1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % Electronic Journal of Mathematics and Technology (eJMT) %
3 % style sheet for LaTeX. Please do not modify sections %
4 % or commands marked 'eJMT'. %
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10 \documentclass[12pt,a4paper]{article}% %
12 \usepackage{amsfonts,amsmath,amssymb} %
13 \usepackage[a4paper]{geometry} %
14 \usepackage{fancyhdr} %
16 \usepackage[pdftex]{hyperref} % see note below %
17 \usepackage{graphicx}% %
23 \newtheorem{theorem}{Theorem} %
24 \newtheorem{acknowledgement}[theorem]{Acknowledgement} %
25 \newtheorem{algorithm}[theorem]{Algorithm} %
26 \newtheorem{axiom}[theorem]{Axiom} %
27 \newtheorem{case}[theorem]{Case} %
28 \newtheorem{claim}[theorem]{Claim} %
29 \newtheorem{conclusion}[theorem]{Conclusion} %
30 \newtheorem{condition}[theorem]{Condition} %
31 \newtheorem{conjecture}[theorem]{Conjecture} %
32 \newtheorem{corollary}[theorem]{Corollary} %
33 \newtheorem{criterion}[theorem]{Criterion} %
34 \newtheorem{definition}[theorem]{Definition} %
35 \newtheorem{example}[theorem]{Example} %
36 \newtheorem{exercise}[theorem]{Exercise} %
37 \newtheorem{lemma}[theorem]{Lemma} %
38 \newtheorem{notation}[theorem]{Notation} %
39 \newtheorem{problem}[theorem]{Problem} %
40 \newtheorem{proposition}[theorem]{Proposition} %
41 \newtheorem{remark}[theorem]{Remark} %
42 \newtheorem{solution}[theorem]{Solution} %
43 \newtheorem{summary}[theorem]{Summary} %
44 \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} } %
45 {\ \rule{0.5em}{0.5em}} %
47 % eJMT page dimensions %
49 \geometry{left=2cm,right=2cm,top=3.2cm,bottom=4cm} %
51 % eJMT header & footer %
53 \newcounter{ejmtFirstpage} %
54 \setcounter{ejmtFirstpage}{1} %
56 \setlength{\headheight}{14pt} %
57 \geometry{left=2cm,right=2cm,top=3.2cm,bottom=4cm} %
58 \pagestyle{fancyplain} %
60 \fancyhead[c]{\small The Electronic Journal of Mathematics%
61 \ and Technology, Volume 1, Number 1, ISSN 1933-2823} %
63 \ifnum\value{ejmtFirstpage}=0% %
64 {\vtop to\hsize{\hrule\vskip .2cm\thepage}}% %
65 \else\setcounter{ejmtFirstpage}{0}\fi% %
68 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
70 % Please place your own definitions here
72 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
73 \def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
76 \definecolor{lgray}{RGB}{238,238,238}
79 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
81 % How to use hyperref %
82 % ------------------- %
84 % Probably the only way you will need to use the hyperref %
85 % package is as follows. To make some text, say %
86 % "My Text Link", into a link to the URL %
87 % http://something.somewhere.com/mystuff, use %
89 % \href{http://something.somewhere.com/mystuff}{My Text Link}
91 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
97 \title{Trials with TP-based Programming
99 for Interactive Course Material}%
101 % Single author. Please supply at least your name,
102 % email address, and affiliation here.
104 \author{\begin{tabular}{c}
105 \textit{Jan Ro\v{c}nik} \\
106 jan.rocnik@student.tugraz.at \\
108 Graz University of Technologie\\
112 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
114 % eJMT commands - do not change these %
119 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
125 Traditional course material in engineering disciplines lacks an
126 important component, interactive support for step-wise problem
127 solving. Theorem-Proving (TP) technology is appropriate for one part
128 of such support, in checking user-input. For the other part of such
129 support, guiding the learner towards a solution, another kind of
130 technology is required. %TODO ... connect to prototype ...
132 A prototype combines TP with a programming language, the latter
133 interpreted in a specific way: certain statements in a program, called
134 tactics, are treated as breakpoints where control is handed over to
135 the user. An input formula is checked by TP (using logical context
136 built up by the interpreter); and if a learner gets stuck, a program
137 describing the steps towards a solution of a problem ``knows the next
138 step''. This kind of interpretation is called Lucas-Interpretation for
139 \emph{TP-based programming languages}.
141 This paper describes the prototype's TP-based programming language
142 within a case study creating interactive material for an advanced
143 course in Signal Processing: implementation of definitions and
144 theorems in TP, formal specification of a problem and step-wise
145 development of the program solving the problem. Experiences with the
146 ork flow in iterative development with testing and identifying errors
147 are described, too. The description clarifies the components missing
148 in the prototype's language as well as deficiencies experienced during
151 These experiences are particularly notable, because the author is the
152 first programmer using the language beyond the core team which
153 developed the prototype's TP-based language interpreter.
157 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
161 \thispagestyle{fancy} %
163 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
165 % Please use the following to indicate sections, subsections,
166 % etc. Please also use \subsubsection{...}, \paragraph{...}
167 % and \subparagraph{...} as necessary.
