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	Special Thanks to\\

	\hfill \\

	\emph{Dr.techn. Walther Neuper}\\

	\emph{Dipl.-Ing. Bernhard Geiger}

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\begin{abstract}

The Baccalaureate Thesis creates interactivee course material for Signal Processing (SP) based on the experimental educational math assistant Isabelle/{\sisac} ({\em Isa}belle for Transparent {\em C}alculations in Applied Mathematics).

\par The content of the course material is defined together with the Institute for Signal Processing and Speech Communication (SPSC) of Graz University of Technology (TUG). The content is planned to be used in specific lectures and labs of the SPSC and thus is thoroughly concerned with underlying mathematical and physical theory.

One challenge of this thesis is, that much theory required for SPSC is not yet mechanized in Computer Theorem Provers (TP); so this thesis will provide preliminary definitions  and theorems (without proofs~!) implemented in Isabelle \emph{theories}.

\par Another challenge is the implementation of interactivee courses: this is done within the educational math assistant Isabelle/{\sisac}, which is under development at Austrian Universities, one of them TU Graz. The present state of {\sisac{}} happens to provide the {\em first} occasion for authoring by a non-member of the {\sisac}-developer team. So this challenge involves  alpha-testing of the underlying \emph{TP-based programing language}, because error messages are still not user-friendly and need frequent contact with {\sisac}-developers.

So the practical outcome of this thesis is twofold:

\begin{enumerate}

\item Interactive course material hopefully useful in education within the SPSC and within \emph{STEOP}\footnote{German: "`Studieneingangs- und Orientierungsphase"'}, the introductory orientation phase at TUG, as a preview for students in Telematics on later application of math knowledge introduced in the first semester and

\item A detailed description of technicalities in programing implemented as an interactivee Isabelle/Isar theory, providing future programers with guidelines and {\sisac}-developers with feedback in usability of the CTP-based program language. 

\end{enumerate}

\end{abstract}

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%This thesis is structured into a fundamental part introducing the motivation, the basic notions concerning the thesis aswell as the {\sisac{}} project and describing the mathematic base. Further a automatically generated practical part representing the work on {\sisac{}} which can be extended.

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\part{Project Fundamentals}

\section{Introduction}

Didactics of mathematics faces a specific issue, a gap between (1) introduction of math concepts and skills and (2) application of these concepts and skills, which usually are separated into different units in curricula (for good reasons). For instance, (1) teaching partial fraction decomposition is separated from (2) application for inverse Z-transform in signal processing.

This gap is an obstacle for applying math as an fundamental thinking technology in engineering: In (1) motivation is lacking because the question ``What is this stuff good for~?'' cannot be treated sufficiently, and in (2) the ``stuff'' is not available to students in higher semesters as widespread experience shows.

\medskip

Motivated by this didactical issue on the one hand, and ongoing R\&D on a novel kind of educational mathematics assistant at Graz University of Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to cope with this issue on the other hand, several institutes are planning to join their expertise: the Institutes for Institute for Information Systems and Computer Media (IICM), the Institute for Software Technology (IST), the Institutes for Mathematics, the Signal Processing and Speech Communication Institute (SPSC), the Institute for Structural Analysis and the Institute of Electrical Measurement and Measurement Signal Processing.

\par This thesis is the first attempt to tackle the above mentioned issue, it focuses on Telematics, because these specific studies focus on mathematics in \emph{STEOP}, the introductory orientation phase. \emph{STEOP} is considered an opportunity to investigate the impact of {\sisac}'s prototype on the issue and others.

\medskip

The thesis is structured as follows: Part~I concerns theory and project controlling, Part~II the implementation work, where the latter is the main work, Part~III the resume including summary, open questions and conclusion and the appendixes in Part~IV.

\par In part I, Section~\ref{isabisac} gives a brief description of the state-of-the-art for educational math assistants (Section~\ref{emas}) and introduces the notions required for the implementation work (Section~\ref{math-auth}). In particular, Section~\ref{user-guid} explains, why math authoring in {\sisac{}} is {\em not} concerned with interaction (and thus not with user guidance etc at all~!). So a concise description of the thesis' goals needs to be postponed to Section~\ref{sec:goals}.

\par Section~\ref{sp} analyzes a problems defined by the SPSC for the knowledge already provided (Section~\ref{know-isab}, Section~\ref{know-isac}), discusses the selection of problems for implementation (Section~\ref{know-missing}) TODO: further structure ?

%(\S\ref{})

\section{Mechanization of Math in Isabelle/ISAC\label{isabisac}}

%\subsubsection*{Notes on Mechanization of Mathematics}

%This thesis tries to \emph{connect} these two worlds and is one of the first guidelines to implement problem classes in {\sisac}. As we are still in a eary part of development, this is the first thesis dealing within this topic and there is \emph{no} related work to guid through. A more detailed description about this fact can be found in Section \ref{sec:related}.

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As mentioned in the introduction, a prototype of an educational math assistant called {\sisac}\footnote{{\sisac}=\textbf{Isa}belle \cite{Nipkow-Paulson-Wenzel:2002} for \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges the gap between (1) introducation and (2) application of mathematics: {\sisac} is based on Computer Theorem Proving (TP), a technology which requires each fact and each action justified by formal logic, so {{\sisac{}}} makes justifications transparent to students in interactivee step-wise problem solving. By that way {\sisac} already can serve both:

\begin{enumerate}

\item Introduction of math stuff (in e.g. partial fraction decomposition) by stepwise explaining and exercising respective symbolic calculations with ``next step guidance (NSG)'' and rigorously checking steps freely input by students  --- this also in context with advanced applications (where the stuff to be taught in higher semesters can be skimmed through by NSG), and

\item Application of math stuff in advanced engineering courses (e.g. problems to be solved by inverse Z-transform in a Signal Processing Lab) --- and now without much ado about basic math techniques (like partial fraction decomposition): ``next step guidance'' supports students in independently (re-)adopting such techniques.

\end{enumerate}

Before the question is answers, how {\sisac} accomplishes this task from a technical point of view, some remarks on the state-of-the-art is given, therefor follow up Section~\ref{emas}.

