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 ('fun' | 'function') target? functionopts? fixes \\ 'where' equations
416 equations: (thmdecl? prop + '|')
418 functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
420 'termination' ( term )?
425 \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
426 functions over datatypes, see also \cite{isabelle-HOL}.
428 \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
429 wellfounded recursion. A detailed description with examples can be
430 found in \cite{isabelle-function}. The function is specified by a
431 set of (possibly conditional) recursive equations with arbitrary
432 pattern matching. The command generates proof obligations for the
433 completeness and the compatibility of patterns.
435 The defined function is considered partial, and the resulting
436 simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
437 (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
438 predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
439 command can then be used to establish that the function is total.
441 \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
442 proof attempts regarding pattern matching and termination. See
443 \cite{isabelle-function} for further details.
445 \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
446 termination proof for the previously defined function \isa{f}. If
447 this is omitted, the command refers to the most recent function
448 definition. After the proof is closed, the recursive equations and
449 the induction principle is established.
453 Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
455 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)
456 refers to a specific induction rule, with parameters named according
457 to the user-specified equations. Cases are numbered (starting from 1).
459 For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the induction principle coincides
460 with structural recursion on the datatype the recursion is carried
463 The equations provided by these packages may be referred later as
464 theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
465 name of the functions defined. Individual equations may be named
468 The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
473 \item \isa{sequential} enables a preprocessor which disambiguates
474 overlapping patterns by making them mutually disjoint. Earlier
475 equations take precedence over later ones. This allows to give the
476 specification in a format very similar to functional programming.
477 Note that the resulting simplification and induction rules
478 correspond to the transformed specification, not the one given
479 originally. This usually means that each equation given by the user
480 may result in several theorems. Also note that this automatic
481 transformation only works for ML-style datatype patterns.
483 \item \isa{domintros} enables the automated generation of
484 introduction rules for the domain predicate. While mostly not
485 needed, they can be helpful in some proofs about partial functions.
487 \item \isa{tailrec} generates the unconstrained recursive
488 equations even without a termination proof, provided that the
489 function is tail-recursive. This currently only works
491 \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
492 (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
493 whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
499 \isamarkupsubsection{Proof methods related to recursive definitions%
503 \begin{isamarkuptext}%
504 \begin{matharray}{rcl}
505 \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
506 \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
507 \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
508 \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}}} & : & \isa{method} \\
514 'lexicographic_order' ( clasimpmod * )
516 'size_change' ( orders ( clasimpmod * ) )
518 orders: ( 'max' | 'min' | 'ms' ) *
523 \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
524 solve goals regarding the completeness of pattern matching, as
525 required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
526 \cite{isabelle-function}).
528 \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
529 proof using the relation \isa{R}. The resulting proof state will
530 contain goals expressing that \isa{R} is wellfounded, and that the
531 arguments of recursive calls decrease with respect to \isa{R}.
532 Usually, this method is used as the initial proof step of manual
535 \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
536 automated termination proof by searching for a lexicographic
537 combination of size measures on the arguments of the function. The
538 method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
539 which it uses internally to prove local descents. The same context
540 modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
541 \secref{sec:clasimp}.
543 In case of failure, extensive information is printed, which can help
544 to analyse the situation (cf.\ \cite{isabelle-function}).
546 \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}} also works on termination goals,
547 using a variation of the size-change principle, together with a
548 graph decomposition technique (see \cite{krauss_phd} for details).
