doc-src/IsarRef/Thy/HOL_Specific.thy
 author haftmann Wed Dec 23 08:31:14 2009 +0100 (2009-12-23) changeset 34172 4301e9ea1c54 parent 33858 0c348f7997f7 child 35331 450ab945c451 permissions -rw-r--r--
updated documentation
     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     @{method_def (HOL) size_change} & : & @{text method} \\

   510   \end{matharray}

   511

   512   \begin{rail}

   513     'relation' term

   514     ;

   515     'lexicographic\_order' ( clasimpmod * )

   516     ;

   517     'size\_change' ( orders ( clasimpmod * ) )

   518     ;

   519     orders: ( 'max' | 'min' | 'ms' ) *

   520   \end{rail}

   521

   522   \begin{description}

   523

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

   525   solve goals regarding the completeness of pattern matching, as

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

   527   \cite{isabelle-function}).

   528

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

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

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

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

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

   534   termination proofs.

   535

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

   537   automated termination proof by searching for a lexicographic

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

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

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

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

   542   \secref{sec:clasimp}.

   543

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

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

   546

   547   \item @{method (HOL) "size_change"} also works on termination goals,

   548   using a variation of the size-change principle, together with a

   549   graph decomposition technique (see \cite{krauss_phd} for details).

   550   Three kinds of orders are used internally: @{text max}, @{text min},

   551   and @{text ms} (multiset), which is only available when the theory

   552   @{text Multiset} is loaded. When no order kinds are given, they are

   553   tried in order. The search for a termination proof uses SAT solving

   554   internally.

   555

   556  For local descent proofs, the same context modifiers as for @{method

   557   auto} are accepted, see \secref{sec:clasimp}.

   558

   559   \end{description}

   560 *}

   561

   562

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

   564

   565 text {*

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

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

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

   569

   570   \begin{matharray}{rcl}

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

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

   573   \end{matharray}

   574

   575   \begin{rail}

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

   577     ;

   578     recdeftc thmdecl? tc

   579     ;

   580     hints: '(' 'hints' ( recdefmod * ) ')'

   581     ;

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

   583     ;

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

   585     ;

   586   \end{rail}

   587

   588   \begin{description}

   589

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

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

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

   593   TFL to recover from failed proof attempts, returning unfinished

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

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

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

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

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

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

   600

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

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

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

   604   constant @{text c}.

   605

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

   607   its internal proofs without manual intervention.

   608

   609   \end{description}

   610

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

   612   globally, using the following attributes.

   613

   614   \begin{matharray}{rcl}

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

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

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

   618   \end{matharray}

   619

   620   \begin{rail}

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

   622     ;

   623   \end{rail}

   624 *}

   625

   626

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

   628

   629 text {*

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

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

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

   633   structural operational semantics is an inductive definition of an

   634   evaluation relation.

   635

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

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

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

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

   640   relations to formalise equivalence of processes and infinite data

   641   structures.

   642

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

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

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

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

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

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

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

   650   to use inference rules for type-checking.

   651

   652   \begin{matharray}{rcl}

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

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

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

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

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

   658   \end{matharray}

   659

   660   \begin{rail}

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

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

   663     ;

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

   665     ;

   666     'mono' (() | 'add' | 'del')

   667     ;

   668   \end{rail}

   669

   670   \begin{description}

   671

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

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

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

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

   676   (co)inductive predicates that remain fixed throughout the

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

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

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

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

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

   682

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

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

   685   allowing the definition of (co)inductive sets.

   686

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

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

   689   (HOL) "inductive"}.

   690

   691   \end{description}

   692 *}

   693

   694

   695 subsection {* Derived rules *}

   696

   697 text {*

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

   699   theory and also proves some theorems:

   700

   701   \begin{description}

   702

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

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

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

   706   theory file;

   707

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

   709

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

   711   rule.

   712

   713   \end{description}

   714

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

   716   defined simultaneously, the list of introduction rules is called

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

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

   719   of mutual induction rules is called @{text

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

   721 *}

   722

   723

   724 subsection {* Monotonicity theorems *}

   725

   726 text {*

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

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

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

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

   731   theorems may be added:

   732

   733   \begin{itemize}

   734

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

   736   monotonicity of inductive definitions whose introduction rules have

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

   738

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

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

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

   742   $  743 \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"}}   744$

   745

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

   747   of expressions, e.g.

