1.1 --- a/doc-src/IsarRef/Thy/document/HOL_Specific.tex Thu May 08 12:27:19 2008 +0200
1.2 +++ b/doc-src/IsarRef/Thy/document/HOL_Specific.tex Thu May 08 12:29:18 2008 +0200
1.3 @@ -11,18 +11,1153 @@
1.4 \isatagtheory
1.5 \isacommand{theory}\isamarkupfalse%
1.6 \ HOL{\isacharunderscore}Specific\isanewline
1.7 -\isakeyword{imports}\ HOL\isanewline
1.8 -\isakeyword{begin}\isanewline
1.9 -\isanewline
1.10 +\isakeyword{imports}\ Main\isanewline
1.11 +\isakeyword{begin}%
1.12 +\endisatagtheory
1.13 +{\isafoldtheory}%
1.14 +%
1.15 +\isadelimtheory
1.16 +%
1.17 +\endisadelimtheory
1.18 +%
1.19 +\isamarkupchapter{HOL specific elements \label{ch:logics}%
1.20 +}
1.21 +\isamarkuptrue%
1.22 +%
1.23 +\isamarkupsection{Primitive types \label{sec:hol-typedef}%
1.24 +}
1.25 +\isamarkuptrue%
1.26 +%
1.27 +\begin{isamarkuptext}%
1.28 +\begin{matharray}{rcl}
1.29 + \indexdef{HOL}{command}{typedecl}\mbox{\isa{\isacommand{typedecl}}} & : & \isartrans{theory}{theory} \\
1.30 + \indexdef{HOL}{command}{typedef}\mbox{\isa{\isacommand{typedef}}} & : & \isartrans{theory}{proof(prove)} \\
1.31 + \end{matharray}
1.32 +
1.33 + \begin{rail}
1.34 + 'typedecl' typespec infix?
1.35 + ;
1.36 + 'typedef' altname? abstype '=' repset
1.37 + ;
1.38 +
1.39 + altname: '(' (name | 'open' | 'open' name) ')'
1.40 + ;
1.41 + abstype: typespec infix?
1.42 + ;
1.43 + repset: term ('morphisms' name name)?
1.44 + ;
1.45 + \end{rail}
1.46 +
1.47 + \begin{descr}
1.48 +
1.49 + \item [\mbox{\isa{\isacommand{typedecl}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}}] is similar to the original \mbox{\isa{\isacommand{typedecl}}} of
1.50 + Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
1.51 + arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an
1.52 + actual HOL type constructor. %FIXME check, update
1.53 +
1.54 + \item [\mbox{\isa{\isacommand{typedef}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}] sets up a goal stating non-emptiness of the set \isa{A}.
1.55 + After finishing the proof, the theory will be augmented by a
1.56 + Gordon/HOL-style type definition, which establishes a bijection
1.57 + between the representing set \isa{A} and the new type \isa{t}.
1.58 +
1.59 + Technically, \mbox{\isa{\isacommand{typedef}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
1.60 + name may be given in parentheses). The injection from type to set
1.61 + is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
1.62 + changed via an explicit \mbox{\isa{\isakeyword{morphisms}}} declaration).
1.63 +
1.64 + 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
1.65 + corresponding injection/surjection pair (in both directions). Rules
1.66 + \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
1.67 + more convenient view on the injectivity part, suitable for automated
1.68 + proof tools (e.g.\ in \mbox{\isa{simp}} or \mbox{\isa{iff}} declarations).
1.69 + Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views on
1.70 + surjectivity; these are already declared as set or type rules for
1.71 + the generic \mbox{\isa{cases}} and \mbox{\isa{induct}} methods.
1.72 +
1.73 + An alternative name may be specified in parentheses; the default is
1.74 + to use \isa{t} as indicated before. The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
1.75 + declaration suppresses a separate constant definition for the
1.76 + representing set.
1.77 +
1.78 + \end{descr}
1.79 +
1.80 + Note that raw type declarations are rarely used in practice; the
1.81 + main application is with experimental (or even axiomatic!) theory
1.82 + fragments. Instead of primitive HOL type definitions, user-level
1.83 + theories usually refer to higher-level packages such as \mbox{\isa{\isacommand{record}}} (see \secref{sec:hol-record}) or \mbox{\isa{\isacommand{datatype}}} (see \secref{sec:hol-datatype}).%
1.84 +\end{isamarkuptext}%
1.85 +\isamarkuptrue%
1.86 +%
1.87 +\isamarkupsection{Adhoc tuples%
1.88 +}
1.89 +\isamarkuptrue%
1.90 +%
1.91 +\begin{isamarkuptext}%
1.92 +\begin{matharray}{rcl}
1.93 + \mbox{\isa{split{\isacharunderscore}format}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
1.94 + \end{matharray}
1.95 +
1.96 + \begin{rail}
1.97 + 'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
1.98 + ;
1.99 + \end{rail}
1.100 +
1.101 + \begin{descr}
1.102 +
1.103 + \item [\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
1.104 + low-level tuple types into canonical form as specified by the
1.105 + arguments given; the \isa{i}-th collection of arguments refers to
1.106 + occurrences in premise \isa{i} of the rule. The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} arguments in function
1.107 + applications to be represented canonically according to their tuple
1.108 + type structure.
1.109 +
1.110 + Note that these operations tend to invent funny names for new local
1.111 + parameters to be introduced.
1.112 +
1.113 + \end{descr}%
1.114 +\end{isamarkuptext}%
1.115 +\isamarkuptrue%
1.116 +%
1.117 +\isamarkupsection{Records \label{sec:hol-record}%
1.118 +}
1.119 +\isamarkuptrue%
1.120 +%
1.121 +\begin{isamarkuptext}%
1.122 +In principle, records merely generalize the concept of tuples, where
1.123 + components may be addressed by labels instead of just position. The
1.124 + logical infrastructure of records in Isabelle/HOL is slightly more
1.125 + advanced, though, supporting truly extensible record schemes. This
1.126 + admits operations that are polymorphic with respect to record
1.127 + extension, yielding ``object-oriented'' effects like (single)
1.128 + inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
1.129 + details on object-oriented verification and record subtyping in HOL.%
1.130 +\end{isamarkuptext}%
1.131 +\isamarkuptrue%
1.132 +%
1.133 +\isamarkupsubsection{Basic concepts%
1.134 +}
1.135 +\isamarkuptrue%
1.136 +%
1.137 +\begin{isamarkuptext}%
1.138 +Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
1.139 + at the level of terms and types. The notation is as follows:
1.140 +
1.141 + \begin{center}
1.142 + \begin{tabular}{l|l|l}
1.143 + & record terms & record types \\ \hline
1.144 + 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}} \\
1.145 + schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
1.146 + \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
1.147 + \end{tabular}
1.148 + \end{center}
1.149 +
1.150 + \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
1.151 +
1.152 + A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
1.153 + \isa{a} and field \isa{y} of value \isa{b}. The corresponding
1.154 + 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}}
1.155 + and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
1.156 +
1.157 + A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
1.158 + \isa{x} and \isa{y} as before, but also possibly further fields
1.159 + as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
1.160 + of the syntax). The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
1.161 + scheme is called the \emph{more part}. Logically it is just a free
1.162 + variable, which is occasionally referred to as ``row variable'' in
1.163 + the literature. The more part of a record scheme may be
1.164 + instantiated by zero or more further components. For example, the
1.165 + 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}{\isachardoublequote}}, where \isa{m{\isacharprime}} refers to a different more part.