549 Three kinds of orders are used internally: \isa{max}, \isa{min},
550 and \isa{ms} (multiset), which is only available when the theory
551 \isa{Multiset} is loaded. When no order kinds are given, they are
552 tried in order. The search for a termination proof uses SAT solving
555 For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}.
561 \isamarkupsubsection{Functions with explicit partiality%
565 \begin{isamarkuptext}%
566 \begin{matharray}{rcl}
567 \indexdef{HOL}{command}{partial\_function}\hypertarget{command.HOL.partial-function}{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isacharunderscore}function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
568 \indexdef{HOL}{attribute}{partial\_function\_mono}\hypertarget{attribute.HOL.partial-function-mono}{\hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}}} & : & \isa{attribute} \\
572 'partial_function' target? '(' mode ')' fixes \\ 'where' thmdecl? prop
577 \item \hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isacharunderscore}function}}}} defines recursive
578 functions based on fixpoints in complete partial orders. No
579 termination proof is required from the user or constructed
580 internally. Instead, the possibility of non-termination is modelled
581 explicitly in the result type, which contains an explicit bottom
584 Pattern matching and mutual recursion are currently not supported.
585 Thus, the specification consists of a single function described by a
586 single recursive equation.
588 There are no fixed syntactic restrictions on the body of the
589 function, but the induced functional must be provably monotonic
590 wrt.\ the underlying order. The monotonicitity proof is performed
591 internally, and the definition is rejected when it fails. The proof
592 can be influenced by declaring hints using the
593 \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}} attribute.
595 The mandatory \isa{mode} argument specifies the mode of operation
596 of the command, which directly corresponds to a complete partial
597 order on the result type. By default, the following modes are
601 \item \isa{option} defines functions that map into the \isa{option} type. Here, the value \isa{None} is used to model a
602 non-terminating computation. Monotonicity requires that if \isa{None} is returned by a recursive call, then the overall result
603 must also be \isa{None}. This is best achieved through the use of
604 the monadic operator \isa{{\isachardoublequote}Option{\isachardot}bind{\isachardoublequote}}.
606 \item \isa{tailrec} defines functions with an arbitrary result
607 type and uses the slightly degenerated partial order where \isa{{\isachardoublequote}undefined{\isachardoublequote}} is the bottom element. Now, monotonicity requires that
608 if \isa{undefined} is returned by a recursive call, then the
609 overall result must also be \isa{undefined}. In practice, this is
610 only satisfied when each recursive call is a tail call, whose result
611 is directly returned. Thus, this mode of operation allows the
612 definition of arbitrary tail-recursive functions.
615 Experienced users may define new modes by instantiating the locale
616 \isa{{\isachardoublequote}partial{\isacharunderscore}function{\isacharunderscore}definitions{\isachardoublequote}} appropriately.
618 \item \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}} declares rules for
619 use in the internal monononicity proofs of partial function
626 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
630 \begin{isamarkuptext}%
631 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.
633 \begin{matharray}{rcl}
634 \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\
635 \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}} \\
639 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
643 hints: '(' 'hints' ( recdefmod * ) ')'
645 recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
647 tc: nameref ('(' nat ')')?
653 \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
654 recursive functions (using the TFL package), see also
655 \cite{isabelle-HOL}. The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
656 TFL to recover from failed proof attempts, returning unfinished
657 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
658 automated proof process of TFL. Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
659 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
660 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
661 Classical reasoner (cf.\ \secref{sec:classical}).
663 \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
664 proof for leftover termination condition number \isa{i} (default
665 1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
668 Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
669 its internal proofs without manual intervention.
673 \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
674 globally, using the following attributes.
676 \begin{matharray}{rcl}
677 \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
678 \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
679 \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
683 ('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del')
689 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
693 \begin{isamarkuptext}%
694 An \textbf{inductive definition} specifies the least predicate (or
695 set) \isa{R} closed under given rules: applying a rule to elements
696 of \isa{R} yields a result within \isa{R}. For example, a
697 structural operational semantics is an inductive definition of an
700 Dually, a \textbf{coinductive definition} specifies the greatest
701 predicate~/ set \isa{R} that is consistent with given rules: every
702 element of \isa{R} can be seen as arising by applying a rule to
703 elements of \isa{R}. An important example is using bisimulation
704 relations to formalise equivalence of processes and infinite data
707 \medskip The HOL package is related to the ZF one, which is
708 described in a separate paper,\footnote{It appeared in CADE
709 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
710 which you should refer to in case of difficulties. The package is
711 simpler than that of ZF thanks to implicit type-checking in HOL.
