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61 \ and Technology, Volume 1, Number 1, ISSN 1933-2823} %
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70 % Please place your own definitions here
72 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
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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 Technology\\
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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.
132 Both kinds of support can be achieved by so-called
133 Lucas-Interpretation which combines deduction and computation and, for
134 the latter, uses a novel kind of programming language. This language
135 is based on (Computer) Theorem Proving (TP), thus called a ``TP-based
136 programming language''.
138 This paper is the experience report of the first ``application
139 programmer'' using this language for creating exercises in step-wise
140 problem solving for an advanced lab in Signal Processing. The tasks
141 involved in TP-based programming are described together with the
142 experience gained from a prototype of the programming language and of
145 The report concludes with a positive proof of concept, states
146 insufficiency usability of the prototype and captures the requirements
147 for further development of both, the programming language and the
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160 % Please use the following to indicate sections, subsections,
161 % etc. Please also use \subsubsection{...}, \paragraph{...}
162 % and \subparagraph{...} as necessary.
165 \section{Introduction}\label{intro}
167 % \paragraph{Didactics of mathematics}
168 %WN: wenn man in einem high-quality paper von 'didactics' spricht,
169 %WN muss man am state-of-the-art ankn"upfen -- siehe
170 %WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
171 % faces a specific issue, a gap
172 % between (1) introduction of math concepts and skills and (2)
173 % application of these concepts and skills, which usually are separated
174 % into different units in curricula (for good reasons). For instance,
175 % (1) teaching partial fraction decomposition is separated from (2)
176 % application for inverse Z-transform in signal processing.
178 % \par This gap is an obstacle for applying math as an fundamental
179 % thinking technology in engineering: In (1) motivation is lacking
180 % because the question ``What is this stuff good for?'' cannot be
181 % treated sufficiently, and in (2) the ``stuff'' is not available to
182 % students in higher semesters as widespread experience shows.
184 % \paragraph{Motivation} taken by this didactic issue on the one hand,
185 % and ongoing research and development on a novel kind of educational
186 % mathematics assistant at Graz University of
187 % Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
188 % scope with this issue on the other hand, several institutes are
189 % planning to join their expertise: the Institute for Information
190 % Systems and Computer Media (IICM), the Institute for Software
191 % Technology (IST), the Institutes for Mathematics, the Institute for
192 % Signal Processing and Speech Communication (SPSC), the Institute for
193 % Structural Analysis and the Institute of Electrical Measurement and
194 % Measurement Signal Processing.
195 %WN diese Information ist f"ur das Paper zu spezielle, zu aktuell
196 %WN und damit zu verg"anglich.
197 % \par This thesis is the first attempt to tackle the above mentioned
198 % issue, it focuses on Telematics, because these specific studies focus
199 % on mathematics in \emph{STEOP}, the introductory orientation phase in
200 % Austria. \emph{STEOP} is considered an opportunity to investigate the
201 % impact of {\sisac}'s prototype on the issue and others.
204 Traditional course material in engineering disciplines lacks an
205 important component, interactive support for step-wise problem
206 solving. Theorem-Proving (TP) technology can provide such support by
207 specific services. An important part of such services is called
208 ``next-step-guidance'', generated by a specific kind of ``TP-based
209 programming language''. In the
210 {\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
211 a language is prototyped in line with~\cite{plmms10} and built upon
212 the theorem prover Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}
213 \footnote{http://isabelle.in.tum.de/}.
214 The TP services are coordinated by a specific interpreter for the
215 programming language, called
216 Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language and the
217 interpreter will be briefly re-introduced in order to make the paper
220 The main part of the paper is an account of first experiences
221 with programming in this TP-based language. The experience was gained
222 in a case study by the author. The author was considered an ideal
223 candidate for this study for the following reasons: as a student in
224 Telematics (computer science with focus on Signal Processing) he had
225 general knowledge in programming as well as specific domain knowledge
226 in Signal Processing; and he was {\em not} involved in the development of
227 {\sisac}'s programming language and interpreter, thus being a novice to the
230 The goals of the case study were: (1) to identify some TP-based programs for
231 interactive course material for a specific ``Advanced Signal
232 Processing Lab'' in a higher semester, (2) respective program
233 development with as little advice as possible from the {\sisac}-team and (3)
234 to document records and comments for the main steps of development in an
235 Isabelle theory; this theory should provide guidelines for future programmers.
236 An excerpt from this theory is the main part of this paper.
239 \medskip The major example resulting from the case study will be used
240 as running example throughout this paper. This example requires a
241 program resembling the size of real-world applications in engineering;
242 such a size was considered essential for the case study, since there
243 are many small programs for a long time (mainly concerned with
244 elementary Computer Algebra like simplification, equation solving,
245 calculus, etc.~\footnote{The programs existing in the {\sisac}
246 prototype are found at
247 http://www.ist.tugraz.at/projects/isac/www/kbase/met/index\_met.html})
249 \paragraph{The mathematical background of the running example} is the
250 following: In Signal Processing, ``the ${\cal Z}$-Transform for
251 discrete-time signals is the counterpart of the Laplace transform for
252 continuous-time signals, and they each have a similar relationship to
253 the corresponding Fourier transform. One motivation for introducing
254 this generalization is that the Fourier transform does not converge
255 for all sequences, and it is useful to have a generalization of the
256 Fourier transform that encompasses a broader class of signals. A
257 second advantage is that in analytic problems, the $z$-transform
258 notation is often more convenient than the Fourier transform
259 notation.'' ~\cite[p. 128]{oppenheim2010discrete}. The $z$-transform
262 X(z)=\sum_{n=-\infty }^{\infty }x[n]z^{-n}
264 where a discrete time sequence $x[n]$ is transformed into the function
265 $X(z)$ where $z$ is a continuous complex variable. The inverse
266 function is addressed in the running example and can be determined by
269 x[n]=\frac{1}{2\pi j} \oint_{C} X(z)\cdot z^{n-1} dz
271 where the letter $C$ represents a contour within the range of
272 convergence of the $z$- transform. The unit circle can be a special
273 case of this contour. Remember that $j$ is the complex number in the
274 domain of engineering. As this transformation requires high effort to
275 be solved, tables of commonly used transform pairs are used in
276 education as well as in engineering practice; such tables can be found
277 at~\cite{wiki:1} or~\cite[Table~3.1]{oppenheim2010discrete} as well.
278 A completely solved and more detailed example can be found at
279 ~\cite[p. 149f]{oppenheim2010discrete}.
281 Following conventions in engineering education and in practice, the
282 running example solves the problem by use of a table.
284 \paragraph{Support for interactive stepwise problem solving} in the
285 {\sisac} prototype is shown in Fig.\ref{fig-interactive}~\footnote{ Fig.\ref{fig-interactive} also shows the prototype status of {\sisac}; for instance,
286 the lack of 2-dimensional presentation and input of formulas is the major obstacle for field-tests in standard classes.}:
287 A student inputs formulas line by line on the \textit{``Worksheet''},
288 and each step (i.e. each formula on completion) is immediately checked
289 by the system such that at most one inconsistent formula can reside on
290 the Worksheet (on the input line, marked by the red $\otimes$).
293 \includegraphics[width=140mm]{fig/isac-Ztrans-math-3}
294 %\includegraphics[width=140mm]{fig/isac-Ztrans-math}
295 \caption{Step-wise problem solving guided by the TP-based program
296 \label{fig-interactive}}
299 If the student gets stuck and does not know the formula to proceed with,
300 there is the button \framebox{NEXT} proceeding to the next step. The
301 button \framebox{AUTO} immediately delivers the final result in case
302 the student is not interested in intermediate steps.
304 Adaptive dialogue guidance is already under
305 construction~\cite{gdaroczy-EP-13} and the two buttons will disappear,
306 since their presence is not wanted in many learning scenarios (in
307 particular, {\em not} in written exams).
309 The buttons \framebox{Theories}, \framebox{Problems} and
310 \framebox{Methods} are the entry points for interactive lookup of the
311 underlying knowledge. For instance, pushing \framebox{Theories} in
312 the configuration shown in Fig.\ref{fig-interactive}, pops up a
313 ``Theory browser'' displaying the theorem(s) justifying the current
314 step. All browsers allow to lookup all other theories, thus
315 supporting indepentend investigation of underlying definitions,
316 theorems, proofs --- where the HTML representation of the browsers is
317 ready for arbitrary multimedia add-ons.
319 % can be explained by having a look at
320 % Fig.\ref{fig-interactive} which shows the beginning of the interactive
321 % construction of a solution for the problem. This construction is done in the
322 % right window named ``Worksheet''.
324 % User-interaction on the Worksheet is {\em checked} and {\em guided} by
327 % \item Formulas input by the user are {\em checked} by TP: such a
328 % formula establishes a proof situation --- the prover has to derive the
329 % formula from the logical context. The context is built up from the
330 % formal specification of the problem (here hidden from the user) by the
332 % \item If the user gets stuck, the program developed below in this
333 % paper ``knows the next step'' and Lucas-Interpretation provides services
334 % featuring so-called ``next-step-guidance''; this is out of scope of this
335 % paper and can be studied in~\cite{gdaroczy-EP-13}.
336 % \end{enumerate} It should be noted that the programmer using the
337 % TP-based language is not concerned with interaction at all; we will
338 % see that the program contains neither input-statements nor
339 % output-statements. Rather, interaction is handled by the interpreter
342 % So there is a clear separation of concerns: Dialogues are adapted by
343 % dialogue authors (in Java-based tools), using TP services provided by
344 % Lucas-Interpretation. The latter acts on programs developed by
345 % mathematics-authors (in Isabelle/ML); their task is concern of this
348 \bigskip The paper is structured as follows: The introduction
349 \S\ref{intro} is followed by a brief re-introduction of the TP-based
350 programming language in \S\ref{PL}, which extends the executable
351 fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
352 play a specific role in Lucas-Interpretation and in providing the TP
353 services (\S\ref{PL-tacs}). The main part \S\ref{trial} describes
354 the main steps in developing the program for the running example:
355 prepare domain knowledge, implement the formal specification of the
356 problem, prepare the environment for the interpreter, implement the
357 program in \S\ref{isabisac} to \S\ref{progr} respectively.
