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-\begin{isabellebody}%
-\def\isabellecontext{Prelim}%
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-\ Prelim\isanewline
-\isakeyword{imports}\ Base\isanewline
-\isakeyword{begin}%
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-\isamarkupchapter{Preliminaries%
-}
-\isamarkuptrue%
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-\isamarkupsection{Contexts \label{sec:context}%
-}
-\isamarkuptrue%
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-\begin{isamarkuptext}%
-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 \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}, which means that a
- proposition \isa{{\isasymphi}} is derivable from hypotheses \isa{{\isasymGamma}}
- within the theory \isa{{\isasymTheta}}. There are logical reasons for
- keeping \isa{{\isasymTheta}} and \isa{{\isasymGamma}} separate: theories can be
- liberal about supporting type constructors and schematic
- polymorphism of constants and axioms, while the inner calculus of
- \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}} 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.\ \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}} implies \isa{{\isasymGamma}{\isacharprime}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\isactrlsub {\isacharprime}\ {\isasymphi}} for contexts \isa{{\isasymTheta}{\isacharprime}\ {\isasymsupseteq}\ {\isasymTheta}} and \isa{{\isasymGamma}{\isacharprime}\ {\isasymsupseteq}\ {\isasymGamma}}.
-
- \item Export: discharge of hypotheses admits results to be exported
- into a \emph{smaller} context, i.e.\ \isa{{\isasymGamma}{\isacharprime}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}
- implies \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymDelta}\ {\isasymLongrightarrow}\ {\isasymphi}} where \isa{{\isasymGamma}{\isacharprime}\ {\isasymsupseteq}\ {\isasymGamma}} and
- \isa{{\isasymDelta}\ {\isacharequal}\ {\isasymGamma}{\isacharprime}\ {\isacharminus}\ {\isasymGamma}}. Note that \isa{{\isasymTheta}} remains unchanged here,
- only the \isa{{\isasymGamma}} part is affected.
-
- \end{itemize}
-
- \medskip By modeling the main characteristics of the primitive
- \isa{{\isasymTheta}} and \isa{{\isasymGamma}} 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 \isa{{\isasymTheta}} and \isa{{\isasymGamma}} 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 \isa{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.%
-\end{isamarkuptext}%
-\isamarkuptrue%
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-\isamarkupsubsection{Theory context \label{sec:context-theory}%
-}
-\isamarkuptrue%
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-\begin{isamarkuptext}%
-A \emph{theory} is a data container with explicit name 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 \isa{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 \isa{begin} operation starts a new theory by importing
- several parent theories and entering a special \isa{draft} mode,
- which is sustained until the final \isa{end} operation. A draft
- theory acts like a linear type, where updates invalidate earlier
- versions. An invalidated draft is called ``stale''.
-
- The \isa{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 \isa{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 \isa{Pure}, with theory \isa{Length}
- importing \isa{Nat} and \isa{List}. The body of \isa{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}
- & & \isa{Pure} \\
- & & \isa{{\isasymdown}} \\
- & & \isa{FOL} \\
- & $\swarrow$ & & $\searrow$ & \\
- \isa{Nat} & & & & \isa{List} \\
- & $\searrow$ & & $\swarrow$ \\
- & & \isa{Length} \\
- & & \multicolumn{3}{l}{~~\hyperlink{keyword.imports}{\mbox{\isa{\isakeyword{imports}}}}} \\
- & & \multicolumn{3}{l}{~~\hyperlink{keyword.begin}{\mbox{\isa{\isakeyword{begin}}}}} \\
- & & $\vdots$~~ \\
- & & \isa{{\isasymbullet}}~~ \\
- & & $\vdots$~~ \\
- & & \isa{{\isasymbullet}}~~ \\
- & & $\vdots$~~ \\
- & & \multicolumn{3}{l}{~~\hyperlink{command.end}{\mbox{\isa{\isacommand{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. Dynamic updating stops after
- an explicit \isa{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.%
-\end{isamarkuptext}%
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-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexdef{}{ML type}{theory}\verb|type theory| \\
- \indexdef{}{ML}{Theory.subthy}\verb|Theory.subthy: theory * theory -> bool| \\
- \indexdef{}{ML}{Theory.merge}\verb|Theory.merge: theory * theory -> theory| \\
- \indexdef{}{ML}{Theory.checkpoint}\verb|Theory.checkpoint: theory -> theory| \\
- \indexdef{}{ML}{Theory.copy}\verb|Theory.copy: theory -> theory| \\
- \end{mldecls}
- \begin{mldecls}
- \indexdef{}{ML type}{theory\_ref}\verb|type theory_ref| \\
- \indexdef{}{ML}{Theory.deref}\verb|Theory.deref: theory_ref -> theory| \\
- \indexdef{}{ML}{Theory.check\_thy}\verb|Theory.check_thy: theory -> theory_ref| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|theory| represents theory contexts. This is
- essentially a linear type! Most operations destroy the original
- version, which then becomes ``stale''.
