doc-src/IsarRef/Thy/HOL_Specific.thy
 author krauss Mon Nov 23 15:06:37 2009 +0100 (2009-11-23) changeset 33857 0cb5002c52db parent 31998 2c7a24f74db9 child 33858 0c348f7997f7 permissions -rw-r--r--
clarified; checked
     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   Recursive definitions introduced by the @{command (HOL) "function"}

   456   command accommodate

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

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

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

   460   to the user-specified equations. Cases are numbered (starting from 1).

   461

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

   463   with structural recursion on the datatype the recursion is carried

   464   out.

   465

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

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

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

   469   explicitly as well.

   470

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

   472   options.

   473

   474   \begin{description}

   475

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

   477   overlapping patterns by making them mutually disjoint.  Earlier

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

   479   specification in a format very similar to functional programming.

   480   Note that the resulting simplification and induction rules

   481   correspond to the transformed specification, not the one given

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

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

   484   transformation only works for ML-style datatype patterns.

   485

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

   487   introduction rules for the domain predicate. While mostly not

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

   489

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

   491   equations even without a termination proof, provided that the

   492   function is tail-recursive. This currently only works

   493

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

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

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

   497

   498   \end{description}

   499 *}

   500

   501

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

   503

   504 text {*

   505   \begin{matharray}{rcl}

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

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

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

   509   \end{matharray}

   510

   511   \begin{rail}

   512     'relation' term

   513     ;

   514     'lexicographic\_order' ( clasimpmod * )

   515     ;

   516   \end{rail}

   517

   518   \begin{description}

   519

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

   521   solve goals regarding the completeness of pattern matching, as

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

   523   \cite{isabelle-function}).

   524

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

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

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

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

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

   530   termination proofs.

   531

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

   533   automated termination proof by searching for a lexicographic

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

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

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

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

   538   \secref{sec:clasimp}.

   539

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

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

   542

   543   \end{description}

   544 *}

   545

   546

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

   548

   549 text {*

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

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

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

   553

   554   \begin{matharray}{rcl}

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

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

   557   \end{matharray}

   558

   559   \begin{rail}

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

   561     ;

   562     recdeftc thmdecl? tc

   563     ;

   564     hints: '(' 'hints' ( recdefmod * ) ')'

   565     ;

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

   567     ;

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

   569     ;

   570   \end{rail}

   571

   572   \begin{description}

   573

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

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

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

   577   TFL to recover from failed proof attempts, returning unfinished

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

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

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

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

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

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

   584

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

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

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

   588   constant @{text c}.

   589

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

   591   its internal proofs without manual intervention.

   592

   593   \end{description}

   594

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

   596   globally, using the following attributes.

   597

   598   \begin{matharray}{rcl}

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

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

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

   602   \end{matharray}

   603

   604   \begin{rail}

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

   606     ;

   607   \end{rail}

   608 *}

   609

   610

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

   612

   613 text {*

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

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

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

   617   structural operational semantics is an inductive definition of an

   618   evaluation relation.

   619

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

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

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

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

   624   relations to formalise equivalence of processes and infinite data

   625   structures.

   626

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

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

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

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

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

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

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

   634   to use inference rules for type-checking.

   635

   636   \begin{matharray}{rcl}

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

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

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

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

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

   642   \end{matharray}

   643

   644   \begin{rail}

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

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

   647     ;

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

   649     ;

   650     'mono' (() | 'add' | 'del')

   651     ;

   652   \end{rail}

   653

   654   \begin{description}

   655

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

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

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

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

   660   (co)inductive predicates that remain fixed throughout the

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

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

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

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

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

   666

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

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

   669   allowing the definition of (co)inductive sets.

   670

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

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

   673   (HOL) "inductive"}.

   674

   675   \end{description}

   676 *}

   677

   678

   679 subsection {* Derived rules *}

   680

   681 text {*

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

   683   theory and also proves some theorems:

   684

   685   \begin{description}

   686

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

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

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

   690   theory file;

   691

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

   693

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

   695   rule.

   696

   697   \end{description}

   698

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

   700   defined simultaneously, the list of introduction rules is called

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

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

   703   of mutual induction rules is called @{text

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

   705 *}

   706

   707

   708 subsection {* Monotonicity theorems *}

   709

   710 text {*

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

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

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

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

   715   theorems may be added:

   716

   717   \begin{itemize}

   718

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

   720   monotonicity of inductive definitions whose introduction rules have

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

   722

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

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

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

   726   $  727 \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"}}   728$

   729

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

   731   of expressions, e.g.

