(that)h(for)f(all)f
FP(i)d FN(2)g(P)1832 2533 y FL(1)1903 2519 y FT(and)j
FP(j)j FN(2)26 b(P)2300 2533 y FL(2)2371 2519 y FT(it)31
b(is)g(the)g(case)i(that)e FP(i)d FN(\030)3234 2533 y
FK(K)3318 2519 y FP(j)5 b FT(,)33 b(and)e(de\014ne)378
2632 y(the)g(collection)f(of)g(sets)h(of)f(coloured)g(sen)m(tences)i
FN(C)j FT(as)c(follo)m(ws:)806 2836 y FN(C)89 b FT(=)82
b FN(f)p FP(S)1197 2850 y FL(1)1257 2836 y FN([)20 b
FP(R)1408 2798 y FO(m)1500 2836 y FN(j)55 b FT(for)31
b(an)m(y)f(sets)h FP(S)2125 2850 y FL(1)2194 2836 y FT(and)f
FP(R)h FT(for)f(whic)m(h)1218 2974 y(there)g(is)g(some)g(set)h
FP(S)1967 2988 y FL(2)2037 2974 y FT(suc)m(h)f(that)1248
3111 y(\()p FP(S)1339 3125 y FL(1)1378 3111 y FP(;)15
b(S)1474 3125 y FL(2)1514 3111 y FT(\))31 b(is)e(a)i(w)m(ell-coloured)e
(partition)g(with)g(resp)s(ect)i(to)g FN(K)q FT(,)1248
3273 y FN(P)1311 3287 y FL(1)1376 3273 y FT(=)25 b(\()p
FP(S)1563 3287 y FL(1)1646 3222 y FK(K)1628 3273 y FN(!)g
FP(S)1800 3287 y FL(2)1839 3273 y FT(\))31 b(and)e FN(P)2144
3287 y FL(2)2209 3273 y FT(=)c(\()p FP(S)2396 3287 y
FL(1)2479 3222 y FK(K)2461 3273 y FN( )g FP(S)2633 3287
y FL(2)2672 3273 y FT(\),)1248 3411 y FP(m)g FN(62)g
Fe(C)p FT(\()p FP(S)1586 3425 y FL(1)1625 3411 y FT(\))31
b(and)e FP(n)c FN(62)g Fe(C)p FT(\()p FP(S)2180 3425
y FL(2)2219 3411 y FT(\),)1248 3549 y FP(S)1304 3563
y FL(1)1363 3549 y FN([)20 b FP(S)1500 3563 y FL(2)1570
3549 y FT(is)29 b FN(K)q FT(-consisten)m(t,)j(and)1248
3687 y FP(S)1304 3701 y FL(2)1363 3687 y FN([)20 b(f:)p
FP(X)1632 3654 y FO(n)1680 3687 y FN(g)30 b FT(is)g FN(K)q
FT(-inconsisten)m(t)g(for)g(all)f FP(X)k FN(2)25 b FP(R)q
FN(g)p FP(:)378 3891 y FT(W)-8 b(e)46 b(no)m(w)g(sho)m(w)f(that)h
FN(C)k FT(is)45 b(a)g FN(K)q FT(-consistency)i(prop)s(ert)m(y)d(b)m(y)h
(deriving)f(all)g(the)h(conditions)f(in)378 4004 y(De\014nition)31
b(7.13.)49 b(Let)34 b FP(S)g FN(2)28 b(C)5 b FT(,)34
b(then)e FP(S)i FT(=)29 b FP(S)1949 4018 y FL(1)2010
4004 y FN([)21 b FP(R)2162 3971 y FO(m)2261 4004 y FT(for)33
b(some)g(sets)g FP(S)2869 4018 y FL(1)2940 4004 y FT(and)g
FP(R)g FT(for)f(whic)m(h)g(there)378 4117 y(is)j(some)i(set)g
FP(S)913 4131 y FL(2)989 4117 y FT(satisfying)e(the)h(requiremen)m(ts)g
(in)f(the)h(ab)s(o)m(v)m(e)i(de\014nition)c(of)i FN(C)5
b FT(.)59 b(Note)38 b(that)f FP(S)3789 4131 y FL(1)378
4230 y FT(and)e FP(R)630 4197 y FO(m)732 4230 y FT(are)h(disjoin)m(t)e
(as)h(otherwise)g FP(i)g FT(=)e FP(j)40 b FT(=)33 b FP(m)j
FT(but)e FP(m)44 b(=)-55 b FN(2)34 b Fe(C)p FT(\()p FP(S)2738
4244 y FL(1)2777 4230 y FT(\),)j(and)e(also)h FP(m)d
FN(6\030)3428 4244 y FK(K)3520 4230 y FP(m)j FT(since)378
4343 y FN(K)q FT(\()p FP(m)p FT(\))26 b(=)f FN(P)783
4357 y FL(1)853 4343 y FT(and)30 b FN(P)1093 4357 y FL(1)1158
4343 y FN(\022)25 b Fe(C)p FT(\()p FP(S)1401 4357 y FL(1)1440
4343 y FT(\).)41 b(Similarly)27 b FP(n)e FN(6\030)2079
4357 y FK(K)2162 4343 y FP(n)p FT(.)489 4506 y(1.)46
b(Supp)s(ose)23 b(that)i(there)f(are)h(t)m(w)m(o)h(literals)c
FP(A)2045 4473 y FO(i)2098 4506 y FT(and)i FN(:)p FP(A)2398
4473 y FO(j)2458 4506 y FT(in)f FP(S)30 b FT(and)23 b(let)i(us)e
(assume)h(that)h FP(i)h FN(\030)3677 4520 y FK(K)3760
4506 y FP(j)5 b FT(.)605 4619 y(Then)26 b(not)i(b)s(oth)e
FP(A)1277 4586 y FO(i)1332 4619 y FT(and)h FN(:)p FP(A)1635
4586 y FO(j)1698 4619 y FT(are)h(in)e FP(S)2006 4633
y FL(1)2072 4619 y FT(as)h FP(S)2236 4633 y FL(1)2289
4619 y FN([)14 b FP(S)2420 4633 y FL(2)2485 4619 y FT(is)26
b FN(K)q FT(-consisten)m(t.)41 b(Also,)28 b(they)f(cannot)605
4732 y(b)s(e)33 b(b)s(oth)h(in)e FP(R)1130 4699 y FO(m)1230
4732 y FT(as)i FP(m)d FN(6\030)1527 4746 y FK(K)1616
4732 y FP(m)p FT(.)51 b(Hence,)36 b(one)e(of)g(them)g(\(sa)m(y)h
FP(A)2849 4699 y FO(i)2878 4732 y FT(\))f(is)f(in)f FP(S)3207
4746 y FL(1)3280 4732 y FT(and)i(the)g(other)605 4845
y(\()p FN(:)p FP(A)769 4812 y FO(j)806 4845 y FT(\))f(is)f(in)f
FP(R)1146 4812 y FO(m)1213 4845 y FT(.)48 b(So)32 b FP(j)j
FT(=)30 b FP(m)i FT(and)g(therefore)i FP(i)29 b FN(2)g(P)2476
4859 y FL(1)2549 4845 y FT(as)k FN(K)q FT(\()p FP(m)p
FT(\))d(=)f FN(P)3076 4859 y FL(1)3116 4845 y FT(.)48
b(Also,)33 b FP(S)3478 4859 y FL(2)3539 4845 y FN([)22
b(f)p FP(A)3735 4812 y FO(n)3782 4845 y FN(g)605 4958
y FT(is)37 b FN(K)q FT(-inconsisten)m(t)h(b)m(y)f(the)h(last)g
(condition)e(in)h(the)h(de\014nition)d(of)j FN(C)43 b
FT(ab)s(o)m(v)m(e)c(as)f FN(:)p FP(A)f FN(2)g FP(R)q
FT(.)605 5071 y(Ho)m(w)m(ev)m(er,)h(this)c(implies)e(that)k(\()p
FP(S)1790 5085 y FL(2)1852 5071 y FN([)23 b(f)p FP(A)2049
5038 y FO(n)2097 5071 y FN(g)p FT(\))2177 5038 y FL(\()p
FK(f)p FO(n;i)p FK(g!)p FO(i)p FL(\))2523 5071 y FT(is)34
b FN(K)q FT(-inconsisten)m(t)h(b)m(y)g(theorem)g(7.5)605
5184 y(as)c FN(K)q FT(\()p FP(i)p FT(\))c(=)e FN(K)q
FT(\()p FP(n)p FT(\))h(=)f FN(P)1391 5198 y FL(2)1461
5184 y FT(and)k FN(f)p FP(i;)15 b(n)p FN(g)22 b([)e(P)2019
5198 y FL(2)2084 5184 y FT(=)25 b FN(fg)p FT(.)41 b(No)m(w)1165
5388 y(\()p FP(S)1256 5402 y FL(2)1315 5388 y FN([)20
b(f)p FP(A)1509 5350 y FO(n)1557 5388 y FN(g)p FT(\))1637
5350 y FL(\()p FK(f)p FO(n;i)p FK(g!)p FO(i)p FL(\))1973
5388 y FT(=)25 b(\()p FP(S)2160 5402 y FL(2)2220 5388
y FN([)20 b(f)p FP(A)2414 5350 y FO(n)2461 5388 y FN(g)p
FT(\))2541 5350 y FL(\()p FO(n)p FK(!)p FO(i)p FL(\))2764
5388 y FT(=)25 b FP(S)2916 5402 y FL(2)2975 5388 y FN([)20
b(f)p FP(A)3169 5350 y FO(i)3198 5388 y FN(g)p FP(;)605
5592 y FT(as)25 b FP(n)35 b(=)-55 b FN(2)25 b Fe(C)p
FT(\()p FP(S)1024 5606 y FL(2)1063 5592 y FT(\).)40 b(Therefore,)26
b FP(S)1652 5606 y FL(1)1700 5592 y FN([)9 b FP(S)1826
5606 y FL(2)1890 5592 y FT(is)24 b FN(K)q FT(-inconsisten)m(t)h(as)h
FP(A)2745 5559 y FO(i)2798 5592 y FN(2)f FP(S)2940 5606
y FL(1)2979 5592 y FT(.)39 b(But)25 b(this)f(con)m(tradicts)605
5705 y(the)31 b(condition)e(in)g(the)h(de\014nition)e(of)j
FN(C)k FT(that)c FP(S)2267 5719 y FL(1)2327 5705 y FN([)20
b FP(S)2464 5719 y FL(2)2533 5705 y FT(is)30 b FN(K)q
FT(-consisten)m(t,)h(and)f(so)h FP(i)25 b FN(6\030)3589
5719 y FK(K)3673 5705 y FP(j)5 b FT(.)p eop
%%Page: 146 156
146 155 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(146)489 396 y(2.)46 b(Let)32
b FN(?)840 363 y FO(i)894 396 y FN(2)26 b FP(S)35 b FT(and)c(that)g
FP(i)c FN(\030)1576 410 y FK(K)1660 396 y FP(j)36 b FT(for)31
b(some)g FP(j)5 b FT(.)43 b(No)m(w,)32 b FN(?)2515 363
y FO(i)2580 396 y FP(=)-56 b FN(2)26 b FP(S)2712 410
y FL(1)2782 396 y FT(as)31 b FP(S)2950 410 y FL(1)3010
396 y FN([)20 b FP(S)3147 410 y FL(2)3217 396 y FT(is)30
b FN(K)q FT(-consisten)m(t.)605 509 y(Also,)e(if)f FN(?)985
476 y FO(i)1038 509 y FN(2)e FP(R)1194 476 y FO(m)1288
509 y FT(then)i FP(S)1548 523 y FL(2)1602 509 y FN([)15
b(f>)1794 476 y FO(n)1840 509 y FN(g)28 b FT(is)f FN(K)q
FT(-inconsisten)m(t)h(and)f(hence)g FP(S)3078 523 y FL(2)3145
509 y FT(is)g FN(K)q FT(-inconsisten)m(t.)605 622 y(As)k(a)f(result)g
FP(S)1125 636 y FL(1)1184 622 y FN([)20 b FP(S)1321 636
y FL(2)1390 622 y FT(is)30 b FN(K)q FT(-inconsisten)m(t)g(whic)m(h)f
(is)h(a)g(con)m(tradiction.)489 810 y(3.)46 b(Let)30
b(\()p FP(A)17 b FN(^)g FP(B)5 b FT(\))25 b FN(2)g FP(S)5
b FT(,)29 b(w)m(e)h(need)e(to)i(sho)m(w)f(that)g FP(S)22
b FN([)17 b(f)p FP(A;)e(B)5 b FN(g)26 b(2)f(C)5 b FT(.)41
b(First)28 b(of)h(all,)f(if)g FP(A)17 b FN(^)g FP(B)30
b FN(2)25 b FP(S)3789 824 y FL(1)605 923 y FT(then)35
b(let)f FP(S)1013 890 y FK(0)1008 947 y FL(1)1080 923
y FT(=)e FP(S)1239 937 y FL(1)1301 923 y FN([)23 b(f)p
FP(A;)15 b(B)5 b FN(g)p FT(,)37 b(and)d(since)g FP(S)2183
937 y FL(1)2245 923 y FN([)23 b FP(S)2385 937 y FL(2)2459
923 y FT(is)34 b FN(K)q FT(-consisten)m(t)i(then)e(so)h(is)f
FP(S)3567 890 y FK(0)3562 947 y FL(1)3624 923 y FN([)23
b FP(S)3764 937 y FL(2)3803 923 y FT(.)605 1036 y(As)36
b(a)f(result)g FP(S)1145 1003 y FK(0)1140 1060 y FL(1)1203
1036 y FN([)23 b FP(R)1357 1003 y FO(m)1458 1036 y FT(whic)m(h)35
b(is)f FP(S)29 b FN([)23 b(f)p FP(A;)15 b(B)5 b FN(g)36
b FT(is)f(in)f FN(C)5 b FT(.)56 b(No)m(w,)37 b(if)e FP(A)23
b FN(^)h FP(B)38 b FN(2)33 b FP(R)3413 1003 y FO(m)3515
1036 y FT(then)i(let)605 1149 y FP(R)675 1116 y FK(0)736
1149 y FT(=)i FP(R)26 b FN([)f(f)p FP(A)1138 1116 y FK(U)1193
1149 y FP(;)15 b(B)1307 1116 y FK(U)1362 1149 y FN(g)p
FT(.)63 b(Since)37 b FP(S)1796 1163 y FL(2)1860 1149
y FN([)25 b(f:)p FP(A)2120 1116 y FK(!)p FO(n)2263 1149
y FN(_)f(:)p FP(B)2483 1116 y FK(!)p FO(n)2600 1149 y
FN(g)38 b FT(is)f FN(K)q FT(-inconsisten)m(t)g(b)m(y)h(the)g(last)605
1262 y(condition)e(in)g(the)h(de\014nition)d(of)j FN(C)5
b FT(,)39 b(then)e(b)s(oth)f FP(S)2415 1276 y FL(2)2479
1262 y FN([)24 b(f:)p FP(A)2738 1229 y FK(!)p FO(n)2856
1262 y FN(g)38 b FT(and)e FP(S)3178 1276 y FL(2)3242
1262 y FN([)24 b(f:)p FP(B)3507 1229 y FK(!)p FO(n)3624
1262 y FN(g)37 b FT(are)605 1374 y FN(K)q FT(-inconsisten)m(t.)i(As)24
b(a)g(result)f(for)h(ev)m(ery)h FP(X)33 b FN(2)25 b FP(R)2313
1341 y FK(0)2336 1374 y FT(,)g(the)g(set)f FP(S)2728
1388 y FL(2)2775 1374 y FN([)8 b(f:)p FP(X)3032 1341
y FO(n)3079 1374 y FN(g)24 b FT(is)g FN(K)q FT(-inconsisten)m(t,)605
1487 y(and)30 b(consequen)m(tly)g FP(S)c FN([)19 b(f)p
FP(A;)c(B)5 b FN(g)p FT(,)32 b(b)s(eing)d FP(S)2112 1501
y FL(1)2171 1487 y FN([)20 b FP(R)2322 1454 y FK(0)p
FO(m)2408 1487 y FT(,)30 b(is)f(in)h FN(C)5 b FT(.)489
1675 y(4.)46 b(Supp)s(ose)36 b(that)j(\()p FP(A)25 b
FN(_)g FP(B)5 b FT(\))38 b FN(2)f FP(S)5 b FT(,)40 b(w)m(e)e(need)g(to)
g(sho)m(w)g(that)g FP(S)31 b FN([)24 b(f)p FP(A)p FN(g)39
b FT(or)f FP(S)30 b FN([)25 b(f)p FP(B)5 b FN(g)38 b
FT(is)f(in)g FN(C)5 b FT(.)605 1788 y(If)41 b FP(A)28
b FN(_)f FP(B)48 b FN(2)c FP(S)1169 1802 y FL(1)1249
1788 y FT(and)d(giv)m(en)h(that)g FP(S)1950 1802 y FL(1)2016
1788 y FN([)28 b FP(S)2161 1802 y FL(2)2241 1788 y FT(is)41
b FN(K)q FT(-consisten)m(t,)k(then)d FP(S)3181 1802 y
FL(1)3247 1788 y FN([)27 b(f)p FP(A)p FN(g)i([)e FP(S)3666
1802 y FL(2)3747 1788 y FT(or)605 1901 y FP(S)661 1915
y FL(1)718 1901 y FN([)18 b(f)p FP(B)5 b FN(g)18 b([)g
FP(S)1114 1915 y FL(2)1182 1901 y FT(is)29 b FN(K)q FT(-consisten)m(t.)
41 b(If)29 b FP(S)1977 1915 y FL(1)2034 1901 y FN([)18
b(f)p FP(A)p FN(g)h([)f FP(S)2425 1915 y FL(2)2493 1901
y FT(is)28 b FN(K)q FT(-consisten)m(t)j(w)m(e)f(de\014ne)e
FP(S)3561 1868 y FK(0)3556 1925 y FL(1)3625 1901 y FT(to)i(b)s(e)605
2014 y FP(S)661 2028 y FL(1)713 2014 y FN([)13 b(f)p
FP(A)p FN(g)p FT(;)29 b(otherwise,)e(if)e FP(S)1561 2028
y FL(1)1613 2014 y FN([)13 b(f)p FP(B)5 b FN(g)13 b([)g
FP(S)1994 2028 y FL(2)2059 2014 y FT(is)26 b FN(K)q FT(-consisten)m(t)i
(w)m(e)f(let)g FP(S)2986 1981 y FK(0)2981 2038 y FL(1)3046
2014 y FT(b)s(e)f FP(S)3222 2028 y FL(1)3274 2014 y FN([)13
b(f)p FP(B)5 b FN(g)p FT(.)40 b(In)26 b(an)m(y)605 2127
y(case,)j FP(S)881 2094 y FK(0)876 2151 y FL(1)927 2127
y FN([)12 b FP(R)1070 2094 y FO(m)1162 2127 y FN(2)25
b(C)31 b FT(and)26 b(therefore)h(one)g(of)g FP(S)17 b
FN([)12 b(f)p FP(A)p FN(g)27 b FT(and)f FP(S)18 b FN([)12
b(f)p FP(B)5 b FN(g)27 b FT(is)e(in)g FN(C)5 b FT(.)40
b(Alternativ)m(ely)-8 b(,)605 2240 y(if)32 b FP(A)23
b FN(_)e FP(B)35 b FN(2)30 b FP(R)1129 2207 y FO(m)1228
2240 y FT(then)j FP(S)1494 2254 y FL(2)1555 2240 y FN([)22
b(f:)p FP(A)1812 2207 y FK(!)p FO(n)1952 2240 y FN(^)g(:)p
FP(B)2170 2207 y FK(!)p FO(n)2286 2240 y FN(g)34 b FT(is)e
FN(K)q FT(-inconsisten)m(t.)49 b(As)34 b(a)f(result,)g(either)605
2352 y FP(S)661 2366 y FL(2)717 2352 y FN([)15 b(f:)p
FP(A)967 2319 y FK(!)p FO(n)1085 2352 y FN(g)29 b FT(or)f
FP(S)1324 2366 y FL(2)1380 2352 y FN([)15 b(f:)p FP(B)1636
2319 y FK(!)p FO(n)1754 2352 y FN(g)28 b FT(is)g FN(K)q
FT(-inconsisten)m(t.)40 b(Similarly)24 b(to)29 b(the)g(previous)e
(case,)j(if)605 2465 y FP(S)661 2479 y FL(2)718 2465
y FN([)18 b(f:)p FP(A)971 2432 y FK(!)p FO(n)1089 2465
y FN(g)29 b FT(is)g FN(K)q FT(-inconsisten)m(t)g(w)m(e)h(de\014ne)e
FP(R)2316 2432 y FK(0)2368 2465 y FT(to)i(b)s(e)f FP(R)19
b FN([)e(f)p FP(A)2880 2432 y FK(U)2936 2465 y FN(g)p
FT(,)30 b(and)e(if)h FP(S)3350 2479 y FL(2)3407 2465
y FN([)17 b(f:)p FP(B)3665 2432 y FK(!)p FO(n)3782 2465
y FN(g)605 2578 y FT(is)27 b FN(K)q FT(-inconsisten)m(t)h(then)g
FP(R)1566 2545 y FK(0)1617 2578 y FT(is)f(de\014ned)f(as)j
FP(R)16 b FN([)f(f)p FP(B)2406 2545 y FK(U)2460 2578
y FN(g)p FT(.)41 b(In)27 b(an)m(y)h(case,)i(the)e(set)h
FP(S)3417 2592 y FL(2)3471 2578 y FN([)15 b(f:)p FP(X)3735
2545 y FO(n)3782 2578 y FN(g)605 2691 y FT(is)32 b FN(K)q
FT(-inconsisten)m(t)h(for)g(ev)m(ery)g FP(X)k FN(2)29
b FP(R)1957 2658 y FK(0)1980 2691 y FT(,)34 b(and)e(therefore)h
FP(S)2658 2705 y FL(1)2719 2691 y FN([)21 b FP(R)2871
2658 y FK(0)p FO(m)2986 2691 y FN(2)29 b(C)5 b FT(.)48
b(So)33 b FP(S)27 b FN([)21 b(f)p FP(A)p FN(g)31 b(2)e(C)605
2804 y FT(or)i FP(S)25 b FN([)20 b(f)p FP(B)5 b FN(g)25
b(2)g(C)5 b FT(.)489 2992 y(5.)46 b(Supp)s(ose)21 b(that)j(\()p
FN(8)p FP(x:A)p FT(\))i FN(2)f FP(S)5 b FT(,)24 b(w)m(e)g(sho)m(w)f
(that)g(for)g(ev)m(ery)h(closed)f(term)g FP(t)p FT(,)h(the)g(set)f
FP(S)11 b FN([)5 b(f)p FP(A)p FN(f)p FP(x)26 b FN(!)605
3105 y FP(t)p FN(gg)42 b FT(is)e(in)g FN(C)5 b FT(.)74
b(Brie\015y)-8 b(,)43 b(if)d FN(8)p FP(x:A)k FN(2)e FP(S)1959
3119 y FL(1)2040 3105 y FT(then)f FP(S)2314 3119 y FL(1)2380
3105 y FN([)27 b FP(S)2524 3119 y FL(2)2591 3105 y FN([)g(f)p
FP(A)p FN(f)p FP(x)44 b FN(!)g FP(t)p FN(gg)d FT(is)g
FN(K)q FT(-consisten)m(t)605 3218 y(and)i(hence)h(w)m(e)g(de\014ne)f
FP(S)1545 3185 y FK(0)1540 3242 y FL(1)1623 3218 y FT(to)i(b)s(e)e
FP(S)1941 3232 y FL(1)2009 3218 y FN([)29 b(f)p FP(A)p
FN(f)p FP(x)48 b FN(!)f FP(t)p FN(gg)e FT(in)d(order)h(that)i
FP(S)3305 3185 y FK(0)3300 3242 y FL(1)3368 3218 y FN([)29
b FP(R)3528 3185 y FO(m)3641 3218 y FN(2)48 b(C)5 b FT(.)605
3330 y(Otherwise,)29 b FN(8)p FP(x:A)c FN(2)g FP(R)1437
3297 y FO(m)1532 3330 y FT(and)j(so)i(since)e FP(S)2095
3344 y FL(2)2152 3330 y FN([)17 b(f9)p FP(x:)p FN(:)p
FP(A)p FN(g)30 b FT(is)e FN(K)q FT(-inconsisten)m(t,)h(w)m(e)h(ha)m(v)m
(e)g(that)605 3443 y FP(S)661 3457 y FL(2)721 3443 y
FN([)20 b(f:)p FP(A)p FN(f)p FP(x)26 b FN(!)f FP(t)p
FN(gg)31 b FT(is)e FN(K)q FT(-inconsisten)m(t)i(for)f(ev)m(ery)h
(closed)g(term)f FP(t)p FT(.)41 b(Therefore)30 b(w)m(e)h(c)m(ho)s(ose)
605 3556 y FP(R)675 3523 y FK(0)729 3556 y FT(to)g(b)s(e)e
FP(R)21 b FN([)f(f)p FP(A)1247 3523 y FK(U)1303 3556
y FN(f)p FP(x)25 b FN(!)g FP(t)p FN(gg)31 b FT(so)g(that)g
FP(S)2060 3570 y FL(1)2119 3556 y FN([)20 b FP(R)2270
3523 y FK(0)p FO(m)2381 3556 y FN(2)25 b(C)5 b FT(.)489
3744 y(6.)46 b(Let)33 b(\()p FN(9)p FP(x:A)p FT(\))c
FN(2)g FP(S)5 b FT(,)33 b(w)m(e)g(sho)m(w)f(that)h FP(S)27
b FN([)21 b(f)p FP(A)p FN(f)p FP(x)30 b FN(!)e FP(p)p
FN(gg)i(2)e(C)37 b FT(for)c(some)f(parameter)h FP(p)p
FT(.)47 b(No)m(w,)605 3857 y(if)36 b FN(9)p FP(x:A)f
FN(2)g FP(S)1078 3871 y FL(1)1154 3857 y FT(then)h FP(S)1423
3871 y FL(1)1487 3857 y FN([)24 b FP(S)1628 3871 y FL(2)1691
3857 y FN([)g(f)p FP(A)p FN(f)p FP(x)37 b FN(!)e FP(p)p
FN(gg)i FT(is)f FN(K)q FT(-consisten)m(t)i(for)e(ev)m(ery)h(parameter)g
FP(p)605 3970 y FT(not)g(found)e(in)h FP(S)1203 3984
y FL(1)1266 3970 y FN([)24 b FP(S)1407 3984 y FL(2)1446
3970 y FT(,)39 b(so)d(w)m(e)h(c)m(ho)s(ose)h FP(S)2123
3937 y FK(0)2118 3994 y FL(1)2194 3970 y FT(to)f(b)s(e)f
FP(S)2497 3984 y FL(1)2560 3970 y FN([)24 b(f)p FP(A)p
FN(f)p FP(x)37 b FN(!)e FP(p)p FN(gg)j FT(for)e(some)h(suc)m(h)f
FP(p)605 4083 y FT(so)g(that)g FP(S)29 b FN([)23 b(f)p
FP(A)p FN(f)p FP(x)35 b FN(!)f FP(p)p FN(gg)g FT(=)g
FP(S)1799 4050 y FK(0)1794 4107 y FL(1)1857 4083 y FN([)23
b FP(R)2011 4050 y FO(m)2077 4083 y FT(.)56 b(Also,)37
b(if)e FN(9)p FP(x:A)e FN(2)h FP(R)q FT(,)j(then)e FP(S)3208
4097 y FL(2)3271 4083 y FN([)23 b(f:)p FT(\()p FN(9)p
FP(x:A)3692 4050 y FK(U)3747 4083 y FT(\))p FN(g)605
4196 y FT(is)36 b FN(K)q FT(-inconsisten)m(t,)i(and)d(therefore)i
FP(S)1961 4210 y FL(2)2025 4196 y FN([)23 b(f:)p FP(A)2283
4163 y FK(U)2339 4196 y FN(f)p FP(x)35 b FN(!)g FP(p)p
FN(gg)i FT(is)f FN(K)q FT(-inconsisten)m(t)g(for)g(ev)m(ery)605
4308 y(parameter)e FP(p)f FT(new)g(to)h FP(S)1482 4322
y FL(2)1521 4308 y FT(.)50 b(Hence,)36 b(w)m(e)d(de\014ne)g
FP(R)2368 4275 y FK(0)2424 4308 y FT(to)i(b)s(e)d FP(R)23
b FN([)f(f:)p FP(A)3014 4275 y FK(U)3069 4308 y FN(f)p
FP(x)31 b FN(!)f FP(p)p FN(gg)k FT(for)f(some)605 4421
y(suc)m(h)d FP(p)p FT(.)41 b(Th)m(us)29 b FP(S)1208 4435
y FL(1)1267 4421 y FN([)20 b FP(R)1418 4388 y FK(0)p
FO(m)1504 4421 y FT(,)30 b(whic)m(h)f(is)h FP(S)25 b
FN([)20 b(f)p FP(A)p FN(f)p FP(x)26 b FN(!)f FP(p)p FN(gg)p
FT(,)31 b(is)f(in)f FN(C)5 b FT(.)519 4609 y(W)-8 b(e)41
b(th)m(us)e(conclude)g(that)h FN(C)k FT(is)38 b(a)i FN(K)q
FT(-consistency)g(prop)s(ert)m(y)-8 b(.)68 b(No)m(w,)43
b(let)c FP(S)3215 4623 y FL(1)3294 4609 y FT(and)f FP(S)3535
4623 y FL(2)3614 4609 y FT(b)s(e)h(as)378 4722 y(required)i(b)m(y)i
(the)g(statemen)m(t)i(of)f(this)e(theorem,)47 b(and)42
b(that)i(for)f(some)g(sen)m(tence)h FP(X)7 b FT(,)47
b(the)c(sets)378 4835 y FP(S)434 4849 y FL(1)480 4835
y FN([)7 b(f)p FP(X)675 4802 y FO(m)741 4835 y FN(g)24
b FT(and)f FP(S)1036 4849 y FL(2)1082 4835 y FN([)7 b(f:)p
FP(X)1338 4802 y FO(n)1385 4835 y FN(g)24 b FT(are)g
FN(K)q FT(-inconsisten)m(t.)38 b(Then)23 b FP(S)2518
4849 y FL(1)2564 4835 y FN([)7 b FP(S)2688 4849 y FL(2)2750
4835 y FT(is)22 b FN(K)q FT(-inconsisten)m(t,)j(otherwise)378
4948 y(the)31 b(set)f FP(S)732 4962 y FL(1)792 4948 y
FN([)20 b(f)p FP(X)1000 4915 y FO(m)1067 4948 y FN(g)31
b FT(w)m(ould)e(b)s(e)h(in)f FN(C)35 b FT(and)30 b(therefore)h
FN(K)q FT(-consisten)m(t.)519 5061 y(The)g(`if)7 b(')31
b(direction)g(follo)m(ws)g(from)g(Theorems)g(7.5)i(and)e(7.7.)45
b(Giv)m(en)32 b(that)g(the)g(set)g FP(S)3498 5075 y FL(1)3558
5061 y FN([)21 b FP(S)3696 5075 y FL(2)3767 5061 y FT(is)378
5174 y FN(K)q FT(-inconsisten)m(t,)33 b(then)f(the)h(partition)e(\()p
FP(S)1845 5188 y FL(1)1885 5174 y FP(;)15 b(S)1981 5188
y FL(2)2020 5174 y FT(\))33 b(has)f(a)h FN(K)q FT(-in)m(terp)s(olan)m
(t)g(\()p FP(Y)2989 5188 y FL(1)3028 5174 y FP(;)15 b(Y)3121
5188 y FL(2)3161 5174 y FT(\).)47 b(W)-8 b(e)34 b(can)f(de\014ne)378
5286 y FP(X)41 b FT(to)35 b(b)s(e)e FP(Y)809 5253 y FK(U)789
5311 y FL(1)863 5286 y FT(,)i(and)e(so)h FN(:)p FP(X)k
FT(=)31 b FP(Y)1567 5253 y FK(U)1547 5311 y FL(2)1622
5286 y FT(.)50 b(No)m(w,)36 b(since)d FP(S)2216 5300
y FL(1)2278 5286 y FN([)22 b(f)p FP(Y)2459 5300 y FL(1)2498
5286 y FN(g)34 b FT(is)f FN(K)q FT(-inconsisten)m(t,)i(w)m(e)f(can)g
(apply)378 5399 y(Theorem)23 b(7.5)h(to)g(recolour)f(all)f(the)h
(colours)g(in)e FP(Y)2061 5413 y FL(1)2124 5399 y FT(to)i
FP(m)p FT(,)i(and)e(th)m(us)f FP(S)2775 5413 y FL(1)2820
5399 y FN([)6 b(f)p FP(X)3014 5366 y FO(m)3081 5399 y
FN(g)23 b FT(is)g FN(K)q FT(-inconsisten)m(t.)378 5512
y(Similarly)-8 b(,)27 b(the)k(set)g FP(S)1137 5526 y
FL(2)1196 5512 y FN([)20 b(f:)p FP(X)1465 5479 y FO(n)1512
5512 y FN(g)31 b FT(is)f FN(K)q FT(-inconsisten)m(t)g(as)h(w)m(ell.)
1186 b Ff(\004)p eop
%%Page: 147 157
147 156 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(147)378 396 y FH(7.6)135
b(An)44 b(Undecidabilit)l(y)j(Result)378 599 y FT(The)24
b(consistency)h(of)g(a)g(coloured)f(\014rst-order)g(problem)f(is)g(in)h
(general)g(undecidable)f(since)h(one)h(can)378 712 y(reduce)e(the)h(v)
-5 b(alidit)m(y)22 b(problem)h(of)g(a)i(\014rst-order)d(sen)m(tence)j
FP(X)32 b FT(to)24 b(the)g(consistency)g(of)f(the)h(coloured)378
825 y(problem)e(\()p FN(f)p FP(X)887 792 y FO(i)916 825
y FN(g)p FP(;)15 b(i)26 b FN($)g FP(i)p FT(\).)39 b(Apart)23
b(from)g(suc)m(h)g(a)g(trivial)f(reduction,)h(the)h(follo)m(wing)d
(theorem)j(sho)m(ws)378 938 y(that)32 b(the)g(v)-5 b(alidit)m(y)29
b(of)j(a)g(\014rst-order)e(sen)m(tence)j FP(X)39 b FT(can)31
b(b)s(e)g(reduced)g(to)h(the)g(consistency)f(of)g(some)378
1051 y(coloured)f(sen)m(tence)h FP(Y)1176 1018 y FO(i)1230
1051 y FN(\))25 b FP(Z)1415 1018 y FO(j)1481 1051 y FT(according)31
b(to)g(the)f(connectabilit)m(y)g(relation)g FP(i)25 b
FN($)h FP(j)36 b FT(where)29 b FP(i)d FN(6)p FT(=)f FP(j)5
b FT(.)378 1264 y FQ(Theorem)34 b(7.9)46 b FI(Given)38
b(a)h(\014rst-or)-5 b(der)40 b(sentenc)-5 b(e)39 b FP(X)7
b FI(,)40 b(then)f(ther)-5 b(e)39 b(ar)-5 b(e)40 b(\014rst-or)-5
b(der)40 b(sentenc)-5 b(es)39 b FP(Y)378 1376 y FI(and)34
b FP(Z)39 b FI(such)32 b(that)i FP(X)40 b FI(is)33 b(valid)g(if)f(and)i
(only)f(if)g FP(Y)2103 1343 y FO(i)2156 1376 y FN(\))25
b FP(Z)2341 1343 y FO(j)2410 1376 y FI(is)33 b FP(i)25
b FN($)g FP(j)5 b FI(-c)-5 b(onsistent.)378 1589 y FQ(Pro)s(of)p
FT(:)33 b(Let)f FP(X)39 b FT(b)s(e)31 b(a)h(\014rst-order)f(sen)m
(tence.)45 b(W)-8 b(e)33 b(can)f(transform)f(the)h(negation)g
FN(:)p FP(X)39 b FT(in)m(to)31 b(a)h(list)378 1702 y(of)e(clauses)h
FP(C)851 1716 y FL(1)890 1702 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(C)1172 1716 y FO(n)1249 1702 y FT(suc)m(h)30
b(that)h FP(X)38 b FT(is)30 b(v)-5 b(alid)28 b(if)i(and)f(only)h(if)
1790 1906 y FN(8)p FP(C)1906 1920 y FL(1)1965 1906 y
FN(^)20 b(\001)15 b(\001)g(\001)21 b(^)f(8)p FP(C)2369
1920 y FO(n)378 2110 y FT(is)35 b(unsatis\014able,)h(where)f
FN(8)p FP(C)1413 2124 y FO(x)1492 2110 y FT(represen)m(ts)h(the)h
(disjunction)c FP(C)2627 2124 y FO(x)2707 2110 y FT(univ)m(ersally)g
(quan)m(ti\014ed)i(b)m(y)h(all)378 2223 y(its)e(free)h(v)-5
b(ariables.)52 b(The)34 b(ab)s(o)m(v)m(e)i(list)e(of)h(clauses)f(can)h
(b)s(e)f(transformed)g(in)m(to)h(an)f(equiv)-5 b(alen)m(t)34
b(list)378 2336 y(in)27 b(whic)m(h)f(eac)m(h)k(clause)d(con)m(tains)i
(either)e(p)s(ositiv)m(e)g(literals)f(only)-8 b(,)29
b(or)f(negativ)m(e)h(literals)d(only)-8 b(.)40 b(This)378
2449 y(can)31 b(b)s(e)e(done)i(b)m(y)f(substituting)e(ev)m(ery)j
(clause)f(of)h(the)f(form)1450 2653 y FN(:)p FP(A)1579
2667 y FL(1)1639 2653 y FN(_)20 b(\001)15 b(\001)g(\001)21
b(_)f(:)p FP(A)2056 2667 y FO(n)2123 2653 y FN(_)f FP(B)2272
2667 y FL(1)2332 2653 y FN(_)h(\001)15 b(\001)g(\001)21
b(_)f FP(B)2689 2667 y FO(m)378 2858 y FT(where)30 b
FP(A)709 2872 y FO(x)783 2858 y FT(and)g FP(B)1029 2872
y FO(y)1101 2858 y FT(are)g(atoms,)i(with)d(the)h(pair)g(of)g(clauses)
1596 3083 y FN(:)p FP(A)1725 3097 y FL(1)1784 3083 y
FN(_)20 b(\001)15 b(\001)g(\001)21 b(_)f(:)p FP(A)2201
3097 y FO(n)2268 3083 y FN(_)g(:)p FP(D)s FT(\()o FP(~)-44
b(x)p FT(\))1676 3239 y FP(B)1745 3253 y FL(1)1805 3239
y FN(_)20 b(\001)15 b(\001)g(\001)21 b(_)f FP(B)2162
3253 y FO(m)2248 3239 y FN(_)g FP(D)s FT(\()o FP(~)-44
b(x)p FT(\))378 3443 y(where)28 b FP(D)i FT(is)e(a)g(new)g(predicate)g
(constan)m(t)h(sym)m(b)s(ol)e(whic)m(h)g(do)s(es)h(not)g(o)s(ccur)g(in)
f(the)h(list)f(of)h(clauses,)378 3556 y(and)h FP(~)-44
b(x)30 b FT(is)f(the)i(list)e(of)i(v)-5 b(ariables)29
b(free)h(in)f(the)i(original)d(clause)1438 3760 y FN(:)p
FP(A)1567 3774 y FL(1)1626 3760 y FN(_)20 b(\001)15 b(\001)g(\001)21
b(_)f(:)p FP(A)2043 3774 y FO(n)2110 3760 y FN(_)g FP(B)2260
3774 y FL(1)2319 3760 y FN(_)g(\001)15 b(\001)g(\001)22
b(_)d FP(B)2676 3774 y FO(m)2743 3760 y FP(:)378 3965
y FT(This)25 b(can)i(b)s(e)f(rep)s(eated)g(un)m(til)f(the)i(original)e
(list)g(of)h(clauses)h FP(C)2534 3979 y FL(1)2573 3965
y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(C)2855 3979 y FO(n)2929
3965 y FT(is)25 b(transformed)h(in)m(to)h(the)378 4077
y(equiv)-5 b(alen)m(t)33 b(list)f FP(N)1041 4091 y FL(1)1081
4077 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(N)1370 4091 y
FO(r)1409 4077 y FP(;)g(P)1507 4091 y FL(1)1547 4077
y FP(;)g(:)g(:)g(:)32 b(;)15 b(P)1822 4091 y FO(s)1893
4077 y FT(where)33 b(all)f(the)i(literals)e(in)g FP(N)2932
4091 y FO(x)3009 4077 y FT(are)i(negativ)m(e,)i(and)d(all)378
4190 y(the)26 b(literals)f(in)g FP(P)985 4204 y FO(y)1053
4190 y FT(are)i(p)s(ositiv)m(e.)38 b(No)m(w,)28 b(it)e(can)h(b)s(e)e
(seen)i(that)f(the)h(only)e(pairs)g(of)i(complemen)m(tary)378
4303 y(literals)37 b(obtained)h(from)g(this)g(list)f(of)i(clauses)f
(con)m(tain)h(one)g(\(instan)m(tiation)f(of)h(a\))g(literal)e(from)378
4416 y(some)32 b(negativ)m(e)g(clause)f FP(N)1308 4430
y FO(x)1352 4416 y FT(,)h(and)f(one)g(\(instan)m(tiation)g(of)g(a\))h
(literal)e(from)h(a)h(p)s(ositiv)m(e)e(clause)h FP(P)3761
4430 y FO(y)3803 4416 y FT(.)378 4529 y(W)-8 b(e)36 b(can)f(explicitly)
d(imp)s(ose)h(the)i(restriction)e(that)i(the)g(only)f(complemen)m(tary)
h(pairs)e(of)i(literals)378 4642 y(in)28 b(whic)m(h)f(one)i(literal)f
(is)g(from)g FP(N)1544 4656 y FL(1)1584 4642 y FP(;)15
b(:)g(:)g(:)31 b(;)15 b(N)1873 4656 y FO(r)1912 4642
y FT(,)29 b(and)f(the)i(other)f(is)f(from)g FP(P)2894
4656 y FL(1)2934 4642 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(P)3209 4656 y FO(s)3275 4642 y FT(are)29 b(allo)m(w)m(ed)g(to)378
4755 y(b)s(e)f(used)f(in)g(sho)m(wing)g(the)i(inconsistency)e(of)h(the)
g(set)h(of)g(clauses.)39 b(Or)28 b(in)f(other)h(w)m(ords,)h(w)m(e)f
(colour)378 4868 y(the)k(negativ)m(e)i(clauses)d(with)g(some)i(colour)e
FP(i)p FT(,)i(and)f(the)g(p)s(ositiv)m(e)f(clauses)h(with)f
FP(j)5 b FT(,)33 b(and)e(c)m(hec)m(k)j(for)378 4981 y
FP(i)26 b FN($)f FP(j)5 b FT(-inconsistency)-8 b(.)40
b(That)31 b(is,)e(the)i(sen)m(tence)h FP(X)37 b FT(is)30
b(v)-5 b(alid)28 b(if)i(and)g(only)f(if)1629 5107 y Fx(^)1572
5303 y FL(0)p FO(<p)p FK(\024)p FO(r)1802 5194 y FN(8)p
FP(N)1936 5156 y FO(i)1926 5216 y(p)2076 5194 y FN(^)2304
5107 y Fx(^)2248 5303 y FL(0)p FO(<p)p FK(\024)p FO(s)2476
5194 y FN(8)p FP(P)2598 5156 y FO(j)2585 5216 y(p)p eop
%%Page: 148 158
148 157 bop 378 5 a FF(CHAPTER)30 b(7.)71 b(A)30 b(COLOURED)g
(FIRST-ORDER)g(LOGIC)1055 b FT(148)378 396 y(is)29 b
FP(i)d FN($)f FP(j)5 b FT(-inconsisten)m(t;)31 b(or)f(whether)1355
517 y Fx(0)1355 681 y(@)1491 614 y(^)1434 810 y FL(0)p
FO(<p)p FK(\024)p FO(r)1664 700 y FN(8)p FP(N)1798 663
y FO(i)1788 723 y(p)1827 517 y Fx(1)1827 681 y(A)2023
700 y FN(\))2230 517 y Fx(0)2230 681 y(@)2309 700 y FN(:)2441
614 y Fx(^)2385 810 y FL(0)p FO(<p)p FK(\024)p FO(s)2614
700 y FN(8)p FP(P)2736 663 y FO(j)2723 723 y(p)2772 517
y Fx(1)2772 681 y(A)378 1002 y FT(is)f FP(i)d FN($)f
FP(j)5 b FT(-consisten)m(t.)2626 b Ff(\004)519 1228 y
FT(As)30 b(a)h(corollary)f(w)m(e)h(get)g(the)g(undecidabilit)m(y)c(of)j
FP(i)c FN($)f FP(j)5 b FT(-consistency)-8 b(.)378 1440
y FQ(Corollary)35 b(7.1)g(\()p FP(i)26 b FN($)f FP(j)5
b FQ(-Consistency)36 b(is)f(Undecidable\))46 b FI(The)h
FP(i)j FN($)g FP(j)5 b FI(-c)-5 b(onsistency)47 b(pr)-5
b(ob-)378 1553 y(lem)33 b(of)g(c)-5 b(olour)g(e)g(d)35
b(\014rst-or)-5 b(der)34 b(sentenc)-5 b(es)33 b(is)g(unde)-5
b(cidable.)378 1766 y FQ(Pro)s(of)p FT(:)33 b(follo)m(ws)f(from)f(the)i
(undecidabilit)m(y)28 b(of)33 b(the)f(v)-5 b(alidit)m(y)31
b(problem)f(of)j(pure)e(\014rst-order)g(logic)378 1879
y(and)f(theorem)h(7.9.)2708 b Ff(\004)378 2165 y FH(7.7)135
b(Summary)378 2368 y FT(This)28 b(c)m(hapter)i(giv)m(es)g(the)g
(de\014nition)e(of)h(a)i(\014rst-order)d(logic)i(whose)f(literals)f
(are)j(annotated)f(with)378 2481 y(colours.)44 b(The)32
b(role)f(of)h(the)g(annotations)g(is)f(to)i(restrict)e(the)h(w)m(a)m(y)
h(literals)d(can)i(b)s(e)f(used)g(to)i(sho)m(w)378 2594
y(the)j(inconsistency)f(of)h(a)g(set)g(of)h(sen)m(tences)g(during)c(a)j
(refutational)g(theorem)g(pro)m(ving)f(pro)s(cess.)378
2707 y(The)29 b(results)f(and)h(de\014nitions)f(giv)m(en)h(in)g(this)f
(c)m(hapter)i(are)g(used)f(in)f(c)m(hapter)j(8)f(to)g(illustrate)e(ho)m
(w)378 2820 y(the)33 b(inferences)f(giv)m(en)g(in)g(structured)f
(straigh)m(tforw)m(ard)i(justi\014cations)e(can)i(b)s(e)f(used)g(to)h
(restrict)378 2933 y(the)38 b(searc)m(h)h(space)g(considered)e(during)f
(pro)s(of)h(c)m(hec)m(king.)65 b(The)38 b(results)f(giv)m(en)h(in)f
(this)g(c)m(hapter)378 3046 y(include:)378 3233 y FQ(Section)e(7.2)46
b FT(con)m(tains)37 b(the)h(basic)e(de\014nitions)e(of)j(the)g
(coloured)g(\014rst-order)f(logic,)i(in)e(partic-)605
3346 y(ular)e(a)h(coloured)g(problem)e(is)i(de\014ned)e(in)h(terms)h
(of)g(a)h(set)f(of)g(coloured)g(sen)m(tences)h(and)f(a)605
3459 y(connectabilit)m(y)28 b(relation)g(b)s(et)m(w)m(een)h(colours.)40
b(A)28 b(notion)g(of)h(coloured)f(consistency)g(is)g(giv)m(en)605
3572 y(in)38 b(whic)m(h)g(complemen)m(tary)h(literals)f(are)h
(considered)f(inconsisten)m(t)g(if)g(and)h(only)f(if)g(their)605
3685 y(colours)30 b(relate)h(with)e(eac)m(h)i(other)g(according)f(to)i
(the)e(connectabilit)m(y)g(relation.)378 3872 y FQ(Section)35
b(7.3)46 b FT(sho)m(ws)29 b(ho)m(w)g(a)g(coloured)f(problem)g(can)h(b)s
(e)f(translated)h(in)m(to)f(an)h(equiv)-5 b(alen)m(t)28
b(set)i(of)605 3985 y(uncoloured)f(\014rst-order)h(sen)m(tences.)378
4173 y FQ(Section)35 b(7.4)46 b FT(sho)m(ws)e(ho)m(w)g(one)g(can)g(c)m
(hange)h(the)f(colours)g(of)g(a)g(coloured)g(problem)e(without)605
4286 y(a\013ecting)31 b(its)f(consistency)g(or)h(inconsistency)-8
b(.)378 4474 y FQ(Section)35 b(7.5)46 b FT(giv)m(es)g(an)f(in)m(terp)s
(olation)e(theorem)j(for)f(the)g(coloured)g(\014rst-order)f(logic)h
(whic)m(h)605 4586 y(generalises)34 b(the)h(in)m(terp)s(olation)d
(theorem)j(due)f(to)h(Craig.)52 b(The)34 b(result)f(giv)m(en)h(here)h
(states)605 4699 y(that)41 b(giv)m(en)f(a)h(v)-5 b(alid)38
b(implication)g FP(X)49 b FN(\))41 b FP(Y)20 b FT(,)43
b(then)d(it)g(has)g(an)g(in)m(terp)s(olan)m(t)f FP(I)47
b FT(suc)m(h)40 b(that)605 4812 y FP(X)45 b FN(\))37
b FP(I)44 b FT(and)37 b FP(I)44 b FN(\))37 b FP(X)45
b FT(can)38 b(b)s(e)e(deriv)m(ed)h(using)f(the)i(same)f(restrictions)g
(imp)s(osed)e(on)j(the)605 4925 y(deriv)-5 b(ation)29
b(of)i FP(X)h FN(\))26 b FP(Y)20 b FT(.)378 5113 y FQ(Section)35
b(7.6)46 b FT(sho)m(ws)c(that)h(the)g(problem)e(of)h(deciding)f(a)h
(coloured)g(problem)f(with)g(only)h(t)m(w)m(o)605 5226
y(colours)31 b(is)f(in)g(general)h(undecidable.)41 b(This)29
b(result)i(is)f(relev)-5 b(an)m(t)31 b(b)s(ecause)g(it)g(is)f(used)h
(in)f(the)605 5339 y(next)g(c)m(hapter)h(to)f(sho)m(w)g(that)h(the)f(v)
-5 b(alidit)m(y)28 b(of)i(the)g(structured)f(straigh)m(tforw)m(ard)h
(justi\014ca-)605 5452 y(tions)g(giv)m(en)g(in)f(c)m(hapter)i(6)g(is)f
(undecidable.)p eop
%%Page: 149 159
149 158 bop 378 1019 a FJ(Chapter)65 b(8)378 1434 y FR(Pro)6
b(of)78 b(Chec)-6 b(king)77 b(Structured)378 1683 y(Straigh)-6
b(tforw)g(ard)77 b(Justi\014cations)378 2165 y FH(8.1)135
b(In)l(tro)t(duction)378 2368 y FT(In)30 b(c)m(hapter)h(6)h(w)m(e)f
(de\014ned)e(the)i(notion)g(of)g(structured)f(justi\014cations)f(whic)m
(h)g(include)g(\(not)i(o)m(v)m(er-)378 2481 y(detailed\))k(information)
f(on)i(whic)m(h)e(inferences)h(are)h(used)f(in)g(the)g(justi\014cation)
g(pro)s(cess.)56 b(These)378 2594 y(justi\014cations)36
b(are)i(in)m(tended)e(to)j(impro)m(v)m(e)e(the)h(readabilit)m(y)e(and)h
(pro)s(of)g(c)m(hec)m(king)h(e\016ciency)g(of)378 2706
y(declarativ)m(e)j(language)f(pro)s(of)g(scripts.)69
b(This)38 b(information)h(is)g(built)f(up)h(b)m(y)i(using)d(the)j(op)s
(era-)378 2819 y(tors)33 b Fw(on)o FT(,)h Fw(then)d FT(and)i
Fw(and)e FT(whic)m(h)h(construct)h(structured)f(expressions)g(from)g
(the)i(premises)d(in)h(the)378 2932 y(justi\014cation.)39
b(F)-8 b(or)31 b(example,)g(one)f(can)h(implemen)m(t)e(the)i(follo)m
(wing)d(justi\014ed)h(conclusion:)473 3105 y FM(")p FT(\()p
FP(b)48 b(>)g(c)p FT(\))g FN(\))g FT(\()p FP(a)g(>)f(c)p
FT(\))p FM(")664 3218 y(by)g(")p FN(8)16 b FP(x;)f(y)s(;)g(z)t
FM(.)p FT(\()p FP(x)48 b(>)g(y)s FT(\))g FN(^)f FT(\()p
FP(y)k(>)c(z)t FT(\))i FN(\))e FP(x)h(>)f(z)t FM(")h(on)f(")p
FP(a)h(>)f(b)p FM(";)378 3391 y FT(The)30 b(ab)s(o)m(v)m(e)h(statemen)m
(t)i(is)c(v)-5 b(alid)29 b(since)g(the)i(sen)m(tence)1396
3595 y FN(8)p FP(x;)15 b(y)s(;)g(z)t(:)p FT(\()p FP(x)27
b(>)e(y)s FT(\))20 b FN(^)g FT(\()p FP(y)28 b(>)d(z)t
FT(\))h FN(\))f FT(\()p FP(x)h(>)f(z)t FT(\))378 3799
y(can)31 b(b)s(e)e(used)h(to)h(deriv)m(e)f(the)h(form)m(ula)1484
4003 y(\()p FP(a)25 b(>)g(b)p FT(\))h FN(\))f(8)p FP(z)t(:)p
FT(\()p FP(b)h(>)f(z)t FT(\))h FN(\))f FT(\()p FP(c)h(>)f(z)t
FT(\))378 4207 y(using)30 b(a)j(n)m(um)m(b)s(er)d(of)i(implicit)d
(inferences)i(\(or)h(trivial)e(manipulations\),)g(so)i(that)h(one)f
(can)g(apply)378 4320 y(the)f(inference)e(rule)g(of)i(Mo)s(dus)e(P)m
(onens)i(on)f(this)f(form)m(ula)h(and)f(the)i(sen)m(tence)h
FP(a)25 b(>)g(b)30 b FT(to)h(deriv)m(e)1677 4525 y FN(8)p
FP(z)t(:)p FT(\()p FP(b)26 b(>)e(z)t FT(\))i FN(\))g
FT(\()p FP(a)f(>)g(z)t FT(\))p FP(:)378 4729 y FT(This)k(sen)m(tence)i
(can)g(then)f(b)s(e)g(used)f(to)i(implicitly)c(deriv)m(e)j(the)g
(conclusion)1745 4933 y(\()p FP(b)25 b(>)g(c)p FT(\))h
FN(\))g FT(\()p FP(a)f(>)g(c)p FT(\))p FP(:)378 5137
y FT(This)k(deriv)-5 b(ation)29 b(can)h(b)s(e)g(represen)m(ted)g(b)m(y)
956 5323 y FN(8)p FP(x;)15 b(y)s(;)g(z)t(:)p FT(\()p
FP(x)27 b(>)e(y)s FT(\))20 b FN(^)g FT(\()p FP(y)28 b(>)d(z)t
FT(\))h FN(\))f FT(\()p FP(x)h(>)f(z)t FT(\))p 956 5365
1413 4 v 1039 5450 a(\()p FP(a)h(>)f(b)p FT(\))h FN(\))f(8)p
FP(z)t(:)p FT(\()p FP(b)g(>)g(z)t FT(\))h FN(\))f FT(\()p
FP(a)h(>)f(z)t FT(\))2411 5388 y(\()p Ff(\032)2547 5355
y FK(\003)2587 5388 y FT(\))2713 5323 y(\()p FP(a)h(>)f(b)p
FT(\))p 2713 5365 279 4 v 2713 5450 a(\()p FP(a)h(>)f(b)p
FT(\))3034 5388 y(\()p Ff(\032)3170 5355 y FK(\003)3209
5388 y FT(\))p 1039 5493 1953 4 v 1476 5578 a FN(8)p
FP(z)t(:)p FT(\()p FP(b)h(>)e(z)t FT(\))i FN(\))g FT(\()p
FP(a)f(>)g(z)t FT(\))p 1476 5620 827 4 v 1544 5705 a(\()p
FP(b)g(>)g(c)p FT(\))h FN(\))g FT(\()p FP(a)f(>)g(c)p
FT(\))2344 5643 y(\()p Ff(\032)2480 5610 y FK(\003)2520
5643 y FT(\))3034 5516 y(\(MP\))2035 5954 y(149)p eop
%%Page: 150 160
150 159 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(150)378
396 y(where)27 b(the)i(rule)d(\()p Ff(\032)1108 363 y
FK(\003)1148 396 y FT(\))i(represen)m(ts)g(the)g(implicit)d(deriv)-5
b(ations)27 b(de\014ned)g(in)f(section)i(6.4.1.)42 b(In)27
b(gen-)378 509 y(eral,)38 b(the)e(deriv)-5 b(ation)35
b(of)h(a)h(conclusion)e(from)h(a)g(structured)g(justi\014cation)f(can)h
(b)s(e)g(represen)m(ted)378 622 y(b)m(y)f(a)g(n)m(um)m(b)s(er)f(of)h
(implicit)c(deriv)-5 b(ations)34 b(and)g(a)h(n)m(um)m(b)s(er)f(of)h
(explicit)e(deriv)-5 b(ations)34 b(whic)m(h)f(corre-)378
735 y(sp)s(ond)c(to)i(the)f(op)s(erators)h(in)e(the)i(justi\014cation.)
39 b(This)29 b(is)g(describ)s(ed)g(in)g(section)i(6.4.2)h(where)e(the)
378 848 y(seman)m(tics)h(of)f(structured)g(justi\014cations)e(is)i(giv)
m(en.)519 961 y(The)42 b(seman)m(tics)g(of)h(structured)e
(justi\014cations)g(is)g(non-deterministic,)i(and)f(in)f(general,)k(a)
378 1074 y(structured)c(justi\014cation)g(can)h(b)s(e)f(used)g(to)i
(deriv)m(e)f(sev)m(eral)g(conclusions.)74 b(F)-8 b(or)43
b(example,)i(the)378 1187 y(structured)29 b(justi\014cation)g(giv)m(en)
i(ab)s(o)m(v)m(e)g(can)g(also)f(b)s(e)g(used)g(to)h(deriv)m(e)f(the)g
(conclusion)473 1350 y FM(")p FT(\()p FP(c)49 b(>)e(a)p
FT(\))h FN(\))g FT(\()p FP(c)g(>)g(b)p FT(\))p FM(")378
1513 y FT(since)956 1589 y FN(8)p FP(x;)15 b(y)s(;)g(z)t(:)p
FT(\()p FP(x)27 b(>)e(y)s FT(\))20 b FN(^)g FT(\()p FP(y)28
b(>)d(z)t FT(\))h FN(\))f FT(\()p FP(x)h(>)f(z)t FT(\))p
956 1632 1413 4 v 1031 1717 a FN(8)p FP(x:)p FT(\()p
FP(x)g(>)g(a)p FT(\))h FN(\))f FT(\()p FP(a)h(>)e(b)p
FT(\))i FN(\))f FT(\()p FP(x)h(>)f(b)p FT(\))2411 1655
y(\()p Ff(\032)2547 1622 y FK(\003)2587 1655 y FT(\))p
1031 1759 1265 4 v 1031 1844 a(\()p FP(a)g(>)g(b)p FT(\))h
FN(\))f(8)p FP(x:)p FT(\()p FP(x)g(>)g(a)p FT(\))h FN(\))f
FT(\()p FP(x)h(>)f(b)p FT(\))2337 1782 y(\()p Ff(\032)2473
1749 y FK(\003)2512 1782 y FT(\))2713 1717 y(\()p FP(a)h(>)f(b)p
FT(\))p 2713 1759 279 4 v 2713 1844 a(\()p FP(a)h(>)f(b)p
FT(\))3034 1782 y(\()p Ff(\032)3170 1749 y FK(\003)3209
1782 y FT(\))p 1031 1887 1962 4 v 1463 1972 a FN(8)p
FP(x:)p FT(\()p FP(x)g(>)g(a)p FT(\))h FN(\))f FT(\()p
FP(x)h(>)f(b)p FT(\))p 1463 2014 844 4 v 1527 2099 a(\()p
FP(c)h(>)f(a)p FT(\))g FN(\))h FT(\()p FP(c)g(>)f(b)p
FT(\))p FP(:)2349 2037 y FT(\()p Ff(\032)2485 2004 y
FK(\003)2524 2037 y FT(\))3034 1909 y(\(MP\))378 2266
y(As)f(a)g(result,)g(one)h(cannot)f(implemen)m(t)f FI(functions)31
b FT(corresp)s(onding)22 b(to)j(the)f(op)s(erators)g
Fw(on)o FT(,)h Fw(and)o FT(,)g(and)378 2379 y Fw(then)g
FT(whic)m(h)h(tak)m(e)j(t)m(w)m(o)g(premises)d(and)g(infer)g(a)i
(conclusion.)38 b(On)26 b(the)i(other)f(hand,)g(it)g(is)f(necessary)378
2492 y(to)f(implemen)m(t)e(c)m(hec)m(king)j(functions)d(\(decision)g
(pro)s(cedures\))h(whic)m(h)f(c)m(hec)m(k)k(whether)d(a)h(particular)
378 2605 y(conclusion)k(follo)m(ws)g(from)h(a)h(giv)m(en)f
(justi\014cation.)519 2717 y(In)23 b(this)g(c)m(hapter)h(w)m(e)g(sho)m
(w)g(ho)m(w)f(one)h(can)g(pro)s(of)f(c)m(hec)m(k)i(structured)e
(justi\014cations)f(b)m(y)i(restrict-)378 2830 y(ing)35
b(the)h(searc)m(h)h(for)f(a)g(pro)s(of)g(of)g(the)g(conclusion)f(from)g
(the)h(premises)f(in)g(a)h(giv)m(en)g(justi\014cation)378
2943 y(according)30 b(to)g(the)g(op)s(erators)g(in)e(the)i
(justi\014cation.)39 b(W)-8 b(e)31 b(use)e(the)h(de\014nitions)d(and)i
(results)g(giv)m(en)378 3056 y(in)37 b(c)m(hapter)j(7)f(to)h(de\014ne)e
(the)h(required)e(restriction,)j(and)e(therefore)h(assume)g(familiarit)
m(y)d(with)378 3169 y(the)h(material)f(in)f(c)m(hapter)i(7,)i(as)d(w)m
(ell)g(as)h(with)e(the)i(material)f(in)f(c)m(hapter)i(6)g(whic)m(h)e
(in)m(tro)s(duces)378 3282 y(the)e(de\014nitions)d(of)j(structured)f
(justi\014cations)f(and)h(implicit)d(and)k(explicit)e(deriv)-5
b(ations.)46 b(In)32 b(the)378 3395 y(approac)m(h)i(giv)m(en)g(in)e
(this)h(c)m(hapter,)j(a)e(coloured)f(problem)f(\()p FP(S;)15
b FN(K)q FT(\))36 b(is)c(constructed)i(from)g(a)g(giv)m(en)378
3508 y(justi\014ed)26 b(assertion)h Fv(C)50 b Fw(by)42
b Fv(P)e FT(suc)m(h)27 b(that)h Fv(P)40 b FT(justi\014es)26
b Fv(C)33 b FT(if)27 b(and)g(only)g(if)f(\()p FP(S;)15
b FN(K)q FT(\))29 b(is)e(inconsisten)m(t.)39 b(It)378
3621 y(should)30 b(b)s(e)i(noted)h(that)g(the)f(colouring)g(and)g(the)g
(connectabilit)m(y)g(relation)g(in)f(a)i(coloured)f(prob-)378
3734 y(lem)h(\()p FP(S;)15 b FN(K)q FT(\))35 b(denote)g(a)f
(restriction)e(on)i(the)g(w)m(a)m(y)h(the)f(sen)m(tences)g(in)f
FP(S)38 b FT(can)c(b)s(e)g(used)e(to)j(sho)m(w)e(its)378
3847 y(inconsistency)-8 b(.)59 b(Therefore)37 b(it)f(is)g(only)g
(necessary)h(to)h(consider)d(a)j(smaller)d(searc)m(h)i(space)h(when)378
3959 y(sho)m(wing)i(the)h(inconsistency)e(of)i(\()p FP(S;)15
b FN(K)q FT(\))42 b(than)e(when)g(sho)m(wing)g(the)h(inconsistency)e
(of)i(the)g(un-)378 4072 y(coloured)34 b(pro)5 b(jection)34
b(of)g(the)h(sen)m(tences)h(in)d FP(S)5 b FT(.)52 b(This)33
b(restriction)g(on)i(the)f(searc)m(h)h(space)g(results)378
4185 y(in)28 b(the)i(pro)s(of)f(c)m(hec)m(king)i(of)f(structured)f
(justi\014cations)f(b)s(eing)g(more)i(e\016cien)m(t)h(than)e(the)h(c)m
(hec)m(king)378 4298 y(of)g(unstructured)f(justi\014cations.)519
4411 y(W)-8 b(e)34 b(stress)e(that)i(the)e(main)g(result)f(giv)m(en)i
(in)e(this)h(c)m(hapter)h FI(is)i(not)42 b FT(an)32 b(algorithm)g(for)g
(c)m(hec)m(k-)378 4524 y(ing)i(structured)f(justi\014cations.)52
b(The)34 b(main)g(result)f(is)h(that)h(a)g(structured)e
(justi\014cation)h(can)h(b)s(e)378 4637 y(c)m(hec)m(k)m(ed)d(b)m(y)e
(restricting)g(the)g(searc)m(h)h(space)g(considered)e(b)m(y)h
(\014rst-order)g(theorem)h(pro)m(v)m(ers.)41 b(This)378
4750 y(restriction)35 b(is)g(giv)m(en)g(in)g(terms)h(of)g(the)g
(coloured)f(\014rst-order)g(logic)h(giv)m(en)f(in)g(c)m(hapter)h(7)h
(and)e(is)378 4863 y FI(indep)-5 b(endent)40 b FT(of)31
b(the)f(particular)f(\014rst-order)g(logic)h(semi-decision)f(pro)s
(cedure)g(used)g(in)g(c)m(hec)m(king)378 4976 y(them.)41
b(The)30 b(fact)h(that)514 5139 y FN(\017)46 b FT(the)28
b(restriction)f(on)h(the)g(pro)s(of)f(searc)m(h)i(required)d(to)i(c)m
(hec)m(k)i(structured)d(justi\014cations)f(do)s(es)605
5252 y(not)31 b(dep)s(end)e(on)h(the)g(algorithm)g(used)f(to)i(c)m(hec)
m(k)h(them,)514 5429 y FN(\017)46 b FT(and)30 b(the)h(fact)g(that)g
(the)f(seman)m(tics)h(of)g(structured)e(justi\014cations)g(is)g
(non-deterministic)378 5592 y(suggest)42 b(that)h(pro)s(ofs)e(in)m(v)m
(olving)f(structured)h(justi\014cations)f(are)i(not)g(pro)s(cedural.)73
b(Although)378 5705 y(structured)37 b(justi\014cations)f(con)m(tain)i
(some)g(information)f(on)g(what)h(inferences)f(are)h(required)e(to)p
eop
%%Page: 151 161
151 160 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(151)378
396 y(justify)33 b(the)h(conclusion,)g(they)g(do)g(not)g(corresp)s(ond)
f(to)i(a)g(sp)s(eci\014c)e(pro)s(cedure)g(for)h(deriving)e(the)378
509 y(conclusion)d(from)h(the)g(justi\014cation.)519
622 y(W)-8 b(e)23 b(recall)e(that)i(the)f(de\014nition)d(of)j(the)g(v)
-5 b(alidit)m(y)20 b(of)i(structured)f(justi\014cations)g
(\(de\014nition)e(6.4\))378 735 y(is)28 b(giv)m(en)i(in)e(terms)h(of)h
(the)f(explicit)f(deriv)-5 b(ations)28 b(relation)g Ff( )p
FT(,)i(whic)m(h)e(is)h(de\014ned)f(in)g(terms)h(of)h(the)378
848 y(implicit)21 b(deriv)-5 b(ations)23 b(relation)h
Ff(\032)1591 815 y FK(\003)1630 848 y FT(.)39 b(Since,)25
b(the)f(main)g(goal)h(of)f(this)f(c)m(hapter)i(is)f(to)h(sho)m(w)f(ho)m
(w)h(one)378 961 y(can)37 b(c)m(hec)m(k)h(the)f(v)-5
b(alidit)m(y)35 b(of)h(structured)g(justi\014cations)f(b)m(y)h
(constructing)g(a)h(coloured)f(problem)378 1074 y(\()p
FP(S;)15 b FN(K)q FT(\))42 b(and)d(then)h(c)m(hec)m(king)h(the)f
FN(K)q FT(-inconsistency)g(of)g FP(S)5 b FT(,)43 b(w)m(e)d(\014rst)g
(sho)m(w)g(in)e(section)j(8.2)g(ho)m(w)378 1187 y(one)35
b(can)h(construct)f(an)g(inconsisten)m(t)f(coloured)h(problem)e(from)h
(an)h(implicit)d(deriv)-5 b(ation.)54 b(This)378 1300
y(construction)44 b(is)g(also)g(sho)m(wn)g(to)h(b)s(e)f(sound)g(and)g
(complete)h(in)e(the)i(sense)f(that)h(an)g(implicit)378
1413 y(deriv)-5 b(ation)39 b(is)h(v)-5 b(alid)39 b(if)h(and)g(only)g
(if)g(the)h(corresp)s(onding)e(coloured)h(problem)g(is)g(inconsisten)m
(t.)378 1526 y(Section)e(8.3)h(then)f(sho)m(ws)g(ho)m(w)g(one)g(can)h
(construct)f(a)h(coloured)f(problem)e(from)i(a)g(conclusion)378
1638 y(justi\014ed)j(b)m(y)i(a)h(structured)e(justi\014cation.)77
b(In)42 b(section)i(8.4,)j(it)c(is)f(sho)m(wn)g(that)i(the)f(coloured)
378 1751 y(problem)36 b(constructed)h(b)m(y)g(the)h(metho)s(d)e(giv)m
(en)i(in)d(section)j(8.3)g(is)e(inconsisten)m(t)h(if)f(and)h(only)f(if)
378 1864 y(the)41 b(giv)m(en)f(justi\014ed)e(conclusion)h(is)h(v)-5
b(alid.)69 b(Section)40 b(8.5)h(illustrates)e(ho)m(w)h(the)h
FN(C)5 b(B)s(S)i(E)48 b FT(deriv)m(ed)378 1977 y(rule)39
b(giv)m(en)h(in)f(c)m(hapter)i(5)g(is)e(mo)s(di\014ed)f(so)j(that)g(it)
f(can)g(b)s(e)g(used)f(to)i(pro)s(of)f(c)m(hec)m(k)i(structured)378
2090 y(justi\014cations.)d(A)31 b(summary)e(of)h(this)g(c)m(hapter)h
(is)e(giv)m(en)h(in)g(section)g(8.6.)378 2377 y FH(8.2)135
b(Pro)t(of)45 b(Chec)l(king)h(Implicit)g(First-Order)e(Inferences)378
2579 y FT(In)c(this)f(section)i(w)m(e)g(sho)m(w)g(ho)m(w)f(one)h(can)g
(c)m(hec)m(k)h(whether)e(a)h(\014rst-order)f(sen)m(tence)i
FP(B)j FT(can)c(b)s(e)378 2692 y(implicitly)23 b(deriv)m(ed)k(from)f
(another)i(sen)m(tence)g FP(A)f FT(\(that)h(is,)g(whether)e
FP(A)g Ff(\032)2976 2659 y FK(\003)3040 2692 y FP(B)5
b FT(;)28 b(see)g(de\014nition)d(6.2)378 2805 y(on)34
b(page)g(105\))i(b)m(y)d(restricting)g(the)h(searc)m(h)g(for)g(a)g(pro)
s(of)f(of)h FP(A)d FN(\))g FP(B)5 b FT(.)50 b(This)32
b(restriction)h(is)g(giv)m(en)378 2918 y(in)g(section)i(8.2.1)h(and)e
(it)h(is)e(sho)m(wn)h(to)h(giv)m(e)h(sound)d(and)h(complete)h(metho)s
(ds)f(for)g(c)m(hec)m(king)i(im-)378 3031 y(plicit)j(\014rst-order)h
(deriv)-5 b(ations)40 b(in)f(sections)i(8.2.2)i(and)e(8.2.3.)74
b(These)41 b(results)f(are)h(then)g(used)378 3144 y(in)31
b(section)i(8.2.4)h(to)f(sho)m(w)f(that)i(the)e(problem)f(of)i(c)m(hec)
m(king)g(implicit)c(\014rst-order)j(deriv)-5 b(ations)31
b(is)378 3257 y(undecidable.)378 3500 y FG(8.2.1)112
b(A)37 b(Restricted)f(Pro)s(of)h(Searc)m(h)h(for)g(Chec)m(king)f
(Implicit)c(Inferences)378 3672 y FT(Giv)m(en)25 b(the)g(\014rst-order)
f(sen)m(tences)i FP(A)f FT(and)f FP(B)5 b FT(,)26 b(the)f(implicit)d
(deriv)-5 b(ation)23 b FP(A)j Ff(\032)3083 3639 y FK(\003)3147
3672 y FP(B)k FT(can)25 b(b)s(e)f(c)m(hec)m(k)m(ed)378
3785 y(b)m(y)33 b(lo)s(oking)f(for)h(a)g(refutational)g(pro)s(of)f(of)i
FP(A)c FN(\))g FP(B)37 b FT(in)32 b(whic)m(h)g(complemen)m(tary)h
(pairs)f(of)i(literals)378 3898 y(are)g(allo)m(w)m(ed)g(to)g(b)s(e)f
(used)g(in)f(refuting)h FN(f:)p FT(\()p FP(A)e FN(\))g
FP(B)5 b FT(\))p FN(g)34 b FT(if)e(one)i(literal)e(in)h(the)g(pair)g
(is)g(tak)m(en)h(from)378 4011 y FP(A)42 b FT(and)f(the)h(other)f(one)h
(from)f FP(B)5 b FT(.)74 b(More)43 b(formally)-8 b(,)43
b(and)e(using)f(the)i(notation)g(in)m(tro)s(duced)e(in)378
4124 y(c)m(hapter)32 b(7,)g(it)e(is)g(the)i(case)g(that)f
FP(A)c Ff(\032)1734 4091 y FK(\003)1800 4124 y FP(B)36
b FT(if)30 b(and)g(only)g(if)g FN(f)p FP(A)2563 4091
y FO(i)2592 4124 y FP(;)15 b FN(:)p FP(B)2767 4091 y
FO(j)2803 4124 y FN(g)32 b FT(is)e FP(i)d FN($)f FP(j)5
b FT(-inconsisten)m(t)31 b(for)378 4237 y(distinct)c(colours)h
FP(i)h FT(and)g FP(j)5 b FT(.)40 b(This)27 b(claim)h(is)g(pro)m(v)m(ed)
h(in)f(the)h(follo)m(wing)e(t)m(w)m(o)j(sections)f(and)f(giv)m(en)h(as)
378 4349 y(theorem)i(8.3)g(on)f(page)i(157.)519 4462
y(This)k(result)g(is)h(used)f(to)i(deriv)m(e)g(the)f(main)g(goal)h(of)f
(this)g(c)m(hapter,)j(whic)m(h)c(is)g(to)j(sho)m(w)e(ho)m(w)378
4575 y(one)c(can)g(c)m(hec)m(k)h(the)e(v)-5 b(alidit)m(y)31
b(of)i(a)g(structured)e(justi\014cation)g(b)m(y)i(\014rst)e
(constructing)h(a)h(coloured)378 4688 y(problem)39 b(\()p
FP(S;)15 b FN(K)q FT(\),)46 b(and)40 b(then)h(sho)m(wing)f(that)h(\()p
FP(S;)15 b FN(K)q FT(\))43 b(is)d(inconsisten)m(t.)72
b(In)40 b(particular,)i(w)m(e)g(can)378 4801 y(already)33
b(see)g(that)h(one)f(can)g(sho)m(w)g(that)h(a)f(conclusion)e
FP(C)40 b FT(can)33 b(b)s(e)f(justi\014ed)f(b)m(y)i FP(P)46
b FT(where)33 b FP(P)45 b FT(is)32 b(a)378 4914 y(single)d(sen)m
(tence,)j(b)m(y)e(sho)m(wing)g(that)h(\()p FP(S;)15 b
FN(K)q FT(\))31 b(is)f(inconsisten)m(t)f(where)1709 5118
y FP(S)h FT(=)25 b FN(f)p FP(P)2007 5081 y FO(i)2035
5118 y FP(;)15 b FN(:)p FP(C)2208 5081 y FO(j)2244 5118
y FN(g)p FP(;)47 b FT(and)1699 5256 y FN(K)27 b FT(=)e
FP(i)g FN($)g FP(j)378 5460 y FT(for)39 b(distinct)e(colours)i
FP(i)g FT(and)f FP(j)5 b FT(.)67 b(This)38 b(follo)m(ws)g(from)g(the)i
(main)d(result)h(of)i(this)d(section)j(\(theo-)378 5573
y(rem)30 b(8.3\))i(and)e(the)g(fact)h(that)p eop
%%Page: 152 162
152 161 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(152)473
396 y FP(C)102 b FM(by)95 b FP(P)378 556 y FT(is)28 b(v)-5
b(alid)27 b(if)g(and)h(only)g(if)f FP(P)39 b Ff( )25
b FP(C)35 b FT(\(b)m(y)29 b(de\014nition)d(6.4\))k(and)e(that)h
FP(P)38 b Ff( )26 b FP(C)35 b FT(if)27 b(and)h(only)g(if)g
FP(P)38 b Ff(\032)3692 523 y FK(\003)3756 556 y FP(C)378
669 y FT(\(b)m(y)31 b(de\014nition)d(6.3\).)378 907 y
FG(8.2.2)112 b(Soundness)40 b(of)d(the)h(Restriction)378
1078 y FT(In)27 b(this)g(section)h(w)m(e)g(sho)m(w)f(that)i(for)e(sen)m
(tences)i FP(A)f FT(and)f FP(B)32 b FT(the)c(restriction)f(giv)m(en)h
(in)e(section)i(8.2.1)378 1191 y(for)38 b(searc)m(hing)g(for)h(a)g(pro)
s(of)e(of)i FP(A)g FN(\))f FP(B)43 b FT(in)37 b(order)h(to)h(c)m(hec)m
(k)h(whether)e FP(A)h Ff(\032)3164 1158 y FK(\003)3242
1191 y FP(B)k FT(is)38 b(sound,)h(in)378 1304 y(the)31
b(sense)g(that)g(whenev)m(er)g(a)g(pro)s(of)f(of)h FP(A)26
b FN(\))f FP(B)36 b FT(is)29 b(found)h(according)h(to)g(the)g(giv)m(en)
g(restrictions,)378 1417 y(it)i(is)g(the)h(case)h(that)g
FP(A)c Ff(\032)1324 1384 y FK(\003)1395 1417 y FP(B)5
b FT(.)51 b(In)33 b(order)g(to)i(sho)m(w)f(this)f(result)f(w)m(e)j
(need)f(the)g(follo)m(wing)e(rather)378 1530 y(straigh)m(tforw)m(ard)e
(prop)s(osition.)378 1708 y FQ(Prop)s(osition)36 b(8.1)46
b FI(L)-5 b(et)29 b FP(A)f FI(and)h FP(B)k FI(b)-5 b(e)28
b(some)h(\014rst-or)-5 b(der)31 b(formulae)e(such)g(that)g
FP(A)d Ff(\032)3359 1675 y FK(\003)3423 1708 y FP(B)5
b FI(.)40 b(F)-7 b(or)29 b(al)5 b(l)378 1821 y(terms)34
b FP(t)g FI(and)g FP(t)910 1788 y FK(0)967 1821 y FI(wher)-5
b(e)35 b FP(t)e FI(is)h(either)g(a)g(c)-5 b(onstant,)35
b(p)-5 b(ar)g(ameter)37 b(or)d(variable,)g(and)h(no)f(fr)-5
b(e)g(e)34 b(variable)378 1934 y(in)e FP(t)522 1901 y
FK(0)578 1934 y FI(b)-5 b(e)g(c)g(omes)34 b(b)-5 b(ound)33
b(in)g FP(A)p FN(f)p FP(t)26 b FN(!)f FP(t)1617 1901
y FK(0)1640 1934 y FN(g)33 b FI(and)h FP(B)5 b FN(f)p
FP(t)25 b FN(!)g FP(t)2221 1901 y FK(0)2244 1934 y FN(g)p
FI(,)33 b(it)f(is)h(the)g(c)-5 b(ase)33 b(that)1615 2139
y FP(A)p FN(f)p FP(t)26 b FN(!)f FP(t)1936 2101 y FK(0)1959
2139 y FN(g)h Ff(\032)2131 2101 y FK(\003)2196 2139 y
FP(B)5 b FN(f)p FP(t)25 b FN(!)g FP(t)2522 2101 y FK(0)2545
2139 y FN(g)378 2343 y FI(wher)-5 b(e)38 b(for)g(any)f(formula)i
FP(C)7 b FI(,)37 b(the)g(expr)-5 b(ession)39 b FP(C)7
b FN(f)p FP(t)33 b FN(!)g FP(t)2385 2310 y FK(0)2408
2343 y FN(g)38 b FI(r)-5 b(epr)g(esents)38 b(the)g(formula)g
FP(C)44 b FI(with)37 b(al)5 b(l)378 2456 y(its)33 b(o)-5
b(c)g(curr)g(enc)g(es)34 b(of)f FP(t)f FI(r)-5 b(eplac)g(e)g(d)35
b(with)f FP(t)1737 2423 y FK(0)1760 2456 y FI(.)378 2634
y FQ(Pro)s(of)p FT(:)f(The)f(fact)h(that)g FP(A)p FN(f)p
FP(t)c FN(!)f FP(t)1585 2601 y FK(0)1609 2634 y FN(g)g
Ff(\032)h FP(B)5 b FN(f)p FP(t)28 b FN(!)h FP(t)2145
2601 y FK(0)2168 2634 y FN(g)j FT(whenev)m(er)g FP(A)d
Ff(\032)g FP(B)36 b FT(can)d(b)s(e)f(easily)f(c)m(hec)m(k)m(ed)378
2747 y(for)26 b(eac)m(h)i(rule)e(in)f(de\014nition)f(6.1.)41
b(The)26 b(statemen)m(t)j(of)d(this)g(prop)s(osition)e(follo)m(ws)i
(from)g(this)g(result)378 2860 y(and)k(the)g(fact)i(that)f
Ff(\032)1189 2827 y FK(\003)1258 2860 y FT(is)f(the)g(re\015exiv)m(e)h
(transitiv)m(e)e(closure)h(of)h Ff(\032)p FT(.)956 b
Ff(\004)519 3086 y FT(W)-8 b(e)34 b(no)m(w)e(sho)m(w)h(that)g(the)g
(sen)m(tence)h FP(B)j FT(can)c(b)s(e)e(implicitly)e(deriv)m(ed)j(from)g
(some)h(sen)m(tence)h FP(A)378 3199 y FT(if)29 b FN(f)p
FP(A)574 3166 y FO(i)603 3199 y FP(;)15 b FN(:)p FP(B)778
3166 y FO(j)814 3199 y FN(g)31 b FT(is)e FP(i)d FN($)f
FP(j)5 b FT(-inconsisten)m(t)30 b(for)h(distinct)d(colours)i
FP(i)h FT(and)f FP(j)5 b FT(.)378 3377 y FQ(Theorem)34
b(8.1)46 b FI(Given)25 b(two)i(sentenc)-5 b(es)25 b FP(X)33
b FI(and)26 b FP(Y)20 b FI(,)27 b(and)f(distinct)h(c)-5
b(olours)27 b FP(i)e FI(and)i FP(j)5 b FI(,)27 b(if)e
FN(f)p FP(X)3543 3344 y FO(i)3572 3377 y FP(;)15 b FN(:)p
FP(Y)3746 3344 y FO(j)3782 3377 y FN(g)378 3490 y FI(is)33
b FP(i)25 b FN($)g FP(j)5 b FI(-inc)-5 b(onsistent)34
b(then)f FP(X)g Ff(\032)1636 3457 y FK(\003)1701 3490
y FP(Y)20 b FI(.)378 3669 y FQ(Pro)s(of)p FT(:)35 b(F)-8
b(or)34 b(an)m(y)g(form)m(ula)e FP(Z)7 b FT(,)34 b(let)g(us)e(de\014ne)
h(the)h(set)g FP(D)2396 3683 y FO(Z)2486 3669 y FT(con)m(taining)f(the)
h(sen)m(tences)h(that)f(can)378 3782 y(b)s(e)c(implicitly)c(deriv)m(ed)
k(from)g FP(Z)7 b FT(:)1707 3986 y FP(D)1782 4000 y FO(Z)1864
3986 y FT(=)25 b FN(f)p FP(\030)30 b FN(j)25 b FP(Z)32
b Ff(\032)2320 3948 y FK(\003)2385 3986 y FP(\030)t FN(g)p
FP(:)378 4190 y FT(No)m(w,)f(let)g(the)f(collection)g(of)h(sets)g
FN(C)k FT(b)s(e)30 b(de\014ned)f(as)h(follo)m(ws:)799
4394 y FN(C)88 b FT(=)83 b FN(f)p FP(P)1205 4357 y FO(i)1254
4394 y FN([)20 b FP(Q)1407 4357 y FO(j)1468 4394 y FN(j)26
b FP(P)38 b FN(\022)25 b FP(D)1786 4408 y FO(X)1884 4394
y FT(and)30 b FP(Q)25 b FN(\022)g FP(D)2329 4408 y FK(:)p
FO(Y)2437 4394 y FP(;)46 b FT(for)30 b(sen)m(tences)h
FP(X)38 b FT(and)30 b FP(Y)1180 4532 y FT(suc)m(h)g(that)h(it)f(is)f
(not)i(the)g(case)g(that)g FP(X)i Ff(\032)2681 4499 y
FK(\003)2745 4532 y FP(Y)20 b FN(g)p FP(:)378 4736 y
FT(Note)32 b(that)f(all)e(the)h(form)m(ulae)g(in)g(the)g(sets)h(in)e
FN(C)35 b FT(are)c(homogeneously)f(coloured)g(b)m(y)h
FP(i)f FT(or)h FP(j)5 b FT(.)519 4849 y(W)-8 b(e)47 b(sho)m(w)e(that)h
FN(C)51 b FT(is)44 b(an)h FP(i)51 b FN($)f FP(j)5 b FT(-consistency)47
b(prop)s(ert)m(y)-8 b(.)85 b(Let)46 b(some)g(set)g FP(S)56
b FN(2)50 b(C)5 b FT(,)49 b(then)378 4962 y FP(S)31 b
FT(=)25 b FP(P)632 4929 y FO(i)681 4962 y FN([)20 b FP(Q)834
4929 y FO(j)900 4962 y FT(where)31 b FP(P)38 b FN(\022)26
b FP(D)1432 4976 y FO(X)1530 4962 y FT(and)k FP(Q)25
b FN(\022)h FP(D)1976 4976 y FK(:)p FO(Y)2114 4962 y
FT(for)31 b(some)g(sen)m(tences)h FP(X)38 b FT(and)30
b FP(Y)50 b FT(suc)m(h)30 b(that)i(it)e(is)378 5075 y(not)i(the)g(case)
h(that)g FP(X)i Ff(\032)1303 5042 y FK(\003)1370 5075
y FP(Y)20 b FT(.)45 b(Note)33 b(that)g(for)e(ev)m(ery)i(form)m(ula)e
FP(')2709 5042 y FO(i)2766 5075 y FN(2)c FP(S)37 b FT(the)32
b(form)m(ula)f FP(')h FT(is)f(in)g FP(P)13 b FT(,)378
5188 y(and)30 b(for)g(ev)m(ery)h FP(')993 5155 y FO(j)1055
5188 y FN(2)25 b FP(S)35 b FT(w)m(e)c(ha)m(v)m(e)h FP(')26
b FN(2)e FP(Q)p FT(.)489 5366 y(1.)46 b(Supp)s(ose)30
b(that)i(there)g(is)f(some)h(literal)e FP(A)p FT(,)j(suc)m(h)e(that)h
(b)s(oth)f FP(A)2795 5333 y FO(i)2855 5366 y FT(and)g
FN(:)p FP(A)3162 5333 y FO(j)3230 5366 y FT(are)h(in)e
FP(S)5 b FT(.)45 b(Then)605 5479 y FP(A)28 b FN(2)f FP(P)41
b FN(\022)27 b FP(D)1061 5493 y FO(X)1160 5479 y FT(and)32
b(\()p FN(:)p FP(A)p FT(\))c FN(2)f FP(Q)g FN(\022)h
FP(D)1927 5493 y FK(:)p FO(Y)2035 5479 y FT(.)45 b(Therefore,)32
b FP(X)j Ff(\032)2755 5446 y FK(\003)2822 5479 y FP(A)d
FT(and)f FN(:)p FP(Y)47 b Ff(\032)3362 5446 y FK(\003)3429
5479 y FN(:)p FP(A)p FT(.)e(Also,)605 5592 y(b)m(y)40
b(prop)s(osition)d(6.2,)44 b FP(A)d Ff(\032)1622 5559
y FK(\003)1702 5592 y FP(Y)20 b FT(.)69 b(Hence)41 b
FP(X)48 b Ff(\032)2373 5559 y FK(\003)2454 5592 y FP(Y)59
b FT(whic)m(h)39 b(is)g(a)h(con)m(tradiction.)69 b(As)39
b(a)605 5705 y(result,)30 b(not)g(b)s(oth)g FP(A)1328
5672 y FO(i)1387 5705 y FT(and)f FN(:)p FP(A)1692 5672
y FO(j)1759 5705 y FT(are)i(in)e FP(S)35 b FT(for)30
b(ev)m(ery)h(literal)e FP(A)i FT(and)f(set)h FP(S)k FT(in)29
b FN(C)5 b FT(.)p eop
%%Page: 153 163
153 162 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(153)489
396 y(2.)46 b(Since)24 b FP(X)32 b Ff(\032)26 b FN(>)d
FT(then)h FP(P)e FN([)8 b(f>g)25 b(\022)g FP(D)1872 410
y FO(X)1964 396 y FT(and)f(therefore)g FP(S)13 b FN([)8
b(f>)2764 363 y FO(i)2792 396 y FN(g)26 b(2)f(C)5 b FT(.)39
b(Hence)25 b(b)m(y)f(the)h(ab)s(o)m(v)m(e)605 509 y(case,)39
b FN(?)901 476 y FO(j)982 509 y FP(=)-55 b FN(2)34 b
FP(S)5 b FT(.)57 b(Similarly)-8 b(,)34 b(as)i FN(:)p
FP(Y)55 b Ff(\032)34 b FN(>)p FT(,)j(it)f(follo)m(ws)f(that)h
FP(Q)24 b FN([)g(f>g)35 b(\022)f FP(D)3335 523 y FK(:)p
FO(Y)3479 509 y FT(and)h(that)605 622 y FP(S)25 b FN([)20
b(f>)883 589 y FO(j)920 622 y FN(g)26 b(2)e(C)5 b FT(.)41
b(And)30 b(again)g FN(?)1708 589 y FO(i)1771 622 y FP(=)-55
b FN(2)25 b FP(S)5 b FT(.)41 b(Therefore)30 b(for)g(an)m(y)h(colour)f
FP(k)j FT(in)c FP(i)d FN($)f FP(j)5 b FT(,)31 b FN(?)3498
589 y FO(k)3576 622 y FP(=)-55 b FN(2)25 b FP(S)5 b FT(.)489
810 y(3.)46 b(Let)41 b(some)g(conjunctiv)m(e)g(sen)m(tence)g(\011)h
FN(2)g FP(S)5 b FT(.)71 b(W)-8 b(e)42 b(consider)d(the)i(t)m(w)m(o)h
(cases)f(where)f(\011)i(=)605 923 y(\()p FP(')21 b FN(^)f
FP( )s FT(\))898 890 y FO(i)927 923 y FT(,)31 b(or)f(\011)25
b(=)g(\()p FP(')c FN(^)f FP( )s FT(\))1579 890 y FO(j)1646
923 y FT(for)31 b(some)f(sen)m(tences)i FP(')f FT(and)e
FP( )s FT(.)605 1073 y(If)39 b(\()p FP(')28 b FN(^)e
FP( )s FT(\))1011 1040 y FO(i)1080 1073 y FN(2)41 b FP(S)j
FT(then)c FP(X)48 b Ff(\032)1723 1040 y FK(\003)1803
1073 y FT(\()p FP(')27 b FN(^)f FP( )s FT(\))42 b Ff(\032)f
FP(')p FT(,)h(and)d(similarly)e FP(X)48 b Ff(\032)3210
1040 y FK(\003)3290 1073 y FP( )s FT(.)69 b(Therefore)605
1186 y FP(P)33 b FN([)20 b(f)p FP(';)15 b( )s FN(g)28
b(\022)d FP(D)1227 1200 y FO(X)1325 1186 y FT(and)k(hence)i
FP(S)25 b FN([)20 b(f)p FP(')2020 1153 y FO(i)2049 1186
y FP(;)15 b( )2151 1153 y FO(i)2180 1186 y FN(g)26 b(2)f(C)5
b FT(.)605 1336 y(F)-8 b(or)32 b(the)f(second)f(case,)i(if)e(\()p
FP(')21 b FN(^)f FP( )s FT(\))1814 1303 y FO(j)1878 1336
y FN(2)25 b FP(S)36 b FT(then)30 b FN(:)p FP(Y)45 b Ff(\032)2523
1303 y FK(\003)2589 1336 y FP(')21 b FN(^)f FP( )29 b
Ff(\032)d FP(')31 b FT(and)f(also)h FN(:)p FP(Y)45 b
Ff(\032)3675 1303 y FK(\003)3740 1336 y FP( )s FT(.)605
1449 y(So)30 b FP(Q)21 b FN([)e(f)p FP(';)c( )s FN(g)28
b(\022)d FP(D)1354 1463 y FO(Y)1445 1449 y FT(and)30
b(so)h FP(S)25 b FN([)20 b(f)p FP(')2000 1416 y FO(j)2037
1449 y FP(;)15 b( )2139 1416 y FO(j)2176 1449 y FN(g)26
b(2)f(C)5 b FT(.)489 1637 y(4.)46 b(W)-8 b(e)35 b(no)m(w)f(assume)f
(that)h(a)g(disjunctiv)m(e)e(sen)m(tence)j(\011)30 b
FN(2)g FP(S)39 b FT(and)33 b(consider)f(the)i(cases)g(where)605
1750 y(\011)25 b(=)g(\()p FP(')c FN(_)f FP( )s FT(\))1090
1717 y FO(i)1149 1750 y FT(and)30 b(\011)25 b(=)g(\()p
FP(')c FN(_)f FP( )s FT(\))1811 1717 y FO(j)1848 1750
y FT(.)605 1900 y(Let)k(\()p FP(')6 b FN(_)g FP( )s FT(\))1025
1867 y FO(i)1080 1900 y FN(2)25 b FP(S)5 b FT(.)39 b(W)-8
b(e)24 b(are)g(required)e(to)i(pro)m(v)m(e)g(that)g(either)f
FP(S)12 b FN([)6 b(f)p FP(')2957 1867 y FO(i)2985 1900
y FN(g)26 b(2)f(C)j FT(or)c FP(S)12 b FN([)6 b(f)p FP( )3565
1867 y FO(i)3593 1900 y FN(g)26 b(2)f(C)5 b FT(.)605
2013 y(In)35 b(other)h(w)m(ords,)g(w)m(e)g(need)f(to)h(sho)m(w)g(that)g
(there)f(are)h(some)g(sen)m(tences)h FP(X)3269 2027 y
FL(1)3344 2013 y FT(and)e FP(Y)3579 2027 y FL(1)3653
2013 y FT(suc)m(h)605 2126 y(that)i FP(P)f FN([)24 b(f)p
FP(')p FN(g)36 b(\022)e FP(D)1352 2140 y FO(X)1410 2149
y FC(1)1449 2126 y FT(,)j FP(Q)d FN(\022)g FP(D)1797
2140 y FK(:)p FO(Y)1885 2149 y FC(1)1960 2126 y FT(and)h(it)h(is)e(not)
i(the)g(case)h(that)g FP(X)3140 2140 y FL(1)3214 2126
y Ff(\032)3315 2093 y FK(\003)3389 2126 y FP(Y)3442 2140
y FL(1)3481 2126 y FT(;)h(or)e(that)605 2239 y(there)26
b(are)g(some)f(sen)m(tences)i FP(X)1670 2253 y FL(2)1735
2239 y FT(and)e FP(Y)1960 2253 y FL(2)2024 2239 y FT(where)g
FP(P)e FN([)10 b(f)p FP( )s FN(g)27 b(\022)e FP(D)2784
2253 y FO(X)2842 2262 y FC(2)2881 2239 y FT(,)h FP(Q)f
FN(\022)g FP(D)3200 2253 y FK(:)p FO(Y)3288 2262 y FC(2)3353
2239 y FT(and)f(it)h(is)g(not)605 2352 y(the)34 b(case)h(that)f
FP(X)1236 2366 y FL(2)1306 2352 y Ff(\032)1407 2319 y
FK(\003)1478 2352 y FP(Y)1531 2366 y FL(2)1570 2352 y
FT(.)50 b(Supp)s(ose)32 b(that)i(this)f(is)g(not)h(true;)h(that)f(is,)g
(for)f(all)g(sen)m(tences)605 2465 y FP(X)680 2479 y
FL(1)720 2465 y FT(,)c FP(Y)827 2479 y FL(1)895 2465
y FT(either)f FP(X)1226 2479 y FL(1)1291 2465 y Ff(\032)1392
2432 y FK(\003)1457 2465 y FP(Y)1510 2479 y FL(1)1549
2465 y FT(,)h(or)g FP(P)g FN([)16 b(f)p FP(')p FN(g)27
b Fl(*)e FP(D)2224 2479 y FO(X)2282 2488 y FC(1)2321
2465 y FT(,)k(or)g(else)f FP(Q)d Fl(*)g FP(D)2923 2479
y FK(:)p FO(Y)3011 2488 y FC(1)3050 2465 y FT(;)30 b(and)e(for)g(all)g
FP(X)3617 2479 y FL(2)3656 2465 y FT(,)h FP(Y)3763 2479
y FL(2)3803 2465 y FT(,)605 2577 y(either)h FP(X)938
2591 y FL(2)1004 2577 y Ff(\032)1105 2544 y FK(\003)1170
2577 y FP(Y)1223 2591 y FL(2)1293 2577 y FT(or)h FP(P)i
FN([)21 b(f)p FP( )s FN(g)27 b Fl(*)e FP(D)1928 2591
y FO(X)1986 2600 y FC(2)2056 2577 y FT(or)31 b FP(Q)26
b Fl(*)f FP(D)2437 2591 y FK(:)p FO(Y)2525 2600 y FC(1)2564
2577 y FT(.)42 b(In)30 b(particular,)f(let)i FP(X)3397
2591 y FL(1)3463 2577 y FT(=)25 b FP(X)j FN(^)20 b FP(')p
FT(,)605 2690 y FP(Y)658 2704 y FL(1)733 2690 y FT(=)36
b FP(Y)20 b FT(,)38 b FP(X)1051 2704 y FL(2)1127 2690
y FT(=)d FP(X)d FN(^)24 b FP( )40 b FT(and)c FP(Y)1760
2704 y FL(2)1835 2690 y FT(=)g FP(Y)20 b FT(.)60 b(Then)35
b FP(X)2418 2704 y FL(1)2494 2690 y Ff(\032)h FP(X)43
b Ff(\032)2850 2657 y FK(\003)2925 2690 y FP(\030)e FT(for)36
b(ev)m(ery)i FP(\030)h FN(2)d FP(P)49 b FT(and)605 2803
y FP(X)680 2817 y FL(1)745 2803 y Ff(\032)25 b FP(')p
FT(,)j(hence)e FP(P)d FN([)11 b(f)p FP(')p FN(g)27 b(\022)d
FP(D)1730 2817 y FO(X)1788 2826 y FC(1)1828 2803 y FT(.)39
b(Also)25 b FP(Q)g FN(\022)g FP(D)2360 2817 y FK(:)p
FO(Y)2448 2826 y FC(1)2487 2803 y FT(,)i(and)e(therefore)h(it)f(m)m
(ust)h(b)s(e)f(the)h(case)605 2916 y(that)33 b FP(X)879
2930 y FL(1)947 2916 y Ff(\032)1048 2883 y FK(\003)1116
2916 y FP(Y)1169 2930 y FL(1)1208 2916 y FT(,)g(i.e.,)16
b FP(X)29 b FN(^)21 b FP(')28 b Ff(\032)1797 2883 y FK(\003)1865
2916 y FP(Y)20 b FT(.)46 b(Similarly)-8 b(,)30 b FP(X)2491
2930 y FL(2)2559 2916 y Ff(\032)2660 2883 y FK(\003)2728
2916 y FP(Y)2781 2930 y FL(2)2820 2916 y FT(,)j(or)f(simply)e
FP(X)f FN(^)21 b FP( )32 b Ff(\032)3662 2883 y FK(\003)3730
2916 y FP(Y)20 b FT(.)605 3029 y(But)33 b(this)f(results)g(in)f(a)i
(con)m(tradiction)g(as)g(since)f FP(X)37 b Ff(\032)2554
2996 y FK(\003)2623 3029 y FP(')22 b FN(_)g FP(\036)33
b FT(\(b)s(ecause)g FP(')22 b FN(_)g FP(\036)29 b FN(2)g
FP(P)13 b FT(\))33 b(w)m(e)605 3142 y(ha)m(v)m(e:)1734
3346 y FP(X)g Ff(\032)25 b FP(X)j FN(^)19 b FP(X)1842
3484 y Ff(\032)1943 3447 y FK(\003)2007 3484 y FP(X)28
b FN(^)20 b FT(\()p FP(')h FN(_)f FP( )s FT(\))1842 3622
y Ff(\032)25 b FT(\()p FP(X)j FN(^)20 b FP(')p FT(\))h
FN(_)f FT(\()p FP(X)28 b FN(^)19 b FP( )s FT(\))1842
3760 y Ff(\032)1943 3722 y FK(\003)2007 3760 y FP(Y)41
b FN(_)19 b FP(Y)1842 3898 y Ff(\032)25 b FP(Y)5 b(:)605
4139 y FT(The)20 b(second)g(case,)k(where)c(\()p FP(')p
FN(_)p FP( )s FT(\))1780 4106 y FO(j)1843 4139 y FN(2)25
b FP(S)5 b FT(,)23 b(pro)s(ceeds)c(similarly)-8 b(.)35
b(W)-8 b(e)21 b(assume)f(that)h FP(S)5 b FN([f)p FP(')3659
4106 y FO(j)3697 4139 y FN(g)35 b FP(=)-55 b FN(2)605
4252 y(C)36 b FT(and)30 b FP(S)25 b FN([)20 b(f)p FP( )1135
4219 y FO(j)1172 4252 y FN(g)36 b FP(=)-55 b FN(2)25
b(C)36 b FT(and)30 b(sho)m(w)g(that)h(this)f(giv)m(es)g(a)h(con)m
(tradiction.)41 b(Therefore,)31 b(w)m(e)g(ha)m(v)m(e)605
4365 y(that)26 b(for)f(all)f(sen)m(tences)i FP(X)1519
4379 y FL(1)1584 4365 y FT(and)f FP(Y)1809 4379 y FL(1)1873
4365 y FT(either)g FP(X)2201 4379 y FL(1)2266 4365 y
Ff(\032)2367 4332 y FK(\003)2432 4365 y FP(Y)2485 4379
y FL(1)2524 4365 y FT(,)h(or)f FP(P)39 b Fl(*)25 b FP(D)2949
4379 y FO(X)3007 4388 y FC(1)3071 4365 y FT(or)g FP(Q)10
b FN([)g(f)p FP(')p FN(g)26 b Fl(*)f FP(D)3676 4379 y
FK(:)p FO(Y)3764 4388 y FC(1)3803 4365 y FT(;)605 4478
y(and)36 b(for)f(all)g FP(X)1139 4492 y FL(2)1215 4478
y FT(and)g FP(Y)1450 4492 y FL(2)1525 4478 y FT(either)g
FP(X)1863 4492 y FL(2)1938 4478 y Ff(\032)2039 4445 y
FK(\003)2113 4478 y FP(Y)2166 4492 y FL(2)2205 4478 y
FT(,)i(or)f FP(P)48 b Fl(*)34 b FP(D)2670 4492 y FK(:)p
FO(Y)2758 4501 y FC(2)2833 4478 y FT(or)i(else)f FP(Q)24
b FN([)g(f)p FP( )s FN(g)36 b Fl(*)e FP(D)3676 4492 y
FK(:)p FO(Y)3764 4501 y FC(2)3803 4478 y FT(.)605 4591
y(No)m(w,)40 b(let)c FP(X)1058 4605 y FL(1)1134 4591
y FT(=)g FP(X)7 b FT(,)39 b FP(Y)1440 4605 y FL(1)1515
4591 y FT(=)d FP(Y)44 b FN(_)24 b(:)p FP(')p FT(,)39
b FP(X)2063 4605 y FL(2)2139 4591 y FT(=)c FP(X)45 b
FT(and)36 b FP(Y)2601 4605 y FL(2)2676 4591 y FT(=)g
FP(Y)44 b FN(_)24 b(:)p FP( )s FT(.)60 b(Then)36 b FP(P)49
b FN(\022)36 b FP(D)3706 4605 y FO(X)3764 4614 y FC(1)3803
4591 y FT(.)605 4704 y(Also,)d(for)g(all)e FP(\030)i
FN(2)c FP(Q)p FT(,)k(it)f(is)g(the)g(case)i(that)f FN(:)p
FP(Y)2254 4718 y FL(1)2322 4704 y Ff(\032)2423 4671 y
FK(\003)2491 4704 y FP(\030)k FT(and)32 b(that)h FN(:)p
FP(Y)3060 4718 y FL(1)3128 4704 y Ff(\032)c FN(:)p FP(Y)48
b Ff(\032)3521 4671 y FK(\003)3590 4704 y FP(')32 b FT(and)605
4817 y(so)f FP(Q)20 b FN([)g(f)p FP(')p FN(g)27 b(\022)f
FP(D)1238 4831 y FK(:)p FO(Y)1326 4840 y FC(1)1365 4817
y FT(.)41 b(So)31 b(w)m(e)g(conclude)f(that)h FP(X)2338
4831 y FL(1)2403 4817 y Ff(\032)2504 4784 y FK(\003)2569
4817 y FP(Y)2622 4831 y FL(1)2692 4817 y FT(and)f(with)f(a)i(similar)d
(argumen)m(t)605 4929 y FP(X)680 4943 y FL(2)745 4929
y Ff(\032)846 4896 y FK(\003)911 4929 y FP(Y)964 4943
y FL(2)1003 4929 y FT(.)41 b(Hence)1663 5134 y FP(X)33
b Ff(\032)25 b FP(X)j FN(^)20 b FP(X)1771 5272 y Ff(\032)1872
5234 y FK(\003)1937 5272 y FT(\()p FP(Y)40 b FN(_)20
b(:)p FP(')p FT(\))g FN(^)g FT(\()p FP(Y)41 b FN(_)20
b(:)p FP( )s FT(\))1771 5409 y Ff(\032)25 b FP(Y)40 b
FN(_)20 b FT(\()p FN(:)p FP(')h FN(^)f(:)p FP( )s FT(\))1771
5547 y Ff(\032)1872 5510 y FK(\003)1937 5547 y FP(Y)40
b FN(_)19 b FP(Y)1771 5685 y Ff(\032)25 b FP(Y)p eop
%%Page: 154 164
154 163 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(154)605
396 y(whic)m(h)29 b(is)h(a)h(con)m(tradiction.)489 575
y(5.)46 b(Let)28 b FN(8)p FP(x:')952 542 y FO(i)1006
575 y FN(2)c FP(S)32 b FT(then)27 b FN(8)p FP(x:')f FN(2)f
FP(P)40 b FT(and)26 b(so)i FP(X)k Ff(\032)2270 542 y
FK(\003)2335 575 y FN(8)p FP(x:')25 b Ff(\032)2648 542
y FK(\003)2713 575 y FP(')p FN(f)p FP(x)h FN(!)f FP(t)p
FN(g)j FT(for)f(all)f(closed)h(term)605 688 y FP(t)p
FT(.)53 b(Therefore,)36 b FP(P)g FN([)23 b(f)p FP(')p
FN(f)p FP(x)33 b FN(!)g FP(t)p FN(gg)g(\022)f FP(D)2029
702 y FO(X)2131 688 y FT(and)i FP(S)28 b FN([)23 b(f)p
FP(')p FN(f)p FP(x)33 b FN(!)f FP(t)p FN(g)2915 655 y
FO(i)2944 688 y FN(g)h(2)f(C)39 b FT(for)c(ev)m(ery)g(closed)605
801 y(term)c FP(t)p FT(.)40 b(Similarly)-8 b(,)27 b(if)j
FN(8)p FP(x:')1596 768 y FO(j)1658 801 y FN(2)25 b FP(S)35
b FT(then)30 b FP(S)25 b FN([)20 b(f)p FP(')p FN(f)p
FP(x)27 b FN(!)e FP(t)p FN(g)2626 768 y FO(j)2662 801
y FN(g)h(2)f(C)36 b FT(for)30 b(ev)m(ery)h(closed)f(term)g
FP(t)p FT(.)489 980 y(6.)46 b(Supp)s(ose)33 b(that)i(some)f(existen)m
(tial)g(form)m(ula)g(\011)d FN(2)h FP(S)5 b FT(.)52 b(W)-8
b(e)36 b(consider)d(the)h(t)m(w)m(o)i(cases)f(where)605
1093 y(\011)25 b(=)g FN(9)p FP(x:')984 1060 y FO(i)1043
1093 y FT(or)30 b(\011)25 b(=)g FN(9)p FP(x:')1533 1060
y FO(j)1600 1093 y FT(separately)-8 b(.)605 1239 y(F)g(or)33
b(the)g(\014rst)f(case,)i(w)m(e)f(are)g(giv)m(en)f(that)h
FN(9)p FP(x:')2254 1206 y FO(i)2311 1239 y FN(2)28 b
FP(S)38 b FT(and)32 b(w)m(e)g(are)h(required)e(to)i(sho)m(w)f(that)605
1352 y FP(S)16 b FN([)11 b(f)p FP(')p FN(f)p FP(x)27
b FN(!)e FP(p)p FN(g)1184 1319 y FO(i)1212 1352 y FN(g)h(2)f(C)31
b FT(for)25 b(some)i(parameter)f FP(p)p FT(.)39 b(Similarly)22
b(to)k(the)g(fourth)f(case)i(ab)s(o)m(v)m(e,)h(w)m(e)605
1465 y(pro)m(v)m(e)g(this)e(b)m(y)h(con)m(tradiction.)40
b(Supp)s(ose)25 b(that)j FP(S)19 b FN([)14 b(f)p FP(')p
FN(f)p FP(x)26 b FN(!)f FP(p)p FN(g)2863 1432 y FO(i)2892
1465 y FN(g)35 b FP(=)-55 b FN(2)25 b(C)32 b FT(for)27
b(all)g(parameters)605 1577 y FP(p)p FT(,)k(then)g(for)g(all)f(sen)m
(tences)i FP(X)1655 1591 y FL(1)1726 1577 y FT(and)e
FP(Y)1956 1591 y FL(1)1996 1577 y FT(,)h(either)g FP(P)i
FN([)21 b(f)p FP(')p FN(f)p FP(x)27 b FN(!)g FP(p)p FN(gg)g
Fl(*)f FP(D)3165 1591 y FO(X)3223 1600 y FC(1)3262 1577
y FT(,)31 b(or)h FP(Q)26 b Fl(*)g FP(D)3701 1591 y FK(:)p
FO(Y)3789 1600 y FC(1)605 1690 y FT(or)j(else)f FP(X)960
1704 y FL(1)1025 1690 y Ff(\032)1126 1657 y FK(\003)1191
1690 y FP(Y)1244 1704 y FL(1)1283 1690 y FT(.)40 b(Let)29
b FP(p)g FT(b)s(e)f(some)h(parameter)g(whic)m(h)e(do)s(es)h(not)h(o)s
(ccur)g(in)e FP(X)36 b FT(or)28 b FP(Y)20 b FT(,)29 b(and)605
1803 y(let)f FP(X)809 1817 y FL(1)874 1803 y FT(=)d FP(X)f
FN(^)16 b FP(')p FN(f)p FP(x)26 b FN(!)f FP(p)p FN(g)j
FT(and)g FP(Y)1791 1817 y FL(1)1856 1803 y FT(=)d FP(Y)19
b FT(.)40 b(No)m(w,)30 b(for)e(all)f FP(\030)j FN(2)24
b FP(P)13 b FT(,)29 b(w)m(e)g(ha)m(v)m(e)h FP(X)i Ff(\032)3409
1770 y FK(\003)3474 1803 y FP(\030)t FT(,)d(and)f(so)605
1916 y FP(X)g FN(^)21 b FP(')p FN(f)p FP(x)27 b FN(!)f
FP(p)p FN(g)g Ff(\032)h FP(X)33 b Ff(\032)1544 1883 y
FK(\003)1610 1916 y FP(\030)t FT(.)43 b(Moreo)m(v)m(er,)33
b FP(X)28 b FN(^)20 b FP(')p FN(f)p FP(x)28 b FN(!)e
FP(p)p FN(g)g Ff(\032)h FP(')p FN(f)p FP(x)g FN(!)f FP(p)p
FN(g)31 b FT(and)f(therefore)605 2029 y FP(P)38 b FN([)25
b(f)p FP(')p FN(f)p FP(x)39 b FN(!)e FP(p)p FN(gg)h(\022)f
FP(D)1512 2043 y FO(X)1570 2052 y FC(1)1609 2029 y FT(.)62
b(Also,)40 b FP(Q)d FN(\022)g FP(D)2228 2043 y FK(:)p
FO(Y)2316 2052 y FC(1)2392 2029 y FT(and)g(hence)h(it)f(m)m(ust)h(b)s
(e)f(the)h(case)g(that)605 2142 y FP(X)680 2156 y FL(1)745
2142 y Ff(\032)846 2109 y FK(\003)911 2142 y FP(Y)964
2156 y FL(1)1003 2142 y FT(,)30 b(or)f(in)f(other)i(w)m(ords)e
FP(X)e FN(^)18 b FP(')p FN(f)p FP(x)26 b FN(!)f FP(p)p
FN(g)g Ff(\032)2464 2109 y FK(\003)2529 2142 y FP(Y)20
b FT(.)40 b(But)30 b(since)f FP(p)g FT(do)s(es)g(not)g(o)s(ccur)g(in)
605 2255 y FP(X)38 b FT(and)30 b FP(Y)50 b FT(w)m(e)31
b(get)g FP(X)d FN(^)20 b FP(')26 b Ff(\032)1653 2222
y FK(\003)1717 2255 y FP(Y)51 b FT(b)m(y)30 b(Prop)s(osition)e(8.1)k
(as)1088 2459 y(\()p FP(X)c FN(^)20 b FP(')p FN(f)p FP(x)26
b FN(!)f FP(p)p FN(g)p FT(\))p FN(f)p FP(p)h FN(!)f FP(x)p
FN(g)h FT(=)f FP(X)i FN(^)20 b FP(')84 b FT(and)e FP(Y)20
b FN(f)p FP(p)26 b FN(!)f FP(x)p FN(g)g FT(=)g FP(Y)5
b(:)605 2663 y FT(But)39 b(this)f(is)g(con)m(tradictory)h(since,)i
(using)c(the)i(fact)h(that)g FP(X)46 b FT(and)38 b FP(Y)59
b FT(are)39 b(sen)m(tences,)j(w)m(e)605 2776 y(deriv)m(e)30
b(the)h(follo)m(wing:)1879 2980 y FP(X)i Ff(\032)25 b
FP(X)j FN(^)19 b FP(X)1987 3118 y Ff(\032)2088 3081 y
FK(\003)2152 3118 y FP(X)28 b FN(^)20 b(9)p FP(x:')1987
3256 y Ff(\032)25 b FN(9)p FP(x:)p FT(\()p FP(X)j FN(^)19
b FP(')p FT(\))1987 3394 y Ff(\032)2088 3356 y FK(\003)2152
3394 y FN(9)p FP(x:Y)1987 3532 y Ff(\032)25 b FP(Y)5
b(:)605 3769 y FT(The)30 b(second)g(case)h(is)e(v)m(ery)h(similar)d(to)
k(the)f(\014rst)g(one.)41 b(If)29 b FN(9)p FP(x:')2799
3736 y FO(j)2861 3769 y FN(2)c FP(S)35 b FT(and)29 b(w)m(e)i(assume)e
(that)605 3882 y FP(S)13 b FN([)8 b(f)p FP(')p FN(f)p
FP(x)27 b FN(!)e FP(p)p FN(g)1178 3849 y FO(j)1215 3882
y FN(g)35 b FP(=)-55 b FN(2)25 b(C)30 b FT(then)24 b(w)m(e)h(get)g
(that)g(for)f(ev)m(ery)h(parameter)g FP(p)f FT(and)g(sen)m(tences)h
FP(X)3617 3896 y FL(1)3681 3882 y FT(and)605 3995 y FP(Y)658
4009 y FL(1)697 3995 y FT(,)31 b(either)f FP(P)j FN([)20
b(f)p FP(')p FN(f)p FP(x)27 b FN(!)f FP(p)p FN(gg)g Fl(*)f
FP(D)1861 4009 y FO(X)1919 4018 y FC(1)1958 3995 y FT(,)31
b(or)f FP(Q)c Fl(*)f FP(D)2394 4009 y FK(:)p FO(Y)2482
4018 y FC(1)2551 3995 y FT(or)31 b(else)f FP(X)2910 4009
y FL(1)2975 3995 y Ff(\032)3076 3962 y FK(\003)3141 3995
y FP(Y)3194 4009 y FL(1)3233 3995 y FT(.)42 b(In)29 b(particular,)605
4108 y(w)m(e)36 b(let)f FP(p)g FT(b)s(e)f(some)i(parameter)f(whic)m(h)f
(do)s(es)h(not)g(o)s(ccur)g(in)f FP(X)42 b FT(or)35 b
FP(Y)20 b FT(,)37 b(and)d FP(X)3387 4122 y FL(1)3460
4108 y FT(=)f FP(X)42 b FT(and)605 4221 y FP(Y)658 4235
y FL(1)727 4221 y FT(=)30 b FP(Y)42 b FN(_)22 b(:)p FP(')p
FN(f)p FP(x)30 b FN(!)g FP(p)p FN(g)p FT(.)49 b(Then)32
b FP(P)43 b FN(\022)30 b FP(D)2056 4235 y FO(X)2114 4244
y FC(1)2153 4221 y FT(,)k(and)f(also)g FP(Q)22 b FN([)f(f)p
FP(')p FN(f)p FP(x)32 b FN(!)e FP(p)p FN(gg)g(\022)g
FP(D)3449 4235 y FO(Y)3490 4244 y FC(1)3529 4221 y FT(.)49
b(Th)m(us,)605 4333 y(w)m(e)29 b(are)f(left)g(with)e
FP(X)1322 4347 y FL(1)1387 4333 y Ff(\032)1488 4301 y
FK(\003)1553 4333 y FP(Y)1606 4347 y FL(1)1645 4333 y
FT(,)j(i.e.,)15 b FP(X)33 b Ff(\032)2064 4301 y FK(\003)2129
4333 y FP(Y)i FN(_)15 b(:)p FP(')p FN(f)p FP(x)26 b FN(!)f
FP(p)p FN(g)p FT(.)40 b(Hence)29 b(b)m(y)f(Prop)s(osition)e(8.1,)605
4446 y FP(X)33 b Ff(\032)814 4413 y FK(\003)879 4446
y FP(Y)40 b FN(_)20 b(:)p FP(')p FT(.)40 b(This)29 b(can)i(b)s(e)e
(used)h(to)h(deduce)f(that)1834 4651 y FP(X)i Ff(\032)26
b FN(8)p FP(x:X)1941 4788 y Ff(\032)2042 4751 y FK(\003)2107
4788 y FN(8)p FP(x:)p FT(\()p FP(Y)40 b FN(_)20 b(:)p
FP(')p FT(\))1941 4926 y Ff(\032)26 b FP(Y)40 b FN(_)20
b(:)p FT(\()p FN(9)p FP(x:')p FT(\))1941 5064 y Ff(\032)2042
5027 y FK(\003)2107 5064 y FP(Y)40 b FN(_)20 b FP(Y)1941
5202 y Ff(\032)26 b FP(Y)605 5406 y FT(whic)m(h)j(con)m(tradicts)i(our)
f(assumption)f(that)i FP(X)i FN(6)p Ff(\032)2382 5373
y FK(\003)2446 5406 y FP(Y)20 b FT(.)519 5592 y(Therefore)k
FN(C)30 b FT(is)23 b(an)h FP(i)i FN($)f FP(j)5 b FT(-consistency)25
b(prop)s(ert)m(y)-8 b(.)38 b(No)m(w,)27 b(if)c(it)h(is)f(not)i(the)f
(case)i(that)f FP(X)32 b Ff(\032)3690 5559 y FK(\003)3755
5592 y FP(Y)378 5705 y FT(then)42 b(the)g(set)g FN(f)p
FP(X)1045 5672 y FO(i)1074 5705 y FP(;)15 b FN(:)p FP(Y)1248
5672 y FO(j)1285 5705 y FN(g)45 b(2)f(C)j FT(and)42 b(is)f(th)m(us)g
FP(i)k FN($)f FP(j)5 b FT(-consisten)m(t.)77 b(With)42
b(this)f(statemen)m(t)i(w)m(e)p eop
%%Page: 155 165
155 164 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(155)378
396 y(conclude)30 b(that)h(if)e FN(f)p FP(X)1159 363
y FO(i)1188 396 y FP(;)15 b FN(:)p FP(Y)1362 363 y FO(j)1399
396 y FN(g)30 b FT(is)g FP(i)25 b FN($)g FP(j)5 b FT(-inconsisten)m(t)
31 b(then)f FP(X)j Ff(\032)2726 363 y FK(\003)2790 396
y FP(Y)20 b FT(.)869 b Ff(\004)378 640 y FG(8.2.3)112
b(Completeness)37 b(of)h(the)f(Restriction)378 812 y
FT(W)-8 b(e)34 b(no)m(w)g(sho)m(w)f(the)g(con)m(v)m(erse)i(of)e
(theorem)h(8.1,)h(or)f(in)d(other)j(w)m(ords)f(whenev)m(er)g
FP(A)d Ff(\032)3449 779 y FK(\003)3519 812 y FP(B)5 b
FT(,)33 b(then)378 924 y(a)e(pro)s(of)f(of)h FP(A)26
b FN(\))f FP(B)35 b FT(can)c(b)s(e)f(found)g(according)g(to)i(the)f
(restriction)e(giv)m(en)i(in)e(section)i(8.2.1.)43 b(The)378
1037 y(main)29 b(part)h(of)h(the)g(pro)s(of)e(of)i(this)e(statemen)m(t)
j(is)e(giv)m(en)g(b)m(y)g(the)h(follo)m(wing)e(lemma.)378
1250 y FQ(Lemma)k(8.1)46 b FI(F)-7 b(or)44 b(al)5 b(l)44
b(formulae)h FP(X)50 b FI(and)45 b FP(Y)63 b FI(such)43
b(that)i FP(X)52 b Ff(\032)45 b FP(Y)20 b FI(,)46 b(and)e(for)g(every)f
(set)g FP(S)49 b FI(of)378 1363 y(c)-5 b(olour)g(e)g(d)38
b(sentenc)-5 b(es)35 b(and)i(substitution)f FP(\022)h
FI(which)f(maps)h(every)e(fr)-5 b(e)g(e)36 b(variable)g(in)f
FP(X)43 b FI(and)36 b FP(Y)56 b FI(to)35 b(a)378 1476
y(close)-5 b(d)34 b(term,)f(if)f FP(S)25 b FN([)20 b(f)p
FP(X)1265 1443 y FO(i)1294 1476 y FP(\022)s FN(g)32 b
FI(is)h FP(i)26 b FN($)f FP(j)5 b FI(-c)-5 b(onsistent)34
b(then)f(so)g(is)g FP(S)25 b FN([)20 b(f)p FP(Y)2882
1443 y FO(i)2910 1476 y FP(\022)s FN(g)p FI(.)378 1688
y FQ(Pro)s(of)p FT(:)39 b(W)-8 b(e)40 b(pro)s(ceed)e(b)m(y)g(rule)f
(induction)f(on)j(the)f(relation)g Ff(\032)p FT(.)65
b(The)37 b(pro)s(ofs)h(of)g(most)h(of)g(the)378 1801
y(cases)31 b(are)g(routine)e(and)h(w)m(e)h(presen)m(t)f(here)h(a)f(few)
h(of)f(the)h(less)e(trivial)g(ones.)41 b(Let)31 b(us)e(de\014ne)1889
2005 y FN(K)e FT(=)e FP(i)h FN($)f FP(j:)378 2210 y FT(W)-8
b(e)31 b(use)e FP(\022)737 2224 y FL(\026)-39 b FO(x)806
2210 y FT(to)30 b(denote)h(the)e(substitution)f(whic)m(h)g(maps)h(the)h
(v)-5 b(ariable)28 b FP(x)i FT(to)g(itself)f(and)g(an)m(y)h(other)378
2323 y(v)-5 b(ariable)25 b FP(y)30 b FT(to)d FP(y)s(\022)s
FT(.)38 b(Also,)28 b(w)m(e)f(represen)m(t)f(the)h(substitution)d
FP(\022)29 b FT(restricted)d(to)h(all)f(the)h(free)f(v)-5
b(ariables)378 2435 y(in)29 b(some)h(term)g FP(t)f FT(b)m(y)h
FP(\022)1159 2454 y FK(j)p FO(t)1208 2435 y FT(.)41 b(Note)31
b(that)f(most)g(of)g(the)h(implications)c(in)h(the)i(pro)s(ofs)f(of)h
(the)g(follo)m(wing)378 2548 y(cases)g(can)f(b)s(e)g(substituted)e
(with)h(a)h(bi-implication)c(\()p FN(,)p FT(\).)41 b(W)-8
b(e)30 b(do)f(not)h(do)f(this)e(since)i(our)f(goal)i(is)378
2661 y(simply)e(to)j(sho)m(w)f(the)h FI(implic)-5 b(ation)970
2865 y FP(S)25 b FN([)20 b(f)p FP(X)1259 2828 y FO(i)1288
2865 y FP(\022)s FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t)57
b FN(\))e FP(S)25 b FN([)20 b(f)p FP(Y)2477 2828 y FO(i)2505
2865 y FP(\022)s FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t)q
FP(:)514 3070 y FN(\017)46 b FT(The)30 b(sen)m(tence)i
FP(X)g FT(=)25 b FP(A)c FN(^)f FT(\()p FP(B)25 b FN(_)19
b FP(C)7 b FT(\))31 b(and)e FP(Y)46 b FT(=)25 b(\()p
FP(A)20 b FN(^)g FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)h
FN(^)f FP(C)7 b FT(\).)1207 3274 y FP(S)25 b FN([)20
b(f)p FT(\()p FP(A)h FN(^)f FT(\()p FP(B)25 b FN(_)19
b FP(C)7 b FT(\)\))1970 3236 y FO(i)1999 3274 y FP(\022)s
FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t)1290 3412
y FN(\))83 b FP(S)25 b FN([)20 b(f)p FP(A)1739 3374 y
FO(i)1767 3412 y FP(\022)j FN(^)d FT(\()p FP(B)2023 3374
y FO(i)2051 3412 y FP(\022)i FN(_)e FP(C)2269 3374 y
FO(i)2296 3412 y FP(\022)s FT(\))p FN(g)31 b FT(is)e
FN(K)q FT(-consisten)m(t)1290 3550 y FN(\))83 b FP(S)25
b FN([)20 b(f)p FP(A)1739 3512 y FO(i)1767 3550 y FP(\022)s(;)15
b(B)1927 3512 y FO(i)1955 3550 y FP(\022)22 b FN(_)e
FP(C)2173 3512 y FO(i)2201 3550 y FP(\022)s FN(g)29 b
FT(is)h FN(K)q FT(-consisten)m(t)1290 3687 y FN(\))83
b FP(S)25 b FN([)20 b(f)p FP(A)1739 3650 y FO(i)1767
3687 y FP(\022)s(;)15 b(B)1927 3650 y FO(i)1955 3687
y FP(\022)s FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t,)j(or)1494
3825 y FP(S)25 b FN([)20 b(f)p FP(A)1769 3788 y FO(i)1798
3825 y FP(\022)s(;)15 b(C)1956 3788 y FO(i)1983 3825
y FP(\022)s FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t)1290
3963 y FN(\))83 b FP(S)25 b FN([)20 b(f)p FP(A)1739 3926
y FO(i)1767 3963 y FP(\022)j FN(^)d FP(B)1988 3926 y
FO(i)2015 3963 y FP(\022)s FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t,)
h(or)1494 4101 y FP(S)25 b FN([)20 b(f)p FP(A)1769 4063
y FO(i)1798 4101 y FP(\022)i FN(^)e FP(C)2016 4063 y
FO(i)2044 4101 y FP(\022)s FN(g)29 b FT(is)h FN(K)q FT(-consisten)m(t)
1290 4239 y FN(\))83 b FP(S)25 b FN([)20 b(f)p FT(\(\()p
FP(A)h FN(^)f FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)g
FN(^)g FP(C)7 b FT(\)\))2467 4201 y FO(i)2496 4239 y
FP(\022)s FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t.)514
4480 y FN(\017)46 b FT(The)23 b(sen)m(tence)i FP(X)33
b FT(=)25 b FN(8)p FP(x:A)e FT(and)g FP(Y)45 b FT(=)25
b FN(8)p FP(x:A)p FN(f)p FP(x)h FN(!)f FP(t)p FN(g)f
FT(where)f(no)g(free)h(v)-5 b(ariable)22 b(in)g FP(t)i
FT(b)s(ecomes)605 4593 y(b)s(ound)29 b(in)g FP(A)p FN(f)p
FP(x)c FN(!)h FP(t)p FN(g)p FT(.)921 4797 y FP(S)f FN([)20
b(f)p FT(\()p FN(8)p FP(x:A)1359 4760 y FO(i)1388 4797
y FT(\))p FP(\022)s FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t)1004
4935 y FN(\))83 b FP(S)25 b FN([)20 b(f8)p FP(x:)p FT(\()p
FP(A)1616 4898 y FO(i)1645 4935 y FP(\022)1692 4949 y
FL(\026)-39 b FO(x)1731 4935 y FT(\))p FN(g)31 b FT(is)f
FN(K)q FT(-consisten)m(t)1004 5073 y FN(\))83 b FP(S)25
b FN([)20 b(f)p FP(A)1453 5036 y FO(i)1482 5073 y FP(\022)1529
5087 y FL(\026)-39 b FO(x)1568 5073 y FN(f)p FP(x)26
b FN(!)f FP(c)p FN(gg)32 b FT(is)d FN(K)q FT(-consisten)m(t)j(for)e(ev)
m(ery)h(closed)f(term)h FP(c)p FT(.)1004 5211 y FN(\))83
b FP(S)25 b FN([)20 b(f)p FP(A)1453 5173 y FO(i)1482
5211 y FN(f)p FP(x)25 b FN(!)h FP(c)p FN(g)p FP(\022)1852
5225 y FL(\026)-39 b FO(x)1892 5211 y FN(g)31 b FT(is)e
FN(K)q FT(-consisten)m(t)j(for)e(ev)m(ery)h(closed)f(term)h
FP(c)p FT(.)p eop
%%Page: 156 166
156 165 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(156)605
396 y(In)26 b(particular,)g FP(S)18 b FN([)13 b(f)p FP(A)1418
363 y FO(i)1446 396 y FN(f)p FP(x)26 b FN(!)f FP(c)p
FN(g)p FP(\022)1816 410 y FL(\026)-39 b FO(x)1856 396
y FN(g)28 b FT(is)d FN(K)q FT(-inconsisten)m(t)i(for)f(all)g(closed)h
(terms)f FP(c)h FT(whic)m(h)e(are)605 509 y(of)31 b(the)f(form)g
FP(t)p FN(f)p FP(x)c FN(!)f FP(c)1391 476 y FK(0)1415
509 y FN(g)p FP(\022)1503 528 y FK(j)p FO(t)1582 509
y FT(where)30 b FP(c)1884 476 y FK(0)1938 509 y FT(is)f(an)m(y)i
(closed)f(term.)41 b(That)30 b(is,)670 726 y FP(S)c FN([)19
b(f)p FP(A)945 688 y FO(i)974 726 y FN(f)p FP(x)26 b
FN(!)f FT(\()p FP(t)p FN(f)p FP(x)h FN(!)f FP(c)1559
688 y FK(0)1583 726 y FN(g)p FP(\022)1671 744 y FK(j)p
FO(t)1720 726 y FT(\))p FN(g)p FP(\022)1847 740 y FL(\026)-39
b FO(x)1887 726 y FN(g)31 b FT(is)f FN(K)q FT(-consisten)m(t)h(for)f
(ev)m(ery)i(closed)e(term)g FP(c)3482 693 y FK(0)753
875 y FN(\))83 b FP(S)26 b FN([)19 b(f)p FP(A)1202 838
y FO(i)1231 875 y FT(\()p FN(f)p FP(x)26 b FN(!)f FP(t)p
FN(gf)p FP(x)h FN(!)g FP(c)1862 838 y FK(0)1885 875 y
FN(g)p FP(\022)1973 894 y FK(j)p FO(t)2023 875 y FT(\))p
FP(\022)2105 889 y FL(\026)-39 b FO(x)2144 875 y FN(g)31
b FT(is)f FN(K)q FT(-consisten)m(t)h(for)f(ev)m(ery)h(closed)g(term)f
FP(c)3739 842 y FK(0)753 1025 y FN(\))83 b FP(S)26 b
FN([)19 b(f)p FP(A)1202 988 y FO(i)1231 1025 y FN(f)p
FP(x)26 b FN(!)f FP(t)p FN(gf)p FP(x)h FN(!)f FP(c)1826
988 y FK(0)1850 1025 y FN(g)p FP(\022)1938 1044 y FK(j)p
FO(t)1987 1025 y FP(\022)2034 1039 y FL(\026)-39 b FO(x)2074
1025 y FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t)h(for)g(ev)m(ery)g
(closed)1079 1163 y(term)f FP(c)1335 1130 y FK(0)1389
1163 y FT(as)h(no)f(free)h(v)-5 b(ariable)29 b(in)g FP(t)h
FT(is)f(b)s(ound)g(in)g FP(A)p FN(f)p FP(x)d FN(!)f FP(t)p
FN(g)p FT(,)1079 1301 y(and)30 b(th)m(us)g(no)g(free)g(v)-5
b(ariable)30 b(in)f FP(t)h FT(is)f(b)s(ound)g(in)g FP(A)p
FN(f)p FP(x)c FN(!)h FP(t)p FN(gf)p FP(x)g FN(!)f FP(c)3408
1268 y FK(0)3431 1301 y FN(g)753 1439 y(\))83 b FP(S)26
b FN([)19 b(f)p FP(A)1202 1401 y FO(i)1231 1439 y FN(f)p
FP(x)26 b FN(!)f FP(t)p FN(gf)p FP(x)h FN(!)f FP(c)1826
1401 y FK(0)1850 1439 y FN(g)p FP(\022)1942 1453 y FL(\026)-39
b FO(x)1982 1439 y FN(g)31 b FT(is)e FN(K)q FT(-consisten)m(t)j(for)e
(ev)m(ery)h(closed)f(term)g FP(c)3576 1406 y FK(0)753
1576 y FN(\))83 b FP(S)26 b FN([)19 b(f)p FP(A)1202 1539
y FO(i)1231 1576 y FN(f)p FP(x)26 b FN(!)f FP(t)p FN(g)p
FP(\022)1595 1590 y FL(\026)-39 b FO(x)1635 1576 y FN(f)p
FP(x)26 b FN(!)f FP(c)1913 1539 y FK(0)1936 1576 y FN(gg)32
b FT(is)d FN(K)q FT(-consisten)m(t)j(for)e(ev)m(ery)h(closed)f(term)g
FP(c)3576 1543 y FK(0)753 1714 y FN(\))83 b FP(S)26 b
FN([)19 b(f8)p FP(x:)p FT(\()p FP(A)1365 1677 y FO(i)1394
1714 y FN(f)p FP(x)26 b FN(!)f FP(t)p FN(g)p FP(\022)1758
1728 y FL(\026)-39 b FO(x)1798 1714 y FT(\))p FN(g)31
b FT(is)e FN(K)q FT(-consisten)m(t)753 1852 y FN(\))83
b FP(S)26 b FN([)19 b(f)p FT(\()p FN(8)p FP(x:A)1365
1814 y FO(i)1394 1852 y FN(f)p FP(x)26 b FN(!)f FP(t)p
FN(g)p FT(\))p FP(\022)s FN(g)31 b FT(is)e FN(K)q FT(-consisten)m(t.)
