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authorKarl Berry <karl@freefriends.org>2008-10-03 23:11:35 +0000
committerKarl Berry <karl@freefriends.org>2008-10-03 23:11:35 +0000
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treec712d16e5f998d9d95286b66750bc442a34380a9 /Master/texmf-dist/doc/latex/logic
parent425881b562f2bbf12c3d8fd24db155db11aabad1 (diff)
update logic, without gene-logic files (cfrees 3 Oct 2008 00:10:49)
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-%*****************************************************************************/
-%* */
-%* Version: 1.00 Date: 21/04/92 File: gn-logic14.tex */
-%* Last Version: File: */
-%* Changes: */
-%* 30/12/90 First version of documentation. */
-%* 21/04/92 new properties */
-%* */
-%* Title: */
-%* Author: Gerd Neugebauer */
-%* */
-%* Usage: latex gn-logic14.tex */
-%* */
-%*****************************************************************************/
-
-\documentstyle[11pt,dina4,gn-logic14]{article}
-
-\setlength{\unitlength}{1pt}
-
-
-\begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newlength{\Width} \Width=\textwidth \advance\Width by -1.5em \divide\Width by 2
-
-\section{The {\tt gn-logic} style option}
-Description of Version 1.4 (5/95) by Gerd Neugebauer \bigskip
-
-The {\tt gn-logic} style option provides a facility to typeset logical
-formulas of a certain kind. This style option provides an environment like
-\verb|eqnarray|, an extended {\tt newtheorem} environment and several macros.
-
-
-
-\subsection{Mathematical Symbols}
-
-The following marcos provide better usage of the junctors and quantifiers.
-Especially the spacing is improved.
-
-
-\noindent\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}*{4}{l}}
-\multicolumn{1}{c}{\small Symbol}
- & \multicolumn{1}{c}{\small Macro}
- & \multicolumn{2}{c}{\small Example} \\
- & & & \\
-$\AND$ & \verb|\AND| & \verb$A\AND B$ & $A\AND B$ \\
-$\OR$ & \verb|\OR| & \verb$A\OR B$ & $A\OR B$ \\
-$\XOR$ & \verb|\XOR| & \verb$A\XOR B$ & $A\XOR B$ \\
-$\IMPLIES$ & \verb|\IMPLIES| & \verb$A\IMPLIES B$ & $A\IMPLIES B$ \\
-$\IMPL$ & \verb|\IMPL| & \verb$A\IMPL B$ & $A\IMPL B$ \\
-$\IF$ & \verb|\IF| & \verb$A\IF B$ & $A\IF B$ \\
-$\IFF$ & \verb|\IFF| & \verb$A\IFF B$ & $A\IFF B$ \\
-$\IFFdef$ & \verb|\IFFdef| & \verb$A\IFFdef B$ & $A\IFFdef B$ \\
-$\ANDdots$ & \verb|\ANDdots| & \verb$A_1\ANDdots A_n$ & $A_1\ANDdots A_n$ \\
-$\ORdots$ & \verb|\ORdots| & \verb$A_1\ORdots A_n$ & $A_1\ORdots A_n$ \\
-$\is$ & \verb|\is| & \verb$x\is y$ & $x\is y$ \\
-$\Nat$ & \verb|\Nat| & \verb$n\in\Nat$ & $n\in\Nat$ \\
-$\Forall$ & \verb|\Forall| & \verb$\Forall x P(x)$ & $\Forall x P(x)$ \\
-$\Exists$ & \verb|\Exists| & \verb$\Exists y P(x)$ & $\Exists y P(x)$ \\
-\end{tabular*}
-
-
-\newcommand{\bs}{{\tt\char"5C}}
-\newcommand{\mac}[1]{The {\tt\char92 #1} Macro}
-\newcommand{\macs}[2]{The {\tt\char92 #1} and the {\tt\char92 #2} Macros}
-\newenvironment{compare}%
-{\noindent\begin{center}%
- \begin{tabular}{@{}l@{\hspace*{1.5em}produces\hspace*{1.5em}}l@{}}}%
-{\end{tabular}\end{center}}
-
-\subsubsection*{\mac{AND}}
-This macro can be used for the logical conjunction. In addition to the
-\verb|\wedge| macro it adds more space and the formulas tend to be better
-readable. Compare
-
-\begin{compare}
-\verb$x=1\AND y=x$ & $x=1\AND y=x$ \\
-\verb$x=1\wedge y=x$ & $x=1\wedge y=x$ \\
-\verb$x=1\land y=x$ & $x=1\land y=x$
-\end{compare}
-
-\subsubsection*{\mac{OR}}
-This macro can be used for the logical disjunction. In addition to the
-\verb|\vee| macro it adds more space. Compare
-
-\begin{compare}
-\verb$x=1\OR y=x$ & $x=1\OR y=x$ \\
-\verb$x=1\vee y=x$ & $x=1\vee y=x$ \\
-\verb$x=1\lor y=x$ & $x=1\lor y=x$
-\end{compare}
-
-
-\subsubsection*{\mac{XOR}}
-This macro can be used for the exclusive disjunction. It has no common
-counterpart. The spacing is like in in all junctor macros.
