doc-src/IsarImplementation/Thy/prelim.thy

tuned;

(* $Id$ *)

theory prelim imports base begin

chapter {* Preliminaries *}

section {* Contexts \label{sec:context} *}

text {*
  A logical context represents the background that is required for
  formulating statements and composing proofs.  It acts as a medium to
  produce formal content, depending on earlier material (declarations,
  results etc.).

  For example, derivations within the Isabelle/Pure logic can be
  described as a judgment @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}, which means that a
  proposition @{text "\<phi>"} is derivable from hypotheses @{text "\<Gamma>"}
  within the theory @{text "\<Theta>"}.  There are logical reasons for
  keeping @{text "\<Theta>"} and @{text "\<Gamma>"} separate: theories can be
  liberal about supporting type constructors and schematic
  polymorphism of constants and axioms, while the inner calculus of
  @{text "\<Gamma> \<turnstile> \<phi>"} is strictly limited to Simple Type Theory (with
  fixed type variables in the assumptions).

  \medskip Contexts and derivations are linked by the following key
  principles:

  \begin{itemize}

  \item Transfer: monotonicity of derivations admits results to be
  transferred into a \emph{larger} context, i.e.\ @{text "\<Gamma> \<turnstile>\<^sub>\<Theta>
  \<phi>"} implies @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta>\<^sub>' \<phi>"} for contexts @{text "\<Theta>'
  \<supseteq> \<Theta>"} and @{text "\<Gamma>' \<supseteq> \<Gamma>"}.

  \item Export: discharge of hypotheses admits results to be exported
  into a \emph{smaller} context, i.e.\ @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta> \<phi>"}
  implies @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<Delta> \<Longrightarrow> \<phi>"} where @{text "\<Gamma>' \<supseteq> \<Gamma>"} and
  @{text "\<Delta> = \<Gamma>' - \<Gamma>"}.  Note that @{text "\<Theta>"} remains unchanged here,
  only the @{text "\<Gamma>"} part is affected.

  \end{itemize}

  \medskip By modeling the main characteristics of the primitive
  @{text "\<Theta>"} and @{text "\<Gamma>"} above, and abstracting over any
  particular logical content, we arrive at the fundamental notions of
  \emph{theory context} and \emph{proof context} in Isabelle/Isar.
  These implement a certain policy to manage arbitrary \emph{context
  data}.  There is a strongly-typed mechanism to declare new kinds of
  data at compile time.

  The internal bootstrap process of Isabelle/Pure eventually reaches a
  stage where certain data slots provide the logical content of @{text
  "\<Theta>"} and @{text "\<Gamma>"} sketched above, but this does not stop there!
  Various additional data slots support all kinds of mechanisms that
  are not necessarily part of the core logic.

  For example, there would be data for canonical introduction and
  elimination rules for arbitrary operators (depending on the
  object-logic and application), which enables users to perform
  standard proof steps implicitly (cf.\ the @{text "rule"} method
  \cite{isabelle-isar-ref}).

  \medskip Thus Isabelle/Isar is able to bring forth more and more
  concepts successively.  In particular, an object-logic like
  Isabelle/HOL continues the Isabelle/Pure setup by adding specific
  components for automated reasoning (classical reasoner, tableau
  prover, structured induction etc.) and derived specification
  mechanisms (inductive predicates, recursive functions etc.).  All of
  this is ultimately based on the generic data management by theory
  and proof contexts introduced here.
*}


subsection {* Theory context \label{sec:context-theory} *}

text {*
  \glossary{Theory}{FIXME}

  A \emph{theory} is a data container with explicit named and unique
  identifier.  Theories are related by a (nominal) sub-theory
  relation, which corresponds to the dependency graph of the original
  construction; each theory is derived from a certain sub-graph of
  ancestor theories.

  The @{text "merge"} operation produces the least upper bound of two
  theories, which actually degenerates into absorption of one theory
  into the other (due to the nominal sub-theory relation).

