doc-src/IsarRef/Thy/HOL_Specific.thy
author wenzelm
Fri Mar 19 00:46:08 2010 +0100 (2010-03-19)
changeset 35841 94f901e4969a
parent 35744 93603d7b8ee9
child 36139 0c2538afe8e8
permissions -rw-r--r--
allow sort constraints in HOL/typedef;
     1 theory HOL_Specific
     2 imports Main
     3 begin
     4 
     5 chapter {* Isabelle/HOL \label{ch:hol} *}
     6 
     7 section {* Typedef axiomatization \label{sec:hol-typedef} *}
     8 
     9 text {*
    10   \begin{matharray}{rcl}
    11     @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
    12   \end{matharray}
    13 
    14   \begin{rail}
    15     'typedef' altname? abstype '=' repset
    16     ;
    17 
    18     altname: '(' (name | 'open' | 'open' name) ')'
    19     ;
    20     abstype: typespecsorts mixfix?
    21     ;
    22     repset: term ('morphisms' name name)?
    23     ;
    24   \end{rail}
    25 
    26   \begin{description}
    27   
    28   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
    29   axiomatizes a Gordon/HOL-style type definition in the background
    30   theory of the current context, depending on a non-emptiness result
    31   of the set @{text A} (which needs to be proven interactively).
    32 
    33   The raw type may not depend on parameters or assumptions of the
    34   context --- this is logically impossible in Isabelle/HOL --- but the
    35   non-emptiness property can be local, potentially resulting in
    36   multiple interpretations in target contexts.  Thus the established
    37   bijection between the representing set @{text A} and the new type
    38   @{text t} may semantically depend on local assumptions.
    39   
    40   By default, @{command (HOL) "typedef"} defines both a type @{text t}
    41   and a set (term constant) of the same name, unless an alternative
    42   base name is given in parentheses, or the ``@{text "(open)"}''
    43   declaration is used to suppress a separate constant definition
    44   altogether.  The injection from type to set is called @{text Rep_t},
    45   its inverse @{text Abs_t} --- this may be changed via an explicit
    46   @{keyword (HOL) "morphisms"} declaration.
    47   
    48   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
    49   Abs_t_inverse} provide the most basic characterization as a
    50   corresponding injection/surjection pair (in both directions).  Rules
    51   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
    52   more convenient view on the injectivity part, suitable for automated
    53   proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
    54   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
    55   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
    56   on surjectivity; these are already declared as set or type rules for
    57   the generic @{method cases} and @{method induct} methods.
    58   
    59   An alternative name for the set definition (and other derived
    60   entities) may be specified in parentheses; the default is to use
    61   @{text t} as indicated before.
    62 
    63   \end{description}
    64 *}
    65 
    66 
    67 section {* Adhoc tuples *}
    68 
    69 text {*
    70   \begin{matharray}{rcl}
    71     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
    72   \end{matharray}
    73 
    74   \begin{rail}
    75     'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
    76     ;
    77   \end{rail}
    78 
    79   \begin{description}
    80   
    81   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
    82   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
    83   canonical form as specified by the arguments given; the @{text i}-th
    84   collection of arguments refers to occurrences in premise @{text i}
    85   of the rule.  The ``@{text "(complete)"}'' option causes \emph{all}
    86   arguments in function applications to be represented canonically
    87   according to their tuple type structure.
    88 
    89   Note that these operations tend to invent funny names for new local
    90   parameters to be introduced.
    91 
    92   \end{description}
    93 *}
    94 
    95 
    96 section {* Records \label{sec:hol-record} *}
    97 
    98 text {*
    99   In principle, records merely generalize the concept of tuples, where
   100   components may be addressed by labels instead of just position.  The
   101   logical infrastructure of records in Isabelle/HOL is slightly more
   102   advanced, though, supporting truly extensible record schemes.  This
   103   admits operations that are polymorphic with respect to record
   104   extension, yielding ``object-oriented'' effects like (single)
   105   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   106   details on object-oriented verification and record subtyping in HOL.
   107 *}
   108 
   109 
   110 subsection {* Basic concepts *}
   111 
   112 text {*
   113   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   114   at the level of terms and types.  The notation is as follows:
   115 
   116   \begin{center}
   117   \begin{tabular}{l|l|l}
   118     & record terms & record types \\ \hline
   119     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
   120     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
   121       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
   122   \end{tabular}
   123   \end{center}
   124 
   125   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
   126   "(| x = a |)"}.
   127 
   128   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
   129   @{text a} and field @{text y} of value @{text b}.  The corresponding
   130   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
   131   and @{text "b :: B"}.
   132 
   133   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
   134   @{text x} and @{text y} as before, but also possibly further fields
   135   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
   136   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
   137   scheme is called the \emph{more part}.  Logically it is just a free
   138   variable, which is occasionally referred to as ``row variable'' in
   139   the literature.  The more part of a record scheme may be
   140   instantiated by zero or more further components.  For example, the
   141   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
   142   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
   143   Fixed records are special instances of record schemes, where
   144   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
   145   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
   146   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
   147   
   148   \medskip Two key observations make extensible records in a simply
   149   typed language like HOL work out:
   150 
   151   \begin{enumerate}
   152 
   153   \item the more part is internalized, as a free term or type
   154   variable,
   155 
   156   \item field names are externalized, they cannot be accessed within
   157   the logic as first-class values.
   158 
   159   \end{enumerate}
   160 
   161   \medskip In Isabelle/HOL record types have to be defined explicitly,
   162   fixing their field names and types, and their (optional) parent
   163   record.  Afterwards, records may be formed using above syntax, while
   164   obeying the canonical order of fields as given by their declaration.
