MyWikiBiz

Changes

Differential Logic and Dynamic Systems • Appendices (view source)

Revision as of 22:21, 12 April 2026

128,276 bytes added ,  12 April
add Differential Logic and Dynamic Systems • Appendices
<b>Author: [[User:Jon Awbrey|Jon Awbrey]]</b>

----
<div align="center">
&bull; [[Differential Logic and Dynamic Systems &bull; Overview|Overview]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 1|Part&nbsp;1]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 2|Part&nbsp;2]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 3|Part&nbsp;3]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 4|Part&nbsp;4]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 5|Part&nbsp;5]]
&bull; [[Differential Logic and Dynamic Systems &bull; Appendices|Appendices]]
&bull; [[Differential Logic and Dynamic Systems &bull; References|References]]
&bull; [[Differential Logic and Dynamic Systems &bull; Document History|Document&nbsp;History]]
&bull;
</div>
----

==Appendices==

===Appendix 1. Propositional Forms and Differential Expansions===

====Table A1. Propositional Forms on Two Variables====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
| &nbsp;
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
| &nbsp; || &nbsp; || &nbsp;
|- style="background:ghostwhite"
| &nbsp;
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
| &nbsp; || &nbsp; || &nbsp;
|-
|
<math>\begin{matrix}
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} ~~ \texttt{)}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)} ~ ~ ~ ~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
~ ~ ~ ~ \texttt{(} y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x ~ ~ ~ y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{false}
\\
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
\text{not}~ x
\\
x ~\text{without}~ y
\\
\text{not}~ y
\\
x ~\text{not equal to}~ y
\\
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
0
\\
\lnot x \land \lnot y
\\
\lnot x \land y
\\
\lnot x
\\
x \land \lnot y
\\
\lnot y
\\
x \ne y
\\
\lnot x \lor \lnot y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
\end{matrix}</math>
|
<math>\begin{matrix}
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
x ~ ~ ~ y
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
~ ~ ~ ~ y
\\
~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\
x ~ ~ ~ ~
\\
\texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} ~~ \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{and}~ y
\\
x ~\text{equal to}~ y
\\
y
\\
\text{not}~ x ~\text{without}~ y
\\
x
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\\
\text{true}
\end{matrix}</math>
|
<math>\begin{matrix}
x \land y
\\
x = y
\\
y
\\
x \Rightarrow y
\\
x
\\
x \Leftarrow y
\\
x \lor y
\\
1
\end{matrix}</math>
|}

<br>

====Table A2. Propositional Forms on Two Variables====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
| &nbsp;
| align="right" | <math>x\colon</math>
| <math>1~1~0~0</math>
| &nbsp; || &nbsp; || &nbsp;
|- style="background:ghostwhite"
| &nbsp;
| align="right" | <math>y\colon</math>
| <math>1~0~1~0</math>
| &nbsp; || &nbsp; || &nbsp;
|-
| <math>f_{0}</math>
| <math>f_{0000}</math>
| <math>0~0~0~0</math>
| <math>\texttt{(} ~~ \texttt{)}</math>
| <math>\text{false}</math>
| <math>0</math>
|-
|
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
x ~\text{without}~ y
\\
x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \land \lnot y
\\
\lnot x \land y
\\
x \land \lnot y
\\
x \land y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0011}\\f_{1100}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1\\1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ x
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x
\\
x
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0110}\\f_{1001}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~0\\1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\
x ~\text{equal to}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
x \ne y
\\
x = y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0101}\\f_{1010}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1\\1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ y
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot y
\\
y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x ~ ~ ~ y \texttt{)}
\\
~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\
\text{not}~ x ~\text{without}~ y
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \lor \lnot y
\\
x \Rightarrow y
\\
x \Leftarrow y
\\
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>f_{1111}</math>
| <math>1~1~1~1</math>
| <math>\texttt{((} ~~ \texttt{))}</math>
| <math>\text{true}</math>
| <math>1</math>
|}