358 The work-flow of programming, debugging and testing is
359 described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
360 give directions identified for future development.
363 \section{\isac's Prototype for a Programming Language}\label{PL}
364 The prototype of the language and of the Lucas-Interpreter is briefly
365 described from the point of view of a programmer. The language extends
366 the executable fragment of Higher-Order Logic (HOL) in the theorem prover
367 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
369 \subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
370 The executable fragment consists of data-type and function
371 definitions. It's usability even suggests that fragment for
372 introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic whose type system resembles that of functional programming
373 languages. Thus there are
375 \item[base types,] in particular \textit{bool}, the type of truth
376 values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
377 natural, integer and complex numbers respectively in mathematics.
378 \item[type constructors] allow to define arbitrary types, from
379 \textit{set}, \textit{list} to advanced data-structures like
380 \textit{trees}, red-black-trees etc.
381 \item[function types,] denoted by $\Rightarrow$.
382 \item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
383 type polymorphism. Isabelle automatically computes the type of each
384 variable in a term by use of Hindley-Milner type inference
385 \cite{pl:hind97,Milner-78}.
388 \textbf{Terms} are formed as in functional programming by applying
389 functions to arguments. If $f$ is a function of type
390 $\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
391 $f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
392 has type $\tau$. There are many predefined infix symbols like $+$ and
393 $\leq$ most of which are overloaded for various types.
395 HOL also supports some basic constructs from functional programming:
396 {\footnotesize\it\label{isabelle-stmts}
397 \begin{tabbing} 123\=\kill
398 01\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
399 02\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
400 03\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
401 \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
403 \noindent The running example's program uses some of these elements
404 (marked by {\tt tt-font} on p.\pageref{s:impl}): for instance {\tt
405 let}\dots{\tt in} in lines {\rm 02} \dots {\rm 13}. In fact, the whole program
406 is an Isabelle term with specific function constants like {\tt
407 program}, {\tt Take}, {\tt Rewrite}, {\tt Subproblem} and {\tt
408 Rewrite\_Set} in lines {\rm 01, 03. 04, 07, 10} and {\rm 11, 12}
411 % Terms may also contain $\lambda$-abstractions. For example, $\lambda
412 % x. \; x$ is the identity function.
414 %JR warum auskommentiert? WN2...
415 %WN2 weil ein Punkt wie dieser in weiteren Zusammenh"angen innerhalb
416 %WN2 des Papers auftauchen m"usste; nachdem ich einen solchen
417 %WN2 Zusammenhang _noch_ nicht sehe, habe ich den Punkt _noch_ nicht
419 %WN2 Wenn der Punkt nicht weiter gebraucht wird, nimmt er nur wertvollen
420 %WN2 Platz f"ur Anderes weg.
422 \textbf{Formulae} are terms of type \textit{bool}. There are the basic
423 constants \textit{True} and \textit{False} and the usual logical
424 connectives (in decreasing order of precedence): $\neg, \land, \lor,
427 \textbf{Equality} is available in the form of the infix function $=$
428 of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
429 formulas, where it means ``if and only if''.
431 \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
432 P$. Quantifiers lead to non-executable functions, so functions do not
433 always correspond to programs, for instance, if comprising \\$(
434 \;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
437 \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
438 The prototype extends Isabelle's language by specific statements
439 called tactics~\footnote{{\sisac}'s. This tactics are different from
440 Isabelle's tactics: the former concern steps in a calculation, the
441 latter concern proofs.} and tactics. For the programmer these
442 statements are functions with the following signatures:
445 \item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
446 term} * {\it term}\;{\it list}$:
447 this tactic applies {\it theorem} to a {\it term} yielding a {\it
448 term} and a {\it term list}, the list are assumptions generated by
449 conditional rewriting. For instance, the {\it theorem}
450 $b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
451 applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
452 $(\frac{2}{3}, [x\not=0])$.
454 \item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
455 term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
456 this tactic applies {\it ruleset} to a {\it term}; {\it ruleset} is
457 a confluent and terminating term rewrite system, in general. If
458 none of the rules ({\it theorem}s) is applicable on interpretation
459 of this tactic, an exception is thrown.
461 % \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
462 % theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
465 % \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
466 % ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
470 \item[Substitute:] ${\it substitution}\Rightarrow{\it
471 term}\Rightarrow{\it term}$: allows to access sub-terms.
474 \item[Take:] ${\it term}\Rightarrow{\it term}$:
475 this tactic has no effect in the program; but it creates a side-effect
476 by Lucas-Interpretation (see below) and writes {\it term} to the
479 \item[Subproblem:] ${\it theory} * {\it specification} * {\it
480 method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
481 this tactic is a generalisation of a function call: it takes an
482 \textit{argument list} as usual, and additionally a triple consisting
483 of an Isabelle \textit{theory}, an implicit \textit{specification} of the
484 program and a \textit{method} containing data for Lucas-Interpretation,
485 last not least a program (as an explicit specification)~\footnote{In
486 interactive tutoring these three items can be determined explicitly
489 The tactics play a specific role in
490 Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
491 break-points where, as a side-effect, a line is added to a calculation
492 as a protocol for proceeding towards a solution in step-wise problem
493 solving. At the same points Lucas-Interpretation serves interactive
494 tutoring and hands over control to the user. The user is free to
495 investigate underlying knowledge, applicable theorems, etc. And the
496 user can proceed constructing a solution by input of a tactic to be
497 applied or by input of a formula; in the latter case the
498 Lucas-Interpreter has built up a logical context (initialised with the
499 precondition of the formal specification) such that Isabelle can
500 derive the formula from this context --- or give feedback, that no
501 derivation can be found.
503 \subsection{Tactics as Control Flow Statements}
504 The flow of control in a program can be determined by {\tt if then else}
505 and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
506 by additional tactics:
508 \item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
509 term}$: iterates over tactics which take a {\it term} as argument as
510 long as a tactic is applicable (for instance, {\tt Rewrite\_Set} might
513 \item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
514 if {\it tactic} is applicable, then it is applied to {\it term},
515 otherwise {\it term} is passed on without changes.
517 \item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
518 term}\Rightarrow{\it term}$: If the first {\it tactic} is applicable,
519 it is applied to the first {\it term} yielding another {\it term},
520 otherwise the second {\it tactic} is applied; if none is applicable an
523 \item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
524 term}\Rightarrow{\it term}$: applies the first {\it tactic} to the
525 first {\it term} yielding an intermediate term (not appearing in the
526 signature) to which the second {\it tactic} is applied.
528 \item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
529 term}\Rightarrow{\it term}$: if the first {\it term} is true, then the
530 {\it tactic} is applied to the first {\it term} yielding an
531 intermediate term (not appearing in the signature); the intermediate
532 term is added to the environment the first {\it term} is evaluated in
533 etc. as long as the first {\it term} is true.
535 The tactics are not treated as break-points by Lucas-Interpretation
536 and thus do neither contribute to the calculation nor to interaction.
538 \section{Concepts and Tasks in TP-based Programming}\label{trial}
539 %\section{Development of a Program on Trial}
541 This section presents all the concepts involved in TP-based
542 programming and all the tasks to be accomplished by programmers. The
543 presentation uses the running example from
544 Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}.
546 \subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
548 %WN was Fachleute unter obigem Titel interessiert findet sich
549 %WN unterhalb des auskommentierten Textes.
551 %WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
552 %WN auf Computer-Mathematiker fokussiert.
553 % \paragraph{As mentioned in the introduction,} a prototype of an
554 % educational math assistant called
555 % {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
556 % \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
557 % the gap between (1) introducation and (2) application of mathematics:
558 % {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
559 % requires each fact and each action justified by formal logic, so
560 % {{{\sisac}{}}} makes justifications transparent to students in
561 % interactive step-wise problem solving. By that way {{\sisac}} already
564 % \item Introduction of math stuff (in e.g. partial fraction
565 % decomposition) by stepwise explaining and exercising respective
566 % symbolic calculations with ``next step guidance (NSG)'' and rigorously
567 % checking steps freely input by students --- this also in context with
568 % advanced applications (where the stuff to be taught in higher
569 % semesters can be skimmed through by NSG), and
570 % \item Application of math stuff in advanced engineering courses
571 % (e.g. problems to be solved by inverse Z-transform in a Signal
572 % Processing Lab) and now without much ado about basic math techniques
573 % (like partial fraction decomposition): ``next step guidance'' supports
574 % students in independently (re-)adopting such techniques.
576 % Before the question is answers, how {{\sisac}}
577 % accomplishes this task from a technical point of view, some remarks on
578 % the state-of-the-art is given, therefor follow up Section~\ref{emas}.
580 % \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
582 % \paragraph{Educational software in mathematics} is, if at all, based
583 % on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
584 % Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
585 % \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
586 % http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
587 % base technologies are used to program math lessons and sometimes even
588 % exercises. The latter are cumbersome: the steps towards a solution of
589 % such an interactive exercise need to be provided with feedback, where
590 % at each step a wide variety of possible input has to be foreseen by
591 % the programmer - so such interactive exercises either require high
592 % development efforts or the exercises constrain possible inputs.
594 % \subparagraph{A new generation} of educational math assistants (EMAs)
595 % is emerging presently, which is based on Theorem Proving (TP). TP, for
596 % instance Isabelle and Coq, is a technology which requires each fact
597 % and each action justified by formal logic. Pushed by demands for
598 % \textit{proven} correctness of safety-critical software TP advances
599 % into software engineering; from these advancements computer
600 % mathematics benefits in general, and math education in particular. Two
601 % features of TP are immediately beneficial for learning:
603 % \paragraph{TP have knowledge in human readable format,} that is in
604 % standard predicate calculus. TP following the LCF-tradition have that
605 % knowledge down to the basic definitions of set, equality,
606 % etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
607 % following the typical deductive development of math, natural numbers
608 % are defined and their properties
609 % proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
610 % etc. Present knowledge mechanized in TP exceeds high-school
611 % mathematics by far, however by knowledge required in software
612 % technology, and not in other engineering sciences.