-
- \item \verb|Theory.subthy|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}}
- compares theories according to the inherent graph structure of the
- construction. This sub-theory relation is a nominal approximation
- of inclusion (\isa{{\isasymsubseteq}}) of the corresponding content.
-
- \item \verb|Theory.merge|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}}
- absorbs one theory into the other. This fails for unrelated
- theories!
-
- \item \verb|Theory.checkpoint|~\isa{thy} produces a safe
- stepping stone in the linear development of \isa{thy}. The next
- update will result in two related, valid theories.
-
- \item \verb|Theory.copy|~\isa{thy} produces a variant of \isa{thy} that holds a copy of the same data. The result is not
- related to the original; the original is unchanged.
-
- \item \verb|theory_ref| represents a sliding reference to an
- always valid theory; updates on the original are propagated
- automatically.
-
- \item \verb|Theory.deref|~\isa{thy{\isacharunderscore}ref} turns a \verb|theory_ref| into an \verb|theory| value. As the referenced
- theory evolves monotonically over time, later invocations of \verb|Theory.deref| may refer to a larger context.
-
- \item \verb|Theory.check_thy|~\isa{thy} produces a \verb|theory_ref| from a valid \verb|theory| value.
-
- \end{description}%
-\end{isamarkuptext}%
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-\isamarkupsubsection{Proof context \label{sec:context-proof}%
-}
-\isamarkuptrue%
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-\begin{isamarkuptext}%
-A proof context is a container for pure data with a back-reference
- to the theory it belongs to. The \isa{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 \isa{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 \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}}, just to make double
- sure. Results could still leak into an alien proof context due to
- programming errors, but Isabelle/Isar includes some extra validity
- checks in critical positions, notably at the end of a 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.%
-\end{isamarkuptext}%
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-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexdef{}{ML type}{Proof.context}\verb|type Proof.context| \\
- \indexdef{}{ML}{ProofContext.init}\verb|ProofContext.init: theory -> Proof.context| \\
- \indexdef{}{ML}{ProofContext.theory\_of}\verb|ProofContext.theory_of: Proof.context -> theory| \\
- \indexdef{}{ML}{ProofContext.transfer}\verb|ProofContext.transfer: theory -> Proof.context -> Proof.context| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Proof.context| represents proof contexts. Elements
- of this type are essentially pure values, with a sliding reference
- to the background theory.
-
- \item \verb|ProofContext.init|~\isa{thy} produces a proof context
- derived from \isa{thy}, initializing all data.
-
- \item \verb|ProofContext.theory_of|~\isa{ctxt} selects the
- background theory from \isa{ctxt}, dereferencing its internal
- \verb|theory_ref|.
-
- \item \verb|ProofContext.transfer|~\isa{thy\ ctxt} promotes the
- background theory of \isa{ctxt} to the super theory \isa{thy}.
-
- \end{description}%
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-\isamarkupsubsection{Generic contexts \label{sec:generic-context}%
-}
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-\begin{isamarkuptext}%
-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 \isa{theory{\isacharunderscore}of} and \isa{proof{\isacharunderscore}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 \isa{init} operation.%
-\end{isamarkuptext}%
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-\begin{mldecls}
- \indexdef{}{ML type}{Context.generic}\verb|type Context.generic| \\
- \indexdef{}{ML}{Context.theory\_of}\verb|Context.theory_of: Context.generic -> theory| \\
- \indexdef{}{ML}{Context.proof\_of}\verb|Context.proof_of: Context.generic -> Proof.context| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Context.generic| is the direct sum of \verb|theory| and \verb|Proof.context|, with the datatype
- constructors \verb|Context.Theory| and \verb|Context.Proof|.
