--- a/doc-src/TutorialI/Misc/natsum.thy Tue Aug 28 18:46:15 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,129 +0,0 @@
-(*<*)
-theory natsum imports Main begin
-(*>*)
-text{*\noindent
-In particular, there are @{text"case"}-expressions, for example
-@{term[display]"case n of 0 => 0 | Suc m => m"}
-primitive recursion, for example
-*}
-
-primrec sum :: "nat \<Rightarrow> nat" where
-"sum 0 = 0" |
-"sum (Suc n) = Suc n + sum n"
-
-text{*\noindent
-and induction, for example
-*}
-
-lemma "sum n + sum n = n*(Suc n)"
-apply(induct_tac n)
-apply(auto)
-done
-
-text{*\newcommand{\mystar}{*%
-}
-\index{arithmetic operations!for \protect\isa{nat}}%
-The arithmetic operations \isadxboldpos{+}{$HOL2arithfun},
-\isadxboldpos{-}{$HOL2arithfun}, \isadxboldpos{\mystar}{$HOL2arithfun},
-\sdx{div}, \sdx{mod}, \cdx{min} and
-\cdx{max} are predefined, as are the relations
-\isadxboldpos{\isasymle}{$HOL2arithrel} and
-\isadxboldpos{<}{$HOL2arithrel}. As usual, @{prop"m-n = (0::nat)"} if
-@{prop"m<n"}. There is even a least number operation
-\sdx{LEAST}\@. For example, @{prop"(LEAST n. 0 < n) = Suc 0"}.
-\begin{warn}\index{overloading}
- The constants \cdx{0} and \cdx{1} and the operations
- \isadxboldpos{+}{$HOL2arithfun}, \isadxboldpos{-}{$HOL2arithfun},
- \isadxboldpos{\mystar}{$HOL2arithfun}, \cdx{min},
- \cdx{max}, \isadxboldpos{\isasymle}{$HOL2arithrel} and
- \isadxboldpos{<}{$HOL2arithrel} are overloaded: they are available
- not just for natural numbers but for other types as well.
- For example, given the goal @{text"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 @{text 0} and @{text"+"} are
- declared. As a consequence, you will be unable to prove the
- goal. To alert you to such pitfalls, Isabelle flags numerals without a
- fixed type in its output: @{prop"x+0 = x"}. (In the absence of a numeral,
- it may take you some time to realize what has happened if \pgmenu{Show
- Types} is not set). In this particular example, you need to include
- an explicit type constraint, for example @{text"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.
-
- For details on overloading see \S\ref{sec:overloading}.
- Table~\ref{tab:overloading} in the appendix shows the most important
- overloaded operations.
-\end{warn}
-\begin{warn}
- The symbols \isadxboldpos{>}{$HOL2arithrel} and
- \isadxboldpos{\isasymge}{$HOL2arithrel} are merely syntax: @{text"x > y"}
- stands for @{prop"y < x"} and similary for @{text"\<ge>"} and
- @{text"\<le>"}.
-\end{warn}
-\begin{warn}
- Constant @{text"1::nat"} is defined to equal @{term"Suc 0"}. This definition
- (see \S\ref{sec:ConstDefinitions}) is unfolded automatically by some
- tactics (like @{text auto}, @{text simp} and @{text arith}) but not by
- others (especially the single step tactics in Chapter~\ref{chap:rules}).
- If you need the full set of numerals, see~\S\ref{sec:numerals}.
- \emph{Novices are advised to stick to @{term"0::nat"} and @{term Suc}.}
-\end{warn}
-
-Both @{text auto} and @{text simp}
-(a method introduced below, \S\ref{sec:Simplification}) prove
-simple arithmetic goals automatically:
-*}
-
-lemma "\<lbrakk> \<not> m < n; m < n + (1::nat) \<rbrakk> \<Longrightarrow> m = n"
-(*<*)by(auto)(*>*)
-
-text{*\noindent
-For efficiency's sake, this built-in prover ignores quantified formulae,
-many logical connectives, and all arithmetic operations apart from addition.
-In consequence, @{text auto} and @{text simp} cannot prove this slightly more complex goal:
-*}
-
-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 \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)
-(*<*)done(*>*)
-
-text{*\noindent
-succeeds because @{term"k*k"} can be treated as atomic. In contrast,
-*}
-
-lemma "n*n = n+1 \<Longrightarrow> n=0"
-(*<*)oops(*>*)
-
-text{*\noindent
-is not proved by @{text arith} because the proof relies
-on properties of multiplication. Only multiplication by numerals (which is
-the same as iterated addition) is taken into account.
-
-\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.
-
-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 quantifiers, @{text arith} may take
-super-exponential time and space.
-\end{warn}
-*}
-
-(*<*)
-end
-(*>*)