3 \def\isabellecontext{HOL{\isacharunderscore}Specific}%
10 \isacommand{theory}\isamarkupfalse%
11 \ HOL{\isacharunderscore}Specific\isanewline
12 \isakeyword{imports}\ Main\isanewline
21 \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
25 \isamarkupsection{Typedef axiomatization \label{sec:hol-typedef}%
29 \begin{isamarkuptext}%
30 \begin{matharray}{rcl}
31 \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
35 'typedef' altname? abstype '=' repset
38 altname: '(' (name | 'open' | 'open' name) ')'
40 abstype: typespecsorts mixfix?
42 repset: term ('morphisms' name name)?
48 \item \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}
49 axiomatizes a Gordon/HOL-style type definition in the background
50 theory of the current context, depending on a non-emptiness result
51 of the set \isa{A} (which needs to be proven interactively).
53 The raw type may not depend on parameters or assumptions of the
54 context --- this is logically impossible in Isabelle/HOL --- but the
55 non-emptiness property can be local, potentially resulting in
56 multiple interpretations in target contexts. Thus the established
57 bijection between the representing set \isa{A} and the new type
58 \isa{t} may semantically depend on local assumptions.
60 By default, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t}
61 and a set (term constant) of the same name, unless an alternative
62 base name is given in parentheses, or the ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
63 declaration is used to suppress a separate constant definition
64 altogether. The injection from type to set is called \isa{Rep{\isacharunderscore}t},
65 its inverse \isa{Abs{\isacharunderscore}t} --- this may be changed via an explicit
66 \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration.
68 Theorems \isa{Rep{\isacharunderscore}t}, \isa{Rep{\isacharunderscore}t{\isacharunderscore}inverse}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inverse} provide the most basic characterization as a
69 corresponding injection/surjection pair (in both directions). Rules
70 \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
71 more convenient view on the injectivity part, suitable for automated
72 proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
73 declarations). Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
74 \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
75 on surjectivity; these are already declared as set or type rules for
76 the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods.
78 An alternative name for the set definition (and other derived
79 entities) may be specified in parentheses; the default is to use
80 \isa{t} as indicated before.
86 \isamarkupsection{Adhoc tuples%
90 \begin{isamarkuptext}%
91 \begin{matharray}{rcl}
92 \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
96 'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
102 \item \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ {\isasymAND}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}} puts expressions of low-level tuple types into
103 canonical form as specified by the arguments given; the \isa{i}-th
104 collection of arguments refers to occurrences in premise \isa{i}
105 of the rule. The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all}
106 arguments in function applications to be represented canonically
107 according to their tuple type structure.
109 Note that these operations tend to invent funny names for new local
110 parameters to be introduced.
116 \isamarkupsection{Records \label{sec:hol-record}%
120 \begin{isamarkuptext}%
121 In principle, records merely generalize the concept of tuples, where
122 components may be addressed by labels instead of just position. The
123 logical infrastructure of records in Isabelle/HOL is slightly more
124 advanced, though, supporting truly extensible record schemes. This
125 admits operations that are polymorphic with respect to record
126 extension, yielding ``object-oriented'' effects like (single)
127 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
128 details on object-oriented verification and record subtyping in HOL.%
132 \isamarkupsubsection{Basic concepts%
136 \begin{isamarkuptext}%
137 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
138 at the level of terms and types. The notation is as follows:
141 \begin{tabular}{l|l|l}
142 & record terms & record types \\ \hline
143 fixed & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}} \\
144 schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
145 \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
149 \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
151 A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
152 \isa{a} and field \isa{y} of value \isa{b}. The corresponding
153 type is \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}}, assuming that \isa{{\isachardoublequote}a\ {\isacharcolon}{\isacharcolon}\ A{\isachardoublequote}}
154 and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
156 A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
157 \isa{x} and \isa{y} as before, but also possibly further fields
158 as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
159 of the syntax). The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
160 scheme is called the \emph{more part}. Logically it is just a free
161 variable, which is occasionally referred to as ``row variable'' in
162 the literature. The more part of a record scheme may be
163 instantiated by zero or more further components. For example, the
164 previous scheme may get instantiated to \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ z\ {\isacharequal}\ c{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isacharprime}{\isasymrparr}{\isachardoublequote}}, where \isa{m{\isacharprime}} refers to a different more part.
