Differential Propositional Calculus • Appendices

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Appendices • References •

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Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables

${\text{Table A1.}}~~{\text{Propositional Forms on Two Variables}}$

${\begin{matrix}{\mathcal {L}}_{1}\\{\text{Decimal}}\\{\text{Index}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{2}\\{\text{Binary}}\\{\text{Index}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{3}\\{\text{Truth}}\\{\text{Table}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{4}\\{\text{Cactus}}\\{\text{Language}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{5}\\{\text{English}}\\{\text{Paraphrase}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{6}\\{\text{Conventional}}\\{\text{Formula}}\end{matrix}}$
	$x\colon$	$1~1~0~0$
	$y\colon$	$1~0~1~0$
${\begin{matrix}f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}\end{matrix}}$	${\begin{matrix}f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}\end{matrix}}$	${\begin{matrix}0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~{\texttt {)}}\\{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\{\texttt {(}}x{\texttt {)}}~~~~~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\~~~~~~{\texttt {(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {(}}x~~~~y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\text{false}}\\{\text{neither}}~x~{\text{nor}}~y\\y~{\text{without}}~x\\{\text{not}}~x\\x~{\text{without}}~y\\{\text{not}}~y\\x~{\text{not equal to}}~y\\{\text{not both}}~x~{\text{and}}~y\end{matrix}}$	${\begin{matrix}0\\\lnot x\land \lnot y\\\lnot x\land y\\\lnot x\\x\land \lnot y\\\lnot y\\x\neq y\\\lnot x\lor \lnot y\end{matrix}}$
${\begin{matrix}f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}\end{matrix}}$	${\begin{matrix}f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}\end{matrix}}$	${\begin{matrix}1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1\end{matrix}}$	${\begin{matrix}x~~~~y\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\~~~~~~y\\~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\x~~~~~~\\{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\{\texttt {((}}~~{\texttt {))}}\end{matrix}}$	${\begin{matrix}x~{\text{and}}~y\\x~{\text{equal to}}~y\\y\\{\text{not}}~x~{\text{without}}~y\\x\\{\text{not}}~y~{\text{without}}~x\\x~{\text{or}}~y\\{\text{true}}\end{matrix}}$	${\begin{matrix}x\land y\\x=y\\y\\x\Rightarrow y\\x\\x\Leftarrow y\\x\lor y\\1\end{matrix}}$

Table A2. Propositional Forms on Two Variables

${\text{Table A2.}}~~{\text{Propositional Forms on Two Variables}}$

${\begin{matrix}{\mathcal {L}}_{1}\\{\text{Decimal}}\\{\text{Index}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{2}\\{\text{Binary}}\\{\text{Index}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{3}\\{\text{Truth}}\\{\text{Table}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{4}\\{\text{Cactus}}\\{\text{Language}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{5}\\{\text{English}}\\{\text{Paraphrase}}\end{matrix}}$	${\begin{matrix}{\mathcal {L}}_{6}\\{\text{Conventional}}\\{\text{Formula}}\end{matrix}}$
	$x\colon$	$1~1~0~0$
	$y\colon$	$1~0~1~0$
$f_{0}$	$f_{0000}$	$0~0~0~0$	${\texttt {(}}~~{\texttt {)}}$	${\text{false}}$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}\end{matrix}}$	${\begin{matrix}0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}{\text{neither}}~x~{\text{nor}}~y\\y~{\text{without}}~x\\x~{\text{without}}~y\\x~{\text{and}}~y\end{matrix}}$	${\begin{matrix}\lnot x\land \lnot y\\\lnot x\land y\\x\land \lnot y\\x\land y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}f_{0011}\\f_{1100}\end{matrix}}$	${\begin{matrix}0~0~1~1\\1~1~0~0\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}{\text{not}}~x\\x\end{matrix}}$	${\begin{matrix}\lnot x\\x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}f_{0110}\\f_{1001}\end{matrix}}$	${\begin{matrix}0~1~1~0\\1~0~0~1\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}x~{\text{not equal to}}~y\\x~{\text{equal to}}~y\end{matrix}}$	${\begin{matrix}x\neq y\\x=y\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}f_{0101}\\f_{1010}\end{matrix}}$	${\begin{matrix}0~1~0~1\\1~0~1~0\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}{\text{not}}~y\\y\end{matrix}}$	${\begin{matrix}\lnot y\\y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}\end{matrix}}$	${\begin{matrix}0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0\end{matrix}}$	${\begin{matrix}{\texttt {(}}x~~~~y{\texttt {)}}\\~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\text{not both}}~x~{\text{and}}~y\\{\text{not}}~x~{\text{without}}~y\\{\text{not}}~y~{\text{without}}~x\\x~{\text{or}}~y\end{matrix}}$	${\begin{matrix}\lnot x\lor \lnot y\\x\Rightarrow y\\x\Leftarrow y\\x\lor y\end{matrix}}$
$f_{15}$	$f_{1111}$	$1~1~1~1$	${\texttt {((}}~~{\texttt {))}}$	${\text{true}}$	$1$

Table A3. Ef Expanded Over Ordinary Features

${\text{Table A3.}}~~\mathrm {E} f~{\text{Expanded Over Ordinary Features}}~\{x,y\}$

