doc-src/IsarRef/Thy/document/HOL_Specific.tex
author haftmann
Mon Jan 19 13:37:24 2009 +0100 (2009-01-19)
changeset 29560 fa6c5d62adf5
parent 29115 6fb7be34506e
child 30172 afdf7808cfd0
permissions -rw-r--r--
"code equation" replaces "defining equation"
     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{HOL{\isacharunderscore}Specific}%
     4 %
     5 \isadelimtheory
     6 %
     7 \endisadelimtheory
     8 %
     9 \isatagtheory
    10 \isacommand{theory}\isamarkupfalse%
    11 \ HOL{\isacharunderscore}Specific\isanewline
    12 \isakeyword{imports}\ Main\isanewline
    13 \isakeyword{begin}%
    14 \endisatagtheory
    15 {\isafoldtheory}%
    16 %
    17 \isadelimtheory
    18 %
    19 \endisadelimtheory
    20 %
    21 \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
    22 }
    23 \isamarkuptrue%
    24 %
    25 \isamarkupsection{Primitive types \label{sec:hol-typedef}%
    26 }
    27 \isamarkuptrue%
    28 %
    29 \begin{isamarkuptext}%
    30 \begin{matharray}{rcl}
    31     \indexdef{HOL}{command}{typedecl}\hypertarget{command.HOL.typedecl}{\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
    32     \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}} \\
    33   \end{matharray}
    34 
    35   \begin{rail}
    36     'typedecl' typespec infix?
    37     ;
    38     'typedef' altname? abstype '=' repset
    39     ;
    40 
    41     altname: '(' (name | 'open' | 'open' name) ')'
    42     ;
    43     abstype: typespec infix?
    44     ;
    45     repset: term ('morphisms' name name)?
    46     ;
    47   \end{rail}
    48 
    49   \begin{description}
    50   
    51   \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
    52   to the original \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} of Isabelle/Pure (see
    53   \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
    54   constructor.  %FIXME check, update
    55   
    56   \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
    57   a goal stating non-emptiness of the set \isa{A}.  After finishing
    58   the proof, the theory will be augmented by a Gordon/HOL-style type
    59   definition, which establishes a bijection between the representing
    60   set \isa{A} and the new type \isa{t}.
    61   
    62   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
    63   name may be given in parentheses).  The injection from type to set
    64   is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
    65   changed via an explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration).
    66   
    67   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
    68   corresponding injection/surjection pair (in both directions).  Rules
    69   \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
    70   more convenient view on the injectivity part, suitable for automated
    71   proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
    72   declarations).  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
    73   \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
    74   on surjectivity; these are already declared as set or type rules for
    75   the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods.
    76   
    77   An alternative name may be specified in parentheses; the default is
    78   to use \isa{t} as indicated before.  The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
    79   declaration suppresses a separate constant definition for the
    80   representing set.
    81 
    82   \end{description}
    83 
    84   Note that raw type declarations are rarely used in practice; the
    85   main application is with experimental (or even axiomatic!) theory
    86   fragments.  Instead of primitive HOL type definitions, user-level
    87   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}).%
    88 \end{isamarkuptext}%
    89 \isamarkuptrue%
    90 %
    91 \isamarkupsection{Adhoc tuples%
    92 }
    93 \isamarkuptrue%
    94 %
    95 \begin{isamarkuptext}%
    96 \begin{matharray}{rcl}
    97     \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
    98   \end{matharray}
    99 
   100   \begin{rail}
   101     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
   102     ;
   103   \end{rail}
   104 
   105   \begin{description}
   106   
   107   \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
   108   canonical form as specified by the arguments given; the \isa{i}-th
   109   collection of arguments refers to occurrences in premise \isa{i}
   110   of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all}
   111   arguments in function applications to be represented canonically
   112   according to their tuple type structure.
   113 
   114   Note that these operations tend to invent funny names for new local
   115   parameters to be introduced.