-
- \item \verb|Context.theory_of|~\isa{context} always produces a
- theory from the generic \isa{context}, using \verb|ProofContext.theory_of| as required.
-
- \item \verb|Context.proof_of|~\isa{context} always produces a
- proof context from the generic \isa{context}, using \verb|ProofContext.init| as required (note that this re-initializes the
- context data with each invocation).
-
- \end{description}%
-\end{isamarkuptext}%
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-\isamarkupsubsection{Context data \label{sec:context-data}%
-}
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-\begin{isamarkuptext}%
-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 SML signature:
-
- \medskip
- \begin{tabular}{ll}
- \isa{{\isasymtype}\ T} & representing type \\
- \isa{{\isasymval}\ empty{\isacharcolon}\ T} & empty default value \\
- \isa{{\isasymval}\ copy{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & refresh impure data \\
- \isa{{\isasymval}\ extend{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & re-initialize on import \\
- \isa{{\isasymval}\ merge{\isacharcolon}\ T\ {\isasymtimes}\ T\ {\isasymrightarrow}\ T} & join on import \\
- \end{tabular}
- \medskip
-
- \noindent The \isa{empty} value acts as initial default for
- \emph{any} theory that does not declare actual data content; \isa{copy} maintains persistent integrity for impure data, it is just
- the identity for pure values; \isa{extend} is acts like a
- unitary version of \isa{merge}, both operations should also
- include the functionality of \isa{copy} for impure data.
-
- \paragraph{Proof context data} is purely functional. A declaration
- needs to implement the following SML signature:
-
- \medskip
- \begin{tabular}{ll}
- \isa{{\isasymtype}\ T} & representing type \\
- \isa{{\isasymval}\ init{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} & produce initial value \\
- \end{tabular}
- \medskip
-
- \noindent The \isa{init} operation is supposed to produce a pure
- value from the given background theory.
-
- \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 \isa{copy}. The \isa{init}
- operation for proof contexts merely selects the current data value
- from the background theory.
-
- \bigskip A data declaration of type \isa{T} results in the
- following interface:
-
- \medskip
- \begin{tabular}{ll}
- \isa{init{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} \\
- \isa{get{\isacharcolon}\ context\ {\isasymrightarrow}\ T} \\
- \isa{put{\isacharcolon}\ T\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\
- \isa{map{\isacharcolon}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\
- \end{tabular}
- \medskip
-
- \noindent Here \isa{init} is only applicable to impure theory
- data to install a fresh copy persistently (destructive update on
- uninitialized has no permanent effect). 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.%
-\end{isamarkuptext}%
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-\begin{isamarkuptext}%
-\begin{mldecls}
- \indexdef{}{ML functor}{TheoryDataFun}\verb|functor TheoryDataFun| \\
- \indexdef{}{ML functor}{ProofDataFun}\verb|functor ProofDataFun| \\
- \indexdef{}{ML functor}{GenericDataFun}\verb|functor GenericDataFun| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|TheoryDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} declares data for
- type \verb|theory| according to the specification provided as
- argument structure. The resulting structure provides data init and
- access operations as described above.
-
- \item \verb|ProofDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
- \verb|TheoryDataFun| for type \verb|Proof.context|.
-
- \item \verb|GenericDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
- \verb|TheoryDataFun| for type \verb|Context.generic|.