   165   The record package provides several standard operations like
   166   selectors and updates.  The common setup for various generic proof
   167   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   168   tutorial \cite{isabelle-hol-book} for further instructions on using
   169   records in practice.
   170 *}
   171 
   172 
   173 subsection {* Record specifications *}
   174 
   175 text {*
   176   \begin{matharray}{rcl}
   177     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
   178   \end{matharray}
   179 
   180   \begin{rail}
   181     'record' typespec '=' (type '+')? (constdecl +)
   182     ;
   183   \end{rail}
   184 
   185   \begin{description}
   186 
   187   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
   188   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
   189   derived from the optional parent record @{text "\<tau>"} by adding new
   190   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
   191 
   192   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
   193   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
   194   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
   195   \<tau>} needs to specify an instance of an existing record type.  At
   196   least one new field @{text "c\<^sub>i"} has to be specified.
   197   Basically, field names need to belong to a unique record.  This is
   198   not a real restriction in practice, since fields are qualified by
   199   the record name internally.
   200 
   201   The parent record specification @{text \<tau>} is optional; if omitted
   202   @{text t} becomes a root record.  The hierarchy of all records
   203   declared within a theory context forms a forest structure, i.e.\ a
   204   set of trees starting with a root record each.  There is no way to
   205   merge multiple parent records!
   206 
   207   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
   208   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
   209   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
   210   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
   211   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
   212   \<zeta>\<rparr>"}.
   213 
   214   \end{description}
   215 *}
   216 
   217 
   218 subsection {* Record operations *}
   219 
   220 text {*
   221   Any record definition of the form presented above produces certain
   222   standard operations.  Selectors and updates are provided for any
   223   field, including the improper one ``@{text more}''.  There are also
   224   cumulative record constructor functions.  To simplify the
   225   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
   226   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
   227   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
   228 
   229   \medskip \textbf{Selectors} and \textbf{updates} are available for
   230   any field (including ``@{text more}''):
   231 
   232   \begin{matharray}{lll}
   233     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   234     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   235   \end{matharray}
   236 
   237   There is special syntax for application of updates: @{text "r\<lparr>x :=
   238   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
   239   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
   240   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
   241   because of postfix notation the order of fields shown here is
   242   reverse than in the actual term.  Since repeated updates are just
   243   function applications, fields may be freely permuted in @{text "\<lparr>x
   244   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
   245   Thus commutativity of independent updates can be proven within the
   246   logic for any two fields, but not as a general theorem.
   247 
   248   \medskip The \textbf{make} operation provides a cumulative record
   249   constructor function:
   250 
   251   \begin{matharray}{lll}
   252     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   253   \end{matharray}
   254 
   255   \medskip We now reconsider the case of non-root records, which are
   256   derived of some parent.  In general, the latter may depend on
   257   another parent as well, resulting in a list of \emph{ancestor
   258   records}.  Appending the lists of fields of all ancestors results in
   259   a certain field prefix.  The record package automatically takes care
   260   of this by lifting operations over this context of ancestor fields.
   261   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
   262   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
   263   the above record operations will get the following types:
   264 
   265   \medskip
   266   \begin{tabular}{lll}
   267     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   268     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
   269       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
   270       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   271     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
   272       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   273   \end{tabular}
   274   \medskip
   275 
   276   \noindent Some further operations address the extension aspect of a
   277   derived record scheme specifically: @{text "t.fields"} produces a
   278   record fragment consisting of exactly the new fields introduced here
   279   (the result may serve as a more part elsewhere); @{text "t.extend"}
   280   takes a fixed record and adds a given more part; @{text
   281   "t.truncate"} restricts a record scheme to a fixed record.
   282 
   283   \medskip
   284   \begin{tabular}{lll}
   285     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   286     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
   287       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   288     @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   289   \end{tabular}
   290   \medskip
   291 
   292   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
   293   for root records.
   294 *}
   295 
   296 
   297 subsection {* Derived rules and proof tools *}
   298 
   299 text {*
   300   The record package proves several results internally, declaring
   301   these facts to appropriate proof tools.  This enables users to
   302   reason about record structures quite conveniently.  Assume that
   303   @{text t} is a record type as specified above.
   304 
   305   \begin{enumerate}
   306   
   307   \item Standard conversions for selectors or updates applied to
   308   record constructor terms are made part of the default Simplifier
   309   context; thus proofs by reduction of basic operations merely require
   310   the @{method simp} method without further arguments.  These rules
   311   are available as @{text "t.simps"}, too.
   312   
   313   \item Selectors applied to updated records are automatically reduced
   314   by an internal simplification procedure, which is also part of the
   315   standard Simplifier setup.
   316 
   317   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
   318   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
   319   Reasoner as @{attribute iff} rules.  These rules are available as
   320   @{text "t.iffs"}.
   321 
   322   \item The introduction rule for record equality analogous to @{text
   323   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
   324   and as the basic rule context as ``@{attribute intro}@{text "?"}''.
   325   The rule is called @{text "t.equality"}.
   326 
   327   \item Representations of arbitrary record expressions as canonical
   328   constructor terms are provided both in @{method cases} and @{method
   329   induct} format (cf.\ the generic proof methods of the same name,
   330   \secref{sec:cases-induct}).  Several variations are available, for
   331   fixed records, record schemes, more parts etc.
   332   
   333   The generic proof methods are sufficiently smart to pick the most
   334   sensible rule according to the type of the indicated record
   335   expression: users just need to apply something like ``@{text "(cases
   336   r)"}'' to a certain proof problem.
   337 
   338   \item The derived record operations @{text "t.make"}, @{text
   339   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
   340   treated automatically, but usually need to be expanded by hand,
   341   using the collective fact @{text "t.defs"}.