170 \section{Introduction}\label{intro}
172 % \paragraph{Didactics of mathematics}
173 %WN: wenn man in einem high-quality paper von 'didactics' spricht,
174 %WN muss man am state-of-the-art ankn"upfen -- siehe
175 %WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
176 % faces a specific issue, a gap
177 % between (1) introduction of math concepts and skills and (2)
178 % application of these concepts and skills, which usually are separated
179 % into different units in curricula (for good reasons). For instance,
180 % (1) teaching partial fraction decomposition is separated from (2)
181 % application for inverse Z-transform in signal processing.
183 % \par This gap is an obstacle for applying math as an fundamental
184 % thinking technology in engineering: In (1) motivation is lacking
185 % because the question ``What is this stuff good for?'' cannot be
186 % treated sufficiently, and in (2) the ``stuff'' is not available to
187 % students in higher semesters as widespread experience shows.
189 % \paragraph{Motivation} taken by this didactic issue on the one hand,
190 % and ongoing research and development on a novel kind of educational
191 % mathematics assistant at Graz University of
192 % Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
193 % scope with this issue on the other hand, several institutes are
194 % planning to join their expertise: the Institute for Information
195 % Systems and Computer Media (IICM), the Institute for Software
196 % Technology (IST), the Institutes for Mathematics, the Institute for
197 % Signal Processing and Speech Communication (SPSC), the Institute for
198 % Structural Analysis and the Institute of Electrical Measurement and
199 % Measurement Signal Processing.
200 %WN diese Information ist f"ur das Paper zu spezielle, zu aktuell
201 %WN und damit zu verg"anglich.
202 % \par This thesis is the first attempt to tackle the above mentioned
203 % issue, it focuses on Telematics, because these specific studies focus
204 % on mathematics in \emph{STEOP}, the introductory orientation phase in
205 % Austria. \emph{STEOP} is considered an opportunity to investigate the
206 % impact of {\sisac}'s prototype on the issue and others.
209 \paragraph{Traditional course material} in engineering disciplines lacks an
210 important component, interactive support for step-wise problem
211 solving. Theorem-Proving (TP) technology can provide such support by
212 specific services. An important part of such services is called
213 ``next-step-guidance'', generated by a specific kind of ``TP-based
214 programming language''. In the
215 {\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
216 a language is prototyped in line with~\cite{plmms10} and built upon
218 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
219 The TP services are coordinated by a specific interpreter for the
220 programming language, called
221 Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language and the
222 interpreter will be briefly re-introduced in order to make the paper
225 \subparagraph{The main part} of the paper is an account of first experiences
226 with programming in this TP-based language. The experience was gained
227 in a case study by the author. The author was considered an ideal
228 candidate for this study for the following reasons: as a student in
229 Telematics (computer science with focus on Signal Processing) he had
230 general knowledge in programming as well as specific domain knowledge
231 in Signal Processing; and he was not involved in the development of
232 {\sisac}'s programming language and interpeter, thus a novice to the
235 \subparagraph{The goal} of the case study was (1) some TP-based programs for
236 interactive course material for a specific ``Adavanced Signal
237 Processing Lab'' in a higher semester, (2) respective program
238 development with as little advice from the {\sisac}-team and (3) records
239 and comments for the main steps of development in an Isabelle theory;
240 this theory should provide guidelines for future programmers. An
241 excerpt from this theory is the main part of this paper.
243 The paper will use the problem in Fig.\ref{fig-interactive} as a
247 \includegraphics[width=140mm]{fig/isac-Ztrans-math}
248 \caption{Step-wise problem solving guided by the TP-based program}
249 \label{fig-interactive}
253 \paragraph{The problem is} from the domain of Signal Processing and requests to
254 determine the inverse Z-transform for a given term. Fig.\ref{fig-interactive}
255 also shows the beginning of the interactive construction of a solution
256 for the problem. This construction is done in the right window named
259 User-interaction on the Worksheet is {\em checked} and {\em guided} by
262 \item Formulas input by the user are {\em checked} by TP: such a
263 formula establishes a proof situation --- the prover has to derive the
264 formula from the logical context. The context is built up from the
265 formal specification of the problem (here hidden from the user) by the
267 \item If the user gets stuck, the program developed below in this
268 paper ``knows the next step'' from behind the scenes. How the latter
269 TP-service is exploited by dialogue authoring is out of scope of this
270 paper and can be studied in~\cite{gdaroczy-EP-13}.
271 \end{enumerate} It should be noted that the programmer using the
272 TP-based language is not concerned with interaction at all; we will
273 see that the program contains neither input-statements nor
274 output-statements. Rather, interaction is handled by services
275 generated automatically.
277 So there is a clear separation of concerns: Dialogues are
278 adapted by dialogue authors (in Java-based tools), using automatically
279 generated TP services, while the TP-based program is written by
280 mathematics experts (in Isabelle/ML). The latter is concern of this
283 \paragraph{The paper is structed} as follows: The introduction
284 \S\ref{intro} is followed by a brief re-introduction of the TP-based
285 programming language in \S\ref{PL}, which extends the executable
286 fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
287 play a specific role in Lucas-Interpretation and in providing the TP
288 services (\S\ref{PL-tacs}). The main part in \S\ref{trial} describes
289 the main steps in developing the program for the running example:
290 prepare domain knowledge, implement the formal specification of the
291 problem, prepare the environment for the program, implement the
292 program. The workflow of programming, debugging and testing is
293 described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
294 give directions identified for future development.
