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cleartomark
%%EndFont
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 82 /R put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 44 /comma put
dup 48 /zero put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 67 /C put
dup 70 /F put
dup 76 /L put
dup 83 /S put
dup 97 /a put
dup 99 /c put
dup 101 /e put
dup 108 /l put
dup 114 /r put
dup 116 /t put
dup 117 /u put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 88 /summationdisplay put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 116 /t put
dup 117 /u put
dup 119 /w put
dup 121 /y put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
currentdict end
currentfile eexec
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%%BeginFont: CMR8
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
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cleartomark
%%EndFont
%%BeginFont: CMMI8
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%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
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end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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dup 61 /slash put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 110 /n put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 41 /parenright put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 58 /colon put
dup 66 /B put
dup 67 /C put
dup 72 /H put
dup 73 /I put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 87 /W put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 121 /y put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 62 /greater put
dup 70 /F put
dup 80 /P put
dup 81 /Q put
dup 83 /S put
dup 97 /a put
dup 98 /b put
dup 102 /f put
dup 105 /i put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 113 /q put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 18 /reflexsubset put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 36 /arrowboth put
dup 41 /arrowdblright put
dup 50 /element put
dup 54 /negationslash put
dup 56 /universal put
dup 57 /existential put
dup 58 /logicalnot put
dup 59 /emptyset put
dup 94 /logicaland put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 33 /exclam put
dup 34 /quotedblright put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
dup 89 /Y put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 124 /emdash put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMDUNH10
%!PS-AdobeFont-1.1: CMDUNH10 1.0
%%CreationDate: 1991 Aug 20 16:37:03
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMDUNH10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMDUNH10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 38 /ampersand put
dup 46 /period put
dup 48 /zero put
dup 50 /two put
dup 51 /three put
dup 54 /six put
dup 55 /seven put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 86 /V put
dup 87 /W put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-40 -528 1010 1028}readonly def
currentdict end
currentfile eexec
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%begin_xml_code
/pdfmark where {pop true} {false} ifelse
/currentdistillerparams where {pop currentdistillerparams
/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:d8888400-2701-11db-8e40-c92e72bd77fb
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
{pop statusdict /processcolors known
{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
/_image systemdict /image get def
/_setgray systemdict /setgray get def
/_currentgray systemdict /currentgray get def
/_settransfer systemdict /settransfer get def
/_currenttransfer systemdict /currenttransfer get def
/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
{0 _currenttransfer exec 0.5 lt}
{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
/inverted? negative def
/level2 systemdict /languagelevel known
{languagelevel 2 ge} {false} ifelse def
/level3 systemdict /languagelevel known
