doc-src/IsarImplementation/Thy/ML.thy

more robust examples: explicit @{assert} instead of unchecked output;

theory "ML"
imports Base
begin

chapter {* Isabelle/ML *}

text {* Isabelle/ML is best understood as a certain culture based on
  Standard ML.  Thus it is not a new programming language, but a
  certain way to use SML at an advanced level within the Isabelle
  environment.  This covers a variety of aspects that are geared
  towards an efficient and robust platform for applications of formal
  logic with fully foundational proof construction --- according to
  the well-known \emph{LCF principle}.  There are specific library
  modules and infrastructure to address the needs for such difficult
  tasks.  For example, the raw parallel programming model of Poly/ML
  is presented as considerably more abstract concept of \emph{future
  values}, which is then used to augment the inference kernel, proof
  interpreter, and theory loader accordingly.

  The main aspects of Isabelle/ML are introduced below.  These
  first-hand explanations should help to understand how proper
  Isabelle/ML is to be read and written, and to get access to the
  wealth of experience that is expressed in the source text and its
  history of changes.\footnote{See
  \url{http://isabelle.in.tum.de/repos/isabelle} for the full
  Mercurial history.  There are symbolic tags to refer to official
  Isabelle releases, as opposed to arbitrary \emph{tip} versions that
  merely reflect snapshots that are never really up-to-date.}  *}


section {* SML embedded into Isabelle/Isar *}

text {* ML and Isar are intertwined via an open-ended bootstrap
  process that provides more and more programming facilities and
  logical content in an alternating manner.  Bootstrapping starts from
  the raw environment of existing implementations of Standard ML
  (mainly Poly/ML, but also SML/NJ).

  Isabelle/Pure marks the point where the original ML toplevel is
  superseded by the Isar toplevel that maintains a uniform environment
  for arbitrary ML values (see also \secref{sec:context}).  This
  formal context holds logical entities as well as ML compiler
  bindings, among many other things.  Raw Standard ML is never
  encountered again after the initial bootstrap of Isabelle/Pure.

  Object-logics such as Isabelle/HOL are built within the
  Isabelle/ML/Isar environment of Pure by introducing suitable
  theories with associated ML text, either inlined or as separate
  files.  Thus Isabelle/HOL is defined as a regular user-space
  application within the Isabelle framework.  Further add-on tools can
  be implemented in ML within the Isar context in the same manner: ML
  is part of the regular user-space repertoire of Isabelle.
*}


subsection {* Isar ML commands *}

text {* The primary Isar source language provides various facilities
  to open a ``window'' to the underlying ML compiler.  Especially see
  @{command_ref "use"} and @{command_ref "ML"}, which work exactly the
  same way, only the source text is provided via a file vs.\ inlined,
  respectively.  Apart from embedding ML into the main theory
  definition like that, there are many more commands that refer to ML
  source, such as @{command_ref setup} or @{command_ref declaration}.
  An example of even more fine-grained embedding of ML into Isar is
  the proof method @{method_ref tactic}, which refines the pending goal state
  via a given expression of type @{ML_type tactic}.
*}

text %mlex {* The following artificial example demonstrates some ML
  toplevel declarations within the implicit Isar theory context.  This
  is regular functional programming without referring to logical
  entities yet.
*}

ML {*
  fun factorial 0 = 1
    | factorial n = n * factorial (n - 1)
*}

text {* Here the ML environment is really managed by Isabelle, i.e.\
  the @{ML factorial} function is not yet accessible in the preceding
  paragraph, nor in a different theory that is independent from the
  current one in the import hierarchy.

  Removing the above ML declaration from the source text will remove
  any trace of this definition as expected.  The Isabelle/ML toplevel
  environment is managed in a \emph{stateless} way: unlike the raw ML
  toplevel or similar command loops of Computer Algebra systems, there
  are no global side-effects involved here.\footnote{Such a stateless
  compilation environment is also a prerequisite for robust parallel
  compilation within independent nodes of the implicit theory
  development graph.}

  \medskip The next example shows how to embed ML into Isar proofs.
  Instead of @{command_ref "ML"} for theory mode, we use @{command_ref
  "ML_prf"} for proof mode.  As illustrated below, its effect on the
  ML environment is local to the whole proof body, while ignoring its
  internal block structure.  *}

example_proof
  ML_prf %"ML" {* val a = 1 *}
  { -- {* Isar block structure ignored by ML environment *}
    ML_prf %"ML" {* val b = a + 1 *}
  } -- {* Isar block structure ignored by ML environment *}
  ML_prf %"ML" {* val c = b + 1 *}
qed

text {* By side-stepping the normal scoping rules for Isar proof
  blocks, embedded ML code can refer to the different contexts
  explicitly, and manipulate corresponding entities, e.g.\ export a
  fact from a block context.

