
wenzelm@26840: (* $Id$ *)
wenzelm@26840: 
wenzelm@26840: theory HOL_Specific
wenzelm@26849: imports Main
wenzelm@26840: begin
wenzelm@26840: 
wenzelm@26852: chapter {* Isabelle/HOL \label{ch:hol} *}
wenzelm@26849: 
wenzelm@26849: section {* Primitive types \label{sec:hol-typedef} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "typedecl"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "typedef"} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'typedecl' typespec infix?
wenzelm@26849:     ;
wenzelm@26849:     'typedef' altname? abstype '=' repset
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     altname: '(' (name | 'open' | 'open' name) ')'
wenzelm@26849:     ;
wenzelm@26849:     abstype: typespec infix?
wenzelm@26849:     ;
wenzelm@26849:     repset: term ('morphisms' name name)?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26849:   \item [@{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)
wenzelm@26849:   t"}] is similar to the original @{command "typedecl"} of
wenzelm@26849:   Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
wenzelm@26849:   arity @{text "t :: (type, \<dots>, type) type"}, making @{text t} an
wenzelm@26849:   actual HOL type constructor.   %FIXME check, update
wenzelm@26849:   
wenzelm@26849:   \item [@{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)
wenzelm@26849:   t = A"}] sets up a goal stating non-emptiness of the set @{text A}.
wenzelm@26849:   After finishing the proof, the theory will be augmented by a
wenzelm@26849:   Gordon/HOL-style type definition, which establishes a bijection
wenzelm@26849:   between the representing set @{text A} and the new type @{text t}.
wenzelm@26849:   
wenzelm@26849:   Technically, @{command (HOL) "typedef"} defines both a type @{text
wenzelm@26849:   t} and a set (term constant) of the same name (an alternative base
wenzelm@26849:   name may be given in parentheses).  The injection from type to set
wenzelm@26849:   is called @{text Rep_t}, its inverse @{text Abs_t} (this may be
wenzelm@26849:   changed via an explicit @{keyword (HOL) "morphisms"} declaration).
wenzelm@26849:   
wenzelm@26849:   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
wenzelm@26849:   Abs_t_inverse} provide the most basic characterization as a
wenzelm@26849:   corresponding injection/surjection pair (in both directions).  Rules
wenzelm@26849:   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
wenzelm@26849:   more convenient view on the injectivity part, suitable for automated
wenzelm@26894:   proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
wenzelm@26894:   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
wenzelm@26894:   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
wenzelm@26894:   on surjectivity; these are already declared as set or type rules for
wenzelm@26849:   the generic @{method cases} and @{method induct} methods.
wenzelm@26849:   
wenzelm@26849:   An alternative name may be specified in parentheses; the default is
wenzelm@26849:   to use @{text t} as indicated before.  The ``@{text "(open)"}''
wenzelm@26849:   declaration suppresses a separate constant definition for the
wenzelm@26849:   representing set.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   Note that raw type declarations are rarely used in practice; the
wenzelm@26849:   main application is with experimental (or even axiomatic!) theory
wenzelm@26849:   fragments.  Instead of primitive HOL type definitions, user-level
wenzelm@26849:   theories usually refer to higher-level packages such as @{command
wenzelm@26849:   (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)
wenzelm@26849:   "datatype"} (see \secref{sec:hol-datatype}).
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Adhoc tuples *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26894:   \item [@{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m
wenzelm@26849:   \<AND> \<dots> \<AND> q\<^sub>1 \<dots> q\<^sub>n"}] puts expressions of
wenzelm@26849:   low-level tuple types into canonical form as specified by the
wenzelm@26849:   arguments given; the @{text i}-th collection of arguments refers to
wenzelm@26849:   occurrences in premise @{text i} of the rule.  The ``@{text
wenzelm@26849:   "(complete)"}'' option causes \emph{all} arguments in function
wenzelm@26849:   applications to be represented canonically according to their tuple
wenzelm@26849:   type structure.
wenzelm@26849: 
wenzelm@26849:   Note that these operations tend to invent funny names for new local
wenzelm@26849:   parameters to be introduced.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Records \label{sec:hol-record} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   In principle, records merely generalize the concept of tuples, where
wenzelm@26849:   components may be addressed by labels instead of just position.  The
wenzelm@26849:   logical infrastructure of records in Isabelle/HOL is slightly more
wenzelm@26849:   advanced, though, supporting truly extensible record schemes.  This
wenzelm@26849:   admits operations that are polymorphic with respect to record
wenzelm@26849:   extension, yielding ``object-oriented'' effects like (single)
wenzelm@26849:   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
wenzelm@26849:   details on object-oriented verification and record subtyping in HOL.
