doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Mon Dec 15 21:41:21 2008 +0100 (2008-12-15)
changeset 29113 fb31b7a6c858
parent 28788 ff9d8a8932e4
child 29115 6fb7be34506e
permissions -rw-r--r--
updated generated files;
     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{HOL{\isacharunderscore}Specific}%
     4 %
     5 \isadelimtheory
     6 \isanewline
     7 \isanewline
     8 %
     9 \endisadelimtheory
    10 %
    11 \isatagtheory
    12 \isacommand{theory}\isamarkupfalse%
    13 \ HOL{\isacharunderscore}Specific\isanewline
    14 \isakeyword{imports}\ Main\isanewline
    15 \isakeyword{begin}%
    16 \endisatagtheory
    17 {\isafoldtheory}%
    18 %
    19 \isadelimtheory
    20 %
    21 \endisadelimtheory
    22 %
    23 \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
    24 }
    25 \isamarkuptrue%
    26 %
    27 \isamarkupsection{Primitive types \label{sec:hol-typedef}%
    28 }
    29 \isamarkuptrue%
    30 %
    31 \begin{isamarkuptext}%
    32 \begin{matharray}{rcl}
    33     \indexdef{HOL}{command}{typedecl}\hypertarget{command.HOL.typedecl}{\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
    34     \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
    35   \end{matharray}
    36 
    37   \begin{rail}
    38     'typedecl' typespec infix?
    39     ;
    40     'typedef' altname? abstype '=' repset
    41     ;
    42 
    43     altname: '(' (name | 'open' | 'open' name) ')'
    44     ;
    45     abstype: typespec infix?
    46     ;
    47     repset: term ('morphisms' name name)?
    48     ;
    49   \end{rail}
    50 
    51   \begin{description}
    52   
    53   \item \hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}} is similar
    54   to the original \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} of Isabelle/Pure (see
    55   \secref{sec:types-pure}), but also declares type arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an actual HOL type
    56   constructor.  %FIXME check, update
    57   
    58   \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}} sets up
    59   a goal stating non-emptiness of the set \isa{A}.  After finishing
    60   the proof, the theory will be augmented by a Gordon/HOL-style type
    61   definition, which establishes a bijection between the representing
    62   set \isa{A} and the new type \isa{t}.
    63   
    64   Technically, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
    65   name may be given in parentheses).  The injection from type to set
    66   is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
    67   changed via an explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration).
    68   
    69   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
    70   corresponding injection/surjection pair (in both directions).  Rules
    71   \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
    72   more convenient view on the injectivity part, suitable for automated
    73   proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
    74   declarations).  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
    75   \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
    76   on surjectivity; these are already declared as set or type rules for
    77   the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods.
    78   
    79   An alternative name may be specified in parentheses; the default is
    80   to use \isa{t} as indicated before.  The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
    81   declaration suppresses a separate constant definition for the
    82   representing set.
    83 
    84   \end{description}
    85 
    86   Note that raw type declarations are rarely used in practice; the
    87   main application is with experimental (or even axiomatic!) theory
    88   fragments.  Instead of primitive HOL type definitions, user-level
    89   theories usually refer to higher-level packages such as \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}} (see \secref{sec:hol-record}) or \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} (see \secref{sec:hol-datatype}).%
    90 \end{isamarkuptext}%
    91 \isamarkuptrue%
    92 %
    93 \isamarkupsection{Adhoc tuples%
    94 }
    95 \isamarkuptrue%
    96 %
    97 \begin{isamarkuptext}%
    98 \begin{matharray}{rcl}
    99     \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
   100   \end{matharray}
   101 
   102   \begin{rail}
   103     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
   104     ;
   105   \end{rail}
   106 
   107   \begin{description}
   108   
   109   \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
   110   canonical form as specified by the arguments given; the \isa{i}-th
   111   collection of arguments refers to occurrences in premise \isa{i}
   112   of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all}
   113   arguments in function applications to be represented canonically
   114   according to their tuple type structure.
   115 
   116   Note that these operations tend to invent funny names for new local
   117   parameters to be introduced.