   116 
   117   \end{description}%
   118 \end{isamarkuptext}%
   119 \isamarkuptrue%
   120 %
   121 \isamarkupsection{Records \label{sec:hol-record}%
   122 }
   123 \isamarkuptrue%
   124 %
   125 \begin{isamarkuptext}%
   126 In principle, records merely generalize the concept of tuples, where
   127   components may be addressed by labels instead of just position.  The
   128   logical infrastructure of records in Isabelle/HOL is slightly more
   129   advanced, though, supporting truly extensible record schemes.  This
   130   admits operations that are polymorphic with respect to record
   131   extension, yielding ``object-oriented'' effects like (single)
   132   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   133   details on object-oriented verification and record subtyping in HOL.%
   134 \end{isamarkuptext}%
   135 \isamarkuptrue%
   136 %
   137 \isamarkupsubsection{Basic concepts%
   138 }
   139 \isamarkuptrue%
   140 %
   141 \begin{isamarkuptext}%
   142 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   143   at the level of terms and types.  The notation is as follows:
   144 
   145   \begin{center}
   146   \begin{tabular}{l|l|l}
   147     & record terms & record types \\ \hline
   148     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}} \\
   149     schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
   150       \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
   151   \end{tabular}
   152   \end{center}
   153 
   154   \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
   155 
   156   A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
   157   \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
   158   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}}
   159   and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
   160 
   161   A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
   162   \isa{x} and \isa{y} as before, but also possibly further fields
   163   as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
   164   of the syntax).  The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
   165   scheme is called the \emph{more part}.  Logically it is just a free
   166   variable, which is occasionally referred to as ``row variable'' in
   167   the literature.  The more part of a record scheme may be
   168   instantiated by zero or more further components.  For example, the
   169   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.
   170   Fixed records are special instances of record schemes, where
   171   ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
   172   element.  In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
   173   for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
   174   
   175   \medskip Two key observations make extensible records in a simply
   176   typed language like HOL work out:
   177 
   178   \begin{enumerate}
   179 
   180   \item the more part is internalized, as a free term or type
   181   variable,
   182 
   183   \item field names are externalized, they cannot be accessed within
   184   the logic as first-class values.
   185 
   186   \end{enumerate}
   187 
   188   \medskip In Isabelle/HOL record types have to be defined explicitly,
   189   fixing their field names and types, and their (optional) parent
   190   record.  Afterwards, records may be formed using above syntax, while
   191   obeying the canonical order of fields as given by their declaration.
   192   The record package provides several standard operations like
   193   selectors and updates.  The common setup for various generic proof
   194   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   195   tutorial \cite{isabelle-hol-book} for further instructions on using
   196   records in practice.%
   197 \end{isamarkuptext}%
   198 \isamarkuptrue%
   199 %
   200 \isamarkupsubsection{Record specifications%
   201 }
   202 \isamarkuptrue%
   203 %
   204 \begin{isamarkuptext}%
   205 \begin{matharray}{rcl}
   206     \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
   207   \end{matharray}
   208 
   209   \begin{rail}
   210     'record' typespec '=' (type '+')? (constdecl +)
   211     ;
   212   \end{rail}
   213 
   214   \begin{description}
   215 
   216   \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}},
   217   derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
   218   field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
   219 
   220   The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
   221   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
   222   least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
   223   Basically, field names need to belong to a unique record.  This is
   224   not a real restriction in practice, since fields are qualified by
   225   the record name internally.
   226 
   227   The parent record specification \isa{{\isasymtau}} is optional; if omitted
   228   \isa{t} becomes a root record.  The hierarchy of all records
   229   declared within a theory context forms a forest structure, i.e.\ a
   230   set of trees starting with a root record each.  There is no way to
   231   merge multiple parent records!
   232 
   233   For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
   234   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
   235   \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}}.
   236 
   237   \end{description}%
   238 \end{isamarkuptext}%
   239 \isamarkuptrue%
   240 %
   241 \isamarkupsubsection{Record operations%
   242 }
   243 \isamarkuptrue%
   244 %
   245 \begin{isamarkuptext}%
   246 Any record definition of the form presented above produces certain
   247   standard operations.  Selectors and updates are provided for any
   248   field, including the improper one ``\isa{more}''.  There are also
   249   cumulative record constructor functions.  To simplify the
   250   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}}.
   251 
   252   \medskip \textbf{Selectors} and \textbf{updates} are available for
   253   any field (including ``\isa{more}''):
   254 
   255   \begin{matharray}{lll}
   256     \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}} \\
   257     \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}} \\
   258   \end{matharray}
   259 
   260   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
   261   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
   262   because of postfix notation the order of fields shown here is
   263   reverse than in the actual term.  Since repeated updates are just
   264   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.
   265   Thus commutativity of independent updates can be proven within the
   266   logic for any two fields, but not as a general theorem.