-
- \end{description}%
-\end{isamarkuptext}%
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-\isamarkupsection{Names \label{sec:names}%
-}
-\isamarkuptrue%
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-\begin{isamarkuptext}%
-In principle, a name is just a string, but there are various
- convention for encoding additional structure. For example, ``\isa{Foo{\isachardot}bar{\isachardot}baz}'' is considered as a qualified name consisting of
- three basic name components. The individual constituents of a name
- may have further substructure, e.g.\ the string
- ``\verb,\,\verb,<alpha>,'' encodes as a single symbol.%
-\end{isamarkuptext}%
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-\isamarkupsubsection{Strings of symbols%
-}
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-\begin{isamarkuptext}%
-A \emph{symbol} constitutes the smallest textual unit in Isabelle
- --- raw characters are normally not encountered at all. Isabelle
- strings consist of a sequence of symbols, represented as a packed
- string or a list of strings. Each symbol is in itself a small
- string, which has either one of the following forms:
-
- \begin{enumerate}
-
- \item a single ASCII character ``\isa{c}'', for example
- ``\verb,a,'',
-
- \item a regular symbol ``\verb,\,\verb,<,\isa{ident}\verb,>,'',
- for example ``\verb,\,\verb,<alpha>,'',
-
- \item a control symbol ``\verb,\,\verb,<^,\isa{ident}\verb,>,'',
- for example ``\verb,\,\verb,<^bold>,'',
-
- \item a raw symbol ``\verb,\,\verb,<^raw:,\isa{text}\verb,>,''
- where \isa{text} constists of printable characters excluding
- ``\verb,.,'' and ``\verb,>,'', for example
- ``\verb,\,\verb,<^raw:$\sum_{i = 1}^n$>,'',
-
- \item a numbered raw control symbol ``\verb,\,\verb,<^raw,\isa{n}\verb,>, where \isa{n} consists of digits, for example
- ``\verb,\,\verb,<^raw42>,''.
-
- \end{enumerate}
-
- \noindent The \isa{ident} syntax for symbol names is \isa{letter\ {\isacharparenleft}letter\ {\isacharbar}\ digit{\isacharparenright}\isactrlsup {\isacharasterisk}}, where \isa{letter\ {\isacharequal}\ A{\isachardot}{\isachardot}Za{\isachardot}{\isachardot}z} and \isa{digit\ {\isacharequal}\ {\isadigit{0}}{\isachardot}{\isachardot}{\isadigit{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 letter, which means it
- may occur within regular Isabelle identifiers.
-
- Since the character set underlying Isabelle symbols is 7-bit ASCII
- and 8-bit characters are passed through transparently, Isabelle may
- also process Unicode/UCS data in UTF-8 encoding. Unicode provides
- its own collection of mathematical symbols, but there is no built-in
- link to the standard collection of Isabelle.
-
- \medskip Output of Isabelle symbols depends on the print mode
- (\secref{print-mode}). For example, the standard {\LaTeX} setup of
- the Isabelle document preparation system would present
- ``\verb,\,\verb,<alpha>,'' as \isa{{\isasymalpha}}, and
- ``\verb,\,\verb,<^bold>,\verb,\,\verb,<alpha>,'' as \isa{\isactrlbold {\isasymalpha}}.%
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-\begin{mldecls}
- \indexdef{}{ML type}{Symbol.symbol}\verb|type Symbol.symbol| \\
- \indexdef{}{ML}{Symbol.explode}\verb|Symbol.explode: string -> Symbol.symbol list| \\
- \indexdef{}{ML}{Symbol.is\_letter}\verb|Symbol.is_letter: Symbol.symbol -> bool| \\
- \indexdef{}{ML}{Symbol.is\_digit}\verb|Symbol.is_digit: Symbol.symbol -> bool| \\
- \indexdef{}{ML}{Symbol.is\_quasi}\verb|Symbol.is_quasi: Symbol.symbol -> bool| \\
- \indexdef{}{ML}{Symbol.is\_blank}\verb|Symbol.is_blank: Symbol.symbol -> bool| \\
- \end{mldecls}
- \begin{mldecls}
- \indexdef{}{ML type}{Symbol.sym}\verb|type Symbol.sym| \\
- \indexdef{}{ML}{Symbol.decode}\verb|Symbol.decode: Symbol.symbol -> Symbol.sym| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Symbol.symbol| represents individual Isabelle
- symbols; this is an alias for \verb|string|.
-
- \item \verb|Symbol.explode|~\isa{str} produces a symbol list
- from the packed form. This function supercedes \verb|String.explode| for virtually all purposes of manipulating text in
- Isabelle!