	$f$	$\mathrm {E} f|_{xy}$	$\mathrm {E} f|_{x{\texttt {(}}y{\texttt {)}}}$	$\mathrm {E} f|_{{\texttt {(}}x{\texttt {)}}y}$	$\mathrm {E} f|_{{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\\{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\{\texttt {(}}\mathrm {d} x{\texttt {)}}\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\{\texttt {(}}\mathrm {d} x{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}\\\mathrm {d} x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {((}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {((}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {))}}\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\{\texttt {(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\{\texttt {(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {((}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} x~~~~\mathrm {d} y~~{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} x~~~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}~~\mathrm {d} x~~~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {((}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~\mathrm {d} x~~~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {(}}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {))}}\\{\texttt {((}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~{\texttt {)}}\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\end{matrix}}$
$f_{15}$	$1$	$1$	$1$	$1$	$1$

Table A4. Ef Expanded Over Differential Features

${\text{Table A4.}}~~\mathrm {E} f~{\text{Expanded Over Differential Features}}~\{\mathrm {d} x,\mathrm {d} y\}$

	$f$	${\begin{matrix}\mathrm {T} _{11}f\\\mathrm {E} f|_{\mathrm {d} x~\mathrm {d} y}\end{matrix}}$	${\begin{matrix}\mathrm {T} _{10}f\\\mathrm {E} f|_{\mathrm {d} x{\texttt {(}}\mathrm {d} y{\texttt {)}}}\end{matrix}}$	${\begin{matrix}\mathrm {T} _{01}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)}}\mathrm {d} y}\end{matrix}}$	${\begin{matrix}\mathrm {T} _{00}f\\\mathrm {E} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}}\end{matrix}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}x~~~~y\\~~x~~{\texttt {(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\\{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}~~y~~\\{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\x~~~~y\\~~x~~{\texttt {(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}x\\{\texttt {(}}x{\texttt {)}}\end{matrix}}$	${\begin{matrix}x\\{\texttt {(}}x{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}y\\{\texttt {(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}y\\{\texttt {(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {(}}~~x~~~~y~~{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$
$f_{15}$	$1$	$1$	$1$	$1$	$1$
${\text{Fixed Point Total}}$		$4$	$4$	$4$	$16$

Table A5. Df Expanded Over Ordinary Features

${\text{Table A5.}}~~\mathrm {D} f~{\text{Expanded Over Ordinary Features}}~\{x,y\}$

	$f$	$\mathrm {D} f|_{xy}$	$\mathrm {D} f|_{x{\texttt {(}}y{\texttt {)}}}$	$\mathrm {D} f|_{{\texttt {(}}x{\texttt {)}}y}$	$\mathrm {D} f|_{{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {,}}~~\mathrm {d} y{\texttt {)}}\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\end{matrix}}$	${\begin{matrix}~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\\mathrm {d} x~~~~\mathrm {d} y\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\end{matrix}}$	${\begin{matrix}\mathrm {d} x~~~~\mathrm {d} y\\~~\mathrm {d} x~~{\texttt {(}}\mathrm {d} y{\texttt {)}}\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~~\mathrm {d} y~~\\{\texttt {((}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {))}}\end{matrix}}$
$f_{15}$	$1$	$0$	$0$	$0$	$0$

Table A6. Df Expanded Over Differential Features

${\text{Table A6.}}~~\mathrm {D} f~{\text{Expanded Over Differential Features}}~\{\mathrm {d} x,\mathrm {d} y\}$

	$f$	$\mathrm {D} f|_{\mathrm {d} x~\mathrm {d} y}$	$\mathrm {D} f|_{\mathrm {d} x{\texttt {(}}\mathrm {d} y{\texttt {)}}}$	$\mathrm {D} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)}}\mathrm {d} y}$	$\mathrm {D} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\\{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}\\x\\x\end{matrix}}$	${\begin{matrix}0\\0\\0\\0\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}y\\{\texttt {(}}y{\texttt {)}}\\y\\{\texttt {(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}x\\x\\{\texttt {(}}x{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}\end{matrix}}$	${\begin{matrix}0\\0\\0\\0\end{matrix}}$
$f_{15}$	$1$	$0$	$0$	$0$	$0$

Appendix 2. Differential Forms

The actions of the difference operator $\mathrm {D}$ and the tangent operator $\mathrm {d}$ on the 16 bivariate propositions are shown in Tables A7 and A8.

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

$\{~{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}},~\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}},~{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y,~\mathrm {d} x~\mathrm {d} y~\}.$

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:

$\partial x~=~\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}$     and     $\partial y~=~{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y.$

