doc-src/IsarRef/Thy/HOL_Specific.thy
author wenzelm
Fri Mar 19 00:46:08 2010 +0100 (2010-03-19)
changeset 35841 94f901e4969a
parent 35744 93603d7b8ee9
child 36139 0c2538afe8e8
permissions -rw-r--r--
allow sort constraints in HOL/typedef;
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theory HOL_Specific
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imports Main
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begin
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chapter {* Isabelle/HOL \label{ch:hol} *}
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section {* Typedef axiomatization \label{sec:hol-typedef} *}
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text {*
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  \begin{matharray}{rcl}
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    @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
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  \end{matharray}
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  \begin{rail}
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    'typedef' altname? abstype '=' repset
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    ;
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    altname: '(' (name | 'open' | 'open' name) ')'
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    ;
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    abstype: typespecsorts mixfix?
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    ;
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    repset: term ('morphisms' name name)?
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    ;
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  \end{rail}
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  \begin{description}
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  \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
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  axiomatizes a Gordon/HOL-style type definition in the background
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  theory of the current context, depending on a non-emptiness result
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  of the set @{text A} (which needs to be proven interactively).
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  The raw type may not depend on parameters or assumptions of the
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  context --- this is logically impossible in Isabelle/HOL --- but the
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  non-emptiness property can be local, potentially resulting in
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  multiple interpretations in target contexts.  Thus the established
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  bijection between the representing set @{text A} and the new type
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  @{text t} may semantically depend on local assumptions.
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  By default, @{command (HOL) "typedef"} defines both a type @{text t}
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  and a set (term constant) of the same name, unless an alternative
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  base name is given in parentheses, or the ``@{text "(open)"}''
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  declaration is used to suppress a separate constant definition
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  altogether.  The injection from type to set is called @{text Rep_t},
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  its inverse @{text Abs_t} --- this may be changed via an explicit
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  @{keyword (HOL) "morphisms"} declaration.
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  Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
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  Abs_t_inverse} provide the most basic characterization as a
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  corresponding injection/surjection pair (in both directions).  Rules
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  @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
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  more convenient view on the injectivity part, suitable for automated
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  proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
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  declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
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  @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
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  on surjectivity; these are already declared as set or type rules for
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  the generic @{method cases} and @{method induct} methods.
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  An alternative name for the set definition (and other derived
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  entities) may be specified in parentheses; the default is to use
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  @{text t} as indicated before.
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  \end{description}
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*}
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section {* Adhoc tuples *}
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text {*
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  \begin{matharray}{rcl}
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    @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
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  \end{matharray}
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  \begin{rail}
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    'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
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    ;
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  \end{rail}
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  \begin{description}
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  \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
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  \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
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  canonical form as specified by the arguments given; the @{text i}-th
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  collection of arguments refers to occurrences in premise @{text i}
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  of the rule.  The ``@{text "(complete)"}'' option causes \emph{all}
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  arguments in function applications to be represented canonically
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  according to their tuple type structure.
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  Note that these operations tend to invent funny names for new local
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  parameters to be introduced.
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  \end{description}
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*}
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section {* Records \label{sec:hol-record} *}
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text {*
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  In principle, records merely generalize the concept of tuples, where
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  components may be addressed by labels instead of just position.  The
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  logical infrastructure of records in Isabelle/HOL is slightly more
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  advanced, though, supporting truly extensible record schemes.  This
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  admits operations that are polymorphic with respect to record
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  extension, yielding ``object-oriented'' effects like (single)
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  inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
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  details on object-oriented verification and record subtyping in HOL.
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*}
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subsection {* Basic concepts *}
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text {*
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  Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
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  at the level of terms and types.  The notation is as follows:
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  \begin{center}
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  \begin{tabular}{l|l|l}
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    & record terms & record types \\ \hline
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    fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
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    schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
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      @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
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  \end{tabular}
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  \end{center}
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  \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
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  "(| x = a |)"}.
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  A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
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  @{text a} and field @{text y} of value @{text b}.  The corresponding
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  type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
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  and @{text "b :: B"}.
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  A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
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  @{text x} and @{text y} as before, but also possibly further fields
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  as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
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  of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
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  scheme is called the \emph{more part}.  Logically it is just a free
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  variable, which is occasionally referred to as ``row variable'' in
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  the literature.  The more part of a record scheme may be
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  instantiated by zero or more further components.  For example, the
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  previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
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  c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
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  Fixed records are special instances of record schemes, where
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  ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
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  element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
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  for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
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  \medskip Two key observations make extensible records in a simply
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  typed language like HOL work out:
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  \begin{enumerate}
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  \item the more part is internalized, as a free term or type
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  variable,
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  \item field names are externalized, they cannot be accessed within
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  the logic as first-class values.
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  \end{enumerate}
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  \medskip In Isabelle/HOL record types have to be defined explicitly,
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  fixing their field names and types, and their (optional) parent
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  record.  Afterwards, records may be formed using above syntax, while
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  obeying the canonical order of fields as given by their declaration.
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  The record package provides several standard operations like
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  selectors and updates.  The common setup for various generic proof
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  tools enable succinct reasoning patterns.  See also the Isabelle/HOL
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  tutorial \cite{isabelle-hol-book} for further instructions on using
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  records in practice.
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*}
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subsection {* Record specifications *}
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text {*
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  \begin{matharray}{rcl}
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    @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
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  \end{matharray}
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  \begin{rail}
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    'record' typespec '=' (type '+')? (constdecl +)
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    ;
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  \end{rail}
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  \begin{description}
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  \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
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  \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
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  derived from the optional parent record @{text "\<tau>"} by adding new
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  field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
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  The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
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  covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
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  \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
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  \<tau>} needs to specify an instance of an existing record type.  At
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  least one new field @{text "c\<^sub>i"} has to be specified.
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  Basically, field names need to belong to a unique record.  This is
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  not a real restriction in practice, since fields are qualified by
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  the record name internally.
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  The parent record specification @{text \<tau>} is optional; if omitted
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  @{text t} becomes a root record.  The hierarchy of all records
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  declared within a theory context forms a forest structure, i.e.\ a
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  set of trees starting with a root record each.  There is no way to
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  merge multiple parent records!
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  For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
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  type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
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  \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
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  "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
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  @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
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  \<zeta>\<rparr>"}.
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  \end{description}
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*}
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subsection {* Record operations *}
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text {*
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  Any record definition of the form presented above produces certain
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  standard operations.  Selectors and updates are provided for any
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  field, including the improper one ``@{text more}''.  There are also
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  cumulative record constructor functions.  To simplify the
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  presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
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  \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
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  \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
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  \medskip \textbf{Selectors} and \textbf{updates} are available for
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  any field (including ``@{text more}''):
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  \begin{matharray}{lll}
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    @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
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    @{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>"} \\
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  \end{matharray}
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  There is special syntax for application of updates: @{text "r\<lparr>x :=
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  a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
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  repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
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  c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
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  because of postfix notation the order of fields shown here is
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  reverse than in the actual term.  Since repeated updates are just
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  function applications, fields may be freely permuted in @{text "\<lparr>x
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  := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
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  Thus commutativity of independent updates can be proven within the
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  logic for any two fields, but not as a general theorem.
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  \medskip The \textbf{make} operation provides a cumulative record
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  constructor function:
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  \begin{matharray}{lll}
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    @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
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  \end{matharray}
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  \medskip We now reconsider the case of non-root records, which are
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  derived of some parent.  In general, the latter may depend on
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  another parent as well, resulting in a list of \emph{ancestor
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  records}.  Appending the lists of fields of all ancestors results in
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  a certain field prefix.  The record package automatically takes care
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  of this by lifting operations over this context of ancestor fields.
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  Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
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  fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
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  the above record operations will get the following types:
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  \medskip
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  \begin{tabular}{lll}
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    @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
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    @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
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      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
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      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
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    @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
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      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
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  \end{tabular}
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  \medskip
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  \noindent Some further operations address the extension aspect of a
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  derived record scheme specifically: @{text "t.fields"} produces a
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  record fragment consisting of exactly the new fields introduced here
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  (the result may serve as a more part elsewhere); @{text "t.extend"}
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  takes a fixed record and adds a given more part; @{text
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  "t.truncate"} restricts a record scheme to a fixed record.