   342 
   343   \end{enumerate}
   344 *}
   345 
   346 
   347 section {* Datatypes \label{sec:hol-datatype} *}
   348 
   349 text {*
   350   \begin{matharray}{rcl}
   351     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   352   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   353   \end{matharray}
   354 
   355   \begin{rail}
   356     'datatype' (dtspec + 'and')
   357     ;
   358     'rep\_datatype' ('(' (name +) ')')? (term +)
   359     ;
   360 
   361     dtspec: parname? typespec mixfix? '=' (cons + '|')
   362     ;
   363     cons: name ( type * ) mixfix?
   364   \end{rail}
   365 
   366   \begin{description}
   367 
   368   \item @{command (HOL) "datatype"} defines inductive datatypes in
   369   HOL.
   370 
   371   \item @{command (HOL) "rep_datatype"} represents existing types as
   372   inductive ones, generating the standard infrastructure of derived
   373   concepts (primitive recursion etc.).
   374 
   375   \end{description}
   376 
   377   The induction and exhaustion theorems generated provide case names
   378   according to the constructors involved, while parameters are named
   379   after the types (see also \secref{sec:cases-induct}).
   380 
   381   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   382   the old-style theory syntax being used there!  Apart from proper
   383   proof methods for case-analysis and induction, there are also
   384   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
   385   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
   386   to refer directly to the internal structure of subgoals (including
   387   internally bound parameters).
   388 *}
   389 
   390 
   391 section {* Recursive functions \label{sec:recursion} *}
   392 
   393 text {*
   394   \begin{matharray}{rcl}
   395     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   396     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   397     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   398     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   399   \end{matharray}
   400 
   401   \begin{rail}
   402     'primrec' target? fixes 'where' equations
   403     ;
   404     equations: (thmdecl? prop + '|')
   405     ;
   406     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   407     ;
   408     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   409     ;
   410     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   411     ;
   412     'termination' ( term )?
   413   \end{rail}
   414 
   415   \begin{description}
   416 
   417   \item @{command (HOL) "primrec"} defines primitive recursive
   418   functions over datatypes, see also \cite{isabelle-HOL}.
   419 
   420   \item @{command (HOL) "function"} defines functions by general
   421   wellfounded recursion. A detailed description with examples can be
   422   found in \cite{isabelle-function}. The function is specified by a
   423   set of (possibly conditional) recursive equations with arbitrary
   424   pattern matching. The command generates proof obligations for the
   425   completeness and the compatibility of patterns.
   426 
   427   The defined function is considered partial, and the resulting
   428   simplification rules (named @{text "f.psimps"}) and induction rule
   429   (named @{text "f.pinduct"}) are guarded by a generated domain
   430   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   431   command can then be used to establish that the function is total.
   432 
   433   \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
   434   (HOL) "function"}~@{text "(sequential)"}, followed by automated
   435   proof attempts regarding pattern matching and termination.  See
   436   \cite{isabelle-function} for further details.
   437 
   438   \item @{command (HOL) "termination"}~@{text f} commences a
   439   termination proof for the previously defined function @{text f}.  If
   440   this is omitted, the command refers to the most recent function
   441   definition.  After the proof is closed, the recursive equations and
   442   the induction principle is established.
   443 
   444   \end{description}
   445 
   446   Recursive definitions introduced by the @{command (HOL) "function"}
   447   command accommodate
   448   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   449   "c.induct"} (where @{text c} is the name of the function definition)
   450   refers to a specific induction rule, with parameters named according
   451   to the user-specified equations. Cases are numbered (starting from 1).
   452 
   453   For @{command (HOL) "primrec"}, the induction principle coincides
   454   with structural recursion on the datatype the recursion is carried
   455   out.
   456 
   457   The equations provided by these packages may be referred later as
   458   theorem list @{text "f.simps"}, where @{text f} is the (collective)
   459   name of the functions defined.  Individual equations may be named
   460   explicitly as well.
   461 
   462   The @{command (HOL) "function"} command accepts the following
   463   options.
   464 
   465   \begin{description}
   466 
   467   \item @{text sequential} enables a preprocessor which disambiguates
   468   overlapping patterns by making them mutually disjoint.  Earlier
   469   equations take precedence over later ones.  This allows to give the
   470   specification in a format very similar to functional programming.
   471   Note that the resulting simplification and induction rules
   472   correspond to the transformed specification, not the one given
   473   originally. This usually means that each equation given by the user
   474   may result in several theroems.  Also note that this automatic
   475   transformation only works for ML-style datatype patterns.
   476 
   477   \item @{text domintros} enables the automated generation of
   478   introduction rules for the domain predicate. While mostly not
   479   needed, they can be helpful in some proofs about partial functions.
   480 
   481   \item @{text tailrec} generates the unconstrained recursive
   482   equations even without a termination proof, provided that the
   483   function is tail-recursive. This currently only works
   484 
   485   \item @{text "default d"} allows to specify a default value for a
   486   (partial) function, which will ensure that @{text "f x = d x"}
   487   whenever @{text "x \<notin> f_dom"}.
