doc-src/IsarRef/Thy/HOL_Specific.thy
 author wenzelm Mon Dec 15 21:41:00 2008 +0100 (2008-12-15) changeset 29112 f2b45eea6dac parent 28761 9ec4482c9201 child 29114 715178f6ae31 permissions -rw-r--r--
added 'atp_messages' command, which displays recent messages synchronously;
     1 (* $Id$ *)

     2

     3 theory HOL_Specific

     4 imports Main

     5 begin

     6

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

     8

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

    10

    11 text {*

    12   \begin{matharray}{rcl}

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

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

    15   \end{matharray}

    16

    17   \begin{rail}

    18     'typedecl' typespec infix?

    19     ;

    20     'typedef' altname? abstype '=' repset

    21     ;

    22

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

    24     ;

    25     abstype: typespec infix?

    26     ;

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

    28     ;

    29   \end{rail}

    30

    31   \begin{description}

    32

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

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

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

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

    37   constructor.  %FIXME check, update

    38

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

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

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

    42   definition, which establishes a bijection between the representing

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

    44

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

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

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

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

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

    50

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

    52   Abs_t_inverse} provide the most basic characterization as a

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

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

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

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

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

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

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

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

    61

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

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

    64   declaration suppresses a separate constant definition for the

    65   representing set.

    66

    67   \end{description}

    68

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

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

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

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

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

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

    75 *}

    76

    77

    78 section {* Adhoc tuples *}

    79

    80 text {*

    81   \begin{matharray}{rcl}

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

    83   \end{matharray}

    84

    85   \begin{rail}

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

    87     ;

    88   \end{rail}

    89

    90   \begin{description}

    91

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

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

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

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

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

    97   arguments in function applications to be represented canonically

    98   according to their tuple type structure.

    99

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

   101   parameters to be introduced.

   102

   103   \end{description}

   104 *}

   105

   106

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

   108

   109 text {*

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

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

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

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

   114   admits operations that are polymorphic with respect to record

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

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

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

   118 *}

   119

   120

   121 subsection {* Basic concepts *}

   122

   123 text {*

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

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

   126

   127   \begin{center}

   128   \begin{tabular}{l|l|l}

   129     & record terms & record types \\ \hline

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

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

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

   133   \end{tabular}

   134   \end{center}

   135

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

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

   138

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

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

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

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

   143

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

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

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

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

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

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

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

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

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

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

   154   Fixed records are special instances of record schemes, where

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

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

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

   158

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

   160   typed language like HOL work out:

   161

   162   \begin{enumerate}

   163

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

   165   variable,

   166

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

   168   the logic as first-class values.

   169

   170   \end{enumerate}

   171

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

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

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

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

   176   The record package provides several standard operations like

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

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

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

   180   records in practice.

   181 *}

   182

   183

   184 subsection {* Record specifications *}

   185

   186 text {*

   187   \begin{matharray}{rcl}

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

   189   \end{matharray}

   190

   191   \begin{rail}

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

   193     ;

   194   \end{rail}

   195

   196   \begin{description}

   197

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

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

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

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

   202

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

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

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

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

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

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

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

   210   the record name internally.

   211

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

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

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

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

   216   merge multiple parent records!

   217

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

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

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

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

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

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

   224

   225   \end{description}

   226 *}

   227

   228

   229 subsection {* Record operations *}

   230

   231 text {*

   232   Any record definition of the form presented above produces certain

   233   standard operations.  Selectors and updates are provided for any

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

   235   cumulative record constructor functions.  To simplify the

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

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

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

   239

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

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

   242

   243   \begin{matharray}{lll}

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

   245     @{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>"} \\

   246   \end{matharray}

   247

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

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

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

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

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

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

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

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

   256   Thus commutativity of independent updates can be proven within the

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

   258

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

   260   constructor function:

   261

   262   \begin{matharray}{lll}

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

   264   \end{matharray}

   265

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

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

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

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

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

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

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

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

   274   the above record operations will get the following types:

   275

   276   \medskip

   277   \begin{tabular}{lll}

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

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

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

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

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

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

   284   \end{tabular}

   285   \medskip

   286

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

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

   289   record fragment consisting of exactly the new fields introduced here

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

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

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

   293

   294   \medskip

   295   \begin{tabular}{lll}

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

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

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

   299     @{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>"} \\

   300   \end{tabular}

   301   \medskip

   302

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

   304   for root records.

