doc-src/IsarRef/Thy/HOL_Specific.thy
 author wenzelm Sat Feb 28 17:08:33 2009 +0100 (2009-02-28) changeset 30171 5989863ffafc parent 30169 9531eaafd781 child 30242 aea5d7fa7ef5 permissions -rw-r--r--
tuned formal markup;
     1 theory HOL_Specific

     2 imports Main

     3 begin

     4

     5 chapter {* Isabelle/HOL \label{ch:hol} *}

     6

     7 section {* Primitive types \label{sec:hol-typedef} *}

     8

     9 text {*

    10   \begin{matharray}{rcl}

    11     @{command_def (HOL) "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\

    12     @{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

    13   \end{matharray}

    14

    15   \begin{rail}

    16     'typedecl' typespec infix?

    17     ;

    18     'typedef' altname? abstype '=' repset

    19     ;

    20

    21     altname: '(' (name | 'open' | 'open' name) ')'

    22     ;

    23     abstype: typespec infix?

    24     ;

    25     repset: term ('morphisms' name name)?

    26     ;

    27   \end{rail}

    28

    29   \begin{description}

    30

    31   \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar

    32   to the original @{command "typedecl"} of Isabelle/Pure (see

    33   \secref{sec:types-pure}), but also declares type arity @{text "t ::

    34   (type, \<dots>, type) type"}, making @{text t} an actual HOL type

    35   constructor.  %FIXME check, update

    36

    37   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up

    38   a goal stating non-emptiness of the set @{text A}.  After finishing

    39   the proof, the theory will be augmented by a Gordon/HOL-style type

    40   definition, which establishes a bijection between the representing

    41   set @{text A} and the new type @{text t}.

    42

    43   Technically, @{command (HOL) "typedef"} defines both a type @{text

    44   t} and a set (term constant) of the same name (an alternative base

    45   name may be given in parentheses).  The injection from type to set

    46   is called @{text Rep_t}, its inverse @{text Abs_t} (this may be

    47   changed via an explicit @{keyword (HOL) "morphisms"} declaration).

    48

    49   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text

    50   Abs_t_inverse} provide the most basic characterization as a

    51   corresponding injection/surjection pair (in both directions).  Rules

    52   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly

    53   more convenient view on the injectivity part, suitable for automated

    54   proof tools (e.g.\ in @{attribute simp} or @{attribute iff}

    55   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and

    56   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views

    57   on surjectivity; these are already declared as set or type rules for

    58   the generic @{method cases} and @{method induct} methods.

    59

    60   An alternative name may be specified in parentheses; the default is

    61   to use @{text t} as indicated before.  The @{text "(open)"}''

    62   declaration suppresses a separate constant definition for the

    63   representing set.

    64

    65   \end{description}

    66

    67   Note that raw type declarations are rarely used in practice; the

    68   main application is with experimental (or even axiomatic!) theory

    69   fragments.  Instead of primitive HOL type definitions, user-level

    70   theories usually refer to higher-level packages such as @{command

    71   (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)

    72   "datatype"} (see \secref{sec:hol-datatype}).

    73 *}

    74

    75

    76 section {* Adhoc tuples *}

    77

    78 text {*

    79   \begin{matharray}{rcl}

    80     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\

    81   \end{matharray}

    82

    83   \begin{rail}

    84     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))

    85     ;

    86   \end{rail}

    87

    88   \begin{description}

    89

    90   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>

    91   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into

    92   canonical form as specified by the arguments given; the @{text i}-th

    93   collection of arguments refers to occurrences in premise @{text i}

    94   of the rule.  The @{text "(complete)"}'' option causes \emph{all}

    95   arguments in function applications to be represented canonically

    96   according to their tuple type structure.

    97

    98   Note that these operations tend to invent funny names for new local

    99   parameters to be introduced.

   100

   101   \end{description}

   102 *}

   103

   104

   105 section {* Records \label{sec:hol-record} *}

   106

   107 text {*

   108   In principle, records merely generalize the concept of tuples, where

   109   components may be addressed by labels instead of just position.  The

   110   logical infrastructure of records in Isabelle/HOL is slightly more

   111   advanced, though, supporting truly extensible record schemes.  This

   112   admits operations that are polymorphic with respect to record

   113   extension, yielding object-oriented'' effects like (single)

   114   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more

   115   details on object-oriented verification and record subtyping in HOL.

   116 *}

   117

   118

   119 subsection {* Basic concepts *}

   120

   121 text {*

   122   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records

   123   at the level of terms and types.  The notation is as follows:

   124

   125   \begin{center}

   126   \begin{tabular}{l|l|l}

   127     & record terms & record types \\ \hline

   128     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\

   129     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &

   130       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\

   131   \end{tabular}

   132   \end{center}

   133

   134   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text

   135   "(| x = a |)"}.

   136

   137   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value

   138   @{text a} and field @{text y} of value @{text b}.  The corresponding

   139   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}

   140   and @{text "b :: B"}.

   141

   142   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields

   143   @{text x} and @{text y} as before, but also possibly further fields

   144   as indicated by the @{text "\<dots>"}'' notation (which is actually part

   145   of the syntax).  The improper field @{text "\<dots>"}'' of a record

   146   scheme is called the \emph{more part}.  Logically it is just a free

   147   variable, which is occasionally referred to as row variable'' in

   148   the literature.  The more part of a record scheme may be

   149   instantiated by zero or more further components.  For example, the

   150   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =

   151   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.

   152   Fixed records are special instances of record schemes, where

   153   @{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}

   154   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation

   155   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.