712 The types of the (co)inductive predicates (or sets) determine the
713 domain of the fixedpoint definition, and the package does not have
714 to use inference rules for type-checking.
716 \begin{matharray}{rcl}
717 \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}} \\
718 \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}} \\
719 \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}} \\
720 \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}} \\
721 \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
725 ('inductive' | 'inductive_set' | 'coinductive' | 'coinductive_set') target? fixes ('for' fixes)? \\
726 ('where' clauses)? ('monos' thmrefs)?
728 clauses: (thmdecl? prop + '|')
730 'mono' (() | 'add' | 'del')
736 \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the
737 introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part. The
738 optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
739 (co)inductive predicates that remain fixed throughout the
740 definition. The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
741 \emph{monotonicity theorems}, which are required for each operator
742 applied to a recursive set in the introduction rules. There
743 \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
744 for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
746 \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,
747 allowing the definition of (co)inductive sets.
749 \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules. These
750 rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
756 \isamarkupsubsection{Derived rules%
760 \begin{isamarkuptext}%
761 Each (co)inductive definition \isa{R} adds definitions to the
762 theory and also proves some theorems:
766 \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
767 theorems, for the recursive predicates (or sets). The rules are
768 also available individually, using the names given them in the
771 \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
773 \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
778 When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
779 defined simultaneously, the list of introduction rules is called
780 \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
781 called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
782 of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
786 \isamarkupsubsection{Monotonicity theorems%
790 \begin{isamarkuptext}%
791 Each theory contains a default set of theorems that are used in
792 monotonicity proofs. New rules can be added to this set via the
793 \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute. The HOL theory \isa{Inductive}
794 shows how this is done. In general, the following monotonicity
795 theorems may be added:
799 \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
800 monotonicity of inductive definitions whose introduction rules have
801 premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
803 \item Monotonicity theorems for logical operators, which are of the
804 general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}. For example, in
805 the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
807 \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}}}
810 \item De Morgan style equations for reasoning about the ``polarity''
813 \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
814 \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
817 \item Equations for reducing complex operators to more primitive
818 ones whose monotonicity can easily be proved, e.g.
820 \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
821 \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
826 %FIXME: Example of an inductive definition%
830 \isamarkupsection{Arithmetic proof support%
834 \begin{isamarkuptext}%
835 \begin{matharray}{rcl}
836 \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
837 \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
838 \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
841 The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
842 (on types \isa{nat}, \isa{int}, \isa{real}). Any current
843 facts are inserted into the goal before running the procedure.
845 The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
846 always supplied to the arithmetic provers implicitly.
848 The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
849 rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
851 Note that a simpler (but faster) arithmetic prover is
852 already invoked by the Simplifier.%
856 \isamarkupsection{Intuitionistic proof search%
860 \begin{isamarkuptext}%
861 \begin{matharray}{rcl}
862 \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
866 'iprover' ( rulemod * )
870 The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
871 search, depending on specifically declared rules from the context,
872 or given as explicit arguments. Chained facts are inserted into the
873 goal before commencing proof search.
875 Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
876 \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
877 ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
878 applied aggressively (without considering back-tracking later).
879 Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
880 single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these). An
881 explicit weight annotation may be given as well; otherwise the
882 number of rule premises will be taken into account here.%
886 \isamarkupsection{Coherent Logic%
890 \begin{isamarkuptext}%
891 \begin{matharray}{rcl}
892 \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
900 The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
901 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
902 applications in confluence theory, lattice theory and projective
903 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
908 \isamarkupsection{Checking and refuting propositions%
912 \begin{isamarkuptext}%
913 Identifying incorrect propositions usually involves evaluation of
914 particular assignments and systematic counter example search. This
915 is supported by the following commands.
