Sunday, December 16, 2012
Math test
This is a test of MathJax.
$$
\newcommand{\zero}{\mathsf{z}}
\newcommand{\succ}[1]{\mathsf{s}({#1})}
\newcommand{\iter}[4]{\mathsf{iter}({#1}, \zero \to {#2}, \succ{#3} \to {#4})}
\begin{array}{lcl}
A & ::= & 1 \bnfalt A \times B \bnfalt A \to B \bnfalt \N \\
e & ::= & () \bnfalt (e, e) \bnfalt \fst{e} \bnfalt \snd{e} \bnfalt \fun{x}{e} \bnfalt e \; e' \\
& | & \zero \bnfalt \succ{e} \bnfalt \iter{e}{e_z}{x}{e_s} \bnfalt x \bnfalt e : A \\
\Gamma & ::= & \cdot \bnfalt \Gamma, x:A
\end{array}
$$
$$
\begin{array}{cc}
\frac{ x:A \in \Gamma }
{ \judge{\Gamma}{x}{A} }
&
\frac{ }
{\judge{\Gamma}{()}{1}}
\\[1em]
\frac{\judge{\Gamma}{e_1}{A_1} \qquad
\judge{\Gamma}{e_2}{A_2}}
{\judge{\Gamma}{(e_1, e_2)}{A_1 \times A_2}}
&
\frac{\judge{\Gamma}{e}{A_1 \times A_2}}
{\judge{\Gamma}{\pi_i(e)}{A_i}}
\\[1em]
\frac{\judge{\Gamma, x:A}{e}{B}}
{\judge{\Gamma}{\fun{x}{e}}{A \to B}}
&
\frac{\judge{\Gamma}{e}{A \to B} \qquad
\judge{\Gamma}{e'}{A}}
{\judge{\Gamma}{e\;e'}{B}}
\\[1em]
\frac{ }
{\judge{\Gamma}{\zero}{\N}}
&
\frac{\judge{\Gamma}{e}{\N} }
{\judge{\Gamma}{\succ{e}}{\N}}
\\[1em]
\frac{\begin{array}{l}
\judge{\Gamma}{e}{\N} \\
\judge{\Gamma}{e_z}{A} \\
\judge{\Gamma, x:A}{e_s}{A}
\end{array}}
{\judge{\Gamma}{\iter{e}{e_z}{x}{e_s}}{A}}
&
\frac{\judge{\Gamma}{e}{A}}
{\judge{\Gamma}{(e:A)}{A}}
\end{array}
$$
It seems to work!
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