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Basic concepts *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
wenzelm@26849:   at the level of terms and types.  The notation is as follows:
wenzelm@26849: 
wenzelm@26849:   \begin{center}
wenzelm@26849:   \begin{tabular}{l|l|l}
wenzelm@26849:     & record terms & record types \\ \hline
wenzelm@26849:     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
wenzelm@26849:     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
wenzelm@26849:       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
wenzelm@26849:   \end{tabular}
wenzelm@26849:   \end{center}
wenzelm@26849: 
wenzelm@26849:   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
wenzelm@26849:   "(| x = a |)"}.
wenzelm@26849: 
wenzelm@26849:   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
wenzelm@26849:   @{text a} and field @{text y} of value @{text b}.  The corresponding
wenzelm@26849:   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
wenzelm@26849:   and @{text "b :: B"}.
wenzelm@26849: 
wenzelm@26849:   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
wenzelm@26849:   @{text x} and @{text y} as before, but also possibly further fields
wenzelm@26849:   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
wenzelm@26849:   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
wenzelm@26849:   scheme is called the \emph{more part}.  Logically it is just a free
wenzelm@26849:   variable, which is occasionally referred to as ``row variable'' in
wenzelm@26849:   the literature.  The more part of a record scheme may be
wenzelm@26849:   instantiated by zero or more further components.  For example, the
wenzelm@26849:   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
wenzelm@26852:   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
wenzelm@26849:   Fixed records are special instances of record schemes, where
wenzelm@26849:   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
wenzelm@26849:   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
wenzelm@26849:   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
wenzelm@26849:   
wenzelm@26849:   \medskip Two key observations make extensible records in a simply
wenzelm@26849:   typed language like HOL work out:
wenzelm@26849: 
wenzelm@26849:   \begin{enumerate}
wenzelm@26849: 
wenzelm@26849:   \item the more part is internalized, as a free term or type
wenzelm@26849:   variable,
wenzelm@26849: 
wenzelm@26852:   \item field names are externalized, they cannot be accessed within
wenzelm@26852:   the logic as first-class values.
wenzelm@26849: 
wenzelm@26849:   \end{enumerate}
wenzelm@26849: 
wenzelm@26849:   \medskip In Isabelle/HOL record types have to be defined explicitly,
wenzelm@26849:   fixing their field names and types, and their (optional) parent
wenzelm@26849:   record.  Afterwards, records may be formed using above syntax, while
wenzelm@26849:   obeying the canonical order of fields as given by their declaration.
wenzelm@26849:   The record package provides several standard operations like
wenzelm@26849:   selectors and updates.  The common setup for various generic proof
wenzelm@26849:   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
wenzelm@26849:   tutorial \cite{isabelle-hol-book} for further instructions on using
wenzelm@26849:   records in practice.
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Record specifications *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "record"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'record' typespec '=' (type '+')? (constdecl +)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t
wenzelm@26849:   = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"}] defines
wenzelm@26849:   extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
wenzelm@26849:   derived from the optional parent record @{text "\<tau>"} by adding new
wenzelm@26849:   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
wenzelm@26849: 
wenzelm@26849:   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
wenzelm@26849:   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
wenzelm@26849:   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
wenzelm@26849:   \<tau>} needs to specify an instance of an existing record type.  At
wenzelm@26849:   least one new field @{text "c\<^sub>i"} has to be specified.
wenzelm@26849:   Basically, field names need to belong to a unique record.  This is
wenzelm@26849:   not a real restriction in practice, since fields are qualified by
wenzelm@26849:   the record name internally.
wenzelm@26849: 
wenzelm@26849:   The parent record specification @{text \<tau>} is optional; if omitted
wenzelm@26849:   @{text t} becomes a root record.  The hierarchy of all records
wenzelm@26849:   declared within a theory context forms a forest structure, i.e.\ a
wenzelm@26849:   set of trees starting with a root record each.  There is no way to
wenzelm@26849:   merge multiple parent records!
wenzelm@26849: 
wenzelm@26849:   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
wenzelm@26849:   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
wenzelm@26849:   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
wenzelm@26849:   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
wenzelm@26849:   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
wenzelm@26849:   \<zeta>\<rparr>"}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Record operations *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   Any record definition of the form presented above produces certain
wenzelm@26849:   standard operations.  Selectors and updates are provided for any
wenzelm@26849:   field, including the improper one ``@{text more}''.  There are also
wenzelm@26849:   cumulative record constructor functions.  To simplify the
wenzelm@26849:   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
wenzelm@26849:   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
wenzelm@26849:   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
wenzelm@26849: 
wenzelm@26849:   \medskip \textbf{Selectors} and \textbf{updates} are available for
wenzelm@26849:   any field (including ``@{text more}''):
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{lll}
wenzelm@26852:     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
wenzelm@26852:     @{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>"} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   There is special syntax for application of updates: @{text "r\<lparr>x :=
wenzelm@26849:   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
wenzelm@26849:   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
wenzelm@26849:   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
wenzelm@26849:   because of postfix notation the order of fields shown here is
wenzelm@26849:   reverse than in the actual term.  Since repeated updates are just
wenzelm@26849:   function applications, fields may be freely permuted in @{text "\<lparr>x
wenzelm@26849:   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
wenzelm@26849:   Thus commutativity of independent updates can be proven within the
wenzelm@26849:   logic for any two fields, but not as a general theorem.
