summary |
shortlog |
changelog |
graph |
tags |
branches |
files |
changeset |
file |
revisions |
annotate |
diff |
raw

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-- |

*** empty log message ***

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 *}

10 consts sum :: "nat \<Rightarrow> nat";

11 primrec "sum 0 = 0"

12 "sum (Suc n) = Suc n + sum n";

14 text{*\noindent

15 and induction, for example

16 *}

18 lemma "sum n + sum n = n*(Suc n)";

19 apply(induct_tac n);

20 apply(auto);

21 done

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}.

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}

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 *}

62 lemma "\<lbrakk> \<not> m < n; m < n+1 \<rbrakk> \<Longrightarrow> m = n"

63 (*<*)by(auto)(*>*)

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.

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 *}

78 lemma "min i (max j (k*k)) = max (min (k*k) i) (min i (j::nat))";

79 apply(arith)

80 (*<*)done(*>*)

82 text{*\noindent

83 succeeds because @{term"k*k"} can be treated as atomic. In contrast,

84 *}

86 lemma "n*n = n \<Longrightarrow> n=0 \<or> n=1"

87 (*<*)oops(*>*)

89 text{*\noindent

90 is not even proved by @{text arith} because the proof relies essentially

91 on properties of multiplication.

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.

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 *}

105 (*<*)

106 end

107 (*>*)