   748   $  749 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad   750 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}   751$

   752

   753   \item Equations for reducing complex operators to more primitive

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

   755   $  756 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad   757 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}   758$

   759

   760   \end{itemize}

   761

   762   %FIXME: Example of an inductive definition

   763 *}

   764

   765

   766 section {* Arithmetic proof support *}

   767

   768 text {*

   769   \begin{matharray}{rcl}

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

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

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

   773   \end{matharray}

   774

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

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

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

   778

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

   780   always supplied to the arithmetic provers implicitly.

   781

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

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

   784

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

   786   already invoked by the Simplifier.

   787 *}

   788

   789

   790 section {* Intuitionistic proof search *}

   791

   792 text {*

   793   \begin{matharray}{rcl}

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

   795   \end{matharray}

   796

   797   \begin{rail}

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

   799     ;

   800   \end{rail}

   801

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

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

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

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

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

   807

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

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

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

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

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

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

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

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

   816 *}

   817

   818

   819 section {* Coherent Logic *}

   820

   821 text {*

   822   \begin{matharray}{rcl}

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

   824   \end{matharray}

   825

   826   \begin{rail}

   827     'coherent' thmrefs?

   828     ;

   829   \end{rail}

   830

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

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

   833   applications in confluence theory, lattice theory and projective

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

   835   examples.

   836 *}

   837

   838

   839 section {* Checking and refuting propositions *}

   840

   841 text {*

   842   Identifying incorrect propositions usually involves evaluation of

   843   particular assignments and systematic counter example search.  This

   844   is supported by the following commands.

   845

   846   \begin{matharray}{rcl}

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

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

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

   850   \end{matharray}

   851

   852   \begin{rail}

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

   854     ;

   855

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

   857     ;

   858

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

   860     ;

   861

   862     modes: '(' (name + ) ')'

   863     ;

   864

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

   866     ;

   867   \end{rail}

   868

   869   \begin{description}

   870

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

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

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

   874     Internally, the evaluation is performed by registered evaluators,

   875     which are invoked sequentially until a result is returned.

   876     Alternatively a specific evaluator can be selected using square

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

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

   879     generation in SML.

   880

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

   882     counter examples using a series of arbitrary assignments for its

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

   884     can be selected explicitly using an optional goal index.

   885     A number of configuration options are supported for

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

   887

   888     \begin{description}

   889

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

   891         assignment values.

   892

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

   894         generated for each particular size.

   895

   896     \end{description}

   897

   898     These option can be given within square brackets.

   899

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

   901     configuration options persitently.

   902

   903   \end{description}

   904 *}

   905

   906

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

   908

   909 text {*

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

   911   external automated reasoning tools to crunch a pending goal.

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

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

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

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

   916   web service.

   917

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

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

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

   921   usually reconstructs the proof within Isabelle, without requiring

   922   external provers again.  The Metis

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

   924   integrated into Isabelle/HOL is being used here.

   925

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

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

   928   works via explicit inferences going through the Isabelle kernel.

   929   Moreover, rechecking Isabelle proof texts with already specified

   930   auxiliary facts is much faster than performing fully automated

   931   search over and over again.

   932

   933   \begin{matharray}{rcl}

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

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

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

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

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

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

   940   \end{matharray}

   941

   942   \begin{rail}

   943   'sledgehammer' ( nameref * )

   944   ;

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

   946   ;

   947

   948   'metis' thmrefs

   949   ;

   950   \end{rail}

   951

   952   \begin{description}

   953

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

   955   invokes the specified automated theorem provers on the first

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

   957   is displayed, and the other attempts are terminated.

   958

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

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

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

   962   ATP provers'' preference by the user interface.

   963

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

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

   966   excessive provers are automatically terminated.

   967

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

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

   970   command.

   971

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

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

   974   until timeout.

   975

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

   977   provers.

   978

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

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

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

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

   983   be specified (default 5).

   984

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

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

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

   988   inferences going through the kernel.  Thus its results are

   989   guaranteed to be correct by construction''.

   990

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

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

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

   994

   995   \end{description}

   996 *}

   997

   998

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

  1000

  1001 text {*

  1002   The following tools of Isabelle/HOL support cases analysis and

  1003   induction in unstructured tactic scripts; see also

  1004   \secref{sec:cases-induct} for proper Isar versions of similar ideas.