   118 
   119   \end{description}%
   120 \end{isamarkuptext}%
   121 \isamarkuptrue%
   122 %
   123 \isamarkupsection{Records \label{sec:hol-record}%
   124 }
   125 \isamarkuptrue%
   126 %
   127 \begin{isamarkuptext}%
   128 In principle, records merely generalize the concept of tuples, where
   129   components may be addressed by labels instead of just position.  The
   130   logical infrastructure of records in Isabelle/HOL is slightly more
   131   advanced, though, supporting truly extensible record schemes.  This
   132   admits operations that are polymorphic with respect to record
   133   extension, yielding ``object-oriented'' effects like (single)
   134   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   135   details on object-oriented verification and record subtyping in HOL.%
   136 \end{isamarkuptext}%
   137 \isamarkuptrue%
   138 %
   139 \isamarkupsubsection{Basic concepts%
   140 }
   141 \isamarkuptrue%
   142 %
   143 \begin{isamarkuptext}%
   144 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   145   at the level of terms and types.  The notation is as follows:
   146 
   147   \begin{center}
   148   \begin{tabular}{l|l|l}
   149     & record terms & record types \\ \hline
   150     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}} \\
   151     schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
   152       \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
   153   \end{tabular}
   154   \end{center}
   155 
   156   \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
   157 
   158   A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
   159   \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
   160   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}}
   161   and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
   162 
   163   A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
   164   \isa{x} and \isa{y} as before, but also possibly further fields
   165   as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
   166   of the syntax).  The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
   167   scheme is called the \emph{more part}.  Logically it is just a free
   168   variable, which is occasionally referred to as ``row variable'' in
   169   the literature.  The more part of a record scheme may be
   170   instantiated by zero or more further components.  For example, the
   171   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.
   172   Fixed records are special instances of record schemes, where
   173   ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
   174   element.  In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
   175   for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
   176   
   177   \medskip Two key observations make extensible records in a simply
   178   typed language like HOL work out:
   179 
   180   \begin{enumerate}
   181 
   182   \item the more part is internalized, as a free term or type
   183   variable,
   184 
   185   \item field names are externalized, they cannot be accessed within
   186   the logic as first-class values.
   187 
   188   \end{enumerate}
   189 
   190   \medskip In Isabelle/HOL record types have to be defined explicitly,
   191   fixing their field names and types, and their (optional) parent
   192   record.  Afterwards, records may be formed using above syntax, while
   193   obeying the canonical order of fields as given by their declaration.
   194   The record package provides several standard operations like
   195   selectors and updates.  The common setup for various generic proof
   196   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   197   tutorial \cite{isabelle-hol-book} for further instructions on using
   198   records in practice.%
   199 \end{isamarkuptext}%
   200 \isamarkuptrue%
   201 %
   202 \isamarkupsubsection{Record specifications%
   203 }
   204 \isamarkuptrue%
   205 %
   206 \begin{isamarkuptext}%
   207 \begin{matharray}{rcl}
   208     \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
   209   \end{matharray}
   210 
   211   \begin{rail}
   212     'record' typespec '=' (type '+')? (constdecl +)
   213     ;
   214   \end{rail}
   215 
   216   \begin{description}
   217 
   218   \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}},
   219   derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
   220   field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
   221 
   222   The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
   223   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
   224   least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
   225   Basically, field names need to belong to a unique record.  This is
   226   not a real restriction in practice, since fields are qualified by
   227   the record name internally.
   228 
   229   The parent record specification \isa{{\isasymtau}} is optional; if omitted
   230   \isa{t} becomes a root record.  The hierarchy of all records
   231   declared within a theory context forms a forest structure, i.e.\ a
   232   set of trees starting with a root record each.  There is no way to
   233   merge multiple parent records!
   234 
   235   For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
   236   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
   237   \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}}.
   238 
   239   \end{description}%
   240 \end{isamarkuptext}%
   241 \isamarkuptrue%
   242 %
   243 \isamarkupsubsection{Record operations%
   244 }
   245 \isamarkuptrue%
   246 %
   247 \begin{isamarkuptext}%
   248 Any record definition of the form presented above produces certain
   249   standard operations.  Selectors and updates are provided for any
   250   field, including the improper one ``\isa{more}''.  There are also
   251   cumulative record constructor functions.  To simplify the
   252   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}}.
   253 
   254   \medskip \textbf{Selectors} and \textbf{updates} are available for
   255   any field (including ``\isa{more}''):
   256 
   257   \begin{matharray}{lll}
   258     \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}} \\
   259     \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}} \\
   260   \end{matharray}
   261 
   262   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
   263   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
   264   because of postfix notation the order of fields shown here is
   265   reverse than in the actual term.  Since repeated updates are just
   266   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.
   267   Thus commutativity of independent updates can be proven within the
   268   logic for any two fields, but not as a general theorem.
   269 
   270   \medskip The \textbf{make} operation provides a cumulative record
   271   constructor function:
   272 
   273   \begin{matharray}{lll}
   274     \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}} \\
   275   \end{matharray}
   276 
   277   \medskip We now reconsider the case of non-root records, which are
   278   derived of some parent.  In general, the latter may depend on
   279   another parent as well, resulting in a list of \emph{ancestor
   280   records}.  Appending the lists of fields of all ancestors results in
   281   a certain field prefix.  The record package automatically takes care
   282   of this by lifting operations over this context of ancestor fields.
