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\begin{isabellebody}%
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\def\isabellecontext{prelim}%
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%
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\isadelimtheory
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\isanewline
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\isanewline
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\isanewline
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\endisadelimtheory
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ prelim\ \isakeyword{imports}\ base\ \isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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\isadelimtheory
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\endisadelimtheory
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%
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\isamarkupchapter{Preliminaries%
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}
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\isamarkuptrue%
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%
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\isamarkupsection{Contexts \label{sec:context}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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A logical context represents the background that is required for
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formulating statements and composing proofs. It acts as a medium to
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produce formal content, depending on earlier material (declarations,
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results etc.).
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For example, derivations within the Isabelle/Pure logic can be
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described as a judgment \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}, which means that a
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proposition \isa{{\isasymphi}} is derivable from hypotheses \isa{{\isasymGamma}}
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within the theory \isa{{\isasymTheta}}. There are logical reasons for
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keeping \isa{{\isasymTheta}} and \isa{{\isasymGamma}} separate: theories can be
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liberal about supporting type constructors and schematic
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polymorphism of constants and axioms, while the inner calculus of
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\isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}} is strictly limited to Simple Type Theory (with
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fixed type variables in the assumptions).
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\medskip Contexts and derivations are linked by the following key
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principles:
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\begin{itemize}
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\item Transfer: monotonicity of derivations admits results to be
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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}}.
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\item Export: discharge of hypotheses admits results to be exported
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into a \emph{smaller} context, i.e.\ \isa{{\isasymGamma}{\isacharprime}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymphi}}
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implies \isa{{\isasymGamma}\ {\isasymturnstile}\isactrlsub {\isasymTheta}\ {\isasymDelta}\ {\isasymLongrightarrow}\ {\isasymphi}} where \isa{{\isasymGamma}{\isacharprime}\ {\isasymsupseteq}\ {\isasymGamma}} and
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\isa{{\isasymDelta}\ {\isacharequal}\ {\isasymGamma}{\isacharprime}\ {\isacharminus}\ {\isasymGamma}}. Note that \isa{{\isasymTheta}} remains unchanged here,
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only the \isa{{\isasymGamma}} part is affected.
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\end{itemize}
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\medskip By modeling the main characteristics of the primitive
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\isa{{\isasymTheta}} and \isa{{\isasymGamma}} above, and abstracting over any
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particular logical content, we arrive at the fundamental notions of
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\emph{theory context} and \emph{proof context} in Isabelle/Isar.
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These implement a certain policy to manage arbitrary \emph{context
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data}. There is a strongly-typed mechanism to declare new kinds of
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data at compile time.
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The internal bootstrap process of Isabelle/Pure eventually reaches a
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stage where certain data slots provide the logical content of \isa{{\isasymTheta}} and \isa{{\isasymGamma}} sketched above, but this does not stop there!
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Various additional data slots support all kinds of mechanisms that
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are not necessarily part of the core logic.
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For example, there would be data for canonical introduction and
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elimination rules for arbitrary operators (depending on the
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object-logic and application), which enables users to perform
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standard proof steps implicitly (cf.\ the \isa{rule} method
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\cite{isabelle-isar-ref}).
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\medskip Thus Isabelle/Isar is able to bring forth more and more
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concepts successively. In particular, an object-logic like
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Isabelle/HOL continues the Isabelle/Pure setup by adding specific
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components for automated reasoning (classical reasoner, tableau
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prover, structured induction etc.) and derived specification
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mechanisms (inductive predicates, recursive functions etc.). All of
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this is ultimately based on the generic data management by theory
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and proof contexts introduced here.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Theory context \label{sec:context-theory}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\glossary{Theory}{FIXME}
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A \emph{theory} is a data container with explicit named and unique
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identifier. Theories are related by a (nominal) sub-theory
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relation, which corresponds to the dependency graph of the original
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construction; each theory is derived from a certain sub-graph of
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ancestor theories.
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The \isa{merge} operation produces the least upper bound of two
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theories, which actually degenerates into absorption of one theory
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into the other (due to the nominal sub-theory relation).
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The \isa{begin} operation starts a new theory by importing
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several parent theories and entering a special \isa{draft} mode,
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which is sustained until the final \isa{end} operation. A draft
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theory acts like a linear type, where updates invalidate earlier
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versions. An invalidated draft is called ``stale''.
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The \isa{checkpoint} operation produces an intermediate stepping
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stone that will survive the next update: both the original and the
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changed theory remain valid and are related by the sub-theory
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relation. Checkpointing essentially recovers purely functional
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theory values, at the expense of some extra internal bookkeeping.
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The \isa{copy} operation produces an auxiliary version that has
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the same data content, but is unrelated to the original: updates of
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the copy do not affect the original, neither does the sub-theory
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relation hold.
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\medskip The example in \figref{fig:ex-theory} below shows a theory
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graph derived from \isa{Pure}, with theory \isa{Length}
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importing \isa{Nat} and \isa{List}. The body of \isa{Length} consists of a sequence of updates, working mostly on
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drafts. Intermediate checkpoints may occur as well, due to the
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history mechanism provided by the Isar top-level, cf.\
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\secref{sec:isar-toplevel}.
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\begin{figure}[htb]
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\begin{center}
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\begin{tabular}{rcccl}
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& & \isa{Pure} \\
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& & \isa{{\isasymdown}} \\
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& & \isa{FOL} \\
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& $\swarrow$ & & $\searrow$ & \\
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$Nat$ & & & & \isa{List} \\
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& $\searrow$ & & $\swarrow$ \\
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& & \isa{Length} \\
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& & \multicolumn{3}{l}{~~$\isarkeyword{imports}$} \\
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& & \multicolumn{3}{l}{~~$\isarkeyword{begin}$} \\
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& & $\vdots$~~ \\
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& & \isa{{\isasymbullet}}~~ \\
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& & $\vdots$~~ \\
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& & \isa{{\isasymbullet}}~~ \\
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& & $\vdots$~~ \\
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& & \multicolumn{3}{l}{~~$\isarkeyword{end}$} \\
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\end{tabular}
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\caption{A theory definition depending on ancestors}\label{fig:ex-theory}
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\end{center}
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\end{figure}
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\medskip There is a separate notion of \emph{theory reference} for
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maintaining a live link to an evolving theory context: updates on
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drafts are propagated automatically. The dynamic stops after an
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explicit \isa{end} only.