1.166 + Fixed records are special instances of record schemes, where
1.167 + ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
1.168 + element. In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
1.169 + for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
1.170 +
1.171 + \medskip Two key observations make extensible records in a simply
1.172 + typed language like HOL work out:
1.173 +
1.174 + \begin{enumerate}
1.175 +
1.176 + \item the more part is internalized, as a free term or type
1.177 + variable,
1.178 +
1.179 + \item field names are externalized, they cannot be accessed within the logic
1.180 + as first-class values.
1.181 +
1.182 + \end{enumerate}
1.183 +
1.184 + \medskip In Isabelle/HOL record types have to be defined explicitly,
1.185 + fixing their field names and types, and their (optional) parent
1.186 + record. Afterwards, records may be formed using above syntax, while
1.187 + obeying the canonical order of fields as given by their declaration.
1.188 + The record package provides several standard operations like
1.189 + selectors and updates. The common setup for various generic proof
1.190 + tools enable succinct reasoning patterns. See also the Isabelle/HOL
1.191 + tutorial \cite{isabelle-hol-book} for further instructions on using
1.192 + records in practice.%
1.193 +\end{isamarkuptext}%
1.194 +\isamarkuptrue%
1.195 +%
1.196 +\isamarkupsubsection{Record specifications%
1.197 +}
1.198 +\isamarkuptrue%
1.199 +%
1.200 +\begin{isamarkuptext}%
1.201 +\begin{matharray}{rcl}
1.202 + \indexdef{HOL}{command}{record}\mbox{\isa{\isacommand{record}}} & : & \isartrans{theory}{theory} \\
1.203 + \end{matharray}
1.204 +
1.205 + \begin{rail}
1.206 + 'record' typespec '=' (type '+')? (constdecl +)
1.207 + ;
1.208 + \end{rail}
1.209 +
1.210 + \begin{descr}
1.211 +
1.212 + \item [\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
1.213 + extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
1.214 + derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
1.215 + field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
1.216 +
1.217 + The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
1.218 + 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
1.219 + least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
1.220 + Basically, field names need to belong to a unique record. This is
1.221 + not a real restriction in practice, since fields are qualified by
1.222 + the record name internally.
1.223 +
1.224 + The parent record specification \isa{{\isasymtau}} is optional; if omitted
1.225 + \isa{t} becomes a root record. The hierarchy of all records
1.226 + declared within a theory context forms a forest structure, i.e.\ a
1.227 + set of trees starting with a root record each. There is no way to
1.228 + merge multiple parent records!
1.229 +
1.230 + For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
1.231 + 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
1.232 + \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}}.
1.233 +
1.234 + \end{descr}%
1.235 +\end{isamarkuptext}%
1.236 +\isamarkuptrue%
1.237 +%
1.238 +\isamarkupsubsection{Record operations%
1.239 +}
1.240 +\isamarkuptrue%
1.241 +%
1.242 +\begin{isamarkuptext}%
1.243 +Any record definition of the form presented above produces certain
1.244 + standard operations. Selectors and updates are provided for any
1.245 + field, including the improper one ``\isa{more}''. There are also
1.246 + cumulative record constructor functions. To simplify the
1.247 + 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}}.
1.248 +
1.249 + \medskip \textbf{Selectors} and \textbf{updates} are available for
1.250 + any field (including ``\isa{more}''):
1.251 +
1.252 + \begin{matharray}{lll}
1.253 + \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \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}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
1.254 + \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\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}\ {\isasymRightarrow}\ {\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}} \\
1.255 + \end{matharray}
1.256 +
1.257 + 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
1.258 + 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
1.259 + because of postfix notation the order of fields shown here is
1.260 + reverse than in the actual term. Since repeated updates are just
1.261 + 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.
1.262 + Thus commutativity of independent updates can be proven within the
1.263 + logic for any two fields, but not as a general theorem.
1.264 +
1.265 + \medskip The \textbf{make} operation provides a cumulative record
1.266 + constructor function:
1.267 +
1.268 + \begin{matharray}{lll}
1.269 + \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}} \\
1.270 + \end{matharray}
1.271 +
1.272 + \medskip We now reconsider the case of non-root records, which are
1.273 + derived of some parent. In general, the latter may depend on
1.274 + another parent as well, resulting in a list of \emph{ancestor
1.275 + records}. Appending the lists of fields of all ancestors results in
1.276 + a certain field prefix. The record package automatically takes care
1.277 + of this by lifting operations over this context of ancestor fields.
1.278 + Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
1.279 + 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}},
1.280 + the above record operations will get the following types:
1.281 +
1.282 + \begin{matharray}{lll}
1.283 + \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
1.284 + \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}} \\
1.285 + \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}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} \\
1.286 + \end{matharray}
1.287 + \noindent
1.288 +
1.289 + \medskip Some further operations address the extension aspect of a
1.290 + derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
1.291 + record fragment consisting of exactly the new fields introduced here
1.292 + (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
1.293 + takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
1.294 +
1.295 + \begin{matharray}{lll}
1.296 + \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} \\
1.297 + \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}\ {\isasymRightarrow}\ {\isasymzeta}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}} \\
1.298 + \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}} \\
1.299 + \end{matharray}
1.300 +
1.301 + \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
1.302 + for root records.%
1.303 +\end{isamarkuptext}%
1.304 +\isamarkuptrue%
1.305 +%
1.306 +\isamarkupsubsection{Derived rules and proof tools%
1.307 +}
1.308 +\isamarkuptrue%
1.309 +%
1.310 +\begin{isamarkuptext}%
1.311 +The record package proves several results internally, declaring
1.312 + these facts to appropriate proof tools. This enables users to
1.313 + reason about record structures quite conveniently. Assume that
1.314 + \isa{t} is a record type as specified above.