   305 *}

   306

   307

   308 subsection {* Derived rules and proof tools *}

   309

   310 text {*

   311   The record package proves several results internally, declaring

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

   313   reason about record structures quite conveniently.  Assume that

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

   315

   316   \begin{enumerate}

   317

   318   \item Standard conversions for selectors or updates applied to

   319   record constructor terms are made part of the default Simplifier

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

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

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

   323

   324   \item Selectors applied to updated records are automatically reduced

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

   326   standard Simplifier setup.

   327

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

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

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

   331   @{text "t.iffs"}.

   332

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

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

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

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

   337

   338   \item Representations of arbitrary record expressions as canonical

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

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

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

   342   fixed records, record schemes, more parts etc.

   343

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

   345   sensible rule according to the type of the indicated record

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

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

   348

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

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

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

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

   353

   354   \end{enumerate}

   355 *}

   356

   357

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

   359

   360 text {*

   361   \begin{matharray}{rcl}

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

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

   364   \end{matharray}

   365

   366   \begin{rail}

   367     'datatype' (dtspec + 'and')

   368     ;

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

   370     ;

   371

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

   373     ;

   374     cons: name (type *) mixfix?

   375   \end{rail}

   376

   377   \begin{description}

   378

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

   380   HOL.

   381

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

   383   inductive ones, generating the standard infrastructure of derived

   384   concepts (primitive recursion etc.).

   385

   386   \end{description}

   387

   388   The induction and exhaustion theorems generated provide case names

   389   according to the constructors involved, while parameters are named

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

   391

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

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

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

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

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

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

   398   internally bound parameters).

   399 *}

   400

   401

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

   403

   404 text {*

   405   \begin{matharray}{rcl}

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

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

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

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

   410   \end{matharray}

   411

   412   \begin{rail}

   413     'primrec' target? fixes 'where' equations

   414     ;

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

   416     ;

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

   418     ;

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

   420     ;

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

   422     ;

   423     'termination' ( term )?

   424   \end{rail}

   425

   426   \begin{description}

   427

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

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

   430

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

   432   wellfounded recursion. A detailed description with examples can be

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

   434   set of (possibly conditional) recursive equations with arbitrary

   435   pattern matching. The command generates proof obligations for the

   436   completeness and the compatibility of patterns.

   437

   438   The defined function is considered partial, and the resulting

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

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

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

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

   443

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

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

   446   proof attempts regarding pattern matching and termination.  See

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

   448

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

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

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

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

   453   the induction principle is established.

   454

   455   \end{description}

   456

   457   %FIXME check

   458

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

   460   command accommodate

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

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

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

   464   to the user-specified equations.

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

   466   with structural recursion on the datatype the recursion is carried

   467   out.

   468   Case names of @{command (HOL)

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

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

   471

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

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

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

   475   explicitly as well.

   476

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

   478   options.

   479

   480   \begin{description}

   481

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

   483   overlapping patterns by making them mutually disjoint.  Earlier

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

   485   specification in a format very similar to functional programming.

   486   Note that the resulting simplification and induction rules

   487   correspond to the transformed specification, not the one given

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

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

   490   transformation only works for ML-style datatype patterns.