514 2092 y FN(\017)46 b FT(The)39 b(sen)m(tence)i FP(X)48
b FT(=)41 b FP(A)26 b FN(^)g FP(C)46 b FT(and)40 b FP(Y)60
b FT(=)41 b FP(B)31 b FN(^)26 b FP(C)7 b FT(,)41 b(where)e
FP(A)i Ff(\032)g FP(B)j FT(with)38 b(the)i(induction)605
2205 y(h)m(yp)s(othesis)23 b(that)i(for)f(all)f FP(S)29
b FT(and)24 b FP(\022)i FT(if)e FP(S)13 b FN([)8 b(f)p
FP(A)2145 2172 y FO(i)2173 2205 y FP(\022)s FN(g)24 b
FT(is)f FP(i)j FN($)f FP(j)5 b FT(-consisten)m(t)26 b(then)d(so)i(is)e
FP(S)13 b FN([)8 b(f)p FP(B)3684 2172 y FO(i)3712 2205
y FP(\022)s FN(g)p FT(.)921 2410 y FP(S)25 b FN([)20
b(f)p FT(\()p FP(A)h FN(^)f FP(C)7 b FT(\))1440 2372
y FO(i)1468 2410 y FP(\022)s FN(g)30 b FT(is)f FN(K)q
FT(-consisten)m(t)1004 2547 y FN(\))83 b FP(S)25 b FN([)20
b(f)p FP(A)1453 2510 y FO(i)1481 2547 y FP(\022)s(;)15
b(C)1639 2510 y FO(i)1667 2547 y FP(\022)s FN(g)30 b
FT(is)f FN(K)q FT(-consisten)m(t)1004 2685 y FN(\))83
b FP(S)25 b FN([)20 b(f)p FP(B)1459 2648 y FO(i)1487
2685 y FP(\022)s(;)15 b(C)1645 2648 y FO(i)1672 2685
y FP(\022)s FN(g)30 b FT(is)f FN(K)q FT(-consisten)m(t)j(b)m(y)e(the)h
(induction)d(h)m(yp)s(othesis)1004 2823 y FN(\))83 b
FP(S)25 b FN([)20 b(f)p FT(\()p FP(B)25 b FN(^)20 b FP(C)7
b FT(\))1702 2785 y FO(i)1730 2823 y FP(\022)s FN(g)30
b FT(is)f FN(K)q FT(-consisten)m(t)r FP(:)514 3063 y
FN(\017)46 b FT(The)37 b(sen)m(tence)h FP(X)44 b FT(=)36
b FN(8)p FP(x:A)h FT(and)g FP(Y)56 b FT(=)36 b FN(8)p
FP(x:B)5 b FT(,)39 b(where)d FP(A)h Ff(\032)f FP(B)42
b FT(with)36 b(the)h(induction)e(h)m(y-)605 3176 y(p)s(othesis)28
b(that)h(for)f(all)g FP(S)33 b FT(and)28 b FP(\022)j
FT(if)d FP(S)22 b FN([)16 b(f)p FP(A)2101 3143 y FO(i)2130
3176 y FP(\022)s FN(g)28 b FT(is)g FP(i)d FN($)g FP(j)5
b FT(-consisten)m(t)30 b(then)f(so)g(is)e FP(S)22 b FN([)16
b(f)p FP(Y)3684 3143 y FO(i)3712 3176 y FP(\022)s FN(g)p
FT(.)931 3381 y FP(S)25 b FN([)20 b(f)p FT(\()p FN(8)p
FP(x:A)1369 3343 y FO(i)1398 3381 y FT(\))p FP(\022)s
FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t)1014 3518
y FN(\))83 b FP(S)25 b FN([)20 b(f8)p FP(x:A)1591 3481
y FO(i)1619 3518 y FP(\022)1666 3532 y FL(\026)-39 b
FO(x)1706 3518 y FN(g)31 b FT(is)e FN(K)q FT(-consisten)m(t)1014
3656 y FN(\))83 b FP(S)25 b FN([)20 b(f)p FP(A)1463 3619
y FO(i)1492 3656 y FP(\022)1539 3670 y FL(\026)-39 b
FO(x)1578 3656 y FN(f)p FP(x)26 b FN(!)f FP(c)p FN(gg)32
b FT(is)d FN(K)q FT(-consisten)m(t)j(for)e(ev)m(ery)h(closed)f(term)h
FP(c)1014 3794 y FN(\))83 b FP(S)25 b FN([)20 b(f)p FP(B)1469
3756 y FO(i)1497 3794 y FP(\022)1544 3808 y FL(\026)-39
b FO(x)1584 3794 y FN(f)p FP(x)25 b FN(!)g FP(c)p FN(gg)32
b FT(is)d FN(K)q FT(-consisten)m(t)j(for)e(ev)m(ery)h(closed)f(term)h
FP(c)1370 3932 y FT(b)m(y)f(the)h(induction)d(h)m(yp)s(othesis)1014
4070 y FN(\))83 b FP(S)25 b FN([)20 b(f)p FT(\()p FN(8)p
FP(x:B)1632 4032 y FO(i)1660 4070 y FT(\))p FP(\022)s
FN(g)30 b FT(is)g FN(K)q FT(-consisten)m(t)q FP(:)605
4274 y FT(W)-8 b(e)32 b(th)m(us)e(conclude)g(that)h(if)e
FP(S)c FN([)20 b(f)p FP(X)1905 4241 y FO(i)1934 4274
y FP(\022)s FN(g)30 b FT(is)f FP(i)d FN($)f FP(j)5 b
FT(-consisten)m(t)32 b(then)e(so)g(is)g FP(S)25 b FN([)20
b(f)p FP(Y)3505 4241 y FO(i)3533 4274 y FP(\022)s FN(g)p
FT(.)108 b Ff(\004)519 4456 y FT(W)-8 b(e)32 b(are)e(no)m(w)h(ready)f
(to)h(pro)m(v)m(e)g(the)g(required)e(result.)378 4661
y FQ(Theorem)34 b(8.2)46 b FI(F)-7 b(or)43 b(every)e(sentenc)-5
b(e)42 b FP(X)49 b FI(and)43 b FP(Y)20 b FI(,)43 b(if)f
FP(X)49 b Ff(\032)2580 4628 y FK(\003)2661 4661 y FP(Y)62
b FI(then)42 b FN(f)p FP(X)3114 4628 y FO(i)3143 4661
y FP(;)15 b FN(:)p FP(Y)3317 4628 y FO(j)3353 4661 y
FN(g)43 b FI(is)e FP(i)h FN($)g FP(j)5 b FI(-)378 4774
y(inc)-5 b(onsistent.)378 4980 y FQ(Pro)s(of)p FT(:)31
b(Supp)s(ose)d(that)i FP(X)j Ff(\032)1448 4947 y FK(\003)1513
4980 y FP(Y)20 b FT(,)30 b(that)g(is,)g(there)g(is)e(a)j(\014nite)d
(sequence)j(of)f(sen)m(tences)h FP(Z)3522 4994 y FO(x)3595
4980 y FT(where)378 5092 y FP(x)25 b FN(2)g(f)p FT(1)p
FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p FN(g)30 b FT(suc)m(h)g(that)1468
5297 y FP(X)i FT(=)25 b FP(Z)1733 5311 y FL(1)1798 5297
y Ff(\032)g FP(Z)1986 5311 y FL(2)2051 5297 y Ff(\032)g
FN(\001)15 b(\001)g(\001)27 b Ff(\032)e FP(Z)2497 5311
y FO(n)2569 5297 y FT(=)g FP(Y)378 5501 y FT(and)30 b(let)g(us)g
(assume)g(that)h FN(f)p FP(X)1441 5468 y FO(i)1470 5501
y FP(;)15 b FN(:)p FP(Y)1644 5468 y FO(j)1680 5501 y
FN(g)31 b FT(is)f FP(i)25 b FN($)g FP(j)5 b FT(-consisten)m(t.)42
b(Note)32 b(that)f(for)f(all)f(substitutions)f FP(\022)1802
5705 y(X)1884 5668 y FO(i)1938 5705 y FT(=)d FP(Z)2103
5668 y FO(i)2096 5728 y FL(1)2161 5705 y FT(=)g FP(Z)2326
5668 y FO(i)2319 5728 y FL(1)2358 5705 y FP(\022)p eop
%%Page: 157 167
157 166 bop 378 5 a FF(CHAPTER)30 b(8.)122 b(CHECKING)29
b(STR)m(UCTURED)h(JUSTIFICA)-8 b(TIONS)650 b FT(157)378
396 y(as)31 b FP(X)37 b FT(is)30 b(a)g(sen)m(tence.)42
b(No)m(w)32 b(if)d FN(f)p FP(Z)1571 363 y FO(i)1564 421
y FL(1)1604 396 y FP(\022)s(;)15 b FN(:)p FP(Y)1823 363
y FO(j)1860 396 y FN(g)30 b FT(is)g FP(i)25 b FN($)g
FP(j)5 b FT(-consisten)m(t)32 b(then)1314 601 y FN(f)p
FP(Z)1428 563 y FO(i)1421 623 y FL(2)1461 601 y FP(\022)s(;)15
b FN(:)p FP(Y)1680 563 y FO(j)1716 601 y FN(g)61 b FT(is)30
b FP(i)25 b FN($)h FP(j)5 b FT(-consisten)m(t)31 b(b)m(y)g(Lemma)f(8.1)
1140 739 y FN(\))83 b(f)p FP(Z)1428 701 y FO(i)1421 761
y FL(3)1461 739 y FP(\022)s(;)15 b FN(:)p FP(Y)1680 701
y FO(j)1716 739 y FN(g)61 b FT(is)30 b FP(i)25 b FN($)h
FP(j)5 b FT(-consisten)m(t)31 b(b)m(y)g(Lemma)f(8.1)1405
854 y(.)1405 887 y(.)1405 920 y(.)1140 1058 y FN(\))83
b(f)p FP(Z)1428 1021 y FO(i)1421 1081 y(n)1468 1058 y
FP(\022)s(;)15 b FN(:)p FP(Y)1687 1021 y FO(j)1724 1058
y FN(g)61 b FT(is)29 b FP(i)d FN($)f FP(j)5 b FT(-consisten)m(t.)378
1262 y(where)24 b FP(\022)j FT(is)d(an)m(y)i(substitution)c(whic)m(h)i
(maps)g(all)g(the)h(free)g(v)-5 b(ariables)24 b(in)f
FP(Z)2920 1276 y FO(x)2989 1262 y FT(to)j(some)f(closed)g(terms.)378
1375 y(Again)30 b(w)m(e)h(note)g(that)1800 1488 y FP(Y)1873
1451 y FO(i)1926 1488 y FT(=)25 b FP(Z)2091 1451 y FO(i)2084
1511 y(n)2156 1488 y FT(=)g FP(Z)2321 1451 y FO(i)2314
1511 y(n)2361 1488 y FP(\022)378 1655 y FT(for)31 b(all)e
FP(\022)k FT(since)d FP(Y)51 b FT(is)30 b(a)h(sen)m(tence.)43
b(But)31 b(the)g(statemen)m(t)i(that)e FN(f)p FP(Y)2690
1622 y FO(i)2718 1655 y FP(;)15 b FN(:)p FP(Y)2892 1622
y FO(j)2929 1655 y FN(g)31 b FT(is)f FP(i)c FN($)g FP(j)5
b FT(-consisten)m(t)32 b(is)378 1768 y(a)f(con)m(tradiction,)f(and)g
(therefore)h FN(f)p FP(X)1718 1735 y FO(i)1747 1768 y
FP(;)15 b FN(:)p FP(Y)1921 1735 y FO(j)1957 1768 y FN(g)31
b FT(m)m(ust)f(b)s(e)g FP(i)c FN($)f FP(j)5 b FT(-inconsisten)m(t.)636
b Ff(\004)519 1994 y FT(F)-8 b(or)27 b(completeness)f(w)m(e)h(giv)m(e)f
(the)g(corresp)s(ondence)g(b)s(et)m(w)m(een)g(implicit)d(deriv)-5
b(ation)25 b(and)h(incon-)378 2107 y(sistency)k(according)g(to)h(the)g
(connectabilit)m(y)f(relation)f FP(i)d FN($)f FP(j)36
b FT(in)29 b(the)i(follo)m(wing)d(theorem.)378 2317 y
FQ(Theorem)34 b(8.3)46 b(\(Chec)m(king)i Ff(\032)1610
2284 y FK(\003)1698 2317 y FQ(b)m(y)g(a)g(Coloured)g(Problem\))42
b FI(Given)h(two)h(sentenc)-5 b(es)44 b FP(A)378 2430
y FI(and)f FP(B)5 b FI(,)45 b(and)e(two)h(distinct)f(c)-5
b(olours)44 b FP(i)f FI(and)h FP(j)5 b FI(,)45 b(then)e
FP(A)h Ff(\032)2533 2397 y FK(\003)2616 2430 y FP(B)j
FI(if)42 b(and)h(only)h(if)e FN(f)p FP(A)3435 2397 y
FO(i)3464 2430 y FP(;)15 b FN(:)p FP(B)3639 2397 y FO(j)3675
2430 y FN(g)43 b FI(is)378 2543 y FP(i)26 b FN($)f FP(j)5
b FI(-inc)-5 b(onsistent.)378 2753 y FQ(Pro)s(of)p FT(:)31
b(By)g(theorems)g(8.1)g(and)f(8.2.)2074 b Ff(\004)378
2996 y FG(8.2.4)112 b(The)36 b(Undecidabilit)m(y)d(of)i(First-Order)g
(Implicit)c(and)36 b(Explicit)c(Deriv)-6 b(a-)720 3112
y(tions)378 3284 y FT(In)27 b(theorem)h(7.9)h(in)d(section)i(7.6)h(w)m
(e)f(ha)m(v)m(e)h(seen)e(that)i(the)f(v)-5 b(alidit)m(y)26
b(of)h(ev)m(ery)i(\014rst-order)e(sen)m(tence)378 3397
y FP(X)49 b FT(is)41 b(equiv)-5 b(alen)m(t)42 b(to)g(the)h
FP(i)h FN($)h FP(j)5 b FT(-consistency)43 b(of)f FP(Y)2302
3364 y FO(i)2375 3397 y FN(\))i FP(Z)2579 3364 y FO(j)2657
3397 y FT(for)e(some)g(sen)m(tences)h FP(Y)62 b FT(and)41
b FP(Z)378 3510 y FT(and)31 b(distinct)f(colours)i FP(i)g
FT(and)f FP(j)5 b FT(.)45 b(By)32 b(theorem)h(8.3,)g(this)e(is)g(in)f
(turn)h(equiv)-5 b(alen)m(t)31 b(to)i(whether)e(the)378
3622 y(sen)m(tence)i FP(Z)k FT(can)32 b(b)s(e)f(implicitly)c(deriv)m
(ed)k(from)g FP(Y)20 b FT(.)43 b(As)32 b(a)f(consequence)h(of)g(these)g
(results)e(w)m(e)i(get)378 3735 y(the)f(undecidabilit)m(y)26
b(of)31 b(implicit)c(deriv)-5 b(ations.)378 3946 y FQ(Theorem)34
b(8.4)h(\(Undecidabilit)m(y)h(of)f Ff(\032)1953 3913
y FK(\003)1992 3946 y FQ(\))45 b FI(The)c(pr)-5 b(oblem)43
b(of)e(che)-5 b(cking)40 b(whether)i FP(X)48 b Ff(\032)3676
3913 y FK(\003)3755 3946 y FP(Y)378 4058 y FI(for)33
b(al)5 b(l)33 b(\014rst-or)-5 b(der)35 b(sentenc)-5 b(es)33
b FP(X)40 b FI(and)33 b FP(Y)53 b FI(is)32 b(unde)-5
b(cidable.)378 4269 y FQ(Pro)s(of)p FT(:)31 b(F)-8 b(ollo)m(ws)31
b(from)e(the)i(undecidabilit)m(y)26 b(of)31 b(the)f(v)-5
b(alidit)m(y)29 b(problem)f(of)j(pure)e(\014rst-order)g(logic)378
4382 y(and)h(theorems)g(7.9)i(and)e(8.3.)2349 b Ff(\004)519
4607 y FT(Since)39 b(the)h(de\014nition)e(of)i(the)h(explicit)d
(\014rst-order)h(deriv)-5 b(ations)39 b(giv)m(en)h(in)f(section)h
(6.4.2)i(is)378 4720 y(based)h(on)g(the)h(de\014nition)d(of)j(implicit)
c(deriv)-5 b(ations,)46 b(it)d(follo)m(ws)f(from)h(the)h(undecidabilit)
m(y)c(of)378 4833 y(implicit)27 b(deriv)-5 b(ations)29
b(that)i(the)g(v)-5 b(alidit)m(y)28 b(of)j(explicit)e(deriv)-5
b(ation)29 b(is)g(also)h(undecidable.)378 5043 y FQ(Theorem)k(8.5)h
(\(Undecidabilit)m(y)h(of)f Ff( )p FQ(\))45 b FI(The)35
b(pr)-5 b(oblem)37 b(of)f(che)-5 b(cking)35 b(whether)h
FP(X)i Ff( )30 b FP(C)41 b FI(for)378 5156 y(an)33 b(arbitr)-5
b(ary)35 b(structur)-5 b(e)g(d)34 b(expr)-5 b(ession)34
b FP(X)40 b FI(and)34 b(\014rst-or)-5 b(der)35 b(sentenc)-5
b(e)32 b FP(C)39 b FI(is)33 b(unde)-5 b(cidable.)378
5366 y FQ(Pro)s(of)p FT(:)31 b(F)-8 b(ollo)m(ws)31 b(from)f(theorem)h
(8.4)g(and)f(de\014nition)e(6.3)j(\(page)h(109\).)862
b Ff(\004)519 5592 y FT(As)25 b(a)g(particular)f(case)h(of)h(theorem)f
(8.5,)i(the)e(v)-5 b(alidit)m(y)24 b(of)h(structured)f(straigh)m(tforw)
m(ard)g(justi\014-)378 5705 y(cations)29 b(\(de\014nition)e(6.4,)j
(page)f(109\))i(is)c(undecidable.)38 b(As)29 b(a)g(result,)f(it)g(is)g
(necessarily)g(to)h(imp)s(ose)p eop
%%Page: 158 168
158 167 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(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
(w)f(ho)m(w)h(a)g(coloured)f(problem)378 1737 y(can)k(b)s(e)e
(constructed)i(from)f(a)g(structured)f(straigh)m(tforw)m(ard)i
(justi\014cation)d(suc)m(h)i(that)h(the)g(result-)378
1850 y(ing)21 b(problem)g(is)g(inconsisten)m(t)g(if)g(and)h(only)f(if)g
(the)h(justi\014cation)f(is)g(v)-5 b(alid.)37 b(This)20
b(construction)i(giv)m(es)378 1963 y(a)31 b(mec)m(hanism)e(for)i
(restricting)e(the)h(pro)s(of)g(searc)m(h)h(required)e(for)h(c)m(hec)m
(king)h(suc)m(h)f(justi\014cations.)519 2076 y(The)i(construction)g(of)
g(a)h(coloured)f(problem)f(giv)m(en)h(in)f(this)g(section)i(requires)d
(the)j(notion)f(of)378 2189 y(structured)k(expressions)g(whose)h(form)m
(ulae)g(are)h(coloured.)61 b(Coloured)36 b(structured)g(expressions)378
2302 y(are)31 b(in)m(tro)s(duced)e(in)g(the)h(follo)m(wing)f
(de\014nition.)378 2489 y FQ(De\014nition)35 b(8.1)46
b(\(Coloured)30 b(Structured)f(Expressions\))e FT(A)f(coloured)f
(structured)g(expres-)378 2602 y(sion)30 b(is)h(a)h(structured)e
(expression)g(constructed)i(from)f(coloured)g(\014rst-order)g(sen)m
(tences.)45 b(W)-8 b(e)32 b(ex-)378 2714 y(tend)25 b(the)h
(de\014nition)d(and)i(notation)g(of)h(the)g(colouring)e(mapping)g(in)g
(de\014nition)f(7.6)k(to)f(structured)378 2827 y(expressions)j(as)i
(follo)m(ws:)1591 3032 y(\()p Fv(X)50 b Fw(on)43 b Fv(Y)19
b FT(\))1978 2994 y FO(i)2089 3032 y FT(=)83 b Fv(X)2319
3001 y Fq(i)2390 3032 y Fw(on)42 b Fv(Y)2587 3001 y Fq(i)1548
3169 y FT(\()p Fv(X)50 b Fw(and)42 b Fv(Y)19 b FT(\))1978
3132 y FO(i)2089 3169 y FT(=)83 b Fv(X)2319 3139 y Fq(i)2390
3169 y Fw(and)42 b Fv(Y)2631 3139 y Fq(i)1504 3307 y
FT(\()p Fv(X)50 b Fw(then)42 b Fv(Y)19 b FT(\))1978 3270
y FO(i)2089 3307 y FT(=)83 b Fv(X)2319 3277 y Fq(i)2390
3307 y Fw(then)42 b Fv(Y)2674 3277 y Fq(i)378 3511 y
FT(for)29 b(ev)m(ery)i(colour)e FP(i)h FT(and)f(structured)g
(expressions)f FP(X)37 b FT(and)29 b FP(Y)20 b FT(.)41
b(The)29 b(notions)g(of)h(recolouring)e(and)378 3624
y(isomorphism)i(b)m(y)j(renaming)f(colours)h(giv)m(en)g(in)e(c)m
(hapter)j(7)g(can)f(also)g(b)s(e)f(extended)i(to)f(coloured)378
3737 y(structured)c(expressions.)2469 b Ff(\003)519 3924
y FT(W)-8 b(e)32 b(also)e(de\014ne)g(a)g(coloured)g(structured)g
(problem)e(as)j(follo)m(ws.)378 4111 y FQ(De\014nition)k(8.2)h
(\(Coloured)e(Structured)h(Problem\))45 b FT(A)26 b(coloured)f
(structured)f(problem)g(is)378 4223 y(a)f(pair)e(\()p
FP(S;)15 b FN(K)q FT(\))24 b(where)e FP(S)27 b FT(is)22
b(a)h(set)g(of)g(coloured)f(structured)f(expressions)g(and)h
FN(K)i FT(is)e(a)g(connectabilit)m(y)378 4336 y(relation.)3050
b Ff(\003)378 4523 y FT(Note)35 b(that)g(since)e(the)h(set)g(of)g
(coloured)g(structured)f(expressions)f(includes)g(the)i(set)g(of)g
(coloured)378 4636 y(sen)m(tences)27 b(\(since)e(a)h(sen)m(tence)h(is)e
(a)h(structured)f(expression\),)h(\014rst-order)f(coloured)g(problems)f
(are)378 4749 y(a)31 b(sp)s(ecial)e(case)i(of)g(coloured)f(structured)f
(problems.)519 4862 y(W)-8 b(e)39 b(no)m(w)e(ha)m(v)m(e)i(a)f(lo)s(ok)f
(at)i(ho)m(w)e(a)h(coloured)f(problem)f(can)i(b)s(e)f(constructed)h
(from)f(a)h(giv)m(en)378 4975 y(structured)i(justi\014cation.)73
b(In)40 b(the)i(ligh)m(t)e(of)i(prop)s(osition)d(6.5)j(whic)m(h)e
(states)j(that)f(structured)378 5087 y(justi\014cations)24
b(in)m(v)m(olving)h(the)h Fw(then)e FT(op)s(erator)j(can)f(b)s(e)f
(transformed)h(in)m(to)g(equiv)-5 b(alen)m(t)25 b(ones)h(whic)m(h)378
5200 y(do)35 b(not)g(con)m(tain)h(it,)g(w)m(e)g(de\014ne)e(the)h
(required)f(construction)g(of)i(coloured)e(problems)g(according)378
5313 y(to)c(justi\014cations)e(whic)m(h)g(con)m(tain)i(only)f(the)g
Fw(on)g FT(and)g Fw(and)f FT(op)s(erators.)41 b(The)29
b(construction)g(is)f(done)378 5426 y(in)h(t)m(w)m(o)j(steps:)514
5592 y FN(\017)46 b FT(A)25 b(coloured)e(structured)g(problem)g(is)g
(constructed)i(from)f(the)g(giv)m(en)g(justi\014cation,)g(as)h(giv)m
(en)605 5705 y(b)m(y)30 b(de\014nition)f(8.3)i(b)s(elo)m(w)f(whic)m(h)f
(in)m(tro)s(duces)g(the)h(relation)g FN(\))2821 5719
y FL(c)2856 5705 y FT(;)p eop
%%Page: 159 169
159 168 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(159)514
396 y FN(\017)46 b FT(The)32 b(coloured)f(structured)g(problem)g(is)g
(transformed)g(in)m(to)h(a)g(\014rst-order)f(coloured)h(prob-)605
509 y(lem.)42 b(This)29 b(transformation)h(is)g(giv)m(en)h(in)f
(de\014nition)e(8.4)k(whic)m(h)e(in)m(tro)s(duces)g(the)h(relations)605
622 y FN(!)696 636 y FL(c)762 622 y FT(and)f FN(!)1030
589 y FK(\003)1030 645 y FL(c)1069 622 y FT(.)378 808
y(The)g(\014rst)f(step)i(is)e(no)m(w)i(giv)m(en)f(in)f(the)i(follo)m
(wing)e(de\014nition.)378 1019 y FQ(De\014nition)35 b(8.3)46
b(\(Structured)35 b(Justi\014cations)g(to)g(Coloured)g(Structured)g
(Problems\))378 1132 y FT(Let)27 b FP(C)34 b FT(b)s(e)26
b(a)i(sen)m(tence)g(and)e FP(P)40 b FT(b)s(e)27 b(a)g(structured)f
(expression.)38 b(The)27 b(justi\014ed)e(conclusion)h
Fv(C)49 b Fw(by)43 b Fv(P)378 1245 y FT(can)31 b(b)s(e)e(transformed)h
(to)h(the)g(coloured)f(structured)f(problem)g(\()p FN(f)p
FP(P)2754 1212 y FO(i)2783 1245 y FP(;)15 b FN(:)p FP(C)2956
1212 y FO(j)2992 1245 y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5
b FT(\),)32 b(and)e(write)1434 1449 y(\()p Fv(C)50 b
Fw(by)43 b Fv(P)12 b FT(\))25 b FN(\))1925 1463 y FL(c)1986
1449 y FT(\()p FN(f)p FP(P)2137 1411 y FO(i)2166 1449
y FP(;)15 b FN(:)p FP(C)2339 1411 y FO(j)2375 1449 y
FT(\))p FN(g)p FP(;)g(i)27 b FN($)e FP(j)5 b FT(\))p
FP(:)986 b Ff(\003)519 1659 y FT(The)31 b(second)g(step)g(is)g(giv)m
(en)g(b)m(y)g(breaking)f(up)g(the)i(coloured)f(structured)f
(expressions)g(in)g(the)378 1772 y(coloured)g(structured)f(problem)g
(according)h(to)h(the)g(follo)m(wing)e(rules.)378 1983
y FQ(De\014nition)35 b(8.4)46 b(\(Con)m(v)m(ergence)c(of)g(Structured)f
(Coloured)h(Problems\))36 b FT(The)g(relation)378 2096
y FN(!)469 2110 y FL(c)542 2096 y FT(on)h(coloured)g(structured)g
(problems)f(is)g(de\014ned)h(as)g(the)h(smallest)f(relation)f
(satisfying)h(the)378 2208 y(follo)m(wing)29 b(rules:)514
2394 y FN(\017)46 b FT(F)-8 b(or)33 b(all)e(structured)h(expressions)e
FP(X)40 b FT(and)32 b FP(Y)20 b FT(,)32 b(colours)g FP(i)p
FT(,)h(sets)g FP(S)k FT(of)c(coloured)e(expressions)605
2507 y(and)f(connectabilit)m(y)g(relations)f FN(K)q FT(:)1204
2712 y(\()p FP(S)d FN([)20 b(f)p FT(\()p Fv(X)50 b Fw(on)43
b Fv(Y)19 b FT(\))1834 2674 y FO(i)1862 2712 y FN(g)p
FP(;)c FN(K)q FT(\))27 b FN(!)2170 2726 y FL(c)2231 2712
y FT(\()p FP(S)e FN([)20 b(f)p Fv(X)2549 2674 y FO(i)2577
2712 y FP(;)15 b Fv(Y)2684 2674 y FO(j)2721 2712 y FN(g)p
FP(;)g FN(K)22 b([)e FP(i)26 b FN($)f FP(j)5 b FT(\))605
2916 y(where)30 b FP(j)36 b FT(is)29 b(new)h(to)h(\()p
FP(S)26 b FN([)20 b(f)p FT(\()p Fv(X)50 b Fw(on)43 b
Fv(Y)19 b FT(\))1960 2883 y FO(i)1988 2916 y FN(g)p FP(;)c
FN(K)q FT(\).)514 3103 y FN(\017)46 b FT(F)-8 b(or)33
b(all)e(structured)h(expressions)e FP(X)40 b FT(and)32
b FP(Y)20 b FT(,)32 b(colours)g FP(i)p FT(,)h(sets)g
FP(S)k FT(of)c(coloured)e(expressions)605 3216 y(and)f(connectabilit)m
(y)g(relations)f FN(K)q FT(:)1162 3420 y(\()p FP(S)c
FN([)20 b(f)p FT(\()p Fv(X)51 b Fw(and)42 b Fv(Y)19 b
FT(\))1835 3382 y FO(i)1863 3420 y FN(g)p FP(;)c FN(K)q
FT(\))27 b FN(!)2171 3434 y FL(c)2232 3420 y FT(\()p
FP(S)f FN([)19 b(f)p Fv(X)2550 3382 y FO(i)2578 3420
y FP(;)c Fv(Y)2685 3382 y FO(j)2722 3420 y FN(g)p FP(;)g
FN(K)23 b([)c(K)3049 3382 y FL(\()p FO(i)p FK(!)p FO(j)t
FL(\))3236 3420 y FT(\))605 3624 y(where)30 b FP(j)36
b FT(is)29 b(new)h(to)h(\()p FP(S)26 b FN([)20 b(f)p
FT(\()p Fv(X)50 b Fw(and)43 b Fv(Y)18 b FT(\))2003 3591
y FO(i)2032 3624 y FN(g)p FP(;)d FN(K)q FT(\).)378 3810
y(W)-8 b(e)33 b(denote)f(the)f(re\015exiv)m(e)h(transitiv)m(e)e
(closure)h(of)h FN(!)2251 3824 y FL(c)2318 3810 y FT(b)m(y)f
FN(!)2536 3777 y FK(\003)2536 3832 y FL(c)2575 3810 y
FT(.)44 b(W)-8 b(e)33 b(sa)m(y)f(that)g(a)g(coloured)f(struc-)378
3923 y(tured)45 b(problem)g(\()p FP(S;)15 b FN(K)q FT(\))47
b(con)m(v)m(erges)h(to)f(the)f(coloured)g(problem)e(\()p
FP(S)2856 3890 y FK(0)2880 3923 y FP(;)15 b FN(K)2990
3890 y FK(0)3014 3923 y FT(\),)50 b(and)c(denote)g(it)g(b)m(y)378
4036 y(\()p FP(S;)15 b FN(K)q FT(\))43 b FN(+)713 4050
y FL(c)789 4036 y FT(\()p FP(S)885 4003 y FK(0)908 4036
y FP(;)15 b FN(K)1018 4003 y FK(0)1042 4036 y FT(\),)43
b(if)c(\()p FP(S;)15 b FN(K)q FT(\))42 b FN(!)1607 4003
y FK(\003)1607 4058 y FL(c)1688 4036 y FT(\()p FP(S)1784
4003 y FK(0)1807 4036 y FP(;)15 b FN(K)1917 4003 y FK(0)1942
4036 y FT(\))40 b(and)f(there)h(is)f(no)g(coloured)h(structured)e
(problem)378 4149 y(\()p FP(S)474 4116 y FK(00)517 4149
y FP(;)15 b FN(K)627 4116 y FK(0)q(0)670 4149 y FT(\))31
b(for)f(whic)m(h)f(\()p FP(S)1231 4116 y FK(0)1255 4149
y FP(;)15 b FN(K)1365 4116 y FK(0)1389 4149 y FT(\))26
b FN(!)1541 4163 y FL(c)1601 4149 y FT(\()p FP(S)1697
4116 y FK(0)q(0)1740 4149 y FP(;)15 b FN(K)1850 4116
y FK(0)q(0)1893 4149 y FT(\).)1804 b Ff(\003)378 4359
y FQ(De\014nition)35 b(8.5)h(\(Breaking)f(Expressions)h(Up\))45
b FT(Let)33 b Fv(X)38 b FT(b)s(e)31 b(some)h Fw(on)o
FT(-expression)f Fv(Y)62 b Fw(on)43 b Fv(Z)6 b FT(,)378
4472 y(or)30 b(some)h Fw(and)o FT(-expression)e Fv(Y)63
b Fw(and)42 b Fv(Z)36 b FT(suc)m(h)30 b(that)1399 4676
y(\()p FP(S)c FN([)20 b(f)p Fv(X)1717 4639 y FO(i)1746
4676 y FN(g)p FP(;)15 b FN(K)q FT(\))27 b FN(!)2054 4690
y FL(c)2114 4676 y FT(\()p FP(S)f FN([)20 b(f)p Fv(Y)2424
4639 y FO(i)2452 4676 y FP(;)15 b Fv(Z)2555 4639 y FO(j)2591
4676 y FN(g)p FP(;)g FN(K)2746 4639 y FK(0)2771 4676
y FT(\))378 4881 y(for)25 b(some)h(set)h(of)e(coloured)g(structured)g
(expressions)f FP(S)5 b FT(,)27 b(connectabilit)m(y)e(relations)g
FN(K)i FT(and)e FN(K)3632 4848 y FK(0)3681 4881 y FT(and)378
4994 y(colours)35 b FP(i)i FT(and)e FP(j)5 b FT(,)38
b(then)e(w)m(e)h(sa)m(y)f(that)h(the)f(coloured)g(structured)f
(expression)g Fv(X)3262 4961 y FO(i)3326 4994 y FT(is)g(brok)m(en)h(up)
378 5106 y(in)m(to)g Fv(Y)635 5073 y FO(i)699 5106 y
FT(and)g Fv(Z)945 5073 y FO(j)1017 5106 y FT(b)m(y)g(the)h(application)
e(of)h(the)h(relation)e FN(!)2490 5120 y FL(c)2525 5106
y FT(.)59 b(W)-8 b(e)37 b(sa)m(y)g(that)g Fv(Y)3204 5073
y FO(i)3268 5106 y FT(\(and)f(similarly)378 5219 y Fv(Z)441
5186 y FO(j)477 5219 y FT(\))30 b(has)e(b)s(een)h(brok)m(en)g(up)f
(from)h Fv(X)1632 5186 y FO(i)1689 5219 y FT(b)m(y)g(the)g(application)
f(of)h FN(!)2631 5233 y FL(c)2667 5219 y FT(.)40 b(W)-8
b(e)30 b(also)f(sa)m(y)h(that)g(a)f(coloured)378 5332
y(structured)g(expression)g Fv(U)40 b FT(has)30 b(b)s(een)f(brok)m(en)i
(up)e(from)h Fv(V)49 b FT(b)m(y)30 b(some)h(applications)e(of)h
FN(!)3525 5346 y FL(c)514 5518 y FN(\017)46 b FT(if)30
b Fv(U)k FT(=)25 b Fv(V)18 b FT(,)31 b(or)514 5705 y
FN(\017)46 b FT(if)30 b Fv(U)39 b FT(has)30 b(b)s(een)f(brok)m(en)i(up)
e(from)h Fv(V)49 b FT(b)m(y)30 b(the)h(application)e(of)h
FN(!)2850 5719 y FL(c)2885 5705 y FT(,)h(or)p eop
%%Page: 160 170
160 169 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(160)1262
509 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 7.34561 3.6728 3.74124
} false /N@Son1 16 {InitRnode } NewNode end end
1262 509 a FP(S)55 b FN([)1484 509 y
tx@Dict begin tx@NodeDict begin {9.26236 2.73749 60.95769 30.47884
3.26244 } false /N@on1 16 {InitRnode } NewNode end end
1484 509 a
FN(f)p FT(\()p Fv(X)1684 509 y
tx@Dict begin tx@NodeDict begin {4.30554 0.0 10.49991 5.24995 2.15277
} false /N@on 16 {InitRnode } NewNode end end
1684 509 a Fw(on)43 b
Fv(Y)18 b FT(\))1916 476 y FO(i)1945 509 y FN(g)2172
509 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 7.34561 3.6728 3.74124
} false /N@Sand1 16 {InitRnode } NewNode end end
2172 509 a FP(S)56 b FN([)2395 509 y
tx@Dict begin tx@NodeDict begin {9.26236 2.73749 66.20764 33.10382
3.26244 } false /N@and1 16 {InitRnode } NewNode end end
2395 509 a
FN(f)p FT(\()p Fv(X)2594 509 y
tx@Dict begin tx@NodeDict begin {6.11111 0.0 15.74986 7.87492 3.05556
} false /N@and 16 {InitRnode } NewNode end end
2594 509 a Fw(and)43 b
Fv(Y)18 b FT(\))2870 476 y FO(i)2899 509 y FN(g)1329
941 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 7.34561 3.6728 3.74124
} false /N@Son2 16 {InitRnode } NewNode end end
1329 941 a FP(S)55 b FN([)1551 941 y
tx@Dict begin tx@NodeDict begin {9.26236 2.73749 44.3564 22.17819
3.26244 } false /N@on2 16 {InitRnode } NewNode end end
1551 941 a
FN(f)1596 941 y
tx@Dict begin tx@NodeDict begin {6.83331 0.0 9.06943 4.53471 3.41666
} false /N@Xon 16 {InitRnode } NewNode end end
1596 941 a Fv(X)1672 908 y FO(i)1700
941 y FP(;)1771 941 y
tx@Dict begin tx@NodeDict begin {6.83331 0.0 8.02779 4.01389 3.41666
} false /N@Yon 16 {InitRnode } NewNode end end
1771 941 a Fv(Y)1838 908 y FO(j)1874
941 y FN(g)2263 941 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 7.34561 3.6728 3.74124
} false /N@Sand2 16 {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
} def /ArrowB { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.0
1.0 0 0 /N@Sand2 /N@Xand 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@Sand2 /N@Yand 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 1179
1339 a FQ(Fig.)18 b(18.)41 b FT(The)29 b(Application)g(of)h(the)h
(Relation)f FN(!)2942 1353 y FL(c)2977 1339 y FT(.)514
1721 y FN(\017)46 b Fv(U)h FT(has)39 b(b)s(een)e(brok)m(en)i(up)e(from)
h(some)h(coloured)f(structured)g(expression)f Fv(W)51
b FT(b)m(y)38 b(the)h(ap-)605 1834 y(plication)c(of)h
FN(!)1185 1848 y FL(c)1256 1834 y FT(and)f Fv(W)48 b
FT(has)36 b(b)s(een)f(brok)m(en)h(up)f(from)g Fv(V)55
b FT(b)m(y)36 b(some)g(applications)f(of)h(the)605 1946
y(relation)30 b FN(!)1030 1960 y FL(c)1065 1946 y FT(.)2667
b Ff(\003)519 2142 y FT(Note)34 b(that)g(in)d(the)i(rule)f(breaking)g
(up)g Fw(and)f FT(expressions,)i(the)g(colours)f(that)i(relate)f(with)f
(the)378 2255 y(new)j(colour)g FP(j)41 b FT(in)34 b FN(K)26
b([)d(K)1286 2222 y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))1508
2255 y FT(are)36 b(exactly)g(the)g(colours)f(that)h(relate)g(with)e
FP(i)i FT(in)e FN(K)q FT(,)j(whic)m(h)d(are)378 2368
y(also)c(the)h(colours)f(that)h(relate)f(with)g FP(i)g
FT(in)f FN(K)22 b([)e(K)2092 2335 y FL(\()p FO(i)p FK(!)p
FO(j)t FL(\))2278 2368 y FT(,)31 b(that)g(is)1050 2572
y FN(K)22 b([)e(K)1292 2535 y FL(\()p FO(i)p FK(!)p FO(j)t
FL(\))1504 2572 y FT(=)25 b FN(K)c([)f(f)p FT(\()p FP(j;)15
b(k)s FT(\))28 b FN(j)d FP(i)h FN(\030)2219 2586 y FK(K)2302
2572 y FP(k)s FN(g)21 b([)f(f)p FT(\()p FP(k)s(;)15 b(j)5
b FT(\))27 b FN(j)f FP(k)i FN(\030)2970 2586 y FK(K)3054
2572 y FP(i)p FN(g)p FP(:)378 2776 y FT(The)39 b(application)f(of)i
(the)g(relation)g FN(!)1767 2790 y FL(c)1842 2776 y FT(is)f
(illustrated)e(in)i(\014gure)g(18.)70 b(W)-8 b(e)41 b(also)f
(illustrate)e(the)378 2889 y(ab)s(o)m(v)m(e)32 b(de\014nitions)27
b(with)j(the)g(follo)m(wing)f(examples.)378 3085 y FQ(Example)34
b(8.1)h(\(Justi\014cations)g(to)g(Coloured)g(Problems\))489
3281 y FT(1.)46 b(A)31 b(coloured)f(problem)e(is)i(constructed)g(from)g
(a)h(justi\014ed)e(conclusion)f(of)j(the)g(form)653 3489
y FP(C)102 b FM(by)47 b FP(A)h FM(on)f FP(B)5 b FM(;)605
3697 y FT(where)30 b FP(A)p FT(,)h FP(B)k FT(and)29 b
FP(C)37 b FT(are)31 b(form)m(ulae)f(as)h(follo)m(ws:)1272
3901 y Fv(C)50 b Fw(by)43 b Fv(A)h Fw(on)f Fv(B)87 b
FN(\))1990 3915 y FL(c)2108 3901 y FT(\()p FN(f)p FT(\()p
Fv(A)45 b Fw(on)e Fv(B)t FT(\))2563 3864 y FO(i)2592
3901 y FP(;)15 b FN(:)p Fv(C)2758 3864 y FO(j)2795 3901
y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5 b FT(\))1899 4039
y FN(!)1990 4053 y FL(c)2108 4039 y FT(\()p FN(f)p Fv(A)2251
4002 y FO(i)2280 4039 y FP(;)15 b Fv(B)2387 4002 y FO(k)2430
4039 y FP(;)g FN(:)p Fv(C)2596 4002 y FO(j)2633 4039
y FN(g)p FP(;)g(k)29 b FN($)c FP(i)h FN($)f FP(j)5 b
FT(\))605 4243 y(where)34 b(the)g(colours)f FP(i)p FT(,)i
FP(j)40 b FT(and)33 b FP(k)k FT(are)e(distinct)d(from)h(eac)m(h)i
(other.)52 b(It)34 b(can)g(b)s(e)f(seen)h(that)h(if)605
4356 y FP(A)26 b FT(=)f(\()p FP(B)30 b FN(\))25 b FP(C)7
b FT(\),)30 b(then)g(the)h(\014nal)e(coloured)h(problem)f(is)g(of)i
(the)f(form)1555 4561 y(\()p FN(f)p FT(\()p Fv(B)f Fu(\))23
b Fv(C)6 b FT(\))1968 4523 y FO(i)1997 4561 y FP(;)15
b Fv(B)2104 4523 y FO(k)2147 4561 y FP(;)g FN(:)p Fv(C)2313
4523 y FO(j)2350 4561 y FN(g)p FP(;)g(k)29 b FN($)c FP(i)h
FN($)f FP(j)5 b FT(\))605 4765 y(whic)m(h)29 b(is)h(inconsisten)m(t.)
489 4947 y(2.)46 b(A)31 b(coloured)f(problem)e(is)i(constructed)g(from)
g(a)h(justi\014ed)e(conclusion)f(of)j(the)g(form)653
5155 y FP(C)102 b FM(by)47 b FP(A)h FM(and)f FP(B)5 b
FM(;)605 5363 y FT(where)30 b FP(A)p FT(,)h FP(B)k FT(and)29
b FP(C)37 b FT(are)31 b(form)m(ulae)f(as)h(follo)m(ws:)1244
5567 y Fv(C)50 b Fw(by)42 b Fv(A)i Fw(and)e Fv(B)88 b
FN(\))2005 5581 y FL(c)2123 5567 y FT(\()p FN(f)p FT(\()p
Fv(A)45 b Fw(and)d Fv(B)5 b FT(\))2622 5530 y FO(i)2650
5567 y FP(;)15 b FN(:)p Fv(C)2816 5530 y FO(j)2853 5567
y FN(g)p FP(;)g(i)26 b FN($)g FP(j)5 b FT(\))1914 5705
y FN(!)2005 5719 y FL(c)2123 5705 y FT(\()p FN(f)p Fv(A)2266
5668 y FO(i)2294 5705 y FP(;)15 b Fv(B)2402 5668 y FO(k)2445
5705 y FP(;)g FN(:)p Fv(C)2611 5668 y FO(j)2647 5705
y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s FN(g)28 b($)d FP(j)5
b FT(\))p eop
%%Page: 161 171
161 170 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(161)605
396 y(where)34 b(the)g(colours)f FP(i)p FT(,)i FP(j)40
b FT(and)33 b FP(k)k FT(are)e(distinct)d(from)h(eac)m(h)i(other.)52
b(It)34 b(can)g(b)s(e)f(seen)h(that)h(if)605 509 y FP(C)d
FT(=)25 b(\()p FP(A)c FN(^)e FP(B)5 b FT(\),)31 b(then)f(the)h(\014nal)
e(coloured)g(problem)g(is)h(of)g(the)h(form)1557 714
y(\()p FN(f)p Fv(A)1700 676 y FO(i)1728 714 y FP(;)15
b Fv(B)1836 676 y FO(k)1878 714 y FP(;)g FT(\()p Fu(:)p
Fv(A)20 b Fu(_)f(:)p Fv(B)t FT(\))2321 676 y FO(j)2358
714 y FN(g)p FP(;)c FN(f)p FP(i;)g(k)s FN(g)28 b($)d
FP(j)5 b FT(\))605 918 y(whic)m(h)29 b(is)h(inconsisten)m(t.)489
1105 y(3.)46 b(The)30 b(follo)m(wing)f(justi\014ed)f(conclusion)653
1330 y FP(C)102 b FM(by)47 b FT(\()p FP(A)h FM(then)f
FP(B)5 b FT(\))48 b FM(on)f FP(D)s FM(;)605 1555 y FT(is)30
b(\014rst)f(transformed)h(in)m(to)g(the)h(equiv)-5 b(alen)m(t)653
1780 y FP(C)102 b FM(by)47 b FP(B)52 b FM(on)47 b FT(\()p
FP(A)i FM(on)e FP(D)s FT(\))p FM(;)605 2005 y FT(and)30
b(then)g(the)h(follo)m(wing)d(coloured)i(problem)f(is)h(constructed:)
959 2210 y Fv(C)50 b Fw(by)43 b Fv(B)48 b Fw(on)43 b
Ft(\()p Fv(A)h Fw(on)e Fv(D)r Ft(\))84 b FN(\))1987 2224
y FL(c)2105 2210 y FT(\()p FN(f)p Fv(B)49 b Fw(on)43
b Ft(\()p Fv(A)h Fw(on)e Fv(D)r Ft(\))2800 2172 y FO(i)2828
2210 y FP(;)15 b FN(:)p Fv(C)2994 2172 y FO(j)3031 2210
y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5 b FT(\))1896 2347
y FN(!)1987 2361 y FL(c)2105 2347 y FT(\()p FN(f)p Fv(B)2253
2310 y FO(i)2281 2347 y FP(;)15 b FT(\()p Fv(A)45 b Fw(on)e
Fv(D)r FT(\))2700 2310 y FO(k)2743 2347 y FP(;)15 b FN(:)p
Fv(C)2909 2310 y FO(j)2946 2347 y FN(g)p FP(;)g(k)29
b FN($)c FP(i)h FN($)f FP(j)5 b FT(\))1896 2485 y FN(!)1987
2499 y FL(c)2105 2485 y FT(\()p FN(f)p Fv(B)2253 2448
y FO(i)2281 2485 y FP(;)15 b Fv(A)2384 2448 y FO(k)2427
2485 y FP(;)g Fv(D)2538 2448 y FO(l)2564 2485 y FP(;)g
FN(:)p Fv(C)2730 2448 y FO(j)2767 2485 y FN(g)p FP(;)g(l)28
b FN($)d FP(k)k FN($)c FP(i)g FN($)g FP(j)5 b FT(\))605
2690 y(where)30 b(the)h(colours)f FP(i)p FT(,)g FP(j)5
b FT(,)32 b FP(k)h FT(and)d FP(l)i FT(are)f(distinct)e(from)h(eac)m(h)h
(other.)489 2877 y(4.)46 b(The)30 b(follo)m(wing)f(justi\014ed)f
(conclusion)653 3102 y FP(C)102 b FM(by)47 b FP(A)h FM(on)f
FT(\()p FP(B)53 b FM(and)47 b FP(D)s FT(\))p FM(;)605
3327 y FT(con)m(v)m(erges)33 b(to:)916 3531 y Fv(C)50
b Fw(by)42 b Fv(A)i Fw(on)f Ft(\()p Fv(B)48 b Fw(and)42
b Fv(D)r Ft(\))84 b FN(\))1987 3545 y FL(c)2105 3531
y FT(\()p FN(f)p FT(\()p Fv(A)45 b Fw(on)e Ft(\()p Fv(B)48
b Fw(and)42 b Fv(D)r Ft(\))q FT(\))2914 3494 y FO(i)2942
3531 y FP(;)15 b FN(:)p Fv(C)3108 3494 y FO(j)3145 3531
y FN(g)p FP(;)g(i)26 b FN($)g FP(j)5 b FT(\))1896 3669
y FN(!)1987 3683 y FL(c)2105 3669 y FT(\()p FN(f)p Fv(A)2248
3632 y FO(i)2276 3669 y FP(;)15 b FT(\()p Fv(B)49 b Fw(and)42
b Fv(D)r FT(\))2743 3632 y FO(k)2786 3669 y FP(;)15 b
FN(:)p Fv(C)2953 3632 y FO(j)2989 3669 y FN(g)p FP(;)g(k)29
b FN($)d FP(i)f FN($)g FP(j)5 b FT(\))1896 3807 y FN(!)1987
3821 y FL(c)2105 3807 y FT(\()p FN(f)p Fv(A)2248 3769
y FO(i)2276 3807 y FP(;)15 b Fv(B)2384 3769 y FO(k)2427
3807 y FP(;)g Fv(D)2538 3769 y FO(l)2564 3807 y FP(;)g
FN(:)p Fv(C)2730 3769 y FO(j)2767 3807 y FN(g)p FP(;)g
FN(f)p FP(k)s(;)g(l)r FN(g)27 b($)f FP(i)f FN($)g FP(j)5
b FT(\))378 4049 y Ff(\003)519 4261 y FT(The)30 b(follo)m(wing)f(prop)s
(osition)f(is)h(straigh)m(tforw)m(ard.)378 4449 y FQ(Prop)s(osition)36
b(8.2)46 b FI(Given)33 b(the)h(c)-5 b(olour)g(e)g(d)36
b(structur)-5 b(e)g(d)34 b(pr)-5 b(oblems)36 b FT(\()p
FP(S;)15 b FN(K)q FT(\))35 b FI(and)f FT(\()p FP(S)3241
4416 y FK(0)3264 4449 y FP(;)15 b FN(K)3374 4416 y FK(0)3399
4449 y FT(\))33 b FI(such)h(that)378 4562 y FT(\()p FP(S;)15
b FN(K)q FT(\))27 b FN(!)732 4529 y FK(\003)732 4584
y FL(c)796 4562 y FT(\()p FP(S)892 4529 y FK(0)916 4562
y FP(;)15 b FN(K)1026 4529 y FK(0)1050 4562 y FT(\))33
b FI(then)485 4749 y(1.)46 b FN(K)27 b(\022)e(K)867 4716
y FK(0)891 4749 y FI(,)32 b(and)485 4937 y(2.)46 b Fe(C)p
FT(\()p FP(S)5 b FT(\))26 b FN(\022)f Fe(C)p FT(\()p
FP(S)1066 4904 y FK(0)1089 4937 y FT(\))p FI(.)378 5125
y FQ(Pro)s(of)p FT(:)39 b(The)f(\014rst)f(part)h(of)g(the)g(curren)m(t)
g(prop)s(osition)e(follo)m(ws)h(from)g(the)h(fact)h(that)g(whenev)m(er)
378 5237 y(\()p FP(S;)15 b FN(K)q FT(\))32 b FN(!)737
5251 y FL(c)803 5237 y FT(\()p FP(S)899 5204 y FK(0)923
5237 y FP(;)15 b FN(K)1033 5204 y FK(0)1057 5237 y FT(\))34
b(then)g FN(K)e(\032)e(K)1609 5204 y FK(0)1633 5237 y
FT(.)51 b(The)33 b(second)h(part)f(follo)m(ws)g(from)g(the)h(fact)h
(that)f(whenev)m(er)378 5350 y(\()p FP(S;)15 b FN(K)q
FT(\))27 b FN(!)732 5364 y FL(c)793 5350 y FT(\()p FP(S)889
5317 y FK(0)912 5350 y FP(;)15 b FN(K)1022 5317 y FK(0)1046
5350 y FT(\))31 b(then)f Fe(C)p FT(\()p FP(S)1471 5317
y FK(0)1494 5350 y FT(\))c(=)f Fe(C)p FT(\()p FP(S)5
b FT(\))21 b FN([)e(f)p FP(j)5 b FN(g)32 b FT(where)e(the)h(colour)f
FP(j)36 b FT(is)29 b(new)h(to)h(\()p FP(S;)15 b FN(K)q
FT(\).)238 b Ff(\004)519 5576 y FT(W)-8 b(e)32 b(also)e(giv)m(e)h(the)f
(follo)m(wing)f(de\014nitions.)p eop
%%Page: 162 172
162 171 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(162)378
396 y FQ(De\014nition)35 b(8.6)h(\(Construction)e(of)i(a)e(Coloured)h
(Problem\))45 b FT(Giv)m(en)27 b(a)g(set)h FP(S)j FT(of)c(coloured)378
509 y(form)m(ulae,)36 b(and)e(a)i(connectabilit)m(y)e(relation)g
FN(K)q FT(,)j(w)m(e)e(sa)m(y)h(that)f(the)h(coloured)e(problem)f(\()p
FP(S;)15 b FN(K)q FT(\))37 b(is)378 622 y(constructed)31
b(from)f(the)g(justi\014ed)f(conclusion)f Fv(C)50 b Fw(by)43
b Fv(P)12 b FT(,)30 b(and)g(write)1726 827 y(\()p Fv(C)50
b Fw(by)43 b Fv(P)12 b FT(\))25 b FN(+)2182 841 y FL(c)2243
827 y FT(\()p FP(S;)15 b FN(K)q FT(\))378 1031 y(if)38
b(\()p Fv(C)50 b Fw(by)43 b Fv(P)12 b FT(\))39 b FN(\))975
1045 y FL(c)1050 1031 y FT(\()p FP(S)1146 998 y FK(0)1170
1031 y FP(;)15 b FN(K)1280 998 y FK(0)1304 1031 y FT(\))39
b(and)g(\()p FP(S)1660 998 y FK(0)1683 1031 y FP(;)15
b FN(K)1793 998 y FK(0)1817 1031 y FT(\))40 b FN(+)1948
1045 y FL(c)2023 1031 y FT(\()p FP(S;)15 b FN(K)q FT(\))40
b(for)f(some)g(coloured)g(structured)e(problem)378 1144
y(\()p FP(S)474 1111 y FK(0)498 1144 y FP(;)15 b FN(K)608
1111 y FK(0)632 1144 y FT(\).)41 b(W)-8 b(e)31 b(also)g(write)1649
1257 y(\()p Fv(C)50 b Fw(by)43 b Fv(P)11 b FT(\))26 b
FN(!)2140 1219 y FK(\003)2140 1279 y FL(c)2205 1257 y
FT(\()p FP(S)2301 1219 y FK(00)2343 1257 y FP(;)15 b
FN(K)2453 1219 y FK(0)q(0)2497 1257 y FT(\))p FP(;)378
1423 y FT(where)37 b(\()p FP(S)744 1390 y FK(00)786 1423
y FP(;)15 b FN(K)896 1390 y FK(0)r(0)940 1423 y FT(\))37
b(is)f(a)i(coloured)f(structured)f(problem,)h(and)g(sa)m(y)h(that)f
Fv(C)50 b Fw(by)43 b Fv(P)49 b FT(can)37 b(b)s(e)g(trans-)378
1536 y(formed)23 b(in)m(to)h(\()p FP(S)951 1503 y FK(0)q(0)994
1536 y FP(;)15 b FN(K)1104 1503 y FK(0)q(0)1147 1536
y FT(\))25 b(if)d(\()p Fv(C)51 b Fw(by)42 b Fv(P)12 b
FT(\))26 b FN(\))1775 1550 y FL(c)1836 1536 y FT(\()p
FP(S)1932 1503 y FK(0)1955 1536 y FP(;)15 b FN(K)2065
1503 y FK(0)2089 1536 y FT(\))25 b(and)e(\()p FP(S)2415
1503 y FK(0)2439 1536 y FP(;)15 b FN(K)2549 1503 y FK(0)2573
1536 y FT(\))25 b FN(!)2724 1503 y FK(\003)2724 1559
y FL(c)2789 1536 y FT(\()p FP(S)2885 1503 y FK(00)2928
1536 y FP(;)15 b FN(K)3038 1503 y FK(0)q(0)3081 1536
y FT(\))24 b(for)g(some)g(coloured)378 1649 y(structured)29
b(problem)g(\()p FP(S)1268 1616 y FK(0)1292 1649 y FP(;)15
b FN(K)1402 1616 y FK(0)1426 1649 y FT(\).)2271 b Ff(\003)378
1862 y FQ(De\014nition)35 b(8.7)h(\(Consistency)f(of)g(Structured)g
(Problems\))45 b FT(Giv)m(en)30 b(a)h(set)f FP(S)35 b
FT(of)30 b(coloured)378 1975 y(structured)25 b(expressions)f(and)h(a)h
(connectabilit)m(y)f(relation)g FN(K)q FT(,)i(the)f(coloured)f
(structured)f(problem)378 2088 y(\()p FP(S;)15 b FN(K)q
FT(\))33 b(is)d(said)g(to)i(b)s(e)f(consisten)m(t)g(if)g(whenev)m(er)g
(\()p FP(S;)15 b FN(K)q FT(\))28 b FN(+)2392 2102 y FL(c)2454
2088 y FT(\()p FP(S)2550 2055 y FK(0)2574 2088 y FP(;)15
b FN(K)2684 2055 y FK(0)2708 2088 y FT(\))31 b(then)g(the)h(coloured)e
(problem)378 2201 y(\()p FP(S)474 2168 y FK(0)498 2201
y FP(;)15 b FN(K)608 2168 y FK(0)632 2201 y FT(\))34
b(is)f(consisten)m(t.)53 b(Similarly)-8 b(,)32 b(\()p
FP(S;)15 b FN(K)q FT(\))36 b(is)d(said)g(to)i(b)s(e)e(inconsisten)m(t)h
(if)f(whenev)m(er)h(\()p FP(S;)15 b FN(K)q FT(\))33 b
FN(+)3793 2215 y FL(c)378 2313 y FT(\()p FP(S)474 2280
y FK(0)498 2313 y FP(;)15 b FN(K)608 2280 y FK(0)632
2313 y FT(\))31 b(then)f(\()p FP(S)1001 2280 y FK(0)1024
2313 y FP(;)15 b FN(K)1134 2280 y FK(0)1158 2313 y FT(\))31
b(is)f(inconsisten)m(t.)1947 b Ff(\003)378 2526 y FT(W)-8
b(e)41 b(will)d(sho)m(w)i(in)f(prop)s(osition)f(8.7)j(b)s(elo)m(w)f
(that)h(giv)m(en)f(some)g(coloured)g(structured)f(problem)378
2639 y(\()p FP(S;)15 b FN(K)q FT(\),)47 b(if)42 b(there)h(is)f(some)h
(coloured)f(problem)f(\()p FP(S)2209 2606 y FK(0)2232
2639 y FP(;)15 b FN(K)2342 2606 y FK(0)2367 2639 y FT(\))43
b(suc)m(h)f(that)h(\()p FP(S;)15 b FN(K)q FT(\))47 b
FN(+)3210 2653 y FL(c)3291 2639 y FT(\()p FP(S)3387 2606
y FK(0)3411 2639 y FP(;)15 b FN(K)3521 2606 y FK(0)3545
2639 y FT(\),)46 b(then)378 2752 y(\()p FP(S)474 2719
y FK(0)498 2752 y FP(;)15 b FN(K)608 2719 y FK(0)632
2752 y FT(\))47 b(is)f(consisten)m(t)h(if)e(and)i(only)e(if)h(\()p
FP(S)1965 2719 y FK(0)q(0)2008 2752 y FP(;)15 b FN(K)2118
2719 y FK(0)q(0)2161 2752 y FT(\))47 b(is)f(consisten)m(t)h(for)g(all)e
(coloured)i(problems)378 2865 y(\()p FP(S)474 2832 y
FK(00)517 2865 y FP(;)15 b FN(K)627 2832 y FK(0)q(0)670
2865 y FT(\))45 b(for)g(whic)m(h)f(\()p FP(S;)15 b FN(K)q
FT(\))51 b FN(+)1522 2879 y FL(c)1607 2865 y FT(\()p
FP(S)1703 2832 y FK(0)q(0)1746 2865 y FP(;)15 b FN(K)1856
2832 y FK(0)q(0)1899 2865 y FT(\).)85 b(As)45 b(a)h(result,)h(a)f
(coloured)e(structured)g(problem)378 2978 y(\()p FP(S;)15
b FN(K)q FT(\))36 b(is)d(consisten)m(t)h(if)f(and)g(only)h(if)f(\()p
FP(S;)15 b FN(K)q FT(\))32 b FN(+)2053 2992 y FL(c)2120
2978 y FT(\()p FP(S)2216 2945 y FK(0)2240 2978 y FP(;)15
b FN(K)2350 2945 y FK(0)2374 2978 y FT(\))34 b(holds)f(for)g
FI(some)42 b FT(consisten)m(t)34 b(coloured)378 3091
y(problem)i(\()p FP(S)835 3058 y FK(0)859 3091 y FP(;)15
b FN(K)969 3058 y FK(0)993 3091 y FT(\).)65 b(Similarly)-8
b(,)37 b(\()p FP(S;)15 b FN(K)q FT(\))39 b(is)f(inconsisten)m(t)f(if)g
(and)g(only)h(if)f(\()p FP(S;)15 b FN(K)q FT(\))40 b
FN(+)3319 3105 y FL(c)3392 3091 y FT(\()p FP(S)3488 3058
y FK(0)3511 3091 y FP(;)15 b FN(K)3621 3058 y FK(0)3646
3091 y FT(\))38 b(for)378 3203 y(some)31 b(inconsisten)m(t)f(coloured)h
(problem)e(\()p FP(S)1920 3170 y FK(0)1944 3203 y FP(;)15
b FN(K)2054 3170 y FK(0)2078 3203 y FT(\).)43 b(It)31
b(th)m(us)f(follo)m(ws)h(that)g(a)g(coloured)g(structured)378
3316 y(problem)e(is)g(inconsisten)m(t)h(if)f(and)h(only)f(if)h(it)f(is)
h(not)h(consisten)m(t.)519 3429 y(It)e(can)f(b)s(e)g(easily)f(c)m(hec)m
(k)m(ed)k(that)e(all)e(conclusions)g(justi\014ed)f(b)m(y)i(a)h
(structured)e(expression)h(can)378 3542 y(b)s(e)39 b(used)g(to)h
(construct)g(some)g(coloured)f(problem,)i(or)f(in)e(other)i(w)m(ords)f
(that)h(all)f(applications)378 3655 y(of)i(the)h(relation)e
FN(!)1095 3669 y FL(c)1172 3655 y FT(terminate)h(to)h(a)f(coloured)g
(problem.)71 b(This)40 b(is)g(giv)m(en)h(b)m(y)g(the)g(follo)m(wing)378
3768 y(prop)s(osition.)378 3981 y FQ(Prop)s(osition)36
b(8.3)g(\(T)-9 b(ermination)33 b(of)i FN(!)1957 3995
y FL(c)1993 3981 y FQ(\))45 b FI(F)-7 b(or)30 b(every)g(c)-5
b(olour)g(e)g(d)31 b(structur)-5 b(e)g(d)31 b(pr)-5 b(oblem)31
b FT(\()p FP(S;)15 b FN(K)q FT(\))378 4093 y FI(wher)-5
b(e)34 b FP(S)j FI(is)c(\014nite,)f(ther)-5 b(e)34 b(is)e(some)i(c)-5
b(olour)g(e)g(d)35 b(pr)-5 b(oblem)34 b FT(\()p FP(S)2433
4060 y FK(0)2457 4093 y FP(;)15 b FN(K)2567 4060 y FK(0)2591
4093 y FT(\))33 b FI(such)g(that)g FT(\()p FP(S;)15 b
FN(K)q FT(\))27 b FN(+)3370 4107 y FL(c)3431 4093 y FT(\()p
FP(S)3527 4060 y FK(0)3550 4093 y FP(;)15 b FN(K)3660
4060 y FK(0)3685 4093 y FT(\))p FI(.)378 4306 y FQ(Pro)s(of)p
FT(:)36 b(F)-8 b(or)35 b(the)h(purp)s(ose)d(of)i(this)e(pro)s(of,)j
(let)f(us)f(de\014ne)g(the)h FI(or)-5 b(der)46 b FT(of)35
b(a)g(coloured)g(structured)378 4419 y(problem)i(\()p
FP(S;)15 b FN(K)q FT(\))41 b(as)e(the)g(n)m(um)m(b)s(er)f(of)h(times)g
(the)g Fw(on)f FT(and)h Fw(and)e FT(op)s(erators)i(o)s(ccur)g(in)f
FP(S)5 b FT(.)66 b(It)40 b(can)378 4532 y(b)s(e)32 b(seen)i(from)e
(de\014nition)f(8.4)j(that)g(the)f(relation)f FN(!)2264
4546 y FL(c)2333 4532 y FT(is)g(applicable)f(to)j(a)f(coloured)g
(structured)378 4645 y(problem)40 b(if)h(and)g(only)f(if)h(its)g(order)
g(is)g(greater)i(than)e(0,)k(and)c(that)i(the)e(order)h(of)f(a)h
(coloured)378 4758 y(problem)27 b(decreases)j(at)g(ev)m(ery)g
(application)d(of)i FN(!)2134 4772 y FL(c)2170 4758 y
FT(.)40 b(As)29 b(a)h(result,)e(all)g(rep)s(etitiv)m(e)g(applications)g
(of)378 4871 y FN(!)469 4885 y FL(c)535 4871 y FT(terminate)k(in)d(a)j
(coloured)e(structured)h(problem)e(whose)i(order)g(is)f(0,)i(that)f
(is,)g(in)f(a)h(coloured)378 4983 y(problem.)3029 b Ff(\004)519
5209 y FT(The)29 b(follo)m(wing)e(prop)s(osition)g(sho)m(ws)i(that)h(a)
f(n)m(um)m(b)s(er)f(of)h(prop)s(erties)f(of)h(structured)f(coloured)378
5322 y(problems)h(are)h(preserv)m(ed)g(when)g(the)g(latter)h(are)g
(transformed)e(in)m(to)i(coloured)f(problems.)378 5535
y FQ(Prop)s(osition)36 b(8.4)46 b FI(F)-7 b(or)48 b(al)5
b(l)46 b(c)-5 b(olour)g(e)g(d)49 b(structur)-5 b(e)g(d)48
b(pr)-5 b(oblems)48 b FT(\()p FP(S;)15 b FN(K)q FT(\))48
b FI(and)f FT(\()p FP(S)3215 5502 y FK(0)3238 5535 y
FP(;)15 b FN(K)3348 5502 y FK(0)3373 5535 y FT(\))46
b FI(such)h(that)378 5648 y FT(\()p FP(S;)15 b FN(K)q
FT(\))27 b FN(!)732 5615 y FK(\003)732 5670 y FL(c)796
5648 y FT(\()p FP(S)892 5615 y FK(0)916 5648 y FP(;)15
b FN(K)1026 5615 y FK(0)1050 5648 y FT(\))p FI(,)33 b(then)p
eop
%%Page: 163 173
163 172 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(163)485
396 y FI(1.)46 b(if)33 b FN(K)26 b FT(=)f FN(K)q(d)p
FP(S)5 b FN(e)34 b FI(then)f FN(K)1402 363 y FK(0)1451
396 y FT(=)25 b FN(K)1617 363 y FK(0)1641 396 y FN(d)p
FP(S)1742 363 y FK(0)1766 396 y FN(e)p FI(;)485 584 y(2.)46
b(if)33 b Fe(C)p FT(\()p FP(S)5 b FT(\))25 b(=)g Fe(C)p
FT(\()p FN(K)q FT(\))34 b FI(then)f Fe(C)p FT(\()p FP(S)1586
551 y FK(0)1609 584 y FT(\))26 b(=)f Fe(C)p FT(\()p FN(K)1927
551 y FK(0)1951 584 y FT(\))p FI(;)485 772 y(3.)46 b(if)33
b(no)g(c)-5 b(olour)34 b(in)e FN(K)i FI(r)-5 b(elates)34
b(with)g(itself,)e(then)h(no)g(c)-5 b(olour)34 b(in)f
FN(K)2831 739 y FK(0)2887 772 y FI(r)-5 b(elates)34 b(with)g(itself.)
378 984 y FQ(Pro)s(of)p FT(:)29 b(The)e(\014rst)h(t)m(w)m(o)h(parts)e
(of)i(this)d(prop)s(osition)g(follo)m(w)h(from)h(the)g(fact)h(that)f
(at)h(eac)m(h)g(applica-)378 1097 y(tion)24 b(of)i(the)f(relation)f
FN(!)1228 1111 y FL(c)1288 1097 y FT(the)h(new)g(colour)f(\()p
FP(j)31 b FT(in)24 b(de\014nition)f(8.4\))j(in)m(tro)s(duced)e(b)m(y)g
(the)i(application)378 1210 y(is)j(in)m(tro)s(duced)f(to)j(b)s(oth)e
(the)h(set)g(and)f(the)h(connectabilit)m(y)g(relation)f(of)h(the)g
(coloured)f(structured)378 1323 y(problem.)58 b(The)37
b(last)f(part)h(follo)m(ws)f(from)g(the)h(fact)h(that)f(the)g(new)g
(colour)f(\()p FP(j)5 b FT(\))38 b(do)s(es)e(not)h(relate)378
1436 y(with)29 b(itself)g(in)g FN(K)981 1403 y FK(0)1035
1436 y FT(whenev)m(er)i(\()p FP(S;)15 b FN(K)q FT(\))27
b FN(!)1789 1450 y FL(c)1849 1436 y FT(\()p FP(S)1945
1403 y FK(0)1969 1436 y FP(;)15 b FN(K)2079 1403 y FK(0)2103
1436 y FT(\).)1594 b Ff(\004)378 1648 y FQ(Prop)s(osition)36
b(8.5)46 b FI(F)-7 b(or)41 b(every)f(formula)i FP(C)k
FI(and)c(structur)-5 b(e)g(d)41 b(expr)-5 b(ession)42
b FP(P)13 b FI(,)42 b(if)e(it)g(is)g(the)g(c)-5 b(ase)378
1761 y(that)34 b Fv(C)50 b Fw(by)42 b Fv(P)c FN(!)985
1728 y FK(\003)985 1784 y FL(c)1049 1761 y FT(\()p FP(S;)15
b FN(K)q FT(\))34 b FI(for)f(some)h(c)-5 b(olour)g(e)g(d)35
b(structur)-5 b(e)g(d)34 b(pr)-5 b(oblem)34 b FT(\()p
FP(S;)15 b FN(K)q FT(\))p FI(,)34 b(then)485 1949 y(1.)46
b FN(K)27 b FT(=)e FN(K)q(d)p FP(S)5 b FN(e)p FI(,)485
2137 y(2.)46 b Fe(C)p FT(\()p FP(S)5 b FT(\))26 b(=)f
Fe(C)p FT(\()p FN(K)q FT(\))p FI(,)33 b(and)485 2324
y(3.)46 b(for)33 b(al)5 b(l)34 b(c)-5 b(olour)34 b FP(i)25
b FN(2)g(K)q FI(,)33 b(we)g(have)g FP(i)25 b FN(6\030)1898
2338 y FK(K)1982 2324 y FP(i)p FI(.)378 2537 y FQ(Pro)s(of)p
FT(:)39 b(F)-8 b(rom)39 b(de\014nition)d(8.3,)42 b(if)c(\()p
Fv(C)50 b Fw(by)43 b Fv(P)11 b FT(\))39 b FN(\))2132
2551 y FL(c)2206 2537 y FT(\()p FP(S)2302 2504 y FK(0)2326
2537 y FP(;)15 b FN(K)2436 2504 y FK(0)2460 2537 y FT(\),)41
b(then)d FP(S)2837 2504 y FK(0)2899 2537 y FT(=)h FN(f)p
FP(P)3125 2504 y FO(i)3153 2537 y FP(;)15 b FN(:)p Fv(C)3320
2504 y FO(j)3356 2537 y FN(g)39 b FT(and)f FN(K)3695
2504 y FK(0)3757 2537 y FT(=)378 2650 y FP(i)27 b FN($)g
FP(j)37 b FT(for)31 b(some)h(colours)f FP(i)h FT(and)f
FP(j)37 b FT(where)30 b FP(i)e FN(6)p FT(=)f FP(j)5 b
FT(.)44 b(Therefore,)32 b(it)f(is)f(the)i(case)g(that)g
FN(K)3395 2617 y FK(0)3446 2650 y FT(=)27 b FN(K)3614
2617 y FK(0)3638 2650 y FN(d)p FP(S)3739 2617 y FK(0)3762
2650 y FN(e)p FT(,)378 2763 y Fe(C)p FT(\()p FP(S)530
2730 y FK(0)553 2763 y FT(\))j(=)g Fe(C)p FT(\()p FN(K)880
2730 y FK(0)904 2763 y FT(\),)k(and)f(that)h(no)f(colour)f(in)g
FP(i)e FN($)g FP(j)39 b FT(relates)33 b(with)f(itself.)48
b(The)33 b(three)g(parts)g(of)g(this)378 2876 y(prop)s(osition)28
b(then)i(follo)m(w)g(b)m(y)g(prop)s(osition)e(8.4)j(ab)s(o)m(v)m(e.)
1430 b Ff(\004)378 3119 y FG(8.3.2)112 b(A)37 b(Con\015uence)i(Prop)s
(ert)m(y)378 3291 y FT(Note)i(that)f(an)g(application)e(of)i(the)g
(relation)f FN(!)2141 3305 y FL(c)2216 3291 y FT(\(as)h(giv)m(en)g(b)m
(y)f(the)h(rules)f(in)f(de\014nition)f(8.4\))378 3403
y(in)m(tro)s(duces)44 b(a)h(new)g(colour)f(nondeterministically)d(to)46
b(a)g(coloured)e(structured)g(problem)f(and)378 3516
y(as)i(a)g(result)f(the)h(relation)f FN(!)1472 3530 y
FL(c)1552 3516 y FT(is)g(not)h(con\015uen)m(t.)85 b(In)44
b(particular,)j(it)d(is)g(not)h(the)g(case)h(that)378
3629 y(if)39 b(\()p FP(S;)15 b FN(K)q FT(\))44 b FN(+)807
3643 y FL(c)883 3629 y FT(\()p FP(S)974 3643 y FL(1)1014
3629 y FP(;)15 b FN(K)1123 3643 y FL(1)1163 3629 y FT(\))40
b(and)g(\()p FP(S;)15 b FN(K)q FT(\))43 b FN(+)1760 3643
y FL(c)1837 3629 y FT(\()p FP(S)1928 3643 y FL(2)1968
3629 y FP(;)15 b FN(K)2077 3643 y FL(2)2117 3629 y FT(\))40
b(for)g(coloured)g(structured)g(problems)e(\()p FP(S;)15
b FN(K)q FT(\),)378 3742 y(\()p FP(S)469 3756 y FL(1)508
3742 y FP(;)g FN(K)617 3756 y FL(1)658 3742 y FT(\))40
b(and)g(\()p FP(S)1011 3756 y FL(2)1051 3742 y FP(;)15
b FN(K)1160 3756 y FL(2)1200 3742 y FT(\))41 b(then)f(\()p
FP(S)1584 3756 y FL(1)1624 3742 y FP(;)15 b FN(K)1733
3756 y FL(1)1773 3742 y FT(\))42 b(=)g(\()p FP(S)2054
3756 y FL(2)2094 3742 y FP(;)15 b FN(K)2203 3756 y FL(2)2243
3742 y FT(\).)72 b(It)40 b(can)h(b)s(e)f(sho)m(wn,)j(ho)m(w)m(ev)m(er,)
i(that)c(the)378 3855 y(relation)35 b FN(!)808 3869 y
FL(c)879 3855 y FT(satis\014es)g(a)g(con\015uence)h(prop)s(ert)m(y)f
(mo)s(dulo)f(isomorphism)e(b)m(y)k(renaming)e(colours.)378
3968 y(In)c(other)g(w)m(ords,)g(whenev)m(er)1115 4172
y(\()p FP(S)1206 4186 y FL(1)1246 4172 y FP(;)15 b FN(K)1355
4186 y FL(1)1395 4172 y FT(\))26 b FN(!)1547 4135 y FK(\003)1547
4195 y FL(c)1611 4172 y FT(\()p FP(S)1702 4186 y FL(2)1742
4172 y FP(;)15 b FN(K)1851 4186 y FL(2)1891 4172 y FT(\))91
b(and)f(\()p FP(S)2345 4186 y FL(3)2385 4172 y FP(;)15
b FN(K)2494 4186 y FL(3)2534 4172 y FT(\))26 b FN(!)2686
4135 y FK(\003)2686 4195 y FL(c)2750 4172 y FT(\()p FP(S)2841
4186 y FL(4)2881 4172 y FP(;)15 b FN(K)2990 4186 y FL(4)3030
4172 y FT(\))p FP(;)378 4376 y FT(if)k(\()p FP(S)542
4390 y FL(1)582 4376 y FP(;)c FN(K)691 4390 y FL(1)731
4376 y FT(\))26 b Fl(u)863 4390 y FL(rc)950 4376 y FT(\()p
FP(S)1041 4390 y FL(3)1081 4376 y FP(;)15 b FN(K)1190
4390 y FL(3)1230 4376 y FT(\))21 b(then)f(there)g(are)h(coloured)f
(structured)f(problems)g(\()p FP(S)3102 4390 y FL(5)3141
4376 y FP(;)c FN(K)3250 4390 y FL(5)3291 4376 y FT(\))20
b(and)g(\()p FP(S)3604 4390 y FL(6)3643 4376 y FP(;)15
b FN(K)3752 4390 y FL(6)3793 4376 y FT(\))378 4489 y(suc)m(h)30
b(that)673 4694 y(\()p FP(S)764 4708 y FL(2)803 4694
y FP(;)15 b FN(K)912 4708 y FL(2)953 4694 y FT(\))25
b FN(!)1104 4656 y FK(\003)1104 4716 y FL(c)1169 4694
y FT(\()p FP(S)1260 4708 y FL(5)1299 4694 y FP(;)15 b
FN(K)1408 4708 y FL(5)1448 4694 y FT(\))p FP(;)107 b
FT(\()p FP(S)1706 4708 y FL(4)1746 4694 y FP(;)15 b FN(K)1855
4708 y FL(4)1895 4694 y FT(\))25 b FN(!)2046 4656 y FK(\003)2046
4716 y FL(c)2111 4694 y FT(\()p FP(S)2202 4708 y FL(6)2241
4694 y FP(;)15 b FN(K)2350 4708 y FL(6)2391 4694 y FT(\))60
b(and)h(\()p FP(S)2785 4708 y FL(5)2824 4694 y FP(;)15
b FN(K)2933 4708 y FL(5)2973 4694 y FT(\))26 b Fl(u)3105
4708 y FL(rc)3193 4694 y FT(\()p FP(S)3284 4708 y FL(6)3323
4694 y FP(;)15 b FN(K)3432 4708 y FL(6)3472 4694 y FT(\))p
FP(:)378 4898 y FT(This)31 b(is)h(deriv)m(ed)h(b)m(y)g(sho)m(wing)f
(that)i FN(!)1773 4912 y FL(c)1841 4898 y FT(is)e(also)h(strongly)g
(con\015uen)m(t)g(mo)s(dulo)f(isomorphism)e(b)m(y)378
5011 y(renaming)f(colours.)378 5223 y FQ(Prop)s(osition)36
b(8.6)46 b(\(Strong)55 b(Con\015uence)g(of)g FN(!)2309
5237 y FL(c)2400 5223 y FQ(mo)s(dulo)f Fl(u)2865 5237
y FL(rc)2928 5223 y FQ(\))48 b FI(Given)h(four)g(c)-5
b(olour)g(e)g(d)378 5336 y(structur)g(e)g(d)34 b(pr)-5
b(oblems)35 b FT(\()p FP(S)1272 5350 y FO(i)1300 5336
y FP(;)15 b FN(K)1409 5350 y FO(i)1438 5336 y FT(\))33
b FI(for)g FP(i)25 b FN(2)g(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33
b(;)15 b FT(4)p FN(g)p FI(,)34 b(such)e(that)i FT(\()p
FP(S)2736 5350 y FL(1)2776 5336 y FP(;)15 b FN(K)2885
5350 y FL(1)2925 5336 y FT(\))26 b Fl(u)3057 5350 y FL(rc)3144
5336 y FT(\()p FP(S)3235 5350 y FL(3)3275 5336 y FP(;)15
b FN(K)3384 5350 y FL(3)3424 5336 y FT(\))33 b FI(and)1141
5540 y FT(\()p FP(S)1232 5554 y FL(1)1272 5540 y FP(;)15
b FN(K)1381 5554 y FL(1)1421 5540 y FT(\))25 b FN(!)1572
5554 y FL(c)1633 5540 y FT(\()p FP(S)1724 5554 y FL(2)1763
5540 y FP(;)15 b FN(K)1872 5554 y FL(2)1913 5540 y FT(\))83
b FI(and)h FT(\()p FP(S)2349 5554 y FL(3)2389 5540 y
FP(;)15 b FN(K)2498 5554 y FL(3)2538 5540 y FT(\))25
b FN(!)2689 5554 y FL(c)2750 5540 y FT(\()p FP(S)2841
5554 y FL(4)2880 5540 y FP(;)15 b FN(K)2989 5554 y FL(4)3030
5540 y FT(\))p eop
%%Page: 164 174
164 173 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(164)1017
534 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@LT1 16 {InitRnode } NewNode end end
1017 534 a FT(\()p FP(S)1108 548 y FL(1)1148 534
y FP(;)15 b FN(K)1257 548 y FL(1)1297 534 y FT(\))25
b Fl(u)1428 548 y FL(rc)1516 534 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@RT1 16 {InitRnode } NewNode end end
1516 534 a FT(\()p
FP(S)1607 548 y FL(3)1647 534 y FP(;)15 b FN(K)1756 548
y FL(3)1796 534 y FT(\))2286 534 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@LT2 16 {InitRnode } NewNode end end
2286 534 a FT(\()p
FP(S)2377 548 y FL(1)2417 534 y FP(;)g FN(K)2526 548
y FL(1)2566 534 y FT(\))25 b Fl(u)2697 548 y FL(rc)2785
534 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@RT2 16 {InitRnode } NewNode end end
2785 534 a FT(\()p FP(S)2876 548 y FL(3)2916 534
y FP(;)15 b FN(K)3025 548 y FL(3)3065 534 y FT(\))1974
672 y(or)1017 810 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@LB1 16 {InitRnode } NewNode end end
1017 810 a FT(\()p FP(S)1108 824
y FL(2)1148 810 y FP(;)g FN(K)1257 824 y FL(2)1297 810
y FT(\))25 b Fl(u)1428 824 y FL(rc)1516 810 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@RB1 16 {InitRnode } NewNode end end
1516 810
a FT(\()p FP(S)1607 824 y FL(4)1647 810 y FP(;)15 b FN(K)1756
824 y FL(4)1796 810 y FT(\))2195 810 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@LM2 16 {InitRnode } NewNode end end
2195 810 a FT(\()p
FP(S)2286 824 y FL(2)2326 810 y FP(;)g FN(K)2435 824
y FL(2)2475 810 y FT(\))2874 810 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@RM2 16 {InitRnode } NewNode end end
2874 810 a FT(\()p
FP(S)2965 824 y FL(4)3004 810 y FP(;)g FN(K)3113 824
y FL(4)3153 810 y FT(\))2286 1086 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@LB2 16 {InitRnode } NewNode end end
2286 1086 a FT(\()p
FP(S)2377 1100 y FL(5)2417 1086 y FP(;)g FN(K)2526 1100
y FL(5)2566 1086 y FT(\))25 b Fl(u)2697 1100 y FL(rc)2785
1086 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.94128 18.97064
2.7375 } false /N@RB2 16 {InitRnode } 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(\()p FP(S)1502 1898 y FL(4)1541
1884 y FP(;)15 b FN(K)1650 1898 y FL(4)1690 1884 y FT(\))46
b FI(or)g(else)g(ther)-5 b(e)46 b(ar)-5 b(e)47 b(two)f(c)-5
b(olour)g(e)g(d)48 b(structur)-5 b(e)g(d)47 b(pr)-5 b(oblems)378
1997 y FT(\()p FP(S)469 2011 y FL(5)508 1997 y FP(;)15
b FN(K)617 2011 y FL(5)658 1997 y FT(\))32 b FI(and)i
FT(\()p FP(S)993 2011 y FL(6)1033 1997 y FP(;)15 b FN(K)1142
2011 y FL(6)1182 1997 y FT(\))33 b FI(such)f(that)1128
2201 y FT(\()p FP(S)1219 2215 y FL(2)1259 2201 y FP(;)15
b FN(K)1368 2215 y FL(2)1408 2201 y FT(\))26 b FN(!)1560
2215 y FL(c)1620 2201 y FT(\()p FP(S)1711 2215 y FL(5)1751
2201 y FP(;)15 b FN(K)1860 2215 y FL(5)1900 2201 y FT(\))83
b FI(and)h FT(\()p FP(S)2336 2215 y FL(4)2376 2201 y
FP(;)15 b FN(K)2485 2215 y FL(4)2525 2201 y FT(\))26
b FN(!)2677 2215 y FL(c)2737 2201 y FT(\()p FP(S)2828
2215 y FL(6)2868 2201 y FP(;)15 b FN(K)2977 2215 y FL(6)3017
2201 y FT(\))p FP(;)378 2405 y FI(and)34 b FT(\()p FP(S)646
2419 y FL(5)685 2405 y FP(;)15 b FN(K)794 2419 y FL(5)834
2405 y FT(\))26 b Fl(u)966 2419 y FL(rc)1054 2405 y FT(\()p
FP(S)1145 2419 y FL(6)1184 2405 y FP(;)15 b FN(K)1293
2419 y FL(6)1333 2405 y FT(\))p FI(.)42 b(\(se)-5 b(e)33
b(\014gur)-5 b(e)33 b(19\))378 2618 y FQ(Pro)s(of)p FT(:)24
b(Giv)m(en)e(that)i(\()p FP(S)1216 2632 y FL(1)1255 2618
y FP(;)15 b FN(K)1364 2632 y FL(1)1404 2618 y FT(\))26
b Fl(u)1536 2632 y FL(rc)1624 2618 y FT(\()p FP(S)1715
2632 y FL(3)1754 2618 y FP(;)15 b FN(K)1863 2632 y FL(3)1903
2618 y FT(\))23 b(then)g(there)g(is)e(a)i(recolouring)f(mapping)f
Fe(R)i FT(suc)m(h)f(that)378 2731 y Fe(R)p FT(\()p FP(S)544
2745 y FL(1)584 2731 y FP(;)15 b FN(K)693 2745 y FL(1)733
2731 y FT(\))29 b(=)f(\()p FP(S)987 2745 y FL(3)1027
2731 y FP(;)15 b FN(K)1136 2745 y FL(3)1176 2731 y FT(\).)47
b(Also,)33 b(since)f(\()p FP(S)1832 2745 y FL(1)1871
2731 y FP(;)15 b FN(K)1980 2745 y FL(1)2021 2731 y FT(\))29
b FN(!)2176 2745 y FL(c)2240 2731 y FT(\()p FP(S)2331
2745 y FL(2)2370 2731 y FP(;)15 b FN(K)2479 2745 y FL(2)2519
2731 y FT(\))33 b(then)f(there)h(is)e(some)i(set)g FP(E)3565
2745 y FL(1)3633 2731 y FN(\022)c FP(S)3789 2745 y FL(1)378
2844 y FT(consisting)c(of)i(one)f(coloured)g(structured)g(expression)f
(and)h(some)h(set)f FP(E)2888 2858 y FL(2)2953 2844 y
FN(\022)f FP(S)3105 2858 y FL(2)3171 2844 y FT(consisting)g(of)i(t)m(w)
m(o)378 2957 y(coloured)j(structured)f(expressions)g(suc)m(h)h(that)
1199 3161 y FP(S)1255 3175 y FL(1)1314 3161 y FN(\000)20
b FP(E)1472 3175 y FL(1)1537 3161 y FT(=)25 b FP(S)1689
3175 y FL(2)1748 3161 y FN(\000)20 b FP(E)1906 3175 y
FL(2)1976 3161 y FT(and)29 b(\()p FP(E)2254 3175 y FL(1)2294
3161 y FP(;)15 b FN(K)2403 3175 y FL(1)2443 3161 y FT(\))26
b FN(!)2595 3175 y FL(c)2656 3161 y FT(\()p FP(E)2758
3175 y FL(2)2798 3161 y FP(;)15 b FN(K)2907 3175 y FL(2)2947
3161 y FT(\))p FP(:)378 3365 y FT(Similarly)-8 b(,)40
b(as)h(\()p FP(S)1008 3379 y FL(3)1047 3365 y FP(;)15
b FN(K)1156 3379 y FL(3)1196 3365 y FT(\))43 b FN(!)1365
3379 y FL(c)1443 3365 y FT(\()p FP(S)1534 3379 y FL(4)1573
3365 y FP(;)15 b FN(K)1682 3379 y FL(4)1722 3365 y FT(\))41
b(then)g(there)f(is)g(some)h(set)g FP(E)2817 3379 y FL(3)2899
3365 y FN(\022)h FP(S)3068 3379 y FL(3)3147 3365 y FT(consisting)e(of)h
(one)378 3478 y(coloured)31 b(structured)g(expression)g(and)g(some)h
(set)g FP(E)2242 3492 y FL(4)2310 3478 y FN(\022)27 b
FP(S)2464 3492 y FL(4)2535 3478 y FT(consisting)j(of)i(t)m(w)m(o)i
(coloured)d(struc-)378 3591 y(tured)f(expressions)f(suc)m(h)h(that)1199
3795 y FP(S)1255 3809 y FL(3)1314 3795 y FN(\000)20 b
FP(E)1472 3809 y FL(3)1537 3795 y FT(=)25 b FP(S)1689
3809 y FL(4)1748 3795 y FN(\000)20 b FP(E)1906 3809 y
FL(4)1976 3795 y FT(and)29 b(\()p FP(E)2254 3809 y FL(3)2294
3795 y FP(;)15 b FN(K)2403 3809 y FL(3)2443 3795 y FT(\))26
b FN(!)2595 3809 y FL(c)2656 3795 y FT(\()p FP(E)2758
3809 y FL(4)2798 3795 y FP(;)15 b FN(K)2907 3809 y FL(4)2947
3795 y FT(\))p FP(:)378 3999 y FT(W)-8 b(e)32 b(consider)d(the)h(t)m(w)
m(o)i(cases)f(where)f Fe(R)p FT(\()p FP(E)1885 4013 y
FL(1)1925 3999 y FT(\))c(=)f FP(E)2149 4013 y FL(3)2219
3999 y FT(or)30 b Fe(R)p FT(\()p FP(E)2507 4013 y FL(1)2547
3999 y FT(\))c FN(6)p FT(=)f FP(E)2771 4013 y FL(3)2810
3999 y FT(.)514 4187 y FN(\017)46 b FT(If)32 b Fe(R)p
FT(\()p FP(E)875 4201 y FL(1)916 4187 y FT(\))d(=)f FP(E)1146
4201 y FL(3)1219 4187 y FT(then)k(w)m(e)h(claim)f(that)h(\()p
FP(S)2099 4201 y FL(2)2138 4187 y FP(;)15 b FN(K)2247
4201 y FL(2)2287 4187 y FT(\))30 b Fl(u)2423 4201 y FL(rc)2514
4187 y FT(\()p FP(S)2605 4201 y FL(4)2645 4187 y FP(;)15
b FN(K)2754 4201 y FL(4)2794 4187 y FT(\).)48 b(W)-8
b(e)33 b(pro)m(v)m(e)h(this)d(claim)h(b)m(y)605 4300
y(considering)g(whether)h FP(E)1503 4314 y FL(1)1576
4300 y FT(con)m(tains)h(an)f FM(on)g FT(expression)f(or)i(whether)f(it)
g(con)m(tains)g(an)h FM(and)605 4413 y FT(expression)29
b(separately)-8 b(.)605 4563 y(If)30 b FP(E)763 4577
y FL(1)828 4563 y FT(=)25 b(\()p FP(A)31 b FM(on)e FP(B)5
b FT(\))1292 4530 y FO(i)1351 4563 y FT(for)30 b(some)h(expressions)e
FP(A)h FT(and)g FP(B)35 b FT(and)30 b(colour)g FP(i)p
FT(,)g(then)1673 4767 y FP(E)1740 4781 y FL(3)1805 4767
y FT(=)25 b Fe(R)p FT(\()p FP(E)2078 4781 y FL(1)2118
4767 y FT(\))h(=)f FN(f)p FT(\()p FP(A)31 b FM(on)f FP(B)5
b FT(\))2689 4730 y FO(l)2715 4767 y FN(g)605 4972 y
FT(where)30 b FP(l)d FT(=)e Fe(R)p FT(\()p FP(i)p FT(\).)42
b(Therefore,)1283 5201 y FP(S)1339 5215 y FL(2)1403 5201
y FT(=)25 b FP(S)1555 5215 y FL(1)1615 5201 y FN(\000)20
b FP(E)1773 5215 y FL(1)1832 5201 y FN([)g(f)p FP(A)2026
5163 y FO(i)2055 5201 y FP(;)15 b(B)2169 5163 y FO(j)2205
5201 y FN(g)211 b(K)2530 5215 y FL(2)2594 5201 y FT(=)25
b FN(K)2759 5215 y FL(1)2819 5201 y FN([)20 b FP(i)26
b FN($)f FP(j)1283 5347 y(S)1339 5361 y FL(4)1403 5347
y FT(=)g FP(S)1555 5361 y FL(3)1615 5347 y FN(\000)20
b FP(E)1773 5361 y FL(3)1832 5347 y FN([)g(f)p FP(A)2026
5310 y FO(l)2053 5347 y FP(;)15 b(B)2167 5310 y FO(m)2233
5347 y FN(g)183 b(K)2530 5361 y FL(4)2594 5347 y FT(=)25
b FN(K)2759 5361 y FL(3)2819 5347 y FN([)20 b FP(l)27
b FN($)e FP(m)p eop
%%Page: 165 175
165 174 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(165)605
396 y(where)44 b FP(j)50 b FT(is)44 b(new)g(to)h(\()p
FP(S)1492 410 y FL(1)1531 396 y FP(;)15 b FN(K)1640 410
y FL(1)1680 396 y FT(\))45 b(and)f FP(m)g FT(is)g(new)g(to)h(\()p
FP(S)2598 410 y FL(3)2637 396 y FP(;)15 b FN(K)2746 410
y FL(3)2787 396 y FT(\).)83 b(No)m(w,)49 b(if)43 b(w)m(e)i(de\014ne)f
(the)605 509 y(recolouring)29 b(mapping)g Fe(R)1524 476
y FK(0)1578 509 y FT(suc)m(h)h(that)1804 714 y Fe(R)1879
676 y FK(0)1902 714 y FT(\()p FP(X)7 b FT(\))27 b(=)e
Fe(R)p FT(\()p FP(X)7 b FT(\))2404 676 y FL(\()p FO(j)t
FK(!)p FO(m)p FL(\))605 918 y FT(for)23 b(all)e FP(X)7
b FT(,)25 b(where)e FP(Y)1316 885 y FL(\()p FO(j)t FK(!)p
FO(m)p FL(\))1563 918 y FT(for)f(some)h(arbitrary)f(coloured)g(ob)5
b(ject)24 b FP(Y)42 b FT(represen)m(ts)23 b(the)g(ob)5
b(ject)605 1031 y FP(Y)54 b FT(with)33 b(all)g(the)h(o)s(ccurrences)g
(of)g(the)g(colour)g FP(j)39 b FT(replaced)34 b(with)e
FP(m)i FT(\(see)h(de\014nition)d(7.19\).)605 1144 y(Therefore)e
Fe(R)1092 1111 y FK(0)1116 1144 y FT(\()p FP(S)1207 1158
y FL(2)1246 1144 y FP(;)15 b FN(K)1355 1158 y FL(2)1396
1144 y FT(\))25 b(=)g(\()p FP(S)1643 1158 y FL(4)1683
1144 y FP(;)15 b FN(K)1792 1158 y FL(4)1832 1144 y FT(\))30
b(and)g(so)h(\()p FP(S)2277 1158 y FL(2)2316 1144 y FP(;)15
b FN(K)2425 1158 y FL(2)2465 1144 y FT(\))26 b Fl(u)2597
1158 y FL(rc)2685 1144 y FT(\()p FP(S)2776 1158 y FL(4)2816
1144 y FP(;)15 b FN(K)2925 1158 y FL(4)2965 1144 y FT(\).)605
1294 y(W)-8 b(e)33 b(no)m(w)e(consider)f(the)i(case)g(where)f
FP(E)1990 1308 y FL(1)2057 1294 y FT(=)26 b FN(f)p FT(\()p
FP(A)32 b FM(and)d FP(B)5 b FT(\))2616 1261 y FO(i)2644
1294 y FN(g)32 b FT(for)f(some)h FP(A)p FT(,)g FP(B)k
FT(and)30 b(colour)h FP(i)p FT(.)605 1407 y(Therefore)f
FP(E)1084 1421 y FL(3)1149 1407 y FT(=)25 b FN(f)p FT(\()p
FP(A)31 b FM(and)f FP(B)5 b FT(\))1707 1374 y FO(l)1733
1407 y FN(g)31 b FT(where)e(the)i(colour)f FP(l)d FT(=)e
Fe(R)p FT(\()p FP(i)p FT(\),)32 b(and)1266 1652 y FP(S)1322
1666 y FL(2)1386 1652 y FT(=)25 b FP(S)1538 1666 y FL(1)1597
1652 y FN(\000)20 b FP(E)1755 1666 y FL(1)1815 1652 y
FN([)g(f)p FP(A)2009 1614 y FO(i)2038 1652 y FP(;)15
b(B)2152 1614 y FO(j)2188 1652 y FN(g)210 b(K)2512 1666
y FL(2)2577 1652 y FT(=)25 b FN(K)2742 1666 y FL(1)2802
1652 y FN([)20 b(K)2953 1604 y FL(\()p FO(i)p FK(!)p
FO(j)t FL(\))2952 1678 y(1)1266 1813 y FP(S)1322 1827
y FL(4)1386 1813 y FT(=)25 b FP(S)1538 1827 y FL(3)1597
1813 y FN(\000)20 b FP(E)1755 1827 y FL(3)1815 1813 y
FN([)g(f)p FP(A)2009 1775 y FO(l)2036 1813 y FP(;)15
b(B)2150 1775 y FO(m)2216 1813 y FN(g)182 b(K)2512 1827
y FL(4)2577 1813 y FT(=)25 b FN(K)2742 1827 y FL(3)2802
1813 y FN([)20 b(K)2953 1765 y FL(\()p FO(l)q FK(!)p
FO(m)p FL(\))2952 1839 y(3)605 2017 y FT(where)30 b FP(j)36
b FT(and)30 b FP(m)g FT(are)h(new)f(to)h(\()p FP(S)1769
2031 y FL(1)1808 2017 y FP(;)15 b FN(K)1917 2031 y FL(1)1957
2017 y FT(\))31 b(and)f(\()p FP(S)2291 2031 y FL(3)2330
2017 y FP(;)15 b FN(K)2439 2031 y FL(3)2479 2017 y FT(\))31
b(resp)s(ectiv)m(ely)-8 b(.)41 b(By)30 b(de\014ning)1804
2221 y Fe(R)1879 2184 y FK(0)1902 2221 y FT(\()p FP(X)7
b FT(\))27 b(=)e Fe(R)p FT(\()p FP(X)7 b FT(\))2404 2184
y FL(\()p FO(j)t FK(!)p FO(m)p FL(\))605 2425 y FT(again,)28
b(w)m(e)g(get)g(that)g Fe(R)1418 2392 y FK(0)1441 2425
y FT(\()p FP(S)1532 2439 y FL(2)1572 2425 y FP(;)15 b
FN(K)1681 2439 y FL(2)1721 2425 y FT(\))26 b(=)e(\()p
FP(S)1968 2439 y FL(4)2008 2425 y FP(;)15 b FN(K)2117
2439 y FL(4)2157 2425 y FT(\))27 b(and)g(hence)g(it)g(is)f(the)h(case)h
(that)g(\()p FP(S)3445 2439 y FL(2)3485 2425 y FP(;)15
b FN(K)3594 2439 y FL(2)3634 2425 y FT(\))25 b Fl(u)3765
2439 y FL(rc)605 2538 y FT(\()p FP(S)696 2552 y FL(4)736
2538 y FP(;)15 b FN(K)845 2552 y FL(4)885 2538 y FT(\).)514
2726 y FN(\017)46 b FT(If)33 b(on)g(the)h(other)f(hand)f
Fe(R)p FT(\()p FP(E)1635 2740 y FL(1)1675 2726 y FT(\))f
FN(6)p FT(=)e FP(E)1908 2740 y FL(3)1981 2726 y FT(then)k(w)m(e)h
(claim)e(that)i(there)f(is)g(some)g(\()p FP(S)3424 2740
y FL(5)3464 2726 y FP(;)15 b FN(K)3573 2740 y FL(5)3613
2726 y FT(\))33 b(and)605 2839 y(\()p FP(S)696 2853 y
FL(6)736 2839 y FP(;)15 b FN(K)845 2853 y FL(6)885 2839
y FT(\))31 b(suc)m(h)f(that)715 3043 y(\()p FP(S)806
3057 y FL(2)845 3043 y FP(;)15 b FN(K)954 3057 y FL(2)994
3043 y FT(\))26 b FN(!)1146 3057 y FL(c)1207 3043 y FT(\()p
FP(S)1298 3057 y FL(5)1337 3043 y FP(;)15 b FN(K)1446
3057 y FL(5)1486 3043 y FT(\))p FP(;)198 b FT(\()p FP(S)1835
3057 y FL(4)1874 3043 y FP(;)15 b FN(K)1983 3057 y FL(4)2023
3043 y FT(\))26 b FN(!)2175 3057 y FL(c)2236 3043 y FT(\()p
FP(S)2327 3057 y FL(6)2366 3043 y FP(;)15 b FN(K)2475
3057 y FL(6)2515 3043 y FT(\))92 b(and)e(\()p FP(S)2970
3057 y FL(5)3010 3043 y FP(;)15 b FN(K)3119 3057 y FL(5)3159
3043 y FT(\))25 b Fl(u)3290 3057 y FL(rc)3378 3043 y
FT(\()p FP(S)3469 3057 y FL(6)3509 3043 y FP(;)15 b FN(K)3618
3057 y FL(6)3658 3043 y FT(\))p FP(:)605 3247 y FT(The)28
b(pro)s(of)g(of)g(this)f(claim)h(can)g(b)s(e)g(done)g(b)m(y)g(case)i
(analysis)d(on)h(whether)f FP(S)3225 3261 y FL(1)3293
3247 y FT(and)h FP(S)3524 3261 y FL(3)3591 3247 y FT(are)h
Fw(on)605 3360 y FT(or)e Fw(and)f FT(expressions)g(\(4)h(cases)h(in)e
(all\).)39 b(Since)26 b(the)h(pro)s(ofs)f(of)h(these)h(cases)g(are)f
(quite)g(similar)605 3473 y(w)m(e)h(presen)m(t)e(only)g(the)h(case)h
(where)f FP(E)1913 3487 y FL(1)1979 3473 y FT(con)m(tains)g(an)g
FM(on)f FT(expression)g(while)f FP(E)3313 3487 y FL(3)3379
3473 y FT(con)m(tains)i(an)605 3586 y FM(and)j FT(expression.)39
b(So,)31 b(w)m(e)g(ha)m(v)m(e)g(that)1384 3790 y FP(E)1451
3804 y FL(1)1516 3790 y FT(=)25 b FN(f)p FT(\()p FP(A)31
b FM(on)f FP(B)5 b FT(\))2026 3753 y FO(i)2054 3790 y
FN(g)182 b FP(E)2348 3804 y FL(3)2413 3790 y FT(=)25
b FN(f)p FT(\()p FP(C)37 b FM(and)30 b FP(D)s FT(\))2978
3753 y FO(l)3004 3790 y FN(g)605 3994 y FT(and)g(therefore)1262
4224 y FP(S)1318 4238 y FL(2)1382 4224 y FT(=)25 b FP(S)1534
4238 y FL(1)1594 4224 y FN(\000)20 b FP(E)1752 4238 y
FL(1)1811 4224 y FN([)g(f)p FP(A)2005 4186 y FO(i)2034
4224 y FP(;)15 b(B)2148 4186 y FO(j)2184 4224 y FN(g)218
b(K)2516 4238 y FL(2)2581 4224 y FT(=)25 b FN(K)2746
4238 y FL(1)2806 4224 y FN([)20 b FP(i)25 b FN($)h FP(j)1262
4374 y(S)1318 4388 y FL(4)1382 4374 y FT(=)f FP(S)1534
4388 y FL(3)1594 4374 y FN(\000)20 b FP(E)1752 4388 y
FL(3)1811 4374 y FN([)g(f)p FP(C)2009 4337 y FO(l)2035
4374 y FP(;)15 b(D)2153 4337 y FO(m)2220 4374 y FN(g)182
b(K)2516 4388 y FL(4)2581 4374 y FT(=)25 b FN(K)2746
4388 y FL(3)2806 4374 y FN([)20 b(K)2957 4337 y FL(\()p
FO(l)q FK(!)p FO(m)p FL(\))605 4578 y FT(where)i FP(j)27
b FT(and)21 b FP(m)h FT(are)g(new)g(to)g(\()p FP(S)1709
4592 y FL(1)1749 4578 y FP(;)15 b FN(K)1858 4592 y FL(1)1898
4578 y FT(\))22 b(and)f(\()p FP(S)2214 4592 y FL(3)2254
4578 y FP(;)15 b FN(K)2363 4592 y FL(3)2403 4578 y FT(\))22
b(resp)s(ectiv)m(ely)-8 b(.)38 b(Let)22 b(us)f(also)h(denote)h(the)605
4691 y(colour)29 b Fe(R)p FT(\()p FP(i)p FT(\))g(b)m(y)g
FP(p)p FT(,)g(and)f Fe(R)1557 4658 y FK(\000)p FL(1)1652
4691 y FT(\()p FP(l)r FT(\))h(b)m(y)f FP(r)s FT(.)40
b(Please)29 b(note)g(that)g FP(E)2753 4705 y FL(3)2818
4691 y FN(\022)c Fe(R)p FT(\()p FP(S)3080 4705 y FL(2)3120
4691 y FT(\))k(and)f Fe(R)p FT(\()p FP(E)3536 4705 y
FL(1)3576 4691 y FT(\))e FN(\022)f FP(S)3789 4705 y FL(4)605
4804 y FT(as)31 b Fe(R)p FT(\()p FP(S)883 4818 y FL(1)923
4804 y FP(;)15 b FN(K)1032 4818 y FL(1)1072 4804 y FT(\))25
b(=)g(\()p FP(S)1319 4818 y FL(3)1359 4804 y FP(;)15
b FN(K)1468 4818 y FL(3)1508 4804 y FT(\).)41 b(If)30
b(w)m(e)h(no)m(w)f(de\014ne)g(the)g(follo)m(wing)1124
5049 y FP(S)1180 5063 y FL(5)1245 5049 y FT(=)25 b(\()p
FP(S)1432 5063 y FL(2)1491 5049 y FN(\000)20 b Fe(R)1657
5012 y FK(\000)p FL(1)1752 5049 y FT(\()p FP(E)1854 5063
y FL(3)1894 5049 y FT(\)\))h FN([)f(f)p FP(C)2183 5012
y FO(r)2220 5049 y FP(;)15 b(D)2338 5012 y FO(s)2375
5049 y FN(g)183 b(K)2672 5063 y FL(5)2737 5049 y FT(=)25
b FN(K)2902 5063 y FL(2)2961 5049 y FN([)20 b(K)3112
5001 y FL(\()p FO(r)r FK(!)p FO(s)p FL(\))3111 5075 y(2)1124
5187 y FP(S)1180 5201 y FL(6)1245 5187 y FT(=)25 b(\()p
FP(S)1432 5201 y FL(4)1491 5187 y FN(\000)20 b Fe(R)p
FT(\()p FP(E)1759 5201 y FL(1)1799 5187 y FT(\)\))h FN([)f(f)p
FP(A)2084 5149 y FO(p)2125 5187 y FP(;)15 b(B)2239 5149
y FO(q)2276 5187 y FN(g)282 b(K)2672 5201 y FL(6)2737
5187 y FT(=)25 b FN(K)2902 5201 y FL(4)2961 5187 y FN([)20
b FP(p)25 b FN($)g FP(q)605 5391 y FT(where)30 b FP(s)g
FT(and)g FP(q)j FT(are)e(new)f(to)h FI(b)-5 b(oth)38
b FT(\()p FP(S)1930 5405 y FL(2)1970 5391 y FP(;)15 b
FN(K)2079 5405 y FL(2)2119 5391 y FT(\))31 b(and)e(\()p
FP(S)2452 5405 y FL(4)2492 5391 y FP(;)15 b FN(K)2601
5405 y FL(4)2641 5391 y FT(\))31 b(then)1228 5595 y(\()p
FP(S)1319 5609 y FL(2)1359 5595 y FP(;)15 b FN(K)1468
5609 y FL(2)1508 5595 y FT(\))25 b FN(!)1659 5609 y FL(c)1720
5595 y FT(\()p FP(S)1811 5609 y FL(5)1851 5595 y FP(;)15
b FN(K)1960 5609 y FL(6)2000 5595 y FT(\))p FP(;)83 b
FT(and)g(\()p FP(S)2464 5609 y FL(4)2503 5595 y FP(;)15
b FN(K)2612 5609 y FL(4)2652 5595 y FT(\))26 b FN(!)2804
5609 y FL(c)2865 5595 y FT(\()p FP(S)2956 5609 y FL(6)2995
5595 y FP(;)15 b FN(K)3104 5609 y FL(6)3144 5595 y FT(\))p
FP(:)p eop
%%Page: 166 176
166 175 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(166)605
396 y(Moreo)m(v)m(er,)33 b(if)c(w)m(e)i(de\014ne)f(the)g(recolouring)g
(mapping)e Fe(R)2585 363 y FK(0)2639 396 y FT(suc)m(h)i(that)1670
601 y Fe(R)1745 563 y FK(0)1769 601 y FT(\()p FP(X)7
b FT(\))26 b(=)f(\()p Fe(R)p FT(\()p FP(X)7 b FT(\))2305
563 y FL(\()p FO(j)t FK(!)p FO(q)r FL(\))2503 601 y FT(\))2538
563 y FL(\()p FO(s)p FK(!)p FO(m)p FL(\))605 805 y FT(for)30
b(all)g FP(X)7 b FT(,)31 b(then)f(it)g(is)f(routine)g(to)j(sho)m(w)e
(that)1792 1009 y Fe(R)1867 972 y FK(0)1890 1009 y FT(\()p
FP(S)1981 1023 y FL(5)2021 1009 y FP(;)15 b FN(K)2130
1023 y FL(5)2170 1009 y FT(\))26 b(=)e(\()p FP(S)2417
1023 y FL(6)2457 1009 y FP(;)15 b FN(K)2566 1023 y FL(6)2606
1009 y FT(\))605 1213 y(and)30 b(therefore)h(\()p FP(S)1255
1227 y FL(5)1294 1213 y FP(;)15 b FN(K)1403 1227 y FL(5)1443
1213 y FT(\))26 b Fl(u)1575 1227 y FL(rc)1663 1213 y
FT(\()p FP(S)1754 1227 y FL(6)1794 1213 y FP(;)15 b FN(K)1903
1227 y FL(6)1943 1213 y FT(\).)1754 b Ff(\004)519 1382
y FT(The)43 b(required)f(con\015uence)i(result)f(follo)m(ws)g(from)g
(the)h(ab)s(o)m(v)m(e)h(prop)s(osition)c(b)m(y)j(Newman's)378
1495 y(Theorem)21 b(\(Newman)g(1942\))i(whic)m(h)d(states)i(that)f(ev)m
(ery)h(strongly)e(con\015uen)m(t)h(relation)g(is)f(con\015uen)m(t.)378
1684 y FQ(Theorem)34 b(8.6)46 b(\(Con\015uence)29 b(of)f
FN(!)1776 1698 y FL(c)1840 1684 y FQ(mo)s(dulo)h Fl(u)2280
1698 y FL(rc)2342 1684 y FQ(\))f FI(Given)f(four)h(c)-5
b(olour)g(e)g(d)30 b(structur)-5 b(e)g(d)29 b(pr)-5 b(ob-)378
1797 y(lems)33 b FT(\()p FP(S)678 1811 y FO(i)706 1797
y FP(;)15 b FN(K)815 1811 y FO(i)844 1797 y FT(\))33
b FI(for)g FP(i)26 b FN(2)f(f)p FT(1)p FP(;)15 b(:)g(:)g(:)32
b(;)15 b FT(4)p FN(g)p FI(,)34 b(such)f(that)681 2001
y FT(\()p FP(S)772 2015 y FL(1)812 2001 y FP(;)15 b FN(K)921
2015 y FL(1)961 2001 y FT(\))26 b FN(!)1113 1964 y FK(\003)1113
2024 y FL(c)1177 2001 y FT(\()p FP(S)1268 2015 y FL(2)1308
2001 y FP(;)15 b FN(K)1417 2015 y FL(2)1457 2001 y FT(\))p
FP(;)108 b FT(\()p FP(S)1716 2015 y FL(3)1756 2001 y
FP(;)15 b FN(K)1865 2015 y FL(3)1905 2001 y FT(\))26
b FN(!)2057 1964 y FK(\003)2057 2024 y FL(c)2121 2001
y FT(\()p FP(S)2212 2015 y FL(4)2252 2001 y FP(;)15 b
FN(K)2361 2015 y FL(4)2401 2001 y FT(\))65 b FI(and)h
FT(\()p FP(S)2801 2015 y FL(1)2841 2001 y FP(;)15 b FN(K)2950
2015 y FL(1)2990 2001 y FT(\))26 b Fl(u)3122 2015 y FL(rc)3210
2001 y FT(\()p FP(S)3301 2015 y FL(3)3340 2001 y FP(;)15
b FN(K)3449 2015 y FL(3)3489 2001 y FT(\))378 2206 y
FI(then)33 b(ther)-5 b(e)34 b(ar)-5 b(e)33 b(c)-5 b(olour)g(e)g(d)35
b(structur)-5 b(e)g(d)34 b(pr)-5 b(oblems)35 b FT(\()p
FP(S)2207 2220 y FL(5)2246 2206 y FP(;)15 b FN(K)2355
2220 y FL(5)2395 2206 y FT(\))33 b FI(and)h FT(\()p FP(S)2731
2220 y FL(6)2770 2206 y FP(;)15 b FN(K)2879 2220 y FL(6)2919
2206 y FT(\))33 b FI(such)g(that)669 2410 y FT(\()p FP(S)760
2424 y FL(2)799 2410 y FP(;)15 b FN(K)908 2424 y FL(2)948
2410 y FT(\))26 b FN(!)1100 2372 y FK(\003)1100 2432
y FL(c)1164 2410 y FT(\()p FP(S)1255 2424 y FL(5)1295
2410 y FP(;)15 b FN(K)1404 2424 y FL(5)1444 2410 y FT(\))p
FP(;)109 b FT(\()p FP(S)1704 2424 y FL(4)1743 2410 y
FP(;)15 b FN(K)1852 2424 y FL(4)1892 2410 y FT(\))26
b FN(!)2044 2372 y FK(\003)2044 2432 y FL(c)2109 2410
y FT(\()p FP(S)2200 2424 y FL(6)2239 2410 y FP(;)15 b
FN(K)2348 2424 y FL(6)2388 2410 y FT(\))66 b FI(and)g
FT(\()p FP(S)2789 2424 y FL(5)2828 2410 y FP(;)15 b FN(K)2937
2424 y FL(5)2977 2410 y FT(\))26 b Fl(u)3109 2424 y FL(rc)3197
2410 y FT(\()p FP(S)3288 2424 y FL(6)3328 2410 y FP(;)15
b FN(K)3437 2424 y FL(6)3477 2410 y FT(\))p FP(:)378
2599 y FQ(Pro)s(of)p FT(:)35 b(F)-8 b(ollo)m(ws)33 b(from)g(prop)s
(osition)e(8.6)k(b)m(y)e(a)h(result)f(of)g(Newman)h(\(1942\))i(\(see)e
(also)g(\(Plaisted)378 2712 y(1993a\)\))f(that)e(ev)m(ery)g(strongly)f
(con\015uen)m(t)h(relation)e(is)h(con\015uen)m(t.)1059
b Ff(\004)519 2938 y FT(The)30 b(follo)m(wing)f(corollary)h(follo)m(ws)
f(easily)h(from)f(theorem)i(8.6.)378 3128 y FQ(Corollary)k(8.1)46
b FI(Given)37 b(the)g(c)-5 b(olour)g(e)g(d)40 b(structur)-5
b(e)g(d)38 b(pr)-5 b(oblems)39 b FT(\()p FP(S)2706 3142
y FL(1)2745 3128 y FP(;)15 b FN(K)2854 3142 y FL(1)2895
3128 y FT(\))37 b FI(and)h FT(\()p FP(S)3239 3142 y FL(3)3278
3128 y FP(;)15 b FN(K)3387 3142 y FL(3)3427 3128 y FT(\))p
FI(,)39 b(and)e(the)378 3241 y(c)-5 b(olour)g(e)g(d)35
b(pr)-5 b(oblems)34 b FT(\()p FP(S)1198 3255 y FL(2)1238
3241 y FP(;)15 b FN(K)1347 3255 y FL(2)1387 3241 y FT(\))33
b FI(and)h FT(\()p FP(S)1723 3255 y FL(4)1762 3241 y
FP(;)15 b FN(K)1871 3255 y FL(4)1911 3241 y FT(\))33
b FI(such)g(that)721 3445 y FT(\()p FP(S)812 3459 y FL(1)851
3445 y FP(;)15 b FN(K)960 3459 y FL(1)1000 3445 y FT(\))26
b FN(+)1117 3459 y FL(c)1177 3445 y FT(\()p FP(S)1268
3459 y FL(2)1308 3445 y FP(;)15 b FN(K)1417 3459 y FL(2)1457
3445 y FT(\))p FP(;)108 b FT(\()p FP(S)1716 3459 y FL(3)1756
3445 y FP(;)15 b FN(K)1865 3459 y FL(3)1905 3445 y FT(\))26
b FN(+)2022 3459 y FL(c)2082 3445 y FT(\()p FP(S)2173
3459 y FL(4)2213 3445 y FP(;)15 b FN(K)2322 3459 y FL(4)2362
3445 y FT(\))65 b FI(and)h FT(\()p FP(S)2762 3459 y FL(1)2802
3445 y FP(;)15 b FN(K)2911 3459 y FL(1)2951 3445 y FT(\))25
b Fl(u)3082 3459 y FL(rc)3170 3445 y FT(\()p FP(S)3261
3459 y FL(3)3301 3445 y FP(;)15 b FN(K)3410 3459 y FL(3)3450
3445 y FT(\))378 3649 y FI(then)33 b FT(\()p FP(S)671
3663 y FL(2)711 3649 y FP(;)15 b FN(K)820 3663 y FL(2)860
3649 y FT(\))25 b Fl(u)991 3663 y FL(rc)1079 3649 y FT(\()p
FP(S)1170 3663 y FL(4)1210 3649 y FP(;)15 b FN(K)1319
3663 y FL(4)1359 3649 y FT(\))p FI(.)378 3839 y FQ(Pro)s(of)p
FT(:)30 b(If)e(\()p FP(S)866 3853 y FL(1)906 3839 y FP(;)15
b FN(K)1015 3853 y FL(1)1055 3839 y FT(\))26 b FN(+)1172
3853 y FL(c)1232 3839 y FT(\()p FP(S)1323 3853 y FL(2)1362
3839 y FP(;)15 b FN(K)1471 3853 y FL(2)1512 3839 y FT(\),)29
b(\()p FP(S)1692 3853 y FL(3)1732 3839 y FP(;)15 b FN(K)1841
3853 y FL(3)1881 3839 y FT(\))26 b FN(+)1998 3853 y FL(c)2058
3839 y FT(\()p FP(S)2149 3853 y FL(4)2188 3839 y FP(;)15
b FN(K)2297 3853 y FL(4)2338 3839 y FT(\))29 b(and)f(\()p
FP(S)2668 3853 y FL(1)2708 3839 y FP(;)15 b FN(K)2817
3853 y FL(1)2857 3839 y FT(\))25 b Fl(u)2988 3853 y FL(rc)3076
3839 y FT(\()p FP(S)3167 3853 y FL(3)3207 3839 y FP(;)15
b FN(K)3316 3853 y FL(3)3356 3839 y FT(\))29 b(then)g(there)378
3951 y(are)i(coloured)f(structured)f(problems)g(\()p
FP(S)1815 3965 y FL(5)1854 3951 y FP(;)15 b FN(K)1963
3965 y FL(5)2003 3951 y FT(\))31 b(and)f(\()p FP(S)2337
3965 y FL(6)2376 3951 y FP(;)15 b FN(K)2485 3965 y FL(6)2525
3951 y FT(\))31 b(suc)m(h)f(that)686 4156 y(\()p FP(S)777
4170 y FL(2)816 4156 y FP(;)15 b FN(K)925 4170 y FL(2)965
4156 y FT(\))26 b FN(!)1117 4118 y FK(\003)1117 4178
y FL(c)1181 4156 y FT(\()p FP(S)1272 4170 y FL(5)1312
4156 y FP(;)15 b FN(K)1421 4170 y FL(5)1461 4156 y FT(\))p
FP(;)107 b FT(\()p FP(S)1719 4170 y FL(4)1758 4156 y
FP(;)15 b FN(K)1867 4170 y FL(4)1907 4156 y FT(\))26
b FN(!)2059 4118 y FK(\003)2059 4178 y FL(c)2124 4156
y FT(\()p FP(S)2215 4170 y FL(6)2254 4156 y FP(;)15 b
FN(K)2363 4170 y FL(6)2403 4156 y FT(\))61 b(and)f(\()p
FP(S)2797 4170 y FL(5)2837 4156 y FP(;)15 b FN(K)2946
4170 y FL(5)2986 4156 y FT(\))25 b Fl(u)3117 4170 y FL(rc)3205
4156 y FT(\()p FP(S)3296 4170 y FL(6)3336 4156 y FP(;)15
b FN(K)3445 4170 y FL(6)3485 4156 y FT(\))378 4360 y(b)m(y)30
b(theorem)h(8.6.)42 b(Ho)m(w)m(ev)m(er,)32 b(since)e(\()p
FP(S)1745 4374 y FL(2)1785 4360 y FP(;)15 b FN(K)1894
4374 y FL(2)1934 4360 y FT(\))31 b(and)e(\()p FP(S)2267
4374 y FL(4)2307 4360 y FP(;)15 b FN(K)2416 4374 y FL(4)2456
4360 y FT(\))31 b(are)f(coloured)g(problems)f(then)1084
4564 y(\()p FP(S)1175 4578 y FL(5)1215 4564 y FP(;)15
b FN(K)1324 4578 y FL(5)1364 4564 y FT(\))25 b(=)g(\()p
FP(S)1611 4578 y FL(2)1651 4564 y FP(;)15 b FN(K)1760
4578 y FL(2)1800 4564 y FT(\))182 b(and)f(\()p FP(S)2436
4578 y FL(6)2476 4564 y FP(;)15 b FN(K)2585 4578 y FL(6)2625
4564 y FT(\))26 b(=)f(\()p FP(S)2873 4578 y FL(4)2912
4564 y FP(;)15 b FN(K)3021 4578 y FL(4)3061 4564 y FT(\))p
FP(:)636 b Ff(\004)519 4768 y FT(W)-8 b(e)26 b(conclude)e(this)g
(section)h(b)m(y)g(sho)m(wing)f(that)h(if)f(a)h(coloured)g(structured)f
(problem)f(con)m(v)m(erges)378 4881 y(to)29 b(some)f(consisten)m(t)h
(coloured)e(problem,)h(then)g(all)f(the)h(coloured)f(problems)g(it)h
(con)m(v)m(erges)i(to)e(are)378 4994 y(consisten)m(t.)378
5184 y FQ(Prop)s(osition)36 b(8.7)46 b FI(F)-7 b(or)49
b(every)f(c)-5 b(olour)g(e)g(d)50 b(structur)-5 b(e)g(d)49
b(pr)-5 b(oblem)50 b FT(\()p FP(S;)15 b FN(K)q FT(\))49
b FI(and)g(for)f(al)5 b(l)49 b(c)-5 b(olour)g(e)g(d)378
5297 y(pr)g(oblems)34 b FT(\()p FP(S)851 5264 y FK(0)875
5297 y FP(;)15 b FN(K)985 5264 y FK(0)1009 5297 y FT(\))33
b FI(and)h FT(\()p FP(S)1350 5264 y FK(00)1393 5297 y
FP(;)15 b FN(K)1503 5264 y FK(0)q(0)1546 5297 y FT(\))33
b FI(such)g(that)1260 5501 y FT(\()p FP(S;)15 b FN(K)q
FT(\))27 b FN(+)1579 5515 y FL(c)1640 5501 y FT(\()p
FP(S)1736 5463 y FK(0)1759 5501 y FP(;)15 b FN(K)1869
5463 y FK(0)1893 5501 y FT(\))84 b FI(and)g FT(\()p FP(S;)15
b FN(K)q FT(\))27 b FN(+)2558 5515 y FL(c)2618 5501 y
FT(\()p FP(S)2714 5463 y FK(0)q(0)2757 5501 y FP(;)15
b FN(K)2867 5463 y FK(0)q(0)2910 5501 y FT(\))378 5705
y FI(then)33 b FP(S)641 5672 y FK(0)697 5705 y FI(is)f
FN(K)864 5672 y FK(0)888 5705 y FI(-c)-5 b(onsistent)34
b(if)e(and)i(only)f(if)f FP(S)1955 5672 y FK(00)2030
5705 y FI(is)h FN(K)2198 5672 y FK(0)q(0)2241 5705 y
FI(-c)-5 b(onsistent.)p eop
%%Page: 167 177
167 176 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(167)378
396 y FQ(Pro)s(of)p FT(:)34 b(If)f(\()p FP(S;)15 b FN(K)q
FT(\))31 b FN(+)1107 410 y FL(c)1172 396 y FT(\()p FP(S)1268
363 y FK(0)1292 396 y FP(;)15 b FN(K)1402 363 y FK(0)1426
396 y FT(\))34 b(and)e(\()p FP(S;)15 b FN(K)q FT(\))32
b FN(+)1998 410 y FL(c)2063 396 y FT(\()p FP(S)2159 363
y FK(00)2201 396 y FP(;)15 b FN(K)2311 363 y FK(0)r(0)2355
396 y FT(\))33 b(then)g(\()p FP(S)2729 363 y FK(0)2753
396 y FP(;)15 b FN(K)2863 363 y FK(0)2887 396 y FT(\))30
b Fl(u)3023 410 y FL(rc)3116 396 y FT(\()p FP(S)3212
363 y FK(00)3255 396 y FP(;)15 b FN(K)3365 363 y FK(0)q(0)3408
396 y FT(\))33 b(b)m(y)g(corol-)378 509 y(lary)d(8.1,)h(and)f
(therefore)h FP(S)1354 476 y FK(0)1408 509 y FT(is)e
FN(K)1569 476 y FK(0)1593 509 y FT(-consisten)m(t)i(if)e(and)h(only)g
(if)f FP(S)2650 476 y FK(00)2723 509 y FT(is)g FN(K)2884
476 y FK(0)q(0)2927 509 y FT(-consisten)m(t)i(b)m(y)f(prop)s(osi-)378
622 y(tion)g(7.10.)3007 b Ff(\004)378 903 y FH(8.4)135
b(Soundness)47 b(and)g(Completeness)i(of)f(the)g(Restricted)h(Pro)t(of)
684 1053 y(Chec)l(king)d(of)f(Structured)f(Justi\014cations)378
1259 y FG(8.4.1)112 b(Soundness)40 b(and)e(Completeness)f(for)g(P)m
(articular)f(Cases)378 1430 y FT(In)42 b(this)f(section)i(w)m(e)g(sho)m
(w)f(the)h(soundness)e(and)g(completeness)i(of)g(the)f(mec)m(hanism)g
(for)g(con-)378 1543 y(structing)29 b(a)i(coloured)e(problem)g(from)g
(a)i(justi\014ed)d(conclusion)g(for)i(t)m(w)m(o)h(particular)e(cases.)