   732   $  733 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad   734 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}   735$

   736

   737   \item Equations for reducing complex operators to more primitive

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

   739   $  740 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad   741 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}   742$

   743

   744   \end{itemize}

   745

   746   %FIXME: Example of an inductive definition

   747 *}

   748

   749

   750 section {* Arithmetic proof support *}

   751

   752 text {*

   753   \begin{matharray}{rcl}

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

   755     @{attribute_def (HOL) arith} & : & @{text attribute} \\

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

   757   \end{matharray}

   758

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

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

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

   762

   763   The @{attribute (HOL) arith} attribute declares facts that are

   764   always supplied to the arithmetic provers implicitly.

   765

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

   767   rules to be expanded before @{method (HOL) arith} is invoked.

   768

   769   Note that a simpler (but faster) arithmetic prover is

   770   already invoked 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 {* Checking and refuting propositions *}

   824

   825 text {*

   826   Identifying incorrect propositions usually involves evaluation of

   827   particular assignments and systematic counter example search.  This

   828   is supported by the following commands.

   829

   830   \begin{matharray}{rcl}

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

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

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

   834   \end{matharray}

   835

   836   \begin{rail}

   837     'value' ( ( '[' name ']' ) ? ) modes? term

   838     ;

   839

   840     'quickcheck' ( ( '[' args ']' ) ? ) nat?

   841     ;

   842

   843     'quickcheck_params' ( ( '[' args ']' ) ? )

   844     ;

   845

   846     modes: '(' (name + ) ')'

   847     ;

   848

   849     args: ( name '=' value + ',' )

   850     ;

   851   \end{rail}

   852

   853   \begin{description}

   854

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

   856     term; optionally @{text modes} can be specified, which are

   857     appended to the current print mode (see also \cite{isabelle-ref}).

   858     Internally, the evaluation is performed by registered evaluators,

   859     which are invoked sequentially until a result is returned.

   860     Alternatively a specific evaluator can be selected using square

   861     brackets; available evaluators include @{text nbe} for

   862     \emph{normalization by evaluation} and \emph{code} for code

   863     generation in SML.

   864

   865   \item @{command (HOL) "quickcheck"} tests the current goal for

   866     counter examples using a series of arbitrary assignments for its

   867     free variables; by default the first subgoal is tested, an other

   868     can be selected explicitly using an optional goal index.

   869     A number of configuration options are supported for

   870     @{command (HOL) "quickcheck"}, notably:

   871

   872     \begin{description}

   873

   874       \item[size] specifies the maximum size of the search space for

   875         assignment values.

   876

   877       \item[iterations] sets how many sets of assignments are

   878         generated for each particular size.

   879

   880     \end{description}

   881

   882     These option can be given within square brackets.

   883

   884   \item @{command (HOL) "quickcheck_params"} changes quickcheck

   885     configuration options persitently.

   886

   887   \end{description}

   888 *}

   889

   890

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

   892

   893 text {*

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

   895   external automated reasoning tools to crunch a pending goal.

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

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

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

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

   900   web service.

   901

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

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

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

   905   usually reconstructs the proof within Isabelle, without requiring

   906   external provers again.  The Metis

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

   908   integrated into Isabelle/HOL is being used here.

   909

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

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

   912   works via explicit inferences going through the Isabelle kernel.

   913   Moreover, rechecking Isabelle proof texts with already specified

   914   auxiliary facts is much faster than performing fully automated

   915   search over and over again.

   916

   917   \begin{matharray}{rcl}

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

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

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

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

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

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

   924   \end{matharray}

   925

   926   \begin{rail}

   927   'sledgehammer' ( nameref * )

   928   ;

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

   930   ;

   931

   932   'metis' thmrefs

   933   ;

   934   \end{rail}

   935

   936   \begin{description}

   937

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

   939   invokes the specified automated theorem provers on the first

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

   941   is displayed, and the other attempts are terminated.

   942

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

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

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

   946   ATP provers'' preference by the user interface.

   947

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

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

   950   excessive provers are automatically terminated.

   951

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

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

   954   command.

   955

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

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

   958   until timeout.