297 \section{\isac's Prototype for a Programming Language}\label{PL}
298 The prototype's language extends the executable fragment in the
299 language of the theorem prover
300 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}
301 by tactics which have a specific role in Lucas-Interpretation.
303 \subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
304 The executable fragment consists of data-type and function
305 definitions. It's usability even suggests that fragment for
306 introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic
307 whose type system resembles that of functional programming
308 languages. Thus there are
310 \item[base types,] in particular \textit{bool}, the type of truth
311 values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
312 natural, integer and complex numbers respectively in mathematics.
313 \item[type constructors] allow to define arbitrary types, from
314 \textit{set}, \textit{list} to advanced data-structures like
315 \textit{trees}, red-black-trees etc.
316 \item[function types,] denoted by $\Rightarrow$.
317 \item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
318 type polymorphism. Isabelle automatically computes the type of each
319 variable in a term by use of Hindley-Milner type inference
320 \cite{pl:hind97,Milner-78}.
323 \textbf{Terms} are formed as in functional programming by applying
324 functions to arguments. If $f$ is a function of type
325 $\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
326 $f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
327 has type $\tau$. There are many predefined infix symbols like $+$ and
328 $\leq$ most of which are overloaded for various types.
330 HOL also supports some basic constructs from functional programming:
331 {\it\label{isabelle-stmts}
332 \begin{tabbing} 123\=\kill
333 \>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
334 \>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
335 \>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
336 \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
338 \noindent \textit{The running example's program uses some of these elements
339 (marked by {\tt tt-font} on p.\pageref{expl-program}): ${\tt
340 let}\dots{\tt in}$ in lines $02 \dots 11$, as well as {\tt last} for
341 lists and {\tt o} for functional (forward) composition in line
342 $10$. In fact, the whole program is an Isabelle term with specific
343 function constants like {\sc program}, {\sc Substitute} and {\sc
344 Rewrite\_Set\_Inst} in lines $01$ and $10$ respectively.}
346 % Terms may also contain $\lambda$-abstractions. For example, $\lambda
347 % x. \; x$ is the identity function.
349 %JR warum auskommentiert?
351 \textbf{Formulae} are terms of type \textit{bool}. There are the basic
352 constants \textit{True} and \textit{False} and the usual logical
353 connectives (in decreasing order of precedence): $\neg, \land, \lor,
356 \textbf{Equality} is available in the form of the infix function $=$
357 of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
358 formulas, where it means ``if and only if''.
360 \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
361 P$. Quantifiers lead to non-executable functions, so functions do not
362 always correspond to programs, for instance, if comprising \\$(
363 \;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
366 \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
367 The prototype extends Isabelle's language by specific statements
368 called tactics~\footnote{{\sisac}'s tactics are different from
369 Isabelle's tactics: the former concern steps in a calculation, the
370 latter concern proof steps.} and tacticals. For the programmer these
371 statements are functions with the following signatures:
374 \item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
375 term} * {\it term}\;{\it list}$:
376 this tactic appplies {\it theorem} to a {\it term} yielding a {\it
377 term} and a {\it term list}, the list are assumptions generated by
378 conditional rewriting. For instance, the {\it theorem}
379 $b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
380 applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
381 $(\frac{2}{3}, [x\not=0])$.
383 \item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
384 term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
385 this tactic appplies {\it ruleset} to a {\it term}; {\it ruleset} is
386 a confluent and terminating term rewrite system, in general. If
387 none of the rules ({\it theorem}s) is applicable on interpretation
388 of this tactic, an exception is thrown.
390 % \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
391 % theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
394 % \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
395 % ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
398 \item[Substitute:] ${\it substitution}\Rightarrow{\it
399 term}\Rightarrow{\it term}$:
401 \item[Take:] ${\it term}\Rightarrow{\it term}$:
402 this tactic has no effect in the program; but it creates a side-effect
403 by Lucas-Interpretation (see below) and writes {\it term} to the
406 \item[Subproblem:] ${\it theory} * {\it specification} * {\it
407 method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
408 this tactic allows to enter a phase of interactive specification
409 of a theory ($\Re$, $\cal C$, etc), a formal specification (for instance,
410 a specific type of equation) and a method (for instance, solving an
411 equation symbolically or numerically).
414 The tactics play a specific role in
415 Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
416 break-points and control is handed over to the user. The user is free
417 to investigate underlying knowledge, applicable theorems, etc. And
418 the user can proceed constructing a solution by input of a tactic to
419 be applied or by input of a formula; in the latter case the
420 Lucas-Interpreter has built up a logical context (initialised with the
421 precondition of the formal specification) such that Isabelle can
422 derive the formula from this context --- or give feedback, that no
423 derivation can be found.
425 \subsection{Tacticals for Control of Interpretation}
426 The flow of control in a program can be determined by {\tt if then else}
427 and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
428 by additional tacticals:
430 \item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
431 term}$: iterates over tactics which take a {\it term} as argument as
432 long as a tactic is applicable (for instance, {\it Rewrite\_Set} might
435 \item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
436 if {\it tactic} is applicable, then it is applied to {\it term},
437 otherwise {\it term} is passed on unchanged.