wenzelm@26849: 
wenzelm@26849:   \medskip The \textbf{make} operation provides a cumulative record
wenzelm@26849:   constructor function:
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{lll}
wenzelm@26852:     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \medskip We now reconsider the case of non-root records, which are
wenzelm@26849:   derived of some parent.  In general, the latter may depend on
wenzelm@26849:   another parent as well, resulting in a list of \emph{ancestor
wenzelm@26849:   records}.  Appending the lists of fields of all ancestors results in
wenzelm@26849:   a certain field prefix.  The record package automatically takes care
wenzelm@26849:   of this by lifting operations over this context of ancestor fields.
wenzelm@26849:   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
wenzelm@26849:   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
wenzelm@26849:   the above record operations will get the following types:
wenzelm@26849: 
wenzelm@26852:   \medskip
wenzelm@26852:   \begin{tabular}{lll}
wenzelm@26852:     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
wenzelm@26849:     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
wenzelm@26852:       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
wenzelm@26852:       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
wenzelm@26852:     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
wenzelm@26852:       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
wenzelm@26852:   \end{tabular}
wenzelm@26852:   \medskip
wenzelm@26849: 
wenzelm@26852:   \noindent Some further operations address the extension aspect of a
wenzelm@26849:   derived record scheme specifically: @{text "t.fields"} produces a
wenzelm@26849:   record fragment consisting of exactly the new fields introduced here
wenzelm@26849:   (the result may serve as a more part elsewhere); @{text "t.extend"}
wenzelm@26849:   takes a fixed record and adds a given more part; @{text
wenzelm@26849:   "t.truncate"} restricts a record scheme to a fixed record.
wenzelm@26849: 
wenzelm@26852:   \medskip
wenzelm@26852:   \begin{tabular}{lll}
wenzelm@26852:     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
wenzelm@26852:     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
wenzelm@26852:       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
wenzelm@26852:     @{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>"} \\
wenzelm@26852:   \end{tabular}
wenzelm@26852:   \medskip
wenzelm@26849: 
wenzelm@26849:   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
wenzelm@26849:   for root records.
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Derived rules and proof tools *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   The record package proves several results internally, declaring
wenzelm@26849:   these facts to appropriate proof tools.  This enables users to
wenzelm@26849:   reason about record structures quite conveniently.  Assume that
wenzelm@26849:   @{text t} is a record type as specified above.
wenzelm@26849: 
wenzelm@26849:   \begin{enumerate}
wenzelm@26849:   
wenzelm@26849:   \item Standard conversions for selectors or updates applied to
wenzelm@26849:   record constructor terms are made part of the default Simplifier
wenzelm@26849:   context; thus proofs by reduction of basic operations merely require
wenzelm@26849:   the @{method simp} method without further arguments.  These rules
wenzelm@26849:   are available as @{text "t.simps"}, too.
wenzelm@26849:   
wenzelm@26849:   \item Selectors applied to updated records are automatically reduced
wenzelm@26849:   by an internal simplification procedure, which is also part of the
wenzelm@26849:   standard Simplifier setup.
wenzelm@26849: 
wenzelm@26849:   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
wenzelm@26849:   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
wenzelm@26849:   Reasoner as @{attribute iff} rules.  These rules are available as
wenzelm@26849:   @{text "t.iffs"}.
wenzelm@26849: 
wenzelm@26849:   \item The introduction rule for record equality analogous to @{text
wenzelm@26849:   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
wenzelm@26849:   and as the basic rule context as ``@{attribute intro}@{text "?"}''.
wenzelm@26849:   The rule is called @{text "t.equality"}.
wenzelm@26849: 
wenzelm@26849:   \item Representations of arbitrary record expressions as canonical
wenzelm@26849:   constructor terms are provided both in @{method cases} and @{method
wenzelm@26849:   induct} format (cf.\ the generic proof methods of the same name,
wenzelm@26849:   \secref{sec:cases-induct}).  Several variations are available, for
wenzelm@26849:   fixed records, record schemes, more parts etc.
wenzelm@26849:   
wenzelm@26849:   The generic proof methods are sufficiently smart to pick the most
wenzelm@26849:   sensible rule according to the type of the indicated record
wenzelm@26849:   expression: users just need to apply something like ``@{text "(cases
wenzelm@26849:   r)"}'' to a certain proof problem.
wenzelm@26849: 
wenzelm@26849:   \item The derived record operations @{text "t.make"}, @{text
wenzelm@26849:   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
wenzelm@26849:   treated automatically, but usually need to be expanded by hand,
wenzelm@26849:   using the collective fact @{text "t.defs"}.
wenzelm@26849: 
wenzelm@26849:   \end{enumerate}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Datatypes \label{sec:hol-datatype} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "datatype"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "rep_datatype"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'datatype' (dtspec + 'and')
wenzelm@26849:     ;
wenzelm@26849:     'rep\_datatype' (name *) dtrules
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     dtspec: parname? typespec infix? '=' (cons + '|')
wenzelm@26849:     ;
wenzelm@26849:     cons: name (type *) mixfix?
wenzelm@26849:     ;
wenzelm@26849:     dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "datatype"}] defines inductive datatypes in
wenzelm@26849:   HOL.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "rep_datatype"}] represents existing types as
wenzelm@26849:   inductive ones, generating the standard infrastructure of derived
wenzelm@26849:   concepts (primitive recursion etc.).