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Derived entities may store a theory reference in order to indicate
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the context they belong to. This implicitly assumes monotonic
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reasoning, because the referenced context may become larger without
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further notice.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimmlref
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\endisadelimmlref
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\isatagmlref
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\begin{isamarkuptext}%
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\begin{mldecls}
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\indexmltype{theory}\verb|type theory| \\
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\indexml{Theory.subthy}\verb|Theory.subthy: theory * theory -> bool| \\
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\indexml{Theory.merge}\verb|Theory.merge: theory * theory -> theory| \\
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\indexml{Theory.checkpoint}\verb|Theory.checkpoint: theory -> theory| \\
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\indexml{Theory.copy}\verb|Theory.copy: theory -> theory| \\[1ex]
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\indexmltype{theory-ref}\verb|type theory_ref| \\
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\indexml{Theory.self-ref}\verb|Theory.self_ref: theory -> theory_ref| \\
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\indexml{Theory.deref}\verb|Theory.deref: theory_ref -> theory| \\
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\end{mldecls}
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\begin{description}
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\item \verb|theory| represents theory contexts. This is
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essentially a linear type! Most operations destroy the original
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version, which then becomes ``stale''.
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\item \verb|Theory.subthy|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}}
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compares theories according to the inherent graph structure of the
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construction. This sub-theory relation is a nominal approximation
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of inclusion (\isa{{\isasymsubseteq}}) of the corresponding content.
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\item \verb|Theory.merge|~\isa{{\isacharparenleft}thy\isactrlsub {\isadigit{1}}{\isacharcomma}\ thy\isactrlsub {\isadigit{2}}{\isacharparenright}}
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absorbs one theory into the other. This fails for unrelated
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theories!
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\item \verb|Theory.checkpoint|~\isa{thy} produces a safe
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stepping stone in the linear development of \isa{thy}. The next
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update will result in two related, valid theories.
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\item \verb|Theory.copy|~\isa{thy} produces a variant of \isa{thy} that holds a copy of the same data. The result is not
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related to the original; the original is unchanched.
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\item \verb|theory_ref| represents a sliding reference to an
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always valid theory; updates on the original are propagated
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automatically.
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\item \verb|Theory.self_ref|~\isa{thy} and \verb|Theory.deref|~\isa{thy{\isacharunderscore}ref} convert between \verb|theory| and \verb|theory_ref|. As the referenced theory
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evolves monotonically over time, later invocations of \verb|Theory.deref| may refer to a larger context.
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\end{description}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\endisatagmlref
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{\isafoldmlref}%
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\isadelimmlref
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%
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\endisadelimmlref
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%
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\isamarkupsubsection{Proof context \label{sec:context-proof}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\glossary{Proof context}{The static context of a structured proof,
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acts like a local ``theory'' of the current portion of Isar proof
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text, generalizes the idea of local hypotheses \isa{{\isasymGamma}} in
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judgments \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}} of natural deduction calculi. There is a
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generic notion of introducing and discharging hypotheses.
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Arbritrary auxiliary context data may be adjoined.}
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A proof context is a container for pure data with a back-reference
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to the theory it belongs to. The \isa{init} operation creates a
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proof context from a given theory. Modifications to draft theories
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are propagated to the proof context as usual, but there is also an
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explicit \isa{transfer} operation to force resynchronization
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with more substantial updates to the underlying theory. The actual
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context data does not require any special bookkeeping, thanks to the
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lack of destructive features.
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Entities derived in a proof context need to record inherent logical
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requirements explicitly, since there is no separate context
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identification as for theories. For example, hypotheses used in
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primitive derivations (cf.\ \secref{sec:thms}) are recorded
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separately within the sequent \isa{{\isasymGamma}\ {\isasymturnstile}\ {\isasymphi}}, just to make double
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sure. Results could still leak into an alien proof context do to
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programming errors, but Isabelle/Isar includes some extra validity
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checks in critical positions, notably at the end of sub-proof.
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Proof contexts may be manipulated arbitrarily, although the common
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discipline is to follow block structure as a mental model: a given
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context is extended consecutively, and results are exported back
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into the original context. Note that the Isar proof states model
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block-structured reasoning explicitly, using a stack of proof
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contexts internally, cf.\ \secref{sec:isar-proof-state}.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimmlref
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%
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\endisadelimmlref
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%
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\isatagmlref
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%
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\begin{isamarkuptext}%
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\begin{mldecls}
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\indexmltype{Proof.context}\verb|type Proof.context| \\
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\indexml{ProofContext.init}\verb|ProofContext.init: theory -> Proof.context| \\
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\indexml{ProofContext.theory-of}\verb|ProofContext.theory_of: Proof.context -> theory| \\
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\indexml{ProofContext.transfer}\verb|ProofContext.transfer: theory -> Proof.context -> Proof.context| \\
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\end{mldecls}
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\begin{description}
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\item \verb|Proof.context| represents proof contexts. Elements
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of this type are essentially pure values, with a sliding reference
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to the background theory.
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\item \verb|ProofContext.init|~\isa{thy} produces a proof context
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derived from \isa{thy}, initializing all data.
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\item \verb|ProofContext.theory_of|~\isa{ctxt} selects the
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background theory from \isa{ctxt}, dereferencing its internal
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\verb|theory_ref|.
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\item \verb|ProofContext.transfer|~\isa{thy\ ctxt} promotes the
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background theory of \isa{ctxt} to the super theory \isa{thy}.
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\end{description}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\endisatagmlref
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{\isafoldmlref}%
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\isadelimmlref
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%
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\endisadelimmlref
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%
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\isamarkupsubsection{Generic contexts \label{sec:generic-context}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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A generic context is the disjoint sum of either a theory or proof
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context. Occasionally, this enables uniform treatment of generic
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context data, typically extra-logical information. Operations on
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generic contexts include the usual injections, partial selections,
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and combinators for lifting operations on either component of the
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disjoint sum.