-
- \item \verb|Symbol.is_letter|, \verb|Symbol.is_digit|, \verb|Symbol.is_quasi|, \verb|Symbol.is_blank| classify standard
- symbols according to fixed syntactic conventions of Isabelle, cf.\
- \cite{isabelle-isar-ref}.
-
- \item \verb|Symbol.sym| is a concrete datatype that represents
- the different kinds of symbols explicitly, with constructors \verb|Symbol.Char|, \verb|Symbol.Sym|, \verb|Symbol.Ctrl|, \verb|Symbol.Raw|.
-
- \item \verb|Symbol.decode| converts the string representation of a
- symbol into the datatype version.
-
- \end{description}%
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-\isamarkupsubsection{Basic names \label{sec:basic-names}%
-}
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-\begin{isamarkuptext}%
-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: one underscore means
- \emph{internal name}, two underscores means \emph{Skolem name},
- three underscores means \emph{internal Skolem name}.
-
- For example, the basic name \isa{foo} has the internal version
- \isa{foo{\isacharunderscore}}, with Skolem versions \isa{foo{\isacharunderscore}{\isacharunderscore}} and \isa{foo{\isacharunderscore}{\isacharunderscore}{\isacharunderscore}}, respectively.
-
- These special versions provide copies of the basic name space, apart
- from anything that normally appears in the user text. For example,
- system generated variables in Isar proof contexts are usually marked
- as internal, which prevents mysterious name references like \isa{xaa} to appear in the text.
-
- \medskip Manipulating binding scopes often requires on-the-fly
- renamings. A \emph{name context} contains a collection of already
- used names. The \isa{declare} operation adds names to the
- context.
-
- The \isa{invents} operation derives a number of fresh names from
- a given starting point. For example, the first three names derived
- from \isa{a} are \isa{a}, \isa{b}, \isa{c}.
-
- The \isa{variants} operation produces fresh names by
- incrementing tentative names as base-26 numbers (with digits \isa{a{\isachardot}{\isachardot}z}) until all clashes are resolved. For example, name \isa{foo} results in variants \isa{fooa}, \isa{foob}, \isa{fooc}, \dots, \isa{fooaa}, \isa{fooab} etc.; each renaming
- step picks the next unused variant from this sequence.%
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-\begin{mldecls}
- \indexdef{}{ML}{Name.internal}\verb|Name.internal: string -> string| \\
- \indexdef{}{ML}{Name.skolem}\verb|Name.skolem: string -> string| \\
- \end{mldecls}
- \begin{mldecls}
- \indexdef{}{ML type}{Name.context}\verb|type Name.context| \\
- \indexdef{}{ML}{Name.context}\verb|Name.context: Name.context| \\
- \indexdef{}{ML}{Name.declare}\verb|Name.declare: string -> Name.context -> Name.context| \\
- \indexdef{}{ML}{Name.invents}\verb|Name.invents: Name.context -> string -> int -> string list| \\
- \indexdef{}{ML}{Name.variants}\verb|Name.variants: string list -> Name.context -> string list * Name.context| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|Name.internal|~\isa{name} produces an internal name
- by adding one underscore.
-
- \item \verb|Name.skolem|~\isa{name} produces a Skolem name by
- adding two underscores.
-
- \item \verb|Name.context| represents the context of already used
- names; the initial value is \verb|Name.context|.
-
- \item \verb|Name.declare|~\isa{name} enters a used name into the
- context.
-
- \item \verb|Name.invents|~\isa{context\ name\ n} produces \isa{n} fresh names derived from \isa{name}.
-
- \item \verb|Name.variants|~\isa{names\ context} produces fresh
- variants of \isa{names}; the result is entered into the context.
-
- \end{description}%
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-}
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-An \emph{indexed name} (or \isa{indexname}) is a pair of a basic
- name and a natural number. This representation allows efficient
- renaming by incrementing the second component only. The canonical
- way to rename two collections of indexnames apart from each other is
- this: determine the maximum index \isa{maxidx} of the first
- collection, then increment all indexes of the second collection by
- \isa{maxidx\ {\isacharplus}\ {\isadigit{1}}}; the maximum index of an empty collection is
- \isa{{\isacharminus}{\isadigit{1}}}.