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   The induction and exhaustion theorems generated provide case names
wenzelm@26849:   according to the constructors involved, while parameters are named
wenzelm@26849:   after the types (see also \secref{sec:cases-induct}).
wenzelm@26849: 
wenzelm@26849:   See \cite{isabelle-HOL} for more details on datatypes, but beware of
wenzelm@26849:   the old-style theory syntax being used there!  Apart from proper
wenzelm@26849:   proof methods for case-analysis and induction, there are also
wenzelm@26849:   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
wenzelm@26849:   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
wenzelm@26849:   to refer directly to the internal structure of subgoals (including
wenzelm@26849:   internally bound parameters).
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Recursive functions \label{sec:recursion} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "primrec"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{command_def (HOL) "fun"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{command_def (HOL) "function"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
wenzelm@26849:     @{command_def (HOL) "termination"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'primrec' target? fixes 'where' equations
wenzelm@26849:     ;
wenzelm@26849:     equations: (thmdecl? prop + '|')
wenzelm@26849:     ;
wenzelm@26985:     ('fun' | 'function') target? functionopts? fixes 'where' clauses
wenzelm@26849:     ;
wenzelm@26849:     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
wenzelm@26849:     ;
wenzelm@26985:     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
wenzelm@26849:     ;
wenzelm@26849:     'termination' ( term )?
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "primrec"}] defines primitive recursive
wenzelm@26849:   functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "function"}] defines functions by general
wenzelm@26849:   wellfounded recursion. A detailed description with examples can be
wenzelm@26849:   found in \cite{isabelle-function}. The function is specified by a
wenzelm@26849:   set of (possibly conditional) recursive equations with arbitrary
wenzelm@26849:   pattern matching. The command generates proof obligations for the
wenzelm@26849:   completeness and the compatibility of patterns.
wenzelm@26849: 
wenzelm@26849:   The defined function is considered partial, and the resulting
wenzelm@26849:   simplification rules (named @{text "f.psimps"}) and induction rule
wenzelm@26849:   (named @{text "f.pinduct"}) are guarded by a generated domain
wenzelm@26849:   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
wenzelm@26849:   command can then be used to establish that the function is total.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "fun"}] is a shorthand notation for
wenzelm@26849:   ``@{command (HOL) "function"}~@{text "(sequential)"}, followed by
wenzelm@26849:   automated proof attempts regarding pattern matching and termination.
wenzelm@26849:   See \cite{isabelle-function} for further details.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "termination"}~@{text f}] commences a
wenzelm@26849:   termination proof for the previously defined function @{text f}.  If
wenzelm@26849:   this is omitted, the command refers to the most recent function
wenzelm@26849:   definition.  After the proof is closed, the recursive equations and
wenzelm@26849:   the induction principle is established.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   %FIXME check
wenzelm@26849: 
wenzelm@26849:   Recursive definitions introduced by both the @{command (HOL)
wenzelm@26849:   "primrec"} and the @{command (HOL) "function"} command accommodate
wenzelm@26849:   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
wenzelm@26849:   "c.induct"} (where @{text c} is the name of the function definition)
wenzelm@26849:   refers to a specific induction rule, with parameters named according
wenzelm@26849:   to the user-specified equations.  Case names of @{command (HOL)
wenzelm@26849:   "primrec"} are that of the datatypes involved, while those of
wenzelm@26849:   @{command (HOL) "function"} are numbered (starting from 1).
wenzelm@26849: 
wenzelm@26849:   The equations provided by these packages may be referred later as
wenzelm@26849:   theorem list @{text "f.simps"}, where @{text f} is the (collective)
wenzelm@26849:   name of the functions defined.  Individual equations may be named
wenzelm@26849:   explicitly as well.
wenzelm@26849: 
wenzelm@26849:   The @{command (HOL) "function"} command accepts the following
wenzelm@26849:   options.
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{text sequential}] enables a preprocessor which
wenzelm@26849:   disambiguates overlapping patterns by making them mutually disjoint.
wenzelm@26849:   Earlier equations take precedence over later ones.  This allows to
wenzelm@26849:   give the specification in a format very similar to functional
wenzelm@26849:   programming.  Note that the resulting simplification and induction
wenzelm@26849:   rules correspond to the transformed specification, not the one given
wenzelm@26849:   originally. This usually means that each equation given by the user
wenzelm@26849:   may result in several theroems.  Also note that this automatic
wenzelm@26849:   transformation only works for ML-style datatype patterns.
wenzelm@26849: 
wenzelm@26849:   \item [@{text domintros}] enables the automated generation of
wenzelm@26849:   introduction rules for the domain predicate. While mostly not
wenzelm@26849:   needed, they can be helpful in some proofs about partial functions.
