Bewijs

4 files changed, 157 insertions, 142 deletions

diff --git a/analyse-proof.tex b/analyse-proof.tex
new file mode 100644
index 0000000..5cc8e6a
--- /dev/null
+++ b/analyse-proof.tex
@@ -0,0 +1,122 @@
+% vim: spelllang=nl:
+\subsection{Bewijs}
+\label{sec:analyse:proof}
+
+We zullen het buitenste programma als beschreven op~\pageref{pgm:reverse} met
+$P_1$ aanduiden, en het recursieve binnenste programma met $P_2$.
+
+Nu laten we zien dat deze code daadwerkelijk alle mogelijke strings omdraait,
+oftewel: er is een afleidingsboom voor
+$$
+\trans
+	{P_1}{[s:\Nil]}{(\Nil,\emptystore)}
+	{\Nil}{[s^R:\Nil]}{\emptystore}
+$$
+
+voor alle $s\in\String$, met
+\begin{align*}
+	\lambda^R &= \lambda, \\
+	(c~s)^R   &= s^R c.
+\end{align*}
+
+Het bewijs gaat in twee stappen. Eerst bewijzen we dat $P_2$ de eerste string
+op de stack achter de omgedraaide versie van de tweede string op de stack
+plaatst, en dat output. Dit wordt bewezen in \autoref{thm:p2}. Hiermee kunnen
+we van $P_1$ bewijzen dat het de eerste string op de stack omgedraaid output.
+Dit doen we in \autoref{thm:p1}.
+
+\begin{thm}
+	{
+		Voor alle $s,t\in\String$ met $t\ne\lambda$ hebben we een afleidingsboom
+		voor
+		\normalfont
+		\belowdisplayskip=-1em
+		$$
+		\trans
+			{P_2}{[s:t:\Nil]}{(\Nil,\emptystore)}
+			{\Nil}{[t^R~s:\Nil]}{\emptystore}.
+		$$
+	}
+	\label{thm:p2}
+	\begin{proof}[Bewijs]
+		Met inductie naar de lengte van $t$.
+
+		Inductiebasis: stel $t=c$, met $c$ een willekeurig karakter. In dit geval
+		krijgen we de boom in \autoref{fig:tree:bootstrap-base}. Hier is $c$ geen
+		variabele maar het werkelijke karakter \lit{c}. We kunnen \emph{CleanSmurf}
+		namelijk helaas nog niet vertellen dat een bepaalde variabele lengte $1$
+		heeft. Uit de afleidingsboom blijkt dat $c$ niet gebruikt wordt als naam in
+		de variable store, waardoor we ons geen zorgen hoeven te maken over
+		situaties waar $c$ gelijk is aan \'e\'en van de variabelenamen die we
+		gebruiken. Uit de boom in \autoref{fig:tree:bootstrap-base} volgt dus dat
+		bovenstaande stelling geldt voor $t$ van lengte $1$.
+
+		\medskip
+		Inductiestap: we nemen aan dat we een afleidingsboom hebben voor
+		\begin{equation}\tag{IH}\label{eq:p2:ih}
+			\trans
+				{P_2}{[s:t:\Nil]}{(\Nil,\emptystore)}