   491

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

   493   introduction rules for the domain predicate. While mostly not

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

   495

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

   497   equations even without a termination proof, provided that the

   498   function is tail-recursive. This currently only works

   499

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

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

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

   503

   504   \end{description}

   505 *}

   506

   507

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

   509

   510 text {*

   511   \begin{matharray}{rcl}

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

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

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

   515   \end{matharray}

   516

   517   \begin{rail}

   518     'relation' term

   519     ;

   520     'lexicographic\_order' (clasimpmod *)

   521     ;

   522   \end{rail}

   523

   524   \begin{description}

   525

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

   527   solve goals regarding the completeness of pattern matching, as

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

   529   \cite{isabelle-function}).

   530

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

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

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

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

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

   536   termination proofs.

   537

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

   539   automated termination proof by searching for a lexicographic

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

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

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

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

   544   \secref{sec:clasimp}.

   545

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

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

   548

   549   \end{description}

   550 *}

   551

   552

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

   554

   555 text {*

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

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

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

   559

   560   \begin{matharray}{rcl}

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

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

   563   \end{matharray}

   564

   565   \begin{rail}

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

   567     ;

   568     recdeftc thmdecl? tc

   569     ;

   570     hints: '(' 'hints' (recdefmod *) ')'

   571     ;

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

   573     ;

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

   575     ;

   576   \end{rail}

   577

   578   \begin{description}

   579

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

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

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

   583   TFL to recover from failed proof attempts, returning unfinished

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

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

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

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

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

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

   590

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

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

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

   594   constant @{text c}.

   595

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

   597   its internal proofs without manual intervention.

   598

   599   \end{description}

   600

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

   602   globally, using the following attributes.

   603

   604   \begin{matharray}{rcl}

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

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

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

   608   \end{matharray}

   609

   610   \begin{rail}

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

   612     ;

   613   \end{rail}

   614 *}

   615

   616

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

   618

   619 text {*

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

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

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

   623   structural operational semantics is an inductive definition of an

   624   evaluation relation.

   625

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

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

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

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

   630   relations to formalise equivalence of processes and infinite data

   631   structures.

   632

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

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

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

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

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

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

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

   640   to use inference rules for type-checking.

   641

   642   \begin{matharray}{rcl}

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

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

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

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

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

   648   \end{matharray}

   649

   650   \begin{rail}

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

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

   653     ;

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

   655     ;

   656     'mono' (() | 'add' | 'del')

   657     ;

   658   \end{rail}

   659

   660   \begin{description}

   661

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

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

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

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

   666   (co)inductive predicates that remain fixed throughout the

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

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

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

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

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

   672

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

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

   675   allowing the definition of (co)inductive sets.

   676

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

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

   679   (HOL) "inductive"}.

   680

   681   \end{description}

   682 *}

   683

   684

   685 subsection {* Derived rules *}

   686

   687 text {*

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

   689   theory and also proves some theorems:

   690

   691   \begin{description}

   692

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

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

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

   696   theory file;

   697

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

   699

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

   701   rule.

   702

   703   \end{description}

   704

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

   706   defined simultaneously, the list of introduction rules is called

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

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

   709   of mutual induction rules is called @{text

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

   711 *}

   712

   713

   714 subsection {* Monotonicity theorems *}

   715

   716 text {*

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

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

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

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

   721   theorems may be added:

   722

   723   \begin{itemize}

   724

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

   726   monotonicity of inductive definitions whose introduction rules have

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

   728

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

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

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

   732   $  733 \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"}}   734$

   735

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

   737   of expressions, e.g.

   738   $  739 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad   740 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}   741$

   742

   743   \item Equations for reducing complex operators to more primitive

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

   745   $  746 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad   747 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}   748$

   749

   750   \end{itemize}

   751

   752   %FIXME: Example of an inductive definition

   753 *}

   754

   755

   756 section {* Arithmetic proof support *}

   757

   758 text {*

   759   \begin{matharray}{rcl}

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

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

   762   \end{matharray}

   763

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

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

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

   767

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

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

   770

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

   772   already performed by the Simplifier.

   773 *}

   774

   775

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

   777

   778 text {*

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

   780   external automated reasoning tools to crunch a pending goal.