<br>

====Table A3. E''f'' Expanded Over Differential Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" | &nbsp;
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
x ~ ~ ~ y
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)} ~~ y ~~
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
x ~ ~ ~ y
\\
~~ 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{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
x
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
x
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\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{))}
\\
\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{))}
\\
\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{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\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{))}
\\
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16</math>
|}

<br>

====Table A4. D''f'' Expanded Over Differential Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" | &nbsp;
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
x
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
x
\\
x
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

<br>

====Table A5. E''f'' Expanded Over Ordinary Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" | &nbsp;
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{xy}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\mathrm{d}x
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ \mathrm{d}x ~ ~ ~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{(} ~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~ \texttt{)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
|}

<br>

====Table A6. D''f'' Expanded Over Ordinary Features====

<br>

{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" | &nbsp;
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>f</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{xy}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,} ~~ \mathrm{d}y \texttt{)}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\mathrm{d}x ~ ~ ~ \mathrm{d}y
\\
~~ \mathrm{d}x ~~ \texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)} ~~ \mathrm{d}y ~~
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0</math>
|}

<br>

===Appendix 2. Differential Forms===

The actions of the difference operator <math>\mathrm{D}</math> and the tangent operator <math>\mathrm{d}</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.

Table A7 expands the differential forms that result over a ''logical basis'':

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.</math>
|}

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

{| align="center" cellpadding="6" style="text-align:center"
|
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.</math>
|}

Table A8 expands the differential forms that result over an ''algebraic basis'':

{| align="center" cellpadding="6" style="text-align:center"
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.</math>
|}

This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.

====Table A7. Differential Forms Expanded on a Logical Basis====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}</math>
|- style="background:ghostwhite; height:40px"
| &nbsp;
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" | <math>\mathrm{D}f</math>
| <math>\mathrm{d}f</math>
|-
| <math>f_{0}</math>
| style="border-right:none" | <math>\texttt{(} ~~ \texttt{)}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
~~ y ~~ ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
~~ x ~~ ~\partial y
\\
~~ y ~~ ~\partial x
& + &
~~ x ~~ ~\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial x
\\
\partial x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial x & + & \partial y
\\
\partial x & + & \partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial y
\\
\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,} ~~ y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,} ~~ y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ ~\partial x
& + &
~~ x ~~ ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
~~ x ~~ ~\partial y
\\
~~ y ~~ ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| style="border-right:none" | <math>\texttt{((} ~~ \texttt{))}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|}

<br>

====Table A8. Differential Forms Expanded on an Algebraic Basis====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}</math>
|- style="background:ghostwhite; height:40px"
| &nbsp;
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" | <math>\mathrm{D}f</math>
| <math>\mathrm{d}f</math>
|-
| <math>f_{0}</math>
| style="border-right:none" | <math>\texttt{(} ~~ \texttt{)}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
| <math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y & + & \mathrm{d}x ~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & ~~ x ~~ ~\mathrm{d}y
\\
~~ y ~~ ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\\
\texttt{(} y \texttt{)} ~\mathrm{d}x & + & \texttt{(} x \texttt{)} ~\mathrm{d}y
\end{matrix}</math>
|-
| <math>f_{15}</math>
| style="border-right:none" | <math>\texttt{((} ~~ \texttt{))}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
|}

<br>

====Table A9. Tangent Proposition as Pointwise Linear Approximation====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}^2\!f =
\\[2pt]
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A10. Taylor Series Expansion D''f'' = d''f'' + d²''f''====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{D}f
\\
= & \mathrm{d}f & + & \mathrm{d}^2\!f
\\
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 0 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 0 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 1 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ 0 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ 0 ~~ \cdot \mathrm{d}x & + &
~~ 1 ~~ \cdot \mathrm{d}y & + &
~~ 0 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
~~ x ~~ \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
~~ 1 ~~ \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}</math>
| style="border-left:4px double black" | <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A11. Partial Differentials and Relative Differentials====

