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\chapter{Object-logic specific elements}\label{ch:logics}

\section{General logic setup}\label{sec:object-logic}


  \isarcmd{judgment} & : & \isartrans{theory}{theory} \\
  atomize & : & \isarmeth \\
  atomize & : & \isaratt \\
  rule_format & : & \isaratt \\
  rulify & : & \isaratt \\

The very starting point for any Isabelle object-logic is a ``truth judgment''
that links object-level statements to the meta-logic (with its minimal
language of $prop$ that covers universal quantification $\Forall$ and
implication $\Imp$).

Common object-logics are sufficiently expressive to internalize rule
statements over $\Forall$ and $\Imp$ within their own language.  This is
useful in certain situations where a rule needs to be viewed as an atomic
statement from the meta-level perspective, e.g.\ $\All x x \in A \Imp P(x)$
versus $\forall x \in A. P(x)$.

From the following language elements, only the $atomize$ method and
$rule_format$ attribute are occasionally required by end-users, the rest is
for those who need to setup their own object-logic.  In the latter case
existing formulations of Isabelle/FOL or Isabelle/HOL may be taken as
realistic examples.

Generic tools may refer to the information provided by object-logic
declarations internally.


  'judgment' constdecl
  'atomize' ('(' 'full' ')')?
  ruleformat ('(' 'noasm' ')')?

\item [$\isarkeyword{judgment}~c::\sigma~~(mx)$] declares constant $c$ as the
  truth judgment of the current object-logic.  Its type $\sigma$ should
  specify a coercion of the category of object-level propositions to $prop$ of
  the Pure meta-logic; the mixfix annotation $(mx)$ would typically just link
  the object language (internally of syntactic category $logic$) with that of
  $prop$.  Only one $\isarkeyword{judgment}$ declaration may be given in any
  theory development.
\item [$atomize$] (as a method) rewrites any non-atomic premises of a
  sub-goal, using the meta-level equations declared via $atomize$ (as an
  attribute) beforehand.  As a result, heavily nested goals become amenable to
  fundamental operations such as resolution (cf.\ the $rule$ method) and
  proof-by-assumption (cf.\ $assumption$).  Giving the ``$(full)$'' option
  here means to turn the whole subgoal into an object-statement (if possible),
  including the outermost parameters and assumptions as well.

  A typical collection of $atomize$ rules for a particular object-logic would
  provide an internalization for each of the connectives of $\Forall$, $\Imp$,
  and $\equiv$.  Meta-level conjunction expressed in the manner of minimal
  higher-order logic as $\All{\PROP\,C} (A \Imp B \Imp \PROP\,C) \Imp PROP\,C$
  should be covered as well (this is particularly important for locales, see

\item [$rule_format$] rewrites a theorem by the equalities declared as
  $rulify$ rules in the current object-logic.  By default, the result is fully
  normalized, including assumptions and conclusions at any depth.  The
  $no_asm$ option restricts the transformation to the conclusion of a rule.

  In common object-logics (HOL, FOL, ZF), the effect of $rule_format$ is to
  replace (bounded) universal quantification ($\forall$) and implication
  ($\imp$) by the corresponding rule statements over $\Forall$ and $\Imp$.



\subsection{Primitive types}\label{sec:hol-typedef}

  \isarcmd{typedecl} & : & \isartrans{theory}{theory} \\
  \isarcmd{typedef} & : & \isartrans{theory}{proof(prove)} \\

  'typedecl' typespec infix?
  'typedef' altname? abstype '=' repset

  altname: '(' (name | 'open' | 'open' name) ')'
  abstype: typespec infix?
  repset: term ('morphisms' name name)?

\item [$\isarkeyword{typedecl}~(\vec\alpha)t$] is similar to the original
  $\isarkeyword{typedecl}$ of Isabelle/Pure (see \S\ref{sec:types-pure}), but
  also declares type arity $t :: (type, \dots, type) type$, making $t$ an
  actual HOL type constructor.
\item [$\isarkeyword{typedef}~(\vec\alpha)t = A$] sets up a goal stating
  non-emptiness of the set $A$.  After finishing the proof, the theory will be
  augmented by a Gordon/HOL-style type definition, which establishes a
  bijection between the representing set $A$ and the new type $t$.
  Technically, $\isarkeyword{typedef}$ defines both a type $t$ and a set (term
  constant) of the same name (an alternative base name may be given in
  parentheses).  The injection from type to set is called $Rep_t$, its inverse
  $Abs_t$ (this may be changed via an explicit $\isarkeyword{morphisms}$
  Theorems $Rep_t$, $Rep_t_inverse$, and $Abs_t_inverse$ provide the most
  basic characterization as a corresponding injection/surjection pair (in both
  directions).  Rules $Rep_t_inject$ and $Abs_t_inject$ provide a slightly
  more convenient view on the injectivity part, suitable for automated proof
  tools (e.g.\ in $simp$ or $iff$ declarations).  Rules
  $Rep_t_cases/Rep_t_induct$, and $Abs_t_cases/Abs_t_induct$ provide
  alternative views on surjectivity; these are already declared as set or type
  rules for the generic $cases$ and $induct$ methods.
  An alternative name may be specified in parentheses; the default is to use
  $t$ as indicated before.  The $open$ declaration suppresses a separate
  constant definition for the representing set.