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

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

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

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

   785   web service.

   786

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

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

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

   790   usually reconstructs the proof within Isabelle, without requiring

   791   external provers again.  The Metis

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

   793   integrated into Isabelle/HOL is being used here.

   794

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

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

   797   works via explicit inferences going through the Isabelle kernel.

   798   Moreover, rechecking Isabelle proof texts with already specified

   799   auxiliary facts is much faster than performing fully automated

   800   search over and over again.

   801

   802   \begin{matharray}{rcl}

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

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

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

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

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

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

   809   \end{matharray}

   810

   811   \begin{rail}

   812   'sledgehammer' (nameref *)

   813   ;

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

   815

   816   'metis' thmrefs

   817   ;

   818   \end{rail}

   819

   820   \begin{description}

   821

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

   823   invokes the specified automated theorem provers on the first

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

   825   is displayed, and the other attempts are terminated.

   826

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

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

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

   830   ATP provers'' preference by the user interface.

   831

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

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

   834   excessive provers are automatically terminated.

   835

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

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

   838   command.

   839

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

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

   842   until timeout.

   843

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

   845   provers.

   846

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

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

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

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

   851   be specified (default 5).

   852

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

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

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

   856   inferences going through the kernel.  Thus its results are

   857   guaranteed to be correct by construction''.

   858

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

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

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

   862

   863   \end{description}

   864 *}

   865

   866

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

   868

   869 text {*

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

   871   induction in unstructured tactic scripts; see also

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

   873

   874   \begin{matharray}{rcl}

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

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

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

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

   879   \end{matharray}

   880

   881   \begin{rail}

   882     'case\_tac' goalspec? term rule?

   883     ;

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

   885     ;

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

   887     ;

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

   889     ;

   890

   891     rule: ('rule' ':' thmref)

   892     ;

   893   \end{rail}

   894

   895   \begin{description}

   896

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

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

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

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

   901   datatype} package already takes care of this.

   902

   903   These unstructured tactics feature both goal addressing and dynamic

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

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

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

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

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

   909   being addressed.

   910

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

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

   913   mk_cases} operation.  Rules are simplified in an unrestricted

   914   forward manner.

   915

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

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

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

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

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

   921   be generalized before applying the resulting rule.

   922

   923   \end{description}

   924 *}

   925

   926

   927 section {* Executable code *}

   928

   929 text {*

   930   Isabelle/Pure provides two generic frameworks to support code

   931   generation from executable specifications.  Isabelle/HOL

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

   933   applications.

   934

   935   One framework generates code from both functional and relational

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

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

   938

   939   \begin{matharray}{rcl}

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

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

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

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

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

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

   946   \end{matharray}

   947

   948   \begin{rail}

   949   'value' term

   950   ;

   951

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

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

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

   955   ;

   956

   957   modespec: '(' ( name * ) ')'

   958   ;

   959

   960   'consts\_code' (codespec +)

   961   ;

   962

   963   codespec: const template attachment ?

   964   ;

   965

   966   'types\_code' (tycodespec +)

   967   ;

   968

   969   tycodespec: name template attachment ?

   970   ;

   971

   972   const: term

   973   ;

   974

   975   template: '(' string ')'

   976   ;

   977

   978   attachment: 'attach' modespec ? verblbrace text verbrbrace

   979   ;

   980

   981   'code' (name)?

   982   ;

   983   \end{rail}

   984

   985   \begin{description}

   986

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

   988   using the code generator.

   989

   990   \end{description}

   991

   992   \medskip The other framework generates code from functional programs

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

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

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

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

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

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

   999   introduction on how to use it.

  1000

  1001   \begin{matharray}{rcl}

  1002     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1003     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1004     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1005     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\

  1006     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\

  1007     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\

  1008     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\

  1009     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\

  1010     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\

  1011     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\

  1012     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\

  1013     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\

  1014     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\

  1015     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1016     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1017   \end{matharray}

  1018

  1019   \begin{rail}

  1020     'export\_code' ( constexpr + ) ? \\

  1021       ( ( 'in' target ( 'module\_name' string ) ? \\

  1022         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

  1023     ;

  1024

  1025     'code\_thms' ( constexpr + ) ?