165 Fixed records are special instances of record schemes, where
166 ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
167 element. In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
168 for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
170 \medskip Two key observations make extensible records in a simply
171 typed language like HOL work out:
175 \item the more part is internalized, as a free term or type
178 \item field names are externalized, they cannot be accessed within
179 the logic as first-class values.
183 \medskip In Isabelle/HOL record types have to be defined explicitly,
184 fixing their field names and types, and their (optional) parent
185 record. Afterwards, records may be formed using above syntax, while
186 obeying the canonical order of fields as given by their declaration.
187 The record package provides several standard operations like
188 selectors and updates. The common setup for various generic proof
189 tools enable succinct reasoning patterns. See also the Isabelle/HOL
190 tutorial \cite{isabelle-hol-book} for further instructions on using
191 records in practice.%
195 \isamarkupsubsection{Record specifications%
199 \begin{isamarkuptext}%
200 \begin{matharray}{rcl}
201 \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
205 'record' typespecsorts '=' (type '+')? (constdecl +)
211 \item \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}\ {\isacharplus}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} defines extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
212 derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
213 field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
215 The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
216 covered by the (distinct) parameters \isa{{\isachardoublequote}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isachardoublequote}}. Type constructor \isa{t} has to be new, while \isa{{\isasymtau}} needs to specify an instance of an existing record type. At
217 least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
218 Basically, field names need to belong to a unique record. This is
219 not a real restriction in practice, since fields are qualified by
220 the record name internally.
222 The parent record specification \isa{{\isasymtau}} is optional; if omitted
223 \isa{t} becomes a root record. The hierarchy of all records
224 declared within a theory context forms a forest structure, i.e.\ a
225 set of trees starting with a root record each. There is no way to
226 merge multiple parent records!
228 For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
229 type abbreviation for the fixed record type \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}}, likewise is \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharcomma}\ {\isasymzeta}{\isacharparenright}\ t{\isacharunderscore}scheme{\isachardoublequote}} made an abbreviation for
230 \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}}.
236 \isamarkupsubsection{Record operations%
240 \begin{isamarkuptext}%
241 Any record definition of the form presented above produces certain
242 standard operations. Selectors and updates are provided for any
243 field, including the improper one ``\isa{more}''. There are also
244 cumulative record constructor functions. To simplify the
245 presentation below, we assume for now that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is a root record with fields \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}}.
247 \medskip \textbf{Selectors} and \textbf{updates} are available for
248 any field (including ``\isa{more}''):
250 \begin{matharray}{lll}
251 \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
252 \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
255 There is special syntax for application of updates: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} abbreviates term \isa{{\isachardoublequote}x{\isacharunderscore}update\ a\ r{\isachardoublequote}}. Further notation for
256 repeated updates is also available: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isasymlparr}y\ {\isacharcolon}{\isacharequal}\ b{\isasymrparr}{\isasymlparr}z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}} may be written \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}. Note that
257 because of postfix notation the order of fields shown here is
258 reverse than in the actual term. Since repeated updates are just
259 function applications, fields may be freely permuted in \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}, as far as logical equality is concerned.
260 Thus commutativity of independent updates can be proven within the
261 logic for any two fields, but not as a general theorem.
263 \medskip The \textbf{make} operation provides a cumulative record
264 constructor function:
266 \begin{matharray}{lll}
267 \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
270 \medskip We now reconsider the case of non-root records, which are
271 derived of some parent. In general, the latter may depend on
272 another parent as well, resulting in a list of \emph{ancestor
273 records}. Appending the lists of fields of all ancestors results in
274 a certain field prefix. The record package automatically takes care
275 of this by lifting operations over this context of ancestor fields.
276 Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
277 fields \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isachardoublequote}},
278 the above record operations will get the following types:
282 \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
283 \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
284 \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymrho}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymrho}\isactrlsub k\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
288 \noindent Some further operations address the extension aspect of a
289 derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
290 record fragment consisting of exactly the new fields introduced here
291 (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
292 takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
296 \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
297 \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymzeta}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
298 \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
302 \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
307 \isamarkupsubsection{Derived rules and proof tools%
311 \begin{isamarkuptext}%
312 The record package proves several results internally, declaring
313 these facts to appropriate proof tools. This enables users to
314 reason about record structures quite conveniently. Assume that
315 \isa{t} is a record type as specified above.