+				{\Nil}{[t^R:s:\Nil]}{\emptystore}
+		\end{equation}
+
+		Vervolgens laten we zien dat we voor elke $c\in\Char$ ook een
+		afleidingsboom hebben voor
+		$$
+		\trans
+			{P_2}{[s:c~t:\Nil]}{(\Nil,\emptystore)}
+			{\Nil}{[t^R~c~s:\Nil]}{\emptystore}.
+		$$
+
+		Ook hiervoor kunnen we \emph{CleanSmurf} gebruiken. Geven
+		we~\eqref{eq:p2:ih} mee als aanname, dan krijgen we de boom
+		in~\autoref{fig:tree:bootstrap-step}. Net zoals bij de inductiebasis moeten
+		we $c$ als letterlijk \lit{c} meegeven. Eenzelfde inspectie van de
+		boom wijst uit dat dit geen problemen oplevert met andere waarden voor $c$.
+
+		\medskip
+		Uit het principe van inductie naar de structuur van de string $t$ volgt nu
+		voor alle $s,t\in\String$ met $t\ne\lambda$ dat we een afleidingsboom
+		hebben voor
+
+		$$
+		\trans
+			{P_2}{[s:t:\Nil]}{(\Nil,\emptystore)}
+			{\Nil}{[t^R~s:\Nil]}{\emptystore}.
+		$$
+	\end{proof}
+\end{thm}
+
+\begin{thm}
+	{
+		Voor alle $s\in\String$ hebben we een afleidingsboom voor
+		\normalfont
+		\belowdisplayskip=-1em
+		$$
+		\trans
+			{P_1}{[s:\Nil]}{(\Nil,\emptystore)}
+			{\Nil}{[s^R:\Nil]}{\emptystore}.
+		$$
+	}
+	\label{thm:p1}
+	\begin{proof}[Bewijs]
+		We maken een gevalsonderscheiding over de structuur van $s$.
+
+		\begin{enumerate}[label={Geval \arabic*.},itemindent=1.5\parindent]
+			\item Stel $s=\lambda$. In dit geval hebben we geen enkele variabele. We
+				kunnen dus simpelweg een boom maken. Dit geeft de boom in
+				\autoref{fig:tree:lambda}, waaruit blijkt dat de stelling inderdaad
+				geldt voor $s=\lambda$:
+				$$
+				\trans
+					{P_1}{[\lambda:\Nil]}{(\Nil,\emptystore)}
+					{\Nil}{[\lambda:\Nil]}{\emptystore}.
+				$$
+
+			\item Stel $s=c~t$. In dit geval gaat $P_1$ het recursieve binnenste
+				programma gebruiken. We maken dus een boom die het resultaat van
+				\autoref{thm:p2} als aanname mag gebruiken. Dit geeft de afleidingsboom
+				in \autoref{fig:tree:bootstrap}, waaruit blijkt dat de bovenstaande
+				stelling geldt voor alle $s\ne\lambda$.
+		\end{enumerate}
+	\end{proof}
+\end{thm}
diff --git a/analyse.tex b/analyse.tex
index e63de78..4ed43f7 100644
--- a/analyse.tex
+++ b/analyse.tex
@@ -15,6 +15,7 @@ programma geschreven voor het omdraaien van een string. Dit programma ziet er
 als volgt uit:
 