   156

   157   \medskip Two key observations make extensible records in a simply

   158   typed language like HOL work out:

   159

   160   \begin{enumerate}

   161

   162   \item the more part is internalized, as a free term or type

   163   variable,

   164

   165   \item field names are externalized, they cannot be accessed within

   166   the logic as first-class values.

   167

   168   \end{enumerate}

   169

   170   \medskip In Isabelle/HOL record types have to be defined explicitly,

   171   fixing their field names and types, and their (optional) parent

   172   record.  Afterwards, records may be formed using above syntax, while

   173   obeying the canonical order of fields as given by their declaration.

   174   The record package provides several standard operations like

   175   selectors and updates.  The common setup for various generic proof

   176   tools enable succinct reasoning patterns.  See also the Isabelle/HOL

   177   tutorial \cite{isabelle-hol-book} for further instructions on using

   178   records in practice.

   179 *}

   180

   181

   182 subsection {* Record specifications *}

   183

   184 text {*

   185   \begin{matharray}{rcl}

   186     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\

   187   \end{matharray}

   188

   189   \begin{rail}

   190     'record' typespec '=' (type '+')? (constdecl +)

   191     ;

   192   \end{rail}

   193

   194   \begin{description}

   195

   196   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1

   197   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},

   198   derived from the optional parent record @{text "\<tau>"} by adding new

   199   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.

   200

   201   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be

   202   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,

   203   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text

   204   \<tau>} needs to specify an instance of an existing record type.  At

   205   least one new field @{text "c\<^sub>i"} has to be specified.

   206   Basically, field names need to belong to a unique record.  This is

   207   not a real restriction in practice, since fields are qualified by

   208   the record name internally.

   209

   210   The parent record specification @{text \<tau>} is optional; if omitted

   211   @{text t} becomes a root record.  The hierarchy of all records

   212   declared within a theory context forms a forest structure, i.e.\ a

   213   set of trees starting with a root record each.  There is no way to

   214   merge multiple parent records!

   215

   216   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a

   217   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::

   218   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text

   219   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for

   220   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::

   221   \<zeta>\<rparr>"}.

   222

   223   \end{description}

   224 *}

   225

   226

   227 subsection {* Record operations *}

   228

   229 text {*

   230   Any record definition of the form presented above produces certain

   231   standard operations.  Selectors and updates are provided for any

   232   field, including the improper one @{text more}''.  There are also

   233   cumulative record constructor functions.  To simplify the

   234   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,

   235   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::

   236   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.

   237

   238   \medskip \textbf{Selectors} and \textbf{updates} are available for

   239   any field (including @{text more}''):

   240

   241   \begin{matharray}{lll}

   242     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

   243     @{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>"} \\

   244   \end{matharray}

   245

   246   There is special syntax for application of updates: @{text "r\<lparr>x :=

   247   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for

   248   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=

   249   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that

   250   because of postfix notation the order of fields shown here is

   251   reverse than in the actual term.  Since repeated updates are just

   252   function applications, fields may be freely permuted in @{text "\<lparr>x

   253   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.

   254   Thus commutativity of independent updates can be proven within the

   255   logic for any two fields, but not as a general theorem.

   256

   257   \medskip The \textbf{make} operation provides a cumulative record

   258   constructor function:

   259

   260   \begin{matharray}{lll}

   261     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   262   \end{matharray}

   263

   264   \medskip We now reconsider the case of non-root records, which are

   265   derived of some parent.  In general, the latter may depend on

   266   another parent as well, resulting in a list of \emph{ancestor

   267   records}.  Appending the lists of fields of all ancestors results in

   268   a certain field prefix.  The record package automatically takes care

   269   of this by lifting operations over this context of ancestor fields.

   270   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor

   271   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},

   272   the above record operations will get the following types:

   273

   274   \medskip

   275   \begin{tabular}{lll}

   276     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

   277     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>

   278       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>

   279       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

   280     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>

   281       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   282   \end{tabular}

   283   \medskip

   284

   285   \noindent Some further operations address the extension aspect of a

   286   derived record scheme specifically: @{text "t.fields"} produces a

   287   record fragment consisting of exactly the new fields introduced here

   288   (the result may serve as a more part elsewhere); @{text "t.extend"}

   289   takes a fixed record and adds a given more part; @{text

   290   "t.truncate"} restricts a record scheme to a fixed record.

   291

   292   \medskip

   293   \begin{tabular}{lll}

   294     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   295     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>

   296       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

   297     @{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>"} \\

   298   \end{tabular}

   299   \medskip

   300

   301   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide

   302   for root records.

   303 *}

   304

   305

   306 subsection {* Derived rules and proof tools *}

   307

   308 text {*

   309   The record package proves several results internally, declaring

   310   these facts to appropriate proof tools.  This enables users to

   311   reason about record structures quite conveniently.  Assume that

   312   @{text t} is a record type as specified above.

   313

   314   \begin{enumerate}

   315

   316   \item Standard conversions for selectors or updates applied to

   317   record constructor terms are made part of the default Simplifier

   318   context; thus proofs by reduction of basic operations merely require

   319   the @{method simp} method without further arguments.  These rules

   320   are available as @{text "t.simps"}, too.

   321

   322   \item Selectors applied to updated records are automatically reduced

   323   by an internal simplification procedure, which is also part of the

   324   standard Simplifier setup.

   325

   326   \item Inject equations of a form analogous to @{prop "(x, y) = (x',

   327   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical

   328   Reasoner as @{attribute iff} rules.  These rules are available as

   329   @{text "t.iffs"}.