  The @{text "begin"} operation starts a new theory by importing
  several parent theories and entering a special @{text "draft"} mode,
  which is sustained until the final @{text "end"} operation.  A draft
  theory acts like a linear type, where updates invalidate earlier
  versions.  An invalidated draft is called ``stale''.

  The @{text "checkpoint"} operation produces an intermediate stepping
  stone that will survive the next update: both the original and the
  changed theory remain valid and are related by the sub-theory
  relation.  Checkpointing essentially recovers purely functional
  theory values, at the expense of some extra internal bookkeeping.

  The @{text "copy"} operation produces an auxiliary version that has
  the same data content, but is unrelated to the original: updates of
  the copy do not affect the original, neither does the sub-theory
  relation hold.

  \medskip The example in \figref{fig:ex-theory} below shows a theory
  graph derived from @{text "Pure"}, with theory @{text "Length"}
  importing @{text "Nat"} and @{text "List"}.  The body of @{text
  "Length"} consists of a sequence of updates, working mostly on
  drafts.  Intermediate checkpoints may occur as well, due to the
  history mechanism provided by the Isar top-level, cf.\
  \secref{sec:isar-toplevel}.

  \begin{figure}[htb]
  \begin{center}
  \begin{tabular}{rcccl}
        &            & @{text "Pure"} \\
        &            & @{text "\<down>"} \\
        &            & @{text "FOL"} \\
        & $\swarrow$ &              & $\searrow$ & \\
  $Nat$ &            &              &            & @{text "List"} \\
        & $\searrow$ &              & $\swarrow$ \\
        &            & @{text "Length"} \\
        &            & \multicolumn{3}{l}{~~$\isarkeyword{imports}$} \\
        &            & \multicolumn{3}{l}{~~$\isarkeyword{begin}$} \\
        &            & $\vdots$~~ \\
        &            & @{text "\<bullet>"}~~ \\
        &            & $\vdots$~~ \\
        &            & @{text "\<bullet>"}~~ \\
        &            & $\vdots$~~ \\
        &            & \multicolumn{3}{l}{~~$\isarkeyword{end}$} \\
  \end{tabular}
  \caption{A theory definition depending on ancestors}\label{fig:ex-theory}
  \end{center}
  \end{figure}

  \medskip There is a separate notion of \emph{theory reference} for
  maintaining a live link to an evolving theory context: updates on
  drafts are propagated automatically.  The dynamic stops after an
  explicit @{text "end"} only.

  Derived entities may store a theory reference in order to indicate
  the context they belong to.  This implicitly assumes monotonic
  reasoning, because the referenced context may become larger without
  further notice.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type theory} \\
  @{index_ML Theory.subthy: "theory * theory -> bool"} \\
  @{index_ML Theory.merge: "theory * theory -> theory"} \\
  @{index_ML Theory.checkpoint: "theory -> theory"} \\
  @{index_ML Theory.copy: "theory -> theory"} \\[1ex]
  @{index_ML_type theory_ref} \\
  @{index_ML Theory.self_ref: "theory -> theory_ref"} \\
  @{index_ML Theory.deref: "theory_ref -> theory"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML_type theory} represents theory contexts.  This is
  essentially a linear type!  Most operations destroy the original
  version, which then becomes ``stale''.

  \item @{ML "Theory.subthy"}~@{text "(thy\<^sub>1, thy\<^sub>2)"}
  compares theories according to the inherent graph structure of the
  construction.  This sub-theory relation is a nominal approximation
  of inclusion (@{text "\<subseteq>"}) of the corresponding content.

  \item @{ML "Theory.merge"}~@{text "(thy\<^sub>1, thy\<^sub>2)"}
  absorbs one theory into the other.  This fails for unrelated
  theories!

  \item @{ML "Theory.checkpoint"}~@{text "thy"} produces a safe
  stepping stone in the linear development of @{text "thy"}.  The next
  update will result in two related, valid theories.