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  \medskip
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  \begin{tabular}{lll}
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    @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
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    @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
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      \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
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    @{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>"} \\
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  \end{tabular}
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   290
  \medskip
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   291
wenzelm@26849
   292
  \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
wenzelm@26849
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  for root records.
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   294
*}
wenzelm@26849
   295
wenzelm@26849
   296
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   297
subsection {* Derived rules and proof tools *}
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   298
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   299
text {*
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   300
  The record package proves several results internally, declaring
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   301
  these facts to appropriate proof tools.  This enables users to
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  reason about record structures quite conveniently.  Assume that
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   303
  @{text t} is a record type as specified above.
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   304
wenzelm@26849
   305
  \begin{enumerate}
wenzelm@26849
   306
  
wenzelm@26849
   307
  \item Standard conversions for selectors or updates applied to
wenzelm@26849
   308
  record constructor terms are made part of the default Simplifier
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   309
  context; thus proofs by reduction of basic operations merely require
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   310
  the @{method simp} method without further arguments.  These rules
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   311
  are available as @{text "t.simps"}, too.
wenzelm@26849
   312
  
wenzelm@26849
   313
  \item Selectors applied to updated records are automatically reduced
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   314
  by an internal simplification procedure, which is also part of the
wenzelm@26849
   315
  standard Simplifier setup.
wenzelm@26849
   316
wenzelm@26849
   317
  \item Inject equations of a form analogous to @{prop "(x, y) = (x',
wenzelm@26849
   318
  y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
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   319
  Reasoner as @{attribute iff} rules.  These rules are available as
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   320
  @{text "t.iffs"}.
wenzelm@26849
   321
wenzelm@26849
   322
  \item The introduction rule for record equality analogous to @{text
wenzelm@26849
   323
  "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
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   324
  and as the basic rule context as ``@{attribute intro}@{text "?"}''.
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   325
  The rule is called @{text "t.equality"}.
wenzelm@26849
   326
wenzelm@26849
   327
  \item Representations of arbitrary record expressions as canonical
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   328
  constructor terms are provided both in @{method cases} and @{method
wenzelm@26849
   329
  induct} format (cf.\ the generic proof methods of the same name,
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   330
  \secref{sec:cases-induct}).  Several variations are available, for
wenzelm@26849
   331
  fixed records, record schemes, more parts etc.
wenzelm@26849
   332
  
wenzelm@26849
   333
  The generic proof methods are sufficiently smart to pick the most
wenzelm@26849
   334
  sensible rule according to the type of the indicated record
wenzelm@26849
   335
  expression: users just need to apply something like ``@{text "(cases
wenzelm@26849
   336
  r)"}'' to a certain proof problem.
wenzelm@26849
   337
wenzelm@26849
   338
  \item The derived record operations @{text "t.make"}, @{text
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   339
  "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
wenzelm@26849
   340
  treated automatically, but usually need to be expanded by hand,
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   341
  using the collective fact @{text "t.defs"}.
wenzelm@26849
   342
wenzelm@26849
   343
  \end{enumerate}
wenzelm@26849
   344
*}
wenzelm@26849
   345
wenzelm@26849
   346
wenzelm@26849
   347
section {* Datatypes \label{sec:hol-datatype} *}
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   348
wenzelm@26849
   349
text {*
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   350
  \begin{matharray}{rcl}
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   351
    @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
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   352
  @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
wenzelm@26849
   353
  \end{matharray}
wenzelm@26849
   354
wenzelm@26849
   355
  \begin{rail}
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   356
    'datatype' (dtspec + 'and')
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   357
    ;
haftmann@27452
   358
    'rep\_datatype' ('(' (name +) ')')? (term +)
wenzelm@26849
   359
    ;
wenzelm@26849
   360
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   361
    dtspec: parname? typespec mixfix? '=' (cons + '|')
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   362
    ;
haftmann@31912
   363
    cons: name ( type * ) mixfix?
wenzelm@26849
   364
  \end{rail}
wenzelm@26849
   365
wenzelm@28760
   366
  \begin{description}
wenzelm@26849
   367
wenzelm@28760
   368
  \item @{command (HOL) "datatype"} defines inductive datatypes in
wenzelm@26849
   369
  HOL.
wenzelm@26849
   370
wenzelm@28760
   371
  \item @{command (HOL) "rep_datatype"} represents existing types as
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   372
  inductive ones, generating the standard infrastructure of derived
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   373
  concepts (primitive recursion etc.).
wenzelm@26849
   374
wenzelm@28760
   375
  \end{description}
wenzelm@26849
   376
wenzelm@26849
   377
  The induction and exhaustion theorems generated provide case names
wenzelm@26849
   378
  according to the constructors involved, while parameters are named
wenzelm@26849
   379
  after the types (see also \secref{sec:cases-induct}).
wenzelm@26849
   380
wenzelm@26849
   381
  See \cite{isabelle-HOL} for more details on datatypes, but beware of
wenzelm@26849
   382
  the old-style theory syntax being used there!  Apart from proper
wenzelm@26849
   383
  proof methods for case-analysis and induction, there are also
wenzelm@26849
   384
  emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
wenzelm@26849
   385
  induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
wenzelm@26849
   386
  to refer directly to the internal structure of subgoals (including
wenzelm@26849
   387
  internally bound parameters).
wenzelm@26849
   388
*}
wenzelm@26849
   389
wenzelm@26849
   390
wenzelm@26849
   391
section {* Recursive functions \label{sec:recursion} *}
wenzelm@26849
   392
wenzelm@26849
   393
text {*
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   394
  \begin{matharray}{rcl}
wenzelm@28761
   395
    @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   396
    @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   397
    @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
wenzelm@28761
   398
    @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
wenzelm@26849
   399
  \end{matharray}
wenzelm@26849
   400
wenzelm@26849
   401
  \begin{rail}
wenzelm@26849
   402
    'primrec' target? fixes 'where' equations
wenzelm@26849
   403
    ;
wenzelm@26849
   404
    equations: (thmdecl? prop + '|')
wenzelm@26849
   405
    ;
wenzelm@26985
   406
    ('fun' | 'function') target? functionopts? fixes 'where' clauses
wenzelm@26849
   407
    ;
wenzelm@26849
   408
    clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
wenzelm@26849
   409
    ;
wenzelm@26985
   410
    functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
wenzelm@26849
   411
    ;
wenzelm@26849
   412
    'termination' ( term )?
wenzelm@26849
   413
  \end{rail}
wenzelm@26849
   414
wenzelm@28760
   415
  \begin{description}
wenzelm@26849
   416
wenzelm@28760
   417
  \item @{command (HOL) "primrec"} defines primitive recursive
wenzelm@26849
   418
  functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849
   419
wenzelm@28760
   420
  \item @{command (HOL) "function"} defines functions by general
wenzelm@26849
   421
  wellfounded recursion. A detailed description with examples can be
wenzelm@26849
   422
  found in \cite{isabelle-function}. The function is specified by a
wenzelm@26849
   423
  set of (possibly conditional) recursive equations with arbitrary
wenzelm@26849
   424
  pattern matching. The command generates proof obligations for the
wenzelm@26849
   425
  completeness and the compatibility of patterns.
wenzelm@26849
   426
wenzelm@26849
   427
  The defined function is considered partial, and the resulting
wenzelm@26849
   428
  simplification rules (named @{text "f.psimps"}) and induction rule
wenzelm@26849
   429
  (named @{text "f.pinduct"}) are guarded by a generated domain
wenzelm@26849
   430
  predicate @{text "f_dom"}. The @{command (HOL) "termination"}
wenzelm@26849
   431
  command can then be used to establish that the function is total.
wenzelm@26849
   432
wenzelm@28760
   433
  \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
wenzelm@28760
   434
  (HOL) "function"}~@{text "(sequential)"}, followed by automated
wenzelm@28760
   435
  proof attempts regarding pattern matching and termination.  See
wenzelm@28760
   436
  \cite{isabelle-function} for further details.
wenzelm@26849
   437
wenzelm@28760
   438
  \item @{command (HOL) "termination"}~@{text f} commences a
wenzelm@26849
   439
  termination proof for the previously defined function @{text f}.  If
wenzelm@26849
   440
  this is omitted, the command refers to the most recent function
wenzelm@26849
   441
  definition.  After the proof is closed, the recursive equations and
wenzelm@26849
   442
  the induction principle is established.