  1005

  1006   \begin{matharray}{rcl}

  1007     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\

  1008     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\

  1009     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\

  1010     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

  1011   \end{matharray}

  1012

  1013   \begin{rail}

  1014     'case\_tac' goalspec? term rule?

  1015     ;

  1016     'induct\_tac' goalspec? (insts * 'and') rule?

  1017     ;

  1018     'ind\_cases' (prop +) ('for' (name +)) ?

  1019     ;

  1020     'inductive\_cases' (thmdecl? (prop +) + 'and')

  1021     ;

  1022

  1023     rule: ('rule' ':' thmref)

  1024     ;

  1025   \end{rail}

  1026

  1027   \begin{description}

  1028

  1029   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit

  1030   to reason about inductive types.  Rules are selected according to

  1031   the declarations by the @{attribute cases} and @{attribute induct}

  1032   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)

  1033   datatype} package already takes care of this.

  1034

  1035   These unstructured tactics feature both goal addressing and dynamic

  1036   instantiation.  Note that named rule cases are \emph{not} provided

  1037   as would be by the proper @{method cases} and @{method induct} proof

  1038   methods (see \secref{sec:cases-induct}).  Unlike the @{method

  1039   induct} method, @{method induct_tac} does not handle structured rule

  1040   statements, only the compact object-logic conclusion of the subgoal

  1041   being addressed.

  1042

  1043   \item @{method (HOL) ind_cases} and @{command (HOL)

  1044   "inductive_cases"} provide an interface to the internal @{ML_text

  1045   mk_cases} operation.  Rules are simplified in an unrestricted

  1046   forward manner.

  1047

  1048   While @{method (HOL) ind_cases} is a proof method to apply the

  1049   result immediately as elimination rules, @{command (HOL)

  1050   "inductive_cases"} provides case split theorems at the theory level

  1051   for later use.  The @{keyword "for"} argument of the @{method (HOL)

  1052   ind_cases} method allows to specify a list of variables that should

  1053   be generalized before applying the resulting rule.

  1054

  1055   \end{description}

  1056 *}

  1057

  1058

  1059 section {* Executable code *}

  1060

  1061 text {*

  1062   Isabelle/Pure provides two generic frameworks to support code

  1063   generation from executable specifications.  Isabelle/HOL

  1064   instantiates these mechanisms in a way that is amenable to end-user

  1065   applications.

  1066

  1067   One framework generates code from both functional and relational

  1068   programs to SML.  See \cite{isabelle-HOL} for further information

  1069   (this actually covers the new-style theory format as well).

  1070

  1071   \begin{matharray}{rcl}

  1072     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\

  1073     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\

  1074     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\

  1075     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\

  1076     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1077   \end{matharray}

  1078

  1079   \begin{rail}

  1080   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\

  1081     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\

  1082     'contains' ( ( name '=' term ) + | term + )

  1083   ;

  1084

  1085   modespec: '(' ( name * ) ')'

  1086   ;

  1087

  1088   'consts\_code' (codespec +)

  1089   ;

  1090

  1091   codespec: const template attachment ?

  1092   ;

  1093

  1094   'types\_code' (tycodespec +)

  1095   ;

  1096

  1097   tycodespec: name template attachment ?

  1098   ;

  1099

  1100   const: term

  1101   ;

  1102

  1103   template: '(' string ')'

  1104   ;

  1105

  1106   attachment: 'attach' modespec ? verblbrace text verbrbrace

  1107   ;

  1108

  1109   'code' (name)?

  1110   ;

  1111   \end{rail}

  1112

  1113   \medskip The other framework generates code from functional programs

  1114   (including overloading using type classes) to SML \cite{SML}, OCaml

  1115   \cite{OCaml} and Haskell \cite{haskell-revised-report}.

  1116   Conceptually, code generation is split up in three steps:

  1117   \emph{selection} of code theorems, \emph{translation} into an

  1118   abstract executable view and \emph{serialization} to a specific

  1119   \emph{target language}.  See \cite{isabelle-codegen} for an

  1120   introduction on how to use it.

