diff -r ab988a7a8a3b -r a994b92ab1ea doc-src/TutorialI/Misc/document/natsum.tex
--- a/doc-src/TutorialI/Misc/document/natsum.tex	Fri May 09 17:19:58 2003 +0200
+++ b/doc-src/TutorialI/Misc/document/natsum.tex	Fri May 09 18:00:30 2003 +0200
@@ -92,15 +92,13 @@
 \isamarkupfalse%
 %
 \begin{isamarkuptext}%
-\noindent
-The method \methdx{arith} is more general.  It attempts to prove
-the first subgoal provided it is a quantifier-free \textbf{linear arithmetic}
-formula.  Such formulas may involve the
-usual logical connectives (\isa{{\isasymnot}}, \isa{{\isasymand}}, \isa{{\isasymor}},
-\isa{{\isasymlongrightarrow}}), the relations \isa{{\isacharequal}}, \isa{{\isasymle}} and \isa{{\isacharless}},
-and the operations
-\isa{{\isacharplus}}, \isa{{\isacharminus}}, \isa{min} and \isa{max}. 
-For example,%
+\noindent The method \methdx{arith} is more general.  It attempts to
+prove the first subgoal provided it is a \textbf{linear arithmetic} formula.
+Such formulas may involve the usual logical connectives (\isa{{\isasymnot}},
+\isa{{\isasymand}}, \isa{{\isasymor}}, \isa{{\isasymlongrightarrow}}, \isa{{\isacharequal}},
+\isa{{\isasymforall}}, \isa{{\isasymexists}}), the relations \isa{{\isacharequal}},
+\isa{{\isasymle}} and \isa{{\isacharless}}, and the operations \isa{{\isacharplus}}, \isa{{\isacharminus}},
+\isa{min} and \isa{max}.  For example,%
 \end{isamarkuptext}%
 \isamarkuptrue%
 \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
@@ -119,16 +117,19 @@
 \begin{isamarkuptext}%
 \noindent
 is not proved even by \isa{arith} because the proof relies 
-on properties of multiplication.
+on properties of multiplication. Only multiplication by numerals (which is
+the same as iterated addition) is allowed.
 
-\begin{warn}
-  The running time of \isa{arith} is exponential in the number of occurrences
-  of \ttindexboldpos{-}{$HOL2arithfun}, \cdx{min} and
+\begin{warn} The running time of \isa{arith} is exponential in the number
+  of occurrences of \ttindexboldpos{-}{$HOL2arithfun}, \cdx{min} and
   \cdx{max} because they are first eliminated by case distinctions.
 
-  Even for linear arithmetic formulae, \isa{arith} is incomplete. If divisibility plays a
-  role, it may fail to prove a valid formula, for example
-  \isa{m{\isacharplus}m\ {\isasymnoteq}\ n{\isacharplus}n{\isacharplus}{\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}}. Fortunately, such examples are rare.
+If \isa{k} is a numeral, \sdx{div}~\isa{k}, \sdx{mod}~\isa{k} and
+\isa{k}~\sdx{dvd} are also supported, where the former two are eliminated
+by case distinctions, again blowing up the running time.
+
+If the formula involves explicit quantifiers, \isa{arith} may take
+super-exponential time and space.
 \end{warn}%
 \end{isamarkuptext}%
 \isamarkuptrue%