614 % \paragraph{TP can model the whole problem solving process} in
615 % mathematical problem solving {\em within} a coherent logical
616 % framework. This is already being done by three projects, by
617 % Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
619 % Having the whole problem solving process within a logical coherent
620 % system, such a design guarantees correctness of intermediate steps and
621 % of the result (which seems essential for math software); and the
622 % second advantage is that TP provides a wealth of theories which can be
623 % exploited for mechanizing other features essential for educational
626 % \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
628 % One essential feature for educational software is feedback to user
629 % input and assistance in coming to a solution.
631 % \paragraph{Checking user input} by ATP during stepwise problem solving
632 % is being accomplished by the three projects mentioned above
633 % exclusively. They model the whole problem solving process as mentioned
634 % above, so all what happens between formalized assumptions (or formal
635 % specification) and goal (or fulfilled postcondition) can be
636 % mechanized. Such mechanization promises to greatly extend the scope of
637 % educational software in stepwise problem solving.
639 % \paragraph{NSG (Next step guidance)} comprises the system's ability to
640 % propose a next step; this is a challenge for TP: either a radical
641 % restriction of the search space by restriction to very specific
642 % problem classes is required, or much care and effort is required in
643 % designing possible variants in the process of problem solving
644 % \cite{proof-strategies-11}.
646 % Another approach is restricted to problem solving in engineering
647 % domains, where a problem is specified by input, precondition, output
648 % and postcondition, and where the postcondition is proven by ATP behind
649 % the scenes: Here the possible variants in the process of problem
650 % solving are provided with feedback {\em automatically}, if the problem
651 % is described in a TP-based programing language: \cite{plmms10} the
652 % programmer only describes the math algorithm without caring about
653 % interaction (the respective program is functional and even has no
654 % input or output statements!); interaction is generated as a
655 % side-effect by the interpreter --- an efficient separation of concern
656 % between math programmers and dialog designers promising application
657 % all over engineering disciplines.
660 % \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
661 % Authoring new mathematics knowledge in {{\sisac}} can be compared with
662 % ``application programing'' of engineering problems; most of such
663 % programing uses CAS-based programing languages (CAS = Computer Algebra
664 % Systems; e.g. Mathematica's or Maple's programing language).
666 % \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
667 % \cite{plmms10} for describing how to construct a solution to an
668 % engineering problem and for calling equation solvers, integration,
669 % etc~\footnote{Implementation of CAS-like functionality in TP is not
670 % primarily concerned with efficiency, but with a didactic question:
671 % What to decide for: for high-brow algorithms at the state-of-the-art
672 % or for elementary algorithms comprehensible for students?} within TP;
673 % TP can ensure ``systems that never make a mistake'' \cite{casproto} -
674 % are impossible for CAS which have no logics underlying.
676 % \subparagraph{Authoring is perfect} by writing such TP based programs;
677 % the application programmer is not concerned with interaction or with
678 % user guidance: this is concern of a novel kind of program interpreter
679 % called Lucas-Interpreter. This interpreter hands over control to a
680 % dialog component at each step of calculation (like a debugger at
681 % breakpoints) and calls automated TP to check user input following
682 % personalized strategies according to a feedback module.
684 % However ``application programing with TP'' is not done with writing a
685 % program: according to the principles of TP, each step must be
686 % justified. Such justifications are given by theorems. So all steps
687 % must be related to some theorem, if there is no such theorem it must
688 % be added to the existing knowledge, which is organized in so-called
689 % \textbf{theories} in Isabelle. A theorem must be proven; fortunately
690 % Isabelle comprises a mechanism (called ``axiomatization''), which
691 % allows to omit proofs. Such a theorem is shown in
692 % Example~\ref{eg:neuper1}.
694 The running example requires to determine the inverse $\cal
695 Z$-transform for a class of functions. The domain of Signal Processing
696 is accustomed to specific notation for the resulting functions, which
697 are absolutely capable of being totalled and are called step-response: $u[n]$, where $u$ is the
698 function, $n$ is the argument and the brackets indicate that the
699 arguments are discrete. Surprisingly, Isabelle accepts the rules for
700 $z^{-1}$ in this traditional notation~\footnote{Isabelle
701 experts might be particularly surprised, that the brackets do not
702 cause errors in typing (as lists).}:
706 {\footnotesize\begin{tabbing}
707 123\=123\=123\=123\=\kill
709 01\>axiomatization where \\
710 02\>\> rule1: ``$z^{-1}\;1 = \delta [n]$'' and\\
711 03\>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow z^{-1}\;z / (z - 1) = u [n]$'' and\\
712 04\>\> rule3: ``$\vert\vert z \vert\vert < 1 \Rightarrow z / (z - 1) = -u [-n - 1]$'' and \\
713 05\>\> rule4: ``$\vert\vert z \vert\vert > \vert\vert$ $\alpha$ $\vert\vert \Rightarrow z / (z - \alpha) = \alpha^n \cdot u [n]$'' and\\
714 06\>\> rule5: ``$\vert\vert z \vert\vert < \vert\vert \alpha \vert\vert \Rightarrow z / (z - \alpha) = -(\alpha^n) \cdot u [-n - 1]$'' and\\
715 07\>\> rule6: ``$\vert\vert z \vert\vert > 1 \Rightarrow z/(z - 1)^2 = n \cdot u [n]$''
719 These 6 rules can be used as conditional rewrite rules, depending on
720 the respective convergence radius. Satisfaction from accordance with traditional notation
721 contrasts with the above word {\em axiomatization}: As TP-based, the
722 programming language expects these rules as {\em proved} theorems, and
723 not as axioms implemented in the above brute force manner; otherwise
724 all the verification efforts envisaged (like proof of the
725 post-condition, see below) would be meaningless.
727 Isabelle provides a large body of knowledge, rigorously proved from
728 the basic axioms of mathematics~\footnote{This way of rigorously
729 deriving all knowledge from first principles is called the
730 LCF-paradigm in TP.}. In the case of the ${\cal Z}$-Transform the most advanced
731 knowledge can be found in the theories on Multivariate
732 Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
733 building up knowledge such that a proof for the above rules would be
734 reasonably short and easily comprehensible, still requires lots of
735 work (and is definitely out of scope of our case study).
737 %REMOVED DUE TO SPACE CONSTRAINTS
738 %At the state-of-the-art in mechanization of knowledge in engineering
739 %sciences, the process does not stop with the mechanization of
740 %mathematics traditionally used in these sciences. Rather, ``Formal
741 %Methods''~\cite{ fm-03} are expected to proceed to formal and explicit
742 %description of physical items. Signal Processing, for instance is
743 %concerned with physical devices for signal acquisition and
744 %reconstruction, which involve measuring a physical signal, storing it,
745 %and possibly later rebuilding the original signal or an approximation
746 %thereof. For digital systems, this typically includes sampling and
747 %quantization; devices for signal compression, including audio
748 %compression, image compression, and video compression, etc. ``Domain
749 %engineering''\cite{db:dom-eng} is concerned with {\em specification}
750 %of these devices' components and features; this part in the process of
751 %mechanization is only at the beginning in domains like Signal
754 %TP-based programming, concern of this paper, is determined to
755 %add ``algorithmic knowledge'' to the mechanised body of knowledge.
756 %% in Fig.\ref{fig:mathuni} on
757 %% p.\pageref{fig:mathuni}. As we shall see below, TP-based programming
758 %% starts with a formal {\em specification} of the problem to be solved.
761 %% \includegraphics[width=110mm]{../../fig/jrocnik/math-universe-small}
762 %% \caption{The three-dimensional universe of mathematics knowledge}
763 %% \label{fig:mathuni}
766 %% The language for both axes is defined in the axis at the bottom, deductive
767 %% knowledge, in {\sisac} represented by Isabelle's theories.
769 \subsection{Preparation of Simplifiers for the Program}\label{simp}
771 All evaluation in the prototype's Lucas-Interpreter is done by term rewriting on
772 Isabelle's terms, see \S\ref{meth} below; in this section some of respective
773 preparations are described. In order to work reliably with term rewriting, the
774 respective rule-sets must be confluent and terminating~\cite{nipk:rew-all-that},
775 then they are called (canonical) simplifiers. These properties do not go without
776 saying, their establishment is a difficult task for the programmer; this task is
777 not yet supported in the prototype.
779 The prototype rewrites using theorems only. Axioms which are theorems as well
780 have been already shown in \S\ref{eg:neuper1} on p.\pageref{eg:neuper1} , we
781 assemble them in a rule-set and apply them in ML as follows:
785 01 val inverse_z = Rls
786 02 {id = "inverse_z",
787 03 rew_ord = dummy_ord,
789 05 rules = [Thm ("rule1", @{thm rule1}), Thm ("rule2", @{thm rule1}),
790 06 Thm ("rule3", @{thm rule3}), Thm ("rule4", @{thm rule4}),
791 07 Thm ("rule5", @{thm rule5}), Thm ("rule6", @{thm rule6})],
796 \noindent The items, line by line, in the above record have the following purpose:
798 \item[01..02] the ML-value \textit{inverse\_z} stores it's identifier
799 as a string for ``reflection'' when switching between the language
800 layers of Isabelle/ML (like in the Lucas-Interpreter) and
801 Isabelle/Isar (like in the example program on p.\pageref{s:impl} on
804 \item[03..04] both, (a) the rewrite-order~\cite{nipk:rew-all-that}
805 \textit{rew\_ord} and (b) the rule-set \textit{erls} are trivial here:
806 (a) the \textit{rules} in {\rm 07..12} don't need ordered rewriting
807 and (b) the assumptions of the \textit{rules} need not be evaluated
808 (they just go into the context during rewriting).
810 \item[05..07] the \textit{rules} are the axioms from p.\pageref{eg:neuper1};
811 also ML-functions (\S\ref{funs}) can come into this list as shown in
812 \S\ref{flow-prep}; so they are distinguished by type-constructors \textit{Thm}
813 and \textit{Calc} respectively; for the purpose of reflection both
814 contain their identifiers.
