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 %---------- --------------------------------------------------- Beginn -----------------------------------------------------------------------

 \title{Greatest Common Divisor \\ for Multivariate Polynomials}

 \author{Diana Meindl\\meindl\_diana@yahoo.com}

 \date{\today}

 \begin{document}

 \maketitle

 %{\w .}\\[12cm]

 %\begin{center}

 %Presented to \\

 %A.Univ.Prof. Dipl.-Ing. Dr. Wolfgang Schreiner (RISC Insitute)\\

 %and\\

 %Dr. techn. Walther Neuper (Institut für Softwaretechnologie, TU Graz)

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 \abstract{

 This is a proposal for a Masters Thesis at RISC, the Research Institute for Symbolic Computation at Linz University.\\

 Calculation with fractions is an important part of Computer Algebra Systems (CAS). This proposal aims at a specific part of such calculations, the greatest common divisor (GCD) used for cancellation, but in the very general context of multivariate polynomials. Cancellation of multivariate polynomials is a settled topic in Computer Algebra, respective algorithms well documented and implementations available in all CASs.

 This proposal claims for novelty with respect to the context of implementation, an implementation as a CAS-feature in Computer Theorem Proving (CTP). On CTP's present development towards industrial use in software and systems verification, specific domain models involve demand on more and more mathematics, and within mathematics involve demand for more and more features. Thus the proposed implementation of GCD and cancellation follows an actual demand.

 If the implementation is successful, it is planned to be included into the distribution of Isabelle, one of the two dominating CTPs in Europe. As part of the Isabelle distribution it will also serve the {\sisac} project aiming at an educational math assistant under development at RISC Linz and Graz University of Technology.

 }

 \newpage

 %WN vorerst zu Zwecken der "Ubersicht lassen ...

 \tableofcontents

 \section{Background}

 The \sisac-project is a research and development project launched at the Institute for Software Technology of the Graz University of Technology (TUG) and now continued at the Research Institute for Symbolic Computation (RISC) of University of Linz and at the Institute for Information Systems and Computer Media (IICM) of TUG. The resulting \sisac{} prototype is a ``transparent single-stepping system for applied mathematics'' based on the computer theorem prover Isabelle. The prototype has been proven useful in field tests at Austrain schools \cite{imst-htl06-SH,imst-htl07-SH,imst-hpts08-SH} and is now extended for wider use.

 Authoring knowledge in \sisac{} provides a strict separation of concerns between authoring math knowledge and authoring dialogues. The latter is pursued at IICM, the former is concern of this thesis. Math authoring is done by use of a CTP-based programming language \cite{plmms10} or by use of SML \cite{pl:milner97} as the meta language and implementation language of Isabelle. Since the code resulting from this thesis shall serve Isabelle, it will be written in SML. Via Isabelle distribution this thesis shall also serve \sisac; a re-implementation in \sisac's CTP-based language is planned as a subsequent project -- this will make cancellation transparent for singe-stepping.

 %The special is an easy readable knowledge base including Isabelles HOL-theories and a transparently working knowledge interpreter (a generalization of 'single stepping' algebra systems).

 %The background to both, development and research, is given by actual needs in math education as well as by foundamental questions about 'the mechanization of thinking' as an essential aspect in mathematics and in technology.

 %The \sisac-system under construction comprises a tutoring-system and an authoring-system. The latter provides for adaption to various needs of individual users and educational institutions and for extensions to arbitrary fields of applied mathematics.

 TODO.WN111107 bitte googeln und je einen Absatz kopieren + zitieren woher (PLAGIATsgefahr):\\

 European provers: Isabelle \cite{Nipkow-Paulson-Wenzel:2002}, Coq \cite{Huet_all:94}\\

 American provers: PVS~\cite{pvs}, ACL2~\footnote{http://userweb.cs.utexas.edu/~moore/acl2/}\\

 \section{Goal of the thesis}

 \subsection{Current situation}

 At the presetn time there is no implimentation for the problem of canceling fractions in Isabelle, and a deficient one in \sisac. But because canceling is important for calculating with fractions a new implimentation is necessary.

 \subsection{Problem} 

 The wish is to handle fractions in \sisac{} not only in one variable also in more. So the goal of this thesis ist to find, assess and evaluate the existing algorithms and methods for finding the GCD. This will be an functional programm with the posibility to include it into Isabelle, where it will be used by \sisac{} as well.