41 b(More)378 1656 y(precisely)-8 b(,)30 b(w)m(e)g(sho)m(w)h(that)g(if)
e(the)i(justi\014ed)d(conclusion)473 1814 y FP(C)102
b FM(by)48 b FP(P)378 1972 y FT(con)m(v)m(erges)39 b(to)e(the)g
(coloured)g(problem)e(\()p FP(S;)15 b FN(K)q FT(\),)40
b(then)c(\()p FP(S;)15 b FN(K)q FT(\))39 b(is)d(inconsisten)m(t)g(if)f
(and)h(only)g(if)g Fv(P)378 2085 y FT(justi\014es)43
b Fv(C)51 b FT(for)44 b(the)h(cases)g(that)g Fv(P)61
b FT(=)48 b Fv(A)c Fw(on)f Fv(B)48 b FT(and)c Fv(P)61
b FT(=)48 b Fv(A)c Fw(and)f Fv(B)48 b FT(where)c FP(A)h
FT(and)f FP(B)k FT(are)378 2198 y(sen)m(tences)744 2165
y FL(1)785 2198 y FT(.)f(The)32 b(`only)h(if)7 b(')31
b(direction)h(states)i(that)f(if)f(a)h(pro)s(of)f(is)g(found)f(using)g
(the)i(restrictions)378 2311 y(giv)m(en)28 b(b)m(y)f(the)h(constructed)
g(coloured)f(problem)f(then)h(it)g(is)g(the)g(case)i(that)f(the)g
(conclusion)e Fv(C)34 b FT(can)378 2424 y(b)s(e)c(justi\014ed)e(b)m(y)j
Fv(P)11 b FT(.)41 b(This)29 b(corresp)s(onds)g(to)i(the)g(soundness)e
(of)h(the)h(mec)m(hanism)f(of)g(constructing)378 2537
y(coloured)36 b(problems)e(in)h(order)h(to)h(pro)s(of)e(c)m(hec)m(k)j
(structured)d(justi\014cations.)57 b(Similarly)-8 b(,)35
b(the)h(`if)7 b(')378 2650 y(direction)33 b(corresp)s(onds)h(to)h(the)g
(completeness)g(of)g(the)g(pro)s(of)f(c)m(hec)m(king)i(mec)m(hanism.)53
b(The)34 b(role)378 2763 y(of)i(the)g(pro)s(ofs)f(giv)m(en)h(in)f(this)
g(section)h(is)f(to)i(giv)m(e)f(an)g(idea)f(of)h(what)g(is)f(required)f
(to)j(deriv)m(e)f(the)378 2876 y(soundness)29 b(and)g(completeness)i
(results)e(for)h(the)h(general)f(case.)378 3053 y FQ(Prop)s(osition)36
b(8.8)46 b FI(F)-7 b(or)34 b(al)5 b(l)33 b(\014rst-or)-5
b(der)35 b(sentenc)-5 b(es)33 b FP(A)p FI(,)f FP(B)5
b FI(,)32 b FP(C)7 b FI(,)32 b(if)1606 3257 y FT(\()p
Fv(C)51 b Fw(by)42 b Fv(A)i Fw(on)f Fv(B)t FT(\))26 b
FN(+)2302 3271 y FL(c)2362 3257 y FT(\()p FP(S;)15 b
FN(K)q FT(\))378 3461 y FI(for)28 b(some)g(c)-5 b(olour)g(e)g(d)29
b(pr)-5 b(oblem)29 b FT(\()p FP(S;)15 b FN(K)q FT(\))p
FI(,)30 b(then)d FP(S)33 b FI(is)27 b FN(K)q FI(-inc)-5
b(onsistent)28 b(if)f(and)h(only)g(if)f Ft(\()p Fv(A)44
b Fw(on)f Fv(B)t Ft(\))26 b Ff( )f Fv(C)6 b FI(.)378
3638 y FQ(Pro)s(of)p FT(:)31 b(As)g(illustrated)d(in)h(example)h
(8.1\(1\),)1189 3842 y Ft(\()p Fv(C)50 b Fw(by)43 b Fv(A)h
Fw(on)f Fv(B)t Ft(\))26 b FN(+)1879 3856 y FL(c)1939
3842 y FT(\()p FN(f)p Fv(A)2082 3805 y FO(i)2110 3842
y FP(;)15 b Fv(B)2218 3805 y FO(k)2260 3842 y FP(;)g
FN(:)p Fv(C)2427 3805 y FO(j)2463 3842 y FN(g)p FP(;)g(k)29
b FN($)d FP(i)f FN($)g FP(j)5 b FT(\))p FP(:)378 4046
y FT(By)31 b(corollary)e(8.1)j(if)d Ft(\()p Fv(C)50 b
Fw(by)43 b Fv(A)h Fw(on)f Fv(B)t Ft(\))25 b FN(+)1816
4060 y FL(c)1877 4046 y FT(\()p FP(S;)15 b FN(K)q FT(\))32
b(then)1366 4251 y(\()p FP(S;)15 b FN(K)q FT(\))27 b
Fl(u)1700 4265 y FL(rc)1788 4251 y FT(\()p FN(f)p Fv(A)1931
4213 y FO(i)1959 4251 y FP(;)15 b Fv(B)2066 4213 y FO(k)2109
4251 y FP(;)g FN(:)p Fv(C)2275 4213 y FO(j)2312 4251
y FN(g)p FP(;)g(k)29 b FN($)c FP(i)h FN($)f FP(j)5 b
FT(\))378 4455 y(and)21 b(therefore)h(\()p FP(S;)15 b
FN(K)q FT(\))23 b(is)e(inconsisten)m(t)g(if)f(and)h(only)g(if)g
FN(f)p Fv(A)2366 4422 y FO(i)2395 4455 y FP(;)15 b Fv(B)2502
4422 y FO(k)2545 4455 y FP(;)g FN(:)p Fv(C)2711 4422
y FO(j)2748 4455 y FN(g)22 b FT(is)e FP(k)29 b FN($)c
FP(i)g FN($)h FP(j)5 b FT(-inconsisten)m(t.)519 4568
y(The)30 b(set)h FN(f)p Fv(A)956 4535 y FO(i)984 4568
y FP(;)15 b Fv(B)1091 4535 y FO(k)1134 4568 y FP(;)g
FN(:)p Fv(C)1300 4535 y FO(j)1337 4568 y FN(g)31 b FT(can)f(b)s(e)g
(partitioned)f(in)m(to)1755 4772 y(\()p FN(f)p Fv(A)1898
4735 y FO(i)1926 4772 y FP(;)15 b FN(:)p Fv(C)2093 4735
y FO(j)2129 4772 y FN(g)p FP(;)g FN(f)p Fv(B)2327 4735
y FO(k)2370 4772 y FN(g)p FT(\))378 4976 y(whic)m(h)24
b(is)g(also)h(w)m(ell-coloured)f(with)f(resp)s(ect)i(to)h
FP(k)i FN($)e FP(i)f FN($)g FP(j)5 b FT(.)40 b(W)-8 b(e)26
b(can)f(therefore)h(use)e(theorem)i(7.8)378 5089 y(to)31
b(deduce)f(that)h FN(f)p Fv(A)1097 5056 y FO(i)1125 5089
y FP(;)15 b Fv(B)1233 5056 y FO(k)1275 5089 y FP(;)g
FN(:)p Fv(C)1441 5056 y FO(j)1478 5089 y FN(g)31 b FT(is)e
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)g(if)1497 5293 y FN(f)p Fv(A)1605 5256
y FO(i)1633 5293 y FP(;)15 b FN(:)p Fv(C)1799 5256 y
FO(j)1836 5293 y FP(;)g Fv(I)1919 5256 y FO(k)1962 5293
y FN(g)91 b FT(and)g FN(f)p Fv(B)2448 5256 y FO(k)2491
5293 y FP(;)15 b FN(:)p Fv(I)2635 5256 y FO(i)2663 5293
y FN(g)p 378 5437 1380 4 v 482 5491 a FC(1)516 5522 y
FB(It)22 b(can)h(b)r(e)f(also)i(noted)e(that)h(the)f(fact)h(that)f(\()p
FA(S;)13 b Fz(K)q FB(\))23 b(is)g(inconsisten)n(t)g(if)g(and)f(only)h
(if)g FA(P)33 b FB(justi\014es)23 b FA(C)28 b FB(can)22
b(b)r(e)h(easily)378 5614 y(sho)n(wn)f(to)f(hold)h(for)g(the)f
(particular)h(case)h(where)e FA(P)33 b FB(is)22 b(a)f(\014rst-order)g
(sen)n(tence)h(b)n(y)e(theorem)h(8.3)h(and)f(de\014nitions)h(6.3)378
5705 y(and)j(6.4.)p eop
%%Page: 168 178
168 177 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(168)378
396 y(are)39 b(for)f(some)i(\014rst-order)d(sen)m(tence)j
FP(I)7 b FT(.)66 b(As)39 b(an)f(aside,)j(w)m(e)e(note)g(that)g
FP(I)46 b FT(can)39 b(b)s(e)f(c)m(hosen)h(suc)m(h)378
509 y(that)29 b(the)g(pair)f(\()p FP(I)996 476 y FO(k)1039
509 y FP(;)15 b FN(:)p FP(I)1187 476 y FO(i)1216 509
y FT(\))29 b(is)f(a)h FP(k)f FN($)d FP(i)h FN($)f FP(j)5
b FT(-in)m(terp)s(olan)m(t)29 b(for)g(\()p FN(f)p Fv(A)2628
476 y FO(i)2656 509 y FP(;)15 b FN(:)p Fv(C)2822 476
y FO(j)2859 509 y FN(g)p FP(;)g FN(f)p Fv(B)3057 476
y FO(k)3100 509 y FN(g)p FT(\))29 b(b)m(y)g(theorem)h(7.7,)378
622 y(although)g(this)f(prop)s(ert)m(y)h(is)f(not)i(required)d(for)i
(the)h(curren)m(t)f(pro)s(of.)378 735 y(No)m(w,)h FN(f)p
Fv(A)718 702 y FO(i)746 735 y FP(;)15 b FN(:)p Fv(C)912
702 y FO(j)949 735 y FP(;)g Fv(I)1032 702 y FO(k)1075
735 y FN(g)31 b FT(is)e FP(k)g FN($)c FP(i)g FN($)h FP(j)5
b FT(-inconsisten)m(t)30 b(if)f(and)h(only)g(if)1006
939 y FN(f)p Fv(A)1114 902 y FO(i)1142 939 y FP(;)15
b FN(:)p Fv(C)1308 902 y FO(j)1345 939 y FP(;)g Fv(I)1428
902 y FO(j)1465 939 y FN(g)91 b FT(is)29 b FP(i)d FN($)f
FP(j)5 b FT(-inconsisten)m(t)30 b(b)m(y)h(thm.)f(7.5)h(and)f(prop.)g
(7.7)832 1077 y FN(,)83 b(f)p Fv(A)1114 1040 y FO(i)1142
1077 y FP(;)15 b FN(:)p Ft(\()p Fv(I)30 b Fu(\))23 b
Fv(C)6 b Ft(\))1545 1040 y FO(j)1581 1077 y FN(g)92 b
FT(is)29 b FP(i)d FN($)f FP(j)5 b FT(-inconsisten)m(t)832
1215 y FN(,)83 b FP(A)25 b Ff(\032)1200 1178 y FK(\003)1265
1215 y FT(\()p FP(I)33 b FN(\))25 b FP(C)7 b FT(\))91
b(b)m(y)30 b(theorem)h(8.3)q FP(:)378 1419 y FT(Also,)f
FN(f)p Fv(B)721 1386 y FO(k)764 1419 y FP(;)15 b FN(:)p
Fv(I)908 1386 y FO(i)936 1419 y FN(g)30 b FT(is)g FP(k)e
FN($)d FP(i)h FN($)f FP(j)5 b FT(-inconsisten)m(t)31
b(if)e(and)h(only)f(if)1338 1624 y FN(f)p Fv(B)1451 1586
y FO(k)1493 1624 y FP(;)15 b FN(:)p Fv(I)1637 1586 y
FO(i)1665 1624 y FN(g)92 b FT(is)29 b FP(k)g FN($)c FP(i)p
FT(-inconsisten)m(t)30 b(b)m(y)g(prop.)g(7.7)1164 1761
y FN(,)83 b FP(B)30 b Ff(\032)1538 1724 y FK(\003)1602
1761 y FP(I)98 b FT(b)m(y)31 b(theorem)f(8.3)r FP(:)378
1966 y FT(Th)m(us,)g(\()p FP(S;)15 b FN(K)q FT(\))31
b(is)f(inconsisten)m(t)f(if)h(and)f(only)h(if)f(there)i(is)e(some)i
FP(I)37 b FT(suc)m(h)30 b(that)1811 2170 y FP(A)25 b
Ff(\032)2005 2132 y FK(\003)2070 2170 y FT(\()p FP(I)33
b FN(\))25 b FP(C)7 b FT(\))1805 2308 y FP(B)30 b Ff(\032)2005
2270 y FK(\003)2070 2308 y FP(I)378 2512 y FT(and)41
b(b)m(y)g(the)g(de\014nition)e(of)j(the)f(relation)g
Ff( )g FT(\(de\014nition)e(6.3\),)46 b(this)40 b(is)h(indeed)e(equiv)-5
b(alen)m(t)41 b(to)378 2625 y(whether)30 b Ft(\()p Fv(A)44
b Fw(on)f Fv(B)t Ft(\))25 b Ff( )g Fv(C)37 b FT(as)31
b(required.)1960 b Ff(\004)378 2837 y FQ(Prop)s(osition)36
b(8.9)46 b FI(F)-7 b(or)34 b(al)5 b(l)33 b(\014rst-or)-5
b(der)35 b(formulae)e FP(A)p FI(,)g FP(B)5 b FI(,)32
b FP(C)7 b FI(,)32 b(if)1588 3042 y Ft(\()p Fv(C)50 b
Fw(by)42 b Fv(A)i Fw(and)f Fv(B)t Ft(\))25 b FN(+)2320
3056 y FL(c)2381 3042 y FT(\()p FP(S;)15 b FN(K)q FT(\))378
3246 y FI(for)24 b(some)h(c)-5 b(olour)g(e)g(d)26 b(pr)-5
b(oblem)25 b FT(\()p FP(S;)15 b FN(K)q FT(\))p FI(,)27
b(then)e FP(S)j FI(is)c FN(K)q FI(-inc)-5 b(onsistent)25
b(if)f(and)g(only)h(if)e Ft(\()p Fv(A)45 b Fw(and)d Fv(B)t
Ft(\))26 b Ff( )f Fv(C)6 b FI(.)378 3458 y FQ(Pro)s(of)p
FT(:)31 b(As)g(illustrated)d(in)h(example)h(8.1\(2\),)1173
3663 y Ft(\()p Fv(C)50 b Fw(by)43 b Fv(A)g Fw(and)g Fv(B)t
Ft(\))25 b FN(+)1905 3677 y FL(c)1966 3663 y FT(\()p
FN(f)p Fv(A)2109 3625 y FO(i)2137 3663 y FP(;)15 b Fv(B)2244
3625 y FO(k)2287 3663 y FP(;)g FN(:)p Fv(C)2453 3625
y FO(j)2490 3663 y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s FN(g)28
b($)d FP(j)5 b FT(\))p FP(:)378 3867 y FT(By)31 b(corollary)e(8.1)j(if)
d Ft(\()p Fv(C)50 b Fw(by)43 b Fv(A)h Fw(and)e Fv(B)t
Ft(\))26 b FN(+)1860 3881 y FL(c)1920 3867 y FT(\()p
FP(S;)15 b FN(K)q FT(\))32 b(then)1371 4071 y(\()p FP(S;)15
b FN(K)q FT(\))27 b Fl(u)1705 4085 y FL(rc)1793 4071
y FT(\()p FN(f)p Fv(A)1936 4033 y FO(i)1964 4071 y FP(;)15
b Fv(B)2072 4033 y FO(k)2114 4071 y FP(;)g FN(:)p Fv(C)2280
4033 y FO(j)2317 4071 y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s
FN(g)28 b($)d FP(j)5 b FT(\))378 4275 y(and)22 b(therefore)h(\()p
FP(S;)15 b FN(K)q FT(\))24 b(is)e(inconsisten)m(t)f(if)h(and)g(only)f
(if)h FN(f)p Fv(A)2375 4242 y FO(i)2403 4275 y FP(;)15
b Fv(B)2510 4242 y FO(k)2553 4275 y FP(;)g FN(:)p Fv(C)2719
4242 y FO(j)2756 4275 y FN(g)23 b FT(is)e FN(f)p FP(i;)15
b(k)s FN(g)27 b($)f FP(j)5 b FT(-inconsisten)m(t.)519
4388 y(The)30 b(set)h FN(f)p Fv(A)956 4355 y FO(i)984
4388 y FP(;)15 b Fv(B)1091 4355 y FO(k)1134 4388 y FP(;)g
FN(:)p Fv(C)1300 4355 y FO(j)1337 4388 y FN(g)31 b FT(can)f(b)s(e)g
(partitioned)f(in)m(to)1755 4592 y(\()p FN(f)p Fv(B)1903
4555 y FO(k)1946 4592 y FP(;)15 b FN(:)p Fv(C)2112 4555
y FO(j)2148 4592 y FN(g)p FP(;)g FN(f)p Fv(A)2342 4555
y FO(i)2370 4592 y FN(g)p FT(\))378 4797 y(whic)m(h)30
b(is)g(w)m(ell-coloured)g(with)g(resp)s(ect)h(to)h FN(f)p
FP(i;)15 b(k)s FN(g)29 b($)d FP(j)5 b FT(.)44 b(W)-8
b(e)32 b(can)g(therefore)f(use)g(theorem)h(7.8)g(to)378
4910 y(deduce)e(that)h FN(f)p Fv(A)986 4877 y FO(i)1014
4910 y FP(;)15 b Fv(B)1121 4877 y FO(k)1164 4910 y FP(;)g
FN(:)p Fv(C)1330 4877 y FO(j)1367 4910 y FN(g)31 b FT(is)e
FN(f)p FP(i;)15 b(k)s FN(g)27 b($)e FP(j)5 b FT(-inconsisten)m(t)31
b(if)e(and)h(only)f(if)1500 5114 y FN(f)p Fv(B)1613 5076
y FO(k)1656 5114 y FP(;)15 b FN(:)p Fv(C)1822 5076 y
FO(j)1858 5114 y FP(;)g Fv(I)1942 5076 y FO(i)1970 5114
y FN(g)91 b FT(and)g FN(f)p Fv(A)2451 5076 y FO(i)2480
5114 y FP(;)15 b FN(:)p Fv(I)2624 5076 y FO(j)2660 5114
y FN(g)378 5318 y FT(are)31 b(for)f(some)h(\014rst-order)e(form)m(ula)h
FP(I)7 b FT(.)40 b(The)30 b(set)h FN(f)p Fv(B)2210 5285
y FO(k)2253 5318 y FP(;)15 b FN(:)p Fv(C)2419 5285 y
FO(j)2455 5318 y FP(;)g Fv(I)2539 5285 y FO(i)2567 5318
y FN(g)31 b FT(can)f(b)s(e)g(partitioned)f(in)m(to)1765
5522 y(\()p FN(f:)p Fv(C)1971 5485 y FO(j)2008 5522 y
FP(;)15 b Fv(I)2091 5485 y FO(i)2119 5522 y FN(g)p FP(;)g
FN(f)p Fv(B)2318 5485 y FO(k)2361 5522 y FN(g)p FT(\))p
eop
%%Page: 169 179
169 178 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(169)378
396 y(whic)m(h)32 b(is)h(w)m(ell-coloured)f(with)g(resp)s(ect)i(to)g
FN(f)p FP(i;)15 b(k)s FN(g)33 b($)d FP(j)39 b FT(as)34
b(w)m(ell.)49 b(Hence,)36 b(b)m(y)d(theorem)h(7.8,)i(it)d(is)378
509 y(the)e(case)g(that)g FN(f)p Fv(B)1037 476 y FO(k)1079
509 y FP(;)15 b FN(:)p Fv(C)1245 476 y FO(j)1282 509
y FP(;)g Fv(I)1365 476 y FO(i)1394 509 y FN(g)30 b FT(is)g
FN(f)p FP(i;)15 b(k)s FN(g)27 b($)e FP(j)5 b FT(-inconsisten)m(t)30
b(if)g(and)g(only)f(if)1492 714 y FN(f:)p Fv(C)1663 676
y FO(j)1700 714 y FP(;)15 b Fv(I)1783 676 y FO(i)1811
714 y FP(;)g Fv(J)1906 676 y FO(k)1948 714 y FN(g)92
b FT(and)e FN(f)p Fv(B)2435 676 y FO(k)2477 714 y FP(;)15
b FN(:)p Fv(J)2632 676 y FO(j)2669 714 y FN(g)378 918
y FT(are)31 b(for)f(some)h(form)m(ula)e FP(J)9 b FT(.)378
1031 y(No)m(w,)31 b FN(f:)p Fv(C)782 998 y FO(j)818 1031
y FP(;)15 b Fv(I)902 998 y FO(i)930 1031 y FP(;)g Fv(J)1024
998 y FO(k)1067 1031 y FN(g)31 b FT(is)e FN(f)p FP(i;)15
b(k)s FN(g)27 b($)e FP(j)5 b FT(-inconsisten)m(t)31 b(if)e(and)h(only)f
(if)1014 1235 y FN(f:)p Fv(C)1186 1197 y FO(j)1222 1235
y FP(;)15 b Fv(I)1305 1197 y FO(i)1334 1235 y FP(;)g
Fv(J)1428 1197 y FO(i)1456 1235 y FN(g)92 b FT(is)29
b FP(i)d FN($)f FP(j)5 b FT(-inconsisten)m(t)30 b(b)m(y)g(thm.)h(7.5)g
(and)f(prop.)f(7.7)840 1373 y FN(,)83 b(f:)p Fv(C)1186
1335 y FO(j)1222 1373 y FP(;)15 b Ft(\()p Fv(I)26 b Fu(^)19
b Fv(J)8 b Ft(\))1516 1335 y FO(i)1544 1373 y FN(g)92
b FT(is)29 b FP(i)d FN($)f FP(j)5 b FT(-inconsisten)m(t)840
1511 y FN(,)83 b FT(\()p FP(I)28 b FN(^)20 b FP(J)9 b
FT(\))25 b Ff(\032)1418 1473 y FK(\003)1483 1511 y FP(C)98
b FT(b)m(y)30 b(theorem)h(8.3)q FP(:)378 1715 y FT(Also,)f
FN(f)p Fv(B)721 1682 y FO(k)764 1715 y FP(;)15 b FN(:)p
Fv(J)919 1682 y FO(j)955 1715 y FN(g)31 b FT(is)e FN(f)p
FP(i;)15 b(k)s FN(g)27 b($)e FP(j)5 b FT(-inconsisten)m(t)31
b(if)e(and)h(only)f(if)1323 1919 y FN(f)p Fv(B)1435 1882
y FO(k)1478 1919 y FP(;)15 b FN(:)p Fv(J)1633 1882 y
FO(j)1669 1919 y FN(g)92 b FT(is)29 b FP(k)g FN($)c FP(j)5
b FT(-inconsisten)m(t)30 b(b)m(y)g(prop.)g(7.7)1149 2057
y FN(,)83 b FP(B)29 b Ff(\032)1522 2019 y FK(\003)1587
2057 y FP(J)100 b FT(b)m(y)30 b(theorem)h(8.3)q FP(:)378
2261 y FT(And)e(also,)i FN(f)p Fv(A)893 2228 y FO(i)921
2261 y FP(;)15 b FN(:)p Fv(I)1065 2228 y FO(j)1101 2261
y FN(g)31 b FT(is)f FN(f)p FP(i;)15 b(k)s FN(g)27 b($)e
FP(j)5 b FT(-inconsisten)m(t)30 b(if)g(and)f(only)h(if)1347
2465 y FN(f)p Fv(A)1455 2428 y FO(i)1483 2465 y FP(;)15
b FN(:)p Fv(I)1627 2428 y FO(j)1664 2465 y FN(g)91 b
FT(is)29 b FP(i)d FN($)f FP(j)5 b FT(-inconsisten)m(t)31
b(b)m(y)f(prop.)f(7.7)1173 2603 y FN(,)83 b FP(A)26 b
Ff(\032)1542 2566 y FK(\003)1606 2603 y FP(I)98 b FT(b)m(y)31
b(theorem)f(8.3)r FP(:)378 2807 y FT(Th)m(us,)g(\()p
FP(S;)15 b FN(K)q FT(\))31 b(is)f(inconsisten)m(t)f(if)h(and)f(only)h
(if)f(there)i(are)g(form)m(ulae)f FP(I)37 b FT(and)30
b FP(J)39 b FT(suc)m(h)30 b(that)2043 3012 y FP(A)25
b Ff(\032)2237 2974 y FK(\003)2302 3012 y FP(I)2037 3149
y(B)30 b Ff(\032)2237 3112 y FK(\003)2302 3149 y FP(J)1833
3287 y FT(\()p FP(I)d FN(^)20 b FP(J)9 b FT(\))26 b Ff(\032)2237
3250 y FK(\003)2302 3287 y FP(C)378 3492 y FT(and)41
b(b)m(y)g(the)g(de\014nition)e(of)j(the)f(relation)g
Ff( )g FT(\(de\014nition)e(6.3\),)46 b(this)40 b(is)h(indeed)e(equiv)-5
b(alen)m(t)41 b(to)378 3604 y(whether)30 b Ft(\()p Fv(A)44
b Fw(and)e Fv(B)t Ft(\))26 b Ff( )f Fv(C)6 b FT(.)2387
b Ff(\004)378 3843 y FQ(An)44 b(Ov)m(erview)e(of)i(the)f(Pro)s(of)i(of)
e(the)g(Soundness)i(and)e(Completeness)g(Result)h(for)378
3956 y(the)34 b(General)h(Case)378 4127 y FT(In)23 b(the)h(previous)e
(t)m(w)m(o)j(prop)s(ositions)c(w)m(e)k(ha)m(v)m(e)g(sho)m(wn)e(that)h
(the)g(metho)s(d)f(of)h(c)m(hec)m(king)h(the)e(v)-5 b(alidit)m(y)378
4240 y(of)30 b(a)h(justi\014ed)e(conclusion)473 4418
y FP(C)102 b FM(by)48 b FP(P)378 4596 y FT(b)m(y)21 b(\014rst)f
(constructing)g(a)h(coloured)f(problem)f(\()p FP(S;)c
FN(K)q FT(\))23 b(and)d(then)g(sho)m(wing)g(that)h FP(S)26
b FT(is)20 b FN(K)q FT(-inconsisten)m(t)378 4708 y(is)k(sound)g(and)h
(complete)h(for)f(the)g(t)m(w)m(o)i(particular)d(cases)i(of)f
FP(P)39 b FT(=)25 b Fv(A)43 b Fw(on)g Fv(B)30 b FT(and)24
b FP(P)39 b FT(=)25 b Fv(A)43 b Fw(and)g Fv(B)29 b FT(for)378
4821 y(sen)m(tences)f FP(A)f FT(and)f FP(B)5 b FT(.)39
b(Our)26 b(goal)h(is)f(to)i(sho)m(w)f(that)g FP(P)40
b FT(justi\014es)25 b FP(C)34 b FT(if)25 b(and)i(only)f(if)g(the)h
(constructed)378 4934 y(coloured)40 b(problem)e(is)h(inconsisten)m(t)g
(for)h FI(any)49 b FT(structured)39 b(justi\014cation)g
FP(P)13 b FT(.)70 b(This)38 b(is)h(giv)m(en)h(in)378
5047 y(theorem)d(8.7)h(b)s(elo)m(w,)f(and)f(its)g(pro)s(of)g(pro)s
(ceeds)h(b)m(y)f(induction)e(on)j(the)g(structure)f(of)h
FP(P)13 b FT(,)38 b(whic)m(h)378 5160 y(requires)29 b(the)h(three)h
(cases:)514 5338 y FN(\017)46 b FT(the)31 b(base)f(case)i(where)e
FP(P)43 b FT(is)29 b(a)i(form)m(ula,)514 5521 y FN(\017)46
b FT(the)31 b(\014rst)e(inductiv)m(e)g(case)j(where)e
FP(P)43 b FT(is)29 b(some)i Fw(on)f FT(expression)f Fv(X)50
b Fw(on)42 b Fv(Y)19 b FT(,)514 5705 y FN(\017)46 b FT(the)31
b(second)f(inductiv)m(e)f(case)j(where)d FP(P)44 b FT(is)29
b(some)i Fw(and)e FT(expression)g Fv(X)50 b Fw(and)42
b Fv(Y)19 b FT(,)p eop
%%Page: 170 180
170 179 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(170)378
396 y(where)27 b FP(X)35 b FT(and)28 b FP(Y)47 b FT(are)29
b FI(structur)-5 b(e)g(d)38 b FT(expressions.)h(The)27
b(pro)s(of)g(of)h(the)g(base)g(case)h(is)e(quite)g(straigh)m(t-)378
509 y(forw)m(ard,)32 b(and)f(the)g(pro)s(ofs)g(of)h(the)g(t)m(w)m(o)h
(inductiv)m(e)d(cases)j(are)f(a)g(generalisation)e(of)i(the)g(pro)s
(ofs)f(of)378 622 y(prop)s(ositions)26 b(8.8)31 b(and)d(8.9)i(resp)s
(ectiv)m(ely)-8 b(,)29 b(where)g(the)g(structured)f(expressions)g
FP(X)36 b FT(and)28 b FP(Y)49 b FT(gener-)378 735 y(alise)32
b(the)g(sen)m(tences)i FP(A)e FT(and)g FP(B)5 b FT(.)47
b(In)32 b(this)f(section)i(w)m(e)g(iden)m(tify)e(the)h(results)g(whic)m
(h)f(are)i(required)378 848 y(in)h(order)h(to)i(generalise)e(the)g(pro)
s(of)g(of)h(prop)s(ositions)d(8.8)k(and)e(8.9)h(in)m(to)g(the)f(pro)s
(ofs)g(of)h(the)g(t)m(w)m(o)378 961 y(inductiv)m(e)29
b(cases.)41 b(Since)30 b(the)g(pro)s(ofs)g(of)g(the)h(t)m(w)m(o)g(prop)
s(ositions)d(are)j(quite)e(similar)f(w)m(e)j(only)e(con-)378
1074 y(sider)38 b(the)i(pro)s(of)e(of)i(the)f(case)i(where)e
FP(P)52 b FT(is)39 b(an)g Fw(on)o FT(-expression)g(here.)67
b(Ho)m(w)m(ev)m(er,)44 b(the)c(pro)s(of)f(of)378 1187
y(theorem)31 b(8.7)g(considers)e(b)s(oth)h(inductiv)m(e)f(cases)i(in)e
(detail.)519 1300 y(The)36 b(k)m(ey)i(step)e(in)g(the)g(pro)s(of)g(of)h
(b)s(oth)f(of)h(the)g(ab)s(o)m(v)m(e)h(prop)s(ositions)c(is)h(the)i
(partitioning)e(of)378 1413 y(the)e(set)h FP(S)k FT(in)m(to)33
b(some)g(appropriate)g(\()p FP(S)1778 1427 y FL(1)1817
1413 y FP(;)15 b(S)1913 1427 y FL(2)1953 1413 y FT(\))33
b(and)g(using)e(theorem)j(7.8)g(to)g(sho)m(wn)e(that)i
FP(S)k FT(is)32 b FN(K)q FT(-)378 1526 y(inconsisten)m(t)26
b(if)g(and)g(only)g(if)g FP(S)1458 1540 y FL(1)1511 1526
y FN([)13 b(f)p FP(I)1677 1493 y FO(m)1744 1526 y FN(g)27
b FT(and)g FP(S)2046 1540 y FL(2)2098 1526 y FN([)13
b(f:)p FP(I)2325 1493 y FO(n)2373 1526 y FN(g)27 b FT(are)g
FN(K)q FT(-inconsisten)m(t)g(for)g(some)g(colours)378
1638 y FP(n)36 b FT(and)g FP(m)p FT(,)i(and)e(sen)m(tence)i
FP(I)7 b FT(.)60 b(This)35 b(step)i(is)e(used)h(once)h(in)f(the)h(pro)s
(of)f(of)g(prop)s(osition)f(8.8)i(and)378 1751 y(t)m(wice)31
b(in)e(the)h(pro)s(of)g(of)h(prop)s(osition)d(8.9.)41
b(In)30 b(the)h(particular)d(case)k(of)e(prop)s(osition)e(8.8,)k(w)m(e)
f(ha)m(v)m(e)1190 1956 y(\()p Fv(C)50 b Fw(by)43 b Fv(A)g
Fw(on)g Fv(B)t FT(\))26 b FN(+)1885 1970 y FL(c)1945
1956 y FT(\()p FN(f)p FP(A)2093 1918 y FO(i)2122 1956
y FP(;)15 b(B)2236 1918 y FO(k)2279 1956 y FP(;)g FN(:)p
FP(C)2452 1918 y FO(j)2488 1956 y FN(g)p FP(;)g(k)29
b FN($)d FP(i)f FN($)g FP(j)5 b FT(\))378 2160 y(and)30
b FN(f)p FP(A)668 2127 y FO(i)697 2160 y FP(;)15 b(B)811
2127 y FO(k)853 2160 y FP(;)g FN(:)p FP(C)1026 2127 y
FO(j)1062 2160 y FN(g)31 b FT(is)e(partitioned)g(in)m(to)1609
2364 y(\()p FN(f)1689 2364 y
tx@Dict begin tx@NodeDict begin {9.80986 0.0 11.60646 5.80322 4.90492
} false /N@A 16 {InitRnode } NewNode end end
1689 2364 a FP(A)1757 2327
y FO(i)1787 2364 y FP(;)106 b FN(:)1979 2364 y
tx@Dict begin tx@NodeDict begin {9.80986 0.0 13.008 6.504 4.90492
} false /N@C 16 {InitRnode } NewNode end end
1979 2364
a FP(C)2051 2327 y FO(j)2086 2364 y FN(g)p FP(;)198 b
FN(f)2399 2364 y
tx@Dict begin tx@NodeDict begin {10.07672 0.0 13.99399 6.997 5.03836
} false /N@B 16 {InitRnode } NewNode end end
2399 2364 a FP(B)2473 2327 y FO(k)2516
2364 y FN(g)p FT(\))2596 2364 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
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
b(STR)m(UCTURED)h(JUSTIFICA)-8 b(TIONS)650 b FT(171)378
396 y(where)30 b FP(S)697 410 y FO(X)794 396 y FT(and)g
FP(S)1027 410 y FO(Y)1118 396 y FT(are)h(sets)f(of)h(coloured)f(sen)m
(tences)h(suc)m(h)f(that)987 601 y(\()p FN(f)p FP(X)1149
563 y FO(i)1179 601 y FP(;)15 b(Y)1292 563 y FO(k)1335
601 y FP(;)g FN(:)p FP(C)1508 563 y FO(j)1544 601 y FN(g)p
FP(;)g(k)29 b FN($)c FP(i)h FN($)f FP(j)5 b FT(\))26
b FN(+)2153 615 y FL(c)2213 601 y FT(\()p FP(S)2304 615
y FO(X)2392 601 y FN([)20 b FP(S)2529 615 y FO(Y)2610
601 y FN([)f(f:)p FP(C)2868 563 y FO(j)2904 601 y FN(g)p
FP(;)c FN(K)3059 563 y FK(0)3058 623 y FO(X)5 b(Y)3183
601 y FT(\))378 805 y(b)m(y)28 b(\014rst)e(breaking)h(up)g(all)f(the)i
(structured)f(expressions)f(in)h FP(X)2556 772 y FO(i)2612
805 y FT(and)g(then)g(those)i(in)d FP(Y)3402 772 y FO(k)3445
805 y FT(.)39 b(In)27 b(order)378 918 y(to)i(generalise)e(the)h(pro)s
(of)f(of)h(prop)s(osition)d(8.8,)30 b(w)m(e)e(need)g(to)h(b)s(e)e(able)
g(to)i(use)e(the)h(sets)g FP(S)3442 932 y FO(X)3537 918
y FT(and)f FP(S)3767 932 y FO(Y)378 1031 y FT(in)32 b(the)i(same)f(w)m
(a)m(y)i(that)f(w)m(e)g(used)e(the)i(sen)m(tences)g FP(A)g
FT(and)e FP(B)38 b FT(ab)s(o)m(v)m(e.)51 b(In)33 b(other)g(w)m(ords,)h
(w)m(e)g(need)378 1144 y(to)d(b)s(e)f(able)g(to)h(partition)e
FP(S)1351 1158 y FO(X)1438 1144 y FN([)20 b FP(S)1575
1158 y FO(Y)1656 1144 y FN([)f(f:)p FP(C)1914 1111 y
FO(j)1951 1144 y FN(g)30 b FT(in)m(to)1702 1348 y(\()p
FP(S)1793 1362 y FO(X)1881 1348 y FN([)19 b(f:)p FP(C)2139
1310 y FO(j)2175 1348 y FN(g)p FP(;)107 b(S)2408 1362
y FO(Y)2469 1348 y FT(\))378 1552 y(and)30 b(sho)m(w)g(that)h(it)f(is)f
FN(K)1229 1519 y FK(0)1228 1579 y FO(X)5 b(Y)1352 1552
y FT(-inconsisten)m(t)30 b(if)g(and)f(only)h(if)f(the)i(sets)1418
1756 y FP(S)1474 1770 y FO(X)1561 1756 y FN([)20 b(f:)p
FP(C)1820 1719 y FO(j)1856 1756 y FP(;)15 b(I)1943 1719
y FO(k)1986 1756 y FN(g)83 b FT(and)g FP(S)2400 1770
y FO(Y)2481 1756 y FN([)19 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)
g(w)m(ell-coloured)f(partitions)g(giv)m(en)h(in)f(section)i(7.5)378
1263 y(\(page)j(143\))h(to)f(consider)f(partitions)e(of)j(more)f(than)g
(t)m(w)m(o)i(elemen)m(ts.)59 b(The)36 b(motiv)-5 b(ation)36
b(for)g(the)378 1376 y(de\014nition)42 b(of)i(this)g(notion)f(is)h(men)
m(tioned)f(to)m(w)m(ards)i(the)g(end)f(of)g(the)h(previous)d(section,)
48 b(and)378 1489 y(\(informally\))28 b(in)m(v)m(olv)m(es)i(the)g
(abilit)m(y)f(to)i(use)f(the)g(sets)h(of)f(coloured)f(sen)m(tences)j
(in)d(a)h(w)m(ell-coloured)378 1602 y(partition)k FN(f)p
FP(S)863 1616 y FL(1)902 1602 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(S)1175 1616 y FO(n)1222 1602 y FN(g)36 b FT(in)e(the)h(same)h
(freedom)f(that)g(individual)c(coloured)k(sen)m(tences)h(can)g(b)s(e)
378 1715 y(used.)519 1828 y(A)d(partition)e FP(P)42 b
FT(=)28 b FN(f)p FP(S)1301 1842 y FL(1)1341 1828 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(S)1614 1842 y FO(n)1661 1828
y FN(g)33 b FT(of)f(a)h(set)g FP(S)38 b FT(of)32 b(coloured)g
(structured)g(expressions)f(is)h(w)m(ell-)378 1941 y(coloured)25
b(if)g(no)h(t)m(w)m(o)h(sets)f(in)e FP(P)39 b FT(ha)m(v)m(e)27
b(a)f(colour)f(in)g(common,)i(and)e(there)h(are)g(some)g(sets)h(of)e
(colours)378 2054 y FN(P)441 2068 y FO(x)510 2054 y FN(\022)g
Fe(C)p FT(\()p FP(S)753 2068 y FO(x)797 2054 y FT(\))k(for)f(ev)m(ery)h
FP(S)1292 2068 y FO(x)1361 2054 y FN(2)c FP(P)41 b FT(suc)m(h)28
b(that)i(if)d FP(S)2082 2068 y FO(x)2151 2054 y FN(\031)2222
2068 y FK(K)2305 2054 y FP(S)2361 2068 y FO(y)2431 2054
y FT(for)h(distinct)f FP(S)2951 2068 y FO(x)3023 2054
y FT(and)h FP(S)3254 2068 y FO(y)3323 2054 y FT(in)f
FP(P)42 b FT(then)28 b(all)378 2166 y(the)33 b(colours)e(in)g
FN(P)1018 2180 y FO(x)1095 2166 y FT(relate)i(with)e(all)g(the)i
(colours)f(in)f FN(P)2328 2180 y FO(y)2369 2166 y FT(,)i(and)f(no)g
(other)h(colour)f(in)f Fe(C)p FT(\()p FP(S)3504 2180
y FO(x)3548 2166 y FT(\))i(apart)378 2279 y(from)h(the)g(colours)g(in)f
FN(P)1243 2293 y FO(x)1322 2279 y FT(relates)h(with)f(the)i(colours)f
(in)f FP(S)39 b FT(that)c(are)g(not)g(in)e FP(S)3193
2293 y FO(x)3236 2279 y FT(.)53 b(This)32 b(is)i(giv)m(en)378
2392 y(more)c(formally)f(b)s(elo)m(w:)378 2605 y FQ(De\014nition)35
b(8.8)h(\(W)-9 b(ell-Coloured)35 b(P)m(artition\))45
b FT(A)25 b(\014nite)g(set)h(of)f(sets)h FP(P)38 b FT(=)25
b FN(f)p FP(S)3287 2619 y FL(1)3327 2605 y FP(;)15 b(S)3423
2619 y FL(2)3463 2605 y FP(;)g(:)g(:)g(:)31 b(;)15 b(S)3735
2619 y FO(n)3782 2605 y FN(g)378 2718 y FT(is)27 b(said)g(to)h(b)s(e)g
(a)g(w)m(ell-coloured)e(partition)h(of)h(a)g(set)h FP(S)j
FT(of)c(coloured)g(structured)f(expressions)f(with)378
2831 y(resp)s(ect)k(to)h(a)g(connectabilit)m(y)f(relation)g
FN(K)q FT(,)h(if)489 3018 y(1.)605 2950 y Fx(S)696 3018
y FP(P)38 b FT(=)25 b FP(S)5 b FT(,)489 3206 y(2.)46
b(for)30 b(all)g FP(x;)15 b(y)28 b FN(2)d(f)p FT(1)p
FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p FN(g)p FT(,)31 b(if)e
FP(x)c FN(6)p FT(=)g FP(y)33 b FT(then)d Fe(C)p FT(\()p
FP(S)2274 3220 y FO(x)2318 3206 y FT(\))21 b FN(\\)e
Fe(C)p FT(\()p FP(S)2601 3220 y FO(y)2643 3206 y FT(\))25
b(=)g FN(fg)p FT(.)489 3421 y(3.)46 b(for)34 b(all)f(distinct)f
FP(x;)15 b(y)s(;)g(z)36 b FN(2)31 b(f)p FT(1)p FP(;)15
b(:)g(:)g(:)33 b(;)15 b(n)p FN(g)p FT(,)35 b(if)e(\()p
FP(S)2206 3435 y FO(x)2300 3370 y FK(K)2281 3421 y FN(!)e
FP(S)2459 3435 y FO(y)2501 3421 y FT(\))g FN(6)p FT(=)g
FN(fg)k FT(and)e(\()p FP(S)3065 3435 y FO(x)3159 3370
y FK(K)3140 3421 y FN(!)e FP(S)3318 3435 y FO(z)3358
3421 y FT(\))g FN(6)p FT(=)g FN(fg)k FT(then)605 3562
y(\()p FP(S)696 3576 y FO(x)784 3511 y FK(K)765 3562
y FN(!)26 b FP(S)938 3576 y FO(y)979 3562 y FT(\))f(=)g(\()p
FP(S)1226 3576 y FO(x)1314 3511 y FK(K)1296 3562 y FN(!)g
FP(S)1468 3576 y FO(z)1507 3562 y FT(\).)489 3778 y(4.)46
b(for)37 b(all)g FP(x;)15 b(y)40 b FN(2)c(f)p FT(1)p
FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p FN(g)p FT(,)40 b(if)c
FP(S)1778 3792 y FO(x)1859 3778 y FN(\031)1930 3792 y
FK(K)2024 3778 y FP(S)2080 3792 y FO(y)2159 3778 y FT(then)h(for)g(ev)m
(ery)h(colour)f FP(i)g FN(2)g FT(\()p FP(S)3303 3792
y FO(x)3402 3726 y FK(K)3384 3778 y FN(!)g FP(S)3568
3792 y FO(y)3609 3778 y FT(\))g(and)605 3919 y FP(j)31
b FN(2)25 b FT(\()p FP(S)850 3933 y FO(x)938 3867 y FK(K)919
3919 y FN( )g FP(S)1091 3933 y FO(y)1133 3919 y FT(\))30
b(it)g(is)g(the)g(case)h(that)g FP(i)26 b FN(\030)2054
3933 y FK(K)2137 3919 y FP(j)5 b FT(.)1553 b Ff(\003)378
4131 y FT(W)-8 b(e)32 b(illustrate)c(the)j(ab)s(o)m(v)m(e)g
(de\014nition)d(with)h(the)i(follo)m(wing)e(example.)378
4344 y FQ(Example)34 b(8.2)h(\(W)-9 b(ell-Coloured)35
b(P)m(artition\))45 b FT(Let)31 b(the)g(sets)g FP(S)2762
4358 y FL(1)2801 4344 y FT(,)f FP(S)2912 4358 y FL(2)2982
4344 y FT(and)g FP(S)3215 4358 y FL(3)3284 4344 y FT(b)s(e)1730
4548 y FP(S)1786 4562 y FL(1)1851 4548 y FT(=)25 b FN(f)p
FP(A)2060 4511 y FO(i)2089 4548 y FP(;)15 b(B)2203 4511
y FO(j)2239 4548 y FN(g)1730 4695 y FP(S)1786 4709 y
FL(2)1851 4695 y FT(=)25 b FN(f)p FP(C)2064 4657 y FO(k)2106
4695 y FP(;)15 b(D)2224 4657 y FO(l)2251 4695 y FP(;)g(E)2363
4657 y FO(m)2430 4695 y FN(g)1730 4833 y FP(S)1786 4847
y FL(3)1851 4833 y FT(=)25 b FN(f)p FP(F)2063 4795 y
FO(n)2110 4833 y FP(;)15 b(G)2222 4795 y FO(o)2261 4833
y FN(g)p FP(;)378 5037 y FT(for)31 b(some)g(form)m(ulae)f
FP(A)p FT(,)i FP(B)5 b FT(,)31 b FP(C)7 b FT(,)30 b FP(D)s
FT(,)h FP(E)5 b FT(,)32 b FP(F)44 b FT(and)30 b FP(G)g
FT(and)h(distinct)e(colours)h FP(i)p FT(,)p FP(j)5 b
FT(,)p FP(k)s FT(,)p FP(l)r FT(,)p FP(m)p FT(,)p FP(n)33
b FT(and)d FP(o)p FT(.)42 b(Let)378 5150 y(the)31 b(connectabilit)m(y)e
(relation)h FN(K)i FT(b)s(e)1268 5354 y FN(K)26 b FT(=)f(\()p
FP(i)h FN($)g FP(j)5 b FT(\))21 b FN([)f FT(\()p FP(l)27
b FN($)e FP(m)p FT(\))c FN([)e FT(\()p FP(j)32 b FN($)25
b(f)p FP(k)s(;)15 b(l)r(;)g(n;)g(o)p FN(g)p FT(\))p FP(;)p
eop
%%Page: 173 183
173 182 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(173)378
396 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. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2428 1373 a 378 1577 a FT(No)m(w,)i(if)f(w)m(e)g(let)h
FP(P)38 b FT(=)25 b FN(f)p FP(S)1252 1591 y FL(1)1292
1577 y FP(;)15 b(S)1388 1591 y FL(2)1427 1577 y FP(;)g(S)1523
1591 y FL(3)1563 1577 y FN(g)p FT(,)31 b(then)489 1729
y(1.)605 1661 y Fx(S)696 1729 y FP(P)38 b FT(=)25 b FP(S)5
b FT(,)489 1902 y(2.)46 b Fe(C)p FT(\()p FP(S)752 1916
y FL(1)791 1902 y FT(\))21 b FN(\\)f Fe(C)p FT(\()p FP(S)1075
1916 y FL(2)1114 1902 y FT(\))56 b(=)f Fe(C)p FT(\()p
FP(S)1478 1916 y FL(1)1517 1902 y FT(\))21 b FN(\\)f
Fe(C)p FT(\()p FP(S)1801 1916 y FL(3)1840 1902 y FT(\))56
b(=)f Fe(C)p FT(\()p FP(S)2204 1916 y FL(2)2243 1902
y FT(\))21 b FN(\\)f Fe(C)p FT(\()p FP(S)2527 1916 y
FL(3)2566 1902 y FT(\))56 b(=)f FN(fg)p FT(,)489 2076
y(3.)46 b(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(follo)m(wing:)1626
2299 y(\()p FP(S)1717 2313 y FL(1)1800 2247 y FK(K)1782
2299 y FN(!)25 b FP(S)1954 2313 y FL(2)1993 2299 y FT(\))h(=)f(\()p
FP(S)2241 2313 y FL(1)2324 2247 y FK(K)2306 2299 y FN(!)g
FP(S)2478 2313 y FL(3)2517 2299 y FT(\))h(=)f FN(f)p
FP(j)5 b FN(g)1626 2468 y FT(\()p FP(S)1717 2482 y FL(2)1800
2416 y FK(K)1782 2468 y FN(!)25 b FP(S)1954 2482 y FL(1)1993
2468 y FT(\))h(=)f FN(f)p FP(k)s(;)15 b(l)r FN(g)1626
2637 y FT(\()p FP(S)1717 2651 y FL(3)1800 2586 y FK(K)1782
2637 y FN(!)25 b FP(S)1954 2651 y FL(1)1993 2637 y FT(\))h(=)f
FN(f)p FP(n;)15 b(o)p FN(g)1626 2806 y FT(\()p FP(S)1717
2820 y FL(2)1800 2755 y FK(K)1782 2806 y FN(!)25 b FP(S)1954
2820 y FL(3)1993 2806 y FT(\))h(=)f(\()p FP(S)2241 2820
y FL(3)2324 2755 y FK(K)2306 2806 y FN(!)g FP(S)2478
2820 y FL(2)2517 2806 y FT(\))h(=)f FN(fg)605 3010 y
FT(and)30 b(therefore)h(the)f(partition)f FP(P)44 b FT(of)30
b FP(S)36 b FT(satis\014es)29 b(the)i(third)d(condition)h(in)h
(de\014nition)e(8.8.)489 3184 y(4.)46 b(It)31 b(is)e(the)i(case)g(that)
1440 3388 y FP(S)1496 3402 y FL(1)1560 3388 y FN(\031)1631
3402 y FK(K)1714 3388 y FP(S)1770 3402 y FL(2)1809 3388
y FP(;)198 b(S)2088 3402 y FL(1)2152 3388 y FN(\031)2223
3402 y FK(K)2306 3388 y FP(S)2362 3402 y FL(3)2401 3388
y FP(;)g(S)2680 3402 y FL(2)2744 3388 y FN(6\031)2815
3402 y FK(K)2898 3388 y FP(S)2954 3402 y FL(3)605 3592
y FT(and)30 b(th)m(us)g(the)h(fourth)e(condition)g(in)g(de\014nition)f
(8.8)k(is)d(also)h(satis\014ed.)378 3744 y(As)g(a)h(result,)f(the)g
(partition)f FP(P)43 b FT(is)30 b(w)m(ell-coloured)f(with)g(resp)s(ect)
i(to)g FN(K)q FT(.)519 3857 y(W)-8 b(e)32 b(also)e(note)h(that)g(if)e
(w)m(e)i(de\014ne)1798 4061 y FN(K)1868 4024 y FK(0)1917
4061 y FT(=)25 b FN(K)d([)e FP(i)25 b FN($)h FP(k)378
4265 y FT(then)k(the)h(partition)e FP(P)43 b FT(is)29
b(not)i(w)m(ell-coloured)e(with)g(resp)s(ect)i(to)g FN(K)2723
4233 y FK(0)2777 4265 y FT(since)1738 4495 y(\()p FP(S)1829
4509 y FL(1)1901 4443 y FK(K)1955 4420 y FD(0)1894 4495
y FN(!)25 b FP(S)2066 4509 y FL(2)2106 4495 y FT(\))g(=)g
FN(f)p FP(i;)15 b(j)5 b FN(g)1738 4669 y FT(\()p FP(S)1829
4683 y FL(1)1901 4618 y FK(K)1955 4594 y FD(0)1894 4669
y FN(!)25 b FP(S)2066 4683 y FL(3)2106 4669 y FT(\))g(=)g
FN(f)p FP(j)5 b FN(g)378 4911 y FT(and)34 b(so)g(\()p
FP(S)765 4925 y FL(1)843 4859 y FK(K)897 4836 y FD(0)836
4911 y FN(!)d FP(S)1014 4925 y FL(2)1054 4911 y FT(\))h
FN(6)p FT(=)f(\()p FP(S)1314 4925 y FL(1)1392 4859 y
FK(K)1446 4836 y FD(0)1385 4911 y FN(!)g FP(S)1563 4925
y FL(3)1603 4911 y FT(\))j(and)g(as)g(a)g(result)g(the)g(third)e
(condition)h(in)g(de\014nition)f(8.8)j(is)378 5024 y(not)c
(satis\014ed.)2872 b Ff(\003)519 5193 y FT(In)33 b(the)g(follo)m(wing)f
(prop)s(osition,)g(it)h(is)f(sho)m(wn)h(that)h(the)f(fourth)g
(condition)f(in)g(de\014nition)f(8.8)378 5306 y(can)g(b)s(e)e(replaced)
h(with)f(the)i(equation)966 5529 y FN(K)q(d)p FP(S)5
b FN(e)27 b FT(=)1363 5443 y Fx([)1300 5639 y FL(1)p
FK(\024)p FO(x)p FK(\024)p FO(n)1542 5529 y FN(K)q(d)p
FP(S)1708 5543 y FO(x)1753 5529 y FN(e)51 b([)2019 5443
y Fx([)1956 5654 y FL(1)p FK(\024)p FO(x)p FK(\024)p
FO(n)1955 5729 y(x<y)r FK(\024)p FO(n)2185 5529 y FT(\(\()p
FP(S)2311 5543 y FO(x)2399 5478 y FK(K)2380 5529 y FN(!)25
b FP(S)2552 5543 y FO(y)2594 5529 y FT(\))g FN($)g FT(\()p
FP(S)2861 5543 y FO(x)2949 5478 y FK(K)2931 5529 y FN( )g
FP(S)3103 5543 y FO(y)3144 5529 y FT(\)\))p FP(:)p eop
%%Page: 174 184
174 183 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(174)378
396 y(In)40 b(other)h(w)m(ords,)i(the)e(subrelation)e(of)i
FN(K)i FT(relev)-5 b(an)m(t)41 b(to)g FP(S)46 b FT(consists)41
b(of)g(the)g(subrelations)d(of)j FN(K)378 509 y FT(relev)-5
b(an)m(t)34 b(to)g(the)g(elemen)m(ts)g(of)g(the)g(partition,)f
(together)i(with)d(the)i(full-connections)e(of)i FN(P)3604
523 y FO(x)3681 509 y FT(and)378 622 y FN(P)441 636 y
FO(y)516 622 y FT(for)f(eac)m(h)i FP(S)922 636 y FO(x)996
622 y FN(\031)1067 636 y FK(K)1156 622 y FP(S)1212 636
y FO(y)1286 622 y FT(where)f FN(P)1616 636 y FO(x)1693
622 y FT(is)f(the)g(set)i(of)e(colours)g(in)g FP(S)2677
636 y FO(x)2754 622 y FT(that)h(relate)g(with)e(an)m(y)i(of)g(the)378
735 y(colours)j(in)g FP(S)30 b FN(\000)25 b FP(S)1046
749 y FO(x)1089 735 y FT(,)40 b(and)d(the)h FN(P)1565
749 y FO(y)1645 735 y FT(is)f(the)h(set)g(of)g(colours)f(in)g
FP(S)2654 749 y FO(y)2732 735 y FT(that)i(relate)f(with)f(an)m(y)h(of)g
(the)378 848 y(colours)f(in)e FP(S)30 b FN(\000)24 b
FP(S)1043 862 y FO(y)1085 848 y FT(.)60 b(Note)39 b(that)e(this)f
(result)g(is)h(a)g(generalisation)f(of)i(prop)s(osition)c(7.12.)63
b(This)378 961 y(c)m(haracterisation)41 b(of)g FN(K)q(d)p
FP(S)5 b FN(e)41 b FT(is)f(an)g(imp)s(ortan)m(t)f(to)s(ol)i(for)f
(manipulating)e(expressions)h(denoting)378 1074 y(connectabilit)m(y)d
(relations)g(during)e(the)j(pro)s(ofs)e(in)g(this)h(section,)i(as)f(w)m
(ell)f(as)g(in)g(visualising)d(the)378 1187 y(w)m(a)m(y)e(particular)e
(subsets)h(of)g(sets)h(in)e(coloured)h(problems)f(connect)i(with)e(eac)
m(h)j(other.)378 1354 y FQ(Prop)s(osition)k(8.10)46 b
FI(F)-7 b(or)34 b(every)f(set)f(of)h(sets)g FP(P)38 b
FT(=)25 b FN(f)p FP(S)2305 1368 y FL(1)2345 1354 y FP(;)15
b(S)2441 1368 y FL(2)2481 1354 y FP(;)g(:)g(:)g(:)31
b(;)15 b(S)2753 1368 y FO(n)2800 1354 y FN(g)33 b FI(such)g(that)485
1504 y(1.)605 1436 y Fx(S)696 1504 y FP(P)38 b FT(=)25
b FP(S)38 b FI(for)33 b(some)g(set)g FP(S)5 b FI(,)485
1676 y(2.)46 b(for)33 b(al)5 b(l)34 b FP(x;)15 b(y)28
b FN(2)d(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33 b(;)15 b(n)p
FN(g)p FI(,)32 b(if)h FP(x)25 b FN(6)p FT(=)g FP(y)35
b FI(then)e Fe(C)p FT(\()p FP(S)2291 1690 y FO(x)2335
1676 y FT(\))20 b FN(\\)g Fe(C)p FT(\()p FP(S)2618 1690
y FO(y)2659 1676 y FT(\))26 b(=)f FN(fg)p FI(.)378 1826
y(then)514 1976 y FN(\017)46 b FI(for)35 b(al)5 b(l)34
b FP(x;)15 b(y)30 b FN(2)d(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33
b(;)15 b(n)p FN(g)p FI(,)34 b(if)g FP(S)1754 1990 y FO(x)1825
1976 y FN(\031)1896 1990 y FK(K)1981 1976 y FP(S)2037
1990 y FO(y)2112 1976 y FI(then)g FP(i)28 b FN(\030)2445
1990 y FK(K)2530 1976 y FP(j)39 b FI(for)c(every)e(c)-5
b(olour)35 b FP(i)28 b FN(2)f FT(\()p FP(S)3506 1990
y FO(x)3596 1924 y FK(K)3577 1976 y FN(!)g FP(S)3751
1990 y FO(y)3793 1976 y FT(\))605 2117 y FI(and)34 b
FP(j)d FN(2)25 b FT(\()p FP(S)1027 2131 y FO(x)1114 2065
y FK(K)1096 2117 y FN( )g FP(S)1268 2131 y FO(y)1309
2117 y FT(\))p FI(,)378 2266 y(if)32 b(and)i(only)f(if)514
2435 y FN(\017)46 b(K)q(d)p FP(S)5 b FN(e)27 b FT(=)1002
2349 y Fx([)939 2545 y FL(1)p FK(\024)p FO(x)p FK(\024)p
FO(n)1182 2435 y FN(K)q(d)p FP(S)1348 2449 y FO(x)1392
2435 y FN(e)53 b([)1663 2349 y Fx([)1600 2560 y FL(1)p
FK(\024)p FO(x)p FK(\024)p FO(n)1599 2635 y(x<y)r FK(\024)p
FO(n)1828 2435 y FT(\(\()p FP(S)1954 2449 y FO(x)2042
2384 y FK(K)2024 2435 y FN(!)25 b FP(S)2196 2449 y FO(y)2237
2435 y FT(\))h FN($)f FT(\()p FP(S)2505 2449 y FO(x)2593
2384 y FK(K)2574 2435 y FN( )g FP(S)2746 2449 y FO(y)2787
2435 y FT(\)\))p FP(:)378 2782 y FQ(Pro)s(of)p FT(:)30
b(First)e(of)h(all)f(w)m(e)h(note)h(that)f(w)m(e)g(do)g(not)g(need)g
(the)g(third)e(condition)g(in)h(de\014nition)e(8.8)k(for)378
2895 y(the)e(conclusion)f(of)i(this)e(prop)s(osition)f(to)i(hold.)39
b(The)28 b(follo)m(wing)f(pro)s(of)g(is)g(similar)f(to)j(the)f(pro)s
(of)g(of)378 3008 y(prop)s(osition)g(7.12)k(giv)m(en)e(on)h(page)g
(144.)42 b(Our)29 b(goal)i(is)e(to)i(sho)m(w)g(that)1442
3145 y Fx([)1358 3342 y FO(S)1401 3350 y Fy(x)1440 3342
y FK(\031)1495 3353 y FD(K)1546 3342 y FO(S)1589 3350
y Fy(y)1626 3232 y FT(\(\()p FP(S)1752 3246 y FO(x)1840
3180 y FK(K)1822 3232 y FN(!)25 b FP(S)1994 3246 y FO(y)2035
3232 y FT(\))h FN($)f FT(\()p FP(S)2303 3246 y FO(x)2390
3180 y FK(K)2372 3232 y FN( )g FP(S)2544 3246 y FO(y)2585
3232 y FT(\)\))h FN(\022)f(K)866 b FT(\(1\))378 3532
y(if)29 b(and)h(only)g(if)966 3755 y FN(K)q(d)p FP(S)5
b FN(e)27 b FT(=)1363 3669 y Fx([)1300 3865 y FL(1)p
FK(\024)p FO(x)p FK(\024)p FO(n)1542 3755 y FN(K)q(d)p
FP(S)1708 3769 y FO(x)1753 3755 y FN(e)51 b([)2019 3669
y Fx([)1956 3880 y FL(1)p FK(\024)p FO(x)p FK(\024)p
FO(n)1955 3955 y(x<y)r FK(\024)p FO(n)2185 3755 y FT(\(\()p
FP(S)2311 3769 y FO(x)2399 3704 y FK(K)2380 3755 y FN(!)25
b FP(S)2552 3769 y FO(y)2594 3755 y FT(\))g FN($)g FT(\()p
FP(S)2861 3769 y FO(x)2949 3704 y FK(K)2931 3755 y FN( )g
FP(S)3103 3769 y FO(y)3144 3755 y FT(\)\))p FP(:)473
b FT(\(2\))378 4144 y(W)-8 b(e)32 b(notice)e(that)h(\(1\))h(is)d(equiv)
-5 b(alen)m(t)30 b(to)1371 4286 y Fx([)1287 4483 y FO(S)1330
4491 y Fy(x)1369 4483 y FK(\031)1424 4494 y FD(K)1475
4483 y FO(S)1518 4491 y Fy(y)1555 4373 y FT(\(\()p FP(S)1681
4387 y FO(x)1769 4321 y FK(K)1751 4373 y FN(!)25 b FP(S)1923
4387 y FO(y)1964 4373 y FT(\))h FN($)f FT(\()p FP(S)2232
4387 y FO(x)2320 4321 y FK(K)2301 4373 y FN( )g FP(S)2473
4387 y FO(y)2514 4373 y FT(\)\))h FN(\022)f(K)q(d)p FP(S)5
b FN(e)378 4716 y FT(since)31 b(if)f(\()p FP(i;)15 b(j)5
b FT(\))29 b FN(2)d FT(\(\()p FP(S)1111 4730 y FO(x)1201
4665 y FK(K)1182 4716 y FN(!)h FP(S)1356 4730 y FO(y)1397
4716 y FT(\))g FN($)g FT(\()p FP(S)1668 4730 y FO(x)1757
4665 y FK(K)1739 4716 y FN( )g FP(S)1913 4730 y FO(y)1954
4716 y FT(\)\))32 b(for)f(some)h FP(S)2481 4730 y FO(x)2556
4716 y FT(and)e FP(S)2789 4730 y FO(y)2862 4716 y FT(then)h(b)s(oth)f
FP(i)i FT(and)f FP(j)36 b FT(are)c(in)378 4829 y(the)f(colours)e(of)i
(the)f(set)h FP(S)5 b FT(.)41 b(This)29 b(is)g(also)h(equiv)-5
b(alen)m(t)30 b(to)1371 4966 y Fx([)1307 5177 y FL(1)p
FK(\024)p FO(x)p FK(\024)p FO(n)1306 5252 y(x<y)r FK(\024)p
FO(n)1536 5052 y FT(\(\()p FP(S)1662 5066 y FO(x)1750
5001 y FK(K)1732 5052 y FN(!)25 b FP(S)1904 5066 y FO(y)1945
5052 y FT(\))g FN($)h FT(\()p FP(S)2213 5066 y FO(x)2300
5001 y FK(K)2282 5052 y FN( )f FP(S)2454 5066 y FO(y)2495
5052 y FT(\)\))h FN(\022)f(K)q(d)p FP(S)5 b FN(e)814
b FT(\(3\))378 5479 y(as)33 b(the)g(relation)f(\(\()p
FP(S)1113 5493 y FO(x)1205 5428 y FK(K)1187 5479 y FN(!)d
FP(S)1363 5493 y FO(y)1405 5479 y FT(\))g FN($)h FT(\()p
FP(S)1681 5493 y FO(x)1773 5428 y FK(K)1754 5479 y FN( )g
FP(S)1931 5493 y FO(y)1972 5479 y FT(\)\))j(for)g FP(S)2273
5493 y FO(x)2346 5479 y FN(6\031)2417 5493 y FK(K)2504
5479 y FP(S)2560 5493 y FO(y)2634 5479 y FT(is)f(empt)m(y)-8
b(.)49 b(No)m(w,)35 b(the)e(statemen)m(t)378 5592 y(\(3\))39
b(follo)m(ws)d(from)h(\(2\))i(b)m(y)f(the)f(standard)g(results)g(in)f
(set)i(theory)-8 b(.)63 b(T)-8 b(o)38 b(sho)m(w)f(the)h(con)m(v)m
(erse,)j(w)m(e)378 5705 y(assume)30 b(that)h(\(3\))h(holds)c(and)i
(deriv)m(e)g(the)h(follo)m(wing)e(t)m(w)m(o)i(statemen)m(ts:)p
eop
%%Page: 175 185
175 184 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(175)514
396 y FN(\017)46 b FT(The)30 b(statemen)m(t)1092 601
y FN(K)q(d)p FP(S)5 b FN(e)27 b(\022)1489 514 y Fx([)1426
710 y FL(1)p FK(\024)p FO(x)p FK(\024)p FO(n)1669 601
y FN(K)q(d)p FP(S)1835 615 y FO(x)1879 601 y FN(e)51
b([)2146 514 y Fx([)2082 726 y FL(1)p FK(\024)p FO(x)p
FK(\024)p FO(n)2081 800 y(x<y)r FK(\024)p FO(n)2311 601
y FT(\(\()p FP(S)2437 615 y FO(x)2525 549 y FK(K)2506
601 y FN(!)26 b FP(S)2679 615 y FO(y)2720 601 y FT(\))f
FN($)h FT(\()p FP(S)2988 615 y FO(x)3075 549 y FK(K)3057
601 y FN( )f FP(S)3229 615 y FO(y)3270 601 y FT(\)\))605
990 y(follo)m(ws)i(from)g(the)h(fact)h(that)f(if)f FP(i)f
FN(\030)1847 1004 y FK(K)1930 990 y FP(j)33 b FT(and)27
b FP(i;)15 b(j)32 b FN(2)25 b Fe(C)p FT(\()p FP(S)5 b
FT(\))28 b(then)f(either)g FP(i)h FT(and)g FP(j)33 b
FT(are)28 b(in)e(some)605 1103 y(set)32 b FP(S)804 1117
y FO(x)878 1103 y FT(in)e(whic)m(h)g(case)i(\()p FP(i;)15
b(j)5 b FT(\))28 b FN(2)1737 1035 y Fx(S)1828 1103 y
FN(K)q(d)p FP(S)1994 1117 y FO(x)2039 1103 y FN(e)j FT(or)g(else)g
(they)g(are)h(in)d(di\013eren)m(t)i(sets,)h FP(S)3479
1117 y FO(x)3553 1103 y FT(and)f FP(S)3787 1117 y FO(y)605
1244 y FT(sa)m(y)-8 b(,)32 b(in)d(whic)m(h)g(case)j(\()p
FP(i;)15 b(j)5 b FT(\))27 b FN(2)1635 1176 y Fx(S)1711
1244 y FT(\(\()p FP(S)1837 1258 y FO(x)1924 1192 y FK(K)1906
1244 y FN(!)e FP(S)2078 1258 y FO(y)2119 1244 y FT(\))h
FN($)f FT(\()p FP(S)2387 1258 y FO(x)2475 1192 y FK(K)2456
1244 y FN( )h FP(S)2629 1258 y FO(y)2670 1244 y FT(\)\).)514
1431 y FN(\017)46 b FT(The)30 b(statemen)m(t)1156 1549
y Fx([)1092 1745 y FL(1)p FK(\024)p FO(x)p FK(\024)p
FO(n)1335 1636 y FN(K)q(d)p FP(S)1501 1650 y FO(x)1546
1636 y FN(e)51 b([)1812 1549 y Fx([)1749 1761 y FL(1)p
FK(\024)p FO(x)p FK(\024)p FO(n)1748 1835 y(x<y)r FK(\024)p
FO(n)1977 1636 y FT(\(\()p FP(S)2103 1650 y FO(x)2191
1584 y FK(K)2173 1636 y FN(!)25 b FP(S)2345 1650 y FO(y)2386
1636 y FT(\))h FN($)f FT(\()p FP(S)2654 1650 y FO(x)2742
1584 y FK(K)2723 1636 y FN( )g FP(S)2895 1650 y FO(y)2937
1636 y FT(\)\))g FN(\022)g(K)q(d)p FP(S)5 b FN(e)605
2025 y FT(follo)m(ws)30 b(from)f(the)i(fact)g(that)f
FN(K)q(d)p FP(S)1816 2039 y FO(x)1861 2025 y FN(e)c(\022)f(K)q(d)p
FP(S)5 b FN(e)31 b FT(for)f(ev)m(ery)h FP(S)2700 2039
y FO(x)2769 2025 y FN(\022)25 b FP(S)35 b FT(and)30 b(from)f(the)i
(assump-)605 2138 y(tion)f(\(3\).)2825 b Ff(\004)378
2350 y FQ(Example)34 b(8.3)46 b FT(Let)31 b FN(f)p FP(S)1253
2364 y FL(1)1293 2350 y FP(;)15 b(S)1389 2364 y FL(2)1429
2350 y FP(;)g(S)1525 2364 y FL(3)1564 2350 y FN(g)32
b FT(b)s(e)e(a)i(w)m(ell-coloured)e(partition)g(of)h(some)h(set)g
FP(S)k FT(with)30 b(resp)s(ect)378 2463 y(to)h(a)g(connectabilit)m(y)f
(relation)f FN(K)j FT(suc)m(h)e(that)1354 2667 y FP(S)1410
2681 y FL(1)1474 2667 y FN(\031)1545 2681 y FK(K)1628
2667 y FP(S)1684 2681 y FL(2)1905 2667 y FP(S)1961 2681
y FL(1)2026 2667 y FN(\031)2097 2681 y FK(K)2180 2667
y FP(S)2236 2681 y FL(3)2457 2667 y FP(S)2513 2681 y
FL(2)2577 2667 y FN(6\031)2648 2681 y FK(K)2732 2667
y FP(S)2788 2681 y FL(3)2827 2667 y FP(:)378 2872 y FT(W)-8
b(e)32 b(can)e(denote)h(the)g(three)f(subsets)g(with)f(the)i(follo)m
(wing)d(\014gure:)1869 3076 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
1869 3076 a FP(S)1925 3090
y FL(1)2055 3076 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 3076 a FP(S)2111 3090 y FL(2)2241
3076 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
2241 3076 a FP(S)2297 3090 y FL(3)2337 3076 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 45. def /AngleB 135. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2337 3076 a 2337 3076 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 45. def /AngleB 135. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2337 3076 a 378 3280 a FT(whic)m(h)33
b(sho)m(ws)h(whic)m(h)f(subsets)h(ha)m(v)m(e)h(colours)f(that)h(relate)
f(with)f(eac)m(h)j(other.)52 b(It)35 b(is)e(the)i(case)g(b)m(y)378
3393 y(prop)s(osition)28 b(8.10)k(that)947 3597 y FN(K)q(d)p
FP(S)5 b FN(e)26 b FT(=)f FN(K)q(d)p FP(S)1446 3611 y
FL(1)1487 3597 y FN(e)20 b([)g(K)q(d)p FP(S)1794 3611
y FL(2)1834 3597 y FN(e)h([)f(K)q(d)p FP(S)2142 3611
y FL(3)2182 3597 y FN(e)g([)g FT(\()p FN(P)2421 3611
y FL(1)2486 3597 y FN($)26 b(P)2666 3611 y FL(2)2705
3597 y FT(\))21 b FN([)f FT(\()p FN(P)2940 3611 y FL(1)3005
3597 y FN($)25 b(P)3184 3611 y FL(3)3224 3597 y FT(\))378
3801 y(where)1528 4006 y FN(P)1591 4020 y FL(1)1656 4006
y FT(=)g(\()p FP(S)1843 4020 y FL(1)1926 3954 y FK(K)1908
4006 y FN(!)g FP(S)2080 4020 y FL(2)2119 4006 y FT(\))h(=)e(\()p
FP(S)2366 4020 y FL(1)2450 3954 y FK(K)2431 4006 y FN(!)h
FP(S)2603 4020 y FL(3)2643 4006 y FT(\))1528 4175 y FN(P)1591
4189 y FL(2)1656 4175 y FT(=)g(\()p FP(S)1843 4189 y
FL(2)1926 4123 y FK(K)1908 4175 y FN(!)g FP(S)2080 4189
y FL(1)2119 4175 y FT(\))1528 4344 y FN(P)1591 4358 y
FL(3)1656 4344 y FT(=)g(\()p FP(S)1843 4358 y FL(3)1926
4292 y FK(K)1908 4344 y FN(!)g FP(S)2080 4358 y FL(1)2119
4344 y FT(\))p FP(:)378 4548 y FT(Note)g(that)e(the)h(connection)f(b)s
(et)m(w)m(een)h(the)g(subsets)e FP(S)2234 4562 y FL(1)2296
4548 y FT(and)h FP(S)2522 4562 y FL(2)2584 4548 y FT(in)f(the)i
(diagram)f(ab)s(o)m(v)m(e)h(represen)m(ts)378 4661 y(the)35
b(full-connection)f FN(P)1221 4675 y FL(1)1294 4661 y
FN($)f(P)1481 4675 y FL(2)1521 4661 y FT(.)55 b(Similarly)-8
b(,)33 b(the)j(connection)f(b)s(et)m(w)m(een)h FP(S)3044
4675 y FL(1)3118 4661 y FT(and)f FP(S)3356 4675 y FL(3)3430
4661 y FT(represen)m(ts)378 4774 y FN(P)441 4788 y FL(1)506
4774 y FN($)25 b(P)685 4788 y FL(3)725 4774 y FT(.)37
b(The)22 b(subrelations)d FN(K)q(d)p FP(S)1629 4788 y
FL(1)1670 4774 y FN(e)p FT(,)24 b FN(K)q(d)p FP(S)1925
4788 y FL(2)1965 4774 y FN(e)e FT(and)f FN(K)q(d)p FP(S)2361
4788 y FL(3)2401 4774 y FN(e)h FT(are)g(not)g(represen)m(ted)g(in)e
(the)i(diagram.)519 4887 y(No)m(w,)31 b(if)1366 5000
y FP(S)1422 5014 y FL(1)1487 5000 y FN(6\031)1558 5014
y FK(K)1641 5000 y FP(S)1697 5014 y FL(2)1918 5000 y
FP(S)1974 5014 y FL(1)2038 5000 y FN(\031)2109 5014 y
FK(K)2193 5000 y FP(S)2249 5014 y FL(3)2470 5000 y FP(S)2526
5014 y FL(2)2590 5000 y FN(\031)2661 5014 y FK(K)2744
5000 y FP(S)2800 5014 y FL(3)378 5167 y FT(as)g(sho)m(wn)e(b)m(y)i(the)
f(follo)m(wing)f(diagram)1869 5371 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
1869 5371 a FP(S)1925
5385 y FL(1)2055 5371 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 5371 a FP(S)2111 5385 y
FL(2)2241 5371 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
2241 5371 a FP(S)2297 5385 y FL(3)2337
5371 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@S3 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
2337 5371 a 2337 5371 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@S2 /N@S3 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
2337 5371 a 378 5575 a
FT(then)947 5688 y FN(K)q(d)p FP(S)5 b FN(e)26 b FT(=)f
FN(K)q(d)p FP(S)1446 5702 y FL(1)1487 5688 y FN(e)20
b([)g(K)q(d)p FP(S)1794 5702 y FL(2)1834 5688 y FN(e)h([)f(K)q(d)p
FP(S)2142 5702 y FL(3)2182 5688 y FN(e)g([)g FT(\()p
FN(P)2421 5702 y FL(1)2486 5688 y FN($)26 b(P)2666 5702
y FL(3)2705 5688 y FT(\))21 b FN([)f FT(\()p FN(P)2940
5702 y FL(2)3005 5688 y FN($)25 b(P)3184 5702 y FL(3)3224
5688 y FT(\))p eop
%%Page: 176 186
176 185 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(176)378
396 y(where)1515 601 y FN(P)1578 615 y FL(1)1643 601
y FT(=)25 b(\()p FP(S)1830 615 y FL(1)1913 549 y FK(K)1895
601 y FN(!)g FP(S)2067 615 y FL(3)2106 601 y FT(\))1515
770 y FN(P)1578 784 y FL(2)1643 770 y FT(=)g(\()p FP(S)1830
784 y FL(2)1913 718 y FK(K)1895 770 y FN(!)g FP(S)2067
784 y FL(3)2106 770 y FT(\))1515 939 y FN(P)1578 953
y FL(3)1643 939 y FT(=)g(\()p FP(S)1830 953 y FL(3)1913
887 y FK(K)1895 939 y FN(!)g FP(S)2067 953 y FL(1)2106
939 y FT(\))h(=)f(\()p FP(S)2354 953 y FL(3)2437 887
y FK(K)2419 939 y FN(!)g FP(S)2591 953 y FL(2)2630 939
y FT(\))p FP(:)1067 b Ff(\003)519 1151 y FT(The)22 b(follo)m(wing)e
(prop)s(osition)g(states)j(that)g(the)f(application)f(of)h(the)g
(relation)f FN(!)3251 1165 y FL(c)3309 1151 y FT(on)h(a)g(coloured)378
1264 y(structured)38 b(problem)f(\()p FP(S;)15 b FN(K)q
FT(\))40 b(where)e FP(S)44 b FT(can)39 b(b)s(e)f(partitioned)f(in)m(to)
h(a)h(w)m(ell-coloured)f(partition,)378 1377 y(results)31
b(in)g(a)i(coloured)f(structured)g(problem)f(\()p FP(S)2116
1344 y FK(0)2139 1377 y FP(;)15 b FN(K)2249 1344 y FK(0)2273
1377 y FT(\))33 b(where)f FP(S)2667 1344 y FK(0)2723
1377 y FT(can)h(also)f(b)s(e)g(partitioned)f(in)m(to)378
1490 y(a)e(w)m(ell-coloured)f(partition.)39 b(F)-8 b(urthermore,)29
b(the)g(subsets)f(in)f(the)i(partition)f(of)h FP(S)3232
1457 y FK(0)3284 1490 y FT(are)g(connected)378 1603 y(with)g(resp)s
(ect)i(to)h FN(K)1079 1570 y FK(0)1133 1603 y FT(in)e(the)h(same)g(w)m
(a)m(y)h(that)f(the)g(subsets)f(in)f(the)i(partition)f(of)h
FP(S)k FT(connect)d(with)378 1716 y(eac)m(h)f(other)g(with)e(resp)s
(ect)i(to)g FN(K)q FT(.)378 1929 y FQ(Prop)s(osition)36
b(8.11)46 b FI(L)-5 b(et)30 b FT(\()p FP(S;)15 b FN(K)q
FT(\))31 b FI(b)-5 b(e)29 b(a)h(c)-5 b(olour)g(e)g(d)32
b(structur)-5 b(e)g(d)31 b(pr)-5 b(oblem,)32 b(and)e(let)g
FN(f)p FP(S)3315 1943 y FL(1)3354 1929 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(S)3627 1943 y FO(n)3674 1929 y FN(g)30 b FI(b)-5
b(e)378 2041 y(a)40 b(wel)5 b(l-c)-5 b(olour)g(e)g(d)42
b(p)-5 b(artition)42 b(of)e FP(S)k FI(with)d(r)-5 b(esp)g(e)g(ct)41
b(to)f FN(K)q FI(.)63 b(If)40 b FT(\()p FP(S;)15 b FN(K)q
FT(\))39 b FN(!)2857 2008 y FK(\003)2857 2064 y FL(c)2935
2041 y FT(\()p FP(S)3031 2008 y FK(0)3054 2041 y FP(;)15
b FN(K)3164 2008 y FK(0)3188 2041 y FT(\))40 b FI(then)g(ther)-5
b(e)41 b(ar)-5 b(e)378 2154 y(some)33 b(sets)g FP(S)850
2121 y FK(0)845 2179 y FL(1)884 2154 y FI(,)g FP(:)15
b(:)g(:)31 b FI(,)p FP(S)1170 2121 y FK(0)1165 2177 y
FO(n)1244 2154 y FI(such)i(that:)485 2342 y(1.)46 b(The)36
b(elements)h(in)f FP(S)1348 2309 y FK(0)1343 2364 y FO(x)1422
2342 y FI(for)g FT(1)c FN(\024)f FP(x)g FN(\024)g FP(n)36
b FI(have)g(b)-5 b(e)g(en)36 b(br)-5 b(oken)37 b(up)f(fr)-5
b(om)37 b(the)f(elements)h(in)e FP(S)3784 2356 y FO(x)605
2455 y FI(by)e(some)g(applic)-5 b(ations)36 b(of)c(the)h(r)-5
b(elation)35 b FN(!)2138 2469 y FL(c)2173 2455 y FI(.)485
2643 y(2.)46 b FP(S)661 2657 y FO(x)730 2643 y FN(\031)801
2657 y FK(K)884 2643 y FP(S)940 2657 y FO(y)1014 2643
y FI(if)32 b(and)i(only)f(if)g FP(S)1626 2610 y FK(0)1621
2665 y FO(x)1689 2643 y FN(\031)1760 2658 y FK(K)1814
2639 y FD(0)1866 2643 y FP(S)1927 2610 y FK(0)1922 2665
y FO(y)1963 2643 y FI(.)485 2830 y(3.)46 b FN(f)p FP(S)711
2797 y FK(0)706 2855 y FL(1)746 2830 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(S)1024 2797 y FK(0)1019 2853 y FO(n)1066 2830
y FN(g)33 b FI(is)f(a)h(wel)5 b(l-c)-5 b(olour)g(e)g(d)35
b(p)-5 b(artition)35 b(with)e(r)-5 b(esp)g(e)g(ct)35
b(to)e FN(K)2904 2797 y FK(0)2928 2830 y FI(.)378 3043
y FQ(Pro)s(of)p FT(:)e(The)e(statemen)m(t)j(of)e(this)e(prop)s(osition)
g(follo)m(ws)h(from)g(the)h(fact)h(that)f FN(!)3186 3010
y FK(\003)3186 3065 y FL(c)3255 3043 y FT(is)f(the)h(re\015exiv)m(e)378
3156 y(transitiv)m(e)c(closure)g(of)h FN(!)1270 3170
y FL(c)1332 3156 y FT(and)g(from)f(the)h(fact)h(that)f(the)g(ab)s(o)m
(v)m(e)h(three)f(results)e(hold)h(if)g(\()p FP(S;)15
b FN(K)q FT(\))27 b FN(!)3793 3170 y FL(c)378 3269 y
FT(\()p FP(S)474 3236 y FK(0)498 3269 y FP(;)15 b FN(K)608
3236 y FK(0)632 3269 y FT(\).)47 b(Without)33 b(loss)e(of)i(generalit)m
(y)g(w)m(e)g(can)f(assume)h(that)g(the)f(application)f(of)i(the)g
(relation)378 3381 y FN(!)469 3395 y FL(c)546 3381 y
FT(breaks)41 b(up)f(a)i(coloured)f(structured)f(expression)g(in)g
FP(S)2520 3395 y FL(1)2601 3381 y FT(as)h(illustrated)f(b)m(y)h(the)g
(follo)m(wing)378 3494 y(diagram.)1649 3699 y(\()1684
3699 y
tx@Dict begin tx@NodeDict begin {7.48248 1.64249 11.46454 5.73227
2.92 } false /N@N1 16 {InitRnode } NewNode end end
1684 3699 a FP(S)1740 3713 y FL(1)1800 3699 y
FN([)20 b FP(S)1937 3713 y FL(2)1996 3699 y FN([)g(\001)15
b(\001)g(\001)21 b([)f FP(S)2340 3713 y FO(n)2387 3699
y FP(;)15 b FN(K)q FT(\))1649 3974 y(\()1684 3974 y
tx@Dict begin tx@NodeDict begin {8.41812 2.93385 11.46454 5.73227
2.74213 } false /N@N2 16 {InitRnode } NewNode end end
1684
3974 a FP(S)1745 3941 y FK(0)1740 3999 y FL(1)1800 3974
y FN([)20 b FP(S)1937 3988 y FL(2)1996 3974 y FN([)g(\001)15
b(\001)g(\001)21 b([)f FP(S)2340 3988 y FO(n)2387 3974
y FP(;)15 b FN(K)2497 3937 y FK(0)2521 3974 y FT(\))2556
3974 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@N1 /N@N2 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
2556 3974 a 2556 3974 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
2556 3974 a 2550 4002
a FL(c)2556 3974 y
tx@Dict begin PutEnd end
2556 3974 a 378 4178 a FT(Therefore)30
b(our)g(goal)h(is)e(to)i(sho)m(w)g(that)g(there)f(is)g(some)g(set)h
FP(S)2523 4145 y FK(0)2518 4203 y FL(1)2588 4178 y FT(suc)m(h)f(that:)
489 4366 y(1.)46 b(There)30 b(is)g(some)g(elemen)m(t)h
FP(X)38 b FT(in)29 b FP(S)1798 4380 y FL(1)1867 4366
y FT(and)h(elemen)m(ts)h FP(X)2491 4380 y FL(1)2531 4366
y FT(,)f FP(X)2661 4380 y FL(2)2731 4366 y FT(in)f FP(S)2898
4333 y FK(0)2893 4390 y FL(1)2963 4366 y FT(suc)m(h)h(that)1683
4570 y FP(S)1739 4584 y FL(1)1798 4570 y FN(\000)20 b(f)p
FP(X)7 b FN(g)27 b FT(=)d FP(S)2244 4533 y FK(0)2239
4593 y FL(1)2299 4570 y FN(\000)c(f)p FP(X)2510 4584
y FL(1)2550 4570 y FP(;)15 b(X)2665 4584 y FL(2)2705
4570 y FN(g)605 4775 y FT(and)31 b(that)h(the)g(structured)f
(expression)f FP(X)39 b FT(is)31 b(brok)m(en)h(up)e(in)m(to)i
FP(X)2920 4789 y FL(1)2991 4775 y FT(and)f FP(X)3244
4789 y FL(2)3315 4775 y FT(b)m(y)h(the)g(appli-)605 4887
y(cation)f(of)f FN(!)1072 4901 y FL(c)1108 4887 y FT(.)489
5075 y(2.)46 b FP(S)661 5089 y FL(1)735 5075 y FN(\031)806
5089 y FK(K)898 5075 y FP(S)954 5089 y FO(x)1033 5075
y FT(if)35 b(and)g(only)g(if)g FP(S)1659 5042 y FK(0)1654
5099 y FL(1)1727 5075 y FN(\031)1798 5090 y FK(K)1852
5071 y FD(0)1913 5075 y FP(S)1969 5089 y FO(x)2048 5075
y FT(for)h(1)e FP(<)g(x)h FN(\024)f FP(n)p FT(;)k(and)d
FP(S)2925 5089 y FO(x)3003 5075 y FN(\031)3074 5089 y
FK(K)3166 5075 y FP(S)3222 5089 y FO(y)3299 5075 y FT(if)g(and)g(only)g
(if)605 5188 y FP(S)661 5202 y FO(x)730 5188 y FN(\031)801
5203 y FK(K)855 5184 y FD(0)907 5188 y FP(S)963 5202
y FO(y)1034 5188 y FT(for)30 b FP(x;)15 b(y)29 b FN(2)c(f)p
FT(2)p FP(;)15 b(:)g(:)g(:)32 b(;)15 b(n)p FN(g)p FT(.)489
5376 y(3.)46 b FN(f)p FP(S)711 5343 y FK(0)706 5400 y
FL(1)746 5376 y FP(;)15 b(S)842 5390 y FL(2)881 5376
y FP(;)g(:)g(:)g(:)32 b(;)15 b(S)1154 5390 y FO(n)1201
5376 y FN(g)31 b FT(is)f(a)g(w)m(ell-coloured)f(partition)h(of)g
FP(S)2536 5343 y FK(0)2590 5376 y FT(with)f(resp)s(ect)h(to)h
FN(K)3289 5343 y FK(0)3313 5376 y FT(.)378 5563 y(T)-8
b(o)23 b(pro)m(v)m(e)g(the)g(required)e(statemen)m(t)j(w)m(e)f
(consider)e(the)i(t)m(w)m(o)h(cases)f(where)f(the)h(coloured)f
(structured)378 5676 y(expression)h FP(X)32 b FT(is)23
b(\()p Fv(Y)63 b Fw(on)42 b Fv(Z)6 b FT(\))1376 5643
y FO(i)1429 5676 y FT(or)24 b(\()p Fv(Y)63 b Fw(and)42
b Fv(Z)6 b FT(\))1952 5643 y FO(i)2005 5676 y FT(for)24
b(some)g(colour)g FP(i)h FT(and)f(structured)f(expressions)g
FP(Y)p eop
%%Page: 177 187
177 186 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(177)378
396 y(and)21 b FP(Z)7 b FT(.)37 b(In)21 b(eac)m(h)i(case)g(the)f(set)g
FP(S)1504 363 y FK(0)1499 421 y FL(1)1560 396 y FT(is)e(\()p
FP(S)1733 410 y FL(1)1776 396 y FN(\000)s(f)p FP(X)7
b FN(g)p FT(\))s FN([)s(f)p Fv(Y)2236 363 y FO(i)2265
396 y FP(;)15 b Fv(Z)2368 363 y FO(j)2404 396 y FN(g)22
b FT(where)f(the)h(colour)g FP(j)27 b FT(is)20 b(new)i(to)g(\()p
FP(S;)15 b FN(K)q FT(\).)378 509 y(If)31 b FP(X)36 b
FT(=)27 b(\()p Fv(Y)63 b Fw(on)43 b Fv(Z)5 b FT(\))1053
476 y FO(i)1113 509 y FT(then)32 b FN(K)1392 476 y FK(0)1444
509 y FT(=)27 b FN(K)c([)e FP(i)28 b FN($)g FP(j)5 b
FT(,)33 b(and)e(if)g FP(X)k FT(=)28 b(\()p Fv(Y)62 b
Fw(and)43 b Fv(Z)5 b FT(\))2883 476 y FO(i)2944 509 y
FT(then)31 b FN(K)3222 476 y FK(0)3274 509 y FT(=)c FN(K)c([)e(K)3616
476 y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))3803 509 y FT(.)378
622 y(The)32 b(\014rst)h(required)e(result)h(\(\(1\))i(ab)s(o)m(v)m
(e\))h(follo)m(ws)d(easily)g(b)m(y)h(c)m(ho)s(osing)f
FP(X)2982 636 y FL(1)3052 622 y FT(=)d Fv(Y)3219 589
y FO(i)3280 622 y FT(and)j FP(X)3534 636 y FL(2)3603
622 y FT(=)e Fv(Z)3766 589 y FO(j)3803 622 y FT(.)378
735 y(The)g(second)g(and)g(third)f(results)g(are)h(also)h(straigh)m
(tforw)m(ard,)f(and)g(follo)m(w)g(from)f(the)i(facts)g(that:)514
923 y FN(\017)46 b FT(for)28 b(the)h(case)g(when)f FP(X)35
b FT(is)28 b(a)h(coloured)e Fw(on)h FT(expression,)g(the)g(colour)g
FP(j)34 b FT(is)28 b(new)f(to)j(\()p FP(S;)15 b FN(K)q
FT(\))29 b(and)605 1036 y(relates)i(only)e(with)g(the)i(colour)f
FP(i)h FT(in)e FN(K)1968 1003 y FK(0)1992 1036 y FT(,)h(and)g
FP(i)g FT(o)s(ccurs)h(only)e(in)g(the)i(subsets)e FP(S)3399
1050 y FL(1)3469 1036 y FT(and)h FP(S)3707 1003 y FK(0)3702
1060 y FL(1)3741 1036 y FT(;)514 1223 y FN(\017)46 b
FT(for)25 b(the)h(case)h(when)d FP(X)33 b FT(is)24 b(a)i(coloured)f
Fw(and)f FT(expression,)i(the)f(colour)g FP(j)31 b FT(is)25
b(also)g(new)g(to)h(\()p FP(S;)15 b FN(K)q FT(\))605
1336 y(and)30 b(relates)h(in)e FN(K)1247 1303 y FK(0)1301
1336 y FT(with)g(all)g(the)i(colours)f(in)f(\()p FP(S;)15
b FN(K)q FT(\))32 b(that)f(relate)f(with)f FP(i)i FT(in)e
FN(K)q FT(.)364 b Ff(\004)378 1549 y FQ(Example)34 b(8.4)46
b FT(Let)31 b FP(X)7 b FT(,)31 b FP(Y)50 b FT(b)s(e)30
b(structured)f(expressions,)g FP(C)37 b FT(b)s(e)30 b(a)h(sen)m(tence,)
g(and)f(let)1351 1753 y(\()p FN(f)p FP(X)1513 1716 y
FO(i)1543 1753 y FP(;)15 b(Y)1656 1716 y FO(k)1699 1753
y FP(;)g FN(:)p FP(C)1872 1716 y FO(j)1908 1753 y FN(g)p
FP(;)g(k)29 b FN($)c FP(i)h FN($)f FP(j)5 b FT(\))26
b FN(!)2552 1716 y FK(\003)2552 1776 y FL(c)2617 1753
y FT(\()p FP(S;)15 b FN(K)q FT(\))378 1957 y(for)42 b(some)g(coloured)f
(structured)g(problem)g(\()p FP(S;)15 b FN(K)q FT(\).)77
b(By)42 b(prop)s(osition)d(8.4,)46 b FN(K)h FT(=)d FN(K)q(d)p
FP(S)5 b FN(e)p FT(.)76 b(The)378 2070 y(partition)1653
2183 y FN(f)31 b(f)p FP(X)1856 2146 y FO(i)1885 2183
y FN(g)p FP(;)15 b FN(f)p FP(Y)2089 2146 y FO(k)2132
2183 y FN(g)p FP(;)g FN(f:)p FP(C)2395 2146 y FO(j)2432
2183 y FN(g)30 b(g)378 2350 y FT(of)k(the)g(set)g FN(f)p
FP(X)917 2317 y FO(i)946 2350 y FP(;)15 b(Y)1060 2317
y FO(k)1102 2350 y FP(;)g FN(:)p FP(C)1275 2317 y FO(j)1311
2350 y FN(g)35 b FT(is)d(w)m(ell-coloured)h(with)g(resp)s(ect)g(to)i
FP(k)f FN($)d FP(i)g FN($)g FP(j)5 b FT(,)36 b(and)d(therefore)h(b)m(y)
378 2463 y(prop)s(osition)28 b(8.11,)k(there)f(are)g(sets)f
FP(S)1691 2477 y FO(X)1758 2463 y FT(,)h FP(S)1870 2477
y FO(Y)1961 2463 y FT(and)f FP(S)2194 2477 y FK(:)p FO(C)2330
2463 y FT(suc)m(h)g(that:)489 2651 y(1.)46 b(The)35 b(elemen)m(ts)i(in)
d FP(S)1342 2665 y FO(X)1409 2651 y FT(,)k FP(S)1528
2665 y FO(Y)1624 2651 y FT(and)d FP(S)1862 2665 y FK(:)p
FO(C)2004 2651 y FT(ha)m(v)m(e)i(b)s(een)e(brok)m(en)g(up)g(b)m(y)h
(some)g(applications)e(of)605 2763 y FN(!)696 2777 y
FL(c)766 2763 y FT(from)g(the)g(elemen)m(ts)h(in)e FP(X)7
b FT(,)36 b FP(Y)20 b FT(,)35 b(and)f FN(f:)p FP(C)2266
2730 y FO(j)2302 2763 y FN(g)h FT(resp)s(ectiv)m(ely)-8
b(.)52 b(Since)33 b FN(:)p FP(C)40 b FT(is)34 b(neither)f(an)605
2876 y Fw(on)o FT(-expression,)d(nor)g(an)g Fw(and)o
FT(-expression,)g(then)g FP(S)2362 2890 y FK(:)p FO(C)2493
2876 y FT(=)25 b FN(f:)p FP(C)2767 2843 y FO(j)2803 2876
y FN(g)p FT(.)489 3064 y(2.)46 b(It)31 b(is)e(the)i(case)g(that)1479
3268 y FP(S)1535 3282 y FO(X)1627 3268 y FN(\031)1698
3282 y FK(K)1781 3268 y FP(S)1837 3282 y FO(Y)2072 3268
y FT(and)173 b FP(S)2448 3282 y FO(X)2541 3268 y FN(\031)2612
3282 y FK(K)2695 3268 y FN(f:)p FP(C)2873 3231 y FO(j)2909
3268 y FN(g)605 3472 y FT(but)30 b FP(S)828 3486 y FO(Y)914
3472 y FN(6\031)985 3486 y FK(K)1068 3472 y FN(f:)p FP(C)1246
3439 y FO(j)1282 3472 y FN(g)p FT(.)489 3660 y(3.)46
b(The)32 b(partition)f FN(f)p FP(S)1276 3674 y FO(X)1344
3660 y FP(;)15 b(S)1440 3674 y FO(Y)1501 3660 y FP(;)g
FN(f:)p FP(C)1719 3627 y FO(j)1755 3660 y FN(gg)34 b
FT(of)e FP(S)38 b FT(is)31 b(w)m(ell-coloured)g(with)g(resp)s(ect)i(to)
g FN(K)q FT(.)48 b(Th)m(us)31 b(no)605 3773 y(t)m(w)m(o)h(distinct)d
(sets)i(in)e(the)h(partition)f(ha)m(v)m(e)j(a)f(colour)f(in)f(common)h
(and)1762 3977 y FP(S)h FT(=)24 b FP(S)2000 3991 y FO(X)2088
3977 y FN([)c FP(S)2225 3991 y FO(Y)2305 3977 y FN([)g(f:)p
FP(C)2564 3940 y FO(j)2600 3977 y FN(g)p FP(:)605 4181
y FT(F)-8 b(rom)37 b(the)g(fact)h(that)f(no)g(colour)f(in)f
FN(K)j FT(relates)f(with)f(itself)f(\(b)m(y)i(prop)s(osition)d
(8.4\(3\)\))39 b(w)m(e)605 4294 y(deduce)30 b(that)1892
4407 y FN(K)q(df:)p FP(C)2180 4370 y FO(j)2217 4407 y
FN(ge)d FT(=)e FN(fg)p FP(;)605 4574 y FT(and)30 b(b)m(y)g(prop)s
(osition)e(8.10)k(w)m(e)f(get)1259 4778 y FN(K)c FT(=)e
FN(K)q(d)p FP(S)1617 4792 y FO(X)1685 4778 y FN(e)c([)f(K)q(d)p
FP(S)1993 4792 y FO(Y)2054 4778 y FN(e)h([)f FT(\()p
FN(P)2294 4792 y FO(X)2387 4778 y FN($)25 b FP(j)5 b
FT(\))21 b FN([)f FT(\()p FN(P)2780 4792 y FO(X)2873
4778 y FN($)25 b(P)3052 4792 y FO(Y)3113 4778 y FT(\))p
FP(;)605 4983 y FT(where)1507 5187 y FN(P)1570 5201 y
FO(X)1663 5187 y FT(=)g(\()p FP(S)1850 5201 y FO(X)1961
5135 y FK(K)1942 5187 y FN(!)h FP(S)2115 5201 y FO(Y)2175
5187 y FT(\))g(=)f(\()p FP(S)2423 5201 y FO(X)2534 5135
y FK(K)2515 5187 y FN(!)g(f:)p FP(C)2809 5149 y FO(j)2846
5187 y FN(g)p FT(\))1513 5356 y FN(P)1576 5370 y FO(Y)1663
5356 y FT(=)g(\()p FP(S)1850 5370 y FO(Y)1954 5304 y
FK(K)1936 5356 y FN(!)g FP(S)2108 5370 y FO(X)2175 5356
y FT(\))605 5560 y(and)30 b(it)g(is)f(the)i(case)g(that)1779
5673 y FN(f)p FP(j)5 b FN(g)27 b FT(=)e(\()p FN(f:)p
FP(C)2247 5636 y FO(j)2283 5673 y FN(g)2372 5622 y FK(K)2354
5673 y FN(!)g FP(S)2526 5687 y FO(X)2593 5673 y FT(\))p
FP(:)p eop
%%Page: 178 188
178 187 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(178)378
396 y(The)30 b(follo)m(wing)f(diagram)1438 593 y(\()1473
593 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.32498 6.66249 4.63118
} false /N@X 16 {InitRnode } NewNode end end
1473 593 a FP(X)1555 560 y FO(i)1765 593 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y 16 {InitRnode } NewNode end end
1765
593 a FP(Y)1838 560 y FO(k)2055 593 y FN(:)2116 593 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C1 16 {InitRnode } NewNode end end
2116 593 a FP(C)2188 560 y FO(j)2223 593 y FP(;)84 b(k)28
b FN($)d FP(i)h FN($)f FP(j)5 b FT(\))1432 932 y(\()1467
932 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX 16 {InitRnode } NewNode end end
1467 932 a FP(S)1523 946 y FO(X)1764 932 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374
} false /N@SY 16 {InitRnode } NewNode end end
1764
932 a FP(S)1820 946 y FO(Y)2055 932 y FN(:)2116 932 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C2 16 {InitRnode } NewNode end end
2116 932 a FP(C)2188 899 y FO(j)2223 932 y FP(;)84 b
FN(K)q FT(\))2437 932 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@X /N@SX InitNC { NCLine } if end gsave 0.8 SLW 0. setgray 0
setlinecap stroke grestore grestore end
2437 932 a 2437 932 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
2437 932
a 2431 959 a FL(c)2437 932 y
tx@Dict begin PutEnd end
2437 932 a 2437 932 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
2437
932 a 2408 959 a FK(\003)2437 932 y
tx@Dict begin PutEnd end
2437 932 a 2437 932
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 6.0
0 0 /N@Y /N@SY InitNC { NCLine } if end gsave 0.8 SLW 0. setgray 0
setlinecap stroke grestore grestore end
2437 932 a 2437 932 a
tx@Dict begin tx@NodeDict begin /t 0.6 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
2437 932 a 2431 959 a FL(c)2437
932 y
tx@Dict begin PutEnd end
2437 932 a 2437 932 a
tx@Dict begin tx@NodeDict begin /t 0.6 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
2437 932 a 2408 959 a FK(\003)2437
932 y
tx@Dict begin PutEnd end
2437 932 a 2437 932 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@X /N@Y InitNC { /AngleA 20. def /AngleB 160. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2437 932 a 2437 932 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@X /N@C1 InitNC { /AngleA 25. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2437
932 a 2437 932 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 20. def /AngleB 160. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2437 932 a 2437 932 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@C2 InitNC { /AngleA 25. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2437 932 a 378
1144 a FT(illustrates)26 b(the)h(application)f(of)i(the)f(relation)g
FN(!)2092 1111 y FK(\003)2092 1167 y FL(c)2159 1144 y
FT(on)g(\()p FN(f)p FP(X)2444 1111 y FO(i)2474 1144 y
FP(;)15 b(Y)2587 1111 y FO(k)2630 1144 y FP(;)g FN(:)p
FP(C)2803 1111 y FO(j)2839 1144 y FN(g)p FP(;)g(k)29
b FN($)c FP(i)h FN($)f FP(j)5 b FT(\).)41 b(The)27 b(curv)m(e)378
1257 y(connecting)41 b(the)h(set)g FP(S)1220 1271 y FO(X)1328
1257 y FT(with)e FN(f:)p FP(C)1724 1224 y FO(j)1760 1257
y FN(g)i FT(represen)m(ts)f(the)g(relation)g(\()p FN(P)2896
1271 y FO(X)3007 1257 y FN($)i FP(j)5 b FT(\))42 b(and)f(the)h(curv)m
(e)378 1370 y(connecting)31 b FP(S)889 1384 y FO(X)986
1370 y FT(with)e FP(S)1249 1384 y FO(Y)1340 1370 y FT(represen)m(ts)h
(\()p FN(P)1866 1384 y FO(X)1959 1370 y FN($)25 b(P)2138
1384 y FO(Y)2199 1370 y FT(\).)1498 b Ff(\003)378 1610
y FG(8.4.3)112 b(Soundness)40 b(and)e(Completeness)f(for)g(the)h
(General)f(Case)378 1782 y FT(In)h(this)g(section)g(w)m(e)i(pro)m(v)m
(e)f(that)h(if)d Fv(C)50 b Fw(by)43 b Fv(P)51 b FN(+)2090
1796 y FL(c)2164 1782 y FT(\()p FP(S;)15 b FN(K)q FT(\))40
b(for)f(ev)m(ery)g(structured)f(expression)g FP(P)13
b FT(,)378 1895 y(conclusion)36 b FP(C)44 b FT(and)36
b(coloured)h(problem)f(\()p FP(S;)15 b FN(K)q FT(\),)41
b(then)c FP(P)50 b FT(justi\014es)36 b FP(C)44 b FT(if)36
b(and)h(only)g(if)f(\()p FP(S;)15 b FN(K)q FT(\))39 b(is)378
2008 y(inconsisten)m(t.)i(This)30 b(result)f(is)h(giv)m(en)h(b)m(y)g
(theorem)g(8.7)h(b)s(elo)m(w,)f(whose)g(pro)s(of)f(uses)g(the)h(follo)m
(wing)378 2121 y(prop)s(osition.)378 2313 y FQ(Prop)s(osition)36
b(8.12)46 b FI(Given)26 b(the)f(sets)h(of)g(c)-5 b(olour)g(e)g(d)27
b(structur)-5 b(e)g(d)27 b(expr)-5 b(essions)27 b FP(S)5
b FI(,)27 b FP(S)3250 2327 y FL(1)3314 2313 y FI(and)f
FP(S)3539 2327 y FL(2)3604 2313 y FI(wher)-5 b(e)378
2426 y FP(S)21 b FN(\\)16 b FP(S)588 2440 y FL(1)653
2426 y FT(=)24 b FN(fg)p FI(,)32 b FP(S)22 b FN(\\)15
b FP(S)1108 2440 y FL(2)1173 2426 y FT(=)25 b FN(fg)p
FI(,)32 b(and)f(the)h(set)f FP(S)k FI(c)-5 b(ontains)33
b(an)e Fw(on)f FI(or)h(an)h Fw(and)d FI(expr)-5 b(ession,)33
b(and)f(given)378 2538 y(the)h(c)-5 b(onne)g(ctability)34
b(r)-5 b(elations)35 b FN(K)1538 2552 y FL(1)1610 2538
y FI(and)e FN(K)1855 2552 y FL(2)1895 2538 y FI(,)g(then)485
2709 y(1.)46 b(ther)-5 b(e)31 b(is)e(some)h(set)f FP(S)1351
2676 y FK(0)1404 2709 y FI(of)h(c)-5 b(olour)g(e)g(d)31
b(structur)-5 b(e)g(d)31 b(expr)-5 b(essions)32 b(and)e(c)-5
b(onne)g(ctability)31 b(r)-5 b(elations)605 2822 y FN(K)675
2789 y FK(0)674 2846 y FL(1)746 2822 y FI(and)34 b FN(K)993
2789 y FK(0)992 2846 y FL(2)1064 2822 y FI(such)f(that)957
3026 y FT(\()p FP(S)25 b FN([)20 b FP(S)1210 3040 y FL(1)1249
3026 y FP(;)15 b FN(K)1358 3040 y FL(1)1398 3026 y FT(\))26
b FN(!)1550 3040 y FL(c)1611 3026 y FT(\()p FP(S)1707
2989 y FK(0)1751 3026 y FN([)19 b FP(S)1887 3040 y FL(1)1927
3026 y FP(;)c FN(K)2037 2989 y FK(0)2036 3049 y FL(1)2076
3026 y FT(\))186 b(\()p FP(S)25 b FN([)20 b FP(S)2550
3040 y FL(2)2589 3026 y FP(;)15 b FN(K)2698 3040 y FL(2)2739
3026 y FT(\))25 b FN(!)2890 3040 y FL(c)2951 3026 y FT(\()p
FP(S)3047 2989 y FK(0)3091 3026 y FN([)19 b FP(S)3227
3040 y FL(2)3267 3026 y FP(;)c FN(K)3377 2989 y FK(0)3376
3049 y FL(2)3416 3026 y FT(\);)485 3264 y FI(2.)46 b(if)33
b FN(K)763 3278 y FL(1)802 3264 y FN(d)p FP(S)5 b FN(e)26
b FT(=)f FN(K)1134 3278 y FL(2)1174 3264 y FN(d)p FP(S)5
b FN(e)33 b FI(then)g FN(K)1620 3231 y FK(0)1619 3289
y FL(1)1659 3264 y FN(d)p FP(S)1760 3231 y FK(0)1784
3264 y FN(e)25 b FT(=)g FN(K)2015 3231 y FK(0)2014 3289
y FL(2)2054 3264 y FN(d)p FP(S)2155 3231 y FK(0)2179
3264 y FN(e)p FI(;)485 3472 y(3.)46 b(if)36 b Fe(C)p
FT(\()p FP(S)5 b FT(\))23 b FN(\\)f Fe(C)p FT(\()p FP(S)1137
3486 y FL(1)1177 3472 y FT(\))32 b(=)f FN(fg)37 b FI(and)g
Fe(C)p FT(\()p FP(S)5 b FT(\))23 b FN(\\)f Fe(C)p FT(\()p
FP(S)2093 3486 y FL(2)2133 3472 y FT(\))32 b(=)f FN(fg)37
b FI(then)g(if)e FT(\()p FP(S)2856 3419 y FK(K)2910 3428
y FC(1)2854 3472 y FN(!)d FP(S)3033 3486 y FL(1)3072
3472 y FT(\))g(=)f(\()p FP(S)3371 3419 y FK(K)3425 3428
y FC(2)3369 3472 y FN(!)h FP(S)3548 3486 y FL(2)3587
3472 y FT(\))k FI(then)605 3627 y FT(\()p FP(S)701 3594
y FK(0)751 3563 y(K)805 3540 y FD(0)805 3584 y FC(1)750
3627 y FN(!)25 b FP(S)922 3641 y FL(1)961 3627 y FT(\))h(=)f(\()p
FP(S)1214 3594 y FK(0)1264 3563 y(K)1318 3540 y FD(0)1318
3584 y FC(2)1263 3627 y FN(!)g FP(S)1435 3641 y FL(2)1474
3627 y FT(\))p FI(.)378 3819 y FQ(Pro)s(of)p FT(:)30
b(Let)g FP(X)37 b FT(b)s(e)28 b(some)i Fw(on)e FT(or)i
Fw(and)e FT(expression)g(in)g FP(S)5 b FT(,)29 b(and)g(let)g
FP(j)35 b FT(b)s(e)29 b(an)m(y)g(colour)g(new)g(to)h
FP(S)5 b FT(,)30 b FP(S)3764 3833 y FL(1)3803 3819 y
FT(,)378 3932 y FP(S)434 3946 y FL(2)473 3932 y FT(,)k
FN(K)601 3946 y FL(1)674 3932 y FT(and)e FN(K)922 3946
y FL(2)962 3932 y FT(.)49 b(The)32 b(pro)s(of)h(of)g(this)f(prop)s
(osition)e(follo)m(ws)j(b)m(y)g(considering)e(the)i(follo)m(wing)f(t)m
(w)m(o)378 4045 y(cases:)514 4216 y FN(\017)46 b FT(If)30
b FP(X)j FT(=)25 b(\()p Fv(Y)62 b Fw(on)43 b Fv(Z)6 b
FT(\))1274 4183 y FO(i)1333 4216 y FT(for)30 b(some)h(colour)e
FP(i)i FT(and)f(structured)f(expressions)g Fv(Y)49 b
FT(and)30 b Fv(Z)36 b FT(then)974 4420 y(\()p FP(S)26
b FN([)20 b FP(S)1228 4434 y FL(1)1267 4420 y FP(;)15
b FN(K)1376 4434 y FL(1)1416 4420 y FT(\))26 b FN(!)1568
4434 y FL(c)1628 4420 y FT(\(\(\()p FP(S)h FN(\000)19
b(f)p FP(X)7 b FN(g)p FT(\))22 b FN([)e(f)p Fv(Y)2328
4383 y FO(i)2356 4420 y FP(;)15 b Fv(Z)2459 4383 y FO(j)2496
4420 y FN(g)p FT(\))21 b FN([)f FP(S)2734 4434 y FL(1)2773
4420 y FP(;)15 b Fv(Z)2876 4383 y FO(j)2913 4420 y FN(g)p
FP(;)g FN(K)3067 4434 y FL(1)3127 4420 y FN([)20 b FP(i)26
b FN($)f FP(j)5 b FT(\))962 4558 y(\()p FP(S)25 b FN([)20
b FP(S)1215 4572 y FL(2)1254 4558 y FP(;)15 b FN(K)1363
4572 y FL(1)1403 4558 y FT(\))26 b FN(!)1555 4572 y FL(c)1616
4558 y FT(\(\(\()p FP(S)g FN(\000)20 b(f)p FP(X)7 b FN(g)p
FT(\))22 b FN([)e(f)p Fv(Y)2316 4520 y FO(i)2344 4558
y FP(;)15 b Fv(Z)2447 4520 y FO(j)2483 4558 y FN(g)p
FT(\))21 b FN([)f FP(S)2721 4572 y FL(2)2760 4558 y FP(;)15
b Fv(Z)2863 4520 y FO(j)2900 4558 y FN(g)p FP(;)g FN(K)3054
4572 y FL(2)3115 4558 y FN([)k FP(i)26 b FN($)f FP(j)5