wenzelm@26849: 
wenzelm@26849:   \item [@{text tailrec}] generates the unconstrained recursive
wenzelm@26849:   equations even without a termination proof, provided that the
wenzelm@26849:   function is tail-recursive. This currently only works
wenzelm@26849: 
wenzelm@26849:   \item [@{text "default d"}] allows to specify a default value for a
wenzelm@26849:   (partial) function, which will ensure that @{text "f x = d x"}
wenzelm@26849:   whenever @{text "x \<notin> f_dom"}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Proof methods related to recursive definitions *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{method_def (HOL) pat_completeness} & : & \isarmeth \\
wenzelm@26849:     @{method_def (HOL) relation} & : & \isarmeth \\
wenzelm@26849:     @{method_def (HOL) lexicographic_order} & : & \isarmeth \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'relation' term
wenzelm@26849:     ;
wenzelm@26849:     'lexicographic\_order' (clasimpmod *)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{method (HOL) pat_completeness}] is a specialized method to
wenzelm@26849:   solve goals regarding the completeness of pattern matching, as
wenzelm@26849:   required by the @{command (HOL) "function"} package (cf.\
wenzelm@26849:   \cite{isabelle-function}).
wenzelm@26849: 
wenzelm@26849:   \item [@{method (HOL) relation}~@{text R}] introduces a termination
wenzelm@26849:   proof using the relation @{text R}.  The resulting proof state will
wenzelm@26849:   contain goals expressing that @{text R} is wellfounded, and that the
wenzelm@26849:   arguments of recursive calls decrease with respect to @{text R}.
wenzelm@26849:   Usually, this method is used as the initial proof step of manual
wenzelm@26849:   termination proofs.
wenzelm@26849: 
wenzelm@26849:   \item [@{method (HOL) "lexicographic_order"}] attempts a fully
wenzelm@26849:   automated termination proof by searching for a lexicographic
wenzelm@26849:   combination of size measures on the arguments of the function. The
wenzelm@26849:   method accepts the same arguments as the @{method auto} method,
wenzelm@26849:   which it uses internally to prove local descents.  The same context
wenzelm@26849:   modifiers as for @{method auto} are accepted, see
wenzelm@26849:   \secref{sec:clasimp}.
wenzelm@26849: 
wenzelm@26849:   In case of failure, extensive information is printed, which can help
wenzelm@26849:   to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Old-style recursive function definitions (TFL) *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
wenzelm@26849:   "recdef_tc"} for defining recursive are mostly obsolete; @{command
wenzelm@26849:   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "recdef"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849:     ;
wenzelm@26849:     recdeftc thmdecl? tc
wenzelm@26849:     ;
wenzelm@26849:     hints: '(' 'hints' (recdefmod *) ')'
wenzelm@26849:     ;
wenzelm@26849:     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849:     ;
wenzelm@26849:     tc: nameref ('(' nat ')')?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26849:   \item [@{command (HOL) "recdef"}] defines general well-founded
wenzelm@26849:   recursive functions (using the TFL package), see also
wenzelm@26849:   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
wenzelm@26849:   TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849:   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
wenzelm@26849:   recdef_wf} hints refer to auxiliary rules to be used in the internal
wenzelm@26849:   automated proof process of TFL.  Additional @{syntax clasimpmod}
wenzelm@26849:   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849:   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849:   Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849:   
wenzelm@26849:   \item [@{command (HOL) "recdef_tc"}~@{text "c (i)"}] recommences the
wenzelm@26849:   proof for leftover termination condition number @{text i} (default
wenzelm@26849:   1) as generated by a @{command (HOL) "recdef"} definition of
wenzelm@26849:   constant @{text c}.
wenzelm@26849:   
wenzelm@26849:   Note that in most cases, @{command (HOL) "recdef"} is able to finish
wenzelm@26849:   its internal proofs without manual intervention.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   \medskip Hints for @{command (HOL) "recdef"} may be also declared
wenzelm@26849:   globally, using the following attributes.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{attribute_def (HOL) recdef_simp} & : & \isaratt \\
wenzelm@26849:     @{attribute_def (HOL) recdef_cong} & : & \isaratt \\
wenzelm@26849:     @{attribute_def (HOL) recdef_wf} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849:   set) @{text R} closed under given rules: applying a rule to elements
wenzelm@26849:   of @{text R} yields a result within @{text R}.  For example, a
wenzelm@26849:   structural operational semantics is an inductive definition of an
wenzelm@26849:   evaluation relation.
wenzelm@26849: 
wenzelm@26849:   Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849:   predicate~/ set @{text R} that is consistent with given rules: every
wenzelm@26849:   element of @{text R} can be seen as arising by applying a rule to
wenzelm@26849:   elements of @{text R}.  An important example is using bisimulation
wenzelm@26849:   relations to formalise equivalence of processes and infinite data
wenzelm@26849:   structures.
wenzelm@26849: 
wenzelm@26849:   \medskip The HOL package is related to the ZF one, which is
wenzelm@26849:   described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849:   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849:   which you should refer to in case of difficulties.  The package is
wenzelm@26849:   simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849:   The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849:   domain of the fixedpoint definition, and the package does not have
wenzelm@26849:   to use inference rules for type-checking.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "inductive"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849:     @{attribute_def (HOL) mono} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849:     ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849:     ;
wenzelm@26849:     clauses: (thmdecl? prop + '|')
wenzelm@26849:     ;
wenzelm@26849:     'mono' (() | 'add' | 'del')
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "inductive"} and @{command (HOL)
wenzelm@26849:   "coinductive"}] define (co)inductive predicates from the
wenzelm@26849:   introduction rules given in the @{keyword "where"} part.  The
wenzelm@26849:   optional @{keyword "for"} part contains a list of parameters of the
wenzelm@26849:   (co)inductive predicates that remain fixed throughout the
wenzelm@26849:   definition.  The optional @{keyword "monos"} section contains
wenzelm@26849:   \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849:   applied to a recursive set in the introduction rules.  There
wenzelm@26849:   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
wenzelm@26849:   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "inductive_set"} and @{command (HOL)
wenzelm@26849:   "coinductive_set"}] are wrappers for to the previous commands,
wenzelm@26849:   allowing the definition of (co)inductive sets.