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Moreover, there are total operations \isa{theory{\isacharunderscore}of} and \isa{proof{\isacharunderscore}of} to convert a generic context into either kind: a theory
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can always be selected from the sum, while a proof context might
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have to be constructed by an ad-hoc \isa{init} operation.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isadelimmlref
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\endisadelimmlref
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%
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\isatagmlref
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\begin{isamarkuptext}%
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\begin{mldecls}
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\indexmltype{Context.generic}\verb|type Context.generic| \\
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\indexml{Context.theory-of}\verb|Context.theory_of: Context.generic -> theory| \\
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\indexml{Context.proof-of}\verb|Context.proof_of: Context.generic -> Proof.context| \\
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\end{mldecls}
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\begin{description}
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\item \verb|Context.generic| is the direct sum of \verb|theory| and \verb|Proof.context|, with the datatype
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constructors \verb|Context.Theory| and \verb|Context.Proof|.
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\item \verb|Context.theory_of|~\isa{context} always produces a
|
|
343 |
theory from the generic \isa{context}, using \verb|ProofContext.theory_of| as required.
|
|
344 |
|
|
345 |
\item \verb|Context.proof_of|~\isa{context} always produces a
|
20451
|
346 |
proof context from the generic \isa{context}, using \verb|ProofContext.init| as required (note that this re-initializes the
|
|
347 |
context data with each invocation).
|
20449
|
348 |
|
|
349 |
\end{description}%
|
20430
|
350 |
\end{isamarkuptext}%
|
|
351 |
\isamarkuptrue%
|
|
352 |
%
|
|
353 |
\endisatagmlref
|
|
354 |
{\isafoldmlref}%
|
|
355 |
%
|
|
356 |
\isadelimmlref
|
|
357 |
%
|
|
358 |
\endisadelimmlref
|
|
359 |
%
|
20477
|
360 |
\isamarkupsubsection{Context data \label{sec:context-data}%
|
20447
|
361 |
}
|
|
362 |
\isamarkuptrue%
|
|
363 |
%
|
|
364 |
\begin{isamarkuptext}%
|
20451
|
365 |
The main purpose of theory and proof contexts is to manage arbitrary
|
|
366 |
data. New data types can be declared incrementally at compile time.
|
|
367 |
There are separate declaration mechanisms for any of the three kinds
|
|
368 |
of contexts: theory, proof, generic.
|
20449
|
369 |
|
|
370 |
\paragraph{Theory data} may refer to destructive entities, which are
|
20451
|
371 |
maintained in direct correspondence to the linear evolution of
|
|
372 |
theory values, including explicit copies.\footnote{Most existing
|
|
373 |
instances of destructive theory data are merely historical relics
|
|
374 |
(e.g.\ the destructive theorem storage, and destructive hints for
|
|
375 |
the Simplifier and Classical rules).} A theory data declaration
|
|
376 |
needs to implement the following specification (depending on type
|
|
377 |
\isa{T}):
|
20449
|
378 |
|
|
379 |
\medskip
|
|
380 |
\begin{tabular}{ll}
|
|
381 |
\isa{name{\isacharcolon}\ string} \\
|
|
382 |
\isa{empty{\isacharcolon}\ T} & initial value \\
|
|
383 |
\isa{copy{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & refresh impure data \\
|
|
384 |
\isa{extend{\isacharcolon}\ T\ {\isasymrightarrow}\ T} & re-initialize on import \\
|
|
385 |
\isa{merge{\isacharcolon}\ T\ {\isasymtimes}\ T\ {\isasymrightarrow}\ T} & join on import \\
|
|
386 |
\isa{print{\isacharcolon}\ T\ {\isasymrightarrow}\ unit} & diagnostic output \\
|
|
387 |
\end{tabular}
|
|
388 |
\medskip
|
|
389 |
|
|
390 |
\noindent The \isa{name} acts as a comment for diagnostic
|
|
391 |
messages; \isa{copy} is just the identity for pure data; \isa{extend} is acts like a unitary version of \isa{merge}, both
|
|
392 |
should also include the functionality of \isa{copy} for impure
|
|
393 |
data.
|
|
394 |
|
20451
|
395 |
\paragraph{Proof context data} is purely functional. A declaration
|
|
396 |
needs to implement the following specification:
|
20449
|
397 |
|
|
398 |
\medskip
|
|
399 |
\begin{tabular}{ll}
|
|
400 |
\isa{name{\isacharcolon}\ string} \\
|
|
401 |
\isa{init{\isacharcolon}\ theory\ {\isasymrightarrow}\ T} & produce initial value \\
|
|
402 |
\isa{print{\isacharcolon}\ T\ {\isasymrightarrow}\ unit} & diagnostic output \\
|
|
403 |
\end{tabular}
|
|
404 |
\medskip
|
|
405 |
|
|
406 |
\noindent The \isa{init} operation is supposed to produce a pure
|
20451
|
407 |
value from the given background theory. The remainder is analogous
|
|
408 |
to theory data.
|
20449
|
409 |
|
20451
|
410 |
\paragraph{Generic data} provides a hybrid interface for both theory
|
|
411 |
and proof data. The declaration is essentially the same as for
|
|
412 |
(pure) theory data, without \isa{copy}, though. The \isa{init} operation for proof contexts merely selects the current data
|
|
413 |
value from the background theory.