Note that raw type declarations are rarely used in practice; the main
application is with experimental (or even axiomatic!) theory fragments.
Instead of primitive HOL type definitions, user-level theories usually refer
to higher-level packages such as $\isarkeyword{record}$ (see
\S\ref{sec:hol-record}) or $\isarkeyword{datatype}$ (see

\subsection{Adhoc tuples}

  split_format^* & : & \isaratt \\


  splitformat (((name *) + 'and') | ('(' 'complete' ')'))

\item [$split_format~\vec p@1 \dots \vec p@n$] puts expressions of low-level
  tuple types into canonical form as specified by the arguments given; $\vec
  p@i$ refers to occurrences in premise $i$ of the rule.  The ``$(complete)$''
  option causes \emph{all} arguments in function applications to be
  represented canonically according to their tuple type structure.

  Note that these operations tend to invent funny names for new local
  parameters to be introduced.



In principle, records merely generalize the concept of tuples, where
components may be addressed by labels instead of just position.  The logical
infrastructure of records in Isabelle/HOL is slightly more advanced, though,
supporting truly extensible record schemes.  This admits operations that are
polymorphic with respect to record extension, yielding ``object-oriented''
effects like (single) inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for
more details on object-oriented verification and record subtyping in HOL.

\subsubsection{Basic concepts}

Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records at the
level of terms and types.  The notation is as follows:

  & record terms & record types \\ \hline
  fixed & $\record{x = a\fs y = b}$ & $\record{x \ty A\fs y \ty B}$ \\
  schematic & $\record{x = a\fs y = b\fs \more = m}$ &
    $\record{x \ty A\fs y \ty B\fs \more \ty M}$ \\

\noindent The ASCII representation of $\record{x = a}$ is \texttt{(| x = a |)}.

A fixed record $\record{x = a\fs y = b}$ has field $x$ of value $a$ and field
$y$ of value $b$.  The corresponding type is $\record{x \ty A\fs y \ty B}$,
assuming that $a \ty A$ and $b \ty B$.

A record scheme like $\record{x = a\fs y = b\fs \more = m}$ contains fields
$x$ and $y$ as before, but also possibly further fields as indicated by the
``$\more$'' notation (which is actually part of the syntax).  The improper
field ``$\more$'' of a record scheme is called the \emph{more part}.
Logically it is just a free variable, which is occasionally referred to as
``row variable'' in the literature.  The more part of a record scheme may be
instantiated by zero or more further components.  For example, the previous
scheme may get instantiated to $\record{x = a\fs y = b\fs z = c\fs \more =
  m'}$, where $m'$ refers to a different more part.  Fixed records are special
instances of record schemes, where ``$\more$'' is properly terminated by the
$() :: unit$ element.  Actually, $\record{x = a\fs y = b}$ is just an
abbreviation for $\record{x = a\fs y = b\fs \more = ()}$.


Two key observations make extensible records in a simply typed language like
HOL feasible:
\item the more part is internalized, as a free term or type variable,
\item field names are externalized, they cannot be accessed within the logic
  as first-class values.


In Isabelle/HOL record types have to be defined explicitly, fixing their field
names and types, and their (optional) parent record.  Afterwards, records may
be formed using above syntax, while obeying the canonical order of fields as
given by their declaration.  The record package provides several standard
operations like selectors and updates.  The common setup for various generic
proof tools enable succinct reasoning patterns.  See also the Isabelle/HOL
tutorial \cite{isabelle-hol-book} for further instructions on using records in

\subsubsection{Record specifications}

  \isarcmd{record} & : & \isartrans{theory}{theory} \\

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

\item [$\isarkeyword{record}~(\vec\alpha)t = \tau + \vec c :: \vec\sigma$]
  defines extensible record type $(\vec\alpha)t$, derived from the optional
  parent record $\tau$ by adding new field components $\vec c :: \vec\sigma$.

  The type variables of $\tau$ and $\vec\sigma$ need to be covered by the
  (distinct) parameters $\vec\alpha$.  Type constructor $t$ has to be new,
  while $\tau$ needs to specify an instance of an existing record type.  At
  least one new field $\vec c$ has to be specified.  Basically, field names
  need to belong to a unique record.  This is not a real restriction in
  practice, since fields are qualified by the record name internally.

  The parent record specification $\tau$ is optional; if omitted $t$ becomes a
  root record.  The hierarchy of all records declared within a theory context
  forms a forest structure, i.e.\ a set of trees starting with a root record
  each.  There is no way to merge multiple parent records!

  For convenience, $(\vec\alpha) \, t$ is made a type abbreviation for the
  fixed record type $\record{\vec c \ty \vec\sigma}$, likewise is
  $(\vec\alpha, \zeta) \, t_scheme$ made an abbreviation for $\record{\vec c
    \ty \vec\sigma\fs \more \ty \zeta}$.


\subsubsection{Record operations}

Any record definition of the form presented above produces certain standard
operations.  Selectors and updates are provided for any field, including the
improper one ``$more$''.  There are also cumulative record constructor
functions.  To simplify the presentation below, we assume for now that
$(\vec\alpha) \, t$ is a root record with fields $\vec c \ty \vec\sigma$.

