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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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