1.315 +
1.316 + \begin{enumerate}
1.317 +
1.318 + \item Standard conversions for selectors or updates applied to
1.319 + record constructor terms are made part of the default Simplifier
1.320 + context; thus proofs by reduction of basic operations merely require
1.321 + the \mbox{\isa{simp}} method without further arguments. These rules
1.322 + are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
1.323 +
1.324 + \item Selectors applied to updated records are automatically reduced
1.325 + by an internal simplification procedure, which is also part of the
1.326 + standard Simplifier setup.
1.327 +
1.328 + \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
1.329 + Reasoner as \mbox{\isa{iff}} rules. These rules are available as
1.330 + \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
1.331 +
1.332 + \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,
1.333 + and as the basic rule context as ``\mbox{\isa{intro}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
1.334 + The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
1.335 +
1.336 + \item Representations of arbitrary record expressions as canonical
1.337 + constructor terms are provided both in \mbox{\isa{cases}} and \mbox{\isa{induct}} format (cf.\ the generic proof methods of the same name,
1.338 + \secref{sec:cases-induct}). Several variations are available, for
1.339 + fixed records, record schemes, more parts etc.
1.340 +
1.341 + The generic proof methods are sufficiently smart to pick the most
1.342 + sensible rule according to the type of the indicated record
1.343 + expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
1.344 +
1.345 + \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}
1.346 + treated automatically, but usually need to be expanded by hand,
1.347 + using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
1.348 +
1.349 + \end{enumerate}%
1.350 +\end{isamarkuptext}%
1.351 +\isamarkuptrue%
1.352 +%
1.353 +\isamarkupsection{Datatypes \label{sec:hol-datatype}%
1.354 +}
1.355 +\isamarkuptrue%
1.356 +%
1.357 +\begin{isamarkuptext}%
1.358 +\begin{matharray}{rcl}
1.359 + \indexdef{HOL}{command}{datatype}\mbox{\isa{\isacommand{datatype}}} & : & \isartrans{theory}{theory} \\
1.360 + \indexdef{HOL}{command}{rep-datatype}\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
1.361 + \end{matharray}
1.362 +
1.363 + \begin{rail}
1.364 + 'datatype' (dtspec + 'and')
1.365 + ;
1.366 + 'rep\_datatype' (name *) dtrules
1.367 + ;
1.368 +
1.369 + dtspec: parname? typespec infix? '=' (cons + '|')
1.370 + ;
1.371 + cons: name (type *) mixfix?
1.372 + ;
1.373 + dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
1.374 + \end{rail}
1.375 +
1.376 + \begin{descr}
1.377 +
1.378 + \item [\mbox{\isa{\isacommand{datatype}}}] defines inductive datatypes in
1.379 + HOL.
1.380 +
1.381 + \item [\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}] represents existing types as
1.382 + inductive ones, generating the standard infrastructure of derived
1.383 + concepts (primitive recursion etc.).
1.384 +
1.385 + \end{descr}
1.386 +
1.387 + The induction and exhaustion theorems generated provide case names
1.388 + according to the constructors involved, while parameters are named
1.389 + after the types (see also \secref{sec:cases-induct}).
1.390 +
1.391 + See \cite{isabelle-HOL} for more details on datatypes, but beware of
1.392 + the old-style theory syntax being used there! Apart from proper
1.393 + proof methods for case-analysis and induction, there are also
1.394 + emulations of ML tactics \mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}} available, see \secref{sec:hol-induct-tac}; these admit
1.395 + to refer directly to the internal structure of subgoals (including
1.396 + internally bound parameters).%
1.397 +\end{isamarkuptext}%
1.398 +\isamarkuptrue%
1.399 +%
1.400 +\isamarkupsection{Recursive functions \label{sec:recursion}%
1.401 +}
1.402 +\isamarkuptrue%
1.403 +%
1.404 +\begin{isamarkuptext}%
1.405 +\begin{matharray}{rcl}
1.406 + \indexdef{HOL}{command}{primrec}\mbox{\isa{\isacommand{primrec}}} & : & \isarkeep{local{\dsh}theory} \\
1.407 + \indexdef{HOL}{command}{fun}\mbox{\isa{\isacommand{fun}}} & : & \isarkeep{local{\dsh}theory} \\
1.408 + \indexdef{HOL}{command}{function}\mbox{\isa{\isacommand{function}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
1.409 + \indexdef{HOL}{command}{termination}\mbox{\isa{\isacommand{termination}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
1.410 + \end{matharray}
1.411 +
1.412 + \railalias{funopts}{function\_opts} %FIXME ??
1.413 +
1.414 + \begin{rail}
1.415 + 'primrec' target? fixes 'where' equations
1.416 + ;
1.417 + equations: (thmdecl? prop + '|')
1.418 + ;
1.419 + ('fun' | 'function') (funopts)? fixes 'where' clauses
1.420 + ;
1.421 + clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
1.422 + ;
1.423 + funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
1.424 + 'default' term) + ',') ')'
1.425 + ;
1.426 + 'termination' ( term )?
1.427 + \end{rail}
1.428 +
1.429 + \begin{descr}
1.430 +
1.431 + \item [\mbox{\isa{\isacommand{primrec}}}] defines primitive recursive
1.432 + functions over datatypes, see also \cite{isabelle-HOL}.
1.433 +
1.434 + \item [\mbox{\isa{\isacommand{function}}}] defines functions by general
1.435 + wellfounded recursion. A detailed description with examples can be
1.436 + found in \cite{isabelle-function}. The function is specified by a
1.437 + set of (possibly conditional) recursive equations with arbitrary
1.438 + pattern matching. The command generates proof obligations for the
1.439 + completeness and the compatibility of patterns.
1.440 +
1.441 + The defined function is considered partial, and the resulting
1.442 + simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
1.443 + (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
1.444 + predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \mbox{\isa{\isacommand{termination}}}
1.445 + command can then be used to establish that the function is total.
1.446 +
1.447 + \item [\mbox{\isa{\isacommand{fun}}}] is a shorthand notation for
1.448 + ``\mbox{\isa{\isacommand{function}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
1.449 + automated proof attempts regarding pattern matching and termination.
1.450 + See \cite{isabelle-function} for further details.