 %WN eine pr"azisere Beschreibung des Problems kann ich mir nicht vorstellen (englische Version der Mail haben wir auch, aber sie passt nicht zur deutschen Antwort von Prof.Nipkow) ...

 \bigskip

 A mail to Prof. Nipkow, leader of the development of Isabelle \cite{Nipkow-Paulson-Wenzel:2002} at TU M\"unchen, Mon, 23 May 2011 08:58:14 +0200 describes the problem as follows:

 \begin{verbatim}

 Eine erste Idee, wie die Integration der Diplomarbeit f"ur

 einen Benutzer von Isabelle aussehen k"onnte, w"are zum 

 Beispiel im

    lemma cancel:

      assumes asm3: "x2 - x*y \<noteq> 0" and asm4: "x \<noteq> 0"

      shows "(x2 - y2) / (x2 - x*y) = (x + y) / (x::real)"

      apply (insert asm3 asm4)

      apply simp

    sorry

 die Assumptions

    asm1: "(x2 - y2) = (x + y) * (x - y)" and asm2: "x2 - x*y = x * (x - y)"

 im Hintergrund automatisch zu erzeugen (mit der Garantie, 

 dass "(x - y)" der GCD ist) und sie dem Simplifier (f"ur die 

 Rule nonzero_mult_divide_mult_cancel_right) zur Verf"ugung zu 

 stellen, sodass anstelle von "sorry" einfach "done" stehen kann.

 Und weiters w"are eventuell asm3 zu "x - y \<noteq> 0" zu vereinfachen, 

 eine Rewriteorder zum Herstellen einer Normalform festzulegen, etc.

 \end{verbatim}

 %WN und eine bessere Motivation f"ur eine Master Thesis kann ich mir auch nicht vorstellen ...

 Response of Prof. Nipkow:

 \begin{verbatim}

 Unser Spezialist fuer die mathematischen Theorien ist Johannes H"olzl. 

 Etwas allgemeinere Fragen sollten auf jeden Fall an isabelle-dev@ 

 gestellt werden.

 Viel Erfolg bei der Arbeit!

 Tobias Nipkow

 \end{verbatim}

 \subsection{Expected results}

 Implementation of algorithms for the different problems, and find out which one will be the best for the specific requirements in Isabelle.\\

 The program should accomplish:

 \begin{itemize}

 \item Real and rational coefficients. Maybe also imaginary coefficients.

 \item Canceling and adding multivariate polynomials, when they are in normal form.

 \end{itemize}

 The program will be written in the functional programming language SML with appropriate comments. The resulting code shall meet the coding standards of Isabelle \cite{isar-impl} p.3-10. The integration of the code into the Isabelle distribution will be done by the Isabelle developer team.

 \section{State of the art}

 In a broad view the context of this thesis can be seen as ``computation and deduction'': simplification and in particular cancellation of rational terms is concern of \textbf{computation} implemented in Computer Algebra Systems (CAS) --- whereas the novelty within the thesis is given by an implementation of cancellation in a computer theorem prover (CTP), i.e. in the domain of \textbf{deduction} with respective logical rigor not addressed in the realm of CAS.

 Below, after a general survey on computation, represented by CAS, and on deduction, represented by CTP, a more narrow view on ``CAS-functionality in CTP'' is pursued.

 \subsection{Computer Algebra and Proof Assistants}

 %WN achtung: diese subsection is fast w"ortlich kopiert aus \cite{plmms10} -- also in der Endfassung bitte "uberarbeiten !!!

 Computer Algebra and Proof Assistants have coexisted for a many years so there is much research trying to bridge the gap between these approaches from both sides. We shall continue to abbreviate Computer Algebra (Systems) by ``CAS'', and in analogy we shall abbreviate proof assistants by CTP, computer theorem provig (comprising both, interactive theorem proving (ITP) and automated theorem proving (ATP), since in CTP there are ATP-tools included today.)

 First, many CTPs already have CAS-like functionality,

 especially for domains like arithmetic. They provide the user

 with conversions, tactics or decision procedures that solve

 problems in a particular domain. Such decision procedures present

 in the standard library of HOL Light~\footnote{http://www.cl.cam.ac.uk/~jrh13/hol-light/} are used inside the

 prototype described in Sect.\ref{cas-funct} on p.\pageref{part-cond} for arithmetic's,

 symbolic differentiation and others.

