theory HOL_Specific

imports Base Main

begin

chapter {* Isabelle/HOL \label{ch:hol} *}

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

text {*

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

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

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

  structural operational semantics is an inductive definition of an

  evaluation relation.

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

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

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

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

  relations to formalise equivalence of processes and infinite data

  structures.

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

  described in a separate paper,\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 that of ZF thanks to implicit type-checking in HOL.

  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.

  \begin{matharray}{rcl}

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

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

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

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

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

  \end{matharray}

  @{rail "

    (@@{command (HOL) inductive} | @@{command (HOL) inductive_set} |

      @@{command (HOL) coinductive} | @@{command (HOL) coinductive_set})

    @{syntax target}? @{syntax \"fixes\"} (@'for' @{syntax \"fixes\"})? \\

    (@'where' clauses)? (@'monos' @{syntax thmrefs})?

    ;

    clauses: (@{syntax thmdecl}? @{syntax prop} + '|')

    ;

    @@{attribute (HOL) mono} (() | 'add' | 'del')

  "}

  \begin{description}

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

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

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

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

  (co)inductive predicates that remain fixed throughout the

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

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

  applied to a recursive set in the introduction rules.  There

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

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

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

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

  allowing the definition of (co)inductive sets.

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

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

  (HOL) "inductive"}.

  \end{description}

*}

subsection {* Derived rules *}

text {*

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

  theory and also proves some theorems:

  \begin{description}

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

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

  also available individually, using the names given them in the

  theory file;

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

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

  rule.

  \end{description}

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

  defined simultaneously, the list of introduction rules is called

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

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

  of mutual induction rules is called @{text

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

*}

subsection {* Monotonicity theorems *}

text {*

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

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

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

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

  theorems may be added:

  \begin{itemize}

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

  monotonicity of inductive definitions whose introduction rules have

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

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

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

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

  \[

  \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"}}

  \]

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

  of expressions, e.g.

  \[

  @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad

  @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}

  \]

  \item Equations for reducing complex operators to more primitive

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

  \[

  @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad

  @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}

  \]

  \end{itemize}

  %FIXME: Example of an inductive definition

*}

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

text {*

  \begin{matharray}{rcl}

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

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

    @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

    @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

  \end{matharray}

  @{rail "

    @@{command (HOL) primrec} @{syntax target}? @{syntax \"fixes\"} @'where' equations

    ;

    (@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts?

      @{syntax \"fixes\"} \\ @'where' equations

    ;

    equations: (@{syntax thmdecl}? @{syntax prop} + '|')

    ;

    functionopts: '(' (('sequential' | 'domintros') + ',') ')'

    ;

    @@{command (HOL) termination} @{syntax term}?

  "}

  \begin{description}

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

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

  \item @{command (HOL) "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 @{text "f.psimps"}) and induction rule

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

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

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

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

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

  proof attempts regarding pattern matching and termination.  See

  \cite{isabelle-function} for further details.

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

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

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

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

  the induction principle is established.

  \end{description}

  Recursive definitions introduced by the @{command (HOL) "function"}

  command accommodate

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

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

  refers to a specific induction rule, with parameters named according

  to the user-specified equations. Cases are numbered (starting from 1).

  For @{command (HOL) "primrec"}, the induction principle coincides

  with structural recursion on the datatype the recursion is carried

  out.

  The equations provided by these packages may be referred later as

  theorem list @{text "f.simps"}, where @{text f} is the (collective)

  name of the functions defined.  Individual equations may be named

  explicitly as well.

  The @{command (HOL) "function"} command accepts the following

  options.

  \begin{description}

  \item @{text 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 theorems.  Also note that this automatic

  transformation only works for ML-style datatype patterns.

  \item @{text 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.

  \end{description}

*}

subsection {* Proof methods related to recursive definitions *}

text {*

  \begin{matharray}{rcl}

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

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

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

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

  \end{matharray}

  @{rail "

    @@{method (HOL) relation} @{syntax term}

    ;

    @@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * )

    ;

    @@{method (HOL) size_change} ( orders (@{syntax clasimpmod} * ) )

    ;

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

  "}

  \begin{description}

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

  solve goals regarding the completeness of pattern matching, as

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

  \cite{isabelle-function}).