   488 
   489   \end{description}
   490 *}
   491 
   492 
   493 subsection {* Proof methods related to recursive definitions *}
   494 
   495 text {*
   496   \begin{matharray}{rcl}
   497     @{method_def (HOL) pat_completeness} & : & @{text method} \\
   498     @{method_def (HOL) relation} & : & @{text method} \\
   499     @{method_def (HOL) lexicographic_order} & : & @{text method} \\
   500     @{method_def (HOL) size_change} & : & @{text method} \\
   501   \end{matharray}
   502 
   503   \begin{rail}
   504     'relation' term
   505     ;
   506     'lexicographic\_order' ( clasimpmod * )
   507     ;
   508     'size\_change' ( orders ( clasimpmod * ) )
   509     ;
   510     orders: ( 'max' | 'min' | 'ms' ) *
   511   \end{rail}
   512 
   513   \begin{description}
   514 
   515   \item @{method (HOL) pat_completeness} is a specialized method to
   516   solve goals regarding the completeness of pattern matching, as
   517   required by the @{command (HOL) "function"} package (cf.\
   518   \cite{isabelle-function}).
   519 
   520   \item @{method (HOL) relation}~@{text R} introduces a termination
   521   proof using the relation @{text R}.  The resulting proof state will
   522   contain goals expressing that @{text R} is wellfounded, and that the
   523   arguments of recursive calls decrease with respect to @{text R}.
   524   Usually, this method is used as the initial proof step of manual
   525   termination proofs.
   526 
   527   \item @{method (HOL) "lexicographic_order"} attempts a fully
   528   automated termination proof by searching for a lexicographic
   529   combination of size measures on the arguments of the function. The
   530   method accepts the same arguments as the @{method auto} method,
   531   which it uses internally to prove local descents.  The same context
   532   modifiers as for @{method auto} are accepted, see
   533   \secref{sec:clasimp}.
   534 
   535   In case of failure, extensive information is printed, which can help
   536   to analyse the situation (cf.\ \cite{isabelle-function}).
   537 
   538   \item @{method (HOL) "size_change"} also works on termination goals,
   539   using a variation of the size-change principle, together with a
   540   graph decomposition technique (see \cite{krauss_phd} for details).
   541   Three kinds of orders are used internally: @{text max}, @{text min},
   542   and @{text ms} (multiset), which is only available when the theory
   543   @{text Multiset} is loaded. When no order kinds are given, they are
   544   tried in order. The search for a termination proof uses SAT solving
   545   internally.
   546 
   547  For local descent proofs, the same context modifiers as for @{method
   548   auto} are accepted, see \secref{sec:clasimp}.
   549 
   550   \end{description}
   551 *}
   552 
   553 
   554 subsection {* Old-style recursive function definitions (TFL) *}
   555 
   556 text {*
   557   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   558   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   559   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   560 
   561   \begin{matharray}{rcl}
   562     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
   563     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   564   \end{matharray}
   565 
   566   \begin{rail}
   567     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   568     ;
   569     recdeftc thmdecl? tc
   570     ;
   571     hints: '(' 'hints' ( recdefmod * ) ')'
   572     ;
   573     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   574     ;
   575     tc: nameref ('(' nat ')')?
   576     ;
   577   \end{rail}
   578 
   579   \begin{description}
   580   
   581   \item @{command (HOL) "recdef"} defines general well-founded
   582   recursive functions (using the TFL package), see also
   583   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   584   TFL to recover from failed proof attempts, returning unfinished
   585   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   586   recdef_wf} hints refer to auxiliary rules to be used in the internal
   587   automated proof process of TFL.  Additional @{syntax clasimpmod}
   588   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   589   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   590   Classical reasoner (cf.\ \secref{sec:classical}).
   591   
   592   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
   593   proof for leftover termination condition number @{text i} (default
   594   1) as generated by a @{command (HOL) "recdef"} definition of
   595   constant @{text c}.
   596   
   597   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   598   its internal proofs without manual intervention.
   599 
   600   \end{description}
   601 
   602   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   603   globally, using the following attributes.
   604 
   605   \begin{matharray}{rcl}
   606     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
   607     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   608     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   609   \end{matharray}
   610 
   611   \begin{rail}
   612     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   613     ;
   614   \end{rail}
   615 *}
   616 
   617 
   618 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   619 
   620 text {*
   621   An \textbf{inductive definition} specifies the least predicate (or
   622   set) @{text R} closed under given rules: applying a rule to elements
   623   of @{text R} yields a result within @{text R}.  For example, a
   624   structural operational semantics is an inductive definition of an
   625   evaluation relation.
   626 
   627   Dually, a \textbf{coinductive definition} specifies the greatest
   628   predicate~/ set @{text R} that is consistent with given rules: every
   629   element of @{text R} can be seen as arising by applying a rule to
   630   elements of @{text R}.  An important example is using bisimulation
   631   relations to formalise equivalence of processes and infinite data
   632   structures.
   633 
   634   \medskip The HOL package is related to the ZF one, which is
   635   described in a separate paper,\footnote{It appeared in CADE
   636   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   637   which you should refer to in case of difficulties.  The package is
   638   simpler than that of ZF thanks to implicit type-checking in HOL.
   639   The types of the (co)inductive predicates (or sets) determine the
   640   domain of the fixedpoint definition, and the package does not have
   641   to use inference rules for type-checking.
   642 
   643   \begin{matharray}{rcl}
   644     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   645     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   646     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   647     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   648     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   649   \end{matharray}
   650 
   651   \begin{rail}
   652     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   653     ('where' clauses)? ('monos' thmrefs)?
   654     ;
   655     clauses: (thmdecl? prop + '|')
   656     ;
   657     'mono' (() | 'add' | 'del')
   658     ;
   659   \end{rail}
   660 
   661   \begin{description}
   662 
   663   \item @{command (HOL) "inductive"} and @{command (HOL)
   664   "coinductive"} define (co)inductive predicates from the
   665   introduction rules given in the @{keyword "where"} part.  The
   666   optional @{keyword "for"} part contains a list of parameters of the
   667   (co)inductive predicates that remain fixed throughout the
   668   definition.  The optional @{keyword "monos"} section contains
   669   \emph{monotonicity theorems}, which are required for each operator
   670   applied to a recursive set in the introduction rules.  There
   671   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   672   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   673 
   674   \item @{command (HOL) "inductive_set"} and @{command (HOL)
   675   "coinductive_set"} are wrappers for to the previous commands,
   676   allowing the definition of (co)inductive sets.