 Similarly some CAS systems provide environments that allow

 logical reasoning and proving properties within the system. Such

 environments are provided either as logical

 extensions (e.g.~\cite{logicalaxiom}) or are implemented within a

 CAS using its language~\cite{theorema00}.

 There are numerous architectures for information exchange between

 CAS and CTP with different levels of \emph{degree of trust}

 between the prover and the CAS. In principle, there are several approaches. 

 If CAS-functionality is not fully embedded in CTP, CAS can be called as ``oracles'' nevertheless (for efficiency reasons, in general) --- their results are regarded like prophecies of Pythia in Delphi. There are three kinds of checking oracles, however:

 \begin{enumerate}

 \item Just adopt the CAS result without any check. Isabelle internally marks such results.

 \item Check the result inside CTP. There are many cases, where such checks are straight forward, for instance, checking the result of factorization by multiplication of the factors, or checking integrals by differentiation.

 \item Generate a derivation of the result within CTP; in Isabelle this is called ``proof reconstruction''.

 \end{enumerate}

 A longer list of frameworks for

 information exchange and bridges between systems can be found

 in~\cite{casproto}.

 There are many approaches to defining partial functions in proof

 assistants. Since we would like the user to define functions

 without being exposed to the underlying logic of the proof

 assistant we only mention some automated mechanisms for defining

 partial functions in the logic of a CTP.

 Krauss~\cite{krauss} has developed a framework for defining

 partial recursive functions in Isabelle/HOL, which formally

 proves termination by searching for lexicographic combinations of

 size measures. Farmer~\cite{farmer} implements a scheme for

 defining partial recursive functions in \textrm{IMPS}.

 \subsection{Motivation for CAS-functionality in CTP}

 In the realm of CTP formuas are dominated by quantifiers $\forall$, $\exists$ and $\epsilon$ (such) and by operations like $\Rightarrow$, $\land$ and $\lor$. Numbers were strangers initially; numerals have been introduced to Isabelle not much before the year 2000~\footnote{In directory src/Provers/Arith/ see the files cancel\_numerals.ML and cancel\_numeral\_factor.ML in the Isabelle distribution 2011. They still use the notation $\#1,\#2,\#3,\dots$ from before 2000~!}. However, then numerals have been implemented with {\em polymorphic type} such that $2\cdot r\cdot\pi$ ($2$ is type \textit{real}) and $\pi_{\it approx}=3.14\,\land\, 2\cdot r\cdot\pi_{\it approx}$ can be written as well as $\sum_i^n i=\frac{n\cdot(n+1)}{2}$ ($2$ is type \textit{nat}). The different types are inferred by Hindle-Milner type inference \cite{damas-milner-82,Milner-78,Hindley-69}.

 1994 was an important year for CTP: the Pentium Bug caused excitement in the IT community all around the world and motivated INTEL to invest greatly into formal verification of circuits (which carried over to verification of software). Not much later John Harrison mechanized real numbers as Dedekind Cuts in HOL Light \footnote{http://www.cl.cam.ac.uk/~jrh13/hol-light/} and derived calculus, derivative and integral from that definition \cite{harr:thesis}, an implementation which has been transferred to Isabelle very soon after that~\footnote{In the directory src/HOL/Multivariate\_Analysis/ see the files Gauge\_Measure.thy, Integration.thy, Derivative.thy, Real\_Integration.thy, Brouwer\_Fixpoint.thy, Fashoda.thy}.

 Harrison also says that ``CAS are ill-defined'' and gives, among others, an example relevant for this thesis on cancellation: TODO.WN111104 search for ... meromorphic functions in http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-428.ps.gz

 \medskip

 The main motivation for further introduction of CAS-functionality to CTP is also technology-driven: In this decade domain engineering is becoming an academic discipline with industrial relevance \cite{db:dom-eng}: vigorous efforts extend the scope of formal specifications even beyond software technology, and thus respective domains of mathematical knowledge are being mechanized in CTP. The Archive of Formal Proofs~\footnote{http://afp.sourceforge.net/} is Isabelle's repository for such work.