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

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

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

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

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

  termination proofs.

  \item @{method (HOL) "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 @{method auto} method,

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

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

  \secref{sec:clasimp}.

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

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

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

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

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

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

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

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

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

  internally.

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

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

  \end{description}

*}

subsection {* Functions with explicit partiality *}

text {*

  \begin{matharray}{rcl}

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

    @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\

  \end{matharray}

  @{rail "

    @@{command (HOL) partial_function} @{syntax target}?

      '(' @{syntax nameref} ')' @{syntax \"fixes\"} \\

      @'where' @{syntax thmdecl}? @{syntax prop}

  "}

  \begin{description}

  \item @{command (HOL) "partial_function"}~@{text "(mode)"} defines

  recursive functions based on fixpoints in complete partial

  orders. No termination proof is required from the user or

  constructed internally. Instead, the possibility of non-termination

  is modelled explicitly in the result type, which contains an

  explicit bottom element.

  Pattern matching and mutual recursion are currently not supported.

  Thus, the specification consists of a single function described by a

  single recursive equation.

  There are no fixed syntactic restrictions on the body of the

  function, but the induced functional must be provably monotonic

  wrt.\ the underlying order.  The monotonicitity proof is performed

  internally, and the definition is rejected when it fails. The proof

  can be influenced by declaring hints using the

  @{attribute (HOL) partial_function_mono} attribute.

  The mandatory @{text mode} argument specifies the mode of operation

  of the command, which directly corresponds to a complete partial

  order on the result type. By default, the following modes are

  defined:

  \begin{description}

  \item @{text option} defines functions that map into the @{type

  option} type. Here, the value @{term None} is used to model a

  non-terminating computation. Monotonicity requires that if @{term

  None} is returned by a recursive call, then the overall result

  must also be @{term None}. This is best achieved through the use of

  the monadic operator @{const "Option.bind"}.

  \item @{text tailrec} defines functions with an arbitrary result

  type and uses the slightly degenerated partial order where @{term

  "undefined"} is the bottom element.  Now, monotonicity requires that

  if @{term undefined} is returned by a recursive call, then the

  overall result must also be @{term undefined}. In practice, this is

  only satisfied when each recursive call is a tail call, whose result

  is directly returned. Thus, this mode of operation allows the

  definition of arbitrary tail-recursive functions.

  \end{description}

  Experienced users may define new modes by instantiating the locale

  @{const "partial_function_definitions"} appropriately.

  \item @{attribute (HOL) partial_function_mono} declares rules for

  use in the internal monononicity proofs of partial function

  definitions.

  \end{description}

*}

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

text {*

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

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

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

  \begin{matharray}{rcl}

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

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

  \end{matharray}

  @{rail "

    @@{command (HOL) recdef} ('(' @'permissive' ')')? \\

      @{syntax name} @{syntax term} (@{syntax prop} +) hints?

    ;

    recdeftc @{syntax thmdecl}? tc

    ;

    hints: '(' @'hints' ( recdefmod * ) ')'

    ;

    recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf')

      (() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod}

    ;

    tc: @{syntax nameref} ('(' @{syntax nat} ')')?

  "}

  \begin{description}

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

  recursive functions (using the TFL package), see also

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

  TFL to recover from failed proof attempts, returning unfinished

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

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

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

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

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

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

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

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

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

  constant @{text c}.

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

  its internal proofs without manual intervention.

  \end{description}

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

  globally, using the following attributes.

  \begin{matharray}{rcl}

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

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

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

  \end{matharray}

  @{rail "

    (@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} |

      @@{attribute (HOL) recdef_wf}) (() | 'add' | 'del')

  "}

*}

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

text {*

  \begin{matharray}{rcl}

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

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

  \end{matharray}

  @{rail "

    @@{command (HOL) datatype} (spec + @'and')

    ;

    @@{command (HOL) rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +)

    ;

    spec: @{syntax parname}? @{syntax typespec} @{syntax mixfix}? '=' (cons + '|')

    ;

    cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}?

