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\def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2013-03/}}}

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{\LARGE\bf Cheat Sheet $\beta$}\hfill
\parbox[b]{4.5in}{\small
  \null\hfill\sheeturl
  \newline\null\hfill initiated 24/3/13; modified \today, \ampmtime; continued \href{http://drorbn.net/AcademicPensieve/2013-04/}{2013-04}
}

\noindent\rule{\textwidth}{1pt}

The original $\beta$-calculus: With $\epsilon:=1+\alpha$, $\langle\alpha\rangle:=\sum_v\alpha_v$,
and $\langle\gamma\rangle:=\sum_{v\neq u}\gamma_v$,
\[
  \begin{array}{c|c}\omega_1&H_1\\\hline T_1&\alpha_1\end{array}
  \ast
  \begin{array}{c|c}\omega_2&H_2\\\hline T_2&\alpha_2\end{array}
  \underset{\beta}{=}
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & \alpha_1 & 0 \\
    T_2 & 0 & \alpha_2
  \end{array}
  \qquad
  \begin{CD}
    \begin{array}{c|c}
      \omega & \cdots \\
      \hline
      u & \alpha \\
      v & \beta \\
      \vdots & \gamma
    \end{array}
    @>tm^{uv}_w>\beta>
    \begin{array}{c|c}
      \omega & \cdots \\
      \hline
      w & \alpha+\beta \\
      \vdots & \gamma
    \end{array}
  \end{CD}
  \qquad R^\pm_{ux}
  \underset{\beta}{=}
  \begin{array}{c|cc}
    1 & x \\
    \hline
    u & t_u^{\pm 1}-1
  \end{array}
\]

\[
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & x & y & \cdots \\
      \hline
      \vdots & \alpha & \beta & \gamma
    \end{array}
    @>{hm^{xy}_z}>\beta>
    \begin{array}{c|cc}
      \omega & z & \cdots \\
      \hline
      \vdots & \alpha+\beta+\langle\alpha\rangle\beta & \gamma
    \end{array}
  \end{CD}
  \qquad
  \begin{CD}
    \begin{array}{c|cc}
      \omega & x & \cdots \\
      \hline
      u & \alpha & \beta \\
      \vdots & \gamma & \delta
    \end{array}
    @>{sw^{ux}_{th}}>\beta>
    \begin{array}{c|cc}
      \omega\epsilon & x & \cdots \\
      \hline
      u & \alpha(1+\langle\gamma\rangle/\epsilon)
        & \beta(1+\langle\gamma\rangle/\epsilon) \\
      \vdots & \gamma/\epsilon & \delta-\gamma\beta/\epsilon
    \end{array}
  \end{CD}
\]

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