# User:Jon Awbrey/APPENDICES

## Appendices (Version 1)

### Appendix 1-A. Operator Maps for the Disjunction “f”

#### Table A7. Computation Summary for Disjunction

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

### Appendix 1-B. Operator Maps for the Equality “g”

#### Table B7. Computation Summary for Equality

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

### Appendix 2

#### Table C9. Tangent Proposition as Pointwise Linear Approximation

 $f\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}^2\!f = \\[2pt] \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{x \, y}$ $\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}\!$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table C10. Taylor Series Expansion Df = df + d2f

 $f\!$ $\begin{matrix} \mathrm{D}f \\ = & \mathrm{d}f & + & \mathrm{d}^2\!f \\ = & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{x \, y}$ $\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table C11. Partial Differentials and Relative Differentials

 $f\!$ $\frac{\partial f}{\partial x}\!$ $\frac{\partial f}{\partial y}\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\left. \frac{\partial x}{\partial y} \right| f\!$ $\left. \frac{\partial y}{\partial x} \right| f\!$ $f_0\!$ $(~)\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ $\begin{matrix} (x) & (y) \\ (x) & y \\ x & (y) \\ x & y \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ (y) \\ y \\ \end{matrix}$ $\begin{matrix} (x) \\ (x) \\ x \\ x \\ \end{matrix}$ $\begin{matrix} (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & x & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ $\begin{matrix} (x) \\ x \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}\!$ $\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ $\begin{matrix} (x, & y) \\ ((x, & y)) \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \mathrm{d}x & + & \mathrm{d}y \\ \mathrm{d}x & + & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}\!$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ $\begin{matrix} (x & y) \\ (x & (y)) \\ ((x) & y) \\ ((x) & (y)) \\ \end{matrix}$ $\begin{matrix} y \\ (y) \\ y \\ (y) \\ \end{matrix}$ $\begin{matrix} x \\ x \\ (x) \\ (x) \\ \end{matrix}$ $\begin{matrix} y & \mathrm{d}x & + & x & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $f_{15}\!$ $((~))\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table C12. Detail of Calculation for Df = Ef + f

$\text{Table C12.} ~~ \text{Detail of Calculation for}~ {\mathrm{D}f = \mathrm{E}f + f}\!$

$\begin{array}{cr} & \mathrm{E}f|_{\mathrm{d}x\ \mathrm{d}y} \\ + & f|_{\mathrm{d}x\ \mathrm{d}y} \\ = & \mathrm{D}f|_{\mathrm{d}x\ \mathrm{d}y} \end{array}$

$\begin{array}{cr} & \mathrm{E}f|_{\mathrm{d}x\ (\mathrm{d}y)} \\ + & f|_{\mathrm{d}x\ (\mathrm{d}y)} \\ = & \mathrm{D}f|_{\mathrm{d}x\ (\mathrm{d}y)} \end{array}$

$\begin{array}{cr} & \mathrm{E}f|_{(\mathrm{d}x)\ \mathrm{d}y} \\ + & f|_{(\mathrm{d}x)\ \mathrm{d}y} \\ = & \mathrm{D}f|_{(\mathrm{d}x)\ \mathrm{d}y} \end{array}$

$\begin{array}{cr} & \mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)} \\ + & f|_{(\mathrm{d}x)(\mathrm{d}y)} \\ = & \mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)} \end{array}$

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

## Appendices (Version 2)

### Appendix 1

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

$\text{Table A1.}~~\text{Propositional Forms on Two Variables}$
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