\subsection{Educational Mathematics Assistants (EMAs)}\label{emas}

Educational software in mathematics is, if at all, based on Computer Algebra Systems (CAS, for instance \cite{progr-mathematica,prog-maple06}), Dynamic Geometry Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org, Cinderella http://www.cinderella.de/, GCLC http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These base technologies are used to program math lessons and sometimes even exercises. The latter are cumbersome: the steps towards a solution of such an interactivee exercise need to be provided with feedback, where at each step a wide variety of possible input has to be foreseen by the programer --- so such interactivee exercises either require high development efforts or the exercises constrain possible inputs.

A new generation of educational math assistants (EMAs) is emerging presently, which is based on Theorem Proving (TP). TP, for instance Isabelle \cite{Nipkow-Paulson-Wenzel:2002} and Coq \cite{Huet_all:94}, is a technology which requires each fact and each action justified by formal logic. Pushed by demands for \textit{proven} correctness of safety-critical software TP advances into software engineering; from these advancements computer mathematics benefits in general, and math education in particular. Two features of TP are immediately beneficial for learning:

\paragraph{TP have knowledge in human readable format,} that is in standard predicate calculus. TP following the LCF-tradition have that knowledge down to the basic definitions of set, equality, etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html}; following the typical deductive development of math, natural numbers are defined and their properties proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html}, etc. Present knowledge mechanized in TP exceeds high-school mathematics by far, however by knowledge required in software technology, and not in other engineering sciences.

\paragraph{TP can model the whole problem solving process} in mathematical problem solving {\em within} a coherent logical framework. This is already being done by three projects, by Ralph-Johan Back \cite{Back-SD09}, by ActiveMath \cite{ActiveMath-MAIN11} and by Carnegie Mellon Tutor \cite{mat-tutor-cmu-MAIN11}.

Having the whole problem solving process within a logical coherent system, such a design guarantees correctness of intermediate steps and of the result (which seems essential for math software); and the second advantage is that TP provides a wealth of theories which can be exploited for mechanizing other features essential for educational software.

\subsection{Generation of User Guidance in EMAs}\label{user-guid}

One essential feature for educational software is feedback to user input and assistance in coming to a solution.

\paragraph{Checking user input} by ATP during stepwise problem solving is being accomplished by the three projects mentioned above \cite{Back-SD09,ActiveMath-MAIN11,mat-tutor-cmu-MAIN11} exclusively. They model the whole problem solving process as mentioned above, so all what happens between formalized assumptions (or formal specification) and goal (or fulfilled postcondition) can be mechanized. Such mechanization promises to greatly extend the scope of educational software in stepwise problem solving.

\paragraph{Next step guidance (NSG)} comprises the system's ability to propose a next step; this is a challenge for TP: either a radical restriction of the search space by restriction to very specific problem classes is required, or much care and effort is required in designing possible variants in the process of problem solving \cite{proof-strategies-11}.

Another approach is restricted to problem solving in engineering domains, where a problem is specified by input, precondition, output and postcondition, and where the postcondition is proven by ATP behind the scenes \cite{wn:lucas-interp-12}: Here the possible variants in the process of problem solving are provided with feedback {\em automatically}, if the problem is described in a TP-based programing language~\cite{plmms10}: the programer only describes the math algorithm without caring about interaction (the respective program is functional and even has no in/output statements~!); interaction is generated as a side-effect by the interpreter --- an efficient separation of concern between math programers and dialog designers promising application all over engineering disciplines.

\subsection{Math Authoring in Isabelle/ISAC\label{math-auth}}

Authoring new mathematics knowledge in {\sisac} can be compared with ``application programing'' of engineering problems; most of such programing uses CAS-based programing languages (CAS = Computer Algebra Systems; e.g. Mathematica's \cite{progr-mathematica} or Maple's programing language \cite{prog-maple06}).

{\sisac}, however, uses a novel type of TP-based language \cite{plmms10} for describing how to construct a solution to an engineering problem and for calling equation solvers, integration, etc~\footnote{Implementation of CAS-like functionality in TP is not primarily concerned with efficiency, but with a didactic question: What to decide for: for high-brow algorithms at the state-of-the-art or for elementary algorithms comprehensible for students~?} within TP; TP can ensure ``systems that never make a mistake'' \cite{casproto} --- are impossible for CAS which have no logics underlying.

With writing such TP-based programs authoring is perfect, the application programer is not concerned with interaction or with user guidance: this is concern of a novel kind of program interpreter called Lucas-Interpreter \cite{wn:lucas-interp-12}. This interpreter hands over control to a dialog component at each step of calculation (like a debugger at breakpoints) and calls automated TP to check user input following personalized strategies according to a feedback module.

\medskip

However ``application programing with TP'' is not done with writing a program: according to the principles of TP, each step must be justified. Such justifications are given by theorems. So all steps must be related to some theorem, if there is no such theorem it must be added to the existing knowledge, which is organized in so-called \textbf{theories} in  Isabelle. A theorem must be proven; fortunately Isabelle comprises a mechanism (called ``axiomatization''), which allows to omit proofs. Such a theorem is shown in Example~\ref{eg:neuper1}.

\begin{example}

{\small\begin{tabbing}

123\=123\=123\=123\=\kill

\hfill \\

\>axiomatization where \\

\>\>  rule1: "1 = $\delta$ [n]" and\\

\>\>  rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and\\

\>\>  rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and \\

\>\>  rule4: "|| z || > || $\alpha$ || ==> z / (z - $\alpha$) = $\alpha^n$ * u [n]" and\\

\>\>  rule5: "|| z || < || $\alpha$ || ==> z / (z - $\alpha$) = -($\alpha^n$) * u [-n - 1]" and\\

\>\>  rule6: "|| z || > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]"

\end{tabbing}

}

\caption{Axiomatization in Isabelle\label{eg:neuper1}}

\end{example}

In order to provide TP with logical facts for checking user input, the Lucas-Interpreter requires a \textbf{specification}. Such a specification is shown in Example~\ref{eg:neuper2}.

\begin{example}

{\small\begin{tabbing}

123,\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill

\hfill \\

Specification no.1:\\

%\>input\>: $\{\;r={\it arbitraryFix}\;\}$  \\

\>input    \>: $\{\;r\;\}$  \\

\>precond  \>: $0 < r$   \\

\>output   \>: $\{\;A,\; u,v\;\}$ \\

\>postcond \>:{\small  $\;A=2uv-u^2 \;\land\; (\frac{u}{2})^2+(\frac{v}{2})^2=r^2 \;\land$}\\