-
-\begin{compare}
-\verb$x=1\XOR y=x$ & $x=1\XOR y=x$
-\end{compare}
-
-
-\subsubsection*{\macs{IMPL}{IMPLIES}}
-These macros can be used for the logical implication. In addition to the
-\verb|\rightarrow| macro it adds more space. Compare
-
-\begin{compare}
-\verb$x=1\IMPL y=x$ & $x=1\IMPL y=x$ \\
-\verb$x=1\IMPLIES y=x$ & $x=1\IMPLIES y=x$ \\
-\verb$x=1\rightarrow y=x$ & $x=1\rightarrow y=x$
-\end{compare}
-
-
-\subsubsection*{\mac{IF}}
- This macro can be used for the logical implication written in reverse order.
-In addition to the \verb|\leftarrow| macro it adds more space. Compare
-
-\begin{compare}
-\verb$x=1\IF y=x$ & $x=1\IF y=x$ \\
-\verb$x=1\lefttarrow y=x$ & $x=1\leftarrow y=x$
-\end{compare}
-
-\subsubsection*{\mac{IFF}}
- This macro can be used for the logical equivalence.
-In addition to the \verb|\leftrightarrow| macro it adds more space. Compare
-
-\begin{compare}
-\verb$x=1\IFF y=x$ & $x=1\IFF y=x$ \\
-\verb$x=1\leftrighttarrow y=x$ & $x=1\leftrightarrow y=x$
-\end{compare}
-
-\subsubsection*{\mac{IFFdef}}
- Like above but with a small ``def'' above the arrow.
-
-\begin{compare}
-\verb$x=1\IFFdef y=x$ & $x=1\IFFdef y=x$
-\end{compare}
-
-\subsubsection*{\mac{is}}
- This macro is for typesetting unifiers. In this case the predefined
-\verb|\setminus| produces to much space.
-
-\begin{compare}
-\verb$\{y\setminus x, z\setminus 4\}$ & $\{y\setminus x, z\setminus 4\}$ \\
-\verb$\{y\is x, z\is 4\}$ & $\{y\is x, z\is 4\}$ \\
-\verb$\{y\backslash x, z\backslash 4P}$ & $\{y\backslash x, z\backslash 4\}$
-\end{compare}
-
-\ifx\AmSTeX\undefined
-\def\AmSTeX{$\cal A$\kern-.1667em\lower.5ex\hbox
- {$\cal M$}\kern-.125em$\cal S$-\kern-.1em\TeX}
-\fi
-
-\subsubsection*{The Number Macros}
-This are macros for those who have no access to the \AmSTeX{} fonts. It makes
-the symbols for the natural numbers, integers, rationals, reals and complex
-numbers. The usual magnification commands apply to it aswell.