816 \item[08..09] are error-patterns not discussed here and \textit{scr}
817 is prepared to get a program, automatically generated by {\sisac} for
818 producing intermediate rewrites when requested by the user.
822 %OUTCOMMENTED DUE TO SPACE RESTRICTIONS
823 % \noindent It is advisable to immediately test rule-sets; for that
824 % purpose an appropriate term has to be created; \textit{parse} takes a
825 % context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
826 % Z}^{-1}$) and creates a term:
831 % 02 val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
833 % 04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1",
834 % 05 "RealDef.real => RealDef.real => RealDef.real") $
835 % 06 (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...)
838 % \noindent The internal representation of the term, as required for
839 % rewriting, consists of \textit{Const}ants, a pair of a string
840 % \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
841 % \textit{Free} and the respective constructor \textit{\$}. Now the
842 % term can be rewritten by the rule-set \textit{inverse\_z}:
847 % 02 val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
851 % 06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
852 % 07 val it = "|| z || > 1 & || z || > </alpha>" : string
855 % \noindent The resulting term \textit{t} and the assumptions
856 % \textit{asm} are converted to readable strings by \textit{term2str}
857 % and \textit{terms2str}.
859 \subsection{Preparation of ML-Functions}\label{funs}
860 Some functionality required in programming, cannot be accomplished by
861 rewriting. So the prototype has a mechanism to call functions within
862 the rewrite-engine: certain redexes in Isabelle terms call these
863 functions written in SML~\cite{pl:milner97}, the implementation {\em
864 and} meta-language of Isabelle. The programmer has to use this
867 In the running example's program on p.\pageref{s:impl} the lines {\rm
868 05} and {\rm 06} contain such functions; we go into the details with
869 \textit{argument\_in X\_z;}. This function fetches the argument from a
870 function application: Line {\rm 03} in the example calculation on
871 p.\pageref{exp-calc} is created by line {\rm 06} of the example
872 program on p.\pageref{s:impl} where the program's environment assigns
873 the value \textit{X z} to the variable \textit{X\_z}; so the function
874 shall extract the argument \textit{z}.
876 \medskip In order to be recognised as a function constant in the
877 program source the constant needs to be declared in a theory, here in
878 \textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
879 the context \textit{ctxt} of that theory:
884 02 argument'_in :: "real => real" ("argument'_in _" 10)
887 %^3.2^ ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
888 %^3.2^ val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real")
889 %^3.2^ $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
890 %^3.2^ \end{verbatim}}
892 %^3.2^ \noindent Parsing produces a term \texttt{t} in internal
893 %^3.2^ representation~\footnote{The attentive reader realizes the
894 %^3.2^ differences between interal and extermal representation even in the
895 %^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
896 %^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
897 %^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
899 The function body below is implemented directly in SML,
900 i.e in an \texttt{ML \{* *\}} block; the function definition provides
901 a unique prefix \texttt{eval\_} to the function name:
906 02 fun eval_argument_in _
907 03 "Build_Inverse_Z_Transform.argument'_in"
908 04 (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
909 05 if is_Free arg (*could be something to be simplified before*)
910 06 then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
912 08 | eval_argument_in _ _ _ _ = NONE;
916 \noindent The function body creates either \texttt{NONE}
917 telling the rewrite-engine to search for the next redex, or creates an
918 ad-hoc theorem for rewriting, thus the programmer needs to adopt many
919 technicalities of Isabelle, for instance, the \textit{Trueprop}
922 \bigskip This sub-task particularly sheds light on basic issues in the
923 design of a programming language, the integration of differential language
924 layers, the layer of Isabelle/Isar and Isabelle/ML.
926 Another point of improvement for the prototype is the rewrite-engine: The
927 program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
930 {\small\it\label{s:impl}
932 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
933 \>{\rm 05/06}\>\>\> (z::real) = argument\_in (lhs X\_eq) ;
936 \noindent because nested function calls would require creating redexes
937 inside-out; however, the prototype's rewrite-engine only works top down
938 from the root of a term down to the leaves.
940 How all these technicalities are to be checked in the prototype is
941 shown in \S\ref{flow-prep} below.
943 % \paragraph{Explicit Problems} require explicit methods to solve them, and within
944 % this methods we have some explicit steps to do. This steps can be unique for
945 % a special problem or refindable in other problems. No mather what case, such
946 % steps often require some technical functions behind. For the solving process
947 % of the Inverse Z Transformation and the corresponding partial fraction it was
948 % neccessary to build helping functions like \texttt{get\_denominator},
949 % \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
950 % to filter the denominator or numerator out of a fraction, last one helps us to
951 % get to know the bound variable in a equation.
953 % By taking \texttt{get\_denominator} as an example, we want to explain how to
954 % implement new functions into the existing system and how we can later use them
957 % \subsubsection{Find a place to Store the Function}
959 % The whole system builds up on a well defined structure of Knowledge. This
960 % Knowledge sets up at the Path:
961 % \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
962 % For implementing the Function \texttt{get\_denominator} (which let us extract
963 % the denominator out of a fraction) we have choosen the Theory (file)
964 % \texttt{Rational.thy}.
966 % \subsubsection{Write down the new Function}
968 % In upper Theory we now define the new function and its purpose:
970 % get_denominator :: "real => real"
972 % This command tells the machine that a function with the name
973 % \texttt{get\_denominator} exists which gets a real expression as argument and
974 % returns once again a real expression. Now we are able to implement the function
975 % itself, upcoming example now shows the implementation of
976 % \texttt{get\_denominator}.
979 % \label{eg:getdenom}
983 % 02 *("get_denominator",
984 % 03 * ("Rational.get_denominator", eval_get_denominator ""))
986 % 05 fun eval_get_denominator (thmid:string) _
987 % 06 (t as Const ("Rational.get_denominator", _) $
988 % 07 (Const ("Rings.inverse_class.divide", _) $num
990 % 09 SOME (mk_thmid thmid ""
991 % 10 (Print_Mode.setmp []
992 % 11 (Syntax.string_of_term (thy2ctxt thy)) denom) "",
993 % 12 Trueprop $ (mk_equality (t, denom)))
994 % 13 | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
997 % Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
998 % there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont)
1000 % into its two parts (\texttt{\$num \$denom}). The lines before are additionals
1001 % commands for declaring the function and the lines after are modeling and
1002 % returning a real variable out of \texttt{\$denom}.
1004 % \subsubsection{Add a test for the new Function}
1006 % \paragraph{Everytime when adding} a new function it is essential also to add
1007 % a test for it. Tests for all functions are sorted in the same structure as the
1008 % knowledge it self and can be found up from the path:
1009 % \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
1010 % This tests are nothing very special, as a first prototype the functionallity
1011 % of a function can be checked by evaluating the result of a simple expression
1012 % passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
1013 % \textit{just} created function \texttt{get\_denominator}.
1016 % \label{eg:getdenomtest}
1019 % 01 val thy = @{theory Isac};
1020 % 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
1021 % 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1022 % 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
1023 % 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
1026 % \begin{description}
1027 % \item[01] checks if the proofer set up on our {\sisac{}} System.
1028 % \item[02] passes a simple expression (fraction) to our suddenly created
1030 % \item[04] checks if the resulting variable is the correct one (in this case
1031 % ``b'' the denominator) and returns.
1032 % \item[05] handels the error case and reports that the function is not able to
1033 % solve the given problem.
1036 \subsection{Specification of the Problem}\label{spec}
1037 %WN <--> \chapter 7 der Thesis
1038 %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
1040 Mechanical treatment requires to translate a textual problem
1041 description like in Fig.\ref{fig-interactive} on
1042 p.\pageref{fig-interactive} into a {\em formal} specification. The
1043 formal specification of the running example could look like is this:
1045 %WN Hier brauchen wir die Spezifikation des 'running example' ...
1046 %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
1047 %JR der post condition - die existiert für uns ja eigentlich nicht aka
1048 %JR haben sie bis jetzt nicht beachtet WN...
1049 %WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
1053 {\small\begin{tabbing}
1054 123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
1057 \> \>input \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
1058 \>\>precond \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
1059 \>\>output \>: stepResponse $x[n]$ \\
1060 \>\>postcond \>: TODO
1063 %JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
1066 % Defining the postcondition requires a high amount mathematical
1067 % knowledge, the difficult part in our case is not to set up this condition
1068 % nor it is more to define it in a way the interpreter is able to handle it.
1069 % Due the fact that implementing that mechanisms is quite the same amount as
1070 % creating the programm itself, it is not avaible in our prototype.
1071 % \label{rm:postcond}
1074 The implementation of the formal specification in the present
1075 prototype, still bar-bones without support for authoring, is done
1077 %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
1079 {\footnotesize\label{exp-spec}
1082 01 store_specification
1083 02 (prepare_specification
1084 03 "pbl_SP_Ztrans_inv"
1087 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
1088 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1089 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1090 09 ("#Find" , ["stepResponse n_eq"]),
1091 10 ("#Post" , [" TODO "])])
1094 13 [["SignalProcessing","Z_Transform","Inverse"]]);
1098 Although the above details are partly very technical, we explain them
1099 in order to document some intricacies of TP-based programming in the
1100 present state of the {\sisac} prototype:
1102 \item[01..02]\textit{store\_specification:} stores the result of the
1103 function \textit{prep\_specification} in a global reference
1104 \textit{Unsynchronized.ref}, which causes principal conflicts with
1105 Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
1106 parallel execution~\cite{Makarius-09:parall-proof} and is under
1107 reconstruction already.
1109 \textit{prep\_specification:} translates the specification to an internal format
1110 which allows efficient processing; see for instance line {\rm 07}
1112 \item[03..04] are a unique identifier for the specification within {\sisac}
1113 and the ``mathematics author'' holding the copy-rights.
1114 \item[05] is the Isabelle \textit{theory} required to parse the
1115 specification in lines {\rm 07..10}.
1116 \item[06] is a key into the tree of all specifications as presented to
1117 the user (where some branches might be hidden by the dialogue
1119 \item[07..10] are the specification with input, pre-condition, output
1120 and post-condition respectively; note that the specification contains
1121 variables to be instantiated with concrete values for a concrete problem ---
1122 thus the specification actually captures a class of problems. The post-condition is not handled in
1123 the prototype presently.