439 \item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
440 term}\Rightarrow{\it term}$:
443 \item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
444 term}\Rightarrow{\it term}$:
446 \item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
447 term}\Rightarrow{\it term}$:
451 no input / output --- Lucas-Interpretation
453 .\\.\\.\\TODO\\.\\.\\
455 \section{Development of a Program on Trial}\label{trial}
456 As mentioned above, {\sisac} is an experimental system for a proof of
457 concept for Lucas-Interpretation~\cite{wn:lucas-interp-12}. The
458 latter interprets a specific kind of TP-based programming language,
459 which is as experimental as the whole prototype --- so programming in
460 this language can be only ``on trial'', presently. However, as a
461 prototype, the language addresses essentials described below.
463 \subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
465 %WN was Fachleute unter obigem Titel interessiert findet
466 %WN unterhalb des auskommentierten Textes.
468 %WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
469 %WN auf Computer-Mathematiker fokussiert.
470 % \paragraph{As mentioned in the introduction,} a prototype of an
471 % educational math assistant called
472 % {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
473 % \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
474 % the gap between (1) introducation and (2) application of mathematics:
475 % {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
476 % requires each fact and each action justified by formal logic, so
477 % {{{\sisac}{}}} makes justifications transparent to students in
478 % interactive step-wise problem solving. By that way {{\sisac}} already
481 % \item Introduction of math stuff (in e.g. partial fraction
482 % decomposition) by stepwise explaining and exercising respective
483 % symbolic calculations with ``next step guidance (NSG)'' and rigorously
484 % checking steps freely input by students --- this also in context with
485 % advanced applications (where the stuff to be taught in higher
486 % semesters can be skimmed through by NSG), and
487 % \item Application of math stuff in advanced engineering courses
488 % (e.g. problems to be solved by inverse Z-transform in a Signal
489 % Processing Lab) and now without much ado about basic math techniques
490 % (like partial fraction decomposition): ``next step guidance'' supports
491 % students in independently (re-)adopting such techniques.
493 % Before the question is answers, how {{\sisac}}
494 % accomplishes this task from a technical point of view, some remarks on
495 % the state-of-the-art is given, therefor follow up Section~\ref{emas}.
497 % \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
499 % \paragraph{Educational software in mathematics} is, if at all, based
500 % on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
501 % Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
502 % \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
503 % http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
504 % base technologies are used to program math lessons and sometimes even
505 % exercises. The latter are cumbersome: the steps towards a solution of
506 % such an interactive exercise need to be provided with feedback, where
507 % at each step a wide variety of possible input has to be foreseen by
508 % the programmer - so such interactive exercises either require high
509 % development efforts or the exercises constrain possible inputs.
511 % \subparagraph{A new generation} of educational math assistants (EMAs)
512 % is emerging presently, which is based on Theorem Proving (TP). TP, for
513 % instance Isabelle and Coq, is a technology which requires each fact
514 % and each action justified by formal logic. Pushed by demands for
515 % \textit{proven} correctness of safety-critical software TP advances
516 % into software engineering; from these advancements computer
517 % mathematics benefits in general, and math education in particular. Two
518 % features of TP are immediately beneficial for learning:
520 % \paragraph{TP have knowledge in human readable format,} that is in
521 % standard predicate calculus. TP following the LCF-tradition have that
522 % knowledge down to the basic definitions of set, equality,
523 % etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
524 % following the typical deductive development of math, natural numbers
525 % are defined and their properties
526 % proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
527 % etc. Present knowledge mechanized in TP exceeds high-school
528 % mathematics by far, however by knowledge required in software
529 % technology, and not in other engineering sciences.
531 % \paragraph{TP can model the whole problem solving process} in
532 % mathematical problem solving {\em within} a coherent logical
533 % framework. This is already being done by three projects, by
534 % Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
536 % Having the whole problem solving process within a logical coherent
537 % system, such a design guarantees correctness of intermediate steps and
538 % of the result (which seems essential for math software); and the
539 % second advantage is that TP provides a wealth of theories which can be
540 % exploited for mechanizing other features essential for educational
543 % \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
545 % One essential feature for educational software is feedback to user
546 % input and assistance in coming to a solution.
548 % \paragraph{Checking user input} by ATP during stepwise problem solving
549 % is being accomplished by the three projects mentioned above
550 % exclusively. They model the whole problem solving process as mentioned
551 % above, so all what happens between formalized assumptions (or formal
552 % specification) and goal (or fulfilled postcondition) can be
553 % mechanized. Such mechanization promises to greatly extend the scope of
554 % educational software in stepwise problem solving.
556 % \paragraph{NSG (Next step guidance)} comprises the system's ability to
557 % propose a next step; this is a challenge for TP: either a radical
558 % restriction of the search space by restriction to very specific
559 % problem classes is required, or much care and effort is required in
560 % designing possible variants in the process of problem solving
561 % \cite{proof-strategies-11}.