|
20449
|
414 |
|
|
415 |
\bigskip In any case, a data declaration of type \isa{T} results
|
|
416 |
in the following interface:
|
|
417 |
|
|
418 |
\medskip
|
|
419 |
\begin{tabular}{ll}
|
|
420 |
\isa{init{\isacharcolon}\ theory\ {\isasymrightarrow}\ theory} \\
|
|
421 |
\isa{get{\isacharcolon}\ context\ {\isasymrightarrow}\ T} \\
|
|
422 |
\isa{put{\isacharcolon}\ T\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\
|
|
423 |
\isa{map{\isacharcolon}\ {\isacharparenleft}T\ {\isasymrightarrow}\ T{\isacharparenright}\ {\isasymrightarrow}\ context\ {\isasymrightarrow}\ context} \\
|
|
424 |
\isa{print{\isacharcolon}\ context\ {\isasymrightarrow}\ unit}
|
|
425 |
\end{tabular}
|
|
426 |
\medskip
|
|
427 |
|
|
428 |
\noindent Here \isa{init} needs to be applied to the current
|
|
429 |
theory context once, in order to register the initial setup. The
|
|
430 |
other operations provide access for the particular kind of context
|
|
431 |
(theory, proof, or generic context). Note that this is a safe
|
|
432 |
interface: there is no other way to access the corresponding data
|
20451
|
433 |
slot of a context. By keeping these operations private, a component
|
|
434 |
may maintain abstract values authentically, without other components
|
|
435 |
interfering.%
|
20447
|
436 |
\end{isamarkuptext}%
|
|
437 |
\isamarkuptrue%
|
|
438 |
%
|
20450
|
439 |
\isadelimmlref
|
|
440 |
%
|
|
441 |
\endisadelimmlref
|
|
442 |
%
|
|
443 |
\isatagmlref
|
|
444 |
%
|
|
445 |
\begin{isamarkuptext}%
|
|
446 |
\begin{mldecls}
|
|
447 |
\indexmlfunctor{TheoryDataFun}\verb|functor TheoryDataFun| \\
|
|
448 |
\indexmlfunctor{ProofDataFun}\verb|functor ProofDataFun| \\
|
|
449 |
\indexmlfunctor{GenericDataFun}\verb|functor GenericDataFun| \\
|
|
450 |
\end{mldecls}
|
|
451 |
|
|
452 |
\begin{description}
|
|
453 |
|
|
454 |
\item \verb|TheoryDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} declares data for
|
|
455 |
type \verb|theory| according to the specification provided as
|
20451
|
456 |
argument structure. The resulting structure provides data init and
|
|
457 |
access operations as described above.
|
20450
|
458 |
|
20471
|
459 |
\item \verb|ProofDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
|
|
460 |
\verb|TheoryDataFun| for type \verb|Proof.context|.
|
20450
|
461 |
|
20471
|
462 |
\item \verb|GenericDataFun|\isa{{\isacharparenleft}spec{\isacharparenright}} is analogous to
|
|
463 |
\verb|TheoryDataFun| for type \verb|Context.generic|.
|
20450
|
464 |
|
|
465 |
\end{description}%
|
|
466 |
\end{isamarkuptext}%
|
|
467 |
\isamarkuptrue%
|
|
468 |
%
|
|
469 |
\endisatagmlref
|
|
470 |
{\isafoldmlref}%
|
|
471 |
%
|
|
472 |
\isadelimmlref
|
|
473 |
%
|
|
474 |
\endisadelimmlref
|
|
475 |
%
|
20477
|
476 |
\isamarkupsection{Names%
|
20438
|
477 |
}
|
|
478 |
\isamarkuptrue%
|
|
479 |
%
|
|
480 |
\begin{isamarkuptext}%
|
20477
|
481 |
In principle, a name is just a string, but there are various
|
|
482 |
convention for encoding additional structure.
|
20471
|
483 |
|
20477
|
484 |
For example, the string ``\isa{Foo{\isachardot}bar{\isachardot}baz}'' is considered as a
|
|
485 |
qualified name. The most basic constituents of names may have their
|
|
486 |
own structure, e.g.\ the string ``\verb,\,\verb,<alpha>,'' is
|
|
487 |
considered as a single symbol (printed as ``\isa{{\isasymalpha}}'').%
|
20438
|
488 |
\end{isamarkuptext}%
|
|
489 |
\isamarkuptrue%
|
|
490 |
%
|
|
491 |
\isamarkupsubsection{Strings of symbols%
|
|
492 |
}
|
|
493 |
\isamarkuptrue%
|
|
494 |
%
|
|
495 |
\begin{isamarkuptext}%
|
20477
|
496 |
\glossary{Symbol}{The smallest unit of text in Isabelle, subsumes
|
|
497 |
plain ASCII characters as well as an infinite collection of named
|
|
498 |
symbols (for greek, math etc.).}
|
20471
|
499 |
|
20477
|
500 |
A \emph{symbol} constitutes the smallest textual unit in Isabelle
|
|
501 |
--- raw characters are normally not encountered. Isabelle strings
|
|
502 |
consist of a sequence of symbols, represented as a packed string or
|
|
503 |
a list of symbols. Each symbol is in itself a small string, which
|
|
504 |
is of one of the following forms:
|
20438
|
505 |
|
20451
|
506 |
\begin{enumerate}
|
|
507 |
|
20477
|
508 |
\item singleton ASCII character ``\isa{c}'' (character code
|
|
509 |
0--127), for example ``\verb,a,'',
|
20438
|
510 |
|
20477
|
511 |
\item regular symbol ``\verb,\,\verb,<,\isa{ident}\verb,>,'',
|
|
512 |
for example ``\verb,\,\verb,<alpha>,'',
|
20438
|
513 |
|
20477
|
514 |
\item control symbol ``\verb,\,\verb,<^,\isa{ident}\verb,>,'',
|
|
515 |
for example ``\verb,\,\verb,<^bold>,'',
|
20438
|
516 |
|
20477
|
517 |
\item raw symbol ``\verb,\,\verb,<^raw:,\isa{text}\verb,>,'' where
|
|
518 |
\isa{text} is constists of printable characters excluding
|
|
519 |
``\verb,.,'' and ``\verb,>,'', for example
|
|
520 |
``\verb,\,\verb,<^raw:$\sum_{i = 1}^n$>,'',
|
20438
|
521 |
|
20477
|
522 |
\item numbered raw control symbol ``\verb,\,\verb,<^raw,\isa{n}\verb,>, where \isa{n} consists of digits, for example
|
20451
|
523 |
``\verb,\,\verb,<^raw42>,''.