   283   Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
   284   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}},
   285   the above record operations will get the following types:
   286 
   287   \medskip
   288   \begin{tabular}{lll}
   289     \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}} \\
   290     \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}} \\
   291     \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}} \\
   292   \end{tabular}
   293   \medskip
   294 
   295   \noindent Some further operations address the extension aspect of a
   296   derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
   297   record fragment consisting of exactly the new fields introduced here
   298   (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
   299   takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
   300 
   301   \medskip
   302   \begin{tabular}{lll}
   303     \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}} \\
   304     \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}} \\
   305     \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}} \\
   306   \end{tabular}
   307   \medskip
   308 
   309   \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
   310   for root records.%
   311 \end{isamarkuptext}%
   312 \isamarkuptrue%
   313 %
   314 \isamarkupsubsection{Derived rules and proof tools%
   315 }
   316 \isamarkuptrue%
   317 %
   318 \begin{isamarkuptext}%
   319 The record package proves several results internally, declaring
   320   these facts to appropriate proof tools.  This enables users to
   321   reason about record structures quite conveniently.  Assume that
   322   \isa{t} is a record type as specified above.
   323 
   324   \begin{enumerate}
   325   
   326   \item Standard conversions for selectors or updates applied to
   327   record constructor terms are made part of the default Simplifier
   328   context; thus proofs by reduction of basic operations merely require
   329   the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments.  These rules
   330   are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
   331   
   332   \item Selectors applied to updated records are automatically reduced
   333   by an internal simplification procedure, which is also part of the
   334   standard Simplifier setup.
   335 
   336   \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
   337   Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules.  These rules are available as
   338   \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
   339 
   340   \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,
   341   and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
   342   The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
   343 
   344   \item Representations of arbitrary record expressions as canonical
   345   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,
   346   \secref{sec:cases-induct}).  Several variations are available, for
   347   fixed records, record schemes, more parts etc.
   348   
   349   The generic proof methods are sufficiently smart to pick the most
   350   sensible rule according to the type of the indicated record
   351   expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
   352 
   353   \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}
   354   treated automatically, but usually need to be expanded by hand,
   355   using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
   356 
   357   \end{enumerate}%
   358 \end{isamarkuptext}%
   359 \isamarkuptrue%
   360 %
   361 \isamarkupsection{Datatypes \label{sec:hol-datatype}%
   362 }
   363 \isamarkuptrue%
   364 %
   365 \begin{isamarkuptext}%
   366 \begin{matharray}{rcl}
   367     \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
   368   \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}} \\
   369   \end{matharray}
   370 
   371   \begin{rail}
   372     'datatype' (dtspec + 'and')
   373     ;
   374     'rep\_datatype' ('(' (name +) ')')? (term +)
   375     ;
   376 
   377     dtspec: parname? typespec infix? '=' (cons + '|')
   378     ;
   379     cons: name (type *) mixfix?
   380   \end{rail}
   381 
   382   \begin{description}
   383 
   384   \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
   385   HOL.
   386 
   387   \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as
   388   inductive ones, generating the standard infrastructure of derived
   389   concepts (primitive recursion etc.).
   390 
   391   \end{description}
   392 
   393   The induction and exhaustion theorems generated provide case names
   394   according to the constructors involved, while parameters are named
   395   after the types (see also \secref{sec:cases-induct}).
   396 
   397   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   398   the old-style theory syntax being used there!  Apart from proper
   399   proof methods for case-analysis and induction, there are also
   400   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
   401   to refer directly to the internal structure of subgoals (including
   402   internally bound parameters).%
   403 \end{isamarkuptext}%
   404 \isamarkuptrue%
   405 %
   406 \isamarkupsection{Recursive functions \label{sec:recursion}%
   407 }
   408 \isamarkuptrue%
   409 %
   410 \begin{isamarkuptext}%
   411 \begin{matharray}{rcl}
   412     \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}} \\
   413     \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}} \\
   414     \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}} \\
   415     \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}} \\
   416   \end{matharray}
   417 
   418   \begin{rail}
   419     'primrec' target? fixes 'where' equations
   420     ;
   421     equations: (thmdecl? prop + '|')
   422     ;
   423     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   424     ;
   425     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   426     ;
   427     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   428     ;
   429     'termination' ( term )?
   430   \end{rail}
   431 
   432   \begin{description}
   433 
   434   \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
   435   functions over datatypes, see also \cite{isabelle-HOL}.
   436 
   437   \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
   438   wellfounded recursion. A detailed description with examples can be
   439   found in \cite{isabelle-function}. The function is specified by a
   440   set of (possibly conditional) recursive equations with arbitrary
   441   pattern matching. The command generates proof obligations for the
   442   completeness and the compatibility of patterns.
   443 
   444   The defined function is considered partial, and the resulting
   445   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
   446   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
   447   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
   448   command can then be used to establish that the function is total.
   449 
   450   \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
   451   proof attempts regarding pattern matching and termination.  See
   452   \cite{isabelle-function} for further details.