   677 
   678   \item @{attribute (HOL) mono} declares monotonicity rules.  These
   679   rule are involved in the automated monotonicity proof of @{command
   680   (HOL) "inductive"}.
   681 
   682   \end{description}
   683 *}
   684 
   685 
   686 subsection {* Derived rules *}
   687 
   688 text {*
   689   Each (co)inductive definition @{text R} adds definitions to the
   690   theory and also proves some theorems:
   691 
   692   \begin{description}
   693 
   694   \item @{text R.intros} is the list of introduction rules as proven
   695   theorems, for the recursive predicates (or sets).  The rules are
   696   also available individually, using the names given them in the
   697   theory file;
   698 
   699   \item @{text R.cases} is the case analysis (or elimination) rule;
   700 
   701   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
   702   rule.
   703 
   704   \end{description}
   705 
   706   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   707   defined simultaneously, the list of introduction rules is called
   708   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   709   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   710   of mutual induction rules is called @{text
   711   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   712 *}
   713 
   714 
   715 subsection {* Monotonicity theorems *}
   716 
   717 text {*
   718   Each theory contains a default set of theorems that are used in
   719   monotonicity proofs.  New rules can be added to this set via the
   720   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   721   shows how this is done.  In general, the following monotonicity
   722   theorems may be added:
   723 
   724   \begin{itemize}
   725 
   726   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   727   monotonicity of inductive definitions whose introduction rules have
   728   premises involving terms such as @{text "M R\<^sub>i t"}.
   729 
   730   \item Monotonicity theorems for logical operators, which are of the
   731   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   732   the case of the operator @{text "\<or>"}, the corresponding theorem is
   733   \[
   734   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   735   \]
   736 
   737   \item De Morgan style equations for reasoning about the ``polarity''
   738   of expressions, e.g.
   739   \[
   740   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   741   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   742   \]
   743 
   744   \item Equations for reducing complex operators to more primitive
   745   ones whose monotonicity can easily be proved, e.g.
   746   \[
   747   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   748   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   749   \]
   750 
   751   \end{itemize}
   752 
   753   %FIXME: Example of an inductive definition
   754 *}
   755 
   756 
   757 section {* Arithmetic proof support *}
   758 
   759 text {*
   760   \begin{matharray}{rcl}
   761     @{method_def (HOL) arith} & : & @{text method} \\
   762     @{attribute_def (HOL) arith} & : & @{text attribute} \\
   763     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   764   \end{matharray}
   765 
   766   The @{method (HOL) arith} method decides linear arithmetic problems
   767   (on types @{text nat}, @{text int}, @{text real}).  Any current
   768   facts are inserted into the goal before running the procedure.
   769 
   770   The @{attribute (HOL) arith} attribute declares facts that are
   771   always supplied to the arithmetic provers implicitly.
   772 
   773   The @{attribute (HOL) arith_split} attribute declares case split
   774   rules to be expanded before @{method (HOL) arith} is invoked.
   775 
   776   Note that a simpler (but faster) arithmetic prover is
   777   already invoked by the Simplifier.
   778 *}
   779 
   780 
   781 section {* Intuitionistic proof search *}
   782 
   783 text {*
   784   \begin{matharray}{rcl}
   785     @{method_def (HOL) iprover} & : & @{text method} \\
   786   \end{matharray}
   787 
   788   \begin{rail}
   789     'iprover' ( rulemod * )
   790     ;
   791   \end{rail}
   792 
   793   The @{method (HOL) iprover} method performs intuitionistic proof
   794   search, depending on specifically declared rules from the context,
   795   or given as explicit arguments.  Chained facts are inserted into the
   796   goal before commencing proof search.
   797 
   798   Rules need to be classified as @{attribute (Pure) intro},
   799   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
   800   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
   801   applied aggressively (without considering back-tracking later).
   802   Rules declared with ``@{text "?"}'' are ignored in proof search (the
   803   single-step @{method rule} method still observes these).  An
   804   explicit weight annotation may be given as well; otherwise the
   805   number of rule premises will be taken into account here.
   806 *}
   807 
   808 
   809 section {* Coherent Logic *}
   810 
   811 text {*
   812   \begin{matharray}{rcl}
   813     @{method_def (HOL) "coherent"} & : & @{text method} \\
   814   \end{matharray}
   815 
   816   \begin{rail}
   817     'coherent' thmrefs?
   818     ;
   819   \end{rail}
   820 
   821   The @{method (HOL) coherent} method solves problems of
   822   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   823   applications in confluence theory, lattice theory and projective
   824   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
   825   examples.
   826 *}
   827 
   828 
   829 section {* Checking and refuting propositions *}
   830 
   831 text {*
   832   Identifying incorrect propositions usually involves evaluation of
   833   particular assignments and systematic counter example search.  This
   834   is supported by the following commands.
   835 
   836   \begin{matharray}{rcl}
   837     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   838     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   839     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
   840   \end{matharray}
   841 
   842   \begin{rail}
   843     'value' ( ( '[' name ']' ) ? ) modes? term
   844     ;
   845 
   846     'quickcheck' ( ( '[' args ']' ) ? ) nat?