\>     \>\>{\small $\;\forall \;A^\prime\; u^\prime \;v^\prime.\;(A^\prime=2u^\prime v^\prime-(u^\prime)^2 \land

(\frac{u^\prime}{2})^2+(\frac{v^\prime}{2})^2=r^2) \Longrightarrow A^\prime \leq A$} \\

\>props\>: $\{\;A=2uv-u^2,\;(\frac{u}{2})^2+(\frac{v}{2})^2=r^2\;\}$

\end{tabbing}

}

\caption{Specification for the Lucas-Interpreter\label{eg:neuper2}}

\end{example}

Such a specification is checked before the execution of a program is started, the same applies for sub-programs. In the following example program (Example~\ref{eg:subprob}) the sub-programs are designated by \ttfamily SubProblem \normalfont:

\begin{example}

\hfill \\

{\ttfamily \begin{tabbing}

``(L\_L::bool list) = (\=SubProblem (\=Test','' \\

``\>\>[linear,univariate,equation,test],'' \\

``\>\>[Test,solve\_linear])'' \\

``\>[BOOL equ, REAL z])'' \\

\end{tabbing}

}

{\small\textit{

	\noindent If a program requires a result which has to be calculated first we can use a subproblem to do so. In our specific case we wanted to calculate the zeros of a fraction and used a subproblem to calculate the zeros of the denominator polynom.

	}}

\caption{Usage of Subproblems in Programs\label{eg:subprob}}

\end{example}

\subsection{Goals of the Thesis}\label{sec:goals}

Imagine a piece of software would be able to support you by understanding every problem class, upcoming in the first years attending university - wouldn't it be great?

\par {{\sisac{}}} tries to do that, but the current state of the art is miles away from this goal and a single implementation of a problem is not enough to change this circumstance. Through this fact it is all the more essential to try, test, research and document the implementation of problem classes from "`real world"' applications. Responding to the abstract at the begin of this document the thesis has two folds; on the one hand certainly to provide interactivee course material for Signal Processing (which means to implement a single problem provided by the Institute of Signal Processing and Speech Communication (SPSC); follow up Calulcations), and to extract experience data respectively help the {{\sisac{}}}-team by setting up a detailed description of technicalities hacking {\sisac{}} on the other hand.

All the notions are in place to describe the task ``Interactive Course Material for Signal Processing based on Isabelle/{\sisac}'', the main task of this thesis, appropriately by the following points:

\begin{enumerate}

\item Analyze the problems given by the SPSC Lab for mathematics \textbf{knowledge required}, search the knowledge already available in Isabelle/{\sisac}, estimate efforts required to fill the gap between knowledge required and knowledge available, and finally select problems for implementation accordingly.

\item Implement the selected problems in Isabelle/{\sisac}, which means, in appropriate Isabelle theories \textbf{for each problem} implement:

  \begin{enumerate}

  \item \textbf{Definitions and theorems} required within the specification (including ``descriptions'' for input variables and output variables) and the program (proofs omitted via ``axiomatization'')

  \item \textbf{A specification} which describes the input variables, the preconditions on the input (a challenge for rigorously exact mathematics~!), the output variables and the postcondition, which relates input to output such that the problem is characterized formally (another challenge for rigorously exact mathematics~!)

  \item \textbf{A program} describing the algorithm which solves the problem, i.e. which constructs output meeting the postcondition. Programming involves identifying the steps (tactics~!) which create the calculation and calling CAS-functions (simplification, equation solvers, etc) appropriately. Modularization of programs into {\tt SubProblems} has to prepare for re-use of code.

  \end{enumerate}

\item Add \textbf{multimedia explanations} to each problem (i.e. to specific definitions, theorems, the specification and the program) such that non-expert students (e.g. within \emph{STEOP}, the introductory orientation phase at TUG) get an idea the problem is about.

\item \textbf{Document the implementation} such that

  \begin{enumerate}

  \item Interactive course material hopefully useful in education within the SPSC and within \emph{STEOP}, the introductory orientation phase at TUG, as a preview for students in Telematics on later application of math knowledge introduced in the first semester and

  \item A detailed description of technicalities in programing implemented as an interactivee Isabelle/Isar theory, providing future programers with guidelines and {\sisac}-developers with feedback in usability of the TP-based program language. 

  \item subsequent application programers have guidelines for further implementation of interactivee course material in SPSC and other engineering sciences

  \item {\sisac{}} developers get feedback for ongoing improvement of the TP-based programing language, the respective development environment and the respective program interpreter (called Lucas-Interpreter)

  \item development of knowledge for engineering sciences is being motivated in the Isabelle community.

  \end{enumerate}

\end{enumerate}

\section{Mechanization of Signal Processing Problems}\label{sp}

\subsection{Relevant Knowledge available in Isabelle}\label{know-isab}

Isabelle is developed now for a long time and so we are able to access a huge range of theories and useful snippets. The main problem according this snip sets is that isabelle still is a theorem proofer and not an algebra system. But due the work of the {\sisac}-development team there are already also many calculation examples provided.

\par The SPSC provided a list of problems which are often done wrong or are misunderstood by students in term of the problem classes. Out of these tasks we tried to extract the core operations and looked up which parts are already implemented or useful. The provided problem classes are:

\begin{itemize}

	\item Fourier-Transformation

	\item Convolution

	\item Inverse Z-Transformation and partial fraction decomposition

	\item Index transformation

\end{itemize}

A survey of these selected Problems on their "`correct"' solution can be found in Appendix~\ref{app:calc}. After manually solving this problems we started checking which of them are able to realizable. As already remarked above, isabelle is just a theorem proover due this fact isabelle only provides some theories which are useful for the provided problem classes.

\par Yet, isabelle also has some problems on traditional notation. For example a theory called FFT is already implemented this theory does not provide methods for solving fourier transformation tasks, it only defines the basic condition a fast Fourier transformed expression has to fulfill.

\par For the problems we had a look-on isabelle did not provide specific theories which can be implemented one by one, so a look at the mechanized knowledge in {\sisac} is the next step, therefore follow up Section~\ref{know-isac}.

