doc-src/IsarRef/Thy/HOL_Specific.thy
 author wenzelm Mon Jun 02 23:11:51 2008 +0200 (2008-06-02) changeset 27045 4e7ecec1b685 parent 27041 22dcf2fc0aa2 child 27103 d8549f4d900b permissions -rw-r--r--
moved (ax_)specification to end;
     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"} & : & \isartrans{theory}{theory} \\

    14     @{command_def (HOL) "typedef"} & : & \isartrans{theory}{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{descr}

    32

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

    34   t"}] is similar to the original @{command "typedecl"} of

    35   Isabelle/Pure (see \secref{sec:types-pure}), but also declares type

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

    37   actual HOL type constructor.   %FIXME check, update

    38

    39   \item [@{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)

    40   t = A"}] sets up a goal stating non-emptiness of the set @{text A}.

    41   After finishing the proof, the theory will be augmented by a

    42   Gordon/HOL-style type definition, which establishes a bijection

    43   between the representing 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{descr}

    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>*"} & : & \isaratt \\

    83   \end{matharray}

    84

    85   \begin{rail}

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

    87     ;

    88   \end{rail}

    89

    90   \begin{descr}

    91

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

    93   \<AND> \<dots> \<AND> q\<^sub>1 \<dots> q\<^sub>n"}] puts expressions of

    94   low-level tuple types into canonical form as specified by the

    95   arguments given; the @{text i}-th collection of arguments refers to

    96   occurrences in premise @{text i} of the rule.  The @{text

    97   "(complete)"}'' option causes \emph{all} arguments in function

    98   applications to be represented canonically according to their tuple

    99   type structure.

   100

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

   102   parameters to be introduced.

   103

   104   \end{descr}

   105 *}

   106

   107

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

   109

   110 text {*

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

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

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

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

   115   admits operations that are polymorphic with respect to record

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

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

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

   119 *}

   120

   121

   122 subsection {* Basic concepts *}

   123

   124 text {*

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

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

   127

   128   \begin{center}

   129   \begin{tabular}{l|l|l}

   130     & record terms & record types \\ \hline

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

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

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

   134   \end{tabular}

   135   \end{center}

   136

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

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

   139

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

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

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

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

   144

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

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

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

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

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

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

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

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

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

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

   155   Fixed records are special instances of record schemes, where

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

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

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

   159

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

   161   typed language like HOL work out:

   162

   163   \begin{enumerate}

   164

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

   166   variable,

   167

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

   169   the logic as first-class values.

   170

   171   \end{enumerate}

   172

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

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

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

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

   177   The record package provides several standard operations like

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

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

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

   181   records in practice.

   182 *}

   183

   184

   185 subsection {* Record specifications *}

   186

   187 text {*

   188   \begin{matharray}{rcl}

   189     @{command_def (HOL) "record"} & : & \isartrans{theory}{theory} \\

   190   \end{matharray}

   191

   192   \begin{rail}

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

   194     ;

   195   \end{rail}

   196

   197   \begin{descr}

   198

   199   \item [@{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t

   200   = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"}] defines

   201   extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},

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

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

   204

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

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

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

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

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

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

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

   212   the record name internally.

   213

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

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

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

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

   218   merge multiple parent records!

   219

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

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

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

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

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

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

   226

   227   \end{descr}

   228 *}

   229

   230

   231 subsection {* Record operations *}

   232

   233 text {*

   234   Any record definition of the form presented above produces certain

   235   standard operations.  Selectors and updates are provided for any

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

   237   cumulative record constructor functions.  To simplify the

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

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

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

   241

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

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

   244

   245   \begin{matharray}{lll}

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

   247     @{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>"} \\

   248   \end{matharray}

   249

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

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

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

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

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

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

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

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

   258   Thus commutativity of independent updates can be proven within the

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

   260

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

   262   constructor function:

   263

   264   \begin{matharray}{lll}

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

   266   \end{matharray}

   267

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

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

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

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

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

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

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

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

   276   the above record operations will get the following types:

   277

   278   \medskip

   279   \begin{tabular}{lll}

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

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

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

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

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

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

   286   \end{tabular}

   287   \medskip

   288

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

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

   291   record fragment consisting of exactly the new fields introduced here

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

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

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

   295

   296   \medskip

   297   \begin{tabular}{lll}

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

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

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

   301     @{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>"} \\

   302   \end{tabular}

   303   \medskip

   304

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

   306   for root records.