   959

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

   961   provers.

   962

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

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

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

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

   967   be specified (default 5).

   968

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

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

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

   972   inferences going through the kernel.  Thus its results are

   973   guaranteed to be correct by construction''.

   974

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

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

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

   978

   979   \end{description}

   980 *}

   981

   982

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

   984

   985 text {*

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

   987   induction in unstructured tactic scripts; see also

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

   989

   990   \begin{matharray}{rcl}

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

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

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

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

   995   \end{matharray}

   996

   997   \begin{rail}

   998     'case\_tac' goalspec? term rule?

   999     ;

  1000     'induct\_tac' goalspec? (insts * 'and') rule?

  1001     ;

  1002     'ind\_cases' (prop +) ('for' (name +)) ?

  1003     ;

  1004     'inductive\_cases' (thmdecl? (prop +) + 'and')

  1005     ;

  1006

  1007     rule: ('rule' ':' thmref)

  1008     ;

  1009   \end{rail}

  1010

  1011   \begin{description}

  1012

  1013   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit

  1014   to reason about inductive types.  Rules are selected according to

  1015   the declarations by the @{attribute cases} and @{attribute induct}

  1016   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)

  1017   datatype} package already takes care of this.

  1018

  1019   These unstructured tactics feature both goal addressing and dynamic

  1020   instantiation.  Note that named rule cases are \emph{not} provided

  1021   as would be by the proper @{method cases} and @{method induct} proof

  1022   methods (see \secref{sec:cases-induct}).  Unlike the @{method

  1023   induct} method, @{method induct_tac} does not handle structured rule

  1024   statements, only the compact object-logic conclusion of the subgoal

  1025   being addressed.

  1026

  1027   \item @{method (HOL) ind_cases} and @{command (HOL)

  1028   "inductive_cases"} provide an interface to the internal @{ML_text

  1029   mk_cases} operation.  Rules are simplified in an unrestricted

  1030   forward manner.

  1031

  1032   While @{method (HOL) ind_cases} is a proof method to apply the

  1033   result immediately as elimination rules, @{command (HOL)

  1034   "inductive_cases"} provides case split theorems at the theory level

  1035   for later use.  The @{keyword "for"} argument of the @{method (HOL)

  1036   ind_cases} method allows to specify a list of variables that should

  1037   be generalized before applying the resulting rule.

  1038

  1039   \end{description}

  1040 *}

  1041

  1042

  1043 section {* Executable code *}

  1044

  1045 text {*

  1046   Isabelle/Pure provides two generic frameworks to support code

  1047   generation from executable specifications.  Isabelle/HOL

  1048   instantiates these mechanisms in a way that is amenable to end-user

  1049   applications.

  1050

  1051   One framework generates code from both functional and relational

  1052   programs to SML.  See \cite{isabelle-HOL} for further information

  1053   (this actually covers the new-style theory format as well).

  1054

  1055   \begin{matharray}{rcl}

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

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

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

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

  1060     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1061   \end{matharray}

  1062

  1063   \begin{rail}

  1064   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\

  1065     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\

  1066     'contains' ( ( name '=' term ) + | term + )

  1067   ;

  1068

  1069   modespec: '(' ( name * ) ')'

  1070   ;

  1071

  1072   'consts\_code' (codespec +)

  1073   ;

  1074

  1075   codespec: const template attachment ?

  1076   ;

  1077

  1078   'types\_code' (tycodespec +)

  1079   ;

  1080

  1081   tycodespec: name template attachment ?

  1082   ;

  1083

  1084   const: term

  1085   ;

  1086

  1087   template: '(' string ')'

  1088   ;

  1089

  1090   attachment: 'attach' modespec ? verblbrace text verbrbrace

  1091   ;

  1092

  1093   'code' (name)?

  1094   ;

  1095   \end{rail}

  1096

  1097   \medskip The other framework generates code from functional programs

  1098   (including overloading using type classes) to SML \cite{SML}, OCaml

  1099   \cite{OCaml} and Haskell \cite{haskell-revised-report}.

  1100   Conceptually, code generation is split up in three steps:

  1101   \emph{selection} of code theorems, \emph{translation} into an

  1102   abstract executable view and \emph{serialization} to a specific

  1103   \emph{target language}.  See \cite{isabelle-codegen} for an

  1104   introduction on how to use it.