563 % Another approach is restricted to problem solving in engineering
564 % domains, where a problem is specified by input, precondition, output
565 % and postcondition, and where the postcondition is proven by ATP behind
566 % the scenes: Here the possible variants in the process of problem
567 % solving are provided with feedback {\em automatically}, if the problem
568 % is described in a TP-based programing language: \cite{plmms10} the
569 % programmer only describes the math algorithm without caring about
570 % interaction (the respective program is functional and even has no
571 % input or output statements!); interaction is generated as a
572 % side-effect by the interpreter --- an efficient separation of concern
573 % between math programmers and dialog designers promising application
574 % all over engineering disciplines.
577 % \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
578 % Authoring new mathematics knowledge in {{\sisac}} can be compared with
579 % ``application programing'' of engineering problems; most of such
580 % programing uses CAS-based programing languages (CAS = Computer Algebra
581 % Systems; e.g. Mathematica's or Maple's programing language).
583 % \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
584 % \cite{plmms10} for describing how to construct a solution to an
585 % engineering problem and for calling equation solvers, integration,
586 % etc~\footnote{Implementation of CAS-like functionality in TP is not
587 % primarily concerned with efficiency, but with a didactic question:
588 % What to decide for: for high-brow algorithms at the state-of-the-art
589 % or for elementary algorithms comprehensible for students?} within TP;
590 % TP can ensure ``systems that never make a mistake'' \cite{casproto} -
591 % are impossible for CAS which have no logics underlying.
593 % \subparagraph{Authoring is perfect} by writing such TP based programs;
594 % the application programmer is not concerned with interaction or with
595 % user guidance: this is concern of a novel kind of program interpreter
596 % called Lucas-Interpreter. This interpreter hands over control to a
597 % dialog component at each step of calculation (like a debugger at
598 % breakpoints) and calls automated TP to check user input following
599 % personalized strategies according to a feedback module.
601 % However ``application programing with TP'' is not done with writing a
602 % program: according to the principles of TP, each step must be
603 % justified. Such justifications are given by theorems. So all steps
604 % must be related to some theorem, if there is no such theorem it must
605 % be added to the existing knowledge, which is organized in so-called
606 % \textbf{theories} in Isabelle. A theorem must be proven; fortunately
607 % Isabelle comprises a mechanism (called ``axiomatization''), which
608 % allows to omit proofs. Such a theorem is shown in
609 % Example~\ref{eg:neuper1}.
611 The running example, introduced by Fig.\ref{fig-interactive} on
612 p.\pageref{fig-interactive}, requires to determine the inverse $\cal
613 Z$-transform for a class of functions. The domain of Signal Processing
614 is accustomed to specific notation for the resulting functions, which
615 are absolutely summable and are called TODO: $u[n]$, where $u$ is the
616 function, $n$ is the argument and the brackets indicate that the
617 arguments are TODO. Surprisingly, Isabelle accepts the rules for
618 ${\cal Z}^{-1}$ in this traditional notation~\footnote{Isabelle
619 experts might be particularly surprised, that the brackets do not
620 cause errors in typing (as lists).}:
624 {\small\begin{tabbing}
625 123\=123\=123\=123\=\kill
627 \>axiomatization where \\
628 \>\> rule1: ``${\cal Z}^{-1}\;1 = \delta [n]$'' and\\
629 \>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow {\cal Z}^{-1}\;z / (z - 1) = u [n]$'' and\\
630 \>\> rule3: ``$\vert\vert$ z $\vert\vert$ < 1 ==> z / (z - 1) = -u [-n - 1]'' and \\
632 \>\> rule4: ``$\vert\vert$ z $\vert\vert$ > $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = $\alpha^n$ $\cdot$ u [n]'' and\\
634 \>\> rule5: ``$\vert\vert$ z $\vert\vert$ < $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = -($\alpha^n$) $\cdot$ u [-n - 1]'' and\\
636 \>\> rule6: ``$\vert\vert$ z $\vert\vert$ > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]''\\
642 These 6 rules can be used as conditional rewrite rules, depending on
643 the respective convergence radius. Satisfaction from accordance with traditional notation
644 contrasts with the above word {\em axiomatization}: As TP-based, the
645 programming language expects these rules as {\em proved} theorems, and
646 not as axioms implemented in the above brute force manner; otherwise
647 all the verification efforts envisaged (like proof of the
648 post-condition, see below) would be meaningless.
650 Isabelle provides a large body of knowledge, rigorously proven from
651 the basic axioms of mathematics~\footnote{This way of rigorously
652 deriving all knowledge from first principles is called the
653 LCF-paradigm in TP.}. In the case of the Z-Transform the most advanced
654 knowledge can be found in the theoris on Multivariate
655 Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
656 building up knowledge such that a proof for the above rules would be
657 reasonably short and easily comprehensible, still requires lots of
658 work (and is definitely out of scope of our case study).
660 \paragraph{At the state-of-the-art in mechanization of knowledge} in
661 engineering sciences, the process does not stop with the mechanization of
662 mathematics traditionally used in these sciences. Rather, ``Formal Methods''~\cite{TODO-formal-methods}
663 are expected to proceed to formal and explicit description of physical items. Signal Processing,
664 for instance is concerned with physical devices for signal acquisition
665 and reconstruction, which involve measuring a physical signal, storing
666 it, and possibly later rebuilding the original signal or an
667 approximation thereof. For digital systems, this typically includes
668 sampling and quantization; devices for signal compression, including
669 audio compression, image compression, and video compression, etc.
