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\noindent 
We already proved that

\[
\text{If}\;nullable(r)\;\text{then}\;POSIX\;(mkeps\; r)\;r
\]

\noindent 
holds. This is essentially the ``base case'' for the
correctness proof of the algorithm. For the ``induction
case'' we need the following main theorem, which we are 
currently after:

\begin{center}
\begin{tabular}{lll}
If & (*) & $POSIX\;v\;(der\;c\;r)$ and $\vdash v : der\;c\;r$\\
then & & $POSIX\;(inj\;r\;c\;v)\;r$
\end{tabular}
\end{center}

\noindent 
That means a POSIX value $v$ is still $POSIX$ after injection.
I am not sure whether this theorem is actually true in this
full generality. Maybe it requires some restrictions.

If we unfold the $POSIX$ definition in the then-part, we 
arrive at

\[
\forall v'.\;
\text{if}\;\vdash v' : r\; \text{and} \;|inj\;r\;c\;v| = |v'|\;
\text{then}\; |inj\;r\;c\;v| \succ_r v' 
\]

\noindent 
which is what we need to prove assuming the if-part (*) in the
theorem above. Since this is a universally quantified formula,
we just need to fix a $v'$. We can then prove the implication
by assuming

\[
\text{(a)}\;\;\vdash v' : r\;\; \text{and} \;\;
\text{(b)}\;\;inj\;r\;c\;v = |v'|
\]

\noindent 
and our goal is

\[
(goal)\;\;inj\;r\;c\;v \succ_r v'
\]

\noindent 
There are already two lemmas proved that can transform 
the assumptions (a) and (b) into

\[
\text{(a*)}\;\;\vdash proj\;r\;c\;v' : der\;c\;r\;\; \text{and} \;\;
\text{(b*)}\;\;c\,\#\,|v| = |v'|
\]

\noindent 
Another lemma shows that

\[
|v'| = c\,\#\,|proj\;r\;c\;v|
\]

\noindent 
Using (b*) we can therefore infer 

\[
\text{(b**)}\;\;|v| = |proj\;r\;c\;v|
\]

\noindent 
The main idea of the proof is now a simple instantiation
of the assumption $POSIX\;v\;(der\;c\;r)$. If we unfold 
the $POSIX$ definition, we get

\[
\forall v'.\;
\text{if}\;\vdash v' : der\;c\;r\; \text{and} \;|v| = |v'|\;
\text{then}\; v \succ_{der\;c\;r}\; v' 
\]

\noindent 
We can instantiate this $v'$ with $proj\;r\;c\;v'$ and can use 
(a*) and (b**) in order to infer

\[
v \succ_{der\;c\;r}\; proj\;r\;c\;v'
\]

\noindent 
The point of the side-lemma below is that we can ``add'' an
$inj$ to both sides to obtain

\[
inj\;r\;c\;v \succ_r\; inj\;r\;c\;(proj\;r\;c\;v')
\]

\noindent Finally there is already a lemma proved that shows
that an injection and projection is the identity, meaning

\[
inj\;r\;c\;(proj\;r\;c\;v') = v'
\]

\noindent 
With this we have shown our goal (pending a proof of the side-lemma 
next).


\subsection*{Side-Lemma}

A side-lemma needed for the theorem above which might be true, but can also be false, is as follows:

\begin{center}
\begin{tabular}{lll}
If   & (1) & $v_1 \succ_{der\;c\;r} v_2$,\\
     & (2) & $\vdash v_1 : der\;c\;r$, and\\ 
     & (3) & $\vdash v_2 : der\;c\;r$ holds,\\
then &     & $inj\;r\;c\;v_1 \succ_r inj\;r\;c\;v_2$ also holds.  
\end{tabular}
\end{center}

\noindent It essentially states that if one value $v_1$ is 
bigger than $v_2$ then this ordering is preserved under 
injections. This is proved by induction (on the definition of 
$der$\ldots this is very similar to an induction on $r$).
\bigskip

\noindent
The case that is still unproved is the sequence case where we 
assume $r = r_1\cdot r_2$ and also $r_1$ being nullable.
The derivative $der\;c\;r$ is then

\begin{center}
$der\;c\;r = ((der\;c\;r_1) \cdot r_2) + (der\;c\;r_2)$
\end{center}

\noindent 
or without the parentheses

\begin{center}
$der\;c\;r = (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$
\end{center}

\noindent 
In this case the assumptions are

\begin{center}
\begin{tabular}{ll}
(a) & $v_1 \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} v_2$\\
(b) & $\vdash v_1 : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
(c) & $\vdash v_2 : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
(d) & $nullable(r_1)$
\end{tabular}
\end{center}

\noindent 
The induction hypotheses are

\begin{center}
\begin{tabular}{ll}
(IH1) & $\forall v_1 v_2.\;v_1 \succ_{der\;c\;r_1} v_2
\;\wedge\; \vdash v_1 : der\;c\;r_1 \;\wedge\; 
\vdash v_2 : der\;c\;r_1\qquad$\\
      & $\hfill\longrightarrow
         inj\;r_1\;c\;v_1 \succ{r_1} \;inj\;r_1\;c\;v_2$\smallskip\\
(IH2) & $\forall v_1 v_2.\;v_1 \succ_{der\;c\;r_2} v_2
\;\wedge\; \vdash v_2 : der\;c\;r_2 \;\wedge\; 
\vdash v_2 : der\;c\;r_2\qquad$\\
      & $\hfill\longrightarrow
         inj\;r_2\;c\;v_1 \succ{r_2} \;inj\;r_2\;c\;v_2$\\
\end{tabular}
\end{center}

