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doc-src/TutorialI/Advanced/simp.thy

author | nipkow |

Thu Sep 14 17:46:00 2000 +0200 (2000-09-14) | |

changeset 9958 | 67f2920862c7 |

child 10186 | 499637e8f2c6 |

permissions | -rw-r--r-- |

*** empty log message ***

1 (*<*)

2 theory simp = Main:

3 (*>*)

5 section{*Simplification*}

7 text{*\label{sec:simplification-II}\index{simplification|(}

8 This section discusses some additional nifty features not covered so far and

9 gives a short introduction to the simplification process itself. The latter

10 is helpful to understand why a particular rule does or does not apply in some

11 situation.

12 *}

14 subsection{*Advanced features*}

16 subsubsection{*Congruence rules*}

18 text{*\label{sec:simp-cong}

19 It is hardwired into the simplifier that while simplifying the conclusion $Q$

20 of $P \isasymImp Q$ it is legal to make uses of the assumptions $P$. This

21 kind of contextual information can also be made available for other

22 operators. For example, @{prop"xs = [] --> xs@xs = xs"} simplifies to @{term

23 True} because we may use @{prop"xs = []"} when simplifying @{prop"xs@xs =

24 xs"}. The generation of contextual information during simplification is

25 controlled by so-called \bfindex{congruence rules}. This is the one for

26 @{text"\<longrightarrow>"}:

27 @{thm[display]imp_cong[no_vars]}

28 It should be read as follows:

29 In order to simplify @{prop"P-->Q"} to @{prop"P'-->Q'"},

30 simplify @{prop P} to @{prop P'}

31 and assume @{prop"P'"} when simplifying @{prop Q} to @{prop"Q'"}.

33 Here are some more examples. The congruence rules for bounded

34 quantifiers supply contextual information about the bound variable:

35 @{thm[display,eta_contract=false,margin=60]ball_cong[no_vars]}

36 The congruence rule for conditional expressions supply contextual

37 information for simplifying the arms:

38 @{thm[display]if_cong[no_vars]}

39 A congruence rule can also \emph{prevent} simplification of some arguments.

40 Here is an alternative congruence rule for conditional expressions:

41 @{thm[display]if_weak_cong[no_vars]}

42 Only the first argument is simplified; the others remain unchanged.

43 This makes simplification much faster and is faithful to the evaluation

44 strategy in programming languages, which is why this is the default

45 congruence rule for @{text if}. Analogous rules control the evaluaton of

46 @{text case} expressions.

48 You can delare your own congruence rules with the attribute @{text cong},

49 either globally, in the usual manner,

50 \begin{quote}

51 \isacommand{declare} \textit{theorem-name} @{text"[cong]"}

52 \end{quote}

53 or locally in a @{text"simp"} call by adding the modifier

54 \begin{quote}

55 @{text"cong:"} \textit{list of theorem names}

56 \end{quote}

57 The effect is reversed by @{text"cong del"} instead of @{text cong}.

59 \begin{warn}

60 The congruence rule @{thm[source]conj_cong}

61 @{thm[display]conj_cong[no_vars]}

62 is occasionally useful but not a default rule; you have to use it explicitly.

63 \end{warn}

64 *}

66 subsubsection{*Permutative rewrite rules*}

68 text{*

69 \index{rewrite rule!permutative|bold}

70 \index{rewriting!ordered|bold}

71 \index{ordered rewriting|bold}

72 \index{simplification!ordered|bold}

73 An equation is a \bfindex{permutative rewrite rule} if the left-hand

74 side and right-hand side are the same up to renaming of variables. The most

75 common permutative rule is commutativity: @{prop"x+y = y+x"}. Other examples

76 include @{prop"(x-y)-z = (x-z)-y"} in arithmetic and @{prop"insert x (insert

77 y A) = insert y (insert x A)"} for sets. Such rules are problematic because

78 once they apply, they can be used forever. The simplifier is aware of this

79 danger and treats permutative rules by means of a special strategy, called

80 \bfindex{ordered rewriting}: a permutative rewrite

81 rule is only applied if the term becomes ``smaller'' (w.r.t.\ some fixed

82 lexicographic ordering on terms). For example, commutativity rewrites

83 @{term"b+a"} to @{term"a+b"}, but then stops because @{term"a+b"} is strictly

84 smaller than @{term"b+a"}. Permutative rewrite rules can be turned into

85 simplification rules in the usual manner via the @{text simp} attribute; the

86 simplifier recognizes their special status automatically.

88 Permutative rewrite rules are most effective in the case of

89 associative-commutative operators. (Associativity by itself is not

90 permutative.) When dealing with an AC-operator~$f$, keep the

91 following points in mind:

92 \begin{itemize}\index{associative-commutative operators}

94 \item The associative law must always be oriented from left to right,

95 namely $f(f(x,y),z) = f(x,f(y,z))$. The opposite orientation, if

96 used with commutativity, can lead to nontermination.

98 \item To complete your set of rewrite rules, you must add not just

99 associativity~(A) and commutativity~(C) but also a derived rule, {\bf

100 left-com\-mut\-ativ\-ity} (LC): $f(x,f(y,z)) = f(y,f(x,z))$.

101 \end{itemize}

102 Ordered rewriting with the combination of A, C, and LC sorts a term

103 lexicographically:

104 \[\def\maps#1{~\stackrel{#1}{\leadsto}~}

105 f(f(b,c),a) \maps{A} f(b,f(c,a)) \maps{C} f(b,f(a,c)) \maps{LC} f(a,f(b,c)) \]

107 Note that ordered rewriting for @{text"+"} and @{text"*"} on numbers is rarely

108 necessary because the builtin arithmetic capabilities often take care of

109 this.

110 *}

112 subsection{*How it works*}

114 text{*\label{sec:SimpHow}

115 Roughly speaking, the simplifier proceeds bottom-up (subterms are simplified

116 first) and a conditional equation is only applied if its condition could be

117 proved (again by simplification). Below we explain some special

118 *}

120 subsubsection{*Higher-order patterns*}

122 subsubsection{*Local assumptions*}

124 subsubsection{*The preprocessor*}

126 text{*

127 \index{simplification|)}

128 *}

129 (*<*)

130 end

131 (*>*)