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\def\isabellecontext{Prelim}%
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\ Prelim\isanewline
\isakeyword{imports}\ Base\isanewline
\isakeyword{begin}%
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\isamarkupchapter{Preliminaries%
}
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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}%
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\isamarkupsubsection{Theory context \label{sec:context-theory}%
}
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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. To this end, the system maintains a set of
symbolic ``identification stamps'' within each theory.
In order to avoid the full-scale overhead of explicit sub-theory
identification of arbitrary intermediate stages, a theory is
switched into \isa{draft} mode under certain circumstances. A
draft theory acts like a linear type, where updates invalidate
earlier versions. An invalidated draft is called \emph{stale}.
The \isa{checkpoint} operation produces a safe stepping stone
that will survive the next update without becoming stale: both the
old and the new 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.
The \isa{merge} operation produces the least upper bound of two
theories, which actually degenerates into absorption of one theory
into the other (according to the nominal sub-theory relation).
The \isa{begin} operation starts a new theory by importing
several parent theories and entering a special mode of nameless
incremental updates, until the final \isa{end} operation is
performed.
\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 internally, while transaction boundaries of Isar top-level
commands (\secref{sec:isar-toplevel}) are guaranteed to be safe
checkpoints.
\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{mldecls}
\indexdef{}{ML type}{theory}\verb|type theory| \\
\indexdef{}{ML}{Theory.subthy}\verb|Theory.subthy: theory * theory -> bool| \\
\indexdef{}{ML}{Theory.checkpoint}\verb|Theory.checkpoint: theory -> theory| \\
\indexdef{}{ML}{Theory.copy}\verb|Theory.copy: theory -> theory| \\
\indexdef{}{ML}{Theory.merge}\verb|Theory.merge: theory * theory -> theory| \\
\indexdef{}{ML}{Theory.begin\_theory}\verb|Theory.begin_theory: string -> theory list -> 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, with explicit runtime checking! Most
internal theory 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 intrinsic graph structure of the construction.
This sub-theory relation is a nominal approximation of inclusion
(\isa{{\isasymsubseteq}}) of the corresponding content (according to the
semantics of the ML modules that implement the data).
\item \verb|Theory.checkpoint|~\isa{thy} produces a safe
stepping stone in the linear development of \isa{thy}. This
changes the old theory, but the next update will result in two
related, valid theories.
\item \verb|Theory.copy|~\isa{thy} produces a variant of \isa{thy} with the same data. The copy is not related to the original,
but the original is unchanged.
\item \verb|Theory.merge|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}} absorbs one theory
into the other, without changing \isa{thy\isactrlsub {\isadigit{1}}} or \isa{thy\isactrlsub {\isadigit{2}}}.
This version of ad-hoc theory merge fails for unrelated theories!
\item \verb|Theory.begin_theory|~\isa{name\ parents} constructs
a new theory based on the given parents. This {\ML} function is
normally not invoked directly.
\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}%
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\isamarkupsubsection{Proof context \label{sec:context-proof}%
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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.
Entities derived in a proof context need to record logical
requirements explicitly, since there is no separate context
identification or symbolic inclusion 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 an Isar proof state models
block-structured reasoning explicitly, using a stack of proof
contexts internally. For various technical reasons, the background
theory of an Isar proof state must not be changed while the proof is
still under construction!%
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\begin{mldecls}
\indexdef{}{ML type}{Proof.context}\verb|type Proof.context| \\
\indexdef{}{ML}{ProofContext.init\_global}\verb|ProofContext.init_global: 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_global|~\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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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, which
incurs a small runtime overhead.%
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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_global| as required (note that this re-initializes the
context data with each invocation).
\end{description}%
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\isamarkupsubsection{Context data \label{sec:context-data}%
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The main purpose of theory and proof contexts is to manage
arbitrary (pure) 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} declarations need 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}\ 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{extend} is acts like a unitary version of \isa{merge}.
Implementing \isa{merge} can be tricky. The general idea is
that \isa{merge\ {\isacharparenleft}data\isactrlsub {\isadigit{1}}{\isacharcomma}\ data\isactrlsub {\isadigit{2}}{\isacharparenright}} inserts those parts of \isa{data\isactrlsub {\isadigit{2}}} into \isa{data\isactrlsub {\isadigit{1}}} that are not yet present, while
keeping the general order of things. The \verb|Library.merge|
function on plain lists may serve as canonical template.