   330

   331   \item The introduction rule for record equality analogous to @{text

   332   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,

   333   and as the basic rule context as @{attribute intro}@{text "?"}''.

   334   The rule is called @{text "t.equality"}.

   335

   336   \item Representations of arbitrary record expressions as canonical

   337   constructor terms are provided both in @{method cases} and @{method

   338   induct} format (cf.\ the generic proof methods of the same name,

   339   \secref{sec:cases-induct}).  Several variations are available, for

   340   fixed records, record schemes, more parts etc.

   341

   342   The generic proof methods are sufficiently smart to pick the most

   343   sensible rule according to the type of the indicated record

   344   expression: users just need to apply something like @{text "(cases

   345   r)"}'' to a certain proof problem.

   346

   347   \item The derived record operations @{text "t.make"}, @{text

   348   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}

   349   treated automatically, but usually need to be expanded by hand,

   350   using the collective fact @{text "t.defs"}.

   351

   352   \end{enumerate}

   353 *}

   354

   355

   356 section {* Datatypes \label{sec:hol-datatype} *}

   357

   358 text {*

   359   \begin{matharray}{rcl}

   360     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\

   361   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   362   \end{matharray}

   363

   364   \begin{rail}

   365     'datatype' (dtspec + 'and')

   366     ;

   367     'rep\_datatype' ('(' (name +) ')')? (term +)

   368     ;

   369

   370     dtspec: parname? typespec infix? '=' (cons + '|')

   371     ;

   372     cons: name (type *) mixfix?

   373   \end{rail}

   374

   375   \begin{description}

   376

   377   \item @{command (HOL) "datatype"} defines inductive datatypes in

   378   HOL.

   379

   380   \item @{command (HOL) "rep_datatype"} represents existing types as

   381   inductive ones, generating the standard infrastructure of derived

   382   concepts (primitive recursion etc.).

   383

   384   \end{description}

   385

   386   The induction and exhaustion theorems generated provide case names

   387   according to the constructors involved, while parameters are named

   388   after the types (see also \secref{sec:cases-induct}).

   389

   390   See \cite{isabelle-HOL} for more details on datatypes, but beware of

   391   the old-style theory syntax being used there!  Apart from proper

   392   proof methods for case-analysis and induction, there are also

   393   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)

   394   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit

   395   to refer directly to the internal structure of subgoals (including

   396   internally bound parameters).

   397 *}

   398

   399

   400 section {* Recursive functions \label{sec:recursion} *}

   401

   402 text {*

   403   \begin{matharray}{rcl}

   404     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   405     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   406     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

   407     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

   408   \end{matharray}

   409

   410   \begin{rail}

   411     'primrec' target? fixes 'where' equations

   412     ;

   413     equations: (thmdecl? prop + '|')

   414     ;

   415     ('fun' | 'function') target? functionopts? fixes 'where' clauses

   416     ;

   417     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')

   418     ;

   419     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'

   420     ;

   421     'termination' ( term )?

   422   \end{rail}

   423

   424   \begin{description}

   425

   426   \item @{command (HOL) "primrec"} defines primitive recursive

   427   functions over datatypes, see also \cite{isabelle-HOL}.

   428

   429   \item @{command (HOL) "function"} defines functions by general

   430   wellfounded recursion. A detailed description with examples can be

   431   found in \cite{isabelle-function}. The function is specified by a

   432   set of (possibly conditional) recursive equations with arbitrary

   433   pattern matching. The command generates proof obligations for the

   434   completeness and the compatibility of patterns.

   435

   436   The defined function is considered partial, and the resulting

   437   simplification rules (named @{text "f.psimps"}) and induction rule

   438   (named @{text "f.pinduct"}) are guarded by a generated domain

   439   predicate @{text "f_dom"}. The @{command (HOL) "termination"}

   440   command can then be used to establish that the function is total.

   441

   442   \item @{command (HOL) "fun"} is a shorthand notation for @{command

   443   (HOL) "function"}~@{text "(sequential)"}, followed by automated

   444   proof attempts regarding pattern matching and termination.  See

   445   \cite{isabelle-function} for further details.

   446

   447   \item @{command (HOL) "termination"}~@{text f} commences a

   448   termination proof for the previously defined function @{text f}.  If

   449   this is omitted, the command refers to the most recent function

   450   definition.  After the proof is closed, the recursive equations and

   451   the induction principle is established.

   452

   453   \end{description}

   454

   455   %FIXME check

   456

   457   Recursive definitions introduced by the @{command (HOL) "function"}

   458   command accommodate

   459   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text

   460   "c.induct"} (where @{text c} is the name of the function definition)

   461   refers to a specific induction rule, with parameters named according

   462   to the user-specified equations.

   463   For the @{command (HOL) "primrec"} the induction principle coincides

   464   with structural recursion on the datatype the recursion is carried

   465   out.

   466   Case names of @{command (HOL)

   467   "primrec"} are that of the datatypes involved, while those of

   468   @{command (HOL) "function"} are numbered (starting from 1).

   469

   470   The equations provided by these packages may be referred later as

   471   theorem list @{text "f.simps"}, where @{text f} is the (collective)

   472   name of the functions defined.  Individual equations may be named

   473   explicitly as well.

   474

   475   The @{command (HOL) "function"} command accepts the following

   476   options.

