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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 18:40, 14 September 2013

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===Appendix B===
 
===Appendix B===
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<br>
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
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|+ style="height:30px" | <math>\text{Table B1.} ~~ \text{Detail of Calculation for}~ {\mathrm{D}f = \mathrm{E}f + f}\!</math>
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|- style="background:ghostwhite"
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| style="width:6%" | &nbsp;
 +
| style="width:14%; border-left:1px solid black"  | <math>f\!</math>
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| style="width:20%; border-left:4px double black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\\[4pt]
 +
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\\[4pt]
 +
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\end{array}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{0}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{1}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{2}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{4}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{8}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{3}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} x \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{12}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>x\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{6}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{9}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{5}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} y \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{10}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>y\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{7}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{11}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{13}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{14}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{15}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
|}
 +
 +
<br>
    
===Appendix C===
 
===Appendix C===
Jon Awbrey
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Retrieved from "https://mywikibiz.com/Special:MobileDiff/214690"

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