stepper-papers/appendix-rules.tex at main · fplab/stepper-papers · GitHub

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\section{Standard Rules}\label{sec:standard_rules}

This appendix collects the standard meta-functions and rules of the
filtered stepper calculus (substitution, decomposition, and
composition) deferred from \autoref{sec:filter}. The syntax of
evaluation contexts and the value judgment are given in
\autoref{fig:filter-syntax} and the accompanying rules in the main
body.

\fbox{\([v / x] e = e'\)} \(e'\) can be obtained by substitution of \(v\) for
\(x\) in expression \(e\).
\[
  \begin{aligned}
    [v / x] \Nat{n} &= \Nat{n} \\
    [v / x] x &= v \\
    [v / x] y &= y && \text{if } x \neq y \\
    [v / x] (e_1(e_2)) &= ([v / x] e_1)([v / x] e_2) \\
    [v / x] (e_1 + e_2) &= ([v / x] e_1) + ([v / x] e_2) \\
    [v / x] \Lam{y}{e} &= \Lam{y}{[v / x] e} && \text{if } x \neq y \\
    [v / x] \Lam{y}{e} &= \Lam{y}{e} && \text{if } x = y \\
    [v / x] \Filter{(p, a, g)}{e} &= \Filter{([v / x]p, a, g)}{[v / x] e} \\
    [v / x] \Residue{a}{g}{l}{e} &= \Residue{a}{g}{l}{[v / x] e} \\
    [v / x] \Fix{x}{e} &= \Fix{x}{e} \\
    [v / x] \Fix{y}{e} &= \Fix{y}{[v / x]e} && \text{if } x \neq y
  \end{aligned}
\]

\fbox{\([v / x] p = p'\)} \(p'\) can be obtained by substitution of \(v\) for
\(x\) in pattern \(p\).
\[
  \begin{aligned}
    [v / x] \$e &= \$e \\
    [v / x] \$v &= \$v \\
    [v / x] \Nat{n} &= \Nat{n} \\
    [v / x] y &= y && \text{if } x \neq y \\
    [v / x] (p_1(p_2)) &= ([v / x] p_1)([v / x] p_2) \\
    [v / x] (p_1 + p_2) &= ([v / x] p_1) + ([v / x] p_2) \\
    [v / x] \Lam{y}{e} &= \Lam{y}{[v / x] e} && \text{if } x \neq y \\
    [v / x] \Lam{y}{e} &= \Lam{y}{e} && \text{if } x = y \\
    [v / x] \Fix{x}{e} &= \Fix{x}{e} \\
    [v / x] \Fix{y}{e} &= \Fix{y}{[v / x]e} && \text{if } x \neq y
  \end{aligned}
\]

\jbox{\(\Decompose{e}{\mathcal{E}}{e'}\)} Expression \(e\) can be decomposed into context \(\mathcal{E}\) and redex \(e'\).
\begin{mathpar}
  % Residue
  \inferrule[D-Residue-T]{
    \Decompose{e}{\mathcal{E}}{e'}
  }{
    \Decompose{\Residue{a}{g}{l}{e}}{\Residue{a}{g}{l}{{\mathcal{E}}}}{e'}
  } \qquad
  \inferrule[D-Residue-E]{
    \Value{v}
  }{
    \Decompose{\Residue{a}{g}{l}{v}}{\circ}{\Residue{a}{g}{l}{v}}
  } \\
  % Filter
  \inferrule[D-Filter-T]{
    \Decompose{e}{\mathcal{E}}{e'}
  }{
    \Decompose{\Filter{f}{e}}{\Filter{f}{\mathcal{E}}}{e'}
  } \qquad
  \inferrule[D-Filter-E]{
    \Value{v}
  }{
    \Decompose{\Filter{f}{v}}{\circ}{\Filter{f}{v}}
  } \\
  % Application
  \inferrule[D-Ap-L]{
    \Decompose{e_1}{\mathcal{E}_1}{e_1'}
  }{
    \Decompose{e_1(e_2)}{\mathcal{E}_1(e_2)}{e_1'}
  } \qquad
  \inferrule[D-Ap-R]{
    \Value{e_1} \\
    \Decompose{e_2}{\mathcal{E}_2}{e_2'}
  }{
    \Decompose{e_1(e_2)}{e_1(\mathcal{E}_2)}{e_2'}
  } \qquad
  \inferrule[D-Ap-E]{
    \Value{e_1} \\
    \Value{e_2}
  }{
    \Decompose{e_1(e_2)}{\circ}{e_1(e_2)}
  } \\
  % Addition
  \inferrule[D-Add-L]{
    \Decompose{e_1}{\mathcal{E}_1}{e_1'}
  }{
    \Decompose{e_1 + e_2}{\mathcal{E}_1 + e_2}{e_1'}
  } \qquad
  \inferrule[D-Add-R]{
    \Value{e_1} \\
    \Decompose{e_2}{\mathcal{E}_2}{e_2'}
  }{
    \Decompose{e_1 + e_2}{e_1 + \mathcal{E}_2}{e_2'}
  } \qquad
  \inferrule[D-Add-E]{
    \Value{e_1} \\
    \Value{e_2}
  }{
    \Decompose{e_1 + e_2}{\circ}{e_1 + e_2}
  } \\
  \inferrule[D-Fix-E]{
    \
  }{
    \Decompose{\Fix{x}{e}}{\circ}{\Fix{x}{e}}
  }
\end{mathpar}

\jbox{\(\Compose{e}{\mathcal{E}}{e'}\)} Expression \(e\) can be obtained by plugging \(e'\) into the mark of \(\mathcal{E}\).
\begin{mathpar}
  \inferrule[C-Top]{
  }{
    \Compose{e}{\circ}{e}
  } \\
  \inferrule[C-Ap-L]{
    \Compose{e_1}{\mathcal{E}_1}{e_1'}
  }{
    \Compose{e_1(e_2)}{\mathcal{E}_1(e_2)}{e_2'}
  } \qquad
  \inferrule[C-Ap-R]{
    \Compose{e_2}{\mathcal{E}}{e_2'}
  }{
    \Compose{e_1(e_2)}{e_1(\mathcal{E}_2)}{e_2'}
  } \\
  \inferrule[C-Add-L]{
    \Compose{e_1}{\mathcal{E}_1}{e_1'}
  }{
    \Compose{e_1 + e_2}{\mathcal{E}_1 + e_2}{e_1'}
  } \qquad
  \inferrule[C-Add-R]{
    \Compose{e_2}{\mathcal{E}_2}{e_2'}
  }{
    \Compose{e_1 + e_2}{e_1 + \mathcal{E}_2}{e_2'}
  } \\
  \inferrule[C-Filter]{
    \Compose{e}{\mathcal{E}}{e'}
  }{
    \Compose{\Filter{f}{e}}{\Filter{f}{\mathcal{E}}}{e'}
  } \qquad
  \inferrule[C-Residue]{
    \Compose{e}{\mathcal{E}}{e'}
  }{
    \Compose{\Residue{a}{g}{l}{e}}{\Residue{a}{g}{l}{\mathcal{E}}}{e'}
  }
\end{mathpar}

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