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

$\{~1,~\mathrm {d} x,~\mathrm {d} y,~\mathrm {d} x~\mathrm {d} y~\}.$

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

${\text{Table A7.}}~~{\text{Differential Forms Expanded on a Logical Basis}}$

	$f$	$\mathrm {D} f$	$\mathrm {d} f$
$f_{0}$	${\texttt {(}}~~{\texttt {)}}$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}x{\texttt {)}}&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}&\mathrm {d} x~\mathrm {d} y\\y&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}x{\texttt {)}}&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&x&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}&\mathrm {d} x~\mathrm {d} y\\y&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&x&{\texttt {(}}\mathrm {d} x)~\mathrm {d} y&+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}~\partial x&+&{\texttt {(}}x{\texttt {)}}~\partial y\\~~y~~~\partial x&+&{\texttt {(}}x{\texttt {)}}~\partial y\\{\texttt {(}}y{\texttt {)}}~\partial x&+&~~x~~~\partial y\\~~y~~~\partial x&+&~~x~~~\partial y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&\mathrm {d} x~\mathrm {d} y\\\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}\partial x\\\partial x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y\\\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y\end{matrix}}$	${\begin{matrix}\partial x&+&\partial y\\\partial x&+&\partial y\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}\partial y\\\partial y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}y&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&x&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&x&{\texttt {(}}\mathrm {d} x)~\mathrm {d} y&+&{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}&\mathrm {d} x~\mathrm {d} y\\y&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}x{\texttt {)}}&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}&\mathrm {d} x~{\texttt {(}}\mathrm {d} y{\texttt {)}}&+&{\texttt {(}}x{\texttt {)}}&{\texttt {(}}\mathrm {d} x{\texttt {)}}~\mathrm {d} y&+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}~~y~~~\partial x&+&~~x~~~\partial y\\{\texttt {(}}y{\texttt {)}}~\partial x&+&~~x~~~\partial y\\~~y~~~\partial x&+&{\texttt {(}}x{\texttt {)}}~\partial y\\{\texttt {(}}y{\texttt {)}}~\partial x&+&{\texttt {(}}x{\texttt {)}}~\partial y\end{matrix}}$
$f_{15}$	${\texttt {((}}~~{\texttt {))}}$	$0$	$0$

Table A8. Differential Forms Expanded on an Algebraic Basis

${\text{Table A8.}}~~{\text{Differential Forms Expanded on an Algebraic Basis}}$

	$f$	$\mathrm {D} f$	$\mathrm {d} f$
$f_{0}$	${\texttt {(}}~~{\texttt {)}}$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&~~x~~~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&~~x~~~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&~~x~~~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&~~x~~~\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}\mathrm {d} x&+&\mathrm {d} y\\\mathrm {d} x&+&\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x&+&\mathrm {d} y\\\mathrm {d} x&+&\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~~y~~~\mathrm {d} x&+&~~x~~~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&~~x~~~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y&+&\mathrm {d} x~\mathrm {d} y\end{matrix}}$	${\begin{matrix}~~y~~~\mathrm {d} x&+&~~x~~~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&~~x~~~\mathrm {d} y\\~~y~~~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}~\mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}~\mathrm {d} y\end{matrix}}$
$f_{15}$	${\texttt {((}}~~{\texttt {))}}$	$0$	$0$

Table A9. Tangent Proposition as Pointwise Linear Approximation

${\text{Table A9.}}~~{\text{Tangent Proposition}}~\mathrm {d} f={\text{Pointwise Linear Approximation to the Difference Map}}~\mathrm {D} f$

$f$	${\begin{matrix}\mathrm {d} f=\\[2pt]\partial _{x}f\cdot \mathrm {d} x~+~\partial _{y}f\cdot \mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} ^{2}\!f=\\[2pt]\partial _{xy}f\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	$\mathrm {d} f|_{x\,y}$	$\mathrm {d} f|_{x\,{\texttt {(}}y{\texttt {)}}}$	$\mathrm {d} f|_{{\texttt {(}}x{\texttt {)}}\,y}$	$\mathrm {d} f|_{{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\0\\\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\\0\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\\\mathrm {d} x\\0\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}\mathrm {d} x&+&\mathrm {d} y\\\mathrm {d} x&+&\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\\\mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\\\mathrm {d} x\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\\0\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\0\\\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$
$f_{15}$	$0$	$0$	$0$	$0$	$0$	$0$

Table A10. Taylor Series Expansion Df = df + d²f

${\text{Table A10.}}~~{\text{Taylor Series Expansion}}~{\mathrm {D} f=\mathrm {d} f+\mathrm {d} ^{2}\!f}$

$f$	${\begin{matrix}\mathrm {D} f\\=&\mathrm {d} f&+&\mathrm {d} ^{2}\!f\\=&\partial _{x}f\cdot \mathrm {d} x~+~\partial _{y}f\cdot \mathrm {d} y&+&\partial _{xy}f\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	$\mathrm {d} f|_{x\,y}$	$\mathrm {d} f|_{x\,{\texttt {(}}y{\texttt {)}}}$	$\mathrm {d} f|_{{\texttt {(}}x{\texttt {)}}\,y}$	$\mathrm {d} f|_{{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}}$
$f_{0}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\0\\\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\\0\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\\\mathrm {d} x\\0\end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}~~1~~\cdot \mathrm {d} x&+&~~0~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~1~~\cdot \mathrm {d} x&+&~~0~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}~~1~~\cdot \mathrm {d} x&+&~~1~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~1~~\cdot \mathrm {d} x&+&~~1~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}~~0~~\cdot \mathrm {d} x&+&~~1~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~0~~\cdot \mathrm {d} x&+&~~1~~\cdot \mathrm {d} y&+&~~0~~\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y&+&~~1~~\cdot \mathrm {d} x\;\mathrm {d} y\end{matrix}}$	${\begin{matrix}\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\\\mathrm {d} x\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\\0\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\0\\\mathrm {d} x+\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}0\\\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} x+\mathrm {d} y\end{matrix}}$
$f_{15}$	$0$	$0$	$0$	$0$	$0$