  1121

  1122   \begin{matharray}{rcl}

  1123     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1124     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1125     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1126     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\

  1127     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\

  1128     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\

  1129     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\

  1130     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\

  1131     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\

  1132     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\

  1133     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\

  1134     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\

  1135     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\

  1136     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1137     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

  1138     @{attribute_def (HOL) code} & : & @{text attribute} \\

  1139   \end{matharray}

  1140

  1141   \begin{rail}

  1142     'export\_code' ( constexpr + ) \\

  1143       ( ( 'in' target ( 'module\_name' string ) ? \\

  1144         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

  1145     ;

  1146

  1147     'code\_thms' ( constexpr + ) ?

  1148     ;

  1149

  1150     'code\_deps' ( constexpr + ) ?

  1151     ;

  1152

  1153     const: term

  1154     ;

  1155

  1156     constexpr: ( const | 'name.*' | '*' )

  1157     ;

  1158

  1159     typeconstructor: nameref

  1160     ;

  1161

  1162     class: nameref

  1163     ;

  1164

  1165     target: 'OCaml' | 'SML' | 'Haskell'

  1166     ;

  1167

  1168     'code\_datatype' const +

  1169     ;

  1170

  1171     'code\_const' (const + 'and') \\

  1172       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1173     ;

  1174

  1175     'code\_type' (typeconstructor + 'and') \\

  1176       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1177     ;

  1178

  1179     'code\_class' (class + 'and') \\

  1180       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )

  1181     ;

  1182

  1183     'code\_instance' (( typeconstructor '::' class ) + 'and') \\

  1184       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )

  1185     ;

  1186

  1187     'code\_monad' const const target

  1188     ;

  1189

  1190     'code\_reserved' target ( string + )

  1191     ;

  1192

  1193     'code\_include' target ( string ( string | '-') )

  1194     ;

  1195

  1196     'code\_modulename' target ( ( string string ) + )

  1197     ;

  1198

  1199     'code\_abort' ( const + )

  1200     ;

  1201

  1202     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string

  1203     ;

  1204

  1205     'code' ( 'del' ) ?

  1206     ;

  1207

  1208     'code_unfold' ( 'del' ) ?

  1209     ;

  1210

  1211     'code_post' ( 'del' ) ?

  1212     ;

  1213   \end{rail}

  1214

  1215   \begin{description}

  1216

  1217   \item @{command (HOL) "export_code"} is the canonical interface for

  1218   generating and serializing code: for a given list of constants, code

  1219   is generated for the specified target languages.  If no serialization

  1220   instruction is given, only abstract code is generated internally.

  1221

  1222   Constants may be specified by giving them literally, referring to

  1223   all executable contants within a certain theory by giving @{text

  1224   "name.*"}, or referring to \emph{all} executable constants currently

  1225   available by giving @{text "*"}.

  1226

  1227   By default, for each involved theory one corresponding name space

  1228   module is generated.  Alternativly, a module name may be specified

  1229   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1230   placed in this module.

  1231

  1232   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1233   single file; for \emph{Haskell}, it refers to a whole directory,

  1234   where code is generated in multiple files reflecting the module

  1235   hierarchy.  The file specification @{text "-"}'' denotes standard

  1236   output.  For \emph{SML}, omitting the file specification compiles

  1237   code internally in the context of the current ML session.

  1238

  1239   Serializers take an optional list of arguments in parentheses.  For

  1240   \emph{SML} and \emph{OCaml}, @{text no_signatures} omits

  1241   explicit module signatures.

  1242

  1243   For \emph{Haskell} a module name prefix may be given using the @{text

  1244   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1245   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1246   declaration.

  1247

  1248   \item @{command (HOL) "code_thms"} prints a list of theorems

  1249   representing the corresponding program containing all given

  1250   constants.

  1251

  1252   \item @{command (HOL) "code_deps"} visualizes dependencies of

  1253   theorems representing the corresponding program containing all given

  1254   constants.

  1255

  1256   \item @{command (HOL) "code_datatype"} specifies a constructor set

  1257   for a logical type.

  1258

  1259   \item @{command (HOL) "code_const"} associates a list of constants

  1260   with target-specific serializations; omitting a serialization

  1261   deletes an existing serialization.

  1262

  1263   \item @{command (HOL) "code_type"} associates a list of type

  1264   constructors with target-specific serializations; omitting a

  1265   serialization deletes an existing serialization.