\noindent 
The goal is

\[
(goal)\qquad
inj\; (r_1 \cdot r_2)\;c\;v_1 \succ_{r_1 \cdot r_2} 
inj\; (r_1 \cdot r_2)\;c\;v_2
\]

\noindent 
If we analyse how (a) could have arisen (that is make a case
distinction), then we will find four cases:

\begin{center}
\begin{tabular}{ll}
LL & $v_1 = Left(w_1)$, $v_2 = Left(w_2)$\\
LR & $v_1 = Left(w_1)$, $v_2 = Right(w_2)$\\
RL & $v_1 = Right(w_1)$, $v_2 = Left(w_2)$\\
RR & $v_1 = Right(w_1)$, $v_2 = Right(w_2)$\\
\end{tabular}
\end{center}


\noindent 
We have to establish our goal in all four cases. 


\subsubsection*{Case LR}

The corresponding rule (instantiated) is:

\begin{center}
\begin{tabular}{c}
$len\,|w_1| \geq len\,|w_2|$\\
\hline
$Left(w_1) \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} Right(w_2)$
\end{tabular}
\end{center}

\noindent 
This means we can also assume in this case

\[
(e)\quad len\,|w_1| \geq len\,|w_2|
\] 

\noindent 
which is the premise of the rule above.
Instantiating $v_1$ and $v_2$ in the assumptions (b) and (c)
gives us

\begin{center}
\begin{tabular}{ll}
(b*) & $\vdash Left(w_1) : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
(c*) & $\vdash Right(w_2) : (der\;c\;r_1) \cdot r_2 + der\;c\;r_2$\\
\end{tabular}
\end{center}

\noindent Since these are assumptions, we can further analyse
how they could have arisen according to the rules of $\vdash
\_ : \_\,$. This gives us two new assumptions

\begin{center}
\begin{tabular}{ll}
(b**) & $\vdash w_1 : (der\;c\;r_1) \cdot r_2$\\
(c**) & $\vdash w_2 : der\;c\;r_2$\\
\end{tabular}
\end{center}

\noindent 
Looking at (b**) we can further analyse how this
judgement could have arisen. This tells us that $w_1$
must have been a sequence, say $u_1\cdot u_2$, with

\begin{center}
\begin{tabular}{ll}
(b***) & $\vdash u_1 : der\;c\;r_1$\\
       & $\vdash u_2 : r_2$\\
\end{tabular}
\end{center}

\noindent 
Instantiating the goal means we need to prove

\[
inj\; (r_1 \cdot r_2)\;c\;(Left(u_1\cdot u_2)) \succ_{r_1 \cdot r_2} 
inj\; (r_1 \cdot r_2)\;c\;(Right(w_2))
\]

\noindent 
We can simplify this according to the rules of $inj$:

\[
(inj\; r_1\;c\;u_1)\cdot u_2 \succ_{r_1 \cdot r_2} 
(mkeps\;r_1) \cdot (inj\; r_2\;c\;w_2)
\]

\noindent
This is what we need to prove. There are only two rules that
can be used to prove this judgement:

\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c}
$v_1 = v_1'$\qquad $v_2 \succ_{r_2} v_2'$\\
\hline
$v_1\cdot v_2 \succ_{r_1\cdot r_2} v_1'\cdot v_2'$
\end{tabular} &
\begin{tabular}{c}
$v_1 \succ_{r_1} v_1'$\\
\hline
$v_1\cdot v_2 \succ_{r_1\cdot r_2} v_1'\cdot v_2'$
\end{tabular}
\end{tabular}
\end{center}

\noindent 
Using the left rule would mean we need to show that

\[
inj\; r_1\;c\;u_1 = mkeps\;r_1
\]

\noindent 
but this can never be the case.\footnote{Actually Isabelle
found this out after analysing its argument. ;o)} Lets assume
it would be true, then also if we flat each side, it must hold
that

\[
|inj\; r_1\;c\;u_1| = |mkeps\;r_1|
\]

\noindent 
But this leads to a contradiction, because the right-hand side
will be equal to the empty list, or empty string. This is 
because we assumed $nullable(r_1)$ and there is a lemma
called \texttt{mkeps\_flat} which shows this. On the other
side we know by assumption (b***) and lemma \texttt{v4} that 
the other side needs to be a string starting with $c$ (since
we inject $c$ into $u_1$). The empty string can never be equal 
to something starting with $c$\ldots therefore there is a 
contradiction.

That means we can only use the rule on the right-hand side to 
prove our goal. This implies we need to prove

\[
inj\; r_1\;c\;u_1 \succ_{r_1} mkeps\;r_1
\]


\subsubsection*{Case RL}

The corresponding rule (instantiated) is:

\begin{center}
\begin{tabular}{c}
$len\,|w_1| > len\,|w_2|$\\
\hline
$Right(w_1) \succ_{(der\;c\;r_1) \cdot r_2 + der\;c\;r_2} Left(w_2)$
\end{tabular}
\end{center}


\end{document}