 \begin{smurf}
+\label{pgm:reverse}
 \footnotesize
 i "input" p
 "input" g
@@ -37,116 +38,5 @@ Het bovenstaande programma is correcte Smurfsyntax en hierdoor niet erg
 leesbaar. We hebben daarom getracht het programma iets leesbaarder te maken: 
 
 \input{reverse2}
-
 \input{explanation}
-
-\subsection{bewijs} 
-Nu laten we zien dat deze code daadwerkelijk alle mogelijke strings omdraait,
-oftewel: er is een afleidingsboom voor
-$$
-\trans
-	{Programma}{[s:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[s^R:\Nil]}{\st}
-$$
-
-voor alle s, waar
-$$(c~s)^R=s^R c$$
-$$\lambda^R=\lambda$$
-
-
-Dit bewijs gaat in twee stappen.
-
-$$
-\trans
-	{Programma}{[c:s:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[s^R:c:\Nil]}{\st}
-$$
-
-
-
-Eerst bewijzen we aan de hand van inductie dat voor het recursieve binnenste programma geldt dat voor de stack met waarde $(s:t:\Nil)$ de output $(t^R: s^R: \Nil)$  zal zijn. 
-
-\bigskip 
-
-Ook wel
-$$
-\trans
-	{Programma}{[cs:g:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[g^R:cs:\Nil]}{\st}
-$$
-
-\begin{proof}[Bewijs]
-\medskip
-	Met inductie naar de lengte van $g$.
-
-	Basisgeval: $g=\lambda$.
-	
-	We moeten laten zien dat 
-	
-	$$
-    \trans
-    	{Programma}{[s:c:\Nil]}{(\Nil,\emptystore)}
-    	{\Nil}{[cs:\Nil]}{\st}
-    $$
-    
-    Dit volgt uit bewijsboom *referentie*
-    \bigskip
-    
-    inductiestap: $g = t $ (t is een variabele)
-    
-    We moeten laten zien dat
-    $$
-    \trans
-	{Programma}{[s:ct:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[t^R:cs:\Nil]}{\st}
-    $$
-
-    Met als inductiehypothese 
-     $$
-    \trans
-	{Programma}{[cs:t:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[t^R:cs:\Nil]}{\st}
-    $$
-    
-    Dit volgt uit bewijsboom * referentie*
-	
-\end{proof}
-
-Vervolgens zullen we laten zien dat voor het buitenste programma geldt dat 
-
-$$
-\trans
-	{Programma}{[s:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[s^R:\Nil]}{\st}
-$$
-
-Hierbij zullen we laten zien dat dit geldt voor zowel een lege input als een niet-lege input. 
-Hierbij mogen we natuurlijk gebruik maken van wat we hierboven bewezen hebben. 
-\medskip
-
-Eerst zullen we laten zien dat het geldt voor een lege input 
-
-We zullen dus moeten laten zien dat:
-$$
-\trans
-	{Programma}{[\Nil:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[\Nil:\Nil]}{\st}
-$$
-
-Dit volgt direct uit bewijsboom *referentie* 
-
-
-Ten tweede zullen we laten zien dat het geldt voor een niet-lege input 
-We zullen dus moeten laten zien dat:
-$$
-\trans
-	{Programma}{[s:\Nil]}{(\Nil,\emptystore)}
-	{\Nil}{[s^t:\Nil]}{\st}
-$$
-
-Dit volgt uit de bewijsbomen *referentie* (3*) 
-
-
-
-
-
+\input{analyse-proof}
diff --git a/tree-gen-bootstrap.tex b/tree-gen-bootstrap.tex
index e13e2fc..b2494d5 100644
--- a/tree-gen-bootstrap.tex
+++ b/tree-gen-bootstrap.tex
@@ -1,59 +1,59 @@
-\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\axjustifies\trans{\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{""}:\texttt{"c$s$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rule{assumption}{}}\]
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-\justifies{}\trans{\StmPush~\texttt{"c$s$"}:\StmPush~\texttt{""}:\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\Nil,\emptyset\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
-\justifies{}\trans{\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
+\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\[\axjustifies\trans{\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{""}:\texttt{"c$t$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rule{assumption}{}}\]
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+\justifies{}\trans{\StmPush~\texttt{"c$t$"}:\StmPush~\texttt{""}:\StmPush~\texttt{"\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx"}:\StmPush~\texttt{"u"}:\StmPut:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmHead:\StmPush~\texttt{"v"}:\StmGet:\StmCat:\StmPush~\texttt{"v"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmTail:\StmPush~\texttt{"w"}:\StmPut:\StmPush~\texttt{"w"}:\StmGet:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"o"}:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmQuotify:\StmPush~\texttt{"v"}:\StmGet:\StmQuotify:\StmPush~\texttt{"u"}:\StmGet:\StmQuotify:\StmCat:\StmCat:\StmPush~\texttt{"u"}:\StmGet:\StmCat:\StmPush~\texttt{"w"}:\StmGet:\StmPut:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
+\justifies{}\trans{\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
 \textquotedblright{}\textquotedblright{}\textquotedblright{}\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gh\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gt\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}g\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}o\textbackslash{}\textquotedblright{}+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}gq++\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gp\textbackslash{}\textquotedblright{}\textbackslash{}\textquotedblright{}pgx\textquotedblright{}\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx
-"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$s$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
+"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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-"},\texttt{"input"}\mapsto\texttt{"c$s$"}\}\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rexecns}\]
-\justifies{}\trans{\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$s$"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$s$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
+"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rexecns}\]
+\justifies{}\trans{\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\{\texttt{""}\mapsto\texttt{"\textquotedblright{}\textquotedblright{}o"},\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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-\justifies{}\trans{\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{""}:\texttt{"\textquotedblright{}\textquotedblright{}o"}:\texttt{"c$s$"}:\Nil,\{\texttt{"c$s$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