\medskip \textbf{Selectors} and \textbf{updates} are available for any field
(including ``$more$''):
  c@i & \ty & \record{\vec c \ty \vec \sigma, \more \ty \zeta} \To \sigma@i \\
  c@i_update & \ty & \sigma@i \To \record{\vec c \ty \vec\sigma, \more \ty \zeta} \To
    \record{\vec c \ty \vec\sigma, \more \ty \zeta}

There is special syntax for application of updates: $r \, \record{x \asn a}$
abbreviates term $x_update \, a \, r$.  Further notation for repeated updates
is also available: $r \, \record{x \asn a} \, \record{y \asn b} \, \record{z
  \asn c}$ may be written $r \, \record{x \asn a\fs y \asn b\fs z \asn c}$.
Note that because of postfix notation the order of fields shown here is
reverse than in the actual term.  Since repeated updates are just function
applications, fields may be freely permuted in $\record{x \asn a\fs y \asn
  b\fs z \asn c}$, as far as logical equality is concerned.  Thus
commutativity of independent updates can be proven within the logic for any
two fields, but not as a general theorem.

\medskip The \textbf{make} operation provides a cumulative record constructor
  t{\dtt}make & \ty & \vec\sigma \To \record{\vec c \ty \vec \sigma} \\

\medskip We now reconsider the case of non-root records, which are derived of
some parent.  In general, the latter may depend on another parent as well,
resulting in a list of \emph{ancestor records}.  Appending the lists of fields
of all ancestors results in a certain field prefix.  The record package
automatically takes care of this by lifting operations over this context of
ancestor fields.  Assuming that $(\vec\alpha) \, t$ has ancestor fields $\vec
b \ty \vec\rho$, the above record operations will get the following types:
  c@i & \ty & \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty
    \zeta} \To \sigma@i \\
  c@i_update & \ty & \sigma@i \To
    \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty \zeta} \To
    \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty \zeta} \\
  t{\dtt}make & \ty & \vec\rho \To \vec\sigma \To
    \record{\vec b \ty \vec\rho, \vec c \ty \vec \sigma} \\

\medskip Some further operations address the extension aspect of a derived
record scheme specifically: $fields$ produces a record fragment consisting of
exactly the new fields introduced here (the result may serve as a more part
elsewhere); $extend$ takes a fixed record and adds a given more part;
$truncate$ restricts a record scheme to a fixed record.

  t{\dtt}fields & \ty & \vec\sigma \To \record{\vec c \ty \vec \sigma} \\
  t{\dtt}extend & \ty & \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma} \To
    \zeta \To \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma, \more \ty \zeta} \\
  t{\dtt}truncate & \ty & \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma, \more \ty \zeta} \To
    \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma} \\

\noindent Note that $t{\dtt}make$ and $t{\dtt}fields$ actually coincide for root records.

\subsubsection{Derived rules and proof tools}

The record package proves several results internally, declaring these facts to
appropriate proof tools.  This enables users to reason about record structures
quite conveniently.  Assume that $t$ is a record type as specified above.

\item Standard conversions for selectors or updates applied to record
  constructor terms are made part of the default Simplifier context; thus
  proofs by reduction of basic operations merely require the $simp$ method
  without further arguments.  These rules are available as $t{\dtt}simps$,
\item Selectors applied to updated records are automatically reduced by an
  internal simplification procedure, which is also part of the standard
  Simplifier setup.

\item Inject equations of a form analogous to $((x, y) = (x', y')) \equiv x=x'
  \conj y=y'$ are declared to the Simplifier and Classical Reasoner as $iff$
  rules.  These rules are available as $t{\dtt}iffs$.

\item The introduction rule for record equality analogous to $x~r = x~r' \Imp
  y~r = y~r' \Imp \dots \Imp r = r'$ is declared to the Simplifier, and as the
  basic rule context as ``$intro?$''.  The rule is called $t{\dtt}equality$.

\item Representations of arbitrary record expressions as canonical constructor
  terms are provided both in $cases$ and $induct$ format (cf.\ the generic
  proof methods of the same name, \S\ref{sec:cases-induct}).  Several
  variations are available, for fixed records, record schemes, more parts etc.
  The generic proof methods are sufficiently smart to pick the most sensible
  rule according to the type of the indicated record expression: users just
  need to apply something like ``$(cases~r)$'' to a certain proof problem.

\item The derived record operations $t{\dtt}make$, $t{\dtt}fields$,
  $t{\dtt}extend$, $t{\dtt}truncate$ are \emph{not} treated automatically, but
  usually need to be expanded by hand, using the collective fact



  \isarcmd{datatype} & : & \isartrans{theory}{theory} \\
  \isarcmd{rep_datatype} & : & \isartrans{theory}{theory} \\


  'datatype' (dtspec + 'and')
  repdatatype (name *) dtrules

  dtspec: parname? typespec infix? '=' (cons + '|')
  cons: name (type *) mixfix?
  dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs

\item [$\isarkeyword{datatype}$] defines inductive datatypes in HOL.
\item [$\isarkeyword{rep_datatype}$] represents existing types as inductive
  ones, generating the standard infrastructure of derived concepts (primitive
  recursion etc.).

The induction and exhaustion theorems generated provide case names according
to the constructors involved, while parameters are named after the types (see
also \S\ref{sec:cases-induct}).