 $x\!$ : $y\!$ :
 1 1 0 0 1 0 1 0

 $f_{0}\!$ $f_{1}\!$ $f_{2}\!$ $f_{3}\!$ $f_{4}\!$ $f_{5}\!$ $f_{6}\!$ $f_{7}\!$
 $f_{0000}\!$ $f_{0001}~\!$ $f_{0010}\!$ $f_{0011}\!$ $f_{0100}\!$ $f_{0101}\!$ $f_{0110}\!$ $f_{0111}\!$
 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
 $(~)\!$ $(x)(y)\!$ $(x)~y\!$ $(x)\!$ $x~(y)\!$ $(y)\!$ $(x,~y)\!$ $(x~y)\!$
 $\operatorname{false}$ $\operatorname{neither}~ x ~\operatorname{nor}~ y$ $y ~\operatorname{without}~ x$ $\operatorname{not}~ x$ $x ~\operatorname{without}~ y$ $\operatorname{not}~ y$ $x ~\operatorname{not~equal~to}~ y$ $\operatorname{not~both}~ x ~\operatorname{and}~ y$
 $0\!$ $\lnot x \land \lnot y$ $\lnot x \land y$ $\lnot x$ $x \land \lnot y$ $\lnot y$ $x \ne y$ $\lnot x \lor \lnot y$
 $f_{8}\!$ $f_{9}\!$ $f_{10}\!$ $f_{11}\!$ $f_{12}\!$ $f_{13}\!$ $f_{14}\!$ $f_{15}\!$
 $f_{1000}\!$ $f_{1001}\!$ $f_{1010}\!$ $f_{1011}\!$ $f_{1100}\!$ $f_{1101}\!$ $f_{1110}\!$ $f_{1111}\!$
 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
 $x~y\!$ $((x,~y))\!$ $y\!$ $(x~(y))\!$ $x\!$ $((x)~y)\!$ $((x)(y))\!$ $((~))\!$
 $x ~\operatorname{and}~ y$ $x ~\operatorname{equal~to}~ y$ $y\!$ $\operatorname{not}~ x ~\operatorname{without}~ y$ $x\!$ $\operatorname{not}~ y ~\operatorname{without}~ x$ $x ~\operatorname{or}~ y$ $\operatorname{true}$
 $x \land y$ $x = y\!$ $y\!$ $x \Rightarrow y$ $x\!$ $x \Leftarrow y$ $x \lor y$ $1\!$

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

$\text{Table A2.}~~\text{Propositional Forms on Two Variables}$
$\mathcal{L}_1$ $\mathcal{L}_2$ $\mathcal{L}_3$ $\mathcal{L}_4$ $\mathcal{L}_5$ $\mathcal{L}_6$

 $x\!$ : $y\!$ :
 1 1 0 0 1 0 1 0

$f_{0}\!$

$f_{0000}\!$

0 0 0 0

$(~)\!$

$\operatorname{false}$

$1\!$

 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $f_{0001}~\!$ $f_{0010}\!$ $f_{0100}\!$ $f_{1000}\!$
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
 $(x)(y)\!$ $(x)~y\!$ $x~(y)\!$ $x~y\!$
 $\operatorname{neither}~ x ~\operatorname{nor}~ y$ $y ~\operatorname{without}~ x$ $x ~\operatorname{without}~ y$ $x ~\operatorname{and}~ y$
 $\lnot x \land \lnot y$ $\lnot x \land y$ $x \land \lnot y$ $x \land y$
 $f_{3}\!$ $f_{12}\!$
 $f_{0011}\!$ $f_{1100}\!$
 0 0 1 1 1 1 0 0
 $(x)\!$ $x\!$
 $\operatorname{not}~ x$ $x\!$
 $\lnot x$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $f_{0110}\!$ $f_{1001}\!$
 0 1 1 0 1 0 0 1
 $(x,~y)\!$ $((x,~y))\!$
 $x ~\operatorname{not~equal~to}~ y$ $x ~\operatorname{equal~to}~ y$
 $x \ne y$ $x = y\!$
 $f_{5}\!$ $f_{10}\!$
 $f_{0101}\!$ $f_{1010}\!$
 0 1 0 1 1 0 1 0
 $(y)\!$ $y\!$
 $\operatorname{not}~ y$ $y\!$
 $\lnot y$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $f_{0111}\!$ $f_{1011}\!$ $f_{1101}\!$ $f_{1110}\!$
 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
 $(x~y)\!$ $(x~(y))\!$ $((x)~y)\!$ $((x)(y))\!$
 $\operatorname{not~both}~ x ~\operatorname{and}~ y$ $\operatorname{not}~ x ~\operatorname{without}~ y$ $\operatorname{not}~ y ~\operatorname{without}~ x$ $x ~\operatorname{or}~ y$
 $\lnot x \lor \lnot y$ $x \Rightarrow y$ $x \Leftarrow y$ $x \lor y$