917 \begin{matharray}{rcl}
918 \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}} \\
919 \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}} \\
920 \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}}
924 'value' ( ( '[' name ']' ) ? ) modes? term
927 'quickcheck' ( ( '[' args ']' ) ? ) nat?
930 'quickcheck_params' ( ( '[' args ']' ) ? )
933 modes: '(' (name + ) ')'
936 args: ( name '=' value + ',' )
942 \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
943 term; optionally \isa{modes} can be specified, which are
944 appended to the current print mode (see also \cite{isabelle-ref}).
945 Internally, the evaluation is performed by registered evaluators,
946 which are invoked sequentially until a result is returned.
947 Alternatively a specific evaluator can be selected using square
948 brackets; typical evaluators use the current set of code equations
949 to normalize and include \isa{simp} for fully symbolic evaluation
950 using the simplifier, \isa{nbe} for \emph{normalization by evaluation}
951 and \emph{code} for code generation in SML.
953 \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
954 counter examples using a series of arbitrary assignments for its
955 free variables; by default the first subgoal is tested, an other
956 can be selected explicitly using an optional goal index.
957 A number of configuration options are supported for
958 \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
962 \item[\isa{size}] specifies the maximum size of the search space
963 for assignment values.
965 \item[\isa{iterations}] sets how many sets of assignments are
966 generated for each particular size.
968 \item[\isa{no{\isacharunderscore}assms}] specifies whether assumptions in
969 structured proofs should be ignored.
971 \item[\isa{timeout}] sets the time limit in seconds.
973 \item[\isa{default{\isacharunderscore}type}] sets the type(s) generally used to
974 instantiate type variables.
976 \item[\isa{report}] if set quickcheck reports how many tests
977 fulfilled the preconditions.
979 \item[\isa{quiet}] if not set quickcheck informs about the
980 current size for assignment values.
982 \item[\isa{expect}] can be used to check if the user's
983 expectation was met (\isa{no{\isacharunderscore}expectation}, \isa{no{\isacharunderscore}counterexample}, or \isa{counterexample}).
985 \item[timeout] sets the time limit in seconds.
987 \item[default\_type] sets the type(s) generally used to instantiate
990 \item[report] if set quickcheck reports how many tests fulfilled
993 \item[quiet] if not set quickcheck informs about the current size
994 for assignment values.
996 \item[expect] can be used to check if the user's expectation was met
997 (no\_expectation, no\_counterexample, or counterexample)
1001 These option can be given within square brackets.
1003 \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}} changes quickcheck
1004 configuration options persitently.
1007 \end{isamarkuptext}%
1010 \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
1014 \begin{isamarkuptext}%
1015 The following tools of Isabelle/HOL support cases analysis and
1016 induction in unstructured tactic scripts; see also
1017 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
1019 \begin{matharray}{rcl}
1020 \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} \\
1021 \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} \\
1022 \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} \\
1023 \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}} \\
1027 'case_tac' goalspec? term rule?
1029 'induct_tac' goalspec? (insts * 'and') rule?
1031 'ind_cases' (prop +) ('for' (name +)) ?
1033 'inductive_cases' (thmdecl? (prop +) + 'and')
1036 rule: ('rule' ':' thmref)
1042 \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit
1043 to reason about inductive types. Rules are selected according to
1044 the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
1045 attributes, cf.\ \secref{sec:cases-induct}. The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
1047 These unstructured tactics feature both goal addressing and dynamic
1048 instantiation. Note that named rule cases are \emph{not} provided
1049 as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
1050 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
1051 statements, only the compact object-logic conclusion of the subgoal
1054 \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
1057 While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
1058 result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
1059 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
1060 be generalized before applying the resulting rule.
1063 \end{isamarkuptext}%
1066 \isamarkupsection{Executable code%
1070 \begin{isamarkuptext}%
1071 Isabelle/Pure provides two generic frameworks to support code
1072 generation from executable specifications. Isabelle/HOL
1073 instantiates these mechanisms in a way that is amenable to end-user
1076 \medskip One framework generates code from functional programs
1077 (including overloading using type classes) to SML \cite{SML}, OCaml
1078 \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
1079 \cite{scala-overview-tech-report}.
