(*<*)

 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

 (*>*)