   307 *}

   308

   309

   310 subsection {* Derived rules and proof tools *}

   311

   312 text {*

   313   The record package proves several results internally, declaring

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

   315   reason about record structures quite conveniently.  Assume that

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

   317

   318   \begin{enumerate}

   319

   320   \item Standard conversions for selectors or updates applied to

   321   record constructor terms are made part of the default Simplifier

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

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

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

   325

   326   \item Selectors applied to updated records are automatically reduced

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

   328   standard Simplifier setup.

   329

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

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

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

   333   @{text "t.iffs"}.

   334

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

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

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

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

   339

   340   \item Representations of arbitrary record expressions as canonical

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

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

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

   344   fixed records, record schemes, more parts etc.

   345

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

   347   sensible rule according to the type of the indicated record

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

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

   350

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

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

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

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

   355

   356   \end{enumerate}

   357 *}

   358

   359

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

   361

   362 text {*

   363   \begin{matharray}{rcl}

   364     @{command_def (HOL) "datatype"} & : & \isartrans{theory}{theory} \\

   365     @{command_def (HOL) "rep_datatype"} & : & \isartrans{theory}{theory} \\

   366   \end{matharray}

   367

   368   \begin{rail}

   369     'datatype' (dtspec + 'and')

   370     ;

   371     'rep\_datatype' (name *) dtrules

   372     ;

   373

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

   375     ;

   376     cons: name (type *) mixfix?

   377     ;

   378     dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs

   379   \end{rail}

   380

   381   \begin{descr}

   382

   383   \item [@{command (HOL) "datatype"}] defines inductive datatypes in

   384   HOL.

   385

   386   \item [@{command (HOL) "rep_datatype"}] represents existing types as

   387   inductive ones, generating the standard infrastructure of derived

   388   concepts (primitive recursion etc.).

   389

   390   \end{descr}

   391

   392   The induction and exhaustion theorems generated provide case names

   393   according to the constructors involved, while parameters are named

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

   395

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

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

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

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

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

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

   402   internally bound parameters).

   403 *}

   404

   405

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

   407

   408 text {*

   409   \begin{matharray}{rcl}

   410     @{command_def (HOL) "primrec"} & : & \isarkeep{local{\dsh}theory} \\

   411     @{command_def (HOL) "fun"} & : & \isarkeep{local{\dsh}theory} \\

   412     @{command_def (HOL) "function"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

   413     @{command_def (HOL) "termination"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

   414   \end{matharray}

   415

   416   \begin{rail}

   417     'primrec' target? fixes 'where' equations

   418     ;

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

   420     ;

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

   422     ;

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

   424     ;

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

   426     ;

   427     'termination' ( term )?

   428   \end{rail}

   429

   430   \begin{descr}

   431

   432   \item [@{command (HOL) "primrec"}] defines primitive recursive

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

   434

   435   \item [@{command (HOL) "function"}] defines functions by general

   436   wellfounded recursion. A detailed description with examples can be

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

   438   set of (possibly conditional) recursive equations with arbitrary

   439   pattern matching. The command generates proof obligations for the

   440   completeness and the compatibility of patterns.

   441

   442   The defined function is considered partial, and the resulting

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

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

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

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

   447

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

   449   @{command (HOL) "function"}~@{text "(sequential)"}, followed by

   450   automated proof attempts regarding pattern matching and termination.

   451   See \cite{isabelle-function} for further details.

   452

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

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

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

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

   457   the induction principle is established.