   453 
   454   \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
   455   termination proof for the previously defined function \isa{f}.  If
   456   this is omitted, the command refers to the most recent function
   457   definition.  After the proof is closed, the recursive equations and
   458   the induction principle is established.
   459 
   460   \end{description}
   461 
   462   %FIXME check
   463 
   464   Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
   465   command accommodate
   466   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)
   467   refers to a specific induction rule, with parameters named according
   468   to the user-specified equations.
   469   For the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} the induction principle coincides
   470   with structural recursion on the datatype the recursion is carried
   471   out.
   472   Case names of \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} are that of the datatypes involved, while those of
   473   \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} are numbered (starting from 1).
   474 
   475   The equations provided by these packages may be referred later as
   476   theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
   477   name of the functions defined.  Individual equations may be named
   478   explicitly as well.
   479 
   480   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
   481   options.
   482 
   483   \begin{description}
   484 
   485   \item \isa{sequential} enables a preprocessor which disambiguates
   486   overlapping patterns by making them mutually disjoint.  Earlier
   487   equations take precedence over later ones.  This allows to give the
   488   specification in a format very similar to functional programming.
   489   Note that the resulting simplification and induction rules
   490   correspond to the transformed specification, not the one given
   491   originally. This usually means that each equation given by the user
   492   may result in several theroems.  Also note that this automatic
   493   transformation only works for ML-style datatype patterns.
   494 
   495   \item \isa{domintros} enables the automated generation of
   496   introduction rules for the domain predicate. While mostly not
   497   needed, they can be helpful in some proofs about partial functions.
   498 
   499   \item \isa{tailrec} generates the unconstrained recursive
   500   equations even without a termination proof, provided that the
   501   function is tail-recursive. This currently only works
   502 
   503   \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
   504   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
   505   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
   506 
   507   \end{description}%
   508 \end{isamarkuptext}%
   509 \isamarkuptrue%
   510 %
   511 \isamarkupsubsection{Proof methods related to recursive definitions%
   512 }
   513 \isamarkuptrue%
   514 %
   515 \begin{isamarkuptext}%
   516 \begin{matharray}{rcl}
   517     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
   518     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
   519     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
   520   \end{matharray}
   521 
   522   \begin{rail}
   523     'relation' term
   524     ;
   525     'lexicographic\_order' (clasimpmod *)
   526     ;
   527   \end{rail}
   528 
   529   \begin{description}
   530 
   531   \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
   532   solve goals regarding the completeness of pattern matching, as
   533   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
   534   \cite{isabelle-function}).
   535 
   536   \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
   537   proof using the relation \isa{R}.  The resulting proof state will
   538   contain goals expressing that \isa{R} is wellfounded, and that the
   539   arguments of recursive calls decrease with respect to \isa{R}.
   540   Usually, this method is used as the initial proof step of manual
   541   termination proofs.
   542 
   543   \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
   544   automated termination proof by searching for a lexicographic
   545   combination of size measures on the arguments of the function. The
   546   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
   547   which it uses internally to prove local descents.  The same context
   548   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
   549   \secref{sec:clasimp}.
   550 
   551   In case of failure, extensive information is printed, which can help
   552   to analyse the situation (cf.\ \cite{isabelle-function}).
   553 
   554   \end{description}%
   555 \end{isamarkuptext}%
   556 \isamarkuptrue%
   557 %
   558 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
   559 }
   560 \isamarkuptrue%
   561 %
   562 \begin{isamarkuptext}%
   563 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.
   564 
   565   \begin{matharray}{rcl}
   566     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\
   567     \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}} \\
   568   \end{matharray}
   569 
   570   \begin{rail}
   571     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   572     ;
   573     recdeftc thmdecl? tc
   574     ;
   575     hints: '(' 'hints' (recdefmod *) ')'
   576     ;
   577     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   578     ;
   579     tc: nameref ('(' nat ')')?
   580     ;
   581   \end{rail}
   582 
   583   \begin{description}
   584   
   585   \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
   586   recursive functions (using the TFL package), see also
   587   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
   588   TFL to recover from failed proof attempts, returning unfinished
   589   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
   590   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
   591   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   592   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   593   Classical reasoner (cf.\ \secref{sec:classical}).
   594   
   595   \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
   596   proof for leftover termination condition number \isa{i} (default
   597   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
   598   constant \isa{c}.
   599   
   600   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
   601   its internal proofs without manual intervention.
   602 
   603   \end{description}
   604 
   605   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
   606   globally, using the following attributes.