670 ``Domain engineering''\cite{db-domain-engineering} is concerned with
671 {\em specification} of these devices' components and features; this
672 part in the process of mechanization is only at the beginning in domains
673 like Signal Processing.
675 \subparagraph{TP-based programming, concern of this paper,} is determined to
676 add ``algorithmic knowledge'' in Fig.\ref{fig:mathuni} on
677 p.\pageref{fig:mathuni}. As we shall see below, TP-based programming
678 starts with a formal {\em specification} of the problem to be solved.
681 \subsection{Specification of the Problem}\label{spec}
682 %WN <--> \chapter 7 der Thesis
683 %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
685 The problem of the running example is textually described in
686 Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}. The {\em
687 formal} specification of this problem, in traditional mathematical
688 notation, could look lik is this:
690 %WN Hier brauchen wir die Spezifikation des 'running example' ...
692 %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
693 %JR der post condition - die existiert für uns ja eigentlich nicht aka
694 %JR haben sie bis jetzt nicht beachtet
697 {\small\begin{tabbing}
698 123,\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
701 \>input \>: filterExpression $X=\frac{3}{(z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}$\\
702 \>precond \>: filterExpression continius on $\mathbb{R}$ \\
703 \>output \>: stepResponse $x[n]$ \\
704 \>postcond \>:{\small $\;A=2uv-u^2 \;\land\; (\frac{u}{2})^2+(\frac{v}{2})^2=r^2 \;\land$}\\
705 \> \>\>{\small $\;\forall \;A^\prime\; u^\prime \;v^\prime.\;(A^\prime=2u^\prime v^\prime-(u^\prime)^2 \land
706 (\frac{u^\prime}{2})^2+(\frac{v^\prime}{2})^2=r^2) \Longrightarrow A^\prime \leq A$} \\
710 And this is the implementation of the formal specification in the present
711 prototype, still bar-bones without support for authoring:
712 %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
715 01 store_specification
716 02 (prepare_specification
718 04 "pbl_SP_Ztrans_inv"
720 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
721 07 [ ("#Given", ["filterExpression X_eq"]),
722 08 ("#Pre" , ["X_eq is_continuous"]),
723 19 ("#Find" , ["stepResponse n_eq"]),
724 10 ("#Post" , [" TODO "])],
725 11 append_rls Erls [Calc ("Atools.is_continuous", eval_is_continuous)],
727 13 [["SignalProcessing","Z_Transform","Inverse"]]));
729 Although the above details are partly very technical, we explain them
730 in order to document some intricacies of TP-based programming in the
731 present state of the {\sisac} prototype:
733 \item[01..02]\textit{store\_specification:} stores the result of the
734 function \textit{prep\_specification} in a global reference
735 \textit{Unsynchronized.ref}, which causes principal conflicts with
736 Isabelle's asyncronous document model~\cite{Wenzel-11:doc-orient} and
737 parallel execution~\cite{Makarius-09:parall-proof} and is under
738 reconstruction already.
740 \textit{prep\_pbt:} translates the specification to an internal format
741 which allows efficient processing; see for instance line {\rm 07}
743 \item[03..04] are the ``mathematics author'' holding the copy-rights
744 and a unique identifier for the specification within {\sisac},
745 complare line {\rm 06}.
746 \item[05] is the Isabelle \textit{theory} required to parse the
747 specification in lines {\rm 07..10}.
748 \item[06] is a key into the tree of all specifications as presented to
749 the user (where some branches might be hidden by the dialog
751 \item[07..10] are the specification with input, pre-condition, output
752 and post-condition respectively; the post-condition is not handled in
753 the prototype presently.
754 \item[11] creates a term rewriting system (abbreviated \textit{rls} in
755 {\sisac}) which evaluates the pre-condition for the actual input data.
756 Only if the evaluation yields \textit{True}, a program con be started.
757 \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
758 problem associated to a function from Computer Algebra (like an
759 equation solver) which is not the case here.
760 \item[13] is the specific key into the tree of programs addressing a
761 method which is able to find a solution which satiesfies the
762 post-condition of the specification.
766 %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
769 % {guh : guh, (*unique within this isac-knowledge*)
770 % mathauthors: string list, (*copyright*)
771 % init : pblID, (*to start refinement with*)
772 % thy : theory, (* which allows to compile that pbt
773 % TODO: search generalized for subthy (ref.p.69*)
774 % (*^^^ WN050912 NOT used during application of the problem,
775 % because applied terms may be from 'subthy' as well as from super;
776 % thus we take 'maxthy'; see match_ags !*)
777 % cas : term option,(*'CAS-command'*)
778 % prls : rls, (* for preds in where_*)
779 % where_: term list, (* where - predicates*)
781 % (*this is the model-pattern;
782 % it contains "#Given","#Where","#Find","#Relate"-patterns
783 % for constraints on identifiers see "fun cpy_nam"*)
784 % met : metID list}; (* methods solving the pbt*)
786 %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
787 %WN oben selbst geschrieben.
792 %WN das w"urde ich in \sec\label{progr} verschieben und
793 %WN das SubProblem partial fractions zum Erkl"aren verwenden.