Table A11. Partial Differentials and Relative Differentials

${\text{Table A11.}}~~{\text{Partial Differentials and Relative Differentials}}$

	$f$	${\frac {\partial f}{\partial x}}$	${\frac {\partial f}{\partial y}}$	${\begin{matrix}\mathrm {d} f=\\[2pt]\partial _{x}f\cdot \mathrm {d} x~+~\partial _{y}f\cdot \mathrm {d} y\end{matrix}}$	$\left.{\frac {\partial x}{\partial y}}\right|f$	$\left.{\frac {\partial y}{\partial x}}\right|f$
$f_{0}$	${\texttt {(}}~{\texttt {)}}$	$0$	$0$	$0$	$0$	$0$
${\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}~~y~~\\~~x~~{\texttt {(}}y{\texttt {)}}\\x~~~~y\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\\{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}\\x\\x\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\end{matrix}}$	${\begin{matrix}\cdots \\\cdots \\\cdots \\\cdots \end{matrix}}$	${\begin{matrix}\cdots \\\cdots \\\cdots \\\cdots \end{matrix}}$
${\begin{matrix}f_{3}\\f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\x\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}\mathrm {d} x\\\mathrm {d} x\end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$
${\begin{matrix}f_{6}\\f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}\\{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}\mathrm {d} x&+&\mathrm {d} y\\\mathrm {d} x&+&\mathrm {d} y\end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$
${\begin{matrix}f_{5}\\f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\y\end{matrix}}$	${\begin{matrix}0\\0\end{matrix}}$	${\begin{matrix}1\\1\end{matrix}}$	${\begin{matrix}\mathrm {d} y\\\mathrm {d} y\end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$	${\begin{matrix}\cdots \\\cdots \end{matrix}}$
${\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {(}}~~x~~~~y~~{\texttt {)}}\\{\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}\\{\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}\\{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}y\\{\texttt {(}}y{\texttt {)}}\\y\\{\texttt {(}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}x\\x\\{\texttt {(}}x{\texttt {)}}\\{\texttt {(}}x{\texttt {)}}\end{matrix}}$	${\begin{matrix}~~y~~\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&~~x~~\cdot \mathrm {d} y\\~~y~~\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\\{\texttt {(}}y{\texttt {)}}\cdot \mathrm {d} x&+&{\texttt {(}}x{\texttt {)}}\cdot \mathrm {d} y\end{matrix}}$	${\begin{matrix}\cdots \\\cdots \\\cdots \\\cdots \end{matrix}}$	${\begin{matrix}\cdots \\\cdots \\\cdots \\\cdots \end{matrix}}$
$f_{15}$	${\texttt {((}}~{\texttt {))}}$	$0$	$0$	$0$	$0$	$0$

Table A12. Detail of Calculation for the Difference Map

${\text{Table A12.}}~~{\text{Detail of Calculation for}}~{\mathrm {E} f+f=\mathrm {D} f}$