wenzelm@26849
   443
wenzelm@28760
   444
  \end{description}
wenzelm@26849
   445
haftmann@27452
   446
  Recursive definitions introduced by the @{command (HOL) "function"}
haftmann@27452
   447
  command accommodate
wenzelm@26849
   448
  reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
wenzelm@26849
   449
  "c.induct"} (where @{text c} is the name of the function definition)
wenzelm@26849
   450
  refers to a specific induction rule, with parameters named according
krauss@33857
   451
  to the user-specified equations. Cases are numbered (starting from 1).
krauss@33857
   452
krauss@33857
   453
  For @{command (HOL) "primrec"}, the induction principle coincides
haftmann@27452
   454
  with structural recursion on the datatype the recursion is carried
haftmann@27452
   455
  out.
wenzelm@26849
   456
wenzelm@26849
   457
  The equations provided by these packages may be referred later as
wenzelm@26849
   458
  theorem list @{text "f.simps"}, where @{text f} is the (collective)
wenzelm@26849
   459
  name of the functions defined.  Individual equations may be named
wenzelm@26849
   460
  explicitly as well.
wenzelm@26849
   461
wenzelm@26849
   462
  The @{command (HOL) "function"} command accepts the following
wenzelm@26849
   463
  options.
wenzelm@26849
   464
wenzelm@28760
   465
  \begin{description}
wenzelm@26849
   466
wenzelm@28760
   467
  \item @{text sequential} enables a preprocessor which disambiguates
wenzelm@28760
   468
  overlapping patterns by making them mutually disjoint.  Earlier
wenzelm@28760
   469
  equations take precedence over later ones.  This allows to give the
wenzelm@28760
   470
  specification in a format very similar to functional programming.
wenzelm@28760
   471
  Note that the resulting simplification and induction rules
wenzelm@28760
   472
  correspond to the transformed specification, not the one given
wenzelm@26849
   473
  originally. This usually means that each equation given by the user
wenzelm@26849
   474
  may result in several theroems.  Also note that this automatic
wenzelm@26849
   475
  transformation only works for ML-style datatype patterns.
wenzelm@26849
   476
wenzelm@28760
   477
  \item @{text domintros} enables the automated generation of
wenzelm@26849
   478
  introduction rules for the domain predicate. While mostly not
wenzelm@26849
   479
  needed, they can be helpful in some proofs about partial functions.
wenzelm@26849
   480
wenzelm@28760
   481
  \item @{text tailrec} generates the unconstrained recursive
wenzelm@26849
   482
  equations even without a termination proof, provided that the
wenzelm@26849
   483
  function is tail-recursive. This currently only works
wenzelm@26849
   484
wenzelm@28760
   485
  \item @{text "default d"} allows to specify a default value for a
wenzelm@26849
   486
  (partial) function, which will ensure that @{text "f x = d x"}
wenzelm@26849
   487
  whenever @{text "x \<notin> f_dom"}.
wenzelm@26849
   488
wenzelm@28760
   489
  \end{description}
wenzelm@26849
   490
*}
wenzelm@26849
   491
wenzelm@26849
   492
wenzelm@26849
   493
subsection {* Proof methods related to recursive definitions *}
wenzelm@26849
   494
wenzelm@26849
   495
text {*
wenzelm@26849
   496
  \begin{matharray}{rcl}
wenzelm@28761
   497
    @{method_def (HOL) pat_completeness} & : & @{text method} \\
wenzelm@28761
   498
    @{method_def (HOL) relation} & : & @{text method} \\
wenzelm@28761
   499
    @{method_def (HOL) lexicographic_order} & : & @{text method} \\
krauss@33858
   500
    @{method_def (HOL) size_change} & : & @{text method} \\
wenzelm@26849
   501
  \end{matharray}
wenzelm@26849
   502
wenzelm@26849
   503
  \begin{rail}
wenzelm@26849
   504
    'relation' term
wenzelm@26849
   505
    ;
haftmann@31912
   506
    'lexicographic\_order' ( clasimpmod * )
wenzelm@26849
   507
    ;
krauss@33858
   508
    'size\_change' ( orders ( clasimpmod * ) )
krauss@33858
   509
    ;
krauss@33858
   510
    orders: ( 'max' | 'min' | 'ms' ) *
wenzelm@26849
   511
  \end{rail}
wenzelm@26849
   512
wenzelm@28760
   513
  \begin{description}
wenzelm@26849
   514
wenzelm@28760
   515
  \item @{method (HOL) pat_completeness} is a specialized method to
wenzelm@26849
   516
  solve goals regarding the completeness of pattern matching, as
wenzelm@26849
   517
  required by the @{command (HOL) "function"} package (cf.\
wenzelm@26849
   518
  \cite{isabelle-function}).
wenzelm@26849
   519
wenzelm@28760
   520
  \item @{method (HOL) relation}~@{text R} introduces a termination
wenzelm@26849
   521
  proof using the relation @{text R}.  The resulting proof state will
wenzelm@26849
   522
  contain goals expressing that @{text R} is wellfounded, and that the
wenzelm@26849
   523
  arguments of recursive calls decrease with respect to @{text R}.
wenzelm@26849
   524
  Usually, this method is used as the initial proof step of manual
wenzelm@26849
   525
  termination proofs.
wenzelm@26849
   526
wenzelm@28760
   527
  \item @{method (HOL) "lexicographic_order"} attempts a fully
wenzelm@26849
   528
  automated termination proof by searching for a lexicographic
wenzelm@26849
   529
  combination of size measures on the arguments of the function. The
wenzelm@26849
   530
  method accepts the same arguments as the @{method auto} method,
wenzelm@26849
   531
  which it uses internally to prove local descents.  The same context
wenzelm@26849
   532
  modifiers as for @{method auto} are accepted, see
wenzelm@26849
   533
  \secref{sec:clasimp}.
wenzelm@26849
   534
wenzelm@26849
   535
  In case of failure, extensive information is printed, which can help
wenzelm@26849
   536
  to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849
   537
krauss@33858
   538
  \item @{method (HOL) "size_change"} also works on termination goals,
krauss@33858
   539
  using a variation of the size-change principle, together with a
krauss@33858
   540
  graph decomposition technique (see \cite{krauss_phd} for details).
krauss@33858
   541
  Three kinds of orders are used internally: @{text max}, @{text min},
krauss@33858
   542
  and @{text ms} (multiset), which is only available when the theory
krauss@33858
   543
  @{text Multiset} is loaded. When no order kinds are given, they are
krauss@33858
   544
  tried in order. The search for a termination proof uses SAT solving
krauss@33858
   545
  internally.
krauss@33858
   546
krauss@33858
   547
 For local descent proofs, the same context modifiers as for @{method
krauss@33858
   548
  auto} are accepted, see \secref{sec:clasimp}.
krauss@33858
   549
wenzelm@28760
   550
  \end{description}
wenzelm@26849
   551
*}
wenzelm@26849
   552
wenzelm@26849
   553
wenzelm@26849
   554
subsection {* Old-style recursive function definitions (TFL) *}
wenzelm@26849
   555
wenzelm@26849
   556
text {*
wenzelm@26849
   557
  The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
wenzelm@26849
   558
  "recdef_tc"} for defining recursive are mostly obsolete; @{command
wenzelm@26849
   559
  (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
wenzelm@26849
   560
wenzelm@26849
   561
  \begin{matharray}{rcl}
wenzelm@28761
   562
    @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
wenzelm@28761
   563
    @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
wenzelm@26849
   564
  \end{matharray}
wenzelm@26849
   565
wenzelm@26849
   566
  \begin{rail}
wenzelm@26849
   567
    'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849
   568
    ;
wenzelm@26849
   569
    recdeftc thmdecl? tc
wenzelm@26849
   570
    ;
haftmann@31912
   571
    hints: '(' 'hints' ( recdefmod * ) ')'
wenzelm@26849
   572
    ;
wenzelm@26849
   573
    recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849
   574
    ;
wenzelm@26849
   575
    tc: nameref ('(' nat ')')?