 \subsection{Simplification within CTP}

 Cancellation, the topic of this thesis, is a specific part of simplification of rationals. In the realm of CAS cancellation is {\em not} an interesting part of the state of the art, because cancellation has been implemented in the prevailing CAS more than thirty years ago --- however, cancellation of multivariate polynomials is {\em not} yet implemted in any of the dominating CTPs.

 %WN: bitte mit Anfragen an die Mailing-Listen nachpr"ufen: Coq, HOL, ACL2, PVS

 As in other CTPs, in Isabelle the simplifier is a powerful software component; the sets of rewrite rules, called \textit{simpsets}, contain several hundreds of elements. Rewriting is still very efficient, because the simpsets are transformed to term nets \cite{term-nets}.

 Rational terms of multivariate polynomials still have a normal form \cite{bb-loos} and thus equivalence of respective terms is decidable. This is not the case, however, with terms containing roots or transcedent functions. Thus, CAS are unreliable by design in these cases.

 In CTP, simplification of non-decidable domains is already an issue, as can be seem in the mail with subject ``simproc divide\_cancel\_factor produces error'' in the mailing-list \textit{isabelle-dev@mailbroy.informatik.tu-muenchen.de} from Thu, 15 Sep 2011 16:34:12 +0200

 {%\footnotesize --- HILFT NICHTS

 \begin{verbatim}

 Hi everyone,

 in the following snippet, applying the simplifier causes an error:

 ------------------------------------------

 theory Scratch

   imports Complex_Main

 begin

 lemma

   shows "(3 / 2) * ln n = ((6 * k * ln n) / n) * ((1 / 2 * n / k) / 2)"

 apply simp

 ------------------------------------------

 outputs

 ------------------------------------------

 Proof failed.

 (if n = 0 then 0 else 6 * (k * ln n) / 1) * 2 / (4 * k) =

 2 * (Numeral1 * (if n = 0 then 0 else 6 * (k * ln n) / 1)) / (2 * (2 * k))

  1. n \not= Numeral0 \rightarrow k * (ln n * (2 * 6)) / (k * 4) = k * (ln n * 12) / (k * 4)

 1 unsolved goal(s)!

 The error(s) above occurred for the goal statement:

 (if n = 0 then 0 else 6 * (k * ln n) / 1) * 2 / (4 * k) =

 2 * (Numeral1 * (if n = 0 then 0 else 6 * (k * ln n) / 1)) / (2 * (2 * k))

 ------------------------------------------

 \end{verbatim}

 }

 Mail ``Re: simproc divide\_cancel\_factor produces error'' on Fri, 16 Sep 2011 22:33:36 +0200

 \begin{verbatim}

 > > After the release, I'll have to think about doing a complete overhaul

 > > of all of the cancellation simprocs.

 You are very welcome to do so.  Before you start, call on me and I will

 write down some ideas I had long ago (other may want to join, too).

 \end{verbatim}

 %WN: bist du schon angemeldet in den Mailing-Listen isabelle-users@ und isabelle-dev@ ? WENN NICHT, DANN WIRD ES H"OCHSTE ZEIT !!!

 \subsection{Open Issues with CAS-functionality in CTP}\label{cas-funct}

 There is at least one effort explicitly dedicated to implement CAS-functionality in CTP \cite{cezary-phd}. %WN bitte unbedingt lesen (kann von mir in Papierform ausgeborgt werden) !!!

 In this work three issues has been identified: partiality conditions, multi-valued functions and real numbers. These issues are addressed in the subsequent paragraphs, followed by a forth issue raised by \sisac{}.

 \paragraph{Partiality conditions}\label{part-cond} are introduced by partial functions or by conditional rewriting. An example of how the CAS-functionality \cite{cezary-phd} looks like is given on p.\pageref{fig:casproto}. 

 \cite{cezary-phd} gives an introductory example (floated to p.\pageref{fig:casproto}) which will be referred to in the sequel.