  "}

  \begin{description}

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

  HOL.

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

  datatypes.

  For foundational reasons, some basic types such as @{typ nat}, @{typ

  "'a \<times> 'b"}, @{typ "'a + 'b"}, @{typ bool} and @{typ unit} are

  introduced by more primitive means using @{command_ref typedef}.  To

  recover the rich infrastructure of @{command datatype} (e.g.\ rules

  for @{method cases} and @{method induct} and the primitive recursion

  combinators), such types may be represented as actual datatypes

  later.  This is done by specifying the constructors of the desired

  type, and giving a proof of the induction rule, distinctness and

  injectivity of constructors.

  For example, see @{file "~~/src/HOL/Sum_Type.thy"} for the

  representation of the primitive sum type as fully-featured datatype.

  \end{description}

  The generated rules for @{method induct} and @{method cases} provide

  case names according to the given constructors, while parameters are

  named after the types (see also \secref{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 @{method (HOL) case_tac} and @{method (HOL)

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

  to refer directly to the internal structure of subgoals (including

  internally bound parameters).

*}

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

text {*

  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.

*}

subsection {* Basic concepts *}

text {*

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

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

  \begin{center}

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

    & record terms & record types \\ \hline

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

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

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

  \end{tabular}

  \end{center}

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

  "(| x = a |)"}.

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

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

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

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

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

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

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

  of the syntax).  The improper field ``@{text "\<dots>"}'' 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 @{text "\<lparr>x = a, y = b, z =

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

  Fixed records are special instances of record schemes, where

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

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

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

  \medskip Two key observations make extensible records in a simply

  typed language like HOL work out:

  \begin{enumerate}

  \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.

  \end{enumerate}

  \medskip 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 practice.

*}

subsection {* Record specifications *}

text {*

  \begin{matharray}{rcl}

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

  \end{matharray}

  @{rail "

    @@{command (HOL) record} @{syntax typespec_sorts} '=' \\

      (@{syntax type} '+')? (@{syntax constdecl} +)

  "}

  \begin{description}

  \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1

  \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},

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

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

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

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

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

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

  least one new field @{text "c\<^sub>i"} 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 @{text \<tau>} is optional; if omitted

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

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

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

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

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

  \<zeta>\<rparr>"}.

  \end{description}

*}

subsection {* Record operations *}

text {*

  Any record definition of the form presented above produces certain

  standard operations.  Selectors and updates are provided for any

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

  cumulative record constructor functions.  To simplify the

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

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

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

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

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

  \begin{matharray}{lll}

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

    @{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>"} \\

  \end{matharray}

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

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

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

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

  := a, y := b, z := c\<rparr>"}, 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 function:

  \begin{matharray}{lll}

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

  \end{matharray}

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

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

  the above record operations will get the following types:

  \medskip

  \begin{tabular}{lll}

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

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

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

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

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

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

  \end{tabular}

  \medskip

  \noindent Some further operations address the extension aspect of a

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

  record fragment consisting of exactly the new fields introduced here

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

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

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

  \medskip

  \begin{tabular}{lll}

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

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

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

    @{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>"} \\

  \end{tabular}

  \medskip

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

  for root records.

*}

subsection {* Derived rules and proof tools *}

text {*

  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

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

  \begin{enumerate}

  \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 @{method simp} method without further arguments.  These rules

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

  \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 @{prop "(x, y) = (x',

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

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

  @{text "t.iffs"}.

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

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

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

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

  \item Representations of arbitrary record expressions as canonical

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

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

  \secref{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 ``@{text "(cases

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

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

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

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

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

  \end{enumerate}

*}

section {* Adhoc tuples *}

text {*

  \begin{matharray}{rcl}

    @{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\

  \end{matharray}

  @{rail "

    @@{attribute (HOL) split_format} ('(' 'complete' ')')?