<br>

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}</math>
|- style="background:ghostwhite; height:50px"
| &nbsp;
| <math>f</math>
| <math>\frac{\partial f}{\partial x}</math>
| <math>\frac{\partial f}{\partial y}</math>
|
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\left. \frac{\partial x}{\partial y} \right| f</math>
| <math>\left. \frac{\partial y}{\partial x} \right| f</math>
|-
| <math>f_0</math>
| <math>\texttt{(} ~ \texttt{)}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)} ~~ y ~~
\\
~~ x ~~ \texttt{(} y \texttt{)}
\\
x ~ ~ ~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
x
\\
x
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{,} ~~ y \texttt{)}
\\
\texttt{((} x \texttt{,} ~~ y \texttt{))}
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} ~~ x ~ ~ ~ y ~~ \texttt{)}
\\
\texttt{(} ~~ x ~~ \texttt{(} y \texttt{))}
\\
\texttt{((} x \texttt{)} ~~ y ~~ \texttt{)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
x
\\
x
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
~~ y ~~ \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & ~~ x ~~ \cdot \mathrm{d}y
\\
~~ y ~~ \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>f_{15}</math>
| <math>\texttt{((} ~ \texttt{))}</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
| <math>0</math>
|}

<br>

====Table A12. Detail of Calculation for the Difference Map====

<br>

{| 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%"
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}</math>
|- style="background:ghostwhite"
| style="width:6%" | &nbsp;
| style="width:14%; border-left:1px solid black" | <math>f</math>
| 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}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[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}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
= & ~~ \texttt{(} x \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 ~~</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & ~~ \texttt{(} x \texttt{,} ~~ y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\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 ~ ~ ~
\\[4pt]
= & ~~ \texttt{(} x \texttt{)} ~ ~ ~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[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>~~ x ~~ \texttt{(} y \texttt{)}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)} ~~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ 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]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~ ~ ~ \texttt{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ 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>x ~ ~ ~ y</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x \texttt{)(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[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 ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~~ \texttt{(} y \texttt{)} ~~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[4pt]
+ & ~ ~ ~ x ~ ~ ~ y ~ ~ ~
\\[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 ~ ~ ~ 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 ~ ~ ~ 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 ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\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 ~ ~ ~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~ ~ ~ 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{(} y \texttt{)} ~~
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & ~~ \texttt{(} x ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & ~~ \texttt{(} x ~~ \texttt{(} y \texttt{))}
\\[4pt]
= & ~ ~ ~ x ~ ~ ~ ~~ ~ ~ ~
\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 ~ ~ ~ y \texttt{)} ~~
\\[4pt]
+ & \texttt{((} x \texttt{)} ~~ y \texttt{)} ~~
\\[4pt]
= & ~ ~ ~ ~~ ~ ~ ~ y ~ ~ ~
\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{)} ~ ~ ~ ~ ~ ~
\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 ~ ~ ~ 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{(} 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{)} ~ ~ ~ ~ ~ ~
\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 3. Computational Details===

====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>&#10216;''u'', ''v''&#10217;====