319 \item Standard conversions for selectors or updates applied to
320 record constructor terms are made part of the default Simplifier
321 context; thus proofs by reduction of basic operations merely require
322 the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments. These rules
323 are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
325 \item Selectors applied to updated records are automatically reduced
326 by an internal simplification procedure, which is also part of the
327 standard Simplifier setup.
329 \item Inject equations of a form analogous to \isa{{\isachardoublequote}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x{\isacharprime}{\isacharcomma}\ y{\isacharprime}{\isacharparenright}\ {\isasymequiv}\ x\ {\isacharequal}\ x{\isacharprime}\ {\isasymand}\ y\ {\isacharequal}\ y{\isacharprime}{\isachardoublequote}} are declared to the Simplifier and Classical
330 Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules. These rules are available as
331 \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
333 \item The introduction rule for record equality analogous to \isa{{\isachardoublequote}x\ r\ {\isacharequal}\ x\ r{\isacharprime}\ {\isasymLongrightarrow}\ y\ r\ {\isacharequal}\ y\ r{\isacharprime}\ {\isasymdots}\ {\isasymLongrightarrow}\ r\ {\isacharequal}\ r{\isacharprime}{\isachardoublequote}} is declared to the Simplifier,
334 and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
335 The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
337 \item Representations of arbitrary record expressions as canonical
338 constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name,
339 \secref{sec:cases-induct}). Several variations are available, for
340 fixed records, record schemes, more parts etc.
342 The generic proof methods are sufficiently smart to pick the most
343 sensible rule according to the type of the indicated record
344 expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
346 \item The derived record operations \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} are \emph{not}
347 treated automatically, but usually need to be expanded by hand,
348 using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
354 \isamarkupsection{Datatypes \label{sec:hol-datatype}%
358 \begin{isamarkuptext}%
359 \begin{matharray}{rcl}
360 \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
361 \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
365 'datatype' (dtspec + 'and')
367 'rep\_datatype' ('(' (name +) ')')? (term +)
370 dtspec: parname? typespec mixfix? '=' (cons + '|')
372 cons: name ( type * ) mixfix?
377 \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
380 \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as
381 inductive ones, generating the standard infrastructure of derived
382 concepts (primitive recursion etc.).
386 The induction and exhaustion theorems generated provide case names
387 according to the constructors involved, while parameters are named
388 after the types (see also \secref{sec:cases-induct}).
390 See \cite{isabelle-HOL} for more details on datatypes, but beware of
391 the old-style theory syntax being used there! Apart from proper
392 proof methods for case-analysis and induction, there are also
393 emulations of ML tactics \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} available, see \secref{sec:hol-induct-tac}; these admit
394 to refer directly to the internal structure of subgoals (including
395 internally bound parameters).%
399 \isamarkupsection{Recursive functions \label{sec:recursion}%
403 \begin{isamarkuptext}%
404 \begin{matharray}{rcl}
405 \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
406 \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
407 \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
408 \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
412 'primrec' target? fixes 'where' equations
414 equations: (thmdecl? prop + '|')
416 ('fun' | 'function') target? functionopts? fixes 'where' clauses
418 clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
420 functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
422 'termination' ( term )?
427 \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
428 functions over datatypes, see also \cite{isabelle-HOL}.
430 \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
431 wellfounded recursion. A detailed description with examples can be
432 found in \cite{isabelle-function}. The function is specified by a
433 set of (possibly conditional) recursive equations with arbitrary
434 pattern matching. The command generates proof obligations for the
435 completeness and the compatibility of patterns.
437 The defined function is considered partial, and the resulting
438 simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
439 (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
440 predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
441 command can then be used to establish that the function is total.
443 \item \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} is a shorthand notation for ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by automated
444 proof attempts regarding pattern matching and termination. See
445 \cite{isabelle-function} for further details.
447 \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
448 termination proof for the previously defined function \isa{f}. If
449 this is omitted, the command refers to the most recent function
450 definition. After the proof is closed, the recursive equations and
451 the induction principle is established.
455 Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
457 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule \isa{{\isachardoublequote}c{\isachardot}induct{\isachardoublequote}} (where \isa{c} is the name of the function definition)
458 refers to a specific induction rule, with parameters named according
459 to the user-specified equations. Cases are numbered (starting from 1).
461 For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the induction principle coincides
462 with structural recursion on the datatype the recursion is carried
465 The equations provided by these packages may be referred later as
466 theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
467 name of the functions defined. Individual equations may be named
470 The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
475 \item \isa{sequential} enables a preprocessor which disambiguates
476 overlapping patterns by making them mutually disjoint. Earlier
477 equations take precedence over later ones. This allows to give the
478 specification in a format very similar to functional programming.