wenzelm@26849
   576
    ;
wenzelm@26849
   577
  \end{rail}
wenzelm@26849
   578
wenzelm@28760
   579
  \begin{description}
wenzelm@26849
   580
  
wenzelm@28760
   581
  \item @{command (HOL) "recdef"} defines general well-founded
wenzelm@26849
   582
  recursive functions (using the TFL package), see also
wenzelm@26849
   583
  \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
wenzelm@26849
   584
  TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849
   585
  results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
wenzelm@26849
   586
  recdef_wf} hints refer to auxiliary rules to be used in the internal
wenzelm@26849
   587
  automated proof process of TFL.  Additional @{syntax clasimpmod}
wenzelm@26849
   588
  declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849
   589
  context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849
   590
  Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849
   591
  
wenzelm@28760
   592
  \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
wenzelm@26849
   593
  proof for leftover termination condition number @{text i} (default
wenzelm@26849
   594
  1) as generated by a @{command (HOL) "recdef"} definition of
wenzelm@26849
   595
  constant @{text c}.
wenzelm@26849
   596
  
wenzelm@26849
   597
  Note that in most cases, @{command (HOL) "recdef"} is able to finish
wenzelm@26849
   598
  its internal proofs without manual intervention.
wenzelm@26849
   599
wenzelm@28760
   600
  \end{description}
wenzelm@26849
   601
wenzelm@26849
   602
  \medskip Hints for @{command (HOL) "recdef"} may be also declared
wenzelm@26849
   603
  globally, using the following attributes.
wenzelm@26849
   604
wenzelm@26849
   605
  \begin{matharray}{rcl}
wenzelm@28761
   606
    @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
wenzelm@28761
   607
    @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
wenzelm@28761
   608
    @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
wenzelm@26849
   609
  \end{matharray}
wenzelm@26849
   610
wenzelm@26849
   611
  \begin{rail}
wenzelm@26849
   612
    ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849
   613
    ;
wenzelm@26849
   614
  \end{rail}
wenzelm@26849
   615
*}
wenzelm@26849
   616
wenzelm@26849
   617
wenzelm@26849
   618
section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
wenzelm@26849
   619
wenzelm@26849
   620
text {*
wenzelm@26849
   621
  An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849
   622
  set) @{text R} closed under given rules: applying a rule to elements
wenzelm@26849
   623
  of @{text R} yields a result within @{text R}.  For example, a
wenzelm@26849
   624
  structural operational semantics is an inductive definition of an
wenzelm@26849
   625
  evaluation relation.
wenzelm@26849
   626
wenzelm@26849
   627
  Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849
   628
  predicate~/ set @{text R} that is consistent with given rules: every
wenzelm@26849
   629
  element of @{text R} can be seen as arising by applying a rule to
wenzelm@26849
   630
  elements of @{text R}.  An important example is using bisimulation
wenzelm@26849
   631
  relations to formalise equivalence of processes and infinite data
wenzelm@26849
   632
  structures.
wenzelm@26849
   633
wenzelm@26849
   634
  \medskip The HOL package is related to the ZF one, which is
wenzelm@26849
   635
  described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849
   636
  \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849
   637
  which you should refer to in case of difficulties.  The package is
wenzelm@26849
   638
  simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849
   639
  The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849
   640
  domain of the fixedpoint definition, and the package does not have
wenzelm@26849
   641
  to use inference rules for type-checking.
wenzelm@26849
   642
wenzelm@26849
   643
  \begin{matharray}{rcl}
wenzelm@28761
   644
    @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   645
    @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   646
    @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   647
    @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   648
    @{attribute_def (HOL) mono} & : & @{text attribute} \\
wenzelm@26849
   649
  \end{matharray}
wenzelm@26849
   650
wenzelm@26849
   651
  \begin{rail}
wenzelm@26849
   652
    ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849
   653
    ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849
   654
    ;
wenzelm@26849
   655
    clauses: (thmdecl? prop + '|')
wenzelm@26849
   656
    ;
wenzelm@26849
   657
    'mono' (() | 'add' | 'del')
wenzelm@26849
   658
    ;
wenzelm@26849
   659
  \end{rail}
wenzelm@26849
   660
wenzelm@28760
   661
  \begin{description}
wenzelm@26849
   662
wenzelm@28760
   663
  \item @{command (HOL) "inductive"} and @{command (HOL)
wenzelm@28760
   664
  "coinductive"} define (co)inductive predicates from the
wenzelm@26849
   665
  introduction rules given in the @{keyword "where"} part.  The
wenzelm@26849
   666
  optional @{keyword "for"} part contains a list of parameters of the
wenzelm@26849
   667
  (co)inductive predicates that remain fixed throughout the
wenzelm@26849
   668
  definition.  The optional @{keyword "monos"} section contains
wenzelm@26849
   669
  \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849
   670
  applied to a recursive set in the introduction rules.  There
wenzelm@26849
   671
  \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
wenzelm@26849
   672
  for each premise @{text "M R\<^sub>i t"} in an introduction rule!
wenzelm@26849
   673
wenzelm@28760
   674
  \item @{command (HOL) "inductive_set"} and @{command (HOL)
wenzelm@28760
   675
  "coinductive_set"} are wrappers for to the previous commands,
wenzelm@26849
   676
  allowing the definition of (co)inductive sets.
wenzelm@26849
   677
wenzelm@28760
   678
  \item @{attribute (HOL) mono} declares monotonicity rules.  These
wenzelm@26849
   679
  rule are involved in the automated monotonicity proof of @{command
wenzelm@26849
   680
  (HOL) "inductive"}.
wenzelm@26849
   681
wenzelm@28760
   682
  \end{description}
wenzelm@26849
   683
*}
wenzelm@26849
   684
wenzelm@26849
   685
wenzelm@26849
   686
subsection {* Derived rules *}
wenzelm@26849
   687
wenzelm@26849
   688
text {*
wenzelm@26849
   689
  Each (co)inductive definition @{text R} adds definitions to the
wenzelm@26849
   690
  theory and also proves some theorems:
wenzelm@26849
   691
wenzelm@26849
   692
  \begin{description}
wenzelm@26849
   693
wenzelm@28760
   694
  \item @{text R.intros} is the list of introduction rules as proven
wenzelm@26849
   695
  theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849
   696
  also available individually, using the names given them in the
wenzelm@26849
   697
  theory file;
wenzelm@26849
   698
wenzelm@28760
   699
  \item @{text R.cases} is the case analysis (or elimination) rule;
wenzelm@26849
   700
wenzelm@28760
   701
  \item @{text R.induct} or @{text R.coinduct} is the (co)induction
wenzelm@26849
   702
  rule.
wenzelm@26849
   703
wenzelm@26849
   704
  \end{description}
wenzelm@26849
   705
wenzelm@26849
   706
  When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
wenzelm@26849
   707
  defined simultaneously, the list of introduction rules is called
wenzelm@26849
   708
  @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
wenzelm@26849
   709
  called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
wenzelm@26849
   710
  of mutual induction rules is called @{text
wenzelm@26849
   711
  "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
wenzelm@26849
   712
*}
wenzelm@26849
   713
wenzelm@26849
   714
wenzelm@26849
   715
subsection {* Monotonicity theorems *}
wenzelm@26849
   716
wenzelm@26849
   717
text {*
wenzelm@26849
   718
  Each theory contains a default set of theorems that are used in
wenzelm@26849
   719
  monotonicity proofs.  New rules can be added to this set via the
wenzelm@26849
   720
  @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
wenzelm@26849
   721
  shows how this is done.  In general, the following monotonicity
wenzelm@26849
   722
  theorems may be added:
wenzelm@26849
   723
wenzelm@26849
   724
  \begin{itemize}
wenzelm@26849
   725
wenzelm@26849
   726
  \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
wenzelm@26849
   727
  monotonicity of inductive definitions whose introduction rules have
wenzelm@26849
   728
  premises involving terms such as @{text "M R\<^sub>i t"}.
wenzelm@26849
   729
wenzelm@26849
   730
  \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849
   731
  general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
wenzelm@26849
   732
  the case of the operator @{text "\<or>"}, the corresponding theorem is
wenzelm@26849
   733
  \[
wenzelm@26849
   734
  \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
wenzelm@26849
   735
  \]
wenzelm@26849
   736
wenzelm@26849
   737
  \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849
   738
  of expressions, e.g.
wenzelm@26849
   739
  \[
wenzelm@26849
   740
  @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
wenzelm@26849
   741
  @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
wenzelm@26849
   742
  \]
wenzelm@26849
   743
wenzelm@26849
   744
  \item Equations for reducing complex operators to more primitive
wenzelm@26849
   745
  ones whose monotonicity can easily be proved, e.g.