|
20438
|
524 |
|
20451
|
525 |
\end{enumerate}
|
20438
|
526 |
|
20477
|
527 |
\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
|
|
528 |
regular symbols and control symbols, but a fixed collection of
|
|
529 |
standard symbols is treated specifically. For example,
|
20451
|
530 |
``\verb,\,\verb,<alpha>,'' is classified as a (non-ASCII) letter,
|
|
531 |
which means it may occur within regular Isabelle identifier syntax.
|
20438
|
532 |
|
20477
|
533 |
Note that the character set underlying Isabelle symbols is plain
|
|
534 |
7-bit ASCII. Since 8-bit characters are passed through
|
|
535 |
transparently, Isabelle may process Unicode/UCS data (in UTF-8
|
|
536 |
encoding) as well. Unicode provides its own collection of
|
|
537 |
mathematical symbols, but there is no built-in link to the ones of
|
|
538 |
Isabelle.
|
20438
|
539 |
|
20477
|
540 |
\medskip Output of Isabelle symbols depends on the print mode
|
|
541 |
(\secref{FIXME}). For example, the standard {\LaTeX} setup of the
|
|
542 |
Isabelle document preparation system would present
|
|
543 |
``\verb,\,\verb,<alpha>,'' as \isa{{\isasymalpha}}, and
|
|
544 |
``\verb,\,\verb,<^bold>,\verb,\,\verb,<alpha>,'' as \isa{\isactrlbold {\isasymalpha}}.%
|
20438
|
545 |
\end{isamarkuptext}%
|
|
546 |
\isamarkuptrue%
|
|
547 |
%
|
|
548 |
\isadelimmlref
|
|
549 |
%
|
|
550 |
\endisadelimmlref
|
|
551 |
%
|
|
552 |
\isatagmlref
|
|
553 |
%
|
|
554 |
\begin{isamarkuptext}%
|
|
555 |
\begin{mldecls}
|
|
556 |
\indexmltype{Symbol.symbol}\verb|type Symbol.symbol| \\
|
|
557 |
\indexml{Symbol.explode}\verb|Symbol.explode: string -> Symbol.symbol list| \\
|
|
558 |
\indexml{Symbol.is-letter}\verb|Symbol.is_letter: Symbol.symbol -> bool| \\
|
|
559 |
\indexml{Symbol.is-digit}\verb|Symbol.is_digit: Symbol.symbol -> bool| \\
|
|
560 |
\indexml{Symbol.is-quasi}\verb|Symbol.is_quasi: Symbol.symbol -> bool| \\
|
20451
|
561 |
\indexml{Symbol.is-blank}\verb|Symbol.is_blank: Symbol.symbol -> bool| \\[1ex]
|
20438
|
562 |
\indexmltype{Symbol.sym}\verb|type Symbol.sym| \\
|
|
563 |
\indexml{Symbol.decode}\verb|Symbol.decode: Symbol.symbol -> Symbol.sym| \\
|
|
564 |
\end{mldecls}
|
|
565 |
|
|
566 |
\begin{description}
|
|
567 |
|
20451
|
568 |
\item \verb|Symbol.symbol| represents Isabelle symbols. This
|
|
569 |
type is an alias for \verb|string|, but emphasizes the
|
20438
|
570 |
specific format encountered here.
|
|
571 |
|
20477
|
572 |
\item \verb|Symbol.explode|~\isa{str} produces a symbol list
|
|
573 |
from the packed form that. This function supercedes \verb|String.explode| for virtually all purposes of manipulating text in
|
|
574 |
Isabelle!
|
20438
|
575 |
|
20477
|
576 |
\item \verb|Symbol.is_letter|, \verb|Symbol.is_digit|, \verb|Symbol.is_quasi|, \verb|Symbol.is_blank| classify standard
|
|
577 |
symbols according to fixed syntactic conventions of Isabelle, cf.\
|
|
578 |
\cite{isabelle-isar-ref}.
|
20438
|
579 |
|
|
580 |
\item \verb|Symbol.sym| is a concrete datatype that represents
|
20451
|
581 |
the different kinds of symbols explicitly with constructors \verb|Symbol.Char|, \verb|Symbol.Sym|, \verb|Symbol.Ctrl|, or \verb|Symbol.Raw|.
|
20438
|
582 |
|
|
583 |
\item \verb|Symbol.decode| converts the string representation of a
|
20451
|
584 |
symbol into the datatype version.
|
20438
|
585 |
|
|
586 |
\end{description}%
|
|
587 |
\end{isamarkuptext}%
|
|
588 |
\isamarkuptrue%
|
|
589 |
%
|
|
590 |
\endisatagmlref
|
|
591 |
{\isafoldmlref}%
|
|
592 |
%
|
|
593 |
\isadelimmlref
|
|
594 |
%
|
|
595 |
\endisadelimmlref
|
|
596 |
%
|
20477
|
597 |
\isamarkupsubsection{Basic names \label{sec:basic-names}%
|
20438
|
598 |
}
|
|
599 |
\isamarkuptrue%
|
|
600 |
%
|
|
601 |
\begin{isamarkuptext}%
|
20477
|
602 |
A \emph{basic name} essentially consists of a single Isabelle
|
|
603 |
identifier. There are conventions to mark separate classes of basic
|
|
604 |
names, by attaching a suffix of underscores (\isa{{\isacharunderscore}}): one
|
|
605 |
underscore means \emph{internal name}, two underscores means
|
|
606 |
\emph{Skolem name}, three underscores means \emph{internal Skolem
|
|
607 |
name}.
|
|
608 |
|
|
609 |
For example, the basic name \isa{foo} has the internal version
|
|
610 |
\isa{foo{\isacharunderscore}}, with Skolem versions \isa{foo{\isacharunderscore}{\isacharunderscore}} and \isa{foo{\isacharunderscore}{\isacharunderscore}{\isacharunderscore}}, respectively.