  \item @{ML "Theory.copy"}~@{text "thy"} produces a variant of @{text
  "thy"} that holds a copy of the same data.  The result is not
  related to the original; the original is unchanched.

  \item @{ML_type theory_ref} represents a sliding reference to an
  always valid theory; updates on the original are propagated
  automatically.

  \item @{ML "Theory.self_ref"}~@{text "thy"} and @{ML
  "Theory.deref"}~@{text "thy_ref"} convert between @{ML_type
  "theory"} and @{ML_type "theory_ref"}.  As the referenced theory
  evolves monotonically over time, later invocations of @{ML
  "Theory.deref"} may refer to a larger context.

  \end{description}
*}


subsection {* Proof context \label{sec:context-proof} *}

text {*
  \glossary{Proof context}{The static context of a structured proof,
  acts like a local ``theory'' of the current portion of Isar proof
  text, generalizes the idea of local hypotheses @{text "\<Gamma>"} in
  judgments @{text "\<Gamma> \<turnstile> \<phi>"} of natural deduction calculi.  There is a
  generic notion of introducing and discharging hypotheses.
  Arbritrary auxiliary context data may be adjoined.}

  A proof context is a container for pure data with a back-reference
  to the theory it belongs to.  The @{text "init"} operation creates a
  proof context from a given theory.  Modifications to draft theories
  are propagated to the proof context as usual, but there is also an
  explicit @{text "transfer"} operation to force resynchronization
  with more substantial updates to the underlying theory.  The actual
  context data does not require any special bookkeeping, thanks to the
  lack of destructive features.

  Entities derived in a proof context need to record inherent logical
  requirements explicitly, since there is no separate context
  identification as for theories.  For example, hypotheses used in
  primitive derivations (cf.\ \secref{sec:thms}) are recorded
  separately within the sequent @{text "\<Gamma> \<turnstile> \<phi>"}, just to make double
  sure.  Results could still leak into an alien proof context do to
  programming errors, but Isabelle/Isar includes some extra validity
  checks in critical positions, notably at the end of sub-proof.

  Proof contexts may be manipulated arbitrarily, although the common
  discipline is to follow block structure as a mental model: a given
  context is extended consecutively, and results are exported back
  into the original context.  Note that the Isar proof states model
  block-structured reasoning explicitly, using a stack of proof
  contexts internally, cf.\ \secref{sec:isar-proof-state}.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type Proof.context} \\
  @{index_ML ProofContext.init: "theory -> Proof.context"} \\
  @{index_ML ProofContext.theory_of: "Proof.context -> theory"} \\
  @{index_ML ProofContext.transfer: "theory -> Proof.context -> Proof.context"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML_type Proof.context} represents proof contexts.  Elements
  of this type are essentially pure values, with a sliding reference
  to the background theory.

  \item @{ML ProofContext.init}~@{text "thy"} produces a proof context
  derived from @{text "thy"}, initializing all data.

  \item @{ML ProofContext.theory_of}~@{text "ctxt"} selects the
  background theory from @{text "ctxt"}, dereferencing its internal
  @{ML_type theory_ref}.

  \item @{ML ProofContext.transfer}~@{text "thy ctxt"} promotes the
  background theory of @{text "ctxt"} to the super theory @{text
  "thy"}.

  \end{description}
*}


subsection {* Generic contexts \label{sec:generic-context} *}

text {*
  A generic context is the disjoint sum of either a theory or proof
  context.  Occasionally, this enables uniform treatment of generic
  context data, typically extra-logical information.  Operations on
  generic contexts include the usual injections, partial selections,
  and combinators for lifting operations on either component of the
  disjoint sum.