1080 Conceptually, code generation is split up in three steps:
1081 \emph{selection} of code theorems, \emph{translation} into an
1082 abstract executable view and \emph{serialization} to a specific
1083 \emph{target language}. Inductive specifications can be executed
1084 using the predicate compiler which operates within HOL.
1085 See \cite{isabelle-codegen} for an introduction.
1087 \begin{matharray}{rcl}
1088 \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}} \\
1089 \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
1090 \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}} \\
1091 \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}} \\
1092 \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}} \\
1093 \indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}}} & : & \isa{attribute} \\
1094 \indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}}} & : & \isa{attribute} \\
1095 \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}} \\
1096 \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}} \\
1097 \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}} \\
1098 \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}} \\
1099 \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}} \\
1100 \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}} \\
1101 \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}} \\
1102 \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}} \\
1103 \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}} \\
1104 \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}} \\
1105 \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}} \\
1106 \indexdef{HOL}{command}{code\_reflect}\hypertarget{command.HOL.code-reflect}{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isacharunderscore}reflect}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}}
1110 'export_code' ( constexpr + ) \\
1111 ( ( 'in' target ( 'module_name' string ) ? \\
1112 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1118 constexpr: ( const | 'name.*' | '*' )
1121 typeconstructor: nameref
1127 target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
1130 'code' ( 'del' | 'abstype' | 'abstract' ) ?
1133 'code_abort' ( const + )
1136 'code_datatype' ( const + )
1139 'code_inline' ( 'del' ) ?
1142 'code_post' ( 'del' ) ?
1145 'code_thms' ( constexpr + ) ?
1148 'code_deps' ( constexpr + ) ?
1151 'code_const' (const + 'and') \\
1152 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1155 'code_type' (typeconstructor + 'and') \\
1156 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1159 'code_class' (class + 'and') \\
1160 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1163 'code_instance' (( typeconstructor '::' class ) + 'and') \\
1164 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1167 'code_reserved' target ( string + )
1170 'code_monad' const const target
1173 'code_include' target ( string ( string | '-') )
1176 'code_modulename' target ( ( string string ) + )
1179 'code_reflect' string ( 'datatypes' ( string '=' ( string + '|' ) + 'and' ) ) ? \\
1180 ( 'functions' ( string + ) ) ? ( 'file' string ) ?
1183 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1190 \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} generates code for a given list
1191 of constants in the specified target language(s). If no
1192 serialization instruction is given, only abstract code is generated
1195 Constants may be specified by giving them literally, referring to
1196 all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
1197 available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
1199 By default, for each involved theory one corresponding name space
1200 module is generated. Alternativly, a module name may be specified
1201 after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
1202 placed in this module.
1204 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
1205 refers to a single file; for \emph{Haskell}, it refers to a whole
1206 directory, where code is generated in multiple files reflecting the
1207 module hierarchy. Omitting the file specification denotes standard
1210 Serializers take an optional list of arguments in parentheses. For
1211 \emph{SML} and \emph{OCaml}, ``\isa{no{\isacharunderscore}signatures}`` omits
1212 explicit module signatures.
1214 For \emph{Haskell} a module name prefix may be given using the
1215 ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a
1216 ``\verb|deriving (Read, Show)|'' clause to each appropriate
1217 datatype declaration.
1219 \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
1220 ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code generation.
1221 Usually packages introducing code equations provide a reasonable
1222 default setup for selection. Variants \isa{{\isachardoublequote}code\ abstype{\isachardoublequote}} and
1223 \isa{{\isachardoublequote}code\ abstract{\isachardoublequote}} declare abstract datatype certificates or
1224 code equations on abstract datatype representations respectively.
1226 \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
1227 required to have a definition by means of code equations; if needed
1228 these are implemented by program abort instead.
1230 \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
1233 \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
1234 selected code equations and code generator datatypes.