   267 
   268   \medskip The \textbf{make} operation provides a cumulative record
   269   constructor function:
   270 
   271   \begin{matharray}{lll}
   272     \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}} \\
   273   \end{matharray}
   274 
   275   \medskip We now reconsider the case of non-root records, which are
   276   derived of some parent.  In general, the latter may depend on
   277   another parent as well, resulting in a list of \emph{ancestor
   278   records}.  Appending the lists of fields of all ancestors results in
   279   a certain field prefix.  The record package automatically takes care
   280   of this by lifting operations over this context of ancestor fields.
   281   Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
   282   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}},
   283   the above record operations will get the following types:
   284 
   285   \medskip
   286   \begin{tabular}{lll}
   287     \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}} \\
   288     \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}} \\
   289     \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}} \\
   290   \end{tabular}
   291   \medskip
   292 
   293   \noindent Some further operations address the extension aspect of a
   294   derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
   295   record fragment consisting of exactly the new fields introduced here
   296   (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
   297   takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
   298 
   299   \medskip
   300   \begin{tabular}{lll}
   301     \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}} \\
   302     \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}} \\
   303     \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}} \\
   304   \end{tabular}
   305   \medskip
   306 
   307   \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
   308   for root records.%
   309 \end{isamarkuptext}%
   310 \isamarkuptrue%
   311 %
   312 \isamarkupsubsection{Derived rules and proof tools%
   313 }
   314 \isamarkuptrue%
   315 %
   316 \begin{isamarkuptext}%
   317 The record package proves several results internally, declaring
   318   these facts to appropriate proof tools.  This enables users to
   319   reason about record structures quite conveniently.  Assume that
   320   \isa{t} is a record type as specified above.
   321 
   322   \begin{enumerate}
   323   
   324   \item Standard conversions for selectors or updates applied to
   325   record constructor terms are made part of the default Simplifier
   326   context; thus proofs by reduction of basic operations merely require
   327   the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments.  These rules
   328   are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
   329   
   330   \item Selectors applied to updated records are automatically reduced
   331   by an internal simplification procedure, which is also part of the
   332   standard Simplifier setup.
   333 
   334   \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
   335   Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules.  These rules are available as
   336   \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
   337 
   338   \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,
   339   and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
   340   The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
   341 
   342   \item Representations of arbitrary record expressions as canonical
   343   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,
   344   \secref{sec:cases-induct}).  Several variations are available, for
   345   fixed records, record schemes, more parts etc.
   346   
   347   The generic proof methods are sufficiently smart to pick the most
   348   sensible rule according to the type of the indicated record
   349   expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
   350 
   351   \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}
   352   treated automatically, but usually need to be expanded by hand,
   353   using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
   354 
   355   \end{enumerate}%
   356 \end{isamarkuptext}%
   357 \isamarkuptrue%
   358 %
   359 \isamarkupsection{Datatypes \label{sec:hol-datatype}%
   360 }
   361 \isamarkuptrue%
   362 %
   363 \begin{isamarkuptext}%
   364 \begin{matharray}{rcl}
   365     \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
   366   \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}} \\
   367   \end{matharray}
   368 
   369   \begin{rail}
   370     'datatype' (dtspec + 'and')
   371     ;
   372     'rep\_datatype' ('(' (name +) ')')? (term +)
   373     ;
   374 
   375     dtspec: parname? typespec infix? '=' (cons + '|')
   376     ;
   377     cons: name (type *) mixfix?
   378   \end{rail}
   379 
   380   \begin{description}
   381 
   382   \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
   383   HOL.
   384 
   385   \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as
   386   inductive ones, generating the standard infrastructure of derived
   387   concepts (primitive recursion etc.).
   388 
   389   \end{description}
   390 
   391   The induction and exhaustion theorems generated provide case names
   392   according to the constructors involved, while parameters are named
   393   after the types (see also \secref{sec:cases-induct}).
   394 
   395   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   396   the old-style theory syntax being used there!  Apart from proper
   397   proof methods for case-analysis and induction, there are also
   398   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
   399   to refer directly to the internal structure of subgoals (including
   400   internally bound parameters).%
   401 \end{isamarkuptext}%
   402 \isamarkuptrue%
   403 %
   404 \isamarkupsection{Recursive functions \label{sec:recursion}%
   405 }
   406 \isamarkuptrue%
   407 %
   408 \begin{isamarkuptext}%
   409 \begin{matharray}{rcl}
   410     \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}} \\
   411     \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}} \\
   412     \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}} \\
   413     \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}} \\
   414   \end{matharray}
   415 
   416   \begin{rail}
   417     'primrec' target? fixes 'where' equations
   418     ;
   419     equations: (thmdecl? prop + '|')
   420     ;
   421     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   422     ;
   423     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   424     ;
   425     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   426     ;
   427     'termination' ( term )?