  Moreover, there are total operations @{text "theory_of"} and @{text
  "proof_of"} to convert a generic context into either kind: a theory
  can always be selected from the sum, while a proof context might
  have to be constructed by an ad-hoc @{text "init"} operation.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type Context.generic} \\
  @{index_ML Context.theory_of: "Context.generic -> theory"} \\
  @{index_ML Context.proof_of: "Context.generic -> Proof.context"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML_type Context.generic} is the direct sum of @{ML_type
  "theory"} and @{ML_type "Proof.context"}, with the datatype
  constructors @{ML "Context.Theory"} and @{ML "Context.Proof"}.

  \item @{ML Context.theory_of}~@{text "context"} always produces a
  theory from the generic @{text "context"}, using @{ML
  "ProofContext.theory_of"} as required.

  \item @{ML Context.proof_of}~@{text "context"} always produces a
  proof context from the generic @{text "context"}, using @{ML
  "ProofContext.init"} as required (note that this re-initializes the
  context data with each invocation).

  \end{description}
*}


subsection {* Context data \label{sec:context-data} *}

text {*
  The main purpose of theory and proof contexts is to manage arbitrary
  data.  New data types can be declared incrementally at compile time.
  There are separate declaration mechanisms for any of the three kinds
  of contexts: theory, proof, generic.

  \paragraph{Theory data} may refer to destructive entities, which are
  maintained in direct correspondence to the linear evolution of
  theory values, including explicit copies.\footnote{Most existing
  instances of destructive theory data are merely historical relics
  (e.g.\ the destructive theorem storage, and destructive hints for
  the Simplifier and Classical rules).}  A theory data declaration
  needs to implement the following specification (depending on type
  @{text "T"}):

  \medskip
  \begin{tabular}{ll}
  @{text "name: string"} \\
  @{text "empty: T"} & initial value \\
  @{text "copy: T \<rightarrow> T"} & refresh impure data \\
  @{text "extend: T \<rightarrow> T"} & re-initialize on import \\
  @{text "merge: T \<times> T \<rightarrow> T"} & join on import \\
  @{text "print: T \<rightarrow> unit"} & diagnostic output \\
  \end{tabular}
  \medskip

  \noindent The @{text "name"} acts as a comment for diagnostic
  messages; @{text "copy"} is just the identity for pure data; @{text
  "extend"} is acts like a unitary version of @{text "merge"}, both
  should also include the functionality of @{text "copy"} for impure
  data.

  \paragraph{Proof context data} is purely functional.  A declaration
  needs to implement the following specification:

  \medskip
  \begin{tabular}{ll}
  @{text "name: string"} \\
  @{text "init: theory \<rightarrow> T"} & produce initial value \\
  @{text "print: T \<rightarrow> unit"} & diagnostic output \\
  \end{tabular}
  \medskip

  \noindent The @{text "init"} operation is supposed to produce a pure
  value from the given background theory.  The remainder is analogous
  to theory data.

  \paragraph{Generic data} provides a hybrid interface for both theory
  and proof data.  The declaration is essentially the same as for
  (pure) theory data, without @{text "copy"}, though.  The @{text
  "init"} operation for proof contexts merely selects the current data
  value from the background theory.

  \bigskip In any case, a data declaration of type @{text "T"} results
  in the following interface:

  \medskip
  \begin{tabular}{ll}
  @{text "init: theory \<rightarrow> theory"} \\
  @{text "get: context \<rightarrow> T"} \\
  @{text "put: T \<rightarrow> context \<rightarrow> context"} \\
  @{text "map: (T \<rightarrow> T) \<rightarrow> context \<rightarrow> context"} \\
  @{text "print: context \<rightarrow> unit"}
  \end{tabular}
  \medskip

  \noindent Here @{text "init"} needs to be applied to the current
  theory context once, in order to register the initial setup.  The
  other operations provide access for the particular kind of context
  (theory, proof, or generic context).  Note that this is a safe
  interface: there is no other way to access the corresponding data
  slot of a context.  By keeping these operations private, a component
  may maintain abstract values authentically, without other components
  interfering.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML_functor TheoryDataFun} \\
  @{index_ML_functor ProofDataFun} \\
  @{index_ML_functor GenericDataFun} \\
  \end{mldecls}

  \begin{description}

  \item @{ML_functor TheoryDataFun}@{text "(spec)"} declares data for
  type @{ML_type theory} according to the specification provided as
  argument structure.  The resulting structure provides data init and
  access operations as described above.