   458

   459   \end{descr}

   460

   461   %FIXME check

   462

   463   Recursive definitions introduced by both the @{command (HOL)

   464   "primrec"} and the @{command (HOL) "function"} command accommodate

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

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

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

   468   to the user-specified equations.  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{descr}

   481

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

   483   disambiguates overlapping patterns by making them mutually disjoint.

   484   Earlier equations take precedence over later ones.  This allows to

   485   give the specification in a format very similar to functional

   486   programming.  Note that the resulting simplification and induction

   487   rules 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{descr}

   505 *}

   506

   507

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

   509

   510 text {*

   511   \begin{matharray}{rcl}

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

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

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

   515   \end{matharray}

   516

   517   \begin{rail}

   518     'relation' term

   519     ;

   520     'lexicographic\_order' (clasimpmod *)

   521     ;

   522   \end{rail}

   523

   524   \begin{descr}

   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{descr}

   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"} & : & \isartrans{theory}{theory} \\

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

   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{descr}

   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} & : & \isaratt \\

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

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

   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"} & : & \isarkeep{local{\dsh}theory} \\

   644     @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\

   645     @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\

   646     @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\

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

   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{descr}

   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{descr}

   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} & : & \isarmeth \\

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

   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 {* Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac} *}

   777

   778 text {*

   779   The following important tactical tools of Isabelle/HOL have been

   780   ported to Isar.  These should be never used in proper proof texts!

   781

   782   \begin{matharray}{rcl}

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

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

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

   786     @{command_def (HOL) "inductive_cases"} & : & \isartrans{theory}{theory} \\

   787   \end{matharray}

   788

   789   \begin{rail}

   790     'case\_tac' goalspec? term rule?

   791     ;

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

   793     ;

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

   795     ;

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

   797     ;

   798

   799     rule: ('rule' ':' thmref)

   800     ;

   801   \end{rail}

   802

   803   \begin{descr}

   804

   805   \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]

   806   admit to reason about inductive datatypes only (unless an

   807   alternative rule is given explicitly).  Furthermore, @{method (HOL)

   808   case_tac} does a classical case split on booleans; @{method (HOL)

   809   induct_tac} allows only variables to be given as instantiation.

   810   These tactic emulations feature both goal addressing and dynamic

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

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

   813   methods (see \secref{sec:cases-induct}).

   814

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

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

   817   mk_cases} operation.  Rules are simplified in an unrestricted

   818   forward manner.

   819

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

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

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

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

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

   825   be generalized before applying the resulting rule.

   826

   827   \end{descr}

   828 *}

   829

   830

   831 section {* Executable code *}

   832

   833 text {*

   834   Isabelle/Pure provides two generic frameworks to support code

   835   generation from executable specifications.  Isabelle/HOL

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

   837   applications.

   838

   839   One framework generates code from both functional and relational

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

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

   842

   843   \begin{matharray}{rcl}

   844     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   845     @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\

   846     @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\

   847     @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\

   848     @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\

   849     @{attribute_def (HOL) code} & : & \isaratt \\

   850   \end{matharray}

   851

   852   \begin{rail}

   853   'value' term

   854   ;

   855

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

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

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

   859   ;

   860

   861   modespec: '(' ( name * ) ')'

   862   ;

   863

   864   'consts\_code' (codespec +)

   865   ;

   866

   867   codespec: const template attachment ?

   868   ;

   869

   870   'types\_code' (tycodespec +)

   871   ;

   872

   873   tycodespec: name template attachment ?

   874   ;

   875

   876   const: term

   877   ;

   878

   879   template: '(' string ')'

   880   ;

   881

   882   attachment: 'attach' modespec ? verblbrace text verbrbrace

   883   ;

   884

   885   'code' (name)?

   886   ;

   887   \end{rail}

   888

   889   \begin{descr}

   890

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

   892   term using the code generator.

   893

   894   \end{descr}

   895

   896   \medskip The other framework generates code from functional programs

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

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

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

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

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

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

   903   introduction on how to use it.