{languagelevel 3 ge} {false} ifelse def
/foureq {4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
/setcmykcolor where {pop
1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
0 1 0 0 foureq {/band 2 store} if
0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 67 def
/cols 171 def
171 67 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
/picstr1 171 string def
/picstr2 171 string def
/picstr3 171 string def
/picstr4 171 string def
/_rowpadstr 171 string def
/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
true 3 colorimage} if
Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
/DataSource load _image end} def} ifelse
/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
%%BeginBinary: 45840
beginimage
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@endspecial 218 x Fj(Theorem:)87 b Fl(8)p Fk(k)28 b
Fl(2)d Fj(N)p Fm(,)1071 5066 y Fi(k)1024 5093 y Ff(X)1033
5288 y Fi(i)p Fh(=0)1171 5179 y Fk(i)g Fm(=)1333 5118
y Fk(k)s Fm(\()p Fk(k)f Fm(+)c(1\))p 1333 5158 328 4
v 1474 5242 a(2)1671 5179 y(.)150 5415 y Fj(Pro)s(of)41
b Fm(\(b)m(y)31 b(induction)f(on)g Fk(k)s Fm(\):)287
5692 y Fe(\210)45 b Fm(Base)35 b(Case:)46 b Fk(P)13 b
Fm(\(0\))35 b(asserts:)1460 5578 y Fh(0)1412 5606 y Ff(X)1420
5801 y Fi(i)p Fh(=0)1558 5692 y Fk(i)30 b Fm(=)1730 5630
y(0\(0)22 b(+)e(1\))p 1730 5671 319 4 v 1867 5754 a(2)2058
5692 y(.)50 b(This)32 b(clearly)j(holds,)e(since)h(the)g(left)g(and)e
(righ)m(t)i(hand)377 5882 y(sides)c(b)s(oth)g(equal)h(0.)2077
6131 y(1)p eop end
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TeXDict begin 2 1 bop 287 650 a Fe(\210)45 b Fm(Inductiv)m(e)31
b(Hyp)s(othesis:)40 b(Assume)30 b Fk(P)13 b Fm(\()p Fk(k)s
Fm(\))31 b(is)g(true.)40 b(That)31 b(is,)2551 537 y Fi(k)2504
564 y Ff(X)2513 759 y Fi(i)p Fh(=0)2651 650 y Fk(i)25
b Fm(=)2813 589 y Fk(k)s Fm(\()p Fk(k)f Fm(+)c(1\))p
2813 629 328 4 v 2954 713 a(2)3151 650 y(.)287 1020 y
Fe(\210)45 b Fm(Inductiv)m(e)31 b(Step:)40 b(W)-8 b(e)32
b(m)m(ust)e(sho)m(w)g Fk(P)13 b Fm(\()p Fk(k)24 b Fm(+)c(1\).)41
b(That)31 b(is,)2388 907 y Fi(k)r Fh(+1)2386 934 y Ff(X)2395
1129 y Fi(i)p Fh(=0)2533 1020 y Fk(i)25 b Fm(=)2695 959
y(\()p Fk(k)f Fm(+)c(1\)\()p Fk(k)k Fm(+)c(2\))p 2695
999 555 4 v 2950 1083 a(2)3260 1020 y(:)606 1293 y Fi(k)r
Fh(+1)605 1321 y Ff(X)613 1516 y Fi(i)p Fh(=0)751 1407
y Fk(i)26 b Fm(=)f(\()985 1293 y Fi(k)939 1321 y Ff(X)948
1516 y Fi(i)p Fh(=0)1085 1407 y Fk(i)p Fm(\))c(+)f(\()p
Fk(k)k Fm(+)c(1\))808 1688 y(=)914 1627 y Fk(k)s Fm(\()p
Fk(k)j Fm(+)d(1\))p 914 1667 328 4 v 1055 1750 a(2)1271
1688 y(+)g(\()p Fk(k)k Fm(+)c(1\))301 b(\(b)m(y)30 b(the)h(inductiv)m
(e)g(h)m(yp)s(othesis\))808 1908 y(=)25 b(\()p Fk(k)e
Fm(+)d(1\)\()1225 1847 y Fk(k)p 1225 1887 51 4 v 1227
1971 a Fm(2)1307 1908 y(+)g(1\))808 2134 y(=)914 2072
y(\()p Fk(k)j Fm(+)d(1\)\()p Fk(k)25 b Fm(+)20 b(2\))p
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%EndPhotoshop
%begin_xml_code
/pdfmark where {pop true} {false} ifelse
/currentdistillerparams where {pop currentdistillerparams
/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:ec7fc453-2743-11db-98bd-c2819b70781c
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
{pop statusdict /processcolors known
{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
/_image systemdict /image get def
/_setgray systemdict /setgray get def
/_currentgray systemdict /currentgray get def
/_settransfer systemdict /settransfer get def
/_currenttransfer systemdict /currenttransfer get def
/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
{0 _currenttransfer exec 0.5 lt}
{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
/inverted? negative def
/level2 systemdict /languagelevel known
{languagelevel 2 ge} {false} ifelse def
/level3 systemdict /languagelevel known
{languagelevel 3 ge} {false} ifelse def
/foureq {4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
/setcmykcolor where {pop
1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