  1266

  1267   \item @{command (HOL) "code_class"} associates a list of classes

  1268   with target-specific class names; omitting a serialization deletes

  1269   an existing serialization.  This applies only to \emph{Haskell}.

  1270

  1271   \item @{command (HOL) "code_instance"} declares a list of type

  1272   constructor / class instance relations as already present'' for a

  1273   given target.  Omitting a @{text "-"}'' deletes an existing

  1274   already present'' declaration.  This applies only to

  1275   \emph{Haskell}.

  1276

  1277   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism

  1278   to generate monadic code for Haskell.

  1279

  1280   \item @{command (HOL) "code_reserved"} declares a list of names as

  1281   reserved for a given target, preventing it to be shadowed by any

  1282   generated code.

  1283

  1284   \item @{command (HOL) "code_include"} adds arbitrary named content

  1285   (include'') to generated code.  A @{text "-"}'' as last argument

  1286   will remove an already added include''.

  1287

  1288   \item @{command (HOL) "code_modulename"} declares aliasings from one

  1289   module name onto another.

  1290

  1291   \item @{command (HOL) "code_abort"} declares constants which are not

  1292   required to have a definition by means of code equations; if

  1293   needed these are implemented by program abort instead.

  1294

  1295   \item @{attribute (HOL) code} explicitly selects (or with option

  1296   @{text "del"}'' deselects) a code equation for code

  1297   generation.  Usually packages introducing code equations provide

  1298   a reasonable default setup for selection.

  1299

  1300   \item @{attribute (HOL) code_inline} declares (or with

  1301   option @{text "del"}'' removes) inlining theorems which are

  1302   applied as rewrite rules to any code equation during

  1303   preprocessing.

  1304

  1305   \item @{attribute (HOL) code_post} declares (or with

  1306   option @{text "del"}'' removes) theorems which are

  1307   applied as rewrite rules to any result of an evaluation.

  1308

  1309   \item @{command (HOL) "print_codesetup"} gives an overview on

  1310   selected code equations and code generator datatypes.

  1311

  1312   \item @{command (HOL) "print_codeproc"} prints the setup

  1313   of the code generator preprocessor.

  1314

  1315   \end{description}

  1316 *}

  1317

  1318

  1319 section {* Definition by specification \label{sec:hol-specification} *}

  1320

  1321 text {*

  1322   \begin{matharray}{rcl}

  1323     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1324     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

  1325   \end{matharray}

  1326

  1327   \begin{rail}

  1328   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

  1329   ;

  1330   decl: ((name ':')? term '(' 'overloaded' ')'?)

  1331   \end{rail}

  1332

  1333   \begin{description}

  1334

  1335   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a

  1336   goal stating the existence of terms with the properties specified to

  1337   hold for the constants given in @{text decls}.  After finishing the

  1338   proof, the theory will be augmented with definitions for the given

  1339   constants, as well as with theorems stating the properties for these

  1340   constants.

  1341

  1342   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up

  1343   a goal stating the existence of terms with the properties specified

  1344   to hold for the constants given in @{text decls}.  After finishing

  1345   the proof, the theory will be augmented with axioms expressing the

  1346   properties given in the first place.

  1347

  1348   \item @{text decl} declares a constant to be defined by the

  1349   specification given.  The definition for the constant @{text c} is

  1350   bound to the name @{text c_def} unless a theorem name is given in

  1351   the declaration.  Overloaded constants should be declared as such.

  1352

  1353   \end{description}

  1354

  1355   Whether to use @{command (HOL) "specification"} or @{command (HOL)

  1356   "ax_specification"} is to some extent a matter of style.  @{command

  1357   (HOL) "specification"} introduces no new axioms, and so by

  1358   construction cannot introduce inconsistencies, whereas @{command

  1359   (HOL) "ax_specification"} does introduce axioms, but only after the

  1360   user has explicitly proven it to be safe.  A practical issue must be

  1361   considered, though: After introducing two constants with the same

  1362   properties using @{command (HOL) "specification"}, one can prove

  1363   that the two constants are, in fact, equal.  If this might be a

  1364   problem, one should use @{command (HOL) "ax_specification"}.

  1365 *}

  1366

  1367 end