  "}

  \begin{description}

  \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes

  arguments in function applications to be represented canonically

  according to their tuple type structure.

  Note that this operation tends to invent funny names for new local

  parameters introduced.

  \end{description}

*}

section {* Typedef axiomatization \label{sec:hol-typedef} *}

text {* A Gordon/HOL-style type definition is a certain axiom scheme

  that identifies a new type with a subset of an existing type.  More

  precisely, the new type is defined by exhibiting an existing type

  @{text \<tau>}, a set @{text "A :: \<tau> set"}, and a theorem that proves

  @{prop "\<exists>x. x \<in> A"}.  Thus @{text A} is a non-empty subset of @{text

  \<tau>}, and the new type denotes this subset.  New functions are

  postulated that establish an isomorphism between the new type and

  the subset.  In general, the type @{text \<tau>} may involve type

  variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} which means that the type definition

  produces a type constructor @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} depending on

  those type arguments.

  The axiomatization can be considered a ``definition'' in the sense

  of the particular set-theoretic interpretation of HOL

  \cite{pitts93}, where the universe of types is required to be

  downwards-closed wrt.\ arbitrary non-empty subsets.  Thus genuinely

  new types introduced by @{command "typedef"} stay within the range

  of HOL models by construction.  Note that @{command_ref

  type_synonym} from Isabelle/Pure merely introduces syntactic

  abbreviations, without any logical significance.

  \begin{matharray}{rcl}

    @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

  \end{matharray}

  @{rail "

    @@{command (HOL) typedef} alt_name? abs_type '=' rep_set

    ;

    alt_name: '(' (@{syntax name} | @'open' | @'open' @{syntax name}) ')'

    ;

    abs_type: @{syntax typespec_sorts} @{syntax mixfix}?

    ;

    rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})?

  "}

  \begin{description}

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

  axiomatizes a type definition in the background theory of the

  current context, depending on a non-emptiness result of the set

  @{text A} that needs to be proven here.  The set @{text A} may

  contain type variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} as specified on the LHS,

  but no term variables.

  Even though a local theory specification, the newly introduced type

  constructor cannot depend on parameters or assumptions of the

  context: this is structurally impossible in HOL.  In contrast, the

  non-emptiness proof may use local assumptions in unusual situations,

  which could result in different interpretations in target contexts:

  the meaning of the bijection between the representing set @{text A}

  and the new type @{text t} may then change in different application

  contexts.

  By default, @{command (HOL) "typedef"} defines both a type

  constructor @{text t} for the new type, and a term constant @{text

  t} for the representing set within the old type.  Use the ``@{text

  "(open)"}'' option to suppress a separate constant definition

  altogether.  The injection from type to set is called @{text Rep_t},

  its inverse @{text Abs_t}, unless explicit @{keyword (HOL)

  "morphisms"} specification provides alternative names.

  The core axiomatization uses the locale predicate @{const

  type_definition} as defined in Isabelle/HOL.  Various basic

  consequences of that are instantiated accordingly, re-using the

  locale facts with names derived from the new type constructor.  Thus

  the generic @{thm type_definition.Rep} is turned into the specific

  @{text "Rep_t"}, for example.

  Theorems @{thm type_definition.Rep}, @{thm

  type_definition.Rep_inverse}, and @{thm type_definition.Abs_inverse}

  provide the most basic characterization as a corresponding

  injection/surjection pair (in both directions).  The derived rules

  @{thm type_definition.Rep_inject} and @{thm

  type_definition.Abs_inject} provide a more convenient version of

  injectivity, suitable for automated proof tools (e.g.\ in

  declarations involving @{attribute simp} or @{attribute iff}).

  Furthermore, the rules @{thm type_definition.Rep_cases}~/ @{thm

  type_definition.Rep_induct}, and @{thm type_definition.Abs_cases}~/

  @{thm type_definition.Abs_induct} provide alternative views on

  surjectivity.  These rules are already declared as set or type rules

  for the generic @{method cases} and @{method induct} methods,

  respectively.