See \cite{isabelle-HOL} for more details on datatypes, but beware of the
old-style theory syntax being used there!  Apart from proper proof methods for
case-analysis and induction, there are also emulations of ML tactics
\texttt{case_tac} and \texttt{induct_tac} available, see
\S\ref{sec:hol-induct-tac}; these admit to refer directly to the internal
structure of subgoals (including internally bound parameters).

\subsection{Recursive functions}\label{sec:recursion}


  \isarcmd{primrec} & : & \isarkeep{local{\dsh}theory} \\
  \isarcmd{fun} & : & \isarkeep{local{\dsh}theory} \\
  \isarcmd{function} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
  \isarcmd{termination} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\


  'primrec' target? fixes 'where' equations
  equations: (thmdecl? prop + '|')
  ('fun' | 'function') (funopts)? fixes 'where' clauses
  clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
  funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
  'default' term) + ',') ')'
  'termination' ( term )?


\item [$\isarkeyword{primrec}$] defines primitive recursive functions over
  datatypes, see also \cite{isabelle-HOL}.

\item [$\isarkeyword{function}$] defines functions by general
  wellfounded recursion. A detailed description with examples can be
  found in \cite{isabelle-function}. The function is specified by a
  set of (possibly conditional) recursive equations with arbitrary
  pattern matching. The command generates proof obligations for the
  completeness and the compatibility of patterns.

  The defined function is considered partial, and the resulting
  simplification rules (named $f.psimps$) and induction rule (named
  $f.pinduct$) are guarded by a generated domain predicate $f_dom$. 
  The $\isarkeyword{termination}$ command can then be used to establish
  that the function is total.

\item [$\isarkeyword{fun}$] is a shorthand notation for
  $\isarkeyword{function}~(\textit{sequential})$, followed by automated
  proof attemts regarding pattern matching and termination. For
  details, see \cite{isabelle-function}.

\item [$\isarkeyword{termination}$~f] commences a termination proof
  for the previously defined function $f$. If no name is given, it
  refers to the most recent function definition. After the proof is
  closed, the recursive equations and the induction principle is established.

Recursive definitions introduced by both the $\isarkeyword{primrec}$
and the $\isarkeyword{function}$ command accommodate reasoning by
induction (cf.\ \S\ref{sec:cases-induct}): rule $c\mathord{.}induct$
(where $c$ is the name of the function definition) refers to a
specific induction rule, with parameters named according to the
user-specified equations.  Case names of $\isarkeyword{primrec}$ are
that of the datatypes involved, while those of
$\isarkeyword{function}$ are numbered (starting from $1$).

The equations provided by these packages may be referred later as theorem list
$f{\dtt}simps$, where $f$ is the (collective) name of the functions defined.
Individual equations may be named explicitly as well.

The $\isarkeyword{function}$ command accepts the following options:

\item [\emph{sequential}] enables a preprocessor which disambiguates
  overlapping patterns by making them mutually disjoint. Earlier
  equations take precedence over later ones. This allows to give the
  specification in a format very similar to functional programming.
  Note that the resulting simplification and induction rules
  correspond to the transformed specification, not the one given
  originally. This usually means that each equation given by the user
  may result in several theroems.
  Also note that this automatic transformation only works
  for ML-style datatype patterns.

\item [\emph{in name}] gives the target for the definition.

\item [\emph{domintros}] enables the automated generation of
  introduction rules for the domain predicate. While mostly not
  needed, they can be helpful in some proofs about partial functions.

\item [\emph{tailrec}] generates the unconstrained recursive equations
  even without a termination proof, provided that the function is
  tail-recursive. This currently only works 

\item [\emph{default d}] allows to specify a default value for a
  (partial) function, which will ensure that $f(x)=d(x)$ whenever $x
  \notin \textit{f\_dom}$. This feature is experimental.

\subsubsection{Proof methods related to recursive definitions}


  pat\_completeness & : & \isarmeth \\
  relation & : & \isarmeth \\
  lexicographic\_order & : & \isarmeth \\

  'relation' term
  'lexicographic\_order' clasimpmod

\item [\emph{pat\_completeness}] Specialized method to solve goals
  regarding the completeness of pattern matching, as required by the
  $\isarkeyword{function}$ package (cf.~\cite{isabelle-function}).

\item [\emph{relation R}] Introduces a termination proof using the
  relation $R$. The resulting proof state will contain goals
  expressing that $R$ is wellfounded, and that the arguments
  of recursive calls decrease with respect to $R$. Usually, this
  method is used as the initial proof step of manual termination

\item [\emph{lexicographic\_order}] Attempts a fully automated
  termination proof by searching for a lexicographic combination of
  size measures on the arguments of the function. The method
  accepts the same arguments as the \emph{auto} method, which it uses
  internally to prove local descents. Hence, modifiers like
  \emph{simp}, \emph{intro} etc.\ can be used to add ``hints'' for the
  automated proofs. In case of failure, extensive information is
  printed, which can help to analyse the failure (cf.~\cite{isabelle-function}).

\subsubsection{Legacy recursion package}

The use of the legacy $\isarkeyword{recdef}$ command is now deprecated
in favour of $\isarkeyword{function}$ and $\isarkeyword{fun}$.