wenzelm@26849: 
wenzelm@26849:   \item [@{attribute (HOL) mono}] declares monotonicity rules.  These
wenzelm@26849:   rule are involved in the automated monotonicity proof of @{command
wenzelm@26849:   (HOL) "inductive"}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Derived rules *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   Each (co)inductive definition @{text R} adds definitions to the
wenzelm@26849:   theory and also proves some theorems:
wenzelm@26849: 
wenzelm@26849:   \begin{description}
wenzelm@26849: 
wenzelm@26849:   \item [@{text R.intros}] is the list of introduction rules as proven
wenzelm@26849:   theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849:   also available individually, using the names given them in the
wenzelm@26849:   theory file;
wenzelm@26849: 
wenzelm@26849:   \item [@{text R.cases}] is the case analysis (or elimination) rule;
wenzelm@26849: 
wenzelm@26849:   \item [@{text R.induct} or @{text R.coinduct}] is the (co)induction
wenzelm@26849:   rule.
wenzelm@26849: 
wenzelm@26849:   \end{description}
wenzelm@26849: 
wenzelm@26849:   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
wenzelm@26849:   defined simultaneously, the list of introduction rules is called
wenzelm@26849:   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
wenzelm@26849:   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
wenzelm@26849:   of mutual induction rules is called @{text
wenzelm@26849:   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: subsection {* Monotonicity theorems *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   Each theory contains a default set of theorems that are used in
wenzelm@26849:   monotonicity proofs.  New rules can be added to this set via the
wenzelm@26849:   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
wenzelm@26849:   shows how this is done.  In general, the following monotonicity
wenzelm@26849:   theorems may be added:
wenzelm@26849: 
wenzelm@26849:   \begin{itemize}
wenzelm@26849: 
wenzelm@26849:   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
wenzelm@26849:   monotonicity of inductive definitions whose introduction rules have
wenzelm@26849:   premises involving terms such as @{text "M R\<^sub>i t"}.
wenzelm@26849: 
wenzelm@26849:   \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849:   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
wenzelm@26849:   the case of the operator @{text "\<or>"}, the corresponding theorem is
wenzelm@26849:   \[
wenzelm@26849:   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849:   of expressions, e.g.
wenzelm@26849:   \[
wenzelm@26849:   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
wenzelm@26849:   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \item Equations for reducing complex operators to more primitive
wenzelm@26849:   ones whose monotonicity can easily be proved, e.g.
wenzelm@26849:   \[
wenzelm@26849:   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
wenzelm@26849:   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \end{itemize}
wenzelm@26849: 
wenzelm@26849:   %FIXME: Example of an inductive definition
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Arithmetic proof support *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{method_def (HOL) arith} & : & \isarmeth \\
wenzelm@26894:     @{attribute_def (HOL) arith_split} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   The @{method (HOL) arith} method decides linear arithmetic problems
wenzelm@26849:   (on types @{text nat}, @{text int}, @{text real}).  Any current
wenzelm@26849:   facts are inserted into the goal before running the procedure.
wenzelm@26849: 
wenzelm@26894:   The @{attribute (HOL) arith_split} attribute declares case split
wenzelm@26894:   rules to be expanded before the arithmetic procedure is invoked.
wenzelm@26849: 
wenzelm@26849:   Note that a simpler (but faster) version of arithmetic reasoning is
wenzelm@26849:   already performed by the Simplifier.
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac} *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   The following important tactical tools of Isabelle/HOL have been
wenzelm@26849:   ported to Isar.  These should be never used in proper proof texts!
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & \isarmeth \\
wenzelm@26849:     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & \isarmeth \\
wenzelm@26849:     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & \isarmeth \\
wenzelm@26849:     @{command_def (HOL) "inductive_cases"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'case\_tac' goalspec? term rule?
wenzelm@26849:     ;
wenzelm@26849:     'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849:     ;
wenzelm@26849:     'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849:     ;
wenzelm@26849:     'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     rule: ('rule' ':' thmref)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]
wenzelm@26849:   admit to reason about inductive datatypes only (unless an
wenzelm@26849:   alternative rule is given explicitly).  Furthermore, @{method (HOL)
wenzelm@26849:   case_tac} does a classical case split on booleans; @{method (HOL)
wenzelm@26849:   induct_tac} allows only variables to be given as instantiation.
wenzelm@26849:   These tactic emulations feature both goal addressing and dynamic
wenzelm@26849:   instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@26849:   as would be by the proper @{method induct} and @{method cases} proof
wenzelm@26849:   methods (see \secref{sec:cases-induct}).
