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Lectures

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Dmitry Boulytchev 2018-12-06 21:36:23 +03:00
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doc/07.tex
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@ -68,9 +68,10 @@ On expression level, abstractly/concretely:
The semantics of enriched expressions is modified as follows. First, we add two additional premises to the rule for binary operators:
\setsubarrow{_{\mathscr E}}
\[\trule{\begin{array}{ccc}
\withenv{\Phi}{\trans{c}{A}{c^\prime}} &\phantom{XXXXX}& \withenv{\Phi}{\trans{c^\prime}{B}{c^{\prime\prime}}}\\
\primi{val}{c^\prime}\in\mathbb Z & & \primi{val}{c^{\prime\prime}}\in\mathbb Z
\arraycolsep=10pt
\[\trule{\begin{array}{cc}
\withenv{\Phi}{\trans{c}{A}{c^\prime}} & \withenv{\Phi}{\trans{c^\prime}{B}{c^{\prime\prime}}}\\
\primi{val}{c^\prime}\in\mathbb Z & \primi{val}{c^{\prime\prime}}\in\mathbb Z
\end{array}
}
{\withenv{\Phi}{\trans{c}{A\otimes B}{\primi{ret}{c^{\prime\prime}\;(\primi{val}{c^\prime}\oplus \primi{val}{c^{\prime\prime}})}}}}
@ -81,11 +82,12 @@ These two premises ensure that both operand expressions are evaluated into integ
kinds of expressions (see Figure~\ref{array_expressions}).
\begin{figure}
\[\trule{\begin{array}{ccc}
\withenv{\Phi}{\trans{c}{e}{c^\prime}} &\phantom{XXXX} &\withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}}\\
l=\primi{val}{c^\prime} & &j=\primi{val}{c^{\prime\prime}}\\
l\in\mathscr L & &j\in\mathbb N\\
(n,\,f)=\primi{mem}{l} & &j<n
\arraycolsep=10pt
\[\trule{\begin{array}{cc}
\withenv{\Phi}{\trans{c}{e}{c^\prime}} &\withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}}\\
l=\primi{val}{c^\prime} &j=\primi{val}{c^{\prime\prime}}\\
l\in\mathscr L &j\in\mathbb N\\
(n,\,f)=\primi{mem}{l} &j<n
\end{array}}
{\withenv{\Phi}{\trans{c}{e\,\mathtt{[}j\mathtt{]}}{\primi{ret}{c^{\prime\prime}(f\;j)}}}}
\ruleno{ArrayElement}
@ -125,13 +127,14 @@ On statement level, we add the single construct:
This construct is interpreted as an assignment to an element of an array. The semantics of this construct is described by the following rule:
\[
\trule{\setsubarrow{_{\mathscr E}}
\begin{array}{ccccc}
\withenv{\Phi}{\trans{c}{e}{c^\prime}} & \phantom{XXX} & \withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}} & \phantom{XXX} & \withenv{\Phi}{\trans{c^{\prime\prime}}{g}{\inbr{s,\mu,l_m,i,o,v}}}\\
l=\primi{val}{c^\prime} & & i=\primi{val}{c^{\prime\prime}} & & \\
l\in\mathscr L & & i\in\mathbb N & & \\[3mm]
\multicolumn{5}{c}{(n,\,f)=\mu\;l}\\
\multicolumn{5}{c}{i<n}\\
\multicolumn{5}{c}{\setsubarrow{}\withenv{\llang{skip},\,\Phi}{\trans{\inbr{s,\,\mu\,[l\gets (n,\,f\,[i\gets x])],\,l_m,\,i,\,o,\,\hbox{---}}}{K}{\widetilde{c}}}}
\arraycolsep=10pt
\begin{array}{ccc}
\withenv{\Phi}{\trans{c}{e}{c^\prime}} & \withenv{\Phi}{\trans{c^\prime}{j}{c^{\prime\prime}}} & \withenv{\Phi}{\trans{c^{\prime\prime}}{g}{\inbr{s,\mu,l_m,i,o,v}}}\\
l=\primi{val}{c^\prime} & i=\primi{val}{c^{\prime\prime}} & \\
l\in\mathscr L & i\in\mathbb N & \\[3mm]
\multicolumn{3}{c}{(n,\,f)=\mu\;l}\\
\multicolumn{3}{c}{i<n}\\
\multicolumn{3}{c}{\setsubarrow{}\withenv{\llang{skip},\,\Phi}{\trans{\inbr{s,\,\mu\,[l\gets (n,\,f\,[i\gets x])],\,l_m,\,i,\,o,\,\hbox{---}}}{K}{\widetilde{c}}}}
\end{array}
}
{\setsubarrow{}\withenv{K,\,\Phi}{\trans{c}{e\mathtt{[}j\mathtt{]}\llang{:=}g}{\widetilde{c}}}}