  \medskip Two further ML commands are useful in certain situations:
  @{command_ref ML_val} and @{command_ref ML_command} are both
  \emph{diagnostic} in the sense that there is no effect on the
  underlying environment, and can thus used anywhere (even outside a
  theory).  The examples below produce long strings of digits by
  invoking @{ML factorial}: @{command ML_val} already takes care of
  printing the ML toplevel result, but @{command ML_command} is silent
  so we produce an explicit output message.  *}

ML_val {* factorial 100 *}
ML_command {* writeln (string_of_int (factorial 100)) *}

example_proof
  ML_val {* factorial 100 *}  (* FIXME check/fix indentation *)
  ML_command {* writeln (string_of_int (factorial 100)) *}
qed


subsection {* Compile-time context *}

text {* Whenever the ML compiler is invoked within Isabelle/Isar, the
  formal context is passed as a thread-local reference variable.  Thus
  ML code may access the theory context during compilation, by reading
  or writing the (local) theory under construction.  Note that such
  direct access to the compile-time context is rare; in practice it is
  typically via some derived ML functions.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML ML_Context.the_generic_context: "unit -> Context.generic"} \\
  @{index_ML "Context.>> ": "(Context.generic -> Context.generic) -> unit"} \\
  @{index_ML bind_thms: "string * thm list -> unit"} \\
  @{index_ML bind_thm: "string * thm -> unit"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML "ML_Context.the_generic_context ()"} refers to the theory
  context of the ML toplevel --- at compile time.  ML code needs to
  take care to refer to @{ML "ML_Context.the_generic_context ()"}
  correctly.  Recall that evaluation of a function body is delayed
  until actual run-time.

  \item @{ML "Context.>>"}~@{text f} applies context transformation
  @{text f} to the implicit context of the ML toplevel.

  \item @{ML bind_thms}~@{text "(name, thms)"} stores a list of
  theorems produced in ML both in the (global) theory context and the
  ML toplevel, associating it with the provided name.  Theorems are
  put into a global ``standard'' format before being stored.

  \item @{ML bind_thm} is similar to @{ML bind_thms} but refers to a
  singleton theorem.

  \end{description}

  It is very important to note that the above functions are really
  restricted to the compile time, even though the ML compiler is
  invoked at run-time.  The majority of ML code either uses static
  antiquotations (\secref{sec:ML-antiq}) or refers to the theory or
  proof context at run-time, by explicit functional abstraction.
*}


subsection {* Antiquotations \label{sec:ML-antiq} *}

text {* A very important consequence of embedding SML into Isar is the
  concept of \emph{ML antiquotation}.  First, the standard token
  language of ML is augmented by special syntactic entities of the
  following form:

  \indexouternonterm{antiquote}
  \begin{rail}
  antiquote: atsign lbrace nameref args rbrace | lbracesym | rbracesym
  ;
  \end{rail}

  Here @{syntax nameref} and @{syntax args} are regular outer syntax
  categories.  Note that attributes and proof methods use similar
  syntax.

  \medskip A regular antiquotation @{text "@{name args}"} processes
  its arguments by the usual means of the Isar source language, and
  produces corresponding ML source text, either as literal
  \emph{inline} text (e.g. @{text "@{term t}"}) or abstract
  \emph{value} (e.g. @{text "@{thm th}"}).  This pre-compilation
  scheme allows to refer to formal entities in a robust manner, with
  proper static scoping and with some degree of logical checking of
  small portions of the code.

  Special antiquotations like @{text "@{let \<dots>}"} or @{text "@{note
  \<dots>}"} augment the compilation context without generating code.  The
  non-ASCII braces @{text "\<lbrace>"} and @{text "\<rbrace>"} allow to delimit the
  effect by introducing local blocks within the pre-compilation
  environment.