wenzelm@26849:   
wenzelm@26849:   \item [@{method (HOL) ind_cases} and @{command (HOL)
wenzelm@26860:   "inductive_cases"}] provide an interface to the internal @{ML_text
wenzelm@26860:   mk_cases} operation.  Rules are simplified in an unrestricted
wenzelm@26860:   forward manner.
wenzelm@26849: 
wenzelm@26849:   While @{method (HOL) ind_cases} is a proof method to apply the
wenzelm@26849:   result immediately as elimination rules, @{command (HOL)
wenzelm@26849:   "inductive_cases"} provides case split theorems at the theory level
wenzelm@26849:   for later use.  The @{keyword "for"} argument of the @{method (HOL)
wenzelm@26849:   ind_cases} method allows to specify a list of variables that should
wenzelm@26849:   be generalized before applying the resulting rule.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@26849: section {* Executable code *}
wenzelm@26849: 
wenzelm@26849: text {*
wenzelm@26849:   Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849:   generation from executable specifications.  Isabelle/HOL
wenzelm@26849:   instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849:   applications.
wenzelm@26849: 
wenzelm@26849:   One framework generates code from both functional and relational
wenzelm@26849:   programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849:   (this actually covers the new-style theory format as well).
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849:     @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\  
wenzelm@26849:     @{attribute_def (HOL) code} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:   'value' term
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849:     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849:     'contains' ( ( name '=' term ) + | term + )
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   modespec: '(' ( name * ) ')'
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'consts\_code' (codespec +)
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   codespec: const template attachment ?
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'types\_code' (tycodespec +)
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   tycodespec: name template attachment ?
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   const: term
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   template: '(' string ')'
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'code' (name)?
wenzelm@26849:   ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "value"}~@{text t}] evaluates and prints a
wenzelm@26849:   term using the code generator.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   \medskip The other framework generates code from functional programs
wenzelm@26849:   (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849:   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849:   Conceptually, code generation is split up in three steps:
wenzelm@26849:   \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849:   abstract executable view and \emph{serialization} to a specific
wenzelm@26849:   \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849:   introduction on how to use it.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26849:     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849:     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849:     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849:     @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "code_exception"} & : & \isartrans{theory}{theory} \\
wenzelm@26849:     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849:     @{attribute_def (HOL) code} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'export\_code' ( constexpr + ) ? \\
wenzelm@26849:       ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849:         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_thms' ( constexpr + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_deps' ( constexpr + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     const: term
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     constexpr: ( const | 'name.*' | '*' )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     typeconstructor: nameref
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     class: nameref
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_datatype' const +
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_const' (const + 'and') \\
wenzelm@26849:       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_type' (typeconstructor + 'and') \\
wenzelm@26849:       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_class' (class + 'and') \\
wenzelm@26849:       ( ( '(' target \\
wenzelm@26849:         ( ( string ('where' \\
wenzelm@26849:           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849:       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_monad' const const target
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_reserved' target ( string + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_include' target ( string ( string | '-') )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_modulename' target ( ( string string ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_exception' ( const + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code' ('func' | 'inline') ( 'del' )?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "export_code"}] is the canonical interface
wenzelm@26849:   for generating and serializing code: for a given list of constants,
wenzelm@26849:   code is generated for the specified target languages.  Abstract code
wenzelm@26849:   is cached incrementally.  If no constant is given, the currently
wenzelm@26849:   cached code is serialized.  If no serialization instruction is
wenzelm@26849:   given, only abstract code is cached.
wenzelm@26849: 
wenzelm@26849:   Constants may be specified by giving them literally, referring to
wenzelm@26849:   all executable contants within a certain theory by giving @{text
wenzelm@26849:   "name.*"}, or referring to \emph{all} executable constants currently
wenzelm@26849:   available by giving @{text "*"}.
wenzelm@26849: 
wenzelm@26849:   By default, for each involved theory one corresponding name space
wenzelm@26849:   module is generated.  Alternativly, a module name may be specified
wenzelm@26849:   after the @{keyword "module_name"} keyword; then \emph{all} code is
wenzelm@26849:   placed in this module.
wenzelm@26849: 
wenzelm@26849:   For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849:   single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849:   where code is generated in multiple files reflecting the module
wenzelm@26849:   hierarchy.  The file specification ``@{text "-"}'' denotes standard
wenzelm@26849:   output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849:   code internally in the context of the current ML session.