	$f$	${\begin{array}{cr}~&\mathrm {E} f|_{\mathrm {d} x~\mathrm {d} y}\\[4pt]+&f|_{\mathrm {d} x~\mathrm {d} y}\\[4pt]=&\mathrm {D} f|_{\mathrm {d} x~\mathrm {d} y}\end{array}}$	${\begin{array}{cr}~&\mathrm {E} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)}}\mathrm {d} y}\\[4pt]+&f|_{{\texttt {(}}\mathrm {d} x{\texttt {)}}\mathrm {d} y}\\[4pt]=&\mathrm {D} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)}}\mathrm {d} y}\end{array}}$	${\begin{array}{cr}~&\mathrm {E} f|_{\mathrm {d} x{\texttt {(}}\mathrm {d} y{\texttt {)}}}\\[4pt]+&f|_{\mathrm {d} x{\texttt {(}}\mathrm {d} y{\texttt {)}}}\\[4pt]=&\mathrm {D} f|_{\mathrm {d} x{\texttt {(}}\mathrm {d} y{\texttt {)}}}\end{array}}$	${\begin{array}{cr}~&\mathrm {E} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}}\\[4pt]+&f|_{{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}}\\[4pt]=&\mathrm {D} f|_{{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} y{\texttt {)}}}\end{array}}$
$f_{0}$	$0$	$0~+~0~=~0$	$0~+~0~=~0$	$0~+~0~=~0$	$0~+~0~=~0$
$f_{1}$	${\texttt {(}}x{\texttt {)(}}y{\texttt {)}}$	${\begin{matrix}~&~~~~x~~~~y~~~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]=&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]=&~~~~~~~~{\texttt {(}}y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]=&~~{\texttt {(}}x{\texttt {)}}~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$
$f_{2}$	${\texttt {(}}x{\texttt {)}}~~y~~$	${\begin{matrix}~&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]=&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~~~x~~~~y~~~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]=&~~~~~~~~~~y~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]=&~~{\texttt {(}}x{\texttt {)}}~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]+&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]=&0\end{matrix}}$
$f_{4}$	$~~x~~{\texttt {(}}y{\texttt {)}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]+&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]=&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]+&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]=&~~~~~~~~{\texttt {(}}y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~~~x~~~~y~~~~\\[4pt]+&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]=&~~~~x~~~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]+&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$
$f_{8}$	$x~~~~y$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}~~\\[4pt]+&~~~~x~~~~y~~~~\\[4pt]=&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {)}}~~y~~~~\\[4pt]+&~~~~x~~~~y~~~~\\[4pt]=&~~~~~~~~~~y~~~~\end{matrix}}$	${\begin{matrix}~&~~~~x~~{\texttt {(}}y{\texttt {)}}~~\\[4pt]+&~~~~x~~~~y~~~~\\[4pt]=&~~~~x~~~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~~~x~~~~y~~~~\\[4pt]+&~~~~x~~~~y~~~~\\[4pt]=&0\end{matrix}}$
$f_{3}$	${\texttt {(}}x{\texttt {)}}$	${\begin{matrix}~&x\\[4pt]+&{\texttt {(}}x{\texttt {)}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&x\\[4pt]+&{\texttt {(}}x{\texttt {)}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}x{\texttt {)}}\\[4pt]+&{\texttt {(}}x{\texttt {)}}\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}x{\texttt {)}}\\[4pt]+&{\texttt {(}}x{\texttt {)}}\\[4pt]=&0\end{matrix}}$
$f_{12}$	$x$	${\begin{matrix}~&{\texttt {(}}x{\texttt {)}}\\[4pt]+&x\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}x{\texttt {)}}\\[4pt]+&x\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&~~x~~\\[4pt]+&x\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&~~x~~\\[4pt]+&x\\[4pt]=&0\end{matrix}}$
$f_{6}$	${\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$
$f_{9}$	${\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\\[4pt]=&0\end{matrix}}$
$f_{5}$	${\texttt {(}}y{\texttt {)}}$	${\begin{matrix}~&y\\[4pt]+&{\texttt {(}}y{\texttt {)}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}y{\texttt {)}}\\[4pt]+&{\texttt {(}}y{\texttt {)}}\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&y\\[4pt]+&{\texttt {(}}y{\texttt {)}}\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}y{\texttt {)}}\\[4pt]+&{\texttt {(}}y{\texttt {)}}\\[4pt]=&0\end{matrix}}$
$f_{10}$	$y$	${\begin{matrix}~&{\texttt {(}}y{\texttt {)}}\\[4pt]+&y\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&~~y~~\\[4pt]+&y\\[4pt]=&0\end{matrix}}$	${\begin{matrix}~&{\texttt {(}}y{\texttt {)}}\\[4pt]+&y\\[4pt]=&1\end{matrix}}$	${\begin{matrix}~&~~y~~\\[4pt]+&y\\[4pt]=&0\end{matrix}}$
$f_{7}$	${\texttt {(}}~~x~~~~y~~{\texttt {)}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]=&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]=&~~~~~~~~~~y~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]=&~~~~x~~~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$
$f_{11}$	${\texttt {(}}~~x~~{\texttt {(}}y{\texttt {))}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]=&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]=&~~~~~~~~{\texttt {(}}y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]+&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]=&~~~~x~~~~~~~~~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]+&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]=&0\end{matrix}}$
$f_{13}$	${\texttt {((}}x{\texttt {)}}~~y~~{\texttt {)}}$	${\begin{matrix}~&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]=&~~{\texttt {(}}x{\texttt {,}}~~y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]=&~~~~~~~~~~y~~~~\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]=&~~{\texttt {(}}x{\texttt {)}}~~~~~~~~\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]=&0\end{matrix}}$
$f_{14}$	${\texttt {((}}x{\texttt {)(}}y{\texttt {))}}$	${\begin{matrix}~&~~{\texttt {(}}x~~~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]=&{\texttt {((}}x{\texttt {,}}~~y{\texttt {))}}\end{matrix}}$	${\begin{matrix}~&~~{\texttt {(}}x~~{\texttt {(}}y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]=&~~~~~~~~{\texttt {(}}y{\texttt {)}}~~\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)}}~~y{\texttt {)}}~~\\[4pt]+&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]=&~~{\texttt {(}}x{\texttt {)}}~~~~~~~~\end{matrix}}$	${\begin{matrix}~&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]+&{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[4pt]=&0\end{matrix}}$
$f_{15}$	$1$	$1~+~1~=~0$	$1~+~1~=~0$	$1~+~1~=~0$	$1~+~1~=~0$

Appendix 3. Computational Details

Operator Maps for the Logical Conjunction f₈⟨u, v⟩

Computation of εf₈

${\text{Table F8.1}}~~{\text{Computation of}}~{\boldsymbol {\varepsilon }}f_{8}$

${\begin{array}{*{10}{l}}{\boldsymbol {\varepsilon }}f_{8}&=&&f_{8}\langle u,v\rangle \\[4pt]&=&&uv\\[4pt]&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&uv\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&uv\cdot \mathrm {d} u~\mathrm {d} v\\[20pt]{\boldsymbol {\varepsilon }}f_{8}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v~~\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~~~\mathrm {d} v~~\end{array}}$

Computation of Ef₈

${\text{Table F8.2-i}}~~{\text{Computation of}}~\mathrm {E} f_{8}~{\text{(Method 1)}}$

${\begin{array}{*{9}{l}}\mathrm {E} f_{8}&=&f_{8}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle \\[4pt]&=&{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]&=&uv\cdot f_{8}\langle {\texttt {(}}\mathrm {d} u{\texttt {)}},{\texttt {(}}\mathrm {d} v{\texttt {)}}\rangle &+&u{\texttt {(}}v{\texttt {)}}\cdot f_{8}\langle {\texttt {(}}\mathrm {d} u{\texttt {)}},\mathrm {d} v\rangle &+&{\texttt {(}}u{\texttt {)}}v\cdot f_{8}\langle \mathrm {d} u,{\texttt {(}}\mathrm {d} v{\texttt {)}}\rangle &+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot f_{8}\langle \mathrm {d} u,\mathrm {d} v\rangle \\[4pt]&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[20pt]\mathrm {E} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v\\[4pt]&&&&&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&&&&&&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

${\text{Table F8.2-ii}}~~{\text{Computation of}}~\mathrm {E} f_{8}~{\text{(Method 2)}}$

${\begin{array}{*{9}{c}}\mathrm {E} f_{8}&=&\{u+\mathrm {d} u\}\cdot \{v+\mathrm {d} v\}\\[6pt]&=&u\cdot v&+&u\cdot \mathrm {d} v&+&v\cdot \mathrm {d} u&+&\mathrm {d} u\cdot \mathrm {d} v\\[6pt]\mathrm {E} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Computation of Df₈