Particularly note that shared parts of the data must not be
duplicated by naive concatenation, or a theory graph that is like a
chain of diamonds would cause an exponential blowup!
\paragraph{Proof context data} declarations need 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 and should be somehow
``immediate''. Whenever a proof context is initialized, which
happens frequently, the the system invokes the \isa{init}
operation of \emph{all} theory data slots ever declared.
\paragraph{Generic data} provides a hybrid interface for both theory
and proof data. The \isa{init} operation for proof contexts is
predefined to select the current data value from the background
theory.
\bigskip Any of these data declaration over type \isa{T} result
in an ML structure with the following signature:
\medskip
\begin{tabular}{ll}
\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 These other operations provide exclusive access for the
particular kind of context (theory, proof, or generic context).
This interface fully observes the ML discipline for types and
scopes: there is no other way to access the corresponding data slot
of a context. By keeping these operations private, an Isabelle/ML
module may maintain abstract values authentically.%
\end{isamarkuptext}%
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\begin{mldecls}
\indexdef{}{ML functor}{Theory\_Data}\verb|functor Theory_Data| \\
\indexdef{}{ML functor}{Proof\_Data}\verb|functor Proof_Data| \\
\indexdef{}{ML functor}{Generic\_Data}\verb|functor Generic_Data| \\
\end{mldecls}
\begin{description}
\item \verb|Theory_Data|\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|Proof_Data|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
\verb|Theory_Data| for type \verb|Proof.context|.
\item \verb|Generic_Data|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
\verb|Theory_Data| for type \verb|Context.generic|.
\end{description}%
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The following artificial example demonstrates theory
data: we maintain a set of terms that are supposed to be wellformed
wrt.\ the enclosing theory. The public interface is as follows:%
\end{isamarkuptext}%
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\ \ signature\ WELLFORMED{\isacharunderscore}TERMS\ {\isacharequal}\isanewline
\ \ sig\isanewline
\ \ \ \ val\ get{\isacharcolon}\ theory\ {\isacharminus}{\isachargreater}\ term\ list\isanewline
\ \ \ \ val\ add{\isacharcolon}\ term\ {\isacharminus}{\isachargreater}\ theory\ {\isacharminus}{\isachargreater}\ theory\isanewline
\ \ end{\isacharsemicolon}\isanewline
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\begin{isamarkuptext}%
\noindent The implementation uses private theory data
internally, and only exposes an operation that involves explicit
argument checking wrt.\ the given theory.%
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\ \ structure\ Wellformed{\isacharunderscore}Terms{\isacharcolon}\ WELLFORMED{\isacharunderscore}TERMS\ {\isacharequal}\isanewline
\ \ struct\isanewline
\isanewline
\ \ structure\ Terms\ {\isacharequal}\ Theory{\isacharunderscore}Data\isanewline
\ \ {\isacharparenleft}\isanewline
\ \ \ \ type\ T\ {\isacharequal}\ term\ OrdList{\isachardot}T{\isacharsemicolon}\isanewline
\ \ \ \ val\ empty\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\isanewline
\ \ \ \ val\ extend\ {\isacharequal}\ I{\isacharsemicolon}\isanewline
\ \ \ \ fun\ merge\ {\isacharparenleft}ts{\isadigit{1}}{\isacharcomma}\ ts{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\isanewline
\ \ \ \ \ \ OrdList{\isachardot}union\ Term{\isacharunderscore}Ord{\isachardot}fast{\isacharunderscore}term{\isacharunderscore}ord\ ts{\isadigit{1}}\ ts{\isadigit{2}}{\isacharsemicolon}\isanewline
\ \ {\isacharparenright}\isanewline
\isanewline
\ \ val\ get\ {\isacharequal}\ Terms{\isachardot}get{\isacharsemicolon}\isanewline
\isanewline
\ \ fun\ add\ raw{\isacharunderscore}t\ thy\ {\isacharequal}\isanewline
\ \ \ \ let\ val\ t\ {\isacharequal}\ Sign{\isachardot}cert{\isacharunderscore}term\ thy\ raw{\isacharunderscore}t\isanewline
\ \ \ \ in\ Terms{\isachardot}map\ {\isacharparenleft}OrdList{\isachardot}insert\ Term{\isacharunderscore}Ord{\isachardot}fast{\isacharunderscore}term{\isacharunderscore}ord\ t{\isacharparenright}\ thy\ end{\isacharsemicolon}\isanewline
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\ \ end{\isacharsemicolon}\isanewline
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\begin{isamarkuptext}%
We use \verb|term OrdList.T| for reasonably efficient
representation of a set of terms: all operations are linear in the
number of stored elements. Here we assume that our users do not
care about the declaration order, since that data structure forces
its own arrangement of elements.