   607 
   608   \begin{matharray}{rcl}
   609     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
   610     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
   611     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
   612   \end{matharray}
   613 
   614   \begin{rail}
   615     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   616     ;
   617   \end{rail}%
   618 \end{isamarkuptext}%
   619 \isamarkuptrue%
   620 %
   621 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
   622 }
   623 \isamarkuptrue%
   624 %
   625 \begin{isamarkuptext}%
   626 An \textbf{inductive definition} specifies the least predicate (or
   627   set) \isa{R} closed under given rules: applying a rule to elements
   628   of \isa{R} yields a result within \isa{R}.  For example, a
   629   structural operational semantics is an inductive definition of an
   630   evaluation relation.
   631 
   632   Dually, a \textbf{coinductive definition} specifies the greatest
   633   predicate~/ set \isa{R} that is consistent with given rules: every
   634   element of \isa{R} can be seen as arising by applying a rule to
   635   elements of \isa{R}.  An important example is using bisimulation
   636   relations to formalise equivalence of processes and infinite data
   637   structures.
   638 
   639   \medskip The HOL package is related to the ZF one, which is
   640   described in a separate paper,\footnote{It appeared in CADE
   641   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   642   which you should refer to in case of difficulties.  The package is
   643   simpler than that of ZF thanks to implicit type-checking in HOL.
   644   The types of the (co)inductive predicates (or sets) determine the
   645   domain of the fixedpoint definition, and the package does not have
   646   to use inference rules for type-checking.
   647 
   648   \begin{matharray}{rcl}
   649     \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}} \\
   650     \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}} \\
   651     \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}} \\
   652     \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}} \\
   653     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
   654   \end{matharray}
   655 
   656   \begin{rail}
   657     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   658     ('where' clauses)? ('monos' thmrefs)?
   659     ;
   660     clauses: (thmdecl? prop + '|')
   661     ;
   662     'mono' (() | 'add' | 'del')
   663     ;
   664   \end{rail}
   665 
   666   \begin{description}
   667 
   668   \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the
   669   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
   670   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
   671   (co)inductive predicates that remain fixed throughout the
   672   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
   673   \emph{monotonicity theorems}, which are required for each operator
   674   applied to a recursive set in the introduction rules.  There
   675   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
   676   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
   677 
   678   \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,
   679   allowing the definition of (co)inductive sets.
   680 
   681   \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules.  These
   682   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
   683 
   684   \end{description}%
   685 \end{isamarkuptext}%
   686 \isamarkuptrue%
   687 %
   688 \isamarkupsubsection{Derived rules%
   689 }
   690 \isamarkuptrue%
   691 %
   692 \begin{isamarkuptext}%
   693 Each (co)inductive definition \isa{R} adds definitions to the
   694   theory and also proves some theorems:
   695 
   696   \begin{description}
   697 
   698   \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
   699   theorems, for the recursive predicates (or sets).  The rules are
   700   also available individually, using the names given them in the
   701   theory file;
   702 
   703   \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
   704 
   705   \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
   706   rule.
   707 
   708   \end{description}
   709 
   710   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
   711   defined simultaneously, the list of introduction rules is called
   712   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
   713   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
   714   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
   715 \end{isamarkuptext}%
   716 \isamarkuptrue%
   717 %
   718 \isamarkupsubsection{Monotonicity theorems%
   719 }
   720 \isamarkuptrue%
   721 %
   722 \begin{isamarkuptext}%
   723 Each theory contains a default set of theorems that are used in
   724   monotonicity proofs.  New rules can be added to this set via the
   725   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
   726   shows how this is done.  In general, the following monotonicity
   727   theorems may be added:
   728 
   729   \begin{itemize}
   730 
   731   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
   732   monotonicity of inductive definitions whose introduction rules have
   733   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
   734 
   735   \item Monotonicity theorems for logical operators, which are of the
   736   general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
   737   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
   738   \[
   739   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
   740   \]
   741 
   742   \item De Morgan style equations for reasoning about the ``polarity''
   743   of expressions, e.g.
   744   \[
   745   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
   746   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
   747   \]
   748 
   749   \item Equations for reducing complex operators to more primitive
   750   ones whose monotonicity can easily be proved, e.g.
   751   \[
   752   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
   753   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
   754   \]
   755 
   756   \end{itemize}
   757 
   758   %FIXME: Example of an inductive definition%
   759 \end{isamarkuptext}%
   760 \isamarkuptrue%
   761 %
   762 \isamarkupsection{Arithmetic proof support%
   763 }
   764 \isamarkuptrue%
   765 %
   766 \begin{isamarkuptext}%
   767 \begin{matharray}{rcl}
   768     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
   769     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
   770   \end{matharray}
   771 
   772   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   773   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   774   facts are inserted into the goal before running the procedure.
   775 
   776   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   777   rules to be expanded before the arithmetic procedure is invoked.
   778 
   779   Note that a simpler (but faster) version of arithmetic reasoning is
   780   already performed by the Simplifier.%
   781 \end{isamarkuptext}%
   782 \isamarkuptrue%
   783 %
   784 \isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
   785 }
   786 \isamarkuptrue%
   787 %
   788 \begin{isamarkuptext}%
   789 Isabelle/HOL includes a generic \emph{ATP manager} that allows
   790   external automated reasoning tools to crunch a pending goal.
