2 \chapter{Defining A Sequent-Based Logic}
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
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%$A%
28 is intended to match a sequence of zero or more items.
30 \item A sequence with unspecified sub-sequences and unspecified or
31 individual items is written as a comma-separated list of regular
32 variables (representing items), particular items, and
33 sequence variables, as in
37 Here both \verb%$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
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.
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
75 \%s. Seq0'(A, SeqO'(B, SeqO'(C, s)))
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
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}:
117 "@Trueprop" :: two_seqe ("((_)/ |- (_))" [6,6] 5)
120 where the types used are defined in {\tt Sequents.thy} as
123 two_seqi = [seq'=>seq', seq'=>seq'] => prop
124 two_seqe = [seq, seq] => prop
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
137 val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
138 val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];
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
146 \section{What's in \texttt{Sequents.thy}}
148 Theory \texttt{Sequents.thy} makes many declarations that you need to know
151 \item The Isabelle types given below, which can be used for the
152 constants that map object-level sequents and meta-level propositions:
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
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:
174 "@Side" :: seq=>(seq'=>seq') ("<<(_)>>")
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:
189 val parse_translation =
190 [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
191 val print_translation =
192 [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];
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