\subsection{Relevant Knowledge available in ISAC}\label{know-isac}

{\sisac} already provides a small registry of useful programs and surveys on using isabelle as for solving problems. These snip sets are things like norming rational numbers, solving variate and univariate equations, substitutions, some simple integrals and so on. For a detailed review on the currently implemented theories a look at the {\sisac} project web-page\footnote{http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html} is strongly recommended, as the appreciation for this thesis is no describing all parts of {\sisac} in detail. This would also not be possible as {\sisac} is permanently growing.

\par After we differenced {\sisac} and isabelle we are able to introduce two axes in the project. This axes are the specifications (``application axis'' represented as isabelle) and methods (``algorithmic axis'' represented as {\sisac}).

\par During the first weeks of work on this thesis we decided on implementing the Z-Transformation and particullary the included partial fraction decomposion first. The algoritmix axis therefor already provides the knowledge for the following parts:

\begin{description}

\item[Partial Fractions] We now how to handle them and we are able to illustrate them.

\item[Cancellation of multivariate rational terms] Simplification is possible.

\item[Equations] The possibility of solving linear and second order equations is provided, also the possibility to get the left or right hand side of an equation.

\item[Substitution] We are able to substitute given variables in terms, we are also able to subsitude a list of variables in terms.

\end{description}

\subsection{Survey on Required Knowledge}\label{know-missing}

Following tables (Table~\ref{tab:eff-four},~\ref{tab:eff-conv},~\ref{tab:eff-ztrans}) are showing the expected development effort for specific problems. The values are only very inaccurately approximations of the real work, but needed as a basis for descieding with which problem to start:

\begin{table}

	\centering

	\begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}

		\textbf{Requirements}   & \textbf{Comments}    &\textbf{Effort}\\ \hline\hline

		solving Integrals		    & simple via properties table     &     20\\

		                        & \emph{real}          &    MT\\ \hline

		transformation table    & simple transform     &    20\\ \hline

		visualisation						& backend							 &    10\\ \hline

		example collection      & with explanations    &    20\\ \hline\hline

		\multicolumn{2}{c|}{}                          & 70-80\\

	\end{tabular}

	\caption{Fourier-Transformation development effort\label{tab:eff-four}}

\end{table}

\begin{table}

	\centering

	\begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}

		\textbf{Requirements}   & \textbf{Comments}    &\textbf{Effort}\\ \hline\hline

		simplify rationals      & {\sisac}             &     0\\ \hline

		define $\sum\limits_{i=0}^{n}i$ & partly {\sisac}  &    10\\ \hline

		simplify sum			      & termorder            &    10\\

		                        & simplify rules       &    20\\

		                        & use simplify rationals&     0\\ \hline

		index adjustments       & with unit step       &      10\\ \hline

		example collection      & with explanations    &    20\\ \hline\hline

		\multicolumn{2}{c|}{}                      & 70-90\\

	\end{tabular}

	\caption{Convolution Operations development effort\label{tab:eff-conv}}

\end{table}

\begin{table}

	\centering

	\begin{tabular}{p{4cm}|p{5cm}|rp{2.5cm}}

		\textbf{Requirements}   & \textbf{Comments}    &\textbf{Effort}\\ \hline\hline

		solve for part.fract.   & {\sisac}: degree 2   &     0\\

		                        & complex nomminators  &    30\\

		                        & degree > 2           &    MT\\ \hline

		simplify polynomial     & {\sisac}             &     0\\

		simplify rational       & {\sisac}             &     0\\ \hline

		partial fraction        & degree 2,            &    20\\

		decomposition           & specification, method&    30\\ \hline

		${\cal Z}^{-1}$ table   & explanations, figures&    20\\ \hline

		example collection      & with explanations    &    20\\ \hline\hline

		\multicolumn{2}{c|}{}                          & 90-120\\

	\end{tabular}

\caption{Z-Transformation development effort\label{tab:eff-ztrans}}

\end{table}

As conclusion of the summarized efforts it is evident that only one topic can be tried to realized as a baccalaureate thesis. In accord with Dr. Neuper we decided after some practical tests to start with the implementation of the (Inverse) Z-Transformation. The Reason is that this topic can mostly be done with knowledge which was already tried to be mechanized in {\sisac}.

\subsection{Formalization of missing knowledge in Isabelle}

A problem behind is the mechanization of mathematic theories in TP-bases languages. There is still a huge gap between these algorithms and this what we want as a solution - in Example Signal Processing. 

\begin{example}

	\[

		X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY

  \]

	{\small\textit{

		\noindent A very simple example on this what we call gap is the simplification above. It is needles to say that it is correct and also Isabelle for fills it correct - \emph{always}. But sometimes we don't want expand such terms, sometimes we want another structure of them. Think of a problem were we now would need only the coefficients of $X$ and $Y$. This is what we call the gap between mechanical simplification and the solution.

	}}

	\caption{Correct but not useful}\label{eg:gap}

\end{example}

Until we are not able to fill this gap we have to live with it but first have a look on the meaning of this statement:

\par Mechanized math starts from mathematical models and \emph{hopefully} proceeds to match physics. Academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modeling and formalization. The process from a physical observance to a mathematical theory is unavoidable bound of setting up a big collection of standards, rules, definition but also exceptions. These are the things making mechanization that difficult.

\begin{example}

	\[

		m,\ kg,\ s,\ldots

  \]

	{\small\textit{

		\noindent Think about some units like that one's above. Behind each unit there is a discerning and very accurate definition: One Meter is the distance the light travels, in a vacuum, through the time of 1 / 299.792.458 second; one kilogram is the weight of a platinum-iridium cylinder in paris; and so on. But are these definitions usable in a computer mechanized world?!

	}}

	\caption{Units in measurement}\label{eg:units}

\end{example}

\par A computer or a TP-System builds on programs with predefined logical rules and does not know any mathematical trick (follow up example \ref{eg:trick}) or recipe to walk around difficult expressions. 

\begin{example}

\[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]

\[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=

	 \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]

\[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]

	{\small\textit{

		\noindent Sometimes it is also useful to be able to apply some \emph{tricks} to get a beautiful and particularly meaningful result, which we are able to interpret. But as seen in this example it can be hard to find out what operations have to be done to transform a result into a meaningful one.

	}}

	\caption{Mathematic tricks}\label{eg:trick}

\end{example}

For such a system the only possibility is to work through its known definitions and stops if none of these fits. Specified on Signal Processing or any other application it is often possible to walk through by doing simple creases. This creases are in generell based on simple math operatiopms but the challenge is to teach the machine \emph{all}\footnote{Its pride to call it \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to reach a high level of \emph{all} but it in real it will still be a survey of knowledge which links to other knowledge and {\sisac{}} a trainer and helper but no human compensating calculator. 