  \isarcmd{recdef} & : & \isartrans{theory}{theory} \\
  \isarcmd{recdef_tc}^* & : & \isartrans{theory}{proof(prove)} \\





  'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
  recdeftc thmdecl? tc
  hints: '(' 'hints' (recdefmod *) ')'
  recdefmod: ((recdefsimp | recdefcong | recdefwf) (() | 'add' | 'del') ':' thmrefs) | clasimpmod
  tc: nameref ('(' nat ')')?

\item [$\isarkeyword{recdef}$] defines general well-founded recursive
  functions (using the TFL package), see also \cite{isabelle-HOL}.  The
  ``$(permissive)$'' option tells TFL to recover from failed proof attempts,
  returning unfinished results.  The $recdef_simp$, $recdef_cong$, and
  $recdef_wf$ hints refer to auxiliary rules to be used in the internal
  automated proof process of TFL.  Additional $clasimpmod$ declarations (cf.\ 
  \S\ref{sec:clasimp}) may be given to tune the context of the Simplifier
  (cf.\ \S\ref{sec:simplifier}) and Classical reasoner (cf.\ 
\item [$\isarkeyword{recdef_tc}~c~(i)$] recommences the proof for leftover
  termination condition number $i$ (default $1$) as generated by a
  $\isarkeyword{recdef}$ definition of constant $c$.
  Note that in most cases, $\isarkeyword{recdef}$ is able to finish its
  internal proofs without manual intervention.


\medskip Hints for $\isarkeyword{recdef}$ may be also declared globally, using
the following attributes.

  recdef_simp & : & \isaratt \\
  recdef_cong & : & \isaratt \\
  recdef_wf & : & \isaratt \\




  (recdefsimp | recdefcong | recdefwf) (() | 'add' | 'del')

\subsection{Definition by specification}\label{sec:hol-specification}

  \isarcmd{specification} & : & \isartrans{theory}{proof(prove)} \\
  \isarcmd{ax_specification} & : & \isartrans{theory}{proof(prove)} \\

('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
decl: ((name ':')? term '(' 'overloaded' ')'?)

\item [$\isarkeyword{specification}~decls~\phi$] sets up a goal stating
  the existence of terms with the properties specified to hold for the
  constants given in $\mathit{decls}$.  After finishing the proof, the
  theory will be augmented with definitions for the given constants,
  as well as with theorems stating the properties for these constants.
\item [$\isarkeyword{ax_specification}~decls~\phi$] sets up a goal stating
  the existence of terms with the properties specified to hold for the
  constants given in $\mathit{decls}$.  After finishing the proof, the
  theory will be augmented with axioms expressing the properties given
  in the first place.
\item[$decl$] declares a constant to be defined by the specification
  given.  The definition for the constant $c$ is bound to the name
  $c$\_def unless a theorem name is given in the declaration.
  Overloaded constants should be declared as such.

Whether to use $\isarkeyword{specification}$ or $\isarkeyword{ax_specification}$
is to some extent a matter of style.  $\isarkeyword{specification}$ introduces no new axioms,
and so by construction cannot introduce inconsistencies, whereas $\isarkeyword{ax_specification}$
does introduce axioms, but only after the user has explicitly proven it to be
safe.  A practical issue must be considered, though: After introducing two constants
with the same properties using $\isarkeyword{specification}$, one can prove
that the two constants are, in fact, equal.  If this might be a problem,
one should use $\isarkeyword{ax_specification}$.

\subsection{Inductive and coinductive definitions}\label{sec:hol-inductive}

An {\bf inductive definition} specifies the least predicate (or set) $R$ closed under given
rules.  (Applying a rule to elements of~$R$ yields a result within~$R$.)  For
example, a structural operational semantics is an inductive definition of an
evaluation relation.  Dually, a {\bf coinductive definition} specifies the
greatest predicate (or set) $R$ consistent with given rules.  (Every element of~$R$ can be
seen as arising by applying a rule to elements of~$R$.)  An important example
is using bisimulation relations to formalise equivalence of processes and
infinite data structures.

This package is related to the ZF one, described in a separate
\footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
  distributed with Isabelle.}  %
which you should refer to in case of difficulties.  The package is simpler
than ZF's thanks to HOL's extra-logical automatic type-checking.  The types of
the (co)inductive predicates (or sets) determine the domain of the fixedpoint definition, and
the package does not have to use inference rules for type-checking.

  \isarcmd{inductive} & : & \isarkeep{local{\dsh}theory} \\
  \isarcmd{inductive_set} & : & \isarkeep{local{\dsh}theory} \\
  \isarcmd{coinductive} & : & \isarkeep{local{\dsh}theory} \\
  \isarcmd{coinductive_set} & : & \isarkeep{local{\dsh}theory} \\
  mono & : & \isaratt \\

  ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
  ('where' clauses)? ('monos' thmrefs)?
  clauses: (thmdecl? prop + '|')
  'mono' (() | 'add' | 'del')

\item [$\isarkeyword{inductive}$ and $\isarkeyword{coinductive}$] define
  (co)inductive predicates from the introduction rules given in the \texttt{where} section.
  The optional \texttt{for} section contains a list of parameters of the (co)inductive
  predicates that remain fixed throughout the definition.
  The optional \texttt{monos} section contains \textit{monotonicity theorems},
  which are required for each operator applied to a recursive set in the introduction rules.
  There {\bf must} be a theorem of the form $A \leq B \Imp M~A \leq M~B$, for each
  premise $M~R@i~t$ in an introduction rule!
\item [$\isarkeyword{inductive_set}$ and $\isarkeyword{coinductive_set}$] are wrappers
  for to the previous commands, allowing the definition of (co)inductive sets.
\item [$mono$] declares monotonicity rules.  These rule are involved in the
  automated monotonicity proof of $\isarkeyword{inductive}$.