1.451 +
1.452 + \item [\mbox{\isa{\isacommand{termination}}}~\isa{f}] commences a
1.453 + termination proof for the previously defined function \isa{f}. If
1.454 + this is omitted, the command refers to the most recent function
1.455 + definition. After the proof is closed, the recursive equations and
1.456 + the induction principle is established.
1.457 +
1.458 + \end{descr}
1.459 +
1.460 + %FIXME check
1.461 +
1.462 + Recursive definitions introduced by both the \mbox{\isa{\isacommand{primrec}}} and the \mbox{\isa{\isacommand{function}}} command accommodate
1.463 + 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)
1.464 + refers to a specific induction rule, with parameters named according
1.465 + to the user-specified equations. Case names of \mbox{\isa{\isacommand{primrec}}} are that of the datatypes involved, while those of
1.466 + \mbox{\isa{\isacommand{function}}} are numbered (starting from 1).
1.467 +
1.468 + The equations provided by these packages may be referred later as
1.469 + theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
1.470 + name of the functions defined. Individual equations may be named
1.471 + explicitly as well.
1.472 +
1.473 + The \mbox{\isa{\isacommand{function}}} command accepts the following
1.474 + options.
1.475 +
1.476 + \begin{descr}
1.477 +
1.478 + \item [\isa{sequential}] enables a preprocessor which
1.479 + disambiguates overlapping patterns by making them mutually disjoint.
1.480 + Earlier equations take precedence over later ones. This allows to
1.481 + give the specification in a format very similar to functional
1.482 + programming. Note that the resulting simplification and induction
1.483 + rules correspond to the transformed specification, not the one given
1.484 + originally. This usually means that each equation given by the user
1.485 + may result in several theroems. Also note that this automatic
1.486 + transformation only works for ML-style datatype patterns.
1.487 +
1.488 + \item [\isa{{\isachardoublequote}{\isasymIN}\ name{\isachardoublequote}}] gives the target for the definition.
1.489 + %FIXME ?!?
1.490 +
1.491 + \item [\isa{domintros}] enables the automated generation of
1.492 + introduction rules for the domain predicate. While mostly not
1.493 + needed, they can be helpful in some proofs about partial functions.
1.494 +
1.495 + \item [\isa{tailrec}] generates the unconstrained recursive
1.496 + equations even without a termination proof, provided that the
1.497 + function is tail-recursive. This currently only works
1.498 +
1.499 + \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
1.500 + (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
1.501 + whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
1.502 +
1.503 + \end{descr}%
1.504 +\end{isamarkuptext}%
1.505 +\isamarkuptrue%
1.506 +%
1.507 +\isamarkupsubsection{Proof methods related to recursive definitions%
1.508 +}
1.509 +\isamarkuptrue%
1.510 +%
1.511 +\begin{isamarkuptext}%
1.512 +\begin{matharray}{rcl}
1.513 + \indexdef{HOL}{method}{pat-completeness}\mbox{\isa{pat{\isacharunderscore}completeness}} & : & \isarmeth \\
1.514 + \indexdef{HOL}{method}{relation}\mbox{\isa{relation}} & : & \isarmeth \\
1.515 + \indexdef{HOL}{method}{lexicographic-order}\mbox{\isa{lexicographic{\isacharunderscore}order}} & : & \isarmeth \\
1.516 + \end{matharray}
1.517 +
1.518 + \begin{rail}
1.519 + 'relation' term
1.520 + ;
1.521 + 'lexicographic\_order' (clasimpmod *)
1.522 + ;
1.523 + \end{rail}
1.524 +
1.525 + \begin{descr}
1.526 +
1.527 + \item [\mbox{\isa{pat{\isacharunderscore}completeness}}] is a specialized method to
1.528 + solve goals regarding the completeness of pattern matching, as
1.529 + required by the \mbox{\isa{\isacommand{function}}} package (cf.\
1.530 + \cite{isabelle-function}).
1.531 +
1.532 + \item [\mbox{\isa{relation}}~\isa{R}] introduces a termination
1.533 + proof using the relation \isa{R}. The resulting proof state will
1.534 + contain goals expressing that \isa{R} is wellfounded, and that the
1.535 + arguments of recursive calls decrease with respect to \isa{R}.
1.536 + Usually, this method is used as the initial proof step of manual
1.537 + termination proofs.
1.538 +
1.539 + \item [\mbox{\isa{lexicographic{\isacharunderscore}order}}] attempts a fully
1.540 + automated termination proof by searching for a lexicographic
1.541 + combination of size measures on the arguments of the function. The
1.542 + method accepts the same arguments as the \mbox{\isa{auto}} method,
1.543 + which it uses internally to prove local descents. The same context
1.544 + modifiers as for \mbox{\isa{auto}} are accepted, see
1.545 + \secref{sec:clasimp}.
1.546 +
1.547 + In case of failure, extensive information is printed, which can help
1.548 + to analyse the situation (cf.\ \cite{isabelle-function}).
1.549 +
1.550 + \end{descr}%
1.551 +\end{isamarkuptext}%
1.552 +\isamarkuptrue%
1.553 +%
1.554 +\isamarkupsubsection{Old-style recursive function definitions (TFL)%
1.555 +}
1.556 +\isamarkuptrue%
1.557 +%
1.558 +\begin{isamarkuptext}%
1.559 +The old TFL commands \mbox{\isa{\isacommand{recdef}}} and \mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}} for defining recursive are mostly obsolete; \mbox{\isa{\isacommand{function}}} or \mbox{\isa{\isacommand{fun}}} should be used instead.
1.560 +
1.561 + \begin{matharray}{rcl}
1.562 + \indexdef{HOL}{command}{recdef}\mbox{\isa{\isacommand{recdef}}} & : & \isartrans{theory}{theory} \\
1.563 + \indexdef{HOL}{command}{recdef-tc}\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{proof(prove)} \\
1.564 + \end{matharray}
1.565 +
1.566 + \begin{rail}
1.567 + 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
1.568 + ;
1.569 + recdeftc thmdecl? tc
1.570 + ;
1.571 + hints: '(' 'hints' (recdefmod *) ')'
1.572 + ;
1.573 + recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
1.574 + ;
1.575 + tc: nameref ('(' nat ')')?
1.576 + ;
1.577 + \end{rail}
1.578 +
1.579 + \begin{descr}
1.580 +
1.581 + \item [\mbox{\isa{\isacommand{recdef}}}] defines general well-founded
1.582 + recursive functions (using the TFL package), see also
1.583 + \cite{isabelle-HOL}. The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
1.584 + TFL to recover from failed proof attempts, returning unfinished
1.585 + 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
1.586 + automated proof process of TFL. Additional \mbox{\isa{clasimpmod}}
1.587 + declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
1.588 + context of the Simplifier (cf.\ \secref{sec:simplifier}) and
1.589 + Classical reasoner (cf.\ \secref{sec:classical}).