  1105

  1106   \begin{matharray}{rcl}

  1107     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1108     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1109     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1110     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\

  1111     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\

  1112     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\

  1113     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\

  1114     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\

  1115     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\

  1116     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\

  1117     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\

  1118     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\

  1119     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\

  1120     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1121     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1122     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1123   \end{matharray}

  1124

  1125   \begin{rail}

  1126     'export\_code' ( constexpr + ) ? \\

  1127       ( ( 'in' target ( 'module\_name' string ) ? \\

  1128         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

  1129     ;

  1130

  1131     'code\_thms' ( constexpr + ) ?

  1132     ;

  1133

  1134     'code\_deps' ( constexpr + ) ?

  1135     ;

  1136

  1137     const: term

  1138     ;

  1139

  1140     constexpr: ( const | 'name.*' | '*' )

  1141     ;

  1142

  1143     typeconstructor: nameref

  1144     ;

  1145

  1146     class: nameref

  1147     ;

  1148

  1149     target: 'OCaml' | 'SML' | 'Haskell'

  1150     ;

  1151

  1152     'code\_datatype' const +

  1153     ;

  1154

  1155     'code\_const' (const + 'and') \\

  1156       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1157     ;

  1158

  1159     'code\_type' (typeconstructor + 'and') \\

  1160       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1161     ;

  1162

  1163     'code\_class' (class + 'and') \\

  1164       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )

  1165     ;

  1166

  1167     'code\_instance' (( typeconstructor '::' class ) + 'and') \\

  1168       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )

  1169     ;

  1170

  1171     'code\_monad' const const target

  1172     ;

  1173

  1174     'code\_reserved' target ( string + )

  1175     ;

  1176

  1177     'code\_include' target ( string ( string | '-') )

  1178     ;

  1179

  1180     'code\_modulename' target ( ( string string ) + )

  1181     ;

  1182

  1183     'code\_abort' ( const + )

  1184     ;

  1185

  1186     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string

  1187     ;

  1188

  1189     'code' ( 'del' ) ?

  1190     ;

  1191

  1192     'code_unfold' ( 'del' ) ?

  1193     ;

  1194

  1195     'code_post' ( 'del' ) ?

  1196     ;

  1197   \end{rail}

  1198

  1199   \begin{description}

  1200

  1201   \item @{command (HOL) "export_code"} is the canonical interface for

  1202   generating and serializing code: for a given list of constants, code

  1203   is generated for the specified target languages.  Abstract code is

  1204   cached incrementally.  If no constant is given, the currently cached

  1205   code is serialized.  If no serialization instruction is given, only

  1206   abstract code is cached.

  1207

  1208   Constants may be specified by giving them literally, referring to

  1209   all executable contants within a certain theory by giving @{text

  1210   "name.*"}, or referring to \emph{all} executable constants currently

  1211   available by giving @{text "*"}.

  1212

  1213   By default, for each involved theory one corresponding name space

  1214   module is generated.  Alternativly, a module name may be specified

  1215   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1216   placed in this module.

  1217

  1218   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1219   single file; for \emph{Haskell}, it refers to a whole directory,

  1220   where code is generated in multiple files reflecting the module

  1221   hierarchy.  The file specification @{text "-"}'' denotes standard

  1222   output.  For \emph{SML}, omitting the file specification compiles

  1223   code internally in the context of the current ML session.

  1224

  1225   Serializers take an optional list of arguments in parentheses.  For

  1226   \emph{Haskell} a module name prefix may be given using the @{text

  1227   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1228   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1229   declaration.

  1230

  1231   \item @{command (HOL) "code_thms"} prints a list of theorems

  1232   representing the corresponding program containing all given

  1233   constants; if no constants are given, the currently cached code

  1234   theorems are printed.

  1235

  1236   \item @{command (HOL) "code_deps"} visualizes dependencies of

  1237   theorems representing the corresponding program containing all given

  1238   constants; if no constants are given, the currently cached code

  1239   theorems are visualized.

  1240

  1241   \item @{command (HOL) "code_datatype"} specifies a constructor set

  1242   for a logical type.

  1243

  1244   \item @{command (HOL) "code_const"} associates a list of constants

  1245   with target-specific serializations; omitting a serialization

  1246   deletes an existing serialization.