wenzelm@26849
   746
  \[
wenzelm@26849
   747
  @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
wenzelm@26849
   748
  @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
wenzelm@26849
   749
  \]
wenzelm@26849
   750
wenzelm@26849
   751
  \end{itemize}
wenzelm@26849
   752
wenzelm@26849
   753
  %FIXME: Example of an inductive definition
wenzelm@26849
   754
*}
wenzelm@26849
   755
wenzelm@26849
   756
wenzelm@26849
   757
section {* Arithmetic proof support *}
wenzelm@26849
   758
wenzelm@26849
   759
text {*
wenzelm@26849
   760
  \begin{matharray}{rcl}
wenzelm@28761
   761
    @{method_def (HOL) arith} & : & @{text method} \\
nipkow@30863
   762
    @{attribute_def (HOL) arith} & : & @{text attribute} \\
wenzelm@28761
   763
    @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
wenzelm@26849
   764
  \end{matharray}
wenzelm@26849
   765
wenzelm@26849
   766
  The @{method (HOL) arith} method decides linear arithmetic problems
wenzelm@26849
   767
  (on types @{text nat}, @{text int}, @{text real}).  Any current
wenzelm@26849
   768
  facts are inserted into the goal before running the procedure.
wenzelm@26849
   769
nipkow@30863
   770
  The @{attribute (HOL) arith} attribute declares facts that are
nipkow@30863
   771
  always supplied to the arithmetic provers implicitly.
wenzelm@26849
   772
nipkow@30863
   773
  The @{attribute (HOL) arith_split} attribute declares case split
wenzelm@30865
   774
  rules to be expanded before @{method (HOL) arith} is invoked.
nipkow@30863
   775
nipkow@30863
   776
  Note that a simpler (but faster) arithmetic prover is
nipkow@30863
   777
  already invoked by the Simplifier.
wenzelm@26849
   778
*}
wenzelm@26849
   779
wenzelm@26849
   780
wenzelm@30169
   781
section {* Intuitionistic proof search *}
wenzelm@30169
   782
wenzelm@30169
   783
text {*
wenzelm@30169
   784
  \begin{matharray}{rcl}
wenzelm@30171
   785
    @{method_def (HOL) iprover} & : & @{text method} \\
wenzelm@30169
   786
  \end{matharray}
wenzelm@30169
   787
wenzelm@30169
   788
  \begin{rail}
wenzelm@35613
   789
    'iprover' ( rulemod * )
wenzelm@30169
   790
    ;
wenzelm@30169
   791
  \end{rail}
wenzelm@30169
   792
wenzelm@30171
   793
  The @{method (HOL) iprover} method performs intuitionistic proof
wenzelm@30171
   794
  search, depending on specifically declared rules from the context,
wenzelm@30171
   795
  or given as explicit arguments.  Chained facts are inserted into the
wenzelm@35613
   796
  goal before commencing proof search.
wenzelm@35613
   797
wenzelm@30169
   798
  Rules need to be classified as @{attribute (Pure) intro},
wenzelm@30169
   799
  @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
wenzelm@30169
   800
  ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
wenzelm@30169
   801
  applied aggressively (without considering back-tracking later).
wenzelm@30169
   802
  Rules declared with ``@{text "?"}'' are ignored in proof search (the
wenzelm@30169
   803
  single-step @{method rule} method still observes these).  An
wenzelm@30169
   804
  explicit weight annotation may be given as well; otherwise the
wenzelm@30169
   805
  number of rule premises will be taken into account here.
wenzelm@30169
   806
*}
wenzelm@30169
   807
wenzelm@30169
   808
wenzelm@30171
   809
section {* Coherent Logic *}
wenzelm@30171
   810
wenzelm@30171
   811
text {*
wenzelm@30171
   812
  \begin{matharray}{rcl}
wenzelm@30171
   813
    @{method_def (HOL) "coherent"} & : & @{text method} \\
wenzelm@30171
   814
  \end{matharray}
wenzelm@30171
   815
wenzelm@30171
   816
  \begin{rail}
wenzelm@30171
   817
    'coherent' thmrefs?
wenzelm@30171
   818
    ;
wenzelm@30171
   819
  \end{rail}
wenzelm@30171
   820
wenzelm@30171
   821
  The @{method (HOL) coherent} method solves problems of
wenzelm@30171
   822
  \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
wenzelm@30171
   823
  applications in confluence theory, lattice theory and projective
wenzelm@30171
   824
  geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
wenzelm@30171
   825
  examples.
wenzelm@30171
   826
*}
wenzelm@30171
   827
wenzelm@30171
   828
haftmann@31912
   829
section {* Checking and refuting propositions *}
haftmann@31912
   830
haftmann@31912
   831
text {*
haftmann@31912
   832
  Identifying incorrect propositions usually involves evaluation of
haftmann@31912
   833
  particular assignments and systematic counter example search.  This
haftmann@31912
   834
  is supported by the following commands.
haftmann@31912
   835
haftmann@31912
   836
  \begin{matharray}{rcl}
haftmann@31912
   837
    @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
haftmann@31912
   838
    @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
haftmann@31912
   839
    @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
haftmann@31912
   840
  \end{matharray}
haftmann@31912
   841
haftmann@31912
   842
  \begin{rail}
haftmann@31912
   843
    'value' ( ( '[' name ']' ) ? ) modes? term
haftmann@31912
   844
    ;
haftmann@31912
   845
haftmann@31912
   846
    'quickcheck' ( ( '[' args ']' ) ? ) nat?
haftmann@31912
   847
    ;
haftmann@31912
   848
haftmann@31912
   849
    'quickcheck_params' ( ( '[' args ']' ) ? )
haftmann@31912
   850
    ;
haftmann@31912
   851
haftmann@31912
   852
    modes: '(' (name + ) ')'
haftmann@31912
   853
    ;
haftmann@31912
   854
haftmann@31912
   855
    args: ( name '=' value + ',' )
haftmann@31912
   856
    ;
haftmann@31912
   857
  \end{rail}
haftmann@31912
   858
haftmann@31912
   859
  \begin{description}
haftmann@31912
   860
haftmann@31912
   861
  \item @{command (HOL) "value"}~@{text t} evaluates and prints a
haftmann@31912
   862
    term; optionally @{text modes} can be specified, which are
haftmann@31912
   863
    appended to the current print mode (see also \cite{isabelle-ref}).
haftmann@31912
   864
    Internally, the evaluation is performed by registered evaluators,
haftmann@31912
   865
    which are invoked sequentially until a result is returned.
haftmann@31912
   866
    Alternatively a specific evaluator can be selected using square
haftmann@31912
   867
    brackets; available evaluators include @{text nbe} for
haftmann@31912
   868
    \emph{normalization by evaluation} and \emph{code} for code
haftmann@31912
   869
    generation in SML.
haftmann@31912
   870
haftmann@31912
   871
  \item @{command (HOL) "quickcheck"} tests the current goal for
haftmann@31912
   872
    counter examples using a series of arbitrary assignments for its
haftmann@31912
   873
    free variables; by default the first subgoal is tested, an other
haftmann@31912
   874
    can be selected explicitly using an optional goal index.
haftmann@31912
   875
    A number of configuration options are supported for
haftmann@31912
   876
    @{command (HOL) "quickcheck"}, notably:
haftmann@31912
   877
haftmann@31912
   878
    \begin{description}
haftmann@31912
   879
haftmann@31912
   880
      \item[size] specifies the maximum size of the search space for
haftmann@31912
   881
        assignment values.
haftmann@31912
   882
haftmann@31912
   883
      \item[iterations] sets how many sets of assignments are
haftmann@31912
   884
        generated for each particular size.
haftmann@31912
   885
blanchet@35331
   886
      \item[no\_assms] specifies whether assumptions in
blanchet@35331
   887
        structured proofs should be ignored.
blanchet@35331
   888
haftmann@31912
   889
    \end{description}
haftmann@31912
   890
haftmann@31912
   891
    These option can be given within square brackets.
haftmann@31912
   892
haftmann@31912
   893
  \item @{command (HOL) "quickcheck_params"} changes quickcheck
haftmann@31912
   894
    configuration options persitently.