1.590 +
1.591 + \item [\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
1.592 + proof for leftover termination condition number \isa{i} (default
1.593 + 1) as generated by a \mbox{\isa{\isacommand{recdef}}} definition of
1.594 + constant \isa{c}.
1.595 +
1.596 + Note that in most cases, \mbox{\isa{\isacommand{recdef}}} is able to finish
1.597 + its internal proofs without manual intervention.
1.598 +
1.599 + \end{descr}
1.600 +
1.601 + \medskip Hints for \mbox{\isa{\isacommand{recdef}}} may be also declared
1.602 + globally, using the following attributes.
1.603 +
1.604 + \begin{matharray}{rcl}
1.605 + \indexdef{HOL}{attribute}{recdef-simp}\mbox{\isa{recdef{\isacharunderscore}simp}} & : & \isaratt \\
1.606 + \indexdef{HOL}{attribute}{recdef-cong}\mbox{\isa{recdef{\isacharunderscore}cong}} & : & \isaratt \\
1.607 + \indexdef{HOL}{attribute}{recdef-wf}\mbox{\isa{recdef{\isacharunderscore}wf}} & : & \isaratt \\
1.608 + \end{matharray}
1.609 +
1.610 + \begin{rail}
1.611 + ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
1.612 + ;
1.613 + \end{rail}%
1.614 +\end{isamarkuptext}%
1.615 +\isamarkuptrue%
1.616 +%
1.617 +\isamarkupsection{Definition by specification \label{sec:hol-specification}%
1.618 +}
1.619 +\isamarkuptrue%
1.620 +%
1.621 +\begin{isamarkuptext}%
1.622 +\begin{matharray}{rcl}
1.623 + \indexdef{HOL}{command}{specification}\mbox{\isa{\isacommand{specification}}} & : & \isartrans{theory}{proof(prove)} \\
1.624 + \indexdef{HOL}{command}{ax-specification}\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} & : & \isartrans{theory}{proof(prove)} \\
1.625 + \end{matharray}
1.626 +
1.627 + \begin{rail}
1.628 + ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1.629 + ;
1.630 + decl: ((name ':')? term '(' 'overloaded' ')'?)
1.631 + \end{rail}
1.632 +
1.633 + \begin{descr}
1.634 +
1.635 + \item [\mbox{\isa{\isacommand{specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
1.636 + goal stating the existence of terms with the properties specified to
1.637 + hold for the constants given in \isa{decls}. After finishing the
1.638 + proof, the theory will be augmented with definitions for the given
1.639 + constants, as well as with theorems stating the properties for these
1.640 + constants.
1.641 +
1.642 + \item [\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
1.643 + up a goal stating the existence of terms with the properties
1.644 + specified to hold for the constants given in \isa{decls}. After
1.645 + finishing the proof, the theory will be augmented with axioms
1.646 + expressing the properties given in the first place.
1.647 +
1.648 + \item [\isa{decl}] declares a constant to be defined by the
1.649 + specification given. The definition for the constant \isa{c} is
1.650 + bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
1.651 + the declaration. Overloaded constants should be declared as such.
1.652 +
1.653 + \end{descr}
1.654 +
1.655 + Whether to use \mbox{\isa{\isacommand{specification}}} or \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} is to some extent a matter of style. \mbox{\isa{\isacommand{specification}}} introduces no new axioms, and so by
1.656 + construction cannot introduce inconsistencies, whereas \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} does introduce axioms, but only after the
1.657 + user has explicitly proven it to be safe. A practical issue must be
1.658 + considered, though: After introducing two constants with the same
1.659 + properties using \mbox{\isa{\isacommand{specification}}}, one can prove
1.660 + that the two constants are, in fact, equal. If this might be a
1.661 + problem, one should use \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}.%
1.662 +\end{isamarkuptext}%
1.663 +\isamarkuptrue%
1.664 +%
1.665 +\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
1.666 +}
1.667 +\isamarkuptrue%
1.668 +%
1.669 +\begin{isamarkuptext}%
1.670 +An \textbf{inductive definition} specifies the least predicate (or
1.671 + set) \isa{R} closed under given rules: applying a rule to elements
1.672 + of \isa{R} yields a result within \isa{R}. For example, a
1.673 + structural operational semantics is an inductive definition of an
1.674 + evaluation relation.
1.675 +
1.676 + Dually, a \textbf{coinductive definition} specifies the greatest
1.677 + predicate~/ set \isa{R} that is consistent with given rules: every
1.678 + element of \isa{R} can be seen as arising by applying a rule to
1.679 + elements of \isa{R}. An important example is using bisimulation
1.680 + relations to formalise equivalence of processes and infinite data
1.681 + structures.
1.682 +
1.683 + \medskip The HOL package is related to the ZF one, which is
1.684 + described in a separate paper,\footnote{It appeared in CADE
1.685 + \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
1.686 + which you should refer to in case of difficulties. The package is
1.687 + simpler than that of ZF thanks to implicit type-checking in HOL.
1.688 + The types of the (co)inductive predicates (or sets) determine the
1.689 + domain of the fixedpoint definition, and the package does not have
1.690 + to use inference rules for type-checking.
1.691 +
1.692 + \begin{matharray}{rcl}
1.693 + \indexdef{HOL}{command}{inductive}\mbox{\isa{\isacommand{inductive}}} & : & \isarkeep{local{\dsh}theory} \\
1.694 + \indexdef{HOL}{command}{inductive-set}\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
1.695 + \indexdef{HOL}{command}{coinductive}\mbox{\isa{\isacommand{coinductive}}} & : & \isarkeep{local{\dsh}theory} \\
1.696 + \indexdef{HOL}{command}{coinductive-set}\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
1.697 + \indexdef{HOL}{attribute}{mono}\mbox{\isa{mono}} & : & \isaratt \\
1.698 + \end{matharray}
1.699 +
1.700 + \begin{rail}
1.701 + ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
1.702 + ('where' clauses)? ('monos' thmrefs)?