1236 \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} declares (or with option
1237 ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are applied as
1238 rewrite rules to any code equation during preprocessing.
1240 \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}} declares (or with option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) theorems which are applied as rewrite rules to any
1241 result of an evaluation.
1243 \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} prints the setup of the code
1244 generator preprocessor.
1246 \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
1247 representing the corresponding program containing all given
1248 constants after preprocessing.
1250 \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
1251 theorems representing the corresponding program containing all given
1252 constants after preprocessing.
1254 \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
1255 with target-specific serializations; omitting a serialization
1256 deletes an existing serialization.
1258 \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
1259 constructors with target-specific serializations; omitting a
1260 serialization deletes an existing serialization.
1262 \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
1263 with target-specific class names; omitting a serialization deletes
1264 an existing serialization. This applies only to \emph{Haskell}.
1266 \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
1267 constructor / class instance relations as ``already present'' for a
1268 given target. Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
1269 ``already present'' declaration. This applies only to
1272 \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
1273 reserved for a given target, preventing it to be shadowed by any
1276 \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
1277 to generate monadic code for Haskell.
1279 \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
1280 (``include'') to generated code. A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
1281 will remove an already added ``include''.
1283 \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
1284 module name onto another.
1286 \item \hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isacharunderscore}reflect}}}} without a ``\isa{{\isachardoublequote}file{\isachardoublequote}}''
1287 argument compiles code into the system runtime environment and
1288 modifies the code generator setup that future invocations of system
1289 runtime code generation referring to one of the ``\isa{{\isachardoublequote}datatypes{\isachardoublequote}}'' or ``\isa{{\isachardoublequote}functions{\isachardoublequote}}'' entities use these precompiled
1290 entities. With a ``\isa{{\isachardoublequote}file{\isachardoublequote}}'' argument, the corresponding code
1291 is generated into that specified file without modifying the code
1296 The other framework generates code from both functional and
1297 relational programs to SML. See \cite{isabelle-HOL} for further
1298 information (this actually covers the new-style theory format as
1301 \begin{matharray}{rcl}
1302 \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}} \\
1303 \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}} \\
1304 \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}} \\
1305 \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}} \\
1306 \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
1310 ( 'code_module' | 'code_library' ) modespec ? name ? \\
1311 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1312 'contains' ( ( name '=' term ) + | term + )
1315 modespec: '(' ( name * ) ')'
1318 'consts_code' (codespec +)
1321 codespec: const template attachment ?
1324 'types_code' (tycodespec +)
1327 tycodespec: name template attachment ?
1333 template: '(' string ')'
1336 attachment: 'attach' modespec ? verblbrace text verbrbrace
1342 \end{isamarkuptext}%
1345 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
1349 \begin{isamarkuptext}%
1350 \begin{matharray}{rcl}
1351 \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}} \\
1352 \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}} \\
1356 ('specification' | 'ax_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1358 decl: ((name ':')? term '(' 'overloaded' ')'?)
1363 \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
1364 goal stating the existence of terms with the properties specified to
1365 hold for the constants given in \isa{decls}. After finishing the
1366 proof, the theory will be augmented with definitions for the given
1367 constants, as well as with theorems stating the properties for these
1370 \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
1371 a goal stating the existence of terms with the properties specified
1372 to hold for the constants given in \isa{decls}. After finishing
1373 the proof, the theory will be augmented with axioms expressing the
1374 properties given in the first place.
1376 \item \isa{decl} declares a constant to be defined by the
1377 specification given. The definition for the constant \isa{c} is
1378 bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
1379 the declaration. Overloaded constants should be declared as such.
1383 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
1384 construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
1385 user has explicitly proven it to be safe. A practical issue must be
1386 considered, though: After introducing two constants with the same
1387 properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
1388 that the two constants are, in fact, equal. If this might be a
1389 problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
1390 \end{isamarkuptext}%
1398 \isacommand{end}\isamarkupfalse%
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