$f_{15}\!$

$f_{1111}\!$

1 1 1 1

$((~))\!$

$\operatorname{true}$

$1\!$

#### Table A3. Ef Expanded Over Differential Features

$\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}$
$f\!$
 $\operatorname{T}_{11}f$ $\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$
 $\operatorname{T}_{10}f$ $\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}$
 $\operatorname{T}_{01}f$ $\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}$
 $\operatorname{T}_{00}f$ $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $x\ y\!$ $x\ (y)\!$ $(x)\ y\!$ $(x)(y)\!$
 $x\ (y)\!$ $x\ y\!$ $(x)(y)\!$ $(x)\ y\!$
 $(x)\ y\!$ $(x)(y)\!$ $x\ y\!$ $x\ (y)\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $x\!$ $(x)\!$
 $x\!$ $(x)\!$
 $(x)\!$ $x\!$
 $(x)\!$ $x\!$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $((x,\ y))\!$ $(x,\ y)\!$
 $((x,\ y))\!$ $(x,\ y)\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $y\!$ $(y)\!$
 $(y)\!$ $y\!$
 $y\!$ $(y)\!$
 $(y)\!$ $y\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((x)(y))\!$ $((x)\ y)\!$ $(x\ (y))\!$ $(x\ y)\!$
 $((x)\ y)\!$ $((x)(y))\!$ $(x\ y)\!$ $(x\ (y))\!$
 $(x\ (y))\!$ $(x\ y)\!$ $((x)(y))\!$ $((x)\ y)\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
$f_{15}\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$
Fixed Point Total : $4\!$ $4\!$ $4\!$ $16\!$

#### Table A4. Df Expanded Over Differential Features

$\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}$
$f\!$ $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ $\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}$ $\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}$ $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $((x,\ y))\!$ $(x,\ y)\!$ $(x,\ y)\!$ $((x,\ y))\!$
 $(y)\!$ $y\!$ $(y)\!$ $y\!$
 $(x)\!$ $(x)\!$ $x\!$ $x\!$
 $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $((~))\!$ $((~))\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $(~)\!$ $(~)\!$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(~)\!$ $(~)\!$
 $((~))\!$ $((~))\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $((~))\!$ $((~))\!$
 $(~)\!$ $(~)\!$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((x,\ y))\!$ $(x,\ y)\!$ $(x,\ y)\!$ $((x,\ y))\!$
 $y\!$ $(y)\!$ $y\!$ $(y)\!$
 $x\!$ $x\!$ $(x)\!$ $(x)\!$
 $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
$f_{15}\!$ $((~))\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$

#### Table A5. Ef Expanded Over Ordinary Features

$\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}$
$f\!$ $\operatorname{E}f|_{xy}$ $\operatorname{E}f|_{x(y)}$ $\operatorname{E}f|_{(x)y}$ $\operatorname{E}f|_{(x)(y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x) (\operatorname{d}y)$
 $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $(\operatorname{d}x) (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$
 $(\operatorname{d}x)\ \operatorname{d}y$ $(\operatorname{d}x) (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
 $(\operatorname{d}x) (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $\operatorname{d}x$ $(\operatorname{d}x)$
 $\operatorname{d}x$ $(\operatorname{d}x)$
 $(\operatorname{d}x)$ $\operatorname{d}x$
 $(\operatorname{d}x)$ $\operatorname{d}x$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$
 $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $((\operatorname{d}x,\ \operatorname{d}y))$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $((\operatorname{d}x,\ \operatorname{d}y))$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $\operatorname{d}y$ $(\operatorname{d}y)$
 $(\operatorname{d}y)$ $\operatorname{d}y$
 $\operatorname{d}y$ $(\operatorname{d}y)$
 $(\operatorname{d}y)$ $\operatorname{d}y$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$
 $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$
 $(\operatorname{d}x\ (\operatorname{d}y))$ $(\operatorname{d}x\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$
 $(\operatorname{d}x\ \operatorname{d}y)$ $(\operatorname{d}x\ (\operatorname{d}y))$ $((\operatorname{d}x)\ \operatorname{d}y)$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$ $((~))\!$