   904

   905   \begin{matharray}{rcl}

   906     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   907     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   908     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   909     @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\

   910     @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\

   911     @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\

   912     @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\

   913     @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\

   914     @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\

   915     @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\

   916     @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\

   917     @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\

   918     @{command_def (HOL) "code_exception"} & : & \isartrans{theory}{theory} \\

   919     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   920     @{attribute_def (HOL) code} & : & \isaratt \\

   921   \end{matharray}

   922

   923   \begin{rail}

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

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

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

   927     ;

   928

   929     'code\_thms' ( constexpr + ) ?

   930     ;

   931

   932     'code\_deps' ( constexpr + ) ?

   933     ;

   934

   935     const: term

   936     ;

   937

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

   939     ;

   940

   941     typeconstructor: nameref

   942     ;

   943

   944     class: nameref

   945     ;

   946

   947     target: 'OCaml' | 'SML' | 'Haskell'

   948     ;

   949

   950     'code\_datatype' const +

   951     ;

   952

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

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

   955     ;

   956

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

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

   959     ;

   960

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

   962       ( ( '(' target \\

   963         ( ( string ('where' \\

   964           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )

   965     ;

   966

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

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

   969     ;

   970

   971     'code\_monad' const const target

   972     ;

   973

   974     'code\_reserved' target ( string + )

   975     ;

   976

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

   978     ;

   979

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

   981     ;

   982

   983     'code\_exception' ( const + )

   984     ;

   985

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

   987     ;

   988

   989     'code' ('func' | 'inline') ( 'del' )?

   990     ;

   991   \end{rail}

   992

   993   \begin{descr}

   994

   995   \item [@{command (HOL) "export_code"}] is the canonical interface

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

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

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

   999   cached code is serialized.  If no serialization instruction is

  1000   given, only abstract code is cached.

  1001

  1002   Constants may be specified by giving them literally, referring to

  1003   all executable contants within a certain theory by giving @{text

  1004   "name.*"}, or referring to \emph{all} executable constants currently

  1005   available by giving @{text "*"}.

  1006

  1007   By default, for each involved theory one corresponding name space

  1008   module is generated.  Alternativly, a module name may be specified

  1009   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1010   placed in this module.

  1011

  1012   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1013   single file; for \emph{Haskell}, it refers to a whole directory,

  1014   where code is generated in multiple files reflecting the module

  1015   hierarchy.  The file specification @{text "-"}'' denotes standard

  1016   output.  For \emph{SML}, omitting the file specification compiles

  1017   code internally in the context of the current ML session.

  1018

  1019   Serializers take an optional list of arguments in parentheses.  For

  1020   \emph{Haskell} a module name prefix may be given using the @{text

  1021   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1022   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1023   declaration.

  1024

  1025   \item [@{command (HOL) "code_thms"}] prints a list of theorems

  1026   representing the corresponding program containing all given

  1027   constants; if no constants are given, the currently cached code

  1028   theorems are printed.

  1029

  1030   \item [@{command (HOL) "code_deps"}] visualizes dependencies of

  1031   theorems representing the corresponding program containing all given

  1032   constants; if no constants are given, the currently cached code

  1033   theorems are visualized.

  1034

  1035   \item [@{command (HOL) "code_datatype"}] specifies a constructor set

  1036   for a logical type.

  1037

  1038   \item [@{command (HOL) "code_const"}] associates a list of constants

  1039   with target-specific serializations; omitting a serialization

  1040   deletes an existing serialization.

  1041

  1042   \item [@{command (HOL) "code_type"}] associates a list of type

  1043   constructors with target-specific serializations; omitting a

  1044   serialization deletes an existing serialization.

  1045

  1046   \item [@{command (HOL) "code_class"}] associates a list of classes

  1047   with target-specific class names; in addition, constants associated

  1048   with this class may be given target-specific names used for instance

  1049   declarations; omitting a serialization deletes an existing

  1050   serialization.  This applies only to \emph{Haskell}.