haftmann@31912
   895
haftmann@31912
   896
  \end{description}
haftmann@31912
   897
*}
haftmann@31912
   898
haftmann@31912
   899
wenzelm@35744
   900
section {* Invoking automated reasoning tools --- The Sledgehammer *}
wenzelm@28603
   901
wenzelm@28603
   902
text {*
wenzelm@28603
   903
  Isabelle/HOL includes a generic \emph{ATP manager} that allows
wenzelm@28603
   904
  external automated reasoning tools to crunch a pending goal.
wenzelm@28603
   905
  Supported provers include E\footnote{\url{http://www.eprover.org}},
wenzelm@28603
   906
  SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
wenzelm@28603
   907
  There is also a wrapper to invoke provers remotely via the
wenzelm@28603
   908
  SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
wenzelm@28603
   909
  web service.
wenzelm@28603
   910
wenzelm@28603
   911
  The problem passed to external provers consists of the goal together
wenzelm@28603
   912
  with a smart selection of lemmas from the current theory context.
wenzelm@28603
   913
  The result of a successful proof search is some source text that
wenzelm@28603
   914
  usually reconstructs the proof within Isabelle, without requiring
wenzelm@28603
   915
  external provers again.  The Metis
wenzelm@28603
   916
  prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
wenzelm@28603
   917
  integrated into Isabelle/HOL is being used here.
wenzelm@28603
   918
wenzelm@28603
   919
  In this mode of operation, heavy means of automated reasoning are
wenzelm@28603
   920
  used as a strong relevance filter, while the main proof checking
wenzelm@28603
   921
  works via explicit inferences going through the Isabelle kernel.
wenzelm@28603
   922
  Moreover, rechecking Isabelle proof texts with already specified
wenzelm@28603
   923
  auxiliary facts is much faster than performing fully automated
wenzelm@28603
   924
  search over and over again.
wenzelm@28603
   925
wenzelm@28603
   926
  \begin{matharray}{rcl}
wenzelm@28761
   927
    @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
wenzelm@28761
   928
    @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@28761
   929
    @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
wenzelm@28761
   930
    @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
wenzelm@29112
   931
    @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
wenzelm@28761
   932
    @{method_def (HOL) metis} & : & @{text method} \\
wenzelm@28603
   933
  \end{matharray}
wenzelm@28603
   934
wenzelm@28603
   935
  \begin{rail}
haftmann@31912
   936
  'sledgehammer' ( nameref * )
wenzelm@28603
   937
  ;
wenzelm@29112
   938
  'atp\_messages' ('(' nat ')')?
wenzelm@29114
   939
  ;
wenzelm@28603
   940
wenzelm@28603
   941
  'metis' thmrefs
wenzelm@28603
   942
  ;
wenzelm@28603
   943
  \end{rail}
wenzelm@28603
   944
wenzelm@28760
   945
  \begin{description}
wenzelm@28603
   946
wenzelm@28760
   947
  \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
wenzelm@28760
   948
  invokes the specified automated theorem provers on the first
wenzelm@28760
   949
  subgoal.  Provers are run in parallel, the first successful result
wenzelm@28760
   950
  is displayed, and the other attempts are terminated.
wenzelm@28603
   951
wenzelm@28603
   952
  Provers are defined in the theory context, see also @{command (HOL)
wenzelm@28603
   953
  print_atps}.  If no provers are given as arguments to @{command
wenzelm@28603
   954
  (HOL) sledgehammer}, the system refers to the default defined as
wenzelm@28603
   955
  ``ATP provers'' preference by the user interface.
wenzelm@28603
   956
wenzelm@28603
   957
  There are additional preferences for timeout (default: 60 seconds),
wenzelm@28603
   958
  and the maximum number of independent prover processes (default: 5);
wenzelm@28603
   959
  excessive provers are automatically terminated.
wenzelm@28603
   960
wenzelm@28760
   961
  \item @{command (HOL) print_atps} prints the list of automated
wenzelm@28603
   962
  theorem provers available to the @{command (HOL) sledgehammer}
wenzelm@28603
   963
  command.
wenzelm@28603
   964
wenzelm@28760
   965
  \item @{command (HOL) atp_info} prints information about presently
wenzelm@28603
   966
  running provers, including elapsed runtime, and the remaining time
wenzelm@28603
   967
  until timeout.
wenzelm@28603
   968
wenzelm@28760
   969
  \item @{command (HOL) atp_kill} terminates all presently running
wenzelm@28603
   970
  provers.
wenzelm@28603
   971
wenzelm@29112
   972
  \item @{command (HOL) atp_messages} displays recent messages issued
wenzelm@29112
   973
  by automated theorem provers.  This allows to examine results that
wenzelm@29112
   974
  might have got lost due to the asynchronous nature of default
wenzelm@29112
   975
  @{command (HOL) sledgehammer} output.  An optional message limit may
wenzelm@29112
   976
  be specified (default 5).
wenzelm@29112
   977
wenzelm@28760
   978
  \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
wenzelm@28760
   979
  with the given facts.  Metis is an automated proof tool of medium
wenzelm@28760
   980
  strength, but is fully integrated into Isabelle/HOL, with explicit
wenzelm@28760
   981
  inferences going through the kernel.  Thus its results are
wenzelm@28603
   982
  guaranteed to be ``correct by construction''.
wenzelm@28603
   983
wenzelm@28603
   984
  Note that all facts used with Metis need to be specified as explicit
wenzelm@28603
   985
  arguments.  There are no rule declarations as for other Isabelle
wenzelm@28603
   986
  provers, like @{method blast} or @{method fast}.
wenzelm@28603
   987
wenzelm@28760
   988
  \end{description}
wenzelm@28603
   989
*}
wenzelm@28603
   990
wenzelm@28603
   991
wenzelm@28752
   992
section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
wenzelm@26849
   993
wenzelm@26849
   994
text {*
wenzelm@27123
   995
  The following tools of Isabelle/HOL support cases analysis and
wenzelm@27123
   996
  induction in unstructured tactic scripts; see also
wenzelm@27123
   997
  \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849
   998
wenzelm@26849
   999
  \begin{matharray}{rcl}
wenzelm@28761
  1000
    @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
  1001
    @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
  1002
    @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
  1003
    @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@26849
  1004
  \end{matharray}
wenzelm@26849
  1005
wenzelm@26849
  1006
  \begin{rail}
wenzelm@26849
  1007
    'case\_tac' goalspec? term rule?
wenzelm@26849
  1008
    ;
wenzelm@26849
  1009
    'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849
  1010
    ;
wenzelm@26849
  1011
    'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849
  1012
    ;
wenzelm@26849
  1013
    'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849
  1014
    ;
wenzelm@26849
  1015
wenzelm@26849
  1016
    rule: ('rule' ':' thmref)
wenzelm@26849
  1017
    ;
wenzelm@26849
  1018
  \end{rail}
wenzelm@26849
  1019
wenzelm@28760
  1020
  \begin{description}
wenzelm@26849
  1021
wenzelm@28760
  1022
  \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
wenzelm@28760
  1023
  to reason about inductive types.  Rules are selected according to
wenzelm@28760
  1024
  the declarations by the @{attribute cases} and @{attribute induct}
wenzelm@28760
  1025
  attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
wenzelm@28760
  1026
  datatype} package already takes care of this.
wenzelm@27123
  1027
wenzelm@27123
  1028
  These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849
  1029
  instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27123
  1030
  as would be by the proper @{method cases} and @{method induct} proof
wenzelm@27123
  1031
  methods (see \secref{sec:cases-induct}).  Unlike the @{method
wenzelm@27123
  1032
  induct} method, @{method induct_tac} does not handle structured rule
wenzelm@27123
  1033
  statements, only the compact object-logic conclusion of the subgoal
wenzelm@27123
  1034
  being addressed.
wenzelm@26849
  1035
  
wenzelm@28760
  1036
  \item @{method (HOL) ind_cases} and @{command (HOL)
wenzelm@28760
  1037
  "inductive_cases"} provide an interface to the internal @{ML_text
wenzelm@26860
  1038
  mk_cases} operation.  Rules are simplified in an unrestricted
wenzelm@26860
  1039
  forward manner.
wenzelm@26849
  1040
wenzelm@26849
  1041
  While @{method (HOL) ind_cases} is a proof method to apply the
wenzelm@26849
  1042
  result immediately as elimination rules, @{command (HOL)
wenzelm@26849
  1043
  "inductive_cases"} provides case split theorems at the theory level
wenzelm@26849
  1044
  for later use.  The @{keyword "for"} argument of the @{method (HOL)
wenzelm@26849
  1045
  ind_cases} method allows to specify a list of variables that should
wenzelm@26849
  1046
  be generalized before applying the resulting rule.