\subsubsection{Derived rules}

Each (co)inductive definition $R$ adds definitions to the theory and also
proves some theorems:
\item[$R{\dtt}intros$] is the list of introduction rules, now proved as theorems, for
the recursive predicates (or sets).  The rules are also available individually,
using the names given them in the theory file.
\item[$R{\dtt}cases$] is the case analysis (or elimination) rule.
\item[$R{\dtt}(co)induct$] is the (co)induction rule.
When several predicates $R@1$, $\ldots$, $R@n$ are defined simultaneously,
the list of introduction rules is called $R@1_\ldots_R@n{\dtt}intros$, the
case analysis rules are called $R@1{\dtt}cases$, $\ldots$, $R@n{\dtt}cases$, and
the list of mutual induction rules is called $R@1_\ldots_R@n{\dtt}inducts$.

\subsubsection{Monotonicity theorems}

Each theory contains a default set of theorems that are used in monotonicity
proofs. New rules can be added to this set via the $mono$ attribute.
Theory \texttt{Inductive} shows how this is done. In general, the following
monotonicity theorems may be added:
\item Theorems of the form $A \leq B \Imp M~A \leq M~B$, for proving
  monotonicity of inductive definitions whose introduction rules have premises
  involving terms such as $M~R@i~t$.
\item Monotonicity theorems for logical operators, which are of the general form
  $\List{\cdots \to \cdots;~\ldots;~\cdots \to \cdots} \Imp
    \cdots \to \cdots$.
  For example, in the case of the operator $\lor$, the corresponding theorem is
  \infer{P@1 \lor P@2 \to Q@1 \lor Q@2}
    {P@1 \to Q@1 & P@2 \to Q@2}
\item De Morgan style equations for reasoning about the ``polarity'' of expressions, e.g.
  (\lnot \lnot P) ~=~ P \qquad\qquad
  (\lnot (P \land Q)) ~=~ (\lnot P \lor \lnot Q)
\item Equations for reducing complex operators to more primitive ones whose
  monotonicity can easily be proved, e.g.
  (P \to Q) ~=~ (\lnot P \lor Q) \qquad\qquad
  \mathtt{Ball}~A~P ~\equiv~ \forall x.~x \in A \to P~x

%FIXME: Example of an inductive definition

\subsection{Arithmetic proof support}

  arith & : & \isarmeth \\
  arith_split & : & \isaratt \\

  'arith' '!'?

The $arith$ method decides linear arithmetic problems (on types $nat$, $int$,
$real$).  Any current facts are inserted into the goal before running the
procedure.  The ``!''~argument causes the full context of assumptions to be
included.  The $arith_split$ attribute declares case split rules to be
expanded before the arithmetic procedure is invoked.

Note that a simpler (but faster) version of arithmetic reasoning is already
performed by the Simplifier.

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

The following important tactical tools of Isabelle/HOL have been ported to
Isar.  These should be never used in proper proof texts!

  case_tac^* & : & \isarmeth \\
  induct_tac^* & : & \isarmeth \\
  ind_cases^* & : & \isarmeth \\
  \isarcmd{inductive_cases} & : & \isartrans{theory}{theory} \\





  casetac goalspec? term rule?
  inducttac goalspec? (insts * 'and') rule?
  indcases (prop +) ('for' (name +)) ?
  inductivecases (thmdecl? (prop +) + 'and')

  rule: ('rule' ':' thmref)

\item [$case_tac$ and $induct_tac$] admit to reason about inductive datatypes
  only (unless an alternative rule is given explicitly).  Furthermore,
  $case_tac$ does a classical case split on booleans; $induct_tac$ allows only
  variables to be given as instantiation.  These tactic emulations feature
  both goal addressing and dynamic instantiation.  Note that named rule cases
  are \emph{not} provided as would be by the proper $induct$ and $cases$ proof
  methods (see \S\ref{sec:cases-induct}).
\item [$ind_cases$ and $\isarkeyword{inductive_cases}$] provide an interface
  to the internal \texttt{mk_cases} operation.  Rules are simplified in an
  unrestricted forward manner.

  While $ind_cases$ is a proof method to apply the result immediately as
  elimination rules, $\isarkeyword{inductive_cases}$ provides case split
  theorems at the theory level for later use.
  The \texttt{for} option of the $ind_cases$ method allows to specify a list
  of variables that should be generalized before applying the resulting rule.

\subsection{Executable code}

Isabelle/Pure provides two generic frameworks to support code
generation from executable specifications. Isabelle/HOL
instantiates these mechanisms in a
way that is amenable to end-user applications.

One framework generates code from both functional and
relational programs to SML.  See
\cite{isabelle-HOL} for further information (this actually covers the
new-style theory format as well).