   847     ;
   848 
   849     'quickcheck_params' ( ( '[' args ']' ) ? )
   850     ;
   851 
   852     modes: '(' (name + ) ')'
   853     ;
   854 
   855     args: ( name '=' value + ',' )
   856     ;
   857   \end{rail}
   858 
   859   \begin{description}
   860 
   861   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
   862     term; optionally @{text modes} can be specified, which are
   863     appended to the current print mode (see also \cite{isabelle-ref}).
   864     Internally, the evaluation is performed by registered evaluators,
   865     which are invoked sequentially until a result is returned.
   866     Alternatively a specific evaluator can be selected using square
   867     brackets; available evaluators include @{text nbe} for
   868     \emph{normalization by evaluation} and \emph{code} for code
   869     generation in SML.
   870 
   871   \item @{command (HOL) "quickcheck"} tests the current goal for
   872     counter examples using a series of arbitrary assignments for its
   873     free variables; by default the first subgoal is tested, an other
   874     can be selected explicitly using an optional goal index.
   875     A number of configuration options are supported for
   876     @{command (HOL) "quickcheck"}, notably:
   877 
   878     \begin{description}
   879 
   880       \item[size] specifies the maximum size of the search space for
   881         assignment values.
   882 
   883       \item[iterations] sets how many sets of assignments are
   884         generated for each particular size.
   885 
   886       \item[no\_assms] specifies whether assumptions in
   887         structured proofs should be ignored.
   888 
   889     \end{description}
   890 
   891     These option can be given within square brackets.
   892 
   893   \item @{command (HOL) "quickcheck_params"} changes quickcheck
   894     configuration options persitently.
   895 
   896   \end{description}
   897 *}
   898 
   899 
   900 section {* Invoking automated reasoning tools --- The Sledgehammer *}
   901 
   902 text {*
   903   Isabelle/HOL includes a generic \emph{ATP manager} that allows
   904   external automated reasoning tools to crunch a pending goal.
   905   Supported provers include E\footnote{\url{http://www.eprover.org}},
   906   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   907   There is also a wrapper to invoke provers remotely via the
   908   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   909   web service.
   910 
   911   The problem passed to external provers consists of the goal together
   912   with a smart selection of lemmas from the current theory context.
   913   The result of a successful proof search is some source text that
   914   usually reconstructs the proof within Isabelle, without requiring
   915   external provers again.  The Metis
   916   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   917   integrated into Isabelle/HOL is being used here.
   918 
   919   In this mode of operation, heavy means of automated reasoning are
   920   used as a strong relevance filter, while the main proof checking
   921   works via explicit inferences going through the Isabelle kernel.
   922   Moreover, rechecking Isabelle proof texts with already specified
   923   auxiliary facts is much faster than performing fully automated
   924   search over and over again.
   925 
   926   \begin{matharray}{rcl}
   927     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   928     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   929     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   930     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   931     @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   932     @{method_def (HOL) metis} & : & @{text method} \\
   933   \end{matharray}
   934 
   935   \begin{rail}
   936   'sledgehammer' ( nameref * )
   937   ;
   938   'atp\_messages' ('(' nat ')')?
   939   ;
   940 
   941   'metis' thmrefs
   942   ;
   943   \end{rail}
   944 
   945   \begin{description}
   946 
   947   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
   948   invokes the specified automated theorem provers on the first
   949   subgoal.  Provers are run in parallel, the first successful result
   950   is displayed, and the other attempts are terminated.
   951 
   952   Provers are defined in the theory context, see also @{command (HOL)
   953   print_atps}.  If no provers are given as arguments to @{command
   954   (HOL) sledgehammer}, the system refers to the default defined as
   955   ``ATP provers'' preference by the user interface.
   956 
   957   There are additional preferences for timeout (default: 60 seconds),
   958   and the maximum number of independent prover processes (default: 5);
   959   excessive provers are automatically terminated.
   960 
   961   \item @{command (HOL) print_atps} prints the list of automated
   962   theorem provers available to the @{command (HOL) sledgehammer}
   963   command.
   964 
   965   \item @{command (HOL) atp_info} prints information about presently
   966   running provers, including elapsed runtime, and the remaining time
   967   until timeout.
   968 
   969   \item @{command (HOL) atp_kill} terminates all presently running
   970   provers.
   971 
   972   \item @{command (HOL) atp_messages} displays recent messages issued
   973   by automated theorem provers.  This allows to examine results that
   974   might have got lost due to the asynchronous nature of default
   975   @{command (HOL) sledgehammer} output.  An optional message limit may
   976   be specified (default 5).
   977 
   978   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
   979   with the given facts.  Metis is an automated proof tool of medium
   980   strength, but is fully integrated into Isabelle/HOL, with explicit
   981   inferences going through the kernel.  Thus its results are
   982   guaranteed to be ``correct by construction''.
   983 
   984   Note that all facts used with Metis need to be specified as explicit
   985   arguments.  There are no rule declarations as for other Isabelle
   986   provers, like @{method blast} or @{method fast}.
   987 
   988   \end{description}
   989 *}
   990 
   991 
   992 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   993 
   994 text {*
   995   The following tools of Isabelle/HOL support cases analysis and
   996   induction in unstructured tactic scripts; see also
   997   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   998 
   999   \begin{matharray}{rcl}
  1000     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
  1001     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
  1002     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
  1003     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  1004   \end{matharray}
  1005 
  1006   \begin{rail}
  1007     'case\_tac' goalspec? term rule?
  1008     ;
  1009     'induct\_tac' goalspec? (insts * 'and') rule?
  1010     ;
  1011     'ind\_cases' (prop +) ('for' (name +)) ?