b FT(\))p FP(:)689 4762 y FT(1.)46 b(The)30 b(\014rst)g(part)g(of)g
(this)g(prop)s(osition)e(follo)m(ws)h(b)m(y)i(c)m(ho)s(osing)928
4966 y FP(S)989 4929 y FK(0)1038 4966 y FT(=)25 b(\()p
FP(S)g FN(\000)20 b(f)p FP(X)7 b FN(g)p FT(\))22 b FN([)e(f)p
Fv(Y)1763 4929 y FO(i)1791 4966 y FP(;)15 b Fv(Z)1894
4929 y FO(j)1931 4966 y FN(g)p FP(;)106 b FN(K)2177 4929
y FK(0)2176 4989 y FL(1)2241 4966 y FT(=)25 b FN(K)2406
4980 y FL(1)2466 4966 y FN([)20 b FP(i)25 b FN($)h FP(j)96
b FT(and)30 b FN(K)3100 4929 y FK(0)3099 4989 y FL(2)3164
4966 y FT(=)25 b FN(K)3329 4980 y FL(2)3389 4966 y FN([)20
b FP(i)25 b FN($)g FP(j:)689 5184 y FT(2.)46 b(Since)30
b(b)s(oth)f(the)i(colours)f FP(i)g FT(and)g FP(j)36 b
FT(are)31 b(in)e FP(S)2353 5151 y FK(0)2406 5184 y FT(then)1836
5388 y FN(K)1906 5350 y FK(0)1905 5410 y FL(1)1944 5388
y FN(d)p FP(S)2045 5350 y FK(0)2069 5388 y FN(e)d FT(=)f
FN(K)2300 5402 y FL(1)2339 5388 y FN(d)p FP(S)5 b FN(e)21
b([)f FP(i)26 b FN($)f FP(j)805 5592 y FT(and)30 b(similarly)1826
5705 y FN(K)1896 5668 y FK(0)1895 5728 y FL(2)1934 5705
y FN(d)p FP(S)2035 5668 y FK(0)2059 5705 y FN(e)c FT(=)f
FN(K)2290 5719 y FL(2)2329 5705 y FN(d)p FP(S)5 b FN(e)21
b([)f FP(i)26 b FN($)f FP(j;)p eop
%%Page: 179 189
179 188 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(179)805
396 y(and)30 b(therefore)h FN(K)1434 363 y FK(0)1433
421 y FL(1)1473 396 y FN(d)p FP(S)1574 363 y FK(0)1597
396 y FN(e)26 b FT(=)f FN(K)1829 363 y FK(0)1828 421
y FL(2)1868 396 y FN(d)p FP(S)1969 363 y FK(0)1992 396
y FN(e)31 b FT(if)e FN(K)2215 410 y FL(1)2255 396 y FN(d)p
FP(S)5 b FN(e)26 b FT(=)f FN(K)2587 410 y FL(2)2627 396
y FN(d)p FP(S)5 b FN(e)p FT(.)689 543 y(3.)46 b(Since)31
b FP(j)h FN(62)27 b Fe(C)p FT(\()p FP(S)1348 557 y FL(1)1387
543 y FT(\))32 b(and)f FP(i)g FT(is)g(in)f Fe(C)p FT(\()p
FP(S)5 b FT(\),)32 b(and)f(therefore)h(not)g(in)e Fe(C)p
FT(\()p FP(S)3116 557 y FL(1)3155 543 y FT(\))i(b)s(ecause)f
Fe(C)p FT(\()p FP(S)5 b FT(\))21 b FN(\\)805 655 y Fe(C)p
FT(\()p FP(S)952 669 y FL(1)991 655 y FT(\))26 b(=)f
FN(fg)p FT(,)31 b(then)1876 811 y(\()p FP(S)1972 773
y FK(0)2022 747 y(K)2076 723 y FD(0)2076 767 y FC(1)2021
811 y FN(!)25 b FP(S)2193 825 y FL(1)2232 811 y FT(\))h(=)f(\()p
FP(S)2512 758 y FK(K)2566 767 y FC(1)2510 811 y FN(!)h
FP(S)2683 825 y FL(1)2722 811 y FT(\))805 978 y(and)k(similarly)1863
1128 y(\()p FP(S)1959 1090 y FK(0)2010 1064 y(K)2064
1040 y FD(0)2064 1084 y FC(2)2008 1128 y FN(!)25 b FP(S)2180
1142 y FL(2)2220 1128 y FT(\))g(=)g(\()p FP(S)2499 1075
y FK(K)2553 1084 y FC(2)2498 1128 y FN(!)g FP(S)2670
1142 y FL(2)2709 1128 y FT(\))p FP(:)805 1337 y FT(Th)m(us)30
b(\()p FP(S)1132 1304 y FK(0)1182 1273 y(K)1236 1250
y FD(0)1236 1294 y FC(1)1180 1337 y FN(!)c FP(S)1353
1351 y FL(1)1392 1337 y FT(\))f(=)g(\()p FP(S)1644 1304
y FK(0)1695 1273 y(K)1749 1250 y FD(0)1749 1294 y FC(2)1693
1337 y FN(!)g FP(S)1865 1351 y FL(2)1905 1337 y FT(\))30
b(if)g(\()p FP(S)2176 1285 y FK(K)2230 1294 y FC(1)2175
1337 y FN(!)25 b FP(S)2347 1351 y FL(1)2386 1337 y FT(\))h(=)f(\()p
FP(S)2666 1285 y FK(K)2720 1294 y FC(2)2665 1337 y FN(!)g
FP(S)2837 1351 y FL(2)2876 1337 y FT(\).)514 1525 y FN(\017)46
b FT(If)30 b FP(X)j FT(=)25 b(\()p Fv(Y)62 b Fw(and)43
b Fv(Z)5 b FT(\))1317 1492 y FO(i)1376 1525 y FT(then)954
1745 y(\()p FP(S)25 b FN([)20 b FP(S)1207 1759 y FL(1)1246
1745 y FP(;)15 b FN(K)1355 1759 y FL(1)1395 1745 y FT(\))26
b FN(!)1547 1759 y FL(c)1608 1745 y FT(\(\(\()p FP(S)g
FN(\000)20 b(f)p FP(X)7 b FN(g)p FT(\))22 b FN([)e(f)p
Fv(Y)2308 1707 y FO(i)2336 1745 y FP(;)15 b Fv(Z)2439
1707 y FO(j)2475 1745 y FN(g)p FT(\))21 b FN([)f FP(S)2713
1759 y FL(1)2752 1745 y FP(;)15 b Fv(Z)2855 1707 y FO(j)2892
1745 y FN(g)p FP(;)g FN(K)3046 1759 y FL(1)3107 1745
y FN([)k(K)3257 1697 y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))3256
1771 y(1)3444 1745 y FT(\))941 1906 y(\()p FP(S)26 b
FN([)19 b FP(S)1194 1920 y FL(2)1234 1906 y FP(;)c FN(K)1343
1920 y FL(2)1383 1906 y FT(\))25 b FN(!)1534 1920 y FL(c)1595
1906 y FT(\(\(\()p FP(S)h FN(\000)20 b(f)p FP(X)7 b FN(g)p
FT(\))22 b FN([)e(f)p Fv(Y)2295 1868 y FO(i)2323 1906
y FP(;)15 b Fv(Z)2426 1868 y FO(j)2463 1906 y FN(g)p
FT(\))21 b FN([)f FP(S)2701 1920 y FL(2)2740 1906 y FP(;)15
b Fv(Z)2843 1868 y FO(j)2879 1906 y FN(g)p FP(;)g FN(K)3033
1920 y FL(2)3094 1906 y FN([)20 b(K)3245 1858 y FL(\()p
FO(i)p FK(!)p FO(j)t FL(\))3244 1932 y(2)3431 1906 y
FT(\))p FP(:)689 2110 y FT(1.)46 b(W)-8 b(e)32 b(c)m(ho)s(ose)952
2314 y FP(S)1013 2277 y FK(0)1062 2314 y FT(=)25 b(\()p
FP(S)h FN(\000)19 b(f)p FP(X)7 b FN(g)p FT(\))22 b FN([)e(f)p
Fv(Y)1787 2277 y FO(i)1815 2314 y FP(;)15 b Fv(Z)1918
2277 y FO(j)1955 2314 y FN(g)p FP(;)107 b FN(K)2202 2277
y FK(0)2201 2337 y FL(1)2266 2314 y FT(=)25 b FN(K)2431
2328 y FL(1)2491 2314 y FN([)19 b(K)2641 2266 y FL(\()p
FO(i)p FK(!)p FO(j)t FL(\))2640 2340 y(1)2828 2314 y
FP(;)106 b FN(K)3029 2277 y FK(0)3028 2337 y FL(2)3093
2314 y FT(=)25 b FN(K)3258 2328 y FL(2)3318 2314 y FN([)20
b(K)3469 2266 y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))3468
2340 y(2)3655 2314 y FP(;)805 2518 y FT(and)30 b(it)g(can)h(b)s(e)e(c)m
(hec)m(k)m(ed)k(that)e(the)f(\014rst)g(part)g(of)h(this)e(prop)s
(osition)f(follo)m(ws)h(easily)-8 b(.)689 2665 y(2.)46
b(Using)33 b(the)h(fact)h(that)f(for)g(ev)m(ery)g(set)h
FP(S)5 b FT(,)34 b(connectabilit)m(y)g(relation)f FN(K)q
FT(,)i(colour)e FP(i)i FT(in)d FP(S)805 2777 y FT(and)e(colour)g
FP(j)36 b FT(new)30 b(to)h(\()p FP(S;)15 b FN(K)q FT(\))32
b(it)e(is)f(the)i(case)g(that)1277 2982 y FN(K)21 b([)f(K)1518
2944 y FL(\()p FO(i)p FK(!)p FO(j)t FL(\))1730 2982 y
FT(=)25 b FN(K)d([)e(f)p FT(\()p FP(j;)15 b(k)s FT(\))27
b FN(j)f FP(i)f FN(\030)2445 2996 y FK(K)2528 2982 y
FP(k)s FN(g)c([)f(f)p FT(\()p FP(k)s(;)15 b(j)5 b FT(\))28
b FN(j)d FP(k)k FN(\030)3197 2996 y FK(K)3280 2982 y
FP(i)p FN(g)805 3186 y FT(w)m(e)i(get)1026 3390 y FN(K)1096
3353 y FK(0)1095 3413 y FL(1)1134 3390 y FN(d)p FP(S)1235
3353 y FK(0)1259 3390 y FN(e)26 b FT(=)f(\()p FN(K)1525
3404 y FL(1)1615 3390 y FN([)50 b(f)p FT(\()p FP(j;)15
b(k)s FT(\))28 b FN(j)d FP(i)h FN(\030)2174 3404 y FK(K)2228
3413 y FC(1)2291 3390 y FP(k)s FN(g)52 b([)e(f)p FT(\()p
FP(k)s(;)15 b(j)5 b FT(\))27 b FN(j)f FP(i)f FN(\030)3001
3404 y FK(K)3055 3413 y FC(1)3119 3390 y FP(k)s FN(g)p
FT(\))p FN(d)p FP(S)3350 3353 y FK(0)3374 3390 y FN(e)1325
3528 y FT(=)g FN(K)1490 3542 y FL(1)1529 3528 y FN(d)p
FP(S)1630 3490 y FK(0)1654 3528 y FN(e)51 b([)f(f)p FT(\()p
FP(j;)15 b(k)s FT(\))28 b FN(j)d FP(i)h FN(\030)2304
3542 y FK(K)2358 3551 y FC(1)2421 3528 y FP(k)s(;)15
b(k)29 b FN(2)c Fe(C)p FT(\()p FP(S)2825 3490 y FK(0)2848
3528 y FT(\))p FN(g)52 b([)1663 3666 y(f)p FT(\()p FP(k)s(;)15
b(j)5 b FT(\))28 b FN(j)d FP(i)h FN(\030)2116 3680 y
FK(K)2170 3689 y FC(1)2233 3666 y FP(k)s(;)15 b(k)29
b FN(2)c Fe(C)p FT(\()p FP(S)2637 3628 y FK(0)2660 3666
y FT(\))p FN(g)1325 3804 y FT(=)g FN(K)1490 3818 y FL(1)1529
3804 y FN(d)p FP(S)1630 3766 y FK(0)1654 3804 y FN(e)51
b([)f(f)p FT(\()p FP(j;)15 b(k)s FT(\))28 b FN(j)d FP(i)h
FN(\030)2304 3822 y FK(K)2358 3831 y FC(1)2392 3822 y
FK(d)p FO(S)2470 3803 y FD(0)2492 3822 y FK(e)2553 3804
y FP(k)s FN(g)51 b([)f(f)p FT(\()p FP(k)s(;)15 b(j)5
b FT(\))28 b FN(j)d FP(i)h FN(\030)3263 3822 y FK(K)3317
3831 y FC(1)3351 3822 y FK(d)p FO(S)3429 3803 y FD(0)3451
3822 y FK(e)3512 3804 y FP(k)s FN(g)805 4008 y FT(No)m(w)i
Fe(C)p FT(\()p FP(S)1161 3975 y FK(0)1184 4008 y FT(\))e(=)f
Fe(C)p FT(\()p FP(S)5 b FT(\))14 b FN([)g(f)p FP(j)5
b FN(g)28 b FT(and)e FP(j)31 b FN(62)25 b Fe(C)p FT(\()p
FN(K)2264 4022 y FL(1)2304 4008 y FT(\))i(and)f(therefore)i
FN(K)2987 4022 y FL(1)3027 4008 y FN(d)p FP(S)3128 3975
y FK(0)3151 4008 y FN(e)e FT(=)f FN(K)3382 4022 y FL(1)3422
4008 y FN(d)p FP(S)5 b FN(e)p FT(.)40 b(Th)m(us)1108
4212 y FN(K)1178 4175 y FK(0)1177 4235 y FL(1)1216 4212
y FN(d)p FP(S)1317 4175 y FK(0)1341 4212 y FN(e)26 b
FT(=)f FN(K)1572 4226 y FL(1)1611 4212 y FN(d)p FP(S)5
b FN(e)21 b([)f(f)p FT(\()p FP(j;)15 b(k)s FT(\))28 b
FN(j)d FP(i)h FN(\030)2302 4231 y FK(K)2356 4240 y FC(1)2390
4231 y FK(d)p FO(S)t FK(e)2529 4212 y FP(k)s FN(g)20
b([)g(f)p FT(\()p FP(k)s(;)15 b(j)5 b FT(\))28 b FN(j)d
FP(i)h FN(\030)3178 4231 y FK(K)3232 4240 y FC(1)3266
4231 y FK(d)p FO(S)t FK(e)3405 4212 y FP(k)s FN(g)p FP(:)805
4416 y FT(Similarly)-8 b(,)1108 4621 y FN(K)1178 4583
y FK(0)1177 4643 y FL(2)1216 4621 y FN(d)p FP(S)1317
4583 y FK(0)1341 4621 y FN(e)26 b FT(=)f FN(K)1572 4635
y FL(2)1611 4621 y FN(d)p FP(S)5 b FN(e)21 b([)f(f)p
FT(\()p FP(j;)15 b(k)s FT(\))28 b FN(j)d FP(i)h FN(\030)2302
4639 y FK(K)2356 4648 y FC(2)2390 4639 y FK(d)p FO(S)t
FK(e)2529 4621 y FP(k)s FN(g)20 b([)g(f)p FT(\()p FP(k)s(;)15
b(j)5 b FT(\))28 b FN(j)d FP(i)h FN(\030)3178 4639 y
FK(K)3232 4648 y FC(2)3266 4639 y FK(d)p FO(S)t FK(e)3405
4621 y FP(k)s FN(g)p FP(;)805 4825 y FT(and)k(therefore)h
FN(K)1434 4792 y FK(0)1433 4849 y FL(1)1473 4825 y FN(d)p
FP(S)1574 4792 y FK(0)1597 4825 y FN(e)26 b FT(=)f FN(K)1829
4792 y FK(0)1828 4849 y FL(2)1868 4825 y FN(d)p FP(S)1969
4792 y FK(0)1992 4825 y FN(e)31 b FT(if)e FN(K)2215 4839
y FL(1)2255 4825 y FN(d)p FP(S)5 b FN(e)26 b FT(=)f FN(K)2587
4839 y FL(2)2627 4825 y FN(d)p FP(S)5 b FN(e)p FT(.)689
4971 y(3.)46 b(F)-8 b(or)28 b(the)f(\014nal)e(case)j(w)m(e)f(use)f(the)
h(fact)h(that)f(the)g(colours)f(that)i(relate)f(with)e
FP(j)33 b FT(in)25 b FN(K)3656 4938 y FK(0)3706 4971
y FT(are)805 5084 y(exactly)31 b(the)g(colours)f(that)h(relate)g(with)e
FP(i)h FT(in)f FN(K)q FT(,)i(and)f(therefore)1372 5365
y FP(S)1433 5327 y FK(0)1483 5301 y(K)1537 5277 y FD(0)1537
5321 y FC(1)1482 5365 y FN(!)25 b FP(S)1654 5379 y FL(1)1719
5365 y FT(=)1815 5209 y Fx(\()1929 5308 y FT(\()p FP(S)2052
5256 y FK(K)2106 5265 y FC(1)2051 5308 y FN(!)g FP(S)2223
5322 y FL(1)2262 5308 y FT(\))c FN([)f(f)p FP(j)5 b FN(g)84
b FT(if)30 b FP(i)25 b FN(2)g FT(\()p FP(S)2964 5256
y FK(K)3018 5265 y FC(1)2963 5308 y FN(!)g FP(S)3135
5322 y FL(1)3174 5308 y FT(\))1929 5449 y FP(S)2017 5397
y FK(K)2071 5406 y FC(1)2016 5449 y FN(!)g FP(S)2188
5463 y FL(1)2615 5449 y FT(otherwise)p eop
%%Page: 180 190
180 189 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(180)805
396 y(and)30 b(similarly)1372 678 y FP(S)1433 640 y FK(0)1483
613 y(K)1537 590 y FD(0)1537 634 y FC(2)1482 678 y FN(!)25
b FP(S)1654 692 y FL(2)1719 678 y FT(=)1815 522 y Fx(\()1929
621 y FT(\()p FP(S)2052 568 y FK(K)2106 577 y FC(2)2051
621 y FN(!)g FP(S)2223 635 y FL(2)2262 621 y FT(\))c
FN([)f(f)p FP(j)5 b FN(g)84 b FT(if)30 b FP(i)25 b FN(2)g
FT(\()p FP(S)2964 568 y FK(K)3018 577 y FC(2)2963 621
y FN(!)g FP(S)3135 635 y FL(2)3174 621 y FT(\))1929 762
y FP(S)2017 709 y FK(K)2071 718 y FC(2)2016 762 y FN(!)g
FP(S)2188 776 y FL(2)2615 762 y FT(otherwise.)805 1020
y(Th)m(us)30 b(\()p FP(S)1132 987 y FK(0)1182 956 y(K)1236
932 y FD(0)1236 976 y FC(1)1180 1020 y FN(!)c FP(S)1353
1034 y FL(1)1392 1020 y FT(\))f(=)g(\()p FP(S)1644 987
y FK(0)1695 956 y(K)1749 932 y FD(0)1749 976 y FC(2)1693
1020 y FN(!)g FP(S)1865 1034 y FL(2)1905 1020 y FT(\))30
b(if)g(\()p FP(S)2176 967 y FK(K)2230 976 y FC(1)2175
1020 y FN(!)25 b FP(S)2347 1034 y FL(1)2386 1020 y FT(\))h(=)f(\()p
FP(S)2666 967 y FK(K)2720 976 y FC(2)2665 1020 y FN(!)g
FP(S)2837 1034 y FL(2)2876 1020 y FT(\).)821 b Ff(\004)519
1207 y FT(The)30 b(follo)m(wing)f(corollary)h(of)g(prop)s(osition)e
(8.12)k(is)d(used)h(in)f(theorem)i(8.7.)378 1420 y FQ(Corollary)k(8.2)
46 b FI(Given)39 b(the)h(sets)f(of)g(c)-5 b(olour)g(e)g(d)42
b(structur)-5 b(e)g(d)41 b(expr)-5 b(essions)41 b FP(S)5
b FI(,)40 b FP(S)3208 1434 y FL(1)3286 1420 y FI(and)h
FP(S)3526 1434 y FL(2)3604 1420 y FI(wher)-5 b(e)378
1533 y FP(S)21 b FN(\\)16 b FP(S)588 1547 y FL(1)653
1533 y FT(=)24 b FN(fg)p FI(,)32 b FP(S)22 b FN(\\)15
b FP(S)1108 1547 y FL(2)1173 1533 y FT(=)25 b FN(fg)p
FI(,)32 b(and)f(the)h(set)f FP(S)k FI(c)-5 b(ontains)33
b(an)e Fw(on)f FI(or)h(an)h Fw(and)d FI(expr)-5 b(ession,)33
b(and)f(given)378 1646 y(the)h(c)-5 b(onne)g(ctability)34
b(r)-5 b(elations)35 b FN(K)1538 1660 y FL(1)1610 1646
y FI(and)e FN(K)1855 1660 y FL(2)1928 1646 y FI(then)485
1833 y(1.)46 b(ther)-5 b(e)36 b(is)e(some)i(set)e FP(S)1372
1800 y FK(0)1430 1833 y FI(of)h(c)-5 b(olour)g(e)g(d)37
b(formulae)f(and)g(c)-5 b(onne)g(ctability)36 b(r)-5
b(elations)36 b FN(K)3467 1800 y FK(0)3466 1858 y FL(1)3541
1833 y FI(and)f FN(K)3789 1800 y FK(0)3788 1858 y FL(2)605
1946 y FI(such)e(that)953 2150 y FT(\()p FP(S)25 b FN([)20
b FP(S)1206 2164 y FL(1)1245 2150 y FP(;)15 b FN(K)1354
2164 y FL(1)1395 2150 y FT(\))25 b FN(!)1546 2113 y FK(\003)1546
2173 y FL(c)1611 2150 y FT(\()p FP(S)1707 2113 y FK(0)1751
2150 y FN([)19 b FP(S)1887 2164 y FL(1)1927 2150 y FP(;)c
FN(K)2037 2113 y FK(0)2036 2173 y FL(1)2076 2150 y FT(\))186
b(\()p FP(S)25 b FN([)20 b FP(S)2550 2164 y FL(2)2589
2150 y FP(;)15 b FN(K)2698 2164 y FL(2)2739 2150 y FT(\))25
b FN(!)2890 2113 y FK(\003)2890 2173 y FL(c)2955 2150
y FT(\()p FP(S)3051 2113 y FK(0)3095 2150 y FN([)19 b
FP(S)3231 2164 y FL(2)3271 2150 y FP(;)c FN(K)3381 2113
y FK(0)3380 2173 y FL(2)3420 2150 y FT(\);)485 2392 y
FI(2.)46 b(if)33 b FN(K)763 2406 y FL(1)802 2392 y FN(d)p
FP(S)5 b FN(e)26 b FT(=)f FN(K)1134 2406 y FL(2)1174
2392 y FN(d)p FP(S)5 b FN(e)33 b FI(then)g FN(K)1620
2359 y FK(0)1619 2416 y FL(1)1659 2392 y FN(d)p FP(S)1760
2359 y FK(0)1784 2392 y FN(e)25 b FT(=)g FN(K)2015 2359
y FK(0)2014 2416 y FL(2)2054 2392 y FN(d)p FP(S)2155
2359 y FK(0)2179 2392 y FN(e)p FI(;)485 2607 y(3.)46
b(if)36 b Fe(C)p FT(\()p FP(S)5 b FT(\))23 b FN(\\)f
Fe(C)p FT(\()p FP(S)1137 2621 y FL(1)1177 2607 y FT(\))32
b(=)f FN(fg)37 b FI(and)g Fe(C)p FT(\()p FP(S)5 b FT(\))23
b FN(\\)f Fe(C)p FT(\()p FP(S)2093 2621 y FL(2)2133 2607
y FT(\))32 b(=)f FN(fg)37 b FI(then)g(if)e FT(\()p FP(S)2856
2554 y FK(K)2910 2563 y FC(1)2854 2607 y FN(!)d FP(S)3033
2621 y FL(1)3072 2607 y FT(\))g(=)f(\()p FP(S)3371 2554
y FK(K)3425 2563 y FC(2)3369 2607 y FN(!)h FP(S)3548
2621 y FL(2)3587 2607 y FT(\))k FI(then)605 2762 y FT(\()p
FP(S)701 2729 y FK(0)751 2698 y(K)805 2674 y FD(0)805
2718 y FC(1)750 2762 y FN(!)25 b FP(S)922 2776 y FL(1)961
2762 y FT(\))h(=)f(\()p FP(S)1214 2729 y FK(0)1264 2698
y(K)1318 2674 y FD(0)1318 2718 y FC(2)1263 2762 y FN(!)g
FP(S)1435 2776 y FL(2)1474 2762 y FT(\))p FI(.)378 2950
y(\(Note)33 b(that)i(the)f(set)f FP(S)1176 2917 y FK(0)1232
2950 y FI(in)h(this)g(c)-5 b(or)g(ol)5 b(lary)36 b(c)-5
b(ontains)34 b(only)g(c)-5 b(olour)g(e)g(d)36 b(formulae,)f(while)f
(the)f(set)h FP(S)3805 2917 y FK(0)378 3062 y FI(in)e(the)h(statement)h
(of)f(pr)-5 b(op)g(osition)36 b(8.12)e(c)-5 b(ontains)34
b(c)-5 b(olour)g(e)g(d)35 b(structur)-5 b(e)g(d)34 b(expr)-5
b(essions.\))378 3275 y FQ(Pro)s(of)p FT(:)30 b(F)-8
b(ollo)m(ws)29 b(from)f(the)h(fact)g(that)h FN(!)1839
3242 y FK(\003)1839 3297 y FL(c)1907 3275 y FT(is)d(the)i(re\015exiv)m
(e)g(transitiv)m(e)f(closure)g(of)h FN(!)3404 3289 y
FL(c)3468 3275 y FT(and)f(from)378 3388 y(prop)s(osition)g(8.12.)2716
b Ff(\004)378 3600 y FQ(Theorem)34 b(8.7)46 b FI(F)-7
b(or)44 b(every)e(structur)-5 b(e)g(d)44 b(expr)-5 b(ession)45
b Fv(P)11 b FI(,)45 b(sentenc)-5 b(e)43 b FP(C)7 b FI(,)45
b(and)e(c)-5 b(olour)g(e)g(d)45 b(pr)-5 b(oblem)378 3713
y FT(\()p FP(S;)15 b FN(K)q FT(\))34 b FI(such)f(that)1726
3826 y FT(\()p Fv(C)50 b Fw(by)43 b Fv(P)12 b FT(\))25
b FN(+)2182 3840 y FL(c)2243 3826 y FT(\()p FP(S;)15
b FN(K)q FT(\))378 3993 y FI(then)33 b FP(S)38 b FI(is)32
b FN(K)q FI(-inc)-5 b(onsistent)34 b(if)e(and)i(only)f(if)g
Fv(P)k Ff( )25 b FP(C)7 b FI(.)378 4206 y FQ(Pro)s(of)p
FT(:)31 b(The)f(pro)s(of)g(pro)s(ceeds)g(b)m(y)g(induction)e(on)i(the)h
(structure)f(of)g Fv(P)12 b FT(:)514 4393 y FN(\017)46
b FT(The)30 b(Base)i(Case)e(\()p Fv(P)43 b FT(is)29 b(some)i(sen)m
(tence)g FP(A)p FT(\):)42 b(F)-8 b(or)31 b(all)e(sen)m(tences)j
FP(A)e FT(and)g FP(C)37 b FT(suc)m(h)30 b(that)1841 4598
y(\()p Fv(C)50 b Fw(by)43 b Fv(A)p FT(\))26 b FN(+)2295
4612 y FL(c)2355 4598 y FT(\()p FP(S;)15 b FN(K)q FT(\))605
4802 y(then)30 b FP(S)36 b FT(is)29 b FN(K)q FT(-inconsisten)m(t)h(if)g
(and)f(only)h(if)f Fv(A)d Ff( )f FP(C)7 b FT(.)605 4952
y FQ(Pro)s(of)p FT(:)32 b(Since)d FP(A)i FT(is)e(a)i(sen)m(tence)1586
5156 y(\()p Fv(C)50 b Fw(by)42 b Fv(A)q FT(\))25 b FN(+)2039
5170 y FL(c)2100 5156 y FT(\()p FN(f)p FP(A)2248 5119
y FO(i)2277 5156 y FP(;)15 b FN(:)p FP(C)2450 5119 y
FO(j)2486 5156 y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5 b
FT(\))p FP(;)605 5361 y FT(and)34 b Fv(A)d Ff( )h FP(C)40
b FT(if)33 b(and)h(only)f(if)g FP(A)f Ff(\032)1866 5328
y FK(\003)1937 5361 y FP(C)7 b FT(.)51 b(Therefore)34
b(the)g(goal)h(of)f(this)f(case)i(follo)m(ws)e(from)605
5473 y(theorem)e(8.3.)2658 b Ff(\002)514 5661 y FN(\017)46
b FT(The)30 b Fw(on)o FT(-Induction)f(Case)i(\()p Fv(P)42
b FT(is)30 b(some)h(expression)e(\()p Fv(X)50 b Fw(on)43
b Fv(Y)18 b FT(\)\):)42 b(Giv)m(en)30 b(the)h(h)m(yp)s(otheses:)p
eop
%%Page: 181 191
181 190 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(181)478
484 y(\()513 484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.32498 6.66249 4.63118
} false /N@X11 16 {InitRnode } NewNode end end
513 484 a FP(X)595 451 y FO(i)744
484 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y11 16 {InitRnode } NewNode end end
744 484 a FP(Y)817 451 y FO(k)974 484 y FN(:)1035
484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C11 16 {InitRnode } NewNode end end
1035 484 a FP(C)1107 451 y FO(j)1142 484 y FP(;)83
b(i)26 b FN($)f FP(k)k FN($)c FP(j)5 b FT(\))272 b(\()1999
484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.32498 6.66249 4.63118
} false /N@X12 16 {InitRnode } NewNode end end
1999 484 a FP(X)2081 451 y FO(i)2229 484 y FN(:)2290
484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118
} false /N@A12 16 {InitRnode } NewNode end end
2290 484 a FP(A)2358 451 y FO(j)2395 484 y FP(;)83
b(i)26 b FN($)f FP(j)5 b FT(\))472 823 y(\()507 823 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX21 16 {InitRnode } NewNode end end
507 823 a FP(S)563 837 y FO(X)744 823 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y21 16 {InitRnode } NewNode end end
744 823 a FP(Y)817
790 y FO(k)974 823 y FN(:)1035 823 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C21 16 {InitRnode } NewNode end end
1035 823 a FP(C)1107
790 y FO(j)1142 823 y FP(;)83 b FN(K)1319 837 y FO(X)5
b(Y)1444 823 y FT(\))479 b(\()1993 823 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX22 16 {InitRnode } NewNode end end
1993 823 a FP(S)2049
837 y FO(X)2229 823 y FN(:)2290 823 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118
} false /N@A22 16 {InitRnode } NewNode end end
2290 823 a FP(A)2358
790 y FO(j)2395 823 y FP(;)83 b FN(K)2572 837 y FO(X)2640
823 y FT(\))265 b(\()2975 823 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y23 16 {InitRnode } NewNode end end
2975 823 a FP(Y)3049
790 y FO(k)3205 823 y FN(:)3266 823 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.249 6.1245 4.63118
} false /N@B23 16 {InitRnode } NewNode end end
3266 823 a FP(B)3340
790 y FO(i)3367 823 y FP(;)84 b(i)25 b FN($)g FP(k)s
FT(\))472 1161 y(\()507 1161 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX31 16 {InitRnode } NewNode end end
507 1161 a FP(S)563 1175
y FO(X)744 1161 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374
} false /N@SY31 16 {InitRnode } NewNode end end
744 1161 a FP(S)800 1175 y FO(Y)974
1161 y FN(:)1035 1161 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C31 16 {InitRnode } NewNode end end
1035 1161 a FP(C)1107 1128 y
FO(j)1142 1161 y FP(;)83 b FN(K)1320 1128 y FK(0)1319
1188 y FO(X)5 b(Y)1444 1161 y FT(\))1461 b(\()2975 1161
y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374
} false /N@SY33 16 {InitRnode } NewNode end end
2975 1161 a FP(S)3031 1175 y FO(Y)3205 1161 y FN(:)3266
1161 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.249 6.1245 4.63118
} false /N@B33 16 {InitRnode } NewNode end end
3266 1161 a FP(B)3340 1128 y FO(i)3367 1161 y
FP(;)84 b FN(K)3545 1175 y FO(Y)3606 1161 y FT(\))3641
1161 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@X11 /N@SX21 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3641 1161 a 3641 1161 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
3641 1161 a 3635 1189
a FL(c)3641 1161 y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3641 1161
a 3612 1189 a FK(\003)3641 1161 y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161
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@X12 /N@SX22 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3641 1161 a 3641 1161 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
3641 1161 a 3635 1189 a FL(c)3641
1161 y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3641 1161 a 3612 1189
a FK(\003)3641 1161 y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 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 6.0
0 0 /N@Y21 /N@SY31 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3641 1161
a 3641 1161 a
tx@Dict begin tx@NodeDict begin /t 0.6 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
3641 1161 a 3635 1189 a FL(c)3641 1161
y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 a
tx@Dict begin tx@NodeDict begin /t 0.6 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3641 1161 a 3612 1189 a FK(\003)3641
1161 y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 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@Y23 /N@SY33 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3641 1161 a 3641 1161
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
3641 1161 a 3635 1189 a FL(c)3641 1161 y
tx@Dict begin PutEnd end
3641 1161
a 3641 1161 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3641 1161 a 3612 1189 a FK(\003)3641 1161
y
tx@Dict begin PutEnd end
3641 1161 a 3641 1161 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@X11 /N@Y11 InitNC { /AngleA 25. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
3641 1161 a 3641 1161 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@X11 /N@C11 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
3641
1161 a 3641 1161 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@SX21 /N@Y21 InitNC { /AngleA 25. def /AngleB 155. def
0.67 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap
stroke grestore grestore end
3641 1161 a 3641 1161 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@SX21 /N@C21 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
3641 1161
a 3641 1161 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@SX31 /N@SY31 InitNC { /AngleA 25. def /AngleB 155. def
0.67 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap
stroke grestore grestore end
3641 1161 a 3641 1161 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@SX31 /N@C31 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
3641 1161 a 3641
1161 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@X12 /N@A12 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
3641 1161 a 3641 1161 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@SX22 /N@A22 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
3641 1161 a 3641 1161
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@Y23 /N@B23 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
3641 1161 a 3641 1161 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@SY33 /N@B33 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
3641 1161 a 1644 1571 a FQ(Fig.)18
b(20.)41 b FT(The)30 b FM(on)g FT(Case.)689 1964 y(1.)46
b(for)27 b(ev)m(ery)h(sen)m(tence)h FP(A)p FT(,)f(if)e(\()p
Fv(A)44 b Fw(by)f Fv(X)6 b FT(\))26 b FN(+)2202 1978
y FL(c)2262 1964 y FT(\()p FP(S)2358 1931 y FK(0)2382
1964 y FP(;)15 b FN(K)2492 1931 y FK(0)2516 1964 y FT(\))27
b(then)g FP(S)2843 1931 y FK(0)2894 1964 y FT(is)f FN(K)3052
1931 y FK(0)3076 1964 y FT(-inconsisten)m(t)g(if)g(and)805
2077 y(only)k(if)f Fv(X)j Ff( )25 b FP(A)p FT(;)689 2223
y(2.)46 b(for)35 b(ev)m(ery)h(sen)m(tence)g FP(B)5 b
FT(,)36 b(if)e(\()p Fv(B)48 b Fw(by)43 b Fv(Y)19 b FT(\))33
b FN(+)2251 2237 y FL(c)2319 2223 y FT(\()p FP(S)2415
2190 y FK(0)q(0)2458 2223 y FP(;)15 b FN(K)2568 2190
y FK(0)q(0)2612 2223 y FT(\))35 b(then)g FP(S)2955 2190
y FK(00)3032 2223 y FT(is)f FN(K)3198 2190 y FK(0)q(0)3241
2223 y FT(-inconsisten)m(t)h(if)805 2336 y(and)30 b(only)f(if)h
Fv(Y)44 b Ff( )25 b FP(B)5 b FT(;)605 2523 y(w)m(e)31
b(are)g(required)d(to)k(sho)m(w)e(that)h(for)f(ev)m(ery)h(sen)m(tence)h
FP(C)k FT(if)1714 2728 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(+)2422 2742 y FL(c)2482
2728 y FT(\()p FP(S;)15 b FN(K)q FT(\))605 2932 y(then)30
b FP(S)36 b FT(is)29 b FN(K)q FT(-inconsisten)m(t)h(if)g(and)f(only)h
(if)f(\()p Fv(X)51 b Fw(on)42 b Fv(Y)19 b FT(\))26 b
Ff( )f FP(C)7 b FT(.)605 3082 y FQ(Pro)s(of)p FT(:)32
b(By)e(the)h(de\014nition)d(of)j FN(!)1813 3049 y FK(\003)1813
3105 y FL(c)1882 3082 y FT(w)m(e)g(get)1250 3286 y(\()p
Fv(C)51 b Fw(by)42 b Fv(X)50 b Fw(on)43 b Fv(Y)19 b FT(\))25
b FN(!)1993 3249 y FK(\003)1993 3309 y FL(c)2058 3286
y FT(\()p FN(f)p FP(X)2220 3249 y FO(i)2249 3286 y FP(;)15
b(Y)2363 3249 y FO(k)2405 3286 y FP(;)g FN(:)p FP(C)2578
3249 y FO(j)2614 3286 y FN(g)p FP(;)g(k)30 b FN($)25
b FP(i)g FN($)g FP(j)5 b FT(\))p FP(;)1494 3424 y FT(\()p
Fv(A)45 b Fw(by)d Fv(X)7 b FT(\))25 b FN(!)1993 3387
y FK(\003)1993 3447 y FL(c)2058 3424 y FT(\()p FN(f)p
FP(X)2220 3387 y FO(i)2249 3424 y FP(;)15 b FN(:)p FP(A)2418
3387 y FO(j)2455 3424 y FN(g)p FP(;)g(i)26 b FN($)g FP(j)5
b FT(\))p FP(;)1498 3571 y FT(\()p Fv(B)48 b Fw(by)43
b Fv(Y)19 b FT(\))25 b FN(!)1993 3533 y FK(\003)1993
3593 y FL(c)2058 3571 y FT(\()p FN(f)p FP(Y)2212 3533
y FO(k)2254 3571 y FP(;)15 b FN(:)p FP(B)2429 3533 y
FO(i)2457 3571 y FN(g)p FP(;)g(k)29 b FN($)c FP(i)p FT(\))p
FP(:)605 3775 y FT(Figure)38 b(20)h(illustrates)d(ho)m(w)i(the)g
(relation)f FN(!)2250 3789 y FL(c)2323 3775 y FT(is)g(applied)f(on)i
(the)g(coloured)g(structured)605 3888 y(problems)804
4092 y(\()p FN(f)p FP(X)966 4055 y FO(i)996 4092 y FP(;)15
b(Y)1109 4055 y FO(k)1152 4092 y FP(;)g FN(:)p FP(C)1325
4055 y FO(j)1361 4092 y FN(g)p FP(;)g(k)29 b FN($)c FP(i)h
FN($)f FP(j)5 b FT(\))p FP(;)107 b FT(\()p FN(f)p FP(X)2182
4055 y FO(i)2212 4092 y FP(;)15 b FN(:)p FP(A)2381 4055
y FO(j)2418 4092 y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5
b FT(\))p FP(;)107 b FT(\()p FN(f)p FP(Y)3039 4055 y
FO(k)3082 4092 y FP(;)15 b FN(:)p FP(B)3257 4055 y FO(i)3285
4092 y FN(g)p FP(;)g(k)29 b FN($)c FP(i)p FT(\))605 4297
y(during)j(the)j(pro)s(of)f(of)g(this)f(case.)605 4447
y(By)i(corollary)f(8.2)h(w)m(e)g(deduce)f(that)h(there)g(is)e(a)i(set)g
FP(S)2503 4461 y FO(X)2600 4447 y FT(of)g(coloured)e(form)m(ulae)h(suc)
m(h)h(that)1097 4651 y(\()p FN(f)p FP(X)1259 4613 y FO(i)1288
4651 y FP(;)46 b(Y)1432 4613 y FO(k)1474 4651 y FP(;)15
b FN(:)p FP(C)1647 4613 y FO(j)1683 4651 y FN(g)p FP(;)g(k)30
b FN($)25 b FP(i)g FN($)h FP(j)5 b FT(\))26 b FN(!)2328
4613 y FK(\003)2328 4674 y FL(c)2392 4651 y FT(\()p FP(S)2483
4665 y FO(X)2571 4651 y FN([)20 b(f)p FP(Y)2770 4613
y FO(k)2813 4651 y FP(;)15 b FN(:)p FP(C)2986 4613 y
FO(j)3022 4651 y FN(g)p FP(;)g FN(K)3176 4665 y FO(X)5
b(Y)3301 4651 y FT(\))1448 4789 y(\()p FN(f)p FP(X)1610
4751 y FO(i)1639 4789 y FP(;)46 b FN(:)p FP(A)1839 4751
y FO(j)1875 4789 y FN(g)p FP(;)15 b(i)26 b FN($)g FP(j)5
b FT(\))26 b FN(!)2328 4751 y FK(\003)2328 4811 y FL(c)2392
4789 y FT(\()p FP(S)2483 4803 y FO(X)2571 4789 y FN([)20
b(f:)p FP(A)2826 4751 y FO(j)2863 4789 y FN(g)p FP(;)15
b FN(K)3017 4803 y FO(X)3085 4789 y FT(\))605 4993 y(where)1443
5197 y FN(K)1512 5211 y FO(X)5 b(Y)1637 5197 y FN(d)p
FP(S)1733 5211 y FO(X)1800 5197 y FN(e)26 b FT(=)f FN(K)2031
5211 y FO(X)2098 5197 y FN(d)p FP(S)2194 5211 y FO(X)2262
5197 y FN(e)h FT(=)f FN(K)2493 5211 y FO(S)2536 5222
y Fy(X)2598 5197 y FP(;)46 b FT(sa)m(y)-8 b(,)31 b(and)1206
5370 y(\()p FP(S)1297 5384 y FO(X)1390 5315 y FK(K)1444
5326 y Fy(X)t(Y)1425 5370 y FN(!)60 b(f)p FP(Y)1694 5332
y FO(k)1737 5370 y FP(;)15 b FN(:)p FP(C)1910 5332 y
FO(j)1946 5370 y FN(g)p FT(\))26 b(=)f(\()p FP(S)2239
5384 y FO(X)2332 5315 y FK(K)2386 5326 y Fy(X)2342 5370
y FN(!)35 b(f:)p FP(A)2642 5332 y FO(j)2679 5370 y FN(g)q
FT(\))25 b(=)g FN(P)2944 5384 y FO(X)3012 5370 y FP(;)46
b FT(sa)m(y)-8 b(.)605 5574 y(Note)33 b(that)e(the)h(colour)e
FP(i)i FT(relates)f(with)f FP(k)k FT(and)d FP(j)37 b
FT(in)29 b FN(K)2522 5588 y FO(X)5 b(Y)2678 5574 y FT(since)30
b(\()p FP(i)d FN($)g FP(k)j FN($)c FP(j)5 b FT(\))27
b FN(\022)g(K)3577 5588 y FO(X)5 b(Y)3732 5574 y FT(b)m(y)605
5687 y(prop)s(osition)28 b(8.2.)42 b(Th)m(us)29 b FP(i)d
FN(2)f(P)1701 5701 y FO(X)1768 5687 y FT(.)p eop
%%Page: 182 192
182 191 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(182)605
396 y(No)m(w,)34 b(since)f FN(ff)p FP(X)1238 363 y FO(i)1267
396 y FN(g)p FP(;)15 b FN(f)p FP(Y)1471 363 y FO(k)1514
396 y FN(g)p FP(;)g FN(f:)p FP(C)1777 363 y FO(j)1814
396 y FN(gg)33 b FT(is)f(w)m(ell-coloured)g(with)f(resp)s(ect)i(to)g
FP(k)g FN($)c FP(i)g FN($)g FP(j)39 b FT(it)32 b(is)605
509 y(the)27 b(case)h(b)m(y)e(prop)s(osition)e(8.11)k(that)f
FN(f)p FP(S)2025 523 y FO(X)2093 509 y FP(;)15 b FN(f)p
FP(Y)2252 476 y FO(k)2294 509 y FN(g)p FP(;)g FN(f:)p
FP(C)2557 476 y FO(j)2594 509 y FN(gg)28 b FT(is)d(w)m(ell-coloured)h
(with)f(resp)s(ect)605 622 y(to)31 b FN(K)785 636 y FO(X)5
b(Y)910 622 y FT(,)30 b(and)g(that)1509 735 y FP(S)1565
749 y FO(X)1657 735 y FN(\031)1728 749 y FK(K)1782 760
y Fy(X)t(Y)1918 735 y FN(f)p FP(Y)2036 698 y FO(k)2079
735 y FN(g)p FP(;)107 b(S)2312 749 y FO(X)2404 735 y
FN(\031)2475 749 y FK(K)2529 760 y Fy(X)t(Y)2665 735
y FN(f:)p FP(C)2843 698 y FO(j)2879 735 y FN(g)605 902
y FT(while)29 b FN(f)p FP(Y)961 869 y FO(k)1004 902 y
FN(g)c(6\031)1145 916 y FK(K)1199 927 y Fy(X)t(Y)1335
902 y FN(f:)p FP(C)1513 869 y FO(j)1549 902 y FN(g)p
FT(,)31 b(and)f(therefore)1469 1106 y FN(K)1538 1120
y FO(X)5 b(Y)1688 1106 y FT(=)25 b FN(K)1853 1120 y FO(S)1896
1131 y Fy(X)1978 1106 y FN([)20 b FT(\()p FN(P)2157 1120
y FO(X)2250 1106 y FN($)26 b FP(k)s FT(\))20 b FN([)g
FT(\()p FN(P)2651 1120 y FO(X)2744 1106 y FN($)25 b FP(j)5
b FT(\))p FP(:)605 1311 y FT(Similarly)-8 b(,)23 b(since)h
FN(ff)p FP(X)1394 1278 y FO(i)1424 1311 y FN(g)p FP(;)15
b FN(f:)p FP(A)1683 1278 y FO(j)1720 1311 y FN(gg)26
b FT(is)e(w)m(ell-coloured)f(with)h(resp)s(ect)h(to)g
FP(i)h FN($)f FP(j)5 b FT(,)27 b(it)d(is)g(the)h(case)605
1423 y(that)31 b FN(f)p FP(S)903 1437 y FO(X)971 1423
y FP(;)15 b FN(f:)p FP(A)1185 1390 y FO(j)1222 1423 y
FN(gg)31 b FT(is)f(w)m(ell-coloured)f(with)g(resp)s(ect)h(to)h
FN(K)2681 1437 y FO(X)2749 1423 y FT(.)41 b(It)30 b(is)g(also)g(the)g
(case)i(that)1908 1628 y FP(S)1964 1642 y FO(X)2057 1628
y FN(\031)2128 1642 y FK(K)2182 1653 y Fy(X)2268 1628
y FN(f:)p FP(A)2442 1590 y FO(j)2479 1628 y FN(g)605
1832 y FT(and)e(that)1745 1945 y FN(K)1814 1959 y FO(X)1907
1945 y FT(=)25 b FN(K)2072 1959 y FO(S)2115 1970 y Fy(X)2197
1945 y FN([)20 b FT(\()p FN(P)2376 1959 y FO(X)2469 1945
y FN($)25 b FP(j)5 b FT(\))p FP(:)605 2112 y FT(W)-8
b(e)32 b(no)m(w)e(use)g(corollary)g(8.2)h(again)g(to)g(deduce)f(that)h
(there)g(is)e(some)i(set)g FP(S)3235 2126 y FO(Y)3325
2112 y FT(suc)m(h)g(that)1151 2316 y(\()p FP(S)1242 2330
y FO(X)1330 2316 y FN([)20 b(f)p FP(Y)1529 2278 y FO(k)1572
2316 y FP(;)15 b FN(:)p FP(C)1745 2278 y FO(j)1781 2316
y FN(g)p FP(;)g FN(K)1935 2330 y FO(X)5 b(Y)2060 2316
y FT(\))26 b FN(!)2212 2278 y FK(\003)2212 2338 y FL(c)2276
2316 y FT(\()p FP(S)2367 2330 y FO(X)2455 2316 y FN([)20
b FP(S)2592 2330 y FO(Y)2672 2316 y FN([)g(f:)p FP(C)2931
2278 y FO(j)2967 2316 y FN(g)p FP(;)15 b FN(K)3122 2278
y FK(0)3121 2338 y FO(X)5 b(Y)3246 2316 y FT(\))1352
2463 y(\()p FN(f)p FP(Y)1506 2425 y FO(k)1549 2463 y
FP(;)15 b FN(:)p FP(B)1724 2425 y FO(i)1751 2463 y FN(g)p
FP(;)g(k)30 b FN($)25 b FP(i)p FT(\))h FN(!)2212 2425
y FK(\003)2212 2485 y FL(c)2276 2463 y FT(\()p FP(S)2367
2477 y FO(Y)2448 2463 y FN([)20 b(f:)p FP(B)2709 2425
y FO(i)2737 2463 y FN(g)p FP(;)15 b FN(K)2891 2477 y
FO(Y)2953 2463 y FT(\))605 2667 y(where)1363 2871 y FN(K)1433
2834 y FK(0)1432 2894 y FO(X)5 b(Y)1557 2871 y FN(d)p
FP(S)1653 2885 y FO(X)1720 2871 y FN(e)26 b FT(=)f FN(K)1951
2885 y FO(Y)2012 2871 y FN(d)p FP(S)2108 2885 y FO(X)2175
2871 y FN(e)h FT(=)f FN(K)2406 2885 y FO(X)5 b(Y)2530
2871 y FN(d)p FP(S)2626 2885 y FO(X)2694 2871 y FN(e)25
b FT(=)g FN(K)2924 2885 y FO(S)2967 2896 y Fy(X)3029
2871 y FP(;)1456 3009 y FN(K)1526 2971 y FK(0)1525 3031
y FO(X)5 b(Y)1649 3009 y FN(d)p FP(S)1745 3023 y FO(Y)1806
3009 y FN(e)25 b FT(=)g FN(K)2036 3023 y FO(Y)2097 3009
y FN(d)p FP(S)2193 3023 y FO(Y)2254 3009 y FN(e)h FT(=)f
FN(K)2485 3023 y FO(S)2528 3034 y Fy(Y)2586 3009 y FP(;)45
b FT(sa)m(y)-8 b(,)32 b(and)1151 3198 y(\()p FP(S)1242
3212 y FO(Y)1328 3132 y FK(K)1382 3109 y FD(0)1382 3155
y Fy(X)t(Y)1363 3198 y FN(!)60 b FT(\()p FP(S)1605 3212
y FO(X)1692 3198 y FN([)20 b(f:)p FP(C)1951 3161 y FO(j)1987
3198 y FN(g)p FT(\))q(\))26 b(=)e(\()p FP(S)2315 3212
y FO(Y)2402 3144 y FK(K)2456 3155 y Fy(Y)2410 3198 y
FN(!)33 b(f:)p FP(B)2714 3161 y FO(i)2741 3198 y FN(g)q
FT(\))25 b(=)g FN(P)3006 3212 y FO(Y)3067 3198 y FP(;)46
b FT(sa)m(y)-8 b(.)605 3403 y(Note)32 b(that)f(the)f(colour)g
FP(k)j FT(is)d(in)f FN(P)1792 3417 y FO(Y)1883 3403 y
FT(since)h(it)f(relates)i(with)e FP(i)h FT(in)f FN(K)2929
3370 y FK(0)2928 3429 y FO(X)5 b(Y)3083 3403 y FT(b)s(ecause)30
b(of)g(the)h(fact)605 3515 y(that)g(\()p FP(i)26 b FN($)f
FP(k)k FN($)c FP(j)5 b FT(\))26 b FN(\022)f(K)1471 3482
y FK(0)1470 3542 y FO(X)5 b(Y)1625 3515 y FT(b)m(y)30
b(prop)s(osition)e(8.2.)605 3666 y(By)e(prop)s(osition)c(8.11)27
b(w)m(e)f(deduce)e(that)i FN(f)p FP(S)2121 3680 y FO(X)2189
3666 y FP(;)15 b(S)2285 3680 y FO(Y)2346 3666 y FP(;)g
FN(f:)p FP(C)2564 3633 y FO(j)2600 3666 y FN(gg)26 b
FT(is)e(w)m(ell-coloured)g(with)g(resp)s(ect)605 3779
y(to)31 b FN(K)786 3746 y FK(0)785 3806 y FO(X)5 b(Y)910
3779 y FT(,)30 b(and)g(that)1145 3983 y FP(S)1201 3997
y FO(X)1293 3983 y FN(\031)1364 4002 y FK(K)1418 3979
y FD(0)1418 4025 y Fy(X)t(Y)1554 3983 y FP(S)1610 3997
y FO(Y)1670 3983 y FP(;)106 b(S)1857 3997 y FO(X)1950
3983 y FN(\031)2021 4002 y FK(K)2075 3979 y FD(0)2075
4025 y Fy(X)t(Y)2211 3983 y FN(f:)p FP(C)2389 3945 y
FO(j)2425 3983 y FN(g)p FP(;)g(S)2657 3997 y FO(Y)2743
3983 y FN(6\031)2814 4002 y FK(K)2868 3979 y FD(0)2868
4025 y Fy(X)t(Y)3004 3983 y FN(f:)p FP(C)3182 3945 y
FO(j)3218 3983 y FN(g)p FT(;)605 4187 y(and)30 b(so)1297
4300 y FN(K)1367 4262 y FK(0)1366 4322 y FO(X)5 b(Y)1516
4300 y FT(=)25 b FN(K)1681 4314 y FO(S)1724 4325 y Fy(X)1806
4300 y FN([)20 b(K)1956 4314 y FO(S)1999 4325 y Fy(Y)2077
4300 y FN([)g FT(\()p FN(P)2256 4314 y FO(X)2349 4300
y FN($)25 b(P)2528 4314 y FO(Y)2589 4300 y FT(\))20 b
FN([)g FT(\()p FN(P)2823 4314 y FO(X)2916 4300 y FN($)26
b FP(j)5 b FT(\))p FP(:)605 4467 y FT(Similarly)-8 b(,)28
b FN(f)p FP(S)1111 4481 y FO(Y)1171 4467 y FP(;)15 b
FN(f:)p FP(B)1391 4434 y FO(i)1420 4467 y FN(gg)31 b
FT(is)e(w)m(ell-coloured)g(with)h(resp)s(ect)g(to)h FN(K)2879
4481 y FO(Y)2970 4467 y FT(and)1347 4671 y FP(S)1403
4685 y FO(Y)1489 4671 y FN(\031)1560 4685 y FK(K)1614
4696 y Fy(Y)1696 4671 y FN(f:)p FP(B)1876 4634 y FO(i)1904
4671 y FN(g)p FP(;)197 b FN(K)2240 4685 y FO(Y)2327 4671
y FT(=)25 b FN(K)2492 4685 y FO(S)2535 4696 y Fy(Y)2612
4671 y FN([)20 b FT(\()p FN(P)2791 4685 y FO(Y)2878 4671
y FN($)25 b FP(i)p FT(\))p FP(:)605 4913 y FT(T)-8 b(o)31
b(summarise)e(\(see)i(also)f(\014gure)g(20\),)1035 5117
y(\()p FN(f)p FP(X)1197 5079 y FO(i)1227 5117 y FP(;)15
b(Y)1340 5079 y FO(k)1383 5117 y FP(;)g FN(:)p FP(C)1556
5079 y FO(j)1592 5117 y FN(g)p FP(;)g(i)26 b FN($)f FP(k)k
FN($)c FP(j)5 b FT(\))26 b FN(+)2201 5131 y FL(c)2261
5117 y FT(\()p FP(S)2352 5131 y FO(X)2440 5117 y FN([)20
b FP(S)2577 5131 y FO(Y)2658 5117 y FN([)f(f:)p FP(C)2916
5079 y FO(j)2952 5117 y FN(g)p FP(;)c FN(K)3107 5079
y FK(0)3106 5139 y FO(X)5 b(Y)3232 5117 y FT(\))1386
5255 y(\()p FN(f)p FP(X)1548 5217 y FO(i)1578 5255 y
FP(;)15 b FN(:)p FP(A)1747 5217 y FO(j)1783 5255 y FN(g)p
FP(;)g(i)27 b FN($)e FP(j)5 b FT(\))26 b FN(+)2201 5269
y FL(c)2261 5255 y FT(\()p FP(S)2352 5269 y FO(X)2440
5255 y FN([)20 b(f:)p FP(A)2695 5217 y FO(j)2732 5255
y FN(g)p FP(;)15 b FN(K)2886 5269 y FO(X)2954 5255 y
FT(\))61 b(for)30 b(all)g FP(A)1377 5401 y FT(\()p FN(f)p
FP(Y)1530 5364 y FO(k)1573 5401 y FP(;)15 b FN(:)p FP(B)1748
5364 y FO(i)1776 5401 y FN(g)p FP(;)g(i)26 b FN($)f FP(k)s
FT(\))h FN(+)2201 5415 y FL(c)2261 5401 y FT(\()p FP(S)2352
5415 y FO(Y)2433 5401 y FN([)20 b(f:)p FP(B)2694 5364
y FO(i)2722 5401 y FN(g)p FP(;)15 b FN(K)2876 5415 y
FO(Y)2938 5401 y FT(\))61 b(for)30 b(all)f FP(B)5 b(:)p
eop
%%Page: 183 193
183 192 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(183)1163
396 y(where)61 b FN(K)1527 359 y FK(0)1526 419 y FO(X)5
b(Y)1675 396 y FT(=)25 b FN(K)1840 410 y FO(S)1883 421
y Fy(X)1965 396 y FN([)20 b(K)2115 410 y FO(S)2158 421
y Fy(Y)2236 396 y FN([)g FT(\()p FN(P)2415 410 y FO(X)2508
396 y FN($)25 b(P)2687 410 y FO(Y)2748 396 y FT(\))c
FN([)e FT(\()p FN(P)2982 410 y FO(X)3076 396 y FN($)25
b FP(j)5 b FT(\))1513 534 y FN(K)1582 548 y FO(X)1675
534 y FT(=)25 b FN(K)1840 548 y FO(S)1883 559 y Fy(X)1965
534 y FN([)20 b FT(\()p FN(P)2144 548 y FO(X)2237 534
y FN($)26 b FP(j)5 b FT(\))1520 672 y FN(K)1589 686 y
FO(Y)1675 672 y FT(=)25 b FN(K)1840 686 y FO(S)1883 697
y Fy(Y)1961 672 y FN([)20 b FT(\()p FN(P)2140 686 y FO(Y)2226
672 y FN($)25 b FP(i)p FT(\))p FP(:)605 876 y FT(The)32
b(rest)g(of)g(this)f(pro)s(of)g(is)g(no)m(w)h(similar)d(to)k(the)f(one)
h(of)f(prop)s(osition)d(8.8.)47 b(The)31 b(set)i FP(S)3679
890 y FO(X)3767 876 y FN([)605 989 y FP(S)661 1003 y
FO(Y)742 989 y FN([)20 b(f:)p FP(C)1001 956 y FO(j)1037
989 y FN(g)31 b FT(can)f(b)s(e)g(partitioned)f(in)m(to)1861
1193 y(\()p FP(S)1952 1207 y FO(X)2040 1193 y FN([)19
b(f:)p FP(C)2298 1156 y FO(j)2335 1193 y FN(g)p FP(;)c(S)2476
1207 y FO(Y)2537 1193 y FT(\))605 1398 y(whic)m(h)29
b(is)h(w)m(ell-coloured)f(with)g(resp)s(ect)h(to)i FN(K)2204
1412 y FO(X)5 b(Y)2358 1398 y FT(as)1324 1602 y FN(K)1394
1564 y FK(0)1393 1624 y FO(X)g(Y)1517 1602 y FN(d)p FP(S)1613
1616 y FO(X)1701 1602 y FN([)19 b(f:)p FP(C)1959 1564
y FO(j)1995 1602 y FN(ge)26 b FT(=)f FN(K)2271 1616 y
FO(S)2314 1627 y Fy(Y)2392 1602 y FN([)20 b FT(\()p FN(P)2571
1616 y FO(X)2664 1602 y FN($)25 b FP(j)5 b FT(\))1951
1791 y FN(P)2014 1805 y FO(X)2106 1791 y FT(=)25 b(\()p
FP(S)2293 1805 y FO(X)2381 1791 y FN([)20 b(f:)p FP(C)2640
1754 y FO(j)2676 1791 y FN(g)p FT(\))2782 1725 y FK(K)2836
1702 y FD(0)2836 1748 y Fy(X)t(Y)2817 1791 y FN(!)60
b FP(S)3024 1805 y FO(Y)1957 1981 y FN(P)2020 1995 y
FO(Y)2106 1981 y FT(=)25 b FP(S)2258 1995 y FO(Y)2344
1915 y FK(K)2398 1891 y FD(0)2398 1937 y Fy(X)t(Y)2379
1981 y FN(!)60 b FT(\()p FP(S)2621 1995 y FO(X)2709 1981
y FN([)20 b(f:)p FP(C)2968 1943 y FO(j)3004 1981 y FN(g)p
FT(\))p FP(;)605 2185 y FT(and)30 b(so)1151 2298 y FN(K)1221
2260 y FK(0)1220 2320 y FO(X)5 b(Y)1369 2298 y FT(=)25
b FN(K)1535 2260 y FK(0)1534 2320 y FO(X)5 b(Y)1659 2298
y FN(d)p FP(S)1755 2312 y FO(X)1842 2298 y FN([)20 b(f:)p
FP(C)2101 2260 y FO(j)2137 2298 y FN(ge)h([)f(K)2394
2260 y FK(0)2393 2320 y FO(X)5 b(Y)2517 2298 y FN(d)p
FP(S)2613 2312 y FO(Y)2674 2298 y FN(e)21 b([)f FT(\()p
FN(P)2914 2312 y FO(X)3007 2298 y FN($)25 b(P)3186 2312
y FO(Y)3247 2298 y FT(\))605 2465 y(Therefore)30 b(b)m(y)h(theorem)f
(7.8,)i FP(S)1725 2479 y FO(X)1812 2465 y FN([)20 b FP(S)1949
2479 y FO(Y)2030 2465 y FN([)g(f:)p FP(C)2289 2432 y
FO(j)2325 2465 y FN(g)31 b FT(is)e FN(K)2562 2432 y FK(0)2561
2492 y FO(X)5 b(Y)2685 2465 y FT(-inconsisten)m(t)30
b(if)f(and)h(only)g(if)1531 2669 y FP(S)1587 2683 y FO(X)1675
2669 y FN([)19 b(f:)p FP(C)1933 2631 y FO(j)1969 2669
y FP(;)c(I)2056 2631 y FO(k)2100 2669 y FN(g)83 b FT(and)g
FP(S)2514 2683 y FO(Y)2594 2669 y FN([)20 b(f:)p FP(I)2828
2631 y FO(i)2856 2669 y FN(g)605 2873 y FT(are)37 b(for)g(some)g
(\014rst-order)f(sen)m(tence)i FP(I)7 b FT(.)61 b(No)m(w)37
b FP(S)2345 2887 y FO(X)2437 2873 y FN([)24 b(f:)p FP(C)2700
2840 y FO(j)2736 2873 y FP(;)15 b(I)2823 2840 y FO(k)2866
2873 y FN(g)38 b FT(is)d FN(K)3116 2840 y FK(0)3115 2900
y FO(X)5 b(Y)3240 2873 y FT(-inconsisten)m(t)36 b(if)605
2986 y(and)30 b(only)f(if)1054 3190 y FP(S)1110 3204
y FO(X)1198 3190 y FN([)19 b(f:)p FP(C)1456 3153 y FO(j)1493
3190 y FP(;)c(I)1580 3153 y FO(k)1623 3190 y FN(g)30
b FT(is)g FN(K)1859 3204 y FO(S)1902 3215 y Fy(X)1984
3190 y FN([)20 b FT(\()p FN(P)2163 3204 y FO(X)2256 3190
y FN($)25 b(f)p FP(j;)15 b(k)s FN(g)p FT(\)-inconsisten)m(t)33
b(b)m(y)d(prop.)g(7.7)880 3328 y FN(,)83 b FP(S)1110
3342 y FO(X)1198 3328 y FN([)19 b(f:)p FP(C)1456 3290
y FO(j)1493 3328 y FP(;)c(I)1580 3290 y FO(j)1617 3328
y FN(g)30 b FT(is)g FN(K)1853 3342 y FO(S)1896 3353 y
Fy(X)1978 3328 y FN([)20 b FT(\()p FN(P)2157 3342 y FO(X)2250
3328 y FN($)25 b FP(j)5 b FT(\)-inconsisten)m(t)31 b(b)m(y)f(theorem)h
(7.5)880 3466 y FN(,)83 b FP(S)1110 3480 y FO(X)1198
3466 y FN([)19 b(f:)p FP(C)1456 3428 y FO(j)1493 3466
y FP(;)c(I)1580 3428 y FO(j)1617 3466 y FN(g)30 b FT(is)g
FN(K)1853 3480 y FO(X)1920 3466 y FT(-inconsisten)m(t)880
3604 y FN(,)83 b FP(S)1110 3618 y FO(X)1198 3604 y FN([)19
b(f:)p FT(\()p FP(I)33 b FN(\))25 b FP(C)7 b FT(\))1715
3566 y FO(j)1752 3604 y FN(g)31 b FT(is)e FN(K)1988 3618
y FO(X)2056 3604 y FT(-inconsisten)m(t)880 3741 y FN(,)83
b FP(X)33 b Ff( )25 b FT(\()p FP(I)33 b FN(\))25 b FP(C)7
b FT(\))30 b(b)m(y)g(the)h(\014rst)f(induction)e(h)m(yp)s(othesis)n
FP(:)605 3946 y FT(Also,)j FP(S)892 3960 y FO(Y)972 3946
y FN([)20 b(f:)p FP(I)1206 3913 y FO(i)1234 3946 y FN(g)31
b FT(is)f FN(K)1472 3913 y FK(0)1471 3973 y FO(X)5 b(Y)1595
3946 y FT(-inconsisten)m(t)30 b(if)f(and)h(only)f(if)1235
4150 y FP(S)1291 4164 y FO(X)1379 4150 y FN([)19 b(f:)p
FP(I)1612 4112 y FO(i)1641 4150 y FN(g)31 b FT(is)e FN(K)1877
4164 y FO(S)1920 4175 y Fy(Y)1998 4150 y FN([)20 b FT(\()p
FN(P)2177 4164 y FO(Y)2263 4150 y FN($)25 b FP(i)p FT(\)-inconsisten)m
(t)31 b(b)m(y)f(prop.)g(7.7)1061 4288 y FN(,)83 b FP(S)1291
4302 y FO(Y)1372 4288 y FN([)20 b(f:)p FP(I)1606 4250
y FO(i)1634 4288 y FN(g)31 b FT(is)e FN(K)1870 4302 y
FO(Y)1931 4288 y FT(-inconsisten)m(t)1061 4426 y FN(,)83
b FP(Y)46 b Ff( )25 b FP(I)37 b FT(b)m(y)30 b(the)h(second)f(induction)
e(h)m(yp)s(othesis.)605 4630 y(Th)m(us,)i(\()p FP(S)952
4644 y FO(X)1039 4630 y FN([)19 b FP(S)1175 4644 y FO(Y)1255
4630 y FN([)g(f:)p FP(C)1513 4597 y FO(j)1549 4630 y
FN(g)p FP(;)c FN(K)1704 4597 y FK(0)1703 4657 y FO(X)5
b(Y)1829 4630 y FT(\))30 b(is)f(inconsisten)m(t)g(if)g(and)h(only)f(if)
g(there)h(is)g(some)g FP(I)37 b FT(suc)m(h)605 4743 y(that)1939
4947 y FP(X)c Ff( )25 b FT(\()p FP(I)33 b FN(\))25 b
FP(C)7 b FT(\))1949 5085 y FP(Y)45 b Ff( )25 b FP(I)605
5289 y FT(and)32 b(b)m(y)g(the)g(inductiv)m(e)e(de\014nition)g(of)i
Ff( )g FT(this)f(is)g(equiv)-5 b(alen)m(t)32 b(to)h(whether)e
Fv(X)50 b Fw(on)43 b Fv(Y)j Ff( )28 b FP(C)7 b FT(.)605
5402 y(Finally)-8 b(,)29 b(b)m(y)i(corollary)e(8.1,)j(whenev)m(er)e(\()
p Fv(X)50 b Fw(on)43 b Fv(Y)19 b FT(\))25 b FN(+)2474
5416 y FL(c)2535 5402 y FT(\()p FP(S;)15 b FN(K)q FT(\))32
b(holds)d(then)1503 5606 y(\()p FP(S;)15 b FN(K)q FT(\))27
b Fl(u)1837 5620 y FL(rc)1925 5606 y FT(\()p FP(S)2016
5620 y FO(X)2103 5606 y FN([)20 b FP(S)2240 5620 y FO(Y)2321
5606 y FN([)g(f:)p FP(C)2580 5569 y FO(j)2616 5606 y
FN(g)p FP(;)15 b FN(K)2771 5569 y FK(0)2770 5629 y FO(X)5
b(Y)2895 5606 y FT(\))p eop
%%Page: 184 194
184 193 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(184)605
396 y(and)26 b(therefore)g(\()p FP(S;)15 b FN(K)q FT(\))28
b(is)d(inconsisten)m(t)h(if)f(and)h(only)f(if)g(\()p
FP(S)2618 410 y FO(X)2697 396 y FN([)12 b FP(S)2826 410
y FO(Y)2898 396 y FN([)g(f:)p FP(C)3149 363 y FO(j)3185
396 y FN(g)p FP(;)j FN(K)3340 363 y FK(0)3339 423 y FO(X)5
b(Y)3464 396 y FT(\))26 b(is.)39 b(This)605 509 y(concludes)30
b(the)g(pro)s(of)g(of)h(this)e(case.)1881 b Ff(\002)514
697 y FN(\017)46 b FT(The)25 b Fw(and)n FT(-Induction)f(Case)h(\()p
Fv(P)37 b FT(is)24 b(some)h(expression)f(\()p Fv(X)50
b Fw(and)43 b Fv(Y)18 b FT(\)\):)39 b(Giv)m(en)25 b(the)g(h)m(yp)s
(otheses:)689 885 y(1.)46 b(for)27 b(ev)m(ery)h(sen)m(tence)h
FP(A)p FT(,)f(if)e(\()p Fv(A)44 b Fw(by)f Fv(X)6 b FT(\))26
b FN(+)2202 899 y FL(c)2262 885 y FT(\()p FP(S)2358 852
y FK(0)2382 885 y FP(;)15 b FN(K)2492 852 y FK(0)2516
885 y FT(\))27 b(then)g FP(S)2843 852 y FK(0)2894 885
y FT(is)f FN(K)3052 852 y FK(0)3076 885 y FT(-inconsisten)m(t)g(if)g
(and)805 998 y(only)k(if)f Fv(X)j Ff( )25 b FP(A)p FT(;)689
1144 y(2.)46 b(for)35 b(ev)m(ery)h(sen)m(tence)g FP(B)5
b FT(,)36 b(if)e(\()p Fv(B)48 b Fw(by)43 b Fv(Y)19 b
FT(\))33 b FN(+)2251 1158 y FL(c)2319 1144 y FT(\()p
FP(S)2415 1111 y FK(0)q(0)2458 1144 y FP(;)15 b FN(K)2568
1111 y FK(0)q(0)2612 1144 y FT(\))35 b(then)g FP(S)2955
1111 y FK(00)3032 1144 y FT(is)f FN(K)3198 1111 y FK(0)q(0)3241
1144 y FT(-inconsisten)m(t)h(if)805 1257 y(and)30 b(only)f(if)h
Fv(Y)44 b Ff( )25 b FP(B)5 b FT(;)605 1444 y(w)m(e)31
b(are)g(required)d(to)k(sho)m(w)e(that)h(for)f(an)m(y)h(sen)m(tence)g
FP(C)37 b FT(if)1714 1648 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(+)2422 1662
y FL(c)2482 1648 y FT(\()p FP(S;)15 b FN(K)q FT(\))605
1853 y(then)30 b FP(S)36 b FT(is)29 b FN(K)q FT(-inconsisten)m(t)h(if)g
(and)f(only)h(if)f(\()p Fv(X)51 b Fw(and)42 b Fv(Y)19
b FT(\))25 b Ff( )g FP(C)7 b FT(.)605 2003 y FQ(Pro)s(of)p
FT(:)32 b(By)e(the)h(de\014nition)d(of)j FN(!)1813 1970
y FK(\003)1813 2025 y FL(c)1882 2003 y FT(w)m(e)g(get)1234
2207 y(\()p Fv(C)50 b Fw(by)43 b Fv(X)49 b Fw(and)43
b Fv(Y)18 b FT(\))26 b FN(!)2020 2170 y FK(\003)2020
2230 y FL(c)2085 2207 y FT(\()p FN(f)p FP(X)2247 2170
y FO(i)2276 2207 y FP(;)15 b(Y)2389 2170 y FO(k)2432
2207 y FP(;)g FN(:)p FP(C)2605 2170 y FO(j)2641 2207
y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s FN(g)28 b($)d FP(j)5
b FT(\))p FP(;)1521 2345 y FT(\()p Fv(A)44 b Fw(by)f
Fv(X)6 b FT(\))26 b FN(!)2020 2307 y FK(\003)2020 2367
y FL(c)2085 2345 y FT(\()p FN(f)p FP(X)2247 2307 y FO(i)2276
2345 y FP(;)15 b FN(:)p FP(A)2445 2307 y FO(j)2482 2345
y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5 b FT(\))p FP(;)1525
2492 y FT(\()p Fv(B)48 b Fw(by)43 b Fv(Y)18 b FT(\))26
b FN(!)2020 2454 y FK(\003)2020 2514 y FL(c)2085 2492
y FT(\()p FN(f)p FP(Y)2238 2454 y FO(k)2281 2492 y FP(;)15
b FN(:)p FP(B)2456 2454 y FO(j)2492 2492 y FN(g)p FP(;)g(k)29
b FN($)d FP(j)5 b FT(\))p FP(:)605 2696 y FT(Figure)38
b(21)h(illustrates)d(ho)m(w)i(the)g(relation)f FN(!)2250
2710 y FL(c)2323 2696 y FT(is)g(applied)f(on)i(the)g(coloured)g
(structured)605 2809 y(problems)800 3013 y(\()p FN(f)p
FP(X)962 2976 y FO(i)991 3013 y FP(;)15 b(Y)1104 2976
y FO(k)1147 3013 y FP(;)g FN(:)p FP(C)1320 2976 y FO(j)1356
3013 y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s FN(g)28 b($)d
FP(j)5 b FT(\))p FP(;)108 b FT(\()p FN(f)p FP(X)2168
2976 y FO(i)2197 3013 y FP(;)15 b FN(:)p FP(A)2366 2976
y FO(j)2403 3013 y FN(g)p FP(;)g(i)26 b FN($)f FP(j)5
b FT(\))p FP(;)108 b FT(\()p FN(f)p FP(Y)3024 2976 y
FO(k)3067 3013 y FP(;)15 b FN(:)p FP(B)3242 2976 y FO(j)3278
3013 y FN(g)p FP(;)g(k)29 b FN($)d FP(j)5 b FT(\))605
3217 y(during)30 b(the)j(pro)s(of)f(of)g(this)g(case,)i(and)e(b)m(y)g
(a)h(similar)d(argumen)m(t)j(to)g(the)g(previous)e(case)i(w)m(e)605
3330 y(deduce)d(that)1040 3535 y(\()p FN(f)p FP(X)1202
3497 y FO(i)1232 3535 y FP(;)15 b(Y)1345 3497 y FO(k)1388
3535 y FP(;)g FN(:)p FP(C)1561 3497 y FO(j)1597 3535
y FN(g)p FP(;)g FN(f)p FP(i;)g(k)s FN(g)28 b($)d FP(j)5
b FT(\))26 b FN(+)2196 3549 y FL(c)2256 3535 y FT(\()p
FP(S)2347 3549 y FO(X)2435 3535 y FN([)20 b FP(S)2572
3549 y FO(Y)2653 3535 y FN([)f(f:)p FP(C)2911 3497 y
FO(j)2947 3535 y FN(g)p FP(;)c FN(K)3102 3497 y FK(0)3101
3557 y FO(X)5 b(Y)3226 3535 y FT(\))1381 3672 y(\()p
FN(f)p FP(X)1543 3635 y FO(i)1573 3672 y FP(;)15 b FN(:)p
FP(A)1742 3635 y FO(j)1778 3672 y FN(g)p FP(;)g(i)27
b FN($)e FP(j)5 b FT(\))26 b FN(+)2196 3686 y FL(c)2256
3672 y FT(\()p FP(S)2347 3686 y FO(X)2435 3672 y FN([)20
b(f:)p FP(A)2690 3635 y FO(j)2727 3672 y FN(g)p FP(;)15
b FN(K)2881 3686 y FO(X)2949 3672 y FT(\))61 b(for)30
b(all)f FP(A)1409 3819 y FT(\()p FN(f)p FP(Y)1562 3782
y FO(k)1605 3819 y FP(;)15 b FN(:)p FP(B)1780 3782 y
FO(j)1816 3819 y FP(;)g(i)26 b FN($)f FP(k)s FT(\))h
FN(+)2196 3833 y FL(c)2256 3819 y FT(\()p FP(S)2347 3833
y FO(Y)2428 3819 y FN([)20 b(f:)p FP(B)2689 3782 y FO(i)2717
3819 y FN(g)p FP(;)15 b FN(K)2871 3833 y FO(Y)2933 3819
y FT(\))61 b(for)30 b(all)f FP(B)5 b(:)605 4023 y FT(where)1354
4228 y FN(K)1424 4190 y FK(0)1423 4250 y FO(X)g(Y)1572
4228 y FT(=)25 b FN(K)1737 4242 y FO(S)1780 4253 y Fy(X)1863
4228 y FN([)20 b(K)2013 4242 y FO(S)2056 4253 y Fy(Y)2133
4228 y FN([)g FT(\()p FN(P)2312 4242 y FO(X)2405 4228
y FN($)26 b FP(j)5 b FT(\))21 b FN([)f FT(\()p FN(P)2799
4242 y FO(Y)2885 4228 y FN($)25 b FP(j)5 b FT(\))1410
4365 y FN(K)1479 4379 y FO(X)1572 4365 y FT(=)25 b FN(K)1737
4379 y FO(S)1780 4390 y Fy(X)1863 4365 y FN([)20 b FT(\()p
FN(P)2042 4379 y FO(X)2135 4365 y FN($)25 b FP(j)5 b
FT(\))1417 4503 y FN(K)1486 4517 y FO(Y)1572 4503 y FT(=)25
b FN(K)1737 4517 y FO(S)1780 4528 y Fy(Y)1858 4503 y
FN([)20 b FT(\()p FN(P)2037 4517 y FO(Y)2124 4503 y FN($)25
b FP(j)5 b FT(\))1373 4641 y FN(K)1442 4655 y FO(S)1485
4666 y Fy(X)1572 4641 y FT(=)25 b FN(K)1738 4603 y FK(0)1737
4663 y FO(X)5 b(Y)1862 4641 y FN(d)p FP(S)1958 4655 y
FO(X)2025 4641 y FN(e)26 b FT(=)f FN(K)2256 4655 y FO(X)2323
4641 y FN(d)p FP(S)2419 4655 y FO(X)2487 4641 y FN(e)1378
4779 y(K)1447 4793 y FO(S)1490 4804 y Fy(Y)1572 4779
y FT(=)g FN(K)1738 4741 y FK(0)1737 4801 y FO(X)5 b(Y)1862
4779 y FN(d)p FP(S)1958 4793 y FO(Y)2018 4779 y FN(e)26
b FT(=)f FN(K)2249 4793 y FO(Y)2310 4779 y FN(d)p FP(S)2406
4793 y FO(Y)2467 4779 y FN(e)1417 4968 y(P)1480 4982
y FO(X)1572 4968 y FT(=)g(\()p FP(S)1759 4982 y FO(X)1852
4902 y FK(K)1906 4879 y FD(0)1906 4925 y Fy(X)t(Y)1887
4968 y FN(!)60 b(:)p FP(C)2171 4931 y FO(j)2207 4968
y FT(\))25 b(=)g(\()p FP(S)2454 4982 y FO(X)2547 4914
y FK(K)2601 4925 y Fy(X)2557 4968 y FN(!)36 b(:)p FP(A)2813
4931 y FO(j)2849 4968 y FT(\))1423 5157 y FN(P)1486 5171
y FO(Y)1572 5157 y FT(=)25 b(\()p FP(S)1759 5171 y FO(Y)1845
5091 y FK(K)1899 5068 y FD(0)1899 5114 y Fy(X)t(Y)1880
5157 y FN(!)60 b(:)p FP(C)2164 5120 y FO(j)2200 5157
y FT(\))25 b(=)g(\()p FP(S)2447 5171 y FO(Y)2534 5103
y FK(K)2588 5114 y Fy(Y)2542 5157 y FN(!)33 b(:)p FP(B)2801
5120 y FO(j)2836 5157 y FT(\))605 5362 y(and)1743 5475
y FP(i)26 b FN(2)e(P)1948 5489 y FO(X)2380 5475 y FP(k)k
FN(2)d(P)2604 5489 y FO(Y)2665 5475 y FP(:)p eop
%%Page: 185 195
185 194 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(185)473
484 y(\()508 484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.32498 6.66249 4.63118
} false /N@X11 16 {InitRnode } NewNode end end
508 484 a FP(X)590 451 y FO(i)739
484 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y11 16 {InitRnode } NewNode end end
739 484 a FP(Y)812 451 y FO(k)969 484 y FN(:)1030
484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C11 16 {InitRnode } NewNode end end
1030 484 a FP(C)1102 451 y FO(j)1137 484 y FP(;)84
b FN(f)p FP(i;)15 b(k)s FN(g)27 b($)e FP(j)5 b FT(\))272
b(\()1984 484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.32498 6.66249 4.63118
} false /N@X12 16 {InitRnode } NewNode end end
1984 484 a FP(X)2066 451 y FO(i)2215
484 y FN(:)2276 484 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118
} false /N@A12 16 {InitRnode } NewNode end end
2276 484 a FP(A)2344 451 y FO(j)2380
484 y FP(;)83 b(i)26 b FN($)f FP(j)5 b FT(\))467 823
y(\()502 823 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX21 16 {InitRnode } NewNode end end
502 823 a FP(S)558 837 y FO(X)739 823
y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y21 16 {InitRnode } NewNode end end
739 823 a FP(Y)812 790 y FO(k)969 823 y FN(:)1030 823
y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C21 16 {InitRnode } NewNode end end
1030 823 a FP(C)1102 790 y FO(j)1137 823 y FP(;)84
b FN(K)1315 837 y FO(X)5 b(Y)1439 823 y FT(\))469 b(\()1978
823 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX22 16 {InitRnode } NewNode end end
1978 823 a FP(S)2034 837 y FO(X)2215 823 y FN(:)2276
823 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 12.6111 6.30554 4.63118
} false /N@A22 16 {InitRnode } NewNode end end
2276 823 a FP(A)2344 790 y FO(j)2380 823 y FP(;)83
b FN(K)2557 837 y FO(X)2625 823 y FT(\))265 b(\()2960
823 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.92937 6.96468 4.7646
} false /N@Y23 16 {InitRnode } NewNode end end
2960 823 a FP(Y)3034 790 y FO(k)3190 823 y FN(:)3251
823 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.25363 6.62682 4.63118
} false /N@B23 16 {InitRnode } NewNode end end
3251 823 a FP(B)3325 790 y FO(j)3361 823 y FP(;)83
b(k)29 b FN($)c FP(j)5 b FT(\))467 1161 y(\()502 1161
y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.82985 7.41492 2.89374
} false /N@SX31 16 {InitRnode } NewNode end end
502 1161 a FP(S)558 1175 y FO(X)739 1161 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374
} false /N@SY31 16 {InitRnode } NewNode end end
739 1161
a FP(S)795 1175 y FO(Y)969 1161 y FN(:)1030 1161 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.008 6.504 4.63118
} false /N@C31 16 {InitRnode } NewNode end end
1030
1161 a FP(C)1102 1128 y FO(j)1137 1161 y FP(;)84 b FN(K)1316
1128 y FK(0)1315 1188 y FO(X)5 b(Y)1439 1161 y FT(\))1451
b(\()2960 1161 y
tx@Dict begin tx@NodeDict begin {7.48248 1.695 14.02986 7.01492 2.89374
} false /N@SY33 16 {InitRnode } NewNode end end
2960 1161 a FP(S)3016 1175 y FO(Y)3190
1161 y FN(:)3251 1161 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.25363 6.62682 4.63118
} false /N@B33 16 {InitRnode } NewNode end end
3251 1161 a FP(B)3325 1128 y
FO(j)3361 1161 y FP(;)83 b FN(K)3538 1175 y FO(Y)3599
1161 y FT(\))3634 1161 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@X11 /N@SX21 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3634 1161 a 3634 1161 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
3634
1161 a 3628 1189 a FL(c)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634
1161 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3634 1161 a 3605 1189 a FK(\003)3634 1161 y
tx@Dict begin PutEnd end
3634
1161 a 3634 1161 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@X12 /N@SX22 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3634 1161 a 3634 1161 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
3634 1161
a 3628 1189 a FL(c)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634 1161
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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3634 1161 a 3605 1189 a FK(\003)3634 1161 y
tx@Dict begin PutEnd end
3634 1161
a 3634 1161 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 6.0
0 0 /N@Y21 /N@SY31 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3634 1161 a 3634 1161 a
tx@Dict begin tx@NodeDict begin /t 0.6 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
3634 1161 a 3628
1189 a FL(c)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634 1161 a
tx@Dict begin tx@NodeDict begin /t 0.6 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3634
1161 a 3605 1189 a FK(\003)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634
1161 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@Y23 /N@SY33 InitNC { NCLine } if end gsave 0.8 SLW 0. setgray
0 setlinecap stroke grestore grestore end
3634 1161 a 3634 1161 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
3634 1161 a 3628 1189
a FL(c)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634 1161 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 0.71065 def
/r 0.71065 def /h 0.42206 def /d 3.30017 def /flag false def HPutAdjust
LPutCoor end PutBegin end
3634 1161
a 3605 1189 a FK(\003)3634 1161 y
tx@Dict begin PutEnd end
3634 1161 a 3634 1161
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@Y11 /N@C11 InitNC { /AngleA 25. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
3634 1161 a 3634 1161 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@X11 /N@C11 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
3634 1161 a 3634 1161 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@Y21 /N@C21 InitNC { /AngleA 25. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
3634
1161 a 3634 1161 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@SX21 /N@C21 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
3634 1161 a 3634 1161 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@SY31 /N@C31 InitNC { /AngleA 25. def /AngleB 155. def
0.67 0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap
stroke grestore grestore end
3634 1161
a 3634 1161 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@SX31 /N@C31 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
3634 1161 a 3634 1161 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@X12 /N@A12 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
3634 1161 a 3634
1161 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@SX22 /N@A22 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
3634 1161 a 3634 1161 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@Y23 /N@B23 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
3634 1161 a 3634 1161
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@SY33 /N@B33 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
3634 1161 a 1621 1571 a FQ(Fig.)17 b(21.)41 b FT(The)30
b FM(and)f FT(Case.)605 1964 y(W)-8 b(e)43 b(no)m(w)f(pro)s(ceed)g(in)e
(a)i(similar)e(fashion)g(to)j(the)f(previous)e(case)j(and)e(to)i(the)f
(pro)s(of)f(of)605 2077 y(prop)s(osition)28 b(8.9.)42
b(The)30 b(set)h FP(S)1650 2091 y FO(X)1737 2077 y FN([)20
b FP(S)1874 2091 y FO(Y)1955 2077 y FN([)f(f:)p FP(C)2213
2044 y FO(j)2250 2077 y FN(g)30 b FT(can)h(b)s(e)f(partitioned)f(in)m
(to)1861 2281 y(\()p FP(S)1952 2295 y FO(Y)2033 2281
y FN([)20 b(f:)p FP(C)2292 2243 y FO(j)2328 2281 y FN(g)p
FP(;)15 b(S)2469 2295 y FO(X)2537 2281 y FT(\))605 2485
y(whic)m(h)32 b(is)g(w)m(ell-coloured)h(with)e(resp)s(ect)j(to)g
FN(K)2222 2452 y FK(0)2221 2512 y FO(X)5 b(Y)2345 2485
y FT(,)34 b(and)f(b)m(y)g(theorem)g(7.8)i(w)m(e)e(deduce)g(that)605
2598 y FP(S)661 2612 y FO(X)749 2598 y FN([)19 b FP(S)885
2612 y FO(Y)966 2598 y FN([)h(f:)p FP(C)1225 2565 y FO(j)1261
2598 y FN(g)31 b FT(is)e FN(K)1498 2565 y FK(0)1497 2625
y FO(X)5 b(Y)1621 2598 y FT(-inconsisten)m(t)30 b(if)f(and)h(only)g(if)
1534 2802 y FP(S)1590 2816 y FO(Y)1671 2802 y FN([)20
b(f:)p FP(C)1930 2765 y FO(j)1966 2802 y FP(;)15 b(I)2053
2765 y FO(i)2082 2802 y FN(g)83 b FT(and)g FP(S)2496
2816 y FO(X)2583 2802 y FN([)20 b(f:)p FP(I)2817 2765
y FO(j)2853 2802 y FN(g)605 3007 y FT(are)28 b(for)e(some)i
(\014rst-order)e(form)m(ula)g FP(I)7 b FT(.)40 b(No)m(w,)28
b(the)f(set)h FP(S)2555 3021 y FO(Y)2629 3007 y FN([)13
b(f:)p FP(C)2881 2974 y FO(j)2917 3007 y FP(;)i(I)3004
2974 y FO(i)3032 3007 y FN(g)28 b FT(can)f(b)s(e)f(partitioned)605
3119 y(in)m(to)1915 3232 y(\()p FN(f:)p FP(C)2128 3195
y FO(j)2165 3232 y FP(;)15 b(I)2252 3195 y FO(i)2280
3232 y FN(g)p FP(;)g(S)2421 3246 y FO(Y)2483 3232 y FT(\))605
3399 y(whic)m(h)33 b(is)h(w)m(ell-coloured)f(with)g(resp)s(ect)h(to)h
FN(K)2228 3366 y FK(0)2227 3426 y FO(X)5 b(Y)2386 3399
y FT(as)34 b(w)m(ell.)52 b(Hence)35 b(b)m(y)g(theorem)f(7.8,)j(it)d(is)
605 3512 y(the)d(case)g(that)g FP(S)1207 3526 y FO(Y)1288
3512 y FN([)20 b(f:)p FP(C)1547 3479 y FO(j)1583 3512
y FP(;)15 b(I)1670 3479 y FO(i)1698 3512 y FN(g)31 b
FT(is)f FN(K)1936 3479 y FK(0)1935 3539 y FO(X)5 b(Y)2059
3512 y FT(-inconsisten)m(t)30 b(if)f(and)h(only)f(if)1566
3716 y FN(f:)p FP(C)1744 3679 y FO(j)1780 3716 y FP(;)15
b(I)1867 3679 y FO(i)1896 3716 y FP(;)g(J)1995 3679 y
FO(k)2038 3716 y FN(g)83 b FT(and)g FP(S)2452 3730 y
FO(X)2539 3716 y FN([)20 b(f:)p FP(J)2785 3679 y FO(j)2822
3716 y FN(g)605 3921 y FT(are)31 b(for)f(some)h(sen)m(tence)h
FP(J)9 b FT(.)605 4071 y(The)30 b(set)h FN(f:)p FP(C)1112
4038 y FO(j)1148 4071 y FP(;)15 b(I)1235 4038 y FO(i)1264
4071 y FP(;)g(J)1363 4038 y FO(k)1406 4071 y FN(g)31
b FT(is)e FN(K)1643 4038 y FK(0)1642 4098 y FO(X)5 b(Y)1766
4071 y FT(-inconsisten)m(t)30 b(if)f(and)h(only)g(if)1128
4275 y FN(f:)p Fv(C)1299 4238 y FO(j)1336 4275 y FP(;)15
b Fv(I)1419 4238 y FO(i)1447 4275 y FP(;)g Fv(J)1542
4238 y FO(i)1570 4275 y FN(g)91 b FT(is)30 b FP(i)25
b FN($)g FP(j)5 b FT(-inconsisten)m(t)31 b(b)m(y)f(thm.)g(7.5)i(and)e
(prop.)f(7.7)954 4413 y FN(,)83 b(f:)p Fv(C)1299 4375
y FO(j)1336 4413 y FP(;)15 b Ft(\()p Fv(I)26 b Fu(^)19
b Fv(J)8 b Ft(\))1630 4375 y FO(i)1658 4413 y FN(g)91
b FT(is)30 b FP(i)25 b FN($)h FP(j)5 b FT(-inconsisten)m(t)954
4551 y FN(,)83 b FT(\()p FP(I)27 b FN(^)20 b FP(J)9 b
FT(\))26 b Ff(\032)1532 4513 y FK(\003)1597 4551 y FP(C)97
b FT(b)m(y)30 b(theorem)h(8.3)q FP(:)605 4755 y FT(Also,)g
FP(S)892 4769 y FO(Y)972 4755 y FN([)20 b(f:)p FP(J)1218
4722 y FO(j)1255 4755 y FN(g)31 b FT(is)e FN(K)1492 4722
y FK(0)1491 4782 y FO(X)5 b(Y)1615 4755 y FT(-inconsisten)m(t)30
b(if)f(and)h(only)g(if)1223 4959 y FP(S)1279 4973 y FO(Y)1359
4959 y FN([)20 b(f:)p FP(J)1605 4922 y FO(j)1642 4959
y FN(g)31 b FT(is)e FN(K)1878 4973 y FO(S)1921 4984 y
Fy(Y)1999 4959 y FN([)20 b FT(\()p FN(P)2178 4973 y FO(Y)2264
4959 y FN($)26 b FP(j)5 b FT(\)-inconsisten)m(t)30 b(b)m(y)h(prop.)e
(7.7)1049 5097 y FN(,)83 b FP(S)1279 5111 y FO(Y)1359
5097 y FN([)20 b(f:)p FP(J)1605 5060 y FO(j)1642 5097
y FN(g)31 b FT(is)e FN(K)1878 5111 y FO(Y)1939 5097 y
FT(-inconsisten)m(t)1049 5235 y FN(,)83 b FP(Y)45 b Ff( )25
b FP(J)40 b FT(b)m(y)30 b(the)g(second)h(induction)d(h)m(yp)s(othesis.)
p eop
%%Page: 186 196
186 195 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(186)605
396 y(And)30 b(also,)g FP(S)1068 410 y FO(Y)1149 396
y FN([)20 b(f:)p FP(I)1383 363 y FO(j)1419 396 y FN(g)31
b FT(is)f FN(K)1657 363 y FK(0)1656 423 y FO(X)5 b(Y)1780
396 y FT(-inconsisten)m(t)30 b(if)f(and)h(only)f(if)1220
601 y FP(S)1276 615 y FO(X)1363 601 y FN([)20 b(f:)p
FP(I)1597 563 y FO(j)1634 601 y FN(g)30 b FT(is)g FN(K)1870
615 y FO(S)1913 626 y Fy(X)1995 601 y FN([)20 b FT(\()p
FN(P)2174 615 y FO(X)2267 601 y FN($)25 b FP(j)5 b FT(\)-inconsisten)m
(t)31 b(b)m(y)f(prop.)g(7.7)1046 739 y FN(,)83 b FP(S)1276
753 y FO(X)1363 739 y FN([)20 b(f:)p FP(I)1597 701 y
FO(j)1634 739 y FN(g)30 b FT(is)g FN(K)1870 753 y FO(X)1937
739 y FT(-inconsisten)m(t)1046 876 y FN(,)83 b FP(X)33
b Ff( )25 b FP(I)37 b FT(b)m(y)30 b(the)h(\014rst)f(induction)e(h)m(yp)
s(othesis.)605 1081 y(Th)m(us,)g(\()p FP(S)950 1095 y
FO(X)1035 1081 y FN([)17 b FP(S)1169 1095 y FO(Y)1246
1081 y FN([)g(f:)p FP(C)1502 1048 y FO(j)1538 1081 y
FN(g)p FP(;)e FN(K)1693 1048 y FK(0)1692 1108 y FO(X)5
b(Y)1817 1081 y FT(\))29 b(is)f(inconsisten)m(t)f(if)h(and)g(only)g(if)
g(there)h(are)g(sen)m(tences)h FP(I)605 1193 y FT(and)g
FP(J)39 b FT(suc)m(h)31 b(that)2142 1398 y FP(X)i Ff( )25
b FP(I)2151 1536 y(Y)46 b Ff( )25 b FP(J)1946 1673 y
FT(\()p FP(I)j FN(^)20 b FP(J)9 b FT(\))26 b Ff(\032)2351
1636 y FK(\003)2415 1673 y FP(C)605 1878 y FT(and)39
b(b)m(y)g(the)h(de\014nition)d(of)i Ff( )h FT(this)e(is)g(equiv)-5
b(alen)m(t)39 b(to)h(whether)f Fv(X)50 b Fw(and)42 b
Fv(Y)59 b Ff( )40 b FP(C)46 b FT(holds.)605 1990 y(Finally)-8
b(,)29 b(b)m(y)i(corollary)e(8.1,)j(whenev)m(er)e(\()p
Fv(X)50 b Fw(and)43 b Fv(Y)18 b FT(\))26 b FN(+)2518
2004 y FL(c)2578 1990 y FT(\()p FP(S;)15 b FN(K)q FT(\))32
b(then)1503 2195 y(\()p FP(S;)15 b FN(K)q FT(\))27 b
Fl(u)1837 2209 y FL(rc)1925 2195 y FT(\()p FP(S)2016
2209 y FO(X)2103 2195 y FN([)20 b FP(S)2240 2209 y FO(Y)2321
2195 y FN([)g(f:)p FP(C)2580 2157 y FO(j)2616 2195 y
FN(g)p FP(;)15 b FN(K)2771 2157 y FK(0)2770 2217 y FO(X)5
b(Y)2895 2195 y FT(\))605 2399 y(and)26 b(therefore)g(\()p
FP(S;)15 b FN(K)q FT(\))28 b(is)d(inconsisten)m(t)h(if)f(and)h(only)f
(if)g(\()p FP(S)2618 2413 y FO(X)2697 2399 y FN([)12
b FP(S)2826 2413 y FO(Y)2898 2399 y FN([)g(f:)p FP(C)3149
2366 y FO(j)3185 2399 y FN(g)p FP(;)j FN(K)3340 2366
y FK(0)3339 2426 y FO(X)5 b(Y)3464 2399 y FT(\))26 b(is.)39
b(This)605 2512 y(concludes)30 b(the)g(pro)s(of)g(of)h(this)e(case.)
1881 b Ff(\002)378 2699 y FT(The)30 b(ab)s(o)m(v)m(e)h(case)h
(concludes)d(the)i(pro)s(of)f(of)g(the)h(curren)m(t)f(theorem.)1014
b Ff(\004)378 3025 y FQ(Example)34 b(8.5)46 b FT(In)29
b(this)h(example,)g(w)m(e)h(sho)m(w)f(that)h(it)f(is)f
FI(not)40 b FT(the)31 b(case)g(that)950 3229 y Ft(\()p
Fv(A)44 b Fw(on)f Ft(\()p Fv(A)h Fu(\))f Fv(B)t Ft(\)\))i
Fw(and)d Ft(\(\()p Fv(B)48 b Fu(\))c Fv(C)6 b Ft(\))44
b Fw(on)f Ft(\()p Fv(A)h Fu(\))g Fv(B)t Ft(\)\))26 b
Ff( )f FT(\()p FP(A)20 b FN(^)g FP(C)7 b FT(\))378 3433
y(for)32 b(distinct)e(literals)g FP(A)p FT(,)i FP(B)37
b FT(and)31 b FP(C)38 b FT(\(see)33 b(also)f(example)f(6.2)i(on)f(page)
g(114\).)47 b(It)32 b(is)e(the)i(case)h(that)378 3546
y(the)e(justi\014ed)d(conclusion)473 3734 y(\()p FP(A)49
b FN(^)e FP(C)7 b FT(\))95 b FM(by)g FT(\()p FP(A)48
b FM(on)g FT(\()p FP(A)g FN(\))g FP(B)5 b FT(\)\))47
b FM(and)g FT(\(\()p FP(B)53 b FN(\))48 b FP(C)7 b FT(\))48
b FM(on)f FT(\()p FP(A)h FN(\))g FP(B)5 b FT(\)\))378
3922 y(con)m(v)m(erges)32 b(to)f(the)g(coloured)f(problem)f(\()p
FP(S;)15 b FN(K)q FT(\))32 b(where)1038 4126 y FP(S)e
FT(=)25 b FN(f)p FP(A)1333 4088 y FO(i)1362 4126 y FP(;)15
b FT(\()p FP(A)26 b FN(\))f FP(B)5 b FT(\))1756 4088
y FO(j)1793 4126 y FP(;)15 b FT(\()p FP(B)30 b FN(\))25
b FP(C)7 b FT(\))2190 4088 y FO(k)2233 4126 y FP(;)15
b FT(\()p FP(A)26 b FN(\))f FP(B)5 b FT(\))2627 4088
y FO(l)2653 4126 y FP(;)15 b FN(:)p FT(\()p FP(A)21 b
FN(^)f FP(C)7 b FT(\))3066 4088 y FO(m)3132 4126 y FN(g)1028
4264 y(K)27 b FT(=)e(\()p FP(m)h FN($)f FP(i)g FN($)g
FP(j)5 b FT(\))22 b FN([)d FT(\()p FP(m)26 b FN($)f FP(k)k
FN($)c FP(l)r FT(\))p FP(:)378 4468 y FT(No)m(w,)38 b(the)e(coloured)f
(problem)f(\()p FP(S;)15 b FN(K)q FT(\))37 b(is)d(inconsisten)m(t)h(if)
f(and)h(only)g(if)g(the)g(follo)m(wing)f(coloured)378
4581 y(matrix)c(is)f(refutable:)1417 4856 y Fx(\024)1465
4927 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 11.60646 5.80322 4.63118
} false /N@Ai 16 {InitRnode } NewNode end end
1465 4927 a FP(A)1533 4894 y FO(i)1675 4927 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 19.91112 9.95555 4.63118
} false /N@nAj 16 {InitRnode } NewNode end end
1675 4927 a FN(:)p FP(A)1804 4894 y FO(j)1954 4927 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 21.294 10.647 4.7646
} false /N@nBk 16 {InitRnode } NewNode end end
1954 4927 a FN(:)p FP(B)2089 4894 y FO(k)2244 4927 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 18.64449 9.32224 4.7646
} false /N@nAl 16 {InitRnode } NewNode end end
2244 4927 a FN(:)p FP(A)2373 4894 y FO(l)2529 4927 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 23.53114 11.76556 3.74124
} false /N@nAm 16 {InitRnode } NewNode end end
2529 4927 a FN(:)p FP(A)2658 4894 y FO(m)1703 5040 y
tx@Dict begin tx@NodeDict begin {9.26236 0.0 13.25363 6.62682 4.63118
} false /N@Bj 16 {InitRnode } NewNode end end
1703 5040 a FP(B)1777 5007 y FO(j)1985 5040 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 13.74835 6.87418 4.7646
} false /N@Ck 16 {InitRnode } NewNode end end
1985 5040
a FP(C)2057 5007 y FO(k)2271 5040 y
tx@Dict begin tx@NodeDict begin {9.52922 0.0 11.987 5.9935 4.7646
} false /N@Bl 16 {InitRnode } NewNode end end
2271 5040 a FP(B)2345
5007 y FO(l)2512 5040 y
tx@Dict begin tx@NodeDict begin {7.48248 0.0 23.92804 11.96402 3.74124
} false /N@nCm 16 {InitRnode } NewNode end end
2512 5040 a FN(:)p FP(C)2645
5007 y FO(m)2741 5040 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@Ai /N@nAm 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
2741 5040 a 2741 5040 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@Ai /N@nAj 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
2741
5040 a 2741 5040 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@nBk /N@nAm 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
2741 5040 a 2741 5040 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@nBk /N@nAl 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
2741 5040
a -184 x Fx(\025)378 5237 y FT(where)41 b(the)i(curv)m(es)f(ab)s(o)m(v)
m(e)h(the)f(matrix)f(illustrate)f(whic)m(h)h(columns)f(ha)m(v)m(e)k
(literals)c(whic)m(h)h(can)378 5350 y(connect)j(with)d(eac)m(h)j(other)
e(according)h(to)g(the)g(connectabilit)m(y)f(relation)g
FN(K)q FT(.)78 b(Note)44 b(that)f(this)378 5463 y(matrix)30
b(cannot)h(b)s(e)e(refuted)h(since)g(the)h(path)1678
5667 y FN(f)p FP(B)1797 5629 y FO(j)1834 5667 y FP(;)15
b FN(:)p FP(B)2009 5629 y FO(k)2051 5667 y FP(;)g FN(:)p
FP(A)2220 5629 y FO(l)2246 5667 y FP(;)g FN(:)p FP(A)2415
5629 y FO(m)2482 5667 y FN(g)p eop
%%Page: 187 197
187 196 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(187)378
396 y(do)s(es)30 b(not)h(ha)m(v)m(e)g(a)g(connection)g(since)e
FP(j)i FN(6\030)1846 410 y FK(K)1929 396 y FP(k)s FT(.)41
b(As)31 b(a)f(result)g(it)f(is)h(not)h(the)f(case)h(that)950
601 y Ft(\()p Fv(A)44 b Fw(on)f Ft(\()p Fv(A)h Fu(\))f
Fv(B)t Ft(\)\))i Fw(and)d Ft(\(\()p Fv(B)48 b Fu(\))c
Fv(C)6 b Ft(\))44 b Fw(on)f Ft(\()p Fv(A)h Fu(\))g Fv(B)t
Ft(\)\))26 b Ff( )f FT(\()p FP(A)20 b FN(^)g FP(C)7 b
FT(\))378 805 y(b)m(y)30 b(theorem)h(8.7.)2759 b Ff(\003)378
1091 y FH(8.5)135 b(Mo)t(difying)68 b(the)f Fc(C)7 b(B)t(S)i(E)78
b FH(Deriv)l(ed)68 b(Rule)g(to)g(Chec)l(k)g(Struc-)684
1241 y(tured)45 b(Justi\014cations)378 1444 y FT(In)27
b(this)h(c)m(hapter)g(and)g(in)f(c)m(hapters)i(6)f(and)g(7)h(w)m(e)f
(illustrated)e(ho)m(w)j(one)f(can)h(use)f(structured)f(justi-)378
1557 y(\014cations)h(in)e(a)i(declarativ)m(e)h(language)f(in)f(order)g
(to)i(giv)m(e)f(more)g(information)e(on)i(what)g(inferences)378
1670 y(are)f(needed)g(to)g(deriv)m(e)g(the)g(conclusion)e(of)i(the)g
(justi\014cation.)38 b(This)25 b(information)g(impro)m(v)m(es)i(b)s
(oth)378 1782 y(the)22 b(readabilit)m(y)f(of)i(pro)s(ofs)e(b)m(y)h
(reducing)f(the)i(e\013ort)g(required)d(in)h(follo)m(wing)g(the)i
(justi\014cation,)f(and)378 1895 y(the)36 b(pro)s(of)g(c)m(hec)m(king)h
(e\016ciency)f(b)m(y)h(restricting)e(the)h(pro)s(of)g(searc)m(h.)59
b(This)35 b(restriction)g(in)m(v)m(olv)m(es)378 2008
y(the)29 b(colouring)e(of)i(sen)m(tences)g(giv)m(en)g(in)e(the)h
(justi\014cation)f(according)i(to)g(de\014nitions)d(8.3)k(and)e(8.4.)
