doc-src/IsarRef/Thy/document/HOL_Specific.tex
author nipkow
Fri Apr 03 16:17:50 2009 +0200 (2009-04-03)
changeset 30863 5dc392a59bb7
parent 30172 afdf7808cfd0
child 30865 5106e13d400f
permissions -rw-r--r--
Finite_Set: lemma
IsarRef: attribute arith
     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}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
   768     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
   769   \end{matharray}
   770 
   771   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   772   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   773   facts are inserted into the goal before running the procedure.
   774 
   775   The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
   776   always supplied to the arithmetic provers implicitly.
   777 
   778   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   779   rules to be expanded before \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} is invoked.
   780 
   781   Note that a simpler (but faster) arithmetic prover is
   782   already invoked by the Simplifier.%
   783 \end{isamarkuptext}%
   784 \isamarkuptrue%
   785 %
   786 \isamarkupsection{Intuitionistic proof search%
   787 }
   788 \isamarkuptrue%
   789 %
   790 \begin{isamarkuptext}%
   791 \begin{matharray}{rcl}
   792     \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
   793   \end{matharray}
   794 
   795   \begin{rail}
   796     'iprover' ('!' ?) (rulemod *)
   797     ;
   798   \end{rail}
   799 
   800   The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
   801   search, depending on specifically declared rules from the context,
   802   or given as explicit arguments.  Chained facts are inserted into the
   803   goal before commencing proof search; ``\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}''  means to include the current \hyperlink{fact.prems}{\mbox{\isa{prems}}} as well.
   804   
   805   Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
   806   \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
   807   ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
   808   applied aggressively (without considering back-tracking later).
   809   Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
   810   single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these).  An
   811   explicit weight annotation may be given as well; otherwise the
   812   number of rule premises will be taken into account here.%
   813 \end{isamarkuptext}%
   814 \isamarkuptrue%
   815 %
   816 \isamarkupsection{Coherent Logic%
   817 }
   818 \isamarkuptrue%
   819 %
   820 \begin{isamarkuptext}%
   821 \begin{matharray}{rcl}
   822     \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
   823   \end{matharray}
   824 
   825   \begin{rail}
   826     'coherent' thmrefs?
   827     ;
   828   \end{rail}
   829 
   830   The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
   831   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   832   applications in confluence theory, lattice theory and projective
   833   geometry.  See \hyperlink{file.~~/src/HOL/ex/Coherent.thy}{\mbox{\isa{\isatt{{\isachartilde}{\isachartilde}{\isacharslash}src{\isacharslash}HOL{\isacharslash}ex{\isacharslash}Coherent{\isachardot}thy}}}} for some
   834   examples.%
   835 \end{isamarkuptext}%
   836 \isamarkuptrue%
   837 %
   838 \isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
   839 }
   840 \isamarkuptrue%
   841 %
   842 \begin{isamarkuptext}%
   843 Isabelle/HOL includes a generic \emph{ATP manager} that allows
   844   external automated reasoning tools to crunch a pending goal.
   845   Supported provers include E\footnote{\url{http://www.eprover.org}},
   846   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   847   There is also a wrapper to invoke provers remotely via the
   848   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   849   web service.
   850 
   851   The problem passed to external provers consists of the goal together
   852   with a smart selection of lemmas from the current theory context.
   853   The result of a successful proof search is some source text that
   854   usually reconstructs the proof within Isabelle, without requiring
   855   external provers again.  The Metis
   856   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   857   integrated into Isabelle/HOL is being used here.
   858 
   859   In this mode of operation, heavy means of automated reasoning are
   860   used as a strong relevance filter, while the main proof checking
   861   works via explicit inferences going through the Isabelle kernel.
   862   Moreover, rechecking Isabelle proof texts with already specified
   863   auxiliary facts is much faster than performing fully automated
   864   search over and over again.
   865 
   866   \begin{matharray}{rcl}
   867     \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}} \\
   868     \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}} \\
   869     \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}} \\
   870     \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}} \\
   871     \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}} \\
   872     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
   873   \end{matharray}
   874 
   875   \begin{rail}
   876   'sledgehammer' (nameref *)
   877   ;
   878   'atp\_messages' ('(' nat ')')?