   477

   478   \begin{description}

   479

   480   \item @{text sequential} enables a preprocessor which disambiguates

   481   overlapping patterns by making them mutually disjoint.  Earlier

   482   equations take precedence over later ones.  This allows to give the

   483   specification in a format very similar to functional programming.

   484   Note that the resulting simplification and induction rules

   485   correspond to the transformed specification, not the one given

   486   originally. This usually means that each equation given by the user

   487   may result in several theroems.  Also note that this automatic

   488   transformation only works for ML-style datatype patterns.

   489

   490   \item @{text domintros} enables the automated generation of

   491   introduction rules for the domain predicate. While mostly not

   492   needed, they can be helpful in some proofs about partial functions.

   493

   494   \item @{text tailrec} generates the unconstrained recursive

   495   equations even without a termination proof, provided that the

   496   function is tail-recursive. This currently only works

   497

   498   \item @{text "default d"} allows to specify a default value for a

   499   (partial) function, which will ensure that @{text "f x = d x"}

   500   whenever @{text "x \<notin> f_dom"}.

   501

   502   \end{description}

   503 *}

   504

   505

   506 subsection {* Proof methods related to recursive definitions *}

   507

   508 text {*

   509   \begin{matharray}{rcl}

   510     @{method_def (HOL) pat_completeness} & : & @{text method} \\

   511     @{method_def (HOL) relation} & : & @{text method} \\

   512     @{method_def (HOL) lexicographic_order} & : & @{text method} \\

   513   \end{matharray}

   514

   515   \begin{rail}

   516     'relation' term

   517     ;

   518     'lexicographic\_order' (clasimpmod *)

   519     ;

   520   \end{rail}

   521

   522   \begin{description}

   523

   524   \item @{method (HOL) pat_completeness} is a specialized method to

   525   solve goals regarding the completeness of pattern matching, as

   526   required by the @{command (HOL) "function"} package (cf.\

   527   \cite{isabelle-function}).

   528

   529   \item @{method (HOL) relation}~@{text R} introduces a termination

   530   proof using the relation @{text R}.  The resulting proof state will

   531   contain goals expressing that @{text R} is wellfounded, and that the

   532   arguments of recursive calls decrease with respect to @{text R}.

   533   Usually, this method is used as the initial proof step of manual

   534   termination proofs.

   535

   536   \item @{method (HOL) "lexicographic_order"} attempts a fully

   537   automated termination proof by searching for a lexicographic

   538   combination of size measures on the arguments of the function. The

   539   method accepts the same arguments as the @{method auto} method,

   540   which it uses internally to prove local descents.  The same context

   541   modifiers as for @{method auto} are accepted, see

   542   \secref{sec:clasimp}.

   543

   544   In case of failure, extensive information is printed, which can help

   545   to analyse the situation (cf.\ \cite{isabelle-function}).

   546

   547   \end{description}

   548 *}

   549

   550

   551 subsection {* Old-style recursive function definitions (TFL) *}

   552

   553 text {*

   554   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)

   555   "recdef_tc"} for defining recursive are mostly obsolete; @{command

   556   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.

   557

   558   \begin{matharray}{rcl}

   559     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\

   560     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   561   \end{matharray}

   562

   563   \begin{rail}

   564     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?

   565     ;

   566     recdeftc thmdecl? tc

   567     ;

   568     hints: '(' 'hints' (recdefmod *) ')'

   569     ;

   570     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod

   571     ;

   572     tc: nameref ('(' nat ')')?

   573     ;

   574   \end{rail}

   575

   576   \begin{description}

   577

   578   \item @{command (HOL) "recdef"} defines general well-founded

   579   recursive functions (using the TFL package), see also

   580   \cite{isabelle-HOL}.  The @{text "(permissive)"}'' option tells

   581   TFL to recover from failed proof attempts, returning unfinished

   582   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text

   583   recdef_wf} hints refer to auxiliary rules to be used in the internal

   584   automated proof process of TFL.  Additional @{syntax clasimpmod}

   585   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the

   586   context of the Simplifier (cf.\ \secref{sec:simplifier}) and

   587   Classical reasoner (cf.\ \secref{sec:classical}).

   588

   589   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the

   590   proof for leftover termination condition number @{text i} (default

   591   1) as generated by a @{command (HOL) "recdef"} definition of

   592   constant @{text c}.

   593

   594   Note that in most cases, @{command (HOL) "recdef"} is able to finish

   595   its internal proofs without manual intervention.

   596

   597   \end{description}

   598

   599   \medskip Hints for @{command (HOL) "recdef"} may be also declared

   600   globally, using the following attributes.

   601

   602   \begin{matharray}{rcl}

   603     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\

   604     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\

   605     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\

   606   \end{matharray}

   607

   608   \begin{rail}

   609     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')

   610     ;

   611   \end{rail}

   612 *}

   613

   614

   615 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}

   616

   617 text {*

   618   An \textbf{inductive definition} specifies the least predicate (or

   619   set) @{text R} closed under given rules: applying a rule to elements

   620   of @{text R} yields a result within @{text R}.  For example, a

   621   structural operational semantics is an inductive definition of an

   622   evaluation relation.

   623

   624   Dually, a \textbf{coinductive definition} specifies the greatest

   625   predicate~/ set @{text R} that is consistent with given rules: every

   626   element of @{text R} can be seen as arising by applying a rule to

   627   elements of @{text R}.  An important example is using bisimulation

   628   relations to formalise equivalence of processes and infinite data

   629   structures.

   630

   631   \medskip The HOL package is related to the ZF one, which is

   632   described in a separate paper,\footnote{It appeared in CADE

   633   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}

   634   which you should refer to in case of difficulties.  The package is

   635   simpler than that of ZF thanks to implicit type-checking in HOL.