1.703 + ;
1.704 + clauses: (thmdecl? prop + '|')
1.705 + ;
1.706 + 'mono' (() | 'add' | 'del')
1.707 + ;
1.708 + \end{rail}
1.709 +
1.710 + \begin{descr}
1.711 +
1.712 + \item [\mbox{\isa{\isacommand{inductive}}} and \mbox{\isa{\isacommand{coinductive}}}] define (co)inductive predicates from the
1.713 + introduction rules given in the \mbox{\isa{\isakeyword{where}}} part. The
1.714 + optional \mbox{\isa{\isakeyword{for}}} part contains a list of parameters of the
1.715 + (co)inductive predicates that remain fixed throughout the
1.716 + definition. The optional \mbox{\isa{\isakeyword{monos}}} section contains
1.717 + \emph{monotonicity theorems}, which are required for each operator
1.718 + applied to a recursive set in the introduction rules. There
1.719 + \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
1.720 + for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
1.721 +
1.722 + \item [\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} and \mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}] are wrappers for to the previous commands,
1.723 + allowing the definition of (co)inductive sets.
1.724 +
1.725 + \item [\mbox{\isa{mono}}] declares monotonicity rules. These
1.726 + rule are involved in the automated monotonicity proof of \mbox{\isa{\isacommand{inductive}}}.
1.727 +
1.728 + \end{descr}%
1.729 +\end{isamarkuptext}%
1.730 +\isamarkuptrue%
1.731 +%
1.732 +\isamarkupsubsection{Derived rules%
1.733 +}
1.734 +\isamarkuptrue%
1.735 +%
1.736 +\begin{isamarkuptext}%
1.737 +Each (co)inductive definition \isa{R} adds definitions to the
1.738 + theory and also proves some theorems:
1.739 +
1.740 + \begin{description}
1.741 +
1.742 + \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
1.743 + theorems, for the recursive predicates (or sets). The rules are
1.744 + also available individually, using the names given them in the
1.745 + theory file;
1.746 +
1.747 + \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
1.748 +
1.749 + \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
1.750 + rule.
1.751 +
1.752 + \end{description}
1.753 +
1.754 + When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
1.755 + defined simultaneously, the list of introduction rules is called
1.756 + \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
1.757 + called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
1.758 + of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
1.759 +\end{isamarkuptext}%
1.760 +\isamarkuptrue%
1.761 +%
1.762 +\isamarkupsubsection{Monotonicity theorems%
1.763 +}
1.764 +\isamarkuptrue%
1.765 +%
1.766 +\begin{isamarkuptext}%
1.767 +Each theory contains a default set of theorems that are used in
1.768 + monotonicity proofs. New rules can be added to this set via the
1.769 + \mbox{\isa{mono}} attribute. The HOL theory \isa{Inductive}
1.770 + shows how this is done. In general, the following monotonicity
1.771 + theorems may be added:
1.772 +
1.773 + \begin{itemize}
1.774 +
1.775 + \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
1.776 + monotonicity of inductive definitions whose introduction rules have
1.777 + premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
1.778 +
1.779 + \item Monotonicity theorems for logical operators, which are of the
1.780 + general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}. For example, in
1.781 + the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
1.782 + \[
1.783 + \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
1.784 + \]
1.785 +
1.786 + \item De Morgan style equations for reasoning about the ``polarity''
1.787 + of expressions, e.g.
1.788 + \[
1.789 + \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
1.790 + \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
1.791 + \]
1.792 +
1.793 + \item Equations for reducing complex operators to more primitive
1.794 + ones whose monotonicity can easily be proved, e.g.
1.795 + \[
1.796 + \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
1.797 + \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
1.798 + \]
1.799 +
1.800 + \end{itemize}
1.801 +
1.802 + %FIXME: Example of an inductive definition%
1.803 +\end{isamarkuptext}%
1.804 +\isamarkuptrue%
1.805 +%
1.806 +\isamarkupsection{Arithmetic proof support%
1.807 +}
1.808 +\isamarkuptrue%
1.809 +%
1.810 +\begin{isamarkuptext}%
1.811 +\begin{matharray}{rcl}
1.812 + \indexdef{HOL}{method}{arith}\mbox{\isa{arith}} & : & \isarmeth \\
1.813 + \indexdef{HOL}{method}{arith-split}\mbox{\isa{arith{\isacharunderscore}split}} & : & \isaratt \\
1.814 + \end{matharray}
1.815 +
1.816 + The \mbox{\isa{arith}} method decides linear arithmetic problems
1.817 + (on types \isa{nat}, \isa{int}, \isa{real}). Any current
1.818 + facts are inserted into the goal before running the procedure.
1.819 +
1.820 + The \mbox{\isa{arith{\isacharunderscore}split}} attribute declares case split rules
1.821 + to be expanded before the arithmetic procedure is invoked.
1.822 +
1.823 + Note that a simpler (but faster) version of arithmetic reasoning is
1.824 + already performed by the Simplifier.%
1.825 +\end{isamarkuptext}%
1.826 +\isamarkuptrue%
1.827 +%
1.828 +\isamarkupsection{Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac}%
1.829 +}
1.830 +\isamarkuptrue%
1.831 +%
1.832 +\begin{isamarkuptext}%
1.833 +The following important tactical tools of Isabelle/HOL have been
1.834 + ported to Isar. These should be never used in proper proof texts!
1.835 +
1.836 + \begin{matharray}{rcl}
1.837 + \indexdef{HOL}{method}{case-tac}\mbox{\isa{case{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
1.838 + \indexdef{HOL}{method}{induct-tac}\mbox{\isa{induct{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
1.839 + \indexdef{HOL}{method}{ind-cases}\mbox{\isa{ind{\isacharunderscore}cases}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
1.840 + \indexdef{HOL}{command}{inductive-cases}\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} & : & \isartrans{theory}{theory} \\
1.841 + \end{matharray}
1.842 +
1.843 + \begin{rail}
1.844 + 'case\_tac' goalspec? term rule?
1.845 + ;
1.846 + 'induct\_tac' goalspec? (insts * 'and') rule?
1.847 + ;
1.848 + 'ind\_cases' (prop +) ('for' (name +)) ?
1.849 + ;
1.850 + 'inductive\_cases' (thmdecl? (prop +) + 'and')
1.851 + ;
1.852 +
1.853 + rule: ('rule' ':' thmref)
1.854 + ;
1.855 + \end{rail}
1.856 +
1.857 + \begin{descr}
1.858 +
1.859 + \item [\mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}}]
1.860 + admit to reason about inductive datatypes only (unless an
1.861 + alternative rule is given explicitly). Furthermore, \mbox{\isa{case{\isacharunderscore}tac}} does a classical case split on booleans; \mbox{\isa{induct{\isacharunderscore}tac}} allows only variables to be given as instantiation.