378 2121 y(In)e(this)f(section)i(w)m(e)g(sho)m(w)f(ho)m(w)g(the)h
FN(C)5 b(B)s(S)i(E)34 b FT(deriv)m(ed)26 b(rule)f(describ)s(ed)f(in)h
(c)m(hapter)j(5)e(is)g(mo)s(di\014ed)e(in)378 2234 y(order)31
b(to)i(c)m(hec)m(k)h(structured)d(justi\014cations.)44
b(The)32 b(mo)s(di\014ed)e(rule)g(illustrated)g(in)h(this)g(section)h
(is)378 2347 y(used)e(in)g(c)m(hec)m(king)i(the)g(pro)s(of)e(scripts)g
(dev)m(elop)s(ed)h(during)d(the)k(mec)m(hanisation)f(of)g(group)g
(theory)378 2460 y(describ)s(ed)d(in)h(c)m(hapter)i(9.)519
2573 y(The)26 b(structured)g(justi\014cations)f(de\014ned)g(in)g(c)m
(hapter)i(6)g(can)g(b)s(e)f(used)f(to)j(deriv)m(e)e(their)f(conclu-)378
2686 y(sion)35 b(according)h(to)g FI(pur)-5 b(e)44 b
FT(\014rst-order)35 b(logic,)i(and)e(section)i(8.3)g(giv)m(es)f(the)g
(restrictions)f(required)378 2799 y(on)e(pure)f(\014rst-order)h(logic)g
(calculi)f(in)g(order)h(to)h(pro)s(of)e(c)m(hec)m(k)j(structured)e
(justi\014cations.)48 b(Ho)m(w-)378 2912 y(ev)m(er,)26
b(for)e(e\016ciency)g(reasons)g(the)g(equalit)m(y)g(predicate)f
(requires)g(sp)s(ecial)f(treatmen)m(t)k(during)c(pro)s(of)378
3024 y(searc)m(h)37 b(and)e(the)i FN(C)5 b(B)s(S)i(E)44
b FT(deriv)m(ed)35 b(rule)g(giv)m(en)h(in)f(c)m(hapter)i(5)g(implemen)m
(ts)d(a)j(pro)s(of)e(calculus)g(for)378 3137 y(\014rst-order)d(logic)g
(with)f(equalit)m(y)-8 b(.)48 b(The)32 b(de\014nition)e(of)j(a)g(syn)m
(tax)g(and)g(seman)m(tics)f(for)h(structured)378 3250
y(justi\014cations)20 b(for)h(\014rst-order)f(logic)i(with)e(equalit)m
(y)h(is)f(not)i(considered)e(in)g(this)g(thesis.)37 b(W)-8
b(e)23 b(b)s(eliev)m(e)378 3363 y(that)30 b(this)e(\(and)i(the)f
(de\014nition)e(of)j(structured)e(justi\014cations)g(for)i(other)f
(logics)g(and)g(theories\))h(is)378 3476 y(an)35 b(in)m(teresting)f
(direction)f(for)h(future)g(w)m(ork)h(since)f(it)g(is)g(not)h(straigh)m
(tforw)m(ard)g(to)g(de\014ne)f(struc-)378 3589 y(tured)f
(justi\014cations)f(whic)m(h)g(are)h(easy)h(to)h(understand)c(and)i
(e\016cien)m(t)h(to)g(pro)s(of)f(c)m(hec)m(k.)51 b(Instead)378
3702 y(of)28 b(giving)f(new)g(op)s(erators)h(on)g(structured)e
(expressions)h(to)h(handle)f(equalit)m(y)-8 b(,)28 b(the)g
FN(C)5 b(B)s(S)i(E)36 b FT(deriv)m(ed)378 3815 y(rule)f(is)g(mo)s
(di\014ed)e(according)j(to)h(the)f(restictions)f(giv)m(en)h(in)f(this)g
(c)m(hapter,)j(and)d(w)m(e)i(discuss)d(the)378 3928 y(e\013ect)d(of)f
(suc)m(h)f(restrictions)f(on)i(pro)s(of)f(c)m(hec)m(king)h
(justi\014cations)e(in)m(v)m(olving)g(form)m(ulae)h(con)m(taining)378
4041 y(the)i(equalit)m(y)e(predicate.)519 4154 y(W)-8
b(e)38 b(recall)e(that)h(during)e(the)i(expansion)e(rule)h(of)h(the)f
FN(C)5 b(B)s(S)i(E)45 b FT(calculus,)37 b(the)g(insertion)e(of)h(a)378
4266 y(literal)25 b(in)g(a)i(branc)m(h)f(ma)m(y)h(result)e(in)g(the)i
(insertion)d(of)j(a)g(n)m(um)m(b)s(er)e(of)h(inequalities)e(whic)m(h)h
(are)i(then)378 4379 y(used)33 b(b)m(y)h(other)g(rules)e(of)i(the)g
(calculus)e(to)i(close)h(the)e(branc)m(h.)51 b(More)34
b(precisely)-8 b(,)34 b(the)g(additional)378 4492 y(inequation)1593
4605 y FN(h)p FP(s)1671 4619 y FL(1)1710 4605 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(s)1970 4619 y FO(n)2017 4605
y FN(i)25 b(6)p FT(=)g FN(h)p FP(t)2241 4619 y FL(1)2281
4605 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(t)2531 4619 y
FO(n)2578 4605 y FN(i)378 4772 y FT(is)22 b(inserted)g(in)f(the)j
(branc)m(h)e FP(B)28 b FT(whenev)m(er)22 b(a)i(literal)d
FP(L)k FT(=)g FP(P)13 b FT(\()p FP(s)2484 4786 y FL(1)2524
4772 y FP(;)i(:)g(:)g(:)32 b(;)15 b(s)2784 4786 y FO(n)2830
4772 y FT(\))24 b(is)e(inserted)g(in)f(the)i(branc)m(h)378
4885 y(and)30 b FN(:)p FP(P)13 b FT(\()p FP(t)755 4899
y FL(1)795 4885 y FP(;)i(:)g(:)g(:)31 b(;)15 b(t)1044
4899 y FO(n)1092 4885 y FT(\))31 b(is)f(in)g FP(B)5 b
FT(,)30 b(and)h(whenev)m(er)g(a)g(literal)f FP(L)c FT(=)g
FN(:)p FP(P)13 b FT(\()p FP(s)2799 4899 y FL(1)2838 4885
y FP(;)i(:)g(:)g(:)32 b(;)15 b(s)3098 4899 y FO(n)3145
4885 y FT(\))31 b(is)f(inserted)g(in)f FP(B)378 4998
y FT(and)37 b FP(P)13 b FT(\()p FP(t)701 5012 y FL(1)741
4998 y FP(;)i(:)g(:)g(:)31 b(;)15 b(t)990 5012 y FO(n)1038
4998 y FT(\))37 b(is)g(in)f FP(B)5 b FT(.)62 b(This)36
b(mec)m(hanism)h(is)g(describ)s(ed)e(in)i(section)h(5.2.2,)j(page)d
(81,)j(b)m(y)378 5111 y(de\014ning)30 b(the)i(op)s(erator)h
FN(\016)f FT(on)g(branc)m(hes)g(and)f(literals.)44 b(The)32
b FN(C)5 b(B)s(S)i(E)40 b FT(rule)30 b(is)h(mo)s(di\014ed)f(so)j(that)f
(it)378 5224 y(tak)m(es)j(coloured)e(form)m(ulae)g(and)g(restricts)g
(the)h(insertion)e(of)h(additional)f(inequalities)f(according)378
5337 y(to)f(the)f(connectabilit)m(y)g(relation)f(considered.)40
b(Giv)m(en)29 b(a)g(connectabilit)m(y)g(relation)g FN(K)q
FT(,)h(a)f(coloured)378 5450 y(literal)19 b FP(L)p FT(,)j(and)e(a)h
(tableau)f(branc)m(h)g FP(B)25 b FT(con)m(taining)20
b(coloured)g(literals)f(and)h(a)h(n)m(um)m(b)s(er)e(of)h(uncoloured)378
5562 y(equations)29 b(and)f(inequations,)g(w)m(e)h(de\014ne)g(the)g
(coloured)f(insertion)f(of)i FP(L)g FT(in)f FP(B)5 b
FT(,)29 b(and)f(denote)i(it)e(b)m(y)p eop
%%Page: 188 198
188 197 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(188)378
396 y FP(B)25 b FN(\016)517 410 y FK(K)596 396 y FP(L)p
FT(,)30 b(as)h(follo)m(ws:)488 601 y FP(B)25 b FN(\016)627
615 y FK(K)706 601 y FP(P)777 563 y FO(i)805 601 y FT(\()p
FP(s)883 615 y FL(1)923 601 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(s)1182 615 y FO(n)1229 601 y FT(\))26 b(=)670
749 y FP(B)5 b(;)15 b(P)855 712 y FO(i)883 749 y FT(\()p
FP(s)961 763 y FL(1)1001 749 y FP(;)g(:)g(:)g(:)31 b(;)15
b(s)1260 763 y FO(n)1307 749 y FT(\))21 b FN([)f(fh)p
FP(s)1567 763 y FL(1)1607 749 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(s)1866 763 y FO(n)1913 749 y FN(i)26 b(6)p FT(=)f
FN(h)p FP(t)2138 763 y FL(1)2178 749 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(t)2427 763 y FO(n)2475 749 y FN(i)25 b(j)h(:)p
FP(P)2718 712 y FO(j)2754 749 y FT(\()p FP(t)2822 763
y FL(1)2861 749 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(t)3111
763 y FO(n)3158 749 y FT(\))26 b FN(2)f FP(B)5 b(;)15
b(i)25 b FN(\030)3546 763 y FK(K)3629 749 y FP(j)5 b
FN(g)488 898 y FP(B)25 b FN(\016)627 912 y FK(K)706 898
y FN(:)p FP(P)838 860 y FO(i)866 898 y FT(\()p FP(s)944
912 y FL(1)983 898 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(s)1243
912 y FO(n)1290 898 y FT(\))25 b(=)670 1046 y FP(B)5
b(;)15 b FN(:)p FP(P)916 1009 y FO(i)944 1046 y FT(\()p
FP(s)1022 1060 y FL(1)1061 1046 y FP(;)g(:)g(:)g(:)32
b(;)15 b(s)1321 1060 y FO(n)1368 1046 y FT(\))21 b FN([)e(fh)p
FP(s)1627 1060 y FL(1)1667 1046 y FP(;)c(:)g(:)g(:)32
b(;)15 b(s)1927 1060 y FO(n)1974 1046 y FN(i)26 b(6)p
FT(=)f FN(h)p FP(t)2199 1060 y FL(1)2238 1046 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(t)2488 1060 y FO(n)2535 1046
y FN(i)26 b(j)f FP(P)2717 1009 y FO(j)2754 1046 y FT(\()p
FP(t)2822 1060 y FL(1)2861 1046 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(t)3111 1060 y FO(n)3158 1046 y FT(\))26 b FN(2)f
FP(B)5 b(;)15 b(i)25 b FN(\030)3546 1060 y FK(K)3629
1046 y FP(j)5 b FN(g)488 1195 y FP(B)25 b FN(\016)627
1209 y FK(K)706 1195 y FT(\()p FP(s)g FT(=)g FP(t)p FT(\))973
1157 y FO(i)1027 1195 y FT(=)g FP(B)5 b(;)15 b FT(\()p
FP(s)25 b FT(=)g FP(t)p FT(\))488 1343 y FP(B)g FN(\016)627
1357 y FK(K)706 1343 y FT(\()p FP(s)g FN(6)p FT(=)g FP(t)p
FT(\))973 1306 y FO(i)1027 1343 y FT(=)g FP(B)5 b(;)15
b FT(\()p FP(s)25 b FN(6)p FT(=)g FP(t)p FT(\))378 1548
y(This)i(de\014nition)f(of)j(the)g FN(\016)1288 1562
y FK(K)1375 1548 y FT(op)s(eration)g(di\013ers)e(from)h(the)h
(de\014nition)e(of)h(the)h FN(\016)g FT(op)s(erator)g(giv)m(en)g(in)378
1660 y(section)23 b(5.2.2)h(in)d(the)i(fact)g(that)g(additional)d
(inequations)h(are)i(inserted)f(in)f(a)i(branc)m(h)e(if)h(the)g
(colours)378 1773 y(of)j(the)g(literals)f(`giving')g(the)h(inequation)f
(\(i.e.,)15 b(the)26 b(literals)d FP(P)2536 1740 y FO(i)2564
1773 y FT(\()p FP(s)2642 1787 y FL(1)2682 1773 y FP(;)15
b(:)g(:)g(:)31 b(;)15 b(s)2941 1787 y FO(n)2988 1773
y FT(\))26 b(and)e FN(:)p FP(P)3352 1740 y FO(j)3388
1773 y FT(\()p FP(t)3456 1787 y FL(1)3496 1773 y FP(;)15
b(:)g(:)g(:)31 b(;)15 b(t)3745 1787 y FO(n)3793 1773
y FT(\))378 1886 y(in)27 b(the)h(\014rst)f(part)h(of)h(the)f
(de\014nition,)e(and)i FN(:)p FP(P)2006 1853 y FO(i)2034
1886 y FT(\()p FP(s)2112 1900 y FL(1)2151 1886 y FP(;)15
b(:)g(:)g(:)32 b(;)15 b(s)2411 1900 y FO(n)2458 1886
y FT(\))28 b(and)g FP(P)2767 1853 y FO(j)2803 1886 y
FT(\()p FP(t)2871 1900 y FL(1)2911 1886 y FP(;)15 b(:)g(:)g(:)32
b(;)15 b(t)3161 1900 y FO(n)3208 1886 y FT(\))28 b(in)f(the)h(second\))
378 1999 y(relate)e(with)e(eac)m(h)i(other)g(according)f(to)h
FN(K)q FT(.)40 b(Note)26 b(that)g(the)g(equations)f(and)f(inequations)g
(inserted)378 2112 y(in)29 b(a)i(branc)m(h)g(using)e(the)i
FN(\016)1302 2126 y FK(K)1391 2112 y FT(op)s(erator)g(are)g(not)g
(coloured.)41 b(The)30 b(Expansion)f(rule)h(in)f(\014gure)h(11)i(is)378
2225 y(then)e(mo)s(di\014ed)e(so)j(that)g(literals)e(are)h(inserted)g
(using)e FN(\016)2342 2239 y FK(K)2431 2225 y FT(rather)j(than)f
FN(\016)p FT(:)1559 2411 y FP(B)1628 2425 y FL(1)1683
2411 y FN(j)h(\001)15 b(\001)g(\001)31 b(j)15 b FP(B)1984
2425 y FO(n)2067 2411 y FN(\001)35 b(C)p 1123 2451 1495
4 v 1123 2534 a FP(B)1192 2548 y FL(1)1251 2534 y FN(\016)1296
2548 y FK(K)1375 2534 y FP(L)1437 2548 y FL(1)1492 2534
y FN(j)30 b(\001)15 b(\001)g(\001)31 b(j)15 b FP(B)1792
2548 y FL(1)1852 2534 y FN(\016)1897 2548 y FK(K)1976
2534 y FP(L)2038 2548 y FO(m)2120 2534 y FN(j)30 b(\001)15
b(\001)g(\001)32 b(j)15 b FP(B)2421 2548 y FO(n)2504
2534 y FN(\001)35 b(C)2658 2472 y FT(\(Expand)3000 2499
y FK(K)3058 2472 y FT(\))378 2725 y(where)41 b FP(B)721
2739 y FL(1)775 2725 y FN(j)31 b(\001)15 b(\001)g(\001)31
b(j)15 b FP(B)1076 2739 y FO(n)1165 2725 y FT(is)41 b(a)h(tableau,)i
FN(C)j FT(is)40 b(a)i(constrain)m(t,)j(and)40 b FP(L)2717
2739 y FL(1)2784 2725 y FN(_)28 b(\001)15 b(\001)g(\001)28
b(_)f FP(L)3156 2739 y FO(m)3264 2725 y FT(is)41 b(an)g(instance)378
2838 y FP(L)440 2805 y FK(0)440 2862 y FL(1)479 2838
y FP(\033)32 b FN(_)c(\001)15 b(\001)g(\001)30 b(_)e
FP(L)938 2805 y FK(0)938 2860 y FO(m)1004 2838 y FP(\033)46
b FT(of)d(some)g(clause)g(in)f(the)h(the)g(set)g(of)g(clauses)f(b)s
(eing)g(refuted,)j(and)e FP(\033)j FT(is)c(a)378 2951
y(substitution)37 b(whic)m(h)g(maps)i(all)f(the)h(free)g(v)-5
b(ariables)37 b(in)h FP(L)2453 2918 y FK(0)2453 2975
y FL(1)2492 2951 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(L)2771
2918 y FK(0)2771 2973 y FO(m)2877 2951 y FT(to)40 b(distinct)d(free)i
(v)-5 b(ariable)378 3064 y(new)33 b(to)h FP(B)751 3078
y FL(1)806 3064 y FN(j)c(\001)15 b(\001)g(\001)32 b(j)15
b FP(B)1107 3078 y FO(n)1191 3064 y FN(\001)38 b(C)5
b FT(.)50 b(The)33 b FN(C)5 b(B)s(S)i(E)41 b FT(calculus)32
b(is)g(mo)s(di\014ed)g(b)m(y)h(replacing)f(the)i(Expand)e(rule)378
3177 y(with)i(the)i(Expand)1059 3198 y FK(K)1153 3177
y FT(rule.)56 b(This)34 b(is)h(the)h(only)f(mo)s(di\014cation)f
(applied)f(to)k(the)f FN(C)5 b(B)s(S)i(E)43 b FT(rule)35
b(used)378 3289 y(to)e(c)m(hec)m(k)g(the)f(coloured)g(inconsistency)e
(of)i(a)h(coloured)e(problem)f(whic)m(h)h(is)g(constructed)h(from)g(a)
378 3402 y(structured)d(justi\014cation)g(as)i(describ)s(ed)d(in)h
(de\014nitions)f(8.3)k(and)d(8.4.)519 3515 y(Giv)m(en)d(the)h
(restriction)e(on)h(the)h FN(C)5 b(B)s(S)i(E)34 b FT(deriv)m(ed)25
b(rule)g(describ)s(ed)f(ab)s(o)m(v)m(e,)29 b(one)d(can)h(use)f(the)g
Fw(and)378 3628 y FT(op)s(erator)21 b(to)g(construct)g(structured)f
(expressions)f(in)h(whic)m(h)f(one)i(expression)f(explicitly)e(deriv)m
(es)i(an)378 3741 y(equation)28 b(and)g(the)h(other)f(requires)f(the)i
(deriv)m(ed)e(equation)h(to)h(deriv)m(e)f(the)h(goal.)41
b(More)29 b(formally)-8 b(,)378 3854 y(if)29 b(a)i(structured)e
(expression)h FP(E)1483 3868 y FL(1)1552 3854 y FT(explicitly)e(deriv)m
(es)i(a)h(conjunctions)e(of)i(equation)f FP(E)5 b FT(,)31
b(that)g(is)1087 4058 y FP(E)1154 4072 y FL(1)1218 4058
y Ff( )26 b FP(E)96 b FT(where)60 b FP(E)31 b FT(=)25
b(\(\()p FP(a)2103 4072 y FL(1)2168 4058 y FT(=)g FP(b)2303
4072 y FL(1)2342 4058 y FT(\))c FN(^)f(\001)15 b(\001)g(\001)21
b(^)f FT(\()p FP(a)2769 4072 y FO(n)2841 4058 y FT(=)25
b FP(b)2976 4072 y FO(n)3023 4058 y FT(\)\))p FP(;)378
4262 y FT(and)37 b(another)g(structured)g(expression,)h
FP(E)1889 4276 y FL(2)1966 4262 y FT(sa)m(y)-8 b(,)40
b(explicitly)35 b(deriv)m(es)i(some)g(form)m(ula)g FP(A)p
FT(,)i(then)e(a)378 4375 y(conclusion)29 b FP(C)37 b
FT(can)30 b(b)s(e)g(justi\014ed)f(as)h(follo)m(ws:)473
4532 y FP(C)55 b FM(by)47 b FP(E)803 4546 y FL(1)890
4532 y FM(and)g FP(E)1148 4546 y FL(2)1188 4532 y FM(;)378
4689 y FT(if)31 b FP(A)d FN(`)615 4703 y FO(E)703 4689
y FP(C)7 b FT(.)45 b(By)33 b FP(A)28 b FN(`)1142 4703
y FO(E)1230 4689 y FP(C)38 b FT(w)m(e)33 b(mean)f(that)h
FP(C)38 b FT(can)33 b(b)s(e)e(deriv)m(ed)h(from)f FP(A)i
FT(b)m(y)f(substituting)d(equals)378 4802 y(for)34 b(equals)f
(according)h(to)h(the)f(equations)g(in)f FP(E)5 b FT(.)52
b(W)-8 b(e)35 b(do)f(not)g(pro)m(v)m(e)h(this)e(claim)g(in)g(this)f
(thesis,)378 4915 y(although)f(w)m(e)g(state)i(that)f(w)m(e)f(ha)m(v)m
(e)i(found)d(no)h(coun)m(terexample)g(to)h(this)e(statemen)m(t)j
(during)c(our)378 5028 y(case)h(studies.)39 b(The)29
b(informal)e(in)m(tuition)g(justifying)f(this)i(statemen)m(t)j(is)d
(that)i(the)f(restrictions)f(on)378 5141 y(the)33 b(pro)s(of)e(searc)m
(h)i(allo)m(w)f(the)h(deriv)-5 b(ation)31 b(\(in)h(pure)f
(\014rst-order)h(logic\))g(of)h FP(E)k FT(from)32 b FP(E)3403
5155 y FL(1)3475 5141 y FT(and)g(of)h FP(A)378 5253 y
FT(from)e FP(E)661 5267 y FL(2)700 5253 y FT(,)h(and)f(th)m(us)g
FP(E)26 b FN(^)21 b FP(A)31 b FT(from)g Fv(E)1687 5265
y Fs(1)1768 5253 y Fw(and)42 b Fv(E)2003 5265 y Fs(2)2041
5253 y FT(.)i(Since)30 b(the)i(rules)d(of)j(rigid)d(basic)i(sup)s(erp)s
(osition,)378 5366 y(equational)c(re\015exivit)m(y)-8
b(,)29 b(simpli\014cation)24 b(and)k(trivial)e(closure)h(are)i(not)f
(restricted)f(b)m(y)h(the)g(colours)378 5479 y(of)i(the)h(literals)e
(in)g(the)h(tableau,)h(the)f(equalities)f(in)h FP(E)35
b FT(can)c(then)f(b)s(e)g(used)f(to)i(deriv)m(e)f FP(C)37
b FT(from)30 b FP(A)p FT(.)519 5592 y(The)36 b(follo)m(wing)e(is)h(an)h
(example)g(of)g(a)g(conclusion)f(justi\014ed)f(with)h(a)h(structured)f
(expression)378 5705 y(whic)m(h)29 b(in)m(v)m(olv)m(es)h(a)h(premise)e
(con)m(taining)h(an)h(equation.)p eop
%%Page: 189 199
189 198 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(189)1920
537 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 44.12328 22.06163
3.30017 } false /N@T-0 16 {InitRnode } NewNode end end
1920 537 a 1920 537 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 44.12328 22.06163
2.7375 } false /N@N1 16 {InitRnode } NewNode end end
1920 537 a FN(:)p FP(P)13
b FT(\()p FP(f)d FT(\()p FP(b)p FT(\)\))1946 786 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.91194 18.95596
3.30017 } false /N@T-0-0 16 {InitRnode } NewNode end end
1946
786 a 1946 786 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 37.91194 18.95596
2.7375 } false /N@N2 16 {InitRnode } NewNode end end
1946 786 a FP(P)j FT(\()p FP(f)d FT(\()p
FP(a)p FT(\)\))2103 758 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
2103 758 a 2008 1035 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 22.85564 11.42781
3.30017 } false /N@T-0-0-0 16 {InitRnode } NewNode end end
2008
1035 a 2008 1035 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 22.85564 11.42781
2.7375 } false /N@N3 16 {InitRnode } NewNode end end
2008 1035 a FP(P)j FT(\()p FP(a)p
FT(\))2103 1008 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
2103 1008 a 1587 1284 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 30.62582 15.31291
3.30017 } false /N@T-0-0-0-0 16 {InitRnode } NewNode end end
1587 1284 a
1587 1284 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 30.62582 15.31291
2.7375 } false /N@N4 16 {InitRnode } NewNode end end
1587 1284 a FN(:)p FP(P)g FT(\()p FP(x)p
FT(\))1715 1257 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
1715 1257 a 1415 1533 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 72.23334 36.11667
3.30017 } false /N@T-0-0-0-0-0 16 {InitRnode } NewNode end end
1415 1533 a
FN(h)p FP(f)d FT(\()p FP(a)p FT(\))p FN(i)26 b(6)p FT(=)f
FN(h)p FP(f)10 b FT(\()p FP(b)p FT(\))p FN(i)1715 1506
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 gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore grestore end
1715 1506 a 1533 1782 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 43.67958 21.83978
3.30017 } false /N@T-0-0-0-0-0-0 16 {InitRnode } NewNode end end
1533 1782 a FN(h)p FP(x)p FN(i)26
b(6)p FT(=)f FN(h)p FP(a)p FN(i)1715 1755 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-0-0-0-0-0 /N@T-0-0-0-0-0-0 InitNC { NCLine } if end gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore grestore end
1715 1755
a 2385 1284 a
tx@Dict begin tx@NodeDict begin {7.60416 0.0 25.55751 12.77875 3.30017
} false /N@T-0-0-0-1 16 {InitRnode } NewNode end end
2385 1284 a 2385 1284 a
tx@Dict begin tx@NodeDict begin {7.60416 0.0 25.55751 12.77875 3.80208
} false /N@N5 16 {InitRnode } NewNode end end
2385 1284 a FP(x)h
FT(=)e FP(b)2491 1257 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-0-0-0-1 InitNC { NCLine } if end gsave 0.8
SLW 0. setgray 0 setlinecap stroke grestore grestore end
2491 1257 a 2192 1533 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 72.23334 36.11667
3.30017 } false /N@T-0-0-0-1-0 16 {InitRnode } NewNode end end
2192
1533 a FN(h)p FP(f)10 b FT(\()p FP(a)p FT(\))p FN(i)26
b(6)p FT(=)f FN(h)p FP(f)10 b FT(\()p FP(b)p FT(\))p
FN(i)2491 1506 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-0-0-0-1 /N@T-0-0-0-1-0 InitNC { NCLine } if end gsave
0.8 SLW 0. setgray 0 setlinecap stroke grestore grestore end
2491 1506 a 2791 509 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@N1 /N@N4 InitNC { /AngleA -155. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2791 509 a 2791
509 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@N3 /N@N4 InitNC { /AngleA -175. def /AngleB 155. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2791 509 a 2791 509 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@N1 /N@N2 InitNC { /AngleA -12. def /AngleB 8. def 0.67
0.67 NCCurve } if end gsave 0.8 SLW 0. setgray 0 setlinecap stroke
grestore grestore end
2791 509 a 1250 2180 a FQ(Fig.)18
b(22.)41 b FT(A)30 b(Coloured)g(First-Order)e(T)-8 b(ableau.)473
2555 y(\()p FP(P)61 b FT(\()p FP(f)d(b)p FT(\)\))48 b
FM(by)f FT(\(\()p FN(8)17 b FP(x)p FM(.)p FP(P)60 b(x)48
b FN(\))f FT(\()p FP(x)h FT(=)g FP(b)p FT(\)\))g FM(on)f
FT(\()p FP(P)62 b(a)p FT(\)\))48 b FM(and)f FT(\()p FP(P)61
b FT(\()p FP(f)c(a)p FT(\)\))p FM(;)378 2720 y FT(Note)32
b(that)1263 2925 y(\()p Ft(\()p Fu(8)15 b Fv(x)p Fw(.)p
Fv(P)34 b(x)44 b Fu(\))g Ft(\()p Fv(x)15 b Ft(=)f Fv(b)p
Ft(\)\))44 b Fw(on)f Ft(\()p Fv(P)35 b(a)p Ft(\))p FT(\))26
b Ff( )f Ft(\()p Fv(a)15 b Ft(=)f Fv(b)p Ft(\))2134 3063
y(\()p Fv(P)35 b Ft(\()p Fv(f)d(a)p Ft(\)\))26 b Ff( )f
Ft(\()p Fv(P)35 b Ft(\()p Fv(f)d(a)p Ft(\)\))378 3267
y FT(and)1638 3380 y Fv(P)j Ft(\()p Fv(f)d(a)p Ft(\))56
b FN(`)2019 3406 y Fv(a)14 b Ft(=)g Fv(b)2282 3380 y(P)34
b Ft(\()p Fv(f)e(b)p Ft(\))p FP(:)378 3547 y FT(Figure)43
b(22)i(illustrates)d(the)i(coloured)f(tableau)g(constructed)i(from)e
(the)h(structured)f(justi\014ca-)378 3659 y(tion)30 b(giv)m(en)h(ab)s
(o)m(v)m(e.)42 b(The)30 b(connections)h(b)s(et)m(w)m(een)g(tableau)g
(no)s(des)f(illustrate)e(whic)m(h)i(literals)f(ha)m(v)m(e)378
3772 y(colours)42 b(whic)m(h)g(relate)h(with)f(eac)m(h)i(other)f
(according)g(to)h(the)f(connectabilit)m(y)f(relation)g(in)g(the)378
3885 y(coloured)36 b(problem)g(constructed)h(from)f(the)h(structured)f
(justi\014cation)g(considered.)59 b(As)36 b(sho)m(wn)378
3998 y(in)30 b(the)i(\014gure,)f(the)g(additional)f(inequalities)e
(inserted)j(in)f(the)h(left)g(and)g(righ)m(t)g(tableau)g(branc)m(hes)
378 4111 y(resp)s(ectiv)m(ely)f(are:)1412 4340 y FN(h)p
FP(f)10 b FT(\()p FP(a)p FT(\))p FN(i)26 b(6)p FT(=)f
FN(h)p FP(f)10 b FT(\()p FP(b)p FT(\))p FN(i)183 b(h)p
FP(f)10 b FT(\()p FP(a)p FT(\))p FN(i)26 b(6)p FT(=)f
FN(h)p FP(f)10 b FT(\()p FP(b)p FT(\))p FN(i)1533 4478
y(h)p FP(x)p FN(i)26 b(6)p FT(=)f FN(h)p FP(a)p FN(i)378
4682 y FT(The)h(left)h(branc)m(h)g(can)g(b)s(e)f(closed)h(b)m(y)g
(equational)f(re\015exivit)m(y)h(on)f(the)i(second)f(additional)e
(inequa-)378 4795 y(tion)k(giving)g(the)h(constrain)m(t)f
FN(f)p FP(x)d FN(')f FP(a)p FN(g)p FT(.)41 b(The)29 b(righ)m(t)h(branc)
m(h)f(can)h(then)f(b)s(e)g(closed)h(b)m(y)f(congruence)378
4908 y(closure)i(after)h(instan)m(tiating)e(the)i(v)-5
b(ariable)30 b FP(x)h FT(with)f FP(a)p FT(.)44 b(Note)33
b(that)f(b)s(ecause)f(of)h(the)f(colouring)g(of)378 5021
y(the)g(tableau,)f(the)h(follo)m(wing)d(inequalities)g(are)j(not)g
(included)c(in)i(the)i(tableau)f(branc)m(hes:)1535 5250
y FN(h)p FP(a)p FN(i)c(6)p FT(=)f FN(h)p FP(f)10 b FT(\()p
FP(b)p FT(\))p FN(i)191 b(h)p FP(a)p FN(i)26 b(6)p FT(=)f
FN(h)p FP(f)10 b FT(\()p FP(b)p FT(\))p FN(i)1531 5388
y(h)p FP(x)p FN(i)26 b(6)p FT(=)f FN(h)p FP(f)10 b FT(\()p
FP(a)p FT(\))p FN(i)378 5592 y FT(and)30 b(as)g(a)h(result,)f(a)g
(smaller)f(searc)m(h)i(space)g(is)f(considered)f(during)f(the)j
(refutational)e(pro)s(cess.)519 5705 y(Finally)-8 b(,)37
b(w)m(e)g(note)g(that)g(the)f(undecidabilit)m(y)d(of)k(the)g(v)-5
b(alidit)m(y)34 b(of)j(structured)e(justi\014cations)p
eop
%%Page: 190 200
190 199 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(190)378
396 y(\(theorem)25 b(8.5\))g(implies)c(that)k(there)f(is)f(no)h
(complete)h(terminating)e(algorithm)g(that)h(c)m(hec)m(ks)i(struc-)378
509 y(tured)e(justi\014cations.)37 b(As)24 b(a)g(result,)h(the)g
(\(implemen)m(tation-based\))e(b)s(ounds)f(on)i(the)h(pro)s(of)e(searc)
m(h)378 622 y(describ)s(ed)36 b(in)h(section)h(5.3.3)i(are)f(used)e(to)
i(restrict)f(the)g(searc)m(h)h(space)g(considered)e(during)f(the)378
735 y(pro)s(of)j(c)m(hec)m(king)i(pro)s(cess)e(to)i(a)f(\014nite)f
(one.)69 b(W)-8 b(e)41 b(recall)e(that,)k(giv)m(en)d(a)g(list)f(of)h
(clauses)f(\000,)k(the)378 848 y(implemen)m(tation)29
b(of)h(the)h FN(C)5 b(B)s(S)i(E)38 b FT(rule)29 b(restricts)514
1036 y FN(\017)46 b FT(the)31 b(n)m(um)m(b)s(er)e(of)h(times)g(the)h
(expansion)e(rule)g(can)i(b)s(e)f(used)f(on)h(eac)m(h)i(clause,)e(and)
514 1223 y FN(\017)46 b FT(the)31 b(n)m(um)m(b)s(er)e(of)h(times)g(the)
h(basic)f(rigid)e(sup)s(erp)s(osition)f(rules)i(can)i(b)s(e)e(applied.)
378 1411 y(The)c(\014rst)h(restriction)e(ma)m(y)j(corresp)s(ond)e(to)h
(a)h(restriction)e(on)g(the)i(n)m(um)m(b)s(er)d(of)i(times)g(the)g
(implicit)378 1524 y(inference)j(rule)f Ff(\032)h FT(is)g(applied)e(to)
j(replicate)f(sub-form)m(ulae)f(in)m(v)m(olving)g(the)i(univ)m(ersal)e
(quan)m(ti\014er.)378 1637 y(F)-8 b(or)31 b(instance,)f(the)h(relation)
f Ff(\032)g FT(replicates)g(a)g(sub-form)m(ula)f(in)g(the)i(follo)m
(wing)e(cases:)1089 1841 y FP(A)d Ff(\032)f FP(A)20 b
FN(^)g FP(A)182 b(A)21 b FN(^)e FT(\()p FP(B)25 b FN(_)20
b FP(C)7 b FT(\))25 b Ff(\032)h FT(\()p FP(A)20 b FN(^)g
FP(B)5 b FT(\))20 b FN(_)g FT(\()p FP(A)h FN(^)f FP(C)7
b FT(\))p FP(:)378 2045 y FT(The)32 b(second)g(restriction)f(ma)m(y)i
(corresp)s(ond)e(to)i(a)g(restriction)e(on)h(the)h(n)m(um)m(b)s(er)e
(of)h(substitutions)378 2158 y(of)d(the)g(equations)g(in)e(the)j
(conjunction)e(of)h(equations)f FP(E)34 b FT(are)c(applied)d(to)i
(deriv)m(e)g(a)g(conclusion)e FP(C)378 2271 y FT(from)j(a)h(form)m(ula)
e FP(A)i FT(in)e FP(A)c FN(`)1356 2285 y FO(E)1441 2271
y FP(C)7 b FT(.)519 2384 y(The)33 b(b)s(ounds)f(giv)m(en)i(in)f
(section)h(5.3.3)h(w)m(ere)g(not)f(found)f(to)h(b)s(e)g(o)m(v)m
(er-restrictiv)m(e)h(during)d(the)378 2497 y(implemen)m(tation)22
b(of)h(the)h(case)g(study)f(describ)s(ed)e(in)h(c)m(hapter)i(9,)h(in)e
(the)g(sense)g(that)h(the)g(structured)378 2610 y(justi\014cations)43
b(that)i(w)m(ere)g(used)f(during)e(the)i(implemen)m(tation)g(of)g(the)h
(case)g(study)f(could)g(b)s(e)378 2723 y(pro)s(of)34
b(c)m(hec)m(k)m(ed)i(according)f(to)g(these)g(b)s(ounds.)51
b(This)32 b(suggests)j(that)h(the)e(explicit)f(and)h(implicit)378
2836 y(deriv)-5 b(ations)33 b(de\014ned)h(in)g(c)m(hapter)h(6)h(are)f
(to)s(o)h(strong)f(and)f(cannot)i(b)s(e)e(considered)g(to)i(represen)m
(t)378 2949 y(trivial)26 b(inferences.)39 b(The)28 b(de\014nition)e(of)
i(w)m(eak)m(er)i(and)e(decidable)e(implicit)g(deriv)-5
b(ations)26 b(should)h(b)s(e)378 3061 y(considered)i(in)g(future.)378
3348 y FH(8.6)135 b(Summary)378 3551 y FT(In)27 b(this)g(c)m(hapter)i
(w)m(e)f(ha)m(v)m(e)h(used)e(the)h(de\014nitions)e(and)h(results)g(on)h
(the)g(coloured)f(\014rst-order)g(logic)378 3664 y(giv)m(en)43
b(in)e(c)m(hapter)i(7)g(to)h(de\014ne)e(a)h(restriction)e(on)i(the)g
(pro)s(of)f(searc)m(h)h(required)e(to)i(c)m(hec)m(k)i(the)378
3777 y(structured)29 b(justi\014cations)f(giv)m(en)h(in)g(c)m(hapter)h
(6.)41 b(In)28 b(particular,)h(it)g(is)g(sho)m(wn)g(that)h(a)g(form)m
(ula)f FP(X)378 3890 y FT(implicitly)d(deriv)m(es)k(a)g(form)m(ula)g
FP(Y)50 b FT(\(i.e.,)16 b FP(X)33 b Ff(\032)2001 3857
y FK(\003)2065 3890 y FP(Y)20 b FT(\))31 b(if)e(and)g(only)h(if)f
FP(X)2828 3857 y FO(i)2882 3890 y FN(\))c FP(Y)3071 3857
y FO(j)3137 3890 y FT(is)30 b(consisten)m(t)g(with)378
4002 y(resp)s(ect)e(to)h(the)g(connectabilit)m(y)f(relation)f
FP(i)f FN($)f FP(j)34 b FT(where)28 b(the)g(colours)g
FP(i)h FT(and)f FP(j)33 b FT(are)c(distinct.)39 b(This)378
4115 y(result)31 b(is)h(used)g(to)h(sho)m(w)f(that)h(the)g(problem)e
(of)h(c)m(hec)m(king)i(implicit)29 b(and)j(explicit)f(deriv)-5
b(ations)31 b(is)378 4228 y(undecidable.)519 4341 y(A)i(metho)s(d)g
(for)g(constructing)f(a)i(coloured)f(problem)e(from)i(a)g(conclusion)f
(and)g(a)i(structured)378 4454 y(justi\014cation)i(is)h(then)g
(illustrated.)60 b(This)35 b(metho)s(d)i(is)g(sho)m(wn)g(to)h(corresp)s
(ond)e(to)i(a)g(sound)e(and)378 4567 y(complete)d(restriction)f(on)g
(the)h(pro)s(of)f(searc)m(h)h(required)e(to)i(c)m(hec)m(k)h(structured)
e(justi\014cations.)46 b(In)378 4680 y(other)35 b(w)m(ords,)g(a)g
(structured)f(justi\014cation)f(is)h(v)-5 b(alid)33 b(if)g(and)i(only)e
(if)h(its)g(constructued)g(coloured)378 4793 y(problem)e(is)h
(inconsisten)m(t.)51 b(The)33 b(pro)s(of)h(of)g(the)g(soundness)e(and)i
(completeness)g(result)f(used)g(the)378 4906 y(results)c(on)h(coloured)
g(in)m(terp)s(olan)m(ts)g(deriv)m(ed)f(in)g(section)i(7.5.)519
5019 y(The)d FN(C)5 b(B)s(S)i(E)35 b FT(rule)27 b(de\014ned)g(in)g(c)m
(hapter)i(5)f(is)f(then)h(mo)s(di\014ed)e(so)j(that)f(it)g(can)h(b)s(e)
e(used)g(to)i(c)m(hec)m(k)378 5132 y(structured)h(justi\014cations.)43
b(The)31 b(mo)s(di\014ed)e FN(C)5 b(B)s(S)i(E)39 b FT(rule)30
b(is)h(used)f(to)j(c)m(hec)m(k)g(the)e(justi\014cations)f(in)378
5244 y(the)i(pro)s(ofs)g(implemen)m(ted)e(during)g(the)j(case)g(study)e
(describ)s(ed)f(in)h(c)m(hapter)i(9.)47 b(W)-8 b(e)33
b(argued)f(that)378 5357 y(although)d(the)h(implicit)d(and)j(explicit)e
(deriv)-5 b(ations)28 b(de\014ned)h(in)f(c)m(hapter)j(6)f(ha)m(v)m(e)h
(an)f(undecidable)378 5470 y(v)-5 b(alidit)m(y)20 b(problem,)h(it)g(is)
g(lik)m(ely)f(that)h(only)g(a)h(small,)g(p)s(ossibly)c(decidable,)k
(subset)f(of)g(these)h(are)g(used)378 5583 y(in)34 b(practice,)k(and)d
(that)h(therefore)h(it)e(is)g(desirable)f(that)i(de\014nitions)d(of)j
(w)m(eak)m(er)h(and)e(decidable)378 5696 y(implicit)22
b(and)j(explicit)f(deriv)-5 b(ations)23 b(should)h(b)s(e)g(considered)h
(as)g(future)g(w)m(ork.)39 b(It)25 b(is)f(also)i(desirable)p
eop
%%Page: 191 201
191 200 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(191)378
396 y(that)26 b(the)g(notion)f(of)h(structured)e(justi\014cations,)i
(whic)m(h)e(are)i(curren)m(tly)e(limited)g(to)i(the)g(pure)e(\014rst-)
378 509 y(order)29 b(logic,)h(should)e(b)s(e)h(extended)h(to)h(the)f
(\014rst-order)f(logic)g(with)g(equalit)m(y)g(as)h(w)m(ell)f(as)h(to)h
(other)378 622 y(logics)f(and)g(theories.)p eop
%%Page: 192 202
192 201 bop 378 1019 a FJ(Chapter)65 b(9)378 1434 y FR(A)77
b(Mec)-6 b(hanisation)77 b(of)h(Group)378 1683 y(Theory)378
2165 y FH(9.1)135 b(In)l(tro)t(duction)378 2368 y FT(This)28
b(c)m(hapter)i(illustrates)d(the)j(mec)m(hanisation)f(of)g(a)h(n)m(um)m
(b)s(er)e(of)i(results)e(of)i(group)f(theory)g(using)378
2481 y(the)41 b(SPL)f(language.)73 b(The)40 b(mec)m(hanisation)h(is)f
(based)g(on)h(the)g(textb)s(o)s(ok)g(b)m(y)g(\(Herstein)g(1975\))378
2594 y(and)28 b(includes)e(results)h(on)i(normal)e(groups,)i(quotien)m
(t)f(groups)g(and)g(the)h(isomorphism)c(theorems.)378
2706 y(The)35 b(mec)m(hanisation)f(also)i(includes)c(the)k(implemen)m
(tation)e(of)h(a)g(n)m(um)m(b)s(er)f(of)i(pro)s(of)e(pro)s(cedures)378
2819 y(in)c(SML)i(whic)m(h)e(are)i(used)f(in)g(automating)h(a)g(n)m(um)
m(b)s(er)f(of)h(inferences)f(omitted)g(from)h(the)g(formal)378
2932 y(pro)s(ofs.)519 3045 y(The)e(motiv)-5 b(ations)30
b(for)g(this)f(mec)m(hanisation)h(include:)514 3233 y
FN(\017)46 b FT(in)m(v)m(estigating)e(the)f(idea)g(that)h(the)g(incorp)
s(oration)d(of)j(pro)s(of)e(pro)s(cedures)g(implemen)m(ted)605
3346 y(during)25 b(the)h(mec)m(hanisation)h(of)f(the)h(theory)g(in)e
(order)i(to)g(automate)h(trivial)d(inferences)h(can)605
3459 y(substan)m(tially)j(reduce)h(the)g(di\013erence)g(b)s(et)m(w)m
(een)h(formal)f(and)f(informal)g(pro)s(ofs;)514 3646
y FN(\017)46 b FT(the)28 b(use)f(of)h(structured)e(straigh)m(tforw)m
(ard)h(justi\014cations)f(in)h(order)g(to)h(c)m(hec)m(k)h(whether)e
(they)605 3759 y(can)k(b)s(e)f(used)f(to)i(dev)m(elop)f(readable)g(pro)
s(of)g(scripts,)f(and)h(whether)f(an)m(y)i(substan)m(tial)e(e\013ort)
605 3872 y(is)h(needed)g(in)f(the)h(implemen)m(tation)f(of)i(pro)s(ofs)
e(using)g(suc)m(h)h(justi\014cations.)519 4060 y(The)d(pro)s(of)g
(scripts)f(dev)m(elop)s(ed)h(during)e(the)i(mec)m(hanisation)g(are)h(m)
m(uc)m(h)g(more)f(readable)g(than)378 4173 y(tactic-based)38
b(pro)s(ofs)e(suc)m(h)h(as)g(the)g(ones)g(describ)s(ed)e(in)g(c)m
(hapter)j(3.)61 b(F)-8 b(urthermore,)38 b(the)f(imple-)378
4286 y(men)m(tation)g(of)g(simpli\014ers)32 b(and)k(query)g(functions)f
(on)i(the)f(facts)i(stored)e(in)g(the)g(SPL)g(database)378
4398 y(of)k(trivial)e(kno)m(wledge)h(are)h(used)f(extensiv)m(ely)h(to)g
(automate)h(the)f(inferences)f(whic)m(h)f(are)i(often)378
4511 y(omited)30 b(from)g(the)g(literature.)519 4624
y(The)f(results)g(on)g(group)g(theory)h(giv)m(en)g(in)e(this)h(c)m
(hapter,)h(as)g(w)m(ell)f(as)h(man)m(y)f(other)h(related)g(re-)378
4737 y(sults,)c(ha)m(v)m(e)i(b)s(een)e(mec)m(hanised)h(in)e(pro)s(of)h
(dev)m(elopmen)m(t)h(systems)g(b)s(efore.)39 b(F)-8 b(or)27
b(instance,)h(Gun)m(ter)378 4850 y(\(1990\))44 b(mec)m(hanised)c(a)i(n)
m(um)m(b)s(er)e(of)h(results)f(on)h(group)g(theory)h(in)d(HOL.)j(Kamm)s
(\177)-48 b(uller)38 b(\(1997\))378 4963 y(pro)m(v)m(ed)k(Sylo)m(w's)f
(theorem)g(in)g(Isab)s(elle,)h(and)f(v)m(on)h(W)-8 b(righ)m(t)42
b(\(1992\))i(and)d(Laibinis)d(\(1996\))44 b(for-)378
5076 y(malised)21 b(lattice)i(theory)g(in)e(HOL.)i(Jac)m(kson)h
(\(1995\))h(formalised)c(a)i(substan)m(tial)e(amoun)m(t)i(of)g(results)
378 5189 y(in)h(abstract)j(algebra,)g(including)c(results)h(on)i
(groups,)g(using)f(the)h(Nuprl)e(pro)s(of)h(dev)m(elopmen)m(t)h(sys-)
378 5302 y(tem.)40 b(Bailey)28 b(\(1998\))i(mec)m(hanised)d(Galois)g
(theory)h(in)e(LEGO)i(using)e(sev)m(eral)i(tec)m(hniques)f(includ-)378
5415 y(ing)e(co)s(ercions)h(and)g(literate)g(programming)f(to)i(impro)m
(v)m(e)g(the)f(presen)m(tation)g(of)h(the)f(implemen)m(ted)378
5528 y(pro)s(of)36 b(scripts.)58 b(Sev)m(eral)37 b(results)e(on)i
(groups,)g(rings,)g(lattices)g(and)f(other)h(algebraic)f(structures)378
5640 y(are)i(also)f(mec)m(hanised)f(in)g(the)i(Mizar)f(system.)61
b(The)37 b(con)m(tribution)f(of)h(the)h(w)m(ork)f(presen)m(ted)g(in)
2035 5954 y(192)p eop
%%Page: 193 203
193 202 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(193)378
396 y(this)36 b(c)m(hapter)i(lies)f(in)f(the)i(use)f(of)g(an)h
FI(extensible)h(de)-5 b(clar)g(ative)41 b(pr)-5 b(o)g(of)41
b(language)k FT(in)36 b(whic)m(h)h(pro)s(of)378 509 y(pro)s(cedures)24
b(are)h(implemen)m(ted)f(during)f(the)i(mec)m(hanisation)f(in)g(order)h
(to)h(b)s(e)e(used)g(in)g(minimising)378 622 y(the)31
b(di\013erence)e(b)s(et)m(w)m(een)i(formal)f(and)g(informal)e(pro)s
(ofs.)519 735 y(This)i(c)m(hapter)i(is)f(organised)g(as)h(follo)m(ws.)
44 b(In)31 b(section)h(9.2)h(w)m(e)f(giv)m(e)g(the)g(de\014nition)e(of)
i(groups)378 848 y(in)j(HOL)i(and)f(describ)s(e)f(the)i(preliminary)c
(results)j(that)h(are)g(deriv)m(ed)f(and)g(ho)m(w)h(they)g(are)g(used)
378 961 y(in)43 b(implemen)m(ting)f(pro)s(of)i(pro)s(cedures)f(that)i
(are)g(then)f(incorp)s(orated)g(in)f(the)i(SPL)e(language.)378
1074 y(Section)29 b(9.3)h(giv)m(es)f(a)g(n)m(um)m(b)s(er)f(of)h
(results)f(on)g(congruences,)i(cosets)g(and)f(the)g(pro)s(duct)f(of)h
(subsets)378 1187 y(of)24 b(groups.)38 b(F)-8 b(urther)24
b(results,)g(suc)m(h)g(as)g(the)g(existence)g(of)h(quotien)m(t)f
(groups)f(and)h(the)g(isomorphism)378 1300 y(theorems)31
b(are)f(giv)m(en)h(in)e(section)h(9.4.)42 b(A)31 b(concluding)d
(discussion)g(is)h(then)h(giv)m(en)h(in)e(section)h(9.5.)378
1586 y FH(9.2)135 b(Group)44 b(Theory)h(in)g(SPL)378
1789 y FT(Groups)e(are)h(one)f(of)h(the)g(most)g(common)f(algebraic)h
(structures)f(in)f(mathematics)i(and)f(ha)m(v)m(e)378
1902 y(b)s(een)d(studied)g(in)m(tensiv)m(ely)f(in)h(the)h(nineteen)m
(th)g(and)f(t)m(w)m(en)m(tieth)i(cen)m(turies.)72 b(Groups)41
b(are)g(also)378 2015 y(extended)32 b(to)g(other)g(algebraic)g
(structures)f(including)e(rings,)i(\014elds)f(and)h(v)m(ector)j
(spaces.)45 b(In)31 b(our)378 2128 y(mec)m(hanisation)k(w)m(e)h(follo)m
(w)f(\(Herstein)g(1975\))j(in)c(de\014ning)g(and)h(reasoning)g(ab)s
(out)g(groups,)h(and)378 2241 y(deriv)m(e)k(all)f(the)h(results)f(up)h
(to)g(and)g(including)d(the)j(second)h(isomorphism)c(theorem)k(with)e
(the)378 2354 y(exception)31 b(of)f(those)h(in)m(v)m(olving)e(\014nite)
g(groups.)378 2597 y FG(9.2.1)112 b(The)38 b(De\014nition)e(of)i
(Groups)378 2769 y FT(A)30 b(group)f(is)g(a)h(pair)f(\()p
FP(G;)15 b FN(\016)p FT(\))31 b(where)e FP(G)g FT(is)g(a)i(nonempt)m(y)
e(set)i(and)e FN(\016)h FT(is)f(a)h(binary)e(op)s(erator)j(o)m(v)m(er)g
(the)378 2882 y(elemen)m(ts)g(in)e FP(G)h FT(suc)m(h)g(that)489
3069 y(1.)46 b FP(G)30 b FT(is)f FI(close)-5 b(d)41 b
FT(under)29 b FN(\016)p FT(:)41 b FN(8)p FP(x;)15 b(y)28
b FN(2)d FP(G:)30 b(x)20 b FN(\016)h FP(y)28 b FN(2)d
FP(G)p FT(.)489 3257 y(2.)46 b FN(\016)31 b FT(is)e FI(asso)-5
b(ciative)7 b FT(:)42 b FN(8)p FP(x;)15 b(y)s(;)g(z)30
b FN(2)25 b FP(G:)30 b(x)20 b FN(\016)h FT(\()p FP(y)i
FN(\016)e FP(z)t FT(\))26 b(=)f(\()p FP(x)20 b FN(\016)h
FP(y)s FT(\))f FN(\016)h FP(z)t FT(.)489 3445 y(3.)46
b FP(G)30 b FT(con)m(tains)h(an)f FI(identity)39 b FT(elemen)m(t:)i
FN(9)p FP(e)25 b FN(2)g FP(G:)30 b FN(8)p FP(x)25 b FN(2)g
FP(G:)30 b(x)20 b FN(\016)g FP(e)26 b FT(=)f FP(e)c FN(\016)f
FP(x)25 b FT(=)g FP(x)p FT(.)489 3632 y(4.)46 b(Ev)m(ery)31
b(elemen)m(t)g(in)e FP(G)h FT(has)g(an)g FI(inverse)7
b FT(:)40 b FN(8)p FP(x)25 b FN(2)g FP(G:)30 b FN(9)p
FP(x)2495 3599 y FK(\000)p FL(1)2614 3632 y FN(2)25 b
FP(G:)30 b(x)20 b FN(\016)h FP(x)3017 3599 y FK(\000)p
FL(1)3136 3632 y FT(=)k FP(x)3284 3599 y FK(\000)p FL(1)3399
3632 y FN(\016)20 b FP(x)26 b FT(=)f FP(e)p FT(.)378
3820 y(T)-8 b(erms)22 b(of)h(the)g(form)f FP(p)5 b FN(\016)g
FP(q)25 b FT(are)e(usually)e(abbreviated)h(to)h FP(pq)i
FT(when)d(the)h(binary)e(op)s(erator)i(concerned)378
3933 y(can)31 b(b)s(e)e(understo)s(o)s(d)g(form)h(the)g(con)m(text.)519
4046 y(It)25 b(is)f(straigh)m(tforw)m(ard)g(to)i(giv)m(e)f(a)g(p)s
(olymorphic)d(de\014nition)h(of)i(groups)f(in)f(HOL)i(b)m(y)g(represen)
m(t-)378 4159 y(ing)k(sets)i(b)m(y)g(their)e(c)m(haracteristic)i
(predicate)f(and)g(the)h(pro)s(duct)e(as)h(a)h(curried)e(binary)f(op)s
(erator:)473 4346 y FN(`)529 4361 y FE(def)686 4346 y
FM(Group)47 b FT(\()p FP(G)p FM(:)p FP(\013)g FN(!)h
FM(bool)p FP(;)f(p)p FM(:)p FP(\013)h FN(!)f FP(\013)h
FN(!)g FP(\013)p FT(\))h FN(\021)855 4459 y FT(\()p FM(GClosed)d
FT(\()p FP(G;)15 b(p)p FT(\)\))49 b FN(^)855 4572 y FT(\()p
FM(GAssoc)e FT(\()p FP(G;)15 b(p)p FT(\)\))48 b FN(^)855
4685 y(9)p FP(e)p FM(:)p FP(\013)p FM(.)g FT(\()p FP(G)f(e)p
FT(\))i FN(^)e FT(\()p FM(GId)g FT(\()p FP(G;)15 b(p)p
FT(\))48 b FP(e)p FT(\))h FN(^)1157 4798 y FT(\()p FN(8)p
FP(x)p FM(.)f FP(G)f(x)h FN(\))f FM(GhasInv)f FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(e)g(x)p FT(\))378 4985 y(where)473
5173 y FN(`)529 5188 y FE(def)686 5173 y FM(GClosed)e
FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\021)g(8)p FP(x)f(y)s
FM(.)g FP(G)g(x)h FN(\))f FP(G)h(y)i FN(\))e FP(G)f FT(\()p
FP(p)h(x)g(y)s FT(\))473 5286 y FN(`)529 5301 y FE(def)686
5286 y FM(GAssoc)e FT(\()p FP(G;)15 b(p)p FT(\))48 b
FN(\021)g(8)p FP(x)f(y)k(z)t FM(.)c FP(G)g(x)h FN(\))g
FP(G)f(y)k FN(\))c FP(G)g(z)52 b FN(\))998 5399 y FT(\(\()p
FP(p)d(x)e FT(\()p FP(p)h(y)j(z)t FT(\)\))74 b(=)e(\()p
FP(p)48 b FT(\()p FP(p)g(x)g(y)s FT(\))g FP(z)t FT(\)\))378
5586 y(and)28 b(the)i(iden)m(tit)m(y)e(predicate)h Fw(GId)f
FT(is)g(de\014ned)g(suc)m(h)h(that)g(giv)m(en)h(a)f(group)g(\()p
FP(G;)15 b(p)p FT(\))29 b(and)f(an)h(elemen)m(t)378 5699
y FP(e)p FT(,)i(it)f(holds)f(if)g FP(e)i FT(is)e(b)s(oth)h(a)g(left)h
(and)e(righ)m(t)h(iden)m(tit)m(y)g(for)g(all)f(the)i(elemen)m(ts)g(in)e
FP(G)p FT(:)p eop
%%Page: 194 204
194 203 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(194)473
396 y FN(`)529 411 y FE(def)686 396 y FM(GLeftId)46 b
FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(e)g FN(\021)g(8)p
FP(x)p FM(.)e FP(G)i(x)f FN(\))h FT(\()p FP(p)g(e)g(x)73
b FT(=)f FP(x)p FT(\))473 509 y FN(`)529 524 y FE(def)686
509 y FM(GRightId)46 b FT(\()p FP(G;)15 b(p)p FT(\))48
b FP(e)g FN(\021)f(8)p FP(x)p FM(.)g FP(G)g(x)h FN(\))g
FT(\()p FP(p)g(x)f(e)74 b FT(=)e FP(x)p FT(\))473 622
y FN(`)529 637 y FE(def)686 622 y FM(GId)47 b FP(Gp)g(e)h
FN(\021)g FM(GLeftId)d FP(Gp)i(e)h FN(^)g FM(GRightId)d
FP(Gp)i(e)378 807 y FT(and)28 b(the)g(predicate)g Fw(GhasInv)d
FT(is)j(de\014ned)f(in)g(terms)h(of)g(the)h(predicate)f
Fw(GInv)e FT(whic)m(h)h(tak)m(es)j(a)e(group)378 920
y(\()p FP(G;)15 b(p)p FT(\))29 b(and)f(the)h(elemen)m(ts)g
FP(e)p FT(,)h FP(x)e FT(and)h FP(x)1740 934 y FL(1)1779
920 y FT(,)g(and)f(holds)g(if)f FP(x)2378 934 y FL(1)2446
920 y FT(is)h(b)s(oth)g(a)h(left)g(and)f(righ)m(t)g(in)m(v)m(erse)h(of)
g FP(x)378 1033 y FT(on)h(the)h(assumption)e(that)i FP(e)f
FT(is)g(an)g(iden)m(tit)m(y)g(elemen)m(t)h(in)e FP(G)p
FT(.)473 1217 y FN(`)529 1232 y FE(def)686 1217 y FM(GLeftInv)46
b FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(e)g(x)g(x)1634
1231 y FL(1)1721 1217 y FN(\021)f FT(\()p FP(p)h(x)2020
1231 y FL(1)2107 1217 y FP(x)73 b FT(=)g FP(e)p FT(\))473
1330 y FN(`)529 1345 y FE(def)686 1330 y FM(GRightInv)45
b FT(\()p FP(G;)15 b(p)p FT(\))49 b FP(e)f(x)f(x)1681
1344 y FL(1)1768 1330 y FN(\021)h FT(\()p FP(p)g(x)f(x)2167
1344 y FL(1)2280 1330 y FT(=)72 b FP(e)p FT(\))473 1443
y FN(`)529 1458 y FE(def)686 1443 y FM(GInv)47 b FP(Gp)g(e)h(x)g(x)1332
1457 y FL(1)1419 1443 y FN(\021)f FM(GLeftInv)f FP(Gp)h(e)h(x)f(x)2373
1457 y FL(1)2460 1443 y FN(^)h FM(GRightInv)d FP(Gp)i(e)h(x)g(x)3453
1457 y FL(1)473 1556 y FN(`)529 1571 y FE(def)686 1556
y FM(GhasInv)e(\()p FP(G)p FM(,)p FP(p)p FM(\))g FP(e)i(x)g
FN(\021)f(9)p FP(x)1787 1570 y FL(1)1826 1556 y FM(.)h
FP(G)f(x)2093 1570 y FL(1)2180 1556 y FN(^)g FM(GInv)g
FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(e)g(x)g(x)3045 1570
y FL(1)378 1741 y FT(The)26 b(de\014nition)e(of)j(groups)f(giv)m(en)g
(ab)s(o)m(v)m(e)i(is)e(equiv)-5 b(alen)m(t)26 b(to)h(a)g(simpler)d(one)
j(in)e(whic)m(h)h(the)g(iden)m(tit)m(y)378 1853 y(elemen)m(t)38
b FP(e)g FT(is)f(only)g(assumed)g(to)h(b)s(e)g(a)g(righ)m(t)f(iden)m
(tit)m(y)g(and)g(the)h(in)m(v)m(erse)g(elemen)m(t)g FP(x)3434
1821 y FK(\000)p FL(1)3566 1853 y FT(of)g FP(x)g FT(is)378
1966 y(only)e(assumed)g(to)h(b)s(e)f(a)h(righ)m(t)f(in)m(v)m(erse.)60
b(Deriving)35 b(the)i(equiv)-5 b(alence)36 b(of)h(these)g(t)m(w)m(o)h
(de\014nitions)378 2079 y(allo)m(ws)27 b(one)g(to)h(sho)m(w)f(that)h(a)
g(structure)f(is)f(a)i(group)f(without)f(sho)m(wing)g(that)i(the)g(c)m
(hosen)f(iden)m(tit)m(y)378 2192 y(elemen)m(t)k(is)e(a)i(left)f(iden)m
(tit)m(y)g(and)g(that)h(the)f(c)m(hosen)h(in)m(v)m(erse)f(is)g(a)h
(left)f(in)m(v)m(erse.)519 2305 y(Giv)m(en)i(a)h(group)f(\()p
FP(G;)15 b(p)p FT(\),)33 b(an)g(iden)m(tit)m(y)f(elemen)m(t)g(can)h(b)s
(e)f(selected)h(b)m(y)f(the)h(function)e Fw(IdG)n FT(,)i(and)378
2418 y(giv)m(en)d(an)g(elemen)m(t)h(in)e FP(G)p FT(,)g(its)h(in)m(v)m
(erse)g(can)h(b)s(e)e(selected)i(b)m(y)f(the)g(function)f
Fw(InvG)n FT(;)i(these)f(functions)378 2531 y(are)h(de\014ned)e(as)h
(follo)m(ws:)473 2716 y FN(`)529 2731 y FE(def)686 2716
y FM(IdG)47 b FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\021)g
FP("e)p FM(.)27 b FP(G)47 b(e)h FN(^)f FM(GId)g FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(e)473 2941 y FN(`)529 2956
y FE(def)686 2941 y FM(InvG)f FT(\()p FP(G;)15 b(p)p
FT(\))48 b FP(x)g FN(\021)f FP("x)1513 2955 y FL(1)1553
2941 y FM(.)26 b FP(G)47 b(x)1798 2955 y FL(1)1885 2941
y FN(^)h FM(GInv)e FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p
FM(IdG)g FT(\()p FP(G;)15 b(p)p FT(\)\))48 b FP(x)g(x)3198
2955 y FL(1)519 3126 y FT(Deriving)42 b(theorems)h(sho)m(wing)f(that)i
Fw(IdG)e Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b FT(is)e(an)h(iden)m(tit)m(y)
f(elemen)m(t)i(in)d FP(G)i FT(and)f(that)378 3239 y Fw(InvG)g
Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(x)33 b FT(is)f(an)g(in)m(v)m(erse)
h(of)g FP(x)f FT(is)g(done)h(b)m(y)f(using)f(the)i Fw(select)d
FT(inference)i(rule)g(describ)s(ed)e(in)378 3352 y(section)g(4.2.5,)j
(page)e(68.)378 3595 y FG(9.2.2)112 b(Preliminary)35
b(Results)378 3766 y FT(Giv)m(en)22 b(the)g(de\014nitions)d(in)i(the)h
(previous)e(section,)k(one)e(is)f(required)f(to)j(deriv)m(e)e(a)i(n)m
(um)m(b)s(er)d(of)i(results)378 3879 y(whic)m(h)34 b(although)h(v)m
(ery)h(simple)e(in)g(nature,)j(will)c(b)s(e)i(extremely)h(useful)e(in)g
(the)i(dev)m(elopmen)m(t)g(of)378 3992 y(the)f(theory)-8
b(.)56 b(In)35 b(\(Herstein)g(1975\))i(the)f(follo)m(wing)e(results)g
(are)h(deriv)m(ed)g(after)g(the)h(de\014nition)d(of)378
4105 y(groups)d(is)f(giv)m(en:)489 4290 y(1.)46 b(The)30
b(iden)m(tit)m(y)g(elemen)m(t)h(is)e(unique)g(and)g(ev)m(ery)j(elemen)m
(t)e(has)h(a)f(unique)f(in)m(v)m(erse;)489 4476 y(2.)46
b(The)30 b(follo)m(wing)f(theorems)i(on)f(in)m(v)m(erses)997
4680 y FN(8)p FP(a)25 b FN(2)g FP(G:)30 b FT(\()p FP(a)1417
4643 y FK(\000)p FL(1)1511 4680 y FT(\))1546 4643 y FK(\000)p
FL(1)1666 4680 y FT(=)25 b FP(a)364 b FN(8)p FP(a;)15
b(b)25 b FN(2)g FP(G:)30 b FT(\()p FP(a)21 b FN(\016)f
FP(b)p FT(\))2833 4643 y FK(\000)p FL(1)2953 4680 y FT(=)25
b FP(b)3088 4643 y FK(\000)p FL(1)3203 4680 y FN(\016)c
FP(a)3317 4643 y FK(\000)p FL(1)3411 4680 y FT(;)489
4921 y(3.)46 b(The)30 b(cancellation)g(la)m(ws:)40 b(for)31
b(ev)m(ery)g FP(a)p FT(,)f FP(u)h FT(and)e FP(w)k FT(in)c
FP(G)1061 5125 y FT(\()p FP(a)20 b FN(\016)h FP(u)k FT(=)g
FP(a)20 b FN(\016)h FP(w)r FT(\))26 b FN(\))f FP(u)h
FT(=)f FP(w)366 b FT(\()p FP(u)20 b FN(\016)h FP(a)k
FT(=)g FP(w)e FN(\016)e FP(a)p FT(\))k FN(\))h FP(u)f
FT(=)g FP(w)r(:)519 5366 y FT(The)30 b(uniqueness)e(of)j(the)f(iden)m
(tit)m(y)g(and)g(in)m(v)m(erse)g(elemen)m(ts)h(allo)m(ws)f(one)h(to)g
(uniquely)c(iden)m(tify)378 5479 y(the)e(iden)m(tit)m(y)g(and)f(the)h
(in)m(v)m(erse)g(of)g(an)g(elemen)m(t)h FP(a)f FT(b)m(y)g(the)g(terms)g
FP(e)g FT(and)f FP(a)2907 5446 y FK(\000)p FL(1)3002
5479 y FT(.)39 b(W)-8 b(e)26 b(deriv)m(e)e(the)h(same)378
5592 y(HOL)i(theorems)h(in)e(SPL)h(whic)m(h)f(allo)m(w)h(us)g(to)h
(iden)m(tify)e(the)i(iden)m(tit)m(y)f(and)f(in)m(v)m(erses)i(b)m(y)f
Fw(IdG)42 b Ft(\()p Fv(G;)14 b(p)p Ft(\))378 5705 y FT(and)41
b Fw(InvG)h Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(a)e
FT(resp)s(ectiv)m(ely)f(throughout)h(the)g(rest)g(of)g(the)g(theory)-8
b(.)75 b(The)42 b(pro)s(ofs)f(of)h(the)p eop
%%Page: 195 205
195 204 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(195)378
396 y(uniqueness)34 b(theorems)j(are)g(similar)c(to)k(the)g(ones)g
(found)e(in)g(the)h(literature)g(and)g(are)g(sho)m(wn)g(in)378
509 y(the)31 b(co)s(de)h(fragmen)m(t)g(in)e(\014gure)g(23.)44
b(The)31 b(pro)s(ofs)f(are)i(rather)f(detailed)f(since)h(the)g(mec)m
(hanisation)378 622 y(of)f(the)g(theory)g(is)f(still)f(at)i(an)g(early)
f(stage.)42 b(The)30 b(length)f(of)h(the)g(pro)s(ofs)f(is)g(sligh)m
(tly)f(decreased)i(b)m(y)378 735 y(sp)s(ecifying)22 b(the)j
(de\014nitions)d(of)j Fw(GId)o FT(,)h Fw(GInv)d FT(and)h
Fw(GAssoc)e FT(as)i(simpli\014ers)d(so)k(that)g(they)g(are)g(unfolded)
378 848 y(automatically)31 b(b)s(efore)g(pro)s(of)f(searc)m(h.)44
b(This)30 b(is)g(sp)s(eci\014ed)g(b)m(y)h(the)g Fw(simplify)41
b(with)29 b FT(statemen)m(t)k(in)378 961 y(the)e(co)s(de.)519
1074 y(The)e(results)f(giv)m(en)i(in)e(the)h(second)h(p)s(oin)m(t)e(ab)
s(o)m(v)m(e)j(allo)m(w)e(the)g(author)h(and)e(the)i(reader)f(to)h(ma-)
378 1187 y(nipulate)d(and)h(simplify)e(terms)j(in)m(v)m(olving)e(in)m
(v)m(erses.)40 b(Suc)m(h)28 b(manipulations)e(are)k(then)e(p)s
(erformed)378 1300 y(without)i(an)m(y)h(justi\014cation)f(once)h(these)
h(results)d(are)j(deriv)m(ed.)41 b(It)31 b(is)f(desirable)f(that)j(at)f
(an)g(early)378 1413 y(stage)d(in)e(the)h(mec)m(hanisation,)g(suc)m(h)g
(results)f(are)h(deriv)m(ed)f(and)g(used)g(in)g(some)h(mec)m(hanism)f
(whic)m(h)378 1526 y(allo)m(ws)32 b(the)i(implemen)m(ter)d(to)j(treat)g
(suc)m(h)f(manipulations)d(as)k(trivial)d(and)h(whic)m(h)g(therefore)i
(can)378 1638 y(b)s(e)27 b(omitted)g(from)g(later)g(formal)g(pro)s
(ofs.)39 b(In)27 b(particular,)f(pro)s(ofs)h(in)f(later)h(sections)h
(of)f(the)h(theory)378 1751 y(do)35 b(not)g(ha)m(v)m(e)h(to)g(con)m
(tain)f(the)g(lev)m(el)g(of)g(detail)f(of)h(those)h(giv)m(en)f(in)e
(\014gure)i(23.)55 b(The)35 b(mec)m(hanism)378 1864 y(w)m(e)30
b(use)f(is)f(a)h(term)h(rewriting)d(system)i(whic)m(h)f(normalises)g
(terms)h(represen)m(ting)g(group)f(elemen)m(ts.)378 1977
y(The)36 b(application)f(of)h(Kn)m(uth-Bendix)f(completion)g(\(Kn)m
(uth)h(and)g(Bendix)f(1970\))k(on)d(the)g(group)378 2090
y(axioms)1804 2294 y FP(e)21 b FN(\016)g FP(x)83 b FT(=)f
FP(x)1804 2432 y(x)21 b FN(\016)f FP(e)84 b FT(=)e FP(x)1700
2570 y(x)1752 2532 y FK(\000)p FL(1)1867 2570 y FN(\016)21
b FP(x)83 b FT(=)f FP(e)1700 2708 y(x)21 b FN(\016)f
FP(x)1890 2670 y FK(\000)p FL(1)2068 2708 y FT(=)82 b
FP(e)1596 2846 y(x)20 b FN(\016)h FT(\()p FP(y)i FN(\016)e
FP(z)t FT(\))84 b(=)e(\()p FP(x)21 b FN(\016)f FP(y)s
FT(\))h FN(\016)g FP(z)378 3050 y FT(giv)m(es)h(the)g(follo)m(wing)e
(strongly)h(normalising)e(term)j(rewriting)e(system)i(\(see)h(for)e
(instance)h(\(Plaisted)378 3163 y(1993a\)\):)1839 3367
y FP(e)e FN(\016)h FP(x)83 b FN(!)g FP(x)1839 3505 y(x)20
b FN(\016)h FP(e)83 b FN(!)g FP(x)1735 3643 y(x)1787
3605 y FK(\000)p FL(1)1901 3643 y FN(\016)21 b FP(x)83
b FN(!)g FP(e)1735 3780 y(x)20 b FN(\016)h FP(x)1925
3743 y FK(\000)p FL(1)2102 3780 y FN(!)83 b FP(e)1630
3918 y FT(\()p FP(x)21 b FN(\016)g FP(y)s FT(\))f FN(\016)h
FP(z)87 b FN(!)c FP(x)20 b FN(\016)h FT(\()p FP(y)i FN(\016)e
FP(z)t FT(\))1708 4056 y(\()p FP(x)1795 4019 y FK(\000)p
FL(1)1889 4056 y FT(\))1924 4019 y FK(\000)p FL(1)2102
4056 y FN(!)83 b FP(x)1882 4194 y(e)1924 4156 y FK(\000)p
FL(1)2102 4194 y FN(!)g FP(e)1530 4332 y(x)1582 4294
y FK(\000)p FL(1)1697 4332 y FN(\016)21 b FT(\()p FP(x)f
FN(\016)h FP(y)s FT(\))83 b FN(!)g FP(y)1530 4470 y(x)21
b FN(\016)f FT(\()p FP(x)1755 4432 y FK(\000)p FL(1)1870
4470 y FN(\016)h FP(y)s FT(\))83 b FN(!)g FP(y)1668 4607
y FT(\()p FP(x)21 b FN(\016)f FP(y)s FT(\))1924 4570
y FK(\000)p FL(1)2102 4607 y FN(!)83 b FP(y)2324 4570
y FK(\000)p FL(1)2438 4607 y FN(\016)21 b FP(x)2556 4570
y FK(\000)p FL(1)2650 4607 y FP(:)378 4812 y FT(Note)37
b(that)e(the)h(orien)m(tation)f(of)g(the)h(asso)s(ciativ)m(e)g(la)m(w)f
(in)f(the)h(rewriting)e(rule)h(is)h(di\013eren)m(t)f(from)378
4924 y(that)d(in)e(the)i(de\014nition)d(of)i Fw(GAssoc)e
FT(giv)m(en)i(in)f(the)i(previous)e(section.)519 5037
y(These)e(rules)g(are)h(deriv)m(ed)e(man)m(ually)h(in)f(SPL)h(as)h(the)
f(theorems)h(giv)m(en)g(in)e(\014gure)h(24,)i(and)e(are)378
5150 y(used)j(in)f(the)h(group)g(theory)h(normaliser)d(\(or)j
(simpli\014er\))c(describ)s(ed)h(in)h(section)i(9.2.3.)519
5263 y(The)i(cancellation)g(theorems)h(are)f(deriv)m(ed)g(after)h(the)f
(normaliser)f(is)g(implemen)m(ted)g(and)h(in-)378 5376
y(corp)s(orated)42 b(in)e(the)i(theory)-8 b(.)76 b(W)-8
b(e)43 b(giv)m(e)f(their)f(pro)s(ofs)g(b)s(elo)m(w)f(to)j(illustrate)d
(the)i(e\013ect)h(of)f(this)378 5489 y(normaliser)26
b(on)j(the)f(length)g(of)g(the)h(pro)s(ofs.)39 b(The)28
b(term)g Fw(inv)f FT(is)h(an)g(abbreviation)f(for)h Fw(InvG)42
b Ft(\()p Fv(G;)14 b(p)p Ft(\))q FT(,)378 5602 y Fw(fol)36
b FT(is)g(the)h(iden)m(ti\014er)f(of)h(the)h(pro)m(v)m(er)f(for)g
(\014rst-order)g(logic)g(with)e(equalit)m(y)i(and)g Fw(groups)d
FT(is)j(the)p eop
%%Page: 196 206
196 205 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(196)p
378 766 3453 4 v 376 5252 4 4487 v 515 1001 a Fw(let)42
b("G:'a)g Fu(!)h Fw(bool")f(and)g("p:'a)g Fu(!)h Fw('a)g
Fu(!)h Fw('a";)515 1201 y(assume)d(GroupG:)g("Group)g(\(G,p\)";)689
1400 y(Closed:)g("GClosed)f(\(G,p\)")515 1500 y(and)i(Assoc:)85
b("GAssoc)41 b(\(G,p\)")g(by)i(<Group>GroupG;)515 1699
y(simplify)d(with)i(GLeftId)f(GRightId)f(GId)260 b(\(*)43
b(Simplifying)c(terms)j(with)g(the)86 b(*\))1125 1798
y(GLeftInv)40 b(GRightInv)g(GInv)129 b(\(*)43 b(definitions)c(of)k
(identity,)171 b(*\))1125 1898 y(GAssoc)41 b(GClosed;)476
b(\(*)43 b(inverse,)d(assoc.)i(and)g(closure)f(*\))2258
1998 y(\(*)i(will)f(be)h(done)f(automatically)82 b(*\))515
2197 y(let)42 b("x:'a")f("x1:'a")g("x2:'a")g("e:'a")g("f:'a";)515
2396 y(assume)g(Gx:)h("G)h(x",)f(Ge:)h("G)g(e")f(and)h(Gf:)f("G)h(f")
820 2496 y(Gx1:)f("G)h(x1")f(and)g(Gx2:)g("G)h(x2";)820
2695 y(GIde:)f("GId)f(\(G,p\))h(e")130 b(\(*)43 b(e)g(is)g(an)f
(identity)f(element)f(*\))820 2795 y(GIdf:)i("GId)f(\(G,p\))h(f")130
b(\(*)43 b(f)g(is)g(an)f(identity)f(element)f(*\))820
2994 y(GInvx1:)h("GInv)g(\(G,p\))h(e)h(x)g(x1")86 b(\(*)43
b(x1)g(is)f(an)h(inverse)e(of)i(x)g(*\))820 3094 y(GInvx2:)e("GInv)g
(\(G,p\))h(e)h(x)g(x2";)f(\(*)h(x2)g(is)f(an)h(inverse)e(of)i(x)g(*\))
515 3392 y(theorem)e(GIds_equal:)e("e)j(=)i(f")515 3492
y(proof)602 3592 y("e)f(=)g(p)g(e)g(f")g(by)g(GIdf)f(on)h(Ge)602
3691 y(.")g(=)g(f")g(by)g(GIde)e(on)i(Gf;)515 3791 y(end;)515
3990 y(theorem)e(GInvs_equal:)d("x1)43 b(=)g(x2")515
4090 y(proof)602 4189 y("x1)86 b(=)h(p)43 b(e)g(x1")f(by)h(GIde)f(on)h
(Gx1)689 4289 y(.")g(=)87 b(p)43 b(\(p)g(x2)f(x\))h(x1")g(by)f(GInvx2)
689 4389 y(.")h(=)87 b(p)43 b(x2)g(\(p)f(x)i(x1\)")e(by)g(Assoc)g(on)h
(\(Gx)f(and)g(Gx1)h(and)f(Gx2\))689 4488 y(.")h(=)87
b(p)43 b(x2)g(e")f(by)h(GInvx1)689 4588 y(.")g(=)87 b(x2")42
b(by)h(GIde)f(on)h(Gx2;)515 4688 y(end;)1194 5083 y FT(Figure)30
b(23:)42 b(Pro)s(ofs)30 b(of)g(the)h(Uniqueness)d(Results.)p
3829 5252 V 378 5256 3453 4 v eop
%%Page: 197 207
197 206 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(197)p
378 416 3453 4 v 376 2512 4 2096 v 471 652 a Fu(`)44
b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p Fv(G;)14 b(p)p
Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p Fv(x)p Fw(.)f Fv(G)h(x)g
Fu(\))f Ft(\()p Fv(p)h Ft(\()p Fw(IdG)f Ft(\()p Fv(G;)14
b(p)p Ft(\)\))44 b Fv(x)67 b Ft(=)f Fv(x)p Ft(\)\))471
751 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p Fv(x)p
Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fv(p)h(x)g Ft(\()p
Fw(IdG)f Ft(\()p Fv(G;)14 b(p)p Ft(\)\))67 b(=)f Fv(x)p
Ft(\)\))471 851 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e
Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p
Fv(x)p Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fv(p)h Ft(\()p
Fw(InvG)e Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(x)p Ft(\))h
Fv(x)67 b Ft(=)43 b Fw(IdG)f Ft(\()p Fv(G;)14 b(p)p Ft(\)\)\))471
951 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p Fv(x)p
Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fv(p)h(x)g Ft(\()p
Fw(InvG)e Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(x)p Ft(\))68
b(=)43 b Fw(IdG)f Ft(\()p Fv(G;)14 b(p)p Ft(\)\)\))471
1050 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p Fv(x)f(y)k(z)t
Fw(.)42 b Fv(G)i(x)g Fu(\))f Fv(G)h(y)j Fu(\))c Fv(G)h(z)j
Fu(\))733 1150 y Ft(\()p Fv(p)d Ft(\()p Fv(p)f(x)h(y)s
Ft(\))g Fv(z)70 b Ft(=)c Fv(p)43 b(x)h Ft(\()p Fv(p)g(y)i(z)t
Ft(\)\)\))471 1249 y Fu(`)e(8)p Fv(G)f(p)p Fw(.)g(Group)e
Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fu(8)p
Fv(x)p Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fw(InvG)g Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b(\()p Fw(InvG)e Ft(\()p Fv(G;)14
b(p)p Ft(\))44 b Fv(x)p Ft(\))67 b(=)g Fv(x)p Ft(\)\))471
1349 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b Fu(\))g Ft(\()p Fw(InvG)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b(\()p Fw(IdG)f Ft(\()p Fv(G;)14
b(p)p Ft(\)\))67 b(=)43 b Fw(IdG)g Ft(\()p Fv(G;)14 b(p)p
Ft(\)\))471 1449 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e
Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fu(\))733 1548 y Ft(\()p
Fu(8)p Fv(x)p Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fu(8)p
Fv(y)s Fw(.)g Fv(G)h(y)i Fu(\))d Ft(\()p Fv(p)h(x)g Ft(\()p
Fv(p)g Ft(\()p Fw(InvG)e Ft(\()p Fv(G;)14 b(p)p Ft(\))44
b Fv(x)p Ft(\))g Fv(y)s Ft(\))67 b(=)f Fv(y)s Ft(\)\)\))471
1648 y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b Fu(\))733 1748 y Ft(\()p Fu(8)p
Fv(x)p Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fu(8)p Fv(y)s
Fw(.)g Fv(G)h(y)i Fu(\))d Ft(\()p Fv(p)h Ft(\()p Fw(InvG)e
Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(x)p Ft(\))h(\()p
Fv(p)e(x)h(y)s Ft(\))67 b(=)f Fv(y)s Ft(\)\)\))471 1847
y Fu(`)44 b(8)p Fv(G)f(p)p Fw(.)g(Group)e Ft(\()p Fv(G;)14
b(p)p Ft(\))44 b Fu(\))733 1947 y Ft(\()p Fu(8)p Fv(x)p
Fw(.)f Fv(G)h(x)g Fu(\))f Ft(\()p Fu(8)p Fv(y)s Fw(.)g
Fv(G)h(y)i Fu(\))733 2046 y Ft(\()p Fw(InvG)c Ft(\()p
Fv(G;)14 b(p)p Ft(\))44 b(\()p Fv(p)g(x)g(y)s Ft(\))66
b(=)h Fv(p)43 b Ft(\()p Fw(InvG)f Ft(\()p Fv(G;)14 b(p)p
Ft(\))44 b Fv(y)s Ft(\))g(\()p Fw(InvG)e Ft(\()p Fv(G;)14
b(p)p Ft(\))44 b Fv(x)p Ft(\)\)\)\))1043 2342 y FT(Figure)30
b(24:)42 b(The)29 b(Rules)h(for)g(Normalising)e(Group)i(T)-8
b(erms.)p 3829 2512 V 378 2515 3453 4 v 378 2872 a(iden)m(ti\014er)29
b(for)h(the)g(group)g(theory)h(simpli\014er.)473 3084
y FM(theorem)46 b(Cancel_left)f(:)i("\(p)g(z)h(x)f(=)h(p)f(z)h(y\))f
FN(\))g FM(\(x)h(=)f(y\)")473 3197 y(proof)569 3310 y(assume)f
(zx_eq_zy:)f("p)i(z)h(x)f(=)h(p)f(z)h(y";)521 3536 y("x)f(=)h(p)f
(\(inv)g(z\))g(\(p)g(z)h(x\)")f(<groups>)e(by)i(fol)569
3649 y(."=)g(p)g(\(inv)g(z\))g(\(p)g(z)h(y\)")f(by)g(zx_eq_zy)569
3762 y(."=)g(y")g(<groups>)e(by)j(fol;)473 3875 y(qed;)473
4101 y(theorem)e(Cancel_right:)e("\(p)j(x)h(z)f(=)h(p)f(y)h(z\))f
FN(\))g FM(\(x)h(=)f(y\)")473 4214 y(proof)569 4326 y(assume)f
(xz_eq_yz:)f("p)i(x)h(z)f(=)h(p)f(y)h(z";)569 4552 y("x)f(=)g(p)h(\(p)f
(x)h(z\))f(\(inv)f(z\)")h(<groups>)f(by)h(fol)569 4665
y(.")g(=)g(p)h(\(p)f(y)h(z\))f(\(inv)f(z\)")h(by)g(xz_eq_yz)569
4778 y(.")g(=)g(y")h(<groups>)d(by)i(fol;)473 4891 y(qed;)378
5247 y FG(9.2.3)112 b(Preliminary)35 b(Simpli\014ers)g(and)j(Database)h
(Query)f(F)-9 b(unctions)378 5419 y FT(A)34 b(simpli\014er)d(for)i
(group)h(terms,)h Fw(groups)n FT(,)g(is)e(implemen)m(ted)f(\(in)i(SML)f
(as)i(a)f(HOL)g(deriv)m(ed)f(rule\))378 5532 y(whic)m(h)20
b(normalises)g(terms)h(b)m(y)g(rewriting)e(with)h(the)h(rules)f(giv)m
(en)h(in)f(\014gure)h(24.)39 b(The)20 b(main)g(di\016cult)m(y)378
5645 y(with)28 b(the)i(implemen)m(tation)e(of)i(the)g(required)e(term)i
(rewriting)d(system)j(lies)f(in)f(the)i(fact)g(that)h(the)p
eop
%%Page: 198 208
198 207 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(198)378
396 y(rewriting)41 b(rules)h(are)h(conditional.)78 b(Eac)m(h)43
b(rule)f(can)h(b)s(e)g(applied)e(to)j(some)f(term)g(only)g(if)f(the)378
509 y(appropriate)28 b(subterms)g(are)i(mem)m(b)s(ers)e(of)i(a)f
(group.)40 b(It)30 b(w)m(ould)e(b)s(e)g(cum)m(b)s(ersome)h(if)f(the)i
(required)378 622 y(conditions)k(ha)m(v)m(e)i(to)g(b)s(e)e(deriv)m(ed)h
(man)m(ually)e(and)i(supplied)d(as)j(parameters)h(to)f(the)h
(normaliser)378 735 y(whenev)m(er)g(they)f(are)i(needed.)56
b(F)-8 b(urthermore,)37 b(suc)m(h)f(conditions)e(are)i(simply)d
(considered)i(to)i(b)s(e)378 848 y(trivial)25 b(in)h(the)i
(mathematical)f(literature,)g(and)g(it)g(is)f(therefore)i(desirable)d
(that)j(they)f(are)h(deriv)m(ed)378 961 y(automatically)-8
b(.)56 b(The)36 b(term)f(rewriting)f(system)h(is)g(therefore)h
(implemen)m(ted)e(so)i(that)g(it)f(queries)378 1074 y(the)h(SPL)f(kno)m
(wledge)i(database)g(\(see)g(section)f(4.4.1\))i(in)d(order)h(to)g
(satisfy)g(a)g(rule's)f(conditions)378 1187 y(b)s(efore)24
b(it)g(is)g(applied.)36 b(A)25 b(rule)e(is)g(not)i(applied)d(if)i(one)h
(of)f(its)g(conditions)f(cannot)i(b)s(e)f(automatically)378
1300 y(deriv)m(ed)29 b(b)m(y)i(the)f(query)g(functions.)519
1413 y(A)h(n)m(um)m(b)s(er)f(of)h(kno)m(wledge)f(categories)j(are)e
(included)d(in)i(the)h(database)g(to)h(store)f(the)g(kno)m(wl-)378
1526 y(edge)25 b(required)d(b)m(y)i(the)g(group)g(theory)g(normaliser.)
37 b(The)23 b(appropriate)h(query)f(functions)g(are)h(then)378
1638 y(implemen)m(ted.)39 b(The)30 b(categories)i(that)f(are)g
(included)c(in)i(the)i(database)g(are)g(as)g(follo)m(ws:)514
1813 y FN(\017)46 b Fw(is_group)m FT(:)36 b(Storing)22
b(facts)h(of)f(the)g(form)g Fw(Group)41 b Fv(Gp)p FT(.)d(Queries)21
b(to)i(this)e(category)j(are)f(satis\014ed)605 1926 y(if)30
b(the)g(giv)m(en)h(pair)e(is)g(a)i(group.)514 2108 y
FN(\017)46 b Fw(is_closed)m FT(:)54 b(Storing)37 b(facts)h(of)g(the)f
(form)g Fw(GClosed)k Fv(Gp)p FT(.)62 b(Queries)36 b(to)i(this)e
(category)k(also)605 2221 y(consult)30 b(the)g Fw(is_group)d
FT(category)33 b(to)e(deriv)m(e)f(the)g(required)f(fact.)514
2403 y FN(\017)46 b Fw(in_set)n FT(:)40 b(Storing)27
b(applications)g Ft(\()p Fv(G)44 b(x)p Ft(\))q FT(.)c(Queries)27
b(of)i(this)f(form)g(are)h(satis\014ed)e(if)h(one)h(of)g(the)605
2516 y(follo)m(wing)g(holds:)689 2699 y(1.)46 b(the)31
b(fact)g Ft(\()p Fv(G)44 b(x)p Ft(\))31 b FT(is)f(stored)g(in)f(the)i
Fw(in_set)d FT(category)-8 b(.)689 2840 y(2.)46 b(the)27
b(term)f FP(x)h FT(is)e(of)i(the)f(form)g Fw(IdG)43 b
Ft(\()p Fv(G;)14 b(p)p Ft(\))26 b FT(and)g Ft(\()p Fv(G;)14
b(p)p Ft(\))27 b FT(is)f(a)g(group.)39 b(The)26 b(fact)h(that)g
Ft(\()p Fv(G;)14 b(p)p Ft(\))805 2952 y FT(is)25 b(a)h(group)f(\(that)i
(is)d Fw(Group)42 b Ft(\()p Fv(G;)14 b(p)p Ft(\))q FT(\))25
b(is)g(deriv)m(ed)g(b)m(y)h(querying)e(the)i Fw(is_group)c
FT(database)805 3065 y(category)-8 b(.)689 3206 y(3.)46
b(the)29 b(term)g FP(x)g FT(is)f(of)h(the)g(form)g Fw(InvG)42
b Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(y)r FT(,)30 b(the)f(pair)f
Ft(\()p Fv(G;)14 b(p)p Ft(\))29 b FT(is)f(a)h(group,)g(and)g
FP(y)i FT(is)d(in)805 3319 y FP(G)p FT(.)689 3460 y(4.)46
b(the)32 b(term)f FP(x)h FT(is)e(of)i(the)g(form)f Fv(p)43
b(y)j(z)s FT(,)32 b(the)g(set)g FP(G)f FT(is)f(closed)i(under)e
FP(p)h FT(and)g(b)s(oth)f FP(y)k FT(and)805 3573 y FP(z)h
FT(are)c(in)e FP(G)p FT(.)378 3755 y(Note)45 b(that)f(in)e(general,)48
b(query)43 b(functions)f(dep)s(end)g(on)h(eac)m(h)i(other.)80
b(This)42 b(in)m(terdep)s(endence)378 3868 y(ev)m(olv)m(es)e(and)d(b)s
(ecomes)i(more)g(complex)f(as)h(new)f(results)f(are)i(used)e(to)i
(implemen)m(t)f(new)f(query)378 3981 y(functions)29 b(and)h(up)s(date)f
(the)i(existing)e(ones.)519 4094 y(As)g(men)m(tioned)g(ab)s(o)m(v)m(e,)
h(the)f Fw(groups)e FT(normaliser)g(rep)s(eatedly)h(applies)f(the)i
(rules)f(in)g(\014gure)g(24)378 4207 y(whose)36 b(conditions)f(can)h(b)
s(e)g(automatically)g(deduced)g(b)m(y)g(querying)f(the)h(kno)m(wledge)h
(database.)378 4320 y(F)-8 b(or)31 b(example,)f(in)f(order)h(to)h
(apply)e(the)i(rule)426 4494 y FN(`)47 b(8)p FP(G)g(p)p
FM(.)g(Group)f FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\))g
FT(\()p FN(8)p FP(x)p FM(.)f FP(G)g(x)h FN(\))g FT(\()p
FP(p)g FT(\()p FM(IdG)f FT(\()p FP(G;)15 b(p)p FT(\)\))48
b FP(x)73 b FT(=)g FP(x)p FT(\)\))378 4669 y(on,)33 b(sa)m(y)-8
b(,)34 b(the)f(term)f Fv(p)44 b Ft(\()p Fw(IdG)e Ft(\()p
Fv(G;)14 b(p)p Ft(\)\))45 b Fv(a)p FT(,)33 b(the)f Fw(is_group)d
FT(category)35 b(is)c(\014rst)h(queried)f(b)m(y)h(the)g
Fw(groups)378 4782 y FT(normaliser)c(to)k(deduce)e(the)g(fact)473
4956 y(\000)530 4970 y FL(1)617 4956 y FN(`)48 b FM(Group)e
FT(\()p FP(G;)15 b(p)p FT(\))378 5130 y(for)30 b(some)h(assumptions)e
Ft(\000)1314 5142 y Fs(1)1351 5130 y FT(.)40 b(The)30
b Fw(in_set)e FT(category)k(is)e(then)g(queried)f(to)i(deduce)473
5305 y(\000)530 5319 y FL(2)617 5305 y FN(`)48 b FP(G)f(a)378
5479 y FT(where)34 b Ft(\000)697 5491 y Fs(2)768 5479
y FT(is)g(the)g(list)f(of)i(assumptions)d(required)h(to)i(deduce)f
(this)f(theorem.)53 b(Giv)m(en)34 b(the)h(ab)s(o)m(v)m(e)378
5592 y(t)m(w)m(o)29 b(theorems,)g(one)f(can)g(then)g(apply)f(the)h
(rewrite)f(rule)f(to)j(simplify)24 b Fv(p)44 b Ft(\()p
Fw(IdG)e Ft(\()p Fv(G;)14 b(p)p Ft(\)\))45 b Fv(a)27
b FT(in)m(to)h Fv(a)g FT(b)m(y)378 5705 y(deriving)g(and)i(rewriting)e
(with)h(the)i(HOL)f(theorem:)p eop
%%Page: 199 209
199 208 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(199)473
396 y(\000)530 410 y FL(1)617 396 y FN([)48 b FT(\000)783
410 y FL(2)870 396 y FN(`)f FP(p)g FT(\()p FM(IdG)h FT(\()p
FP(G;)15 b(p)p FT(\)\))48 b FP(a)g FM(=)f FP(a)378 573
y FT(The)e(other)g(rules)f(in)g(\014gure)g(24)i(are)g(treated)g
(similarly)-8 b(.)82 b(A)46 b(rule)e(is)g(applied)f(only)h(if)g(all)g
(its)378 686 y(conditions)22 b(can)i(b)s(e)f(deduced)f(automatically)i
(b)m(y)f(the)h(relev)-5 b(an)m(t)24 b(queries)e(to)j(the)e(trivial)f
(kno)m(wledge)378 799 y(database.)519 912 y(When)32 b(using)f(the)h
Fw(groups)d FT(normaliser)i(to)h(simplify)d(the)j(implemen)m(tation)f
(of)h(formal)g(pro)s(ofs)378 1025 y(one)38 b(needs)g(to)h(supply)d
(enough)i(kno)m(wledge)g(in)f(the)h(database)h(so)g(that)f(the)h
(conditions)d(of)j(the)378 1138 y(rewriting)29 b(rules)h(can)i(b)s(e)f
(deriv)m(ed)f(automatically)-8 b(.)45 b(This)29 b(is)i(done)g(through)g
(the)g Fw(consider)d FT(state-)378 1251 y(men)m(t)c(as)f(illustrated)e
(b)s(elo)m(w.)38 b(The)23 b(terms)g Fw(id)g FT(and)f
Fw(inv)g FT(abbreviate)i(the)f(iden)m(tit)m(y)g(elemen)m(t)h(and)f(the)
378 1364 y(in)m(v)m(erse)35 b(function)e(resp)s(ectiv)m(ely)-8
b(,)36 b(and)e(it)h(is)f(assumed)g(that)h(they)g(are)g(declared)g(as)g
(default)f(sim-)378 1476 y(pli\014ers)29 b(in)h(the)h(section)h(con)m
(taining)f(the)g(follo)m(wing)f(pro)s(of)h(segmen)m(t)h(so)g(that)g
(they)f(are)h(unfolded)378 1589 y(automatically)e(during)e(pro)s(of)i
(c)m(hec)m(king.)473 1902 y FM(assume)46 b(GroupG:)g("Group)g(\(G,p\)")
807 2015 y(Gx:)h("G)h(x";)473 2240 y(\(*)g(the)e(facts)h("G)g(x")g(and)
g("Group)f(\(G,p\)")g(are)h(stored)617 2353 y(as)g(trivial)f(facts)g
(in)h(the)g(appropriate)e(categories)g(*\))473 2466 y(consider)h
(in_set)g(Gx)903 2579 y(is_group)f(GroupG;)473 2805 y(theorem)h
(Idem_id:)g("\(p)h(x)g(x)h(=)f(x\))g FN(\))h FM(\(x)f(=)g(id\)")473
2918 y(proof)569 3031 y(assume)f(pxx_eq_x:)f("p)i(x)h(x)f(=)h(x";)569
3257 y("x)95 b(=)47 b(p)h(\(inv)e(x\))h(\(p)h(x)f(x\)")g(<groups>)f(by)
h(fol)664 3370 y(."=)g(p)h(\(inv)e(x\))h(x")h(by)f(pxx_eq_x)664
3482 y(."=)g(id")g(<groups>)f(by)h(fol;)473 3595 y(end;)519
3908 y FT(It)36 b(is)f(not)h(hard)f(to)i(see)f(that)h(the)f(same)g
(query)f(can)h(b)s(e)g(applied)d(sev)m(eral)k(times)e(during)f(the)378
4021 y(normalisation)29 b(pro)s(cess.)41 b(F)-8 b(or)32
b(instance,)e(the)h(condition)f Fw(Group)41 b Ft(\()p
Fv(G;)14 b(p)p Ft(\))31 b FT(is)f(found)f(in)h(all)f(the)i(rules)378
4134 y(in)k(\014gure)h(24)h(and)f(is)f(therefore)i(queried)e(at)i(eac)m
(h)h(application)d(of)h(the)h(rules.)57 b(This)35 b(led)h(to)h(the)378
4246 y(decision)29 b(to)i(cac)m(he)h(the)f(output)f(of)g(query)g
(functions,)f(as)i(men)m(tioned)f(in)f(section)h(4.4.1.)378
4488 y FG(9.2.4)112 b(Subgroups)378 4659 y FT(A)34 b(subgroup)e(is)g(a)
i(subset)f(of)h(a)g(group)f(whic)m(h)f(is)h(also)h(a)g(group)f(under)f
(the)h(same)h(pro)s(duct.)50 b(W)-8 b(e)378 4772 y(de\014ne)30
b(subgroups)e(b)m(y)426 4949 y FN(`)482 4964 y FE(def)686
4949 y FM(SubGroup)46 b FP(p)h(H)55 b(G)47 b FN(\021)h
FT(\()p FM(Subset)e FP(H)55 b(G)p FT(\))47 b FN(^)h FT(\()p
FM(Group)e FT(\()p FP(H)7 b(;)15 b(p)p FT(\)\))378 5126
y(where)30 b(the)g(predicate)h Fw(Subset)41 b Fv(H)50
b(G)31 b FT(is)e(de\014ned)g(as)i(follo)m(ws)426 5303
y FN(`)482 5318 y FE(def)686 5303 y FM(Subset)46 b FP(X)55
b(Y)68 b FN(\021)48 b FT(\()p FN(8)p FP(x)p FM(.)f FP(X)55
b(x)48 b FN(\))f FP(Y)68 b(x)p FT(\))378 5479 y(Note,)30
b(ho)m(w)m(ev)m(er)f(that)f(in)f(the)g(ab)s(o)m(v)m(e)i(de\014nition)d
(of)i Fw(SubGroup)c FT(w)m(e)k(do)g(not)g(imp)s(ose)e(the)i
(restriction)378 5592 y(that)j(the)g(set)h FP(G)e FT(has)h(to)g(b)s(e)g
(a)g(group)f(under)f(the)i(pro)s(duct)f FP(p)p FT(,)h(and)f(therefore)i
(terms)e(of)h(the)g(form)378 5705 y Fw(SubGroup)40 b
Fv(p)k(H)50 b(G)31 b FT(should)d(b)s(e)i(used)f(together)j(with)d(some)
i(assumption)e Fw(Group)41 b Ft(\()p Fv(G;)14 b(p)p Ft(\))q
FT(.)p eop
%%Page: 200 210
200 209 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(200)519
396 y(The)22 b(in)m(tro)s(duction)e(of)i(this)f(de\014nition)e(allo)m
(ws)j(one)g(to)h(extend)f(the)g(query)f(functions)g(describ)s(ed)378
509 y(in)29 b(the)i(previous)d(section.)41 b(First)30
b(of)h(all)e(w)m(e)i(include)d(the)i(follo)m(wing)f(categories)j(in)d
(the)i(database)514 697 y FN(\017)46 b Fw(is_subset)m
FT(:)38 b(Storing)24 b(facts)j(of)e(the)h(form)f Fw(Subset)41
b Fv(X)50 b(Y)19 b FT(.)39 b(A)25 b(query)g(of)h(this)e(form)h(is)g
(satis\014ed)605 810 y(if)33 b(the)h(required)e(fact)j(is)e(stored)g
(in)g(the)h Fw(is_subset)c FT(category)-8 b(,)37 b(or)d(if)f(there)h
(is)f(some)h(set)g FP(Z)605 923 y FT(suc)m(h)26 b(that)g
Fw(Subset)41 b Fv(X)50 b(Z)32 b FT(and)25 b Fw(Subset)41
b Fv(Z)49 b(Y)c FT(hold,)25 b(or)h(there)g(is)f(some)h(pro)s(duct)f
FP(p)h FT(for)f(whic)m(h)605 1036 y Fw(SubGroup)41 b
Fv(p)i(X)50 b(Y)18 b FT(.)514 1223 y FN(\017)46 b Fw(is_subgroup)l
FT(:)39 b(Storing)26 b Fw(SubGroup)40 b Fv(p)k(H)50 b(G)p
FT(.)40 b(Suc)m(h)27 b(query)f(is)g(satis\014ed)h(if)f(the)i(required)d
(fact)605 1336 y(is)33 b(stored)h(in)e(this)h(category)-8
b(,)37 b(or)d(there)g(is)f(some)h(set)g FP(Z)40 b FT(suc)m(h)34
b(that)g Fw(SubGroup)40 b Fv(p)k(H)50 b(Z)39 b FT(and)605
1449 y Fw(SubGroup)i Fv(p)i(Z)49 b(G)31 b FT(hold.)378
1637 y(The)f(follo)m(wing)f(query)g(functions)g(can)i(no)m(w)f(b)s(e)g
(up)s(dated:)514 1824 y FN(\017)46 b Fw(is_group)m FT(:)36
b Fw(Group)41 b Ft(\()p Fv(G;)14 b(p)p Ft(\))21 b FT(is)f(satis\014ed)g
(if)f(there)i(is)f(some)h(set)g FP(X)28 b FT(suc)m(h)21
b(that)g Fw(SubGroup)40 b Fv(p)k(G)f(X)7 b FT(.)514 2012
y FN(\017)46 b Fw(in_set)n FT(:)40 b Fv(G)k(x)31 b FT(is)e(satis\014ed)
h(if)f(there)i(is)e(some)i(subset)f FP(H)37 b FT(of)31
b FP(G)e FT(con)m(taining)h FP(x)p FT(.)378 2200 y(The)23
b(initial)e(implemen)m(tation)i(of)h(these)g(query)g(functions)e(supp)s
(orts)g(the)i(ab)s(o)m(v)m(e)h(deriv)-5 b(ations)23 b(since)378
2313 y(these)g(only)f(require)g(the)h(de\014nition)e(of)i
Fw(Subset)d FT(and)i Fw(SubGroup)e FT(and)i(a)h(n)m(um)m(b)s(er)f(of)h
(straigh)m(tforw)m(ard)378 2426 y(results)29 b(\(transitivit)m(y)g(of)i
Fw(Subset)d FT(and)i Fw(SubGroup)l FT(,)h(etc.)16 b(\))41
b(whic)m(h)29 b(are)i(pro)m(v)m(ed)g(in)e(SPL.)519 2538
y(A)34 b(result)f(whic)m(h)f(is)h(tak)m(en)i(for)e(gran)m(ted)i(in)d
(\(Herstein)i(1975\))i(is)c(the)i(fact)h(that)f(the)g(iden)m(tit)m(y)
378 2651 y(elemen)m(t)j FP(e)762 2665 y FO(H)865 2651
y FT(of)g(a)f(subgroup)f FP(H)43 b FT(of)36 b FP(G)g
FT(is)f(the)h(same)h(as)f(the)h(iden)m(tit)m(y)e(elemen)m(t)i
FP(e)3291 2665 y FO(G)3387 2651 y FT(of)f FP(G)p FT(.)58
b(This)378 2764 y(follo)m(ws)31 b(from)h(the)h(fact)g(that)g
FP(e)1476 2778 y FO(H)1565 2764 y FN(\016)22 b FP(e)1674
2778 y FO(H)1770 2764 y FT(=)28 b FP(e)1911 2778 y FO(H)2010
2764 y FT(and)k(from)g(the)g(theorem)h Fw(Idem_id)c FT(whose)j(pro)s
(of)g(is)378 2877 y(giv)m(en)e(in)f(section)i(9.2.3)h(whic)m(h)d
(states)j(that)f FN(8)p FP(x)24 b FN(2)h FP(G:)30 b FT(\()p
FP(x)21 b FN(\016)f FP(x)25 b FT(=)g FP(x)p FT(\))h FN(\))f
FP(x)g FT(=)g FP(e)3088 2891 y FO(G)3148 2877 y FT(.)40
b(The)30 b(uniqueness)378 2990 y(of)e(the)g(in)m(v)m(erse)g(elemen)m(t)
g(is)f(used)g(to)h(deriv)m(e)g(the)g(fact)g(that)h(the)f(in)m(v)m(erse)
f(in)g FP(H)35 b FT(is)26 b(the)i(same)h(as)f(the)378
3103 y(in)m(v)m(erse)34 b(in)e FP(G)p FT(.)50 b(Since)33
b(these)h(results)f(are)h(tak)m(en)h(for)e(gran)m(ted)h(in)f(the)h
(literature,)g(a)g(simpli\014er)c(is)378 3216 y(implemen)m(ted)f(whic)m
(h)g(rewrites)g(terms)i(using)e(the)h(follo)m(wing)f(rules)473
3404 y FN(`)48 b(8)p FP(G)e(p)p FM(.)i(Group)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FN(\))951 3516 y FT(\()p FN(8)p
FP(H)7 b FM(.)47 b(SubGroup)e FP(p)j(H)55 b(G)47 b FN(\))g
FT(\()p FM(IdG)h FT(\()p FP(H)7 b(;)15 b(p)p FT(\))74
b(=)47 b FM(IdG)g FT(\()p FP(G;)15 b(p)p FT(\)\)\))473
3742 y FN(`)48 b(8)p FP(G)e(p)p FM(.)i(Group)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FN(\))951 3855 y FT(\()p FN(8)p
FP(H)7 b FM(.)47 b(SubGroup)e FP(p)j(H)55 b(G)47 b FN(\))1094
3968 y FT(\()p FN(8)p FP(x)p FM(.)g FP(H)55 b(x)47 b
FN(\))h FT(\()p FM(InvG)f FT(\()p FP(H)7 b(;)15 b(p)p
FT(\))49 b FP(x)73 b FT(=)47 b FM(InvG)g FT(\()p FP(G;)15
b(p)p FT(\))48 b FP(x)p FT(\)\)\))378 4156 y(substituting)30
b(the)j(iden)m(tit)m(y)f(and)f(in)m(v)m(erses)i(in)e
FP(H)39 b FT(to)33 b(those)g(in)e FP(G)p FT(.)47 b(The)32
b Fw(in_set)d FT(query)j(functions)378 4269 y(are)f(up)s(dated)e(so)h
(that)h(a)g(query)f(of)g(the)h(form)f Fv(H)50 b(x)31
b FT(is)e(satis\014ed)h(if)f(the)i(term)f FP(x)g FT(is)g(of)g(the)h
(form)514 4456 y FN(\017)46 b Fw(IdG)d Ft(\()p Fv(G;)14
b(p)p Ft(\))30 b FT(where)g FP(G)g FT(is)f(a)i(group)f(and)g
FP(H)37 b FT(is)29 b(a)i(subgroup)e(of)h FP(G)p FT(,)h(or)514
4644 y FN(\017)46 b FT(of)28 b(the)f(form)g Fw(InvG)42
b Ft(\()p Fv(G;)14 b(p)p Ft(\))44 b Fv(x)27 b FT(where)g
FP(x)g FT(is)f(in)g FP(H)7 b FT(,)28 b FP(G)f FT(is)f(a)h(group)g(and)g
FP(H)34 b FT(is)26 b(a)h(subgroup)f(of)h FP(G)p FT(.)519
4832 y(The)42 b(implemen)m(tation)e(of)i(the)h(simpli\014er)38
b(and)j(the)i(ab)s(o)m(v)m(e)g(deriv)-5 b(ations)40 b(up)s(dating)g
Fw(in_set)378 4944 y FT(queries)f(pro)m(v)m(ed)i(to)g(b)s(e)e(useful)g
(in)g(our)h(case)h(study)-8 b(.)70 b(A)m(t)41 b(this)e(stage,)45
b(it)39 b(is)h(b)s(ecoming)f(eviden)m(t)378 5057 y(that)f(the)g(dev)m
(elopmen)m(t)g(of)g(this)e(theory)i(in)m(v)m(olv)m(es)g(b)s(oth)f(the)h
(deriv)-5 b(ation)36 b(of)i(theorems)f(in)g(SPL)378 5170
y(and)42 b(the)g(implemen)m(tation)f(of)i(HOL)f(pro)s(of)g(pro)s
(cedures)f(in)g(SML.)h(Queries)f(to)i(the)g(database)378
5283 y(category)f Fw(in_set)c FT(are)j(made)g(v)m(ery)f(often)h(during)
e(the)h(implemen)m(tation)f(suggesting)i(that)g(the)378
5396 y(abilit)m(y)25 b(to)i(automate)h(set)f(con)m(tainmen)m(t)g(is)e
(v)m(ery)i(useful)d(in)h(the)i(mec)m(hanisation)f(of)g(group)g(theory)
-8 b(.)378 5509 y(A)35 b(simpli\014er)30 b Fw(inset)j
FT(is)g(also)i(implemen)m(ted)e(whic)m(h)g(substitutes)g(an)i
(application)e Fv(X)50 b(x)34 b FT(with)g(the)378 5622
y(truth)c(v)-5 b(alue)29 b Fw(T)h FT(if)g FP(x)g FT(can)h(b)s(e)e(sho)m
(wn)h(to)h(b)s(e)f(a)h(mem)m(b)s(er)e(of)i FP(X)38 b
FT(b)m(y)30 b(querying)f Fw(in_set)m FT(.)p eop
%%Page: 201 211
201 210 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(201)378
396 y FH(9.3)135 b(Congruences,)46 b(Cosets)g(and)e(Subgroup)g(Pro)t
(ducts)378 599 y FT(Giv)m(en)31 b(a)h(subgroup)e FP(H)39
b FT(of)32 b(a)g(group)f FP(G)g FT(w)m(e)h(sa)m(y)g(that)g
FP(a)c FN(\021)f FP(b)j FT(mo)s(d)f FP(H)39 b FT(for)31
b FP(a;)15 b(b)28 b FN(2)f FP(G)p FT(,)32 b(if)e FP(ab)3510
566 y FK(\000)p FL(1)3632 599 y FN(2)d FP(H)7 b FT(.)378
712 y(The)25 b(`congruen)m(t)i(mo)s(d')e(relation)g(is)g(an)g(equiv)-5
b(alence)26 b(relation)f(and)g(therefore)h(partitions)e(a)i(group)378
825 y(in)m(to)20 b(distinct)f(equiv)-5 b(alence)20 b(classes.)38
b(Eac)m(h)21 b(equiv)-5 b(alence)20 b(class)g(is)f(equal)h(to)i(some)e
(set)h FN(f)p FP(ha)27 b FN(j)e FP(h)h FN(2)f FP(H)7
b FN(g)378 938 y FT(where)29 b FP(a)h FT(is)f(some)h(represen)m(tativ)m
(e)g(mem)m(b)s(er)f(of)h(the)g(class)f(\(as)i FP(ea)25
b FT(=)g FP(a)30 b FT(is)f(in)f(the)i(class\).)40 b(This)28
b(set)378 1051 y(is)35 b(denoted)h(b)m(y)g FP(H)7 b(a)35
b FT(and)h(is)f(called)g(a)h(righ)m(t)f(coset)j(of)e
FP(H)42 b FT(in)35 b FP(G)p FT(.)57 b(Similarly)-8 b(,)34
b(a)i(left)g(coset)h FP(aH)43 b FT(of)378 1164 y FP(H)e
FT(in)32 b FP(G)h FT(is)g(de\014ned)f(b)m(y)i FP(aH)k
FT(=)31 b FN(f)p FP(ah)g FN(j)g FP(h)h FN(2)e FP(H)7
b FN(g)p FT(.)51 b(It)34 b(can)g(b)s(e)f(sho)m(wn)h(that)g(there)g(is)f
(a)h(one-to-one)378 1277 y(corresp)s(ondence)j(b)s(et)m(w)m(een)h(an)m
(y)g(t)m(w)m(o)h(righ)m(t)e(cosets)i(in)d FP(G)p FT(,)j(and)e
(therefore)h(if)e FP(G)h FT(is)g(\014nite)g(it)g(can)378
1390 y(b)s(e)30 b(partitioned)e(in)m(to)i(a)h(\014nite)e(n)m(um)m(b)s
(er)g(of)h(righ)m(t)g(cosets)i(of)e(the)g(same)h(size.)40
b(Hence,)32 b(the)e(n)m(um)m(b)s(er)378 1503 y(of)40
b(elemen)m(ts)f(in)f(some)i(righ)m(t)f(coset)i(m)m(ust)e(divide)f(the)h
(n)m(um)m(b)s(er)f(of)i(elemen)m(ts)g(in)e FP(G)p FT(,)j(whic)m(h)d(w)m
(e)378 1616 y(denote)c(b)m(y)f FP(o)p FT(\()p FP(G)p
FT(\).)50 b(Since)33 b FP(H)7 b(e)31 b FT(=)f FP(H)40
b FT(is)33 b(a)g(righ)m(t)g(coset)i(in)d FP(G)p FT(,)i
FP(o)p FT(\()p FP(H)7 b FT(\))35 b(m)m(ust)e(b)s(e)g(a)h(divisor)d(of)i
FP(o)p FT(\()p FP(G)p FT(\).)378 1728 y(This)f(result)g(is)g(due)h(to)h
(Lagrange)h(and)d(is)h(usually)e(referred)i(to)h(as)f(Lagrange's)i
(Theorem.)49 b(The)378 1841 y(reasoning)22 b(deriving)f(it)i(is)f
(implemen)m(ted)f(as)i(SPL)f(pro)s(ofs.)38 b(All)21 b(the)j(results)d
(leading)h(to)i(Lagrange's)378 1954 y(theorem)i(are)f(pro)m(v)m(ed)h
(in)d(SPL)i(in)f(m)m(uc)m(h)h(the)g(same)h(w)m(a)m(y)g(as)f(they)h(are)
f(pro)m(v)m(ed)h(in)e(\(Herstein)h(1975\).)378 2067 y(Ho)m(w)m(ev)m
(er,)34 b(the)e(SPL)f(pro)s(ofs)g(of)h(Lagrange's)h(theorem)f(itself)e
(attempted)j(b)m(y)f(the)g(author)f(turned)378 2180 y(out)40
b(to)g(b)s(e)e(m)m(uc)m(h)i(more)f(detailed)g(and)g(tedious)f(than)h
(the)h(one)g(giv)m(en)f(in)f(the)i(literature.)67 b(W)-8
b(e)378 2293 y(attribute)30 b(this)f(to)i(a)g(lac)m(k)g(of)g(pro)s(of)e
(pro)s(cedures)g(and)h(results)f(concerning)h(\014nite)f(sets.)519
2406 y(The)h(HOL)g(de\014nition)e(of)j(the)f(congruence)h(mo)s(d)f
(relation)f(is)h(giv)m(en)g(b)m(y)473 2568 y FN(`)529
2583 y FE(def)686 2568 y FM(CongruentMod)45 b FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(H)54 b(a)48 b(b)g FN(\021)f
FP(H)55 b FT(\()p FP(p)48 b(a)g FT(\()p FM(InvG)f FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(b)p FT(\)\))378 2730 y(and)39
b(it)g(is)g(sho)m(wn)f(in)h(SPL)f(that)i(this)f(relation)g(is)f
(re\015exiv)m(e,)k(symmetric)d(and)g(transitiv)m(e,)i(and)378
2843 y(hence)30 b(an)h(equiv)-5 b(alence)30 b(relation:)473
3004 y FN(`)48 b(8)p FP(G)e(H)55 b(p)p FM(.)47 b(Group)g
FT(\()p FP(G;)15 b(p)p FT(\))48 b FN(\))998 3117 y FM(SubGroup)e
FP(p)h(H)55 b(G)47 b FN(\))998 3230 y FM(GEquivalence)e
FP(G)i FT(\()p FM(CongruentMod)e FT(\()p FP(G;)15 b(p)p
FT(\))48 b FP(H)7 b FT(\))378 3392 y(A)33 b(sen)m(tence)g(of)g(the)g
(form)f Fw(GEquivalence)39 b Fv(X)49 b(R)34 b FT(holds)d(if)g
FP(R)i FT(is)f(an)g(equiv)-5 b(alence)32 b(relation)g(on)h(the)378
3505 y(elemen)m(ts)e(of)f(the)h(set)g(c)m(haracterised)g(b)m(y)f
FP(X)7 b FT(.)473 3667 y FN(`)529 3682 y FE(def)686 3667
y FM(GEquivalence)45 b FP(X)55 b(R)48 b FN(\021)951 3780
y FM(GReflexive)d FP(X)55 b(R)48 b FN(^)g FM(GSymmetric)d
FP(X)55 b(R)48 b FN(^)g FM(GTransitive)c FP(X)55 b(R)473
4006 y FN(`)529 4021 y FE(def)686 4006 y FM(GReflexive)45
b FP(X)55 b(R)49 b FN(\021)e FT(\()p FN(8)p FP(a)p FM(.)h
FP(X)55 b(a)48 b FN(\))f FP(R)i(a)e(a)p FT(\))473 4232
y FN(`)529 4247 y FE(def)686 4232 y FM(GSymmetric)e FP(X)55
b(R)49 b FN(\021)903 4344 y FT(\()p FN(8)p FP(a)f(b)p
FM(.)f FP(X)55 b(a)48 b FN(\))g FP(X)55 b(b)48 b FN(\))f
FP(R)i(a)e(b)h FN(\))g FP(R)g(b)g(a)p FT(\))473 4570
y FN(`)529 4585 y FE(def)686 4570 y FM(GTransitive)d
FP(X)55 b(R)49 b FN(\021)855 4683 y FT(\()p FN(8)p FP(a)f(b)p
FM(.)f FP(X)56 b(a)47 b FN(\))h FP(X)55 b(b)48 b FN(\))f
FP(R)i(a)f(b)f FN(\))855 4796 y FT(\()p FN(8)p FP(c)p
FM(.)h FP(X)55 b(c)48 b FN(\))g FP(R)g(b)g(c)g FN(\))g
FP(R)g(a)g(c)p FT(\)\))519 4958 y(In)25 b(the)i(literature)e(righ)m(t)g
(and)h(left)g(cosets)h(of)f(some)h(subgroup)d FP(H)33
b FT(of)26 b FP(G)g FT(are)g(denoted)g(b)m(y)g(terms)378
5071 y(of)g(the)g(form)f FP(H)7 b(a)26 b FT(and)f FP(aH)33
b FT(resp)s(ectiv)m(ely)-8 b(,)26 b(for)g(some)g(elemen)m(t)g
FP(a)g FN(2)f FP(G)p FT(.)38 b(Juxtap)s(osition)24 b(is)h(also)h(used)
378 5184 y(in)j(the)i(notation)f(for)g(the)h(pro)s(duct)e(of)i(t)m(w)m
(o)g(subgroups)e FP(H)37 b FT(and)30 b FP(X)38 b FT(de\014ned)29
b(as)h(follo)m(ws:)1328 5388 y FP(H)7 b(X)32 b FT(=)25
b FN(f)p FP(a)h FN(2)f FP(G)g FN(j)g FP(a)g FT(=)g FP(hx;)15
b(h)27 b FN(2)e FP(H)r(;)15 b(x)25 b FN(2)g FP(X)7 b
FN(g)p FP(:)519 5592 y FT(Although)32 b(cosets)h(and)f(pro)s(ducts)f
(are)i(de\014ned)e(on)h(subgroups,)f(the)i(notation)f(of)h(juxtap)s
(osi-)378 5705 y(tioning)27 b(subgroups)g(and)h(group)g(elemen)m(ts)h
(is)f(also)h(used)f(for)g(arbitrary)g(subsets)g(of)g(a)i(group.)39
b(F)-8 b(or)p eop
%%Page: 202 212
202 211 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(202)378
396 y(example,)27 b(although)g(it)f(is)g(not)h(men)m(tioned)f
(explicitly)f(in)g(\(Herstein)i(1975\),)j(the)d(term)g
FP(aS)32 b FT(is)26 b(used)378 509 y(to)33 b(denote)h(the)f(set)g
FN(f)p FP(as)c FN(j)g FP(s)g FN(2)g FP(S)5 b FN(g)33
b FT(where)f FP(S)38 b FT(is)32 b(an)g(arbitrary)g FI(subset)41
b FT(of)33 b(some)g(group)f FP(G)p FT(,)h(rather)378
622 y(than)g(a)h(subgroup.)48 b(This)32 b(is)g(eviden)m(t)i(when)e
(terms)h(lik)m(e)g FP(a)p FT(\()p FP(H)7 b(b)p FT(\))34
b(are)g(used)f(where)g FP(H)7 b(b)33 b FT(is)f(a)i(righ)m(t)378
735 y(coset)e(whic)m(h,)d(although)h(it)g(is)f(a)i(subset)f(of)g
FP(G)p FT(,)g(it)g(is)g(not)g(a)h(subgroup.)519 848 y(The)25
b(HOL)g(de\014nitions)e(for)j(righ)m(t)f(cosets,)j(left)d(cosets)i(and)
e(pro)s(ducts)f(of)i(subgroups)e(are)i(giv)m(en)378 961
y(b)m(y)426 1145 y FN(`)482 1160 y FE(def)638 1145 y
FM(RightCoset)45 b FT(\()p FP(H)7 b(;)15 b(p)p FT(\))49
b FP(a)f FN(\021)f FT(\()p FP(\025b)p FM(.)h FN(9)p FP(h)p
FM(.)f FP(H)55 b(h)48 b FN(^)g FT(\()p FP(b)73 b FT(=)g
FP(p)47 b(h)h(a)p FT(\)\))426 1371 y FN(`)482 1386 y
FE(def)638 1371 y FM(LeftCoset)e FP(a)i FT(\()p FP(H)7
b(;)15 b(p)p FT(\))48 b FN(\021)f FT(\()p FP(\025b)p
FM(.)h FN(9)p FP(h)p FM(.)g FP(H)54 b(h)48 b FN(^)g FT(\()p
FP(b)73 b FT(=)g FP(p)47 b(a)h(h)p FT(\)\))426 1597 y
FN(`)482 1612 y FE(def)638 1597 y FM(SProd)f FP(p)g(X)55
b(Y)68 b FN(\021)48 b FT(\()p FP(\025x)p FM(.)f FN(9)p
FP(h)p FM(.)h FP(X)55 b(h)48 b FN(^)f(9)p FP(k)s FM(.)h
FP(Y)67 b(k)51 b FN(^)c FT(\()p FP(x)74 b FT(=)e FP(p)48
b(h)g(k)s FT(\)\))378 1781 y(W)-8 b(e)32 b(also)e(include)e(the)j
(follo)m(wing)d(de\014nition)g(to)k(represen)m(t)e(sets)h(of)f(the)h
(form)f FP(a)p FT(\()p FP(H)7 b(b)p FT(\))426 1965 y
FN(`)482 1980 y FE(def)638 1965 y FM(LRCoset)46 b FP(a)i
FT(\()p FP(H)7 b(;)15 b(p)p FT(\))49 b FP(b)e FN(\021)h
FT(\()p FP(\025x)p FM(.)g FN(9)p FP(h)p FM(.)f FP(H)55
b(h)48 b FN(^)f FT(\()p FP(x)73 b FT(=)g FP(p)47 b(a)h
FT(\()p FP(p)g(h)g(b)p FT(\)\)\))519 2148 y(These)c(de\014nitions)e(do)
j(not)g(imp)s(ose)e(an)m(y)i(restrictions)e(on)i(the)f(sets)h
FP(H)7 b FT(,)48 b FP(X)k FT(and)44 b FP(Y)20 b FT(,)48
b(and)378 2261 y(therefore)37 b(results)d(in)m(v)m(olving)h(them)h
(need)g(to)h(sp)s(ecify)e(whether)g FP(H)7 b FT(,)38
b FP(X)43 b FT(and)36 b FP(Y)56 b FT(are)36 b(subgroups,)378
2374 y(subsets)h(of)i(some)g(group)e(or)h(arbitrary)f(sets.)65
b(In)38 b(general,)i(these)f(four)e(functions)g(are)i(used)e(to)378
2487 y(construct)21 b(subsets)e(of)i(a)f(group)g(in)f(the)i(same)f(w)m
(a)m(y)i(that)f(the)f(notation)h(of)f(juxtap)s(osing)f(subsets)g(and)
378 2600 y(group)33 b(elemen)m(ts)i(is)e(used)g(to)h(denote)h(subsets.)