+"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rgetns}\]
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-\justifies{}\trans{\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}\textquotedblright{}o"}:\texttt{"c$s$"}:\Nil,\{\texttt{"c$s$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
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-\justifies{}\trans{\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$s$"}:\Nil,\{\texttt{"c$s$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
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+\justifies{}\trans{\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\{\texttt{"c$t$"}\mapsto\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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-\justifies{}\trans{\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$s$"}:\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
+"},\texttt{"input"}\mapsto\texttt{"c$t$"}\}\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
+\justifies{}\trans{\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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-"}:\texttt{"c$s$"}:\Nil,\{\texttt{"input"}\mapsto\texttt{"c$s$"}\}\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rputns}\]
-\justifies{}\trans{\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"input"}:\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
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-"}:\texttt{"c$s$"}:\Nil,\{\texttt{"input"}\mapsto\texttt{"c$s$"}\}\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rgetns}\]
-\justifies{}\trans{\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}c\textit{q}($s$)\textquotedblright{}
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+\justifies{}\trans{\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"\textquotedblright{}c\textit{q}($t$)\textquotedblright{}
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-"}:\texttt{"c$s$"}:\Nil,\{\texttt{"input"}\mapsto\texttt{"c$s$"}\}\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
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+"}:\StmCat:\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"input"}:\texttt{"c$t$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rputns}\]
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 \textquotedblright{}\textquotedblright{}\textquotedblright{}\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gh\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gt\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}g\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}o\textbackslash{}\textquotedblright{}+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}gq++\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gp\textbackslash{}\textquotedblright{}\textbackslash{}\textquotedblright{}pgx\textquotedblright{}\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx
-"}:\StmCat:\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$s$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
+"}:\StmCat:\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\Nil}{\left(\texttt{"c$t$"}:\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rpushns}\]
 \justifies{}\trans{\StmInput:\StmPush~\texttt{"input"}:\StmPut:\StmPush~\texttt{"input"}:\StmGet:\StmPush~\texttt{"input"}:\StmGet:\StmQuotify:\StmPush~\texttt{"
 \textquotedblright{}\textquotedblright{}\textquotedblright{}\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gh\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gt\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}p\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}g\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}o\textbackslash{}\textquotedblright{}+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}v\textbackslash{}\textquotedblright{}gq\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}gq++\textbackslash{}\textquotedblright{}u\textbackslash{}\textquotedblright{}g+\textbackslash{}\textquotedblright{}w\textbackslash{}\textquotedblright{}gp\textbackslash{}\textquotedblright{}\textbackslash{}\textquotedblright{}pgx\textquotedblright{}\textquotedblright{}u\textquotedblright{}p\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gh\textquotedblright{}v\textquotedblright{}g+\textquotedblright{}v\textquotedblright{}p\textquotedblright{}w\textquotedblright{}gt\textquotedblright{}w\textquotedblright{}p\textquotedblright{}w\textquotedblright{}g\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}o\textquotedblright{}+\textquotedblright{}w\textquotedblright{}gq\textquotedblright{}v\textquotedblright{}gq\textquotedblright{}u\textquotedblright{}gq++\textquotedblright{}u\textquotedblright{}g+\textquotedblright{}w\textquotedblright{}gp\textquotedblright{}\textquotedblright{}pgx
-"}:\StmCat:\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\texttt{"c$s$"}:\Nil}{\left(\Nil,\emptyset\right)}{\Nil}{\texttt{"$s^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rinputns}
\ No newline at end of file
+"}:\StmCat:\StmPush~\texttt{"input"}:\StmGet:\StmPut:\StmPush~\texttt{"\textquotedblright{}\textquotedblright{}o"}:\StmPush~\texttt{""}:\StmPut:\StmGet:\StmExec}{\texttt{"c$t$"}:\Nil}{\left(\Nil,\emptyset\right)}{\Nil}{\texttt{"$t^R$c"}:\Nil}{\left(\Nil,\emptyset\right)}\using{\rinputns}
diff --git a/werkstuk.tex b/werkstuk.tex
index c4488f6..6935ea0 100644
--- a/werkstuk.tex
+++ b/werkstuk.tex
@@ -36,6 +36,7 @@
 \addto\extrasdutch{%
 	\renewcommand{\sectionautorefname}{Sectie}
 	\renewcommand{\subsectionautorefname}{Paragraaf}
+	\renewcommand{\figureautorefname}{Figuur}
 }
 
 % Eigen commando's en environments, niet specifiek voor Smurf
@@ -48,6 +49,8 @@
 ]{lined}
 \declaretheorem[style=lined]{exmp}
 
+\newtheorem{thm}{Stelling}
+
 \begin{document}
 
 \maketitle