1.862 + These tactic emulations feature both goal addressing and dynamic
1.863 + instantiation. Note that named rule cases are \emph{not} provided
1.864 + as would be by the proper \mbox{\isa{induct}} and \mbox{\isa{cases}} proof
1.865 + methods (see \secref{sec:cases-induct}).
1.866 +
1.867 + \item [\mbox{\isa{ind{\isacharunderscore}cases}} and \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}] provide an interface to the internal
1.868 + \texttt{mk_cases} operation. Rules are simplified in an
1.869 + unrestricted forward manner.
1.870 +
1.871 + While \mbox{\isa{ind{\isacharunderscore}cases}} is a proof method to apply the
1.872 + result immediately as elimination rules, \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} provides case split theorems at the theory level
1.873 + for later use. The \mbox{\isa{\isakeyword{for}}} argument of the \mbox{\isa{ind{\isacharunderscore}cases}} method allows to specify a list of variables that should
1.874 + be generalized before applying the resulting rule.
1.875 +
1.876 + \end{descr}%
1.877 +\end{isamarkuptext}%
1.878 +\isamarkuptrue%
1.879 +%
1.880 +\isamarkupsection{Executable code%
1.881 +}
1.882 +\isamarkuptrue%
1.883 +%
1.884 +\begin{isamarkuptext}%
1.885 +Isabelle/Pure provides two generic frameworks to support code
1.886 + generation from executable specifications. Isabelle/HOL
1.887 + instantiates these mechanisms in a way that is amenable to end-user
1.888 + applications.
1.889 +
1.890 + One framework generates code from both functional and relational
1.891 + programs to SML. See \cite{isabelle-HOL} for further information
1.892 + (this actually covers the new-style theory format as well).
1.893 +
1.894 + \begin{matharray}{rcl}
1.895 + \indexdef{HOL}{command}{value}\mbox{\isa{\isacommand{value}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
1.896 + \indexdef{HOL}{command}{code-module}\mbox{\isa{\isacommand{code{\isacharunderscore}module}}} & : & \isartrans{theory}{theory} \\
1.897 + \indexdef{HOL}{command}{code-library}\mbox{\isa{\isacommand{code{\isacharunderscore}library}}} & : & \isartrans{theory}{theory} \\
1.898 + \indexdef{HOL}{command}{consts-code}\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\
1.899 + \indexdef{HOL}{command}{types-code}\mbox{\isa{\isacommand{types{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\
1.900 + \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
1.901 + \end{matharray}
1.902 +
1.903 + \begin{rail}
1.904 + 'value' term
1.905 + ;
1.906 +
1.907 + ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
1.908 + ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1.909 + 'contains' ( ( name '=' term ) + | term + )
1.910 + ;
1.911 +
1.912 + modespec: '(' ( name * ) ')'
1.913 + ;
1.914 +
1.915 + 'consts\_code' (codespec +)
1.916 + ;
1.917 +
1.918 + codespec: const template attachment ?
1.919 + ;
1.920 +
1.921 + 'types\_code' (tycodespec +)
1.922 + ;
1.923 +
1.924 + tycodespec: name template attachment ?
1.925 + ;
1.926 +
1.927 + const: term
1.928 + ;
1.929 +
1.930 + template: '(' string ')'
1.931 + ;
1.932 +
1.933 + attachment: 'attach' modespec ? verblbrace text verbrbrace
1.934 + ;
1.935 +
1.936 + 'code' (name)?
1.937 + ;
1.938 + \end{rail}
1.939 +
1.940 + \begin{descr}
1.941 +
1.942 + \item [\mbox{\isa{\isacommand{value}}}~\isa{t}] evaluates and prints a
1.943 + term using the code generator.
1.944 +
1.945 + \end{descr}
1.946 +
1.947 + \medskip The other framework generates code from functional programs
1.948 + (including overloading using type classes) to SML \cite{SML}, OCaml
1.949 + \cite{OCaml} and Haskell \cite{haskell-revised-report}.
1.950 + Conceptually, code generation is split up in three steps:
1.951 + \emph{selection} of code theorems, \emph{translation} into an
1.952 + abstract executable view and \emph{serialization} to a specific
1.953 + \emph{target language}. See \cite{isabelle-codegen} for an
1.954 + introduction on how to use it.
1.955 +
1.956 + \begin{matharray}{rcl}
1.957 + \indexdef{HOL}{command}{export-code}\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
1.958 + \indexdef{HOL}{command}{code-thms}\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
1.959 + \indexdef{HOL}{command}{code-deps}\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
1.960 + \indexdef{HOL}{command}{code-datatype}\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
1.961 + \indexdef{HOL}{command}{code-const}\mbox{\isa{\isacommand{code{\isacharunderscore}const}}} & : & \isartrans{theory}{theory} \\
1.962 + \indexdef{HOL}{command}{code-type}\mbox{\isa{\isacommand{code{\isacharunderscore}type}}} & : & \isartrans{theory}{theory} \\
1.963 + \indexdef{HOL}{command}{code-class}\mbox{\isa{\isacommand{code{\isacharunderscore}class}}} & : & \isartrans{theory}{theory} \\
1.964 + \indexdef{HOL}{command}{code-instance}\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}} & : & \isartrans{theory}{theory} \\
1.965 + \indexdef{HOL}{command}{code-monad}\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}} & : & \isartrans{theory}{theory} \\
1.966 + \indexdef{HOL}{command}{code-reserved}\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}} & : & \isartrans{theory}{theory} \\
1.967 + \indexdef{HOL}{command}{code-include}\mbox{\isa{\isacommand{code{\isacharunderscore}include}}} & : & \isartrans{theory}{theory} \\
1.968 + \indexdef{HOL}{command}{code-modulename}\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}} & : & \isartrans{theory}{theory} \\
1.969 + \indexdef{HOL}{command}{code-exception}\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}} & : & \isartrans{theory}{theory} \\
1.970 + \indexdef{HOL}{command}{print-codesetup}\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
1.971 + \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
1.972 + \end{matharray}
1.973 +
1.974 + \begin{rail}
1.975 + 'export\_code' ( constexpr + ) ? \\
1.976 + ( ( 'in' target ( 'module\_name' string ) ? \\
1.977 + ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1.978 + ;
1.979 +
1.980 + 'code\_thms' ( constexpr + ) ?
1.981 + ;
1.982 +
1.983 + 'code\_deps' ( constexpr + ) ?