${\text{Table F8.3-i}}~~{\text{Computation of}}~\mathrm {D} f_{8}~{\text{(Method 1)}}$

${\begin{array}{*{10}{l}}\mathrm {D} f_{8}&=&&\mathrm {E} f_{8}&+&{\boldsymbol {\varepsilon }}f_{8}\\[4pt]&=&&f_{8}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle &+&f_{8}\langle u,v\rangle \\[4pt]&=&&{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&uv\\[20pt]\mathrm {D} f_{8}&=&&0&+&0&+&0&+&0\\[4pt]&&+&uv\cdot ~~{\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v~~~~&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v~~&+&0&+&0\\[4pt]&&+&uv\cdot ~~~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {}}~~&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0\\[4pt]&&+&uv\cdot ~~~~\mathrm {d} u~~~~\mathrm {d} v~~~~&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~\mathrm {d} v~~\\[20pt]\mathrm {D} f_{8}&=&&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v~~&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~\mathrm {d} v~~\end{array}}$

${\text{Table F8.3-ii}}~~{\text{Computation of}}~\mathrm {D} f_{8}~{\text{(Method 2)}}$

${\begin{array}{*{9}{l}}\mathrm {D} f_{8}&=&{\boldsymbol {\varepsilon }}f_{8}&+&\mathrm {E} f_{8}\\[6pt]&=&f_{8}\langle u,v\rangle &+&f_{8}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle \\[6pt]&=&uv&+&{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]&=&0&+&u\cdot \mathrm {d} v&+&v\cdot \mathrm {d} u&+&\mathrm {d} u~\mathrm {d} v\\[6pt]\mathrm {D} f_{8}&=&0&+&u\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&v\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

${\text{Table F8.3-iii}}~~{\text{Computation of}}~\mathrm {D} f_{8}~{\text{(Method 3)}}$

${\begin{array}{c*{9}{l}}\mathrm {D} f_{8}&=&{\boldsymbol {\varepsilon }}f_{8}~+~\mathrm {E} f_{8}\\[20pt]{\boldsymbol {\varepsilon }}f_{8}&=&~~u~~v~~\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&~~u~~v~~\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&~~u~~v~~\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&~~~~u~~~~v~~~~\cdot \mathrm {d} u~~\mathrm {d} v\\[6pt]\mathrm {E} f_{8}&=&~~u~~v~~\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&~~u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v~~\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&~~{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}~~\cdot \mathrm {d} u~~\mathrm {d} v\\[20pt]\mathrm {D} f_{8}&=&~~~~0~~~~\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&~~~~u~~~~\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&~~~~v~~~~\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\cdot \mathrm {d} u~~\mathrm {d} v\end{array}}$

Computation of df₈

${\text{Table F8.4}}~~{\text{Computation of}}~\mathrm {d} f_{8}$

${\begin{array}{c*{8}{l}}\mathrm {D} f_{8}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[6pt]\Downarrow \\[6pt]\mathrm {d} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\end{array}}$

Computation of rf₈

${\text{Table F8.5}}~~{\text{Computation of}}~\mathrm {r} f_{8}$

${\begin{array}{c*{8}{l}}\mathrm {r} f_{8}&=&\mathrm {D} f_{8}~+~\mathrm {d} f_{8}\\[20pt]\mathrm {D} f_{8}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[6pt]\mathrm {d} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[20pt]\mathrm {r} f_{8}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Computation Summary for Conjunction

${\text{Table F8.6}}~~{\text{Computation Summary for}}~f_{8}\langle u,v\rangle =uv$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}f_{8}&=&uv\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {E} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[6pt]\mathrm {D} f_{8}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[6pt]\mathrm {d} f_{8}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {r} f_{8}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Operator Maps for the Logical Equality f₉⟨u, v⟩

Computation of εf₉

${\text{Table F9.1}}~~{\text{Computation of}}~{\boldsymbol {\varepsilon }}f_{9}$

${\begin{array}{*{10}{l}}{\boldsymbol {\varepsilon }}f_{9}&=&&f_{9}\langle u,v\rangle \\[4pt]&=&&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\\[4pt]&=&&uv\cdot f_{9}\langle 1,1\rangle &+&u{\texttt {(}}v{\texttt {)}}\cdot f_{9}\langle 1,0\rangle &+&{\texttt {(}}u{\texttt {)}}v\cdot f_{9}\langle 0,1\rangle &+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot f_{9}\langle 0,0\rangle \\[4pt]&=&&uv&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\[20pt]{\boldsymbol {\varepsilon }}f_{9}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~~~\mathrm {d} v\end{array}}$

Computation of Ef₉

${\text{Table F9.2}}~~{\text{Computation of}}~\mathrm {E} f_{9}$

${\begin{array}{*{10}{l}}\mathrm {E} f_{9}&=&&f_{9}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle \\[4pt]&=&&{\texttt {(((}}u{\texttt {,}}\mathrm {d} u{\texttt {),(}}v{\texttt {,}}\mathrm {d} v{\texttt {)))}}\\[4pt]&=&&uv\cdot f_{9}\langle {\texttt {(}}\mathrm {d} u{\texttt {)}},{\texttt {(}}\mathrm {d} v{\texttt {)}}\rangle &+&u{\texttt {(}}v{\texttt {)}}\cdot f_{9}\langle {\texttt {(}}\mathrm {d} u{\texttt {)}},\mathrm {d} v\rangle &+&{\texttt {(}}u{\texttt {)}}v\cdot f_{9}\langle \mathrm {d} u,{\texttt {(}}\mathrm {d} v{\texttt {)}}\rangle &+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot f_{9}\langle \mathrm {d} u,\mathrm {d} v\rangle \\[4pt]&=&&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~\mathrm {d} v{\texttt {))}}\\[20pt]\mathrm {E} f_{9}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&0&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&0\\[4pt]&&+&0&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~~~\mathrm {d} v\end{array}}$