   791   Supported provers include E\footnote{\url{http://www.eprover.org}},
   792   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   793   There is also a wrapper to invoke provers remotely via the
   794   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   795   web service.
   796 
   797   The problem passed to external provers consists of the goal together
   798   with a smart selection of lemmas from the current theory context.
   799   The result of a successful proof search is some source text that
   800   usually reconstructs the proof within Isabelle, without requiring
   801   external provers again.  The Metis
   802   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   803   integrated into Isabelle/HOL is being used here.
   804 
   805   In this mode of operation, heavy means of automated reasoning are
   806   used as a strong relevance filter, while the main proof checking
   807   works via explicit inferences going through the Isabelle kernel.
   808   Moreover, rechecking Isabelle proof texts with already specified
   809   auxiliary facts is much faster than performing fully automated
   810   search over and over again.
   811 
   812   \begin{matharray}{rcl}
   813     \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}{\isachardoublequote}} \\
   814     \indexdef{HOL}{command}{print\_atps}\hypertarget{command.HOL.print-atps}{\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
   815     \indexdef{HOL}{command}{atp\_info}\hypertarget{command.HOL.atp-info}{\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
   816     \indexdef{HOL}{command}{atp\_kill}\hypertarget{command.HOL.atp-kill}{\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
   817     \indexdef{HOL}{command}{atp\_messages}\hypertarget{command.HOL.atp-messages}{\hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
   818     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
   819   \end{matharray}
   820 
   821   \begin{rail}
   822   'sledgehammer' (nameref *)
   823   ;
   824   'atp\_messages' ('(' nat ')')?
   825 
   826   'metis' thmrefs
   827   ;
   828   \end{rail}
   829 
   830   \begin{description}
   831 
   832   \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}
   833   invokes the specified automated theorem provers on the first
   834   subgoal.  Provers are run in parallel, the first successful result
   835   is displayed, and the other attempts are terminated.
   836 
   837   Provers are defined in the theory context, see also \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}.  If no provers are given as arguments to \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}, the system refers to the default defined as
   838   ``ATP provers'' preference by the user interface.
   839 
   840   There are additional preferences for timeout (default: 60 seconds),
   841   and the maximum number of independent prover processes (default: 5);
   842   excessive provers are automatically terminated.
   843 
   844   \item \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}} prints the list of automated
   845   theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
   846   command.
   847 
   848   \item \hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}} prints information about presently
   849   running provers, including elapsed runtime, and the remaining time
   850   until timeout.
   851 
   852   \item \hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}} terminates all presently running
   853   provers.
   854 
   855   \item \hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}} displays recent messages issued
   856   by automated theorem provers.  This allows to examine results that
   857   might have got lost due to the asynchronous nature of default
   858   \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} output.
   859 
   860   \item \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}} invokes the Metis prover
   861   with the given facts.  Metis is an automated proof tool of medium
   862   strength, but is fully integrated into Isabelle/HOL, with explicit
   863   inferences going through the kernel.  Thus its results are
   864   guaranteed to be ``correct by construction''.
   865 
   866   Note that all facts used with Metis need to be specified as explicit
   867   arguments.  There are no rule declarations as for other Isabelle
   868   provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
   869 
   870   \end{description}%
   871 \end{isamarkuptext}%
   872 \isamarkuptrue%
   873 %
   874 \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
   875 }
   876 \isamarkuptrue%
   877 %
   878 \begin{isamarkuptext}%
   879 The following tools of Isabelle/HOL support cases analysis and
   880   induction in unstructured tactic scripts; see also
   881   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   882 
   883   \begin{matharray}{rcl}
   884     \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} \\
   885     \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} \\
   886     \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} \\
   887     \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}} \\
   888   \end{matharray}
   889 
   890   \begin{rail}
   891     'case\_tac' goalspec? term rule?
   892     ;
   893     'induct\_tac' goalspec? (insts * 'and') rule?
   894     ;
   895     'ind\_cases' (prop +) ('for' (name +)) ?
   896     ;
   897     'inductive\_cases' (thmdecl? (prop +) + 'and')
   898     ;
   899 
   900     rule: ('rule' ':' thmref)
   901     ;
   902   \end{rail}
   903 
   904   \begin{description}
   905 
   906   \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit
   907   to reason about inductive types.  Rules are selected according to
   908   the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
   909   attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
   910 
   911   These unstructured tactics feature both goal addressing and dynamic
   912   instantiation.  Note that named rule cases are \emph{not} provided
   913   as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
   914   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
   915   statements, only the compact object-logic conclusion of the subgoal
   916   being addressed.