  \medskip See also \cite{Wenzel-Chaieb:2007b} for a slightly broader
  perspective on Isabelle/ML antiquotations.  *}

text %mlantiq {*
  \begin{matharray}{rcl}
  @{ML_antiquotation_def "let"} & : & @{text ML_antiquotation} \\
  @{ML_antiquotation_def "note"} & : & @{text ML_antiquotation} \\
  \end{matharray}

  \begin{rail}
  'let' ((term + 'and') '=' term + 'and')
  ;

  'note' (thmdef? thmrefs + 'and')
  ;
  \end{rail}

  \begin{description}

  \item @{text "@{let p = t}"} binds schematic variables in the
  pattern @{text "p"} by higher-order matching against the term @{text
  "t"}.  This is analogous to the regular @{command_ref let} command
  in the Isar proof language.  The pre-compilation environment is
  augmented by auxiliary term bindings, without emitting ML source.

  \item @{text "@{note a = b\<^sub>1 \<dots> b\<^sub>n}"} recalls existing facts @{text
  "b\<^sub>1, \<dots>, b\<^sub>n"}, binding the result as @{text a}.  This is analogous to
  the regular @{command_ref note} command in the Isar proof language.
  The pre-compilation environment is augmented by auxiliary fact
  bindings, without emitting ML source.

  \end{description}
*}

text %mlex {* The following artificial example gives a first
  impression about using the antiquotation elements introduced so far,
  together with the basic @{text "@{thm}"} antiquotation defined
  later.
*}

ML {*
  \<lbrace>
    @{let ?t = my_term}
    @{note my_refl = reflexive [of ?t]}
    fun foo th = Thm.transitive th @{thm my_refl}
  \<rbrace>
*}

text {*
  The extra block delimiters do not affect the compiled code itself, i.e.\
  function @{ML foo} is available in the present context.
*}


section {* Message output channels \label{sec:message-channels} *}

text {* Isabelle provides output channels for different kinds of
  messages: regular output, high-volume tracing information, warnings,
  and errors.

  Depending on the user interface involved, these messages may appear
  in different text styles or colours.  The standard output for
  terminal sessions prefixes each line of warnings by @{verbatim
  "###"} and errors by @{verbatim "***"}, but leaves anything else
  unchanged.

  Messages are associated with the transaction context of the running
  Isar command.  This enables the front-end to manage commands and
  resulting messages together.  For example, after deleting a command
  from a given theory document version, the corresponding message
  output can be retracted from the display.  *}

text %mlref {*
  \begin{mldecls}
  @{index_ML writeln: "string -> unit"} \\
  @{index_ML tracing: "string -> unit"} \\
  @{index_ML warning: "string -> unit"} \\
  @{index_ML error: "string -> 'a"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML writeln}~@{text "text"} outputs @{text "text"} as regular
  message.  This is the primary message output operation of Isabelle
  and should be used by default.

  \item @{ML tracing}~@{text "text"} outputs @{text "text"} as special
  tracing message, indicating potential high-volume output to the
  front-end (hundreds or thousands of messages issued by a single
  command).  The idea is to allow the user-interface to downgrade the
  quality of message display to achieve higher throughput.

  Note that the user might have to take special actions to see tracing
  output, e.g.\ switch to a different output window.  So this channel
  should not be used for regular output.

  \item @{ML warning}~@{text "text"} outputs @{text "text"} as
  warning, which typically means some extra emphasis on the front-end
  side (color highlighting, icon).

  \item @{ML error}~@{text "text"} raises exception @{ML ERROR}~@{text
  "text"} and thus lets the Isar toplevel print @{text "text"} on the
  error channel, which typically means some extra emphasis on the
  front-end side (color highlighting, icon).

  This assumes that the exception is not handled before the command
  terminates.  Handling exception @{ML ERROR}~@{text "text"} is a
  perfectly legal alternative: it means that the error is absorbed
  without any message output.

  \begin{warn}
  The actual error channel is accessed via @{ML Output.error_msg}, but
  the interaction protocol of Proof~General \emph{crashes} if that
  function is used in regular ML code: error output and toplevel
  command failure always need to coincide here.
  \end{warn}

  \end{description}

  \begin{warn}
  Regular Isabelle/ML code should output messages exclusively by the
  official channels.  Using raw I/O on \emph{stdout} or \emph{stderr}
  instead (e.g.\ via @{ML TextIO.output}) is apt to cause problems in
  the presence of parallel and asynchronous processing of Isabelle
  theories.  Such raw output might be displayed by the front-end in
  some system console log, with a low chance that the user will ever
  see it.  Moreover, as a genuine side-effect on global process
  channels, there is no proper way to retract output when Isar command
  transactions are reset.
  \end{warn}
*}


section {* Exceptions *}

text {* The Standard ML semantics of strict functional evaluation
  together with exceptions is rather well defined, but some delicate
  points need to be observed to avoid that ML programs go wrong
  despite static type-checking.  Exceptions in Isabelle/ML are
  subsequently categorized as follows.