  \isarcmd{value}^* & : & \isarkeep{theory~|~proof} \\
  \isarcmd{code_module} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_library} & : & \isartrans{theory}{theory} \\
  \isarcmd{consts_code} & : & \isartrans{theory}{theory} \\
  \isarcmd{types_code} & : & \isartrans{theory}{theory} \\  
  code & : & \isaratt \\


'value' term;

( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  'contains' ( ( name '=' term ) + | term + );

modespec : '(' ( name * ) ')';

'consts\_code' (codespec +);

codespec : const template attachment ?;

'types\_code' (tycodespec +);

tycodespec : name template attachment ?;

const: term;

template: '(' string ')';

attachment: 'attach' modespec ? verblbrace text verbrbrace;

'code' (name)?;

\item [$\isarkeyword{value}~t$] reads, evaluates and prints a term
  using the code generator.

The other framework generates code from functional programs
(including overloading using type classes) to SML \cite{SML},
OCaml \cite{OCaml} and Haskell \cite{haskell-revised-report}.
Conceptually, code generation is split up in three steps: \emph{selection}
of code theorems, \emph{translation} into an abstract executable view
and \emph{serialization} to a specific \emph{target language}.
See \cite{isabelle-codegen} for an introduction on how to use it.

\indexisaratt{code func}
\indexisaratt{code inline}

  \isarcmd{export_code}^* & : & \isarkeep{theory~|~proof} \\
  \isarcmd{code_thms}^* & : & \isarkeep{theory~|~proof} \\
  \isarcmd{code_deps}^* & : & \isarkeep{theory~|~proof} \\
  \isarcmd{code_datatype} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_const} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_type} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_class} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_instance} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_monad} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_reserved} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_include} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_modulename} & : & \isartrans{theory}{theory} \\
  \isarcmd{code_exception} & : & \isartrans{theory}{theory} \\
  \isarcmd{print_codesetup}^* & : & \isarkeep{theory~|~proof} \\
  code\ func & : & \isaratt \\
  code\ inline & : & \isaratt \\

'export\_code' ( constexpr + ) ? \\
  ( ( 'in' target ( 'module\_name' string ) ? \\
      ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?;

'code\_thms' ( constexpr + ) ?;

'code\_deps' ( constexpr + ) ?;

const : term;

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

typeconstructor : nameref;

class : nameref;

target : 'OCaml' | 'SML' | 'Haskell';

'code\_datatype' const +;

'code\_const' (const + 'and') \\
  ( ( '(' target ( syntax ? + 'and' ) ')' ) + );

'code\_type' (typeconstructor + 'and') \\
  ( ( '(' target ( syntax ? + 'and' ) ')' ) + );

'code\_class' (class + 'and') \\
  ( ( '(' target \\
    ( ( string ('where' \\
      ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + );

'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  ( ( '(' target ( '-' ? + 'and' ) ')' ) + );

'code\_monad' const const target;

'code\_reserved' target ( string + );

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

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

'code\_exception' ( const + );

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


'code\ func' ( 'del' ) ?;

'code\ inline' ( 'del' ) ?;


\item [$\isarcmd{export_code}$] is the canonical interface for generating and
  serializing code: for a given list of constants, code is generated for the specified
  target language(s).  Abstract code is cached incrementally.  If no constant is given,
  the currently cached code is serialized.  If no serialization instruction
  is given, only abstract code is cached.

  Constants may be specified by giving them literally, referring
  to all exeuctable contants within a certain theory named ``name''
  by giving (``name.*''), or referring to \emph{all} executable
  constants currently available (``*'').

  By default, for each involved theory one corresponding name space module
  is generated.  Alternativly, a module name may be specified after the
  (``module_name'') keyword; then \emph{all} code is placed in this module.

  For \emph{SML} and \emph{OCaml}, the file specification refers to
  a single file;  for \emph{Haskell}, it refers to a whole directory,
  where code is generated in multiple files reflecting the module hierarchy.
  The file specification ``-'' denotes standard output.  For \emph{SML},
  omitting the file specification compiles code internally
  in the context of the current ML session.

  Serializers take an optional list of arguments in parentheses. 
  For \emph{Haskell} a module name prefix may be given using the ``root:''
  argument;  ``string\_classes'' adds a ``deriving (Read, Show)'' clause
  to each appropriate datatype declaration.

\item [$\isarcmd{code_thms}$] prints a list of theorems representing the
  corresponding program containing all given constants; if no constants are
  given, the currently cached code theorems are printed.

\item [$\isarcmd{code_deps}$] visualizes dependencies of theorems representing the
  corresponding program containing all given constants; if no constants are
  given, the currently cached code theorems are visualized.

\item [$\isarcmd{code_datatype}$] specifies a constructor set for a logical type.

\item [$\isarcmd{code_const}$] associates a list of constants
  with target-specific serializations; omitting a serialization
  deletes an existing serialization.

\item [$\isarcmd{code_type}$] associates a list of type constructors
  with target-specific serializations; omitting a serialization
  deletes an existing serialization.

\item [$\isarcmd{code_class}$] associates a list of classes
  with target-specific class names; in addition, constants associated
  with this class may be given target-specific names used for instance
  declarations; omitting a serialization
  deletes an existing serialization.  Applies only to \emph{Haskell}.

\item [$\isarcmd{code_instance}$] declares a list of type constructor / class
  instance relations as ``already present'' for a given target.
  Omitting a ``-'' deletes an existing ``already present'' declaration.
  Applies only to \emph{Haskell}.

\item [$\isarcmd{code_monad}$] provides an auxiliary mechanism
  to generate monadic code.