  An alternative name for the set definition (and other derived

  entities) may be specified in parentheses; the default is to use

  @{text t} directly.

  \end{description}

  \begin{warn}

  If you introduce a new type axiomatically, i.e.\ via @{command_ref

  typedecl} and @{command_ref axiomatization}, the minimum requirement

  is that it has a non-empty model, to avoid immediate collapse of the

  HOL logic.  Moreover, one needs to demonstrate that the

  interpretation of such free-form axiomatizations can coexist with

  that of the regular @{command_def typedef} scheme, and any extension

  that other people might have introduced elsewhere (e.g.\ in HOLCF

  \cite{MuellerNvOS99}).

  \end{warn}

*}

subsubsection {* Examples *}

text {* Type definitions permit the introduction of abstract data

  types in a safe way, namely by providing models based on already

  existing types.  Given some abstract axiomatic description @{text P}

  of a type, this involves two steps:

  \begin{enumerate}

  \item Find an appropriate type @{text \<tau>} and subset @{text A} which

  has the desired properties @{text P}, and make a type definition

  based on this representation.

  \item Prove that @{text P} holds for @{text \<tau>} by lifting @{text P}

  from the representation.

  \end{enumerate}

  You can later forget about the representation and work solely in

  terms of the abstract properties @{text P}.

  \medskip The following trivial example pulls a three-element type

  into existence within the formal logical environment of HOL. *}

typedef three = "{(True, True), (True, False), (False, True)}"

  by blast

definition "One = Abs_three (True, True)"

definition "Two = Abs_three (True, False)"

definition "Three = Abs_three (False, True)"

lemma three_distinct: "One \<noteq> Two"  "One \<noteq> Three"  "Two \<noteq> Three"

  by (simp_all add: One_def Two_def Three_def Abs_three_inject three_def)

lemma three_cases:

  fixes x :: three obtains "x = One" | "x = Two" | "x = Three"

  by (cases x) (auto simp: One_def Two_def Three_def Abs_three_inject three_def)

text {* Note that such trivial constructions are better done with

  derived specification mechanisms such as @{command datatype}: *}

datatype three' = One' | Two' | Three'

text {* This avoids re-doing basic definitions and proofs from the

  primitive @{command typedef} above. *}

section {* Functorial structure of types *}

text {*

  \begin{matharray}{rcl}

    @{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}

  \end{matharray}

  @{rail "

    @@{command (HOL) enriched_type} (@{syntax name} ':')? @{syntax term}

    ;

  "}

  \begin{description}

  \item @{command (HOL) "enriched_type"}~@{text "prefix: m"} allows to

  prove and register properties about the functorial structure of type

  constructors.  These properties then can be used by other packages

  to deal with those type constructors in certain type constructions.

  Characteristic theorems are noted in the current local theory.  By

  default, they are prefixed with the base name of the type

  constructor, an explicit prefix can be given alternatively.

  The given term @{text "m"} is considered as \emph{mapper} for the

  corresponding type constructor and must conform to the following

  type pattern:

  \begin{matharray}{lll}

    @{text "m"} & @{text "::"} &

      @{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\

  \end{matharray}

  \noindent where @{text t} is the type constructor, @{text

  "\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct

  type variables free in the local theory and @{text "\<sigma>\<^isub>1"},

  \ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow>

  \<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots,

  @{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow>

  \<alpha>\<^isub>n"}.

  \end{description}

*}

section {* Arithmetic proof support *}

text {*

  \begin{matharray}{rcl}

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

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

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

  \end{matharray}

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

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

  facts are inserted into the goal before running the procedure.

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

  always supplied to the arithmetic provers implicitly.

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

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

  Note that a simpler (but faster) arithmetic prover is

  already invoked by the Simplifier.

*}

section {* Intuitionistic proof search *}

text {*

  \begin{matharray}{rcl}

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

  \end{matharray}

  @{rail "

    @@{method (HOL) iprover} ( @{syntax rulemod} * )