794 % Such a specification is checked before the execution of a program is
795 % started, the same applies for sub-programs. In the following example
796 % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
802 % {\ttfamily \begin{tabbing}
803 % ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
804 % ``\>\>[linear,univariate,equation,test],'' \\
805 % ``\>\>[Test,solve\_linear])'' \\
806 % ``\>[BOOL equ, REAL z])'' \\
810 % \noindent If a program requires a result which has to be
811 % calculated first we can use a subproblem to do so. In our specific
812 % case we wanted to calculate the zeros of a fraction and used a
813 % subproblem to calculate the zeros of the denominator polynom.
818 \subsection{Implementation of the Method}\label{meth}
819 %WN <--> \chapter 7 der Thesis
821 \subsection{Preparation of Simplifiers for the Program}\label{simp}
823 \subsection{Preparation of ML-Functions}\label{funs}
824 %WN <--> Thesis 6.1 -- 6.3: jene ausw"ahlen, die Du f"ur \label{progr}
827 \subsection{Implementation of the TP-based Program}\label{progr}
828 %WN <--> \chapter 8 der Thesis
831 \section{Workflow of Programming in the Prototype}\label{workflow}
832 %WN ``workflow'' heisst: das mache ich zuerst, dann das ...
833 \subsection{Preparations and Trials}\label{flow-prep}
834 \subsubsection{Trials on Notation and Termination}
836 \paragraph{Technical notations} are a big problem for our piece of software,
837 but the reason for that isn't a fault of the software itself, one of the
838 troubles comes out of the fact that different technical subtopics use different
839 symbols and notations for a different purpose. The most famous example for such
840 a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
841 math). In the specific part of signal processing one of this notation issues is
842 the use of brackets --- we use round brackets for analoge signals and squared
843 brackets for digital samples. Also if there is no problem for us to handle this
844 fact, we have to tell the machine what notation leads to wich meaning and that
845 this purpose seperation is only valid for this special topic - signal
847 \subparagraph{In the programming language} itself it is not possible to declare
848 fractions, exponents, absolutes and other operators or remarks in a way to make
849 them pretty to read; our only posssiblilty were ASCII characters and a handfull
850 greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
852 With the upper collected knowledge it is possible to check if we were able to
853 donate all required terms and expressions.
855 \subsubsection{Definition and Usage of Rules}
857 \paragraph{The core} of our implemented problem is the Z-Transformation, due
858 the fact that the transformation itself would require higher math which isn't
859 yet avaible in our system we decided to choose the way like it is applied in
860 labratory and problem classes at our university - by applying transformation
861 rules (collected in transformation tables).
862 \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
863 use of axiomatizations like shown in Example~\ref{eg:ruledef}
870 rule1: ``1 = $\delta$[n]'' and
871 rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
872 rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
876 This rules can be collected in a ruleset and applied to a given expression as
877 follows in Example~\ref{eg:ruleapp}.
883 \item Store rules in ruleset:
885 val inverse_Z = append_rls "inverse_Z" e_rls
886 [ Thm ("rule1",num_str @{thm rule1}),
887 Thm ("rule2",num_str @{thm rule2}),
888 Thm ("rule3",num_str @{thm rule3})
890 \item Define exression:
892 val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
895 val SOME (sample_term', asm) =
896 rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
900 The use of rulesets makes it much easier to develop our designated applications,
901 but the programmer has to be careful and patient. When applying rulesets
902 two important issues have to be mentionend:
903 \subparagraph{How often} the rules have to be applied? In case of
904 transformations it is quite clear that we use them once but other fields
905 reuqire to apply rules until a special condition is reached (e.g.
906 a simplification is finished when there is nothing to be done left).
907 \subparagraph{The order} in which rules are applied often takes a big effect
908 and has to be evaluated for each purpose once again.
910 In our special case of Signal Processing and the rules defined in
911 Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
912 constants. After this step has been done it no mather which rule fit's next.
914 \subsubsection{Helping Functions}
915 %get denom, argument in
916 \subsubsection{Trials on equation solving}
917 %simple eq and problem with double fractions/negative exponents
920 \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
921 TODO Build\_Inverse\_Z\_Transform.thy ... ``imports Isac''
923 \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
924 TODO http://www.ist.tugraz.at/isac/index.php/Extend\_ISAC\_Knowledge\#Add\_an\_example ?
926 -------------------------------------------------------------------
928 Das unterhalb hab' ich noch nicht durchgearbeitet; einiges w\"are
929 vermutlich auf andere sections aufzuteilen.
931 -------------------------------------------------------------------
933 \subsection{Formalization of missing knowledge in Isabelle}
935 \paragraph{A problem} behind is the mechanization of mathematic
936 theories in TP-bases languages. There is still a huge gap between
937 these algorithms and this what we want as a solution - in Example
944 X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
947 \noindent A very simple example on this what we call gap is the
948 simplification above. It is needles to say that it is correct and also
949 Isabelle for fills it correct - \emph{always}. But sometimes we don't
950 want expand such terms, sometimes we want another structure of
951 them. Think of a problem were we now would need only the coefficients
952 of $X$ and $Y$. This is what we call the gap between mechanical
953 simplification and the solution.
