doc-src/TutorialI/Misc/natsum.thy

--- a/doc-src/TutorialI/Misc/natsum.thy	Wed Nov 29 10:22:38 2000 +0100
+++ b/doc-src/TutorialI/Misc/natsum.thy	Wed Nov 29 13:44:26 2000 +0100
@@ -7,7 +7,7 @@
 primitive recursion, for example
 *}
 
-consts sum :: "nat \\<Rightarrow> nat";
+consts sum :: "nat \<Rightarrow> nat";
 primrec "sum 0 = 0"
         "sum (Suc n) = Suc n + sum n";
 
@@ -20,6 +20,85 @@
 apply(auto);
 done
 
+text{*\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{<}{$HOL2arithrel}. There is even a least number operation
+\isaindexbold{LEAST}. For example, @{prop"(LEAST n. 1 < n) = 2"}, although
+Isabelle does not prove this completely automatically. Note that @{term 1}
+and @{term 2} are available as abbreviations for the corresponding
+@{term Suc}-expressions. If you need the full set of numerals,
+see~\S\ref{nat-numerals}.
+
+\begin{warn}
+  The constant \ttindexbold{0} and the operations
+  \ttindexboldpos{+}{$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 @{prop"x+0 = x"},
+  there is nothing to indicate that you are talking about natural numbers.
+  Hence Isabelle can only infer that @{term x} is of some arbitrary type where
+  @{term 0} and @{text"+"} 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 @{text show_types} is not set).  In this particular example,
+  you need to include an explicit type constraint, for example
+  @{prop"x+0 = (x::nat)"}. If there is enough contextual information this
+  may not be necessary: @{prop"Suc x = x"} automatically implies
+  @{text"x::nat"} because @{term Suc} is not overloaded.
+\end{warn}
+
+Simple arithmetic goals are proved automatically by both @{term auto} and the
+simplification methods introduced in \S\ref{sec:Simplification}.  For
+example,
+*}
+
+lemma "\<lbrakk> \<not> m < n; m < n+1 \<rbrakk> \<Longrightarrow> m = n"
+(*<*)by(auto)(*>*)
+
+text{*\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 (@{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"},
+@{text"\<longrightarrow>"}), the relations @{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)
+(*<*)done(*>*)
+
+text{*\noindent
+succeeds because @{term"k*k"} can be treated as atomic. In contrast,
+*}
+
+lemma "n*n = n \<Longrightarrow> n=0 \<or> n=1"
+(*<*)oops(*>*)
+
+text{*\noindent
+is not even proved by @{text arith} because the proof relies essentially
+on properties of multiplication.
+
+\begin{warn}
+  The running time of @{text 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
+  @{prop"m+m \<noteq> n+n+1"}. Fortunately, such examples are rare in practice.
+\end{warn}
+*}
+
 (*<*)
 end
 (*>*)