   428   \end{rail}
   429 
   430   \begin{description}
   431 
   432   \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
   433   functions over datatypes, see also \cite{isabelle-HOL}.
   434 
   435   \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
   436   wellfounded recursion. A detailed description with examples can be
   437   found in \cite{isabelle-function}. The function is specified by a
   438   set of (possibly conditional) recursive equations with arbitrary
   439   pattern matching. The command generates proof obligations for the
   440   completeness and the compatibility of patterns.
   441 
   442   The defined function is considered partial, and the resulting
   443   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
   444   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
   445   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
   446   command can then be used to establish that the function is total.
   447 
   448   \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
   449   proof attempts regarding pattern matching and termination.  See
   450   \cite{isabelle-function} for further details.
   451 
   452   \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
   453   termination proof for the previously defined function \isa{f}.  If
   454   this is omitted, the command refers to the most recent function
   455   definition.  After the proof is closed, the recursive equations and
   456   the induction principle is established.
   457 
   458   \end{description}
   459 
   460   %FIXME check
   461 
   462   Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
   463   command accommodate
   464   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)
   465   refers to a specific induction rule, with parameters named according
   466   to the user-specified equations.
   467   For the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} the induction principle coincides
   468   with structural recursion on the datatype the recursion is carried
   469   out.
   470   Case names of \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} are that of the datatypes involved, while those of
   471   \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} are numbered (starting from 1).
   472 
   473   The equations provided by these packages may be referred later as
   474   theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
   475   name of the functions defined.  Individual equations may be named
   476   explicitly as well.
   477 
   478   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
   479   options.
   480 
   481   \begin{description}
   482 
   483   \item \isa{sequential} enables a preprocessor which disambiguates
   484   overlapping patterns by making them mutually disjoint.  Earlier
   485   equations take precedence over later ones.  This allows to give the
   486   specification in a format very similar to functional programming.
   487   Note that the resulting simplification and induction rules
   488   correspond to the transformed specification, not the one given
   489   originally. This usually means that each equation given by the user
   490   may result in several theroems.  Also note that this automatic
   491   transformation only works for ML-style datatype patterns.
   492 
   493   \item \isa{domintros} enables the automated generation of
   494   introduction rules for the domain predicate. While mostly not
   495   needed, they can be helpful in some proofs about partial functions.
   496 
   497   \item \isa{tailrec} generates the unconstrained recursive
   498   equations even without a termination proof, provided that the
   499   function is tail-recursive. This currently only works
   500 
   501   \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
   502   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
   503   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
   504 
   505   \end{description}%
   506 \end{isamarkuptext}%
   507 \isamarkuptrue%
   508 %
   509 \isamarkupsubsection{Proof methods related to recursive definitions%
   510 }
   511 \isamarkuptrue%
   512 %
   513 \begin{isamarkuptext}%
   514 \begin{matharray}{rcl}
   515     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
   516     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
   517     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
   518   \end{matharray}
   519 
   520   \begin{rail}
   521     'relation' term
   522     ;
   523     'lexicographic\_order' (clasimpmod *)
   524     ;
   525   \end{rail}
   526 
   527   \begin{description}
   528 
   529   \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
   530   solve goals regarding the completeness of pattern matching, as
   531   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
   532   \cite{isabelle-function}).
   533 
   534   \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
   535   proof using the relation \isa{R}.  The resulting proof state will
   536   contain goals expressing that \isa{R} is wellfounded, and that the
   537   arguments of recursive calls decrease with respect to \isa{R}.
   538   Usually, this method is used as the initial proof step of manual
   539   termination proofs.
   540 
   541   \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
   542   automated termination proof by searching for a lexicographic
   543   combination of size measures on the arguments of the function. The
   544   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
   545   which it uses internally to prove local descents.  The same context
   546   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
   547   \secref{sec:clasimp}.
   548 
   549   In case of failure, extensive information is printed, which can help
   550   to analyse the situation (cf.\ \cite{isabelle-function}).
   551 
   552   \end{description}%
   553 \end{isamarkuptext}%
   554 \isamarkuptrue%
   555 %
   556 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
   557 }
   558 \isamarkuptrue%
   559 %
   560 \begin{isamarkuptext}%
   561 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.