0 1 0 0 foureq {/band 2 store} if
0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 140 def
/cols 157 def
157 140 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
/picstr1 157 string def
/picstr2 157 string def
/picstr3 157 string def
/picstr4 157 string def
/_rowpadstr 157 string def
/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
true 3 colorimage} if
Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
/DataSource load _image end} def} ifelse
/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
%%BeginBinary: 87932
beginimage
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%begin_xml_code
/pdfmark where {pop true} {false} ifelse
/currentdistillerparams where {pop currentdistillerparams
/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:db7a7b95-2e39-11db-a5a3-c9f5ce905a2d
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
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{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
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/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
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{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
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{languagelevel 3 ge} {false} ifelse def
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4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
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1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
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0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 114 def
/cols 134 def
134 114 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
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/picstr2 134 string def
/picstr3 134 string def
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/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
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Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
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/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
%%BeginBinary: 61116
beginimage
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@endspecial 151 x Fm(Wh)m(y)36 b(do)s(es)e(this)h(w)m(ork?)55
b(W)-8 b(e)36 b(can)f(sa)m(y)h(with)f(certain)m(t)m(y)i(that)e(regions)
h(whic)m(h)f(do)g(not)g(b)s(order)e(the)j(line)377 1416
y(are)c(prop)s(erly)e(t)m(w)m(o-colored.)46 b(But)32
b(what)f(ab)s(out)g(regions)g(that)h(do)f(share)g(a)h(b)s(order)e(with)
h(the)g(line?)44 b(W)-8 b(e)377 1528 y(m)m(ust)31 b(b)s(e)g(certain)h
(that)g(an)m(y)g(t)m(w)m(o)h(suc)m(h)e(regions)h(ha)m(v)m(e)g(opp)s
(osite)g(coloring.)45 b(But)31 b(an)m(y)h(t)m(w)m(o)h(regions)e(that)
377 1641 y(b)s(order)h(the)i(line)f(m)m(ust)g(ha)m(v)m(e)i(b)s(een)d
(the)i(same)g(region)g(when)e(the)h(line)h(w)m(as)g(remo)m(v)m(ed,)h
(so)e(the)h(rev)m(ersal)377 1754 y(of)d(color)g(on)g(one)f(side)h(of)f
(the)h(line)f(guaran)m(tees)i(an)e(opp)s(osite)h(coloring.)150
1983 y(No)m(w)24 b(that)h(w)m(e)f(kno)m(w)f(ho)m(w)h(to)h(pro)m(v)m(e)f
(prop)s(ositions)f(b)m(y)h(induction,)h(let)f(us)f(try)h(to)g(pro)m(v)m
(e)h(the)f(follo)m(wing)h(amazing)150 2096 y(fact.)150
2251 y Fj(Theorem:)87 b Fm(All)31 b(horses)f(are)h(the)f(same)h(color.)
150 2405 y Fj(Pro)s(of)g Fm(\(b)m(y)g(induction)f(on)g(the)h(n)m(um)m
(b)s(er)e(of)h(horses\):)287 2634 y Fe(\210)45 b Fm(Base)32
b(Case:)41 b Fk(P)13 b Fm(\(1\))31 b(is)g(certainly)g(true,)g(since)f
(with)g(just)g(one)h(horse,)f(all)h(horses)f(ha)m(v)m(e)i(the)f(same)f
(color.)287 2822 y Fe(\210)45 b Fm(Inductiv)m(e)31 b(Hyp)s(othesis:)41
b(Assume)30 b Fk(P)13 b Fm(\()p Fk(n)p Fm(\),)31 b(whic)m(h)f(is)h(the)
g(statemen)m(t)h(that)f Fk(n)f Fm(horses)h(all)g(ha)m(v)m(e)h(the)f
(same)377 2935 y(color.)287 3122 y Fe(\210)45 b Fm(Inductiv)m(e)27
b(Step:)38 b(Giv)m(en)27 b(a)f(set)h(of)f Fk(n)12 b Fm(+)g(1)25