 \input{thol.tex}

 %WN das nachfolgende Format-Problem l"osen wir sp"ater ...

 \begin{figure}[hbt]

 \begin{center}

 \begin{holnb}

   In1 := vector [\&2; \&2] - vector [\&1; \&0] + vec 1

  Out1 := vector [\&2; \&3]

   In2 := diff (diff (\Lam{}x. \&3 * sin (\&2 * x) +

          \&7 + exp (exp x)))

  Out2 := \Lam{}x. exp x pow 2 * exp (exp x) +

          exp x * exp (exp x) + -- \&12 * sin (\&2 * x)

   In3 := N (exp (\&1)) 10

  Out3 := #2.7182818284 + ... (exp (\&1)) 10 F

   In4 := x + \&1 - x / \&1 + \&7 * (y + x) pow 2

  Out4 := \&7 * x pow 2 + \&14 * x * y + \&7 * y pow 2 + \&1

   In5 := sum (0,5) (\Lam{}x. \&x * \&x)

  Out5 := \&30

   In6 := sqrt (x * x) assuming x > &1

  Out6 := x

 \end{holnb}

 \end{center}

 \caption{\label{fig:casproto}Example interaction with the prototype

   CAS-like input-response loop. For the user input given in the

   \texttt{In} lines, the system produces the output in \texttt{Out}

   lines together with HOL Light theorems that state the equality

   between the input and the output.}

 \end{figure}

 In lines {\tt In6, Out6} this examples shows how to reliably simplify $\sqrt{x}$. \cite{caspartial} gives more details on handling side conditions in formalized partial functions.

 Analoguous to this example, cancellations (this thesis is concerned with) like

 $$\frac{x^2-y^2}{x^2-x\cdot y}=\frac{x+y}{x}\;\;\;\;{\it assuming}\;x-y\not=0\land x\not=0$$

 produce assumptions, $x-y\not=0, x\not=0$ here. Since the code produced in the framework of this thesis will be implemented in Isabelle's simplifier (outside this thesis), the presentation to the user will be determined by Isabelle and \sisac{}{} using the respective component of Isabelle. Also reliable handling of assumptions like $x-y\not=0, x\not=0$ is up to these systems.

 \paragraph{Multi-valued functions:}\label{multi-valued}

 \cite{seeingroots,davenp-multival-10} discuss cases where CAS are error prone when dropping a branch of a multi-valued function~\footnote{``Multivalued \textit{function}'' is a misnomer, since the value of a function applied to a certain argument is unique by definition of function.}. Familiar examples are ...

 %TODO.WN111104 ... zur Erkl"arung ein paar Beispiele von http://en.wikipedia.org/wiki/Multivalued_function

 \paragraph{Real numbers} cannot be represented by numerals. In engineering applications, however, approximation by floating-point numbers are frequently useful. In CTP floating-point numbers must be handled rigorously as approximations. Already \cite{harr:thesis} introduced operations on real numerals accompanied by rigorous calculation of precision. \cite{russellphd} describes efficient implementation of infinite precision real numbers in Coq.

 \paragraph{All solutions for equations} must be guaranted, if equation solving is embedded within CTP. So, given an equation $f(x)=0$ and the set of solutions $S$ of this equation, we want to have both,

 \begin{eqnarray}

    \exists x_s.\;x_s\in S &\Rightarrow& f(x_s) = 0 \\\label{is-solut}

   x_s\in S &\Leftarrow& \exists x_s.\;f(x_s) = 0    \label{all-solut}

 \end{eqnarray}

 where (\ref{all-solut}) ensures that $S$ contains {\em all} solutions of the equation. The \sisac{}-project has implemented a prototype of an equation solver~\footnote{See \textit{equations} in the hierarchy of specifications at http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html}.

 There is demand for fullfledged equation solving in CTP, including equational systems and differential equations, because \sisac{}{} has a prototype of a CTP-based programming language calling CAS functions; and Lucas-Interpretation \cite{wn:lucas-interp-12} makes these functions accessible by single-stepping and ``next step guidance'', which would automatically generate a learning system for equation solving.

 \subsection{Algorithms for cancellation of multivariate polynomials}

 The most appropriate book for implementing the required algorithms in this thesis is \cite{Winkler:96}. TODO.WN111104 welche noch ?

 \section{Thesis structure}

 The proposed table of contents of the thesis on the chapter level is as follows:

 \begin{enumerate}

 	\item Introduction (2-3 pages)

 	\item Computer Algebra Systems (CAS) (5 - 7 pages)\\

 	Which different CAS exists and whats the focus of them.

 	\item The \sisac{}-Project (5 - 7 pages)\\

 	This chapter will describe the \sisac{}-Project and the goals of the project.