479 Note that the resulting simplification and induction rules
480 correspond to the transformed specification, not the one given
481 originally. This usually means that each equation given by the user
482 may result in several theorems. Also note that this automatic
483 transformation only works for ML-style datatype patterns.
485 \item \isa{domintros} enables the automated generation of
486 introduction rules for the domain predicate. While mostly not
487 needed, they can be helpful in some proofs about partial functions.
489 \item \isa{tailrec} generates the unconstrained recursive
490 equations even without a termination proof, provided that the
491 function is tail-recursive. This currently only works
493 \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
494 (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
495 whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
501 \isamarkupsubsection{Proof methods related to recursive definitions%
505 \begin{isamarkuptext}%
506 \begin{matharray}{rcl}
507 \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
508 \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
509 \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
510 \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}}} & : & \isa{method} \\
516 'lexicographic\_order' ( clasimpmod * )
518 'size\_change' ( orders ( clasimpmod * ) )
520 orders: ( 'max' | 'min' | 'ms' ) *
525 \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
526 solve goals regarding the completeness of pattern matching, as
527 required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
528 \cite{isabelle-function}).
530 \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
531 proof using the relation \isa{R}. The resulting proof state will
532 contain goals expressing that \isa{R} is wellfounded, and that the
533 arguments of recursive calls decrease with respect to \isa{R}.
534 Usually, this method is used as the initial proof step of manual
537 \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
538 automated termination proof by searching for a lexicographic
539 combination of size measures on the arguments of the function. The
540 method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
541 which it uses internally to prove local descents. The same context
542 modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
543 \secref{sec:clasimp}.
545 In case of failure, extensive information is printed, which can help
546 to analyse the situation (cf.\ \cite{isabelle-function}).
548 \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}} also works on termination goals,
549 using a variation of the size-change principle, together with a
550 graph decomposition technique (see \cite{krauss_phd} for details).
551 Three kinds of orders are used internally: \isa{max}, \isa{min},
552 and \isa{ms} (multiset), which is only available when the theory
553 \isa{Multiset} is loaded. When no order kinds are given, they are
554 tried in order. The search for a termination proof uses SAT solving
557 For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}.
563 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
567 \begin{isamarkuptext}%
568 The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead.
570 \begin{matharray}{rcl}
571 \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\
572 \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
576 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
580 hints: '(' 'hints' ( recdefmod * ) ')'
582 recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
584 tc: nameref ('(' nat ')')?
590 \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
591 recursive functions (using the TFL package), see also
592 \cite{isabelle-HOL}. The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
593 TFL to recover from failed proof attempts, returning unfinished
594 results. The \isa{recdef{\isacharunderscore}simp}, \isa{recdef{\isacharunderscore}cong}, and \isa{recdef{\isacharunderscore}wf} hints refer to auxiliary rules to be used in the internal
595 automated proof process of TFL. Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
596 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
597 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
598 Classical reasoner (cf.\ \secref{sec:classical}).
600 \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
601 proof for leftover termination condition number \isa{i} (default
602 1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
605 Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
606 its internal proofs without manual intervention.
610 \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
611 globally, using the following attributes.
613 \begin{matharray}{rcl}
614 \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
615 \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
616 \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
620 ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
626 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
630 \begin{isamarkuptext}%
631 An \textbf{inductive definition} specifies the least predicate (or
632 set) \isa{R} closed under given rules: applying a rule to elements
633 of \isa{R} yields a result within \isa{R}. For example, a
634 structural operational semantics is an inductive definition of an
637 Dually, a \textbf{coinductive definition} specifies the greatest
638 predicate~/ set \isa{R} that is consistent with given rules: every
639 element of \isa{R} can be seen as arising by applying a rule to
640 elements of \isa{R}. An important example is using bisimulation
641 relations to formalise equivalence of processes and infinite data
644 \medskip The HOL package is related to the ZF one, which is
645 described in a separate paper,\footnote{It appeared in CADE
646 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
647 which you should refer to in case of difficulties. The package is
648 simpler than that of ZF thanks to implicit type-checking in HOL.
649 The types of the (co)inductive predicates (or sets) determine the
650 domain of the fixedpoint definition, and the package does not have
651 to use inference rules for type-checking.