#### Table A6. Df Expanded Over Ordinary Features

$\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}$
$f\!$ $\operatorname{D}f|_{xy}$ $\operatorname{D}f|_{x(y)}$ $\operatorname{D}f|_{(x)y}$ $\operatorname{D}f|_{(x)(y)}$
$f_{0}\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$
 $f_{1}\!$ $f_{2}\!$ $f_{4}\!$ $f_{8}\!$
 $(x)(y)\!$ $(x)\ y\!$ $x\ (y)\!$ $x\ y\!$
 $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
 $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
 $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
 $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
 $f_{3}\!$ $f_{12}\!$
 $(x)\!$ $x\!$
 $\operatorname{d}x$ $\operatorname{d}x$
 $\operatorname{d}x$ $\operatorname{d}x$
 $\operatorname{d}x$ $\operatorname{d}x$
 $\operatorname{d}x$ $\operatorname{d}x$
 $f_{6}\!$ $f_{9}\!$
 $(x,\ y)\!$ $((x,\ y))\!$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $(\operatorname{d}x,\ \operatorname{d}y)$ $(\operatorname{d}x,\ \operatorname{d}y)$
 $f_{5}\!$ $f_{10}\!$
 $(y)\!$ $y\!$
 $\operatorname{d}y$ $\operatorname{d}y$
 $\operatorname{d}y$ $\operatorname{d}y$
 $\operatorname{d}y$ $\operatorname{d}y$
 $\operatorname{d}y$ $\operatorname{d}y$
 $f_{7}\!$ $f_{11}\!$ $f_{13}\!$ $f_{14}\!$
 $(x\ y)\!$ $(x\ (y))\!$ $((x)\ y)\!$ $((x)(y))\!$
 $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$
 $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$
 $\operatorname{d}x\ (\operatorname{d}y)$ $\operatorname{d}x\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$ $(\operatorname{d}x)\ \operatorname{d}y$
 $\operatorname{d}x\ \operatorname{d}y$ $\operatorname{d}x\ (\operatorname{d}y)$ $(\operatorname{d}x)\ \operatorname{d}y$ $((\operatorname{d}x)(\operatorname{d}y))$
$f_{15}\!$ $((~))\!$ $(~)\!$ $(~)\!$ $(~)\!$ $(~)\!$

### 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

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

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

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

### Appendix 3

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

 $f\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}^2\!f = \\[2pt] \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{x \, y}$ $\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}\!$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table A10. Taylor Series Expansion Df = df + d2f

 $\begin{matrix} \mathrm{d}f ~= \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}\!$ $\begin{matrix} \mathrm{d}^2\!f ~= \\[2pt] \partial_{xy} f \cdot \mathrm{d}x ~ \mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{ x \; y}\!$ $\mathrm{d}f|_{ x \;(y)}\!$ $\mathrm{d}f|_{(x)\; y}\!$ $\mathrm{d}f|_{(x)(y)}\!$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{smallmatrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{smallmatrix}\!$ $\begin{smallmatrix} (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & x & \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} 0 \\ \mathrm{d}x \\ \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ 0 \\ \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \\ 0 \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}y \\ \mathrm{d}x \\ 0 \end{smallmatrix}$ $\begin{smallmatrix}f_{3}\\f_{12}\end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix}f_{6}\\f_{9}\end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix}f_{5}\\f_{10}\end{smallmatrix}\!$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}\!$ $\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}\!$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}\!$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}\!$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}\!$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}\!$ $\begin{smallmatrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{smallmatrix}$ $\begin{smallmatrix} y & \mathrm{d}x & + & x & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \\ \mathrm{d}x ~\mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}y \\ \mathrm{d}x \\ 0 \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \\ 0 \\ \mathrm{d}x \end{smallmatrix}$ $\begin{smallmatrix} \mathrm{d}x \\ 0 \\ \mathrm{d}x + \mathrm{d}y \\ \mathrm{d}y \end{smallmatrix}$ $\begin{smallmatrix} 0 \\ \mathrm{d}x \\ \mathrm{d}y \\ \mathrm{d}x + \mathrm{d}y \end{smallmatrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table A11. Partial Differentials and Relative Differentials

