theory HOL_Specific

imports Main

begin

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

section {* Primitive types \label{sec:hol-typedef} *}

text {*

  \begin{matharray}{rcl}

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

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

  \end{matharray}

  \begin{rail}

    'typedecl' typespec infix?

    ;

    'typedef' altname? abstype '=' repset

    ;

    altname: '(' (name | 'open' | 'open' name) ')'

    ;

    abstype: typespec infix?

    ;

    repset: term ('morphisms' name name)?

    ;

  \end{rail}

  \begin{description}

  \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar

  to the original @{command "typedecl"} of Isabelle/Pure (see

  \secref{sec:types-pure}), but also declares type arity @{text "t ::

  (type, \<dots>, type) type"}, making @{text t} an actual HOL type

  constructor.  %FIXME check, update

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

  a goal stating non-emptiness of the set @{text 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 @{text A} and the new type @{text t}.

  Technically, @{command (HOL) "typedef"} defines both a type @{text

  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 @{text Rep_t}, its inverse @{text Abs_t} (this may be

  changed via an explicit @{keyword (HOL) "morphisms"} declaration).

  Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text

  Abs_t_inverse} provide the most basic characterization as a

  corresponding injection/surjection pair (in both directions).  Rules

  @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly

  more convenient view on the injectivity part, suitable for automated

  proof tools (e.g.\ in @{attribute simp} or @{attribute iff}

  declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and

  @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views

  on surjectivity; these are already declared as set or type rules for

  the generic @{method cases} and @{method induct} methods.

  An alternative name may be specified in parentheses; the default is

  to use @{text t} as indicated before.  The ``@{text "(open)"}''

  declaration suppresses a separate constant definition for the

  representing set.

  \end{description}

  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 @{command

  (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)

  "datatype"} (see \secref{sec:hol-datatype}).

*}

section {* Adhoc tuples *}

text {*

  \begin{matharray}{rcl}

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

  \end{matharray}

  \begin{rail}

    'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))

    ;

  \end{rail}

  \begin{description}

  \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>

  \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into

  canonical form as specified by the arguments given; the @{text i}-th

  collection of arguments refers to occurrences in premise @{text i}

  of the rule.  The ``@{text "(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.

  \end{description}

*}

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}

  \begin{rail}

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

    ;

  \end{rail}

  \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 {* 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}

  \begin{rail}

    'datatype' (dtspec + 'and')

    ;

    'rep\_datatype' ('(' (name +) ')')? (term +)

    ;

    dtspec: parname? typespec infix? '=' (cons + '|')

    ;

    cons: name ( type * ) mixfix?

  \end{rail}

  \begin{description}

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

  HOL.

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

  inductive ones, generating the standard infrastructure of derived

  concepts (primitive recursion etc.).

  \end{description}

  The induction and exhaustion theorems generated provide case names

  according to the constructors involved, 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 {* 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}

  \begin{rail}

    'primrec' target? fixes 'where' equations

    ;

    equations: (thmdecl? prop + '|')

    ;

    ('fun' | 'function') target? functionopts? fixes 'where' clauses

    ;

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

    ;

    functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'

    ;

    'termination' ( term )?

  \end{rail}

  \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 theroems.  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.

  \item @{text tailrec} generates the unconstrained recursive

  equations even without a termination proof, provided that the

  function is tail-recursive. This currently only works

  \item @{text "default d"} allows to specify a default value for a

  (partial) function, which will ensure that @{text "f x = d x"}

  whenever @{text "x \<notin> f_dom"}.

  \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}

  \begin{rail}

    'relation' term

    ;

    'lexicographic\_order' ( clasimpmod * )

    ;

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

    ;

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

  \end{rail}

  \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 {* 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}

  \begin{rail}

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

    ;

    recdeftc thmdecl? tc

    ;

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

    ;

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

    ;

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

    ;

  \end{rail}

  \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}

  \begin{rail}

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

    ;

  \end{rail}

*}

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}

  \begin{rail}

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

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

    ;

    clauses: (thmdecl? prop + '|')

    ;

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

    ;

  \end{rail}

  \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 {* 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}

  \begin{rail}

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

    ;

  \end{rail}

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

  search, depending on specifically declared rules from the context,

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

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

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

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

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

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

  applied aggressively (without considering back-tracking later).

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

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

  explicit weight annotation may be given as well; otherwise the

  number of rule premises will be taken into account here.

*}

section {* Coherent Logic *}

text {*

  \begin{matharray}{rcl}

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

  \end{matharray}

  \begin{rail}

    'coherent' thmrefs?

    ;

  \end{rail}

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

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

  applications in confluence theory, lattice theory and projective

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

  examples.

*}

section {* Checking and refuting propositions *}

text {*

  Identifying incorrect propositions usually involves evaluation of

  particular assignments and systematic counter example search.  This

  is supported by the following commands.

  \begin{matharray}{rcl}

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

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

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

  \end{matharray}

  \begin{rail}

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

    ;

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

    ;

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

    ;

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

    ;

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

    ;

  \end{rail}

  \begin{description}

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

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

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

    Internally, the evaluation is performed by registered evaluators,

    which are invoked sequentially until a result is returned.

    Alternatively a specific evaluator can be selected using square

    brackets; available evaluators include @{text nbe} for

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

    generation in SML.

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

    counter examples using a series of arbitrary assignments for its

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

    can be selected explicitly using an optional goal index.

    A number of configuration options are supported for

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

    \begin{description}

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

        assignment values.

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

        generated for each particular size.

      \item[no\_assms] specifies whether assumptions in

        structured proofs should be ignored.

    \end{description}

    These option can be given within square brackets.

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

    configuration options persitently.

  \end{description}

*}

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

text {*

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

  external automated reasoning tools to crunch a pending goal.

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

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

  There is also a wrapper to invoke provers remotely via the

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

  web service.

  The problem passed to external provers consists of the goal together

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

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

  usually reconstructs the proof within Isabelle, without requiring

  external provers again.  The Metis

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

  integrated into Isabelle/HOL is being used here.

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

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

  works via explicit inferences going through the Isabelle kernel.

  Moreover, rechecking Isabelle proof texts with already specified

  auxiliary facts is much faster than performing fully automated

  search over and over again.

  \begin{matharray}{rcl}

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

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

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

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

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

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

  \end{matharray}

  \begin{rail}

  'sledgehammer' ( nameref * )

  ;

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

  ;

  'metis' thmrefs

  ;

  \end{rail}

  \begin{description}

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

  invokes the specified automated theorem provers on the first

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

  is displayed, and the other attempts are terminated.

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

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

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

  ``ATP provers'' preference by the user interface.

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

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

  excessive provers are automatically terminated.

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

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

  command.

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

  running provers, including elapsed runtime, and the remaining time

  until timeout.

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

  provers.

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

  by automated theorem provers.  This allows to examine results that

  might have got lost due to the asynchronous nature of default

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

  be specified (default 5).

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

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

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

  inferences going through the kernel.  Thus its results are

  guaranteed to be ``correct by construction''.