  1026     ;

  1027

  1028     'code\_deps' ( constexpr + ) ?

  1029     ;

  1030

  1031     const: term

  1032     ;

  1033

  1034     constexpr: ( const | 'name.*' | '*' )

  1035     ;

  1036

  1037     typeconstructor: nameref

  1038     ;

  1039

  1040     class: nameref

  1041     ;

  1042

  1043     target: 'OCaml' | 'SML' | 'Haskell'

  1044     ;

  1045

  1046     'code\_datatype' const +

  1047     ;

  1048

  1049     'code\_const' (const + 'and') \\

  1050       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1051     ;

  1052

  1053     'code\_type' (typeconstructor + 'and') \\

  1054       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1055     ;

  1056

  1057     'code\_class' (class + 'and') \\

  1058       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )

  1059     ;

  1060

  1061     'code\_instance' (( typeconstructor '::' class ) + 'and') \\

  1062       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )

  1063     ;

  1064

  1065     'code\_monad' const const target

  1066     ;

  1067

  1068     'code\_reserved' target ( string + )

  1069     ;

  1070

  1071     'code\_include' target ( string ( string | '-') )

  1072     ;

  1073

  1074     'code\_modulename' target ( ( string string ) + )

  1075     ;

  1076

  1077     'code\_abort' ( const + )

  1078     ;

  1079

  1080     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string

  1081     ;

  1082

  1083     'code' ( 'inline' ) ? ( 'del' ) ?

  1084     ;

  1085   \end{rail}

  1086

  1087   \begin{description}

  1088

  1089   \item @{command (HOL) "export_code"} is the canonical interface for

  1090   generating and serializing code: for a given list of constants, code

  1091   is generated for the specified target languages.  Abstract code is

  1092   cached incrementally.  If no constant is given, the currently cached

  1093   code is serialized.  If no serialization instruction is given, only

  1094   abstract code is cached.

  1095

  1096   Constants may be specified by giving them literally, referring to

  1097   all executable contants within a certain theory by giving @{text

  1098   "name.*"}, or referring to \emph{all} executable constants currently

  1099   available by giving @{text "*"}.

  1100

  1101   By default, for each involved theory one corresponding name space

  1102   module is generated.  Alternativly, a module name may be specified

  1103   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1104   placed in this module.

  1105

  1106   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1107   single file; for \emph{Haskell}, it refers to a whole directory,

  1108   where code is generated in multiple files reflecting the module

  1109   hierarchy.  The file specification @{text "-"}'' denotes standard

  1110   output.  For \emph{SML}, omitting the file specification compiles

  1111   code internally in the context of the current ML session.

  1112

  1113   Serializers take an optional list of arguments in parentheses.  For

  1114   \emph{Haskell} a module name prefix may be given using the @{text

  1115   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1116   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1117   declaration.

  1118

  1119   \item @{command (HOL) "code_thms"} prints a list of theorems

  1120   representing the corresponding program containing all given

  1121   constants; if no constants are given, the currently cached code

  1122   theorems are printed.

  1123

  1124   \item @{command (HOL) "code_deps"} visualizes dependencies of

  1125   theorems representing the corresponding program containing all given

  1126   constants; if no constants are given, the currently cached code

  1127   theorems are visualized.

  1128

  1129   \item @{command (HOL) "code_datatype"} specifies a constructor set

  1130   for a logical type.

  1131

  1132   \item @{command (HOL) "code_const"} associates a list of constants

  1133   with target-specific serializations; omitting a serialization

  1134   deletes an existing serialization.

  1135

  1136   \item @{command (HOL) "code_type"} associates a list of type

  1137   constructors with target-specific serializations; omitting a

  1138   serialization deletes an existing serialization.