653 \begin{matharray}{rcl}
654 \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
655 \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
656 \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
657 \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
658 \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
662 ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
663 ('where' clauses)? ('monos' thmrefs)?
665 clauses: (thmdecl? prop + '|')
667 'mono' (() | 'add' | 'del')
673 \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the
674 introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part. The
675 optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
676 (co)inductive predicates that remain fixed throughout the
677 definition. The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
678 \emph{monotonicity theorems}, which are required for each operator
679 applied to a recursive set in the introduction rules. There
680 \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
681 for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
683 \item \hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}} are wrappers for to the previous commands,
684 allowing the definition of (co)inductive sets.
686 \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules. These
687 rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
693 \isamarkupsubsection{Derived rules%
697 \begin{isamarkuptext}%
698 Each (co)inductive definition \isa{R} adds definitions to the
699 theory and also proves some theorems:
703 \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
704 theorems, for the recursive predicates (or sets). The rules are
705 also available individually, using the names given them in the
708 \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
710 \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
715 When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
716 defined simultaneously, the list of introduction rules is called
717 \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
718 called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
719 of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
723 \isamarkupsubsection{Monotonicity theorems%
727 \begin{isamarkuptext}%
728 Each theory contains a default set of theorems that are used in
729 monotonicity proofs. New rules can be added to this set via the
730 \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute. The HOL theory \isa{Inductive}
731 shows how this is done. In general, the following monotonicity
732 theorems may be added:
736 \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
737 monotonicity of inductive definitions whose introduction rules have
738 premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
740 \item Monotonicity theorems for logical operators, which are of the
741 general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}. For example, in
742 the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
744 \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
747 \item De Morgan style equations for reasoning about the ``polarity''
750 \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
751 \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
754 \item Equations for reducing complex operators to more primitive
755 ones whose monotonicity can easily be proved, e.g.
757 \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
758 \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
763 %FIXME: Example of an inductive definition%
767 \isamarkupsection{Arithmetic proof support%
771 \begin{isamarkuptext}%
772 \begin{matharray}{rcl}
773 \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
774 \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
775 \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
778 The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
779 (on types \isa{nat}, \isa{int}, \isa{real}). Any current
780 facts are inserted into the goal before running the procedure.
782 The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
783 always supplied to the arithmetic provers implicitly.
785 The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
786 rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
788 Note that a simpler (but faster) arithmetic prover is
789 already invoked by the Simplifier.%
793 \isamarkupsection{Intuitionistic proof search%
797 \begin{isamarkuptext}%
798 \begin{matharray}{rcl}
799 \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
803 'iprover' ( rulemod * )
807 The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
808 search, depending on specifically declared rules from the context,
809 or given as explicit arguments. Chained facts are inserted into the
810 goal before commencing proof search.
812 Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
813 \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
814 ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
815 applied aggressively (without considering back-tracking later).
816 Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
817 single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these). An
818 explicit weight annotation may be given as well; otherwise the
819 number of rule premises will be taken into account here.%
823 \isamarkupsection{Coherent Logic%
827 \begin{isamarkuptext}%
828 \begin{matharray}{rcl}
829 \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
837 The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
838 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
839 applications in confluence theory, lattice theory and projective
840 geometry. See \hyperlink{file.~~/src/HOL/ex/Coherent.thy}{\mbox{\isa{\isatt{{\isachartilde}{\isachartilde}{\isacharslash}src{\isacharslash}HOL{\isacharslash}ex{\isacharslash}Coherent{\isachardot}thy}}}} for some
845 \isamarkupsection{Checking and refuting propositions%
849 \begin{isamarkuptext}%
850 Identifying incorrect propositions usually involves evaluation of
851 particular assignments and systematic counter example search. This
852 is supported by the following commands.
854 \begin{matharray}{rcl}
855 \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
856 \indexdef{HOL}{command}{quickcheck}\hypertarget{command.HOL.quickcheck}{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}{\isachardoublequote}} \\
857 \indexdef{HOL}{command}{quickcheck\_params}\hypertarget{command.HOL.quickcheck-params}{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}}
861 'value' ( ( '[' name ']' ) ? ) modes? term
864 'quickcheck' ( ( '[' args ']' ) ? ) nat?
867 'quickcheck_params' ( ( '[' args ']' ) ? )
870 modes: '(' (name + ) ')'
873 args: ( name '=' value + ',' )
879 \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
880 term; optionally \isa{modes} can be specified, which are
881 appended to the current print mode (see also \cite{isabelle-ref}).
