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\section{Properties of $\backslash c$}

To have a clear idea of what we can and 
need to prove about the algorithms involving
Brzozowski's derivatives, there are a few 
properties we need to be clear about 
it.
\subsection{function $\backslash c$ is not 1-to-1}
\begin{center}
The derivative $w.r.t$ character $c$ is not one-to-one.
Formally,
	$\exists r_1 \;r_2. r_1 \neq r_2 \mathit{and} r_1 \backslash c = r_2 \backslash c$
\end{center}
This property is trivially true for the
character regex example:
\begin{center}
	$r_1 = e; \; r_2 = d;\; r_1 \backslash c = \ZERO = r_2 \backslash c$
\end{center}
But apart from the cases where the derivative
output is $\ZERO$, are there non-trivial results
of derivatives which contain strings?
The answer is yes.
For example,
\begin{center}
	Let $r_1 = a^*b\;\quad r_2 = (a\cdot a^*)\cdot b + b$.\\
	where $a$ is not nullable.\\
	$r_1 \backslash c = ((a \backslash c)\cdot a^*)\cdot c + b \backslash c$\\
	$r_2 \backslash c = ((a \backslash c)\cdot a^*)\cdot c + b \backslash c$
\end{center}
We start with two syntactically different regexes,
and end up with the same derivative result, which is
a "meaningful" regex because it contains strings.
We have rediscovered Arden's lemma:\\
\begin{center}
	$A^*B = A\cdot A^* \cdot B + B$
\end{center}

	
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\subsection{Subsection 1}

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\subsection{Subsection 2}
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\section{Main Section 2}

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