51 b(W)-8 b(e)35 b(call)e(these)h(functions)f(subset)g(constructing)378
2713 y(functions.)56 b(The)36 b Fw(in_set)d FT(category)38
b(query)d(function)g(of)h(the)g(kno)m(wledge)g(database)h(is)e(up)s
(dated)378 2826 y(suc)m(h)30 b(that)h(a)g(query)f Fv(H)50
b Ft(\()p Fv(p)43 b(a)h(b)p Ft(\)\))30 b FT(is)g(satis\014ed)f(if)h
FP(H)37 b FT(is)29 b(of)i(the)f(form:)514 3010 y FN(\017)46
b Fw(RightCoset)40 b Ft(\()p Fv(X)r(;)14 b(p)p Ft(\))43
b Fv(b)30 b FT(where)g FP(a)25 b FN(2)g FP(X)7 b FT(,)31
b(or)514 3196 y FN(\017)46 b Fw(LeftCoset)40 b Fv(a)k
Ft(\()p Fv(Y)5 b(;)14 b(p)p Ft(\))30 b FT(where)g FP(b)25
b FN(2)g FP(Y)20 b FT(,)30 b(or)514 3382 y FN(\017)46
b Fw(SProd)c Fv(p)h(X)50 b(Y)f FT(where)30 b FP(a)25
b FN(2)g FP(X)38 b FT(and)29 b FP(b)d FN(2)f FP(Y)20
b FT(.)378 3566 y(Similarly)-8 b(,)27 b(a)k(query)f Fv(H)50
b Ft(\()p Fv(p)44 b(a)f Ft(\()p Fv(p)h(b)f(c)p Ft(\))31
b FT(is)e(satis\014ed)h(if)f FP(H)37 b FT(is)30 b(of)g(the)h(form)514
3750 y FN(\017)46 b Fw(LRCoset)41 b Fv(a)i Ft(\()p Fv(Y)5
b(;)14 b(p)p Ft(\))44 b Fv(c)30 b FT(where)g FP(b)25
b FN(2)g FP(Y)20 b FT(.)378 3934 y(The)31 b Fw(subset)e
FT(category)k(is)d(also)i(up)s(dated)e(so)h(that)h(a)g(query)f
Fw(Subset)41 b Fv(H)50 b(G)32 b FT(is)e(satis\014ed)h(if)f
FP(H)38 b FT(is)31 b(of)378 4047 y(the)g(form:)514 4231
y FN(\017)46 b Fw(RightCoset)40 b Ft(\()p Fv(X)r(;)14
b(p)p Ft(\))43 b Fv(b)30 b FT(where)g FP(X)j FN(\022)25
b FP(G)k FT(and)h FP(b)25 b FN(2)g FP(G)p FT(,)30 b(or)514
4417 y FN(\017)46 b Fw(LeftCoset)40 b Fv(a)k Ft(\()p
Fv(Y)5 b(;)14 b(p)p Ft(\))30 b FT(where)g FP(Y)45 b FN(\022)25
b FP(G)30 b FT(and)g FP(a)25 b FN(2)g FP(G)p FT(,)30
b(or)514 4603 y FN(\017)46 b Fw(SProd)c Fv(p)h(X)50 b(Y)f
FT(where)30 b FP(X)j FN(\022)24 b FP(G)30 b FT(and)g
FP(Y)45 b FN(\022)25 b FP(G)p FT(,)30 b(or)514 4789 y
FN(\017)46 b Fw(LRCoset)41 b Fv(a)i Ft(\()p Fv(X)r(;)14
b(p)p Ft(\))44 b Fv(b)30 b FT(where)g FP(X)i FN(\022)25
b FP(G)p FT(,)30 b(and)g FP(a;)15 b(b)26 b FN(2)f FP(G)p
FT(.)519 4973 y(The)45 b(notation)g(of)h(juxtap)s(ositioning)c(subsets)
j(and)f(group)h(elemen)m(ts)h(do)s(es)f(not)g(result)f(in)378
5086 y(am)m(biguities)29 b(if)g(paren)m(theses)i(are)g(omitted)f(since)
g(it)g(can)g(b)s(e)g(sho)m(wn)g(that)661 5283 y(\()p
FP(ab)p FT(\))p FP(c)d FT(=)e FP(a)p FT(\()p FP(bc)p
FT(\))214 b(\()p FP(ab)p FT(\))p FP(H)33 b FT(=)25 b
FP(a)p FT(\()p FP(bH)7 b FT(\))171 b(\()p FP(H)7 b(a)p
FT(\))p FP(b)26 b FT(=)f FP(H)7 b FT(\()p FP(ab)p FT(\))195
b(\()p FP(aH)7 b FT(\))p FP(b)26 b FT(=)f FP(a)p FT(\()p
FP(H)7 b(b)p FT(\))574 5396 y(\()p FP(H)g(X)g FT(\))p
FP(a)27 b FT(=)e FP(H)7 b FT(\()p FP(X)g(a)p FT(\))84
b(\()p FP(aH)7 b FT(\))p FP(X)33 b FT(=)25 b FP(a)p FT(\()p
FP(H)7 b(X)g FT(\))85 b(\()p FP(H)7 b(a)p FT(\))p FP(X)33
b FT(=)25 b FP(H)7 b FT(\()p FP(aX)g FT(\))84 b(\()p
FP(H)7 b(X)g FT(\))p FP(Y)46 b FT(=)25 b FP(H)7 b FT(\()p
FP(X)g(Y)21 b FT(\))378 5592 y(where)k FP(a)p FT(,)h
FP(b)g FT(and)f FP(c)g FT(are)h(elemen)m(ts)g(of)f(some)h(group)f
FP(G)g FT(and)f FP(H)7 b FT(,)27 b FP(X)33 b FT(and)24
b FP(Y)46 b FT(are)25 b(subsets)g(of)h FP(G)p FT(.)38
b(These)378 5705 y(results)29 b(are)h(deriv)m(ed)g(in)e(SPL)i(so)g
(that)h(they)f(are)g(used,)g(man)m(ually)f(or)h(otherwise,)g(to)h
(manipulate)p eop
%%Page: 203 213
203 212 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(203)378
396 y(expressions)39 b(in)m(v)m(olving)f(the)i(subset)g(constructing)f
(functions)g Fw(RightCoset)l FT(,)j Fw(LeftCoset)l FT(,)h
Fw(SProd)378 509 y FT(and)38 b Fw(LRCoset)n FT(.)66 b(Other)39
b(simple)e(results)h(whic)m(h)g(ha)m(v)m(e)i(b)s(een)f(deriv)m(ed)f(in)
f(SPL)i(for)g(this)f(purp)s(ose)378 622 y(include)1683
735 y FP(H)7 b(e)25 b FT(=)g FP(H)189 b(eH)32 b FT(=)25
b FP(H)378 902 y FT(where)30 b FP(H)37 b FT(is)30 b(a)g(subset)g(of)h
(some)f(group)g FP(G)p FT(,)g(and)g FP(e)h FT(is)e(the)i(iden)m(tit)m
(y)e(elemen)m(t)i(in)e FP(G)p FT(.)519 1015 y(Since)22
b(one)h(of)f(our)h(motiv)-5 b(ations)22 b(of)g(this)g(mec)m(hanisation)
g(is)g(to)h(try)g(to)g(minimise)d(the)i(di\013erence)378
1128 y(b)s(et)m(w)m(een)29 b(the)f(length)f(of)h(formal)f(and)h
(informal)e(pro)s(ofs)h(b)m(y)h(automating)g(the)g(calculations)f(whic)
m(h)378 1241 y(authors)i(of)g(informal)f(pro)s(ofs)g(consider)g(to)i(b)
s(e)f(trivial,)f(w)m(e)h(ha)m(v)m(e)i(included)26 b(in)i(the)i(system)f
(a)h(sim-)378 1354 y(pli\014er)d(whic)m(h)i(normalises)f(terms)i(in)m
(v)m(olving)e(the)j(subset)e(constructing)g(functions.)40
b(The)29 b(normal)378 1467 y(form)34 b(for)f(suc)m(h)h(terms)g
(according)g(to)h(the)g(implemen)m(ted)d(normaliser)g(consists)i(of)g
(a)h(pro)s(duct)e(of)378 1580 y(subsets)d(asso)s(ciated)g(to)i(the)e
(righ)m(t:)1232 1784 y FP(S)1288 1798 y FL(1)1328 1784
y FT(\()p FP(S)1419 1798 y FL(2)1458 1784 y FT(\()p FN(\001)15
b(\001)g(\001)i FP(S)1671 1798 y FO(n)1717 1784 y FT(\))e
FN(\001)g(\001)g(\001)i FT(\))84 b(or)f(\()p FP(bS)2302
1798 y FL(1)2341 1784 y FT(\)\()p FP(S)2467 1798 y FL(2)2507
1784 y FT(\()p FN(\001)15 b(\001)g(\001)i FP(S)2720 1798
y FO(n)2766 1784 y FT(\))e FN(\001)g(\001)g(\001)i FT(\))378
1988 y(where)23 b(eac)m(h)i(set)f FP(S)1023 2002 y FO(i)1051
1988 y FT(,)h(for)f(0)h FP(<)g(i)h FN(\024)f FP(n)e FT(is)g(of)g(the)h
(form)f FP(X)31 b FT(or)24 b FP(X)7 b(a)p FT(,)26 b(where)d
FP(X)31 b FT(is)23 b(a)h(set)g(not)g(constructed)378
2101 y(b)m(y)g(an)m(y)g(of)g(the)g(subset)f(constructing)h(functions,)g
(and)f FP(a)h FT(and)f FP(b)h FT(are)g(non-iden)m(tit)m(y)f(group)h
(elemen)m(ts)378 2214 y(normalised)k(using)h(the)i(rules)e(in)g
(\014gure)h(24.)41 b(F)-8 b(or)31 b(example,)g(the)f(normal)f(form)h
(of)h(the)f(set)1373 2418 y(\(\(\()p FP(ae)p FT(\))p
FP(H)7 b FT(\)\()p FP(X)g FT(\()p FP(ba)1960 2381 y FK(\000)p
FL(1)2057 2418 y FT(\)\)\)\(\(\()p FP(aY)23 b FT(\))p
FP(a)p FT(\)\(\()p FP(bZ)7 b FT(\))p FP(c)p FT(\)\))378
2622 y(is)1618 2735 y(\()p FP(aH)g FT(\)\(\()p FP(X)g(b)p
FT(\)\(\()p FP(Y)23 b(ab)p FT(\)\()p FP(Z)7 b(c)p FT(\)\)\))p
FP(:)378 2902 y FT(W)-8 b(e)26 b(orien)m(t)f(the)g(rules)f(whic)m(h)f
(manipulate)h(the)h(sets)g(constructed)g(using)f(the)h(subset)f
(constructing)378 3015 y(functions)29 b(as)h(follo)m(ws:)999
3194 y FP(a)p FT(\()p FP(bH)7 b FT(\))26 b FN(!)f FT(\()p
FP(ab)p FT(\))p FP(H)202 b FT(\()p FP(H)7 b(a)p FT(\))p
FP(b)26 b FN(!)g FP(H)7 b FT(\()p FP(ab)p FT(\))170 b(\()p
FP(aH)7 b FT(\))p FP(b)26 b FN(!)f FP(a)p FT(\()p FP(H)7
b(b)p FT(\))955 3307 y(\()p FP(H)g(X)g FT(\))p FP(a)27
b FN(!)e FP(H)7 b FT(\()p FP(X)g(a)p FT(\))109 b FP(a)p
FT(\()p FP(H)7 b(X)g FT(\))26 b FN(!)g FT(\()p FP(aH)7
b FT(\))p FP(X)91 b(H)7 b FT(\()p FP(aX)g FT(\))26 b
FN(!)f FT(\()p FP(H)7 b(a)p FT(\))p FP(X)930 3419 y FT(\()p
FP(H)g(X)g FT(\))p FP(Y)47 b FN(!)25 b FP(H)7 b FT(\()p
FP(X)g(Y)21 b FT(\))263 b FP(H)7 b(e)26 b FN(!)f FP(H)450
b(eH)32 b FN(!)26 b FP(H)378 3616 y FT(and)k(add)f(the)i(extra)g(rule)
1592 3729 y FP(H)7 b FT(\(\()p FP(aX)g FT(\))p FP(Y)21
b FT(\))26 b FN(!)f FT(\()p FP(H)7 b(a)p FT(\)\()p FP(X)g(Y)22
b FT(\))p FP(:)378 3896 y FT(It)37 b(can)g(b)s(e)f(c)m(hec)m(k)m(ed)i
(\(using,)g(for)e(instance,)i(Kn)m(uth-Bendix)d(completion\))i(that)g
(the)g(ab)s(o)m(v)m(e)h(ten)378 4009 y(rules)21 b(with)g(the)i(ten)f
(rules)f(in)g(section)i(9.2.2)h(de\014ne)e(a)h(con\015uen)m(t)f(and)g
(terminating)f(term)i(rewriting)378 4122 y(system.)59
b(Note)38 b(that)g(the)e(rules)g(normalising)d(group)j(elemen)m(ts)h
(are)g(needed)g(for)f(con\015uence)h(as)378 4235 y(illustrated)28
b(b)m(y)i(the)h(examples)f(in)f(\014gure)h(25.)519 4348
y(The)d(ab)s(o)m(v)m(e)i(ten)f(rules)f(are)h(represen)m(ted)g(b)m(y)f
(conditional)g(equalities)f(since)h(eac)m(h)i(rule)e(is)g(v)-5
b(alid)378 4460 y(if)28 b(there)h(is)g(some)g(group)g
FP(G)f FT(suc)m(h)h(that)h(all)e(group)g(elemen)m(ts)i(in)e(the)h(rule)
f(are)h(elemen)m(ts)h(of)f FP(G)f FT(and)378 4573 y(all)i(the)i(sets)f
(in)f(the)i(rule)e(are)i(subsets)e(of)i FP(G)p FT(.)43
b(F)-8 b(or)32 b(example,)f(the)h(rule)e(\()p FP(H)7
b(a)p FT(\))p FP(b)27 b FN(!)g FP(H)7 b FT(\()p FP(ab)p
FT(\))32 b(is)f(v)-5 b(alid)378 4686 y(if)26 b(there)i(is)e(some)i
(group)e FP(G)h FT(suc)m(h)g(that)h FP(H)k FN(\022)25
b FP(G)p FT(,)i(and)g FP(a;)15 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 FP(N)10 b(g)28 b FT(=)d FP(g)s(N)378
1032 y FT(for)30 b(ev)m(ery)h FP(g)e FN(2)c FP(G)p FT(,)30
b(as)h(sho)m(wn)e(b)m(y)i(the)f(follo)m(wing)f(pro)s(of)h(fragmen)m(t:)
378 1216 y FM("RightCoset)45 b(\(N,p\))h(g)760 1329 y(=)95
b(RightCoset)45 b(\(LRCoset)g(g)j(\(N,p\))e(\(invG)h(g\),p\))f(g")h(by)
g(gNg'=N)664 1442 y(."=)95 b(LeftCoset)45 b(g)j(\(N,p\)")e(<cos>)g(by)h
(fol;)378 1626 y FT(where)39 b Fw(gNg'=N)e FT(is)h(the)i(lab)s(el)e(of)
h(the)h(theorem)g(stating)f(that)i FP(g)s(N)10 b(g)2781
1593 y FK(\000)p FL(1)2916 1626 y FT(=)41 b FP(N)49 b
FT(if)38 b FP(N)50 b FT(is)38 b(a)i(normal)378 1739 y(subgroup)34
b(of)i(a)g(group)f FP(G)g FT(and)g FP(g)j FN(2)c FP(G)p
FT(.)56 b(It)36 b(can)g(b)s(e)f(seen)h(that)g(the)g(ab)s(o)m(v)m(e)h
(SPL)e(pro)s(of)g(is)f(quite)378 1852 y(similar)28 b(\(in)h(terms)h(of)
h(the)f(n)m(um)m(b)s(er)g(of)g FI(pr)-5 b(o)g(of)35 b(steps)7
b FT(\))32 b(to)f(the)g(informal)1190 2056 y(b)m(y)f
FP(g)s(N)10 b(g)1491 2023 y FK(\000)p FL(1)1612 2056
y FT(=)25 b FP(N)40 b FT(w)m(e)31 b(get)g FP(N)10 b(g)29
b FT(=)c(\()p FP(g)s(N)10 b(g)2567 2019 y FK(\000)p FL(1)2663
2056 y FT(\))p FP(g)29 b FT(=)c FP(g)s(N)5 b(:)378 2260
y FT(The)39 b(simpli\014cation)d(of)j(\()p FP(g)s(N)10
b(g)1466 2227 y FK(\000)p FL(1)1562 2260 y FT(\))p FP(g)43
b FT(in)m(to)c FP(g)s(N)50 b FT(whic)m(h)38 b(is)g(unjusti\014ed)e(in)i
(the)h(informal)e(pro)s(of)i(is)378 2373 y(automatically)i(deriv)m(ed)f
(b)m(y)h(the)g Fw(cos)f FT(simpli\014er.)70 b(The)40
b(use)h(of)g(appropriate)f(notation)i(in)e(the)378 2486
y(informal)c(pro)s(of,)k(ho)m(w)m(ev)m(er,)h(mak)m(es)e(it)f(m)m(uc)m
(h)g(shorter)g(\(in)f(terms)h(of)g(the)g(n)m(um)m(b)s(er)f(of)h
FI(symb)-5 b(ols)7 b FT(\))378 2599 y(than)40 b(the)g(one)g(implemen)m
(ted)f(in)f(SPL.)i(The)f(problem)g(of)h(reducing)e(the)i(n)m(um)m(b)s
(er)f(of)h(sym)m(b)s(ols)378 2712 y(through)35 b(the)g(abilit)m(y)f(to)
j(in)m(tro)s(duce)d(notation)i(safely)f(during)e(the)j(mec)m
(hanisation)f(of)h(a)g(theory)378 2825 y(are)41 b(not)f(discussed)e(in)
h(this)g(thesis.)70 b(It)40 b(is)f(ho)m(w)m(ev)m(er)j(eviden)m(t)e
(that)g(e\013orts)h(on)f(impro)m(ving)f(the)378 2938
y(notation)31 b(of)f(terms)g(used)g(in)f(mec)m(hanised)h(pro)s(ofs)f
(is)h(quite)g(desirable.)519 3051 y(The)j(fact)g(that)h(the)f(left)g
(cosets)h(of)f(a)h(subgroup)d FP(N)43 b FT(are)33 b(equal)g(to)g(its)g
(righ)m(t)f(cosets)j(is)d(also)h(a)378 3164 y(su\016cien)m(t)d
(condition)f(for)h FP(N)40 b FT(to)31 b(b)s(e)e(normal.)40
b(If)30 b(for)g(ev)m(ery)h FP(g)e FN(2)c FP(G)k FT(it)h(is)f(the)i
(case)g(that)g FP(g)s(N)36 b FT(=)25 b FP(N)10 b(a)378
3277 y FT(for)25 b(some)h FP(a)f FN(2)g FP(G)p FT(,)h(then)g(since)f
FP(g)k FT(is)24 b(in)g FP(g)s(N)36 b FT(it)25 b(m)m(ust)h(also)f(b)s(e)
g(in)f FP(N)10 b(a)p FT(.)39 b(The)25 b(group)g(elemen)m(t)h
FP(g)j FT(is)c(also)378 3390 y(in)i FP(N)10 b(g)31 b
FT(and)d(th)m(us)g FP(N)10 b(a)28 b FT(and)f FP(N)10
b(g)32 b FT(ha)m(v)m(e)d(an)f(elemen)m(t)h(in)e(common.)40
b(No)m(w,)30 b(since)d(the)i(righ)m(t)e(cosets)j(of)378
3502 y(a)35 b(subgroup)d(partition)h(the)h(whole)g(group,)h(then)f
FP(N)10 b(g)37 b FT(and)d FP(N)10 b(a)34 b FT(m)m(ust)g(b)s(e)g(equal,)
h(and)e(therefore)378 3615 y FP(g)s(N)j FT(=)25 b FP(N)10
b(g)s FT(.)41 b(This)29 b(is)g(enough)h(to)h(sho)m(w)f(that:)378
3799 y FM("LRCoset)45 b(g)j(\(N,p\))e(\(invG)h(g\))903
3912 y(=)95 b(RightCoset)45 b(\(LeftCoset)g(g)j(\(N,p\),p\))d(\(invG)h
(g\)")h(<cos>)g(by)g(fol)807 4025 y(."=)95 b(RightCoset)45
b(\(RightCoset)g(\(N,p\))h(g,p\))h(\(invG)f(g\)")h(by)g(gN=Ng)807
4138 y(."=)95 b(N")47 b(<cos>)g(by)g(fol;)378 4322 y
FT(and)28 b(that)i(therefore)g FP(N)39 b FT(is)28 b(normal)g(in)g
FP(G)p FT(.)39 b(The)29 b(lo)s(cal)f(fact)i(lab)s(elled)d(b)m(y)i
Fw(gN=Ng)e FT(is)h(the)h(result)f(that)378 4435 y FP(g)s(N)42
b FT(=)30 b FP(N)10 b(g)s FT(.)52 b(The)33 b(equation)h
FP(g)s(N)42 b FT(=)30 b FP(N)10 b(g)38 b FT(can)c(also)g(b)s(e)f(used)g
(to)i(sho)m(w)e(that)i(the)f(pro)s(duct)f(of)h(t)m(w)m(o)378
4548 y(righ)m(t)c(cosets)i(is)d(itself)g(a)i(righ)m(t)f(coset:)1128
4752 y(\()p FP(N)10 b(a)p FT(\)\()p FP(N)g(b)p FT(\))27
b(=)e FP(N)10 b FT(\()p FP(aN)g FT(\))p FP(b)26 b FT(=)f
FP(N)10 b FT(\()p FP(N)g(a)p FT(\))p FP(b)26 b FT(=)f
FP(N)10 b(N)g(ab)25 b FT(=)g FP(N)10 b(ab)378 4957 y
FT(whic)m(h)29 b(is)g(deriv)m(ed)h(in)f(SPL)g(b)m(y:)378
5141 y FM("SProd)46 b(p)h(\(RightCoset)e(\(N,p\))i(a\))g(\(RightCoset)d
(\(N,p\))j(b\))712 5253 y(=)g(SProd)g(p)g(N)h(\(RightCoset)c
(\(LeftCoset)h(a)j(\(N,p\),p\))d(b\)"<cos>)h(by)h(fol)617
5366 y(."=)f(SProd)h(p)g(N)h(\(RightCoset)c(\(RightCoset)h(\(N,p\))i
(a,p\))f(b\)")1046 5479 y(by)h(Normal_gN_Ng)e(on)i(Ga)617
5592 y(."=)f(RightCoset)f(\(SProd)i(p)g(N)g(N,p\))g(\(p)g(a)h
(b\)"<cos>)d(by)i(fol)617 5705 y(."=)f(RightCoset)f(\(N,p\))i(\(p)g(a)g
(b\)")g(by)h(SProd_Idem)d(on)i(GroupG)f(&)h(NsgG;)p eop
%%Page: 206 216
206 215 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(206)378
396 y(where)39 b Fw(Normal_gN_Ng)34 b FT(is)39 b(the)h(theorem)g
(stating)f(that)i FP(g)s(N)51 b FT(=)40 b FP(N)10 b(g)43
b FT(for)c FP(g)44 b FN(2)d FP(G)p FT(,)g Fw(Ga)e FT(is)f(the)i(fact)
378 509 y FP(a)25 b FN(2)g FP(G)p FT(,)k Fw(GroupG)e
FT(is)h(the)i(fact)g(that)g(\()p FP(G;)15 b(p)p FT(\))29
b(is)f(a)i(group,)f(and)g Fw(NsgG)e FT(the)j(fact)g(that)g
FP(N)39 b FT(is)28 b(a)i(subgroup)378 622 y(of)k FP(G)p
FT(.)52 b(The)34 b(theorem)g Fw(SProd_Idem)c FT(states)36
b(that)e(the)h(pro)s(duct)e FP(H)7 b(H)41 b FT(of)34
b(a)h(subgroup)d FP(H)41 b FT(is)33 b(equal)378 735 y(to)e
FP(H)7 b FT(.)519 848 y(The)34 b(result)g(that)h FP(N)10
b(aN)g(b)32 b FT(=)h FP(N)10 b(ab)34 b FT(is)g(quite)g(imp)s(ortan)m(t)
g(since)g(it)g(is)g(used)g(to)h(sho)m(w)f(that)i(the)378
961 y(set)30 b(of)f(righ)m(t)g(cosets)i(of)e(a)h(normal)e(subgroup)f
FP(N)39 b FT(of)30 b(a)g(group)e FP(G)h FT(is)f(itself)g(a)i(group.)40
b(This)27 b(group)i(is)378 1074 y(called)i(the)g(quotien)m(t)h(group)f
(of)g FP(G)g FT(b)m(y)g FP(N)10 b FT(,)32 b(and)f(is)g(denoted)g(b)m(y)
g FP(G=)-5 b(N)10 b FT(.)44 b(The)31 b(iden)m(tit)m(y)g(elemen)m(t)h
(of)378 1187 y FP(G=)-5 b(N)40 b FT(is)30 b FP(N)40 b
FT(and)30 b(the)g(in)m(v)m(erse)g(elemen)m(t)h(of)g(a)g(coset)g
FP(N)10 b(a)31 b FT(in)e FP(G=)-5 b(N)40 b FT(is)29 b
FP(N)10 b(a)2900 1154 y FK(\000)p FL(1)2995 1187 y FT(.)519
1300 y(The)46 b(quotien)m(t)g(group)g(of)g(a)h(subgroup)d
FP(N)56 b FT(of)47 b FP(G)e FT(is)h(denoted)g(in)f(HOL)h(b)m(y)g(the)g
(function)378 1413 y Fw(QuotientGp)26 b FT(de\014ned)j(b)m(y)473
1600 y FN(`)529 1615 y FE(def)686 1600 y FM(QuotientSet)45
b FT(\()p FP(G)p FM(:'a)i FN(!)g FM(bool)p FP(;)g(p)p
FT(\))h FP(H)55 b FN(\021)760 1713 y FP(\025X)7 b FM(.)48
b FN(9)p FT(\()p FP(a)p FM(:'a)p FT(\))p FM(.)f FP(G)g(a)h
FN(^)f FT(\()p FP(X)81 b FT(=)47 b FM(RightCoset)e FT(\()p
FP(H)7 b(;)15 b(p)p FT(\))49 b FP(a)p FT(\))473 1939
y FN(`)529 1954 y FE(def)686 1939 y FM(QuotientGp)c FT(\()p
FP(G)p FM(:'a)i FN(!)h FM(bool)p FP(;)e(p)p FT(\))i FP(N)58
b FN(\021)47 b FT(\()p FM(QuotientSet)e FT(\()p FP(G;)15
b(p)p FT(\))49 b FP(N)10 b(;)47 b FM(SProd)g FP(p)p FT(\))378
2127 y(and)37 b(it)f(can)i(b)s(e)e(sho)m(wn)h(that)g(all)f(the)i
(conditions)d(making)i Fw(QuotientGp)i Ft(\()p Fv(G;)14
b(p)p Ft(\))45 b Fv(N)g FT(a)38 b(group)e(are)378 2240
y(satis\014ed)29 b(if)h FP(N)40 b FT(is)29 b(normal.)473
2427 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(\))f FM(NormalSG)f
FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(N)58 b FN(\))998
2540 y FM(Group)47 b FT(\()p FM(QuotientGp)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(N)10 b FT(\))473 2766 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(\))f FM(NormalSG)f FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(N)58 b FN(\))998 2879 y
FT(\()p FM(IdG)48 b FT(\()p FM(QuotientGp)d FT(\()p FP(G;)15
b(p)p FT(\))48 b FP(N)10 b FT(\))73 b(=)g FP(N)10 b FT(\))473
3105 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(\))f FM(NormalSG)f
FT(\()p FP(G;)15 b(p)p FT(\))48 b FP(N)58 b FN(\))47
b FT(\()p FN(8)p FP(a)p FM(.)h FP(G)f(a)h FN(\))1142
3218 y FT(\()p FM(InvG)f FT(\()p FM(QuotientGp)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(N)10 b FT(\))48 b(\()p FM(RightCoset)d
FT(\()p FP(N)10 b(;)15 b(p)p FT(\))49 b FP(a)p FT(\))73
b(=)1619 3330 y FM(RightCoset)45 b FT(\()p FP(N)10 b(;)15
b(p)p FT(\))48 b(\()p FM(InvG)f FT(\()p FP(G;)15 b(p)p
FT(\))48 b FP(a)p FT(\)\)\))519 3518 y(The)33 b(e\013orts)h(required)e
(in)h(implemen)m(ting)e(the)j(pro)s(ofs)e(of)i(the)g(results)e(giv)m
(en)i(in)e(this)h(section)378 3631 y(are)41 b(not)f(m)m(uc)m(h)h
(greater)h(than)e(understanding)e(the)i(pro)s(ofs)g(in)f(the)i
(literature,)h(and)e(rewriting)378 3744 y(them)30 b(in)f(SPL)h(and)g
(\014lling)d(a)k(few)f(gaps)h(in)e(the)h(informal)f(argumen)m(ts.)378
3987 y FG(9.4.2)112 b(Homomorphisms)36 b(and)i(Isomorphisms)378
4159 y FT(A)30 b(homomorphism)d(is)i(a)g(structure-preserving)f
(mapping)g(from)h(one)h(group)f(in)m(to)h(another.)40
b(The)378 4272 y(notion)25 b(of)i(a)f(structure-preserving)f(function)f
(b)s(et)m(w)m(een)j(groups)e(is)g(giv)m(en)h(b)m(y)g(the)h(HOL)e
(de\014nition)426 4460 y FN(`)482 4475 y FE(def)686 4460
y FM(Str_Pres)46 b FT(\()p FP(G)p FM(:'a)g FN(!)i FM(bool)p
FP(;)f(p)p FT(\))h(\()p FP(H)7 b FM(:'b)47 b FN(!)h FM(bool)p
FP(;)f(q)s FT(\))g(\()p FP(f)10 b FM(:'a)47 b FN(!)h
FM('b)p FT(\))f FN(\021)569 4572 y FT(\()p FN(8)p FP(x)g(y)s
FM(.)h FP(G)f(x)h FN(\))f FP(G)g(y)k FN(\))d FT(\()p
FP(f)57 b FT(\()p FP(p)48 b(x)f(y)s FT(\))74 b(=)e FP(q)51
b FT(\()p FP(f)57 b(x)p FT(\))48 b(\()p FP(f)58 b(y)s
FT(\)\)\))378 4760 y(or)23 b(in)g(other)g(w)m(ords,)i
FP(\036)g FT(:)h FP(G)f FN(!)g FP(H)30 b FT(is)23 b
(structure-preserving)f(if)g FP(\036)p FT(\()p FP(x)6
b FN(\016)2716 4774 y FO(G)2783 4760 y FP(y)s FT(\))25
b(=)g FP(\036)p FT(\()p FP(x)p FT(\))6 b FN(\016)3214
4774 y FO(H)3290 4760 y FP(\036)p FT(\()p FP(y)s FT(\))24
b(for)f(ev)m(ery)378 4873 y FP(x;)15 b(y)28 b FN(2)d
FP(G)p FT(,)k(where)g FN(\016)1062 4887 y FO(G)1150 4873
y FT(and)g FN(\016)1371 4887 y FO(H)1468 4873 y FT(are)g(the)g(pro)s
(ducts)f(of)i FP(G)e FT(and)h FP(H)35 b FT(resp)s(ectiv)m(ely)-8
b(.)41 b(Homomorphisms)378 4986 y(are)31 b(de\014ned)e(in)g(HOL)h(b)m
(y)426 5174 y FN(`)482 5189 y FE(def)686 5174 y FM(Homomorphism)45
b FT(\()p FP(G)p FM(:'a)h FN(!)i FM(bool)p FP(;)f(p)p
FT(\))h(\()p FP(H)7 b FM(:'b)47 b FN(!)h FM(bool)p FP(;)e(q)s
FT(\))i FP(f)57 b FN(\021)712 5286 y FT(\()p FM(fInto)47
b FP(G)g(H)55 b(f)10 b FT(\))47 b FN(^)g FM(Str_Pres)f
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(H)7 b(;)15
b(q)s FT(\))49 b FP(f)378 5474 y FT(where)30 b Fw(fInto)41
b Fv(G)j(H)51 b(f)38 b FT(holds)29 b(if)h FP(f)39 b FT(maps)30
b(ev)m(ery)h(elemen)m(t)g(in)e FP(G)h FT(in)m(to)g FP(H)7
b FT(:)473 5662 y FN(`)529 5677 y FE(def)686 5662 y FM(fInto)47
b FP(X)55 b(Y)67 b FT(\()p FP(f)10 b FM(:'a)47 b FN(!)h
FM('b)p FT(\))f FN(\021)h FT(\()p FN(8)p FP(x)p FM(.)f
FP(X)55 b(x)48 b FN(\))f FP(Y)68 b FT(\()p FP(f)57 b(x)p
FT(\)\))p eop
%%Page: 207 217
207 216 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(207)378
396 y(Since)25 b(it)h(is)f(quite)g(tedious)h(to)g(sho)m(w)g(that)h
Ft(\()p Fv(f)53 b(x)p Ft(\))26 b FT(is)f(in)g(some)i(set)f
FP(Y)46 b FT(whenev)m(er)26 b Fw(fInto)42 b Fv(X)50 b(Y)62
b(f)34 b FT(and)378 509 y FP(x)d FN(2)g FP(X)7 b FT(,)36
b(a)e(database)h(category)h Fw(fun_into)30 b FT(is)j(used)h(to)g(store)
h(facts)g(of)f(the)g(form)f Fw(fInto)42 b Fv(X)50 b(Y)62
b(f)9 b FT(,)378 622 y(and)30 b(the)g(function)f(querying)g
Fw(in_set)f FT(is)i(up)s(dated)f(suc)m(h)h(that)h(a)f(query)g
Fv(Y)62 b Ft(\()p Fv(f)53 b(x)p Ft(\))31 b FT(is)e(satis\014ed)h(if)514
810 y FN(\017)46 b Fw(fInto)c Fv(X)50 b(Y)62 b(f)39 b
FT(and)30 b Ft(\()p Fv(X)50 b(x)p Ft(\))31 b FT(hold)e(for)h(some)h
(set)g Fv(X)6 b FT(.)519 998 y(Examples)21 b(of)i(homomorphisms)d
(include)g(the)i(iden)m(tit)m(y)g(function)f(and)h(the)h(function)e
(mapping)378 1110 y(ev)m(ery)31 b(elemen)m(t)g(in)m(to)f(the)h(iden)m
(tit)m(y)-8 b(.)473 1298 y FN(`)48 b(8)p FP(G)e(p)p FM(.)i
(Homomorphism)c FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p
FP(G;)15 b(p)p FT(\))48 b FM(I)473 1524 y FN(`)g(8)p
FP(G)e(p)i(G)912 1491 y FK(0)982 1524 y FP(q)s FM(.)g(Group)e
FT(\()p FP(G)1515 1491 y FK(0)1538 1524 y FP(;)15 b(q)s
FT(\))48 b FN(\))903 1637 y FM(Homomorphism)c FT(\()p
FP(G;)15 b(p)p FT(\))48 b(\()p FP(G)1906 1604 y FK(0)1930
1637 y FP(;)15 b(q)s FT(\))48 b(\()p FM(K)g FT(\()p FM(IdG)f
FT(\()p FP(G)2561 1604 y FK(0)2584 1637 y FP(;)15 b(q)s
FT(\)\)\))378 1824 y(where)30 b Fw(K)g FT(and)f Fw(I)h
FT(are)h(the)g(usual)e(com)m(binators:)473 2012 y FN(`)48
b(8)p FP(x)f(y)s FM(.)g(K)h FP(x)f(y)k FT(=)c FP(x)473
2125 y FN(`)h(8)p FP(x)p FM(.)f(I)g FP(x)h FT(=)f FP(x)519
2313 y FT(It)39 b(can)g(b)s(e)f(sho)m(wn)g(that)h(for)f(ev)m(ery)i
(homomorphism)c FP(\036)j FT(of)g FP(G)f FT(in)m(to)g
FP(H)46 b FT(it)38 b(is)g(the)h(case)g(that)378 2426
y FP(\036)p FT(\()p FP(e)509 2440 y FO(G)569 2426 y FT(\))28
b(=)f FP(e)772 2440 y FO(H)871 2426 y FT(and)k FP(\036)p
FT(\()p FP(x)1190 2393 y FK(\000)1245 2404 y Fy(G)1297
2393 y FL(1)1337 2426 y FT(\))c(=)h FP(\036)p FT(\()p
FP(x)p FT(\))1674 2393 y FK(\000)1729 2404 y Fy(H)1787
2393 y FL(1)1858 2426 y FT(where)j FP(e)2164 2440 y FO(X)2264
2426 y FT(represen)m(ts)g(the)h(iden)m(tit)m(y)f(elemen)m(t)i(of)e
(some)378 2538 y(arbitrary)e(group)g FP(X)38 b FT(and)29
b FP(x)1363 2505 y FK(\000)1418 2516 y Fy(X)1476 2505
y FL(1)1545 2538 y FT(is)g(the)h(in)m(v)m(erse)g(of)g
FP(x)g FT(in)f FP(X)7 b FT(.)41 b(These)30 b(results)f(are)h(deriv)m
(ed)f(in)g(SPL)378 2651 y(and)38 b(are)h(used)f(with)f(the)i(fact)h
(that)f(homomorphisms)e(are)i(structure-preserving)e(to)i(simplify)378
2764 y(terms)28 b(in)m(v)m(olving)e(some)i(homomorphism.)38
b(Basically)-8 b(,)28 b(a)g(simpli\014er)c(named)j Fw(hom)g
FT(is)f(implemen)m(ted)378 2877 y(whic)m(h)j(rewrites)h(terms)g(b)m(y)g
(the)h(rules:)473 3065 y FN(`)48 b(8)p FP(G)e(p)i(G)912
3032 y FK(0)982 3065 y FP(q)j(f)10 b FM(.)47 b(Homomorphism)d
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(G)2227 3032
y FK(0)2250 3065 y FP(;)15 b(q)s FT(\))49 b FP(f)57 b
FN(\))617 3178 y FT(\()p FN(8)p FP(x)47 b(y)s FM(.)g
FP(G)h(x)f FN(\))h FP(G)f(y)k FN(\))c FT(\()p FP(f)58
b FT(\()p FP(p)48 b(x)f(y)s FT(\))73 b(=)g FP(q)51 b
FT(\()p FP(f)57 b(x)p FT(\))48 b(\()p FP(f)57 b(y)s FT(\)\)\))473
3404 y FN(`)48 b(8)p FP(G)e(p)p FM(.)i(Group)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FN(\))g FT(\()p FN(8)p FP(G)1747
3371 y FK(0)1817 3404 y FP(q)s FM(.)f(Group)g FT(\()p
FP(G)2350 3371 y FK(0)2373 3404 y FP(;)15 b(q)s FT(\))48
b FN(\))617 3516 y FT(\()p FN(8)p FP(f)10 b FM(.)46 b(Homomorphism)f
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(G)1856 3483
y FK(0)1879 3516 y FP(;)15 b(q)s FT(\))48 b FP(f)57 b
FN(\))951 3629 y FT(\()p FP(f)g FT(\()p FM(IdG)47 b FT(\()p
FP(G;)15 b(p)p FT(\)\))74 b(=)47 b FM(IdG)g FT(\()p FP(G)2067
3596 y FK(0)2090 3629 y FP(;)15 b(q)s FT(\)\)\)\))473
3855 y FN(`)48 b(8)p FP(G)e(p)p FM(.)i(Group)e FT(\()p
FP(G;)15 b(p)p FT(\))48 b FN(\))g FT(\()p FN(8)p FP(G)1747
3822 y FK(0)1817 3855 y FP(q)s FM(.)f(Group)g FT(\()p
FP(G)2350 3822 y FK(0)2373 3855 y FP(;)15 b(q)s FT(\))48
b FN(\))617 3968 y FT(\()p FN(8)p FP(f)10 b FM(.)46 b(Homomorphism)f
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(G)1856 3935
y FK(0)1879 3968 y FP(;)15 b(q)s FT(\))48 b FP(f)57 b
FN(\))48 b FT(\()p FN(8)p FP(x)p FM(.)f FP(G)g(x)h FN(\))951
4081 y FT(\()p FP(f)57 b FT(\()p FM(InvG)47 b FT(\()p
FP(G;)15 b(p)p FT(\))48 b FP(x)p FT(\))73 b(=)48 b FM(InvG)e
FT(\()p FP(G)2262 4048 y FK(0)2286 4081 y FP(;)15 b(q)s
FT(\))48 b(\()p FP(f)57 b(x)p FT(\)\)\)\)\))378 4269
y(Similarly)20 b(to)k(the)g(other)g(simpli\014ers)c(\(suc)m(h)k(as)g
Fw(groups)d FT(and)i Fw(cos)o FT(\))h(men)m(tioned)f(in)f(this)h(c)m
(hapter,)j(the)378 4382 y(conditions)32 b(in)g(eac)m(h)j(rule)d(are)i
(deriv)m(ed)e(automatically)h(b)m(y)h(querying)e(the)h(kno)m(wledge)h
(database)378 4494 y(b)s(efore)e(it)g(is)g(applied.)45
b(A)33 b(database)g(category)i Fw(is_homomorphism)26
b FT(is)31 b(used)h(to)h(store)h(facts)f(of)g(the)378
4607 y(form)23 b Fw(Homomorphism)38 b Ft(\()p Fv(G;)14
b(p)p Ft(\))45 b Fw(\()p Fv(G)1514 4577 y Fr(0)1537 4607
y Fv(;)14 b(q)s Ft(\))44 b Fv(f)8 b FT(.)38 b(The)23
b(function)f(querying)g(the)i Fw(fun_into)c FT(is)i(up)s(dated)g(suc)m
(h)378 4720 y(that)31 b Fw(fInto)41 b Fv(G)j(G)1010 4690
y Fr(0)1077 4720 y Fv(f)39 b FT(is)30 b(satis\014ed)f(if)h
FP(f)39 b FT(is)30 b(a)g(homomorphism)f(of)h FP(G)g FT(in)m(to)g
FP(G)3037 4687 y FK(0)3060 4720 y FT(.)519 4833 y(Giv)m(en)g(a)h
(homomorphism)d FP(f)40 b FT(of)30 b FP(G)g FT(in)m(to)h
FP(H)7 b FT(,)30 b(w)m(e)h(de\014ne)f(its)f FI(kernel)40
b FT(b)m(y)30 b(the)h(set)1534 5037 y FP(K)1611 5052
y FO(f)1681 5037 y FT(=)25 b FN(f)p FP(x)h FN(2)f FP(G)55
b FN(j)h FP(f)10 b FT(\()p FP(x)p FT(\))25 b(=)g FP(e)2534
5051 y FO(H)2602 5037 y FN(g)p FP(:)473 5242 y FN(`)529
5257 y FE(def)686 5242 y FM(Kernel)46 b FP(G)h FT(\()p
FP(H)7 b(;)15 b(q)s FT(\))49 b(\()p FP(f)10 b FM(:'a)47
b FN(!)g FM('b\))g FN(\021)h FP(\025)16 b(x)p FM(.)47
b FP(G)g(x)h FN(^)f FT(\()p FP(f)58 b(x)47 b FT(=)h FM(IdG)f
FT(\()p FP(H)7 b(;)15 b(q)s FT(\)\))378 5429 y(The)30
b(k)m(ernel)g(is)f(a)i(subgroup)e(of)h FP(G)g FT(and)g(if)f
FP(k)f FN(2)d FP(K)2098 5444 y FO(f)2174 5429 y FT(then)p
eop
%%Page: 208 218
208 217 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(208)473
396 y FM("f)48 b(\(p)f(g)g(\(p)g(k)h(\(invG)e(g\)\)\))855
509 y(=)i(q)f(\(f)g(g\))g(\(q)h(\(f)f(k\))g(\(f)g(\(invG)g
(g\)\)\)"<hom>)e(by)i(fol)712 622 y(.")g(=)h(q)f(\(f)g(g\))g(\(f)h
(\(invG)e(g\)\)"<groups,)e(fk_i>)j(by)g(fol)712 735 y(.")g(=)h(q)f(\(f)
g(g\))g(\(invH)g(\(f)g(g\)\)"<hom>)e(by)i(fol)712 848
y(.")g(=)h(iH"<groups>)c(by)k(fol;)378 1032 y FT(where)33
b Fw(iG)g FT(and)g Fw(iH)g FT(abbreviate)h(the)g(terms)f(represen)m
(ting)h(the)g(iden)m(tit)m(y)f(elemen)m(ts)h(in)e FP(G)i
FT(and)f FP(H)7 b FT(,)378 1144 y Fw(invG)28 b FT(and)i
Fw(invH)e FT(abbreviate)i(the)h(in)m(v)m(erse)f(functions)f(of)h
FP(G)g FT(and)f FP(H)7 b FT(,)31 b(and)e Fw(fk_i)g FT(is)g(the)h
(result)f(that)378 1257 y Fv(f)52 b(k)47 b Fw(=)c(iH)o
FT(.)e(Therefore)30 b FP(g)s(k)s(g)1355 1224 y FK(\000)p
FL(1)1476 1257 y FN(2)25 b FP(K)1639 1272 y FO(f)1684
1257 y FT(,)31 b(and)e(hence)i FP(K)2246 1272 y FO(f)2322
1257 y FT(is)e(a)i(normal)e(subgroup)g(of)i FP(G)p FT(.)519
1370 y(A)k(homomorphism)e(is)h(called)h(an)g(isomorphism)d(if)i(it)g
(is)h(one-to-one,)j(and)d(t)m(w)m(o)h(groups)e(are)378
1483 y(said)f(to)i(b)s(e)f(isomorphic)e(if)h(there)i(is)e(an)h
(isomorphism)e(from)i(one)g(group)g FI(onto)41 b FT(the)35
b(other.)52 b(The)378 1596 y(notation)38 b FP(G)g FN(\031)f
FP(H)45 b FT(is)37 b(used)g(to)i(denote)f(the)g(fact)h(that)g
FP(G)e FT(is)g(isomorphic)f(to)j FP(H)7 b FT(.)63 b(W)-8
b(e)39 b(giv)m(e)g(the)378 1709 y(follo)m(wing)29 b(HOL)h
(de\014nitions:)473 1892 y FN(`)529 1907 y FE(def)686
1892 y FM(Isomorphism)45 b FT(\()p FP(G;)15 b(p)p FT(\))48
b(\()p FP(H)7 b(;)15 b(q)s FT(\))49 b(\()p FP(f)10 b
FM(:'a)46 b FN(!)i FM('b)p FT(\))f FN(\021)998 2005 y
FM(Injective)f FP(G)h(f)57 b FN(^)47 b FM(Homomorphism)e
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(H)7 b(;)15
b(q)s FT(\))48 b FP(f)473 2231 y FN(`)529 2246 y FE(def)686
2231 y FM(Isomorphic)d FT(\()p FP(G;)15 b(p)p FT(\))48
b(\()p FP(H)7 b(;)15 b(q)s FT(\))49 b FN(\021)712 2344
y(9)p FT(\()p FP(f)10 b FM(:'a)46 b FN(!)i FM('b)p FT(\))p
FM(.)f(Bijective)e FP(G)i(H)55 b(f)i FN(^)48 b FM(Homomorphism)c
FT(\()p FP(G;)15 b(p)p FT(\))48 b(\()p FP(H)7 b(;)15
b(q)s FT(\))49 b FP(f)473 2570 y FN(`)529 2585 y FE(def)686
2570 y FM(Injective)c FP(X)56 b FT(\()p FP(f)10 b FM(:'a)46
b FN(!)i FM('b)p FT(\))f FN(\021)951 2683 y(8)p FP(x)1054
2697 y FL(1)1093 2683 y FM(.)g FP(X)55 b(x)1370 2697
y FL(1)1457 2683 y FN(\))48 b(8)p FP(x)1699 2697 y FL(2)1738
2683 y FM(.)f FP(X)55 b(x)2015 2697 y FL(2)2103 2683
y FN(\))47 b FT(\()p FP(f)58 b(x)2431 2697 y FL(1)2543
2683 y FT(=)73 b FP(f)57 b(x)2841 2697 y FL(2)2880 2683
y FT(\))48 b FN(\))g FT(\()p FP(x)3189 2697 y FL(1)3302
2683 y FT(=)72 b FP(x)3497 2697 y FL(2)3537 2683 y FT(\))473
2909 y FN(`)529 2924 y FE(def)686 2909 y FM(Surjective)45
b FP(X)55 b(Y)68 b FT(\()p FP(f)10 b FM(:'a)47 b FN(\))g
FM('b)p FT(\))h FN(\021)951 3021 y(8)p FP(y)s FM(.)e
FP(Y)68 b(y)51 b FN(\))c(9)p FP(x)p FM(.)g FP(X)55 b(x)48
b FN(^)f FT(\()p FP(f)58 b(x)73 b FT(=)f FP(y)s FT(\))473
3247 y FN(`)529 3262 y FE(def)686 3247 y FM(Bijective)45
b FP(X)56 b(Y)67 b FT(\()p FP(f)10 b FM(:'a)47 b FN(!)h
FM('b)p FT(\))f FN(\021)951 3360 y FM(fInto)f FP(X)55
b(Y)68 b(f)57 b FN(^)47 b FM(Injective)f FP(X)55 b(f)i
FN(^)47 b FM(Surjective)e FP(X)55 b(Y)68 b(f)519 3544
y FT(The)32 b(isomorphism)e(relation)i(is)g(an)h(equiv)-5
b(alence,)33 b(ho)m(w)m(ev)m(er)h(this)e(cannot)h(b)s(e)f(represen)m
(ted)h(in)378 3657 y(HOL)d(b)m(y)g(the)h(term)473 3840
y FM(GEquivalence)45 b FT(\()p FM(Group:'a)h FN(!)h FM(bool)p
FT(\))g FM(Isomorphic)378 4023 y FT(as)28 b(this)e(w)m(ould)h(infer)f
(the)h(t)m(yp)s(e)h(of)g Fw(Isomorphic)23 b FT(to)29
b(b)s(e)d Fw(:'a)43 b Fu(!)g Fw('a)g Fu(!)h Fw(bool)n
FT(,)28 b(instead)f(of)h(the)g(more)378 4136 y(general)41
b Fw(:'a)h Fu(!)i Fw('b)f Fu(!)g Fw(bool)o FT(.)72 b(\(Recall)41
b(that)g Fw(GEquivalence)e(X)k(R)e FT(denotes)g(the)g(fact)h(that)g
(the)378 4249 y(relation)34 b Fw(R:'a)42 b Fu(!)i Fw('a)e
Fu(!)i Fw(bool)33 b FT(on)i(the)g(elemen)m(ts)g(in)f
Fw(X:'a)42 b Fu(!)i Fw(bool)33 b FT(is)h(an)h(equiv)-5
b(alence.\))54 b(This)378 4362 y(is)27 b(an)i(example)f(of)h(the)f
(di\016culties)e(resulting)h(b)m(y)h(represen)m(ting)g(sets)h(in)e(Ch)m
(urc)m(h's)g(Higher)h(Order)378 4475 y(Logic)33 b(b)m(y)f(their)g(p)s
(olymorphic)e(c)m(haracteristic)j(predicates.)47 b(Suc)m(h)32
b(problems)f(can)i(b)s(e)f(a)m(v)m(oided)h(if)378 4588
y(one)40 b(formalises)e(an)h(axiomatic)h(set)g(theory)f(in)g(HOL)g(as)g
(suggested)h(for)g(instance)f(b)m(y)g(Gordon)378 4701
y(\(1996\).)519 4814 y(The)48 b(follo)m(wing)e(t)m(w)m(o)k(in)m
(teresting)d(results)g(on)h(homomorphisms)e(and)h(isomorphisms)e(are)
378 4927 y(pro)m(v)m(ed)31 b(in)e(SPL:)514 5110 y FN(\017)46
b FT(If)30 b FP(\036)h FT(is)e(a)i(homomorphism)d(of)i
FP(G)g FT(on)m(to)i FP(G)2072 5077 y FK(0)2125 5110 y
FT(with)d(k)m(ernel)h FP(K)7 b FT(,)30 b(then)g FP(G=K)i
FN(\031)25 b FP(G)3340 5077 y FK(0)3363 5110 y FT(.)514
5296 y FN(\017)46 b FT(If)28 b FP(\036)g FT(is)f(a)i(homomorphism)c(of)
k FP(G)e FT(on)m(to)i FP(G)2053 5263 y FK(0)2104 5296
y FT(with)e(k)m(ernel)h FP(K)34 b FT(and)28 b FP(N)2944
5263 y FK(0)2995 5296 y FT(is)f(a)i(normal)e(subgroup)605
5409 y(of)k FP(G)781 5376 y FK(0)804 5409 y FT(,)f(then)g(if)f
FP(N)36 b FT(=)25 b FN(f)p FP(x)g FN(2)g FP(G)g FN(j)g
FP(\036)p FT(\()p FP(x)p FT(\))h FN(2)f FP(N)2080 5376
y FK(0)2104 5409 y FN(g)30 b FT(it)g(is)g(the)g(case)i(that)f
FP(G=)-5 b(N)35 b FN(\031)25 b FP(G)3296 5376 y FK(0)3318
5409 y FP(=)-5 b(N)3441 5376 y FK(0)3465 5409 y FT(.)519
5592 y(Similarly)22 b(to)27 b(the)g(other)f(results)f(describ)s(ed)f
(in)h(this)g(c)m(hapter,)j(not)e(m)m(uc)m(h)g(e\013ort)h(w)m(as)g
(required)378 5705 y(in)39 b(implemen)m(ting)f(the)j(required)e(SPL)g
(pro)s(ofs)h(once)h(the)g(informal)d(pro)s(ofs)i(w)m(ere)g(understo)s
(o)s(d.)p eop
%%Page: 209 219
209 218 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(209)378
396 y(Ho)m(w)m(ev)m(er,)35 b(attempts)e(at)g(the)f(implemen)m(tation)f
(of)h(pro)s(ofs)f(of)i(results)e(on)h(\014nite)f(groups)g(resulted)378
509 y(in)36 b(rather)g(longer)h(and)f(more)h(detailed)f(pro)s(ofs)h
(than)f(those)i(found)d(in)h(the)h(literature.)60 b(This)35
b(is)378 622 y(b)s(ecause)i(of)f(the)h(fact)h(that)f(not)g(enough)f
(e\013ort)i(w)m(as)f(put)f(in)f(implemen)m(ting)f(pro)s(of)i(pro)s
(cedures)378 735 y(whic)m(h)31 b(automate)j(the)e(inferences)f
(considered)g(trivial)f(while)g(reasoning)i(ab)s(out)f(\014nite)g
(sets.)47 b(W)-8 b(e)378 848 y(b)s(eliev)m(e)25 b(that)h(the)g
(implemen)m(tation)e(of)i(suc)m(h)f(pro)s(of)g(pro)s(cedures)f(is)h
(not)h(a)g(trivial)d(task)k(since)e(most)378 961 y(\(trivial,)43
b(or)f(otherwise\))g(results)f(on)h(\014nite)e(sets)j(require)d
(mathematical)j(induction,)f(and)g(the)378 1074 y(automation)31
b(of)f(pro)s(ofs)g(in)m(v)m(olving)f(induction)f(requires)h(substan)m
(tial)g(e\013ort.)378 1360 y FH(9.5)135 b(Discussion)378
1563 y FT(This)25 b(c)m(hapter)i(illustrated)e(the)h(mec)m(hanisation)h
(of)g(a)g(n)m(um)m(b)s(er)e(of)i(results)e(of)i(group)f(theory)h(in)f
(the)378 1676 y(pro)s(of)35 b(language)i(SPL.)e(The)g(mec)m(hanisation)
h(follo)m(w)m(ed)g(the)g(exp)s(osition)e(of)i(Herstein)g(\(1975\))i(in)
378 1789 y(the)31 b(de\014nitions)c(and)j(results)f(deriv)m(ed.)519
1902 y(The)43 b(pro)s(of)f(scripts)g(implemen)m(ted)f(during)g(the)i
(mec)m(hanisation)g(are)g(quite)g(readable)f(and)378
2015 y(m)m(uc)m(h)c(easier)g(to)h(follo)m(w)f(than)g(tactic-based)h
(pro)s(ofs.)64 b(The)38 b(readabilit)m(y)e(of)j(the)f(SPL)f(pro)s(ofs)h
(is)378 2128 y(attributed)30 b(to)h(the)f(follo)m(wing)f(factors.)514
2315 y FN(\017)46 b FT(The)32 b(pro)s(ofs)g(con)m(tain)h(information)d
(whic)m(h)i(is)f(relev)-5 b(an)m(t)33 b(for)f(a)h(h)m(uman)f(reader)g
(who)g(is)g(try-)605 2428 y(ing)39 b(to)h(follo)m(w)f(the)h(pro)s(ofs.)
67 b(The)39 b(SPL)g(language)h(is)f(based)g(on)h(\(a)g(small)e(fragmen)
m(t)i(of)7 b(\))605 2541 y(Mizar)35 b(whic)m(h)f(has)g(a)i(rather)f
(easy)g(to)h(follo)m(w)e(syn)m(tax)i(and)e(supp)s(orts)f(a)i
(declarativ)m(e)h(st)m(yle)605 2654 y(of)e(pro)s(of)f(dev)m(elopmen)m
(t.)51 b(F)-8 b(urthermore,)34 b(structured)f(straigh)m(tforw)m(ard)g
(justi\014cations)f(are)605 2767 y(used)38 b(to)h(pro)m(v)m(e)g(simple)
d(results.)63 b(Suc)m(h)37 b(justi\014cations)g(con)m(tain)h(some)h(of)
f(the)h(inferences)605 2880 y(used)34 b(in)g(the)i(deriv)-5
b(ation)33 b(pro)s(cess)i(and)f(omit)h(all)f(tedious)g(inferences)h
(suc)m(h)f(as)i(particular)605 2993 y(instan)m(tiations)f(of)g(v)-5
b(ariables.)55 b(The)36 b(e\013ort)g(required)e(for)h(the)h(implemen)m
(tation)e(of)i(pro)s(ofs)605 3106 y(using)h(structured)f
(justi\014cations)h(w)m(as)h(not)g(m)m(uc)m(h)g(greater)h(than)f(the)g
(e\013ort)g(required)e(in)605 3219 y(implemen)m(ting)23
b(unstructured)g(ones.)39 b(It)24 b(w)m(as)i(actually)e(noticed)g(that)
i(b)m(y)e(explicitly)f(stating)605 3332 y(the)34 b(inferences)f(in)f
(structured)h(justi\014cations,)g(one)h(can)g(ha)m(v)m(e)h(a)f(b)s
(etter)g(idea)f(of)h(whether)605 3445 y(the)g(justi\014cations)d(used)i
(con)m(tain)g(all)g(the)g(necessarily)f(premises)g(and)h(whether)f
(they)i(can)605 3557 y(b)s(e)c(mac)m(hine)g(c)m(hec)m(k)m(ed)i(b)m(y)f
(the)f(pro)m(v)m(er)h(of)g(the)f(system.)514 3745 y FN(\017)46
b FT(The)34 b(scripts)g(are)h(organised)f(in)m(to)g(sections)h(suc)m(h)
f(that)i(theorems)e(whic)m(h)g(ha)m(v)m(e)i(the)e(same)605
3858 y(h)m(yp)s(otheses)27 b(are)g(group)s(ed)f(together.)41
b(This)25 b(has)h(the)h(e\013ect)i(of)e(shortening)e(the)i(statemen)m
(ts)605 3971 y(of)36 b(the)f(theorems)g(as)h(w)m(ell)e(as)h(the)g
(formal)g(pro)s(ofs,)h(whic)m(h)d(also)j(results)d(in)h(scripts)g(whic)
m(h)605 4084 y(are)d(relativ)m(ely)f(easy)h(to)g(follo)m(w.)514
4271 y FN(\017)46 b FT(Lo)s(cal)30 b(abbreviations)f(are)i(used)f(to)h
(abbreviate)f(commonly)g(used)f(subterms.)514 4459 y
FN(\017)46 b FT(Appropriate)24 b(simpli\014ers)e(whic)m(h)i(are)i(able)
e(to)j(query)d(the)i(SPL)e(database)i(of)g(trivial)d(kno)m(wl-)605
4572 y(edge)j(are)g(implemen)m(ted)d(and)i(incorp)s(orated)f(in)g(the)i
(SPL)e(language)h(as)h(the)f(mec)m(hanisation)605 4685
y(of)32 b(the)g(theory)h(progresses.)45 b(The)32 b(use)f(of)h
(simpli\014ers)c(greatly)33 b(reduced)e(the)h(length)g(of)g(the)605
4798 y(formal)d(pro)s(ofs.)39 b(The)29 b(database)h(of)f(trivial)f(kno)
m(wledge)h(is)f(used)h(to)h(store)g(and)e(deriv)m(e)h(facts)605
4911 y(whic)m(h)21 b(are)i(considered)e(to)i(b)s(e)f(trivial)e(b)m(y)j
(the)f(author)g(of)h(the)f(pro)s(ofs.)37 b(As)23 b(a)f(result,)i(m)m
(uc)m(h)e(te-)605 5024 y(dious)j(inferences)h(are)h(omitted)f(from)g
(the)h(formal)e(pro)s(ofs)h(and)g(are)h(deriv)m(ed)e(automatically)605
5137 y(during)j(pro)s(of)i(c)m(hec)m(king.)514 5324 y
FN(\017)46 b FT(Meaningful)27 b(iden)m(ti\014er)g(names)i(are)g(giv)m
(en)g(to)g(assumptions)e(and)i(pro)s(of)f(step)g(results.)39
b(The)605 5437 y(parser)c(of)h(the)h(SPL)d(language)j(allo)m(ws)e
(certain)h(c)m(haracters,)j(whic)m(h)34 b(are)i(usually)e(used)h(to)605
5550 y(denote)41 b(op)s(erators)f(suc)m(h)g(as)g Fw(=)g
FT(and)f Fw(+)p FT(,)k(to)d(b)s(e)g(used)f(in)g(the)h(name)g(of)h(iden)
m(ti\014ers.)67 b(As)40 b(a)605 5663 y(result,)i(the)e(iden)m(ti\014er)
e(names)h(used)g(can)h(b)s(e)g(quite)f(expressiv)m(e)g(and)g(close)i
(to)f(the)g(facts)p eop
%%Page: 210 220
210 219 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(210)605
396 y(they)27 b(are)f(represen)m(ting.)39 b(F)-8 b(or)27
b(instance,)g(an)f(iden)m(ti\014er)e(name)j Fw(gN=Ng)d
FT(w)m(as)i(used)g(for)g(the)g(fact)605 509 y FP(g)s(N)36
b FT(=)25 b FP(N)10 b(g)34 b FT(in)29 b(the)i(pro)s(of)e(fragmen)m(t)i
(giv)m(en)g(in)e(page)i(205.)519 696 y(Figure)25 b(26)h(illustrates)d
(an)i(SPL)g(pro)s(of)f(of)h(one)h(of)f(the)h(results)e(deriv)m(ed)g(in)
g(the)h(mec)m(hanisation.)378 809 y(The)c(result)g(states)j(that)e(the)
h(function)d FP(\025x:N)10 b(x)22 b FT(is)g(a)g(homomorphism)e(if)h
FP(N)32 b FT(is)21 b(a)h(normal)f(subgroup.)378 922 y(It)35
b(is)e(practically)h(a)h(rew)m(ording)e(of)i(the)g(fact)h(that)f(\()p
FP(N)10 b(a)p FT(\)\()p FP(N)g(b)p FT(\))34 b(=)e FP(N)10
b(ab)p FT(,)36 b(for)e(all)g(elemen)m(ts)h FP(a)f FT(and)378
1035 y FP(b)44 b FT(of)h(some)g(group)f FP(G)f FT(and)h(where)g
FP(N)55 b FT(is)43 b(normal)g(in)g FP(G)p FT(,)48 b(whic)m(h)43
b(is)h(deriv)m(ed)f(as)i(the)f(theorem)378 1148 y Fw(Normal_NaNb_Nab)-6
b FT(.)72 b(It)42 b(can)f(b)s(e)f(seen)i(that)f(the)g(pro)s(of)g(of)g
(the)g(theorem)h(deriv)m(ed)e(in)g(\014gure)g(26)378
1261 y(uses)29 b(only)f(the)h(theorem)h Fw(Normal_NaNb_Nab)23
b FT(together)30 b(with)e(lo)s(cally)g(declared)g(assumptions)f(and)378
1373 y(the)38 b(necessarily)e(de\014nitions,)h(most)h(of)f(whic)m(h)f
(are)i(sp)s(eci\014ed)e(as)i(simpli\014ers)33 b(so)38
b(that)g(they)f(are)378 1486 y(unfolded)26 b(implicitly)f(during)i(pro)
s(of)h(searc)m(h.)41 b(The)28 b(same)h(theorem)g(can)g(b)s(e)f(deriv)m
(ed)g(b)m(y)g(the)h(HOL)378 1599 y(tactic)i(pro)s(of:)378
1811 y FM(val)47 b(Homo_RightCoset)c(=)48 b(prove)473
1923 y(\(--`)p FN(8)p FM(\(G:'a)d FN(!)j FM(bool\))e(p.)h(Group)g
(\(G,p\))f FN(\))712 2036 y FM(\()p FN(8)p FM(N.)g(NormalSG)g(\(G,p\))g
(N)i FN(\))760 2149 y FM(Homomorphism)c(\(G,p\))j(\(QuotientGp)d
(\(G,p\))j(N\))g(\(RightCoset)e(\(N,p\)\)\)`--,)473 2262
y(REWRITE_TAC)g([Homomorphism,QuotientGp,)o(fIn)o(to,)1094
2375 y(Str_Pres,QuotientSet])d(THEN)473 2488 y(REPEAT)k(STRIP_TAC)g
(THENL)473 2601 y([BETA_TAC)g(THEN)521 2714 y(EXISTS_TAC)f
(\(--`x:'a`--\))g(THEN)521 2827 y(ASM_REWRITE_TAC)f([],)521
2940 y(CONV_TAC)i(SYM_CONV)f(THEN)521 3053 y(IMP_RES_TAC)g
(Normal_NaNb_Nab]\);)378 3264 y FT(The)e(ab)s(o)m(v)m(e)j(pro)s(of)d
(is)g(shorter)g(than)h(the)g(\(relev)-5 b(an)m(t)45 b(fragmen)m(t)f(of)
h(the\))f(SPL)f(pro)s(of)g(giv)m(en)h(in)378 3377 y(\014gure)35
b(26,)j(ho)m(w)m(ev)m(er)f(it)e(is)f(harder)h(to)h(follo)m(w)f(b)s
(ecause)g(it)g(is)g(not)g(targeted)i(to)f(a)g(h)m(uman)f(reader)378
3490 y(but)30 b(to)i(the)f(HOL)f(pro)s(of)h(c)m(hec)m(k)m(er.)44
b(The)30 b(complexit)m(y)h(of)g(the)g(pro)s(of)f(steps)h(in)f(the)h
(tactic)h(pro)s(of)e(is)378 3603 y(non-homogeneous)25
b(as)g(the)g(pro)s(of)e(includes)g(rather)h(rather)g(trivial)f
(inferences,)i(suc)m(h)f(as)h Fw(BETA_TAC)378 3715 y
FT(and)j Fw(CONV_TAC)40 b(SYM_CONV)m FT(,)29 b(as)g(w)m(ell)e(as)i(the)
f(relev)-5 b(an)m(t)29 b(inference)f Fw(IMP_RES_TAC)39
b(Normal_NaNb_Nab)-6 b FT(.)519 3828 y(T)e(able)31 b(3)h(lists)e(the)i
(lengths)f(of)g(di\013eren)m(t)g(fragmen)m(ts)h(of)g(the)g(source)g(co)
s(de)f(dev)m(elop)s(ed)g(during)378 3941 y(the)c(mec)m(hanisation)g(of)
g(group)f(theory)-8 b(.)41 b(F)-8 b(or)27 b(eac)m(h)i(part)e(of)g(the)g
(mec)m(hanisation,)g(the)h(total)f(length)378 4054 y(of)j(the)h(source)
g(co)s(de)f(is)g(divided)d(as)k(follo)m(ws:)378 4241
y FQ(ML)k(declarations)46 b FT(whic)m(h)25 b(include)f(the)i
(de\014nitions)e(of)i(ML)g(functions)f(corresp)s(onding)f(to)j(sim-)605
4354 y(pli\014ers)h(and)i(the)g(query)g(functions)f(of)h(the)h
(database)g(of)g(trivial)d(kno)m(wledge.)378 4541 y FQ(HOL)34
b(de\014nitions)46 b FT(whic)m(h)36 b(in)m(v)m(olv)m(e)h(the)g
(de\014nition)e(of)i(HOL)g(constan)m(ts)h(using)d(the)i(functions)605
4654 y(supplied)27 b(with)i(the)i(system.)378 4841 y
FQ(SPL)k(pro)s(ofs)46 b FT(whic)m(h)30 b(are)g(basically)f(the)i(pro)s
(ofs)e(of)i(results)e(in)g(SPL.)519 5028 y(The)i(lengths)f(in)g(table)g
(3)i(can)f(b)s(e)f(compared)h(with)f(the)h(lengths)g(of)g(the)g
(di\013eren)m(t)f(fragmen)m(ts)378 5141 y(of)i(the)g(source)g(co)s(de)g
(of)g(the)g(mec)m(hanisation)g(of)g(the)g(theory)g(of)g(computation)g
(in)f(HOL)g(giv)m(en)h(in)378 5253 y(table)25 b(1,)h(page)g(34.)39
b(It)25 b(can)g(b)s(e)f(seen)h(that)h(a)f(substan)m(tial)e(amoun)m(t)i
(of)g(the)g(mec)m(hanisation)g(of)f(group)378 5366 y(theory)41
b(is)f(dedicated)g(to)i(the)e(implemen)m(tation)g(of)h(pro)s(of)f(pro)s
(cedures.)70 b(On)40 b(the)h(other)g(hand,)378 5479 y(almost)f(all)f
(of)i(the)f(implemen)m(tation)f(of)h(the)h(mec)m(hanisation)e(of)i(the)
f(theory)h(of)f(computation)378 5592 y(consists)28 b(of)g(tactic)i(pro)
s(ofs.)39 b(Th)m(us,)28 b(although)g(it)f(is)h(noticed)g(that)h(not)f
(m)m(uc)m(h)h(e\013ort)g(w)m(as)g(required)378 5705 y(during)f(the)j
(implemen)m(tation)e(of)i(the)g(SPL)e(pro)s(ofs)h(of)h(the)f(results)g
(giv)m(en)g(in)g(this)f(c)m(hapter,)i(quite)p eop
%%Page: 211 221
211 220 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(211)p
378 417 3453 4 v 376 5601 4 5184 v 515 653 a Fw(let)42
b("G:)h('a)f Fu(!)i Fw(bool")689 752 y("p:)f('a)f Fu(!)i
Fw('a)f Fu(!)g Fw('a";)515 952 y(assume)e(GroupG:)84
b("Group)41 b(\(G,p\)";)515 1051 y(consider)f(is_group)g(GroupG;)515
1250 y(let)i("N:'a)g Fu(!)h Fw(bool";)515 1350 y(assume)e(NorN:)h
("NormalSG)d(\(G,p\))j(N";)820 1450 y(NsgG:)g("SubGroup)d(p)44
b(N)f(G")g(by)f(<NormalSG>NorN;)515 1549 y(consider)e(is_subgroup)f
(NsgG;)515 1749 y(define)85 b(GN_def:)40 b("GN)86 b(=)43
b(QuotientSet)c(\(G,p\))j(N")907 1848 y(P_def:)f("P)130
b(=)43 b(SProd)f(p")820 1948 y(GNP_def:)e("GNP)i(=)h(QuotientGp)d
(\(G,p\))h(N";)602 2047 y(then)h(GNP:)g("GNP)g(=)h(\(GN,P\)"<GN_def,P)o
(_de)o(f,)o(GNP)o(_d)o(ef)o(,Qu)o(ot)o(ien)o(tG)o(p>)37
b(by)43 b(fol;)515 2147 y(simplify)d(with)i(GNP;)515
2346 y(theorem)f(Homo_RightCoset)o(:)d("Homomorphism)g(\(G,p\))j(GNP)i
(\(RightCoset)c(\(N,p\)\)")515 2446 y(proof)602 2546
y(into:)j("fInto)f(G)i(GN)g(\(RightCoset)c(\(N,p\)\)")602
2645 y(proof)689 2745 y(let)k("a:'a";)689 2844 y(assume)e(Ga:)i("G)f
(a";)689 2944 y(consider)f(in_set)g(Ga;)689 3044 y(then)h("GN)h
(\(RightCoset)c(\(N,p\))i(a\)"<GN_def,inset)o(>)d(by)k(fol;)689
3143 y(simplify)f(with)g(fInto;)602 3243 y(end;)602 3442
y(strpr:)g("Str_Pres)f(\(G,p\))i(GNP)g(\(RightCoset)d(\(N,p\)\)")602
3542 y(proof)689 3642 y(let)k("a:'a")e("b:'a";)689 3741
y(assume)g(Ga:)i("G)f(a")820 3841 y(and)g(Gb:)h("G)f(b";)689
3940 y(consider)f(in_set)g(Ga)h(and)h(Gb;)689 4140 y("RightCoset)c
(\(N,p\))j(\(p)h(a)g(b\))g(=)907 4239 y(P)g(\(RightCoset)c(\(N,p\))j
(a\))h(\(RightCoset)c(\(N,p\))i(b\)"<P_def>)1125 4339
y(by)i(Normal_NaNb_Nab)37 b(on)43 b(GroupG)e(&)i(NorN)f(&)h(Ga)g(&)g
(Gb;)689 4538 y(simplify)e(with)g(Str_Pres;)602 4638
y(end;)602 4837 y("Homomorphism)d(\(G,p\))k(GNP)g(\(RightCoset)d
(\(N,p\)\)"<Homomorph)o(is)o(m>)820 4937 y(by)k(into)f(and)g(strpr;)515
5136 y(qed;)899 5431 y FT(Figure)30 b(26:)41 b(A)31 b(SPL)e(Pro)s(of)h
(of)h(a)f(Theorem)g(on)h(Homomorphisms.)p 3829 5601 V
378 5604 3453 4 v eop
%%Page: 212 222
212 221 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(212)p
378 918 3453 4 v 376 5101 4 4183 v 1073 1142 a Fb(Sets,)31
b(Relations)f(and)j(F)-8 b(unctions)1291 1241 y Ft(ML)28
b(declarations:)194 b(160)26 b(lines)1291 1341 y(HOL)h(de\014nitions:)
206 b(230)26 b(lines)1291 1441 y(SPL)h(pro)r(ofs:)381
b(420)26 b(lines)1346 1540 y Fb(T)-8 b(otal:)462 b(810)31
b(lines)1073 1739 y(In)m(tro)s(ducing)h(Groups)1291 1839
y Ft(ML)c(declarations:)194 b(400)26 b(lines)1291 1939
y(HOL)h(de\014nitions:)247 b(80)27 b(lines)1291 2038
y(SPL)g(pro)r(ofs:)381 b(230)26 b(lines)1346 2138 y Fb(T)-8
b(otal:)462 b(710)31 b(lines)1073 2337 y(Subgroups)1291
2437 y Ft(ML)d(declarations:)194 b(230)26 b(lines)1291
2536 y(HOL)h(de\014nitions:)160 b Fv(<)22 b Ft(10)27
b(lines)1291 2636 y(SPL)g(pro)r(ofs:)381 b(260)26 b(lines)1346
2736 y Fb(T)-8 b(otal:)462 b(490)31 b(lines)1073 2935
y(Congruences,)g(Cosets)g(and)h(Pro)s(ducts)g(of)g(Subgroups)1291
3035 y Ft(ML)c(declarations:)194 b(420)26 b(lines)1291
3134 y(HOL)h(de\014nitions:)247 b(20)27 b(lines)1291
3234 y(SPL)g(pro)r(ofs:)339 b(1530)26 b(lines)1346 3333
y Fb(T)-8 b(otal:)415 b(1970)30 b(lines)1073 3533 y(Normal)g(Subgroups)
i(and)g(Quotien)m(t)f(Groups)1291 3632 y Ft(ML)d(declarations:)342
b(-)1291 3732 y(HOL)27 b(de\014nitions:)247 b(10)27 b(lines)1291
3832 y(SPL)g(pro)r(ofs:)381 b(630)26 b(lines)1346 3931
y Fb(T)-8 b(otal:)462 b(640)31 b(lines)1073 4130 y(Homomorphisms)26
b(and)32 b(Isomorphisms)1291 4230 y Ft(ML)c(declarations:)194
b(130)26 b(lines)1291 4330 y(HOL)h(de\014nitions:)247
b(30)27 b(lines)1291 4429 y(SPL)g(pro)r(ofs:)339 b(1120)26
b(lines)1346 4529 y Fb(T)-8 b(otal:)415 b(1280)30 b(lines)709
4931 y FT(T)-8 b(able)30 b(3:)41 b(On)30 b(the)h(Source)f(Co)s(de)g(of)
g(the)h(Mec)m(hanisation)f(of)h(Group)e(Theory)-8 b(.)p
3829 5101 V 378 5104 3453 4 v eop
%%Page: 213 223
213 222 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(213)378
396 y(a)29 b(lot)g(of)g(e\013ort)h(w)m(as)f(needed)f(in)g(the)h
(implemen)m(tation)e(of)i(the)g(pro)s(of)f(pro)s(cedures)g(that)h
(automate)378 509 y(the)36 b(trivial)e(inferences)h(omited)g(from)g
(the)h(formal)f(SPL)g(pro)s(ofs.)56 b(The)35 b(p)s(ossibilit)m(y)d(of)k
(reducing)378 622 y(the)27 b(e\013ort)h(required)d(in)h(the)h(implemen)
m(tation)f(of)h(pro)s(of)g(pro)s(cedures)f(\(esp)s(ecially)f
(simpli\014ers)e(and)378 735 y(database)34 b(query)f(functions\),)h(b)m
(y)g(dev)m(eloping)e(sp)s(ecialised)f(high-lev)m(el)i(languages)h(for)f
(instance,)378 848 y(is)c(an)i(in)m(teresting)e(direction)g(for)i
(future)e(researc)m(h.)519 961 y(Although)g(the)g(SPL)g(pro)s(ofs)g(of)
g(the)h(results)e(giv)m(en)i(in)e(this)g(section)i(are)g(quite)f
(similar)d(to)31 b(the)378 1074 y(pro)s(ofs)22 b(giv)m(en)g(in)g(the)g
(literature,)i(the)f(SPL)f(pro)s(ofs)f(of)i(the)g(results)e(on)i
(\014nite)e(groups)h(attempted)i(b)m(y)378 1187 y(the)i(author)f(w)m
(ere)h(not)f(as)h(clear)g(as)f(the)h(informal)d(ones.)40
b(The)24 b(reason)i(for)f(this)g(is)f(that)i(substan)m(tial)378
1300 y(automation)j(ma)m(y)f(b)s(e)g(required)e(to)j(deriv)m(e)f(the)g
(inferences)g(on)g(\014nite)f(sets)i(whic)m(h)d(are)j(considered)378
1413 y(to)f(b)s(e)f(trivial)e(b)m(y)j(a)g(h)m(uman)e(reader.)40
b(In)26 b(particular,)h(sev)m(eral)h(of)f(the)h(results)e(that)i(are)g
(considered)378 1526 y(to)i(b)s(e)f(rather)h(trivial)e(in)g(the)i
(informal)e(literature)h(ma)m(y)h(require)e(some)j(form)e(of)h
(induction)d(to)k(b)s(e)378 1638 y(deriv)m(ed)g(formally)-8
b(.)43 b(The)31 b(automation)h(of)g(pro)s(ofs)f(in)m(v)m(olving)f
(induction)f(is)i(not)h(straigh)m(tforw)m(ard,)378 1751
y(since)25 b(for)f(instance,)j(one)e(often)h(requires)e(the)h(disco)m
(v)m(ery)g(of)h(lemmata)f(whic)m(h)f(are)i(general)f(enough)378
1864 y(for)i(their)f(induction)f(h)m(yp)s(othesis)h(to)i(b)s(e)e(used)g
(in)g(the)i(inductiv)m(e)d(pro)s(of.)39 b(The)27 b(implemen)m(tation)f
(of)378 1977 y(the)31 b(necessarily)e(pro)s(of)h(pro)s(cedures)g(that)h
(w)m(ould)f(mak)m(e)h(reasoning)f(ab)s(out)h(\014nite)e(sets)j(relativ)
m(ely)378 2090 y(straigh)m(tforw)m(ard)e(is)g(also)g(an)g(in)m
(teresting)g(direction)f(for)h(future)f(w)m(ork.)p eop
%%Page: 214 224
214 223 bop 378 1019 a FJ(Chapter)65 b(10)378 1434 y
FR(Conclusions)378 1879 y FT(The)27 b(w)m(ork)h(presen)m(ted)f(in)f
(this)h(thesis)g(in)m(v)m(estigates)h(the)g(implemen)m(tation)e(of)i
(mac)m(hine-c)m(hec)m(k)-5 b(able)378 1992 y(pro)s(ofs)40
b(in)g(a)h(format)g(that)h(is)d(more)i(easily)g(follo)m(w)m(ed)f(b)m(y)
h(a)g(h)m(uman)f(reader.)72 b(In)40 b(this)g(c)m(hapter)378
2105 y(w)m(e)32 b(\014rst)e(summarise)g(the)h(main)f(con)m(tributions)g
(of)h(this)f(thesis,)h(and)g(then)g(discuss)e(a)j(n)m(um)m(b)s(er)e(of)
378 2218 y(directions)f(for)h(future)f(w)m(ork)i(in)e(this)g(area)i(of)
g(researc)m(h.)378 2504 y FH(10.1)136 b(Summary)44 b(of)h(the)h(Main)f
(Con)l(tributions)378 2707 y FT(In)36 b(this)g(section)g(w)m(e)i
(summarise)d(the)i(main)e(con)m(tributions)g(of)i(this)f(thesis)g(whic)
m(h)f(aims)h(at)i(the)378 2820 y(implemen)m(tation)k(of)h(mac)m(hine-c)
m(hec)m(k)-5 b(able)44 b(pro)s(ofs)f(in)e(a)j(readable)e(format.)80
b(The)42 b(motiv)-5 b(ations)378 2933 y(for)39 b(this)f(researc)m(h)h
(are)h(discussed)d(in)h(section)h(2.5)h(and)f(include)e(the)i(fact)h
(that)f(it)g(is)f(easier)h(to)378 3046 y(implemen)m(t,)27
b(correct,)j(and)e(mo)s(dify)e(pro)s(ofs)h(if)g(they)i(can)f(b)s(e)g
(follo)m(w)m(ed)f(easily)-8 b(.)40 b(The)27 b(con)m(tributions)378
3159 y(summerised)h(in)h(this)h(section)g(are)h(categorised)g(as)g
(follo)m(ws:)514 3347 y FN(\017)46 b FI(Case)f(studies)f(involving)h
(tactic-b)-5 b(ase)g(d)45 b(pr)-5 b(o)g(of)47 b(envir)-5
b(onments)7 b FT(:)67 b(Mec)m(hanised)43 b(pro)s(ofs)f(are)605
3460 y(usually)36 b(found)h(using)f(a)j(tactic-based)g(en)m(vironmen)m
(t,)h(and)d(in)g(c)m(hapter)i(3)f(w)m(e)h(study)e(the)605
3572 y(st)m(yle)d(of)g(tactic-based)g(pro)s(of)f(disco)m(v)m(ery)h(and)
f(argue)g(that)h(pro)s(ofs)f(found)f(in)g(this)h(manner)605
3685 y(are)e(v)m(ery)g(hard)e(to)i(follo)m(w.)514 3873
y FN(\017)46 b FI(The)39 b(implementation)i(of)e(the)h(SPL)e(pr)-5
b(o)g(of)41 b(che)-5 b(cker)10 b FT(:)54 b(The)37 b(SPL)f(pro)s(of)h
(language,)i(whic)m(h)605 3986 y(is)d(based)h(on)f(the)i(Mizar)e
(language)i(is)e(discussed)f(in)g(c)m(hapter)j(4.)60
b(SPL)36 b(pro)s(ofs)g(are)i(more)605 4099 y(readable)44
b(than)h(tactic)h(pro)s(ofs)e(b)s(ecause)g(of)h(their)f(declarativ)m(e)
h(nature.)84 b(F)-8 b(urthermore,)605 4212 y(the)30 b(SPL)e(language)i
(is)f(extensible,)g(in)f(the)h(sense)h(that)g(the)f(deductiv)m(e)h(p)s
(o)m(w)m(er)f(of)h(its)f(pro)s(of)605 4325 y(c)m(hec)m(k)m(er)e(can)d
(b)s(e)g(extended)g(in)f(a)i(disciplined)19 b(w)m(a)m(y)26
b(during)c(the)i(mec)m(hanisation)g(of)g(a)h(theory)-8
b(.)514 4512 y FN(\017)46 b FI(Structur)-5 b(e)g(d)29
b(str)-5 b(aightforwar)g(d)31 b(justi\014c)-5 b(ations)7
b FT(:)40 b(The)24 b(notion)f(of)i(structured)e(straigh)m(tforw)m(ard)
605 4625 y(justi\014cations)29 b(is)h(studied)f(in)g(c)m(hapter)i(6.)42
b(These)31 b(justi\014cations)e(di\013er)g(from)h(the)h(unstruc-)605
4738 y(tured)h(justi\014cation)e(of)j(Mizar)f(and)g(similar)d
(languages)j(b)m(y)h(including)28 b(more)33 b(information)605
4851 y(on)d(whic)m(h)e(inferences)h(are)h(used)e(to)j(deriv)m(e)e(the)h
(conclusion)e(of)h(the)h(justi\014cation.)39 b(It)30
b(is)f(ar-)605 4964 y(gued)h(that)h(structured)e(justi\014cations)g
(are)h(easier)g(to)h(follo)m(w)e(and)h(more)g(e\016cien)m(t)h(to)g(pro)
s(of)605 5077 y(c)m(hec)m(k)43 b(than)f(unstructured)d(ones.)74
b(Chapter)41 b(8)h(discusses)e(ho)m(w)h(the)h(searc)m(h)g(space)g(con-)
605 5190 y(sidered)34 b(for)h(c)m(hec)m(king)g(structured)g
(justi\014cation)e(can)j(b)s(e)e(restricted.)54 b(The)35
b(results)f(giv)m(en)605 5303 y(in)e(c)m(hapter)h(8)g(use)g(a)g(v)m
(ersion)g(of)g(\014rst-order)e(logic)i(whose)g(form)m(ulae)f(are)h
(annotated)h(with)605 5416 y FI(c)-5 b(olours)51 b FT(in)41
b(order)g(to)i(restrict)f(the)g(pro)s(of)f(searc)m(h.)76
b(This)41 b(coloured)g(\014rst-order)g(logic)h(is)605
5528 y(studied)29 b(in)g(c)m(hapter)i(7.)2035 5954 y(214)p
eop
%%Page: 215 225
215 224 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(215)514 396 y FN(\017)46 b FI(The)29 b(implementation)i(of)e(the)g
FN(C)5 b(B)s(S)i(E)36 b FI(derive)-5 b(d)29 b(rule)7
b FT(:)39 b(The)25 b FN(C)5 b(B)s(S)i(E)34 b FT(tableau)25
b(calculus,)h(whic)m(h)605 509 y(is)43 b(complete)h(for)f
(\014rst-order)f(logic)h(with)g(equalit)m(y)-8 b(,)47
b(is)42 b(describ)s(ed)f(in)h(c)m(hapter)i(5.)81 b(This)605
622 y(calculus)22 b(is)h(implemen)m(ted)f(as)i(a)g(HOL)f(deriv)m(ed)f
(rule)h(and)f(is)h(used)g(in)f(c)m(hec)m(king)i(SPL)f(scripts.)514
810 y FN(\017)46 b FI(The)30 b(Me)-5 b(chanisation)32
b(of)e(Gr)-5 b(oup)32 b(The)-5 b(ory)31 b(in)f(SPL)p
FT(:)e(The)f(pro)s(ofs)f(of)i(a)g(n)m(um)m(b)s(er)e(of)i(results)e(in)
605 923 y(group)34 b(theory)h(are)f(implemen)m(ted)f(in)g(SPL,)h(and)g
(discussed)e(in)h(c)m(hapter)i(9.)53 b(This)33 b(mec)m(ha-)605
1036 y(nisation)d(is)g(a)i(case)g(study)f(in)e(the)j(use)f(of)g(an)g
(extensible)f(declarativ)m(e)i(pro)s(of)e(language)i(for)605
1149 y(the)f(implemen)m(tation)e(of)h(readable,)g(mac)m(hine-c)m(hec)m
(k)-5 b(able)32 b(pro)s(ofs.)519 1336 y(These)e(con)m(tributions)f(are)
i(discussed)d(in)h(more)i(detail)e(b)s(elo)m(w.)378 1580
y FG(Case)38 b(Studies)f(In)m(v)m(olving)f(T)-9 b(actic-Based)37
b(Pro)s(of)g(En)m(vironmen)m(ts)378 1751 y FT(Chapter)28
b(3)g(discusses)f(t)m(w)m(o)j(case)f(studies)e(in)m(v)m(olving)g
(tactic-based)i(pro)s(of)f(dev)m(elopmen)m(t)h(systems.)378
1864 y(The)38 b(\014rst)h(case)h(study)e(in)m(v)m(olv)m(es)h(the)g(mec)
m(hanisation)f(of)h(a)h(n)m(um)m(b)s(er)d(of)i(results)f(in)g(the)h
(theory)378 1977 y(of)33 b(computation)g(using)e(the)i(HOL)f(system.)48
b(This)31 b(mec)m(hanisation)i(is)e(based)i(on)g(the)f(Unlimited)378
2090 y(Register)i(Mac)m(hine)h(\(URM\))g(mo)s(del)e(of)h(computation)g
(as)h(discussed)d(in)h(the)h(textb)s(o)s(ok)h(b)m(y)f(Cut-)378
2203 y(land)j(\(1980\),)43 b(and)38 b(includes)f(the)h(pro)s(of)g(of)h
(the)g(result)e(that)i(partial)f(recursiv)m(e)f(functions)h(can)378
2316 y(b)s(e)33 b(computed)g(b)m(y)h(URM)f(programs.)50
b(The)33 b(second)h(case)g(study)f(in)m(v)m(olv)m(es)h(the)f(pro)s(of)g
(of)h(the)f FP(S)3761 2283 y FO(m)3756 2338 y(n)378 2429
y FT(theorem)g(in)d(the)j(Co)s(q)f(system.)46 b(The)32
b(pro)s(of)f(of)h(this)f(theorem)i(is)e(based)h(on)g(a)h(mo)s(del)e(of)
h(compu-)378 2542 y(tation)e(similar)c(to)k(the)g(partial)e(recursiv)m
(e)h(functions)f(mo)s(del.)39 b(The)29 b(pro)s(ofs)f(implemen)m(ted)g
(during)378 2655 y(these)38 b(case)h(studies)e(w)m(ere)h(found)f(in)m
(teractiv)m(ely)h(using)e(the)i(tactic-based)h(en)m(vironmen)m(t)f(of)g
(the)378 2768 y(t)m(w)m(o)d(systems.)51 b(Unfortunately)-8
b(,)35 b(as)f(discussed)e(in)g(section)i(3.5,)i(it)e(is)e(extremely)i
(hard)f(to)i(follo)m(w)378 2880 y(tactic)e(pro)s(ofs)d(without)h(the)h
(appropriate)f(feedbac)m(k)h(from)f(the)h(theorem)g(pro)m(ving)f
(system.)45 b(In)30 b(a)378 2993 y(tactic-based)36 b(pro)s(of)e(en)m
(vironmen)m(t,)i(tactics)f(are)h(applied)c(in)m(teractiv)m(ely)j(to)h
(solv)m(e)f(certain)f(goals)378 3106 y(automatically)-8
b(,)44 b(or)d(to)h(break)f(goals)h(in)m(to)f(simpler)e(subgoals.)72
b(A)41 b(tactic)h(pro)s(of)f(of)g(a)g(theorem)378 3219
y(con)m(tains)31 b(the)g(sequence)g(of)g(tactics)g(required)e(to)j(pro)
m(v)m(e)f(the)g(theorem,)g(and)g(it)f(is)g(hard)f(to)j(follo)m(w)378
3332 y(since)h(it)h(do)s(es)g(not)g(state)i(the)e(e\013ect)h(of)g(the)f
(application)e(of)j(eac)m(h)g(tactic)g(on)f(the)g(goal.)53
b(Similar)378 3445 y(argumen)m(ts)28 b(on)g(the)g(unreadabilit)m(y)d
(of)j(tactic)g(pro)s(ofs)f(can)h(b)s(e)f(found,)h(for)f(instance,)h(in)
f(\(Harrison)378 3558 y(1997\))i(and)d(\(Syme)h(1998\).)42
b(As)26 b(a)i(result,)e(other)h(pro)s(of)g(st)m(yles)g(are)g(required)e
(for)h(the)i(implemen)m(ta-)378 3671 y(tion)k(of)h(mac)m(hine-c)m(hec)m
(k)-5 b(able)34 b(pro)s(ofs)e(if)g(the)h(readabilit)m(y)e(of)i(the)g
(pro)s(ofs)e(is)h(a)h(requiremen)m(t.)47 b(The)378 3784
y(t)m(w)m(o)30 b(case)f(studies)e(are)h(also)g(used)g(in)f(section)h
(3.4)h(to)g(compare)g(the)g(di\013eren)m(t)e(w)m(a)m(ys)i(that)g
(theories)378 3897 y(are)i(mec)m(hanised)f(in)f(the)h(HOL)g(and)g(Co)s
(q)g(systems.)378 4140 y FG(The)38 b(Implemen)m(tation)c(of)k(the)f
(SPL)h(Pro)s(of)f(Chec)m(k)m(er)378 4312 y FT(One)f(of)g(the)g(main)f
(con)m(tributions)g(of)h(this)f(thesis)g(is)h(the)g(implemen)m(tation)f
(of)h(a)g(pro)s(of)g(c)m(hec)m(k)m(er)378 4425 y(for)c(a)g(declarativ)m
(e)h(pro)s(of)e(language.)47 b(W)-8 b(e)33 b(call)e(this)g(language)i
(SPL)e(whic)m(h)g(is)g(short)h(for)g(`Simple)378 4538
y(Pro)s(of)f(Language'.)45 b(Pro)s(ofs)31 b(implemen)m(ted)e(in)h(a)i
(declarativ)m(e)g(language)g(do)f(not)h(explicitly)d(state)378
4650 y(all)k(the)h(details)e(ab)s(out)i FI(how)45 b FT(a)34
b(theorem)g(is)f(pro)m(v)m(ed,)i(but)e(rather)h(state)h
FI(what)44 b FT(is)32 b(required.)49 b(The)378 4763 y(SPL)35
b(language)i(is)e(based)h(on)g(the)g(theorem)h(pro)m(ving)e(fragmen)m
(t)i(of)f(the)h(Mizar)f(language.)58 b(The)378 4876 y(pro)s(of)31
b(c)m(hec)m(k)m(er)k(of)d(the)g(SPL)f(language)i(deriv)m(es)e(HOL)h
(theorems)h(from)e(SPL)h(pro)s(of)f(scripts,)g(and)378
4989 y(therefore)i(the)f(pro)s(of)g(c)m(hec)m(k)m(er)i(is)e
FI(ful)5 b(ly-exp)-5 b(ansive)p FT(.)47 b(In)32 b(other)g(w)m(ords,)h
(all)e(theorems)i(are)g(deriv)m(ed)378 5102 y(b)m(y)24
b(the)g(primitiv)m(e)d(inferences)i(of)h(the)g(HOL)g(core)g(inference)f
(engine)h(in)e(order)h(to)i(minimise)c(h)m(uman)378 5215
y(errors)30 b(in)f(the)h(pro)s(ofs.)519 5328 y(A)c(sectioning)f(mec)m
(hanism,)h(similar)e(to)i(that)h(of)f(the)g(Co)s(q)f(system,)i(is)e
(used)g(to)h(structure)g(SPL)378 5441 y(scripts)i(in)f(a)j(mo)s(dular)d
(fashion.)39 b(SPL)28 b(scripts)g(are)h(divided)d(in)m(to)j(p)s
(ossibly)d(nested)j(sections.)41 b(As-)378 5554 y(sumptions,)26
b(abbreviations,)h(and)f(other)i(information)e(can)h(b)s(e)g(declared)f
(lo)s(cally)g(to)i(eac)m(h)h(section)378 5667 y(in)21
b(m)m(uc)m(h)h(the)h(same)f(fashion)f(that)i(v)-5 b(ariables)21
b(and)h(functions)f(can)h(b)s(e)g(declared)g(lo)s(cally)e(to)j
(di\013eren)m(t)p eop
%%Page: 216 226
216 225 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(216)378 396 y(program)28 b(mo)s(dules)f(in)g(a)i(structured)e
(programming)h(language.)40 b(As)29 b(discussed)d(in)h(section)i
(4.2.2,)378 509 y(b)m(y)c(sectioning)f(pro)s(of)g(scripts)g(one)h(can)g
(impro)m(v)m(e)g(the)g(readabilit)m(y)e(and)i(pro)s(of-c)m(hec)m(king)g
(e\016ciency)378 622 y(of)30 b(SPL)g(scripts.)519 735
y(In)42 b(this)g(thesis)g(\(and)g(esp)s(ecially)f(in)h(c)m(hapter)h
(2\))h(w)m(e)f(argue)g(that)g(pro)s(of)f(steps)h(whic)m(h)e(are)378
848 y(considered)29 b(to)h(b)s(e)g(ob)m(vious,)f(or)h(trivial,)f(b)m(y)
g(h)m(uman)h(readers)f(should)f(b)s(e)h(omitted)h(during)e(mec)m(h-)378
961 y(anisation)j(in)g(order)g(to)i(impro)m(v)m(e)e(the)i(readabilit)m
(y)-8 b(,)31 b(as)h(w)m(ell)f(as)h(the)g(ease)h(of)g(implemen)m
(tation,)e(of)378 1074 y(mac)m(hine-c)m(hec)m(k)-5 b(able)33
b(pro)s(ofs.)43 b(This)30 b(in)m(v)m(olv)m(es)i(the)g(implemen)m
(tation)e(of)i(pro)s(of)f(c)m(hec)m(k)m(ers)i(that)f(are)378
1187 y(able)25 b(to)i(deriv)m(e)e(theorems)h(whose)g(pro)s(ofs)f(are)h
(implemen)m(ted)e(at)j(a)f(lev)m(el)g(of)g(detail)f(similar)e(to)j
(that)378 1300 y(found)36 b(in)g(mathematical)h(literature.)61
b(An)37 b(imp)s(ortan)m(t)f(issue)g(discussed)g(in)f(this)i(thesis)f
(is)g(that)378 1413 y(what)30 b(a)g(reader)g(considers)e(to)j(b)s(e)e
(ob)m(vious)g(dep)s(ends)f(on)i(her)f(familiarit)m(y)f(and)h(kno)m
(wledge)h(of)g(the)378 1526 y(sub)5 b(ject,)28 b(and)e(therefore)h(v)-5
b(aries)26 b(during)f(the)i(dev)m(elopmen)m(t)g(of)g(a)g(theory)h(|)e
(pro)s(of)g(steps)h(that)g(are)378 1638 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
(b)s(e)e(used)h(to)h(store)g(facts)g(whic)m(h)e(are)h(considered)g(to)h
(b)s(e)378 2429 y(trivial)22 b(b)m(y)i(the)g(dev)m(elop)s(er)g(of)g
(the)g(mec)m(hanised)g(theory)-8 b(.)39 b(The)24 b(kno)m(wledge)g
(stored)g(in)f(the)h(database)378 2542 y(is)34 b(organised)h(in)m(to)g
(categories,)j(and)c(the)i(dev)m(elop)s(er)e(of)h(the)h(theory)f(is)f
(required)f(to)j(implemen)m(t)378 2655 y(functions)d(\(in)h(ML\))h
(whic)m(h)e(query)h(eac)m(h)h(database)h(category)-8
b(.)56 b(These)34 b(query)g(functions)f(should)378 2768
y(b)s(e)g(able)h(to)g(deriv)m(e)g(HOL)f(theorems)h(from)g(the)g(kno)m
(wledge)g(stored)g(in)e(the)i(database)h(using)e(the)378
2880 y(results)25 b(deriv)m(ed)g(in)f(the)j(curren)m(t)e(state)j(of)e
(the)g(theory)-8 b(.)40 b(The)26 b(database)g(is)f(queried)g
(automatically)378 2993 y(b)m(y)35 b(certain)g(comp)s(onen)m(ts)h(of)f
(the)h(pro)s(of)f(c)m(hec)m(k)m(er,)j(so)e(that)g(trivial)d(facts)j
(need)g(not)f(b)s(e)g(justi\014ed)378 3106 y(explicitly)27
b(in)h(the)i(mec)m(hanisation.)40 b(The)29 b(database)h(and)f(its)g
(query)g(functions)f(are)h(implemen)m(ted)378 3219 y(in)e(suc)m(h)h(a)h
(manner)e(that)i(the)g(user)f(can)g(impro)m(v)m(e)g(the)h(deductiv)m(e)
f(p)s(o)m(w)m(er)g(of)h(the)g(query)e(functions)378 3332
y(during)36 b(the)j(mec)m(hanisation)e(of)i(the)f(theory)-8
b(.)66 b(This)36 b(is)h(done)h(b)m(y)h(including)c(new)i(categories)j
(in)378 3445 y(the)33 b(database,)h(implemen)m(ting)c(new)i(query)g
(functions,)g(and)g(up)s(dating)e(the)j(implemen)m(tation)e(of)378
3558 y(existing)24 b(query)g(functions.)37 b(The)24 b(sectioning)g(mec)
m(hanism)g(of)h(SPL)f(allo)m(ws)g(the)h(kno)m(wledge)g(stored)378
3671 y(in)k(the)i(database)g(to)g(b)s(e)f(lo)s(cal)f(to)i(particular)e
(sections)i(only)-8 b(.)519 3784 y(The)28 b(SPL)g(pro)s(of)f(c)m(hec)m
(k)m(er)k(is)d(extensible)f(in)g(man)m(y)i(other)f(w)m(a)m(ys.)41
b(During)28 b(the)g(mec)m(hanisation)378 3897 y(of)i(a)h(particular)e
(theory)-8 b(,)31 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)
j(on)d(top)h(of)g(the)378 509 y(HOL)c(system)g(b)s(ecause)h(of)f(the)h
(w)m(a)m(y)g(the)g(HOL)f(system)g(is)g(designed.)39 b(In)30
b(particular,)489 691 y(1.)46 b(a)22 b(T)-8 b(uring-complete)21
b(metalanguage)j(is)c(a)m(v)-5 b(ailable)22 b(to)g(allo)m(w)f(the)h
(user)f(to)i(extend)f(the)g(system)605 804 y(with)29
b(new)h(pro)s(of)g(pro)s(cedures)f(and)h(pro)s(of)f(en)m(vironmen)m
(ts,)h(and)489 989 y(2.)46 b(the)38 b(fact)g(that)g(all)f(HOL)g
(theorems)g(are)h(constructed)g(using)e(the)i(core)g(inference)e
(engine)605 1102 y(ensures)30 b(that)h(suc)m(h)f(extensions)g(are)g
(safe.)378 1284 y(It)i(is)e(p)s(ossible)f(to)k(implemen)m(t)d(pro)s(of)
h(c)m(hec)m(k)m(ers)i(of)f(declarativ)m(e)g(languages)g(suc)m(h)f(as)h
(SPL)e(on)i(top)378 1397 y(of)e(other)h(theorem)g(pro)s(of)f(en)m
(vironmen)m(ts)f(giv)m(en)i(that)g(they)f(pro)m(vide)g(these)h(t)m(w)m
(o)g(features.)378 1639 y FG(Structured)37 b(Straigh)m(tforw)m(ard)g
(Justi\014cations)378 1811 y FT(In)26 b(this)f(thesis)g(w)m(e)i(also)f
(study)g(the)g(notion)g(of)h(structured)e(straigh)m(tforw)m(ard)h
(justi\014cations)f(whic)m(h)378 1923 y(are)35 b(in)m(tro)s(duced)d(in)
h(c)m(hapter)i(6.)52 b(Simple)32 b(Mizar)i(statemen)m(ts)i(are)f
(justi\014ed)d(b)m(y)i(straigh)m(tforw)m(ard)378 2036
y(justi\014cations)29 b(whic)m(h)g(consist)h(of)g(the)h
Fw(by)e FT(k)m(eyw)m(ord)i(and)f(a)h(list)e(of)h(premises;)g(for)g
(example:)473 2218 y FM(")p FP(a)48 b FM(<)f FP(b)p FM(")h(by)f(")p
FN(8)15 b FP(x;)g(y)s(;)g(z)t FM(.)i FT(\()p FP(x)48
b FM(<)f FP(y)s FT(\))h FN(\))g FT(\()p FP(y)j FM(<)c
FP(z)t FT(\))h FN(\))g FT(\()p FP(x)g FM(<)g FP(z)t FT(\))p
FM(",)f(")p FP(a)h FM(<)f FP(c)p FM(",)h(")p FP(c)g FM(<)f
FP(b)p FM(";)378 2400 y FT(The)30 b(Mizar)g(pro)s(of)g(c)m(hec)m(k)m
(er)i(then)e(deriv)m(es)g(the)h(conclusion)e Fw(")p Fv(a)43
b Fw(<)g Fv(b)p Fw(")29 b FT(from)h(the)h(premises)1280
2604 y Fw(")p Fu(8)13 b Fv(x;)h(y)s(;)g(z)t Fw(.)f Ft(\()p
Fv(x)44 b Fw(<)g Fv(y)s Ft(\))f Fu(\))h Ft(\()p Fv(y)i
Fw(<)e Fv(z)t Ft(\))f Fu(\))g Ft(\()p Fv(x)i Fw(<)e Fv(z)t
Ft(\))p Fw(")o FP(;)1280 2742 y Fw(")p Fv(a)g Fw(<)g
Fv(c)p Fw(")o FP(;)j FT(and)1280 2879 y Fw(")p Fv(c)d
Fw(<)g Fv(b)p Fw(")o FP(:)378 3084 y FT(In)32 b(structured)g(straigh)m
(tforw)m(ard)g(justi\014cations,)g(one)h(giv)m(es)g(more)g(information)
e(on)i(what)f(infer-)378 3197 y(ences)g(are)f(required)f(to)i(deriv)m
(e)f(the)g(conclusion)f(from)h(the)g(premises)f(in)g(the)i
(justi\014cation.)42 b(This)378 3310 y(is)30 b(done)g(through)g(the)h
(op)s(erators)g Fw(on)p FT(,)g Fw(and)e FT(and)h Fw(then)f
FT(whic)m(h)h(corresp)s(ond)f(to)j(high-lev)m(el,)e(or)h
FI(gen-)378 3422 y(er)-5 b(alise)g(d)p FT(,)39 b(v)m(ersions)34
b(of)i(the)f(rules)f(of)i(implication)d(elimination,)h(in)m(tro)s
(duction)f(of)j(conjunctions,)378 3535 y(and)d(transitivit)m(y)f(of)h
(implication)e(resp)s(ectiv)m(ely)-8 b(.)50 b(F)-8 b(or)34
b(example,)g(the)f(conclusion)f(ab)s(o)m(v)m(e)j(can)f(b)s(e)378
3648 y(justi\014ed)28 b(b)m(y:)473 3830 y FM(")p FP(a)48
b FM(<)f FP(b)p FM(")h(by)f(")p FN(8)15 b FP(x;)g(y)s(;)g(z)t
FM(.)i FT(\()p FP(x)48 b FM(<)f FP(y)s FT(\))h FN(\))g
FT(\()p FP(y)j FM(<)c FP(z)t FT(\))h FN(\))g FT(\()p
FP(x)g FM(<)g FP(z)t FT(\))p FM(")g(on)1142 3943 y(")p
FP(a)f FM(<)h FP(c)p FM(")f(and)g(")p FP(c)h FM(<)f FP(b)p
FM(";)378 4124 y FT(Structured)25 b(straigh)m(tforw)m(ard)h
(justi\014cations)f(are)h(ho)m(w)m(ev)m(er)i(not)f(o)m(v)m(er-detailed)
g(and)e(omit)i(sev)m(eral)378 4237 y(simple)33 b(inferences)i(suc)m(h)g
(as)h(the)g(instan)m(tiation)e(of)i(univ)m(ersally)d(quan)m(ti\014ed)h
(v)-5 b(ariables)34 b(and)h(cer-)378 4350 y(tain)e(manipulations)d(on)j
(the)g(structure)g(of)g(form)m(ulae)g(as)g(describ)s(ed)f(in)f(section)
j(6.4.1.)51 b(Most)34 b(of)378 4463 y(the)i(justi\014cations)f
(implemen)m(ted)g(during)f(the)i(mec)m(hanisation)g(of)g(group)g
(theory)g(describ)s(ed)e(in)378 4576 y(c)m(hapter)d(9)g(are)g
(structured)f(justi\014cations.)40 b(The)31 b(implemen)m(tation)e(of)i
(structured)e(justi\014cations)378 4689 y(during)j(this)g(case)j(study)
e(did)f(not)i(need)g(m)m(uc)m(h)g(more)g(e\013ort)h(than)e(the)h
(implemen)m(tation)f(of)h(un-)378 4802 y(structured)26
b(ones)i(since)e(the)i(detailed)e(inferences)h(whic)m(h)f(w)m(ould)g
(mak)m(e)i(the)g(justi\014cation)e(tedious)378 4915 y(to)31
b(implemen)m(t)e(are)i(omitted.)519 5028 y(The)i(role)g(of)h(the)g(op)s
(erators)f(in)g(structured)f(justi\014cations)g(is)h(to)h(giv)m(e)g
(the)g(reader)f(more)h(in-)378 5141 y(formation)f(whic)m(h)g(is)g
(relev)-5 b(an)m(t)34 b(to)h(the)f(understanding)d(of)j(the)g(pro)s
(of.)51 b(This)32 b(mak)m(es)j(structured)378 5254 y(straigh)m(tforw)m
(ard)28 b(justi\014cations)g(easier)g(to)i(follo)m(w)e(than)g
(unstructured)f(ones.)40 b(The)29 b(seman)m(tics)g(of)378
5366 y(structured)i(justi\014cations)f(giv)m(en)i(in)f(section)h(6.4)h
(is)e(non-deterministic,)f(and)h(therefore)i(sev)m(eral)378
5479 y(conclusions)e(can)h(b)s(e)g(justi\014ed)f(b)m(y)h(the)h(same)f
(structured)g(justi\014cation.)45 b(As)33 b(a)g(result,)f(one)g(can-)
378 5592 y(not)h(implemen)m(t)f(forw)m(ard)h(inference)f(rules)f(whic)m
(h)h(deriv)m(e)h(a)g(conclusion)f(from)h(its)f(justi\014cation,)378
5705 y(but)k(rather)h(pro)s(of)f(c)m(hec)m(king)h(functions)e(whic)m(h)
h(c)m(hec)m(k)i(that)f(the)g(conclusion)f(follo)m(ws)f(from)i(the)p
eop
%%Page: 218 228
218 227 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(218)378 396 y(structured)31 b(justi\014cation.)45
b(Ho)m(w)m(ev)m(er,)35 b(c)m(hapter)d(8)h(illustrates)d(ho)m(w)i(one)h
(can)f(restrict)g(the)g(searc)m(h)378 509 y(space)d(considered)d
(during)g(the)j(pro)s(of)e(c)m(hec)m(king)i(of)f(structured)f
(justi\014cations.)38 b(As)28 b(a)h(result,)f(less)378
622 y(e\013ort)35 b(is)e(required)f(in)h(c)m(hec)m(king)i(structured)e
(justi\014cations)f(than)i(unstructured)e(justi\014cations.)378
735 y(The)i(material)h(on)g(pro)s(of)f(c)m(hec)m(king)i(structured)e
(justi\014cations)f(giv)m(en)i(in)e(c)m(hapter)j(8)f(mak)m(es)h(use)378
848 y(of)e(a)h(theory)f(of)g(coloured)g(\014rst-order)f(logic)h(in)e
(whic)m(h)h(form)m(ulae)h(are)g(annotated)h(with)e(colours.)378
961 y(The)26 b(colours)g(are)g(used)g(to)h(restrict)f(the)h(notion)f
(of)g(the)h(inconsistency)e(of)h(a)h(\014rst-order)f(sen)m(tences)378
1074 y(and)j(are)g(used)g(to)h(restrict)f(the)g(searc)m(h)h(space)g
(required)d(in)h(the)h(automated)i(theorem)e(pro)m(ving)g(of)378
1187 y(coloured)h(form)m(ulae.)40 b(The)30 b(theory)h(of)f(coloured)g
(\014rst-order)g(logic)g(is)f(describ)s(ed)f(in)h(c)m(hapter)i(7.)519
1300 y(It)36 b(is)f(sho)m(wn)g(in)f(section)i(8.2.4)i(that)e(the)g(v)-5
b(alidit)m(y)34 b(of)i(\014rst-order)f(structured)f(justi\014cations)
378 1413 y(de\014ned)k(in)h(c)m(hapter)h(6)g(is)f(undecidable.)66
b(As)40 b(a)g(result,)h(the)f(pro)s(of)f(c)m(hec)m(k)m(er)j(used)d(in)f
(c)m(hec)m(king)378 1526 y(the)26 b(structured)g(justi\014cations)e
(implemen)m(ted)h(in)g(the)h(mec)m(hanisation)g(of)g(group)g(theory)g
(restricts)378 1638 y(the)37 b(searc)m(h)g(space)g(considered)f(to)h(a)
g(\014nite)e(one.)60 b(The)36 b(fact)i(that)f(these)g(restrictions)f(w)
m(ere)h(not)378 1751 y(considered)28 b(to)j(b)s(e)e(to)s(o)h(strong)g
(during)e(the)h(mec)m(hanisation)h(suggests)g(that)g(only)f(a)h(small,)
f(prob-)378 1864 y(ably)h(decidable,)f(fragmen)m(t)j(of)f(the)g(set)g
(of)g(v)-5 b(alid)29 b(\014rst-order)h(structured)f(justi\014cations)g
(giv)m(en)i(in)378 1977 y(c)m(hapter)g(6)g(is)e(required)g(in)g
(practiced.)378 2221 y FG(The)38 b(Implemen)m(tation)c(of)k(the)f
Fo(C)6 b(B)s(S)h(E)48 b FG(Deriv)m(ed)37 b(Rule)378 2392
y FT(The)32 b(implemen)m(tation)f(of)h(a)h(tableau)f(pro)m(v)m(er)h
(for)f(\014rst-order)f(logic)i(with)e(equalit)m(y)h(as)g(a)h(deriv)m
(ed)378 2505 y(rule)i(in)h(the)g(HOL)h(system)f(is)g(describ)s(ed)f(in)
g(c)m(hapter)i(5.)60 b(The)36 b(pro)m(v)m(er)h(is)f(based)g(on)h(the)f
FN(C)5 b(B)s(S)i(E)378 2618 y FT(tableau)35 b(calculus,)g(whic)m(h)f
(refutes)g(a)i(giv)m(en)f(list)f(of)h(clauses)f(and)h(uses)f(the)i
(rules)d(of)i(rigid)e(basic)378 2731 y(sup)s(erp)s(osition)g(\(Degt)m
(y)m(arev)40 b(and)c(V)-8 b(oronk)m(o)m(v)39 b(1998\))f(with)e
(equational)g(re\015exivit)m(y)f(to)j(close)f(the)378
2844 y(tableau)g(branc)m(hes.)61 b(Congruence)37 b(closure)g(is)f(also)
h(used)g(to)h(close)f(redundan)m(t)f(branc)m(hes)h(\(that)378
2957 y(is,)h(branc)m(hes)g(whic)m(h)e(do)h(not)h(need)f(the)g(instan)m
(tiation)g(of)g(their)g(free)g(v)-5 b(ariables)37 b(to)h(b)s(e)e
(closed\).)378 3070 y(During)30 b(the)h(pro)s(of)g(searc)m(h)h(stage)g
(of)g(the)f(HOL)g(deriv)m(ed)f(rule,)g(the)i(expansion)e(of)h(clauses)g
(whic)m(h)378 3183 y(can)h(b)s(e)f(immediately)f(follo)m(w)m(ed)h(b)m
(y)h(the)g(closure)f(of)h(a)g(tableau)g(branc)m(h)f(are)h(giv)m(en)f
(priorit)m(y)g(o)m(v)m(er)378 3296 y(other)37 b(expansions)f(in)g
(order)h(to)h(gain)f(some)g(of)h(the)f(e\016ciency)h(of)f(connection)g
(tableau)g(calculi)378 3408 y(\(see)26 b(\(Letz)g(1993\)\).)41
b(The)24 b FN(C)5 b(B)s(S)i(E)33 b FT(deriv)m(ed)24 b(rule)f(deriv)m
(es)h(a)i(HOL)e(theorem)h(when)f(a)h(closed)g(tableau)378
3521 y(is)k(found.)519 3634 y(The)38 b FN(C)5 b(B)s(S)i(E)46
b FT(deriv)m(ed)37 b(rule)g(is)g(mo)s(di\014ed)f(to)j(pro)s(of)f(c)m
(hec)m(k)h(structured)f(justi\014cations)e(as)j(de-)378
3747 y(scrib)s(ed)34 b(in)g(section)i(8.5.)57 b(It)36
b(is)f(used)g(as)h(the)g(main)e(pro)m(v)m(er)i(during)e(the)i(pro)s(of)
e(c)m(hec)m(king)j(of)f(the)378 3860 y(SPL)e(scripts)g(implemen)m(ted)f
(during)f(the)k(case)f(study)f(describ)s(ed)f(in)h(c)m(hapter)h(9.)55
b(Although)34 b(the)378 3973 y FN(C)5 b(B)s(S)i(E)48
b FT(calculus)39 b(is)h(complete)h(for)f(\014rst-order)f(logic)i(with)e
(equalit)m(y)-8 b(,)43 b(the)e(searc)m(h)g(for)f(a)h(closed)378
4086 y(tableau)33 b(is)f(restricted)g(to)i(a)f(small)f(\014nite)g
(searc)m(h)h(space)h(b)s(ecause)f(of)g(the)g(simplicit)m(y)d(of)j(the)h
(jus-)378 4199 y(ti\014cations.)47 b(F)-8 b(urthermore,)34
b(the)f(searc)m(h)h(strategy)g(used)e(for)h(lo)s(oking)e(for)i(closed)f
(tableaux)h(\(and)378 4312 y(its)d(implemen)m(tation\))f(is)g
(unsuitable)f(for)j(\014nding)d(long)i(and)f(complex)h(pro)s(ofs.)378
4555 y FG(The)38 b(Mec)m(hanisation)f(of)h(Group)g(Theory)f(in)g(SPL)
378 4727 y FT(Chapter)23 b(9)h(describ)s(es)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
(the)i(SPL)378 4840 y(declarativ)m(e)34 b(language.)51
b(The)34 b(mec)m(hanisation)f(is)g(based)g(on)h(the)f(textb)s(o)s(ok)i
(b)m(y)e(Herstein)g(\(1975\))378 4953 y(and)28 b(includes)e(all)h(the)h
(results)g(leading)f(to,)i(and)f(including,)e(the)i(second)h
(isomorphism)c(theorem,)378 5066 y(with)k(the)i(exception)f(of)h(those)
g(in)m(v)m(olving)d(\014nite)i(groups.)519 5178 y(As)c(discussed)e(in)h
(more)h(detail)f(in)g(section)h(9.5,)i(the)e(pro)s(ofs)f(implemen)m
(ted)g(during)e(this)i(mec)m(h-)378 5291 y(anisation)35
b(are)i(quite)f(readable)g(and)g(m)m(uc)m(h)h(easier)f(to)h(follo)m(w)f
(than)g(tactic-based)i(pro)s(ofs.)58 b(The)378 5404 y(reasons)30
b(for)h(this)e(impro)m(v)m(emen)m(t)i(in)e(the)h(readabilit)m(y)f(of)h
(the)h(pro)s(ofs)f(include)e(the)i(follo)m(wing:)514
5592 y FN(\017)46 b FT(The)25 b(pro)s(ofs)f(are)h(declarativ)m(e)h(in)e
(nature,)i(and)e(con)m(tain)i(information)d(whic)m(h)h(is)g(relev)-5
b(an)m(t)25 b(for)605 5705 y(a)36 b(h)m(uman)f(reader)g(to)h
(understand)e(them.)55 b(The)35 b(use)h(of)f(explicit)f(v)-5
b(ariable)34 b(instan)m(tiations)p eop
%%Page: 219 229
219 228 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(219)605 396 y(and)28 b(the)i(use)e(of)h(forw)m(ard)f(inference)g
(rules)g(is)g(a)m(v)m(oided)h(\(with)f(the)h(exception)g(of)g(the)g
(use)g(of)605 509 y(the)i Fw(select)d FT(rule)h(describ)s(ed)f(in)h
(page)i(68\).)514 697 y FN(\017)46 b FT(Structured)33
b(justi\014cations,)g(whic)m(h)g(con)m(tain)h(more)g(information)e(on)i
(what)g(t)m(yp)s(e)g(of)g(infer-)605 810 y(ences)40 b(are)h(used)d(in)h
(the)h(deriv)-5 b(ation)38 b(of)i(the)g(conclusion)e(of)i(the)g
(justi\014cation,)h(are)f(used)605 923 y(instead)30 b(of)g
(unstructured)f(ones.)514 1110 y FN(\017)46 b FT(Scripts)29
b(are)i(organised)f(in)f(a)h(mo)s(dular)f(fashion)g(in)m(to)h
(sections.)514 1298 y FN(\017)46 b FT(Simpli\014ers)38
b(whic)m(h)j(are)h(able)g(to)h(query)e(the)h(SPL)f(database)i(of)f
(trivial)e(kno)m(wledge)i(are)605 1411 y(implemen)m(ted)30
b(and)g(included)e(in)h(the)j(SPL)e(language)h(throughout)f(the)h(mec)m
(hanisation)g(of)605 1524 y(the)g(theory)-8 b(.)514 1712
y FN(\017)46 b FT(The)24 b(deductiv)m(e)f(p)s(o)m(w)m(er)h(of)g(the)g
(kno)m(wledge)g(database)h(is)e(up)s(dated)f(and)h(extended)h(through-)
605 1824 y(out)31 b(the)f(mec)m(hanisation)g(of)h(the)f(theory)-8
b(.)378 2012 y(In)39 b(particular,)i(the)f(inhomogeneit)m(y)g(in)f(the)
h(complexit)m(y)f(of)i(the)f(pro)s(of)f(steps)h(whic)m(h)e(is)h(often)
378 2125 y(noticed)27 b(in)f(mec)m(hanised)h(pro)s(ofs)f(is)g(greatly)i
(reduced)e(b)m(y)h(regularly)f(up)s(dating)f(and)i(querying)f(the)378
2238 y(database)40 b(of)g(trivial)d(kno)m(wledge.)67
b(By)40 b(the)f(inhomogeneit)m(y)g(in)f(the)i(complexit)m(y)f(of)g(the)
h(pro)s(of)378 2351 y(steps)26 b(w)m(e)g(refer)f(to)i(the)e(fact)i
(that)f(the)g(complexit)m(y)g(of)g(the)f(pro)s(of)g(steps)h(in)e(the)i
(same)g(pro)s(of)f(di\013ers)378 2464 y(greatly)-8 b(,)26
b(and)d(simple)f(results)h(deriv)m(ed)g(during)e(the)j(early)g(stages)h
(of)f(a)g(mec)m(hanisation)g(can)g(b)s(e)f(still)378
2577 y(used)30 b(quite)f(often)i(in)e(the)i(pro)s(ofs)e(implemen)m(ted)
g(during)f(later)j(stages)g(of)g(the)f(mec)m(hanisation.)378
2863 y FH(10.2)136 b(F)-11 b(uture)44 b(W)-11 b(ork)378
3066 y FT(In)38 b(this)f(section)i(w)m(e)g(discuss)e(a)i(n)m(um)m(b)s
(er)e(of)i(directions)e(for)h(future)g(w)m(ork)g(aimed)g(primarily)d
(at)378 3179 y(in)m(v)m(estigating)26 b(p)s(ossible)e(w)m(a)m(ys)j(of)g
(impro)m(ving)d(the)i(readabilit)m(y)f(of)i(mec)m(hanised)e(pro)s(ofs.)
39 b(Both)27 b(im-)378 3292 y(pro)m(v)m(emen)m(ts)d(on)e(the)h(w)m(ork)
g(presen)m(ted)f(in)g(the)h(previous)e(c)m(hapters,)k(as)e(w)m(ell)e
(as)i(researc)m(h)g(directions)378 3405 y(not)31 b(considered)e(in)g
(this)g(thesis,)h(are)h(discussed)d(b)s(elo)m(w.)519
3518 y(The)h(declarativ)m(e)g(st)m(yle)h(of)f(pro)s(of)g(implemen)m
(tation)f(results)g(in)f(m)m(uc)m(h)j(more)f(readable)g(pro)s(ofs)378
3631 y(than)34 b(the)g(tactic-based,)i(and)d(other)h(pro)s(cedural,)g
(st)m(yles.)51 b(The)33 b(w)m(ork)h(presen)m(ted)g(in)f(this)g(thesis)
378 3743 y(suggests)41 b(that)g(the)g(extensibilit)m(y)e(of)h(a)h(pro)s
(of)f(language)h(results)f(in)f(an)h(impro)m(v)m(emen)m(t)h(in)f(the)
378 3856 y(readabilit)m(y)31 b(of)i(its)g(pro)s(of)f(scripts.)47
b(An)33 b(imp)s(ortan)m(t)f(direction)g(of)h(researc)m(h)g(is)f
(therefore)i(the)f(de-)378 3969 y(sign)26 b(of)h(extensible)f(pro)s(of)
g(languages.)40 b(The)27 b(SPL)f(pro)s(of)g(c)m(hec)m(k)m(er)j(is)d
(extensible)g(since)g(the)i(theory)378 4082 y(dev)m(elop)s(er)36
b(can)h(implemen)m(t)f(new)g(HOL)g(pro)s(of)g(pro)s(cedures)g(in)f(ML)i
(and)f(incorp)s(orate)g(them)h(in)378 4195 y(the)e(SPL)f(language)h
(during)e(mec)m(hanisation.)53 b(Ho)m(w)m(ev)m(er,)39
b(the)c(curren)m(t)f(implemen)m(tation)g(of)h(the)378
4308 y(pro)s(of)29 b(c)m(hec)m(k)m(er)i(allo)m(ws)e(only)g
FI(glob)-5 b(al)40 b FT(mo)s(di\014cations)28 b(to)i(the)g(pro)s(of)f
(language,)h(and)f(it)g(is)g(desirable)378 4421 y(that)i(certain)f(mo)s
(di\014cations)f(b)s(e)g FI(lo)-5 b(c)g(al)42 b FT(to)31
b(certain)f(theories,)h(sections)f(and)g(pro)s(ofs.)39
b(This)29 b(highly)378 4534 y(desirable)c(feature)h(ma)m(y)h(require)e
(substan)m(tial)h(c)m(hanges)h(to)g(the)g(o)m(v)m(erall)f(design)g(and)
g(implemen)m(ta-)378 4647 y(tion)h(of)g(the)h(pro)s(of)f(c)m(hec)m(k)m
(er.)41 b(It)28 b(should)d(also)i(b)s(e)g(noticed)g(that)h(the)g(pro)s
(of)e(pro)s(cedures)h(dev)m(elop)s(ed)378 4760 y(during)41
b(the)j(mec)m(hanisations)e(are)i(implemen)m(ted)e(in)g(a)i(highly)d
(pro)s(cedural)h(fashion)g(in)g(SML.)378 4873 y(The)33
b(p)s(ossibilit)m(y)e(of)j(dev)m(eloping)f(p)s(ossibly)e(declarativ)m
(e)j(languages)g(for)g(the)g(implemen)m(tation)f(of)378
4985 y(simpli\014ers,)c(database)j(query)g(functions,)f(and)g(other)h
(pro)s(of)f(pro)s(cedures)g(is)g(also)h(an)g(in)m(teresting)378
5098 y(direction)d(for)h(future)g(researc)m(h.)519 5211
y(The)i(case)h(study)f(describ)s(ed)e(in)h(c)m(hapter)i(9)g(in)m(v)m
(estigated)g(the)f(e\013ect)i(of)f(extending)e(the)i(sim-)378
5324 y(pli\014ers)26 b(and)i(the)g(SPL)g(kno)m(wledge)g(database)i
(during)c(the)j(dev)m(elopmen)m(t)f(of)h(a)g(theory)-8
b(.)41 b(Ho)m(w)m(ev)m(er,)378 5437 y(the)27 b(implemen)m(tation)f(of)i
(the)f(SPL)g(pro)s(of)f(c)m(hec)m(k)m(er)j(also)f(allo)m(ws)e(the)i
(extensibilit)m(y)d(of)i(the)h(pro)m(v)m(ers)378 5550
y(used)f(in)f(justifying)f(pro)s(of)i(statemen)m(ts,)j(as)e(w)m(ell)e
(as)i(the)f(syn)m(tax)h(and)f(seman)m(tics)h(of)g(the)f(language)378
5663 y(as)34 b(a)g(whole.)51 b(F)-8 b(or)35 b(instance,)f(one)g(is)f
(able)h(to)g(extend)g(the)h(SPL)e(language)h(with)e(theory-sp)s
(eci\014c)p eop
%%Page: 220 230
220 229 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(220)378 396 y(constructs)38 b(during)e(theory)i(dev)m(elopmen)m
(t.)65 b(Case)38 b(studies)f(on)h(mec)m(hanisations)f(in)m(v)m(olving)g
(the)378 509 y(use)30 b(of)h(suc)m(h)f(extensibilit)m(y)e(are)j
(required)d(in)h(order)h(to)h(ev)-5 b(aluate)31 b(their)f(e\013ect)i
(in)d(practice.)519 622 y(Another)36 b(imp)s(ortan)m(t)f(area)i(of)f
(researc)m(h)g(is)f(the)h(in)m(v)m(estigation)g(of)g(the)g(t)m(yp)s(e)g
(of)g(automation)378 735 y(required)c(b)m(y)i(pro)s(of)g(c)m(hec)m(k)m
(ers)i(of)e(declarativ)m(e)h(languages.)53 b(The)33 b(main)g(comp)s
(onen)m(t)i(of)f(the)h(SPL)378 848 y(pro)s(of)30 b(c)m(hec)m(k)m(er)j
(is)d(the)h FN(C)5 b(B)s(S)i(E)38 b FT(deriv)m(ed)30
b(rule)g(describ)s(ed)e(in)i(c)m(hapter)h(5.)43 b(This)29
b(pro)s(of)h(pro)s(cedure)g(is)378 961 y(e\013ectiv)m(e)24
b(for)e(\014nding)e(simple)g(pro)s(ofs)h(in)g(the)i(classical)e
(\014rst-order)h(logic)g(with)f(equalit)m(y)-8 b(.)38
b(Ho)m(w)m(ev)m(er,)378 1074 y(one)j(often)g(requires)f(the)h(pro)s(of)
f(pro)s(cedures)f(for)i(other)g(logics,)i(including)38
b(higher-order)h(logic)378 1187 y(whic)m(h)27 b(is)g(treated)i(in)e
(SPL)h(through)f(an)h(incomplete)g(transformation)f(from)h
(higher-order)f(terms)378 1300 y(in)m(to)37 b(\014rst-order)g(ones,)i
(as)f(w)m(ell)e(as)h(in)f(other)i(theories)f(suc)m(h)g(as)g(natural)g
(and)f(real)h(arithmetic.)378 1413 y(Automated)26 b(reasoning)e(in)f
(particular)g(theories)i(in)f(SPL)f(is)h(done)h(through)f
(simpli\014ers)d(whic)m(h)i(are)378 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 b(of)k(greatly)g(reducing)e(the)h
(di\013erence)g(b)s(et)m(w)m(een)i(formal)d(and)h(informal)f(pro)s(ofs)
h(since)378 2316 y(the)i(authors)g(of)g(informal)e(pro)s(ofs)h(omit)g
(the)i(justi\014cations)d(of)i(facts)h(considered)e(to)h(b)s(e)g
(trivial.)378 2429 y(The)28 b(p)s(ossibilit)m(y)d(of)j(sp)s(ecifying)f
(searc)m(h)i(strategy)h(heuristics)c(sp)s(eci\014c)h(to)i(particular)e
(theories,)i(or)378 2542 y(sections,)35 b(can)e(also)h(result)f(in)f(a)
i(substan)m(tial)e(impro)m(v)m(emen)m(t)i(to)g(the)g(curren)m(t)f
(system.)51 b(Another)378 2655 y(direction)30 b(for)i(future)f(researc)
m(h)h(is)f(the)h(use)f(of)h(automated)h(inductiv)m(e)d(theorem)i(pro)m
(ving)f(b)m(y)h(the)378 2768 y(pro)s(of)k(c)m(hec)m(k)m(er)j(of)e(a)h
(declarativ)m(e)f(language,)i(since)d(it)h(is)f(observ)m(ed)h(in)f(c)m
(hapter)h(9)g(that)h(certain)378 2880 y(results)32 b(on)i(\014nite)e
(sets)i(that)g(are)g(considered)e(trivial)g(b)m(y)h(the)h(authors)f(of)
g(informal)f(pro)s(ofs)g(ma)m(y)378 2993 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
(an)h(in)m(teresting)f(task)i(whic)m(h)d(requires)h(substan)m(tial)f(w)
m(ork)i(and)f(researc)m(h.)39 b(This)378 3897 y(p)s(ossibilit)m(y)26
b(has)j(b)s(een)f(studied)g(recen)m(tly)h(b)m(y)g(Syme)g(\(1998\))j
(during)27 b(the)i(dev)m(elopmen)m(t)h(of)f(the)h(in-)378
4010 y(teractiv)m(e)44 b(IDECLARE)d(system.)75 b(One)42
b(can)g(also)g(consider)f(future)g(w)m(ork)h(in)f(the)h(automated)378
4122 y(disco)m(v)m(ery)h(of)g(declarativ)m(e)g(pro)s(ofs,)i(and)d(in)f
(the)i(transformation)f(of)h(non-declarativ)m(e)g(pro)s(ofs,)378
4235 y(suc)m(h)d(as)h(pro)s(ofs)f(in)f(a)i(searc)m(h-orien)m(ted)h
(format)f(and)f(tactic)i(pro)s(ofs,)g(in)m(to)f(mac)m(hine-c)m(hec)m(k)
-5 b(able)378 4348 y(declarativ)m(e)31 b(ones.)519 4461
y(Chapter)36 b(6)i(in)m(tro)s(duces)d(the)j(notion)e(of)h(structured)f
(straigh)m(tforw)m(ard)h(justi\014cations)e(based)378
4574 y(on)29 b(explicitly)e(stated)i(inferences)g(and)f(implicitly)d
(assumed)k(trivial)e(inferences.)39 b(Chapter)28 b(6)i(also)378
4687 y(giv)m(es)g(the)g(de\014nition)d(of)j(structured)e
(justi\014cations)h(based)g(on)g(implicit)e(and)i(explicit)f
(inferences)378 4800 y(for)36 b(the)h(pure)e(\014rst-order)h(logic.)59
b(It)36 b(is)g(argued)g(\(in)g(c)m(hapters)h(6)g(and)f(8\))h(that)g
(less)f(e\013ort)h(is)f(re-)378 4913 y(quired)26 b(in)g(follo)m(wing)h
(and)g(pro)s(of)g(c)m(hec)m(king)h(structured)f(justi\014cations)f
(than)i(unstructured)d(ones.)378 5026 y(Ho)m(w)m(ev)m(er,)42
b(the)c(v)-5 b(alidit)m(y)37 b(of)h(the)g(structured)f
(justi\014cations)f(giv)m(en)i(in)f(section)h(6.4)h(is)e(sho)m(wn)g(to)
378 5139 y(b)s(e)h(undecidable,)h(and)f(it)h(is)f(observ)m(ed)h(in)e(c)
m(hapter)j(9)f(that)h(probably)d(only)h(a)h(small,)h(p)s(ossibly)378
5252 y(decidable,)28 b(subsets)h(of)g(suc)m(h)g(justi\014cations)f(are)
i(used)e(in)g(practice.)41 b(More)30 b(w)m(ork)g(is)e(therefore)i(re-)
378 5365 y(quired)24 b(in)g(restricting)g(the)i(de\014nition)d(of)j
(the)g(structured)e(justi\014cations)g(giv)m(en)i(in)e(this)g(thesis.)