1124 \item[11] is a rule-set (defined elsewhere) for evaluation of the pre-condition: \textit{(rhs X\_eq) is\_continuous\_in D}, instantiated with the values of a concrete problem, evaluates to true or false --- and all evaluation is done by
1125 rewriting determined by rule-sets.
1126 \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
1127 problem associated to a function from Computer Algebra (like an
1128 equation solver) which is not the case here.
1129 \item[13] is a list of methods solving the specified problem (here
1130 only one list item) represented analogously to {\rm 06}.
1134 %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
1137 % {guh : guh, (*unique within this isac-knowledge*)
1138 % mathauthors: string list, (*copyright*)
1139 % init : pblID, (*to start refinement with*)
1140 % thy : theory, (* which allows to compile that pbt
1141 % TODO: search generalized for subthy (ref.p.69*)
1142 % (*^^^ WN050912 NOT used during application of the problem,
1143 % because applied terms may be from 'subthy' as well as from super;
1144 % thus we take 'maxthy'; see match_ags !*)
1145 % cas : term option,(*'CAS-command'*)
1146 % prls : rls, (* for preds in where_*)
1147 % where_: term list, (* where - predicates*)
1149 % (*this is the model-pattern;
1150 % it contains "#Given","#Where","#Find","#Relate"-patterns
1151 % for constraints on identifiers see "fun cpy_nam"*)
1152 % met : metID list}; (* methods solving the pbt*)
1154 %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
1155 %WN oben selbst geschrieben.
1160 %WN das w"urde ich in \sec\label{progr} verschieben und
1161 %WN das SubProblem partial fractions zum Erkl"aren verwenden.
1162 % Such a specification is checked before the execution of a program is
1163 % started, the same applies for sub-programs. In the following example
1164 % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1168 % \label{eg:subprob}
1170 % {\ttfamily \begin{tabbing}
1171 % ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
1172 % ``\>\>[linear,univariate,equation,test],'' \\
1173 % ``\>\>[Test,solve\_linear])'' \\
1174 % ``\>[BOOL equ, REAL z])'' \\
1178 % \noindent If a program requires a result which has to be
1179 % calculated first we can use a subproblem to do so. In our specific
1180 % case we wanted to calculate the zeros of a fraction and used a
1181 % subproblem to calculate the zeros of the denominator polynom.
1186 \subsection{Implementation of the Method}\label{meth}
1187 A method collects all data required to interpret a certain program by
1188 Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
1189 the running example is embedded on the last line in the following method:
1190 %The methods represent the different ways a problem can be solved. This can
1191 %include mathematical tactics as well as tactics taught in different courses.
1192 %Declaring the Method itself gives us the possibilities to describe the way of
1193 %calculation in deep, as well we get the oppertunities to build in different
1201 03 "SP_InverseZTransformation_classic"
1204 06 ( ["SignalProcessing", "Z_Transform", "Inverse"],
1205 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1206 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1207 09 ("#Find" , ["stepResponse n_eq"]),
1215 \noindent The above code stores the whole structure analogously to a
1216 specification as described above:
1218 \item[01..06] are identical to those for the example specification on
1219 p.\pageref{exp-spec}.
1221 \item[07..09] show something looking like the specification; this is a
1222 {\em guard}: as long as not all \textit{Given} items are present and
1223 the \textit{Pre}-conditions is not true, interpretation of the program
1226 \item[10..11] all concern rewriting (the respective data are defined elsewhere): \textit{rew\_ord} is the rewrite order~\cite{nipk:rew-all-that} in case
1227 \textit{program} contains a \textit{Rewrite} tactic; and in case the respective rule is a conditional rewrite-rule, \textit{erls} features evaluating the conditions. The rule-sets
1228 \textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
1229 \textit{lhs, argument\_in, rhs} in the program on p.\pageref{s:impl}, (b) the pre-condition analogous to the specification in line 11 on p.\pageref{exp-spec}
1230 and (c) is required for the derivation-machinery checking user-input formulas.
1232 \item[12..13] \textit{errpats} are error-patterns~\cite{gdaroczy-EP-13} for this method and \textit{program} is the variable holding the example from p.\pageref {s:impl}.
1234 The many rule-sets above cause considerable efforts for the
1235 programmers, in particular, because there are no tools for checking
1236 essential features of rule-sets.
1238 % is again very technical and goes hard in detail. Unfortunataly
1239 % most declerations are not essential for a basic programm but leads us to a huge
1240 % range of powerful possibilities.
1242 % \begin{description}
1243 % \item[01..02] stores the method with the given name into the system under a global
1245 % \item[03] specifies the topic within which context the method can be found.
1246 % \item[04..05] as the requirements for different methods can be deviant we
1247 % declare what is \emph{given} and and what to \emph{find} for this specific method.
1248 % The code again helds on the topic of the case studie, where the inverse
1249 % z-transformation does a switch between a term describing a electrical filter into
1250 % its step response. Also the datatype has to be declared (bool - due the fact that
1251 % we handle equations).
1252 % \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one
1253 % theorem of it is used for rewriting one single step.
1254 % \item[07] \texttt{rls} is the currently used ruleset for this method. This set
1255 % has already been defined before.
1256 % \item[08] we would have the possiblitiy to add this method to a predefined tree of
1257 % calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
1259 % \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in
1261 % \item[10] \emph{predicates ruleset} can be used to indicates predicates within
1263 % \item[11] The \emph{check ruleset} summarizes rules for checking formulas
1265 % \item[12] \emph{error patterns} which are expected in this kind of method can be
1266 % pre-specified to recognize them during the method.
1267 % \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier
1268 % of the specific method.
1269 % \item[14] for this code snipset we don't specify the programm itself and keep it
1270 % empty. Follow up \S\ref{progr} for informations on how to implement this
1271 % \textit{main} part.
1274 \subsection{Implementation of the TP-based Program}\label{progr}
1275 So finally all the prerequisites are described and the final task can
1276 be addressed. The program below comes back to the running example: it
1277 computes a solution for the problem from Fig.\ref{fig-interactive} on
1278 p.\pageref{fig-interactive}. The reader is reminded of
1279 \S\ref{PL-isab}, the introduction of the programming language:
1281 {\footnotesize\it\label{s:impl}
1283 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1284 \>{\rm 00}\>ML \{*\\
1285 \>{\rm 00}\>val program =\\
1286 \>{\rm 01}\> "{\tt Program} InverseZTransform (X\_eq::bool) = \\
1287 \>{\rm 02}\>\> {\tt let} \\
1288 \>{\rm 03}\>\>\> X\_eq = {\tt Take} X\_eq ; \\
1289 \>{\rm 04}\>\>\> X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
1290 \>{\rm 05}\>\>\> (X\_z::real) = lhs X\_eq ; \\ %no inside-out evaluation
1291 \>{\rm 06}\>\>\> (z::real) = argument\_in X\_z; \\
1292 \>{\rm 07}\>\>\> (part\_frac::real) = {\tt SubProblem} \\
1293 \>{\rm 08}\>\>\>\>\>\>\>\> ( Isac, [partial\_fraction, rational, simplification], [] )\\
1294 %\>{\rm 10}\>\>\>\>\>\>\>\>\> [simplification, of\_rationals, to\_partial\_fraction] ) \\
1295 \>{\rm 09}\>\>\>\>\>\>\>\> [ (rhs X\_eq)::real, z::real ]; \\
1296 \>{\rm 10}\>\>\> (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
1297 \>{\rm 11}\>\>\> X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@ \\
1298 \>{\rm 12}\>\>\>\>\> $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
1299 \>{\rm 13}\>\> {\tt in } \\
1300 \>{\rm 14}\>\>\> X'\_eq"\\
1303 % ORIGINAL FROM Inverse_Z_Transform.thy
1304 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1305 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1306 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1307 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1308 % " (zzz::real) = argument_in X'_z; "^(* z *)
1309 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1311 % " (pbz::real) = (SubProblem (Isac', "^(**)
1312 % " [partial_fraction,rational,simplification], "^
1313 % " [simplification,of_rationals,to_partial_fraction]) "^
1314 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1316 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1317 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1318 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1319 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1320 % " n_eq = (Rewrite_Set inverse_z False) X_zeq; "^(*([4], Res), X_z = 4 * (1 / 2) ^^^ ?n * ?u [?n] + -4 * (-1 / 4) ^^^ ?n * ?u [?n]*)
1321 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1322 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1323 The program is represented as a string and part of the method in
1324 \S\ref{meth}. As mentioned in \S\ref{PL} the program is purely
1325 functional and lacks any input statements and output statements. So
1326 the steps of calculation towards a solution (and interactive tutoring
1327 in step-wise problem solving) are created as a side-effect by
1328 Lucas-Interpretation. The side-effects are triggered by the tactics
1329 \texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
1330 \texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
1331 {\rm 12} respectively. These tactics produce the respective lines in the
1332 calculation on p.\pageref{flow-impl}.
1334 The above lines {\rm 05, 06} do not contain a tactics, so they do not
1335 immediately contribute to the calculation on p.\pageref{flow-impl};
1336 rather, they compute actual arguments for the \texttt{SubProblem} in
1337 line {\rm 09}~\footnote{The tactics also are break-points for the
1338 interpreter, where control is handed over to the user in interactive
1339 tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
1341 \medskip The above program also indicates the dominant role of interactive
1342 selection of knowledge in the three-dimensional universe of
1343 mathematics. The \texttt{SubProblem} in the above lines
1344 {\rm 07..09} is more than a function call with the actual arguments
1345 \textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
1349 \item the theory, in the example \textit{Isac} because different
1350 methods can be selected in Pt.3 below, which are defined in different
1351 theories with \textit{Isac} collecting them.
1352 \item the specification identified by \textit{[partial\_fraction,
1353 rational, simplification]} in the tree of specifications; this
1354 specification is analogous to the specification of the main program
1355 described in \S\ref{spec}; the problem is to find a ``partial fraction
1356 decomposition'' for a univariate rational polynomial.