|
20471
|
611 |
|
20477
|
612 |
Such special versions are required for bookkeeping of names that are
|
|
613 |
apart from anything that may appear in the text given by the user.
|
|
614 |
In particular, system generated variables in high-level Isar proof
|
|
615 |
contexts are usually marked as internal, which prevents mysterious
|
|
616 |
name references such as \isa{xaa} in the text.
|
|
617 |
|
|
618 |
\medskip Basic manipulations of binding scopes requires names to be
|
|
619 |
modified. A \emph{name context} contains a collection of already
|
|
620 |
used names, which is maintained by the \isa{declare} operation.
|
20438
|
621 |
|
20477
|
622 |
The \isa{invents} operation derives a number of fresh names
|
|
623 |
derived from a given starting point. For example, three names
|
|
624 |
derived from \isa{a} are \isa{a}, \isa{b}, \isa{c},
|
|
625 |
provided there are no clashes with already used names.
|
20438
|
626 |
|
20477
|
627 |
The \isa{variants} operation produces fresh names by
|
|
628 |
incrementing given names as to base-26 numbers (with digits \isa{a{\isachardot}{\isachardot}z}). For example, name \isa{foo} results in variants
|
|
629 |
\isa{fooa}, \isa{foob}, \isa{fooc}, \dots, \isa{fooaa}, \isa{fooab}, \dots; each renaming step picks the next
|
|
630 |
unused variant from this list.%
|
20438
|
631 |
\end{isamarkuptext}%
|
|
632 |
\isamarkuptrue%
|
|
633 |
%
|
20451
|
634 |
\isadelimmlref
|
|
635 |
%
|
|
636 |
\endisadelimmlref
|
|
637 |
%
|
|
638 |
\isatagmlref
|
|
639 |
%
|
|
640 |
\begin{isamarkuptext}%
|
20477
|
641 |
\begin{mldecls}
|
|
642 |
\indexml{Name.internal}\verb|Name.internal: string -> string| \\
|
|
643 |
\indexml{Name.skolem}\verb|Name.skolem: string -> string| \\[1ex]
|
|
644 |
\indexmltype{Name.context}\verb|type Name.context| \\
|
|
645 |
\indexml{Name.context}\verb|Name.context: Name.context| \\
|
|
646 |
\indexml{Name.declare}\verb|Name.declare: string -> Name.context -> Name.context| \\
|
|
647 |
\indexml{Name.invents}\verb|Name.invents: Name.context -> string -> int -> string list| \\
|
|
648 |
\indexml{Name.variants}\verb|Name.variants: string list -> Name.context -> string list * Name.context| \\
|
|
649 |
\end{mldecls}
|
|
650 |
|
|
651 |
\begin{description}
|
|
652 |
|
|
653 |
\item \verb|Name.internal|~\isa{name} produces an internal name
|
|
654 |
by adding one underscore.
|
|
655 |
|
|
656 |
\item \verb|Name.skolem|~\isa{name} produces a Skolem name by
|
|
657 |
adding two underscores.
|
|
658 |
|
|
659 |
\item \verb|Name.context| represents the context of already used
|
|
660 |
names; the initial value is \verb|Name.context|.
|
|
661 |
|
|
662 |
\item \verb|Name.declare|~\isa{name} declares \isa{name} as
|
|
663 |
being used.
|
|
664 |
|
|
665 |
\item \verb|Name.invents|~\isa{context\ base\ n} produces \isa{n} fresh names derived from \isa{base}.
|
|
666 |
|
|
667 |
\end{description}%
|
|
668 |
\end{isamarkuptext}%
|
|
669 |
\isamarkuptrue%
|
|
670 |
%
|
|
671 |
\endisatagmlref
|
|
672 |
{\isafoldmlref}%
|
|
673 |
%
|
|
674 |
\isadelimmlref
|
|
675 |
%
|
|
676 |
\endisadelimmlref
|
|
677 |
%
|
|
678 |
\isamarkupsubsection{Indexed names%
|
|
679 |
}
|
|
680 |
\isamarkuptrue%
|
|
681 |
%
|
|
682 |
\begin{isamarkuptext}%
|
|
683 |
An \emph{indexed name} (or \isa{indexname}) is a pair of a basic
|
|
684 |
name with a natural number. This representation allows efficient
|
|
685 |
renaming by incrementing the second component only. To rename two
|
|
686 |
collections of indexnames apart from each other, first determine the
|
|
687 |
maximum index \isa{maxidx} of the first collection, then
|
|
688 |
increment all indexes of the second collection by \isa{maxidx\ {\isacharplus}\ {\isadigit{1}}}. Note that the maximum index of an empty collection is \isa{{\isacharminus}{\isadigit{1}}}.
|
|
689 |
|
|
690 |
Isabelle syntax observes the following rules for representing an
|
|
691 |
indexname \isa{{\isacharparenleft}x{\isacharcomma}\ i{\isacharparenright}} as a packed string:
|
|
692 |
|
|
693 |
\begin{itemize}
|
|
694 |
|
20481
|
695 |
\item \isa{{\isacharquery}x} if \isa{x} does not end with a digit and \isa{i\ {\isacharequal}\ {\isadigit{0}}},
|
20477
|
696 |
|
|
697 |
\item \isa{{\isacharquery}xi} if \isa{x} does not end with a digit,
|
|
698 |
|
|
699 |
\item \isa{{\isacharquery}x{\isachardot}i} else.
|
|
700 |
|
|
701 |
\end{itemize}
|
|
702 |
|
|
703 |
Occasionally, basic names and indexed names are injected into the
|
|
704 |
same pair type: the (improper) indexname \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}} is used
|
|
705 |
to encode basic names.