   636   The types of the (co)inductive predicates (or sets) determine the

   637   domain of the fixedpoint definition, and the package does not have

   638   to use inference rules for type-checking.

   639

   640   \begin{matharray}{rcl}

   641     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   642     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   643     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   644     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   645     @{attribute_def (HOL) mono} & : & @{text attribute} \\

   646   \end{matharray}

   647

   648   \begin{rail}

   649     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\

   650     ('where' clauses)? ('monos' thmrefs)?

   651     ;

   652     clauses: (thmdecl? prop + '|')

   653     ;

   654     'mono' (() | 'add' | 'del')

   655     ;

   656   \end{rail}

   657

   658   \begin{description}

   659

   660   \item @{command (HOL) "inductive"} and @{command (HOL)

   661   "coinductive"} define (co)inductive predicates from the

   662   introduction rules given in the @{keyword "where"} part.  The

   663   optional @{keyword "for"} part contains a list of parameters of the

   664   (co)inductive predicates that remain fixed throughout the

   665   definition.  The optional @{keyword "monos"} section contains

   666   \emph{monotonicity theorems}, which are required for each operator

   667   applied to a recursive set in the introduction rules.  There

   668   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},

   669   for each premise @{text "M R\<^sub>i t"} in an introduction rule!

   670

   671   \item @{command (HOL) "inductive_set"} and @{command (HOL)

   672   "coinductive_set"} are wrappers for to the previous commands,

   673   allowing the definition of (co)inductive sets.

   674

   675   \item @{attribute (HOL) mono} declares monotonicity rules.  These

   676   rule are involved in the automated monotonicity proof of @{command

   677   (HOL) "inductive"}.

   678

   679   \end{description}

   680 *}

   681

   682

   683 subsection {* Derived rules *}

   684

   685 text {*

   686   Each (co)inductive definition @{text R} adds definitions to the

   687   theory and also proves some theorems:

   688

   689   \begin{description}

   690

   691   \item @{text R.intros} is the list of introduction rules as proven

   692   theorems, for the recursive predicates (or sets).  The rules are

   693   also available individually, using the names given them in the

   694   theory file;

   695

   696   \item @{text R.cases} is the case analysis (or elimination) rule;

   697

   698   \item @{text R.induct} or @{text R.coinduct} is the (co)induction

   699   rule.

   700

   701   \end{description}

   702

   703   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are

   704   defined simultaneously, the list of introduction rules is called

   705   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are

   706   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list

   707   of mutual induction rules is called @{text

   708   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.

   709 *}

   710

   711

   712 subsection {* Monotonicity theorems *}

   713

   714 text {*

   715   Each theory contains a default set of theorems that are used in

   716   monotonicity proofs.  New rules can be added to this set via the

   717   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}

   718   shows how this is done.  In general, the following monotonicity

   719   theorems may be added:

   720

   721   \begin{itemize}

   722

   723   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving

   724   monotonicity of inductive definitions whose introduction rules have

   725   premises involving terms such as @{text "M R\<^sub>i t"}.

   726

   727   \item Monotonicity theorems for logical operators, which are of the

   728   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in

   729   the case of the operator @{text "\<or>"}, the corresponding theorem is

   730   $  731 \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"}}   732$

   733

   734   \item De Morgan style equations for reasoning about the polarity''

   735   of expressions, e.g.

   736   $  737 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad   738 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}   739$

   740

   741   \item Equations for reducing complex operators to more primitive

   742   ones whose monotonicity can easily be proved, e.g.

   743   $  744 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad   745 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}   746$

   747

   748   \end{itemize}

   749

   750   %FIXME: Example of an inductive definition

   751 *}

   752

   753

   754 section {* Arithmetic proof support *}

   755

   756 text {*

   757   \begin{matharray}{rcl}

   758     @{method_def (HOL) arith} & : & @{text method} \\

   759     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\

   760   \end{matharray}

   761

   762   The @{method (HOL) arith} method decides linear arithmetic problems

   763   (on types @{text nat}, @{text int}, @{text real}).  Any current

   764   facts are inserted into the goal before running the procedure.

   765

   766   The @{attribute (HOL) arith_split} attribute declares case split

   767   rules to be expanded before the arithmetic procedure is invoked.

   768

   769   Note that a simpler (but faster) version of arithmetic reasoning is

   770   already performed by the Simplifier.

   771 *}

   772

   773

   774 section {* Intuitionistic proof search *}

   775

   776 text {*

   777   \begin{matharray}{rcl}

   778     @{method_def (HOL) iprover} & : & @{text method} \\

   779   \end{matharray}

   780

   781   \begin{rail}

   782     'iprover' ('!' ?) (rulemod *)

   783     ;

   784   \end{rail}

   785

   786   The @{method (HOL) iprover} method performs intuitionistic proof

   787   search, depending on specifically declared rules from the context,

   788   or given as explicit arguments.  Chained facts are inserted into the

   789   goal before commencing proof search; @{method (HOL) iprover}@{text

   790   "!"}''  means to include the current @{fact prems} as well.

   791

   792   Rules need to be classified as @{attribute (Pure) intro},

   793   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the

   794   @{text "!"}'' indicator refers to safe'' rules, which may be

   795   applied aggressively (without considering back-tracking later).

   796   Rules declared with @{text "?"}'' are ignored in proof search (the

   797   single-step @{method rule} method still observes these).  An

   798   explicit weight annotation may be given as well; otherwise the

   799   number of rule premises will be taken into account here.

