# HG changeset patch
# User nipkow
# Date 1052496030 -7200
# Node ID a994b92ab1eac014caa84b97fd19004e5629a9bd
# Parent  ab988a7a8a3b9416b5aa3092f844cb7161b1b429
*** empty log message ***

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%
diff -r ab988a7a8a3b -r a994b92ab1ea doc-src/TutorialI/Misc/natsum.thy
--- a/doc-src/TutorialI/Misc/natsum.thy	Fri May 09 17:19:58 2003 +0200
+++ b/doc-src/TutorialI/Misc/natsum.thy	Fri May 09 18:00:30 2003 +0200
@@ -81,16 +81,13 @@
 lemma "m \<noteq> (n::nat) \<Longrightarrow> m < n \<or> n < m"
 (*<*)by(arith)(*>*)
 
-text{*\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 (@{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"},
-@{text"\<longrightarrow>"}), the relations @{text"="}, @{text"\<le>"} and @{text"<"},
-and the operations
-@{text"+"}, @{text"-"}, @{term min} and @{term max}. 
-For example,
-*}
+text{*\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 (@{text"\<not>"},
+@{text"\<and>"}, @{text"\<or>"}, @{text"\<longrightarrow>"}, @{text"="},
+@{text"\<forall>"}, @{text"\<exists>"}), the relations @{text"="},
+@{text"\<le>"} and @{text"<"}, and the operations @{text"+"}, @{text"-"},
+@{term min} and @{term max}.  For example, *}
 
 lemma "min i (max j (k*k)) = max (min (k*k) i) (min i (j::nat))";
 apply(arith)
@@ -105,16 +102,19 @@
 
 text{*\noindent
 is not proved even by @{text 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 @{text arith} is exponential in the number of occurrences
-  of \ttindexboldpos{-}{$HOL2arithfun}, \cdx{min} and
+\begin{warn} The running time of @{text arith} is exponential in the number
+  of occurrences of \ttindexboldpos{-}{$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
-  @{text"m+m \<noteq> n+n+(1::nat)"}. Fortunately, such examples are rare.
+If @{text k} is a numeral, \sdx{div}~@{text k}, \sdx{mod}~@{text k} and
+@{text 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, @{text arith} may take
+super-exponential time and space.
 \end{warn}
 *}
 
diff -r ab988a7a8a3b -r a994b92ab1ea doc-src/TutorialI/Types/numerics.tex
--- a/doc-src/TutorialI/Types/numerics.tex	Fri May 09 17:19:58 2003 +0200
+++ b/doc-src/TutorialI/Types/numerics.tex	Fri May 09 18:00:30 2003 +0200
@@ -28,7 +28,9 @@
 and multiplication by constant factors; subterms involving other operators
 are regarded as variables.  The procedure can be slow, especially if the
 subgoal to be proved involves subtraction over type \isa{nat}, which 
-causes case splits.  
+causes case splits.  On types \isa{nat} and \isa{int}, \methdx{arith}
+can deal with quantifiers (this is known as ``Presburger Arithmetic''),
+whereas on type \isa{real} it cannot.
 
 The simplifier reduces arithmetic expressions in other
 ways, such as dividing through by common factors.  For problems that lie
@@ -480,4 +482,4 @@
 Theory \isa{Hyperreal} also defines transcendental functions such as sine,
 cosine, exponential and logarithm --- even the versions for type
 \isa{real}, because they are defined using nonstandard limits.%
-\index{numbers|)}
\ No newline at end of file
+\index{numbers|)}