Observe how the \verb|merge| operation joins the data slots of
the two constituents: \verb|OrdList.union| prevents duplication of
common data from different branches, thus avoiding the danger of
exponential blowup. (Plain list append etc.\ must never be used for
theory data merges.)
\medskip Our intended invariant is achieved as follows:
\begin{enumerate}
\item \verb|Wellformed_Terms.add| only admits terms that have passed
the \verb|Sign.cert_term| check of the given theory at that point.
\item Wellformedness in the sense of \verb|Sign.cert_term| is
monotonic wrt.\ the sub-theory relation. So our data can move
upwards in the hierarchy (via extension or merges), and maintain
wellformedness without further checks.
\end{enumerate}
Note that all basic operations of the inference kernel (which
includes \verb|Sign.cert_term|) observe this monotonicity principle,
but other user-space tools don't. For example, fully-featured
type-inference via \verb|Syntax.check_term| (cf.\
\secref{sec:term-check}) is not necessarily monotonic wrt.\ the
background theory, since constraints of term constants can be
strengthened by later declarations, for example.
In most cases, user-space context data does not have to take such
invariants too seriously. The situation is different in the
implementation of the inference kernel itself, which uses the very
same data mechanisms for types, constants, axioms etc.%
\end{isamarkuptext}%
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\isamarkupsection{Names \label{sec:names}%
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\begin{isamarkuptext}%
In principle, a name is just a string, but there are various
conventions for representing additional structure. For example,
``\isa{Foo{\isachardot}bar{\isachardot}baz}'' is considered as a long name consisting of
qualifier \isa{Foo{\isachardot}bar} and base name \isa{baz}. The
individual constituents of a name may have further substructure,
e.g.\ the string ``\verb,\,\verb,<alpha>,'' encodes as a single
symbol.
\medskip Subsequently, we shall introduce specific categories of
names. Roughly speaking these correspond to logical entities as
follows:
\begin{itemize}
\item Basic names (\secref{sec:basic-name}): free and bound
variables.
\item Indexed names (\secref{sec:indexname}): schematic variables.
\item Long names (\secref{sec:long-name}): constants of any kind
(type constructors, term constants, other concepts defined in user
space). Such entities are typically managed via name spaces
(\secref{sec:name-space}).
\end{itemize}%
\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 ML characters are normally not encountered at all!
Isabelle strings consist of a sequence of symbols, represented as a
packed string or an exploded 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}'' or raw byte in the
range of 128\dots 255, 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} consists 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 can
also process Unicode/UCS data in UTF-8 encoding.\footnote{When
counting precise source positions internally, bytes in the range of
128\dots 191 are ignored. In UTF-8 encoding, this interval covers
the additional trailer bytes, so Isabelle happens to count Unicode
characters here, not bytes in memory. In ISO-Latin encoding, the
ignored range merely includes some extra punctuation characters that
even have replacements within the standard collection of Isabelle
symbols; the accented letters range is counted properly.} Unicode
provides its own collection of mathematical symbols, but within the
core Isabelle/ML world there is no link to the standard collection
of Isabelle regular symbols.
\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}}. On-screen rendering usually works by mapping a finite
subset of Isabelle symbols to suitable Unicode characters.%
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\begin{mldecls}
\indexdef{}{ML type}{Symbol.symbol}\verb|type Symbol.symbol = string| \\
\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.
\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!\footnote{The runtime overhead for exploded strings is
mainly that of the list structure: individual symbols that happen to
be a singleton string --- which is the most common case --- do not
require extra memory in Poly/ML.}
\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}
\paragraph{Historical note.} In the original SML90 standard the
primitive ML type \verb|char| did not exists, and the basic \verb|explode: string -> string list| operation would produce a list of
singleton strings as in Isabelle/ML today. When SML97 came out,
Isabelle did not adopt its slightly anachronistic 8-bit characters,
but the idea of exploding a string into a list of small strings was
extended to ``symbols'' as explained above. Thus Isabelle sources
can refer to an infinite store of user-defined symbols, without
having to worry about the multitude of Unicode encodings.%
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\isamarkupsubsection{Basic names \label{sec:basic-name}%
}
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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 names like \isa{xaa} to
appear in human-readable 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{mldecls}
\indexdef{}{ML}{Variable.names\_of}\verb|Variable.names_of: Proof.context -> 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.