39 b(In)378 5477 y(particular,)25 b(the)g(implicit)e(\014rst-order)h
(inferences)g(de\014ned)h(in)f(section)h(6.4.1)i(should)d(b)s(e)g
(restricted)378 5590 y(in)29 b(some)i(w)m(a)m(y)-8 b(.)519
5703 y(It)33 b(is)e(also)h(desirable)f(that)i(one)g(extends)f(the)h
(notion)f(of)g(structured)g(justi\014cations)e(to)k(other)p
eop
%%Page: 221 231
221 230 bop 378 5 a FF(CHAPTER)30 b(10.)71 b(CONCLUSIONS)1969
b FT(221)378 396 y(logics)20 b(and)h(theories.)37 b(This)19
b(is)h(an)h(in)m(teresting)f(direction)f(for)i(future)f(w)m(ork)h
(since)f(it)g(is)g(not)h(straigh)m(t-)378 509 y(forw)m(ard)j(to)i
(de\014ne)e(structured)g(justi\014cations)f(whic)m(h)h(ha)m(v)m(e)i(an)
f(in)m(tuitiv)m(e)e(seman)m(tics)i(and)g(y)m(et)h(can)378
622 y(also)g(b)s(e)f(pro)s(of)g(c)m(hec)m(k)m(ed)i(e\016cien)m(tly)-8
b(.)39 b(One)26 b(also)f(requires)f(that)j(the)f(e\013ort)g(required)e
(to)i(implemen)m(t)378 735 y(pro)s(ofs)i(in)m(v)m(olving)f(structured)h
(justi\014cations)g(is)g(not)h(m)m(uc)m(h)g(greater)h(than)f(implemen)m
(ting)e(pro)s(ofs)378 848 y(in)m(v)m(olving)h(unstructured)f(ones.)40
b(In)29 b(c)m(hapters)h(7)f(and)g(8)h(it)e(is)h(sho)m(wn)f(ho)m(w)h
(the)h(inferences)e(\(or)i(op-)378 961 y(erators\))f(giv)m(en)e(in)g
(structured)f(justi\014cations)g(can)i(b)s(e)f(used)g(to)h(restrict)g
(the)g(searc)m(h)g(space)g(whic)m(h)378 1074 y(needs)e(to)g(b)s(e)g
(considered)f(b)m(y)h(existing)f(\014rst-order)g(deductiv)m(e)h
(systems.)39 b(This)25 b(\(implemen)m(tation-)378 1187
y(indep)s(enden)m(t\))36 b(restriction)g(is)g(giv)m(en)i(in)e(terms)h
(of)h(annotations,)i(or)d(colours,)i(on)e(form)m(ulae.)62
b(It)378 1300 y(ma)m(y)34 b(b)s(e)e(p)s(ossible)f(to)j(use)f(the)h
(same)f(tec)m(hnique)g(during)e(the)j(dev)m(elopmen)m(t)f(of)h(mec)m
(hanisms)e(for)378 1413 y(pro)s(of)24 b(c)m(hec)m(king)h(the)f
(structured)g(justi\014cations)f(for)h(other)h(logics)f(and)f
(theories.)39 b(In)23 b(other)i(w)m(ords,)378 1526 y(structured)30
b(justi\014cations)f(of)i(a)g(particular)e(theory)i(can)g(b)s(e)f(c)m
(hec)m(k)m(ed)i(b)m(y)f(restricting)e(the)i(searc)m(h)378
1638 y(space)42 b(of)g(existing)f(decision)f(pro)s(cedures)g(for)i
(that)g(theory)-8 b(.)75 b(Inciden)m(tally)-8 b(,)44
b(the)e(use)f(of)h(anno-)378 1751 y(tations,)j(also)d(called)f
(colours,)j(on)d(expressions)g(are)h(used)f(b)m(y)g(Hutter)i(and)e
(Kohlhase)f(\(1997\))378 1864 y(to)c(restrict)g(the)g(uni\014cation)d
(of)j(higher-order)f(terms,)i(and)e(also)g(b)m(y)h(Hutter)g(\(1997\))i
(to)e(con)m(trol)378 1977 y(equational)30 b(reasoning)g(esp)s(ecially)e
(during)g(inductiv)m(e)h(automated)j(theorem)f(pro)m(ving.)519
2090 y(Finally)i(w)m(e)i(note)h(that)g(the)f(readabilit)m(y)e(of)i(mec)
m(hanised)g(pro)s(ofs)f(relies)f(on)i(the)g(readabilit)m(y)378
2203 y(of)e(the)h(terms)f(and)g(sen)m(tences)i(used)d(in)g(the)i(pro)s
(ofs.)49 b(This)31 b(issue)h(is)h(not)g(studied)f(in)g(this)g(thesis,)
378 2316 y(and)i(w)m(e)h(noticed)f(in)f(c)m(hapter)i(9)g(that)g
(although)f(a)g(n)m(um)m(b)s(er)g(of)g(pro)s(ofs)g(mec)m(hanised)g
(during)e(the)378 2429 y(case)26 b(study)f(are)h(observ)m(ed)f(to)h(b)s
(e)f(similar)e(to)j(their)e(informal)g(coun)m(terparts)i(when)e(the)i
(n)m(um)m(b)s(er)e(of)378 2542 y FI(steps)39 b FT(in)29
b(the)h(pro)s(ofs)g(are)g(compared,)h(the)g(length)e(of)i(the)f
FI(symb)-5 b(ols)40 b FT(in)29 b(the)i(formal)e(pro)s(ofs)h(is)f(still)
378 2655 y(m)m(uc)m(h)38 b(higher)e(than)i(that)g(of)g(the)g(informal)e
(pro)s(ofs.)62 b(The)37 b(authors)h(of)g(informal)d(mathematics)378
2768 y(v)m(ery)j(often)f(c)m(hange)i(the)e(syn)m(tax)h(of)g(their)e
(language)i(b)m(y)f(in)m(tro)s(ducing)e(appropriate)i(notations.)378
2880 y(It)e(is)e(therefore)i(desirable)e(that)i(one)g(is)f(able)g(to)h
(safely)f(mo)s(dify)f(the)i(term)f(parser)g(of)h(the)g(pro)s(of)378
2993 y(c)m(hec)m(k)m(er)d(during)d(the)h(mec)m(hanisation)g(of)h(a)f
(theory)-8 b(.)p eop
%%Page: 222 232
222 231 bop 378 1019 a FJ(App)5 b(endix)65 b(A)378 1434
y FR(The)77 b(Syn)-6 b(tax)77 b(of)h(SPL)378 1879 y FT(In)40
b(this)f(App)s(endix)f(w)m(e)i(giv)m(e)h(the)g(syn)m(tax)g(of)f(the)h
(SPL)e(language)i(describ)s(ed)d(in)h(c)m(hapter)i(4)g(in)378
1992 y(Extended)30 b(BNF.)378 2276 y FH(A.1)135 b(Reasoning)46
b(Items)473 2479 y FI(SPL)p 649 2479 29 4 v 35 w(Script)57
b FT(=)47 b FI(Se)-5 b(ction)56 b FN(f)48 b FI(Se)-5
b(ction)55 b FN(g)473 2672 y FI(Se)-5 b(ction)56 b FT(=)569
2785 y FM(section)46 b FI(Se)-5 b(ction)p 1249 2785 V
42 w(Name)569 2898 y(R)g(e)g(asoning)p 982 2898 V 44
w(Item)569 3011 y FM(end)47 b FT([)h FI(Se)-5 b(ction)p
1131 3011 V 42 w(Name)55 b FT(])48 b FM(;)378 3205 y
FT(\(The)30 b FI(Se)-5 b(ction)p 892 3220 31 4 v 38 w(Name)38
b FT(follo)m(wing)28 b Fw(end)h FT(is)h(the)g(same)h(as)g(the)f(one)h
(follo)m(wing)e Fw(section)m FT(.\))473 3398 y FI(L)-5
b(o)g(c)g(al)p 687 3398 29 4 v 46 w(De)g(clar)g(ations)58
b FT(=)569 3511 y FM(local)664 3624 y FI(R)-5 b(e)g(asoning)p
1077 3624 V 45 w(Items)569 3737 y FM(in)664 3850 y FI(R)g(e)g(asoning)p
1077 3850 V 45 w(Items)569 3963 y FM(end)47 b(;)473 4157
y FI(R)-5 b(e)g(asoning)p 886 4157 V 45 w(Items)55 b
FT(=)48 b FN(f)g FI(R)-5 b(e)g(asoning)p 1815 4157 V
44 w(Item)55 b FN(g)473 4350 y FI(R)-5 b(e)g(asoning)p
886 4350 V 45 w(Item)55 b FT(=)47 b([)h FP(R)q(easoning)p
1784 4350 V 37 w(S)5 b(epar)s(ator)50 b FT(])569 4463
y(\()143 b FI(T)-7 b(yp)i(e)p 945 4463 V 42 w(Gener)g(alisation)664
4576 y FN(j)48 b FI(Gener)-5 b(alisation)664 4689 y FN(j)48
b FI(Assumption)664 4802 y FN(j)g FI(Existential)p 1171
4802 V 45 w(Assumption)664 4915 y FN(j)g FI(Step)p 917
4915 V 41 w(R)-5 b(esult)664 5028 y FN(j)48 b FI(Existential)p
1171 5028 V 45 w(R)-5 b(esult)664 5141 y FN(j)48 b FI(The)-5
b(or)g(em)664 5253 y FN(j)48 b FI(A)n(bbr)-5 b(eviation)p
1246 5253 V 42 w(De)g(clar)g(ation)664 5366 y FN(j)48
b FI(Simpli\014c)-5 b(ation)p 1289 5366 V 44 w(De)g(clar)g(ation)664
5479 y FN(j)48 b FI(Know)5 b(le)-5 b(dge)p 1172 5479
V 43 w(De)g(clar)g(ation)664 5592 y FN(j)48 b FI(Se)-5
b(ction)664 5705 y FN(j)48 b FI(L)-5 b(o)g(c)g(al)p 951
5705 V 46 w(De)g(clar)g(ation)57 b FT(\))2035 5954 y(222)p
eop
%%Page: 223 233
223 232 bop 378 5 a FF(APPENDIX)31 b(A.)61 b(THE)30 b(SYNT)-8
b(AX)30 b(OF)h(SPL)1669 b FT(223)473 396 y FI(R)-5 b(e)g(asoning)p
886 396 29 4 v 45 w(Sep)g(ar)g(ator)61 b FT(=)664 509
y FM(and)47 b FN(j)h FM(but)f FN(j)h FM(hence)e FN(j)i
FM(now)f FN(j)h FM(so)f FN(j)664 622 y FM(then)g FN(j)h
FM(therefore)d FN(j)j FM(thus)f FN(j)h FM(==>)473 834
y FI(T)-7 b(yp)i(e)p 671 834 V 42 w(Gener)g(alisation)58
b FT(=)47 b FI(T)-7 b(yp)i(e)p 1647 834 V 42 w(Gener)g(alisation)p
2262 834 V 44 w(Constructor)60 b(T)-7 b(yp)i(e)p 3023
834 V 42 w(V)e(ars)55 b FM(;)473 1046 y FI(T)-7 b(yp)i(e)p
671 1046 V 42 w(Gener)g(alisation)p 1286 1046 V 44 w(Constructor)60
b FT(=)47 b([)h Fw(given)e FT(])i([)g Fw(new)e FT(])i(\()g
Fw(type)f FN(j)g Fw(types)f FT(\))473 1257 y FI(Gener)-5
b(alisation)58 b FT(=)47 b FI(Gener)-5 b(alisation)p
1808 1257 V 44 w(Constructor)60 b(V)-7 b(ar)p 2531 1257
V 44 w(T)g(erms)56 b FM(;)473 1469 y FI(Gener)-5 b(alisation)p
1059 1469 V 44 w(Constructor)60 b FT(=)569 1582 y FM(let)473
1695 y FN(j)48 b FT([)g Fw(given)e FT(])i([)g Fw(new)e
FT(])664 1807 y(\()i Fw(var)f FN(j)h Fw(vars)e FN(j)i
Fw(variable)c FN(j)k Fw(variables)c FT(\))473 2019 y
FI(Assumption)56 b FT(=)48 b FI(Assumption)p 1593 2019
V 42 w(Constructor)60 b(L)-5 b(ab)g(el)5 b(le)-5 b(d)p
2480 2019 V 45 w(Statements)57 b FM(;)473 2231 y FI(Assumption)p
952 2231 V 43 w(Constructor)i FT(=)48 b(\()g Fw(suppose)d
FN(j)j Fw(assume)d FN(j)j Fw(given)e FT(\))i([)g Fw(that)e
FT(])473 2442 y FI(Existential)p 907 2442 V 45 w(Assumption)56
b FT(=)569 2555 y FI(Existential)p 1003 2555 V 44 w(Assumption)p
1511 2555 V 43 w(Constructor)569 2668 y(V)-7 b(ar)p 730
2668 V 44 w(T)g(erms)56 b(Such)p 1264 2668 V 42 w(That)p
1494 2668 V 44 w(Constructor)569 2781 y(L)-5 b(ab)g(el)5
b(le)-5 b(d)p 895 2781 V 45 w(Statements)57 b FM(;)473
2993 y FI(Existential)p 907 2993 V 45 w(Assumption)p
1416 2993 V 42 w(Constructor)j FT(=)47 b FM(given)473
3204 y FI(Step)p 653 3204 V 41 w(R)-5 b(esult)57 b FT(=)48
b([)g FI(Step)p 1354 3204 V 40 w(R)-5 b(esult)p 1639
3204 V 44 w(Constructor)60 b FT(])48 b FI(L)-5 b(ab)g(el)5
b(le)-5 b(d)p 2600 3204 V 45 w(Statement)58 b(Justi\014c)-5
b(ation)56 b FM(;)473 3416 y FI(Step)p 653 3416 V 41
w(R)-5 b(esult)p 939 3416 V 44 w(Constructor)59 b FT(=)48
b FM(fact)e FN(j)i FM(result)473 3627 y FI(Existential)p
907 3627 V 45 w(R)-5 b(esult)57 b FT(=)569 3740 y FI(Existential)p
1003 3740 V 44 w(R)-5 b(esult)p 1288 3740 V 44 w(Constructor)569
3853 y(V)e(ar)p 730 3853 V 44 w(T)g(erms)56 b(Such)p
1264 3853 V 42 w(That)p 1494 3853 V 44 w(Constructor)664
3966 y(L)-5 b(ab)g(el)5 b(le)-5 b(d)p 990 3966 V 46 w(Statements)569
4079 y(Justi\014c)g(ation)56 b FM(;)473 4291 y FI(Existential)p
907 4291 V 45 w(R)-5 b(esult)p 1193 4291 V 44 w(Constructor)59
b FT(=)48 b FM(there)e(is)h FT([)h Fw(some)e FT(])473
4502 y FI(The)-5 b(or)g(em)57 b FT(=)617 4615 y FI(The)-5
b(or)g(em)p 973 4615 V 43 w(Constructor)59 b(L)-5 b(ab)g(el)5
b(le)-5 b(d)p 1860 4615 V 46 w(Statements)617 4728 y(Justi\014c)g
(ation)55 b FM(;)473 4940 y FI(The)-5 b(or)g(em)p 829
4940 V 44 w(Constructor)59 b FT(=)48 b FM(theorem)d FN(j)j
FM(lemma)f FN(j)h FM(proposition)c FN(j)k FM(corollary)473
5151 y FI(A)n(bbr)-5 b(eviation)55 b FT(=)48 b FI(A)n(bbr)-5
b(eviation)p 1652 5151 V 41 w(Constructor)60 b(L)-5 b(ab)g(el)5
b(le)-5 b(d)p 2538 5151 V 45 w(Statements)57 b FM(;)473
5363 y FI(A)n(bbr)-5 b(eviation)p 982 5363 V 42 w(Constructor)59
b FT(=)48 b FM(define)e FN(j)i FM(set)473 5575 y FI(Simpli\014c)-5
b(ation)p 1025 5575 V 44 w(De)g(clar)g(ations)58 b FT(=)569
5687 y FI(Simpli\014c)-5 b(ation)p 1121 5687 V 44 w(Constructor)59
b(Simpli\014c)-5 b(ation)p 2235 5687 V 44 w(Lines)55
b FM(;)p eop
%%Page: 224 234
224 233 bop 378 5 a FF(APPENDIX)31 b(A.)61 b(THE)30 b(SYNT)-8
b(AX)30 b(OF)h(SPL)1669 b FT(224)473 396 y FI(Simpli\014c)-5
b(ation)p 1025 396 29 4 v 44 w(Constructor)60 b FT(=)47
b FM(simplify)473 609 y FI(Simpli\014c)-5 b(ation)p 1025
609 V 44 w(Lines)55 b FT(=)569 722 y FI(Simpli\014c)-5
b(ation)p 1121 722 V 44 w(Line)54 b FN(f)48 b FT([)g
FI(Sep)-5 b(ar)g(ator)61 b FT(])48 b FI(Simpli\014c)-5
b(ation)p 2609 722 V 44 w(Line)54 b FN(g)48 b FM(;)521
934 y FI(Simpli\014c)-5 b(ation)p 1073 934 V 44 w(Line)54
b FT(=)48 b(\()g Fw(with)e FN(j)i Fw(without)d FT(\))j
FI(Simpli\014er)p 2666 934 V 45 w(Identi\014ers)473 1147
y(Know)5 b(le)-5 b(dge)p 908 1147 V 43 w(De)g(clar)g(ation)56
b FT(=)569 1260 y FI(Know)5 b(le)-5 b(dge)p 1004 1260
V 42 w(Constructor)60 b(Know)5 b(le)-5 b(dge)p 2000 1260
V 42 w(Lines)55 b FM(;)473 1472 y FI(Know)5 b(le)-5 b(dge)p
908 1472 V 43 w(Constructor)59 b FT(=)48 b FM(consider)473
1685 y FI(Know)5 b(le)-5 b(dge)p 908 1685 V 43 w(Lines)55
b FT(=)47 b FI(Know)5 b(le)-5 b(dge)p 1761 1685 V 42
w(Line)55 b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61
b FT(])48 b FI(Know)5 b(le)-5 b(dge)p 3131 1685 V 42
w(Line)54 b FN(g)473 1898 y FI(Know)5 b(le)-5 b(dge)p
908 1898 V 43 w(Line)54 b FT(=)47 b FI(Cate)-5 b(gory)p
1650 1898 V 44 w(Identi\014er)59 b(Sentenc)-5 b(e)p 2461
1898 V 41 w(List)473 2110 y(L)g(ab)g(el)5 b(le)-5 b(d)p
799 2110 V 46 w(Statements)56 b FT(=)48 b FI(L)-5 b(ab)g(el)5
b(le)-5 b(d)p 1764 2110 V 45 w(Statement)58 b FN(f)48
b FT([)g FI(Sep)-5 b(ar)g(ator)61 b FT(])48 b FI(L)-5
b(ab)g(el)5 b(le)-5 b(d)p 3247 2110 V 45 w(Statement)58
b FN(g)473 2323 y FI(L)-5 b(ab)g(el)5 b(le)-5 b(d)p 799
2323 V 46 w(Statement)57 b FT(=)569 2435 y([)48 b Fw(case)e
FT(])i([)g FI(L)-5 b(ab)g(el)p 1225 2435 V 45 w(Identi\014er)83
b FT(:)48 b(])g FI(Statement)473 2648 y(Such)p 674 2648
V 42 w(That)p 904 2648 V 44 w(Constructor)60 b FT(=)47
b FM(such)g(that)f FN(j)i FM(st)f FN(j)h FM(where)378
2934 y FH(A.2)135 b(Justi\014cations)473 3137 y FI(Justi\014c)-5
b(ation)56 b FT(=)569 3250 y FI(Pr)-5 b(o)g(of)p 805
3250 V 55 w(Justi\014c)g(ation)473 3363 y FN(j)48 b FI(Case)p
749 3363 V 42 w(Splitting)p 1121 3363 V 43 w(Justi\014c)-5
b(ation)473 3476 y FN(j)48 b FI(Iter)-5 b(ative)p 887
3476 V 43 w(Ine)g(qualities)p 1378 3476 V 42 w(Justi\014c)g(ation)473
3589 y FN(j)48 b FI(Simple)p 823 3589 V 42 w(Justi\014c)-5
b(ation)473 3802 y(Pr)g(o)g(of)p 710 3802 V 56 w(Justi\014c)g(ation)55
b FT(=)569 3914 y FI(Pr)-5 b(o)g(of)p 805 3914 V 55 w(Start)664
4027 y(R)g(e)g(asoning)p 1077 4027 V 45 w(Items)569 4140
y(Pr)g(o)g(of)p 805 4140 V 55 w(Ending)473 4353 y(Pr)g(o)g(of)p
710 4353 V 56 w(Start)57 b FT(=)569 4466 y FM(proof)46
b FT([)i([)g Fw(proceed)d FT(])j FI(Simple)p 1704 4466
V 42 w(Justi\014c)-5 b(ation)56 b FT(;)48 b(])473 4678
y FI(Pr)-5 b(o)g(of)p 710 4678 V 56 w(Ending)55 b FT(=)569
4791 y FI(Backwar)-5 b(d)p 963 4791 V 46 w(Pr)g(o)g(of)p
1230 4791 V 55 w(Ending)473 4904 y FN(j)48 b FT(\()g
Fw(qed)43 b Fu(j)g Fw(end)k FT(\))h([)g FI(Simple)p 1482
4904 V 42 w(Justi\014c)-5 b(ation)56 b FT(])473 5117
y FI(Backwar)-5 b(d)p 867 5117 V 46 w(Pr)g(o)g(of)p 1135
5117 V 55 w(Ending)56 b FT(=)569 5230 y FI(Backwar)-5
b(d)p 963 5230 V 46 w(Pr)g(o)g(of)p 1230 5230 V 55 w(Constructor)59
b(L)-5 b(ab)g(el)5 b(le)-5 b(d)p 2116 5230 V 46 w(Statements)56
b(Simple)p 2913 5230 V 42 w(Justi\014c)-5 b(ation)56
b FM(;)473 5442 y FI(Backwar)-5 b(d)p 867 5442 V 46 w(Pr)g(o)g(of)p
1135 5442 V 55 w(Constructor)60 b FT(=)47 b(\()h Fw(sufficient)40
b(to)j(show)j FN(j)i Fw(sts)e FT(\))p eop
%%Page: 225 235
225 234 bop 378 5 a FF(APPENDIX)31 b(A.)61 b(THE)30 b(SYNT)-8
b(AX)30 b(OF)h(SPL)1669 b FT(225)473 396 y FI(Case)p
676 396 29 4 v 42 w(Splitting)p 1048 396 V 43 w(Justi\014c)-5
b(ation)56 b FT(=)569 509 y FI(Case)p 772 509 V 42 w(Splitting)p
1144 509 V 43 w(Constructor)j FT([)48 b FI(Simple)p 2054
509 V 42 w(Justi\014c)-5 b(ation)56 b FT(;)48 b(])569
622 y FI(Case)p 772 622 V 42 w(Items)569 735 y(End)p
743 735 V 44 w(Cases)p 1012 735 V 42 w(Constructor)60
b FT([)48 b FP(S)5 b(impl)r(e)p 1941 735 V 34 w(J)k(ustif)h(ication)48
b FT(])473 948 y FI(Case)p 676 948 V 42 w(Splitting)p
1048 948 V 43 w(Constructor)60 b FT(=)47 b(\()h Fw(per)f
FN(j)h Fw(consider)c FT(\))k FM(cases)473 1160 y FI(End)p
647 1160 V 44 w(Cases)p 916 1160 V 43 w(Constructor)59
b FT(=)48 b(\()g Fw(end)e FT([)i Fw(cases)e FT(])i FN(j)g
Fw(qed)e FT(\))473 1373 y FI(Case)p 676 1373 V 42 w(Items)56
b FT(=)47 b FI(Case)p 1300 1373 V 42 w(Item)55 b FN(f)48
b FI(Case)p 1861 1373 V 42 w(Item)55 b FN(g)473 1585
y FI(Case)p 676 1585 V 42 w(Item)g FT(=)48 b([)g FP(S)5
b(upposition)p 1625 1585 V 33 w(C)i(onstr)s(uctor)49
b FT(])f FI(L)-5 b(ab)g(el)5 b(le)-5 b(d)p 2600 1585
V 45 w(Statement)58 b(Justi\014c)-5 b(ation)473 1798
y(Supp)g(osition)p 939 1798 V 44 w(Constructor)60 b FT(=)47
b(\()h Fw(suppose)d FN(j)j Fw(case)e FT(\))473 2010 y
FI(Iter)-5 b(ative)p 814 2010 V 43 w(Ine)g(qualities)p
1305 2010 V 42 w(Justi\014c)g(ation)56 b FT(=)569 2123
y FI(Simple)p 846 2123 V 42 w(Justi\014c)-5 b(ation)473
2236 y FN(f)49 b FP(:)f FI(Part)p 830 2236 V 43 w(F)-7
b(ormula)57 b(Simple)p 1521 2236 V 42 w(Justi\014c)-5
b(ation)56 b FN(g)473 2449 y FI(Simple)p 750 2449 V 42
w(Justi\014c)-5 b(ation)56 b FT(=)569 2562 y([)48 b Fw(<)f
FI(Simpli\014ers)57 b Fw(>)47 b FT(])h FM(by)f FT([)h
FI(Flags)56 b FT(])760 2675 y([)48 b FI(Pr)-5 b(over)p
1112 2675 V 45 w(Identi\014er)58 b FT(])48 b([)g FI(Flags)56
b FT(])48 b FI(Pr)-5 b(over)p 2323 2675 V 45 w(Par)g(ams)2659
2690 y FE(Pr)l(over)p 2879 2690 V 43 w(Identi\014er)378
2887 y FT(\()p FI(Pr)g(over)p 692 2887 28 4 v 44 w(Par)g(ams)1027
2902 y FE(Pr)l(over)p 1246 2902 V 41 w(Identi\014er)1603
2887 y FT(dep)s(ends)31 b(on)i(the)h FI(Pr)-5 b(over)p
2525 2887 V 43 w(Identi\014er)44 b FT(follo)m(wing)31
b(the)j(optional)378 3000 y FI(Flags)8 b FT(.\))473 3213
y FI(Flags)57 b FT(=)47 b FI(Flag)p 1039 3213 29 4 v
43 w(Identi\014er)59 b FN(f)48 b FI(Flag)p 1767 3213
V 43 w(Identi\014er)58 b FN(g)473 3425 y FI(Flag)p 657
3425 V 44 w(Identi\014er)g FT(=)47 b FM(pure)473 3638
y FI(Pr)-5 b(over)p 753 3638 V 45 w(Identi\014er)59 b
FT(=)47 b(\()h Fw(cfol)e FN(j)i Fw(fol)f FN(j)h Fw(taut)e
FN(j)i Fw(tab)e FT(\))473 3850 y FI(Pr)-5 b(over)p 753
3850 V 45 w(Par)g(ams)1089 3870 y Fw(cfol)1315 3850 y
FT(=)47 b FI(Structur)-5 b(e)g(d)p 1853 3850 V 46 w(Expr)g(ession)473
3963 y(Pr)g(over)p 753 3963 V 45 w(Par)g(ams)1089 3983
y Fw(fol)1271 3963 y FT(=)48 b([)g FI(Sentenc)-5 b(e)p
1822 3963 V 41 w(List)57 b FT(])473 4076 y FI(Pr)-5 b(over)p
753 4076 V 45 w(Par)g(ams)1089 4091 y Fw(taut)1315 4076
y FT(=)47 b([)h FI(Sentenc)-5 b(e)p 1865 4076 V 42 w(List)56
b FT(])473 4189 y FI(Pr)-5 b(over)p 753 4189 V 45 w(Par)g(ams)1089
4208 y Fw(tab)1271 4189 y FT(=)48 b([)g FI(Sentenc)-5
b(e)p 1822 4189 V 41 w(List)57 b FT(])473 4401 y FI(Structur)-5
b(e)g(d)p 893 4401 V 46 w(Expr)g(ession)56 b FT(=)47
b FN(f)i FI(Then)p 1833 4401 V 41 w(Expr)-5 b(ession)57
b Fw(on)47 b FN(g)h FI(A)n(nd)p 2748 4401 V 43 w(Expr)-5
b(ession)473 4614 y(A)n(nd)p 651 4614 V 44 w(Expr)g(ession)56
b FT(=)48 b FI(Sentenc)-5 b(e)54 b FN(f)49 b Fw(and)d
FI(Sentenc)-5 b(e)55 b FN(g)1189 4727 y(j)48 b FM(\()p
FI(Structur)-5 b(e)g(d)p 1730 4727 V 45 w(Sentenc)g(e)7
b FM(\))473 4939 y FI(Then)p 690 4939 V 42 w(Expr)-5
b(ession)56 b FT(=)48 b FI(Sentenc)-5 b(e)55 b FN(f)48
b Fw(then)e FI(Sentenc)-5 b(e)55 b FN(g)1237 5052 y(j)48
b FM(\()p FI(Structur)-5 b(e)g(d)p 1778 5052 V 45 w(Sentenc)g(e)7
b FM(\))p eop
%%Page: 226 236
226 235 bop 378 5 a FF(APPENDIX)31 b(A.)61 b(THE)30 b(SYNT)-8
b(AX)30 b(OF)h(SPL)1669 b FT(226)378 396 y FH(A.3)135
b(Sen)l(tences)473 599 y FI(Sentenc)-5 b(e)p 832 599
29 4 v 42 w(List)56 b FT(=)48 b FI(Sentenc)-5 b(e)p 1547
599 V 41 w(Item)55 b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61
b FT(])48 b FI(Sentenc)-5 b(e)p 2843 599 V 42 w(Item)55
b FN(g)473 812 y FI(Sentenc)-5 b(e)p 832 812 V 42 w(Item)55
b FT(=)569 925 y([)48 b Fw(<)f FI(Simpli\014ers)57 b
Fw(>)47 b FT(])h(\()g Fw(\()f FI(Sentenc)-5 b(e)p 1905
925 V 42 w(List)56 b Fw(\))48 b FN(j)g FI(Unsimpli\014e)-5
b(d)p 2811 925 V 45 w(Sentenc)g(e)55 b FT(\))473 1137
y FI(Simpli\014ers)105 b(Simpli\014er)58 b FN(f)49 b
FT([)f FI(Sep)-5 b(ar)g(ator)60 b FT(])48 b FI(Simpli\014er)59
b FN(g)473 1350 y FI(Simpli\014er)g FT(=)664 1463 y FI(Simpli\014er)p
1062 1463 V 46 w(Identi\014er)569 1576 y FN(j)48 b FI(L)-5
b(ab)g(el)p 857 1576 V 45 w(Identi\014er)569 1689 y FN(j)48
b FI(Sentenc)-5 b(e)473 1901 y(Sentenc)g(e)55 b FT(=)48
b([)g Fw(<)f FI(Simpli\014ers)56 b Fw(>)48 b FT(])g FI(Unsimpli\014e)-5
b(d)p 2300 1901 V 45 w(Sentenc)g(e)473 2114 y(Unsimpli\014e)g(d)p
977 2114 V 46 w(Sentenc)g(e)54 b FT(=)664 2227 y([)48
b Fw([)g FI(A)n(bstr)-5 b(actions)56 b Fw(])47 b FT(])h(\()g
FI(L)-5 b(ab)g(el)p 1835 2227 V 45 w(Identi\014er)59
b FN(j)47 b FI(F)-7 b(ormula)57 b FT(\))48 b([)g Fw([)g
FI(Applic)-5 b(ations)57 b Fw(])47 b FT(])569 2340 y
FN(j)h FI(Comp)-5 b(ound)p 1075 2340 V 46 w(Sentenc)g(e)473
2552 y(Comp)g(ound)p 906 2552 V 47 w(Sentenc)g(e)54 b
FT(=)664 2665 y FM(\()48 b FI(Comp)-5 b(ound)p 1193 2665
V 46 w(Sentenc)g(e)55 b FM(\))569 2778 y FN(j)48 b FI(R)n(ule)p
833 2778 V 41 w(Identi\014er)58 b(R)n(ule)p 1473 2778
V 41 w(Par)-5 b(ams)1808 2793 y FE(R)n(ule)p 1958 2793
V 40 w(Identi\014er)519 2990 y FT(\()p FI(R)n(ule)p 744
2990 28 4 v 40 w(Par)g(ams)1079 3005 y FE(R)n(ule)p 1228
3005 V 38 w(Identi\014er)1583 2990 y FT(dep)s(ends)29
b(on)h(the)g FI(R)n(ule)p 2407 2990 V 40 w(Identi\014er)10
b FT(.\))473 3203 y FI(R)n(ule)p 664 3203 29 4 v 41 w(Identi\014er)59
b FT(=)47 b FM(select)473 3416 y FI(R)n(ule)p 664 3416
V 41 w(Par)-5 b(ams)1000 3435 y Fw(select)1313 3416 y
FT(=)48 b FI(T)-7 b(erm)55 b(Sentenc)-5 b(e)473 3628
y(A)n(bstr)g(actions)57 b FT(=)47 b FI(A)n(bstr)-5 b(action)56
b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61 b FT(])48
b FI(A)n(bstr)-5 b(action)56 b FN(g)473 3841 y FI(A)n(bstr)-5
b(action)56 b FT(=)664 3953 y FI(T)-7 b(yp)i(e)p 862
3953 V 42 w(A)n(bstr)g(action)569 4066 y FN(j)48 b FI(V)-7
b(ar)p 803 4066 V 44 w(A)n(bstr)i(action)569 4179 y FN(j)48
b FI(T)-7 b(erm)p 867 4179 V 42 w(A)n(bstr)i(action)473
4392 y(T)e(yp)i(e)p 671 4392 V 42 w(A)n(bstr)g(action)56
b FT(=)48 b FI(T)-7 b(yp)i(e)p 1524 4392 V 42 w(V)e(ar)473
4604 y(V)g(ar)p 634 4604 V 45 w(A)n(bstr)i(action)56
b FT(=)47 b FI(V)-7 b(ar)p 1449 4604 V 45 w(T)g(erm)473
4817 y(T)g(erm)p 698 4817 V 42 w(A)n(bstr)i(action)56
b FT(=)48 b FI(L)-5 b(ab)g(el)p 1568 4817 V 44 w(Identi\014er)473
5029 y(Applic)g(ations)58 b FT(=)47 b FI(Applic)-5 b(ation)57
b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61 b FT(])48
b FI(Applic)-5 b(ation)56 b FN(g)473 5242 y FI(Applic)-5
b(ation)57 b FT(=)664 5355 y FI(T)-7 b(yp)i(e)p 862 5355
V 42 w(Applic)g(ation)569 5468 y FN(j)48 b FI(V)-7 b(ar)p
803 5468 V 44 w(Applic)i(ation)473 5680 y(T)e(yp)i(e)p
671 5680 V 42 w(Applic)g(ation)57 b FT(=)47 b FI(HOL)p
1525 5680 V 34 w(T)-7 b(yp)i(e)p 1751 5680 V 42 w(V)e(ar)58
b FM(=)48 b FI(T)-7 b(yp)i(e)p eop
%%Page: 227 237
227 236 bop 378 5 a FF(APPENDIX)31 b(A.)61 b(THE)30 b(SYNT)-8
b(AX)30 b(OF)h(SPL)1669 b FT(227)473 396 y FI(V)-7 b(ar)p
634 396 29 4 v 45 w(Applic)i(ation)57 b FT(=)664 509
y FI(Explicit)p 979 509 V 44 w(V)-7 b(ar)p 1170 509 V
44 w(Applic)i(ation)569 622 y FN(j)48 b FI(Implicit)p
962 622 V 44 w(V)-7 b(ar)p 1153 622 V 44 w(Applic)i(ation)473
835 y(Explicit)p 788 835 V 44 w(V)e(ar)p 979 835 V 44
w(Applic)i(ation)57 b FT(=)47 b FI(HOL)p 1832 835 V 34
w(V)-7 b(ar)p 2021 835 V 45 w(T)g(erm)7 b FT([)48 b Fw(.)f
FI(Inte)-5 b(ger)58 b FT(])48 b FM(=)g FI(T)-7 b(erm)473
1047 y(Implicit)p 793 1047 V 44 w(V)g(ar)p 984 1047 V
45 w(Applic)i(ation)56 b FT(=)48 b FI(T)-7 b(erm)473
1260 y(T)g(erms)56 b(T)-7 b(erm)56 b FN(f)48 b FT([)g
FI(Sep)-5 b(ar)g(ator)61 b FT(])48 b FI(T)-7 b(erm)55
b FN(g)473 1472 y FI(T)-7 b(yp)i(e)56 b FT(=)47 b FM(")p
FI(HOL)p 1081 1472 V 34 w(T)-7 b(erm)7 b FM(")473 1685
y FI(T)-7 b(yp)i(es)57 b FT(=)47 b FI(T)-7 b(yp)i(e)55
b FN(f)49 b FT([)f FI(Sep)-5 b(ar)g(ator)61 b FT(])47
b FI(T)-7 b(yp)i(e)56 b FN(g)473 1898 y FI(T)-7 b(yp)i(e)56
b FT(=)47 b FM(")p FI(HOL)p 1081 1898 V 34 w(T)-7 b(yp)i(e)7
b FM(")473 2110 y FI(T)-7 b(yp)i(e)p 671 2110 V 42 w(V)e(ars)56
b FT(=)48 b FI(T)-7 b(yp)i(e)p 1255 2110 V 42 w(V)e(ar)58
b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61 b FT(])48
b FI(T)-7 b(yp)i(e)p 2358 2110 V 42 w(V)e(ar)58 b FN(g)473
2323 y FI(T)-7 b(yp)i(e)p 671 2323 V 42 w(V)e(ar)58 b
FT(=)48 b FM(")p FI(HOL)p 1271 2323 V 33 w(T)-7 b(yp)i(e)p
1496 2323 V 42 w(V)e(ar)10 b FM(")473 2535 y FI(V)-7
b(ar)p 634 2535 V 45 w(T)g(erms)56 b FT(=)47 b FI(V)-7
b(ar)p 1247 2535 V 45 w(T)g(erm)55 b FN(f)48 b FT([)g
FI(Sep)-5 b(ar)g(ator)61 b FT(])48 b FI(V)-7 b(ar)p 2377
2535 V 45 w(T)g(erm)55 b FN(g)473 2748 y FI(V)-7 b(ar)p
634 2748 V 45 w(T)g(erm)55 b FT(=)48 b FM(")p FI(HOL)p
1298 2748 V 33 w(V)-7 b(ar)p 1486 2748 V 45 w(T)g(erm)7
b FM(")473 2960 y FI(F)-7 b(ormulas)58 b FT(=)47 b FI(F)-7
b(ormula)57 b FN(f)48 b FT([)g FI(Sep)-5 b(ar)g(ator)61
b FT(])48 b FI(F)-7 b(ormula)57 b FN(g)473 3173 y FI(F)-7
b(ormula)57 b FT(=)48 b FM(")p FI(HOL)p 1226 3173 V 33
w(F)-7 b(ormula)9 b FM(")473 3385 y FI(Part)p 663 3385
V 44 w(F)-7 b(ormula)57 b FT(=)47 b FM(")h FI(HOL)p 1493
3385 V 33 w(in\014x)59 b(HOL)p 1952 3385 V 34 w(T)-7
b(erm)55 b FM(")473 3598 y FI(Sep)-5 b(ar)g(ator)61 b
FT(=)48 b FM(,)f FN(j)h FM(and)f FN(j)h FM(&)p eop
%%Page: 228 238
228 237 bop 378 1019 a FJ(App)5 b(endix)65 b(B)378 1434
y FR(Seman)-6 b(tic)77 b(T)-19 b(ableaux)76 b(for)378
1683 y(First-Order)h(Logic)g(With)h(and)378 1932 y(Without)g(Equalit)-6
b(y)378 2377 y FT(Seman)m(tic)36 b(tableau)f(calculi)f(ha)m(v)m(e)j(b)s
(ecome)f(v)m(ery)g(p)s(opular)d(recen)m(tly)j(in)f(the)h(automated)g
(deduc-)378 2490 y(tion)25 b(comm)m(unit)m(y)g(since)f(they)h(can)h(b)s
(e)e(used)h(for)f(a)i(v)-5 b(ariet)m(y)25 b(of)h(di\013eren)m(t)e
(logics)h(including)d(classical)378 2603 y(\014rst-order)35
b(logic)g(\(Fitting)h(1996\),)j(higher-order)c(logics)g(\(Kohlhase)g
(1995;)41 b(Konrad)34 b(1998\),)40 b(in-)378 2716 y(tuitionistic)e
(logic)i(\(Bittel)g(1992\))i(and)e(mo)s(dal)f(logics)g(\(Fitting)h
(1972\).)72 b(This)39 b(in)m(terest)h(is)f(also)378 2829
y(attributed)g(to)h(the)f(success)h(of)f(mo)s(del)g(elimination)d(\(Lo)
m(v)m(eland)k(1968\))i(based)d(pro)s(cedures)f(for)378
2942 y(classical)k(\014rst-order)h(logic)g(whic)m(h)f(represen)m(t)i(a)
f(comp)s(etitiv)m(e)h(alternativ)m(e)g(to)g(the)f(resolution)378
3055 y(paradigm.)74 b(The)41 b(main)g(motiv)-5 b(ation)42
b(of)g(this)e(app)s(endix)g(is)h(to)h(in)m(tro)s(duce)f(the)h(notions)f
(of)h(se-)378 3168 y(man)m(tic)28 b(tableaux)g(and)g(tableau-based)g
(calculi)f(and)g(the)h(problems)f(in)m(v)m(olv)m(ed)h(in)f(reasoning)g
(with)378 3281 y(equalit)m(y)j(in)f(suc)m(h)h(framew)m(orks.)378
3567 y FH(B.1)135 b(The)44 b(Structure)h(of)g(T)-11 b(ableaux)378
3770 y FT(In)43 b(general,)k(a)c(tableau)h(can)g(b)s(e)e(visualised)f
(as)j(a)g(tree)g(whose)f(no)s(des)g(can)g(b)s(e)g(lab)s(elled)e(with)
378 3883 y(form)m(ulae.)f(That)31 b(is,)514 4071 y FN(\017)46
b FT(The)30 b(empt)m(y)h(tree)g(is)e(a)i(tableau)514
4258 y FN(\017)46 b FT(One)40 b(or)g(more)g(tableaux)g(branc)m(hing)f
(from)h(a)g(no)s(de)g(p)s(ossibly)d(lab)s(elled)g(with)i(a)i(form)m
(ula)605 4371 y(constructs)31 b(another)f(tableau.)378
4559 y(Usually)-8 b(,)29 b(all)g(the)h(non-ro)s(ot)g(no)s(des)g(of)g
(the)g(tableau)g(are)g(lab)s(elled)e(with)g(some)j(form)m(ula.)40
b(Concep-)378 4672 y(tually)29 b(a)i(tableau)f(represen)m(ts)g(a)h
(form)m(ula)f(according)g(to)h(the)g(follo)m(wing)e(rules:)514
4859 y FN(\017)46 b FT(A)41 b(tableau)g(whic)m(h)f(do)s(es)h(not)g(con)
m(tain)g(an)m(y)h(no)s(des)e(whic)m(h)g(are)h(lab)s(elled)e(with)g
(form)m(ulae)605 4972 y(represen)m(ts)31 b FN(>)p FT(,)514
5160 y FN(\017)46 b FT(A)31 b(tableau)f(consisting)f(of)i(one)f(no)s
(de)g(lab)s(elled)e(with)h(a)i(form)m(ula)e FP(A)i FT(represen)m(ts)f
FP(A)p FT(.)514 5347 y FN(\017)46 b FT(The)32 b(tableau)g(constructed)g
(from)g(some)g(no)s(de)g(and)f(the)h(tableaux)g FP(T)3041
5361 y FL(1)3081 5347 y FP(;)15 b(:)g(:)g(:)31 b(;)15
b(T)3350 5361 y FO(n)3430 5347 y FT(represen)m(ts)605
5460 y(the)23 b(form)m(ula)f FP(P)1138 5474 y FL(1)1183
5460 y FN(_)5 b(\001)15 b(\001)g(\001)6 b(_)f FP(P)1484
5474 y FO(n)1554 5460 y FT(if)22 b(the)h(no)s(de)f(is)g(not)h(lab)s
(elled)d(with)h(a)j(form)m(ula,)f(and)g(it)f(represen)m(ts)605
5573 y FP(A)5 b FN(^)g FT(\()p FP(P)837 5587 y FL(1)882
5573 y FN(_)g(\001)15 b(\001)g(\001)5 b(_)g FP(P)1182
5587 y FO(n)1229 5573 y FT(\))23 b(if)e(the)i(no)s(de)f(is)g(lab)s
(elled)e(with)h FP(A)p FT(,)k(where)d FP(P)2765 5587
y FO(i)2816 5573 y FT(is)g(the)h(form)m(ula)f(represen)m(ted)605
5686 y(b)m(y)30 b(the)h(tableau)f FP(T)1264 5700 y FO(i)1293
5686 y FT(.)2035 5954 y(228)p eop
%%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 FN(^)20 b FT(\(\()p
FP(A)h FN(^)f FT(\()p FN(:)p FP(A)g FN(_)g(:)p FP(C)7
b FT(\)\))20 b FN(_)g FT(\()p FN(:)p FP(B)k FN(^)c FP(B)5
b FT(\)\))p FP(:)519 2115 y FT(Eac)m(h)22 b(branc)m(h)f(of)h(a)g
(tableau)g(is)f(said)f(to)j(represen)m(t)e(the)h(conjunction)f(of)h
(the)g(form)m(ulae)f(lab)s(elling)378 2228 y(its)28 b(no)s(des.)39
b(By)29 b(distributing)d(the)i(conjunctions)g(o)m(v)m(er)i(the)f
(disjunctions,)e(w)m(e)i(can)g(sho)m(w)f(that)i(the)378
2341 y(form)m(ula)39 b(represen)m(ted)g(b)m(y)h(a)g(tableau)f(is)g
(equiv)-5 b(alen)m(t)39 b(to)h(the)g(disjunction)c(of)k(all)f(the)g
(form)m(ulae)378 2454 y(represen)m(ted)32 b(b)m(y)h(the)g(tableau)f
(branc)m(hes.)47 b(W)-8 b(e)34 b(denote)f(the)f(form)m(ula)g(represen)m
(ting)g(a)h(tableau)f FP(T)378 2567 y FT(b)m(y)e FN(h)p
FP(T)13 b FN(i)p FT(.)519 2680 y(A)43 b(no)s(de)f(lab)s(elled)d(with)j
(a)h(sp)s(ecial)e(sym)m(b)s(ol)g(\()p FN(\002)p FT(\))i(called)e
FI(mark)54 b FT(or)42 b FI(close)50 b FT(can)43 b(b)s(e)e(used)h(in)378
2793 y(the)33 b(construction)f(of)h(tableaux)f(during)e(pro)s(of)i
(searc)m(h.)48 b(A)33 b(tableau)f(branc)m(h)g(con)m(taining)h(a)g(no)s
(de)378 2906 y(mark)m(ed)40 b(with)f(suc)m(h)h(a)g(sym)m(b)s(ol)f(is)g
(said)h(to)g(b)s(e)g(closed,)j(otherwise)c(it)h(is)f(said)g(to)i(b)s(e)
e(op)s(en.)70 b(A)378 3018 y(tableau)30 b(is)g(closed)g(if)f(all)g(its)
h(branc)m(hes)g(are)h(closed.)519 3131 y(Some)c(of)h(the)f(literature)g
(con)m(tains)g(a)h(di\013eren)m(t)e(de\014nition)f(of)j(tableaux)f(in)m
(v)m(olving)f(m)m(ultisets.)378 3244 y(Basically)-8 b(,)34
b(a)g(branc)m(h)f(is)f(de\014ned)g(as)i(a)g(m)m(ultiset)f(of)g(form)m
(ulae)g(\(corresp)s(onding)f(to)i(the)g(m)m(ultiset)378
3357 y(of)d(form)m(ulae)f(lab)s(elling)d(the)j(no)s(des)g(in)f(the)i
(branc)m(h\),)f(and)g(a)h(tableau)f(is)f(de\014ned)g(as)i(the)g(m)m
(ultiset)378 3470 y(of)f(the)h(op)s(en)f(branc)m(hes.)40
b(F)-8 b(or)31 b(instance,)g(the)f(tableau)g(in)f(\014gure)h(27)h(can)g
(b)s(e)f(represen)m(ted)g(b)m(y)1345 3674 y FN(ff)p FP(C)q(;)15
b(A;)g FN(:)p FP(A)p FN(g)p FP(;)g FN(f)p FP(C)q(;)g(A;)g
FN(:)p FP(C)7 b FN(g)p FP(;)15 b FN(f)p FP(C)q(;)g FN(:)p
FP(B)5 b(;)15 b(B)5 b FN(gg)p FP(:)378 3879 y FT(W)-8
b(e)28 b(use)f(the)g(notation)h FP(B)5 b(;)15 b(')27
b FT(to)h(represen)m(t)f FP(B)18 b FN([)c(f)p FP(')p
FN(g)p FT(,)29 b(where)d FP(B)32 b FT(is)26 b(a)h(branc)m(h)g(and)f
FP(')i FT(is)e(a)h(form)m(ula;)378 3992 y(and)e FP(B)619
4006 y FL(1)674 3992 y FN(j)15 b FP(B)783 4006 y FL(2)838
3992 y FN(j)30 b FP(:)15 b(:)g(:)32 b FN(j)15 b FP(B)1139
4006 y FO(n)1212 3992 y FT(to)26 b(represen)m(t)g(the)g(tableau)g
FN(f)p FP(B)2291 4006 y FL(1)2331 3992 y FP(;)15 b(B)2440
4006 y FL(2)2479 3992 y FP(;)g(:)g(:)g(:)32 b(;)15 b(B)2765
4006 y FO(n)2812 3992 y FN(g)27 b FT(where)e FP(B)3211
4006 y FO(i)3265 3992 y FT(is)g(a)h(branc)m(h)f(for)378
4104 y FP(i)h FN(2)f(f)p FT(1)p FP(;)15 b(:)g(:)g(:)j(;)d(n)p
FN(g)p FT(.)41 b(W)-8 b(e)32 b(also)e(use)45 b(\026)-59
b FP(')31 b FT(to)g(denote)g(the)g(form)m(ula)f FP( )k
FT(if)29 b FP(')i FT(is)f(a)h(negated)g(form)m(ula)f
FN(:)p FP( )s FT(,)h(or)378 4217 y FN(:)p FP(')f FT(otherwise.)519
4330 y(W)-8 b(e)31 b(will)c(use)j(the)g(tree)h(represen)m(tation)f(for)
f(visualising)e(tableaux,)j(and)f(the)h(m)m(ultiset)f(repre-)378
4443 y(sen)m(tation)i(for)f(the)h(formal)e(de\014nition)f(of)j(the)f
(inference)g(rules)f(of)h(tableau)h(calculi.)378 4730
y FH(B.2)135 b(T)-11 b(ableaux-Based)45 b(Pro)t(of)g(Pro)t(cedures)378
4933 y FT(T)-8 b(ableaux)33 b(are)g(constructed)h(b)m(y)f(a)g(n)m(um)m
(b)s(er)f(of)i(refutational)e(pro)s(of)g(pro)s(cedures,)h(referred)g
(to)h(as)378 5045 y(tableau)g(calculi.)51 b(Giv)m(en)34
b(a)g(\014nite)f(set)i(of)f(sen)m(tences)h(\000)f(to)h(b)s(e)f
(refuted,)h(tableau)f(calculi)e(consist)378 5158 y(of)e(the)h(follo)m
(wing)e(t)m(yp)s(es)h(of)h(inference)e(rules:)378 5346
y FQ(Start)45 b FT(Select)31 b(an)f(initial)e(tableau)j
FP(T)1686 5360 y FL(0)1756 5346 y FT(whose)g(represen)m(tativ)m(e)g
(form)m(ula)f(is)g(w)m(eak)m(er)i(than)e(\000,)h(that)605
5459 y(is)36 b(\000)h FN(j)-15 b FT(=)36 b FN(h)p FP(T)1002
5473 y FL(0)1042 5459 y FN(i)p FT(.)61 b(The)37 b(seman)m(tics)g(of)h
(the)f(double)f(turnstile)f(sym)m(b)s(ol,)j FN(j)-15
b FT(=,)38 b(dep)s(ends)e(on)h(the)605 5572 y(logic)30
b(concerned.)p eop
%%Page: 230 240
230 239 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(230)378 396 y
FQ(Expansion)46 b FT(Giv)m(en)27 b(a)g(tableau)g FP(T)1600
410 y FO(i)1628 396 y FT(,)h(it)f(is)f(expanded)g(to)i(a)f(tableau)g
FP(T)2813 410 y FO(i)p FL(+1)2958 396 y FT(b)m(y)g(adding)f(more)h
(struc-)605 509 y(ture)c(to)g(it,)h(with)e(the)h(restriction)e(that)j
(\000)h FN(j)-15 b FT(=)25 b FN(h)p FP(T)2245 523 y FO(i)p
FL(+1)2363 509 y FN(i)e FT(giv)m(en)g(the)g(assumption)e(that)j(\000)h
FN(j)-15 b FT(=)25 b FN(h)p FP(T)3739 523 y FO(i)3767
509 y FN(i)p FT(.)378 697 y FQ(Substitution)45 b FT(Apply)29
b(some)i(substitution)d(to)j FI(al)5 b(l)40 b FT(the)31
b(no)s(des)e(in)h(a)g(tableau.)378 885 y FQ(Close)45
b FT(A)35 b(branc)m(h)e(of)h(a)h(tableau)f(is)f(mark)m(ed)h(as)g
(closed)g(\(b)m(y)g(lab)s(elling)d(the)k(leaf)f(no)s(de)f(with)g(the)
605 998 y(close)28 b(sym)m(b)s(ol\))e(if)g(the)h(set)h(of)g(form)m
(ulae)e(lab)s(elling)e(its)j(no)s(des)f(is)g(sho)m(wn)h(to)h(b)s(e)f
(inconsisten)m(t.)605 1110 y(Giv)m(en)40 b(a)g(tableau)f
FP(T)1348 1125 y FO(f)1394 1110 y FT(,)j(since)d FN(h)p
FP(T)1781 1125 y FO(f)1826 1110 y FN(i)h FT(is)f(equiv)-5
b(alen)m(t)39 b(to)h(the)g(disjunction)d(of)j(the)f(form)m(ulae)605
1223 y(represen)m(ting)d(the)i(branc)m(hes,)g(a)g(closed)e(tableau)h
(represen)m(ts)g(an)g(inconsisten)m(t)f(form)m(ulae,)605
1336 y(i.e.,)16 b FN(h)p FP(T)850 1351 y FO(f)896 1336
y FN(i)25 b(j)-15 b FT(=)25 b FN(?)p FT(,)31 b(if)e FP(T)1325
1351 y FO(f)1401 1336 y FT(is)g(closed.)378 1524 y(A)g(closed)f
(tableau)h(deriv)m(ed)f(using)f(these)i(inferences)f(is)f(therefore)i
(a)g(formal)f(pro)s(of)g(ob)5 b(ject)30 b(of)f(the)378
1637 y(in)m(v)-5 b(alidit)m(y)27 b(of)j(the)f(form)m(ula)g(it)g
(represen)m(ts,)h(and)f(if)f(it)h(is)g(constructed)h(b)m(y)f(this)g
(metho)s(d)g(it)g(giv)m(es)g(a)378 1750 y(pro)s(of)h(of)g(the)h
(inconsistency)e(of)h(\000.)519 1863 y(Note)e(that)e(the)h
(substitution)c(rule)i(is)g FI(non-lo)-5 b(c)g(al)p FT(,)30
b(in)24 b(the)j(sense)f(that)h(it)e(a\013ects)j(all)d(the)h(form)m(u-)
378 1976 y(lae)35 b(lab)s(elling)c(the)k(no)s(des)f(in)g(the)h
(tableau.)54 b(Because)36 b(of)f(this,)g(tableaux)g(and)f(related)h
(metho)s(ds)378 2088 y(are)26 b(usually)d(referred)h(to)j(as)e(rigid)f
(v)-5 b(ariable)24 b(metho)s(ds.)38 b(One)25 b(can)h(reduce)f(this)f
(rigidit)m(y)f(of)j(tableau)378 2201 y(v)-5 b(ariables)28
b(b)m(y)h(in)m(tro)s(ducing)f(univ)m(ersal)f(v)-5 b(ariables)29
b(whic)m(h)f(need)h(not)h(b)s(e)f(instan)m(tiated)g(b)m(y)g(the)h(sub-)
378 2314 y(stitution)f(rule)g(\(see)j(\(Bec)m(k)m(ert)h(and)c(H\177)-45
b(ahnle)30 b(1992\)\).)519 2427 y(In)40 b(a)g(tableau)g(implemen)m
(tation,)i(it)d(is)g(more)i(practical)f(to)h(k)m(eep)f(the)h(global)e
(substitution)378 2540 y(applied)28 b(to)i(the)g(tableau)g(in)f(a)h
(separate)h(data)g(structure)e(instead)g(of)h(applying)e(it)i(to)g(all)
f(tableau)378 2653 y(no)s(des.)42 b(In)30 b(suc)m(h)h(case)h(the)f
(global)f(substitution)f(can)i(b)s(e)g(seen)g(as)g(a)h
FI(c)-5 b(onstr)g(aint)42 b FT(on)31 b(the)g(tableau.)378
2766 y(More)39 b(precisely)-8 b(,)39 b(a)f(substitution)d
FN(f)p FP(x)1713 2780 y FL(1)1791 2766 y FN(!)j FP(t)1953
2780 y FL(1)1992 2766 y FP(;)15 b(:)g(:)g(:)32 b(;)15
b(x)2261 2780 y FO(n)2346 2766 y FN(!)37 b FP(t)2507
2780 y FO(n)2554 2766 y FN(g)h FT(can)g(b)s(e)g(seen)g(as)g(the)g
(constrain)m(t)378 2879 y FP(x)430 2893 y FL(1)495 2879
y FN(')26 b FP(t)625 2893 y FL(1)685 2879 y FN(^)20 b(\001)15
b(\001)g(\001)22 b(^)e FP(x)1026 2893 y FO(n)1099 2879
y FN(')26 b FP(t)1229 2893 y FO(n)1307 2879 y FT(where)k(constrain)m
(ts)h(of)g(the)g(form)f FP(s)c FN(')g FP(t)p FT(,)31
b(called)f(equalit)m(y)h(constrain)m(ts,)378 2992 y(signify)e(the)h
(fact)i(that)f(the)g(term)g FP(s)f FT(m)m(ust)g(b)s(e)g(equal)h(to)g
(the)g(term)g FP(t)p FT(.)41 b(F)-8 b(urthermore,)31
b(the)f(v)-5 b(alidit)m(y)378 3105 y(of)30 b(the)h(constrain)m(t)f
FN(9)o FP(~)-44 b(x:s)1236 3119 y FL(1)1300 3105 y FN(')25
b FP(t)1429 3119 y FL(1)1489 3105 y FN(^)19 b(\001)c(\001)g(\001)21
b(^)f FP(s)1819 3119 y FO(n)1891 3105 y FN(')25 b FP(t)2020
3119 y FO(n)2066 3105 y FT(,)31 b(where)d FP(~)-43 b(x)30
b FT(represen)m(ts)g(the)g(list)f(of)i(v)-5 b(ariables)29
b(free)378 3218 y(in)h FP(s)528 3232 y FO(i)582 3218
y FN(')c FP(t)712 3232 y FO(i)771 3218 y FT(for)k FP(i)d
FN(\024)f FP(n)p FT(,)31 b(is)f(equiv)-5 b(alen)m(t)30
b(to)i(whether)e(there)h(is)f(a)i(substitution)c(whic)m(h)i(syn)m
(tactically)378 3330 y FI(uni\014es)37 b FT(\(Robinson)29
b(1971\))j(\(see)f(also)f(\(Jouannaud)f(and)g(Kirc)m(hner)f(1991\)\))33
b(the)d(terms)g FP(s)3508 3344 y FO(i)3565 3330 y FT(and)g
FP(t)3775 3344 y FO(i)3803 3330 y FT(.)378 3443 y(This)g(constrain)m(t)
i(is)f(represen)m(ted)h(b)m(y)g(the)g(set)h FN(f)p FP(s)2112
3457 y FL(1)2180 3443 y FN(')27 b FP(t)2311 3457 y FL(1)2351
3443 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(s)2610 3457 y
FO(n)2685 3443 y FN(')28 b FP(t)2817 3457 y FO(n)2864
3443 y FN(g)p FT(,)33 b(and)e(a)i(solution)d(to)j(this)378
3556 y(constrain)m(t)f(is)g(a)h(substitution)d FP(\033)35
b FT(suc)m(h)d(that)h FP(s)2023 3570 y FO(i)2051 3556
y FP(\033)f FN(')c FP(t)2267 3570 y FO(i)2295 3556 y
FP(\033)35 b FT(for)d FP(i)d FN(\024)g FP(n)p FT(.)46
b(A)32 b(constrain)m(t)h(is)e(said)g(to)j(b)s(e)378 3669
y(satis\014able)g(if)h(it)g(has)g(a)h(solution.)55 b(Uni\014cation)34
b(is)h(often)h(the)g(mec)m(hanism)e(used)h(in)f(\014nding)g(the)378
3782 y(appropriate)c(substitutions)f(to)i(use)g(in)f(the)h(tableau)g
(substitution)e(rule.)41 b(Therefore,)31 b(the)h(global)378
3895 y(substitution)25 b(applied)g(to)j(the)g(tableau)f(is)g(a)h
(solution)e(\(or)h(rather)h(the)f(most)h(general)f(solution\))g(of)378
4008 y(some)22 b(constrain)m(t)h(set.)38 b(The)22 b(m)m(ultiset)f
(notation)h(of)g(tableaux)g(is)f(extended)h(to)h(include)d(constrain)m
(ts)378 4121 y(b)m(y)j(de\014ning)f(a)i FI(c)-5 b(onstr)g(aint)28
b(table)-5 b(au)32 b FT(as)24 b(a)f(pair)g FP(T)34 b
FN(\001)22 b(C)5 b FT(,)25 b(where)e FP(T)36 b FT(is)23
b(a)h(tableau)f(and)g FN(C)28 b FT(is)23 b(a)h(constrain)m(t)378
4234 y(set.)41 b(One)30 b(can)h(then)f(rephrase)g(the)g(substitution)e
(rule)h(in)m(to)i(a)f(constrain)g(rule:)378 4421 y FQ(Constrain)45
b FT(Giv)m(en)39 b(a)f(constrain)m(t)h(tableau)f FP(T)54
b FN(\001)41 b(C)5 b FT(,)41 b(the)d(constrain)m(t)h
FN(C)44 b FT(can)38 b(b)s(e)g(replaced)g(with)605 4534
y(some)31 b(stronger)g(satis\014able)e(constrain)m(t)h
FN(C)2088 4501 y FK(0)2112 4534 y FT(.)378 4722 y(W)-8
b(e)24 b(can)f(also)g(extend)g(equalit)m(y)f(constrain)m(ts)h(to)h
(form)m(ulae)e(b)m(y)h(considering)e(predicates,)j(the)f(unary)378
4835 y(op)s(erator)35 b FN(:)p FT(,)g(and)f(the)g(binary)f(op)s
(erators)h FN(^)g FT(and)g FN(_)g FT(as)h(function)e(sym)m(b)s(ols.)51
b(F)-8 b(or)35 b(example,)h(the)378 4948 y(constrain)m(t)1366
5061 y(\()p FN(:)p FP(P)13 b FT(\()p FP(f)d FT(\()p FP(x)p
FT(\)\))21 b FN(^)f FP(Q)p FT(\()p FP(y)s FT(\)\))26
b FN(')f FT(\()p FN(:)p FP(P)13 b FT(\()p FP(y)s FT(\))21
b FN(^)e FP(Q)p FT(\()p FP(z)t FT(\)\))378 5228 y(is)29
b(satis\014ed)h(b)m(y)g FN(f)p FP(y)f FN(!)c FP(f)10
b FT(\()p FP(x)p FT(\))p FP(;)15 b(z)30 b FN(!)25 b FP(f)10
b FT(\()p FP(x)p FT(\))p FN(g)p FT(,)31 b(and)1397 5432
y(\()p FN(:)p FP(P)13 b FT(\()p FP(f)d FT(\()p FP(x)p
FT(\)\))20 b FN(^)g FP(Q)p FT(\()p FP(y)s FT(\)\))26
b FN(')f FT(\()p FP(P)13 b FT(\()p FP(y)s FT(\))21 b
FN(^)f FP(Q)p FT(\()p FP(z)t FT(\)\))378 5636 y(is)29
b(not)i(satis\014able.)p eop
%%Page: 231 241
231 240 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(231)p 849 408
937 4 v 847 521 4 113 v 1067 487 a FP(\013)p 1341 521
V 279 w(\013)1462 501 y FL(1)1625 487 y FP(\013)1683
501 y FL(2)p 1784 521 V 849 524 937 4 v 847 637 4 113
v 984 603 a FP(')21 b FN(^)f FP( )p 1341 637 V 219 w(')161
b( )p 1784 637 V 847 750 V 919 716 a FN(:)p FT(\()p FP(')20
b FN(_)g FP( )s FT(\))p 1341 750 V 121 w FN(:)p FP(')100
b FN(:)p FP( )p 1784 750 V 847 863 V 899 829 a FN(:)p
FT(\()p FP(')25 b FN(\))g FP( )s FT(\))p 1341 863 V 131
w FP(')131 b FN(:)p FP( )p 1784 863 V 849 866 937 4 v
1885 408 897 4 v 1883 521 4 113 v 2084 487 a(\014)p 2337
521 V 268 w(\014)2454 501 y FL(1)2625 487 y FP(\014)2676
501 y FL(2)p 2780 521 V 1885 524 897 4 v 1883 637 4 113
v 2001 603 a FP(')20 b FN(_)g FP( )p 2337 637 V 199 w(')161
b( )p 2780 637 V 1883 750 V 1935 716 a FN(:)p FT(\()p
FP(')20 b FN(^)g FP( )s FT(\))p 2337 750 V 101 w FN(:)p
FP(')99 b FN(:)p FP( )p 2780 750 V 1883 863 V 1980 829
a(')26 b FN(\))f FP( )p 2337 863 V 149 w FN(:)p FP(')130
b( )p 2780 863 V 1885 866 897 4 v 2881 408 477 4 v 2879
521 4 113 v 2993 487 a(\037)p 3160 521 V 161 w(\037)3268
501 y FL(1)p 3356 521 V 2881 524 477 4 v 2879 637 4 113
v 2931 603 a FN(::)p FP(')p 3160 637 V 118 w(')p 3356
637 V 2879 750 V 2955 716 a FN(:>)p 3160 750 V 137 w(?)p
3356 750 V 2879 863 V 2955 829 a(:?)p 3160 863 V 137
w(>)p 3356 863 V 2881 866 477 4 v 885 983 865 4 v 883
1096 4 113 v 1129 1062 a FP(\015)p 1424 1096 V 316 w(\015)1539
1076 y FL(1)1579 1062 y FT(\()p FP(t)p FT(\))p 1747 1096
V 885 1099 865 4 v 883 1212 4 113 v 1001 1178 a FN(8)p
FP(x:')p FT(\()p FP(x)p FT(\))p 1424 1212 V 196 w FP(')p
FT(\()p FP(t)p FT(\))p 1747 1212 V 883 1325 V 935 1291
a FN(:)p FT(\()p FN(9)p FP(x:')p FT(\()p FP(x)p FT(\)\))p
1424 1325 V 101 w FN(:)p FP(')p FT(\()p FP(t)p FT(\))p
1747 1325 V 885 1328 865 4 v 1901 983 V 1899 1096 4 113
v 2150 1062 a FP(\016)p 2440 1096 V 322 w(\016)2552 1076
y FL(1)2592 1062 y FT(\()p FP(t)p FT(\))p 2764 1096 V
1901 1099 865 4 v 1899 1212 4 113 v 2017 1178 a FN(9)p
FP(x:')p FT(\()p FP(x)p FT(\))p 2440 1212 V 196 w FP(')p
FT(\()p FP(t)p FT(\))p 2764 1212 V 1899 1325 V 1951 1291
a FN(:)p FT(\()p FN(8)p FP(x:')p FT(\()p FP(x)p FT(\)\))p
2440 1325 V 101 w FN(:)p FP(')p FT(\()p FP(t)p FT(\))p
2764 1325 V 1901 1328 865 4 v 995 1582 a(T)-8 b(able)30
b(4:)41 b(A)30 b(Uniform)f(Notation)i(for)g(First-Order)d(F)-8
b(orm)m(ulae.)519 1839 y(Apart)32 b(from)f(equalit)m(y)g(constrain)m
(ts,)h(whic)m(h)e(represen)m(t)i(the)g(global)f(substitution)e(applied)
g(to)378 1952 y(the)34 b(tableau,)h(one)f(can)h(de\014ne)e(other)h(t)m
(yp)s(es)g(of)g(constrain)m(ts)g(whose)g(purp)s(ose)e(is)h(to)i
(restrict)f(the)378 2065 y(searc)m(h)43 b(space)g(during)c(theorem)k
(pro)m(ving.)75 b(These)42 b(include)e FI(or)-5 b(dering)45
b(c)-5 b(onstr)g(aints)53 b FT(whic)m(h)40 b(are)378
2178 y(often)31 b(used)e(in)g(equalit)m(y)h(reasoning.)378
2422 y FG(B.2.1)112 b(F)-9 b(ree)38 b(V)-9 b(ariable)36
b(T)-9 b(ableaux)378 2593 y FT(A)24 b(rather)g(simple)e(tableau)i
(calculus)f(for)g(\014rst-order)g(logic)h(is)f(the)i(free)f(v)-5
b(ariable)23 b(seman)m(tic)h(tableau)378 2706 y(calculus)40
b(whose)i(branc)m(hes)g(con)m(tain)g(no)s(des)f(lab)s(elled)e(b)m(y)j
(form)m(ulae)f(whic)m(h)g(ma)m(y)i(con)m(tain)f(free)378
2819 y(v)-5 b(ariables.)58 b(T)-8 b(ableau)36 b(branc)m(hes)g(are)h
(closed)g(b)m(y)f(unifying)e(complemen)m(tary)j(form)m(ulae)f(lab)s
(elling)378 2932 y(the)e(branc)m(h)f(no)s(des.)49 b(As)33
b(with)g(other)g(tableau)h(calculi,)f(w)m(e)h(de\014ne)e(this)h
(calculus)f(b)m(y)h(giving)g(the)378 3045 y(start,)e(expansion,)f
(constrain)f(and)h(closure)g(rules)f(for)h(refuting)f(a)i(\014nite)e
(set)i(of)g(sen)m(tences)g(\000.)378 3232 y FQ(Start)45
b FT(Initialise)27 b(b)m(y)i(constructing)g(the)h(constrain)m(t)f
(tableau)h(ha)m(ving)f(one)h(branc)m(h)e(whose)i(no)s(des)605
3345 y(are)h(lab)s(elled)d(b)m(y)i(the)h(form)m(ulae)f(in)f(\000,)h
(and)g(an)g(empt)m(y)h(constrain)m(t)f(set.)378 3533
y FQ(Expansion)46 b FT(Select)30 b(a)g(branc)m(h)g(in)e(the)i(tableau,)
g(and)g(a)g(form)m(ula)f FP(')h FT(lab)s(elling)d(one)j(of)g(its)f(no)s
(des.)605 3646 y(Expand)f(the)h(selected)g(branc)m(h)f(according)h(to)g
(the)g(structure)f(of)h FP(')g FT(using)f(one)h(of)f(the)h(rules)605
3759 y(b)s(elo)m(w.)40 b(T)-8 b(able)29 b(4)h(sho)m(ws)g(the)f(t)m(yp)s
(es)h(of)g(form)m(ulae)f(classi\014ed)f(b)m(y)i(their)e(structure)i
(using)e(the)605 3872 y(uniform)e(notation)j(in)m(tro)s(duced)e(b)m(y)h
(Sm)m(ully)m(an)f(\(1995\))k(with)c(the)i(addition)d(of)j(a)g
FP(\037)f FT(class)g(to)605 3985 y(con)m(tain)j(certain)f(negated)i
(form)m(ulae.)40 b(The)30 b(expansion)f(rules)g(are:)605
4197 y FP(\013)46 b FT(Add)30 b FP(\013)967 4211 y FL(1)1037
4197 y FT(and)f FP(\013)1271 4211 y FL(2)1341 4197 y
FT(to)i(the)g(selected)g(branc)m(h,)605 4343 y FP(\014)51
b FT(Branc)m(h)31 b(the)f(last)g(no)s(de)g(of)h(the)f(selected)h(branc)
m(h)f(with)f FP(\014)2728 4357 y FL(1)2798 4343 y FT(and)h
FP(\014)3026 4357 y FL(2)3066 4343 y FT(,)605 4489 y
FP(\015)51 b FT(Add)29 b FP(\015)949 4503 y FL(1)989
4489 y FT(\()p FP(y)s FT(\))i(to)g(the)f(selected)h(branc)m(h)f(where)g
FP(y)j FT(is)c(a)i(free)g(v)-5 b(ariable,)605 4636 y
FP(\016)50 b FT(Add)28 b FP(\016)933 4650 y FL(1)973
4636 y FT(\()p FP(f)10 b FT(\()o FP(~)-44 b(x)p FT(\)\))31
b(to)f(the)g(selected)g(branc)m(h)f(where)h FP(f)39 b
FT(is)29 b(a)h(new)f(Sk)m(olem)g(function)g(sym)m(b)s(ol,)805
4748 y(and)g FP(~)-44 b(x)30 b FT(is)g(the)g(list)f(of)i(v)-5
b(ariables)29 b(free)h(in)f(the)i FP(\016)j FT(form)m(ula)2760
4715 y FL(1)2799 4748 y FT(.)605 4895 y FP(\037)46 b
FT(Add)29 b FP(\037)964 4909 y FL(1)1034 4895 y FT(to)i(the)f(selected)
h(branc)m(h.)p 378 5119 1380 4 v 482 5173 a FC(1)516
5204 y FB(This)e(v)n(ersion)f(of)h(the)f FA(\016)j FB(expansion)d(rule)
h(is)g(called)g(the)e(lib)r(eralised)j FA(\016)h FB(rule)e(\(H\177)-38
b(ahnle)28 b(and)f(Sc)n(hmitt)g(1994\).)43 b(It)378 5296
y(di\013ers)34 b(from)f(the)g FA(\016)j FB(rule)d(giv)n(en)h(in)f
(\(Fitting)h(1996\))h(whic)n(h)e(includes)h(all)g(the)f(free)h(v)l
(ariables)g(in)g(the)f(branc)n(h)f(as)378 5387 y(argumen)n(ts)25
b(to)i(the)f(sk)n(olem)g(function)g FA(f)8 b FB(.)36
b(T)-6 b(ableau)27 b(calculi)h(using)e(the)g(lib)r(eralised)i(rule)f
(are)f(more)g(e\016cien)n(t.)36 b(Ev)n(en)378 5478 y(more)25
b(lib)r(eralised)j FA(\016)g FB(rules)e(are)g(giv)n(en)g(in)f(\(Bec)n
(k)n(ert,)h(H\177)-38 b(ahnle,)26 b(and)f(Sc)n(hmitt)f(1993;)k(Baaz)f
(and)e(F)-6 b(erm)r(\177)-41 b(uller)26 b(1995\).)p eop
%%Page: 232 242
232 241 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(232)p 378 416
3453 4 v 376 2845 4 2429 v 1799 616 334 4 v 1854 692
a Ft(\000)33 b Fu(\001)f(fg)2170 635 y Ft(\(Start\))702
866 y Fv(B)765 878 y Fs(1)802 866 y Fv(;)14 b(\013)g
Fu(j)28 b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1182
878 y Fq(n)1259 866 y Fu(\001)32 b(C)p 620 903 826 4
v 620 979 a Fv(B)683 991 y Fs(1)720 979 y Fv(;)14 b(\013)810
991 y Fs(1)847 979 y Fv(;)g(\013)937 991 y Fs(2)988 979
y Fu(j)28 b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1264
991 y Fq(n)1341 979 y Fu(\001)32 b(C)1483 922 y Ft(\()p
Fv(\013)p Ft(-expansion\))2248 866 y Fv(B)2311 878 y
Fs(1)2348 866 y Fv(;)14 b(\014)k Fu(j)28 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)2726 878 y Fq(n)2803 866 y Fu(\001)32 b(C)p
2096 903 964 4 v 2096 979 a Fv(B)2159 991 y Fs(1)2196
979 y Fv(;)14 b(\014)2280 991 y Fs(1)2331 979 y Fu(j)g
Fv(B)2431 991 y Fs(1)2468 979 y Fv(;)g(\014)2552 991
y Fs(2)2603 979 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)2878 991 y Fq(n)2956 979 y Fu(\001)32 b(C)3097
922 y Ft(\()p Fv(\014)t Ft(-expansion\))741 1170 y Fv(B)804
1182 y Fs(1)842 1170 y Fv(;)14 b(\015)k Fu(j)28 b(\001)14
b(\001)g(\001)27 b(j)14 b Fv(B)1215 1182 y Fq(n)1293
1170 y Fu(\001)32 b(C)p 629 1207 881 4 v 629 1284 a Fv(B)692
1296 y Fs(1)729 1284 y Fv(;)14 b(\015)5 b(;)14 b(\015)894
1296 y Fs(1)931 1284 y Ft(\()p Fv(y)s Ft(\))g Fu(j)27
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1328 1296 y
Fq(n)1405 1284 y Fu(\001)33 b(C)1547 1227 y Ft(\()p Fv(\015)5
b Ft(-expansion\))2284 1170 y Fv(B)2347 1182 y Fs(1)2384
1170 y Fv(;)14 b(\016)j Fu(j)28 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)2751 1182 y Fq(n)2828 1170 y Fu(\001)32
b(C)p 2154 1207 908 4 v 2154 1284 a Fv(B)2217 1296 y
Fs(1)2254 1284 y Fv(;)14 b(\016)2328 1296 y Fs(1)2365
1284 y Ft(\()p Fv(f)9 b Ft(\()o Fv(~)-41 b(x)q Ft(\)\))14
b Fu(j)28 b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)2881
1296 y Fq(n)2958 1284 y Fu(\001)32 b(C)3100 1227 y Ft(\()p
Fv(\016)s Ft(-expansion\))769 1458 y Fv(B)832 1470 y
Fs(1)869 1458 y Fv(;)14 b(P)e Ft(\()p Fv(s)1042 1470
y Fs(1)1079 1458 y Fv(;)i(:)g(:)g(:)g(;)g(s)1303 1470
y Fq(m)1366 1458 y Ft(\))p Fv(;)g Fu(:)p Fv(P)e Ft(\()p
Fv(t)1617 1470 y Fs(1)1654 1458 y Fv(;)i(:)g(:)g(:)g(;)g(t)1869
1470 y Fq(m)1932 1458 y Ft(\))g Fu(j)g Fv(B)2078 1470
y Fs(2)2129 1458 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)2404 1470 y Fq(n)2482 1458 y Fu(\001)32
b(C)p 769 1495 1817 4 v 925 1571 a Fv(B)988 1583 y Fs(2)1039
1571 y Fu(j)c(\001)14 b(\001)g(\001)27 b(j)14 b Fv(B)1314
1583 y Fq(n)1392 1571 y Fu(\001)32 b(C)23 b([)c(f)p Fv(s)1669
1583 y Fs(1)1729 1571 y Fu(')j Fv(t)1846 1583 y Fs(1)1884
1571 y Fv(;)14 b(:)g(:)g(:)27 b(;)14 b(s)2121 1583 y
Fq(m)2207 1571 y Fu(')23 b Fv(t)2325 1583 y Fq(m)2388
1571 y Fu(g)2623 1514 y Ft(\(Substitution)29 b(+)e(Close\))545
1757 y(1.)45 b(The)33 b(substitution)g(+)f(close)g(rule)g(is)h
(applicable)f(only)g(if)i(the)f(constrain)n(t)e(in)i(the)g(conclusion)f
(is)655 1857 y(satis\014able.)545 2006 y(2.)45 b(In)28
b(the)g(start)f(rule,)g(\000)h(is)f(the)h(set)g(of)f(sen)n(tences)g(to)
h(b)r(e)g(refuted.)545 2156 y(3.)45 b(In)28 b(the)g Fv(\015)k
Ft(expansion)27 b(rule,)g Fv(y)j Ft(is)e(a)f(new)h(free)f(v)-5
b(ariable,)545 2305 y(4.)45 b(In)24 b(the)g Fv(\016)i
Ft(expansion)d(rule)g Fv(f)32 b Ft(is)24 b(a)f(new)g(Sk)n(olem)g
(function)h(sym)n(b)r(ol,)g(and)e Fv(~)-41 b(x)25 b Ft(is)e(the)h(list)
g(of)f(v)-5 b(ariables)655 2405 y(free)27 b(in)h Fv(\016)s
Ft(.)1135 2676 y FT(Figure)i(28:)42 b(The)29 b(F)-8 b(ree)32
b(V)-8 b(ariable)30 b(T)-8 b(ableau)30 b(Calculus.)p
3829 2845 4 2429 v 378 2849 3453 4 v 378 3206 a FQ(Constrain)45
b FT(Giv)m(en)30 b(a)h(tableau)f FP(T)48 b FN(\001)36
b(C)f FT(ha)m(ving)30 b(a)h(branc)m(h)e(with)g(form)m(ulae)h
FP( )k FT(and)c FP(')h FT(include)d(the)605 3318 y(constrain)m(t)38
b FP( )i FN(')50 b FT(\026)-59 b FP(')37 b FT(in)f FN(C)5
b FT(.)61 b(This)36 b(rules)g(fails)f(if)h FN(C)30 b([)25
b(f)p FP( )40 b FN(')51 b FT(\026)-60 b FP(')q FN(g)37
b FT(is)g(unsatis\014able.)58 b(This)36 b(is)605 3431
y(equiv)-5 b(alen)m(t)39 b(to)i(applying)d(the)i(most)g(general)g
(uni\014er)d(of)j FP( )k FT(and)53 b(\026)-59 b FP(')40
b FT(to)h(all)d(the)i(form)m(ulae)605 3544 y(lab)s(elling)24
b(the)k(no)s(des)e(of)i(the)f(tableau)g(substituted)f(with)g(the)h
(most)h(general)f(solution)f(of)i FN(C)5 b FT(.)378 3732
y FQ(Close)45 b FT(Giv)m(en)31 b(a)h(tableau)f FP(T)48
b FN(\001)36 b(C)5 b FT(,)32 b(a)f(branc)m(h)f(is)g(closed)h(if)f(it)h
(con)m(tains)g FN(?)p FT(,)g(or)g(a)g(pair)f(of)h(form)m(ulae)605
3845 y(whic)m(h)26 b(b)s(ecome)i(complemen)m(tary)g(when)e(substituted)
g(with)g(the)i(most)f(general)h(solution)e(of)605 3958
y FN(C)5 b FT(.)519 4145 y(The)33 b(ab)s(o)m(v)m(e)i(calculus)e(is)f
(refutationally)h(complete)h(but)f(is)g(highly)e(nondeterministic.)49
b(This)378 4258 y(nondeterminism)20 b(can)j(b)s(e)f(reduced)g(b)m(y)g
(adding)g(sev)m(eral)h(restrictions)e(to)j(the)f(ab)s(o)m(v)m(e)h
(rules)d(without)378 4371 y(impairing)31 b(the)k(calculus')e
(completeness.)52 b(F)-8 b(or)35 b(instance,)g(the)f
FP(\013)p FT(,)i FP(\014)5 b FT(,)36 b FP(\016)i FT(and)33
b FP(\037)h FT(expansion)f(rules)378 4484 y(can)23 b(b)s(e)g(applied)e
(only)h(once)i(on)e(eac)m(h)j(no)s(de,)f(and)e(the)h(constrain)g(rule)f
(can)h(b)s(e)f(resticted)h(to)h(literals,)378 4597 y(and)i(can)h(b)s(e)
e(immediately)g(follo)m(w)m(ed)h(b)m(y)h(a)f(closure)g(rule.)38
b(The)26 b(sen)m(tences)i(in)d(\000)h(can)h(b)s(e)f(simpli\014ed)378
4710 y(b)m(y)k(pushing)d(the)k(negation)f(to)h(the)f(literals)e(and)i
(remo)m(ving)f(the)i FN(>)e FT(and)h FN(?)f FT(literals.)39
b(As)30 b(a)h(result,)378 4823 y(the)37 b FP(\037)f FT(expansion)g
(rule)f(will)f(nev)m(er)k(b)s(e)e(applicable.)57 b(One)37
b(can)g(also)f(use)h(fair)e(strategies)j(whic)m(h)378
4936 y(mak)m(e)30 b(sure)e(that)i(eac)m(h)g(no)s(de)e(in)g(the)h
(tableau)g(will)e(ev)m(en)m(tually)i(b)s(e)f(used)g(for)h(expansion.)39
b(Finally)-8 b(,)378 5049 y(one)31 b(can)f(apply)f(a)i(b)s(ound)d(on)i
(the)g(tableaux)h(considered)e(during)f(the)i(pro)s(of)g(searc)m(h)h
(to)g(limit)d(the)378 5162 y(searc)m(h)h(space)f(to)h(a)f(\014nite)f
(one)i(\(e.g.,)17 b(tableau)28 b(size,)g(branc)m(h)g(length,)g(the)g(n)
m(um)m(b)s(er)f(of)h(times)g(the)g FP(\015)378 5274 y
FT(rule)h(can)i(b)s(e)e(used,)h(etc.)17 b(\).)41 b(This)28
b(b)s(ound)h(is)g(increased)h(un)m(til)e(a)j(closed)f(tableau)h(is)e
(found.)519 5387 y(The)43 b(free)h(v)-5 b(ariable)42
b(tableau)i(calculus)e(is)h(giv)m(en)h(in)e(\014gure)h(28.)81
b(The)43 b(implemen)m(tation)g(of)378 5500 y Fa(lean)p
FP(T)573 5467 y(A)615 5500 y(P)g FT(giv)m(en)31 b(in)e(\(Bec)m(k)m(ert)
k(and)d(P)m(osegga)i(1995\))h(is)c(an)i(example)f(of)g(this)f
(calculus.)p eop
%%Page: 233 243
233 242 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(233)378 396 y
FG(B.2.2)112 b(Connection)37 b(T)-9 b(ableaux)38 b(Calculus)378
568 y FT(The)28 b(free)h(v)-5 b(ariable)28 b(tableau)h(calculus)e(can)i
(b)s(e)g(mo)s(di\014ed)d(to)k(refute)f(a)g(set)h(of)f(form)m(ulae)f(in)
g(clausal)378 681 y(form.)61 b(Since)36 b(clauses)h(are)g(sk)m
(olemised)g(disjunctions)d(univ)m(ersally)h(quan)m(ti\014ed)h
(implicitly)-8 b(,)36 b(only)378 794 y FP(\015)k FT(and)35
b FP(\014)40 b FT(expansion)34 b(rules)f(are)j(applicable.)53
b(One)34 b(can)i(also)f(de\014ne)f(an)h(expansion)f(step)h(corre-)378
907 y(sp)s(onding)f(to)i(a)h(n)m(um)m(b)s(er)e(of)h FP(\013)g
FT(and)g FP(\014)41 b FT(expansion)35 b(steps)h(so)g(that)h(the)f(leaf)
g(no)s(de)g(of)g(a)g(selected)378 1020 y(branc)m(h)31
b(can)h(b)s(e)f(branc)m(hed)g(with)f(all)h(the)h(literals)e(in)g(a)i
(clause)g(in)e(a)i(single)f(rule.)43 b(That)32 b(is,)f(giv)m(en)378
1133 y(the)h(clause)g FP(L)868 1147 y FL(1)929 1133 y
FN(_)21 b(\001)15 b(\001)g(\001)22 b(_)f FP(L)1282 1147
y FO(m)1349 1133 y FT(,)33 b(an)f(inference)f(rule)g(can)h(b)s(e)g
(de\014ned)f(to)i(represen)m(t)f(the)g(sequence)h(of)378
1246 y(expansions:)1180 1431 y FP(B)1249 1445 y FL(1)1288
1431 y FP(;)15 b FN(8)p FP(x)1431 1445 y FL(1)1470 1431
y FP(;)g(:)g(:)g(:)32 b(;)15 b(x)1739 1445 y FO(n)1786
1431 y FP(:L)1873 1445 y FL(1)1933 1431 y FN(_)20 b(\001)15
b(\001)g(\001)21 b(_)f FP(L)2283 1445 y FO(m)2365 1431
y FN(j)30 b(\001)15 b(\001)g(\001)32 b(j)15 b FP(B)2666
1445 y FO(n)2748 1431 y FN(\001)36 b(C)p 940 1474 2161
4 v 940 1558 a FP(B)1009 1572 y FL(1)1049 1558 y FP(;)15
b FN(8)p FP(x)1192 1572 y FL(2)1231 1558 y FP(;)g(:)g(:)g(:)32
b(;)15 b(x)1500 1572 y FO(n)1547 1558 y FP(:)p FT(\()p
FP(L)1669 1572 y FL(1)1729 1558 y FN(_)20 b(\001)15 b(\001)g(\001)21
b(_)f FP(L)2079 1572 y FO(m)2146 1558 y FT(\))p FN(f)p
FP(x)2278 1572 y FL(1)2343 1558 y FN(!)25 b FP(y)2504
1572 y FL(1)2543 1558 y FN(g)15 b(j)31 b(\001)15 b(\001)g(\001)32
b(j)15 b FP(B)2905 1572 y FO(n)2987 1558 y FN(\001)36
b(C)3143 1496 y FT(\()p FP(\015)5 b FT(\))p 940 1601
V 2008 1677 a(.)2008 1710 y(.)2008 1744 y(.)3143 1624
y(\()p FP(\015)g FT(\))p 1366 1827 1310 4 v 1366 1912
a FP(B)1435 1926 y FL(1)1474 1912 y FP(;)15 b(L)1576
1926 y FL(1)1616 1912 y FP(\033)24 b FN(_)19 b(\001)c(\001)g(\001)22
b(_)d FP(L)2041 1926 y FO(m)2108 1912 y FP(\033)f FN(j)31
b(\001)15 b(\001)g(\001)31 b(j)15 b FP(B)2479 1926 y
FO(n)2562 1912 y FN(\001)35 b(C)2717 1850 y FT(\()p FP(\015)5
b FT(\))p 1186 1955 1671 4 v 1186 2039 a FP(B)1255 2053
y FL(1)1294 2039 y FP(;)15 b(L)1396 2053 y FL(1)1436
2039 y FP(\033)j FN(j)d FP(B)1615 2053 y FL(1)1655 2039
y FP(;)g(L)1757 2053 y FL(2)1797 2039 y FP(\033)23 b
FN(_)d(\001)15 b(\001)g(\001)21 b(_)f FP(L)2222 2053
y FO(m)2288 2039 y FP(\033)f FN(j)30 b(\001)15 b(\001)g(\001)32
b(j)15 b FP(B)2660 2053 y FO(n)2742 2039 y FN(\001)36
b(C)2898 1977 y FT(\()p FP(\014)5 b FT(\))p 1186 2082
V 2008 2158 a(.)2008 2191 y(.)2008 2224 y(.)2898 2105
y(\()p FP(\014)g FT(\))p 1322 2308 1398 4 v 1322 2393
a FP(B)1391 2407 y FL(1)1430 2393 y FP(;)15 b(L)1532
2407 y FL(1)1572 2393 y FP(\033)j FN(j)31 b(\001)15 b(\001)g(\001)31
b(j)15 b FP(B)1943 2407 y FL(1)1983 2393 y FP(;)g(L)2085
2407 y FO(m)2152 2393 y FP(\033)j FN(j)31 b(\001)15 b(\001)g(\001)31
b(j)15 b FP(B)2523 2407 y FO(n)2606 2393 y FN(\001)35
b(C)2761 2331 y FT(\()p FP(\014)5 b FT(\))378 2597 y(where)32
b FP(\033)k FT(is)31 b(the)i(substitution)d FN(f)p FP(x)1588
2611 y FL(1)1657 2597 y FN(!)f FP(y)1822 2611 y FL(1)1861
2597 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(x)2130 2611 y
FO(n)2206 2597 y FN(!)29 b FP(y)2371 2611 y FO(n)2417
2597 y FN(g)k FT(and)f(the)h(\(distinct\))f(v)-5 b(ariables)31
b FP(y)3659 2611 y FO(i)3719 2597 y FT(for)378 2710 y(1)26
b FN(\024)f FP(i)g FN(\024)g FP(n)30 b FT(do)g(not)h(o)s(ccur)f(in)f
FP(B)1490 2724 y FL(1)1545 2710 y FN(j)h(\001)15 b(\001)g(\001)32
b(j)15 b FP(B)1846 2724 y FO(n)1928 2710 y FN(\001)36
b(C)5 b FT(.)519 2823 y(One)31 b(can)h(construct)g(a)g(tableau)f
(consisting)f(only)h(of)g(literals)f(using)g(this)h(expansion)f(rule)g
(on)378 2936 y(the)37 b(giv)m(en)f(set)i(of)e(clauses.)60
b(In)35 b(this)h(case,)k(the)c(refutational)g(completeness)h(of)g(the)g
(calculus)e(is)378 3049 y(preserv)m(ed)c(if)e(the)j(pro)s(of)e(searc)m
(h)h(is)f(restricted)h(to)h(tableaux)e(satisfying)g(a)h(n)m(um)m(b)s
(er)f(of)h(structural)378 3162 y(prop)s(erties)24 b(whic)m(h)h(include)
e FI(c)-5 b(onne)g(cte)g(dness)35 b FT(\(see)27 b(the)f(thesis)f(of)h
(Letz)h(\(1993\))h(in)c(whic)m(h)h(a)h(n)m(um)m(b)s(er)378
3274 y(of)41 b(suc)m(h)g(prop)s(erties)f(are)i(de\014ned)e(and)g
(compared\).)74 b(A)41 b(tableau)h(is)e(said)g(to)i(b)s(e)f(connected)h
(if)378 3387 y(eac)m(h)29 b(inner)e(no)s(de)h(lab)s(elled)d(with)j(a)g
(literal)f FP(L)h FT(has)g(an)h(immediate)e(successiv)m(e)i(leaf)f(no)s
(de)f(lab)s(elled)378 3500 y(with)i(its)g(complemen)m(t)1229
3477 y(\026)1218 3500 y FP(L)p FT(.)41 b(F)-8 b(or)31
b(instance,)f(the)g(tableau)g(in)e(\014gure)i(27)h(is)e(not)h
(connected)h(b)s(ecause)378 3613 y(the)45 b(no)s(de)g(lab)s(elled)d
(with)i FP(C)51 b FT(do)s(es)45 b(not)h(ha)m(v)m(e)g(an)f(immediate)f
(successor)h(lab)s(elled)e(with)h FN(:)p FP(C)7 b FT(.)378
3726 y(The)40 b(pro)s(of)g(searc)m(h)h(space)h(in)d(the)i(connected)h
(tableaux)e(calculus)f(can)i(therefore)h(b)s(e)e(reduced)378
3839 y(b)m(y)d(restricting)f(expansion)f(rules)h(to)h(those)h(whic)m(h)
d(yield)h(a)h(successiv)m(e)g(constrain)f(and)g(closure)378
3952 y(rule.)71 b(This)39 b(restriction)h(mak)m(es)h(tableau)g(pro)s
(of)f(searc)m(h)h(on)g(connection)g(tableau)g(m)m(uc)m(h)g(more)378
4065 y(e\016cien)m(t)25 b(than)f(that)h(of)g(the)g(free)f(v)-5
b(ariable)24 b(tableau)g(illustrated)e(earlier,)j(and)f(o\013ers)h(a)g
(high)e(degree)378 4178 y(of)45 b(goal-directedness.)86
b(The)45 b(expansion-constrain-closure)e(sequence)j(of)f(inferences)g
(de\014nes)378 4291 y(the)f FI(extension)52 b FT(rule)42
b(of)i(the)h(connection)f(tableau)f(calculus.)80 b(The)44
b(other)g(inference)f(rules)g(of)378 4404 y(the)g(calculus)e(are)i(the)
f(start)i(rule)d(whic)m(h)g(constructs)i(the)g(connection)f(tableau)h
FP(L)3413 4418 y FL(1)3467 4404 y FN(j)31 b(\001)15 b(\001)g(\001)31
b(j)15 b FP(L)3761 4418 y FO(m)378 4516 y FT(giv)m(en)33
b(a)h(clause)f FP(L)1030 4530 y FL(1)1092 4516 y FN(_)22
b(\001)15 b(\001)g(\001)23 b(_)f FP(L)1448 4530 y FO(m)1514
4516 y FT(,)35 b(and)d(the)i(reduction)e(rule)g(whic)m(h)h(corresp)s
(onds)f(to)i(a)g(constrain)378 4629 y(rule)44 b(follo)m(w)m(ed)h(b)m(y)
g(a)h(closure)f(rule.)85 b(Note)46 b(that)g(all)f(inferences)f(of)i
(the)f(connection)h(tableau)378 4742 y(calculus)31 b(with)g(the)h
(exception)h(of)f(the)h(start)f(rule)f(result)g(in)g(the)i(closure)f
(of)g(some)h(branc)m(h.)45 b(The)378 4855 y(mo)s(del)35
b(elimination)f(calculus)h(of)h(Lo)m(v)m(eland)h(\(1968\))i(is)c(a)i
(connection)f(tableau)g(calculus)f(where)378 4968 y(the)29
b(branc)m(h)f(to)h(b)s(e)f(expanded)g(is)f(selected)i(in)f(a)g
(depth-\014rst)g(left)g(\(or)h(righ)m(t\))g(most)f(strategy)-8
b(.)42 b(The)378 5081 y Fw(MESON)29 b FT(theorem)j(pro)m(v)m(er)g
(implemen)m(ted)e(in)g(the)i(HOL)f(system)h(is)e(a)i(mo)s(del)e
(elimination)f(calculus)378 5194 y(with)g(an)h(optimised)f(pro)s(of)h
(searc)m(h)h(strategy)g(\(Harrison)f(1996c\).)519 5307
y(Figure)42 b(29)g(giv)m(es)h(the)f(rules)e(for)i(the)g(connection)h
(tableau)e(calculus.)75 b(W)-8 b(e)43 b(illustrate)d(this)p
eop
%%Page: 234 244
234 243 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(234)p 378 416
3453 4 v 376 2394 4 1978 v 1661 616 610 4 v 1661 692
a Fv(L)1718 704 y Fs(1)1768 692 y Fu(j)28 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(L)2038 704 y Fq(m)2132 692 y Fu(\001)33 b(fg)2308
635 y Ft(\(Start\))1547 949 y Fv(B)1610 961 y Fs(1)1647
949 y Fv(;)14 b(L)g Fu(j)27 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)2029 961 y Fq(n)2107 949 y Fu(\001)32 b(C)p
629 986 2499 4 v 629 1065 a Fv(B)692 1077 y Fs(1)729
1065 y Fv(;)14 b(L)823 1077 y Fs(1)874 1065 y Fu(j)28
b(\001)14 b(\001)g(\001)27 b(j)14 b Fv(B)1149 1077 y
Fs(1)1186 1065 y Fv(;)g(L)1280 1077 y Fq(i)p Fr(\000)p
Fs(1)1406 1065 y Fu(j)g Fv(B)1506 1077 y Fs(1)1543 1065
y Fv(;)g(L)1637 1077 y Fq(i)p Fs(+1)1762 1065 y Fu(j)28
b(\001)14 b(\001)g(\001)27 b(j)14 b Fv(B)2037 1077 y
Fs(1)2075 1065 y Fv(;)g(L)2169 1077 y Fq(m)2245 1065
y Fu(j)28 b(\001)14 b(\001)g(\001)27 b(j)14 b Fv(B)2520
1077 y Fq(n)2597 1065 y Fu(\001)33 b(C)23 b([)c(f)p Fv(L)j
Fu(')3024 1044 y Ft(\026)3002 1065 y Fv(L)3059 1077 y
Fq(i)3086 1065 y Fu(g)3166 1006 y Ft(\(Extension\))1002
1322 y Fv(B)1065 1334 y Fs(1)1102 1322 y Fv(;)14 b(P)e
Ft(\()p Fv(s)1275 1334 y Fs(1)1313 1322 y Fv(;)i(:)g(:)g(:)f(;)h(s)1536
1334 y Fq(l)1561 1322 y Ft(\))p Fv(;)g Fu(:)p Fv(P)e
Ft(\()p Fv(t)1812 1334 y Fs(1)1850 1322 y Fv(;)i(:)g(:)g(:)g(;)g(t)2065
1334 y Fq(l)2090 1322 y Ft(\))g Fu(j)g Fv(B)2236 1334
y Fs(2)2287 1322 y Fu(j)28 b(\001)14 b(\001)g(\001)28
b(j)14 b Fv(B)2563 1334 y Fq(n)2640 1322 y Fu(\001)32
b(C)p 1002 1359 1742 4 v 1159 1435 a Fv(B)1222 1447 y
Fs(2)1273 1435 y Fu(j)27 b(\001)14 b(\001)g(\001)28 b(j)14
b Fv(B)1548 1447 y Fq(n)1625 1435 y Fu(\001)33 b(C)23
b([)18 b(f)p Fv(s)1902 1447 y Fs(1)1962 1435 y Fu(')23
b Fv(t)2080 1447 y Fs(1)2117 1435 y Fv(;)14 b(:)g(:)g(:)27
b(;)14 b(s)2354 1447 y Fq(l)2403 1435 y Fu(')22 b Fv(t)2520
1447 y Fq(l)2546 1435 y Fu(g)2782 1378 y Ft(\(Reduction\))545
1605 y(1.)45 b(The)28 b(rules)f(are)f(applicable)h(only)h(if)g(the)g
(constrain)n(t)e(in)i(their)f(conclusion)g(is)h(satis\014able.)545
1754 y(2.)45 b(In)26 b(the)h(start)e(and)h(extension)g(rules,)g
Fv(L)1900 1766 y Fs(1)1952 1754 y Fu(_)15 b(\001)f(\001)g(\001)i(_)f
Fv(L)2262 1766 y Fq(m)2351 1754 y Ft(is)26 b(an)g(instance)g(of)g(a)f
(clause)h Fv(C)32 b Ft(in)26 b(the)h(set)f(\000)655 1854
y(of)f(clauses)g(b)r(eing)g(refuted,)h(where)f(eac)n(h)g(free)g(v)-5
b(ariable)24 b(in)i Fv(C)32 b Ft(is)25 b(instan)n(tiated)g(to)g(a)g
(new)h(v)-5 b(ariable)655 1953 y(whic)n(h)28 b(do)r(es)f(not)g(o)r
(ccur)g(in)h(the)g(tableau.)1174 2224 y FT(Figure)i(29:)41
b(The)30 b(Connection)g(T)-8 b(ableau)30 b(Calculus.)p
3829 2394 4 1978 v 378 2397 3453 4 v 378 2730 a(calculus)f(b)m(y)h
(refuting)f(the)i(follo)m(wing)e(set)i(of)f(clauses:)1519
2926 y FP(P)13 b FT(\()p FP(x)p FT(\))21 b FN(_)f FP(Q)p
FT(\()p FP(x)p FT(\))g FN(_)g(:)p FP(R)q FT(\()p FP(x)p
FT(\))83 b FN(:)p FP(P)13 b FT(\()p FP(c)p FT(\))1670
3039 y FP(P)g FT(\()p FP(y)s FT(\))21 b FN(_)e(:)p FP(Q)p
FT(\()p FP(y)s FT(\))265 b FP(R)q FT(\()p FP(c)p FT(\))378
3235 y(as)31 b(follo)m(ws)p 866 3381 1034 4 v 866 3465
a FP(P)13 b FT(\()p FP(v)1016 3479 y FL(1)1055 3465 y
FT(\))i FN(j)g FP(Q)p FT(\()p FP(v)1296 3479 y FL(1)1337
3465 y FT(\))g FN(j)g(:)p FP(R)q FT(\()p FP(v)1637 3479
y FL(1)1677 3465 y FT(\))36 b FN(\001)f(fg)1941 3403
y FT(Start)30 b(\()p FP(P)13 b FT(\()p FP(x)p FT(\))22
b FN(_)e FP(Q)p FT(\()p FP(x)p FT(\))g FN(_)g(:)p FP(R)q
FT(\()p FP(x)p FT(\)\))p FN(f)p FP(x)26 b FN(!)f FP(v)3370
3417 y FL(1)3410 3403 y FN(g)p 866 3508 V 929 3593 a
FP(Q)p FT(\()p FP(v)1080 3607 y FL(1)1120 3593 y FT(\))15
b FN(j)g(:)p FP(R)q FT(\()p FP(v)1420 3607 y FL(1)1460
3593 y FT(\))36 b FN(\001)g FP(v)1636 3607 y FL(1)1700
3593 y FN(')25 b FP(c)1941 3531 y FT(Extension)k FN(:)p
FP(P)13 b FT(\()p FP(c)p FT(\))p 609 3635 1548 4 v 609
3720 a FP(Q)p FT(\()p FP(v)760 3734 y FL(1)800 3720 y
FT(\))p FP(;)i(P)e FT(\()p FP(v)1025 3734 y FL(2)1065
3720 y FT(\))i FN(j)g(:)p FP(R)q FT(\()p FP(v)1365 3734
y FL(1)1405 3720 y FT(\))36 b FN(\001)g(f)p FP(v)1626
3734 y FL(1)1691 3720 y FN(')25 b FP(c;)15 b(v)1910 3734
y FL(2)1975 3720 y FN(')25 b FP(c)p FN(g)2198 3658 y
FT(Extension)k(\()p FP(P)13 b FT(\()p FP(y)s FT(\))21
b FN(_)f(:)p FP(Q)p FT(\()p FP(y)s FT(\)\))p FN(f)p FP(y)29
b FN(!)c FP(v)3512 3672 y FL(2)3552 3658 y FN(g)p 609
3763 V 883 3848 a(:)p FP(R)q FT(\()p FP(v)1093 3862 y
FL(1)1132 3848 y FT(\))35 b FN(\001)h(f)p FP(v)1352 3862
y FL(1)1417 3848 y FN(')25 b FP(c;)15 b(v)1636 3862 y
FL(2)1702 3848 y FN(')25 b FP(c)p FN(g)2198 3786 y FT(Extension)k
FN(:)p FP(P)13 b FT(\()p FP(c)p FT(\))p 883 3890 1000
4 v 979 3975 a FN(fg)37 b(\001)e(f)p FP(v)1255 3989 y
FL(1)1320 3975 y FN(')25 b FP(c;)15 b(v)1539 3989 y FL(2)1605
3975 y FN(')25 b FP(c)p FN(g)1924 3913 y FT(Extension)k
FP(R)q FT(\()p FP(c)p FT(\))378 4179 y(to)i(\014nd)e(the)h(closed)h
(connected)g(tableau)f(in)f(\014gure)h(30.)378 4417 y
FG(B.2.3)112 b(T)-9 b(ableaux)38 b(Calculi)d(for)j(First-Order)f(Logic)
f(with)h(Equalit)m(y)378 4589 y FT(The)32 b(metho)s(ds)g(for)h
(handling)d(the)j(equalit)m(y)f(predicate)h(in)e(tableau)i(calculi)e
(for)h(\014rst-order)g(logic)378 4702 y(include:)489
4859 y(1.)46 b(Eliminating)33 b(equalit)m(y)j(b)m(y)g(transforming)e
(the)i(set)h(of)f(sen)m(tences)h(in)m(to)f(an)g(equiv)-5
b(alen)m(t)35 b(set)605 4971 y(whic)m(h)20 b(do)s(es)h(not)h(in)m(v)m
(olv)m(e)g(equalit)m(y)-8 b(,)23 b(and)e(applying)e(a)j(tableau)f
(calculus)f(for)h(pure)g(\014rst-order)605 5084 y(logic.)489
5260 y(2.)46 b(Adding)29 b(new)h(expansion)f(and)h(closure)g(rules)f
(to)i(the)f(tableau)g(calculus.)489 5435 y(3.)46 b(Closing)29
b(branc)m(hes)h(b)m(y)g FP(E)5 b FT(-uni\014cation.)378
5592 y(The)24 b(\014rst)g(metho)s(d)h(in)m(v)m(olv)m(es)g(adding)e(the)
i(equalit)m(y)f(axioms)h(of)g(re\015exivit)m(y)-8 b(,)26
b(symmetry)-8 b(,)26 b(transitiv-)378 5705 y(it)m(y)k(and)g(congruence)
h(on)g(the)f(function)f(sym)m(b)s(ols)g(in)m(v)m(olv)m(ed)h(in)g(the)g
(set)h(of)g(sen)m(tences.)42 b(Examples)p eop
%%Page: 235 245
235 244 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(235)2092 527
y
tx@Dict begin tx@NodeDict begin {0.0 0.0 0.0 0.0 3.30017 } false /N@T-0
16 {InitRnode } NewNode end end
2092 527 a 1333 776 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 27.12537 13.56268
3.30017 } false /N@T-0-0 16 {InitRnode } NewNode end end
1333 776 a FP(P)13 b FT(\()p
FP(v)1483 790 y FL(1)1523 776 y FT(\))1446 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-0-0 InitNC { NCLine } if end gsave 0.8 SLW 0.
setgray 0 setlinecap stroke grestore grestore end
1446
748 a 1325 1025 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 29.10628 14.55313
3.30017 } false /N@T-0-0-0 16 {InitRnode } NewNode end end
1325 1025 a FN(:)p FP(P)g FT(\()p FP(c)p
FT(\))1446 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-0-0 /N@T-0-0-0 InitNC { NCLine } if end gsave 0.8 SLW
0. setgray 0 setlinecap stroke grestore grestore end
1446 997 a 1325 1133 a
tx@Dict begin tx@NodeDict begin {6.3875 11.8325 28.98734 14.49367
3.30017 } false /N@T-0-0-0-0 16 {InitRnode } NewNode end end
1325 1133 a 85
w FN(\002)1325 1205 y FK(f)p FO(v)1394 1214 y FC(1)1430
1205 y FK(!)p FO(c)p FK(g)1446 1105 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 grestore
end
1446 1105 a 1970
776 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 27.23105 13.61552
3.30017 } false /N@T-0-1 16 {InitRnode } NewNode end end
1970 776 a FP(Q)p FT(\()p FP(v)2121 790 y FL(1)2161
776 y FT(\))2083 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
2083 748 a 1752 1025 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 27.12537 13.56268
3.30017 } false /N@T-0-1-0 16 {InitRnode } NewNode end end
1752 1025
a FP(P)g FT(\()p FP(v)1902 1039 y FL(2)1941 1025 y FT(\))1864
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
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 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 grestore
end
1864 1354 a 2158 1025 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 34.53107 17.26553
3.30017 } false /N@T-0-1-1 16 {InitRnode } NewNode end end
2158 1025 a FN(:)p FP(Q)p FT(\()p
FP(v)2370 1039 y FL(2)2410 1025 y FT(\))2302 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
2302
997 a 2162 1133 a
tx@Dict begin tx@NodeDict begin {6.3875 11.8325 33.61136 16.80568
3.30017 } false /N@T-0-1-1-0 16 {InitRnode } NewNode end end
2162 1133 a 104 w FN(\002)2162 1205
y FK(f)p FO(v)2231 1214 y FC(1)2266 1205 y FK(!)p FO(v)2371
1214 y FC(2)2406 1205 y FK(g)2302 1105 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 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 { } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 4.0
4.0 0 0 /N@T-0-2 /N@T-0-2-0 InitNC { NCLine } if end gsave 0.8 SLW
0. setgray 0 setlinecap stroke grestore grestore end
2739 997 a 2618 1133 a
tx@Dict begin tx@NodeDict begin {6.3875 11.8325 28.98734 14.49367
3.30017 } false /N@T-0-2-0-0 16 {InitRnode } NewNode end end
2618 1133 a 85 w FN(\002)2618
1205 y FK(f)p FO(v)2687 1214 y FC(1)2723 1205 y FK(!)p
FO(c)p FK(g)2739 1105 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-0-2-0 /N@T-0-2-0-0 InitNC { NCLine } if end grestore
end
2739 1105 a 951 1734 a FT(Figure)30
b(30:)41 b(An)30 b(Example)g(of)h(a)f(Closed)g(Connection)g(T)-8
b(ableau.)p 378 2081 3453 4 v 376 3273 4 1193 v 1253
2270 a Fv(B)1316 2282 y Fs(1)1353 2270 y Fv(;)14 b(t)23
b Fu(\031)g Fv(s;)14 b(')p Ft([)p Fv(t)1714 2240 y Fr(0)1737
2270 y Ft(])g Fu(j)28 b(\001)14 b(\001)g(\001)27 b(j)14
b Fv(B)2049 2282 y Fq(n)2127 2270 y Fu(\001)32 b(C)p
1075 2308 1333 4 v 1075 2384 a Fv(B)1138 2396 y Fs(1)1176
2384 y Fv(;)14 b(t)22 b Fu(\031)h Fv(s;)14 b(')p Ft([)p
Fv(s)p Ft(])g Fu(j)28 b(\001)14 b(\001)g(\001)27 b(j)14
b Fv(B)1857 2396 y Fq(n)1935 2384 y Fu(\001)32 b(C)23
b([)c(f)p Fv(t)j Fu(')h Fv(t)2343 2360 y Fr(0)2366 2384
y Fu(g)2445 2327 y Ft(\(Fitting)29 b Fu(\031)p Ft(-expand\))1220
2558 y Fv(B)1283 2570 y Fs(1)1320 2558 y Fv(;)14 b(t)23
b Fu(6\031)g Fv(t)1528 2528 y Fr(0)1565 2558 y Fu(j)14
b Fv(B)1665 2570 y Fs(2)1716 2558 y Fu(j)27 b(\001)14
b(\001)g(\001)28 b(j)14 b Fv(B)1991 2570 y Fq(n)2068
2558 y Fu(\001)33 b(C)p 1220 2595 953 4 v 1226 2671 a
Fv(B)1289 2683 y Fs(2)1340 2671 y Fu(j)28 b(\001)14 b(\001)g(\001)27
b(j)14 b Fv(B)1615 2683 y Fq(n)1693 2671 y Fu(\001)32
b(C)23 b([)c(f)p Fv(t)k Fu(')f Fv(t)2101 2647 y Fr(0)2124
2671 y Fu(g)2210 2614 y Ft(\(Equalit)n(y)27 b(Re\015exivit)n(y\))568
2833 y Fu(\017)45 b Ft(The)28 b(rules)f(are)f(applicable)h(only)h(if)g
(the)g(constrain)n(t)e(in)i(their)f(conclusion)g(is)h(satis\014able.)
856 3103 y FT(Figure)h(31:)42 b(Fitting's)30 b(Additional)e(Expansion)h
(and)g(Closure)g(Rules.)p 3829 3273 4 1193 v 378 3276
3453 4 v 378 3633 a(of)d(the)h(second)f(metho)s(d)g(include)e
(Fitting's)i(approac)m(h)h(in)e(\(Fitting)h(1996\))j(whic)m(h)c(is)g
(an)h(extension)378 3746 y(of)k(the)g(tec)m(hnique)f(of)h(Je\013rey)g
(\(1967\))i(for)e(adding)e(equalit)m(y)i(to)g(ground)f(tableau)h
(calculi.)39 b(In)29 b(Fit-)378 3859 y(ting's)g(approac)m(h)g(the)g
(rules)f(giv)m(en)h(in)e(\014gure)i(31)h(are)f(added)f(to)i(the)f(free)
g(v)-5 b(ariable)28 b(tableau)h(rules.)378 3972 y(W)-8
b(e)39 b(use)e(the)h(notation)g FP(x)f FN(\031)h FP(y)i
FT(to)e(am)m(biguously)f(represen)m(t)g(the)h(equalit)m(y)f(literals)f
FP(x)i FT(=)f FP(y)j FT(and)378 4085 y FP(y)28 b FT(=)d
FP(x)p FT(.)39 b(Similarly)-8 b(,)23 b(w)m(e)j(use)f
FP(x)g FN(6\031)g FP(y)j FT(for)d(b)s(oth)f FN(:)p FT(\()p
FP(x)i FT(=)e FP(y)s FT(\))i(and)e FN(:)p FT(\()p FP(y)k
FT(=)d FP(x)p FT(\).)40 b(The)24 b(main)h(problem)e(with)378
4198 y(suc)m(h)35 b(metho)s(ds)f(is)g(that)i(the)f(use)g(of)g(equalit)m
(y)g(is)f(undirected,)h(and)g(the)g(addition)e(of)j(suc)m(h)e(rules)378
4311 y(results)f(in)h(a)h(v)m(ery)g(large)g(searc)m(h)h(space)f(and)f
(the)h(un)m(tractabilit)m(y)f(of)h(solving)e(ev)m(en)j(v)m(ery)f
(simple)378 4424 y(problems.)519 4537 y(The)22 b(success)g(of)h
(completion-based)e(metho)s(ds)h(\(Kn)m(uth)f(and)h(Bendix)f(1970\))j
(for)e(solving)f(equa-)378 4649 y(tions,)k(often)f(called)f
FP(E)5 b FT(-uni\014cation)23 b(problems,)g(inspired)e(the)j(dev)m
(elopmen)m(t)g(of)g(the)g(third)e(metho)s(d)378 4762
y(men)m(tioned)35 b(ab)s(o)m(v)m(e,)k(where)d(a)g(tableau)g(branc)m(h)f
(is)g(treated)i(as)f(an)g FP(E)5 b FT(-uni\014cation)34
b(problem)g(and)378 4875 y(solv)m(ed)d(usually)e(using)h(calculi)g
(based)h(on)g(unfailing)d(completion)j(\(Bac)m(hmair,)h(Dersho)m(witz,)
g(and)378 4988 y(Plaisted)d(1989\).)43 b(More)31 b(formally)-8
b(,)30 b(a)g(\(general\))i FP(E)5 b FT(-uni\014cation)29
b(problem)f(is)i(of)g(the)h(form)1761 5192 y FP(E)1828
5206 y FL(1)1868 5192 y FP(;)15 b(:)g(:)g(:)31 b(;)15
b(E)2151 5206 y FO(n)2239 5192 y FN(`)2295 5155 y FL(?)2373
5192 y FP(E)378 5397 y FT(where)40 b(the)g(form)m(ulae)g
FP(E)1268 5411 y FO(i)1336 5397 y FT(for)g FP(i)i FN(2)f(f)p
FT(1)p FP(;)15 b(:)g(:)g(:)32 b(;)15 b(n)p FN(g)41 b
FT(are)f(equations)g(whose)g(free)g(v)-5 b(ariables)39
b(are)i(im-)378 5510 y(plicitly)29 b(univ)m(ersally)h(quan)m(ti\014ed,)
h(and)h FP(E)37 b FT(is)31 b(an)h(equation)g(whose)g(free)g(v)-5
b(ariables)31 b(are)i(implicitly)378 5622 y(existen)m(tially)g(quan)m
(ti\014ed.)53 b(A)35 b(solution)e(to)j(a)f(problem)e(of)i(this)f(form)g
(is)g(a)h(substitution)d FP(\033)38 b FT(suc)m(h)p eop
%%Page: 236 246
236 245 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(236)378 396 y(that)1757
509 y FP(E)1824 523 y FL(1)1864 509 y FP(;)15 b(:)g(:)g(:)31
b(;)15 b(E)2147 523 y FO(n)2220 509 y FN(`)25 b FP(E)5
b(\033)n(:)378 676 y FT(Note)24 b(that)g(the)f(substitution)e
FP(\033)26 b FT(is)c(applied)f(only)h(to)i(the)f(conclusion)e
FP(E)5 b FT(.)39 b(Ho)m(w)m(ev)m(er,)27 b(w)m(e)c(recall)f(that)378
789 y(free)34 b(v)-5 b(ariables)33 b(in)f(tableaux)i(are)h(treated)g
(rigidly)c(and)i(that)i(substitutions)d(are)i(applied)e(to)j(the)378
902 y(whole)c(tableau,)i(and)e(therefore)i(the)f(closure)g(of)g(a)h
(tableau)f(branc)m(h)f(cannot)i(corresp)s(ond)e(to)i(the)378
1015 y(solution)22 b(of)i(a)h FP(E)5 b FT(-uni\014cation)22
b(problem.)37 b(This)22 b(lead)i(to)g(the)h(de\014nition)c(of)j(the)g
(rigid)e FP(E)5 b FT(-uni\014cation)378 1128 y(problem)30
b(b)m(y)i(Gallier,)e(Raatz,)k(and)d(Sn)m(yder)g(\(1987\).)47
b(A)32 b(rigid)e FP(E)5 b FT(-uni\014cation)30 b(problem)g(is)h(of)h
(the)378 1241 y(form)1761 1354 y FP(E)1828 1368 y FL(1)1868
1354 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(E)2151 1368 y
FO(n)2239 1354 y FN(`)2295 1316 y FL(?)2295 1376 y(r)2373
1354 y FP(E)378 1521 y FT(where)32 b FP(E)37 b FT(and)32
b(the)h(form)m(ulae)e FP(E)1527 1535 y FO(i)1588 1521
y FT(for)h FP(i)d FN(2)f(f)p FT(1)p FP(;)15 b(:)g(:)g(:)33
b(;)15 b(n)p FN(g)32 b FT(are)h(equations)f(whose)g(free)h(v)-5
b(ariables)30 b(are)378 1633 y(treated)h(rigidly)-8 b(.)39
b(A)30 b(solution)f(to)i(this)f(problem)e(is)i(a)g(substitution)f
FP(\033)k FT(suc)m(h)d(that)1704 1838 y FP(E)1771 1852
y FL(1)1811 1838 y FP(\033)n(;)15 b(:)g(:)g(:)32 b(;)15
b(E)2145 1852 y FO(n)2192 1838 y FP(\033)29 b FN(`)c
FP(E)5 b(\033)n(:)378 2042 y FT(This)20 b(di\013ers)g(from)h(the)g
(de\014nition)f(of)h(the)h(solution)e(for)h(the)h(general)f
FP(E)5 b FT(-uni\014cation)21 b(problem)f(since)378 2155
y(the)31 b(substitution)f FP(\033)k FT(is)c(applied)f(to)k(b)s(oth)d
(the)i(assumptions)d(\()p FP(E)2645 2169 y FL(1)2685
2155 y FP(;)15 b(:)g(:)g(:)32 b(;)15 b(E)2969 2169 y
FO(n)3016 2155 y FT(\))32 b(and)f(the)g(conclusion)378
2268 y(\()p FP(E)5 b FT(\))31 b(of)g(the)f(ab)s(o)m(v)m(e)i(problem.)