-
-
-\def\BB#1{\csname bb#1\endcsname}
-\def\Line#1{\LINE{#1}\(\BB{#1}_{\BB{#1}}\)}
-\def\LINE#1{{\tt \char92bb#1}&%
- {\tiny\BB{#1}}&%
- {\scriptsize\BB{#1}}&%
- {\footnotesize\BB{#1}}&%
- {\small\BB{#1}}&%
- {\normalsize\BB{#1}}&%
- {\large\BB{#1}}&%
- {\Large\BB{#1}}&%
- {\LARGE\BB{#1}}&%
- {\huge\BB{#1}}&%
- {\Huge\BB{#1}}&%
- }
-\begin{center}
- \begin{tabular}{c|cccccccccc|c}
- &\multicolumn{10}{|c|}{{\tt \char92tiny \hfill...\hfill\char92normalsize \hfill...\hfill\char92Huge}}&\verb|X_X|
- \\\hline
- \Line B\\
- \verb|\Complex|\Line C\\
- \Line D\\
- \Line E\\
- \Line F\\
- \Line G\\
- \Line H\\
- \Line I\\
- \Line J\\
- \Line K\\
- \Line L\\
- \Line M\\
- \verb|\Nat| \Line N\\
- \Line O\\
- \Line P\\
- \verb|\Rat| \Line Q\\
- \verb|\Real| \Line R\\
- \verb|\Int| \Line Z\\
- \Line{One}
- \end{tabular}
-\end{center}
-
-Unfortunately the macros \verb|\bbC|, \verb|\bbG|, \verb|\bbO|, and
-\verb|\bbQ| do not scale properly when used in subscripts or superscripts of
-formulae. The following examples shows how the sizing can be achieved manually
-
-\noindent\begin{compare}
-\verb$\bbQ_{\mbox{\scriptsize \bbQ}}$ & $\bbQ_{\mbox{\scriptsize \bbQ}}$
-\end{compare}
-
-
-
-\subsubsection*{\macs{Forall}{Exists}}
- The general problem with quantifies is that after the quantified variable the
-following formula is not automatically seperated with a small space. This can
-be overcome by the following macros.
-
-The \verb|\Forall| and the \verb|\Exists| macros take one argument. They
-typeset the respective quantifier followed by the argument (i.e.\ the variable)
-and finally a small space. As usual the argument has to be enclosed in braces
-if it consists of more than one character. Otherwise the braces can be omitted.
-This allows a elegant notation of short quantified formulas.
-
-\noindent\begin{compare}
-\verb$\Forall x P(x)$ & $\Forall x P(x)$\\
-\verb$\Forall{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$%
-&$\Forall{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$\\
-\verb$\Exists x P(x)$ & $\Exists x P(x)$\\
-\verb$\Exists{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$%
-& $\Exists{x_1,\ldots,x_n}P(x_1,\ldots,x_n)$
-\end{compare}
-
-
-
-\subsection{The {\tt Formula} Environment}
-
-This environment allows to typeset logical formulas. The main problem with the
-\verb|eqnarray| environment was the numbering. In multiline formulas my
-intention was to have the number in the middle of the formula. Inside this
-environment several macros are valid.
-
-\begin{description}
- \item[{\tt\bs begin\{Formula\}[{\em label}] \bs end\{Formula\}}] \ \\
- Start the list of formulas. Optionally a label can be given. This label
- is used to reference the first formula.
- \item[{\tt\bs =}] \ \\
- Start a new line.
- \item[{\tt\bs >}{\em level}] \ \\
- Start a new line and indent to the given {\em level}. This indentation
- is done in quantities of \verb|\FormulaIndent| which can be set with
- the \verb|\setlength| command. The default value is {\tt 3em}.
- \item[{\tt\bs Form[{\em label}]}] \ \\
- Start a new formula. Optionally a {\em label} can be given. This {\em
- label} can be used to reference to the formula (see \verb|\ref|).
-\end{description}
-
-Now lets have a look at some examples. First, we see a single two-line formula.
-Note that the number at the right side is centered between the two lines.
-\medskip
-
-\noindent
-\begin{minipage}{\Width}
-\small\begin{verbatim}
-\begin{Formula}
- P(X) \IMPL
-\= Q(X) \IFF R_1(X) \OR R_2(X)
-\end{Formula}
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{\Width}
-\begin{Formula}
- P(X) \IMPL
-\= Q(X) \IFF R_1(X) \OR R_2(X)
-\end{Formula}
-\end{minipage}\medskip
-
-Next we will see an example of several formulas. The first formula is split to
-three lines and the third line is indented to level 1. Remark: \verb|\=| is in
-reality an abbrevation for \verb|\>0|.