1.984 + ;
1.985 +
1.986 + const: term
1.987 + ;
1.988 +
1.989 + constexpr: ( const | 'name.*' | '*' )
1.990 + ;
1.991 +
1.992 + typeconstructor: nameref
1.993 + ;
1.994 +
1.995 + class: nameref
1.996 + ;
1.997 +
1.998 + target: 'OCaml' | 'SML' | 'Haskell'
1.999 + ;
1.1000 +
1.1001 + 'code\_datatype' const +
1.1002 + ;
1.1003 +
1.1004 + 'code\_const' (const + 'and') \\
1.1005 + ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1.1006 + ;
1.1007 +
1.1008 + 'code\_type' (typeconstructor + 'and') \\
1.1009 + ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1.1010 + ;
1.1011 +
1.1012 + 'code\_class' (class + 'and') \\
1.1013 + ( ( '(' target \\
1.1014 + ( ( string ('where' \\
1.1015 + ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
1.1016 + ;
1.1017 +
1.1018 + 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1.1019 + ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1.1020 + ;
1.1021 +
1.1022 + 'code\_monad' const const target
1.1023 + ;
1.1024 +
1.1025 + 'code\_reserved' target ( string + )
1.1026 + ;
1.1027 +
1.1028 + 'code\_include' target ( string ( string | '-') )
1.1029 + ;
1.1030 +
1.1031 + 'code\_modulename' target ( ( string string ) + )
1.1032 + ;
1.1033 +
1.1034 + 'code\_exception' ( const + )
1.1035 + ;
1.1036 +
1.1037 + syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1.1038 + ;
1.1039 +
1.1040 + 'code' ('func' | 'inline') ( 'del' )?
1.1041 + ;
1.1042 + \end{rail}
1.1043 +
1.1044 + \begin{descr}
1.1045 +
1.1046 + \item [\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}] is the canonical interface
1.1047 + for generating and serializing code: for a given list of constants,
1.1048 + code is generated for the specified target languages. Abstract code
1.1049 + is cached incrementally. If no constant is given, the currently
1.1050 + cached code is serialized. If no serialization instruction is
1.1051 + given, only abstract code is cached.
1.1052 +
1.1053 + Constants may be specified by giving them literally, referring to
1.1054 + all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
1.1055 + available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
1.1056 +
1.1057 + By default, for each involved theory one corresponding name space
1.1058 + module is generated. Alternativly, a module name may be specified
1.1059 + after the \mbox{\isa{\isakeyword{module{\isacharunderscore}name}}} keyword; then \emph{all} code is
1.1060 + placed in this module.
1.1061 +
1.1062 + For \emph{SML} and \emph{OCaml}, the file specification refers to a
1.1063 + single file; for \emph{Haskell}, it refers to a whole directory,
1.1064 + where code is generated in multiple files reflecting the module
1.1065 + hierarchy. The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
1.1066 + output. For \emph{SML}, omitting the file specification compiles
1.1067 + code internally in the context of the current ML session.
1.1068 +
1.1069 + Serializers take an optional list of arguments in parentheses. For
1.1070 + \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
1.1071 + declaration.
1.1072 +
1.1073 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}] prints a list of theorems
1.1074 + representing the corresponding program containing all given
1.1075 + constants; if no constants are given, the currently cached code
1.1076 + theorems are printed.
1.1077 +
1.1078 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}] visualizes dependencies of
1.1079 + theorems representing the corresponding program containing all given
1.1080 + constants; if no constants are given, the currently cached code
1.1081 + theorems are visualized.
1.1082 +
1.1083 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}] specifies a constructor set
1.1084 + for a logical type.
1.1085 +
1.1086 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}] associates a list of constants
1.1087 + with target-specific serializations; omitting a serialization
1.1088 + deletes an existing serialization.
1.1089 +
1.1090 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}] associates a list of type
1.1091 + constructors with target-specific serializations; omitting a
1.1092 + serialization deletes an existing serialization.
1.1093 +
1.1094 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}] associates a list of classes
1.1095 + with target-specific class names; in addition, constants associated
1.1096 + with this class may be given target-specific names used for instance
1.1097 + declarations; omitting a serialization deletes an existing
1.1098 + serialization. This applies only to \emph{Haskell}.
1.1099 +
1.1100 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}] declares a list of type
1.1101 + constructor / class instance relations as ``already present'' for a
1.1102 + given target. Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
1.1103 + ``already present'' declaration. This applies only to
1.1104 + \emph{Haskell}.
1.1105 +
1.1106 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}] provides an auxiliary
1.1107 + mechanism to generate monadic code.
1.1108 +
1.1109 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}] declares a list of names as
1.1110 + reserved for a given target, preventing it to be shadowed by any
1.1111 + generated code.
1.1112 +
1.1113 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}] adds arbitrary named content
1.1114 + (``include'') to generated code. A as last argument ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}''
1.1115 + will remove an already added ``include''.
1.1116 +
1.1117 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}] declares aliasings from
1.1118 + one module name onto another.
1.1119 +
1.1120 + \item [\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}] declares constants which
1.1121 + are not required to have a definition by a defining equations; these
1.1122 + are mapped on exceptions instead.
1.1123 +
1.1124 + \item [\mbox{\isa{code}}~\isa{func}] explicitly selects (or
1.1125 + with option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' deselects) a defining equation for
1.1126 + code generation. Usually packages introducing defining equations
1.1127 + provide a resonable default setup for selection.
1.1128 +
1.1129 + \item [\mbox{\isa{code}}\isa{inline}] declares (or with
1.1130 + option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' removes) inlining theorems which are
1.1131 + applied as rewrite rules to any defining equation during
1.1132 + preprocessing.
1.1133 +
1.1134 + \item [\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}] gives an overview on
1.1135 + selected defining equations, code generator datatypes and
1.1136 + preprocessor setup.
1.1137 +
1.1138 + \end{descr}%
1.1139 +\end{isamarkuptext}%
1.1140 +\isamarkuptrue%
1.1141 +%
1.1142 +\isadelimtheory
1.1143 +%
1.1144 +\endisadelimtheory
1.1145 +%
1.1146 +\isatagtheory
1.1147 \isacommand{end}\isamarkupfalse%
1.1148 %
1.1149 \endisatagtheory
1.1150 {\isafoldtheory}%
1.1151 %
1.1152 \isadelimtheory
1.1153 -\isanewline
1.1154 %
1.1155 \endisadelimtheory
1.1156 +\isanewline
1.1157 +\isanewline
1.1158 \end{isabellebody}%
1.1159 %%% Local Variables:
1.1160 %%% mode: latex