 $f\!$ $\frac{\partial f}{\partial x}\!$ $\frac{\partial f}{\partial y}\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\left. \frac{\partial x}{\partial y} \right| f\!$ $\left. \frac{\partial y}{\partial x} \right| f\!$ $f_0\!$ $(~)\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix} f_{1} \\ f_{2} \\ f_{4} \\ f_{8} \\ \end{matrix}$ $\begin{matrix} (x) & (y) \\ (x) & y \\ x & (y) \\ x & y \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ (y) \\ y \\ \end{matrix}$ $\begin{matrix} (x) \\ (x) \\ x \\ x \\ \end{matrix}$ $\begin{matrix} (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & x & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{3} \\ f_{12} \\ \end{matrix}$ $\begin{matrix} (x) \\ x \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}\!$ $\begin{matrix} \mathrm{d}x \\ \mathrm{d}x \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{6} \\ f_{9} \\ \end{matrix}$ $\begin{matrix} (x, & y) \\ ((x, & y)) \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \mathrm{d}x & + & \mathrm{d}y \\ \mathrm{d}x & + & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{5} \\ f_{10} \\ \end{matrix}$ $\begin{matrix} (y) \\ y \\ \end{matrix}$ $\begin{matrix} 0 \\ 0 \\ \end{matrix}\!$ $\begin{matrix} 1 \\ 1 \\ \end{matrix}$ $\begin{matrix} \mathrm{d}y \\ \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} f_{7} \\ f_{11} \\ f_{13} \\ f_{14} \\ \end{matrix}$ $\begin{matrix} (x & y) \\ (x & (y)) \\ ((x) & y) \\ ((x) & (y)) \\ \end{matrix}$ $\begin{matrix} y \\ (y) \\ y \\ (y) \\ \end{matrix}$ $\begin{matrix} x \\ x \\ (x) \\ (x) \\ \end{matrix}$ $\begin{matrix} y & \mathrm{d}x & + & x & \mathrm{d}y \\ (y) & \mathrm{d}x & + & x & \mathrm{d}y \\ y & \mathrm{d}x & + & (x) & \mathrm{d}y \\ (y) & \mathrm{d}x & + & (x) & \mathrm{d}y \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $\begin{matrix} ~ \\ ~ \\ ~ \\ ~ \\ \end{matrix}$ $f_{15}\!$ $((~))\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