wenzelm@26849
  1047
wenzelm@28760
  1048
  \end{description}
wenzelm@26849
  1049
*}
wenzelm@26849
  1050
wenzelm@26849
  1051
wenzelm@26849
  1052
section {* Executable code *}
wenzelm@26849
  1053
wenzelm@26849
  1054
text {*
wenzelm@26849
  1055
  Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849
  1056
  generation from executable specifications.  Isabelle/HOL
wenzelm@26849
  1057
  instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849
  1058
  applications.
wenzelm@26849
  1059
wenzelm@26849
  1060
  One framework generates code from both functional and relational
wenzelm@26849
  1061
  programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849
  1062
  (this actually covers the new-style theory format as well).
wenzelm@26849
  1063
wenzelm@26849
  1064
  \begin{matharray}{rcl}
wenzelm@28761
  1065
    @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1066
    @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1067
    @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1068
    @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
wenzelm@28761
  1069
    @{attribute_def (HOL) code} & : & @{text attribute} \\
wenzelm@26849
  1070
  \end{matharray}
wenzelm@26849
  1071
wenzelm@26849
  1072
  \begin{rail}
wenzelm@26849
  1073
  ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849
  1074
    ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849
  1075
    'contains' ( ( name '=' term ) + | term + )
wenzelm@26849
  1076
  ;
wenzelm@26849
  1077
wenzelm@26849
  1078
  modespec: '(' ( name * ) ')'
wenzelm@26849
  1079
  ;
wenzelm@26849
  1080
wenzelm@26849
  1081
  'consts\_code' (codespec +)
wenzelm@26849
  1082
  ;
wenzelm@26849
  1083
wenzelm@26849
  1084
  codespec: const template attachment ?
wenzelm@26849
  1085
  ;
wenzelm@26849
  1086
wenzelm@26849
  1087
  'types\_code' (tycodespec +)
wenzelm@26849
  1088
  ;
wenzelm@26849
  1089
wenzelm@26849
  1090
  tycodespec: name template attachment ?
wenzelm@26849
  1091
  ;
wenzelm@26849
  1092
wenzelm@26849
  1093
  const: term
wenzelm@26849
  1094
  ;
wenzelm@26849
  1095
wenzelm@26849
  1096
  template: '(' string ')'
wenzelm@26849
  1097
  ;
wenzelm@26849
  1098
wenzelm@26849
  1099
  attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849
  1100
  ;
wenzelm@26849
  1101
wenzelm@26849
  1102
  'code' (name)?
wenzelm@26849
  1103
  ;
wenzelm@26849
  1104
  \end{rail}
wenzelm@26849
  1105
wenzelm@26849
  1106
  \medskip The other framework generates code from functional programs
wenzelm@26849
  1107
  (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849
  1108
  \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849
  1109
  Conceptually, code generation is split up in three steps:
wenzelm@26849
  1110
  \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849
  1111
  abstract executable view and \emph{serialization} to a specific
wenzelm@26849
  1112
  \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849
  1113
  introduction on how to use it.
wenzelm@26849
  1114
wenzelm@26849
  1115
  \begin{matharray}{rcl}
wenzelm@28761
  1116
    @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@28761
  1117
    @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@28761
  1118
    @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@28761
  1119
    @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1120
    @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1121
    @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1122
    @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1123
    @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1124
    @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1125
    @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1126
    @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1127
    @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1128
    @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1129
    @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
haftmann@31254
  1130
    @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@28761
  1131
    @{attribute_def (HOL) code} & : & @{text attribute} \\
wenzelm@26849
  1132
  \end{matharray}
wenzelm@26849
  1133
wenzelm@26849
  1134
  \begin{rail}
haftmann@34172
  1135
    'export\_code' ( constexpr + ) \\
wenzelm@26849
  1136
      ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849
  1137
        ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849
  1138
    ;
wenzelm@26849
  1139
wenzelm@26849
  1140
    'code\_thms' ( constexpr + ) ?
wenzelm@26849
  1141
    ;
wenzelm@26849
  1142
wenzelm@26849
  1143
    'code\_deps' ( constexpr + ) ?
wenzelm@26849
  1144
    ;
wenzelm@26849
  1145
wenzelm@26849
  1146
    const: term
wenzelm@26849
  1147
    ;
wenzelm@26849
  1148
wenzelm@26849
  1149
    constexpr: ( const | 'name.*' | '*' )
wenzelm@26849
  1150
    ;
wenzelm@26849
  1151
wenzelm@26849
  1152
    typeconstructor: nameref
wenzelm@26849
  1153
    ;
wenzelm@26849
  1154
wenzelm@26849
  1155
    class: nameref
wenzelm@26849
  1156
    ;
wenzelm@26849
  1157
wenzelm@26849
  1158
    target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849
  1159
    ;
wenzelm@26849
  1160
wenzelm@26849
  1161
    'code\_datatype' const +
wenzelm@26849
  1162
    ;
wenzelm@26849
  1163
wenzelm@26849
  1164
    'code\_const' (const + 'and') \\
wenzelm@26849
  1165
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1166
    ;
wenzelm@26849
  1167
wenzelm@26849
  1168
    'code\_type' (typeconstructor + 'and') \\
wenzelm@26849
  1169
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1170
    ;
wenzelm@26849
  1171
wenzelm@26849
  1172
    'code\_class' (class + 'and') \\
haftmann@28687
  1173
      ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
wenzelm@26849
  1174
    ;
wenzelm@26849
  1175
wenzelm@26849
  1176
    'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849
  1177
      ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849
  1178
    ;
wenzelm@26849
  1179
wenzelm@26849
  1180
    'code\_monad' const const target
wenzelm@26849
  1181
    ;
wenzelm@26849
  1182
wenzelm@26849
  1183
    'code\_reserved' target ( string + )
wenzelm@26849
  1184
    ;
wenzelm@26849
  1185
wenzelm@26849
  1186
    'code\_include' target ( string ( string | '-') )
wenzelm@26849
  1187
    ;
wenzelm@26849
  1188
wenzelm@26849
  1189
    'code\_modulename' target ( ( string string ) + )
wenzelm@26849
  1190
    ;
wenzelm@26849
  1191
haftmann@27452
  1192
    'code\_abort' ( const + )
wenzelm@26849
  1193
    ;
wenzelm@26849
  1194
wenzelm@26849
  1195
    syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849
  1196
    ;
wenzelm@26849
  1197
haftmann@31998
  1198
    'code' ( 'del' ) ?
haftmann@31998
  1199
    ;
haftmann@31998
  1200
haftmann@31998
  1201
    'code_unfold' ( 'del' ) ?
haftmann@31998
  1202
    ;
haftmann@31998
  1203
haftmann@31998
  1204
    'code_post' ( 'del' ) ?
wenzelm@26849
  1205
    ;
wenzelm@26849
  1206
  \end{rail}
wenzelm@26849
  1207
wenzelm@28760
  1208
  \begin{description}
wenzelm@26849
  1209
wenzelm@28760
  1210
  \item @{command (HOL) "export_code"} is the canonical interface for
wenzelm@28760
  1211
  generating and serializing code: for a given list of constants, code
haftmann@34172
  1212
  is generated for the specified target languages.  If no serialization
haftmann@34172
  1213
  instruction is given, only abstract code is generated internally.
wenzelm@26849
  1214
wenzelm@26849
  1215
  Constants may be specified by giving them literally, referring to
wenzelm@26849
  1216
  all executable contants within a certain theory by giving @{text
wenzelm@26849
  1217
  "name.*"}, or referring to \emph{all} executable constants currently
wenzelm@26849
  1218
  available by giving @{text "*"}.
wenzelm@26849
  1219
wenzelm@26849
  1220
  By default, for each involved theory one corresponding name space
wenzelm@26849
  1221
  module is generated.  Alternativly, a module name may be specified
wenzelm@26849
  1222
  after the @{keyword "module_name"} keyword; then \emph{all} code is
wenzelm@26849
  1223
  placed in this module.
wenzelm@26849
  1224
wenzelm@26849
  1225
  For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849
  1226
  single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849
  1227
  where code is generated in multiple files reflecting the module
wenzelm@26849
  1228
  hierarchy.  The file specification ``@{text "-"}'' denotes standard
wenzelm@26849
  1229
  output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849
  1230
  code internally in the context of the current ML session.