-\medskip
-
-\noindent
-\begin{minipage}{\Width}
-\small\begin{verbatim}
-\begin{Formula}[form:1]
- P(X) \IMPL
-\= Q(X) \IFF R_1(X)
-\>1 \OR R_2(X)
-\Form[form:2]
- S(X) \IMPL
-\= \neg Q(X) \IFF R_1(X) \OR R_2(X)
-\end{Formula}
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{\Width}
-\begin{Formula}[form:1]
- P(X) \IMPL
-\= Q(X) \IFF R_1(X)
-\>1 \OR R_2(X)
-\Form[form:2]
- S(X) \IMPL
-\= \neg Q(X) \IFF R_1(X) \OR R_2(X)
-\end{Formula}
-\end{minipage}\medskip
-
-
-\subsection{The {\tt NewTheorem} Environment}
-
-My experience with the {\tt newtheorem} environment was that I had a certain
-scheme to use it. First, every theorem got a label. Thus, every {\em theorem}
-was followed by a {\tt label} command. Optionally a {\em theorem} may have a
-name. This name is typeset right after the number. The body of the {\em
- theorem} allways started in the next line. This let to the definition of an
-extended {\tt NewTheorem} environment. The arguments are the same as those of
-the {\tt newtheorem} environment. But the environment defined by this extended
-command take two optional arguments. The first optional argument is a label to
-be assigned to the {\em theorem}. This argument has to be enclosed in
-parentheses. The second type of optional argument has to be enclosed in
-brakets. It is typeset in \verb|\small| after the title text. The third
-optional argument is enclosed in \verb|<>|. It is typeset in \verb|\small\bf|
-and surrounded by parentheses.
-\medskip
-
-\noindent\begin{minipage}{\Width}
-\small\begin{verbatim}
-
-\NewTheorem{guess}{Conjecture}
-
-\begin{guess}[Fermat](thm:fermat)
- There do not exist integers $n>2$,
- $x$, $y$, and $z$ such that
- $x^n+y^n=z^n$.
-\end{guess}
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{\Width}
-\NewTheorem{guess}{Conjecture}
-
-\begin{guess}[Fermat](thm:fermat)
- There do not exist integers $n>2$,
- $x$, $y$, and $z$ such that
- $x^n+y^n=z^n$.
-\end{guess}
-\end{minipage}
-\medskip
-
-The commands used to typeset some of the optional argument can be customized
-in the following way. The macros \verb|\TheoremTitle| and \verb|\TheoremName|
-are used to typeset their argument in \verb|\small| and \verb|\small\bf| and
-enclosed in parentheses respectively. This macros can be redefined using
-\verb|\renewcommand| as shown in the following example:
-\medskip
-
-\noindent\begin{minipage}{\Width}
-\footnotesize\begin{verbatim}
-\NewTheorem{theorem}{Theorem}
-\renewcommand{\TheoremTitle}[1]{{\sf [#1]}}
-\renewcommand{\TheoremName}[1]{{\small(#1)}}
-\begin{theorem}[Fermat]<conjecture>(thm:f2)
- There do not exist integers ...
-\end{theorem}
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{\Width}
-\NewTheorem{theorem}{Theorem}
-\renewcommand{\TheoremTitle}[1]{{\sf #1}}
-\renewcommand{\TheoremName}[1]{{\small(#1)}}
-\begin{theorem}[Fermat]<conjecture>(thm:f2)
- There do not exist integers $n>2$,
- $x$, $y$, and $z$ such that
- $x^n+y^n=z^n$.
-\end{theorem}
-\end{minipage}
-
-\end{document}
-
-
-
-\newcommand{\ENTRY}[1]{{#1 \Nat}& {#1 \Int}& {#1 \Rat}&{#1 \Real}& {#1 \Complex}}
-
-\begin{center}
-\begin{tabular}{l|ccccc}
-&\verb|\Nat|&\verb|\Int|&\verb|\Rat|&\verb|\Real|&\verb|\Complex|\\
-&&\\\hline&&\\
-\verb$\tiny$ & \ENTRY{\tiny} \\
-\verb$\scriptsize$ & \ENTRY{\scriptsize} \\
-\verb$\footnotesize$ & \ENTRY{\footnotesize} \\
-\verb$\small$ & \ENTRY{\small} \\
-\verb$\normalsize$ & \ENTRY{\normalsize} \\
-\verb$\large$ & \ENTRY{\large} \\
-\verb$\Large$ & \ENTRY{\Large} \\
-\verb$\LARGE$ & \ENTRY{\LARGE} \\
-\verb$\huge$ & \ENTRY{\huge} \\
-\verb$\Huge$ & \ENTRY{\Huge}
-\end{tabular}\end{center}