### Appendix 4

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

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

## Appendices (Version 3)

### Appendix 1. Propositional Forms and Differential Expansions

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

 $\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}$ $\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}$ $\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}$ $\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}$ $\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}$ $\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}$ $x\colon\!$ $1~1~0~0\!$ $y\colon\!$ $1~0~1~0\!$ $\begin{matrix} f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7} \end{matrix}$ $\begin{matrix} f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111} \end{matrix}$ $\begin{matrix} 0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1 \end{matrix}\!$ $\begin{matrix} \texttt{(~)} \\ \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)~} y \texttt{~} \\ \texttt{(} x \texttt{)~~~} \\ \texttt{~} x \texttt{~(} y \texttt{)} \\ \texttt{~~~(} y \texttt{)} \\ \texttt{(} x \texttt{,~} y \texttt{)} \\ \texttt{(} x \texttt{~~} y \texttt{)} \end{matrix}$ $\begin{matrix} \text{false} \\ \text{neither}~ x ~\text{nor}~ y \\ y ~\text{without}~ x \\ \text{not}~ x \\ x ~\text{without}~ y \\ \text{not}~ y \\ x ~\text{not equal to}~ y \\ \text{not both}~ x ~\text{and}~ y \end{matrix}$ $\begin{matrix} 0 \\ \lnot x \land \lnot y \\ \lnot x \land y \\ \lnot x \\ x \land \lnot y \\ \lnot y \\ x \ne y \\ \lnot x \lor \lnot y \end{matrix}$ $\begin{matrix} f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15} \end{matrix}$ $\begin{matrix} f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111} \end{matrix}\!$ $\begin{matrix} 1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1 \end{matrix}$ $\begin{matrix} \texttt{~~} x \texttt{~~} y \texttt{~~} \\ \texttt{((} x \texttt{,~} y \texttt{))} \\ \texttt{~~~~~} y \texttt{~~} \\ \texttt{~(} x \texttt{~(} y \texttt{))} \\ \texttt{~~} x \texttt{~~~~~} \\ \texttt{((} x \texttt{)~} y \texttt{)~} \\ \texttt{((} x \texttt{)(} y \texttt{))} \\ \texttt{((~))} \end{matrix}$ $\begin{matrix} x ~\text{and}~ y \\ x ~\text{equal to}~ y \\ y \\ \text{not}~ x ~\text{without}~ y \\ x \\ \text{not}~ y ~\text{without}~ x \\ x ~\text{or}~ y \\ \text{true} \end{matrix}$ $\begin{matrix} x \land y \\ x = y \\ y \\ x \Rightarrow y \\ x \\ x \Leftarrow y \\ x \lor y \\ 1 \end{matrix}$

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