\par {{\sisac{}}} itself aims to adds an \emph{application} axis (formal specifications of problems out of topics from Signal Processing, etc.) and an \emph{algorithmic} axis to the \emph{deductive} axis of physical knowledge. The result is a three-dimensional universe of mathematics seen in Figure~\ref{fig:mathuni}.

\begin{figure}

\hfill \\

  \begin{center}

    \includegraphics[scale=0.7]{math-universe}

  \end{center}

  \caption{Three-dimensional universe of mathematics\label{fig:mathuni}}

\end{figure}

\subsection{Notes on Problems with Traditional Notation}

Due the thesis work we discovers sever ell problems of traditional notations. Some of them have been known in computer science for many years now and are still unsolved, one of them aggregates with the so called \emph{Lambda Calculus}, Example~\ref{eg:lamda} provides a look on the problem that embarrassed us.

\begin{example}

\[ f(x)=\ldots\;	\quad R \rightarrow \quad R \]

\[ f(p)=\ldots\;	p \in \quad R \]

	{\small\textit{

		\noindent Above we see two equations. The first equation aims to be a mapping of an function from the reell range to the reell one, but when we change only one letter we get the second equation which usually aims to insert a reel point $p$ into the reell function. In computer science now we have the problem to tell the machine (TP) the difference between this two notations. This Problem is called Lambda Calculus.

	}}

	\caption{Towards the Lambda Calculus\label{eg:lamda}}

\end{example}

An other problem is that terms are not full simplified in traditional notations, in {\sisac} we have to simplify them complete to check weather results are compatible or not. in e.g. the solutions of an second order linear equation is an rational in {\sisac} but in tradition we keep fractions as long as possible and as long as they aim to be 'beautiful' (1/8, 5/16,...).

The math which should be mechanized in Computer Theorem Provers (\emph{TP}) has (almost) a problem with traditional notations (predicate calculus) for axioms, definitions, lemmas, theorems as a computer program or script is not able to interpret every Greek or Latin letter and every Greek, Latin or whatever calculations symbol. Also if we would be able to handle these symbols we still have a problem to interpret them at all. (Follow up \hbox{Example \ref{eg:symbint1}})

\begin{example}

	\[

		u\left[n\right] \ \ldots \ unitstep

	\]

	{\small\textit{

		\noindent The unitstep is something we need to solve Signal Processing problem classes. But in {{\sisac{}}} the 	rectangular brackets have a different meaning. So we abuse them for our requirements. We get something which is not defined, but usable. The Result is syntax only without semantic.

	}}

	\caption{Expression Interpretation}\label{eg:symbint1}

\end{example}

\noindent In different problems, symbols and letters have different meanings and ask for different ways to get through. (Follow up \hbox{Example \ref{eg:symbint2}}) 

\begin{example}

	\[

		\widehat{\ }\ \widehat{\ }\ \widehat{\ } \  \ldots \  exponent

	\]

	{\small\textit{

	\noindent For using exponents the three widehat symbols are required. The reason for that is due the development of {{\sisac{}}} the single widehat and also the double were already in use for different operations.

	}}

	\caption{Symbol Interpretation}\label{eg:symbint2}

\end{example}

Exclusive from the input, also the output can be a problem. We are familiar with a specified notations and style taught in university but a computer program has no knowledge of the form proved by a professor and the machines themselves also have not yet the possibilities to print every symbol (correct) Recent developments provide proofs in a human readable format but according to the fact that there is no money for good working formel editors yet, the style is one thing we have to live with.

\section{Project Controlling}

We decided to split the thesis into five Iteration defined in Section~\ref{sec:milesurv}. As there is also a lot of work to do outer the thesis we accord on an increased contact by mail. For the coordination of the whole  {\sisac} files i got access to the mercurial repository. We also appointed on periodic team meetings.

\subsection{Survey on Milestones\label{sec:milesurv}}

Doing something completely new requires a good controlling, the thesis itself also needs it. After the first meetings and the definition of the intrinsic work we decided on splitting the thesis into the following iterations.

\begin{description}

	\item[1st Iteration] Information Collection

	\item[2nd Iteration] Problem Selection

	\item[3rd Iteration] Implementation

	\item[4th Iteration] Thesis Writing

	\item[5th Iteration] Finalization

\end{description}

A more detailed description of this milestones can be found in Section~\ref{sec:detmile}.

\subsection{Milestone Details\label{sec:detmile}}

\begin{description}

	\item[Information Collection] The first iteration starts by an introduction to the {\sisac} System and ends up with the first presentation. Listeners of the first presentation were \em Dr. Walther Neuper \normalfont and \em DI Bernhard Geiger\normalfont. We talked about common SPSC problems and the possibilities of realize them in the {\sisac} System. In preparation of the Presentation \em DI Geiger \normalfont sent us a few example problems and we had a experimental survey about the realization effort.

	\item[Problem Selection] In the second iteration we collected informations about the knowledge mechanized in {\sisac} (cf. Section~\ref{know-isab}). After the first iteration it was clear that implementing of problems in {\sisac} requires a higher effort than originally expected due this fact the second iteration ends up on the decision which of the provided problems is going to be implemented. We wrote and collected a handful of experimental scripts regarding sums, Fourier transformation and partial fraction decomposition.

	\item[Implementation] Unfortunately the biggest and most important part is the implementation. The iteration started with the decision on the problem and ends up by finishing the test Script (seen in Part~\ref{part:impl}) as well as the integration of this work into the {\sisac}-Knowledge. For a better controlling and result of this iteration we had severe ll regular meetings (\em Dr. Neuper \normalfont and \em Jan Ro\v{c}nik\normalfont) and contact over e-mail to assess the state of the {\sisac}-developers work.

	\item[Thesis Writing] One part of this thesis is generated automatically out of \ttfamily Build\_Inverse\_Z\_Transform\normalfont. Maybe this part well be the most important result of the thesis as it will be used as a documentation for the upcoming developers. Due this fact this iteration started also contemporaneous with the implementation but ends up separate after finishing the implementation with describing the needed theory and background.