  1051

  1052   \item [@{command (HOL) "code_instance"}] declares a list of type

  1053   constructor / class instance relations as already present'' for a

  1054   given target.  Omitting a @{text "-"}'' deletes an existing

  1055   already present'' declaration.  This applies only to

  1056   \emph{Haskell}.

  1057

  1058   \item [@{command (HOL) "code_monad"}] provides an auxiliary

  1059   mechanism to generate monadic code.

  1060

  1061   \item [@{command (HOL) "code_reserved"}] declares a list of names as

  1062   reserved for a given target, preventing it to be shadowed by any

  1063   generated code.

  1064

  1065   \item [@{command (HOL) "code_include"}] adds arbitrary named content

  1066   (include'') to generated code.  A as last argument @{text "-"}''

  1067   will remove an already added include''.

  1068

  1069   \item [@{command (HOL) "code_modulename"}] declares aliasings from

  1070   one module name onto another.

  1071

  1072   \item [@{command (HOL) "code_exception"}] declares constants which

  1073   are not required to have a definition by a defining equations; these

  1074   are mapped on exceptions instead.

  1075

  1076   \item [@{attribute (HOL) code}~@{text func}] explicitly selects (or

  1077   with option @{text "del:"}'' deselects) a defining equation for

  1078   code generation.  Usually packages introducing defining equations

  1079   provide a resonable default setup for selection.

  1080

  1081   \item [@{attribute (HOL) code}@{text inline}] declares (or with

  1082   option @{text "del:"}'' removes) inlining theorems which are

  1083   applied as rewrite rules to any defining equation during

  1084   preprocessing.

  1085

  1086   \item [@{command (HOL) "print_codesetup"}] gives an overview on

  1087   selected defining equations, code generator datatypes and

  1088   preprocessor setup.

  1089

  1090   \end{descr}

  1091 *}

  1092

  1093

  1094 section {* Definition by specification \label{sec:hol-specification} *}

  1095

  1096 text {*

  1097   \begin{matharray}{rcl}

  1098     @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\

  1099     @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\

  1100   \end{matharray}

  1101

  1102   \begin{rail}

  1103   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

  1104   ;

  1105   decl: ((name ':')? term '(' 'overloaded' ')'?)

  1106   \end{rail}

  1107

  1108   \begin{descr}

  1109

  1110   \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a

  1111   goal stating the existence of terms with the properties specified to

  1112   hold for the constants given in @{text decls}.  After finishing the

  1113   proof, the theory will be augmented with definitions for the given

  1114   constants, as well as with theorems stating the properties for these

  1115   constants.

  1116

  1117   \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets

  1118   up a goal stating the existence of terms with the properties

  1119   specified to hold for the constants given in @{text decls}.  After

  1120   finishing the proof, the theory will be augmented with axioms

  1121   expressing the properties given in the first place.

  1122

  1123   \item [@{text decl}] declares a constant to be defined by the

  1124   specification given.  The definition for the constant @{text c} is

  1125   bound to the name @{text c_def} unless a theorem name is given in

  1126   the declaration.  Overloaded constants should be declared as such.

  1127

  1128   \end{descr}

  1129

  1130   Whether to use @{command (HOL) "specification"} or @{command (HOL)

  1131   "ax_specification"} is to some extent a matter of style.  @{command

  1132   (HOL) "specification"} introduces no new axioms, and so by

  1133   construction cannot introduce inconsistencies, whereas @{command

  1134   (HOL) "ax_specification"} does introduce axioms, but only after the

  1135   user has explicitly proven it to be safe.  A practical issue must be

  1136   considered, though: After introducing two constants with the same

  1137   properties using @{command (HOL) "specification"}, one can prove

  1138   that the two constants are, in fact, equal.  If this might be a

  1139   problem, one should use @{command (HOL) "ax_specification"}.

  1140 *}

  1141

  1142 end