   879   ;
   880 
   881   'metis' thmrefs
   882   ;
   883   \end{rail}
   884 
   885   \begin{description}
   886 
   887   \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}
   888   invokes the specified automated theorem provers on the first
   889   subgoal.  Provers are run in parallel, the first successful result
   890   is displayed, and the other attempts are terminated.
   891 
   892   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
   893   ``ATP provers'' preference by the user interface.
   894 
   895   There are additional preferences for timeout (default: 60 seconds),
   896   and the maximum number of independent prover processes (default: 5);
   897   excessive provers are automatically terminated.
   898 
   899   \item \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}} prints the list of automated
   900   theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
   901   command.
   902 
   903   \item \hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}} prints information about presently
   904   running provers, including elapsed runtime, and the remaining time
   905   until timeout.
   906 
   907   \item \hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}} terminates all presently running
   908   provers.
   909 
   910   \item \hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}} displays recent messages issued
   911   by automated theorem provers.  This allows to examine results that
   912   might have got lost due to the asynchronous nature of default
   913   \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} output.  An optional message limit may
   914   be specified (default 5).
   915 
   916   \item \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}} invokes the Metis prover
   917   with the given facts.  Metis is an automated proof tool of medium
   918   strength, but is fully integrated into Isabelle/HOL, with explicit
   919   inferences going through the kernel.  Thus its results are
   920   guaranteed to be ``correct by construction''.
   921 
   922   Note that all facts used with Metis need to be specified as explicit
   923   arguments.  There are no rule declarations as for other Isabelle
   924   provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
   925 
   926   \end{description}%
   927 \end{isamarkuptext}%
   928 \isamarkuptrue%
   929 %
   930 \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
   931 }
   932 \isamarkuptrue%
   933 %
   934 \begin{isamarkuptext}%
   935 The following tools of Isabelle/HOL support cases analysis and
   936   induction in unstructured tactic scripts; see also
   937   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   938 
   939   \begin{matharray}{rcl}
   940     \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} \\
   941     \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} \\
   942     \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} \\
   943     \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}} \\
   944   \end{matharray}
   945 
   946   \begin{rail}
   947     'case\_tac' goalspec? term rule?
   948     ;
   949     'induct\_tac' goalspec? (insts * 'and') rule?
   950     ;
   951     'ind\_cases' (prop +) ('for' (name +)) ?
   952     ;
   953     'inductive\_cases' (thmdecl? (prop +) + 'and')
   954     ;
   955 
   956     rule: ('rule' ':' thmref)
   957     ;
   958   \end{rail}
   959 
   960   \begin{description}
   961 
   962   \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit
   963   to reason about inductive types.  Rules are selected according to
   964   the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
   965   attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
   966 
   967   These unstructured tactics feature both goal addressing and dynamic
   968   instantiation.  Note that named rule cases are \emph{not} provided
   969   as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
   970   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
   971   statements, only the compact object-logic conclusion of the subgoal
   972   being addressed.
   973   
   974   \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
   975   forward manner.
   976 
   977   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
   978   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
   979   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
   980   be generalized before applying the resulting rule.
   981 
   982   \end{description}%
   983 \end{isamarkuptext}%
   984 \isamarkuptrue%
   985 %
   986 \isamarkupsection{Executable code%
   987 }
   988 \isamarkuptrue%
   989 %
   990 \begin{isamarkuptext}%
   991 Isabelle/Pure provides two generic frameworks to support code
   992   generation from executable specifications.  Isabelle/HOL
   993   instantiates these mechanisms in a way that is amenable to end-user
   994   applications.
   995 
   996   One framework generates code from both functional and relational
   997   programs to SML.  See \cite{isabelle-HOL} for further information
   998   (this actually covers the new-style theory format as well).
   999 
  1000   \begin{matharray}{rcl}
  1001     \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}} \\
  1002     \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}} \\
  1003     \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}} \\
  1004     \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}} \\
  1005     \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}} \\  
  1006     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
  1007   \end{matharray}
  1008 
  1009   \begin{rail}
  1010   'value' term
  1011   ;
  1012 
  1013   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1014     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1015     'contains' ( ( name '=' term ) + | term + )
  1016   ;
  1017 
  1018   modespec: '(' ( name * ) ')'
  1019   ;
  1020 
  1021   'consts\_code' (codespec +)
  1022   ;
  1023 
  1024   codespec: const template attachment ?