   562 
   563   \begin{matharray}{rcl}
   564     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\
   565     \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}} \\
   566   \end{matharray}
   567 
   568   \begin{rail}
   569     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   570     ;
   571     recdeftc thmdecl? tc
   572     ;
   573     hints: '(' 'hints' (recdefmod *) ')'
   574     ;
   575     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   576     ;
   577     tc: nameref ('(' nat ')')?
   578     ;
   579   \end{rail}
   580 
   581   \begin{description}
   582   
   583   \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
   584   recursive functions (using the TFL package), see also
   585   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
   586   TFL to recover from failed proof attempts, returning unfinished
   587   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
   588   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
   589   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   590   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   591   Classical reasoner (cf.\ \secref{sec:classical}).
   592   
   593   \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
   594   proof for leftover termination condition number \isa{i} (default
   595   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
   596   constant \isa{c}.
   597   
   598   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
   599   its internal proofs without manual intervention.
   600 
   601   \end{description}
   602 
   603   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
   604   globally, using the following attributes.
   605 
   606   \begin{matharray}{rcl}
   607     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
   608     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
   609     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
   610   \end{matharray}
   611 
   612   \begin{rail}
   613     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   614     ;
   615   \end{rail}%
   616 \end{isamarkuptext}%
   617 \isamarkuptrue%
   618 %
   619 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
   620 }
   621 \isamarkuptrue%
   622 %
   623 \begin{isamarkuptext}%
   624 An \textbf{inductive definition} specifies the least predicate (or
   625   set) \isa{R} closed under given rules: applying a rule to elements
   626   of \isa{R} yields a result within \isa{R}.  For example, a
   627   structural operational semantics is an inductive definition of an
   628   evaluation relation.
   629 
   630   Dually, a \textbf{coinductive definition} specifies the greatest
   631   predicate~/ set \isa{R} that is consistent with given rules: every
   632   element of \isa{R} can be seen as arising by applying a rule to
   633   elements of \isa{R}.  An important example is using bisimulation
   634   relations to formalise equivalence of processes and infinite data
   635   structures.
   636 
   637   \medskip The HOL package is related to the ZF one, which is
   638   described in a separate paper,\footnote{It appeared in CADE
   639   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   640   which you should refer to in case of difficulties.  The package is
   641   simpler than that of ZF thanks to implicit type-checking in HOL.
   642   The types of the (co)inductive predicates (or sets) determine the
   643   domain of the fixedpoint definition, and the package does not have
   644   to use inference rules for type-checking.
   645 
   646   \begin{matharray}{rcl}
   647     \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}} \\
   648     \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}} \\
   649     \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}} \\
   650     \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}} \\
   651     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
   652   \end{matharray}
   653 
   654   \begin{rail}
   655     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   656     ('where' clauses)? ('monos' thmrefs)?
   657     ;
   658     clauses: (thmdecl? prop + '|')
   659     ;
   660     'mono' (() | 'add' | 'del')
   661     ;
   662   \end{rail}
   663 
   664   \begin{description}
   665 
   666   \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the
   667   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
   668   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
   669   (co)inductive predicates that remain fixed throughout the
   670   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
   671   \emph{monotonicity theorems}, which are required for each operator
   672   applied to a recursive set in the introduction rules.  There
   673   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
   674   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
   675 
   676   \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,
   677   allowing the definition of (co)inductive sets.
   678 
   679   \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules.  These
   680   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
   681 
   682   \end{description}%
   683 \end{isamarkuptext}%
   684 \isamarkuptrue%
   685 %
   686 \isamarkupsubsection{Derived rules%
   687 }
   688 \isamarkuptrue%
   689 %
   690 \begin{isamarkuptext}%
   691 Each (co)inductive definition \isa{R} adds definitions to the
   692   theory and also proves some theorems:
   693 
   694   \begin{description}
   695 
   696   \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
   697   theorems, for the recursive predicates (or sets).  The rules are
   698   also available individually, using the names given them in the
   699   theory file;
   700 
   701   \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
   702 
   703   \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
   704   rule.