Computation of Df₉

${\text{Table F9.3-i}}~~{\text{Computation of}}~\mathrm {D} f_{9}~{\text{(Method 1)}}$

${\begin{array}{*{10}{l}}\mathrm {D} f_{9}&=&&\mathrm {E} f_{9}&+&{\boldsymbol {\varepsilon }}f_{9}\\[4pt]&=&&f_{9}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle &+&f_{9}\langle u,v\rangle \\[4pt]&=&&{\texttt {(((}}u{\texttt {,}}\mathrm {d} u{\texttt {),(}}v{\texttt {,}}\mathrm {d} v{\texttt {)))}}&+&{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\\[20pt]\mathrm {D} f_{9}&=&&0&+&0&+&0&+&0\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&0&+&0&+&0&+&0\\[20pt]\mathrm {D} f_{9}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{array}}$

${\text{Table F9.3-ii}}~~{\text{Computation of}}~\mathrm {D} f_{9}~{\text{(Method 2)}}$

${\begin{array}{*{9}{l}}\mathrm {D} f_{9}&=&0\cdot \mathrm {d} u~\mathrm {d} v&+&1\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&1\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&0\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\end{array}}$

Computation of df₉

${\text{Table F9.4}}~~{\text{Computation of}}~\mathrm {d} f_{9}$

${\begin{array}{c*{8}{l}}\mathrm {D} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\Downarrow \\[6pt]\mathrm {d} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{array}}$

Computation of rf₉

${\text{Table F9.5}}~~{\text{Computation of}}~\mathrm {r} f_{9}$

${\begin{array}{c*{8}{l}}\mathrm {r} f_{9}&=&\mathrm {D} f_{9}~+~\mathrm {d} f_{9}\\[20pt]\mathrm {D} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {d} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[20pt]\mathrm {r} f_{9}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\end{array}}$

Computation Summary for Equality

${\text{Table F9.6}}~~{\text{Computation Summary for}}~f_{9}\langle u,v\rangle ={\texttt {((}}u{\texttt {,}}v{\texttt {))}}$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}f_{9}&=&uv\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 1\\[6pt]\mathrm {E} f_{9}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {D} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {d} f_{9}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {r} f_{9}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\end{array}}$

Operator Maps for the Logical Implication f₁₁⟨u, v⟩

Computation of εf₁₁

${\text{Table F11.1}}~~{\text{Computation of}}~{\boldsymbol {\varepsilon }}f_{11}$

${\begin{array}{*{10}{l}}{\boldsymbol {\varepsilon }}f_{11}&=&&f_{11}\langle u,v\rangle \\[4pt]&=&&{\texttt {(}}u{\texttt {(}}v{\texttt {))}}\\[4pt]&=&&uv\cdot f_{11}\langle 1,1\rangle &+&u{\texttt {(}}v{\texttt {)}}\cdot f_{11}\langle 1,0\rangle &+&{\texttt {(}}u{\texttt {)}}v\cdot f_{11}\langle 0,1\rangle &+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot f_{11}\langle 0,0\rangle \\[4pt]&=&&uv&+&0&+&{\texttt {(}}u{\texttt {)}}v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\[20pt]{\boldsymbol {\varepsilon }}f_{11}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~~~\mathrm {d} v\end{array}}$

Computation of Ef₁₁

${\text{Table F11.2}}~~{\text{Computation of}}~\mathrm {E} f_{11}$

${\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]&=&&uv\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]&=&&uv\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}&=&&uv\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]&&+&uv\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]&&+&uv\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}}$

Computation of Df₁₁

${\text{Table F11.3-i}}~~{\text{Computation of}}~\mathrm {D} f_{11}~{\text{(Method 1)}}$

${\begin{array}{*{10}{l}}\mathrm {D} f_{11}&=&&\mathrm {E} f_{11}&+&{\boldsymbol {\varepsilon }}f_{11}\\[4pt]&=&&f_{11}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle &+&f_{11}\langle u,v\rangle \\[4pt]&=&&{\texttt {(}}{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)}}{\texttt {(}}{\texttt {(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}{\texttt {))}}&+&{\texttt {(}}u{\texttt {(}}v{\texttt {))}}\\[20pt]\mathrm {D} f_{11}&=&&0&+&0&+&0&+&0\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~{\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&0&+&0\\[4pt]&&+&0&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&0&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~~~\mathrm {d} u~~~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~~\mathrm {d} v&+&0\\[20pt]\mathrm {D} f_{11}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\end{array}}$

${\text{Table F11.3-ii}}~~{\text{Computation of}}~\mathrm {D} f_{11}~{\text{(Method 2)}}$

${\begin{array}{c*{9}{l}}\mathrm {D} f_{11}&=&{\boldsymbol {\varepsilon }}f_{11}~+~\mathrm {E} f_{11}\\[20pt]{\boldsymbol {\varepsilon }}f_{11}&=&uv\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 1&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 1\\[6pt]\mathrm {E} f_{11}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}\\[20pt]\mathrm {D} f_{11}&=&uv\cdot ~~{\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\end{array}}$