\item \verb|Variable.names_of|~\isa{ctxt} retrieves the context
of declared type and term variable names. Projecting a proof
context down to a primitive name context is occasionally useful when
invoking lower-level operations. Regular management of ``fresh
variables'' is done by suitable operations of structure \verb|Variable|, which is also able to provide an official status of
``locally fixed variable'' within the logical environment (cf.\
\secref{sec:variables}).
\end{description}%
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\isamarkupsubsection{Indexed names \label{sec:indexname}%
}
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\begin{isamarkuptext}%
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 are injected into the same pair type of
indexed names: then \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}} is used to encode the basic
name \isa{x}.
\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 after several maxidx
shifts have been applied. Results are usually 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 inject basic names into this type. Other negative
indexes should not be used.
\end{description}%
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\isamarkupsubsection{Long names \label{sec:long-name}%
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\begin{isamarkuptext}%
A \emph{long name} consists of a sequence of non-empty 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 is called \emph{qualifier} (which may be
empty). The qualifier can be understood as the access path to the
named entity while passing through some nested block-structure,
although our free-form long names do not really enforce any strict
discipline.
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}. In practice, long names
usually represent 1--3 levels of qualification. User ML code should
not make any assumptions about the particular structure of long
names!
The empty name is commonly used as an indication of unnamed
entities, or entities that are not entered into the corresponding
name space, whenever this makes any sense. The basic operations on
long names map empty names again to empty names.%
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\begin{mldecls}
\indexdef{}{ML}{Long\_Name.base\_name}\verb|Long_Name.base_name: string -> string| \\
\indexdef{}{ML}{Long\_Name.qualifier}\verb|Long_Name.qualifier: string -> string| \\
\indexdef{}{ML}{Long\_Name.append}\verb|Long_Name.append: string -> string -> string| \\
\indexdef{}{ML}{Long\_Name.implode}\verb|Long_Name.implode: string list -> string| \\
\indexdef{}{ML}{Long\_Name.explode}\verb|Long_Name.explode: string -> string list| \\
\end{mldecls}
\begin{description}
\item \verb|Long_Name.base_name|~\isa{name} returns the base name
of a long name.
\item \verb|Long_Name.qualifier|~\isa{name} returns the qualifier
of a long name.
\item \verb|Long_Name.append|~\isa{name\isactrlisub {\isadigit{1}}\ name\isactrlisub {\isadigit{2}}} appends two long
names.
\item \verb|Long_Name.implode|~\isa{names} and \verb|Long_Name.explode|~\isa{name} convert between the packed string
representation and the explicit list form of long names.
\end{description}%
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\isamarkupsubsection{Name spaces \label{sec:name-space}%
}
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A \isa{name\ space} manages a collection of long names,
together with a mapping between partially qualified external names
and fully qualified internal names (in both directions). Note that
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 a given binding, and a naming policy from the context.
\medskip A \isa{binding} specifies details about the prospective
long name of a newly introduced formal entity. It consists of a
base name, prefixes for qualification (separate ones for system
infrastructure and user-space mechanisms), a slot for the original
source position, and some additional flags.