 	\item Univariate Polynomials (15-20 pages)\\

 	This chapter will describe different Algorithms for univariate polynomials, with different coefficients.

 	\item Multivariate Polynomials (20-25 pages)\\

 	This chapter will describe different Algorithms for multivariate polynomials,  with different coefficients

 	\item Functional programming and SML(2-5 pages)\\

 	The basic idea of this programming languages.

 	\item Implimentation in \sisac{}-Project (15-20 pages)

 	\item Conclusion (2-3 pages)

 \end{enumerate}

 %\newpage

 \section{Timeline}

 %Werd nie fertig.\\

 \begin{center}

 		\begin{tabular}{|l|l|l|}

 		\hline

 			 \textbf{Time}&\textbf{Thesis}&\textbf{Project}\\

 			 \hline

 			 & Functional programming & Learning the basics and the idea\\

 			 & & of funcional programming\\

 			 \hline

 			 & Different CAS & Can they handle the problem \\

 			 & &and which algorithm do they use?\\ \hline

 			 & Univariate Polynomials & Implementation of the Algorithm\\

 			 & & for univariate Polynomials \\ \hline

 		   & Multivariate Polynomials &  Implementation of the Algorithm\\

 			 & & for multivariate Polynomials \\ \hline 

 		   & The \sisac-Project &\\ \hline

 		   & Conclusion and Introduction & Find good examples for testing\\

 			\hline

 		\end{tabular}

 	\end{center}

 %WN oben an passender stelle einf"ugen

 \cite{einf-funct-progr}

 \bibliography{bib/math-eng,bib/didact,bib/bk,bib/RISC_2,bib/isac,bib/pl,bib/math,references}

 %\section{Bibliography}

 %%mindestens 10

 %\begin{enumerate}

 % \item Bird/Wadler, \textit{Einführung in die funktionale Programmierung}, Carl Hanser and Prentice-Hall International, 1992

 % \item Franz Winkler, \textit{Polynomial Algorithms in Computer Algebra}, Springer,1996

 % \item %M. Mignotte, \textit{An inequality about factors of polynomial}

 % \item %M. Mignotte, \textit{Some useful bounds}

 % \item %W. S. Brown and J. F. Traub. \textit{On euclid's algorithm and the theory of subresultans}, Journal of the ACM (JACM), 1971

 % \item %Bruno Buchberger, \textit{Algorhimic mathematics: Problem types, data types, algorithm types}, Lecture notes, RISC Jku A-4040 Linz, 1982

 % 

 % \item %Tateaki Sasaki and Masayuki Suzuki, \textit{Thre new algorithms for multivariate polynomial GCD}, J. Symbolic Combutation, 1992

 % \item

 % \item

 % \item

 %\end{enumerate} 

 \end{document}

 ALLES UNTERHALB \end{document} WIRD VON LATEX NICHT BERUECKSICHTIGT

 WN110916 grep-ing through Isabelle code:

 neuper@neuper:/usr/local/isabisac/src$ find -name "*umeral*"

 ./HOL/ex/Numeral.thy

 ./HOL/Tools/nat_numeral_simprocs.ML

 ./HOL/Tools/numeral_syntax.ML

 ./HOL/Tools/numeral.ML

 ./HOL/Tools/numeral_simprocs.ML

 ./HOL/Matrix/ComputeNumeral.thy

 ./HOL/Library/Numeral_Type.thy

 ./HOL/Numeral_Simprocs.thy

 ./HOL/Import/HOL/numeral.imp

 ./HOL/Code_Numeral.thy

 ./HOL/Nat_Numeral.thy

 ./ZF/Tools/numeral_syntax.ML

 ./Provers/Arith/cancel_numeral_factor.ML

 ./Provers/Arith/cancel_numerals.ML

 ./Provers/Arith/combine_numerals.ML

 neuper@neuper:/usr/local/isabisac/src$ find -name "*ancel*"

 ./HOL/Tools/abel_cancel.ML                 

 ./Provers/Arith/cancel_div_mod.ML

 ./Provers/Arith/cancel_numeral_factor.ML  Paulson 2000 !!!

 ./Provers/Arith/cancel_sums.ML            

 ./Provers/Arith/cancel_numerals.ML        Paulson 2000