   917   
   918   \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
   919   forward manner.
   920 
   921   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
   922   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
   923   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
   924   be generalized before applying the resulting rule.
   925 
   926   \end{description}%
   927 \end{isamarkuptext}%
   928 \isamarkuptrue%
   929 %
   930 \isamarkupsection{Executable code%
   931 }
   932 \isamarkuptrue%
   933 %
   934 \begin{isamarkuptext}%
   935 Isabelle/Pure provides two generic frameworks to support code
   936   generation from executable specifications.  Isabelle/HOL
   937   instantiates these mechanisms in a way that is amenable to end-user
   938   applications.
   939 
   940   One framework generates code from both functional and relational
   941   programs to SML.  See \cite{isabelle-HOL} for further information
   942   (this actually covers the new-style theory format as well).
   943 
   944   \begin{matharray}{rcl}
   945     \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}} \\
   946     \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}} \\
   947     \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}} \\
   948     \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}} \\
   949     \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}} \\  
   950     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
   951   \end{matharray}
   952 
   953   \begin{rail}
   954   'value' term
   955   ;
   956 
   957   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   958     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   959     'contains' ( ( name '=' term ) + | term + )
   960   ;
   961 
   962   modespec: '(' ( name * ) ')'
   963   ;
   964 
   965   'consts\_code' (codespec +)
   966   ;
   967 
   968   codespec: const template attachment ?
   969   ;
   970 
   971   'types\_code' (tycodespec +)
   972   ;
   973 
   974   tycodespec: name template attachment ?
   975   ;
   976 
   977   const: term
   978   ;
   979 
   980   template: '(' string ')'
   981   ;
   982 
   983   attachment: 'attach' modespec ? verblbrace text verbrbrace
   984   ;
   985 
   986   'code' (name)?
   987   ;
   988   \end{rail}
   989 
   990   \begin{description}
   991 
   992   \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a term
   993   using the code generator.
   994 
   995   \end{description}
   996 
   997   \medskip The other framework generates code from functional programs
   998   (including overloading using type classes) to SML \cite{SML}, OCaml
   999   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
  1000   Conceptually, code generation is split up in three steps:
  1001   \emph{selection} of code theorems, \emph{translation} into an
  1002   abstract executable view and \emph{serialization} to a specific
  1003   \emph{target language}.  See \cite{isabelle-codegen} for an
  1004   introduction on how to use it.
  1005 
  1006   \begin{matharray}{rcl}
  1007     \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}} \\
  1008     \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}} \\
  1009     \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}} \\
  1010     \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}} \\
  1011     \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}} \\
  1012     \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}} \\
  1013     \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}} \\
  1014     \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}} \\
  1015     \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}} \\
  1016     \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}} \\
  1017     \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}} \\
  1018     \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}} \\
  1019     \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}} \\
  1020     \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}} \\
  1021     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
  1022   \end{matharray}
  1023 
  1024   \begin{rail}
  1025     'export\_code' ( constexpr + ) ? \\
  1026       ( ( 'in' target ( 'module\_name' string ) ? \\
  1027         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1028     ;
  1029 
  1030     'code\_thms' ( constexpr + ) ?
  1031     ;
  1032 
  1033     'code\_deps' ( constexpr + ) ?
  1034     ;
  1035 
  1036     const: term
  1037     ;
  1038 
  1039     constexpr: ( const | 'name.*' | '*' )
  1040     ;
  1041 
  1042     typeconstructor: nameref
  1043     ;
  1044 
  1045     class: nameref
  1046     ;
  1047 
  1048     target: 'OCaml' | 'SML' | 'Haskell'
  1049     ;
  1050 
  1051     'code\_datatype' const +
  1052     ;
  1053 
  1054     'code\_const' (const + 'and') \\
  1055       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1056     ;
  1057 
  1058     'code\_type' (typeconstructor + 'and') \\
  1059       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1060     ;
  1061 
  1062     'code\_class' (class + 'and') \\
  1063       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1064     ;
  1065 
  1066     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1067       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1068     ;
  1069 
  1070     'code\_monad' const const target
  1071     ;
  1072 
  1073     'code\_reserved' target ( string + )
  1074     ;
  1075 
  1076     'code\_include' target ( string ( string | '-') )
  1077     ;
  1078 
  1079     'code\_modulename' target ( ( string string ) + )
  1080     ;
  1081 
  1082     'code\_abort' ( const + )
  1083     ;
  1084 
  1085     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1086     ;
  1087 
  1088     'code' ( 'inline' ) ? ( 'del' ) ?
  1089     ;
  1090   \end{rail}
  1091 
  1092   \begin{description}
  1093 
  1094   \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} is the canonical interface for
  1095   generating and serializing code: for a given list of constants, code
  1096   is generated for the specified target languages.  Abstract code is
  1097   cached incrementally.  If no constant is given, the currently cached
  1098   code is serialized.  If no serialization instruction is given, only
  1099   abstract code is cached.