	\item[Finalization] The work ends up with the last iteration - finalization. It is started by completing the written thesis and the preparation of the second presentation which concludes this project. In the second presentation we will have demonstrated our realized problem embedded in the new {\sisac}-frontend as well as the work, hiding behind. We will also want to give a clear view about the power of {\sisac} and animate the attending agents to go on working within this project.

\end{description}

%\clearpage

%----------// PART 2 \\----------%

\newevenside

\part{Implementation\label{part:impl}}

\input{./preambleForGeneratedDocuments.tex}

\par Please be aware that this is only a part of the thesis output, also follow up the theories and according tests that have been improved or created by me and are involved to the work on this thesis:

\begin{itemize}

\item \ttfamily Partial\_Fractions.thy \normalfont - \ttfamily partial\_fractions.sml \normalfont

\item \ttfamily Inverse\_Z\_Transformation.thy \normalfont - \ttfamily inverse\_z\_transformation.sml \normalfont

\item \ttfamily Rational.thy \normalfont - \ttfamily rational.sml \normalfont

\end{itemize}

\HRule

%\setcounter{section}{0}

\input{../../../test/Tools/isac/ADDTESTS/course/SignalProcess/document/Build_Inverse_Z_Transform}

\clearpage

%----------// PART 3 \\----------%

\newevenside

\part{Thesis Summary and Review}

\section{Related Work}\label{sec:related}

Unusual for a Baccalaureate Thesis, there is {\em no} related work; this requires explanation.

Of course, this thesis relies on front-of-the wave computer mathematics, on TP. But {{\sisac{}}} uses TP in a very specific way, which is too weakly related to other work: programing in the TP-based language and rigorous formal specification of problems in Signal Processing where the main tasks in the practical part of this thesis. The major challenge for the practical work was given by the fact, that the work concerned alpha-testing of the TP-based programing environment.

\par Another  area of work could be considered as related work: authoring of e-learning content. However, {{\sisac{}}} provides division of concern such that the practical part of this thesis could focus on computer mathematics; this work was not concerned with interaction (the TP-based programing language has neither input statements nor output statements), nor with dialog guidance nor with any kind of learning theory.

\par These two reasons are given for the unusual statement, that there is no related work to be discussed in this thesis. 

\section{Issues and Open Questions}

At the end of the first Iteration we agreed on a view issues that might have to be accomplished during the thesis. This Issues were:

\begin{itemize}

	\item What knowledge is already mechanized in \emph{Isabelle}?

	\item How can missing theorems and definitions be mechanized?

	\item What is the effort for such mechanization?

	\item How do calculations look like, by using mechanized knowledge?

	\item What problems and subproblems have to be solved?

	\item Which problems are already implemented in {\sisac}?

	\item How are the new problems specified ({\sisac})?

	\item Which variants of programs in {\sisac} solve the problems?

	\item What is the contents of the interactive course material (Figures, etc.)?

\end{itemize}

The significant part of this list was declared during the thesis. Next sections will review the most important items, regardless they where accomplished or not.

\subsection{Accomplished}

We discovered the process how missing theorems and definitions can be mechanized and used this knowledge to implement new material. Now we ware able to solve partial fractions, we built theorems, specification and program. We completed the background work for interactive solving them. The same thing applies to the inverse Z-transformation with partial fractions; we built theorems, specification and program to step through an amount of exercises. But the most important accomplished part is the Isabelle theory (\ttfamily Build\_Inverse\_Z\_Transform.thy\normalfont) indicating issues for the coming {\sisac}-developers. Until we were not able to fulfill all the things we wanted, this is the most recent work and will be the headstone for the further work on {\sisac}.

\par Expect the necessary theories for solving the mentioned part and the guideline we also put tests for the new theories to the system and added some new methods to existing theories with their according tests. Also we were able to answer the questions about the mechanized knowledge in \emph{isabelle} and {\sisac}. We checked the required effort to add new material to the system and we walked through the question what subproblems have to be solved and how they are declared, when adding new material. Now we also know how problems are declared and which variants exist to solve them. All this can be found in this thesis.

\subsection{Partially Accomplished}

As already told upwards, we accomplished a guideline for upcoming {\sisac}-developers this was a very important and necessary task; but through upcoming changes in the development environment and programing language there is a need for more descriptions and informations. This changes are not yet fulfilled and so it was not possible to include a correct guideline fot them in our example theory or this thesis.

\par Also we were not able to provide a representative count of example problems for partial fractions or the inverse Z-Transformation. We are able to say that our accomplished material can handle one problem each, exactly as we wanted and it is toughed but due to less time we have no more experience data.

\subsection{Not Accomplished}

The properly longest item is the list with the not accomplished issues, but there is no need to count this as a negative aspect due the fact that we collected experience data for following needs:

\par We were not able to put explanations, figures to examples, theorem descriptions and problem description to our course material. The reason for that was the time at one side and the ongoing restructuration of the system on the other side.

\par Only a sufficient number of topics was implemented. There were more problems to solve than we expected. We were only able to implement one from the SPSC given tasks m entered in Appendix~\ref{app:calc}. Due this fact the material we got as an output of the thesis will be no support for labs or lectures at the SPSC and also no material for the STEOP, which also was a big wish at the beginning.

\subsection{Open Questions and Work}

At the end of this thesis there is now part left unattended, but many parts that require beauty corrections and many parts that require a few more manpower. This upcoming open tasks are:

\begin{itemize}

	\item What is the contents of the interactive course material (Figures, etc.)?

	\item What are the pre and post conditions of the problems?

	\item How would a correct implementation of the Z-Transformation look like?

	\item What do students expect of this implementation?

\end{itemize}

To complete the task of creating course material describing the needed steps to do a partial fraction or inverse Z-Transformation in {\sisac} the real effort is not conceivable. But for a first usable version only a few steps are left to do this steps would be:

\begin{enumerate}

	\item Cleanup the\ttfamily InverseZTransformation.thy\normalfont file and put it into the {\sisac} knowledge tree like it was done with\ttfamily PartialFractions.thy\normalfont.

	\item Create a good example program and test it.

	\item Describe the used theories with the mathematic definitions and background.

	\item Add figures and small examples.

\end{enumerate}

There are also needs on finishing other parts to get a complete part of course material but this would be the steps to get a first usable version.

