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doc-src/Logics/Sequents.tex

author | haftmann |

Wed Dec 27 19:10:06 2006 +0100 (2006-12-27) | |

changeset 21915 | 4e63c55f4cb4 |

parent 7160 | 1135f3f8782c |

child 42637 | 381fdcab0f36 |

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

different handling of type variable names

1 %% $Id$

2 \chapter{Defining A Sequent-Based Logic}

3 \label{chap:sequents}

5 \underscoreon %this file contains the @ character

7 The Isabelle theory \texttt{Sequents.thy} provides facilities for using

8 sequent notation in users' object logics. This theory allows users to

9 easily interface the surface syntax of sequences with an underlying

10 representation suitable for higher-order unification.

12 \section{Concrete syntax of sequences}

14 Mathematicians and logicians have used sequences in an informal way

15 much before proof systems such as Isabelle were created. It seems

16 sensible to allow people using Isabelle to express sequents and

17 perform proofs in this same informal way, and without requiring the

18 theory developer to spend a lot of time in \ML{} programming.

20 By using {\tt Sequents.thy}

21 appropriately, a logic developer can allow users to refer to sequences

22 in several ways:

23 %

24 \begin{itemize}

25 \item A sequence variable is any alphanumeric string with the first

26 character being a \verb%$% sign.

27 So, consider the sequent \verb%$A |- B%, where \verb%$A%

28 is intended to match a sequence of zero or more items.

30 \item A sequence with unspecified sub-sequences and unspecified or

31 individual items is written as a comma-separated list of regular

32 variables (representing items), particular items, and

33 sequence variables, as in

34 \begin{ttbox}

35 $A, B, C, $D(x) |- E

36 \end{ttbox}

37 Here both \verb%$A% and \verb%$D(x)%

38 are allowed to match any subsequences of items on either side of the

39 two items that match $B$ and $C$. Moreover, the sequence matching

40 \verb%$D(x)% may contain occurrences of~$x$.

42 \item An empty sequence can be represented by a blank space, as in

43 \verb? |- true?.

44 \end{itemize}

46 These syntactic constructs need to be assimilated into the object

47 theory being developed. The type that we use for these visible objects

48 is given the name {\tt seq}.

49 A {\tt seq} is created either by the empty space, a {\tt seqobj} or a

50 {\tt seqobj} followed by a {\tt seq}, with a comma between them. A

51 {\tt seqobj} is either an item or a variable representing a

52 sequence. Thus, a theory designer can specify a function that takes

53 two sequences and returns a meta-level proposition by giving it the

54 Isabelle type \verb|[seq, seq] => prop|.

56 This is all part of the concrete syntax, but one may wish to

57 exploit Isabelle's higher-order abstract syntax by actually having a

58 different, more powerful {\em internal} syntax.

62 \section{ Basis}

64 One could opt to represent sequences as first-order objects (such as

65 simple lists), but this would not allow us to use many facilities

66 Isabelle provides for matching. By using a slightly more complex

67 representation, users of the logic can reap many benefits in

68 facilities for proofs and ease of reading logical terms.

70 A sequence can be represented as a function --- a constructor for

71 further sequences --- by defining a binary {\em abstract} function

72 \verb|Seq0'| with type \verb|[o,seq']=>seq'|, and translating a

73 sequence such as \verb|A, B, C| into

74 \begin{ttbox}

75 \%s. Seq0'(A, SeqO'(B, SeqO'(C, s)))

76 \end{ttbox}

77 This sequence can therefore be seen as a constructor

78 for further sequences. The constructor \verb|Seq0'| is never given a

79 value, and therefore it is not possible to evaluate this expression

80 into a basic value.

82 Furthermore, if we want to represent the sequence \verb|A, $B, C|,

83 we note that \verb|$B| already represents a sequence, so we can use

84 \verb|B| itself to refer to the function, and therefore the sequence

85 can be mapped to the internal form:

86 \verb|%s. SeqO'(A, B(SeqO'(C, s)))|.

88 So, while we wish to continue with the standard, well-liked {\em

89 external} representation of sequences, we can represent them {\em

90 internally} as functions of type \verb|seq'=>seq'|.

93 \section{Object logics}

95 Recall that object logics are defined by mapping elements of

96 particular types to the Isabelle type \verb|prop|, usually with a

97 function called {\tt Trueprop}. So, an object

98 logic proposition {\tt P} is matched to the Isabelle proposition

99 {\tt Trueprop(P)}\@. The name of the function is often hidden, so the

100 user just sees {\tt P}\@. Isabelle is eager to make types match, so it

101 inserts {\tt Trueprop} automatically when an object of type {\tt prop}

102 is expected. This mechanism can be observed in most of the object

103 logics which are direct descendants of {\tt Pure}.