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b(It)29 b(follo)m(ws)377 3461 y(that)g(all)g Fk(n)15
b Fm(+)h(1)28 b(horses)g(ha)m(v)m(e)h(the)g(same)f(color.)41
b(Th)m(us,)28 b(b)m(y)g(the)g(principle)g(of)g(induction,)h(all)g
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g(a)g(missing)f(1)18 b Fl(\002)e Fm(1)29 b(square\),)h(and)150
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f(2)1493 5073 y Fi(n)1556 5106 y Fl(\002)15 b Fm(2)1687
5073 y Fi(n)1762 5106 y Fm(court)m(y)m(ard)29 b(with)f(a)g(missing)g(1)
15 b Fl(\002)g Fm(1)29 b(square)e(in)h(the)g(middle.)39
b(Here)150 5219 y(is)30 b(an)h(example)g(of)f(a)h(successful)f(trial)h
(in)f(the)h(case)g(that)g Fk(n)25 b Fm(=)g(2:)150 5869
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/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:52eca9e2-2e3c-11db-987d-fec23c49fd2c
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
{pop statusdict /processcolors known
{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
/_image systemdict /image get def
/_setgray systemdict /setgray get def
/_currentgray systemdict /currentgray get def
/_settransfer systemdict /settransfer get def
/_currenttransfer systemdict /currenttransfer get def
/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
{0 _currenttransfer exec 0.5 lt}
{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
/inverted? negative def
/level2 systemdict /languagelevel known
{languagelevel 2 ge} {false} ifelse def
/level3 systemdict /languagelevel known
{languagelevel 3 ge} {false} ifelse def
/foureq {4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
/setcmykcolor where {pop
1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
0 1 0 0 foureq {/band 2 store} if
0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 66 def
/cols 66 def
66 66 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
/picstr1 66 string def
/picstr2 66 string def
/picstr3 66 string def
/picstr4 66 string def
/_rowpadstr 66 string def
/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
true 3 colorimage} if
Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
/DataSource load _image end} def} ifelse
/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
%%BeginBinary: 17436
beginimage
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@endspecial 287 1545 a Fe(\210)45 b Fm(Inductiv)m(e)32
b(Hyp)s(othesis:)42 b(Assume)31 b Fk(P)13 b Fm(\()p Fk(n)p
Fm(\))31 b(is)h(true.)43 b(That)31 b(is,)h(w)m(e)g(can)f(tile)i(a)e(2)
3100 1512 y Fi(n)3168 1545 y Fl(\002)21 b Fm(2)3305 1512
y Fi(n)3384 1545 y Fm(court)m(y)m(ard)32 b(with)f(a)377
1658 y(missing)g(1)20 b Fl(\002)g Fm(1)31 b(square)f(in)g(the)h
(middle.)287 1846 y Fe(\210)45 b Fm(Inductiv)m(e)36 b(Step:)51
b(W)-8 b(e)36 b(w)m(an)m(t)g(to)h(sho)m(w)e(that)h(w)m(e)g(can)g(tile)g
(a)g(2)2572 1813 y Fi(n)p Fh(+1)2734 1846 y Fl(\002)23
b Fm(2)2873 1813 y Fi(n)p Fh(+1)3046 1846 y Fm(court)m(y)m(ard)36
b(with)f(a)h(missing)377 1958 y(1)23 b Fl(\002)f Fm(1)35
b(square)e(in)h(the)g(middle.)51 b(Let's)34 b(try)g(to)g(reduce)g(this)
g(problem)f(so)h(w)m(e)g(can)h(apply)e(our)g(inductiv)m(e)377
2071 y(h)m(yp)s(othesis.)54 b(A)34 b(2)1021 2038 y Fi(n)p
Fh(+1)1182 2071 y Fl(\002)23 b Fm(2)1321 2038 y Fi(n)p
Fh(+1)1493 2071 y Fm(court)m(y)m(ard)36 b(can)f(b)s(e)f(brok)m(en)g(up)
g(in)m(to)i(four)e(smaller)h(court)m(y)m(ards)g(of)g(size)377
2184 y(2)422 2151 y Fi(n)490 2184 y Fl(\002)20 b Fm(2)626
2151 y Fi(n)673 2184 y Fm(,)31 b(eac)m(h)g(with)f(a)h(missing)f(1)21
b Fl(\002)f Fm(1)31 b(square)f(as)g(follo)m(ws:)377 2889
y @beginspecial -30 @llx 0 @lly 0 @urx 110 @ury 210 @rwi
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y Fl(\002)23 b Fm(2)2422 3007 y Fi(n)2504 3040 y Fm(court)m(y)m(ard,)37
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3833 y(trade-o\013:)44 b(w)m(e)32 b(ha)m(v)m(e)g(to)g(pro)m(v)m(e)h
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(othesis)f(and)g(step.)287 4175 y Fe(\210)45 b Fm(Inductiv)m(e)34