\section{Conclusion and Personal Experience}

After working on {\sisac} for month, it is hard do find the right words about the system. For sure the system itself is a great idea and it would have been a big relaxation if i would have been able to use this system in the last years for sever ell subjects. But for me the biggest disadvantage of {\sisac} is the frontend until there is no mathematical notation and visualisation of simply fractions i don't think that i or other students would be able to use it for learning. I also think that for learning there is no need of complex examples or complex solutions, examples have to be simple but also have to cover all necessary steps of a calculation. Unfortunately there exist different learning types and I'm a visual one, what is also the explanation for my maybe drastic explanation.

\par About the backend, what finally was my main work there is not much left to say. The programing language requires a long time to work in and to find out how decelerations work and how isabelle and {\sisac} is build up, maybe this period of collecting informations is to long for a Bakk. Thesis. Also it is hard to find informations about the theories until there was no related work and no usable documentation wether for {\sisac} nor for isabelle or ML.

\par When working within the {\sisac} sources you often hung up on problems which would be simple to solve but are hard to discover, therefor it was often necessary to make breaks and define concrete steps on problem solving. Also when having a plan of the upcoming implementation I had to break myself out to do not to much step at once. Often its better to implement small parts and test, write tests and wait then hurrying up. An other aspect is that it was necessary to build {\sisac} from scratch after every change; this and the fact that my finally theories had over 1000 lines often made the development process to a stop and go dilemma.

\par Finally I have to say a big \bfseries thank you \normalfont to Dr.~Neuper who invested a lot of his time in my thesis and the {\sisac} project, I remember some hard and long meetings but finally I look back glad.

\subsection{Anecdotes}

At the very beginning of the thesis i made some calculations and thoughed of every step more than twice. This calculations were the ground stone of my work, we build every step in the calculation following up the real calculations - when doing them by hand. Time went on and after adding the last commands to the programm for applying the transformation ruleset i compared the results - oh wonder there was a mistake in my calculation :)

\begin{center}

Thank you {\sisac}!

\end{center}

%----------// BIB \\-----------%

\renewcommand{\refname}{\section{References}}

\bibliographystyle{alpha}

\bibliography{references}

\clearpage

%----------// APPENDIX \\-----------%

\newevenside

\part{Appendix}

\appendix

%----------// WORK TIME \\-----------%

\section{Record of Working Time}

\begin{footnotesize}

\begin{longtable}{l p{6.5cm} c c r}

{\bf Date} & {\bf Description} & {\bf Begin} & {\bf End} & {\bf Dur.}\\

\hline \hline

\endhead

\hline 

  \multicolumn{5}{r}{{Continued on next page}} \\

\hline

\endfoot

\hline 

\hline

\endlastfoot

\multicolumn{5}{l}{Start 1st Iteration} \\

\hline

29.06.2011 & Meeting Dr. Neuper and DI Geiger & 15:00 & 17:30 & 2,50 \\

02.07.2011 & Working on Examples from DI Geiger & 20:00 & 21:30 & 1,50 \\

03.07.2011 & Working on Examples from DI Geiger & 21:00 & 22:45 & 1,75 \\

05.07.2011 & Meeting Dr. Neuper, Information Exchange & 10:00 & 13:00 & 3,00 \\

06.07.2011 & Installing Isabelle & 20:00 & 22:30 & 2,50 \\

07.07.2011 & Meeting Dr. Neuper, present-1 & 14:45 & 16:15 & 1,50 \\

18.07.2011 & Meeting Dr. Neuper, present-1 Structure & 14:15 & 16:00 & 1,75 \\

19.07.2011 & Meeting Dr. Neuper, present-1 Content & 07:20 & 09:20 & 2,00\\

19.07.2011 & Meeting Dr. Neuper & 10:00 & 12:00 & 2,00\\

21.07.2011 & HG Error Correction, Latex for Thesis & 11:10 & 14:00 & 2,83\\

22.07.2011 & Meeting Dr. Neuper & 10:00 & 12:00 & 2,00\\

23.07.2011 & Latex for Calculations & 13:45 & 16:30 & 2,75\\

24.07.2011 & Meeting Dr. Neuper, present-1 & 20:10 & 20:40 & 0,50\\

25.07.2011 & Meeting Dr. Neuper, present-1 & 15:15 & 17:55 & 2,67\\

26.07.2011 & Work on Test\_Complex.thy & 10:45 & 12:10 & 1,42\\

27.07.2011 & present-1 (Dr. Neuper, DI Geiger) & 10:00 & 12:00 & 2,00\\

\hline

\multicolumn{5}{l}{End 1st Iteration} \\

\multicolumn{5}{l}{Start 2nd Iteration} \\

\hline

02.09.2011 & Meeting Dr. Neuper, Latex for Thesis & 08:30 & 10:20 & 1,83\\

05.09.2011 & Meeting Dr. Neuper, Partial\_Frations.thy & 09:30 & 12:45 & 3,25\\

05.09.2011 & Partial\_Fractions.thy & 17:10 & 18:30 & 1,33\\

06.09.2011 & Documentation Partial\_Fractions.thy & 10:00 & 13:15 & 3,25\\

07.09.2011 & Meeting Dr. Neuper, ML Introduction & 10:00 & 12:50 & 2,83\\

08.09.2011 & Preparing Scripts for Latex Output & 19:00 & 22:45 & 3,75\\

09.09.2011 & Preparing Scripts for Latex Output & 11:40 & 15:00 & 3,33\\

10.09.2011 & Meeting Dr. Neuper, Latex Output, HG Errors & 10:00 & 12:00 & 2,00\\

\hline

\multicolumn{5}{l}{End 2nd Iteration} \\

\multicolumn{5}{l}{Start 3rd and 4th Iteration} \\

\hline

14.09.2011 & Start on Inverse\_Z\_Transf. Script & 09:10 & 12:25 & 3,25\\

16.09.2011 & Information Exchange Sums & 13:15 & 16:00 & 2,75\\

19.09.2011 & Programming on Inverse\_Z\_Transf. & 10:00 & 13:10 & 3,17\\

20.09.2011 & Meeting Dr. Neuper, Inverse\_Z\_Transf. Script & 15:30 & 18:10 & 2,67\\

23.09.2011 & New IsaMakefile for Latex generation & 13:00 & 14:30 & 1,50\\

23.09.2011 & Meeting Dr. Neuper, Inverse\_Z\_Transf. Script & 14:30 & 17:30 & 3,00\\