wenzelm@26849: 
wenzelm@26849:   Serializers take an optional list of arguments in parentheses.  For
wenzelm@26849:   \emph{Haskell} a module name prefix may be given using the ``@{text
wenzelm@26849:   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
wenzelm@26849:   "deriving (Read, Show)"}'' clause to each appropriate datatype
wenzelm@26849:   declaration.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_thms"}] prints a list of theorems
wenzelm@26849:   representing the corresponding program containing all given
wenzelm@26849:   constants; if no constants are given, the currently cached code
wenzelm@26849:   theorems are printed.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_deps"}] visualizes dependencies of
wenzelm@26849:   theorems representing the corresponding program containing all given
wenzelm@26849:   constants; if no constants are given, the currently cached code
wenzelm@26849:   theorems are visualized.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_datatype"}] specifies a constructor set
wenzelm@26849:   for a logical type.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_const"}] associates a list of constants
wenzelm@26849:   with target-specific serializations; omitting a serialization
wenzelm@26849:   deletes an existing serialization.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_type"}] associates a list of type
wenzelm@26849:   constructors with target-specific serializations; omitting a
wenzelm@26849:   serialization deletes an existing serialization.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_class"}] associates a list of classes
wenzelm@26849:   with target-specific class names; in addition, constants associated
wenzelm@26849:   with this class may be given target-specific names used for instance
wenzelm@26849:   declarations; omitting a serialization deletes an existing
wenzelm@26849:   serialization.  This applies only to \emph{Haskell}.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_instance"}] declares a list of type
wenzelm@26849:   constructor / class instance relations as ``already present'' for a
wenzelm@26849:   given target.  Omitting a ``@{text "-"}'' deletes an existing
wenzelm@26849:   ``already present'' declaration.  This applies only to
wenzelm@26849:   \emph{Haskell}.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_monad"}] provides an auxiliary
wenzelm@26849:   mechanism to generate monadic code.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_reserved"}] declares a list of names as
wenzelm@26849:   reserved for a given target, preventing it to be shadowed by any
wenzelm@26849:   generated code.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_include"}] adds arbitrary named content
wenzelm@26849:   (``include'') to generated code.  A as last argument ``@{text "-"}''
wenzelm@26849:   will remove an already added ``include''.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_modulename"}] declares aliasings from
wenzelm@26849:   one module name onto another.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "code_exception"}] declares constants which
wenzelm@26849:   are not required to have a definition by a defining equations; these
wenzelm@26849:   are mapped on exceptions instead.
wenzelm@26849: 
wenzelm@26849:   \item [@{attribute (HOL) code}~@{text func}] explicitly selects (or
wenzelm@26849:   with option ``@{text "del:"}'' deselects) a defining equation for
wenzelm@26849:   code generation.  Usually packages introducing defining equations
wenzelm@26849:   provide a resonable default setup for selection.
wenzelm@26849: 
wenzelm@26849:   \item [@{attribute (HOL) code}@{text inline}] declares (or with
wenzelm@26849:   option ``@{text "del:"}'' removes) inlining theorems which are
wenzelm@26849:   applied as rewrite rules to any defining equation during
wenzelm@26849:   preprocessing.
wenzelm@26849: 
wenzelm@26849:   \item [@{command (HOL) "print_codesetup"}] gives an overview on
wenzelm@26849:   selected defining equations, code generator datatypes and
wenzelm@26849:   preprocessor setup.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: *}
wenzelm@26849: 
wenzelm@27045: 
wenzelm@27045: section {* Definition by specification \label{sec:hol-specification} *}
wenzelm@27045: 
wenzelm@27045: text {*
wenzelm@27045:   \begin{matharray}{rcl}
wenzelm@27045:     @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@27045:     @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@27045:   \end{matharray}
wenzelm@27045: 
wenzelm@27045:   \begin{rail}
wenzelm@27045:   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27045:   ;
wenzelm@27045:   decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27045:   \end{rail}
wenzelm@27045: 
wenzelm@27045:   \begin{descr}
wenzelm@27045: 
wenzelm@27045:   \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a
wenzelm@27045:   goal stating the existence of terms with the properties specified to
wenzelm@27045:   hold for the constants given in @{text decls}.  After finishing the
wenzelm@27045:   proof, the theory will be augmented with definitions for the given
wenzelm@27045:   constants, as well as with theorems stating the properties for these
wenzelm@27045:   constants.
wenzelm@27045: 
wenzelm@27045:   \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets
wenzelm@27045:   up a goal stating the existence of terms with the properties
wenzelm@27045:   specified to hold for the constants given in @{text decls}.  After
wenzelm@27045:   finishing the proof, the theory will be augmented with axioms
wenzelm@27045:   expressing the properties given in the first place.
wenzelm@27045: 
wenzelm@27045:   \item [@{text decl}] declares a constant to be defined by the
wenzelm@27045:   specification given.  The definition for the constant @{text c} is
wenzelm@27045:   bound to the name @{text c_def} unless a theorem name is given in
wenzelm@27045:   the declaration.  Overloaded constants should be declared as such.
wenzelm@27045: 
wenzelm@27045:   \end{descr}
wenzelm@27045: 
wenzelm@27045:   Whether to use @{command (HOL) "specification"} or @{command (HOL)
wenzelm@27045:   "ax_specification"} is to some extent a matter of style.  @{command
wenzelm@27045:   (HOL) "specification"} introduces no new axioms, and so by
wenzelm@27045:   construction cannot introduce inconsistencies, whereas @{command
wenzelm@27045:   (HOL) "ax_specification"} does introduce axioms, but only after the
wenzelm@27045:   user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27045:   considered, though: After introducing two constants with the same
wenzelm@27045:   properties using @{command (HOL) "specification"}, one can prove
wenzelm@27045:   that the two constants are, in fact, equal.  If this might be a
wenzelm@27045:   problem, one should use @{command (HOL) "ax_specification"}.
wenzelm@27045: *}
wenzelm@27045: 
wenzelm@26840: end