   800 *}

   801

   802

   803 section {* Coherent Logic *}

   804

   805 text {*

   806   \begin{matharray}{rcl}

   807     @{method_def (HOL) "coherent"} & : & @{text method} \\

   808   \end{matharray}

   809

   810   \begin{rail}

   811     'coherent' thmrefs?

   812     ;

   813   \end{rail}

   814

   815   The @{method (HOL) coherent} method solves problems of

   816   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers

   817   applications in confluence theory, lattice theory and projective

   818   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some

   819   examples.

   820 *}

   821

   822

   823 section {* Invoking automated reasoning tools -- The Sledgehammer *}

   824

   825 text {*

   826   Isabelle/HOL includes a generic \emph{ATP manager} that allows

   827   external automated reasoning tools to crunch a pending goal.

   828   Supported provers include E\footnote{\url{http://www.eprover.org}},

   829   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.

   830   There is also a wrapper to invoke provers remotely via the

   831   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}

   832   web service.

   833

   834   The problem passed to external provers consists of the goal together

   835   with a smart selection of lemmas from the current theory context.

   836   The result of a successful proof search is some source text that

   837   usually reconstructs the proof within Isabelle, without requiring

   838   external provers again.  The Metis

   839   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is

   840   integrated into Isabelle/HOL is being used here.

   841

   842   In this mode of operation, heavy means of automated reasoning are

   843   used as a strong relevance filter, while the main proof checking

   844   works via explicit inferences going through the Isabelle kernel.

   845   Moreover, rechecking Isabelle proof texts with already specified

   846   auxiliary facts is much faster than performing fully automated

   847   search over and over again.

   848

   849   \begin{matharray}{rcl}

   850     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

   851     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

   852     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\

   853     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\

   854     @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\

   855     @{method_def (HOL) metis} & : & @{text method} \\

   856   \end{matharray}

   857

   858   \begin{rail}

   859   'sledgehammer' (nameref *)

   860   ;

   861   'atp\_messages' ('(' nat ')')?

   862   ;

   863

   864   'metis' thmrefs

   865   ;

   866   \end{rail}

   867

   868   \begin{description}

   869

   870   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}

   871   invokes the specified automated theorem provers on the first

   872   subgoal.  Provers are run in parallel, the first successful result

   873   is displayed, and the other attempts are terminated.

   874

   875   Provers are defined in the theory context, see also @{command (HOL)

   876   print_atps}.  If no provers are given as arguments to @{command

   877   (HOL) sledgehammer}, the system refers to the default defined as

   878   ATP provers'' preference by the user interface.

   879

   880   There are additional preferences for timeout (default: 60 seconds),

   881   and the maximum number of independent prover processes (default: 5);

   882   excessive provers are automatically terminated.

   883

   884   \item @{command (HOL) print_atps} prints the list of automated

   885   theorem provers available to the @{command (HOL) sledgehammer}

   886   command.

   887

   888   \item @{command (HOL) atp_info} prints information about presently

   889   running provers, including elapsed runtime, and the remaining time

   890   until timeout.

   891

   892   \item @{command (HOL) atp_kill} terminates all presently running

   893   provers.

   894

   895   \item @{command (HOL) atp_messages} displays recent messages issued

   896   by automated theorem provers.  This allows to examine results that

   897   might have got lost due to the asynchronous nature of default

   898   @{command (HOL) sledgehammer} output.  An optional message limit may

   899   be specified (default 5).

   900

   901   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover

   902   with the given facts.  Metis is an automated proof tool of medium

   903   strength, but is fully integrated into Isabelle/HOL, with explicit

   904   inferences going through the kernel.  Thus its results are

   905   guaranteed to be correct by construction''.

   906

   907   Note that all facts used with Metis need to be specified as explicit

   908   arguments.  There are no rule declarations as for other Isabelle

   909   provers, like @{method blast} or @{method fast}.

   910

   911   \end{description}

   912 *}

   913

   914

   915 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}

   916

   917 text {*

   918   The following tools of Isabelle/HOL support cases analysis and

   919   induction in unstructured tactic scripts; see also

   920   \secref{sec:cases-induct} for proper Isar versions of similar ideas.

   921

   922   \begin{matharray}{rcl}

   923     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\

   924     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\

   925     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\

   926     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   927   \end{matharray}

   928

   929   \begin{rail}

   930     'case\_tac' goalspec? term rule?

   931     ;

   932     'induct\_tac' goalspec? (insts * 'and') rule?

   933     ;

   934     'ind\_cases' (prop +) ('for' (name +)) ?

   935     ;

   936     'inductive\_cases' (thmdecl? (prop +) + 'and')

   937     ;

   938

   939     rule: ('rule' ':' thmref)

   940     ;

   941   \end{rail}

   942

   943   \begin{description}

   944

   945   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit

   946   to reason about inductive types.  Rules are selected according to

   947   the declarations by the @{attribute cases} and @{attribute induct}

   948   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)

   949   datatype} package already takes care of this.

   950

   951   These unstructured tactics feature both goal addressing and dynamic

   952   instantiation.  Note that named rule cases are \emph{not} provided

   953   as would be by the proper @{method cases} and @{method induct} proof

   954   methods (see \secref{sec:cases-induct}).  Unlike the @{method

   955   induct} method, @{method induct_tac} does not handle structured rule

   956   statements, only the compact object-logic conclusion of the subgoal

   957   being addressed.