1357 \item the method in the above example is \textit{[ ]}, i.e. empty,
1358 which supposes the interpreter to select one of the methods predefined
1359 in the specification, for instance in line {\rm 13} in the running
1360 example's specification on p.\pageref{exp-spec}~\footnote{The freedom
1361 (or obligation) for selection carries over to the student in
1362 interactive tutoring.}.
1365 The program code, above presented as a string, is parsed by Isabelle's
1366 parser --- the program is an Isabelle term. This fact is expected to
1367 simplify verification tasks in the future; on the other hand, this
1368 fact causes troubles in error detection which are discussed as part
1369 of the work-flow in the subsequent section.
1371 \section{Work-flow of Programming in the Prototype}\label{workflow}
1372 The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
1373 step forward for interactive theory and proof development. The
1374 {\sisac}-prototype re-uses this IDE as a programming environment. The
1375 experiences from this re-use show, that the essential components are
1376 available from Isabelle/jEdit. However, additional tools and features
1377 are required to achieve acceptable usability.
1379 So notable experiences are reported here, also as a requirement
1380 capture for further development of TP-based languages and respective
1383 \subsection{Preparations and Trials}\label{flow-prep}
1384 The many sub-tasks to be accomplished {\em before} the first line of
1385 program code can be written and tested suggest an approach which
1386 step-wise establishes the prerequisites. The case study underlying
1387 this paper~\cite{jrocnik-bakk} documents the approach in a separate
1389 \textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
1390 II in the study comprises this theory, \LaTeX ed from the theory by
1391 use of Isabelle's document preparation system. This paper resembles
1392 the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
1393 implementation work involves several iterations.
1395 \bigskip For instance, only the last step, implementing the program
1396 described in \S\ref{meth}, reveals details required. Let us assume,
1397 this is the ML-function \textit{argument\_in} required in line {\rm 06}
1398 of the example program on p.\pageref{s:impl}; how this function needs
1399 to be implemented in the prototype has been discussed in \S\ref{funs}
1402 Now let us assume, that calling this function from the program code
1403 does not work; so testing this function is required in order to find out
1404 the reason: type errors, a missing entry of the function somewhere or
1405 even more nasty technicalities \dots
1410 02 val SOME t = parseNEW ctxt "argument_in (X (z::real))";
1411 03 val SOME (str, t') = eval_argument_in ""
1412 04 "Build_Inverse_Z_Transform.argument'_in" t 0;
1415 07 val it = "(argument_in X z) = z": string\end{verbatim}}
1417 \noindent So, this works: we get an ad-hoc theorem, which used in
1418 rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
1419 reduction and create a rule-set \texttt{rls} for that purpose:
1424 02 val rls = append_rls "test" e_rls
1425 03 [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
1426 04 val SOME (t', asm) = rewrite_set_ @{theory} rls t;
1428 06 val t' = Free ("z", "RealDef.real"): term
1429 07 val asm = []: term list\end{verbatim}}
1431 \noindent The resulting term \texttt{t'} is \texttt{Free ("z",
1432 "RealDef.real")}, i.e the variable \texttt{z}, so all is
1433 perfect. Probably we have forgotten to store this function correctly~?
1434 We review the respective \texttt{calclist} (again an
1435 \textit{Unsynchronized.ref} to be removed in order to adjust to
1436 Isabelle/Isar's asynchronous document model):
1440 01 calclist:= overwritel (! calclist,
1442 03 ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
1444 05 ]);\end{verbatim}}
1446 \noindent The entry is perfect. So what is the reason~? Ah, probably there
1447 is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
1448 right, the function \texttt{argument\_in} is not contained in the respective
1449 rule-set \textit{srls} \dots this just as an example of the intricacies in
1450 debugging a program in the present state of the prototype.
1452 \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1453 Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
1454 usually developed within several iterations, the program can be
1455 assembled; on p.\pageref{s:impl} there is the complete program of the
1458 The completion of this program required efforts for several weeks
1459 (after some months of familiarisation with {\sisac}), caused by the
1460 abundance of intricacies indicated above. Also writing the program is
1461 not pleasant, given Isabelle/Isar/ without add-ons for
1462 programming. Already writing and parsing a few lines of program code
1463 is a challenge: the program is an Isabelle term; Isabelle's parser,
1464 however, is not meant for huge terms like the program of the running
1465 example. So reading out the specific error (usually type errors) from
1466 Isabelle's message is difficult.
1468 \medskip Testing the evaluation of the program has to rely on very
1469 simple tools. Step-wise execution is modeled by a function
1470 \texttt{me}, short for mathematics-engine~\footnote{The interface used
1471 by the front-end which created the calculation on
1472 p.\pageref{fig-interactive} is different from this function}:
1473 %the following is a simplification of the actual function
1478 02 val it = tac -> ctree * pos -> mout * tac * ctree * pos\end{verbatim}}
1480 \noindent This function takes as arguments a tactic \texttt{tac} which
1481 determines the next step, the step applied to the interpreter-state
1482 \texttt{ctree * pos} as last argument taken. The interpreter-state is
1483 a pair of a tree \texttt{ctree} representing the calculation created
1484 (see the example below) and a position \texttt{pos} in the
1485 calculation. The function delivers a quadruple, beginning with the new
1486 formula \texttt{mout} and the next tactic followed by the new
1489 This function allows to stepwise check the program:
1491 {\footnotesize\label{ml-check-program}
1495 03 ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
1496 04 "stepResponse (x[n::real]::bool)"];
1499 07 ["Inverse", "Z_Transform", "SignalProcessing"],
1500 08 ["SignalProcessing","Z_Transform","Inverse"]);
1501 09 val (mout, tac, ctree, pos) = CalcTreeTEST [(fmz, (dI, pI, mI))];
1502 10 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1503 11 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1504 12 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1508 \noindent Several dozens of calls for \texttt{me} are required to
1509 create the lines in the calculation below (including the sub-problems
1510 not shown). When an error occurs, the reason might be located
1511 many steps before: if evaluation by rewriting, as done by the prototype,
1512 fails, then first nothing happens --- the effects come later and
1513 cause unpleasant checks.
1515 The checks comprise watching the rewrite-engine for many different
1516 kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
1517 particular the environment and the context at the states position ---
1518 all checks have to rely on simple functions accessing the
1519 \texttt{ctree}. So getting the calculation below (which resembles the
1520 calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
1521 is the result of several weeks of development:
1523 {\small\it\label{exp-calc}
1525 123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
1526 \>{\rm 01}\> $\bullet$ \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing]) \`\\
1527 \>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$ \`{\footnotesize {\tt Take} X\_eq}\\
1528 \>{\rm 03}\>\> $X z = \frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}$ \`{\footnotesize {\tt Rewrite} prep\_for\_part\_frac X\_eq}\\
1529 \>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification] \`{\footnotesize {\tt SubProblem} \dots}\\
1530 \>{\rm 05}\>\>\> $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$ \`- - -\\
1531 \>{\rm 06}\>\>\> $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$ \`- - -\\
1532 \>{\rm 07}\>\>\> $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ ) \`- - -\\
1533 \>{\rm 08}\>\>\>\> $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
1534 \>{\rm 09}\>\>\>\> $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$ \`- - -\\
1535 \>{\rm 10}\>\>\>\> $z = \frac{1}{2}\;\lor\;z =$ \_\_\_ \`- - -\\
1536 \> \>\>\>\> \_\_\_ \`- - -\\
1537 \>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$ \`\\
1538 \>{\rm 12}\>\> $X^\prime z = {\cal z}^{-1} (\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}})$ \`{\footnotesize {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac)}\\
1539 \>{\rm 13}\>\> $X^\prime z = {\cal z}^{-1} (4\cdot\frac{z}{z - \frac{1}{2}} + -4\cdot\frac{z}{z - \frac{-1}{4}})$ \`{\footnotesize{\tt Rewrite\_Set} prep\_for\_inverse\_z X'\_eq }\\
1540 \>{\rm 14}\>\> $X^\prime z = 4\cdot(\frac{1}{2})^n \cdot u [n] + -4\cdot(\frac{-1}{4})^n \cdot u [n]$ \`{\footnotesize {\tt Rewrite\_Set} inverse\_z X'\_eq}\\
1541 \>{\rm 15}\> \dots\> $X^\prime z = 4\cdot(\frac{1}{2})^n \cdot u [n] + -4\cdot(\frac{-1}{4})^n \cdot u [n]$ \`{\footnotesize {\tt Check\_Postcond}}
1543 The tactics on the right margin of the above calculation are those in
1544 the program on p.\pageref{s:impl} which create the respective formulas
1546 % ORIGINAL FROM Inverse_Z_Transform.thy
1547 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1548 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1549 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1550 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1551 % " (zzz::real) = argument_in X'_z; "^(* z *)
1552 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1554 % " (pbz::real) = (SubProblem (Isac', "^(**)
1555 % " [partial_fraction,rational,simplification], "^
1556 % " [simplification,of_rationals,to_partial_fraction]) "^
1557 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1559 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1560 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1561 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1562 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1563 % " n_eq = (Rewrite_Set inverse_z False) X_zeq; "^(*([4], Res), X_z = 4 * (1 / 2) ^^^ ?n * ?u [?n] + -4 * (-1 / 4) ^^^ ?n * ?u [?n]*)
1564 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1565 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1567 \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1568 Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
1569 and the knowledge accumulated in it can be distributed to appropriate
1570 theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
1571 sub-problem accomplishing the partial fraction decomposition to
1572 \textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
1573 internals, this kind of distribution is not trivial. For instance, the
1574 function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
1575 string with the theory it has been defined in, so this string needs to
1576 be updated from \texttt{Build\_Inverse\_Z\_Transform} to
1577 \texttt{Atools} if that function is transferred to theory
1578 \textit{Atools.thy}.
1580 In order to obtain the functionality presented in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive} data must be exported from SML-structures to XML.
1581 This process is also rather bare-bones without authoring tools and is
1582 described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
1585 % -------------------------------------------------------------------
1587 % Material, falls noch Platz bleibt ...