=====Computation of &epsilon;''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{8}
& = && f_{8} \langle u, v \rangle
\\[4pt]
& = && u v
\\[4pt]
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & u v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u v \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~~
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{E}f_{8}
& = & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
& = & u v \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{8} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{8} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&&& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
\\[4pt]
&&&&& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{c}}
\mathrm{E}f_{8}
& = & \{ u + \mathrm{d}u \} \cdot \{ v + \mathrm{d}v \}
\\[6pt]
& = & u \cdot v
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{8}
& = && \mathrm{E}f_{8}
& + & \boldsymbol\varepsilon f_{8}
\\[4pt]
& = && f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{8} \langle u, v \rangle
\\[4pt]
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u v
\\[20pt]
\mathrm{D}f_{8}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~ ~ ~
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{} ~~
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v ~ ~ ~
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~
\\[20pt]
\mathrm{D}f_{8}
& = && u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v ~~
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v ~~
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8}
& + & \mathrm{E}f_{8}
\\[6pt]
& = & f_{8} \langle u, v \rangle
& + & f_{8} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[6pt]
& = & u v
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & 0
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~~ u ~~ v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ ~ ~ u ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u ~~ \mathrm{d}v
\\[6pt]
\mathrm{E}f_{8}
& = & ~~ u ~~ v ~~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v ~~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ \texttt{(} u \texttt{)(} v \texttt{)} ~~ \cdot \mathrm{d}u ~~ \mathrm{d}v
\\[20pt]
\mathrm{D}f_{8}
& = & ~ ~ ~ 0 ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ ~ ~ u ~ ~ ~ \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~ ~ ~ v ~ ~ ~ \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} ~~ v \texttt{))} \cdot \mathrm{d}u ~~ \mathrm{d}v
\end{array}</math>
|}

=====Computation of d''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>8</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}
\\[20pt]
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[20pt]
\mathrm{r}f_{8}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Conjunction=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8} \langle u, v \rangle = u v</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{8}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{r}f_{8}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Equality ''f''<sub>9</sub>&#10216;''u'', ''v''&#10217;====

=====Computation of &epsilon;''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{9}
& = && f_{9} \langle u, v \rangle
\\[4pt]
& = && \texttt{((} u \texttt{,} ~~ v \texttt{))}
\\[4pt]
& = && u v \cdot f_{9} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle 0, 0 \rangle
\\[4pt]
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{9}
& = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
\\[4pt]
& = && u v \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{9} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{9} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = && u v \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} ~ \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{9}
& = && \mathrm{E}f_{9}
& + & \boldsymbol\varepsilon f_{9}
\\[4pt]
& = && f_{9} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{9} \langle u, v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{,} v \texttt{))}
\\[20pt]
\mathrm{D}f_{9}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & 0
& + & 0
& + & 0
\\[20pt]
\mathrm{D}f_{9}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} ~~ \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{9}
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>9</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}
\\[20pt]
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{9}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

=====Computation Summary for Equality=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9} \langle u, v \rangle = \texttt{((} u \texttt{,} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{9}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{9}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{9}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{r}f_{9}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Implication ''f''<sub>11</sub>&#10216;''u'', ''v''&#10217;====

=====Computation of &epsilon;''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{11}
& = && f_{11} \langle u, v \rangle
\\[4pt]
& = && \texttt{(} u \texttt{(} v \texttt{))}
\\[4pt]
& = && u v \cdot f_{11} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{11} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{11} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11} \langle 0, 0 \rangle
\\[4pt]
& = && u v
& + & 0
& + & \texttt{(} u \texttt{)} v
& + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{11}
& = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = &&
\texttt{(}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = &&
u v
\cdot
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}
& + &
u \texttt{(} v \texttt{)}
\cdot
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[4pt]
& = &&
u v
\cdot
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + &
u \texttt{(} v \texttt{)}
\cdot
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{11}
& = && \mathrm{E}f_{11}
& + & \boldsymbol\varepsilon f_{11}
\\[4pt]
& = && f_{11} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{11} \langle u, v \rangle
\\[4pt]
& = &&
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\texttt{))}
& + &
\texttt{(} u \texttt{(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v
& + & 0
\\[20pt]
\mathrm{D}f_{11}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{11}
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = &
u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + &
u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = &
u v \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + &
u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \mathrm{d}v
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>11</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
\\[20pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[20pt]
\mathrm{r}f_{11}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Implication=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11} \langle u, v \rangle = \texttt{(} u \texttt{(} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[6pt]
\mathrm{r}f_{11}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>&#10216;''u'', ''v''&#10217;====