\item [$\isarcmd{code_reserved}$] declares a list of names
  as reserved for a given target, preventing it to be shadowed
  by any generated code.

\item [$\isarcmd{code_include}$] adds arbitrary named content (''include``)
  to generated code. A as last argument ``-'' will remove an already added ''include``.

\item [$\isarcmd{code_modulename}$] declares aliasings from one module name
  onto another.

\item [$\isarcmd{code_exception}$] declares constants which are not required
  to have a definition by a defining equations; these are mapped on exceptions

\item [$code\ func$] selects (or with option ''del``, deselects) explicitly
  a defining equation for code generation.  Usually packages introducing
  defining equations provide a resonable default setup for selection.

\item [$code\ inline$] declares (or with option ''del``, removes)
  inlining theorems which are applied as rewrite rules to any defining equation
  during preprocessing.

\item [$\isarcmd{print_codesetup}$] gives an overview on selected
  defining equations, code generator datatypes and preprocessor setup.



\subsection{Mixfix syntax for continuous operations}


  \isarcmd{consts} & : & \isartrans{theory}{theory} \\

HOLCF provides a separate type for continuous functions $\alpha \to
\beta$, with an explicit application operator $f \cdot x$.  Isabelle mixfix
syntax normally refers directly to the pure meta-level function type $\alpha
\To \beta$, with application $f\,x$.

The HOLCF variant of $\CONSTS$ modifies that of Pure Isabelle (cf.\ 
\S\ref{sec:consts}) such that declarations involving continuous function types
are treated specifically.  Any given syntax template is transformed
internally, generating translation rules for the abstract and concrete
representation of continuous application.  Note that mixing of HOLCF and Pure
application is \emph{not} supported!

\subsection{Recursive domains}

  \isarcmd{domain} & : & \isartrans{theory}{theory} \\

  'domain' parname? (dmspec + 'and')

  dmspec: typespec '=' (cons + '|')
  cons: name (type *) mixfix?
  dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs

Recursive domains in HOLCF are analogous to datatypes in classical HOL (cf.\ 
\S\ref{sec:hol-datatype}).  Mutual recursion is supported, but no nesting nor
arbitrary branching.  Domain constructors may be strict (default) or lazy, the
latter admits to introduce infinitary objects in the typical LCF manner (e.g.\ 
lazy lists).  See also \cite{MuellerNvOS99} for a general discussion of HOLCF


\subsection{Type checking}

The ZF logic is essentially untyped, so the concept of ``type checking'' is
performed as logical reasoning about set-membership statements.  A special
method assists users in this task; a version of this is already declared as a
``solver'' in the standard Simplifier setup.


  \isarcmd{print_tcset}^* & : & \isarkeep{theory~|~proof} \\
  typecheck & : & \isarmeth \\
  TC & : & \isaratt \\

  'TC' (() | 'add' | 'del')

\item [$\isarcmd{print_tcset}$] prints the collection of typechecking rules of
  the current context.
  Note that the component built into the Simplifier only knows about those
  rules being declared globally in the theory!
\item [$typecheck$] attempts to solve any pending type-checking problems in
\item [$TC$] adds or deletes type-checking rules from the context.


\subsection{(Co)Inductive sets and datatypes}

\subsubsection{Set definitions}

In ZF everything is a set.  The generic inductive package also provides a
specific view for ``datatype'' specifications.  Coinductive definitions are
available in both cases, too.

  \isarcmd{inductive} & : & \isartrans{theory}{theory} \\
  \isarcmd{coinductive} & : & \isartrans{theory}{theory} \\
  \isarcmd{datatype} & : & \isartrans{theory}{theory} \\
  \isarcmd{codatatype} & : & \isartrans{theory}{theory} \\




  ('inductive' | 'coinductive') domains intros hints

  domains: 'domains' (term + '+') ('<=' | subseteq) term
  intros: 'intros' (thmdecl? prop +)
  hints: monos? condefs? typeintros? typeelims?
  monos: ('monos' thmrefs)?
  condefs: (CONDEFS thmrefs)?
  typeintros: (TYPEINTROS thmrefs)?
  typeelims: (TYPEELIMS thmrefs)?

In the following diagram $monos$, $typeintros$, and $typeelims$ are the same
as above.

  ('datatype' | 'codatatype') domain? (dtspec + 'and') hints

  domain: ('<=' | subseteq) term
  dtspec: term '=' (con + '|')
  con: name ('(' (term ',' +) ')')?  
  hints: monos? typeintros? typeelims?

See \cite{isabelle-ZF} for further information on inductive definitions in
HOL, but note that this covers the old-style theory format.

\subsubsection{Primitive recursive functions}

  \isarcmd{primrec} & : & \isartrans{theory}{theory} \\

  'primrec' (thmdecl? prop +)

\subsubsection{Cases and induction: emulating tactic scripts}

The following important tactical tools of Isabelle/ZF have been ported to
Isar.  These should be never used in proper proof texts!

  case_tac^* & : & \isarmeth \\
  induct_tac^* & : & \isarmeth \\
  ind_cases^* & : & \isarmeth \\
  \isarcmd{inductive_cases} & : & \isartrans{theory}{theory} \\

  (casetac | inducttac) goalspec? name
  indcases (prop +)
  inductivecases (thmdecl? (prop +) + 'and')

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