  1247

  1248   \item @{command (HOL) "code_type"} associates a list of type

  1249   constructors with target-specific serializations; omitting a

  1250   serialization deletes an existing serialization.

  1251

  1252   \item @{command (HOL) "code_class"} associates a list of classes

  1253   with target-specific class names; omitting a serialization deletes

  1254   an existing serialization.  This applies only to \emph{Haskell}.

  1255

  1256   \item @{command (HOL) "code_instance"} declares a list of type

  1257   constructor / class instance relations as already present'' for a

  1258   given target.  Omitting a @{text "-"}'' deletes an existing

  1259   already present'' declaration.  This applies only to

  1260   \emph{Haskell}.

  1261

  1262   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism

  1263   to generate monadic code for Haskell.

  1264

  1265   \item @{command (HOL) "code_reserved"} declares a list of names as

  1266   reserved for a given target, preventing it to be shadowed by any

  1267   generated code.

  1268

  1269   \item @{command (HOL) "code_include"} adds arbitrary named content

  1270   (include'') to generated code.  A @{text "-"}'' as last argument

  1271   will remove an already added include''.

  1272

  1273   \item @{command (HOL) "code_modulename"} declares aliasings from one

  1274   module name onto another.

  1275

  1276   \item @{command (HOL) "code_abort"} declares constants which are not

  1277   required to have a definition by means of code equations; if

  1278   needed these are implemented by program abort instead.

  1279

  1280   \item @{attribute (HOL) code} explicitly selects (or with option

  1281   @{text "del"}'' deselects) a code equation for code

  1282   generation.  Usually packages introducing code equations provide

  1283   a reasonable default setup for selection.

  1284

  1285   \item @{attribute (HOL) code_inline} declares (or with

  1286   option @{text "del"}'' removes) inlining theorems which are

  1287   applied as rewrite rules to any code equation during

  1288   preprocessing.

  1289

  1290   \item @{attribute (HOL) code_post} declares (or with

  1291   option @{text "del"}'' removes) theorems which are

  1292   applied as rewrite rules to any result of an evaluation.

  1293

  1294   \item @{command (HOL) "print_codesetup"} gives an overview on

  1295   selected code equations and code generator datatypes.

  1296

  1297   \item @{command (HOL) "print_codeproc"} prints the setup

  1298   of the code generator preprocessor.

  1299

  1300   \end{description}

  1301 *}

  1302

  1303

  1304 section {* Definition by specification \label{sec:hol-specification} *}

  1305

  1306 text {*

  1307   \begin{matharray}{rcl}

  1308     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1309     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1310   \end{matharray}

  1311

  1312   \begin{rail}

  1313   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

  1314   ;

  1315   decl: ((name ':')? term '(' 'overloaded' ')'?)

  1316   \end{rail}

  1317

  1318   \begin{description}

  1319

  1320   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a

  1321   goal stating the existence of terms with the properties specified to

  1322   hold for the constants given in @{text decls}.  After finishing the

  1323   proof, the theory will be augmented with definitions for the given

  1324   constants, as well as with theorems stating the properties for these

  1325   constants.

  1326

  1327   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up

  1328   a goal stating the existence of terms with the properties specified

  1329   to hold for the constants given in @{text decls}.  After finishing

  1330   the proof, the theory will be augmented with axioms expressing the

  1331   properties given in the first place.

  1332

  1333   \item @{text decl} declares a constant to be defined by the

  1334   specification given.  The definition for the constant @{text c} is

  1335   bound to the name @{text c_def} unless a theorem name is given in

  1336   the declaration.  Overloaded constants should be declared as such.

  1337

  1338   \end{description}

  1339

  1340   Whether to use @{command (HOL) "specification"} or @{command (HOL)

  1341   "ax_specification"} is to some extent a matter of style.  @{command

  1342   (HOL) "specification"} introduces no new axioms, and so by

  1343   construction cannot introduce inconsistencies, whereas @{command

  1344   (HOL) "ax_specification"} does introduce axioms, but only after the

  1345   user has explicitly proven it to be safe.  A practical issue must be

  1346   considered, though: After introducing two constants with the same

  1347   properties using @{command (HOL) "specification"}, one can prove

  1348   that the two constants are, in fact, equal.  If this might be a

  1349   problem, one should use @{command (HOL) "ax_specification"}.

  1350 *}

  1351

  1352 end