wenzelm@26849
  1231
wenzelm@26849
  1232
  Serializers take an optional list of arguments in parentheses.  For
haftmann@34172
  1233
  \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
haftmann@34172
  1234
  explicit module signatures.
haftmann@34172
  1235
  
haftmann@34172
  1236
  For \emph{Haskell} a module name prefix may be given using the ``@{text
wenzelm@26849
  1237
  "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
wenzelm@26849
  1238
  "deriving (Read, Show)"}'' clause to each appropriate datatype
wenzelm@26849
  1239
  declaration.
wenzelm@26849
  1240
wenzelm@28760
  1241
  \item @{command (HOL) "code_thms"} prints a list of theorems
wenzelm@26849
  1242
  representing the corresponding program containing all given
haftmann@34172
  1243
  constants.
wenzelm@26849
  1244
wenzelm@28760
  1245
  \item @{command (HOL) "code_deps"} visualizes dependencies of
wenzelm@26849
  1246
  theorems representing the corresponding program containing all given
haftmann@34172
  1247
  constants.
wenzelm@26849
  1248
wenzelm@28760
  1249
  \item @{command (HOL) "code_datatype"} specifies a constructor set
wenzelm@26849
  1250
  for a logical type.
wenzelm@26849
  1251
wenzelm@28760
  1252
  \item @{command (HOL) "code_const"} associates a list of constants
wenzelm@26849
  1253
  with target-specific serializations; omitting a serialization
wenzelm@26849
  1254
  deletes an existing serialization.
wenzelm@26849
  1255
wenzelm@28760
  1256
  \item @{command (HOL) "code_type"} associates a list of type
wenzelm@26849
  1257
  constructors with target-specific serializations; omitting a
wenzelm@26849
  1258
  serialization deletes an existing serialization.
wenzelm@26849
  1259
wenzelm@28760
  1260
  \item @{command (HOL) "code_class"} associates a list of classes
wenzelm@28760
  1261
  with target-specific class names; omitting a serialization deletes
wenzelm@28760
  1262
  an existing serialization.  This applies only to \emph{Haskell}.
wenzelm@26849
  1263
wenzelm@28760
  1264
  \item @{command (HOL) "code_instance"} declares a list of type
wenzelm@26849
  1265
  constructor / class instance relations as ``already present'' for a
wenzelm@26849
  1266
  given target.  Omitting a ``@{text "-"}'' deletes an existing
wenzelm@26849
  1267
  ``already present'' declaration.  This applies only to
wenzelm@26849
  1268
  \emph{Haskell}.
wenzelm@26849
  1269
wenzelm@28760
  1270
  \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
wenzelm@28760
  1271
  to generate monadic code for Haskell.
wenzelm@26849
  1272
wenzelm@28760
  1273
  \item @{command (HOL) "code_reserved"} declares a list of names as
wenzelm@26849
  1274
  reserved for a given target, preventing it to be shadowed by any
wenzelm@26849
  1275
  generated code.
wenzelm@26849
  1276
wenzelm@28760
  1277
  \item @{command (HOL) "code_include"} adds arbitrary named content
haftmann@27706
  1278
  (``include'') to generated code.  A ``@{text "-"}'' as last argument
wenzelm@26849
  1279
  will remove an already added ``include''.
wenzelm@26849
  1280
wenzelm@28760
  1281
  \item @{command (HOL) "code_modulename"} declares aliasings from one
wenzelm@28760
  1282
  module name onto another.
wenzelm@26849
  1283
wenzelm@28760
  1284
  \item @{command (HOL) "code_abort"} declares constants which are not
haftmann@29560
  1285
  required to have a definition by means of code equations; if
wenzelm@28760
  1286
  needed these are implemented by program abort instead.
wenzelm@26849
  1287
wenzelm@28760
  1288
  \item @{attribute (HOL) code} explicitly selects (or with option
haftmann@29560
  1289
  ``@{text "del"}'' deselects) a code equation for code
haftmann@29560
  1290
  generation.  Usually packages introducing code equations provide
wenzelm@28760
  1291
  a reasonable default setup for selection.
wenzelm@26849
  1292
haftmann@31998
  1293
  \item @{attribute (HOL) code_inline} declares (or with
haftmann@28562
  1294
  option ``@{text "del"}'' removes) inlining theorems which are
haftmann@29560
  1295
  applied as rewrite rules to any code equation during
wenzelm@26849
  1296
  preprocessing.
wenzelm@26849
  1297
haftmann@31998
  1298
  \item @{attribute (HOL) code_post} declares (or with
haftmann@31998
  1299
  option ``@{text "del"}'' removes) theorems which are
haftmann@31998
  1300
  applied as rewrite rules to any result of an evaluation.
haftmann@31998
  1301
wenzelm@28760
  1302
  \item @{command (HOL) "print_codesetup"} gives an overview on
haftmann@31254
  1303
  selected code equations and code generator datatypes.
haftmann@31254
  1304
haftmann@31254
  1305
  \item @{command (HOL) "print_codeproc"} prints the setup
haftmann@31254
  1306
  of the code generator preprocessor.
wenzelm@26849
  1307
wenzelm@28760
  1308
  \end{description}
wenzelm@26849
  1309
*}
wenzelm@26849
  1310
wenzelm@27045
  1311
wenzelm@27045
  1312
section {* Definition by specification \label{sec:hol-specification} *}
wenzelm@27045
  1313
wenzelm@27045
  1314
text {*
wenzelm@27045
  1315
  \begin{matharray}{rcl}
wenzelm@28761
  1316
    @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
wenzelm@28761
  1317
    @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
wenzelm@27045
  1318
  \end{matharray}
wenzelm@27045
  1319
wenzelm@27045
  1320
  \begin{rail}
wenzelm@27045
  1321
  ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27045
  1322
  ;
wenzelm@27045
  1323
  decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27045
  1324
  \end{rail}
wenzelm@27045
  1325
wenzelm@28760
  1326
  \begin{description}
wenzelm@27045
  1327
wenzelm@28760
  1328
  \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
wenzelm@27045
  1329
  goal stating the existence of terms with the properties specified to
wenzelm@27045
  1330
  hold for the constants given in @{text decls}.  After finishing the
wenzelm@27045
  1331
  proof, the theory will be augmented with definitions for the given
wenzelm@27045
  1332
  constants, as well as with theorems stating the properties for these
wenzelm@27045
  1333
  constants.
wenzelm@27045
  1334
wenzelm@28760
  1335
  \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
wenzelm@28760
  1336
  a goal stating the existence of terms with the properties specified
wenzelm@28760
  1337
  to hold for the constants given in @{text decls}.  After finishing
wenzelm@28760
  1338
  the proof, the theory will be augmented with axioms expressing the
wenzelm@28760
  1339
  properties given in the first place.
wenzelm@27045
  1340
wenzelm@28760
  1341
  \item @{text decl} declares a constant to be defined by the
wenzelm@27045
  1342
  specification given.  The definition for the constant @{text c} is
wenzelm@27045
  1343
  bound to the name @{text c_def} unless a theorem name is given in
wenzelm@27045
  1344
  the declaration.  Overloaded constants should be declared as such.
wenzelm@27045
  1345
wenzelm@28760
  1346
  \end{description}
wenzelm@27045
  1347
wenzelm@27045
  1348
  Whether to use @{command (HOL) "specification"} or @{command (HOL)
wenzelm@27045
  1349
  "ax_specification"} is to some extent a matter of style.  @{command
wenzelm@27045
  1350
  (HOL) "specification"} introduces no new axioms, and so by
wenzelm@27045
  1351
  construction cannot introduce inconsistencies, whereas @{command
wenzelm@27045
  1352
  (HOL) "ax_specification"} does introduce axioms, but only after the
wenzelm@27045
  1353
  user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27045
  1354
  considered, though: After introducing two constants with the same
wenzelm@27045
  1355
  properties using @{command (HOL) "specification"}, one can prove
wenzelm@27045
  1356
  that the two constants are, in fact, equal.  If this might be a
wenzelm@27045
  1357
  problem, one should use @{command (HOL) "ax_specification"}.
wenzelm@27045
  1358
*}
wenzelm@27045
  1359
wenzelm@26840
  1360
end