  1025   ;
  1026 
  1027   'types\_code' (tycodespec +)
  1028   ;
  1029 
  1030   tycodespec: name template attachment ?
  1031   ;
  1032 
  1033   const: term
  1034   ;
  1035 
  1036   template: '(' string ')'
  1037   ;
  1038 
  1039   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1040   ;
  1041 
  1042   'code' (name)?
  1043   ;
  1044   \end{rail}
  1045 
  1046   \begin{description}
  1047 
  1048   \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a term
  1049   using the code generator.
  1050 
  1051   \end{description}
  1052 
  1053   \medskip The other framework generates code from functional programs
  1054   (including overloading using type classes) to SML \cite{SML}, OCaml
  1055   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
  1056   Conceptually, code generation is split up in three steps:
  1057   \emph{selection} of code theorems, \emph{translation} into an
  1058   abstract executable view and \emph{serialization} to a specific
  1059   \emph{target language}.  See \cite{isabelle-codegen} for an
  1060   introduction on how to use it.
  1061 
  1062   \begin{matharray}{rcl}
  1063     \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}} \\
  1064     \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}} \\
  1065     \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}} \\
  1066     \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}} \\
  1067     \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}} \\
  1068     \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}} \\
  1069     \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}} \\
  1070     \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}} \\
  1071     \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}} \\
  1072     \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}} \\
  1073     \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}} \\
  1074     \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}} \\
  1075     \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}} \\
  1076     \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}} \\
  1077     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
  1078   \end{matharray}
  1079 
  1080   \begin{rail}
  1081     'export\_code' ( constexpr + ) ? \\
  1082       ( ( 'in' target ( 'module\_name' string ) ? \\
  1083         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1084     ;
  1085 
  1086     'code\_thms' ( constexpr + ) ?
  1087     ;
  1088 
  1089     'code\_deps' ( constexpr + ) ?
  1090     ;
  1091 
  1092     const: term
  1093     ;
  1094 
  1095     constexpr: ( const | 'name.*' | '*' )
  1096     ;
  1097 
  1098     typeconstructor: nameref
  1099     ;
  1100 
  1101     class: nameref
  1102     ;
  1103 
  1104     target: 'OCaml' | 'SML' | 'Haskell'
  1105     ;
  1106 
  1107     'code\_datatype' const +
  1108     ;
  1109 
  1110     'code\_const' (const + 'and') \\
  1111       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1112     ;
  1113 
  1114     'code\_type' (typeconstructor + 'and') \\
  1115       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1116     ;
  1117 
  1118     'code\_class' (class + 'and') \\
  1119       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1120     ;
  1121 
  1122     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1123       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1124     ;
  1125 
  1126     'code\_monad' const const target
  1127     ;
  1128 
  1129     'code\_reserved' target ( string + )
  1130     ;
  1131 
  1132     'code\_include' target ( string ( string | '-') )
  1133     ;
  1134 
  1135     'code\_modulename' target ( ( string string ) + )
  1136     ;
  1137 
  1138     'code\_abort' ( const + )
  1139     ;
  1140 
  1141     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1142     ;
  1143 
  1144     'code' ( 'inline' ) ? ( 'del' ) ?
  1145     ;
  1146   \end{rail}
  1147 
  1148   \begin{description}
  1149 
  1150   \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} is the canonical interface for
  1151   generating and serializing code: for a given list of constants, code
  1152   is generated for the specified target languages.  Abstract code is
  1153   cached incrementally.  If no constant is given, the currently cached
  1154   code is serialized.  If no serialization instruction is given, only
  1155   abstract code is cached.
  1156 
  1157   Constants may be specified by giving them literally, referring to
  1158   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
  1159   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
  1160 
  1161   By default, for each involved theory one corresponding name space
  1162   module is generated.  Alternativly, a module name may be specified
  1163   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
  1164   placed in this module.
  1165 
  1166   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1167   single file; for \emph{Haskell}, it refers to a whole directory,
  1168   where code is generated in multiple files reflecting the module
  1169   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
  1170   output.  For \emph{SML}, omitting the file specification compiles
  1171   code internally in the context of the current ML session.