b(Hyp)s(othesis)f(\(second)g(attempt\):)48 b(Assume)33
b Fk(P)13 b Fm(\()p Fk(n)p Fm(\))33 b(is)g(true,)h(whic)m(h)f(states)i
(that)f(w)m(e)f(can)h(tile)377 4288 y(a)d(2)498 4255
y Fi(n)566 4288 y Fl(\002)20 b Fm(2)702 4255 y Fi(n)779
4288 y Fm(court)m(y)m(ard)31 b(with)f(a)h(missing)f(1)21
b Fl(\002)f Fm(1)31 b(square)f(an)m(ywhere.)287 4476
y Fe(\210)45 b Fm(Inductiv)m(e)34 b(Step)f(\(second)g(attempt\):)48
b(As)33 b(b)s(efore,)h(w)m(e)g(can)g(break)f(up)f(the)h(2)3085
4443 y Fi(n)p Fh(+1)3245 4476 y Fl(\002)22 b Fm(2)3383
4443 y Fi(n)p Fh(+1)3554 4476 y Fm(court)m(y)m(ard)34
b(as)377 4589 y(follo)m(ws.)377 5276 y @beginspecial
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f(get)h(four)e(2)1939 5393 y Fi(n)1993 5426 y Fl(\002)7
b Fm(2)2116 5393 y Fi(n)2187 5426 y Fm(court)m(y)m(ards,)26
b(eac)m(h)e(with)g(a)g(1)7 b Fl(\002)g Fm(1)24 b(hole.)39
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%EndPhotoshop
%begin_xml_code
/pdfmark where {pop true} {false} ifelse
/currentdistillerparams where {pop currentdistillerparams
/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:a2a03d9b-2828-11db-9cf9-c15b669b20e2
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
{pop statusdict /processcolors known
{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
/_image systemdict /image get def
/_setgray systemdict /setgray get def
/_currentgray systemdict /currentgray get def
/_settransfer systemdict /settransfer get def
/_currenttransfer systemdict /currenttransfer get def
/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
{0 _currenttransfer exec 0.5 lt}
{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
/inverted? negative def
/level2 systemdict /languagelevel known
{languagelevel 2 ge} {false} ifelse def
/level3 systemdict /languagelevel known
{languagelevel 3 ge} {false} ifelse def
/foureq {4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
/setcmykcolor where {pop
1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
0 1 0 0 foureq {/band 2 store} if
0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 91 def
/cols 86 def
86 91 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
/picstr1 86 string def
/picstr2 86 string def
/picstr3 86 string def
/picstr4 86 string def
/_rowpadstr 86 string def
/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
true 3 colorimage} if
Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
/DataSource load _image end} def} ifelse
/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
%%BeginBinary: 31316
beginimage
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@endspecial 154 x Fm(Let)34 b(us)f(lo)s(ok)h(at)g(the)f(game)i(b)s(et)
m(w)m(een)f Fk(p)1550 1204 y Fh(1)1622 1190 y Fm(v)m(erus)f
Fk(p)1909 1204 y Fh(3)1948 1190 y Fm(.)50 b(W)-8 b(e)35
b(ha)m(v)m(e)f(t)m(w)m(o)h(cases:)48 b(either)34 b Fk(p)3147
1204 y Fh(3)3219 1190 y Fm(b)s(eats)g Fk(p)3509 1204
y Fh(1)3548 1190 y Fm(,)g(or)f Fk(p)3767 1204 y Fh(1)3840
1190 y Fm(b)s(eats)150 1303 y Fk(p)196 1317 y Fh(3)235
1303 y Fm(.)41 b(In)30 b(the)h(\014rst)f(case)h(where)g
Fk(p)1262 1317 y Fh(3)1331 1303 y Fm(b)s(eats)g Fk(p)1618
1317 y Fh(1)1657 1303 y Fm(,)g(then)f(w)m(e)h(are)g(done)f(b)s(ecause)h
(w)m(e)g(ha)m(v)m(e)h(a)f(3-cycle.)43 b(In)30 b(the)h(second)150
1416 y(case)e(where)e Fk(p)646 1430 y Fh(1)712 1416 y
Fm(b)s(eats)h Fk(p)996 1430 y Fh(3)1035 1416 y Fm(,)g(w)m(e)g(ha)m(v)m
(e)h(a)f(shorter)f(cycle)i(and)e(th)m(us,)h(a)f(con)m(tradiction.)42
b(Therefore,)28 b(if)g(there)f(exists)150 1529 y(a)k(cycle,)h(then)e
(there)g(m)m(ust)h(exist)g(a)g(3-cycle)h(as)f(w)m(ell.)150
2296 y @beginspecial -40 @llx 0 @lly 0 @urx 120 @ury
280 @rwi @setspecial
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%EndPhotoshop
%begin_xml_code
/pdfmark where {pop true} {false} ifelse
/currentdistillerparams where {pop currentdistillerparams
/CoreDistVersion get 5000 ge } {false} ifelse
and not {userdict /pdfmark /cleartomark load put} if
[/NamespacePush pdfmark
