design/tpl (Notational Convetions): Clean up unneeded bicompi
This was originally going to be used to define @yields for the classifier, but I took a very different approach which doesn't require reasoning about the system in terms of recursion.master
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@ -312,33 +312,10 @@ The symbol~$\_$ is used to denote a variable that matches anything but is
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and is often referred to as a ``wildcard'' (since it matches anything)
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or a ``hole'' (since its value goes nowhere).
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\indexsym{\bicompi{R}}{function, binary composition, recursive}
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\index{function!binary composition (\ensuremath{\bicomp{R}})!recursive (\ensuremath{\bicompi{R}})}
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For convenience,
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we also define $\bicompi{R}$,
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which recursively handles combinations of function and scalar values.
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This notation is used to simplify definitions of the classification system
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(see \secpref{class})
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when dealing with vectors
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(see \secref{vec}).
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\mremark{$\bicompi{R}$ may be removed depending on how $\Classify$ is defined.}
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\begin{equation}\label{eq:bicompi}
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\alpha \bicompi{R} \beta =
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\begin{cases}
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\gamma \mapsto \alpha_\gamma \bicompi{R} \beta_\gamma
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&\text{if } (\alpha : A\rightarrow B) \land (\beta : A\rightarrow D),\\
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\gamma \mapsto \alpha_\gamma \bicompi{R} (\_ \mapsto \beta)
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&\text{if } (\alpha : A\rightarrow B) \land (\beta \in\Real),\\
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\alpha R \beta &\text{otherwise}.
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\end{cases}
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\end{equation}
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Note that we consider the bracket notation for the image of a function
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$(f:A\rightarrow B)[A]$ to itself be a binary function.
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Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
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functions (such as vectors of vectors in \secref{vec}),
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noting that $\bicompi{[]}$ is \emph{not} a sensible construction.
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functions (such as vectors of vectors in \secref{vec}).
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\subsection{Monoids and Sequences}\seclabel{monoids}
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@ -505,16 +482,6 @@ Since a vector is a function,
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&= &&\Set{0,1,2}.
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\end{alignat*}
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We can also add two vectors, and scale them:
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\begin{align*}
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1 \bicompi{+} \Vector{1,2,3} \bicompi{+} \Vector{4,5,6}
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&= \Vector{1+1,\, 2+1,\, 3+1} \bicomp{+} \Vector{4,5,6} \\
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&= \Vector{2,3,4} \bicomp{+} \Vector{4,5,6} \\
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&= \Vector{2+4,\, 3+5,\, 4+6} \\
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&= \Vector{6, 8, 10}.
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\end{align*}
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\subsection{XML Notation}
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\indexsym{\xml{<>}}{XML}
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@ -137,7 +137,6 @@
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% Binary function composition
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\newcommand\bicomp[1]{{#1}^\circ}
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\newcommand\bicompi[1]{{#1}^\bullet}
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\let\xml\texttt
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