|
|
706 |
|
|
707 |
\medskip Indexnames may acquire arbitrary large index numbers over
|
|
708 |
time. Results are usually normalized towards \isa{{\isadigit{0}}} at certain
|
|
709 |
checkpoints, such that the very end of a proof. This works by
|
|
710 |
producing variants of the corresponding basic names
|
|
711 |
(\secref{sec:basic-names}). For example, the collection \isa{{\isacharquery}x{\isachardot}{\isadigit{1}}{\isacharcomma}\ {\isacharquery}x{\isachardot}{\isadigit{7}}{\isacharcomma}\ {\isacharquery}x{\isachardot}{\isadigit{4}}{\isadigit{2}}} then becomes \isa{{\isacharquery}x{\isacharcomma}\ {\isacharquery}xa{\isacharcomma}\ {\isacharquery}xb}.%
|
|
712 |
\end{isamarkuptext}%
|
|
713 |
\isamarkuptrue%
|
|
714 |
%
|
|
715 |
\isadelimmlref
|
|
716 |
%
|
|
717 |
\endisadelimmlref
|
|
718 |
%
|
|
719 |
\isatagmlref
|
|
720 |
%
|
|
721 |
\begin{isamarkuptext}%
|
|
722 |
\begin{mldecls}
|
|
723 |
\indexmltype{indexname}\verb|type indexname| \\
|
|
724 |
\end{mldecls}
|
|
725 |
|
|
726 |
\begin{description}
|
|
727 |
|
|
728 |
\item \verb|indexname| represents indexed names. This is an
|
|
729 |
abbreviation for \verb|string * int|. The second component is
|
|
730 |
usually non-negative, except for situations where \isa{{\isacharparenleft}x{\isacharcomma}\ {\isacharminus}{\isadigit{1}}{\isacharparenright}}
|
|
731 |
is used to embed plain names.
|
|
732 |
|
|
733 |
\end{description}%
|
20451
|
734 |
\end{isamarkuptext}%
|
|
735 |
\isamarkuptrue%
|
|
736 |
%
|
|
737 |
\endisatagmlref
|
|
738 |
{\isafoldmlref}%
|
|
739 |
%
|
|
740 |
\isadelimmlref
|
|
741 |
%
|
|
742 |
\endisadelimmlref
|
|
743 |
%
|
20477
|
744 |
\isamarkupsubsection{Qualified names and name spaces%
|
20438
|
745 |
}
|
|
746 |
\isamarkuptrue%
|
|
747 |
%
|
|
748 |
\begin{isamarkuptext}%
|
20477
|
749 |
A \emph{qualified name} consists of a non-empty sequence of basic
|
|
750 |
name components. The packed representation a dot as separator, for
|
|
751 |
example in ``\isa{A{\isachardot}b{\isachardot}c}''. The last component is called
|
|
752 |
\emph{base} name, the remaining prefix \emph{qualifier} (which may
|
20481
|
753 |
be empty). The basic idea of qualified names is to encode a
|
|
754 |
hierarchically structured name spaces by recording the access path
|
|
755 |
as part of the name. For example, \isa{A{\isachardot}b{\isachardot}c} may be understood
|
|
756 |
as a local entity \isa{c} within a local structure \isa{b}
|
|
757 |
of the enclosing structure \isa{A}. Typically, name space
|
|
758 |
hierarchies consist of 1--3 levels, but this need not be always so.
|
20477
|
759 |
|
|
760 |
The empty name is commonly used as an indication of unnamed
|
|
761 |
entities, if this makes any sense. The operations on qualified
|
|
762 |
names are smart enough to pass through such improper names
|
|
763 |
unchanged.
|
|
764 |
|
|
765 |
\medskip A \isa{naming} policy tells how to turn a name
|
|
766 |
specification into a fully qualified internal name (by the \isa{full} operation), and how to fully qualified names may be accessed
|
20481
|
767 |
externally. For example, the default naming prefixes an implicit
|
|
768 |
path from the context: \isa{x} is becomes \isa{path{\isachardot}x}
|
|
769 |
internally; the standard accesses include \isa{x}, \isa{path{\isachardot}x}, and further partial \isa{path} specifications.
|
20477
|
770 |
Normally, the naming is implicit in the theory or proof context.
|
20481
|
771 |
There are separate versions of the corresponding operations for
|
|
772 |
these context types.
|
20477
|
773 |
|
|
774 |
\medskip A \isa{name\ space} manages a collection of fully
|
|
775 |
internalized names, together with a mapping between external names
|
|
776 |
and internal names (in both directions). The corresponding \isa{intern} and \isa{extern} operations are mostly used for
|
|
777 |
parsing and printing only! The \isa{declare} operation augments
|
|
778 |
a name space according to a given naming policy.
|
|
779 |
|
|
780 |
By general convention, there are separate name spaces for each kind
|
|
781 |
of formal entity, such as logical constant, type, type class,
|
|
782 |
theorem etc. It is usually clear from the occurrence in concrete
|
|
783 |
syntax (or from the scope) which kind of entity a name refers to.
|
|
784 |
For example, the very same name \isa{c} may be used uniformly
|
|
785 |
for a constant, type, type class, which are from separate syntactic
|
20481
|
786 |
categories.
|
20477
|
787 |
|
20481
|
788 |
There are common schemes to name theorems systematically, according
|
|
789 |
to the name of the main logical entity being involved, such as
|
|
790 |
\isa{c{\isachardot}intro} and \isa{c{\isachardot}elim} for theorems related to
|
|
791 |
constant \isa{c}. This technique of mapping names from one
|
|
792 |
space into another requires some care in order to avoid conflicts.