26.09.2011 & Partial\_Fractions.thy, get\_Argument & 13:30 & 16:15 & 2,75\\

27.09.2011 & Meeting Dr. Neuper, HG Error & 09:00 & 12:20 & 3,33\\

28.09.2011 & Meeting Dr. Neuper, Hierarchy Reconstruction & 10:00 & 12:30 & 2,50\\

01.10.2011 & Some Testing & 10:00 & 11:00 & 1,00\\

02.10.2011 & Inverse\_Z\_Transf. Errors & 15:00 & 16:10 & 1,17\\

06.10.2011 & Meeting Dr. Neuper & 15:00 & 17:50 & 2,83\\

07.10.2011 & Meeting Dr. Neuper, Inverse\_Z\_Transf. Script & 15:00 & 16:50 & 1,83\\

09.10.2011 & Work on Thesis & 16:30 & 18:45 & 2,25\\

11.10.2011 & Meeting Dr. Neuper, Abstract & 14:10 & 17:10 & 3,00\\

13.10.2011 & Work on Thesis, Implemention part & 08:30 & 10:00 & 1,50\\

18.10.2011 & Work on Thesis & 20:30 & 22:50 & 2,33\\

19.10.2011 & Work on Thesis & 21:20 & 23:40 & 2,33\\

20.10.2011 & Work on Thesis & 08:45 & 10:45 & 2,00\\

25.10.2011 & Meeting Dr. Neuper: thesis review & 08:00 & 10:00 & 2,00\\

25.10.2011 & Reorganising after meeting & 20:10 & 23:50 & 3,67\\

26.10.2011 & Reorganising after meeting, examples in thesis & 08:30 & 11:10 & 2,67\\

07.11.2011 & Meeting Preparation & 16:30 & 17:15 & 0,75\\

08.11.2011 & Meeting Dr. Neuper: thesis addons & 16:00 & 17:30 & 1,50\\

12.11.2011 & Reorganization after meeting & 21:10 & 22:30 & 1,33\\

20.11.2011 & Review of Inv\_Z\_Trans & 10:15 & 13:30 & 3,25\\

21.11.2011 & get\_numerator & 12:15 & 14:30 & 2,25\\

23.11.2011 & get\_numerator & 20:30 & 21:15 & 0,75\\

24.11.2011 & get\_numerator final & 14:10 & 15:30 & 1,33\\

28.11.2011 & tried to go on in build\_inverse\_ & 11:00 & 13:10 & 2,17\\

01.12.2012 & Tuning Invers\_Z\_Tansf. Script & 16:15 & 19:30 & 3,25\\

04.12.2012 & Moved Some Tests to Knowledge & 15:30 & 18:15 & 2,75\\

05.12.2012 & Tuning Invers\_Z\_Tansf. Script & 13:10 & 14:15 & 1,08\\

08.12.2012 & Inverse\_Z\_Tansf. Script, Factors from Solution & 14:30 & 16:15 & 1,75\\

10.12.2012 & Moved Some Tests to Knowledge & 20:10 & 23:00 & 2,83\\

11.12.2012 & Inverse\_Z\_Tansf. Script, PBZ Ansatz & 10:10 & 12:30 & 2,33\\

12.12.2012 & Inverse\_Z\_Tansf. Script, Subsitution & 09:00 & 11:30 & 2,50\\

16.12.2012 & Tuning Invers\_Z\_Tansf. Script & 17:15 & 18:45 & 1,50\\

19.12.2012 & Tuning Invers\_Z\_Tansf. Script & 19:00 & 20:30 & 1,50\\

14.01.2012 & Inverse\_Z\_Tansf. Script, Substitution & 18:45 & 20:30 & 1,75\\

22.01.2012 & Inverse\_Z\_Transf. Script, Calc Coeff. & 20:10 & 21:45 & 1,58\\

12.02.2012 & Inverse\_Z\_Transf. Script & 17:30 & 19:00 & 1,50\\

13.02.2012 & Finished Work on Inverse\_Z\_Transf. Script & 19:30 & 22:10 & 2,67\\

14.02.2012 & Formatting for Latex output & 09:00 & 14:45 & 5,75\\

15.02.2012 & Formatting for Latex output & 10:00 & 15:30 & 5,50\\

16.02.2012 & Going on Work on Thesis & 14:45 & 18:30 & 3,75\\

17.02.2012 & Tuning Invers\_Z\_Tansf. Script & 18:30 & 19:45 & 1,25\\

19.02.2012 & Meeting Dr. Neuper, present-2 & 09:00 & 13:30 & 4,50\\

19.02.2012 & Meeting Dr. Neuper, Integration of work to Isac & 15:00 & 19:10 & 4,17\\

20.02.2012 & Meeting Dr. Neuper, Integration of work to Isac & 09:00 & 12:45 & 3,75\\

20.02.2012 & Meeting Dr. Neuper, Integration of work to Isac & 14:10 & 18:30 & 4,33\\

\hline

\multicolumn{5}{l}{End 3rd Iteration}\\

\hline

20.02.2012 & Finishing Thesis & 20:30 & 22:50 & 2,33\\

21.02.2012 & Finishing Thesis & 13:30 & 14:45 & 1,25\\

02.03.2012 & Finishing Thesis & 15:45 & 19:00 & 3,25\\

06.03.2012 & Finishing Thesis & 09:15 & 10:30 & 1,25\\

07.03.2012 & Finishing Thesis & 13:15 & 16:00 & 2,75\\

\hline

\multicolumn{5}{l}{End 4th Iteration} \\

\multicolumn{5}{l}{Start 5th Iteration} \\

\hline

26.02.2012 & Preparing present-2 & 09:30 & 13:00 & 3,5 \\

20.03.2012 & Preparing present-2 & 14:00 & 16:30 & 2,5 \\

08.05.2012 & Meeting Dr. Neuper, present-2, work on isac & 08:00 & 12:00 & 4,0 \\

08.05.2012 & present-2 & 14:00 & 15:00 & 1,0 \\

\hline

\multicolumn{5}{l}{End 5th Iteration} \\

\hline

\multicolumn{4}{l}{Total working time:} & 203 \\

\end{longtable}

\end{footnotesize}

%----------// CALCULATIONS \\-----------%

\newevenside

\section{Calculations\label{app:calc}}

\input{calulations}

\end{document}