 $\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}$ $\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}$ $\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}$ $\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}$ $\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}$ $\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}$ $x\colon\!$ $1~1~0~0\!$ $y\colon\!$ $1~0~1~0\!$ $f_{0}\!$ $f_{0000}\!$ $0~0~0~0$ $\texttt{(~)}\!$ $\text{false}\!$ $0\!$ $\begin{matrix} f_{1}\\f_{2}\\f_{4}\\f_{8} \end{matrix}$ $\begin{matrix} f_{0001}\\f_{0010}\\f_{0100}\\f_{1000} \end{matrix}$ $\begin{matrix} 0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0 \end{matrix}$ $\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)~} y \texttt{~} \\ \texttt{~} x \texttt{~(} y \texttt{)} \\ \texttt{~} x \texttt{~~} y \texttt{~} \end{matrix}$ $\begin{matrix} \text{neither}~ x ~\text{nor}~ y \\ y ~\text{without}~ x \\ x ~\text{without}~ y \\ x ~\text{and}~ y \end{matrix}$ $\begin{matrix} \lnot x \land \lnot y \\ \lnot x \land y \\ x \land \lnot y \\ x \land y \end{matrix}$ $\begin{matrix} f_{3}\\f_{12} \end{matrix}$ $\begin{matrix} f_{0011}\\f_{1100} \end{matrix}$ $\begin{matrix} 0~0~1~1\\1~1~0~0 \end{matrix}$ $\begin{matrix} \texttt{(} x \texttt{)} \\ \texttt{~} x \texttt{~} \end{matrix}$ $\begin{matrix} \text{not}~ x \\ x \end{matrix}\!$ $\begin{matrix} \lnot x \\ x \end{matrix}$ $\begin{matrix} f_{6}\\f_{9} \end{matrix}$ $\begin{matrix} f_{0110}\\f_{1001} \end{matrix}\!$ $\begin{matrix} 0~1~1~0\\1~0~0~1 \end{matrix}$ $\begin{matrix} \texttt{~(} x \texttt{,~} y \texttt{)~} \\ \texttt{((} x \texttt{,~} y \texttt{))} \end{matrix}$ $\begin{matrix} x ~\text{not equal to}~ y \\ x ~\text{equal to}~ y \end{matrix}$ $\begin{matrix} x \ne y \\ x = y \end{matrix}$ $\begin{matrix} f_{5}\\f_{10} \end{matrix}$ $\begin{matrix} f_{0101}\\f_{1010} \end{matrix}$ $\begin{matrix} 0~1~0~1\\1~0~1~0 \end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \\ \texttt{~} y \texttt{~} \end{matrix}$ $\begin{matrix} \text{not}~ y \\ y \end{matrix}$ $\begin{matrix} \lnot y \\ y \end{matrix}$ $\begin{matrix} f_{7}\\f_{11}\\f_{13}\\f_{14} \end{matrix}$ $\begin{matrix} f_{0111}\\f_{1011}\\f_{1101}\\f_{1110} \end{matrix}$ $\begin{matrix} 0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0 \end{matrix}$ $\begin{matrix} \texttt{~(} x \texttt{~~} y \texttt{)~} \\ \texttt{~(} x \texttt{~(} y \texttt{))} \\ \texttt{((} x \texttt{)~} y \texttt{)~} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}$ $\begin{matrix} \text{not both}~ x ~\text{and}~ y \\ \text{not}~ x ~\text{without}~ y \\ \text{not}~ y ~\text{without}~ x \\ x ~\text{or}~ y \end{matrix}$ $\begin{matrix} \lnot x \lor \lnot y \\ x \Rightarrow y \\ x \Leftarrow y \\ x \lor y \end{matrix}$ $f_{15}\!$ $f_{1111}\!$ $1~1~1~1\!$ $\texttt{((~))}\!$ $\text{true}\!$ $1\!$