  1012     ;
  1013     'inductive\_cases' (thmdecl? (prop +) + 'and')
  1014     ;
  1015 
  1016     rule: ('rule' ':' thmref)
  1017     ;
  1018   \end{rail}
  1019 
  1020   \begin{description}
  1021 
  1022   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
  1023   to reason about inductive types.  Rules are selected according to
  1024   the declarations by the @{attribute cases} and @{attribute induct}
  1025   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
  1026   datatype} package already takes care of this.
  1027 
  1028   These unstructured tactics feature both goal addressing and dynamic
  1029   instantiation.  Note that named rule cases are \emph{not} provided
  1030   as would be by the proper @{method cases} and @{method induct} proof
  1031   methods (see \secref{sec:cases-induct}).  Unlike the @{method
  1032   induct} method, @{method induct_tac} does not handle structured rule
  1033   statements, only the compact object-logic conclusion of the subgoal
  1034   being addressed.
  1035   
  1036   \item @{method (HOL) ind_cases} and @{command (HOL)
  1037   "inductive_cases"} provide an interface to the internal @{ML_text
  1038   mk_cases} operation.  Rules are simplified in an unrestricted
  1039   forward manner.
  1040 
  1041   While @{method (HOL) ind_cases} is a proof method to apply the
  1042   result immediately as elimination rules, @{command (HOL)
  1043   "inductive_cases"} provides case split theorems at the theory level
  1044   for later use.  The @{keyword "for"} argument of the @{method (HOL)
  1045   ind_cases} method allows to specify a list of variables that should
  1046   be generalized before applying the resulting rule.
  1047 
  1048   \end{description}
  1049 *}
  1050 
  1051 
  1052 section {* Executable code *}
  1053 
  1054 text {*
  1055   Isabelle/Pure provides two generic frameworks to support code
  1056   generation from executable specifications.  Isabelle/HOL
  1057   instantiates these mechanisms in a way that is amenable to end-user
  1058   applications.
  1059 
  1060   One framework generates code from both functional and relational
  1061   programs to SML.  See \cite{isabelle-HOL} for further information
  1062   (this actually covers the new-style theory format as well).
  1063 
  1064   \begin{matharray}{rcl}
  1065     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
  1066     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
  1067     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
  1068     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
  1069     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1070   \end{matharray}
  1071 
  1072   \begin{rail}
  1073   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1074     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1075     'contains' ( ( name '=' term ) + | term + )
  1076   ;
  1077 
  1078   modespec: '(' ( name * ) ')'
  1079   ;
  1080 
  1081   'consts\_code' (codespec +)
  1082   ;
  1083 
  1084   codespec: const template attachment ?
  1085   ;
  1086 
  1087   'types\_code' (tycodespec +)
  1088   ;
  1089 
  1090   tycodespec: name template attachment ?
  1091   ;
  1092 
  1093   const: term
  1094   ;
  1095 
  1096   template: '(' string ')'
  1097   ;
  1098 
  1099   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1100   ;
  1101 
  1102   'code' (name)?
  1103   ;
  1104   \end{rail}
  1105 
  1106   \medskip The other framework generates code from functional programs
  1107   (including overloading using type classes) to SML \cite{SML}, OCaml
  1108   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
  1109   Conceptually, code generation is split up in three steps:
  1110   \emph{selection} of code theorems, \emph{translation} into an
  1111   abstract executable view and \emph{serialization} to a specific
  1112   \emph{target language}.  See \cite{isabelle-codegen} for an
  1113   introduction on how to use it.
  1114 
  1115   \begin{matharray}{rcl}
  1116     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1117     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1118     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1119     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
  1120     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
  1121     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
  1122     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
  1123     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
  1124     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
  1125     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
  1126     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
  1127     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
  1128     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
  1129     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1130     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1131     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1132   \end{matharray}
  1133 
  1134   \begin{rail}
  1135     'export\_code' ( constexpr + ) \\
  1136       ( ( 'in' target ( 'module\_name' string ) ? \\
  1137         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1138     ;
  1139 
  1140     'code\_thms' ( constexpr + ) ?
  1141     ;
  1142 
  1143     'code\_deps' ( constexpr + ) ?
  1144     ;
  1145 
  1146     const: term
  1147     ;
  1148 
  1149     constexpr: ( const | 'name.*' | '*' )
  1150     ;
  1151 
  1152     typeconstructor: nameref
  1153     ;
  1154 
  1155     class: nameref
  1156     ;
  1157 
  1158     target: 'OCaml' | 'SML' | 'Haskell'
  1159     ;
  1160 
  1161     'code\_datatype' const +
  1162     ;
  1163 
  1164     'code\_const' (const + 'and') \\
  1165       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1166     ;
  1167 
  1168     'code\_type' (typeconstructor + 'and') \\
  1169       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1170     ;
  1171 
  1172     'code\_class' (class + 'and') \\
  1173       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1174     ;
  1175 
  1176     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1177       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1178     ;
  1179 
  1180     'code\_monad' const const target
  1181     ;
  1182 
  1183     'code\_reserved' target ( string + )
  1184     ;
  1185 
  1186     'code\_include' target ( string ( string | '-') )
  1187     ;
  1188 
  1189     'code\_modulename' target ( ( string string ) + )
  1190     ;
  1191 
  1192     'code\_abort' ( const + )
  1193     ;
  1194 
  1195     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1196     ;
  1197 
  1198     'code' ( 'del' ) ?
  1199     ;
  1200 
  1201     'code_unfold' ( 'del' ) ?
  1202     ;
  1203 
  1204     'code_post' ( 'del' ) ?