\medskip A \isa{naming} provides some additional details for
producing a long name from a binding. Normally, the naming is
implicit in the theory or proof context. The \isa{full}
operation (and its variants for different context types) produces a
fully qualified internal name to be entered into a name space. The
main equation of this ``chemical reaction'' when binding new
entities in a context is as follows:
\smallskip
\begin{tabular}{l}
\isa{binding\ {\isacharplus}\ naming\ {\isasymlongrightarrow}\ long\ name\ {\isacharplus}\ name\ space\ accesses}
\end{tabular}
\smallskip
\medskip As a general principle, there is a separate name space for
each kind of formal entity, e.g.\ fact, logical constant, type
constructor, type class. 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 derived entities systematically
according to the name of the main logical entity involved, e.g.\
fact \isa{c{\isachardot}intro} for a canonical introduction rule 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 type}{binding}\verb|type binding| \\
\indexdef{}{ML}{Binding.empty}\verb|Binding.empty: binding| \\
\indexdef{}{ML}{Binding.name}\verb|Binding.name: string -> binding| \\
\indexdef{}{ML}{Binding.qualify}\verb|Binding.qualify: bool -> string -> binding -> binding| \\
\indexdef{}{ML}{Binding.prefix}\verb|Binding.prefix: bool -> string -> binding -> binding| \\
\indexdef{}{ML}{Binding.conceal}\verb|Binding.conceal: binding -> binding| \\
\indexdef{}{ML}{Binding.str\_of}\verb|Binding.str_of: binding -> string| \\
\end{mldecls}
\begin{mldecls}
\indexdef{}{ML type}{Name\_Space.naming}\verb|type Name_Space.naming| \\
\indexdef{}{ML}{Name\_Space.default\_naming}\verb|Name_Space.default_naming: Name_Space.naming| \\
\indexdef{}{ML}{Name\_Space.add\_path}\verb|Name_Space.add_path: string -> Name_Space.naming -> Name_Space.naming| \\
\indexdef{}{ML}{Name\_Space.full\_name}\verb|Name_Space.full_name: Name_Space.naming -> binding -> string| \\
\end{mldecls}
\begin{mldecls}
\indexdef{}{ML type}{Name\_Space.T}\verb|type Name_Space.T| \\
\indexdef{}{ML}{Name\_Space.empty}\verb|Name_Space.empty: string -> Name_Space.T| \\
\indexdef{}{ML}{Name\_Space.merge}\verb|Name_Space.merge: Name_Space.T * Name_Space.T -> Name_Space.T| \\
\indexdef{}{ML}{Name\_Space.declare}\verb|Name_Space.declare: bool -> Name_Space.naming -> binding -> Name_Space.T ->|\isasep\isanewline%
\verb| string * Name_Space.T| \\
\indexdef{}{ML}{Name\_Space.intern}\verb|Name_Space.intern: Name_Space.T -> string -> string| \\
\indexdef{}{ML}{Name\_Space.extern}\verb|Name_Space.extern: Name_Space.T -> string -> string| \\
\indexdef{}{ML}{Name\_Space.is\_concealed}\verb|Name_Space.is_concealed: Name_Space.T -> string -> bool|
\end{mldecls}
\begin{description}
\item \verb|binding| represents the abstract concept of name
bindings.
\item \verb|Binding.empty| is the empty binding.
\item \verb|Binding.name|~\isa{name} produces a binding with base
name \isa{name}.
\item \verb|Binding.qualify|~\isa{mandatory\ name\ binding}
prefixes qualifier \isa{name} to \isa{binding}. The \isa{mandatory} flag tells if this name component always needs to be
given in name space accesses --- this is mostly \isa{false} in
practice. Note that this part of qualification is typically used in
derived specification mechanisms.
\item \verb|Binding.prefix| is similar to \verb|Binding.qualify|, but
affects the system prefix. This part of extra qualification is
typically used in the infrastructure for modular specifications,
notably ``local theory targets'' (see also \chref{ch:local-theory}).
\item \verb|Binding.conceal|~\isa{binding} indicates that the
binding shall refer to an entity that serves foundational purposes
only. This flag helps to mark implementation details of
specification mechanism etc. Other tools should not depend on the
particulars of concealed entities (cf.\ \verb|Name_Space.is_concealed|).
\item \verb|Binding.str_of|~\isa{binding} produces a string
representation for human-readable output, together with some formal
markup that might get used in GUI front-ends, for example.
\item \verb|Name_Space.naming| represents the abstract concept of
a naming policy.
\item \verb|Name_Space.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|Name_Space.add_path|~\isa{path\ naming} augments the
naming policy by extending its path component.
\item \verb|Name_Space.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|Name_Space.T| represents name spaces.
\item \verb|Name_Space.empty|~\isa{kind} and \verb|Name_Space.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}); \isa{kind} is a formal comment
to characterize the purpose of a name space.
\item \verb|Name_Space.declare|~\isa{strict\ 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|Name_Space.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|Name_Space.full_name| and \verb|Name_Space.declare|
(or their derivatives for \verb|theory| and
\verb|Proof.context|).
\item \verb|Name_Space.extern|~\isa{space\ name} externalizes a
(fully qualified) internal name.
This operation is mostly for printing! User code should not rely on
the precise result too much.
\item \verb|Name_Space.is_concealed|~\isa{space\ name} indicates
whether \isa{name} refers to a strictly private entity that
other tools are supposed to ignore!
\end{description}%
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