519 2381 y(The)25 b(problem)g(of)h(closing)f(a)h(tableau)g(branc)m(h)f
(reduces)g(to)i(that)f(of)h(solving)d(a)i(n)m(um)m(b)s(er)f(of)h(rigid)
378 2494 y FP(E)5 b FT(-uni\014cation)29 b(problems.)39
b(F)-8 b(or)31 b(instance,)g(closing)e(the)i(branc)m(h)881
2698 y FN(f)p FP(x)978 2712 y FL(1)1038 2698 y FN(\003)21
b FT(1)26 b(=)f FP(x)1323 2712 y FL(1)1362 2698 y FP(;)46
b(x)1485 2712 y FL(2)1544 2698 y FT(+)20 b FP(x)1687
2712 y FL(3)1752 2698 y FT(=)25 b FP(x)1900 2712 y FL(3)1959
2698 y FT(+)20 b FP(x)2102 2712 y FL(2)2142 2698 y FP(;)45
b(P)13 b FT(\(3)21 b(+)f(\()p FP(x)2562 2712 y FL(2)2622
2698 y FN(\003)h FT(1\)\))p FP(;)47 b FN(:)p FP(P)13
b FT(\(4)20 b(+)g(3\))p FN(g)378 2902 y FT(is)29 b(equiv)-5
b(alen)m(t)30 b(to)h(the)g(rigid)d FP(E)5 b FT(-uni\014cation)29
b(problem)1022 3106 y FP(x)1074 3120 y FL(1)1134 3106
y FN(\003)20 b FT(1)26 b(=)f FP(x)1418 3120 y FL(1)1457
3106 y FP(;)46 b(x)1580 3120 y FL(2)1640 3106 y FT(+)19
b FP(x)1782 3120 y FL(3)1847 3106 y FT(=)25 b FP(x)1995
3120 y FL(3)2055 3106 y FT(+)20 b FP(x)2198 3120 y FL(2)2277
3106 y FN(`)2333 3069 y FL(?)2333 3129 y(r)2411 3106
y FT(3)g(+)g(\()p FP(x)2654 3120 y FL(2)2714 3106 y FN(\003)h
FT(1\))26 b(=)f(4)c(+)f(3)378 3311 y(and)37 b(can)h(b)s(e)g(solv)m(ed)f
(with)g(the)h(substitution)e FN(f)p FP(x)2138 3325 y
FL(1)2215 3311 y FN(!)i FT(4)p FP(;)15 b(x)2481 3325
y FL(2)2559 3311 y FN(!)37 b FT(4)p FP(;)15 b(x)2824
3325 y FL(3)2902 3311 y FN(!)38 b FT(3)p FN(g)p FT(.)64
b(The)37 b(general)h FP(E)5 b FT(-)378 3423 y(uni\014cation)29
b(problem)h(is)g(undecidable,)f(ev)m(en)j(for)e(v)m(ery)i(simple)d
(equational)h(theories)h(\(see)h(\(Siek-)378 3536 y(mann)f(1989\)\),)k
(but)c(the)h(rigid)e FP(E)5 b FT(-uni\014cation)30 b(problem)h(has)g(b)
s(een)g(sho)m(wn)h(to)g(b)s(e)f FN(N)13 b(P)8 b FT(-complete)378
3649 y(b)m(y)44 b(Gallier,)i(Narendran,)g(Plaisted,)h(and)c(Sn)m(yder)g
(\(1990\).)83 b(E\016cien)m(t)44 b(completion)f(based)h(al-)378
3762 y(gorithms)36 b(for)g(solving)g(the)g(rigid)f FP(E)5
b FT(-uni\014cation)36 b(problem)f(ha)m(v)m(e)i(b)s(een)f(dev)m(elop)s
(ed)g(in)g(\(Gallier,)378 3875 y(Narendran,)k(Plaisted,)f(and)e(Sn)m
(yder)h(1990;)44 b(Goubault)37 b(1993;)44 b(Bec)m(her)c(and)d(P)m
(etermann)i(1994;)378 3988 y(Kogel)33 b(1995\))j(and)c(prop)s(osed)g
(to)i(b)s(e)f(used)f(in)g(closing)h(tableau)g(branc)m(hes)g(during)e
(pro)s(of)h(searc)m(h.)378 4101 y(Although,)j(in)e(general)h(a)h(rigid)
d FP(E)5 b FT(-uni\014cation)33 b(problem)g(can)i(ha)m(v)m(e)g(an)g
(in\014nite)d(n)m(um)m(b)s(er)h(of)h(so-)378 4214 y(lutions)42
b(these)i(algorithms)e(yield)g(a)i(\014nite)f FI(c)-5
b(omplete)46 b(set)f(of)g(solutions)53 b FT(b)m(y)43
b(en)m(umerating)h(the)378 4327 y(substitutions)32 b(whic)m(h)h(are)i
(not)g(equiv)-5 b(alen)m(t)33 b(to)j(eac)m(h)f(other)g(according)f(to)h
(the)g(rigid)d(equational)378 4440 y(theory)40 b(considered.)66
b(F)-8 b(or)41 b(example,)g(the)f(problem)e FP(f)10 b
FT(\()p FP(a)p FT(\))40 b(=)g FP(a)56 b FN(`)2758 4407
y FL(?)2758 4462 y(r)2851 4440 y FP(x)40 b FT(=)g FP(a)g
FT(has)f(the)g(solutions)378 4553 y FN(f)p FP(x)45 b
FN(!)g FP(f)711 4520 y FO(n)758 4553 y FT(\()p FP(a)p
FT(\))p FN(g)e FT(for)f FP(n)i FT(=)h(0)p FP(;)15 b FT(1)p
FP(;)g FT(2)p FP(;)g(:)g(:)g(:)k FT(,)45 b(but)d(the)g(set)h
FN(ff)p FP(x)j FN(!)f FP(a)p FN(gg)e FT(is)e(a)i(complete)f(set)h(of)g
(solu-)378 4665 y(tions)28 b(b)s(ecause)h(all)f(the)i(p)s(ossible)c
(solutions)i(are)h(equiv)-5 b(alen)m(t)29 b(to)h FN(f)p
FP(x)25 b FN(!)h FP(a)p FN(g)j FT(giv)m(en)g(the)g(assumption)378
4778 y FP(f)10 b FT(\()p FP(a)p FT(\))25 b(=)g FP(a)p
FT(.)519 4891 y(Ho)m(w)m(ev)m(er,)h(as)d(can)f(b)s(e)g(seen)g(in)f(the)
i(tableau)f(in)f(\014gure)g(32,)k(a)e(complete)g(set)g(of)f(solutions)f
(closing)378 5004 y(one)31 b(branc)m(h)f(ma)m(y)h(not)g(b)s(e)f(enough)
g(to)i(close)f(a)g(refutable)e(tableau.)42 b(The)30 b(tableau)g(can)h
(b)s(e)f(closed)378 5117 y(b)m(y)39 b(the)g(substitution)d
FN(f)p FP(x)k FN(!)f FP(f)1514 5084 y FL(3)1553 5117
y FT(\()p FP(a)p FT(\))p FN(g)p FT(,)j(but)c(a)i(complete)f(set)g(of)g
(solutions)f(closing)f(the)i(branc)m(h)378 5230 y FN(f)p
FP(P)13 b FT(\()p FP(a)p FT(\))p FP(;)i FN(:)p FP(P)e
FT(\()p FP(x)p FT(\))p FP(;)i(f)10 b FT(\()p FP(a)p FT(\))41
b(=)e FP(a)p FN(g)g FT(giv)m(en)g(b)m(y)f FN(ff)p FP(x)i
FN(!)f FP(a)p FN(gg)g FT(and)f(cannot)i(b)s(e)e(used)g(to)h(close)g
(the)g(other)378 5343 y(branc)m(h)32 b FN(f:)p FP(Q)p
FT(\()p FP(x)p FT(\))p FP(;)15 b(Q)p FT(\()p FP(f)1183
5310 y FL(3)1223 5343 y FT(\()p FP(a)p FT(\)\))p FN(g)p
FT(.)51 b(The)32 b(reasons)h(for)g(this)f(is)g(that)i(di\013eren)m(t)e
(branc)m(hes)h(of)g(the)g(same)378 5456 y(tableau)d(yield)f(di\013eren)
m(t)g(rigid)g(equational)g(theories,)i(and)e(therefore)i(the)g(notion)e
(of)i(a)f(complete)378 5569 y(set)40 b(of)h(solutions)d(is)h(only)g(lo)
s(cal)g(to)i(one)f(branc)m(h)g(rather)g(than)f(global)h(to)g(the)h
(whole)e(tableau.)378 5682 y(In)e(general,)j(one)e(cannot)g(close)g(a)g
(tableau)g(b)m(y)f(treating)h(its)f(branc)m(hes)h(as)f(rigid)f
FP(E)5 b FT(-uni\014cation)p eop
%%Page: 237 247
237 246 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(237)2005 567
y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 22.85564 11.42781
3.30017 } false /N@T-0 16 {InitRnode } NewNode end end
2005 567 a 2005 567 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 22.85564 11.42781
2.7375 } false /N@Pa 16 {InitRnode } NewNode end end
2005 567 a FP(P)13 b FT(\()p
FP(a)p FT(\))1710 817 y
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 30.62582 15.31291
3.30017 } false /N@T-0-0 16 {InitRnode } NewNode end end
1710 817 a FN(:)p FP(P)g FT(\()p
FP(x)p FT(\))1837 789 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
1837 789 a 1666 1066 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 41.23233 20.61617
3.30017 } false /N@T-0-0-0 16 {InitRnode } NewNode end end
1666 1066
a FP(f)d FT(\()p FP(a)p FT(\))25 b(=)g FP(a)1837 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-0-0 /N@T-0-0-0 InitNC { NCLine } if end gsave 0.8 SLW
0. setgray 0 setlinecap stroke grestore grestore end
1837 1038 a 2235 817 a
tx@Dict begin tx@NodeDict begin {8.2125 2.73749 30.7315 15.36575 3.30017
} false /N@T-0-1 16 {InitRnode } NewNode end end
2235 817 a FN(:)p FP(Q)p FT(\()p
FP(x)p FT(\))2363 789 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
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 y(problems)g(and)g(solving)g(them)i(one)f(b)m(y)h(one)f(using)f
(complete)i(list)e(of)i(solutions.)519 1848 y(This)h(led)g(to)i(the)g
(disco)m(v)m(ery)f(of)h(the)f(sim)m(ultaneous)f(rigid)g
FP(E)5 b FT(-uni\014cation)32 b(problem,)h(where)f(a)378
1961 y(n)m(um)m(b)s(er)21 b(of)i(rigid)d FP(E)5 b FT(-uni\014cation)21
b(problems)g(\(represen)m(ting)h(the)g(di\013eren)m(t)g(branc)m(hes)g
(of)h(a)g(tableau\))1717 2183 y FP(E)1784 2197 y FL(11)1859
2183 y FP(;)15 b(:)g(:)g(:)31 b(;)15 b(E)2142 2197 y
FL(1)p FO(n)2220 2206 y FC(1)2300 2183 y FN(`)2356 2146
y FL(?)2356 2206 y(r)2434 2183 y FP(E)2501 2197 y FL(1)1717
2339 y FP(E)1784 2353 y FL(21)1859 2339 y FP(;)g(:)g(:)g(:)31
b(;)15 b(E)2142 2353 y FL(2)p FO(n)2220 2362 y FC(2)2300
2339 y FN(`)2356 2301 y FL(?)2356 2362 y(r)2434 2339
y FP(E)2501 2353 y FL(2)2290 2455 y FT(.)2290 2488 y(.)2290
2521 y(.)1639 2659 y FP(E)1706 2673 y FO(m)p FL(1)1808
2659 y FP(;)g(:)g(:)g(:)31 b(;)15 b(E)2091 2673 y FO(mn)2196
2681 y Fy(m)2300 2659 y FN(`)2356 2621 y FL(?)2356 2681
y(r)2434 2659 y FP(E)2501 2673 y FO(m)378 2863 y FT(need)33
b(to)h(b)s(e)e(solv)m(ed)h(sim)m(ultaneously)-8 b(,)32
b(that)i(is,)f(\014nding)e(a)i(substitution)e FP(\033)37
b FT(whic)m(h)31 b(solv)m(es)j(all)e(the)378 2976 y(ab)s(o)m(v)m(e)i
(rigid)d FP(E)5 b FT(-uni\014cation)31 b(problems.)46
b(This)31 b(problem)h(turns)f(out)i(to)h(b)s(e)e(quite)g(di\013eren)m
(t)g(from)378 3089 y(the)26 b(single)e(rigid)g FP(E)5
b FT(-uni\014cation)25 b(problem)f(and)h(w)m(as)h(sho)m(wn)f(to)i(b)s
(e)e(undecidable)e(\(Degt)m(y)m(arev)29 b(and)378 3202
y(V)-8 b(oronk)m(o)m(v)38 b(1996\),)j(ev)m(en)c(for)f(surprisingly)c
(small)j(fragmen)m(ts)i(of)g(the)f(problem)f(\(Plaisted)h(1995;)378
3315 y(V)-8 b(eanes)36 b(1997\).)54 b(As)35 b(a)g(result,)f(the)h
(problem)e(of)h(deciding)f(whether)g(an)i(expanded)e(tableau)i(can)378
3428 y(b)s(e)30 b(closed)g(with)f(resp)s(ect)h(to)h(the)g(theory)g(of)f
(\014rst-order)g(logic)g(with)f(equalit)m(y)h(is)f(undecidable.)519
3540 y(Degt)m(y)m(arev)38 b(and)c(V)-8 b(oronk)m(o)m(v)36
b(\(1998\))i(prop)s(osed)33 b(the)i(rigid)e(basic)h(sup)s(erp)s
(osition)d(\()p FN(B)s(S)7 b(E)h FT(\))35 b(cal-)378
3653 y(culus)42 b(whic)m(h)g(en)m(umerates)i(a)g(\014nite)e(set)i(of)g
FI(answer)h(c)-5 b(onstr)g(aints)54 b FT(em)m(b)s(edding)42
b(solutions)f(to)j(a)378 3766 y(giv)m(en)31 b(rigid)f
FP(E)5 b FT(-uni\014cation)30 b(problem.)43 b(When)31
b(used)g(to)h(solv)m(e)g(a)g(n)m(um)m(b)s(er)e(of)i(sim)m(ultaneous)e
(rigid)378 3879 y FP(E)5 b FT(-uni\014cation)37 b(problems,)h(it)f(giv)
m(es)h(a)g(terminating,)h(and)e(therefore)h(incomplete,)i(sequence)e
(of)378 3992 y(solutions)25 b(to)i(the)f(problem.)38
b(Ho)m(w)m(ev)m(er,)30 b(it)25 b(giv)m(es)i(a)g(complete)g(calculus)e
(for)h(\014rst-order)f(logic)h(with)378 4105 y(equalit)m(y)20
b(when)g(used)g(for)h(closing)f(tableau)h(branc)m(hes.)37
b(That)21 b(is,)h(although)e(the)i FN(B)s(S)7 b(E)28
b FT(calculus)19 b(ma)m(y)378 4218 y(not)33 b(close)g(all)f(the)h
(branc)m(hes)g(in)e(a)j(refutable)e(\(in)g(principle\))e(tableau,)j(ev)
m(ery)h(refutable)e(tableau)378 4331 y(can)k(b)s(e)f(expanded)g(\(b)m
(y)h(the)h(application)d(of)i(the)g(expand)f(rules\))g(to)h(one)g
(whose)g(branc)m(hes)g(can)378 4444 y(b)s(e)c(closed)g(b)m(y)g(the)g
FN(B)s(S)7 b(E)40 b FT(calculus.)k(Figure)32 b(33)h(illustrates)d(the)j
(three)f(additional)e(tableau)i(rules)378 4557 y(whic)m(h)37
b(solv)m(e)i(the)g(rigid)d FP(E)5 b FT(-uni\014cation)37
b(problem)g(inheren)m(t)g(in)h(the)g(tableau)h(branc)m(hes.)64
b(These)378 4670 y(rules)36 b(are)i(applied)e(to)i(constrain)m(t)g
(tableaux)g(of)g(the)g(form)f FP(T)53 b FN(\001)41 b(C)i
FT(where)37 b FP(T)50 b FT(is)37 b(a)h(free)g(v)-5 b(ariable)378
4782 y(tableau)27 b(and)g FN(C)32 b FT(is)27 b(an)g FI(or)-5
b(dering)31 b(e)-5 b(quality)31 b(c)-5 b(onstr)g(aint)p
FT(.)42 b(Ordering)25 b(equalit)m(y)i(constrain)m(ts)h(are)f(\014rst-)
378 4895 y(order)41 b(form)m(ulae)f(o)m(v)m(er)j(the)e(t)m(w)m(o)i
(binary)c(sym)m(b)s(ols)h FN(')h FT(\(for)g(equalit)m(y)g(constrain)m
(ts\))g(and)g FN(\037)g FT(\(for)378 5008 y(ordering)29
b(constrain)m(ts\),)i(where)f FN(\037)g FT(is)f(a)i(reduction)e
(ordering,)h(that)h(is)514 5196 y FN(\017)46 b FT(it)30
b(is)g(a)g(w)m(ell-founded)f(partial)g(ordering)g(on)h(terms,)514
5384 y FN(\017)46 b FT(it)30 b(is)g(monotonic,)g(i.e.,)16
b(if)29 b FP(a)c FN(\037)g FP(b)31 b FT(then)f FP(s)p
FT([)p FP(a)p FT(])25 b FN(\037)g FP(s)p FT([)p FP(b)p
FT(],)31 b(and)514 5571 y FN(\017)46 b FT(it)30 b(is)g(closed)g(under)f
(substitutions,)f(i.e.,)15 b(if)30 b FP(s)25 b FN(\037)f
FP(t)31 b FT(then)f FP(s\033)e FN(\037)d FP(t\033)33
b FT(for)d(all)f(substitutions)f FP(\033)s FT(,)p eop
%%Page: 238 248
238 247 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(238)p 378 416
3453 4 v 376 2821 4 2406 v 1000 606 a Fv(B)1063 618 y
Fs(1)1100 606 y Fv(;)14 b(l)24 b Fu(\031)f Fv(r)n(;)14
b(s)p Ft([)p Fv(p)p Ft(])23 b Fu(\031)f Fv(t)14 b Fu(j)28
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1903 618 y Fq(n)1980
606 y Fu(\001)32 b(C)p 567 643 1950 4 v 567 719 a Fv(B)630
731 y Fs(1)667 719 y Fv(;)14 b(l)24 b Fu(\031)f Fv(r)n(;)14
b(s)p Ft([)p Fv(r)r Ft(])24 b Fu(\031)f Fv(t)14 b Fu(j)27
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1468 731 y Fq(n)1545
719 y Fu(\001)32 b(C)23 b([)c(f)p Fv(l)24 b Fu(\037)f
Fv(r)n(;)14 b(s)p Ft([)p Fv(p)p Ft(])23 b Fu(\037)f Fv(t;)14
b(l)25 b Fu(')d Fv(p)p Fu(g)2554 662 y Ft(\(left)29 b(rigid)e(basic)g
(sup)r(erp)r(osition\))972 893 y Fv(B)1035 905 y Fs(1)1072
893 y Fv(;)14 b(l)25 b Fu(\031)d Fv(r)n(;)14 b(s)p Ft([)p
Fv(p)p Ft(])23 b Fu(6\031)g Fv(t)14 b Fu(j)27 b(\001)14
b(\001)g(\001)28 b(j)14 b Fv(B)1875 905 y Fq(n)1952 893
y Fu(\001)32 b(C)p 539 931 V 539 1007 a Fv(B)602 1019
y Fs(1)639 1007 y Fv(;)14 b(l)25 b Fu(\031)d Fv(r)n(;)14
b(s)p Ft([)p Fv(r)r Ft(])24 b Fu(6\031)f Fv(t)14 b Fu(j)27
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1440 1019 y
Fq(n)1517 1007 y Fu(\001)33 b(C)23 b([)18 b(f)p Fv(l)24
b Fu(\037)f Fv(r)n(;)14 b(s)p Ft([)p Fv(p)p Ft(])23 b
Fu(\037)g Fv(t;)14 b(l)24 b Fu(')f Fv(p)p Fu(g)2526 950
y Ft(\(righ)n(t)28 b(rigid)f(basic)g(sup)r(erp)r(osition\))1251
1148 y Fv(B)1314 1160 y Fs(1)1351 1148 y Fv(;)14 b(s)23
b Fu(6\031)g Fv(t)14 b Fu(j)g Fv(B)1682 1160 y Fs(2)1733
1148 y Fu(j)27 b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)2008
1160 y Fq(n)2085 1148 y Fu(\001)33 b(C)p 1251 1185 939
4 v 1257 1261 a Fv(B)1320 1273 y Fs(2)1371 1261 y Fu(j)28
b(\001)14 b(\001)g(\001)28 b(j)14 b Fv(B)1647 1273 y
Fq(n)1724 1261 y Fu(\001)32 b(C)23 b([)c(f)p Fv(s)k Fu(')f
Fv(t)p Fu(g)2227 1204 y Ft(\(equalit)n(y)27 b(re\015exivit)n(y\))568
1422 y Fu(\017)45 b Ft(The)28 b(rules)f(can)g(b)r(e)h(applied)g(only)f
(if)h(the)g(follo)n(wing)e(conditions)h(hold)745 1584
y(1.)45 b(the)28 b(constrain)n(t)e(at)i(the)g(conclusion)f(of)g(eac)n
(h)g(rule)g(is)h(satis\014able.)745 1717 y(2.)45 b(in)32
b(the)f(basic)g(sup)r(erp)r(osition)g(rules,)h(the)f(term)h
Fv(p)f Ft(is)g(not)g(a)g(v)-5 b(ariable.)47 b(This)31
b(is)g(called)g(the)855 1817 y(basic)h(restriction)f(whic)n(h)h
(results)g(in)h(a)e(m)n(uc)n(h)i(restrictiv)n(e)e(searc)n(h)f(space)i
(without)h(losing)855 1916 y(the)28 b(completeness)f(of)h(the)g
(calculus.)745 2049 y(3.)45 b(the)28 b(righ)n(t-hand)e(side)i(of)f(the)
h(rigid)f(equation)g(at)g(the)h(premise)f(of)h(eac)n(h)f(rule)g(is)g
(not)h(of)f(the)855 2149 y(form)g Fv(q)f Fu(\031)d Fv(q)31
b Ft(\(to)d(a)n(v)n(oid)e(the)i(substitution)g(of)f(a)g(term)h(b)n(y)f
(itself)6 b(\).)745 2282 y(4.)45 b(in)34 b(the)g(left)g(basic)f(sup)r
(erp)r(osition)g(rule,)h Fv(s)p Ft([)p Fv(r)r Ft(])g
Fu(6)p Ft(=)f Fv(t)g Ft(\(otherwise)g(the)h(literal)f
Fv(t)g Fu(\031)f Fv(t)i Ft(will)f(b)r(e)855 2381 y(included)28
b(in)g(the)g(tableau)f(branc)n(h\).)761 2652 y FT(Figure)j(33:)41
b(Additional)28 b(T)-8 b(ableau)30 b(Rules)g(for)g(Rigid)e(Basic)j(Sup)
s(erp)s(osition.)p 3829 2821 4 2406 v 378 2825 3453 4
v 378 3182 a(whic)m(h)k(is)f(also)i(total)h(on)e(ground)g(terms.)57
b(Suc)m(h)35 b(orderings)g(are)h(describ)s(ed)e(in)g(\(Klop)h(1992\))j
(for)378 3295 y(instance.)55 b(Nieu)m(w)m(enh)m(uis)34
b(and)h(Rubio)e(\(1995\))38 b(giv)m(e)e(e\016cien)m(t)g(algorithms)e
(for)h(solving)f(ordering)378 3408 y(equalit)m(y)c(constrain)m(ts.)519
3520 y(The)i(three)h(rules)d(in)h(\014gure)h(33)h(together)h(with)d
(the)i(start)g(and)f(expansion)f(rules)g(of)h(the)h(free)378
3633 y(v)-5 b(ariable)21 b(tableau)h(giv)m(e)h(a)g(refutationally)e
(complete)i(calculus)e(for)h(\014rst-order)f(logic)i(with)e(equalit)m
(y)-8 b(.)378 3746 y(Note)36 b(that)f(these)g(rules)e(are)i(de\014ned)e
(on)h(equations)g(and)g(inequations)f(only;)j(a)f(p)s(ositiv)m(e)e
(literal)378 3859 y FP(P)43 b FT(is)30 b(treated)h(as)g
FP(P)38 b FN(\031)25 b(>)30 b FT(and)g(a)g(negativ)m(e)i(literal)d
FN(:)p FP(P)43 b FT(as)30 b FP(P)39 b FN(6\031)25 b(>)p
FT(.)p eop
%%Page: 239 249
239 248 bop 378 1019 a FJ(App)5 b(endix)65 b(C)378 1434
y FR(A)77 b(Long)g(Pro)6 b(of)378 1916 y FH(C.1)135 b
Fc(K)r FH(-Consistency)45 b(Implies)h Fc(K)r FH(-Satis\014abilit)l(y)
378 2119 y FT(Let)38 b FN(K)h FT(b)s(e)e(a)i(connectabilit)m(y)e
(relation)g(o)m(v)m(er)i(a)f(coun)m(table)g(set)g(of)g(colours)f
FN(P)46 b FT(whic)m(h)36 b(has)h(some)378 2232 y(total)c(order)f
FN(\024)p FT(.)46 b(Let)32 b FN(C)38 b FT(b)s(e)31 b(a)i
FN(K)q FT(-consistency)g(prop)s(ert)m(y)e(and)h(let)g
FN(C)2751 2199 y Fd(x)p FK(K)2908 2232 y FT(=)c FN(f)p
FP(S)3113 2199 y Fd(x)p FK(K)3238 2211 y FD(\024)3322
2232 y FN(j)h FP(S)k FN(2)28 b(C)5 b(g)p FT(.)47 b(As)378
2345 y(illustrated)31 b(in)h(example)h(7.6,)j FN(C)1513
2312 y Fd(x)p FK(K)1675 2345 y FT(is)d(in)f(general)h(not)h(a)g
(consistency)f(prop)s(ert)m(y)-8 b(.)50 b(Ho)m(w)m(ev)m(er,)36
b(w)m(e)378 2457 y(can)31 b(alw)m(a)m(ys)g(construct)g(a)g(consistency)
f(prop)s(ert)m(y)g(\()p FN(C)2250 2424 y FK(\003K)2375
2457 y FT(de\014ned)f(b)s(elo)m(w\))h(con)m(taining)g
FN(C)3476 2424 y Fd(x)p FK(K)3605 2457 y FT(.)41 b(The)378
2570 y(aim)27 b(of)g(this)f(app)s(endix)f(is)h(to)i(giv)m(e)g(a)f
(detailed)g(pro)s(of)f(of)i(this)e(statemen)m(t.)41 b(A)28
b(consequence)g(of)f(this)378 2683 y(result)i(is)h(that)h(ev)m(ery)g
FN(K)q FT(-consisten)m(t)g(set)g(is)f FN(K)q FT(-satis\014able.)40
b(\(Theorem)31 b(C.1)f(b)s(elo)m(w\).)378 2896 y FQ(De\014nition)35
b(C.1)45 b FT(Giv)m(en)31 b(a)g(\014nite)f(list)g FP(l)i
FT(and)f(an)f(expression)g FP( )s FT([)p FP(j)5 b FT(])32
b(represen)m(ting)e(some)i(form)m(ula)378 3086 y(for)e(ev)m(ery)h
FP(j)36 b FT(in)29 b FP(l)r FT(,)i(let)1191 2973 y FK(^)1164
3000 y Fx([)1152 3198 y FO(j)t FK( )p FO(l)1292 3086
y FP( )s FT([)p FP(j)5 b FT(])32 b(b)s(e)e(de\014ned)f(as)h(follo)m
(ws:)808 3365 y FK(^)781 3393 y Fx([)761 3594 y FO(j)t
FK( )p FL([])918 3479 y FP( )s FT([)p FP(j)5 b FT(])85
b(=)d FN(fg)761 3699 y FK(^)734 3726 y Fx([)666 3927
y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))918
3812 y FP( )s FT([)p FP(j)5 b FT(])85 b(=)d FN(ff)1469
3726 y Fx(^)1400 3927 y FO(j)t FK( )p FL(\()p FO(a)p
FL(:)p FO(l)q FL(\))1653 3812 y FP( )s FT([)p FP(j)5
b FT(])p FN(gg)23 b([)d(ff)p FP( )s FT([)p FP(a)p FT(])p
FP(;)2361 3726 y Fx(^)2291 3927 y FO(j)t FK( )p FL(\()p
FO(a)p FL(:)p FO(l)q FL(\))2545 3812 y FP( )s FT([)p
FP(j)5 b FT(])p FN(g)22 b([)e FP(X)33 b FN(j)25 b FP(X)33
b FN(2)3238 3699 y FK(^)3211 3726 y Fx([)3199 3924 y
FO(j)t FK( )p FO(l)3339 3812 y FP( )s FT([)p FP(j)5 b
FT(])p FN(g)378 4116 y FT(where)1489 4234 y Fx(^)1335
4435 y FO(j)t FK( )p FL([)p FO(x)1499 4444 y FC(1)1532
4435 y FO(;:::)10 b(;x)1682 4443 y Fy(n)1724 4435 y FL(])1758
4320 y FP(P)j FT(\()p FP(j)5 b FT(\))27 b(=)e FP(P)13
b FT(\()p FP(x)2222 4334 y FL(1)2262 4320 y FT(\))20
b FN(^)g(\001)15 b(\001)g(\001)21 b(^)f FP(P)13 b FT(\()p
FP(x)2763 4334 y FO(n)2810 4320 y FT(\))p FP(:)887 b
Ff(\003)378 4632 y FQ(De\014nition)35 b(C.2)45 b FT(Giv)m(en)36
b(a)f(set)h FP(S)k FT(of)c(coloured)f(sen)m(tences)h(and)f(a)h
(connectabilit)m(y)f(relation)f FN(K)q FT(,)378 4745
y(w)m(e)d(de\014ne)1109 4939 y FP(S)1170 4902 y FK(\002K)1308
4939 y FT(=)1404 4853 y Fx([)1520 4753 y(8)1520 4835
y(<)1520 4999 y(:)1715 4826 y FK(^)1688 4853 y Fx([)1601
5054 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])1891
4939 y FP(A)1959 4902 y FO(i)p Fd(x)p FO(j)2116 4939
y FN(j)25 b FP(A)2234 4902 y FO(i)2288 4939 y FN(2)g
FP(S)5 b FT(,)30 b FP(A)2558 4906 y FO(i)2617 4939 y
FT(is)f(a)i(literal)3017 4753 y Fx(9)3017 4835 y(=)3017
4999 y(;)378 5211 y FT(where)24 b([)p FN(K)q FT(\()p
FP(i)p FT(\)])j(is)d(the)h(\014nite)f(list)f(con)m(taining)i(the)g
(colours)f(in)g(the)h(range)g FN(K)q FT(\()p FP(i)p FT(\))h(sorted)f
(in)f(ascending)378 5324 y(order)30 b(according)g(to)h(the)g(ordering)e
FN(\024)p FT(.)2009 b Ff(\003)2035 5954 y FT(239)p eop
%%Page: 240 250
240 249 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(240)378 396 y FQ(De\014nition)35 b(C.3)45 b
FT(Giv)m(en)36 b(a)f(connectabilit)m(y)g(relation)g FN(K)i
FT(and)d(a)i(set)g FP(S)k FT(of)c(coloured)f(sen)m(tences,)378
509 y(w)m(e)c(de\014ne)1413 714 y FP(S)1474 676 y FK(\003K)1593
714 y FT(=)1689 613 y Fx(n)1749 714 y FP(S)1810 676 y
Fd(x)p FK(K)1959 714 y FN([)2040 627 y Fx([)2156 714
y FP(X)i FN(j)25 b FP(X)33 b FN(\022)25 b FP(S)2579 676
y FK(\002K)2692 613 y Fx(o)2768 714 y FP(:)964 b Ff(\003)378
911 y FQ(De\014nition)35 b(C.4)45 b FT(Giv)m(en)33 b(a)g(connectabilit)
m(y)f(relation)g FN(K)i FT(and)e(a)h(collection)g(of)f(sets)i(of)e
(coloured)378 1023 y(sen)m(tences)f FN(C)5 b FT(,)31
b(then)f(w)m(e)h(de\014ne)1633 1236 y FN(C)1686 1199
y FK(\003K)1805 1236 y FT(=)1901 1150 y Fx([)2002 1236
y FN(f)p FP(S)2108 1199 y FK(\003K)2227 1236 y FN(j)25
b FP(S)31 b FN(2)24 b(C)5 b(g)p FP(:)1185 b Ff(\003)378
1440 y FQ(Example)34 b(C.1)45 b FT(Let)d(the)g(set)g
FP(S)49 b FT(=)44 b FN(f)p FP(A)1840 1407 y FO(i)1868
1440 y FP(;)15 b(B)1982 1407 y FO(j)2047 1440 y FN(_)27
b(:)p FP(A)2264 1407 y FO(k)2306 1440 y FP(;)15 b(B)2420
1407 y FO(j)2457 1440 y FN(g)p FT(,)45 b(and)c(the)h(connectabilit)m(y)
f(relation)378 1553 y FN(K)27 b FT(=)e FP(i)g FN($)g
FP(j)31 b FN($)26 b FP(k)33 b FT(with)c FP(i)d(<)f(j)31
b(<)24 b(k)34 b FT(\(as)d(in)e(example)h(7.6\).)42 b(Then,)1128
1721 y FK(^)1101 1749 y Fx([)999 1950 y FO(m)p FK( )p
FL([)p FK(K)p FL(\()p FO(i)p FL(\)])1319 1835 y FP(A)1387
1797 y FO(i)p Fd(x)p FO(m)1632 1835 y FT(=)82 b FN(ff)p
FP(A)1943 1797 y FO(ij)2005 1835 y FN(gg)1110 2055 y
FK(^)1083 2082 y Fx([)977 2283 y FO(m)p FK( )p FL([)p
FK(K)p FL(\()p FO(j)t FL(\)])1306 2168 y FP(B)1380 2131
y FO(j)t Fd(x)p FO(m)1632 2168 y FT(=)g FN(ff)p FP(B)1949
2131 y FO(j)t(i)2030 2168 y FN(^)20 b FP(B)2185 2131
y FO(j)t(k)2260 2168 y FN(g)p FP(;)15 b FN(f)p FP(B)2464
2131 y FO(j)t(i)2545 2168 y FN(^)20 b FP(B)2700 2131
y FO(j)t(k)2775 2168 y FP(;)15 b(B)2889 2131 y FO(j)t(i)2949
2168 y FP(;)g(B)3063 2131 y FO(j)t(k)3138 2168 y FN(gg)378
2472 y FT(And)29 b(so)1091 2585 y FP(S)1152 2547 y FK(\002K)1290
2585 y FT(=)c FN(ff)p FP(A)1544 2547 y FO(ij)1606 2585
y FN(g)p FP(;)15 b FN(f)p FP(B)1810 2547 y FO(j)t(i)1891
2585 y FN(^)20 b FP(B)2046 2547 y FO(j)t(k)2121 2585
y FN(g)p FP(;)15 b FN(f)p FP(B)2325 2547 y FO(j)t(i)2406
2585 y FN(^)20 b FP(B)2561 2547 y FO(j)t(k)2636 2585
y FP(;)15 b(B)2750 2547 y FO(j)t(i)2810 2585 y FP(;)g(B)2924
2547 y FO(j)t(k)2999 2585 y FN(gg)p FP(:)378 2752 y FT(No)m(w)813
2956 y FP(S)874 2918 y FK(\003K)1050 2956 y FT(=)83 b
FN(f)p FP(S)1310 2918 y Fd(x)p FK(K)1459 2956 y FN([)20
b(fg)p FP(;)15 b(S)1731 2918 y Fd(x)p FK(K)1881 2956
y FN([)20 b(f)p FP(A)2075 2918 y FO(ij)2136 2956 y FN(g)p
FP(;)15 b(S)2282 2918 y Fd(x)p FK(K)2432 2956 y FN([)20
b(f)p FP(B)2632 2918 y FO(j)t(i)2713 2956 y FN(^)g FP(B)2868
2918 y FO(j)t(k)2942 2956 y FN(g)p FP(;)1249 3094 y(S)1310
3056 y Fd(x)p FK(K)1459 3094 y FN([)g(f)p FP(B)1659 3056
y FO(j)t(i)1740 3094 y FN(^)g FP(B)1895 3056 y FO(j)t(k)1969
3094 y FP(;)15 b(B)2083 3056 y FO(j)t(i)2144 3094 y FP(;)g(B)2258
3056 y FO(j)t(k)2333 3094 y FN(g)p FP(;)g(S)2479 3056
y Fd(x)p FK(K)2628 3094 y FN([)20 b(f)p FP(A)2822 3056
y FO(ij)2883 3094 y FP(;)15 b(B)2997 3056 y FO(j)t(i)3078
3094 y FN(^)20 b FP(B)3233 3056 y FO(j)t(k)3307 3094
y FN(g)p FP(;)1249 3231 y(S)1310 3194 y Fd(x)p FK(K)1459
3231 y FN([)g(f)p FP(A)1653 3194 y FO(ij)1714 3231 y
FP(;)15 b(B)1828 3194 y FO(j)t(i)1909 3231 y FN(^)20
b FP(B)2064 3194 y FO(j)t(k)2138 3231 y FP(;)15 b(B)2252
3194 y FO(j)t(i)2313 3231 y FP(;)g(B)2427 3194 y FO(j)t(k)2502
3231 y FN(gg)1050 3369 y FT(=)83 b FN(ff)p FP(A)1362
3332 y FO(ij)1423 3369 y FP(;)15 b FT(\()p FP(B)1572
3332 y FO(j)t(i)1654 3369 y FN(^)k FP(B)1808 3332 y FO(j)t(k)1883
3369 y FT(\))h FN(_)g(:)p FP(A)2148 3332 y FO(j)t(k)2223
3369 y FP(;)15 b(B)2337 3332 y FO(j)t(i)2418 3369 y FN(^)20
b FP(B)2573 3332 y FO(j)t(k)2647 3369 y FN(g)p FP(;)1249
3507 y FN(f)p FP(A)1362 3470 y FO(ij)1423 3507 y FP(;)15
b FT(\()p FP(B)1572 3470 y FO(j)t(i)1654 3507 y FN(^)k
FP(B)1808 3470 y FO(j)t(k)1883 3507 y FT(\))h FN(_)g(:)p
FP(A)2148 3470 y FO(j)t(k)2223 3507 y FP(;)15 b(B)2337
3470 y FO(j)t(i)2418 3507 y FN(^)20 b FP(B)2573 3470
y FO(j)t(k)2647 3507 y FP(;)15 b(B)2761 3470 y FO(j)t(i)2822
3507 y FP(;)g(B)2936 3470 y FO(j)t(k)3010 3507 y FN(gg)p
FP(:)378 3711 y FT(If)30 b FN(C)g FT(=)25 b FN(f)p FP(S)5
b FN(g)p FT(,)32 b(then)e FN(C)1111 3678 y FK(\003K)1230
3711 y FT(=)1326 3643 y Fx(S)1401 3711 y FN(f)p FP(S)1507
3678 y FK(\003K)1627 3711 y FN(j)25 b FP(S)30 b FN(2)25
b(C)5 b(g)26 b FT(=)f FP(S)2130 3678 y FK(\003K)2254
3711 y FT(whic)m(h)k(is)g(a)i(consistency)f(prop)s(ert)m(y)-8
b(.)239 b Ff(\003)378 3891 y FQ(Prop)s(osition)36 b(C.1)45
b FI(F)-7 b(or)41 b(every)e(liter)-5 b(al)40 b FP(B)i
FN(2)37 b FP(S)2104 3858 y Fd(x)p FK(K)2232 3891 y FI(,)k(if)e
FP(B)i FN(6)p FT(=)c FN(>)i FI(then)h(ther)-5 b(e)40
b(is)f(some)h(c)-5 b(olour)g(e)g(d)378 4004 y(liter)g(al)34
b FP(A)705 3971 y FO(i)759 4004 y FN(2)24 b FP(S)38 b
FI(such)33 b(that)h FP(B)29 b FT(=)c FP(A)1593 3971 y
FO(i)p Fd(x)p FO(j)1757 4004 y FI(and)34 b FP(i)25 b
FN(\030)2061 4018 y FK(K)2144 4004 y FP(j)5 b FI(.)378
4183 y FQ(Pro)s(of)p FT(:)31 b(If)f FP(B)g FN(2)25 b
FP(S)1024 4150 y Fd(x)p FK(K)1183 4183 y FT(then)30 b(there)h(is)e
(some)i(coloured)f(literal)f FP(A)2637 4150 y FO(i)2690
4183 y FN(2)c FP(S)35 b FT(suc)m(h)30 b(that)1317 4387
y FP(B)g FT(=)25 b(\()p FP(A)1615 4350 y FO(i)1643 4387
y FT(\))1678 4350 y Fd(x)p FK(K)1833 4387 y FT(=)g FN(>)p
FP(;)45 b FT(if)30 b FP(i)35 b(=)-55 b FN(2)25 b Fe(C)p
FT(\()p FN(K)q FT(\))1833 4550 y(=)2016 4463 y Fx(^)1929
4665 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])2219
4550 y FP(A)2287 4512 y FO(i)p Fd(x)p FO(j)2418 4550
y FP(;)46 b FT(otherwise.)378 4867 y(If)28 b FN(K)q FT(\()p
FP(i)p FT(\))f(=)d FN(fg)29 b FT(then)f FP(B)i FT(=)25
b FN(>)p FT(.)40 b(Otherwise,)27 b(if)g FN(K)q FT(\()p
FP(i)p FT(\))g FN(6)p FT(=)e FN(fg)k FT(then)f FP(B)i
FT(=)2762 4799 y Fx(V)2838 4894 y FO(j)t FK( )p FL([)p
FK(K)p FL(\()p FO(i)p FL(\)])3132 4867 y FP(A)3200 4834
y FO(i)p Fd(x)p FO(j)3359 4867 y FT(and)e(since)f FP(B)378
4980 y FT(is)32 b(a)h(literal,)e(and)h(therefore)i(not)e(a)h
(conjunction,)g(then)f FN(K)q FT(\()p FP(i)p FT(\))i(m)m(ust)f(b)s(e)f
(some)h(singleton)e(set)i FN(f)p FP(j)5 b FN(g)378 5093
y FT(with)29 b FP(i)d FN(\030)713 5107 y FK(K)796 5093
y FP(j)5 b FT(.)41 b(Therefore)30 b FP(B)g FT(=)25 b
FP(A)1579 5060 y FO(i)p Fd(x)p FO(j)1710 5093 y FT(,)31
b FP(A)1834 5060 y FO(i)1887 5093 y FN(2)25 b FP(S)36
b FT(and)29 b FP(i)d FN(\030)2369 5107 y FK(K)2452 5093
y FP(j)36 b FT(as)31 b(required.)767 b Ff(\004)378 5272
y FQ(Prop)s(osition)36 b(C.2)45 b FI(Given)26 b(a)f(list)h
FP(l)h FI(and)f(an)g(expr)-5 b(ession)27 b FP( )s FT([)p
FP(j)5 b FT(])26 b FI(r)-5 b(epr)g(esenting)27 b(a)f(formula)g(for)g
(every)378 5385 y FP(j)38 b FI(in)33 b FP(l)r FI(,)f(then)485
5618 y(1.)46 b(If)33 b FP(l)h FI(is)f(non-empty)g(then)h
FN(f)1570 5531 y Fx(^)1558 5729 y FO(j)t FK( )p FO(l)1698
5618 y FP( )s FT([)p FP(j)5 b FT(])p FN(g)27 b(2)2049
5504 y FK(^)2022 5531 y Fx([)2010 5729 y FO(j)t FK( )p
FO(l)2150 5618 y FP( )s FT([)p FP(j)5 b FT(])p FI(.)p
eop
%%Page: 241 251
241 250 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(241)485 465 y FI(2.)46 b(F)-7 b(or)34 b(every)e
FP(S)f FN(2)1226 351 y FK(^)1199 378 y Fx([)1187 576
y FO(j)t FK( )p FO(l)1327 465 y FP( )s FT([)p FP(j)5
b FT(])p FI(,)34 b(it)f(is)g(the)g(c)-5 b(ase)33 b(that)2276
378 y Fx(^)2264 576 y FO(j)t FK( )p FO(l)2404 465 y FP( )s
FT([)p FP(j)5 b FT(])26 b FN(2)f FP(S)5 b FI(.)485 836
y(3.)46 b(F)-7 b(or)34 b(every)e FP(S)f FN(2)1226 723
y FK(^)1199 750 y Fx([)1187 948 y FO(j)t FK( )p FO(l)1327
836 y FP( )s FT([)p FP(j)5 b FT(])p FI(,)34 b(if)f FP(')25
b FN(2)g FP(S)38 b FI(then)714 1117 y FN(\017)46 b FP(')26
b FT(=)f FP( )s FT([)p FP(j)5 b FT(])p FI(,)34 b(for)f(some)g
FP(j)39 b FI(in)32 b FP(l)r FI(,)g(or)714 1272 y FN(\017)46
b FP(')34 b FT(=)1025 1186 y Fx(^)1002 1384 y FO(j)t
FK( )p FO(l)1128 1365 y FD(0)1165 1272 y FP( )s FT([)p
FP(j)5 b FT(])p FI(,)39 b(for)f(some)g(list)f FP(l)1957
1239 y FK(0)1980 1272 y FI(,)i(such)e(that)h FP(l)2477
1239 y FK(0)2537 1272 y FI(is)g(a)f(tail)h(sublist)f(of)g
FP(l)i FI(and)g FP(]l)3594 1239 y FK(0)3650 1272 y FP(>)34
b FT(1)p FI(,)805 1477 y(wher)-5 b(e)35 b FP(l)1090 1491
y FL(1)1164 1477 y FI(is)e(a)i(tail)f(sublist)g(of)h
FP(l)1920 1491 y FL(2)1993 1477 y FI(if)f(ther)-5 b(e)34
b(is)g(some)h(list)f FP(l)2822 1491 y FL(3)2896 1477
y FI(such)g(that)h FP(l)3318 1491 y FL(2)3385 1477 y
FT(=)27 b FP(l)3510 1491 y FL(3)3571 1477 y FT(+)21 b(+)p
FP(l)3761 1491 y FL(1)3800 1477 y FI(,)805 1590 y(and)34
b FP(]l)1046 1557 y FK(0)1102 1590 y FI(is)e(the)h(length)h(of)e(the)i
(list)f FP(l)2056 1557 y FK(0)2079 1590 y FI(.)378 1790
y FQ(Pro)s(of)p FT(:)46 b(The)f(\014rst)g(t)m(w)m(o)h(statemen)m(ts)h
(follo)m(w)e(directly)f(from)g(De\014nition)g(C.1,)50
b(and)44 b(the)i(third)378 1976 y(one)c(pro)s(ceeds)f(b)m(y)g
(induction)e(on)j FP(l)r FT(.)74 b(The)41 b(base)h(case)g(is)f(trivial)
e(since)i(no)g(set)i FP(S)48 b FN(2)3537 1862 y FK(^)3510
1889 y Fx([)3490 2091 y FO(j)t FK( )p FL([])3647 1976
y FP( )s FT([)p FP(j)5 b FT(].)378 2284 y(No)m(w)42 b(for)f(the)h
(induction)d(case,)46 b(if)40 b FP(S)49 b FN(2)1959 2171
y FK(^)1932 2198 y Fx([)1864 2399 y FO(j)t FK( )p FL(\()p
FO(a)p FL(:)p FO(l)q FL(\))2116 2284 y FP( )s FT([)p
FP(j)5 b FT(],)46 b(then)c FP(S)48 b FT(=)c FN(f)2892
2198 y Fx(^)2824 2399 y FO(j)t FK( )p FL(\()p FO(a)p
FL(:)p FO(l)q FL(\))3076 2284 y FP( )s FT([)p FP(j)5
b FT(])p FN(g)p FT(,)47 b(or)41 b(else)g FP(S)49 b FT(=)378
2593 y FN(f)p FP( )s FT([)p FP(a)p FT(])p FP(;)693 2507
y Fx(^)623 2708 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p
FO(l)q FL(\))877 2593 y FP( )s FT([)p FP(j)5 b FT(])p
FN(g)22 b([)e FP(X)7 b FT(,)31 b(for)f(some)h FP(X)h
FN(2)1916 2479 y FK(^)1889 2507 y Fx([)1877 2704 y FO(j)t
FK( )p FO(l)2017 2593 y FP( )s FT([)p FP(j)5 b FT(].)43
b(W)-8 b(e)31 b(consider)e(these)i(cases)h(separately:)514
2901 y FN(\017)46 b FT(If)33 b FP(S)h FT(=)29 b FN(f)1003
2815 y Fx(^)934 3016 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p
FO(l)q FL(\))1187 2901 y FP( )s FT([)p FP(j)5 b FT(])p
FN(g)35 b FT(then)d FP(')e FT(=)1887 2815 y Fx(^)1819
3016 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))2071
2901 y FP( )s FT([)p FP(j)5 b FT(],)36 b(and)c(if)g FP(l)i
FT(is)e(not)h(empt)m(y)h(then)e FP(])p FT(\()p FP(a)f
FT(:)e FP(l)r FT(\))h FP(>)f FT(1.)605 3146 y(Ho)m(w)m(ev)m(er,)k(if)c
FP(l)j FT(is)e(empt)m(y)h(then)1784 3060 y Fx(^)1716
3261 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1968
3146 y FP( )s FT([)p FP(j)5 b FT(])26 b(=)f FP( )s FT([)p
FP(a)p FT(])32 b(and)e FP(a)g FT(is)f(in)g(\()p FP(a)d
FT(:)g FP(l)r FT(\).)514 3525 y FN(\017)46 b FT(If)38
b FP(S)43 b FT(=)37 b FN(f)1025 3439 y Fx(^)956 3640
y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1209
3525 y FP( )s FT([)p FP(j)5 b FT(])p FP(;)15 b( )s FT([)p
FP(a)p FT(])p FN(g)29 b([)24 b FP(X)7 b FT(,)41 b(for)d(some)g
FP(X)45 b FN(2)2510 3411 y FK(^)2483 3439 y Fx([)2471
3636 y FO(j)t FK( )p FO(l)2611 3525 y FP( )s FT([)p FP(j)5
b FT(],)42 b(then)37 b(either)h FP(')g FT(=)g FP( )s
FT([)p FP(a)p FT(])h(\(in)605 3769 y(whic)m(h)d(case)j(w)m(e)f(are)g
(done)f(since)g FP(a)g FT(is)g(in)f(\()p FP(a)i FT(:)f
FP(l)r FT(\)\),)j(or)d FP(')h FT(=)2861 3683 y Fx(^)2793
3884 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3045
3769 y FP( )s FT([)p FP(j)5 b FT(])39 b(\(and)e(the)h(pro)s(of)605
3987 y(pro)s(ceeds)d(as)h(in)d(the)j(previous)e(case\),)k(or)d(else)g
FP(')f FN(2)f FP(X)7 b FT(.)56 b(No)m(w)36 b(if)e FP(')g
FN(2)f FP(X)42 b FT(it)35 b(follo)m(ws)f(from)605 4099
y(the)h(induction)c(h)m(yp)s(othesis)i(that)i FP(')d
FT(=)f FP( )s FT([)p FP(j)5 b FT(])36 b(for)e(some)g
FP(j)40 b FT(in)33 b FP(l)j FT(\(and)e(hence)g(in)f(\()p
FP(a)f FT(:)g FP(l)r FT(\)\),)k(or)605 4226 y FP(')26
b FT(=)809 4140 y Fx(^)786 4338 y FO(j)t FK( )p FO(l)912
4319 y FD(0)948 4226 y FP( )s FT([)p FP(j)5 b FT(])32
b(for)e(some)h FP(l)1530 4193 y FK(0)1584 4226 y FT(tail)e(sublist)f
(of)j FP(l)h FT(\(and)e(hence)g(of)h(\()p FP(a)26 b FT(:)f
FP(l)r FT(\)\))31 b(with)e FP(]l)3325 4193 y FK(0)3374
4226 y FP(>)c FT(1.)217 b Ff(\004)378 4523 y FQ(Prop)s(osition)36
b(C.3)45 b FI(Given)30 b(a)h(list)f FP(l)i FI(and)f(an)g(expr)-5
b(ession)32 b FP( )s FT([)p FP(j)5 b FT(])32 b FI(r)-5
b(epr)g(esenting)31 b(a)g(non-c)-5 b(onjunctive)378 4714
y(formula)39 b(for)g(every)e FP(j)44 b FI(in)37 b FP(l)r
FI(,)i(then)g(for)f(every)g FP(S)h FN(2)2244 4600 y FK(^)2217
4627 y Fx([)2205 4825 y FO(j)t FK( )p FO(l)2345 4714
y FP( )s FT([)p FP(j)5 b FT(])p FI(,)40 b(if)e FP(')24
b FN(^)g FP(#)34 b FN(2)g FP(S)43 b FI(then)c FP(S)29
b FN([)23 b(f)p FP(';)15 b(#)p FN(g)36 b(2)417 4901 y
FK(^)390 4929 y Fx([)378 5126 y FO(j)t FK( )p FO(l)518
5015 y FP( )s FT([)p FP(j)5 b FT(])p FI(.)378 5312 y
FQ(Pro)s(of)p FT(:)35 b(As)f(in)f(the)h(pro)s(of)f(of)h(prop)s(osition)
e(C.2\(3\))j(w)m(e)g(pro)s(ceed)e(b)m(y)h(induction)e(on)i
FP(l)r FT(.)51 b(The)34 b(base)378 5425 y(case)j(is)e(once)h(again)g
(trivial)e(and)h(for)h(the)g(induction)e(case)j(w)m(e)f(consider)f(the)
h(cases)g(of)h(whether)378 5610 y FP(S)30 b FT(=)25 b
FN(f)673 5524 y Fx(^)605 5725 y FO(j)t FK( )p FL(\()p
FO(a)p FL(:)p FO(l)q FL(\))858 5610 y FP( )s FT([)p FP(j)5
b FT(])p FN(g)32 b FT(or)e FP(S)h FT(=)24 b FN(f)1496
5524 y Fx(^)1427 5725 y FO(j)t FK( )p FL(\()p FO(a)p
FL(:)p FO(l)q FL(\))1680 5610 y FP( )s FT([)p FP(j)5
b FT(])p FP(;)15 b( )s FT([)p FP(a)p FT(])p FN(g)24 b([)19
b FP(X)38 b FT(for)30 b(some)h FP(X)i FN(2)2895 5497
y FK(^)2868 5524 y Fx([)2857 5721 y FO(j)t FK( )p FO(l)2997
5610 y FP( )s FT([)p FP(j)5 b FT(])31 b(separately:)p
eop
%%Page: 242 252
242 251 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(242)514 401 y FN(\017)46 b FT(F)-8 b(or)39
b(the)e(\014rst)g(case)i(w)m(e)f(ha)m(v)m(e)h FP(')26
b FN(^)e FP(#)37 b FT(=)2129 314 y Fx(^)2062 516 y FO(j)t
FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))2313 401 y
FP( )s FT([)p FP(j)5 b FT(])40 b(and)d(w)m(e)h(can)g(assume)f(that)h
FP(l)i FT(is)d(not)605 645 y(empt)m(y)42 b(otherwise)1378
559 y Fx(^)1310 760 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p
FO(l)q FL(\))1562 645 y FP( )s FT([)p FP(j)5 b FT(])43
b(w)m(ould)e(not)g(b)s(e)g(a)h(conjunction.)73 b(Th)m(us,)44
b FP(')g FT(=)g FP( )s FT([)p FP(a)p FT(])e(and)605 954
y FP(#)25 b FT(=)792 867 y Fx(^)780 1065 y FO(j)t FK( )p
FO(l)920 954 y FP( )s FT([)p FP(j)5 b FT(].)41 b(No)m(w,)29
b(b)m(y)e(prop)s(osition)e(C.2\(1\))k FN(f)2305 867 y
Fx(^)2293 1065 y FO(j)t FK( )p FO(l)2433 954 y FP( )s
FT([)p FP(j)5 b FT(])p FN(g)27 b(2)2784 840 y FK(^)2757
867 y Fx([)2745 1065 y FO(j)t FK( )p FO(l)2885 954 y
FP( )s FT([)p FP(j)5 b FT(])29 b(and)e(th)m(us)f(b)m(y)i(De\014ni-)605
1255 y(tion)e(C.1,)h FN(f)1089 1168 y Fx(^)1021 1370
y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1273
1255 y FP( )s FT([)p FP(j)5 b FT(])p FP(;)15 b( )s FT([)p
FP(a)p FT(])p FP(;)1683 1168 y Fx(^)1667 1366 y FO(j)t
FK( )p FO(l)1811 1255 y FP( )s FT([)p FP(j)5 b FT(])p
FN(g)27 b FT(\(whic)m(h)e(is)g(equal)h(to)g FP(S)16 b
FN([)11 b(f)p FP(';)k(#)p FN(g)p FT(\))28 b(is)d(in)3490
1141 y FK(^)3463 1168 y Fx([)3395 1370 y FO(j)t FK( )p
FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3647 1255 y FP( )s
FT([)p FP(j)5 b FT(].)514 1636 y FN(\017)46 b FT(If)32
b FP(S)h FT(=)c FN(f)1000 1549 y Fx(^)932 1751 y FO(j)t
FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1184 1636 y
FP( )s FT([)p FP(j)5 b FT(])p FP(;)15 b( )s FT([)p FP(a)p
FT(])p FN(g)25 b([)c FP(X)40 b FT(\(and)32 b FP(X)k FN(2)2257
1522 y FK(^)2230 1549 y Fx([)2219 1747 y FO(j)t FK( )p
FO(l)2358 1636 y FP( )s FT([)p FP(j)5 b FT(]\),)35 b(then)d(either)g
FP(')22 b FN(^)f FP(#)28 b FT(=)3488 1549 y Fx(^)3420
1751 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3672
1636 y FP( )s FT([)p FP(j)5 b FT(])605 1853 y(and)44
b FP(l)j FT(is)c(non-empt)m(y)-8 b(,)49 b(or)c FP(')30
b FN(^)f FP(#)48 b FN(2)h FP(X)7 b FT(.)83 b(\(Note)47
b(that)e FP(')30 b FN(^)f FP(#)49 b FN(6)p FT(=)f FP( )s
FT([)p FP(a)p FT(])e(as)f FP( )s FT([)p FP(a)p FT(])g(is)f(not)605
1966 y(conjunctiv)m(e.\))605 2128 y(If)23 b FP(')6 b
FN(^)g FP(#)25 b FT(=)1064 2042 y Fx(^)996 2243 y FO(j)t
FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1248 2128 y
FP( )s FT([)p FP(j)5 b FT(],)27 b(then)c FP(')i FT(=)g
FP( )s FT([)p FP(a)p FT(])g(and)d FP(#)j FT(=)2375 2042
y Fx(^)2363 2239 y FO(j)t FK( )p FO(l)2503 2128 y FP( )s
FT([)p FP(j)5 b FT(].)40 b(No)m(w,)26 b(b)m(y)d(Prop)s(osition)e
(C.2\(2\),)617 2286 y Fx(^)605 2484 y FO(j)t FK( )p FO(l)745
2373 y FP( )s FT([)p FP(j)5 b FT(])39 b(is)d(in)g FP(X)44
b FT(and)36 b(th)m(us)h(in)f FP(S)5 b FT(.)60 b(And)36
b(since)h FP( )s FT([)p FP(a)p FT(])h(is)e(also)h(in)f
FP(S)5 b FT(,)38 b(it)f(follo)m(ws)f(that)i FP(S)29 b
FN([)605 2674 y(f)p FP(';)15 b(#)p FN(g)27 b FT(=)e FP(S)35
b FT(whic)m(h)29 b(is)h(in)1615 2560 y FK(^)1588 2587
y Fx([)1520 2789 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p
FO(l)q FL(\))1772 2674 y FP( )s FT([)p FP(j)5 b FT(].)605
2922 y(No)m(w,)32 b(if)d FP(')21 b FN(^)e FP(#)25 b FN(2)g
FP(X)7 b FT(,)848 3199 y FP(X)28 b FN([)20 b(f)p FP(';)15
b(#)p FN(g)27 b(2)1426 3085 y FK(^)1400 3113 y Fx([)1388
3310 y FO(j)t FK( )p FO(l)1528 3199 y FP( )s FT([)p FP(j)5
b FT(])62 b(b)m(y)30 b(the)h(induction)d(h)m(yp)s(othesis)874
3525 y FN(\))55 b(f)p FP( )s FT([)p FP(a)p FT(])p FP(;)1335
3439 y Fx(^)1265 3640 y FO(j)t FK( )p FL(\()p FO(a)p
FL(:)p FO(l)q FL(\))1519 3525 y FP( )s FT([)p FP(j)5
b FT(])p FN(g)22 b([)e FT(\()p FP(X)28 b FN([)20 b(f)p
FP(';)15 b(#)p FN(g)p FT(\))27 b FN(2)2526 3411 y FK(^)2499
3439 y Fx([)2431 3640 y FO(j)t FK( )p FL(\()p FO(a)p
FL(:)p FO(l)q FL(\))2683 3525 y FP( )s FT([)p FP(j)5
b FT(])62 b(b)m(y)30 b(De\014nition)f(C.1)874 3858 y
FN(\))55 b FP(S)25 b FN([)20 b(f)p FP(';)15 b(#)p FN(g)27
b(2)1633 3745 y FK(^)1606 3772 y Fx([)1538 3973 y FO(j)t
FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1790 3858 y
FP( )s FT([)p FP(j)5 b FT(])p FP(:)1788 b Ff(\004)378
4203 y FQ(Prop)s(osition)36 b(C.4)45 b FI(Given)26 b(a)f(list)h
FP(l)h FI(and)f(an)g(expr)-5 b(ession)27 b FP( )s FT([)p
FP(j)5 b FT(])26 b FI(r)-5 b(epr)g(esenting)27 b(a)f(formula)g(for)g
(every)378 4394 y FP(j)38 b FI(in)33 b FP(l)r FI(,)f(then)h(for)g
(every)g FP(i)g FI(in)f FP(l)r FI(,)g(ther)-5 b(e)34
b(is)e(some)i(set)f FP(S)d FN(2)2415 4280 y FK(^)2389
4307 y Fx([)2377 4505 y FO(j)t FK( )p FO(l)2517 4394
y FP( )s FT([)p FP(j)5 b FT(])34 b FI(with)f FP( )s FT([)p
FP(i)p FT(])27 b FN(2)e FP(S)5 b FI(.)378 4690 y FQ(Pro)s(of)p
FT(:)24 b(W)-8 b(e)24 b(pro)s(ceed)f(b)m(y)g(induction)e(on)i
FP(l)r FT(.)38 b(The)22 b(base)i(case)g(is)e(trivial,)h(and)f(for)h
(the)g(induction)e(case)378 4876 y(w)m(e)32 b(need)f(to)i(sho)m(w)e
(that)h(for)g(all)e FP(i)i FT(in)e(\()p FP(a)e FT(:)f
FP(l)r FT(\))32 b(there)g(is)f(some)h(set)g FP(S)g FN(2)2930
4762 y FK(^)2903 4790 y Fx([)2835 4991 y FO(j)t FK( )p
FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3087 4876 y FP( )s
FT([)p FP(j)5 b FT(])33 b(with)d FP( )s FT([)p FP(i)p
FT(])f FN(2)e FP(S)5 b FT(.)378 5184 y(If)37 b FP(i)g
FT(=)g FP(a)p FT(,)i(then)e FN(f)1092 5098 y Fx(^)1023
5299 y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))1276
5184 y FP( )s FT([)p FP(j)5 b FT(])p FP(;)15 b( )s FT([)p
FP(a)p FT(])p FP(;)1685 5098 y Fx(^)1670 5296 y FO(j)t
FK( )p FO(l)1813 5184 y FP( )s FT([)p FP(j)5 b FT(])p
FN(g)39 b(2)2244 5071 y FK(^)2217 5098 y Fx([)2149 5299
y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))2401
5184 y FP( )s FT([)p FP(j)5 b FT(])39 b(for)e(non-empt)m(y)g
FP(l)r FT(,)j(and)c FN(f)p FP( )s FT([)p FP(a)p FT(])p
FN(g)j(2)473 5379 y FK(^)446 5407 y Fx([)378 5608 y FO(j)t
FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))630 5493 y
FP( )s FT([)p FP(j)5 b FT(])36 b(if)e FP(l)j FT(is)c(empt)m(y)-8
b(.)55 b(In)34 b(an)m(y)h(case,)i(there)e(is)f(some)h(set)h
FP(S)h FN(2)2896 5379 y FK(^)2869 5407 y Fx([)2801 5608
y FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3053
5493 y FP( )s FT([)p FP(j)5 b FT(])36 b(with)e FP( )s
FT([)p FP(a)p FT(])g FN(2)e FP(S)5 b FT(.)378 5705 y(On)29
b(the)h(other)g(hand,)f(if)g FP(i)c FN(6)p FT(=)g FP(a)30
b FT(then)f FP(i)h FT(m)m(ust)g(b)s(e)f(in)g FP(l)r FT(,)h(and)f(b)m(y)
h(the)f(induction)f(h)m(yp)s(othesis,)h(there)p eop
%%Page: 243 253
243 252 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(243)378 465 y(is)28 b(some)i(set)g FP(S)897
432 y FK(0)946 465 y FN(2)1071 351 y FK(^)1044 378 y
Fx([)1032 576 y FO(j)t FK( )p FO(l)1172 465 y FP( )s
FT([)p FP(j)5 b FT(])31 b(suc)m(h)e(that)h FP( )s FT([)p
FP(i)p FT(])d FN(2)d FP(S)2073 432 y FK(0)2097 465 y
FT(.)40 b(No)m(w)30 b(let)g FP(S)g FT(=)25 b FN(f)p FP( )s
FT([)p FP(a)p FT(])p FP(;)2996 378 y Fx(^)2926 580 y
FO(j)t FK( )p FL(\()p FO(a)p FL(:)p FO(l)q FL(\))3180
465 y FP( )s FT([)p FP(j)5 b FT(])p FN(g)20 b([)e FP(S)3539
427 y FK(0)3563 465 y FT(.)40 b(Th)m(us)378 773 y FP( )s
FT([)p FP(i)p FT(])27 b FN(2)d FP(S)36 b FT(and)29 b(it)h(follo)m(ws)g
(from)g(De\014nition)f(C.1)h(that)h FP(S)g FN(2)2559
660 y FK(^)2532 687 y Fx([)2464 888 y FO(j)t FK( )p FL(\()p
FO(a)p FL(:)p FO(l)q FL(\))2716 773 y FP( )s FT([)p FP(j)5
b FT(].)862 b Ff(\004)378 1106 y FQ(Prop)s(osition)36
b(C.5)45 b FI(If)33 b FP(S)1297 1120 y FL(1)1361 1106
y FN(\022)25 b FP(S)1513 1120 y FL(2)1585 1106 y FI(then)33
b FP(S)1848 1067 y FK(\002K)1843 1132 y FL(1)1986 1106
y FN(\022)25 b FP(S)2143 1067 y FK(\002K)2138 1132 y
FL(2)2257 1106 y FI(.)378 1399 y FQ(Pro)s(of)p FT(:)35
b(Let)g FP(X)j FN(2)31 b FP(S)1124 1361 y FK(\002K)1119
1425 y FL(1)1237 1399 y FT(,)k(then)f(b)m(y)g(De\014nition)f(C.2,)i
(the)f(form)m(ula)g FP(X)k FN(2)3078 1286 y FK(^)3051
1313 y Fx([)2964 1514 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p
FO(i)p FL(\)])3254 1399 y FP(A)3322 1362 y FO(i)p Fd(x)p
FO(j)3488 1399 y FT(for)33 b(some)378 1632 y FP(A)446
1599 y FO(i)500 1632 y FN(2)24 b FP(S)641 1646 y FL(1)681
1632 y FT(,)30 b(and)g(since)g FP(S)1192 1646 y FL(1)1256
1632 y FN(\022)25 b FP(S)1408 1646 y FL(2)1478 1632 y
FT(then)30 b FP(A)1753 1599 y FO(i)1806 1632 y FN(2)25
b FP(S)1948 1646 y FL(2)2018 1632 y FT(and)k(th)m(us)h
FP(X)j FN(2)25 b FP(S)2649 1593 y FK(\002K)2644 1658
y FL(2)2762 1632 y FT(.)970 b Ff(\004)378 1844 y FQ(Prop)s(osition)36
b(C.6)45 b FI(Given)32 b(a)f(c)-5 b(onne)g(ctability)33
b(r)-5 b(elation)33 b FN(K)g FI(and)f(a)g(set)f(of)h(c)-5
b(olour)g(e)g(d)33 b(sentenc)-5 b(es)32 b FP(S)5 b FI(,)378
1957 y(then)485 2145 y(1.)46 b(F)-7 b(or)35 b(every)e(liter)-5
b(al)35 b FP(A)1345 2112 y FO(i)1400 2145 y FN(2)27 b
FP(S)38 b FI(wher)-5 b(e)35 b FP(i)27 b FN(2)f Fe(C)p
FT(\()p FN(K)q FT(\))34 b FI(ther)-5 b(e)35 b(is)e(some)h(set)g
FP(X)g FN(2)27 b FP(S)3175 2112 y FK(\002K)3321 2145
y FI(with)35 b FP(A)3588 2112 y FO(i)p Fd(x)p FK(K)3767
2145 y FN(2)605 2258 y FP(X)7 b FI(.)485 2445 y(2.)46
b(F)-7 b(or)31 b(every)e(liter)-5 b(al)31 b FP(A)1333
2412 y FO(i)1386 2445 y FN(2)25 b FP(S)35 b FI(and)30
b(c)-5 b(olour)31 b FP(j)k FI(such)29 b(that)i FP(i)25
b FN(\030)2591 2459 y FK(K)2675 2445 y FP(j)5 b FI(,)30
b(ther)-5 b(e)30 b(is)g(some)g(set)f FP(X)k FN(2)25 b
FP(S)3715 2412 y FK(\002K)605 2558 y FI(with)34 b FP(A)871
2525 y FO(i)p Fd(x)p FO(j)1027 2558 y FN(2)25 b FP(X)7
b FI(.)485 2746 y(3.)46 b(F)-7 b(or)34 b(every)e(set)h
FP(X)g FN(2)25 b FP(S)1412 2713 y FK(\002K)1525 2746
y FI(,)32 b(if)g(the)h(liter)-5 b(al)34 b FP(A)2151 2713
y FO(i)p Fd(x)p FO(j)2308 2746 y FN(2)25 b FP(X)40 b
FI(then)33 b FP(A)2779 2713 y FO(i)2833 2746 y FN(2)24
b FP(S)38 b FI(and)c FP(i)25 b FN(\030)3316 2760 y FK(K)3399
2746 y FP(j)5 b FI(.)378 2958 y FQ(Pro)s(of)p FT(:)489
3201 y(1.)46 b(F)-8 b(or)36 b(all)d(sets)i FP(X)40 b
FN(2)1406 3087 y FK(^)1379 3115 y Fx([)1292 3316 y FO(j)t
FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])1583 3201
y FP(A)1651 3164 y FO(i)p Fd(x)p FO(j)1782 3201 y FT(,)c(w)m(e)f(ha)m
(v)m(e)2282 3115 y Fx(^)2195 3316 y FO(j)t FK( )p FL([)p
FK(K)p FL(\()p FO(i)p FL(\)])2485 3201 y FP(A)2553 3164
y FO(i)p Fd(x)p FO(j)2716 3201 y FN(2)d FP(X)42 b FT(b)m(y)35
b(Prop)s(osition)d(C.2\(2\).)605 3510 y(No)m(w)24 b FP(A)873
3477 y FO(i)p Fd(x)p FK(K)1051 3510 y FT(=)1234 3423
y Fx(^)1147 3625 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p
FO(i)p FL(\)])1438 3510 y FP(A)1506 3472 y FO(i)p Fd(x)p
FO(j)1637 3510 y FT(,)h(and)d(b)m(y)h(De\014nition)f(C.2)h(it)g(is)f
(the)h(case)i(that)3356 3396 y FK(^)3329 3423 y Fx([)3242
3625 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])3533
3510 y FP(A)3601 3472 y FO(i)p Fd(x)p FO(j)3757 3510
y FN(\022)605 3742 y FP(S)666 3704 y FK(\002K)810 3742
y FT(since)k FP(A)1100 3709 y FO(i)1154 3742 y FN(2)c
FP(S)5 b FT(,)30 b(and)g(th)m(us)g FP(X)j FN(2)25 b FP(S)1988
3709 y FK(\002K)2101 3742 y FT(.)489 4002 y(2.)46 b(There)30
b(is)f(some)h(set)g FP(X)j FN(2)1635 3888 y FK(^)1608
3916 y Fx([)1521 4117 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p
FO(i)p FL(\)])1811 4002 y FP(A)1879 3964 y FO(i)p Fd(x)p
FO(j)2040 4002 y FT(with)c FP(A)2315 3969 y FO(i)p Fd(x)p
FO(j)2472 4002 y FN(2)24 b FP(X)38 b FT(b)m(y)30 b(prop)s(osition)d
(C.4,)k(and)e(as)h(in)605 4310 y(the)h(previous)e(case,)i(since)1678
4197 y FK(^)1651 4224 y Fx([)1564 4425 y FO(j)t FK( )p
FL([)p FK(K)p FL(\()p FO(i)p FL(\)])1854 4310 y FP(A)1922
4273 y FO(i)p Fd(x)p FO(j)2079 4310 y FN(\022)25 b FP(S)2236
4273 y FK(\002K)2379 4310 y FT(it)30 b(follo)m(ws)g(that)h
FP(X)h FN(2)25 b FP(S)3220 4277 y FK(\002K)3333 4310
y FT(.)489 4694 y(3.)46 b(Let)30 b FP(A)835 4661 y FO(i)p
Fd(x)p FO(j)991 4694 y FN(2)25 b FP(X)37 b FT(for)28
b(some)i FP(X)j FN(2)24 b FP(S)1807 4661 y FK(\002K)1920
4694 y FT(,)30 b(then)f FP(X)j FN(2)2513 4580 y FK(^)2486
4607 y Fx([)2374 4809 y FO(m)p FK( )p FL([)p FK(K)p FL(\()p
FO(n)p FL(\)])2713 4694 y FP(B)2787 4656 y FO(n)p Fd(x)p
FO(m)2996 4694 y FT(for)d(some)g(literal)f FP(B)3696
4661 y FO(n)3767 4694 y FN(2)605 4920 y FP(S)5 b FT(.)52
b(And)33 b(since)g FP(A)1240 4887 y FO(i)p Fd(x)p FO(j)1405
4920 y FT(is)g(not)h(a)h(conjunction,)f(it)f(follo)m(ws)g(from)h(Prop)s
(osition)e(C.2\(3\))j(that)605 5033 y FP(A)673 5000 y
FO(i)p Fd(x)p FO(j)833 5033 y FT(=)28 b FP(B)1006 5000
y FO(n)p Fd(x)p FO(m)1218 5033 y FT(for)k(some)h FP(m)f
FT(in)f FN(K)q FT(\()p FP(n)p FT(\).)48 b(Hence,)34 b(since)e
FP(m)g FT(is)f(in)g FN(K)q FT(\()p FP(n)p FT(\))j(then)e
FP(B)3425 5000 y FO(n)p Fd(x)p FO(m)3633 5033 y FN(6)p
FT(=)c FN(>)p FT(,)605 5145 y(and)i(therefore)h FP(A)25
b FT(=)g FP(B)5 b FT(,)30 b FP(i)c FT(=)f FP(n)30 b FT(and)f
FP(j)i FT(=)25 b FP(m)p FT(,)31 b(and)e(th)m(us)h FP(A)2640
5112 y FO(i)2694 5145 y FN(2)25 b FP(S)35 b FT(and)30
b FP(i)c FN(\030)3176 5159 y FK(K)3259 5145 y FP(j)5
b FT(.)3757 5296 y Ff(\004)378 5508 y FQ(Prop)s(osition)36
b(C.7)45 b FI(F)-7 b(or)34 b(al)5 b(l)33 b(sets)g FP(X)g
FN(2)25 b FP(S)1879 5475 y FK(\002K)1992 5508 y FI(,)32
b(if)g FP(')21 b FN(^)f FP(#)25 b FN(2)g FP(X)40 b FI(then)33
b FP(X)28 b FN([)19 b(f)p FP(';)c(#)p FN(g)27 b(2)e FP(S)3383
5475 y FK(\002K)3496 5508 y FI(.)p eop
%%Page: 244 254
244 253 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(244)378 465 y FQ(Pro)s(of)p FT(:)31 b(Since)f
FP(X)i FN(2)25 b FP(S)1179 432 y FK(\002K)1292 465 y
FT(,)31 b(then)f FP(X)j FN(2)1863 351 y FK(^)1836 378
y Fx([)1749 580 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p
FO(i)p FL(\)])2039 465 y FP(A)2107 427 y FO(i)p Fd(x)p
FO(j)2269 465 y FT(for)d(some)g FP(A)2703 432 y FO(i)2757
465 y FN(2)25 b FP(S)5 b FT(.)41 b(No)m(w)857 865 y FP(')21
b FN(^)f FP(#)25 b FN(2)g FP(X)32 b FN(\))56 b FP(X)27
b FN([)20 b(f)p FP(';)15 b(#)p FN(g)27 b(2)2090 751 y
FK(^)2063 778 y Fx([)1976 980 y FO(j)t FK( )p FL([)p
FK(K)p FL(\()p FO(i)p FL(\)])2266 865 y FP(A)2334 827
y FO(i)p Fd(x)p FO(j)2526 865 y FT(\(b)m(y)k(Prop)s(osition)d(C.3\))
1290 1121 y FN(\))56 b FP(X)27 b FN([)20 b(f)p FP(';)15
b(#)p FN(g)27 b(2)e FP(S)2037 1084 y FK(\002K)2211 1121
y FT(\(b)m(y)30 b(De\014nition)f(C.2\).)766 b Ff(\004)378
1314 y FQ(Prop)s(osition)36 b(C.8)45 b FI(Given)32 b(a)f(c)-5
b(onne)g(ctability)33 b(r)-5 b(elation)33 b FN(K)g FI(and)f(a)g(set)f
FP(S)36 b FI(of)c(c)-5 b(olour)g(e)g(d)34 b(sentenc)-5
b(es,)378 1427 y(then)485 1597 y(1.)46 b(F)-7 b(or)31
b(every)f(liter)-5 b(al)31 b FP(A)1334 1564 y FO(i)1388
1597 y FN(2)25 b FP(S)34 b FI(and)d(c)-5 b(olour)32 b
FP(j)j FI(such)30 b(that)h FP(i)26 b FN(\030)2596 1611
y FK(K)2679 1597 y FP(j)35 b FI(ther)-5 b(e)31 b(is)f(a)g(set)g
FP(X)j FN(2)25 b FP(S)3540 1564 y FK(\003K)3663 1597
y FI(with)605 1710 y FP(A)673 1677 y FO(i)p Fd(x)p FO(j)830
1710 y FN(2)g FP(X)7 b FI(.)485 1891 y(2.)46 b(F)-7 b(or)43
b(every)f(liter)-5 b(al)44 b FP(B)i FN(2)c FP(X)50 b
FI(wher)-5 b(e)43 b FP(X)50 b FN(2)41 b FP(S)2200 1858
y FK(\003K)2294 1891 y FI(,)j(if)e FP(B)47 b FN(6)p FT(=)42
b FN(>)f FI(then)i(ther)-5 b(e)43 b(is)f(some)g(liter)-5
b(al)605 2004 y FP(A)673 1971 y FO(i)727 2004 y FN(2)25
b FP(S)37 b FI(such)c(that)h FP(B)c FT(=)25 b FP(A)1562
1971 y FO(i)p Fd(x)p FO(j)1725 2004 y FI(and)34 b FP(i)26
b FN(\030)2030 2018 y FK(K)2113 2004 y FP(j)5 b FI(.)378
2196 y FQ(Pro)s(of)p FT(:)489 2367 y(1.)46 b(If)27 b
FP(A)761 2334 y FO(i)814 2367 y FN(2)e FP(S)32 b FT(and)26
b FP(i)g FN(\030)1289 2381 y FK(K)1372 2367 y FP(j)33
b FT(then)26 b(b)m(y)h(Prop)s(osition)e(C.6\(2\))j(there)f(is)f(some)i
FP(Y)45 b FN(2)25 b FP(S)3320 2334 y FK(\002K)3460 2367
y FT(suc)m(h)h(that)605 2479 y FP(A)673 2446 y FO(i)p
Fd(x)p FO(j)830 2479 y FN(2)f FP(Y)20 b FT(.)39 b(Th)m(us,)26
b FP(A)1373 2446 y FO(i)p Fd(x)p FO(j)1529 2479 y FN(2)f
FP(S)1676 2446 y Fd(x)p FK(K)1816 2479 y FN([)1888 2411
y Fx(S)1963 2479 y FN(f)p FP(Y)20 b FN(g)27 b FT(and)e(since)g
FN(f)p FP(Y)20 b FN(g)26 b(\022)f FP(S)2889 2446 y FK(\002K)3028
2479 y FT(then)g FP(S)3291 2446 y Fd(x)p FK(K)3431 2479
y FN([)3503 2411 y Fx(S)3578 2479 y FN(f)p FP(Y)c FN(g)k(2)605
2592 y FP(S)666 2559 y FK(\003K)760 2592 y FT(.)489 2773
y(2.)46 b(If)39 b FP(X)47 b FN(2)40 b FP(S)989 2740 y
FK(\003K)1082 2773 y FT(,)i(then)d FP(X)47 b FT(=)40
b FP(S)1659 2740 y Fd(x)p FK(K)1813 2773 y FN([)1900
2705 y Fx(S)1991 2773 y FP(Z)46 b FT(for)39 b(some)g
FP(Z)47 b FN(\022)39 b FP(S)2763 2740 y FK(\002K)2876
2773 y FT(.)67 b(Hence,)43 b(since)38 b FP(B)45 b FN(2)39
b FP(X)7 b FT(,)605 2886 y(either)28 b FP(B)i FN(2)25
b FP(S)1107 2853 y Fd(x)p FK(K)1264 2886 y FT(or)k FP(B)h
FN(2)1559 2818 y Fx(S)1649 2886 y FP(Z)7 b FT(.)40 b(F)-8
b(or)29 b(the)g(\014rst)f(case,)i FP(B)g FN(2)25 b FP(S)2747
2853 y Fd(x)p FK(K)2904 2886 y FT(and)j(b)m(y)h(Prop)s(osition)e(C.1)
605 2999 y FP(B)k FT(=)25 b FP(A)869 2966 y FO(i)p Fd(x)p
FO(j)1031 2999 y FT(for)31 b(some)g FP(A)1467 2966 y
FO(i)1521 2999 y FN(2)26 b FP(S)36 b FT(and)30 b(where)g
FP(i)d FN(\030)2269 3013 y FK(K)2352 2999 y FP(j)5 b
FT(.)43 b(Alternativ)m(ely)-8 b(,)31 b(if)f FP(B)g FN(2)3301
2931 y Fx(S)3392 2999 y FP(Z)37 b FT(for)30 b(some)605
3112 y FP(Z)i FN(\022)25 b FP(S)856 3079 y FK(\002K)969
3112 y FT(,)31 b(it)f(follo)m(ws)f(that)i FP(B)f FN(2)25
b FP(Y)50 b FT(for)30 b(some)h FP(Y)45 b FN(2)25 b FP(Z)7
b FT(,)30 b(and)g(therefore)h FP(Y)45 b FN(2)25 b FP(S)3378
3079 y FK(\002K)3491 3112 y FT(.)40 b(Th)m(us)1880 3389
y FP(Y)45 b FN(2)2178 3275 y FK(^)2151 3303 y Fx([)2064
3504 y FO(j)t FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])2354
3389 y FP(A)2422 3351 y FO(i)p Fd(x)p FO(j)605 3706 y
FT(for)c(some)g(literal)e FP(A)1334 3673 y FO(i)1405
3706 y FN(2)k FP(S)5 b FT(.)72 b(As)40 b(a)h(result,)i
FP(B)k FT(=)c FP(A)2487 3673 y FO(i)p Fd(x)p FO(j)2659
3706 y FT(for)d(some)h FP(j)47 b FT(where)40 b FP(i)j
FN(\030)3548 3720 y FK(K)3649 3706 y FP(j)j FT(b)m(y)605
3819 y(Prop)s(osition)29 b(C.2\(3\).)2386 b Ff(\004)519
3990 y FT(W)-8 b(e)32 b(are)e(no)m(w)h(ready)f(to)h(sho)m(w)f(that)h
FN(C)1853 3957 y FK(\003K)1977 3990 y FT(is)f(a)h(consistency)f(prop)s
(ert)m(y)-8 b(.)378 4182 y FQ(Lemma)33 b(C.1)45 b FI(If)32
b FN(C)38 b FI(is)33 b(a)g FN(K)q FI(-c)-5 b(onsistency)33
b(pr)-5 b(op)g(erty)36 b(then)d FN(C)2501 4149 y FK(\003K)2627
4182 y FI(is)g(a)g(c)-5 b(onsistency)33 b(pr)-5 b(op)g(erty.)378
4374 y FQ(Pro)s(of)p FT(:)31 b(W)-8 b(e)31 b(pro)m(v)m(e)f(that)g
FN(C)1336 4341 y FK(\003K)1460 4374 y FT(is)f(a)h(consistency)f(prop)s
(ert)m(y)g(b)m(y)h(sho)m(wing)f(that)h(all)f(the)g(conditions)378
4487 y(in)g(De\014nition)g(7.1)i(are)g(satis\014ed.)489
4657 y(1.)46 b(Let)32 b FP(X)j FN(2)27 b(C)1020 4624
y FK(\003K)1114 4657 y FT(,)32 b(then)f FP(X)k FN(2)26
b FP(S)1637 4624 y FK(\003K)1762 4657 y FT(for)32 b(some)g
FP(S)g FN(2)27 b(C)5 b FT(.)44 b(No)m(w,)33 b(if)e(a)h(literal)e
FP(B)h FN(2)c FP(X)39 b FT(then)31 b(either)605 4770
y FP(B)f FT(=)25 b FN(>)p FT(,)j(or)g(else)g FP(B)i FT(=)25
b FP(A)1466 4737 y FO(i)p Fd(x)p FO(j)1597 4770 y FT(,)j
FP(A)1718 4737 y FO(i)1772 4770 y FN(2)d FP(S)33 b FT(and)27
b FP(i)f FN(\030)2249 4784 y FK(K)2332 4770 y FP(j)5
b FT(,)29 b(b)m(y)f(Prop)s(osition)e(C.8\(2\).)41 b(F)-8
b(or)29 b(the)f(\014rst)605 4883 y(case)1103 5087 y FP(A)1171
5050 y FO(i)1224 5087 y FN(2)d FP(S)35 b FT(and)30 b
FP(i)c FN(\030)1706 5101 y FK(K)1789 5087 y FP(j)89 b
FN(\))83 b(:)p FP(A)2218 5050 y FO(j)2289 5087 y FP(=)-55
b FN(2)25 b FP(S)35 b FT(as)c FP(S)f FN(2)25 b(C)1915
5225 y(\))83 b FT(\()p FN(:)p FP(A)2253 5188 y FO(j)2289
5225 y FT(\))2324 5188 y Fd(x)p FO(i)2459 5225 y FP(=)-55
b FN(2)24 b FP(X)7 b FT(,)31 b(b)m(y)g(Prop.)f(C.8\(1\))1915
5363 y FN(\))83 b(:)p FT(\()p FP(A)2253 5326 y FO(i)p
Fd(x)p FO(j)2384 5363 y FT(\))36 b FP(=)-56 b FN(2)25
b FP(X)1915 5501 y FN(\))83 b(:)p FP(B)39 b(=)-55 b FN(2)25
b FP(X)r(:)605 5705 y FT(F)-8 b(or)31 b(the)g(second)f(case,)i(if)d
FP(B)h FT(=)25 b FN(>)p FT(,)30 b(then)g FN(:>)25 b FT(=)g
FN(?)30 b FT(and)g FN(?)35 b FP(=)-55 b FN(2)25 b FP(X)37
b FT(b)m(y)31 b(case)g(2)g(b)s(elo)m(w.)p eop
%%Page: 245 255
245 254 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(245)489 396 y(2.)46 b(Let)c FP(X)50 b FN(2)42
b(C)1060 363 y FK(\003K)1154 396 y FT(,)h(then)e FP(X)50
b FN(2)42 b FP(S)1729 363 y FK(\003K)1863 396 y FT(for)f(some)g
FP(S)48 b FN(2)42 b(C)5 b FT(.)72 b(W)-8 b(e)42 b(are)f(required)e(to)j
(sho)m(w)e(that)605 509 y FN(?)52 b FP(=)-55 b FN(2)42
b FP(X)7 b FT(.)72 b(Supp)s(ose)39 b(that)i FN(?)h(2)g
FP(X)7 b FT(,)43 b(then)e FN(?)h FT(=)f FP(A)2451 476
y FO(i)p Fd(x)p FO(j)2623 509 y FT(for)g(some)g FP(A)3079
476 y FO(i)3149 509 y FN(2)h FP(S)k FT(and)40 b FP(i)i
FN(\030)3685 523 y FK(K)3785 509 y FP(j)605 622 y FT(b)m(y)36
b(Prop)s(osition)f(C.8\(2\).)59 b(Therefore)36 b FP(A)f
FT(=)g FN(?)h FT(b)m(y)g(de\014nition)e(7.16)k(on)e(page)h(128)g(and)f
(so)605 735 y FN(?)676 702 y FO(i)737 735 y FN(2)d FP(S)5
b FT(.)54 b(It)36 b(is)e(also)h(the)g(case)h(that)g FP(i)f
FT(is)f(in)g FN(K)i FT(as)g FP(i)d FN(\030)2548 749 y
FK(K)2639 735 y FP(j)5 b FT(.)55 b(But)36 b(this)d(is)i(a)g(con)m
(tradiction)605 848 y(since)30 b FP(S)g FN(2)25 b(C)36
b FT(and)29 b FN(C)36 b FT(is)29 b(a)i(coloured)f(consistency)g(prop)s
(ert)m(y)-8 b(.)489 1025 y(3.)46 b(Let)31 b FP(X)i FN(2)25
b(C)1015 992 y FK(\003K)1139 1025 y FT(and)k FP(')21
b FN(^)f FP( )29 b FN(2)24 b FP(X)7 b FT(,)31 b(then)1181
1230 y FP(X)i FN(2)24 b(C)1427 1192 y FK(\003K)1604 1230
y FN(\))83 b FP(X)33 b FN(2)25 b FP(S)2033 1192 y FK(\003K)2156
1230 y FT(for)31 b(some)f FP(S)h FN(2)24 b(C)1604 1367
y(\))83 b FP(X)33 b FT(=)25 b FP(S)2043 1330 y Fd(x)p
FK(K)2192 1367 y FN([)2272 1281 y Fx([)2389 1367 y FP(Y)50
b FT(for)30 b(some)h FP(Y)45 b FN(\022)25 b FP(S)3114
1334 y FK(\002K)3227 1367 y FT(.)605 1595 y(W)-8 b(e)32
b(no)m(w)e(consider)f(the)i(cases)g(of)g(whether)f FP(')20
b FN(^)g FP( )29 b FN(2)2477 1527 y Fx(S)2568 1595 y
FP(Y)50 b FT(or)30 b(whether)g FP(')21 b FN(^)e FP( )29
b FN(2)c FP(S)3526 1562 y Fd(x)p FK(K)3655 1595 y FT(.)714
1773 y FN(\017)46 b FT(If)30 b FP(')21 b FN(^)f FP( )28
b FN(2)1230 1704 y Fx(S)1321 1773 y FP(Y)50 b FT(where)30
b FP(Y)46 b FN(\022)24 b FP(S)1942 1740 y FK(\002K)2056
1773 y FT(,)30 b(then)1443 1977 y FP(')21 b FN(^)e FP( )29
b FN(2)c FP(Z)37 b FT(for)30 b(some)h FP(Z)h FN(2)24
b FP(Y)c FT(,)31 b(i.e.,)16 b FP(Z)31 b FN(2)25 b FP(S)2948
1944 y FK(\002K)1526 2115 y FN(\))83 b FP(Z)27 b FN([)19
b(f)p FP(';)c( )s FN(g)28 b(2)d FP(S)2295 2077 y FK(\002K)2438
2115 y FT(b)m(y)30 b(Prop)s(osition)f(C.7)1526 2252 y
FN(\))83 b FP(Y)40 b FN([)20 b(f)p FP(Z)27 b FN([)20
b(f)p FP(';)15 b( )s FN(gg)28 b(\022)d FP(S)2570 2215
y FK(\002K)1526 2404 y FN(\))83 b FP(S)1761 2366 y Fd(x)p
FK(K)1910 2404 y FN([)1991 2317 y Fx([)2092 2404 y FT(\()p
FP(Y)40 b FN([)20 b(f)p FP(Z)27 b FN([)20 b(f)p FP(';)15
b( )s FN(gg)p FT(\))28 b FN(2)d FP(S)3022 2366 y FK(\003K)3116
2404 y FP(:)1493 2824 y FT(No)m(w,)31 b FP(S)1786 2786
y Fd(x)p FK(K)1935 2824 y FN([)2016 2737 y Fx([)2117
2824 y FT(\()p FP(Y)41 b FN([)19 b(f)p FP(Z)28 b FN([)19
b(f)p FP(';)c( )s FN(gg)p FT(\))1576 3002 y(=)83 b FP(S)1791
2965 y Fd(x)p FK(K)1940 3002 y FN([)2020 2916 y Fx([)2137
3002 y FP(Y)40 b FN([)20 b FT(\()p FP(Z)27 b FN([)20
b(f)p FP(';)15 b( )s FN(g)p FT(\))1576 3181 y(=)83 b
FP(S)1791 3143 y Fd(x)p FK(K)1940 3181 y FN([)2020 3095
y Fx([)2137 3181 y FP(Y)40 b FN([)20 b(f)p FP(';)15 b( )s
FN(g)32 b FT(\(since)e FP(Z)i FN(2)25 b FP(Y)20 b FT(\))1576
3332 y(=)83 b FP(X)27 b FN([)20 b(f)p FP(';)15 b( )s
FN(g)p FP(:)805 3537 y FT(And)30 b(therefore,)h FP(X)c
FN([)20 b(f)p FP(';)15 b( )s FN(g)28 b(2)d(C)2013 3504
y FK(\003K)2106 3537 y FT(.)714 3672 y FN(\017)46 b FT(If)33
b FP(')23 b FN(^)f FP( )33 b FN(2)d FP(S)1308 3639 y
Fd(x)p FK(K)1471 3672 y FT(then)j(there)g(is)g(some)h(form)m(ula)e
FP(\037)e FN(2)g FP(S)39 b FT(suc)m(h)33 b(that)h FP(')22
b FN(^)g FP( )34 b FT(=)c FP(\037)3674 3639 y Fd(x)p
FK(K)3803 3672 y FT(.)805 3785 y(No)m(w,)j FP(\037)f
FT(is)f(either)g(a)i(literal)d(and)h FP(])p FT(\()p FN(K)q
FT(\()p FP(i)p FT(\)\))g FP(>)c FT(1,)33 b(or)f(else)g
FP(\037)f FT(is)g(a)h(conjunction.)45 b(If)31 b FP(\037)h
FT(is)805 3898 y(some)f(literal)e FP(A)1364 3865 y FO(i)1417
3898 y FN(2)c FP(S)36 b FT(then)1729 4102 y FP(')20 b
FN(^)g FP( )29 b FT(=)c FP(A)2141 4065 y FO(i)p Fd(x)p
FK(K)2319 4102 y FT(=)2502 4016 y Fx(^)2415 4217 y FO(j)t
FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])2705 4102
y FP(A)2773 4065 y FO(i)p Fd(x)p FO(j)805 4502 y FT(and)30
b(therefore)h FN(f)p FP(')21 b FN(^)f FP( )s FN(g)26
b(2)1903 4389 y FK(^)1876 4416 y Fx([)1789 4617 y FO(j)t
FK( )p FL([)p FK(K)p FL(\()p FO(i)p FL(\)])2079 4502
y FP(A)2147 4465 y FO(i)p Fd(x)p FO(j)2309 4502 y FT(b)m(y)k(Prop)s
(osition)e(C.2\(1\).)43 b(Hence)1758 4902 y FN(f)p FP(')21
b FN(^)f FP( )s(;)15 b(';)g( )s FN(g)28 b(2)2500 4789
y FK(^)2473 4816 y Fx([)2386 5017 y FO(j)t FK( )p FL([)p
FK(K)p FL(\()p FO(i)p FL(\)])2676 4902 y FP(A)2744 4865
y FO(i)p Fd(x)p FO(j)805 5221 y FT(and)i(th)m(us)g FN(f)p
FP(')21 b FN(^)f FP( )s(;)15 b(';)g( )s FN(g)28 b(2)d
FP(S)1871 5188 y FK(\002K)1984 5221 y FT(.)805 5345 y(Since)30
b FP(Y)45 b FN(\022)25 b FP(S)1298 5312 y FK(\002K)1411
5345 y FT(,)30 b(w)m(e)h(get)1421 5549 y FP(Y)40 b FN([)20
b(ff)p FP(')h FN(^)f FP( )s(;)15 b(';)g( )s FN(gg)28
b(\022)d FP(S)2384 5512 y FK(\002K)1504 5701 y FN(\))83
b FP(S)1739 5663 y Fd(x)p FK(K)1888 5701 y FN([)1968
5614 y Fx([)2069 5701 y FT(\()p FP(Y)41 b FN([)20 b(ff)p
FP(')h FN(^)f FP( )s(;)15 b(';)g( )s FN(gg)p FT(\))29
b FN(2)c FP(S)3094 5663 y FK(\003K)3187 5701 y FP(:)p
eop
%%Page: 246 256
246 255 bop 378 5 a FF(APPENDIX)31 b(C.)60 b(A)31 b(LONG)f(PR)m(OOF)
1917 b FT(246)1357 601 y(No)m(w,)32 b FP(S)1651 563 y
Fd(x)p FK(K)1800 601 y FN([)1880 514 y Fx([)1981 601
y FT(\()p FP(Y)41 b FN([)20 b(ff)p FP(')h FN(^)f FP( )s(;)15
b(';)g( )s FN(gg)p FT(\))1440 779 y(=)83 b FP(S)1655
742 y Fd(x)p FK(K)1804 779 y FN([)1885 693 y Fx([)2001
779 y FP(Y)40 b FN([)20 b(f)p FP(')h FN(^)f FP( )s(;)15
b(';)g( )s FN(g)1440 958 y FT(=)83 b FP(S)1655 920 y
Fd(x)p FK(K)1804 958 y FN([)1885 872 y Fx([)2001 958
y FP(Y)40 b FN([)20 b(f)p FP(';)15 b( )s FN(g)33 b FT(\(since)d
FP(')20 b FN(^)g FP( )29 b FN(2)c FP(S)3112 925 y Fd(x)p
FK(K)3241 958 y FT(\))1440 1109 y(=)83 b FP(X)28 b FN([)20
b(f)p FP(';)15 b( )s FN(g)p FP(:)805 1314 y FT(And)30
b(therefore,)h FP(X)c FN([)20 b(f)p FP(';)15 b( )s FN(g)28
b(2)d(C)2013 1281 y FK(\003K)2106 1314 y FT(.)805 1443
y(W)-8 b(e)34 b(no)m(w)e(consider)g(the)h(case)g(where)f
FP(\037)g FT(is)g(not)h(a)g(literal,)e(and)h(therefore)h(w)m(e)g
(assume)805 1556 y(that)45 b(it)f(is)f(some)i(conjunctiv)m(e)f(form)m
(ula)g FP(\026)29 b FN(^)g FP(\032)49 b FN(2)f FP(S)5
b FT(,)48 b(and)c(that)g FP(\026)3281 1523 y Fd(x)p FK(K)3459
1556 y FT(=)k FP(')c FT(and)805 1669 y FP(\032)852 1636
y Fd(x)p FK(K)1006 1669 y FT(=)25 b FP( )s FT(.)41 b(But)31
b(since)f FP(S)g FN(2)25 b(C)5 b FT(,)30 b(then)h FP(S)25
b FN([)20 b(f)p FP(\026;)15 b(\032)p FN(g)26 b(2)f(C)35
b FT(as)c(w)m(ell.)40 b(No)m(w,)1053 1887 y(\()p FP(S)25
b FN([)20 b(f)p FP(\026;)15 b(\032)p FN(g)p FT(\))1517
1849 y Fd(x)p FK(K)1668 1887 y FN([)1748 1800 y Fx([)1865
1887 y FP(V)45 b FN(2)25 b(C)2102 1849 y FK(\003K)2226
1887 y FT(for)30 b(all)f FP(V)46 b FN(\022)25 b FT(\()p
FP(S)g FN([)20 b(f)p FP(\026;)15 b(\032)p FN(g)p FT(\))3150
1854 y FK(\002K)1136 2065 y FN(\))83 b FT(\()p FP(S)25
b FN([)20 b(f)p FP(\026;)15 b(\032)p FN(g)p FT(\))1774
2028 y Fd(x)p FK(K)1925 2065 y FN([)2005 1979 y Fx([)2122
2065 y FP(Y)45 b FN(2)25 b(C)2359 2028 y FK(\003K)1370
2227 y FT(\(as)31 b FP(Y)46 b FN(\022)25 b FP(S)1773
2194 y FK(\002K)1911 2227 y FN(\022)g FT(\()p FP(S)g
FN([)20 b(f)p FP(\026;)15 b(\032)p FN(g)p FT(\))2471
2194 y FK(\002K)2616 2227 y FT(and)30 b(b)m(y)g(Prop)s(osition)f(C.5\))
1136 2378 y FN(\))83 b FP(S)1371 2341 y Fd(x)p FK(K)1520
2378 y FN([)20 b(f)p FP(\026)1701 2341 y Fd(x)p FK(K)1830
2378 y FP(;)15 b(\032)1917 2341 y Fd(x)p FK(K)2046 2378
y FN(g)21 b([)2192 2292 y Fx([)2309 2378 y FP(Y)45 b
FN(2)25 b(C)2546 2341 y FK(\003K)1136 2545 y FN(\))83
b FP(X)27 b FN([)20 b(f)p FP(\026)1593 2507 y Fd(x)p
FK(K)1722 2545 y FP(;)15 b(\032)1809 2507 y Fd(x)p FK(K)1939
2545 y FN(g)25 b(2)g(C)2148 2507 y FK(\003K)2272 2545
y FT(\(as)31 b FP(X)i FT(=)25 b FP(S)2684 2512 y Fd(x)p
FK(K)2833 2545 y FN([)2914 2476 y Fx(S)3004 2545 y FP(Y)20
b FT(\))1136 2682 y FN(\))83 b FP(X)27 b FN([)20 b(f)p
FP(';)15 b( )s FN(g)28 b(2)d(C)1911 2645 y FK(\003K)2004
2682 y FP(:)378 2903 y FT(The)k(remaining)e(cases)j(follo)m(w)f(easily)
f(from)h(the)g(fact)h(that)g FN(C)k FT(is)28 b(a)i FN(K)q
FT(-consistency)g(prop)s(ert)m(y)-8 b(,)29 b(and)378
3016 y(w)m(e)i(consider)e(only)h(the)g(fourth)g(case)h(for)f
(illustration.)489 3204 y(4.)46 b(Let)27 b FP(X)33 b
FN(2)25 b(C)1011 3171 y FK(\003K)1131 3204 y FT(and)h
FP(')12 b FN(_)g FP( )29 b FN(2)c FP(X)7 b FT(.)40 b(No)m(w)27
b FP(X)33 b FT(=)24 b FP(S)2236 3171 y Fd(x)p FK(K)2378
3204 y FN([)2451 3136 y Fx(S)2541 3204 y FP(Y)47 b FT(for)26
b(some)h FP(Y)45 b FN(\022)25 b FP(S)3255 3171 y FK(\002K)3368
3204 y FT(.)39 b(No)m(w)28 b(since)605 3317 y FP(')23
b FN(_)e FP( )33 b FN(2)d FP(X)40 b FT(is)32 b(neither)g(a)i(literal)d
(nor)i(a)g(conjunction,)g FP(')23 b FN(_)e FP( )33 b
FN(2)d FP(S)2988 3284 y Fd(x)p FK(K)3149 3317 y FT(and)j(th)m(us)g
(there)g(is)605 3430 y(some)f FP(\026)20 b FN(_)g FP(\032)26
b FN(2)g FP(S)36 b FT(and)30 b FP(\026)1474 3397 y Fd(x)p
FK(K)1629 3430 y FT(=)c FP(')31 b FT(and)g FP(\032)2041
3397 y Fd(x)p FK(K)2196 3430 y FT(=)26 b FP( )s FT(.)42
b(Hence,)32 b FP(S)26 b FN([)20 b(f)p FP(\026)p FN(g)27
b(2)f(C)36 b FT(or)31 b FP(S)25 b FN([)c(f)p FP(\032)p
FN(g)27 b(2)f(C)5 b FT(.)605 3542 y(If)30 b FP(S)25 b
FN([)20 b(f)p FP(\026)p FN(g)26 b(2)f(C)35 b FT(then,)999
3760 y(\()p FP(S)26 b FN([)19 b(f)p FP(\026)p FN(g)p
FT(\))1376 3723 y Fd(x)p FK(K)1526 3760 y FN([)1607 3674
y Fx([)1723 3760 y FP(V)46 b FN(2)25 b(C)1961 3723 y
FK(\003K)2085 3760 y FT(for)30 b(all)f FP(V)46 b FN(\022)25
b FT(\()p FP(S)g FN([)20 b(f)p FP(\026)p FN(g)p FT(\))2922
3727 y FK(\002K)1082 3939 y FN(\))83 b FT(\()p FP(S)26
b FN([)19 b(f)p FP(\026)p FN(g)p FT(\))1633 3901 y Fd(x)p
FK(K)1783 3939 y FN([)1864 3853 y Fx([)1980 3939 y FP(Y)46
b FN(2)24 b(C)2217 3901 y FK(\003K)2341 3939 y FT(as)31
b FP(Y)45 b FN(\022)25 b FP(S)2708 3906 y FK(\002K)2847
3939 y FN(\022)g FT(\()p FP(S)g FN([)20 b(f)p FP(\026)p
FN(g)p FT(\))3320 3906 y FK(\002K)1082 4118 y FN(\))83
b FP(S)1317 4080 y Fd(x)p FK(K)1466 4118 y FN([)20 b(f)p
FP(\026)1647 4080 y Fd(x)p FK(K)1776 4118 y FN(g)h([)1922
4031 y Fx([)2038 4118 y FP(Y)46 b FN(2)25 b(C)2276 4080
y FK(\003K)1082 4284 y FN(\))83 b FP(X)28 b FN([)19 b(f)p
FP(\026)1539 4246 y Fd(x)p FK(K)1669 4284 y FN(g)25 b(2)g(C)1878
4246 y FK(\003K)1082 4422 y FN(\))83 b FP(X)28 b FN([)19
b(f)p FP(')p FN(g)27 b(2)e(C)1754 4384 y FK(\003K)1848
4422 y FP(:)605 4626 y FT(Similarly)-8 b(,)41 b(if)g
FP(S)33 b FN([)27 b(f)p FP(\032)p FN(g)45 b(2)e(C)k FT(then)41
b FP(X)35 b FN([)27 b(f)p FP( )s FN(g)45 b(2)f(C)2447
4593 y FK(\003K)2541 4626 y FT(,)g(and)d(hence)h FP(X)35
b FN([)27 b(f)p FP(')p FN(g)45 b(2)f(C)3612 4593 y FK(\003K)3747
4626 y FT(or)605 4739 y FP(X)28 b FN([)20 b(f)p FP( )s
FN(g)26 b(2)f(C)1106 4706 y FK(\003K)1200 4739 y FT(.)2532
b Ff(\004)378 4951 y FQ(Theorem)34 b(C.1)45 b FI(If)33
b FN(C)k FI(is)c(a)g FN(K)q FI(-c)-5 b(onsistency)34
b(pr)-5 b(op)g(erty,)35 b(then)e(every)f(set)h FP(S)d
FN(2)25 b(C)38 b FI(is)33 b FN(K)q FI(-satis\014able.)378
5164 y FQ(Pro)s(of)p FT(:)39 b(If)f FP(S)44 b FN(2)38
b(C)43 b FT(then)38 b FP(S)1360 5131 y FK(\003K)1493
5164 y FN(\022)g(C)1655 5131 y FK(\003K)1748 5164 y FT(.)65
b(By)39 b(de\014nition,)f FP(S)2489 5131 y FK(\003K)2621
5164 y FT(=)g FN(f)p FP(S)2836 5131 y Fd(x)p FK(K)2991
5164 y FN([)3077 5096 y Fx(S)3168 5164 y FP(X)46 b FN(j)39
b FP(X)46 b FN(\022)38 b FP(S)3644 5131 y FK(\002K)3757
5164 y FN(g)p FT(,)378 5277 y(and)32 b(th)m(us)h FP(S)821
5244 y Fd(x)p FK(K)979 5277 y FN(2)c FP(S)1130 5244 y
FK(\003K)1257 5277 y FT(as)k FN(fg)e(\022)e FP(S)1653
5244 y FK(\002K)1766 5277 y FT(,)34 b(and)e(so)h FP(S)2179
5244 y Fd(x)p FK(K)2338 5277 y FN(2)c(C)2481 5244 y FK(\003K)2575
5277 y FT(.)48 b(No)m(w,)35 b(b)m(y)e(Lemma)g(C.1)g FN(C)3562
5244 y FK(\003K)3688 5277 y FT(is)f(a)378 5390 y(consistency)g(prop)s
(ert)m(y)g(and)g(b)m(y)g(the)h(Mo)s(del)e(Existence)i(theorem)f(it)g
(follo)m(ws)g(that)h(the)f(set)h FP(S)3699 5357 y Fd(x)p
FK(K)378 5503 y FT(is)c(satis\014able.)40 b(Th)m(us)29
b FP(S)36 b FT(is)29 b FN(K)q FT(-satis\014able.)1903
b Ff(\004)p eop
%%Page: 247 257
247 256 bop 378 1061 a FR(Bibliograph)-6 b(y)474 1506
y FT(Andrews,)34 b(P)-8 b(.)35 b(B.)g(\(1981,)j(April\).)32
b(Theorem)j(pro)m(ving)e(via)h(general)g(matings.)h FI(Journal)i(of)f
(the)610 1619 y(A)n(CM)44 b(28)12 b FT(\(2\),)33 b(193{214.)474
1768 y(Anon)m(ymous)g(\(1994,)38 b(June)33 b(26{July)g(1,\).)j(The)d
(QED)h(manifesto.)g(In)f(A.)h(Bundy)f(\(Ed.\),)i FI(12th)610
1880 y(International)g(Confer)-5 b(enc)g(e)34 b(on)f(A)n(utomate)-5
b(d)34 b(De)-5 b(duction)p FT(,)32 b(V)-8 b(olume)30
b(814)i(of)f FI(LNAI)p FT(,)f(Nancy)-8 b(,)610 1993 y(F)g(rance,)32
b(pp.)e(238{251.)j(Springer-V)-8 b(erlag.)474 2142 y(Baaz,)39
b(M.)d(and)e(C.)i(G.)g(F)-8 b(erm)s(\177)-48 b(uller)34
b(\(1995,)39 b(Ma)m(y\).)e(Non-elemen)m(tary)g(sp)s(eedups)c(b)s(et)m
(w)m(een)j(dif-)610 2255 y(feren)m(t)27 b(v)m(ersions)f(of)h(tableaux.)
g(In)f(P)-8 b(.)27 b(Baumgartner,)h(R.)f(H\177)-45 b(ahnle,)27
b(and)f(J.)g(P)m(osegga)k(\(Eds.\),)610 2368 y FI(Pr)-5
b(o)g(c)g(e)g(e)g(dings)45 b(of)e(the)h(4th)g(International)h(Workshop)
f(on)g(The)-5 b(or)g(em)45 b(Pr)-5 b(oving)43 b(with)h(A)n(na-)610
2481 y(lytic)c(T)-7 b(able)i(aux)41 b(and)g(R)-5 b(elate)g(d)41
b(Metho)-5 b(ds)p FT(,)42 b(V)-8 b(olume)38 b(918)h(of)f
FI(LNAI)p FT(,)g(Berlin,)g(pp.)g(217{230.)610 2594 y(Springer.)474
2743 y(Bac)m(hmair,)29 b(L.,)f(N.)g(Dersho)m(witz,)h(and)e(D.)h(A.)h
(Plaisted)d(\(1989\).)31 b(Completion)26 b(without)g(failure.)610
2856 y(In)31 b(H.)i(A)-10 b(\177)-35 b(\020t-Kaci)31
b(and)g(M.)i(Niv)-5 b(at)32 b(\(Eds.\),)g FI(R)-5 b(esolution)36
b(of)e(Equations)h(in)f(A)n(lgebr)-5 b(aic)33 b(Struc-)610
2969 y(tur)-5 b(es)p FT(,)25 b(V)-8 b(olume)22 b(2:)37
b(Rewriting)20 b(T)-8 b(ec)m(hniques,)24 b(Chapter)d(1,)k(pp.)c(1{30.)j
(New)e(Y)-8 b(ork:)38 b(Academic)610 3082 y(Press.)474
3231 y(Bac)m(k,)d(R.,)e(J.)f(Grundy)-8 b(,)32 b(and)g(J.)g(v)m(on)h(W)
-8 b(righ)m(t)33 b(\(1996,)i(No)m(v)m(em)m(b)s(er\).)f(Structured)d
(calculational)610 3344 y(pro)s(of.)43 b(TUCS)f(T)-8
b(ec)m(hnical)42 b(Rep)s(ort)h(65,)k(T)-8 b(urku)42 b(Cen)m(tre)h(for)g
(Computer)f(Science,)47 b(Lem-)610 3456 y(mink\177)-45
b(aisenk)-5 b(atu)30 b(14A,)k(20520)g(T)-8 b(urku,)31
b(Finland.)f(Also)i(a)m(v)-5 b(ailable)31 b(as)h(ANU)h(T)-8
b(ec)m(hnical)31 b(Re-)610 3569 y(p)s(ort)f(TR-CS-96-09.)474
3718 y(Bailey)-8 b(,)34 b(A.)f(\(1998,)j(Jan)m(uary\).)d
FI(The)i(Machine-Che)-5 b(cke)g(d)36 b(Liter)-5 b(ate)35
b(F)-7 b(ormalisation)39 b(of)c(A)n(lgebr)-5 b(a)610
3831 y(in)28 b(T)-7 b(yp)i(e)28 b(The)-5 b(ory)p FT(.)27
b(Ph.)e(D.)g(thesis,)h(F)-8 b(acult)m(y)26 b(of)f(Science)g(and)g
(Engineering,)f(The)h(Univ)m(ersit)m(y)610 3944 y(of)31
b(Manc)m(hester.)474 4093 y(Barras)21 b(et)g(al.,)i(B.)e(\(1996,)k(No)m
(v)m(em)m(b)s(er\).)e FI(The)h(Co)-5 b(q)24 b(Pr)-5 b(o)g(of)25
b(Assistant)g(Refer)-5 b(enc)g(e)24 b(Manual)p FT(.)d(Pro)5
b(jet)610 4206 y(Co)s(q)30 b(|)g(INRIA-Ro)s(cquencourt,)h(CNRS-ENS)f
(Ly)m(ons.)g(\(V)-8 b(ersion)31 b(6.1\).)474 4355 y(Bec)m(her,)26
b(G.)e(and)e(U.)i(P)m(etermann)g(\(1994,)j(Septem)m(b)s(er\).)c(Rigid)e
(uni\014cation)h(b)m(y)h(completion)f(and)610 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 b(of)e(the)g(11th)610
5330 y(International)k(Confer)-5 b(enc)g(e)44 b(on)f(A)n(utomate)-5
b(d)45 b(De)-5 b(duction)43 b(\(CADE-11\))p FT(,)i(V)-8
b(olume)42 b(607)h(of)610 5443 y FI(LNAI)p FT(,)30 b(Saratoga)i
(Springs,)c(NY,)j(pp.)f(507{521.)k(Springer.)474 5592
y(Bec)m(k)m(ert,)d(B.,)e(R.)e(H\177)-45 b(ahnle,)28 b(and)f(P)-8
b(.)28 b(H.)g(Sc)m(hmitt)f(\(1993,)k(August\).)d(The)f(ev)m(en)h(more)g
(lib)s(eralized)610 5705 y FP(\016)s FT(-rule)20 b(in)f(free)i(v)-5
b(ariable)19 b(seman)m(tic)i(tableaux.)f(In)f(G.)i(Gottlob,)i(A.)e
(Leitsc)m(h,)i(and)c(D.)i(Mundici)2035 5954 y(247)p eop
%%Page: 248 258
248 257 bop 378 5 a FF(BIBLIOGRAPHY)2588 b FT(248)610
396 y(\(Eds.\),)27 b FI(3r)-5 b(d)30 b(Kurt)f(G\177)-46
b(odel)30 b(Col)5 b(lo)-5 b(quium)29 b(\(K)n(GC\))p FT(,)d(LNCS)f(713,)
k(Brno,)d(Czec)m(h)h(Republic,)e(pp.)610 509 y(108{119.)34
b(Springer.)474 660 y(Bec)m(k)m(ert,)i(B.)d(and)f(J.)g(P)m(osegga)j
(\(1995\).)g Fa(lean)p FP(T)2092 627 y(A)2134 660 y(P)13
b FT(:)45 b(Lean)33 b(tableau-based)f(deduction.)g FI(Journal)610
773 y(of)h(A)n(utomate)-5 b(d)34 b(R)-5 b(e)g(asoning)40
b(15)12 b FT(\(3\),)33 b(339{358.)474 923 y(Benzm)s(\177)-48
b(uller)36 b(et)i(al.,)h(C.)f(\(1997,)j(July13{17)e(\).)f(\012MEGA:)g
(T)-8 b(o)m(w)m(ards)38 b(a)g(mathematical)f(assis-)610
1036 y(tan)m(t.)f(In)e(W.)h(McCune)f(\(Ed.\),)i FI(Pr)-5
b(o)g(c)g(e)g(e)g(dings)39 b(of)e(the)f(14th)i(International)h(Confer)
-5 b(enc)g(e)37 b(on)610 1149 y(A)n(utomate)-5 b(d)34
b(de)-5 b(duction)p FT(,)32 b(V)-8 b(olume)30 b(1249)i(of)f
FI(LNAI)p FT(,)f(Berlin,)f(pp.)g(252{255.)34 b(Springer.)474
1299 y(Bib)s(el,)23 b(W.)h(\(1981,)k(Octob)s(er\).)c(On)e(matrices)i
(with)e(connections.)i FI(Journal)j(of)f(the)h(A)n(CM)37
b(28)12 b FT(\(4\),)610 1412 y(633{645.)474 1562 y(Birkho\013,)48
b(G.)e(\(1935\).)i(On)d(the)g(structure)g(of)h(abstract)g(algebras.)g
(In)e FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)49 b(of)d(the)610
1675 y(Cambridge)34 b(Philosophic)-5 b(al)35 b(So)-5
b(ciety)34 b(31\(4\))p FT(,)e(pp.)d(433{454.)474 1825
y(Bittel,)63 b(O.)56 b(\(1992,)65 b(Septem)m(b)s(er\).)57
b(T)-8 b(ableau-based)56 b(theorem)g(pro)m(ving)g(and)f(syn)m(thesis)h
(of)610 1938 y(lam)m(b)s(da-terms)27 b(in)f(the)i(in)m(tuitionistic)d
(logic.)i(In)g(D.)i(P)m(earce)g(and)e(D.)h(W)-8 b(agner)29
b(\(Eds.\),)g FI(Pr)-5 b(o-)610 2051 y(c)g(e)g(e)g(dings)40
b(of)g(the)f(Eur)-5 b(op)g(e)g(an)42 b(Workshop)f(JELIA)d('92)i(on)f(L)
-5 b(o)g(gics)40 b(in)f(AI)p FT(,)f(V)-8 b(olume)37 b(633)i(of)610
2164 y FI(LNAI)p FT(,)30 b(Berlin,)f(FR)m(G,)j(pp.)d(262{278.)34
b(Springer)28 b(V)-8 b(erlag.)474 2314 y(Bj\034rner,)41
b(N.)e(S.,)h(M.)g(E.)e(Stic)m(k)m(el,)j(and)e(T.)f(E.)h(Urib)s(e)e
(\(1997,)43 b(July13{17)d(\).)f(A)g(practical)f(in-)610
2427 y(tegration)d(of)f(\014rst-order)e(reasoning)i(and)f(decision)f
(pro)s(cedures.)h(In)g(W.)h(McCune)f(\(Ed.\),)610 2540
y FI(Pr)-5 b(o)g(c)g(e)g(e)g(dings)27 b(of)e(the)h(14th)g
(International)h(Confer)-5 b(enc)g(e)26 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(\(1992,)28 b(August\).)d(Reasoning)e(with)g
(inductiv)m(ely)f(de\014ned)h(rela-)610 3819 y(tions)k(in)g(the)g(HOL)h
(theorem)g(pro)m(v)m(er.)g(T)-8 b(ec)m(hnical)27 b(Rep)s(ort)h(265,)h
(Univ)m(ersit)m(y)e(of)h(Cam)m(bridge)610 3932 y(Computer)i(Lab)s
(oratory)-8 b(.)474 4082 y(Chang,)28 b(C.)f(C.)h(and)f(H.)h(J.)f
(Keisler)f(\(1990\).)k FI(Mo)-5 b(del)31 b(The)-5 b(ory)38
b FT(\(3rd)27 b(ed.\),)i(V)-8 b(olume)27 b(73)i(of)f
FI(Studies)610 4195 y(in)33 b(L)-5 b(o)g(gic)33 b(and)h(the)f(F)-7
b(oundations)35 b(of)d(Mathematics)p FT(.)g(Amsterdam:)41
b(North-Holland.)474 4345 y(Ch)m(urc)m(h,)35 b(A.)h(\(1936\).)i(An)c
(unsolv)-5 b(able)34 b(problem)f(of)i(elemen)m(tary)h(n)m(um)m(b)s(er)e
(theory)-8 b(.)36 b FI(A)n(meric)-5 b(an)610 4458 y(Journal)34
b(of)f(Mathematics)39 b(58)p FT(,)31 b(345{363.)474 4608
y(Ch)m(urc)m(h,)38 b(A.)f(\(1940\).)i(A)e(form)m(ulation)f(of)h(a)g
(simple)e(theory)i(of)g(t)m(yp)s(es.)g FI(Journal)j(of)f(Symb)-5
b(olic)610 4721 y(L)g(o)g(gic)36 b(5)p FT(,)31 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
(Workshop)g(on)f(The)-5 b(or)g(em)610 622 y(Pr)g(oving)26
b(with)g(A)n(nalytic)g(T)-7 b(able)i(aux)26 b(and)g(R)-5
b(elate)g(d)28 b(Metho)-5 b(ds)p FT(,)25 b(V)-8 b(olume)23
b(918)h(of)e FI(LNAI)p FT(,)g(Berlin,)610 735 y(pp.)30
b(294{309.)k(Springer.)474 885 y(Konrad,)70 b(K.)62 b(\(1998\).)k(Hot:)
106 b(A)62 b(concurren)m(t)h(automated)h(theorem)f(pro)m(v)m(er)g
(based)f(on)610 998 y(higher-order)25 b(tableaux.)h(Seki)f(Rep)s(ort)h
(SR-98-03,)j(F)-8 b(ac)m(h)m(b)s(ereic)m(h)27 b(Informatik,)f(Univ)m
(ersit\177)-45 b(at)610 1111 y(Saarbr)s(\177)d(uc)m(k)m(en.)31
b(accepted)g(for)g(TPHOLs'98.)474 1262 y(Kreisel,)e(G.)i(\(1958\).)i
(Hilb)s(ert's)28 b(programme.)j FI(Diale)-5 b(ctic)g(a)38
b(12)p FT(,)32 b(346{372.)474 1412 y(Laibinis,)22 b(L.)h(\(1996,)k
(August\).)c(Using)g(lattice)g(theory)h(in)e(higher)f(order)i(logic.)g
(See)h(v)m(on)f(W)-8 b(righ)m(t,)610 1525 y(Grundy)g(,)30
b(and)f(Harrison)h(\(1996\),)j(pp.)c(315{330.)474 1675
y(Lak)-5 b(atos,)49 b(I.)43 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