  1205     ;
  1206   \end{rail}
  1207 
  1208   \begin{description}
  1209 
  1210   \item @{command (HOL) "export_code"} is the canonical interface for
  1211   generating and serializing code: for a given list of constants, code
  1212   is generated for the specified target languages.  If no serialization
  1213   instruction is given, only abstract code is generated internally.
  1214 
  1215   Constants may be specified by giving them literally, referring to
  1216   all executable contants within a certain theory by giving @{text
  1217   "name.*"}, or referring to \emph{all} executable constants currently
  1218   available by giving @{text "*"}.
  1219 
  1220   By default, for each involved theory one corresponding name space
  1221   module is generated.  Alternativly, a module name may be specified
  1222   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1223   placed in this module.
  1224 
  1225   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1226   single file; for \emph{Haskell}, it refers to a whole directory,
  1227   where code is generated in multiple files reflecting the module
  1228   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1229   output.  For \emph{SML}, omitting the file specification compiles
  1230   code internally in the context of the current ML session.
  1231 
  1232   Serializers take an optional list of arguments in parentheses.  For
  1233   \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
  1234   explicit module signatures.
  1235   
  1236   For \emph{Haskell} a module name prefix may be given using the ``@{text
  1237   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1238   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1239   declaration.
  1240 
  1241   \item @{command (HOL) "code_thms"} prints a list of theorems
  1242   representing the corresponding program containing all given
  1243   constants.
  1244 
  1245   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1246   theorems representing the corresponding program containing all given
  1247   constants.
  1248 
  1249   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1250   for a logical type.
  1251 
  1252   \item @{command (HOL) "code_const"} associates a list of constants
  1253   with target-specific serializations; omitting a serialization
  1254   deletes an existing serialization.
  1255 
  1256   \item @{command (HOL) "code_type"} associates a list of type
  1257   constructors with target-specific serializations; omitting a
  1258   serialization deletes an existing serialization.
  1259 
  1260   \item @{command (HOL) "code_class"} associates a list of classes
  1261   with target-specific class names; omitting a serialization deletes
  1262   an existing serialization.  This applies only to \emph{Haskell}.
  1263 
  1264   \item @{command (HOL) "code_instance"} declares a list of type
  1265   constructor / class instance relations as ``already present'' for a
  1266   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1267   ``already present'' declaration.  This applies only to
  1268   \emph{Haskell}.
  1269 
  1270   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1271   to generate monadic code for Haskell.
  1272 
  1273   \item @{command (HOL) "code_reserved"} declares a list of names as
  1274   reserved for a given target, preventing it to be shadowed by any
  1275   generated code.
  1276 
  1277   \item @{command (HOL) "code_include"} adds arbitrary named content
  1278   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1279   will remove an already added ``include''.
  1280 
  1281   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1282   module name onto another.
  1283 
  1284   \item @{command (HOL) "code_abort"} declares constants which are not
  1285   required to have a definition by means of code equations; if
  1286   needed these are implemented by program abort instead.
  1287 
  1288   \item @{attribute (HOL) code} explicitly selects (or with option
  1289   ``@{text "del"}'' deselects) a code equation for code
  1290   generation.  Usually packages introducing code equations provide
  1291   a reasonable default setup for selection.
  1292 
  1293   \item @{attribute (HOL) code_inline} declares (or with
  1294   option ``@{text "del"}'' removes) inlining theorems which are
  1295   applied as rewrite rules to any code equation during
  1296   preprocessing.
  1297 
  1298   \item @{attribute (HOL) code_post} declares (or with
  1299   option ``@{text "del"}'' removes) theorems which are
  1300   applied as rewrite rules to any result of an evaluation.
  1301 
  1302   \item @{command (HOL) "print_codesetup"} gives an overview on
  1303   selected code equations and code generator datatypes.
  1304 
  1305   \item @{command (HOL) "print_codeproc"} prints the setup
  1306   of the code generator preprocessor.
  1307 
  1308   \end{description}
  1309 *}
  1310 
  1311 
  1312 section {* Definition by specification \label{sec:hol-specification} *}
  1313 
  1314 text {*
  1315   \begin{matharray}{rcl}
  1316     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1317     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1318   \end{matharray}
  1319 
  1320   \begin{rail}
  1321   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1322   ;
  1323   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1324   \end{rail}
  1325 
  1326   \begin{description}
  1327 
  1328   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1329   goal stating the existence of terms with the properties specified to
  1330   hold for the constants given in @{text decls}.  After finishing the
  1331   proof, the theory will be augmented with definitions for the given
  1332   constants, as well as with theorems stating the properties for these
  1333   constants.
  1334 
  1335   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1336   a goal stating the existence of terms with the properties specified
  1337   to hold for the constants given in @{text decls}.  After finishing
  1338   the proof, the theory will be augmented with axioms expressing the
  1339   properties given in the first place.
  1340 
  1341   \item @{text decl} declares a constant to be defined by the
  1342   specification given.  The definition for the constant @{text c} is
  1343   bound to the name @{text c_def} unless a theorem name is given in
  1344   the declaration.  Overloaded constants should be declared as such.
  1345 
  1346   \end{description}
  1347 
  1348   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1349   "ax_specification"} is to some extent a matter of style.  @{command
  1350   (HOL) "specification"} introduces no new axioms, and so by
  1351   construction cannot introduce inconsistencies, whereas @{command
  1352   (HOL) "ax_specification"} does introduce axioms, but only after the
  1353   user has explicitly proven it to be safe.  A practical issue must be
  1354   considered, though: After introducing two constants with the same
  1355   properties using @{command (HOL) "specification"}, one can prove
  1356   that the two constants are, in fact, equal.  If this might be a
  1357   problem, one should use @{command (HOL) "ax_specification"}.
  1358 *}
  1359 
  1360 end