#### Table A3. Ef Expanded Over Differential Features

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

#### Table A4. Df Expanded Over Differential Features

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

#### Table A5. Ef Expanded Over Ordinary Features

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

#### Table A6. Df Expanded Over Ordinary Features

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

### 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

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

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

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

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

 $f\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}^2\!f = \\[2pt] \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{x \, y}$ $\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}\!$ $\begin{matrix} \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \\ \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table A10. Taylor Series Expansion Df = df + d2f

 $f\!$ $\begin{matrix} \mathrm{D}f \\ = & \mathrm{d}f & + & \mathrm{d}^2\!f \\ = & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\mathrm{d}f|_{x \, y}$ $\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}$ $\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}$ $f_0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix} 0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0 \end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix} \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}$ $\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}$ $\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}$ $\begin{matrix} \mathrm{d}x\\\mathrm{d}x \end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix} \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}$ $\begin{matrix} \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y & + & \texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y \end{matrix}$ $\begin{matrix} \mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0 \end{matrix}$ $\begin{matrix} \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x \end{matrix}$ $\begin{matrix} \mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y \end{matrix}$ $\begin{matrix} 0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y \end{matrix}$ $f_{15}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

#### Table A11. Partial Differentials and Relative Differentials

 $f\!$ $\frac{\partial f}{\partial x}\!$ $\frac{\partial f}{\partial y}\!$ $\begin{matrix} \mathrm{d}f = \\[2pt] \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y \end{matrix}$ $\left. \frac{\partial x}{\partial y} \right| f\!$ $\left. \frac{\partial y}{\partial x} \right| f\!$ $f_0\!$ $\texttt{(~)}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$ $\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}$ $\begin{matrix} \texttt{(} x \texttt{)(} y \texttt{)} \\ \texttt{(} x \texttt{)~} y \texttt{~} \\ \texttt{~} x \texttt{~(} y \texttt{)} \\ \texttt{~} x \texttt{~~} y \texttt{~} \end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \\ \texttt{~} y \texttt{~} \\ \texttt{(} y \texttt{)} \\ \texttt{~} y \texttt{~} \end{matrix}$ $\begin{matrix} \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \\ \texttt{~} x \texttt{~} \\ \texttt{~} x \texttt{~} \end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}$ $\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}$ $\begin{matrix}f_{3}\\f_{12}\end{matrix}$ $\begin{matrix} \texttt{(} x \texttt{)} \\ \texttt{~} x \texttt{~} \end{matrix}$ $\begin{matrix}1\\1\end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}f_{6}\\f_{9}\end{matrix}$ $\begin{matrix} \texttt{~(} x \texttt{,~} y \texttt{)~} \\ \texttt{((} x \texttt{,~} y \texttt{))} \end{matrix}$ $\begin{matrix}1\\1\end{matrix}$ $\begin{matrix}1\\1\end{matrix}$ $\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}f_{5}\\f_{10}\end{matrix}$ $\begin{matrix} \texttt{(} y \texttt{)} \\ \texttt{~} y \texttt{~} \end{matrix}$ $\begin{matrix}0\\0\end{matrix}$ $\begin{matrix}1\\1\end{matrix}$ $\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}\cdots\\\cdots\end{matrix}$ $\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}$ $\begin{matrix} \texttt{(~} x \texttt{~~} y \texttt{~)} \\ \texttt{(~} x \texttt{~(} y \texttt{))} \\ \texttt{((} x \texttt{)~} y \texttt{~)} \\ \texttt{((} x \texttt{)(} y \texttt{))} \end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \\ \texttt{(} y \texttt{)} \\ \texttt{~} y \texttt{~} \\ \texttt{(} y \texttt{)} \end{matrix}$ $\begin{matrix} \texttt{~} x \texttt{~} \\ \texttt{~} x \texttt{~} \\ \texttt{(} x \texttt{)} \\ \texttt{(} x \texttt{)} \end{matrix}$ $\begin{matrix} \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y \\ \texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \\ \texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y \end{matrix}$ $\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}$ $\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}$ $f_{15}\!$ $\texttt{((~))}\!$ $0\!$ $0\!$ $0\!$ $0\!$ $0\!$

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

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

### Appendix 3. Computational Details

#### Operator Maps for the Logical Conjunction f8(u, v)

##### Computation of εf8

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

##### Computation of Ef8

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

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

##### Computation of Df8

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

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

 $\begin{array}{c*{9}{l}} \mathrm{D}f_{8} & = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8} \\[20pt] \boldsymbol\varepsilon f_{8} & = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v \\[6pt] \mathrm{E}f_{8} & = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v \\[20pt] \mathrm{D}f_{8} & = & ~~~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} & + & ~~~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v & + & ~~~~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} & + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v \end{array}\!$
##### Computation of df8

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

##### Computation of rf8

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

##### Computation Summary for Conjunction

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

#### Operator Maps for the Logical Equality f9(u, v)

##### Computation of εf9

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

##### Computation of Ef9

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

##### Computation of Df9

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

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

##### Computation of df9

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

##### Computation of rf9

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

##### Computation Summary for Equality

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

#### Operator Maps for the Logical Disjunction f14(u, v)

##### Computation of εf14

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

##### Computation of Ef14

 $\begin{array}{*{10}{l}} \mathrm{E}f_{14} & = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v) \\[4pt] & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))} \\[4pt] & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)}) & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ }) & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)}) & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ }) \\[4pt] & = && \texttt{ } u \texttt{ } v \texttt{ } \!\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{))} & + & \texttt{(} u \texttt{) } v \texttt{ } \!\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} & = && \texttt{ } u \texttt{ } v \texttt{ } \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{)} & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} & + & 0 \\[4pt] & = && \texttt{ } u \texttt{ } v \texttt{ } \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{~} & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} \\[4pt] & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} & + & 0 & + & \texttt{(} u \texttt{) } v \texttt{ } \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 & + & \texttt{ } u \texttt{ (} v \texttt{)} \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{~} & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} \end{array}$

##### Computation of Df14

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

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

##### Computation of df14

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

##### Computation of rf14

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

##### Computation Summary for Disjunction

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