  \paragraph{Regular user errors.}  These are meant to provide
  informative feedback about malformed input etc.

  The \emph{error} function raises the corresponding \emph{ERROR}
  exception, with a plain text message as argument.  \emph{ERROR}
  exceptions can be handled internally, in order to be ignored, turned
  into other exceptions, or cascaded by appending messages.  If the
  corresponding Isabelle/Isar command terminates with an \emph{ERROR}
  exception state, the toplevel will print the result on the error
  channel (see \secref{sec:message-channels}).

  It is considered bad style to refer to internal function names or
  values in ML source notation in user error messages.

  Grammatical correctness of error messages can be improved by
  \emph{omitting} final punctuation: messages are often concatenated
  or put into a larger context (e.g.\ augmented with source position).
  By not insisting in the final word at the origin of the error, the
  system can perform its administrative tasks more easily and
  robustly.

  \paragraph{Program failures.}  There is a handful of standard
  exceptions that indicate general failure situations, or failures of
  core operations on logical entities (types, terms, theorems,
  theories, see \chref{ch:logic}).

  These exceptions indicate a genuine breakdown of the program, so the
  main purpose is to determine quickly what has happened where.
  Traditionally, the (short) exception message would include the name
  of an ML function, although this no longer necessary, because the ML
  runtime system prints a detailed source position of the
  corresponding @{verbatim raise} keyword.

  \medskip User modules can always introduce their own custom
  exceptions locally, e.g.\ to organize internal failures robustly
  without overlapping with existing exceptions.  Exceptions that are
  exposed in module signatures require extra care, though, and should
  \emph{not} be introduced by default.  Surprise by end-users or ML
  users of a module can be often minimized by using plain user errors.

  \paragraph{Interrupts.}  These indicate arbitrary system events:
  both the ML runtime system and the Isabelle/ML infrastructure signal
  various exceptional situations by raising the special
  \emph{Interrupt} exception in user code.

  This is the one and only way that physical events can intrude an
  Isabelle/ML program.  Such an interrupt can mean out-of-memory,
  stack overflow, timeout, internal signaling of threads, or the user
  producing a console interrupt manually etc.  An Isabelle/ML program
  that intercepts interrupts becomes dependent on physical effects of
  the environment.  Even worse, exception handling patterns that are
  too general by accident, e.g.\ by mispelled exception constructors,
  will cover interrupts unintentionally, and thus render the program
  semantics ill-defined.

  Note that the Interrupt exception dates back to the original SML90
  language definition.  It was excluded from the SML97 version to
  avoid its malign impact on ML program semantics, but without
  providing a viable alternative.  Isabelle/ML recovers physical
  interruptibility (which an indispensable tool to implement managed
  evaluation of Isar command transactions), but requires user code to
  be strictly transparent wrt.\ interrupts.

  \begin{warn}
  Isabelle/ML user code needs to terminate promptly on interruption,
  without guessing at its meaning to the system infrastructure.
  Temporary handling of interrupts for cleanup of global resources
  etc.\ needs to be followed immediately by re-raising of the original
  exception.
  \end{warn}
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML try: "('a -> 'b) -> 'a -> 'b option"} \\
  @{index_ML can: "('a -> 'b) -> 'a -> bool"} \\
  @{index_ML ERROR: "string -> exn"} \\
  @{index_ML Fail: "string -> exn"} \\
  @{index_ML Exn.is_interrupt: "exn -> bool"} \\
  @{index_ML reraise: "exn -> 'a"} \\
  @{index_ML exception_trace: "(unit -> 'a) -> 'a"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML try}~@{text "f x"} makes the partiality of evaluating
  @{text "f x"} explicit via the option datatype.  Interrupts are
  \emph{not} handled here, i.e.\ this form serves as safe replacement
  for the \emph{unsafe} version @{verbatim "(SOME"}~@{text "f
  x"}~@{verbatim "handle _ => NONE)"} that is occasionally seen in
  books about SML.

  \item @{ML can} is similar to @{ML try} with more abstract result.

  \item @{ML ERROR}~@{text "msg"} represents user errors; this
  exception is always raised via the @{ML error} function (see
  \secref{sec:message-channels}).

  \item @{ML Fail}~@{text "msg"} represents general program failures.

