design/tpl (Classification System): Add always and never figure
This demonstrates the vacuity lemma.master
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@ -122,6 +122,14 @@ This is due to the commutativity of~$\odot$ as proved by
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and not only affords great ease of use to users of~\tame{},
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but also great flexibility to compiler writers.
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For notational convenience,
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we will let
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\begin{align}
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\odot^\land &= \Monoid\Bool\land\true, \\
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\odot^\lor &= \Monoid\Bool\lor\false.
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\end{align}
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\def\cpredmatseq{{M^0_j}_k \bullet\cdots\bullet {M^l_j}_k}
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\def\cpredvecseq{v^0_j\bullet\cdots\bullet v^m_j}
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@ -251,6 +259,36 @@ This is due to the commutativity of~$\odot$ as proved by
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\end{proof}
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\begin{figure}[h]\label{fig:always-never}
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\begin{alignat*}{3}
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\begin{aligned}
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\xml{<classify }&\xml{as="always" yields="alwaysTrue"} \xmlnl
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&\xml{desc="Always true" />}
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\end{aligned}
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\quad&=\quad
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\Classify^\texttt{always}_\texttt{alwaysTrue}
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&&\left(\odot^\land,\emptyset,\emptyset,\emptyset\right). \\
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%
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\begin{aligned}
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\xml{<classify }&\xml{as="never" yields="neverTrue"} \xmlnl
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&\xml{any="true"} \xmlnl
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&\xml{desc="Never true" />}
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\end{aligned}
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\quad&=\quad
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\Classify^\texttt{never}_\texttt{neverTrue}
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&&\left(\odot^\lor,\emptyset,\emptyset,\emptyset\right).
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\end{alignat*}
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\caption{\tameclass{always} and \tameclass{never} from package
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\tamepkg{core/base}.}
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\end{figure}
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Figure~\ref{fig:always-never} demonstrates \lemref{class-pred-vacu} in the
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definitions of the classifications \tameclass{always} and~\tameclass{never}.
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These classifications are typically referenced directly for clarity rather
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than creating other vacuous classifications,
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encapsulating \lemref{class-pred-vacu}.
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\begin{theorem}[Classification Rank Independence]\thmlabel{class-rank-indep}
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Let $\odot=\Monoid\Bool\bullet e$.
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Then,
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@ -139,6 +139,8 @@
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\newcommand\bicomp[1]{{#1}^\circ}
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\let\xml\texttt
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\newcommand\xmlnl{\\[-3mm]}
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\let\tamepkg\texttt
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% Definitions (introduction of terms)
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\let\dfn\textsl
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