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
Mike Gerwitz 2021-05-18 10:14:56 -04:00
parent 2d268f2a55
commit 1ec0fc0c7b
2 changed files with 1 additions and 35 deletions

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

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@ -137,7 +137,6 @@
% Binary function composition
\newcommand\bicomp[1]{{#1}^\circ}
\newcommand\bicompi[1]{{#1}^\bullet}
\let\xml\texttt