Computation of df₁₁

${\text{Table F11.4}}~~{\text{Computation of}}~\mathrm {d} f_{11}$

${\begin{array}{c*{8}{l}}\mathrm {D} f_{11}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[6pt]\Downarrow \\[6pt]\mathrm {d} f_{11}&=&uv\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\end{array}}$

Computation of rf₁₁

${\text{Table F11.5}}~~{\text{Computation of}}~\mathrm {r} f_{11}$

${\begin{array}{c*{8}{l}}\mathrm {r} f_{11}&=&\mathrm {D} f_{11}~+~\mathrm {d} f_{11}\\[20pt]\mathrm {D} f_{11}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {d} f_{11}&=&uv\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\\[20pt]\mathrm {r} f_{11}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Computation Summary for Implication

${\text{Table F11.6}}~~{\text{Computation Summary for}}~f_{11}\langle u,v\rangle ={\texttt {(}}u{\texttt {(}}v{\texttt {))}}$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}f_{11}&=&uv\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 1&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 1\\[6pt]\mathrm {E} f_{11}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {D} f_{11}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {d} f_{11}&=&uv\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\\[6pt]\mathrm {r} f_{11}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Operator Maps for the Logical Disjunction f₁₄⟨u, v⟩

Computation of εf₁₄

${\text{Table F14.1}}~~{\text{Computation of}}~{\boldsymbol {\varepsilon }}f_{14}$

${\begin{array}{*{10}{l}}{\boldsymbol {\varepsilon }}f_{14}&=&&f_{14}\langle u,v\rangle \\[4pt]&=&&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[4pt]&=&&uv\cdot f_{14}\langle 1,1\rangle &+&u{\texttt {(}}v{\texttt {)}}\cdot f_{14}\langle 1,0\rangle &+&{\texttt {(}}u{\texttt {)}}v\cdot f_{14}\langle 0,1\rangle &+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot f_{14}\langle 0,0\rangle \\[4pt]&=&&uv&+&u{\texttt {(}}v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v&+&0\\[20pt]{\boldsymbol {\varepsilon }}f_{14}&=&&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&0\\[4pt]&&+&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v&+&0\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0\\[4pt]&&+&uv\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot ~~\mathrm {d} u~~~~\mathrm {d} v&+&0\end{array}}$

Computation of Ef₁₄

${\text{Table F14.2}}~~{\text{Computation of}}~\mathrm {E} f_{14}$

${\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]&=&&uv\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]&=&&uv\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}&=&&uv\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]&&+&uv\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]&&+&uv\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}}$

Computation of Df₁₄

${\text{Table F14.3-i}}~~{\text{Computation of}}~\mathrm {D} f_{14}~{\text{(Method 1)}}$

${\begin{array}{*{10}{l}}\mathrm {D} f_{14}&=&&\mathrm {E} f_{14}&+&{\boldsymbol {\varepsilon }}f_{14}\\[4pt]&=&&f_{14}\langle u+\mathrm {d} u,v+\mathrm {d} v\rangle &+&f_{14}\langle u,v\rangle \\[4pt]&=&&{\texttt {(((}}u{\texttt {,}}\mathrm {d} u{\texttt {))((}}v{\texttt {,}}\mathrm {d} v{\texttt {)))}}&+&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[20pt]\mathrm {D} f_{14}&=&&0&+&0&+&0&+&0\\[4pt]&&+&0&+&0&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~{\texttt {(}}\mathrm {d} u{\texttt {)}}~~\mathrm {d} v\\[4pt]&&+&0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~~~\mathrm {d} u~~{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&uv\cdot \mathrm {d} u~\mathrm {d} v&+&0&+&0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot ~~~~\mathrm {d} u~~~~\mathrm {d} v\\[20pt]\mathrm {D} f_{14}&=&&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\end{array}}$

${\text{Table F14.3-ii}}~~{\text{Computation of}}~\mathrm {D} f_{14}~{\text{(Method 2)}}$

${\begin{array}{*{9}{l}}\mathrm {D} f_{14}&=&{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&0\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}\end{array}}$

Computation of df₁₄

${\text{Table F14.4}}~~{\text{Computation of}}~\mathrm {d} f_{14}$

${\begin{array}{c*{8}{l}}\mathrm {D} f_{14}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\Downarrow \\[6pt]\mathrm {d} f_{14}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{array}}$

Computation of rf₁₄

${\text{Table F14.5}}~~{\text{Computation of}}~\mathrm {r} f_{14}$

${\begin{array}{c*{8}{l}}\mathrm {r} f_{14}&=&\mathrm {D} f_{14}~+~\mathrm {d} f_{14}\\[20pt]\mathrm {D} f_{14}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {d} f_{14}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[20pt]\mathrm {r} f_{14}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Computation Summary for Disjunction

${\text{Table F14.6}}~~{\text{Computation Summary for}}~f_{14}\langle u,v\rangle ={\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}f_{14}&=&uv\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 1&+&{\texttt {(}}u{\texttt {)}}v\cdot 1&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {E} f_{14}&=&uv\cdot {\texttt {(}}\mathrm {d} u~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {D} f_{14}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {d} f_{14}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {r} f_{14}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\end{array}}$

Appendix 5. Various Definitions of the Tangent Vector

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• Overview • Part 1 • Part 2 • Appendices • References •

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