[/_objdef {photoshop_metadata_stream} /type /stream /OBJ pdfmark
/MetadataString 4649 string def % exact length of metadata
/TempString 100 string def
/ConsumeMetadata {
currentfile TempString readline pop pop
currentfile MetadataString readstring pop pop
} bind def
ConsumeMetadata
%begin_xml_packet: 4649
adobe:docid:photoshop:51c3b1ad-2e56-11db-a841-c72ea1d71917
%end_xml_packet
[{photoshop_metadata_stream} 2 dict begin /Type /Metadata def /Subtype /XML def currentdict end /PUT pdfmark
[{photoshop_metadata_stream} MetadataString /PUT pdfmark
[/_objdef {nextImage} /NI pdfmark
%end_xml_code
gsave % EPS gsave
/hascolor
/deviceinfo where
{pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}
{false} ifelse}
{/statusdict where
{pop statusdict /processcolors known
{statusdict /processcolors get exec 1 gt}
{false} ifelse}
{false} ifelse}
ifelse
def
40 dict begin
/_image systemdict /image get def
/_setgray systemdict /setgray get def
/_currentgray systemdict /currentgray get def
/_settransfer systemdict /settransfer get def
/_currenttransfer systemdict /currenttransfer get def
/blank 0 _currenttransfer exec
1 _currenttransfer exec eq def
/negative blank
{0 _currenttransfer exec 0.5 lt}
{0 _currenttransfer exec 1 _currenttransfer exec gt}
ifelse def
/inverted? negative def
/level2 systemdict /languagelevel known
{languagelevel 2 ge} {false} ifelse def
/level3 systemdict /languagelevel known
{languagelevel 3 ge} {false} ifelse def
/foureq {4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
4 index eq 8 1 roll
pop pop pop pop and and and} def
hascolor {/band 0 def} {/band 5 def} ifelse
/setcmykcolor where {pop
1 0 0 0 setcmykcolor _currentgray 1 exch sub
0 1 0 0 setcmykcolor _currentgray 1 exch sub
0 0 1 0 setcmykcolor _currentgray 1 exch sub
0 0 0 1 setcmykcolor _currentgray 1 exch sub
4 {4 copy} repeat
1 0 0 0 foureq {/band 1 store} if
0 1 0 0 foureq {/band 2 store} if
0 0 1 0 foureq {/band 3 store} if
0 0 0 1 foureq {/band 4 store} if
0 0 0 0 foureq {/band 6 store} if} if
blank {/band 6 store} if
gsave % Image Header gsave
/rows 111 def
/cols 184 def
184 111 scale
level2 {
band 0 eq {
/DeviceRGB
} {/DeviceGray} ifelse
setcolorspace currentdict /PhotoshopDuotoneColorSpace undef currentdict /PhotoshopDuotoneAltColorSpace undef } if
/picstr1 184 string def
/picstr2 184 string def
/picstr3 184 string def
/picstr4 184 string def
/_rowpadstr 184 string def
/rawreaddata {currentfile exch readstring pop} def
/padreaddata { _topPad 0 gt { /_topPad _topPad 1 sub def pop _rowpadstr }
{ _subImageRows 0 gt { /_subImageRows _subImageRows 1 sub def
dup _leftPad _picsubstr rawreaddata putinterval }
{ pop _rowpadstr } ifelse } ifelse } def
/image2 level2 {/image load def} {{begin
Width Height BitsPerComponent ImageMatrix
Decode length 2 eq
{/DataSource load image} if
Decode length 6 eq
{DataSource 0 get DataSource 1 get DataSource 2 get
true 3 colorimage} if
Decode length 8 eq
{DataSource 0 get DataSource 1 get
DataSource 2 get DataSource 3 get
true 4 colorimage} if
end} def} ifelse
/_image2 level2 {/_image load def} {{begin
Width Height BitsPerComponent ImageMatrix
/DataSource load _image end} def} ifelse
/beginimage {
band 0 eq band 4 eq or band 5 eq or
{image2}
{negative {{pop 0}} {{pop 1}} ifelse
_settransfer _image2} ifelse
} def
/readdata /rawreaddata load bind def
12 dict begin
/ImageType 1 def
/Width cols def
/Height rows def
/ImageMatrix [cols 0 0 rows neg 0 rows] def
/BitsPerComponent 8 def
band 0 eq
{/Decode [0 1 0 1 0 1] def
/MultipleDataSources true def
/DataSource [
{picstr1 readdata}
{picstr2 readdata}
{picstr3 readdata picstr4 readdata pop}
] def}
{/Decode [0 1] def
/DataSource {
picstr1 readdata pop
picstr2 readdata pop
picstr3 readdata pop
picstr4 readdata
} def}
ifelse
currentdict end
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beginimage
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%%Trailer
userdict /end-hook known{end-hook}if
%%EOF