doc-src/TutorialI/Misc/natsum.thy
author nipkow
Wed Dec 13 09:39:53 2000 +0100 (2000-12-13)
changeset 10654 458068404143
parent 10608 620647438780
child 10788 ea48dd8b0232
permissions -rw-r--r--
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     1 (*<*)
     2 theory natsum = Main:;
     3 (*>*)
     4 text{*\noindent
     5 In particular, there are @{text"case"}-expressions, for example
     6 @{term[display]"case n of 0 => 0 | Suc m => m"}
     7 primitive recursion, for example
     8 *}
     9 
    10 consts sum :: "nat \<Rightarrow> nat";
    11 primrec "sum 0 = 0"
    12         "sum (Suc n) = Suc n + sum n";
    13 
    14 text{*\noindent
    15 and induction, for example
    16 *}
    17 
    18 lemma "sum n + sum n = n*(Suc n)";
    19 apply(induct_tac n);
    20 apply(auto);
    21 done
    22 
    23 text{*\newcommand{\mystar}{*%
    24 }
    25 The usual arithmetic operations \ttindexboldpos{+}{$HOL2arithfun},
    26 \ttindexboldpos{-}{$HOL2arithfun}, \ttindexboldpos{\mystar}{$HOL2arithfun},
    27 \isaindexbold{div}, \isaindexbold{mod}, \isaindexbold{min} and
    28 \isaindexbold{max} are predefined, as are the relations
    29 \indexboldpos{\isasymle}{$HOL2arithrel} and
    30 \ttindexboldpos{<}{$HOL2arithrel}. As usual, @{prop"m-n = 0"} if
    31 @{prop"m<n"}. There is even a least number operation
    32 \isaindexbold{LEAST}. For example, @{prop"(LEAST n. 1 < n) = 2"}, although
    33 Isabelle does not prove this completely automatically. Note that @{term 1}
    34 and @{term 2} are available as abbreviations for the corresponding
    35 @{term Suc}-expressions. If you need the full set of numerals,
    36 see~\S\ref{sec:numerals}.
    37 
    38 \begin{warn}
    39   The constant \ttindexbold{0} and the operations
    40   \ttindexboldpos{+}{$HOL2arithfun}, \ttindexboldpos{-}{$HOL2arithfun},
    41   \ttindexboldpos{\mystar}{$HOL2arithfun}, \isaindexbold{min},
    42   \isaindexbold{max}, \indexboldpos{\isasymle}{$HOL2arithrel} and
    43   \ttindexboldpos{<}{$HOL2arithrel} are overloaded, i.e.\ they are available
    44   not just for natural numbers but at other types as well (see
    45   \S\ref{sec:overloading}). For example, given the goal @{prop"x+0 = x"},
    46   there is nothing to indicate that you are talking about natural numbers.
    47   Hence Isabelle can only infer that @{term x} is of some arbitrary type where
    48   @{term 0} and @{text"+"} are declared. As a consequence, you will be unable
    49   to prove the goal (although it may take you some time to realize what has
    50   happened if @{text show_types} is not set).  In this particular example,
    51   you need to include an explicit type constraint, for example
    52   @{prop"x+0 = (x::nat)"}. If there is enough contextual information this
    53   may not be necessary: @{prop"Suc x = x"} automatically implies
    54   @{text"x::nat"} because @{term Suc} is not overloaded.
    55 \end{warn}
    56 
    57 Simple arithmetic goals are proved automatically by both @{term auto} and the
    58 simplification methods introduced in \S\ref{sec:Simplification}.  For
    59 example,
    60 *}
    61 
    62 lemma "\<lbrakk> \<not> m < n; m < n+1 \<rbrakk> \<Longrightarrow> m = n"
    63 (*<*)by(auto)(*>*)
    64 
    65 text{*\noindent
    66 is proved automatically. The main restriction is that only addition is taken
    67 into account; other arithmetic operations and quantified formulae are ignored.
    68 
    69 For more complex goals, there is the special method \isaindexbold{arith}
    70 (which attacks the first subgoal). It proves arithmetic goals involving the
    71 usual logical connectives (@{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"},
    72 @{text"\<longrightarrow>"}), the relations @{text"\<le>"} and @{text"<"}, and the operations
    73 @{text"+"}, @{text"-"}, @{term min} and @{term max}. Technically, this is
    74 known as the class of (quantifier free) \bfindex{linear arithmetic} formulae.
    75 For example,
    76 *}
    77 
    78 lemma "min i (max j (k*k)) = max (min (k*k) i) (min i (j::nat))";
    79 apply(arith)
    80 (*<*)done(*>*)
    81 
    82 text{*\noindent
    83 succeeds because @{term"k*k"} can be treated as atomic. In contrast,
    84 *}
    85 
    86 lemma "n*n = n \<Longrightarrow> n=0 \<or> n=1"
    87 (*<*)oops(*>*)
    88 
    89 text{*\noindent
    90 is not even proved by @{text arith} because the proof relies essentially
    91 on properties of multiplication.
    92 
    93 \begin{warn}
    94   The running time of @{text arith} is exponential in the number of occurrences
    95   of \ttindexboldpos{-}{$HOL2arithfun}, \isaindexbold{min} and
    96   \isaindexbold{max} because they are first eliminated by case distinctions.
    97 
    98   \isa{arith} is incomplete even for the restricted class of
    99   linear arithmetic formulae. If divisibility plays a
   100   role, it may fail to prove a valid formula, for example
   101   @{prop"m+m \<noteq> n+n+1"}. Fortunately, such examples are rare in practice.
   102 \end{warn}
   103 *}
   104 
   105 (*<*)
   106 end
   107 (*>*)