|
|
793 |
In particular, theorem names derived from type or class names are
|
|
794 |
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
|
|
795 |
type \isa{c} and class \isa{c}, respectively.%
|
20438
|
796 |
\end{isamarkuptext}%
|
|
797 |
\isamarkuptrue%
|
|
798 |
%
|
20477
|
799 |
\isadelimmlref
|
|
800 |
%
|
|
801 |
\endisadelimmlref
|
|
802 |
%
|
|
803 |
\isatagmlref
|
20438
|
804 |
%
|
|
805 |
\begin{isamarkuptext}%
|
20477
|
806 |
\begin{mldecls}
|
|
807 |
\indexml{NameSpace.base}\verb|NameSpace.base: string -> string| \\
|
|
808 |
\indexml{NameSpace.drop-base}\verb|NameSpace.drop_base: string -> string| \\
|
|
809 |
\indexml{NameSpace.append}\verb|NameSpace.append: string -> string -> string| \\
|
|
810 |
\indexml{NameSpace.pack}\verb|NameSpace.pack: string list -> string| \\
|
|
811 |
\indexml{NameSpace.unpack}\verb|NameSpace.unpack: string -> string list| \\[1ex]
|
|
812 |
\indexmltype{NameSpace.naming}\verb|type NameSpace.naming| \\
|
|
813 |
\indexml{NameSpace.default-naming}\verb|NameSpace.default_naming: NameSpace.naming| \\
|
|
814 |
\indexml{NameSpace.add-path}\verb|NameSpace.add_path: string -> NameSpace.naming -> NameSpace.naming| \\
|
|
815 |
\indexml{NameSpace.full}\verb|NameSpace.full: NameSpace.naming -> string -> string| \\[1ex]
|
|
816 |
\indexmltype{NameSpace.T}\verb|type NameSpace.T| \\
|
|
817 |
\indexml{NameSpace.empty}\verb|NameSpace.empty: NameSpace.T| \\
|
|
818 |
\indexml{NameSpace.merge}\verb|NameSpace.merge: NameSpace.T * NameSpace.T -> NameSpace.T| \\
|
|
819 |
\indexml{NameSpace.declare}\verb|NameSpace.declare: NameSpace.naming -> string -> NameSpace.T -> NameSpace.T| \\
|
|
820 |
\indexml{NameSpace.intern}\verb|NameSpace.intern: NameSpace.T -> string -> string| \\
|
|
821 |
\indexml{NameSpace.extern}\verb|NameSpace.extern: NameSpace.T -> string -> string| \\
|
|
822 |
\end{mldecls}
|
|
823 |
|
|
824 |
\begin{description}
|
|
825 |
|
|
826 |
\item \verb|NameSpace.base|~\isa{name} returns the base name of a
|
|
827 |
qualified name.
|
|
828 |
|
|
829 |
\item \verb|NameSpace.drop_base|~\isa{name} returns the qualifier
|
|
830 |
of a qualified name.
|
|
831 |
|
|
832 |
\item \verb|NameSpace.append|~\isa{name\isactrlisub {\isadigit{1}}\ name\isactrlisub {\isadigit{2}}}
|
|
833 |
appends two qualified names.
|
|
834 |
|
|
835 |
\item \verb|NameSpace.pack|~\isa{name} and \isa{NameSpace{\isachardot}unpack}~\isa{names} convert between the packed
|
|
836 |
string representation and explicit list form of qualified names.
|
|
837 |
|
|
838 |
\item \verb|NameSpace.naming| represents the abstract concept of
|
|
839 |
a naming policy.
|
|
840 |
|
|
841 |
\item \verb|NameSpace.default_naming| is the default naming policy.
|
|
842 |
In a theory context, this is usually augmented by a path prefix
|
|
843 |
consisting of the theory name.
|
|
844 |
|
|
845 |
\item \verb|NameSpace.add_path|~\isa{path\ naming} augments the
|
|
846 |
naming policy by extending its path.
|
|
847 |
|
|
848 |
\item \verb|NameSpace.full|\isa{naming\ name} turns a name
|
|
849 |
specification (usually a basic name) into the fully qualified
|
|
850 |
internal version, according to the given naming policy.
|
|
851 |
|
|
852 |
\item \verb|NameSpace.T| represents name spaces.
|
|
853 |
|
|
854 |
\item \verb|NameSpace.empty| and \verb|NameSpace.merge|~\isa{{\isacharparenleft}space\isactrlisub {\isadigit{1}}{\isacharcomma}\ space\isactrlisub {\isadigit{2}}{\isacharparenright}} provide basic operations for
|
|
855 |
building name spaces in accordance to the usual theory data
|
|
856 |
management (\secref{sec:context-data}).
|
|
857 |
|
|
858 |
\item \verb|NameSpace.declare|~\isa{naming\ name\ space} enters a
|
|
859 |
fully qualified name into the name space, with partial accesses
|
|
860 |
being derived from the given policy.
|
|
861 |
|
|
862 |
\item \verb|NameSpace.intern|~\isa{space\ name} internalizes a
|
|
863 |
(partially qualified) external name.
|
|
864 |
|
|
865 |
This operation is mostly for parsing. Note that fully qualified
|
|
866 |
names stemming from declarations are produced via \verb|NameSpace.full| (or derivatives for \verb|theory| or \verb|Proof.context|).
|
|
867 |
|
|
868 |
\item \verb|NameSpace.extern|~\isa{space\ name} externalizes a
|
|
869 |
(fully qualified) internal name.
|
|
870 |
|
|
871 |
This operation is mostly for printing. Note unqualified names are
|
|
872 |
produced via \verb|NameSpace.base|.
|
|
873 |
|
|
874 |
\end{description}%
|
20438
|
875 |
\end{isamarkuptext}%
|
|
876 |
\isamarkuptrue%
|
|
877 |
%
|
20477
|
878 |
\endisatagmlref
|
|
879 |
{\isafoldmlref}%
|
20438
|
880 |
%
|
20477
|
881 |
\isadelimmlref
|
|
882 |
%
|
|
883 |
\endisadelimmlref
|
20438
|
884 |
%
|
18537
|
885 |
\isadelimtheory
|
|
886 |
%
|
|
887 |
\endisadelimtheory
|
|
888 |
%
|
|
889 |
\isatagtheory
|
|
890 |
\isacommand{end}\isamarkupfalse%
|
|
891 |
%
|
|
892 |
\endisatagtheory
|
|
893 |
{\isafoldtheory}%
|
|
894 |
%
|
|
895 |
\isadelimtheory
|
|
896 |
%
|
|
897 |
\endisadelimtheory
|
|
898 |
\isanewline
|
|
899 |
\end{isabellebody}%
|
|
900 |
%%% Local Variables:
|
|
901 |
%%% mode: latex
|
|
902 |
%%% TeX-master: "root"
|
|
903 |
%%% End:
|