958 \paragraph{We are not able to fill this gap,} until we have to live
959 with it but first have a look on the meaning of this statement:
960 Mechanized math starts from mathematical models and \emph{hopefully}
961 proceeds to match physics. Academic engineering starts from physics
962 (experimentation, measurement) and then proceeds to mathematical
963 modeling and formalization. The process from a physical observance to
964 a mathematical theory is unavoidable bound of setting up a big
965 collection of standards, rules, definition but also exceptions. These
966 are the things making mechanization that difficult.
975 \noindent Think about some units like that one's above. Behind
976 each unit there is a discerning and very accurate definition: One
977 Meter is the distance the light travels, in a vacuum, through the time
978 of 1 / 299.792.458 second; one kilogram is the weight of a
979 platinum-iridium cylinder in paris; and so on. But are these
980 definitions usable in a computer mechanized world?!
985 \paragraph{A computer} or a TP-System builds on programs with
986 predefined logical rules and does not know any mathematical trick
987 (follow up example \ref{eg:trick}) or recipe to walk around difficult
993 \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
994 \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
995 \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
996 \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
998 \noindent Sometimes it is also useful to be able to apply some
999 \emph{tricks} to get a beautiful and particularly meaningful result,
1000 which we are able to interpret. But as seen in this example it can be
1001 hard to find out what operations have to be done to transform a result
1002 into a meaningful one.
1007 \paragraph{The only possibility,} for such a system, is to work
1008 through its known definitions and stops if none of these
1009 fits. Specified on Signal Processing or any other application it is
1010 often possible to walk through by doing simple creases. This creases
1011 are in general based on simple math operational but the challenge is
1012 to teach the machine \emph{all}\footnote{Its pride to call it
1013 \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1014 reach a high level of \emph{all} but it in real it will still be a
1015 survey of knowledge which links to other knowledge and {{\sisac}{}} a
1016 trainer and helper but no human compensating calculator.
1018 {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1019 specifications of problems out of topics from Signal Processing, etc.)
1020 and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1021 physical knowledge. The result is a three-dimensional universe of
1022 mathematics seen in Figure~\ref{fig:mathuni}.
1026 \includegraphics{fig/universe}
1027 \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1028 combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1029 leads to a three dimensional math universe.\label{fig:mathuni}}
1033 %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1034 %WN bitte folgende Bezeichnungen nehmen:
1036 %WN axis 1: Algorithmic Knowledge (Programs)
1037 %WN axis 2: Application-oriented Knowledge (Specifications)
1038 %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1040 %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1041 %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1042 %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1044 %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1045 %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1046 %JR gefordert werden...
1048 \subsection{Notes on Problems with Traditional Notation}
1050 \paragraph{During research} on these topic severely problems on
1051 traditional notations have been discovered. Some of them have been
1052 known in computer science for many years now and are still unsolved,
1053 one of them aggregates with the so called \emph{Lambda Calculus},
1054 Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1061 \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1064 \[ f(p)=\ldots\; p \in \quad R \]
1067 \noindent Above we see two equations. The first equation aims to
1068 be a mapping of an function from the reel range to the reel one, but
1069 when we change only one letter we get the second equation which
1070 usually aims to insert a reel point $p$ into the reel function. In
1071 computer science now we have the problem to tell the machine (TP) the
1072 difference between this two notations. This Problem is called
1073 \emph{Lambda Calculus}.
1078 \paragraph{An other problem} is that terms are not full simplified in
1079 traditional notations, in {{\sisac}} we have to simplify them complete
1080 to check weather results are compatible or not. in e.g. the solutions
1081 of an second order linear equation is an rational in {{\sisac}} but in
1082 tradition we keep fractions as long as possible and as long as they
1083 aim to be \textit{beautiful} (1/8, 5/16,...).
1084 \subparagraph{The math} which should be mechanized in Computer Theorem
1085 Provers (\emph{TP}) has (almost) a problem with traditional notations
1086 (predicate calculus) for axioms, definitions, lemmas, theorems as a
1087 computer program or script is not able to interpret every Greek or
1088 Latin letter and every Greek, Latin or whatever calculations
1089 symbol. Also if we would be able to handle these symbols we still have
1090 a problem to interpret them at all. (Follow up \hbox{Example
1097 u\left[n\right] \ \ldots \ unitstep
1100 \noindent The unitstep is something we need to solve Signal
1101 Processing problem classes. But in {{{\sisac}{}}} the rectangular
1102 brackets have a different meaning. So we abuse them for our
1103 requirements. We get something which is not defined, but usable. The
1104 Result is syntax only without semantic.
1109 In different problems, symbols and letters have different meanings and
1110 ask for different ways to get through. (Follow up \hbox{Example
1117 \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1120 \noindent For using exponents the three \texttt{widehat} symbols
1121 are required. The reason for that is due the development of
1122 {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1123 already in use for different operations.
1128 \paragraph{Also the output} can be a problem. We are familiar with a
1129 specified notations and style taught in university but a computer
1130 program has no knowledge of the form proved by a professor and the
1131 machines themselves also have not yet the possibilities to print every
1132 symbol (correct) Recent developments provide proofs in a human
1133 readable format but according to the fact that there is no money for
1134 good working formal editors yet, the style is one thing we have to
1137 \section{Problems rising out of the Development Environment}
1139 fehlermeldungen! TODO
1141 \section{Conclusion}\label{conclusion}
1145 \bibliographystyle{alpha}
1146 \bibliography{references}