1589 % -------------------------------------------------------------------
1592 % \subsubsection{Trials on Notation and Termination}
1594 % \paragraph{Technical notations} are a big problem for our piece of software,
1595 % but the reason for that isn't a fault of the software itself, one of the
1596 % troubles comes out of the fact that different technical subtopics use different
1597 % symbols and notations for a different purpose. The most famous example for such
1598 % a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
1599 % math). In the specific part of signal processing one of this notation issues is
1600 % the use of brackets --- we use round brackets for analoge signals and squared
1601 % brackets for digital samples. Also if there is no problem for us to handle this
1602 % fact, we have to tell the machine what notation leads to wich meaning and that
1603 % this purpose seperation is only valid for this special topic - signal
1605 % \subparagraph{In the programming language} itself it is not possible to declare
1606 % fractions, exponents, absolutes and other operators or remarks in a way to make
1607 % them pretty to read; our only posssiblilty were ASCII characters and a handfull
1608 % greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
1610 % With the upper collected knowledge it is possible to check if we were able to
1611 % donate all required terms and expressions.
1613 % \subsubsection{Definition and Usage of Rules}
1615 % \paragraph{The core} of our implemented problem is the Z-Transformation, due
1616 % the fact that the transformation itself would require higher math which isn't
1617 % yet avaible in our system we decided to choose the way like it is applied in
1618 % labratory and problem classes at our university - by applying transformation
1619 % rules (collected in transformation tables).
1620 % \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
1621 % use of axiomatizations like shown in Example~\ref{eg:ruledef}
1624 % \label{eg:ruledef}
1627 % axiomatization where
1628 % rule1: ``1 = $\delta$[n]'' and
1629 % rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
1630 % rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
1634 % This rules can be collected in a ruleset and applied to a given expression as
1635 % follows in Example~\ref{eg:ruleapp}.
1639 % \label{eg:ruleapp}
1641 % \item Store rules in ruleset:
1643 % val inverse_Z = append_rls "inverse_Z" e_rls
1644 % [ Thm ("rule1",num_str @{thm rule1}),
1645 % Thm ("rule2",num_str @{thm rule2}),
1646 % Thm ("rule3",num_str @{thm rule3})
1648 % \item Define exression:
1650 % val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
1651 % \item Apply ruleset:
1653 % val SOME (sample_term', asm) =
1654 % rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
1658 % The use of rulesets makes it much easier to develop our designated applications,
1659 % but the programmer has to be careful and patient. When applying rulesets
1660 % two important issues have to be mentionend:
1661 % \subparagraph{How often} the rules have to be applied? In case of
1662 % transformations it is quite clear that we use them once but other fields
1663 % reuqire to apply rules until a special condition is reached (e.g.
1664 % a simplification is finished when there is nothing to be done left).
1665 % \subparagraph{The order} in which rules are applied often takes a big effect
1666 % and has to be evaluated for each purpose once again.
1668 % In our special case of Signal Processing and the rules defined in
1669 % Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
1670 % constants. After this step has been done it no mather which rule fit's next.
1672 % \subsubsection{Helping Functions}
1674 % \paragraph{New Programms require,} often new ways to get through. This new ways
1675 % means that we handle functions that have not been in use yet, they can be
1676 % something special and unique for a programm or something famous but unneeded in
1677 % the system yet. In our dedicated example it was for example neccessary to split
1678 % a fraction into numerator and denominator; the creation of such function and
1679 % even others is described in upper Sections~\ref{simp} and \ref{funs}.
1681 % \subsubsection{Trials on equation solving}
1682 % %simple eq and problem with double fractions/negative exponents
1683 % \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
1684 % equations degree one and two. Solving equations in the first degree is no
1685 % problem, wether for a student nor for our machine; but even second degree
1686 % equations can lead to big troubles. The origin of this troubles leads from
1687 % the build up process of our equation solving functions; they have been
1688 % implemented some time ago and of course they are not as good as we want them to
1689 % be. Wether or not following we only want to show how cruel it is to build up new
1690 % work on not well fundamentials.
1691 % \subparagraph{A simple equation solving,} can be set up as shown in the next
1698 % ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
1702 % val (dI',pI',mI') =
1704 % ["abcFormula","degree_2","polynomial","univariate","equation"],
1705 % ["no_met"]);\end{verbatim}
1708 % Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
1709 % a short overview on the commands; at first we set up the equation and tell the
1710 % machine what's the bound variable and where to store the solution. Second step
1711 % is to define the equation type and determine if we want to use a special method
1712 % to solve this type.) Simple checks tell us that the we will get two results for
1713 % this equation and this results will be real.
1714 % So far it is easy for us and for our machine to solve, but
1715 % mentioned that a unvariate equation second order can have three different types
1716 % of solutions it is getting worth.
1717 % \subparagraph{The solving of} all this types of solutions is not yet supported.
1718 % Luckily it was needed for us; but something which has been needed in this
1719 % context, would have been the solving of an euation looking like:
1720 % $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
1721 % before (remember that befor it was no problem to handle for the machine) but
1722 % now, after a simple equivalent transformation, we are not able to solve
1724 % \subparagraph{Error messages} we get when we try to solve something like upside
1725 % were very confusing and also leads us to no special hint about a problem.
1726 % \par The fault behind is, that we have no well error handling on one side and
1727 % no sufficient formed equation solving on the other side. This two facts are
1728 % making the implemention of new material very difficult.
1730 % \subsection{Formalization of missing knowledge in Isabelle}
1732 % \paragraph{A problem} behind is the mechanization of mathematic
1733 % theories in TP-bases languages. There is still a huge gap between
1734 % these algorithms and this what we want as a solution - in Example
1735 % Signal Processing.
1741 % X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1744 % \noindent A very simple example on this what we call gap is the
1745 % simplification above. It is needles to say that it is correct and also
1746 % Isabelle for fills it correct - \emph{always}. But sometimes we don't
1747 % want expand such terms, sometimes we want another structure of
1748 % them. Think of a problem were we now would need only the coefficients
1749 % of $X$ and $Y$. This is what we call the gap between mechanical
1750 % simplification and the solution.
1755 % \paragraph{We are not able to fill this gap,} until we have to live
1756 % with it but first have a look on the meaning of this statement:
1757 % Mechanized math starts from mathematical models and \emph{hopefully}
1758 % proceeds to match physics. Academic engineering starts from physics
1759 % (experimentation, measurement) and then proceeds to mathematical
1760 % modeling and formalization. The process from a physical observance to
1761 % a mathematical theory is unavoidable bound of setting up a big
1762 % collection of standards, rules, definition but also exceptions. These
1763 % are the things making mechanization that difficult.
1772 % \noindent Think about some units like that one's above. Behind
1773 % each unit there is a discerning and very accurate definition: One
1774 % Meter is the distance the light travels, in a vacuum, through the time
1775 % of 1 / 299.792.458 second; one kilogram is the weight of a
1776 % platinum-iridium cylinder in paris; and so on. But are these
1777 % definitions usable in a computer mechanized world?!
1782 % \paragraph{A computer} or a TP-System builds on programs with
1783 % predefined logical rules and does not know any mathematical trick
1784 % (follow up example \ref{eg:trick}) or recipe to walk around difficult
1790 % \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
1791 % \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
1792 % \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1793 % \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1795 % \noindent Sometimes it is also useful to be able to apply some
1796 % \emph{tricks} to get a beautiful and particularly meaningful result,
1797 % which we are able to interpret. But as seen in this example it can be
1798 % hard to find out what operations have to be done to transform a result
1799 % into a meaningful one.
1804 % \paragraph{The only possibility,} for such a system, is to work
1805 % through its known definitions and stops if none of these
1806 % fits. Specified on Signal Processing or any other application it is
1807 % often possible to walk through by doing simple creases. This creases
1808 % are in general based on simple math operational but the challenge is
1809 % to teach the machine \emph{all}\footnote{Its pride to call it
1810 % \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1811 % reach a high level of \emph{all} but it in real it will still be a
1812 % survey of knowledge which links to other knowledge and {{\sisac}{}} a
1813 % trainer and helper but no human compensating calculator.
1815 % {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1816 % specifications of problems out of topics from Signal Processing, etc.)
1817 % and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1818 % physical knowledge. The result is a three-dimensional universe of
1819 % mathematics seen in Figure~\ref{fig:mathuni}.
1823 % \includegraphics{fig/universe}
1824 % \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1825 % combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1826 % leads to a three dimensional math universe.\label{fig:mathuni}}
1830 % %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1831 % %WN bitte folgende Bezeichnungen nehmen:
1833 % %WN axis 1: Algorithmic Knowledge (Programs)
1834 % %WN axis 2: Application-oriented Knowledge (Specifications)
1835 % %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1837 % %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1838 % %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1839 % %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1841 % %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1842 % %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1843 % %JR gefordert werden WN2...
1844 % %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
1845 % %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
1846 % %WN2 zusammenschneiden um die R"ander weg zu bekommen)
1847 % %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
1848 % %WN2 png + pdf figures mitzuschicken.
1850 % \subsection{Notes on Problems with Traditional Notation}
1852 % \paragraph{During research} on these topic severely problems on
1853 % traditional notations have been discovered. Some of them have been
1854 % known in computer science for many years now and are still unsolved,
1855 % one of them aggregates with the so called \emph{Lambda Calculus},
1856 % Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1863 % \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1866 % \[ f(p)=\ldots\; p \in \quad R \]
1869 % \noindent Above we see two equations. The first equation aims to
1870 % be a mapping of an function from the reel range to the reel one, but
1871 % when we change only one letter we get the second equation which
1872 % usually aims to insert a reel point $p$ into the reel function. In
1873 % computer science now we have the problem to tell the machine (TP) the
1874 % difference between this two notations. This Problem is called
1875 % \emph{Lambda Calculus}.
1880 % \paragraph{An other problem} is that terms are not full simplified in
1881 % traditional notations, in {{\sisac}} we have to simplify them complete
1882 % to check weather results are compatible or not. in e.g. the solutions
1883 % of an second order linear equation is an rational in {{\sisac}} but in
1884 % tradition we keep fractions as long as possible and as long as they
1885 % aim to be \textit{beautiful} (1/8, 5/16,...).