  \item @{ML_functor ProofDataFun}@{text "(spec)"} is analogous to
  @{ML_functor TheoryDataFun} for type @{ML_type Proof.context}.

  \item @{ML_functor GenericDataFun}@{text "(spec)"} is analogous to
  @{ML_functor TheoryDataFun} for type @{ML_type Context.generic}.

  \end{description}
*}


section {* Names *}

text {*
  In principle, a name is just a string, but there are various
  convention for encoding additional structure.

  For example, the string ``@{text "Foo.bar.baz"}'' is considered as a
  qualified name.  The most basic constituents of names may have their
  own structure, e.g.\ the string ``\verb,\,\verb,<alpha>,'' is
  considered as a single symbol (printed as ``@{text "\<alpha>"}'').
*}


subsection {* Strings of symbols *}

text {*
  \glossary{Symbol}{The smallest unit of text in Isabelle, subsumes
  plain ASCII characters as well as an infinite collection of named
  symbols (for greek, math etc.).}

  A \emph{symbol} constitutes the smallest textual unit in Isabelle
  --- raw characters are normally not encountered.  Isabelle strings
  consist of a sequence of symbols, represented as a packed string or
  a list of symbols.  Each symbol is in itself a small string, which
  is of one of the following forms:

  \begin{enumerate}

  \item singleton ASCII character ``@{text "c"}'' (character code
  0--127), for example ``\verb,a,'',

  \item regular symbol ``\verb,\,\verb,<,@{text "ident"}\verb,>,'',
  for example ``\verb,\,\verb,<alpha>,'',

  \item control symbol ``\verb,\,\verb,<^,@{text "ident"}\verb,>,'',
  for example ``\verb,\,\verb,<^bold>,'',

  \item raw symbol ``\verb,\,\verb,<^raw:,@{text text}\verb,>,'' where
  @{text text} is constists of printable characters excluding
  ``\verb,.,'' and ``\verb,>,'', for example
  ``\verb,\,\verb,<^raw:$\sum_{i = 1}^n$>,'',

  \item numbered raw control symbol ``\verb,\,\verb,<^raw,@{text
  n}\verb,>, where @{text n} consists of digits, for example
  ``\verb,\,\verb,<^raw42>,''.

  \end{enumerate}

  \noindent The @{text "ident"} syntax for symbol names is @{text
  "letter (letter | digit)\<^sup>*"}, where @{text "letter =
  A..Za..z"} and @{text "digit = 0..9"}.  There are infinitely many
  regular symbols and control symbols, but a fixed collection of
  standard symbols is treated specifically.  For example,
  ``\verb,\,\verb,<alpha>,'' is classified as a (non-ASCII) letter,
  which means it may occur within regular Isabelle identifier syntax.

  Note that the character set underlying Isabelle symbols is plain
  7-bit ASCII.  Since 8-bit characters are passed through
  transparently, Isabelle may process Unicode/UCS data (in UTF-8
  encoding) as well.  Unicode provides its own collection of
  mathematical symbols, but there is no built-in link to the ones of
  Isabelle.