  1100 
  1101   Constants may be specified by giving them literally, referring to
  1102   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
  1103   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
  1104 
  1105   By default, for each involved theory one corresponding name space
  1106   module is generated.  Alternativly, a module name may be specified
  1107   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
  1108   placed in this module.
  1109 
  1110   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1111   single file; for \emph{Haskell}, it refers to a whole directory,
  1112   where code is generated in multiple files reflecting the module
  1113   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
  1114   output.  For \emph{SML}, omitting the file specification compiles
  1115   code internally in the context of the current ML session.
  1116 
  1117   Serializers take an optional list of arguments in parentheses.  For
  1118   \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype
  1119   declaration.
  1120 
  1121   \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
  1122   representing the corresponding program containing all given
  1123   constants; if no constants are given, the currently cached code
  1124   theorems are printed.
  1125 
  1126   \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
  1127   theorems representing the corresponding program containing all given
  1128   constants; if no constants are given, the currently cached code
  1129   theorems are visualized.
  1130 
  1131   \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
  1132   for a logical type.
  1133 
  1134   \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
  1135   with target-specific serializations; omitting a serialization
  1136   deletes an existing serialization.
  1137 
  1138   \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
  1139   constructors with target-specific serializations; omitting a
  1140   serialization deletes an existing serialization.
  1141 
  1142   \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
  1143   with target-specific class names; omitting a serialization deletes
  1144   an existing serialization.  This applies only to \emph{Haskell}.
  1145 
  1146   \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
  1147   constructor / class instance relations as ``already present'' for a
  1148   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
  1149   ``already present'' declaration.  This applies only to
  1150   \emph{Haskell}.
  1151 
  1152   \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
  1153   to generate monadic code for Haskell.
  1154 
  1155   \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
  1156   reserved for a given target, preventing it to be shadowed by any
  1157   generated code.
  1158 
  1159   \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
  1160   (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
  1161   will remove an already added ``include''.
  1162 
  1163   \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
  1164   module name onto another.
  1165 
  1166   \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
  1167   required to have a definition by means of defining equations; if
  1168   needed these are implemented by program abort instead.
  1169 
  1170   \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
  1171   ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a defining equation for code
  1172   generation.  Usually packages introducing defining equations provide
  1173   a reasonable default setup for selection.
  1174 
  1175   \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}~\isa{inline} declares (or with
  1176   option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
  1177   applied as rewrite rules to any defining equation during
  1178   preprocessing.
  1179 
  1180   \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
  1181   selected defining equations, code generator datatypes and
  1182   preprocessor setup.
  1183 
  1184   \end{description}%
  1185 \end{isamarkuptext}%
  1186 \isamarkuptrue%
  1187 %
  1188 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
  1189 }
  1190 \isamarkuptrue%
  1191 %
  1192 \begin{isamarkuptext}%
  1193 \begin{matharray}{rcl}
  1194     \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}} \\
  1195     \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}} \\
  1196   \end{matharray}
  1197 
  1198   \begin{rail}
  1199   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1200   ;
  1201   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1202   \end{rail}
  1203 
  1204   \begin{description}
  1205 
  1206   \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
  1207   goal stating the existence of terms with the properties specified to
  1208   hold for the constants given in \isa{decls}.  After finishing the
  1209   proof, the theory will be augmented with definitions for the given
  1210   constants, as well as with theorems stating the properties for these
  1211   constants.
  1212 
  1213   \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
  1214   a goal stating the existence of terms with the properties specified
  1215   to hold for the constants given in \isa{decls}.  After finishing
  1216   the proof, the theory will be augmented with axioms expressing the
  1217   properties given in the first place.
  1218 
  1219   \item \isa{decl} declares a constant to be defined by the
  1220   specification given.  The definition for the constant \isa{c} is
  1221   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
  1222   the declaration.  Overloaded constants should be declared as such.
  1223 
  1224   \end{description}
  1225 
  1226   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
  1227   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
  1228   user has explicitly proven it to be safe.  A practical issue must be
  1229   considered, though: After introducing two constants with the same
  1230   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
  1231   that the two constants are, in fact, equal.  If this might be a
  1232   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
  1233 \end{isamarkuptext}%
  1234 \isamarkuptrue%
  1235 %
  1236 \isadelimtheory
  1237 %
  1238 \endisadelimtheory
  1239 %
  1240 \isatagtheory
  1241 \isacommand{end}\isamarkupfalse%
  1242 %
  1243 \endisatagtheory
  1244 {\isafoldtheory}%
  1245 %
  1246 \isadelimtheory
  1247 %
  1248 \endisadelimtheory
  1249 \isanewline
  1250 \end{isabellebody}%
  1251 %%% Local Variables:
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