105 In order to provide the desired syntactic facilities for sequent

106 calculi, rather than use just one function that maps object-level

107 propositions to meta-level propositions, we use two functions, and

108 separate internal from the external representation.

110 These functions need to be given a type that is appropriate for the particular

111 form of sequents required: single or multiple conclusions. So

112 multiple-conclusion sequents (used in the LK logic) can be

113 specified by the following two definitions, which are lifted from the inbuilt

114 {\tt Sequents/LK.thy}:

115 \begin{ttbox}

116 Trueprop :: two_seqi

117 "@Trueprop" :: two_seqe ("((_)/ |- (_))" [6,6] 5)

118 \end{ttbox}

119 %

120 where the types used are defined in {\tt Sequents.thy} as

121 abbreviations:

122 \begin{ttbox}

123 two_seqi = [seq'=>seq', seq'=>seq'] => prop

124 two_seqe = [seq, seq] => prop

125 \end{ttbox}

127 The next step is to actually create links into the low-level parsing

128 and pretty-printing mechanisms, which map external and internal

129 representations. These functions go below the user level and capture

130 the underlying structure of Isabelle terms in \ML{}\@. Fortunately the

131 theory developer need not delve in this level; {\tt Sequents.thy}

132 provides the necessary facilities. All the theory developer needs to

133 add in the \ML{} section is a specification of the two translation

134 functions:

135 \begin{ttbox}

136 ML

137 val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];

138 val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];

139 \end{ttbox}

141 In summary: in the logic theory being developed, the developer needs

142 to specify the types for the internal and external representation of

143 the sequences, and use the appropriate parsing and pretty-printing

144 functions.

146 \section{What's in \texttt{Sequents.thy}}

148 Theory \texttt{Sequents.thy} makes many declarations that you need to know

149 about:

150 \begin{enumerate}

151 \item The Isabelle types given below, which can be used for the

152 constants that map object-level sequents and meta-level propositions:

153 %

154 \begin{ttbox}

155 single_seqe = [seq,seqobj] => prop

156 single_seqi = [seq'=>seq',seq'=>seq'] => prop

157 two_seqi = [seq'=>seq', seq'=>seq'] => prop

158 two_seqe = [seq, seq] => prop

159 three_seqi = [seq'=>seq', seq'=>seq', seq'=>seq'] => prop

160 three_seqe = [seq, seq, seq] => prop

161 four_seqi = [seq'=>seq', seq'=>seq', seq'=>seq', seq'=>seq'] => prop

162 four_seqe = [seq, seq, seq, seq] => prop

163 \end{ttbox}

165 The \verb|single_| and \verb|two_| sets of mappings for internal and

166 external representations are the ones used for, say single and

167 multiple conclusion sequents. The other functions are provided to

168 allow rules that manipulate more than two functions, as can be seen in

169 the inbuilt object logics.

171 \item An auxiliary syntactic constant has been

172 defined that directly maps a sequence to its internal representation:

173 \begin{ttbox}

174 "@Side" :: seq=>(seq'=>seq') ("<<(_)>>")

175 \end{ttbox}

176 Whenever a sequence (such as \verb|<< A, $B, $C>>|) is entered using this

177 syntax, it is translated into the appropriate internal representation. This

178 form can be used only where a sequence is expected.

180 \item The \ML{} functions \texttt{single\_tr}, \texttt{two\_seq\_tr},

181 \texttt{three\_seq\_tr}, \texttt{four\_seq\_tr} for parsing, that is, the

182 translation from external to internal form. Analogously there are

183 \texttt{single\_tr'}, \texttt{two\_seq\_tr'}, \texttt{three\_seq\_tr'},

184 \texttt{four\_seq\_tr'} for pretty-printing, that is, the translation from

185 internal to external form. These functions can be used in the \ML{} section

186 of a theory file to specify the translations to be used. As an example of

187 use, note that in {\tt LK.thy} we declare two identifiers:

188 \begin{ttbox}

189 val parse_translation =

190 [("@Trueprop",Sequents.two_seq_tr "Trueprop")];

191 val print_translation =

192 [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];

193 \end{ttbox}

194 The given parse translation will be applied whenever a \verb|@Trueprop|

195 constant is found, translating using \verb|two_seq_tr| and inserting the

196 constant \verb|Trueprop|. The pretty-printing translation is applied

197 analogously; a term that contains \verb|Trueprop| is printed as a

198 \verb|@Trueprop|.

199 \end{enumerate}