  \medskip Output of Isabelle symbols depends on the print mode
  (\secref{FIXME}).  For example, the standard {\LaTeX} setup of the
  Isabelle document preparation system would present
  ``\verb,\,\verb,<alpha>,'' as @{text "\<alpha>"}, and
  ``\verb,\,\verb,<^bold>,\verb,\,\verb,<alpha>,'' as @{text
  "\<^bold>\<alpha>"}.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML_type "Symbol.symbol"} \\
  @{index_ML Symbol.explode: "string -> Symbol.symbol list"} \\
  @{index_ML Symbol.is_letter: "Symbol.symbol -> bool"} \\
  @{index_ML Symbol.is_digit: "Symbol.symbol -> bool"} \\
  @{index_ML Symbol.is_quasi: "Symbol.symbol -> bool"} \\
  @{index_ML Symbol.is_blank: "Symbol.symbol -> bool"} \\[1ex]
  @{index_ML_type "Symbol.sym"} \\
  @{index_ML Symbol.decode: "Symbol.symbol -> Symbol.sym"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML_type "Symbol.symbol"} represents Isabelle symbols.  This
  type is an alias for @{ML_type "string"}, but emphasizes the
  specific format encountered here.

  \item @{ML "Symbol.explode"}~@{text "str"} produces a symbol list
  from the packed form that.  This function supercedes @{ML
  "String.explode"} for virtually all purposes of manipulating text in
  Isabelle!

  \item @{ML "Symbol.is_letter"}, @{ML "Symbol.is_digit"}, @{ML
  "Symbol.is_quasi"}, @{ML "Symbol.is_blank"} classify standard
  symbols according to fixed syntactic conventions of Isabelle, cf.\
  \cite{isabelle-isar-ref}.

  \item @{ML_type "Symbol.sym"} is a concrete datatype that represents
  the different kinds of symbols explicitly with constructors @{ML
  "Symbol.Char"}, @{ML "Symbol.Sym"}, @{ML "Symbol.Ctrl"}, or @{ML
  "Symbol.Raw"}.

  \item @{ML "Symbol.decode"} converts the string representation of a
  symbol into the datatype version.

  \end{description}
*}


subsection {* Basic names \label{sec:basic-names} *}

text {*
  A \emph{basic name} essentially consists of a single Isabelle
  identifier.  There are conventions to mark separate classes of basic
  names, by attaching a suffix of underscores (@{text "_"}): one
  underscore means \emph{internal name}, two underscores means
  \emph{Skolem name}, three underscores means \emph{internal Skolem
  name}.

  For example, the basic name @{text "foo"} has the internal version
  @{text "foo_"}, with Skolem versions @{text "foo__"} and @{text
  "foo___"}, respectively.

  Such special versions are required for bookkeeping of names that are
  apart from anything that may appear in the text given by the user.
  In particular, system generated variables in high-level Isar proof
  contexts are usually marked as internal, which prevents mysterious
  name references such as @{text "xaa"} in the text.

  \medskip Basic manipulations of binding scopes requires names to be
  modified.  A \emph{name context} contains a collection of already
  used names, which is maintained by the @{text "declare"} operation.

  The @{text "invents"} operation derives a number of fresh names
  derived from a given starting point.  For example, three names
  derived from @{text "a"} are @{text "a"}, @{text "b"}, @{text "c"},
  provided there are no clashes with already used names.

  The @{text "variants"} operation produces fresh names by
  incrementing given names as to base-26 numbers (with digits @{text
  "a..z"}).  For example, name @{text "foo"} results in variants
  @{text "fooa"}, @{text "foob"}, @{text "fooc"}, \dots, @{text
  "fooaa"}, @{text "fooab"}, \dots; each renaming step picks the next
  unused variant from this list.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML Name.internal: "string -> string"} \\
  @{index_ML Name.skolem: "string -> string"} \\[1ex]
  @{index_ML_type Name.context} \\
  @{index_ML Name.context: Name.context} \\
  @{index_ML Name.declare: "string -> Name.context -> Name.context"} \\
  @{index_ML Name.invents: "Name.context -> string -> int -> string list"} \\
  @{index_ML Name.variants: "string list -> Name.context -> string list * Name.context"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML Name.internal}~@{text "name"} produces an internal name
  by adding one underscore.

  \item @{ML Name.skolem}~@{text "name"} produces a Skolem name by
  adding two underscores.

  \item @{ML_type Name.context} represents the context of already used
  names; the initial value is @{ML "Name.context"}.