   958

   959   \item @{method (HOL) ind_cases} and @{command (HOL)

   960   "inductive_cases"} provide an interface to the internal @{ML_text

   961   mk_cases} operation.  Rules are simplified in an unrestricted

   962   forward manner.

   963

   964   While @{method (HOL) ind_cases} is a proof method to apply the

   965   result immediately as elimination rules, @{command (HOL)

   966   "inductive_cases"} provides case split theorems at the theory level

   967   for later use.  The @{keyword "for"} argument of the @{method (HOL)

   968   ind_cases} method allows to specify a list of variables that should

   969   be generalized before applying the resulting rule.

   970

   971   \end{description}

   972 *}

   973

   974

   975 section {* Executable code *}

   976

   977 text {*

   978   Isabelle/Pure provides two generic frameworks to support code

   979   generation from executable specifications.  Isabelle/HOL

   980   instantiates these mechanisms in a way that is amenable to end-user

   981   applications.

   982

   983   One framework generates code from both functional and relational

   984   programs to SML.  See \cite{isabelle-HOL} for further information

   985   (this actually covers the new-style theory format as well).

   986

   987   \begin{matharray}{rcl}

   988     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

   989     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\

   990     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\

   991     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\

   992     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\

   993     @{attribute_def (HOL) code} & : & @{text attribute} \\

   994   \end{matharray}

   995

   996   \begin{rail}

   997   'value' term

   998   ;

   999

  1000   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\

  1001     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\

  1002     'contains' ( ( name '=' term ) + | term + )

  1003   ;

  1004

  1005   modespec: '(' ( name * ) ')'

  1006   ;

  1007

  1008   'consts\_code' (codespec +)

  1009   ;

  1010

  1011   codespec: const template attachment ?

  1012   ;

  1013

  1014   'types\_code' (tycodespec +)

  1015   ;

  1016

  1017   tycodespec: name template attachment ?

  1018   ;

  1019

  1020   const: term

  1021   ;

  1022

  1023   template: '(' string ')'

  1024   ;

  1025

  1026   attachment: 'attach' modespec ? verblbrace text verbrbrace

  1027   ;

  1028

  1029   'code' (name)?

  1030   ;

  1031   \end{rail}

  1032

  1033   \begin{description}

  1034

  1035   \item @{command (HOL) "value"}~@{text t} evaluates and prints a term

  1036   using the code generator.

  1037

  1038   \end{description}

  1039

  1040   \medskip The other framework generates code from functional programs

  1041   (including overloading using type classes) to SML \cite{SML}, OCaml

  1042   \cite{OCaml} and Haskell \cite{haskell-revised-report}.

  1043   Conceptually, code generation is split up in three steps:

  1044   \emph{selection} of code theorems, \emph{translation} into an

  1045   abstract executable view and \emph{serialization} to a specific

  1046   \emph{target language}.  See \cite{isabelle-codegen} for an

  1047   introduction on how to use it.

  1048

  1049   \begin{matharray}{rcl}

  1050     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1051     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1052     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1053     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\

  1054     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\

  1055     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\

  1056     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\

  1057     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\

  1058     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\

  1059     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\

  1060     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\

  1061     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\

  1062     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\

  1063     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1064     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1065   \end{matharray}

  1066

  1067   \begin{rail}

  1068     'export\_code' ( constexpr + ) ? \\

  1069       ( ( 'in' target ( 'module\_name' string ) ? \\

  1070         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

  1071     ;

  1072

  1073     'code\_thms' ( constexpr + ) ?

  1074     ;

  1075

  1076     'code\_deps' ( constexpr + ) ?

  1077     ;

  1078

  1079     const: term

  1080     ;

  1081

  1082     constexpr: ( const | 'name.*' | '*' )

  1083     ;

  1084

  1085     typeconstructor: nameref

  1086     ;

  1087

  1088     class: nameref

  1089     ;

  1090

  1091     target: 'OCaml' | 'SML' | 'Haskell'

  1092     ;

  1093

  1094     'code\_datatype' const +

  1095     ;

  1096

  1097     'code\_const' (const + 'and') \\

  1098       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1099     ;

  1100

  1101     'code\_type' (typeconstructor + 'and') \\

  1102       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1103     ;

  1104

  1105     'code\_class' (class + 'and') \\

  1106       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )

  1107     ;

  1108

  1109     'code\_instance' (( typeconstructor '::' class ) + 'and') \\

  1110       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )

  1111     ;

  1112

  1113     'code\_monad' const const target

  1114     ;

  1115

  1116     'code\_reserved' target ( string + )

  1117     ;

  1118

  1119     'code\_include' target ( string ( string | '-') )

  1120     ;

  1121

  1122     'code\_modulename' target ( ( string string ) + )

  1123     ;

  1124

  1125     'code\_abort' ( const + )

  1126     ;

  1127

  1128     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string

  1129     ;

  1130

  1131     'code' ( 'inline' ) ? ( 'del' ) ?

  1132     ;

  1133   \end{rail}

  1134

  1135   \begin{description}

  1136

  1137   \item @{command (HOL) "export_code"} is the canonical interface for

  1138   generating and serializing code: for a given list of constants, code

  1139   is generated for the specified target languages.  Abstract code is

  1140   cached incrementally.  If no constant is given, the currently cached

  1141   code is serialized.  If no serialization instruction is given, only

  1142   abstract code is cached.

  1143

  1144   Constants may be specified by giving them literally, referring to

  1145   all executable contants within a certain theory by giving @{text

  1146   "name.*"}, or referring to \emph{all} executable constants currently

  1147   available by giving @{text "*"}.