  1172 
  1173   Serializers take an optional list of arguments in parentheses.  For
  1174   \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
  1175   declaration.
  1176 
  1177   \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
  1178   representing the corresponding program containing all given
  1179   constants; if no constants are given, the currently cached code
  1180   theorems are printed.
  1181 
  1182   \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
  1183   theorems representing the corresponding program containing all given
  1184   constants; if no constants are given, the currently cached code
  1185   theorems are visualized.
  1186 
  1187   \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
  1188   for a logical type.
  1189 
  1190   \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
  1191   with target-specific serializations; omitting a serialization
  1192   deletes an existing serialization.
  1193 
  1194   \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
  1195   constructors with target-specific serializations; omitting a
  1196   serialization deletes an existing serialization.
  1197 
  1198   \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
  1199   with target-specific class names; omitting a serialization deletes
  1200   an existing serialization.  This applies only to \emph{Haskell}.
  1201 
  1202   \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
  1203   constructor / class instance relations as ``already present'' for a
  1204   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
  1205   ``already present'' declaration.  This applies only to
  1206   \emph{Haskell}.
  1207 
  1208   \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
  1209   to generate monadic code for Haskell.
  1210 
  1211   \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
  1212   reserved for a given target, preventing it to be shadowed by any
  1213   generated code.
  1214 
  1215   \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
  1216   (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
  1217   will remove an already added ``include''.
  1218 
  1219   \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
  1220   module name onto another.
  1221 
  1222   \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
  1223   required to have a definition by means of code equations; if
  1224   needed these are implemented by program abort instead.
  1225 
  1226   \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
  1227   ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code
  1228   generation.  Usually packages introducing code equations provide
  1229   a reasonable default setup for selection.
  1230 
  1231   \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}~\isa{inline} declares (or with
  1232   option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
  1233   applied as rewrite rules to any code equation during
  1234   preprocessing.
  1235 
  1236   \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
  1237   selected code equations, code generator datatypes and
  1238   preprocessor setup.
  1239 
  1240   \end{description}%
  1241 \end{isamarkuptext}%
  1242 \isamarkuptrue%
  1243 %
  1244 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
  1245 }
  1246 \isamarkuptrue%
  1247 %
  1248 \begin{isamarkuptext}%
  1249 \begin{matharray}{rcl}
  1250     \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}} \\
  1251     \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}} \\
  1252   \end{matharray}
  1253 
  1254   \begin{rail}
  1255   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1256   ;
  1257   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1258   \end{rail}
  1259 
  1260   \begin{description}
  1261 
  1262   \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
  1263   goal stating the existence of terms with the properties specified to
  1264   hold for the constants given in \isa{decls}.  After finishing the
  1265   proof, the theory will be augmented with definitions for the given
  1266   constants, as well as with theorems stating the properties for these
  1267   constants.
  1268 
  1269   \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
  1270   a goal stating the existence of terms with the properties specified
  1271   to hold for the constants given in \isa{decls}.  After finishing
  1272   the proof, the theory will be augmented with axioms expressing the
  1273   properties given in the first place.
  1274 
  1275   \item \isa{decl} declares a constant to be defined by the
  1276   specification given.  The definition for the constant \isa{c} is
  1277   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
  1278   the declaration.  Overloaded constants should be declared as such.
  1279 
  1280   \end{description}
  1281 
  1282   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
  1283   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
  1284   user has explicitly proven it to be safe.  A practical issue must be
  1285   considered, though: After introducing two constants with the same
  1286   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
  1287   that the two constants are, in fact, equal.  If this might be a
  1288   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
  1289 \end{isamarkuptext}%
  1290 \isamarkuptrue%
  1291 %
  1292 \isadelimtheory
  1293 %
  1294 \endisadelimtheory
  1295 %
  1296 \isatagtheory
  1297 \isacommand{end}\isamarkupfalse%
  1298 %
  1299 \endisatagtheory
  1300 {\isafoldtheory}%
  1301 %
  1302 \isadelimtheory
  1303 %
  1304 \endisadelimtheory
  1305 \isanewline
  1306 \end{isabellebody}%
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