  \item @{ML Exn.is_interrupt} identifies interrupts robustly, without
  mentioning concrete exception constructors in user code.  Handled
  interrupts need to be re-raised promptly!

  \item @{ML reraise}~@{text "exn"} raises exception @{text "exn"}
  while preserving its implicit position information (if possible,
  depending on the ML platform).

  \item @{ML exception_trace}~@{verbatim "(fn _ =>"}~@{text
  "e"}@{verbatim ")"} evaluates expression @{text "e"} while printing
  a full trace of its stack of nested exceptions (if possible,
  depending on the ML platform).\footnote{In various versions of
  Poly/ML the trace will appear on raw stdout of the Isabelle
  process.}

  Inserting @{ML exception_trace} into ML code temporarily is useful
  for debugging, but not suitable for production code.

  \end{description}
*}

text %mlantiq {*
  \begin{matharray}{rcl}
  @{ML_antiquotation_def "assert"} & : & @{text ML_antiquotation} \\
  \end{matharray}

  \begin{description}

  \item @{text "@{assert}"} inlines a function @{text "bool \<Rightarrow> unit"}
  that raises @{ML Fail} if the argument is @{ML false}.  Due to
  inlining the source position of failed assertions is included in the
  error output.

  \end{description}
*}


section {* Basic data types *}

text {* The basis library proposal of SML97 need to be treated with
  caution.  Many of its operations simply do not fit with important
  Isabelle/ML conventions (like ``canonical argument order'', see
  \secref{sec:canonical-argument-order}), others can cause serious
  problems with the parallel evaluation model of Isabelle/ML (such as
  @{ML TextIO.print} or @{ML OS.Process.system}).

  Subsequently we give a brief overview of important operations on
  basic ML data types.
*}


subsection {* Characters *}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type char} \\
  \end{mldecls}

  \begin{description}

  \item Type @{ML_type char} is \emph{not} used.  The smallest textual
  unit in Isabelle is represented a ``symbol'' (see
  \secref{sec:symbols}).

  \end{description}
*}


subsection {* Integers *}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type int} \\
  \end{mldecls}

  \begin{description}

  \item Type @{ML_type int} represents regular mathematical integers,
  which are \emph{unbounded}.  Overflow never happens in
  practice.\footnote{The size limit for integer bit patterns in memory
  is 64\,MB for 32-bit Poly/ML, and much higher for 64-bit systems.}
  This works uniformly for all supported ML platforms (Poly/ML and
  SML/NJ).

  Literal integers in ML text (e.g.\ @{ML
  123456789012345678901234567890}) are forced to be of this one true
  integer type --- overloading of SML97 is disabled.

  Structure @{ML_struct IntInf} of SML97 is obsolete, always use
  @{ML_struct Int}.  Structure @{ML_struct Integer} in @{"file"
  "~~/src/Pure/General/integer.ML"} provides some additional
  operations.

  \end{description}
*}


subsection {* Options *}

text %mlref {*
  \begin{mldecls}
  @{index_ML Option.map: "('a -> 'b) -> 'a option -> 'b option"} \\
  @{index_ML is_some: "'a option -> bool"} \\
  @{index_ML is_none: "'a option -> bool"} \\
  @{index_ML the: "'a option -> 'a"} \\
  @{index_ML these: "'a list option -> 'a list"} \\
  @{index_ML the_list: "'a option -> 'a list"} \\
  @{index_ML the_default: "'a -> 'a option -> 'a"} \\
  \end{mldecls}
*}

text {* Apart from @{ML Option.map} most operations defined in
  structure @{ML_struct Option} are alien to Isabelle/ML.  The
  operations shown above are defined in @{"file"
  "~~/src/Pure/General/basics.ML"}, among others.  *}


subsection {* Lists *}

text {* Lists are ubiquitous in ML as simple and light-weight
  ``collections'' for many everyday programming tasks.  Isabelle/ML
  provides some important refinements over the predefined operations
  in SML97. *}

text %mlref {*
  \begin{mldecls}
  @{index_ML cons: "'a -> 'a list -> 'a list"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML cons}~@{text "x xs"} evaluates to @{text "x :: xs"}.

  Tupled infix operators are a historical accident in Standard ML.
  The curried @{ML cons} amends this, but it should be only used when
  partial application is required.