  \item @{ML Name.declare}~@{text "name"} declares @{text "name"} as
  being used.

  \item @{ML Name.invents}~@{text "context base n"} produces @{text
  "n"} fresh names derived from @{text "base"}.

  \end{description}
*}


subsection {* Indexed names *}

text {*
  An \emph{indexed name} (or @{text "indexname"}) is a pair of a basic
  name with a natural number.  This representation allows efficient
  renaming by incrementing the second component only.  To rename two
  collections of indexnames apart from each other, first determine the
  maximum index @{text "maxidx"} of the first collection, then
  increment all indexes of the second collection by @{text "maxidx +
  1"}.  Note that the maximum index of an empty collection is @{text
  "-1"}.

  Isabelle syntax observes the following rules for representing an
  indexname @{text "(x, i)"} as a packed string:

  \begin{itemize}

  \item @{text "?x"} if @{text "x"} does not end with a digit and @{text "i = 0"},

  \item @{text "?xi"} if @{text "x"} does not end with a digit,

  \item @{text "?x.i"} else.

  \end{itemize}

  Occasionally, basic names and indexed names are injected into the
  same pair type: the (improper) indexname @{text "(x, -1)"} is used
  to encode basic names.

  \medskip Indexnames may acquire arbitrary large index numbers over
  time.  Results are usually normalized towards @{text "0"} at certain
  checkpoints, such that the very end of a proof.  This works by
  producing variants of the corresponding basic names
  (\secref{sec:basic-names}).  For example, the collection @{text
  "?x.1, ?x.7, ?x.42"} then becomes @{text "?x, ?xa, ?xb"}.
*}

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

  \begin{description}

  \item @{ML_type indexname} represents indexed names.  This is an
  abbreviation for @{ML_type "string * int"}.  The second component is
  usually non-negative, except for situations where @{text "(x, -1)"}
  is used to embed plain names.

  \end{description}
*}


subsection {* Qualified names and name spaces *}

text {*
  A \emph{qualified name} consists of a non-empty sequence of basic
  name components.  The packed representation a dot as separator, for
  example in ``@{text "A.b.c"}''.  The last component is called
  \emph{base} name, the remaining prefix \emph{qualifier} (which may
  be empty).  The basic idea of qualified names is to encode a
  hierarchically structured name spaces by recording the access path
  as part of the name.  For example, @{text "A.b.c"} may be understood
  as a local entity @{text "c"} within a local structure @{text "b"}
  of the enclosing structure @{text "A"}.  Typically, name space
  hierarchies consist of 1--3 levels, but this need not be always so.

  The empty name is commonly used as an indication of unnamed
  entities, if this makes any sense.  The operations on qualified
  names are smart enough to pass through such improper names
  unchanged.

  \medskip A @{text "naming"} policy tells how to turn a name
  specification into a fully qualified internal name (by the @{text
  "full"} operation), and how to fully qualified names may be accessed
  externally.  For example, the default naming prefixes an implicit
  path from the context: @{text "x"} is becomes @{text "path.x"}
  internally; the standard accesses include @{text "x"}, @{text
  "path.x"}, and further partial @{text "path"} specifications.
  Normally, the naming is implicit in the theory or proof context.
  There are separate versions of the corresponding operations for
  these context types.

  \medskip A @{text "name space"} manages a collection of fully
  internalized names, together with a mapping between external names
  and internal names (in both directions).  The corresponding @{text
  "intern"} and @{text "extern"} operations are mostly used for
  parsing and printing only!  The @{text "declare"} operation augments
  a name space according to a given naming policy.

  By general convention, there are separate name spaces for each kind
  of formal entity, such as logical constant, type, type class,
  theorem etc.  It is usually clear from the occurrence in concrete
  syntax (or from the scope) which kind of entity a name refers to.
  For example, the very same name @{text "c"} may be used uniformly
  for a constant, type, type class, which are from separate syntactic
  categories.

