
diff -r 1d2f15504d38 -r d1bf9ca9008d doc-src/TutorialI/Misc/document/natsum.tex
--- a/doc-src/TutorialI/Misc/document/natsum.tex	Wed Nov 29 10:22:38 2000 +0100
+++ b/doc-src/TutorialI/Misc/document/natsum.tex	Wed Nov 29 13:44:26 2000 +0100
@@ -20,7 +20,80 @@
 \isacommand{lemma}\ {\isachardoublequote}sum\ n\ {\isacharplus}\ sum\ n\ {\isacharequal}\ n{\isacharasterisk}{\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequote}\isanewline
 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ n{\isacharparenright}\isanewline
 \isacommand{apply}{\isacharparenleft}auto{\isacharparenright}\isanewline
-\isacommand{done}\isanewline
+\isacommand{done}%
+\begin{isamarkuptext}%
+\newcommand{\mystar}{*%
+}
+The usual arithmetic operations \ttindexboldpos{+}{$HOL2arithfun},
+\ttindexboldpos{-}{$HOL2arithfun}, \ttindexboldpos{\mystar}{$HOL2arithfun},
+\isaindexbold{div}, \isaindexbold{mod}, \isaindexbold{min} and
+\isaindexbold{max} are predefined, as are the relations
+\indexboldpos{\isasymle}{$HOL2arithrel} and
+\ttindexboldpos{<}{$HOL2arithrel}. There is even a least number operation
+\isaindexbold{LEAST}. For example, \isa{{\isacharparenleft}LEAST\ n{\isachardot}\ {\isadigit{1}}\ {\isacharless}\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}}, although
+Isabelle does not prove this completely automatically. Note that \isa{{\isadigit{1}}}
+and \isa{{\isadigit{2}}} are available as abbreviations for the corresponding
+\isa{Suc}-expressions. If you need the full set of numerals,
+see~\S\ref{nat-numerals}.
+
+\begin{warn}
+  The constant \ttindexbold{0} and the operations
+  \ttindexboldpos{+}{$HOL2arithfun}, \ttindexboldpos{-}{$HOL2arithfun},
+  \ttindexboldpos{\mystar}{$HOL2arithfun}, \isaindexbold{min},
+  \isaindexbold{max}, \indexboldpos{\isasymle}{$HOL2arithrel} and
+  \ttindexboldpos{<}{$HOL2arithrel} are overloaded, i.e.\ they are available
+  not just for natural numbers but at other types as well (see
+  \S\ref{sec:overloading}). For example, given the goal \isa{x\ {\isacharplus}\ {\isadigit{0}}\ {\isacharequal}\ x},
+  there is nothing to indicate that you are talking about natural numbers.
+  Hence Isabelle can only infer that \isa{x} is of some arbitrary type where
+  \isa{{\isadigit{0}}} and \isa{{\isacharplus}} are declared. As a consequence, you will be unable
+  to prove the goal (although it may take you some time to realize what has
+  happened if \isa{show{\isacharunderscore}types} is not set).  In this particular example,
+  you need to include an explicit type constraint, for example
+  \isa{x\ {\isacharplus}\ {\isadigit{0}}\ {\isacharequal}\ x}. If there is enough contextual information this
+  may not be necessary: \isa{Suc\ x\ {\isacharequal}\ x} automatically implies
+  \isa{x{\isacharcolon}{\isacharcolon}nat} because \isa{Suc} is not overloaded.
+\end{warn}
+
+Simple arithmetic goals are proved automatically by both \isa{auto} and the
+simplification methods introduced in \S\ref{sec:Simplification}.  For
+example,%
+\end{isamarkuptext}%
+\isacommand{lemma}\ {\isachardoublequote}{\isasymlbrakk}\ {\isasymnot}\ m\ {\isacharless}\ n{\isacharsemicolon}\ m\ {\isacharless}\ n{\isacharplus}{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n{\isachardoublequote}%
+\begin{isamarkuptext}%
+\noindent
+is proved automatically. The main restriction is that only addition is taken
+into account; other arithmetic operations and quantified formulae are ignored.
+
+For more complex goals, there is the special method \isaindexbold{arith}
+(which attacks the first subgoal). It proves arithmetic goals involving the
+usual logical connectives (\isa{{\isasymnot}}, \isa{{\isasymand}}, \isa{{\isasymor}},
+\isa{{\isasymlongrightarrow}}), the relations \isa{{\isasymle}} and \isa{{\isacharless}}, and the operations
+\isa{{\isacharplus}}, \isa{{\isacharminus}}, \isa{min} and \isa{max}. For example,%
+\end{isamarkuptext}%
+\isacommand{lemma}\ {\isachardoublequote}min\ i\ {\isacharparenleft}max\ j\ {\isacharparenleft}k{\isacharasterisk}k{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ max\ {\isacharparenleft}min\ {\isacharparenleft}k{\isacharasterisk}k{\isacharparenright}\ i{\isacharparenright}\ {\isacharparenleft}min\ i\ {\isacharparenleft}j{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
+\isacommand{apply}{\isacharparenleft}arith{\isacharparenright}%
+\begin{isamarkuptext}%
+\noindent
+succeeds because \isa{k\ {\isacharasterisk}\ k} can be treated as atomic. In contrast,%
+\end{isamarkuptext}%
+\isacommand{lemma}\ {\isachardoublequote}n{\isacharasterisk}n\ {\isacharequal}\ n\ {\isasymLongrightarrow}\ n{\isacharequal}{\isadigit{0}}\ {\isasymor}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequote}%
+\begin{isamarkuptext}%
+\noindent
+is not even proved by \isa{arith} because the proof relies essentially
+on properties of multiplication.
+
+\begin{warn}
+  The running time of \isa{arith} is exponential in the number of occurrences
+  of \ttindexboldpos{-}{$HOL2arithfun}, \isaindexbold{min} and
+  \isaindexbold{max} because they are first eliminated by case distinctions.
+
+  \isa{arith} is incomplete even for the restricted class of formulae
+  described above (known as ``linear arithmetic''). If divisibility plays a
+  role, it may fail to prove a valid formula, for example
+  \isa{m\ {\isacharplus}\ m\ {\isasymnoteq}\ n\ {\isacharplus}\ n\ {\isacharplus}\ {\isadigit{1}}}. Fortunately, such examples are rare in practice.
+\end{warn}%
+\end{isamarkuptext}%
 \end{isabellebody}%
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