=====Computation of &epsilon;''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{14}
& = && f_{14} \langle u, v \rangle
\\[4pt]
& = && \texttt{((} u \texttt{)(} v \texttt{))}
\\[4pt]
& = && u v \cdot f_{14} \langle 1, 1 \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle 1, 0 \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle 0, 1 \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle 0, 0 \rangle
\\[4pt]
& = && u v
& + & u \texttt{(} v \texttt{)}
& + & \texttt{(} u \texttt{)} v
& + & 0
\\[20pt]
\boldsymbol\varepsilon f_{14}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & 0
\end{array}</math>
|}

<br>

=====Computation of E''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{14}
& = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
\\[4pt]
& = &&
\texttt{((}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{)(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = && u v \cdot
f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & u \texttt{(} v \texttt{)} \cdot f_{14} \langle \texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v \rangle
& + & \texttt{(} u \texttt{)} v \cdot f_{14} \langle \mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)} \rangle
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14} \langle \mathrm{d}u, \mathrm{d}v \rangle
\\[4pt]
& = && u v \cdot \texttt{(} \mathrm{d}u ~~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{14}
& = && u v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & u v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation of D''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{14}
& = && \mathrm{E}f_{14}
& + & \boldsymbol\varepsilon f_{14}
\\[4pt]
& = && f_{14} \langle u + \mathrm{d}u, v + \mathrm{d}v \rangle
& + & f_{14} \langle u, v \rangle
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{)(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{14}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & 0
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~~ \texttt{(} \mathrm{d}u \texttt{)} ~~ \mathrm{d}v
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot ~ ~ ~ \mathrm{d}u ~ ~ ~ \mathrm{d}v
\\[20pt]
\mathrm{D}f_{14}
& = && u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v
\texttt{))}
\end{array}</math>
|}

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{14}
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of d''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{array}</math>
|}

<br>

=====Computation of r''f''<sub>14</sub>=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}
\\[20pt]
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

=====Computation Summary for Disjunction=====

<br>

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14} \langle u, v \rangle = \texttt{((} u \texttt{)(} v \texttt{))}</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{14}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 1
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{14}
& = & u v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{d}f_{14}
& = & u v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{r}f_{14}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}

<br>

===Appendix 4. Source Materials===

<br>

===Appendix 5. Various Definitions of the Tangent Vector===

<br>

----
<div align="center">
&bull; [[Differential Logic and Dynamic Systems &bull; Overview|Overview]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 1|Part&nbsp;1]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 2|Part&nbsp;2]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 3|Part&nbsp;3]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 4|Part&nbsp;4]]
&bull; [[Differential Logic and Dynamic Systems &bull; Part 5|Part&nbsp;5]]
&bull; [[Differential Logic and Dynamic Systems &bull; Appendices|Appendices]]
&bull; [[Differential Logic and Dynamic Systems &bull; References|References]]
&bull; [[Differential Logic and Dynamic Systems &bull; Document History|Document&nbsp;History]]
&bull;
</div>
----

[[Category:Adaptive systems]]
[[Category:Artificial intelligence]]
[[Category:Boolean algebra]]
[[Category:Boolean functions]]
[[Category:Category theory]]
[[Category:Combinatorics]]
[[Category:Computation theory]]
[[Category:Cybernetics]]
[[Category:Differential logic]]
[[Category:Discrete systems]]
[[Category:Dynamical systems]]
[[Category:Formal languages]]
[[Category:Formal sciences]]
[[Category:Formal systems]]
[[Category:Functional logic]]
[[Category:Graph theory]]
[[Category:Group theory]]
[[Category:Logic]]
[[Category:Logical graphs]]
[[Category:Neural networks]]
[[Category:Peirce, Charles Sanders]]
[[Category:Semiotics]]
[[Category:Systems theory]]
[[Category:Visualization]]
Jon Awbrey
12,224

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/481308"