1886 % \subparagraph{The math} which should be mechanized in Computer Theorem
1887 % Provers (\emph{TP}) has (almost) a problem with traditional notations
1888 % (predicate calculus) for axioms, definitions, lemmas, theorems as a
1889 % computer program or script is not able to interpret every Greek or
1890 % Latin letter and every Greek, Latin or whatever calculations
1891 % symbol. Also if we would be able to handle these symbols we still have
1892 % a problem to interpret them at all. (Follow up \hbox{Example
1893 % \ref{eg:symbint1}})
1897 % \label{eg:symbint1}
1899 % u\left[n\right] \ \ldots \ unitstep
1902 % \noindent The unitstep is something we need to solve Signal
1903 % Processing problem classes. But in {{{\sisac}{}}} the rectangular
1904 % brackets have a different meaning. So we abuse them for our
1905 % requirements. We get something which is not defined, but usable. The
1906 % Result is syntax only without semantic.
1911 % In different problems, symbols and letters have different meanings and
1912 % ask for different ways to get through. (Follow up \hbox{Example
1913 % \ref{eg:symbint2}})
1917 % \label{eg:symbint2}
1919 % \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1922 % \noindent For using exponents the three \texttt{widehat} symbols
1923 % are required. The reason for that is due the development of
1924 % {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1925 % already in use for different operations.
1930 % \paragraph{Also the output} can be a problem. We are familiar with a
1931 % specified notations and style taught in university but a computer
1932 % program has no knowledge of the form proved by a professor and the
1933 % machines themselves also have not yet the possibilities to print every
1934 % symbol (correct) Recent developments provide proofs in a human
1935 % readable format but according to the fact that there is no money for
1936 % good working formal editors yet, the style is one thing we have to
1939 % \section{Problems rising out of the Development Environment}
1941 % fehlermeldungen! TODO
1943 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
1945 \section{Summary and Conclusions}\label{conclusion}
1949 %This paper gives a first experience report about programming with a
1950 %TP-based programming language.
1952 A brief re-introduction of the novel kind of programming
1953 language by example of the {\sisac}-prototype makes the paper
1954 self-contained. The main section describes all the main concepts
1955 involved in TP-based programming and all the sub-tasks concerning
1956 respective implementation in the {\sisac} prototype: mechanisation of mathematics and domain
1957 modeling, implementation of term rewriting systems for the
1958 rewriting-engine, formal (implicit) specification of the problem to be
1959 (explicitly) described by the program, implementation of the many components
1960 required for Lucas-Interpretation and finally implementation of the
1963 The many concepts and sub-tasks involved in programming require a
1964 comprehensive work-flow; first experiences with the work-flow as
1965 supported by the present prototype are described as well: Isabelle +
1966 Isar + jEdit provide appropriate components for establishing an
1967 efficient development environment integrating computation and
1968 deduction. However, the present state of the prototype is far off a
1969 state appropriate for wide-spread use: the prototype of the program
1970 language lacks expressiveness and elegance, the prototype of the
1971 development environment is hardly usable: error messages still address
1972 the developer of the prototype's interpreter rather than the
1973 application programmer, implementation of the many settings for the
1974 Lucas-Interpreter is cumbersome.
1976 \subsection{Conclusions and Expectations to the Future}
1977 From the above mentioned experiences a successful proof of concept can be concluded:
1978 programming arbitrary problems from engineering sciences is possible,
1979 in principle even in the prototype. Furthermore the experiences allow
1980 to conclude detailed requirements for further development:
1982 \item Clarify underlying logics such that programming is smoothly
1983 integrated with verification of the program; the post-condition should
1984 be proved more or less automatically, otherwise working engineers
1985 would not encounter such programming.
1986 \item Combine the prototype's programming language with Isabelle's
1987 powerful function package and probably with more of SML's
1988 pattern-matching features; include parallel execution on multi-core
1989 machines into the language design.
1990 \item Extend the prototype's Lucas-Interpreter such that it also
1991 handles functions defined by use of Isabelle's functions package; and
1992 generalize Isabelle's code generator such that efficient code for the
1993 whole definition of the programming language can be generated (for
1994 multi-core machines).
1995 \item Develop an efficient development environment with
1996 integration of programming and proving, with management not only of
1997 Isabelle theories, but also of large collections of specifications and
1999 \item\label{CAS} Extend Isabelle's computational features in direction of
2000 \textit{verfied} Computer Algebra: simplification extended by
2001 algorithms beyond rewriting (cancellation of multivariate rationals,
2002 factorisation, partial fraction decomposition, etc), equation solving
2005 Provided successful accomplishment, these points provide distinguished
2006 components for virtual workbenches appealing to practitioners of
2007 engineering in the near future.
2009 \subsection{Preview to Development of Course Material}
2010 Interactive course material, as addressed by the title,
2011 can comprise step-wise problem solving created as a side-effect of a
2012 TP-based program: The introduction \S\ref{intro} briefly shows that Lucas-Interpretation not only provides an
2013 interactive programming environment, Lucas-Interpretation also can
2014 provide TP-based services for a flexible dialogue component with
2015 adaptive user guidance for independent and inquiry-based learning.
2017 However, the {\sisac} prototype is not ready for use in field-tests,
2018 not only due to the above five requirements not sufficiently
2019 accomplished, but also due to usability of the fron-end, in particular
2020 the lack of an editor for formulas in 2-dimension representation.
2022 Nevertheless, the experiences from the case study described in this
2023 paper, allow to give a preview to the development of course material,
2024 if based on Lucas-Interpretation:
2026 \paragraph{Development of material from scratch} is too much effort
2027 just for e-learning; this has become clear with the case study. For
2028 getting support for stepwise problem solving just in {\em one} example
2029 class, the one presented in this paper, involved the following tasks:
2031 \item Adapt the equation solver; since that was too laborous, the
2032 program has been adapted in an unelegant way.
2033 \item Implement an algorithms for partial fraction decomposition,
2034 which is considered a standard normal form in Computer Algebra.
2035 \item Implement a specification for partial fraction decomposition and
2036 locate it appropriately in the hierarchy of specification.
2037 \item Declare definitions and theorems within the theory of ${\cal
2038 Z}$-Transformation, and prove the theorems (which was not done in the
2041 On the other hand, for the one the class of problems implemented,
2042 adding an arbitrary number of examples within this class requires a
2043 few minutes~\footnote{As shown in Fig.\ref{fig-interactive}, an
2044 example is called from an HTML-file by an URL, which addresses an
2045 XML-structure holding the respective data as shown on
2046 p.\pageref{ml-check-program}.} and the support for individual stepwise
2047 problem solving comes for free.
2049 \paragraph{E-learning benefits from Formal Domain Engineering} which can be
2050 expected for various domains in the near future. In order to cope with
2051 increasing complexity in domain of technology, specific domain
2052 knowledge is beeing mechanised, not only for software technology
2053 \footnote{For instance, the Archive of Formal Proofs
2054 http://afp.sourceforge.net/} but also for other engineering domains
2055 \cite{Dehbonei&94,Hansen94b,db:dom-eng}. This fairly new part of
2056 engineering sciences is called ``domain engineering'' in
2057 \cite{db:SW-engIII}.
2059 Given this kind of mechanised knowledge including mathematical
2060 theories, domain specific definitions, specifications and algorithms,
2061 theorems and proofs, then e-learning with support for individual
2062 stepwise problem solving will not be much ado anymore; then e-learning
2063 media in technology education can be derived from this knowledge with
2066 \paragraph{Development differentiates into tasks} more separated than
2067 without Lucas-Interpretation and more challenginging in specific
2068 expertise. These are the kinds of experts expected to cooperate in
2071 \item ``Domain engineers'', who accomplish fairly novel tasks described
2073 \item Course designers, who provide the instructional design according
2074 to curricula, together with usability experts and media designers, are
2075 indispensable in production of e-learning media at the state-of-the
2077 \item ``Dialog designers'', whose part of development is clearly
2078 separated from the part of domain engineers as a consequence of Lucas-Interpretation: TP-based programs are functional, as mentioned, and are only concerned with describing mathematics --- and not at all interaction, psychology, learning theory and the like, because there are no in/output statements. Dialog designers can expect a high-level rule-based language~\cite{gdaroczy-EP-13} for describing their part.
2081 % response-to-referees:
2082 % (2.1) details of novel technology in order to estimate the impact
2083 % (2.2) which kinds of expertise are required for production of e-learning media (instructional design, math authoring, dialog authoring, media design)
2084 % (2.3) what in particular is required for programming new exercises supported by next-step-guidance (expertise / efforts)
2085 % (2.4) estimation of break-even points for development of next-step-guidance
2086 % (2.5) usability of ISAC prototype at the present state
2088 % The points (1.*) seem to be well covered in the paper, the points (2.*) are not. So I decided to address the points (2.*) in a separate section §5.1."".
2090 For this decade there seems to be a window of opportunity opening from
2091 one side inreasing demand for formal domain engineering and from the
2092 other side from TP more and more gaining industrial relevance. Within
2093 this window, development of TP-based educational software can take
2094 benefit from the fact, that the TPs leading in Europe, Coq and
2095 Isabelle are still open source together with the major part of
2096 mechanised knowledge.%~\footnote{NICTA}.
2098 \bibliographystyle{alpha}
2099 {\small\bibliography{references}}
2102 % LocalWords: TP IST SPSC Telematics Dialogues dialogue HOL bool nat Hindley
2103 % LocalWords: Milner tt Subproblem Formulae ruleset generalisation initialised
2104 % LocalWords: axiomatization LCF Simplifiers simplifiers Isar rew Thm Calc SML
2105 % LocalWords: recognised hoc Trueprop redexes Unsynchronized pre rhs ord erls
2106 % LocalWords: srls prls nrls lhs errpats InverseZTransform SubProblem IDE IDEs
2107 % LocalWords: univariate jEdit rls RealDef calclist familiarisation ons pos eq
2108 % LocalWords: mout ctree SignalProcessing frac ZZ Postcond Atools wiki SML's
2109 % LocalWords: mechanisation multi