  \end{description}
*}


subsubsection {* Canonical iteration *}

text {* A function @{text "f: \<alpha> \<rightarrow> \<beta> \<rightarrow> \<beta>"} can be understood as update
  on a configuration of type @{text "\<beta>"} that is parametrized by
  arguments of type @{text "\<alpha>"}.  Given @{text "a: \<alpha>"} the partial
  application @{text "(f a): \<beta> \<rightarrow> \<beta>"} operates homogeneously on @{text
  "\<beta>"}.  This can be iterated naturally over a list of parameters
  @{text "[a\<^sub>1, \<dots>, a\<^sub>n]"} as @{text "f a\<^sub>1; \<dots>; f a\<^bsub>n\<^esub>\<^bsub>\<^esub>"} (where the
  semicolon represents left-to-right composition).  The latter
  expression is again a function @{text "\<beta> \<rightarrow> \<beta>"}.  It can be applied
  to an initial configuration @{text "b: \<beta>"} to start the iteration
  over the given list of arguments: each @{text "a"} in @{text "a\<^sub>1, \<dots>,
  a\<^sub>n"} is applied consecutively by updating a cumulative
  configuration.

  The @{text fold} combinator in Isabelle/ML lifts a function @{text
  "f"} as above to its iterated version over a list of arguments.
  Lifting can be repeated, e.g.\ @{text "(fold \<circ> fold) f"} iterates
  over a list of lists as expected.

  The variant @{text "fold_rev"} works inside-out over the list of
  arguments, such that @{text "fold_rev f \<equiv> fold f \<circ> rev"} holds.

  The @{text "fold_map"} combinator essentially performs @{text
  "fold"} and @{text "map"} simultaneously: each application of @{text
  "f"} produces an updated configuration together with a side-result;
  the iteration collects all such side-results as a separate list.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML fold: "('a -> 'b -> 'b) -> 'a list -> 'b -> 'b"} \\
  @{index_ML fold_rev: "('a -> 'b -> 'b) -> 'a list -> 'b -> 'b"} \\
  @{index_ML fold_map: "('a -> 'b -> 'c * 'b) -> 'a list -> 'b -> 'c list * 'b"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML fold}~@{text f} lifts the parametrized update function
  @{text "f"} to a list of parameters.

  \item @{ML fold_rev}~@{text "f"} is similar to @{ML fold}~@{text
  "f"}, but works inside-out.

  \item @{ML fold_map}~@{text "f"} lifts the parametrized update
  function @{text "f"} (with side-result) to a list of parameters and
  cumulative side-results.

  \end{description}

  \begin{warn}
  The literature on functional programming provides a multitude of
  combinators called @{text "foldl"}, @{text "foldr"} etc.  SML97
  provides its own variations as @{ML List.foldl} and @{ML
  List.foldr}, while the classic Isabelle library still has the
  slightly more convenient historic @{ML Library.foldl} and @{ML
  Library.foldr}.  To avoid further confusion, all of this should be
  ignored, and @{ML fold} (or @{ML fold_rev}) used exclusively.
  \end{warn}
*}

text %mlex {* Using canonical @{ML fold} together with canonical @{ML
  cons}, or similar standard operations, alternates the orientation of
  data.  The is quite natural and should not altered forcible by
  inserting extra applications @{ML rev}.  The alternative @{ML
  fold_rev} can be used in the relatively few situations, where
  alternation should be prevented.
*}

ML {*
  val items = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

  val list1 = fold cons items [];
  @{assert} (list1 = rev items);

  val list2 = fold_rev cons items [];
  @{assert} (list2 = items);
*}


subsection {* Unsynchronized references *}

text %mlref {*
  \begin{mldecls}
  @{index_ML Unsynchronized.ref: "'a -> 'a Unsynchronized.ref"} \\
  @{index_ML "!": "'a Unsynchronized.ref -> 'a"} \\
  @{index_ML "op :=": "'a Unsynchronized.ref * 'a -> unit"} \\
  \end{mldecls}
*}

text {* Due to ubiquitous parallelism in Isabelle/ML (see also
  \secref{sec:multi-threading}), the mutable reference cells of
  Standard ML are notorious for causing problems.  In a highly
  parallel system, both correctness \emph{and} performance are easily
  degraded when using mutable data.

  The unwieldy name of @{ML Unsynchronized.ref} for the constructor
  for references in Isabelle/ML emphasizes the inconveniences caused by
  mutability.  Existing operations @{ML "!"}  and @{ML "op :="} are
  unchanged, but should be used with special precautions, say in a
  strictly local situation that is guaranteed to be restricted to
  sequential evaluation -- now and in the future.  *}

end