-
- Occasionally, basic names and indexed names are injected into the
- same pair type: the (improper) indexname \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}} is used
- to encode basic names.
-
- \medskip Isabelle syntax observes the following rules for
- representing an indexname \isa{{\isacharparenleft}x{\isacharcomma}\ i{\isacharparenright}} as a packed string:
-
- \begin{itemize}
-
- \item \isa{{\isacharquery}x} if \isa{x} does not end with a digit and \isa{i\ {\isacharequal}\ {\isadigit{0}}},
-
- \item \isa{{\isacharquery}xi} if \isa{x} does not end with a digit,
-
- \item \isa{{\isacharquery}x{\isachardot}i} otherwise.
-
- \end{itemize}
-
- Indexnames may acquire large index numbers over time. Results are
- normalized towards \isa{{\isadigit{0}}} at certain checkpoints, notably at
- the end of a proof. This works by producing variants of the
- corresponding basic name components. For example, the collection
- \isa{{\isacharquery}x{\isadigit{1}}{\isacharcomma}\ {\isacharquery}x{\isadigit{7}}{\isacharcomma}\ {\isacharquery}x{\isadigit{4}}{\isadigit{2}}} becomes \isa{{\isacharquery}x{\isacharcomma}\ {\isacharquery}xa{\isacharcomma}\ {\isacharquery}xb}.%
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-\begin{mldecls}
- \indexdef{}{ML type}{indexname}\verb|type indexname| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|indexname| represents indexed names. This is an
- abbreviation for \verb|string * int|. The second component is
- usually non-negative, except for situations where \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}}
- is used to embed basic names into this type.
-
- \end{description}%
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-A \emph{qualified name} consists of a non-empty sequence of basic
- name components. The packed representation uses a dot as separator,
- as in ``\isa{A{\isachardot}b{\isachardot}c}''. The last component is called \emph{base}
- name, the remaining prefix \emph{qualifier} (which may be empty).
- The idea of qualified names is to encode nested structures by
- recording the access paths as qualifiers. For example, an item
- named ``\isa{A{\isachardot}b{\isachardot}c}'' may be understood as a local entity \isa{c}, within a local structure \isa{b}, within a global
- structure \isa{A}. Typically, name space hierarchies consist of
- 1--2 levels of qualification, but this need not be always so.
-
- The empty name is commonly used as an indication of unnamed
- entities, whenever this makes any sense. The basic operations on
- qualified names are smart enough to pass through such improper names
- unchanged.
-
- \medskip A \isa{naming} policy tells how to turn a name
- specification into a fully qualified internal name (by the \isa{full} operation), and how fully qualified names may be accessed
- externally. For example, the default naming policy is to prefix an
- implicit path: \isa{full\ x} produces \isa{path{\isachardot}x}, and the
- standard accesses for \isa{path{\isachardot}x} include both \isa{x} and
- \isa{path{\isachardot}x}. Normally, the naming is implicit in the theory or
- proof context; there are separate versions of the corresponding.
-
- \medskip A \isa{name\ space} manages a collection of fully
- internalized names, together with a mapping between external names
- and internal names (in both directions). The corresponding \isa{intern} and \isa{extern} operations are mostly used for
- parsing and printing only! The \isa{declare} operation augments
- a name space according to the accesses determined by the naming
- policy.
-
- \medskip As a general principle, there is a separate name space for
- each kind of formal entity, e.g.\ logical constant, type
- constructor, type class, theorem. 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 \isa{c} may be used uniformly for a constant, type constructor, and
- type class.