   705 
   706   \end{description}
   707 
   708   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
   709   defined simultaneously, the list of introduction rules is called
   710   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
   711   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
   712   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
   713 \end{isamarkuptext}%
   714 \isamarkuptrue%
   715 %
   716 \isamarkupsubsection{Monotonicity theorems%
   717 }
   718 \isamarkuptrue%
   719 %
   720 \begin{isamarkuptext}%
   721 Each theory contains a default set of theorems that are used in
   722   monotonicity proofs.  New rules can be added to this set via the
   723   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
   724   shows how this is done.  In general, the following monotonicity
   725   theorems may be added:
   726 
   727   \begin{itemize}
   728 
   729   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
   730   monotonicity of inductive definitions whose introduction rules have
   731   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
   732 
   733   \item Monotonicity theorems for logical operators, which are of the
   734   general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
   735   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
   736   \[
   737   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
   738   \]
   739 
   740   \item De Morgan style equations for reasoning about the ``polarity''
   741   of expressions, e.g.
   742   \[
   743   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
   744   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
   745   \]
   746 
   747   \item Equations for reducing complex operators to more primitive
   748   ones whose monotonicity can easily be proved, e.g.
   749   \[
   750   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
   751   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
   752   \]
   753 
   754   \end{itemize}
   755 
   756   %FIXME: Example of an inductive definition%
   757 \end{isamarkuptext}%
   758 \isamarkuptrue%
   759 %
   760 \isamarkupsection{Arithmetic proof support%
   761 }
   762 \isamarkuptrue%
   763 %
   764 \begin{isamarkuptext}%
   765 \begin{matharray}{rcl}
   766     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
   767     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
   768   \end{matharray}
   769 
   770   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   771   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   772   facts are inserted into the goal before running the procedure.
   773 
   774   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   775   rules to be expanded before the arithmetic procedure is invoked.
   776 
   777   Note that a simpler (but faster) version of arithmetic reasoning is
   778   already performed by the Simplifier.%
   779 \end{isamarkuptext}%
   780 \isamarkuptrue%
   781 %
   782 \isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
   783 }
   784 \isamarkuptrue%
   785 %
   786 \begin{isamarkuptext}%
   787 Isabelle/HOL includes a generic \emph{ATP manager} that allows
   788   external automated reasoning tools to crunch a pending goal.
   789   Supported provers include E\footnote{\url{http://www.eprover.org}},
   790   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   791   There is also a wrapper to invoke provers remotely via the
   792   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   793   web service.
   794 
   795   The problem passed to external provers consists of the goal together
   796   with a smart selection of lemmas from the current theory context.
   797   The result of a successful proof search is some source text that
   798   usually reconstructs the proof within Isabelle, without requiring
   799   external provers again.  The Metis
   800   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   801   integrated into Isabelle/HOL is being used here.
   802 
   803   In this mode of operation, heavy means of automated reasoning are
   804   used as a strong relevance filter, while the main proof checking
   805   works via explicit inferences going through the Isabelle kernel.
   806   Moreover, rechecking Isabelle proof texts with already specified
   807   auxiliary facts is much faster than performing fully automated
   808   search over and over again.
   809 
   810   \begin{matharray}{rcl}
   811     \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}} \\
   812     \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}} \\
   813     \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}} \\
   814     \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}} \\
   815     \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}} \\
   816     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
   817   \end{matharray}
   818 
   819   \begin{rail}
   820   'sledgehammer' (nameref *)
   821   ;
   822   'atp\_messages' ('(' nat ')')?
   823   ;
   824 
   825   'metis' thmrefs
   826   ;
   827   \end{rail}
   828 
   829   \begin{description}
   830 
   831   \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}
   832   invokes the specified automated theorem provers on the first
   833   subgoal.  Provers are run in parallel, the first successful result
   834   is displayed, and the other attempts are terminated.
   835 
   836   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
   837   ``ATP provers'' preference by the user interface.
   838 
   839   There are additional preferences for timeout (default: 60 seconds),
   840   and the maximum number of independent prover processes (default: 5);
   841   excessive provers are automatically terminated.
   842 
   843   \item \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}} prints the list of automated
   844   theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
   845   command.
   846 
   847   \item \hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}} prints information about presently
   848   running provers, including elapsed runtime, and the remaining time
   849   until timeout.
   850 
   851   \item \hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}} terminates all presently running
   852   provers.
   853 
   854   \item \hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}} displays recent messages issued
   855   by automated theorem provers.  This allows to examine results that
   856   might have got lost due to the asynchronous nature of default
   857   \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} output.  An optional message limit may
   858   be specified (default 5).
   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 code 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 code equation for code
  1172   generation.  Usually packages introducing code 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 code equation during
  1178   preprocessing.
  1179 
  1180   \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
  1181   selected code 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}%
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