  1148

  1149   By default, for each involved theory one corresponding name space

  1150   module is generated.  Alternativly, a module name may be specified

  1151   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1152   placed in this module.

  1153

  1154   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1155   single file; for \emph{Haskell}, it refers to a whole directory,

  1156   where code is generated in multiple files reflecting the module

  1157   hierarchy.  The file specification @{text "-"}'' denotes standard

  1158   output.  For \emph{SML}, omitting the file specification compiles

  1159   code internally in the context of the current ML session.

  1160

  1161   Serializers take an optional list of arguments in parentheses.  For

  1162   \emph{Haskell} a module name prefix may be given using the @{text

  1163   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1164   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1165   declaration.

  1166

  1167   \item @{command (HOL) "code_thms"} prints a list of theorems

  1168   representing the corresponding program containing all given

  1169   constants; if no constants are given, the currently cached code

  1170   theorems are printed.

  1171

  1172   \item @{command (HOL) "code_deps"} visualizes dependencies of

  1173   theorems representing the corresponding program containing all given

  1174   constants; if no constants are given, the currently cached code

  1175   theorems are visualized.

  1176

  1177   \item @{command (HOL) "code_datatype"} specifies a constructor set

  1178   for a logical type.

  1179

  1180   \item @{command (HOL) "code_const"} associates a list of constants

  1181   with target-specific serializations; omitting a serialization

  1182   deletes an existing serialization.

  1183

  1184   \item @{command (HOL) "code_type"} associates a list of type

  1185   constructors with target-specific serializations; omitting a

  1186   serialization deletes an existing serialization.

  1187

  1188   \item @{command (HOL) "code_class"} associates a list of classes

  1189   with target-specific class names; omitting a serialization deletes

  1190   an existing serialization.  This applies only to \emph{Haskell}.

  1191

  1192   \item @{command (HOL) "code_instance"} declares a list of type

  1193   constructor / class instance relations as already present'' for a

  1194   given target.  Omitting a @{text "-"}'' deletes an existing

  1195   already present'' declaration.  This applies only to

  1196   \emph{Haskell}.

  1197

  1198   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism

  1199   to generate monadic code for Haskell.

  1200

  1201   \item @{command (HOL) "code_reserved"} declares a list of names as

  1202   reserved for a given target, preventing it to be shadowed by any

  1203   generated code.

  1204

  1205   \item @{command (HOL) "code_include"} adds arbitrary named content

  1206   (include'') to generated code.  A @{text "-"}'' as last argument

  1207   will remove an already added include''.

  1208

  1209   \item @{command (HOL) "code_modulename"} declares aliasings from one

  1210   module name onto another.

  1211

  1212   \item @{command (HOL) "code_abort"} declares constants which are not

  1213   required to have a definition by means of code equations; if

  1214   needed these are implemented by program abort instead.

  1215

  1216   \item @{attribute (HOL) code} explicitly selects (or with option

  1217   @{text "del"}'' deselects) a code equation for code

  1218   generation.  Usually packages introducing code equations provide

  1219   a reasonable default setup for selection.

  1220

  1221   \item @{attribute (HOL) code}~@{text inline} declares (or with

  1222   option @{text "del"}'' removes) inlining theorems which are

  1223   applied as rewrite rules to any code equation during

  1224   preprocessing.

  1225

  1226   \item @{command (HOL) "print_codesetup"} gives an overview on

  1227   selected code equations, code generator datatypes and

  1228   preprocessor setup.

  1229

  1230   \end{description}

  1231 *}

  1232

  1233

  1234 section {* Definition by specification \label{sec:hol-specification} *}

  1235

  1236 text {*

  1237   \begin{matharray}{rcl}

  1238     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1239     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1240   \end{matharray}

  1241

  1242   \begin{rail}

  1243   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

  1244   ;

  1245   decl: ((name ':')? term '(' 'overloaded' ')'?)

  1246   \end{rail}

  1247

  1248   \begin{description}

  1249

  1250   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a

  1251   goal stating the existence of terms with the properties specified to

  1252   hold for the constants given in @{text decls}.  After finishing the

  1253   proof, the theory will be augmented with definitions for the given

  1254   constants, as well as with theorems stating the properties for these

  1255   constants.

  1256

  1257   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up

  1258   a goal stating the existence of terms with the properties specified

  1259   to hold for the constants given in @{text decls}.  After finishing

  1260   the proof, the theory will be augmented with axioms expressing the

  1261   properties given in the first place.

  1262

  1263   \item @{text decl} declares a constant to be defined by the

  1264   specification given.  The definition for the constant @{text c} is

  1265   bound to the name @{text c_def} unless a theorem name is given in

  1266   the declaration.  Overloaded constants should be declared as such.

  1267

  1268   \end{description}

  1269

  1270   Whether to use @{command (HOL) "specification"} or @{command (HOL)

  1271   "ax_specification"} is to some extent a matter of style.  @{command

  1272   (HOL) "specification"} introduces no new axioms, and so by

  1273   construction cannot introduce inconsistencies, whereas @{command

  1274   (HOL) "ax_specification"} does introduce axioms, but only after the

  1275   user has explicitly proven it to be safe.  A practical issue must be

  1276   considered, though: After introducing two constants with the same

  1277   properties using @{command (HOL) "specification"}, one can prove

  1278   that the two constants are, in fact, equal.  If this might be a

  1279   problem, one should use @{command (HOL) "ax_specification"}.

  1280 *}

  1281

  1282 end