  1139

  1140   \item @{command (HOL) "code_class"} associates a list of classes

  1141   with target-specific class names; omitting a serialization deletes

  1142   an existing serialization.  This applies only to \emph{Haskell}.

  1143

  1144   \item @{command (HOL) "code_instance"} declares a list of type

  1145   constructor / class instance relations as already present'' for a

  1146   given target.  Omitting a @{text "-"}'' deletes an existing

  1147   already present'' declaration.  This applies only to

  1148   \emph{Haskell}.

  1149

  1150   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism

  1151   to generate monadic code for Haskell.

  1152

  1153   \item @{command (HOL) "code_reserved"} declares a list of names as

  1154   reserved for a given target, preventing it to be shadowed by any

  1155   generated code.

  1156

  1157   \item @{command (HOL) "code_include"} adds arbitrary named content

  1158   (include'') to generated code.  A @{text "-"}'' as last argument

  1159   will remove an already added include''.

  1160

  1161   \item @{command (HOL) "code_modulename"} declares aliasings from one

  1162   module name onto another.

  1163

  1164   \item @{command (HOL) "code_abort"} declares constants which are not

  1165   required to have a definition by means of defining equations; if

  1166   needed these are implemented by program abort instead.

  1167

  1168   \item @{attribute (HOL) code} explicitly selects (or with option

  1169   @{text "del"}'' deselects) a defining equation for code

  1170   generation.  Usually packages introducing defining equations provide

  1171   a reasonable default setup for selection.

  1172

  1173   \item @{attribute (HOL) code}~@{text inline} declares (or with

  1174   option @{text "del"}'' removes) inlining theorems which are

  1175   applied as rewrite rules to any defining equation during

  1176   preprocessing.

  1177

  1178   \item @{command (HOL) "print_codesetup"} gives an overview on

  1179   selected defining equations, code generator datatypes and

  1180   preprocessor setup.

  1181

  1182   \end{description}

  1183 *}

  1184

  1185

  1186 section {* Definition by specification \label{sec:hol-specification} *}

  1187

  1188 text {*

  1189   \begin{matharray}{rcl}

  1190     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1191     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1192   \end{matharray}

  1193

  1194   \begin{rail}

  1195   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

  1196   ;

  1197   decl: ((name ':')? term '(' 'overloaded' ')'?)

  1198   \end{rail}

  1199

  1200   \begin{description}

  1201

  1202   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a

  1203   goal stating the existence of terms with the properties specified to

  1204   hold for the constants given in @{text decls}.  After finishing the

  1205   proof, the theory will be augmented with definitions for the given

  1206   constants, as well as with theorems stating the properties for these

  1207   constants.

  1208

  1209   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up

  1210   a goal stating the existence of terms with the properties specified

  1211   to hold for the constants given in @{text decls}.  After finishing

  1212   the proof, the theory will be augmented with axioms expressing the

  1213   properties given in the first place.

  1214

  1215   \item @{text decl} declares a constant to be defined by the

  1216   specification given.  The definition for the constant @{text c} is

  1217   bound to the name @{text c_def} unless a theorem name is given in

  1218   the declaration.  Overloaded constants should be declared as such.

  1219

  1220   \end{description}

  1221

  1222   Whether to use @{command (HOL) "specification"} or @{command (HOL)

  1223   "ax_specification"} is to some extent a matter of style.  @{command

  1224   (HOL) "specification"} introduces no new axioms, and so by

  1225   construction cannot introduce inconsistencies, whereas @{command

  1226   (HOL) "ax_specification"} does introduce axioms, but only after the

  1227   user has explicitly proven it to be safe.  A practical issue must be

  1228   considered, though: After introducing two constants with the same

  1229   properties using @{command (HOL) "specification"}, one can prove

  1230   that the two constants are, in fact, equal.  If this might be a

  1231   problem, one should use @{command (HOL) "ax_specification"}.

  1232 *}

  1233

  1234 end