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doc-src/TutorialI/Advanced/simp.thy

author | nipkow |

Wed Oct 11 09:09:06 2000 +0200 (2000-10-11) | |

changeset 10186 | 499637e8f2c6 |

parent 9958 | 67f2920862c7 |

child 10281 | 9554ce1c2e54 |

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 features of the rewriting process.

118 *}

120 subsubsection{*Higher-order patterns*}

122 text{*\index{simplification rule|(}

123 So far we have pretended the simplifier can deal with arbitrary

124 rewrite rules. This is not quite true. Due to efficiency (and

125 potentially also computability) reasons, the simplifier expects the

126 left-hand side of each rule to be a so-called \emph{higher-order

127 pattern}~\cite{nipkow-patterns}\indexbold{higher-order

128 pattern}\indexbold{pattern, higher-order}. This restricts where

129 unknowns may occur. Higher-order patterns are terms in $\beta$-normal

130 form (this will always be the case unless you have done something

131 strange) where each occurrence of an unknown is of the form

132 $\Var{f}~x@1~\dots~x@n$, where the $x@i$ are distinct bound

133 variables. Thus all ``standard'' rewrite rules, where all unknowns are

134 of base type, for example @{thm add_assoc}, are OK: if an unknown is

135 of base type, it cannot have any arguments. Additionally, the rule

136 @{text"(\<forall>x. ?P x \<and> ?Q x) = ((\<forall>x. ?P x) \<and> (\<forall>x. ?Q x))"} is also OK, in

137 both directions: all arguments of the unknowns @{text"?P"} and

138 @{text"?Q"} are distinct bound variables.

140 If the left-hand side is not a higher-order pattern, not all is lost

141 and the simplifier will still try to apply the rule, but only if it

142 matches ``directly'', i.e.\ without much $\lambda$-calculus hocus

143 pocus. For example, @{text"?f ?x \<in> range ?f = True"} rewrites

144 @{term"g a \<in> range g"} to @{term True}, but will fail to match

145 @{text"g(h b) \<in> range(\<lambda>x. g(h x))"}. However, you can

146 replace the offending subterms (in our case @{text"?f ?x"}, which

147 is not a pattern) by adding new variables and conditions: @{text"?y =

148 ?f ?x \<Longrightarrow> ?y \<in> range ?f = True"} is fine

149 as a conditional rewrite rule since conditions can be arbitrary

150 terms. However, this trick is not a panacea because the newly

151 introduced conditions may be hard to prove, which has to take place

152 before the rule can actually be applied.

154 There is basically no restriction on the form of the right-hand

155 sides. They may not contain extraneous term or type variables, though.

156 *}

158 subsubsection{*The preprocessor*}

160 text{*

161 When some theorem is declared a simplification rule, it need not be a

162 conditional equation already. The simplifier will turn it into a set of

163 conditional equations automatically. For example, given @{prop"f x =

164 g x & h x = k x"} the simplifier will turn this into the two separate

165 simplifiction rules @{prop"f x = g x"} and @{prop"h x = k x"}. In

166 general, the input theorem is converted as follows:

167 \begin{eqnarray}

168 \neg P &\mapsto& P = False \nonumber\\

169 P \longrightarrow Q &\mapsto& P \Longrightarrow Q \nonumber\\

170 P \land Q &\mapsto& P,\ Q \nonumber\\

171 \forall x.~P~x &\mapsto& P~\Var{x}\nonumber\\

172 \forall x \in A.\ P~x &\mapsto& \Var{x} \in A \Longrightarrow P~\Var{x} \nonumber\\

173 @{text if}\ P\ @{text then}\ Q\ @{text else}\ R &\mapsto&

174 P \Longrightarrow Q,\ \neg P \Longrightarrow R \nonumber

175 \end{eqnarray}

176 Once this conversion process is finished, all remaining non-equations

177 $P$ are turned into trivial equations $P = True$.

178 For example, the formula @{prop"(p \<longrightarrow> q \<and> r) \<and> s"} is converted into the three rules

179 \begin{center}

180 @{prop"p \<Longrightarrow> q = True"},\quad @{prop"p \<Longrightarrow> r = True"},\quad @{prop"s = True"}.

181 \end{center}

182 \index{simplification rule|)}

183 \index{simplification|)}

184 *}

185 (*<*)

186 end

187 (*>*)