-
- There are common schemes to name theorems systematically, according
- to the name of the main logical entity involved, e.g.\ \isa{c{\isachardot}intro} for a canonical theorem related to constant \isa{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 a type constructor or type class are better suffixed in
- addition to the usual qualification, e.g.\ \isa{c{\isacharunderscore}type{\isachardot}intro}
- and \isa{c{\isacharunderscore}class{\isachardot}intro} for theorems related to type \isa{c}
- and class \isa{c}, respectively.%
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-\begin{mldecls}
- \indexdef{}{ML}{NameSpace.base}\verb|NameSpace.base: string -> string| \\
- \indexdef{}{ML}{NameSpace.qualifier}\verb|NameSpace.qualifier: string -> string| \\
- \indexdef{}{ML}{NameSpace.append}\verb|NameSpace.append: string -> string -> string| \\
- \indexdef{}{ML}{NameSpace.implode}\verb|NameSpace.implode: string list -> string| \\
- \indexdef{}{ML}{NameSpace.explode}\verb|NameSpace.explode: string -> string list| \\
- \end{mldecls}
- \begin{mldecls}
- \indexdef{}{ML type}{NameSpace.naming}\verb|type NameSpace.naming| \\
- \indexdef{}{ML}{NameSpace.default\_naming}\verb|NameSpace.default_naming: NameSpace.naming| \\
- \indexdef{}{ML}{NameSpace.add\_path}\verb|NameSpace.add_path: string -> NameSpace.naming -> NameSpace.naming| \\
- \indexdef{}{ML}{NameSpace.full\_name}\verb|NameSpace.full_name: NameSpace.naming -> binding -> string| \\
- \end{mldecls}
- \begin{mldecls}
- \indexdef{}{ML type}{NameSpace.T}\verb|type NameSpace.T| \\
- \indexdef{}{ML}{NameSpace.empty}\verb|NameSpace.empty: NameSpace.T| \\
- \indexdef{}{ML}{NameSpace.merge}\verb|NameSpace.merge: NameSpace.T * NameSpace.T -> NameSpace.T| \\
- \indexdef{}{ML}{NameSpace.declare}\verb|NameSpace.declare: NameSpace.naming -> binding -> NameSpace.T -> string * NameSpace.T| \\
- \indexdef{}{ML}{NameSpace.intern}\verb|NameSpace.intern: NameSpace.T -> string -> string| \\
- \indexdef{}{ML}{NameSpace.extern}\verb|NameSpace.extern: NameSpace.T -> string -> string| \\
- \end{mldecls}
-
- \begin{description}
-
- \item \verb|NameSpace.base|~\isa{name} returns the base name of a
- qualified name.
-
- \item \verb|NameSpace.qualifier|~\isa{name} returns the qualifier
- of a qualified name.
-
- \item \verb|NameSpace.append|~\isa{name\isactrlisub {\isadigit{1}}\ name\isactrlisub {\isadigit{2}}}
- appends two qualified names.
-
- \item \verb|NameSpace.implode|~\isa{name} and \verb|NameSpace.explode|~\isa{names} convert between the packed string
- representation and the explicit list form of qualified names.
-
- \item \verb|NameSpace.naming| represents the abstract concept of
- a naming policy.
-
- \item \verb|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 \verb|NameSpace.add_path|~\isa{path\ naming} augments the
- naming policy by extending its path component.
-
- \item \verb|NameSpace.full_name|\isa{naming\ binding} turns a name
- binding (usually a basic name) into the fully qualified
- internal name, according to the given naming policy.
-
- \item \verb|NameSpace.T| represents name spaces.
-
- \item \verb|NameSpace.empty| and \verb|NameSpace.merge|~\isa{{\isacharparenleft}space\isactrlisub {\isadigit{1}}{\isacharcomma}\ space\isactrlisub {\isadigit{2}}{\isacharparenright}} are the canonical operations for
- maintaining name spaces according to theory data management
- (\secref{sec:context-data}).
-
- \item \verb|NameSpace.declare|~\isa{naming\ bindings\ space} enters a
- name binding as fully qualified internal name into the name space,
- with external accesses determined by the naming policy.
-
- \item \verb|NameSpace.intern|~\isa{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 \verb|NameSpace.full_name| and \verb|NameSpace.declare|
- (or their derivatives for \verb|theory| and
- \verb|Proof.context|).
-
- \item \verb|NameSpace.extern|~\isa{space\ name} externalizes a
- (fully qualified) internal name.
-
- This operation is mostly for printing! Note unqualified names are
- produced via \verb|NameSpace.base|.
-
- \end{description}%
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