882 Internally, the evaluation is performed by registered evaluators,
883 which are invoked sequentially until a result is returned.
884 Alternatively a specific evaluator can be selected using square
885 brackets; available evaluators include \isa{nbe} for
886 \emph{normalization by evaluation} and \emph{code} for code
889 \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
890 counter examples using a series of arbitrary assignments for its
891 free variables; by default the first subgoal is tested, an other
892 can be selected explicitly using an optional goal index.
893 A number of configuration options are supported for
894 \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
898 \item[size] specifies the maximum size of the search space for
901 \item[iterations] sets how many sets of assignments are
902 generated for each particular size.
904 \item[no\_assms] specifies whether assumptions in
905 structured proofs should be ignored.
909 These option can be given within square brackets.
911 \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}} changes quickcheck
912 configuration options persitently.
918 \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
922 \begin{isamarkuptext}%
923 The following tools of Isabelle/HOL support cases analysis and
924 induction in unstructured tactic scripts; see also
925 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
927 \begin{matharray}{rcl}
928 \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
929 \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
930 \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
931 \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
935 'case\_tac' goalspec? term rule?
937 'induct\_tac' goalspec? (insts * 'and') rule?
939 'ind\_cases' (prop +) ('for' (name +)) ?
941 'inductive\_cases' (thmdecl? (prop +) + 'and')
944 rule: ('rule' ':' thmref)
950 \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit
951 to reason about inductive types. Rules are selected according to
952 the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
953 attributes, cf.\ \secref{sec:cases-induct}. The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
955 These unstructured tactics feature both goal addressing and dynamic
956 instantiation. Note that named rule cases are \emph{not} provided
957 as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
958 methods (see \secref{sec:cases-induct}). Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} does not handle structured rule
959 statements, only the compact object-logic conclusion of the subgoal
962 \item \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provide an interface to the internal \verb|mk_cases| operation. Rules are simplified in an unrestricted
965 While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
966 result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
967 for later use. The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} method allows to specify a list of variables that should
968 be generalized before applying the resulting rule.
974 \isamarkupsection{Executable code%
978 \begin{isamarkuptext}%
979 Isabelle/Pure provides two generic frameworks to support code
980 generation from executable specifications. Isabelle/HOL
981 instantiates these mechanisms in a way that is amenable to end-user
984 \medskip One framework generates code from functional programs
985 (including overloading using type classes) to SML \cite{SML}, OCaml
986 \cite{OCaml} and Haskell \cite{haskell-revised-report}.
987 Conceptually, code generation is split up in three steps:
988 \emph{selection} of code theorems, \emph{translation} into an
989 abstract executable view and \emph{serialization} to a specific
990 \emph{target language}. Inductive specifications can be executed
991 using the predicate compiler which operates within HOL.
992 See \cite{isabelle-codegen} for an introduction.
994 \begin{matharray}{rcl}
995 \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
996 \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
997 \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
998 \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
999 \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
1000 \indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}}} & : & \isa{attribute} \\
1001 \indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}}} & : & \isa{attribute} \\
1002 \indexdef{HOL}{command}{print\_codeproc}\hypertarget{command.HOL.print-codeproc}{\hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
1003 \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
1004 \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
1005 \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1006 \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1007 \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1008 \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1009 \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1010 \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1011 \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1012 \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1016 'export\_code' ( constexpr + ) \\
1017 ( ( 'in' target ( 'module\_name' string ) ? \\
1018 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1024 constexpr: ( const | 'name.*' | '*' )
1027 typeconstructor: nameref
1033 target: 'OCaml' | 'SML' | 'Haskell'
1039 'code\_abort' ( const + )
1042 'code\_datatype' ( const + )
1045 'code_inline' ( 'del' ) ?
1048 'code_post' ( 'del' ) ?
1051 'code\_thms' ( constexpr + ) ?
1054 'code\_deps' ( constexpr + ) ?
1057 'code\_const' (const + 'and') \\
1058 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1061 'code\_type' (typeconstructor + 'and') \\
1062 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1065 'code\_class' (class + 'and') \\
1066 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1069 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1070 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1073 'code\_reserved' target ( string + )
1076 'code\_monad' const const target
1079 'code\_include' target ( string ( string | '-') )
1082 'code\_modulename' target ( ( string string ) + )
1085 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1092 \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} generates code for a given list
1093 of constants in the specified target language(s). If no serialization
1094 instruction is given, only abstract code is generated internally.
