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\chapter{Common Identities In Simplification-Related Functions} % Main chapter title

\label{ChapterX} % Change X to a consecutive number; for referencing this chapter elsewhere, use \ref{ChapterX}



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\section{Idempotency of $\simp$}

\begin{equation}
	\simp \;r = \simp\; \simp \; r 
\end{equation}
This property means we do not have to repeatedly
apply simplification in each step, which justifies
our definition of $\blexersimp$.
It will also be useful in future proofs where properties such as 
closed forms are needed.
The proof is by structural induction on $r$.

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\subsection{Syntactic Equivalence Under $\simp$}
We prove that minor differences can be annhilated
by $\simp$.
For example,
\begin{center}
$\simp \;(\simpALTs\; (\map \;(\_\backslash \; x)\; (\distinct \; \mathit{rs}\; \phi))) = 
 \simp \;(\simpALTs \;(\distinct \;(\map \;(\_ \backslash\; x) \; \mathit{rs}) \; \phi))$
\end{center}


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

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