  There are common schemes to name theorems systematically, according
  to the name of the main logical entity being involved, such as
  @{text "c.intro"} and @{text "c.elim"} for theorems related to
  constant @{text "c"}.  This technique of mapping names from one
  space into another requires some care in order to avoid conflicts.
  In particular, theorem names derived from type or class names are
  better suffixed in addition to the usual qualification, e.g.\ @{text
  "c_type.intro"} and @{text "c_class.intro"} for theorems related to
  type @{text "c"} and class @{text "c"}, respectively.
*}

text %mlref {*
  \begin{mldecls}
  @{index_ML NameSpace.base: "string -> string"} \\
  @{index_ML NameSpace.drop_base: "string -> string"} \\
  @{index_ML NameSpace.append: "string -> string -> string"} \\
  @{index_ML NameSpace.pack: "string list -> string"} \\
  @{index_ML NameSpace.unpack: "string -> string list"} \\[1ex]
  @{index_ML_type NameSpace.naming} \\
  @{index_ML NameSpace.default_naming: NameSpace.naming} \\
  @{index_ML NameSpace.add_path: "string -> NameSpace.naming -> NameSpace.naming"} \\
  @{index_ML NameSpace.full: "NameSpace.naming -> string -> string"} \\[1ex]
  @{index_ML_type NameSpace.T} \\
  @{index_ML NameSpace.empty: NameSpace.T} \\
  @{index_ML NameSpace.merge: "NameSpace.T * NameSpace.T -> NameSpace.T"} \\
  @{index_ML NameSpace.declare: "NameSpace.naming -> string -> NameSpace.T -> NameSpace.T"} \\
  @{index_ML NameSpace.intern: "NameSpace.T -> string -> string"} \\
  @{index_ML NameSpace.extern: "NameSpace.T -> string -> string"} \\
  \end{mldecls}

  \begin{description}

  \item @{ML NameSpace.base}~@{text "name"} returns the base name of a
  qualified name.

  \item @{ML NameSpace.drop_base}~@{text "name"} returns the qualifier
  of a qualified name.

  \item @{ML NameSpace.append}~@{text "name\<^isub>1 name\<^isub>2"}
  appends two qualified names.

  \item @{ML NameSpace.pack}~@{text "name"} and @{text
  "NameSpace.unpack"}~@{text "names"} convert between the packed
  string representation and explicit list form of qualified names.

  \item @{ML_type NameSpace.naming} represents the abstract concept of
  a naming policy.

  \item @{ML NameSpace.default_naming} is the default naming policy.
  In a theory context, this is usually augmented by a path prefix
  consisting of the theory name.

  \item @{ML NameSpace.add_path}~@{text "path naming"} augments the
  naming policy by extending its path.

  \item @{ML NameSpace.full}@{text "naming name"} turns a name
  specification (usually a basic name) into the fully qualified
  internal version, according to the given naming policy.

  \item @{ML_type NameSpace.T} represents name spaces.

  \item @{ML NameSpace.empty} and @{ML NameSpace.merge}~@{text
  "(space\<^isub>1, space\<^isub>2)"} provide basic operations for
  building name spaces in accordance to the usual theory data
  management (\secref{sec:context-data}).

  \item @{ML NameSpace.declare}~@{text "naming name space"} enters a
  fully qualified name into the name space, with partial accesses
  being derived from the given policy.

  \item @{ML NameSpace.intern}~@{text "space name"} internalizes a
  (partially qualified) external name.

  This operation is mostly for parsing.  Note that fully qualified
  names stemming from declarations are produced via @{ML
  "NameSpace.full"} (or derivatives for @{ML_type theory} or @{ML_type
  Proof.context}).

  \item @{ML NameSpace.extern}~@{text "space name"} externalizes a
  (fully qualified) internal name.

  This operation is mostly for printing.  Note unqualified names are
  produced via @{ML NameSpace.base}.

  \end{description}
*}

end