1096 Constants may be specified by giving them literally, referring to
1097 all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
1098 available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
1100 By default, for each involved theory one corresponding name space
1101 module is generated. Alternativly, a module name may be specified
1102 after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
1103 placed in this module.
1105 For \emph{SML} and \emph{OCaml}, the file specification refers to a
1106 single file; for \emph{Haskell}, it refers to a whole directory,
1107 where code is generated in multiple files reflecting the module
1108 hierarchy. The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
1109 output. For \emph{SML}, omitting the file specification compiles
1110 code internally in the context of the current ML session.
1112 Serializers take an optional list of arguments in parentheses. For
1113 \emph{SML} and \emph{OCaml}, ``\isa{no{\isacharunderscore}signatures}`` omits
1114 explicit module signatures.
1116 For \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype
1119 \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
1120 ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code
1121 generation. Usually packages introducing code equations provide
1122 a reasonable default setup for selection.
1124 \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
1125 required to have a definition by means of code equations; if
1126 needed these are implemented by program abort instead.
1128 \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
1131 \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
1132 selected code equations and code generator datatypes.
1134 \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} declares (or with
1135 option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
1136 applied as rewrite rules to any code equation during
1139 \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}} declares (or with
1140 option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) theorems which are
1141 applied as rewrite rules to any result of an evaluation.
1143 \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} prints the setup
1144 of the code generator preprocessor.
1146 \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
1147 representing the corresponding program containing all given
1148 constants after preprocessing.
1150 \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
1151 theorems representing the corresponding program containing all given
1152 constants after preprocessing.
1154 \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
1155 with target-specific serializations; omitting a serialization
1156 deletes an existing serialization.
1158 \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
1159 constructors with target-specific serializations; omitting a
1160 serialization deletes an existing serialization.
1162 \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
1163 with target-specific class names; omitting a serialization deletes
1164 an existing serialization. This applies only to \emph{Haskell}.
1166 \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
1167 constructor / class instance relations as ``already present'' for a
1168 given target. Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
1169 ``already present'' declaration. This applies only to
1172 \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
1173 reserved for a given target, preventing it to be shadowed by any
1176 \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
1177 to generate monadic code for Haskell.
1179 \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
1180 (``include'') to generated code. A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
1181 will remove an already added ``include''.
1183 \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
1184 module name onto another.
1188 The other framework generates code from both functional and relational
1189 programs to SML. See \cite{isabelle-HOL} for further information
1190 (this actually covers the new-style theory format as well).
1192 \begin{matharray}{rcl}
1193 \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1194 \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1195 \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1196 \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
1197 \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
1201 ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
1202 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1203 'contains' ( ( name '=' term ) + | term + )
1206 modespec: '(' ( name * ) ')'
1209 'consts\_code' (codespec +)
1212 codespec: const template attachment ?
1215 'types\_code' (tycodespec +)
1218 tycodespec: name template attachment ?
1224 template: '(' string ')'
1227 attachment: 'attach' modespec ? verblbrace text verbrbrace
1233 \end{isamarkuptext}%
1236 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
1240 \begin{isamarkuptext}%
1241 \begin{matharray}{rcl}
1242 \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
1243 \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
1247 ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1249 decl: ((name ':')? term '(' 'overloaded' ')'?)
1254 \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
1255 goal stating the existence of terms with the properties specified to
1256 hold for the constants given in \isa{decls}. After finishing the
1257 proof, the theory will be augmented with definitions for the given
1258 constants, as well as with theorems stating the properties for these
1261 \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
1262 a goal stating the existence of terms with the properties specified
1263 to hold for the constants given in \isa{decls}. After finishing
1264 the proof, the theory will be augmented with axioms expressing the
1265 properties given in the first place.
1267 \item \isa{decl} declares a constant to be defined by the
1268 specification given. The definition for the constant \isa{c} is
1269 bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
1270 the declaration. Overloaded constants should be declared as such.
1274 Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} is to some extent a matter of style. \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by
1275 construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
1276 user has explicitly proven it to be safe. A practical issue must be
1277 considered, though: After introducing two constants with the same
1278 properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
1279 that the two constants are, in fact, equal. If this might be a
1280 problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
1281 \end{isamarkuptext}%
1289 \isacommand{end}\isamarkupfalse%
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