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\isamarkupchapter{Local theory specifications%
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\begin{isamarkuptext}%
A \emph{local theory} combines aspects of both theory and proof
context (cf.\ \secref{sec:context}), such that definitional
specifications may be given relatively to parameters and
assumptions. A local theory is represented as a regular proof
context, augmented by administrative data about the \emph{target
context}.
The target is usually derived from the background theory by adding
local \isa{{\isasymFIX}} and \isa{{\isasymASSUME}} elements, plus
suitable modifications of non-logical context data (e.g.\ a special
type-checking discipline). Once initialized, the target is ready to
absorb definitional primitives: \isa{{\isasymDEFINE}} for terms and
\isa{{\isasymNOTE}} for theorems. Such definitions may get
transformed in a target-specific way, but the programming interface
hides such details.
Isabelle/Pure provides target mechanisms for locales, type-classes,
type-class instantiations, and general overloading. In principle,
users can implement new targets as well, but this rather arcane
discipline is beyond the scope of this manual. In contrast,
implementing derived definitional packages to be used within a local
theory context is quite easy: the interfaces are even simpler and
more abstract than the underlying primitives for raw theories.
Many definitional packages for local theories are available in
Isabelle. Although a few old packages only work for global
theories, the local theory interface is already the standard way of
implementing definitional packages in Isabelle.%
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\isamarkupsection{Definitional elements%
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There are separate elements \isa{{\isasymDEFINE}\ c\ {\isasymequiv}\ t} for terms, and
\isa{{\isasymNOTE}\ b\ {\isacharequal}\ thm} for theorems. Types are treated
implicitly, according to Hindley-Milner discipline (cf.\
\secref{sec:variables}). These definitional primitives essentially
act like \isa{let}-bindings within a local context that may
already contain earlier \isa{let}-bindings and some initial
\isa{{\isasymlambda}}-bindings. Thus we gain \emph{dependent definitions}
that are relative to an initial axiomatic context. The following
diagram illustrates this idea of axiomatic elements versus
definitional elements:
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
& \isa{{\isasymlambda}}-binding & \isa{let}-binding \\
\hline
types & fixed \isa{{\isasymalpha}} & arbitrary \isa{{\isasymbeta}} \\
terms & \isa{{\isasymFIX}\ x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}} & \isa{{\isasymDEFINE}\ c\ {\isasymequiv}\ t} \\
theorems & \isa{{\isasymASSUME}\ a{\isacharcolon}\ A} & \isa{{\isasymNOTE}\ b\ {\isacharequal}\ \isactrlBG B\isactrlEN } \\
\hline
\end{tabular}
\end{center}
A user package merely needs to produce suitable \isa{{\isasymDEFINE}}
and \isa{{\isasymNOTE}} elements according to the application. For
example, a package for inductive definitions might first \isa{{\isasymDEFINE}} a certain predicate as some fixed-point construction,
then \isa{{\isasymNOTE}} a proven result about monotonicity of the
functor involved here, and then produce further derived concepts via
additional \isa{{\isasymDEFINE}} and \isa{{\isasymNOTE}} elements.
The cumulative sequence of \isa{{\isasymDEFINE}} and \isa{{\isasymNOTE}}
produced at package runtime is managed by the local theory
infrastructure by means of an \emph{auxiliary context}. Thus the
system holds up the impression of working within a fully abstract
situation with hypothetical entities: \isa{{\isasymDEFINE}\ c\ {\isasymequiv}\ t}
always results in a literal fact \isa{\isactrlBG c\ {\isasymequiv}\ t\isactrlEN }, where
\isa{c} is a fixed variable \isa{c}. The details about
global constants, name spaces etc. are handled internally.
So the general structure of a local theory is a sandwich of three
layers:
\begin{center}
\framebox{\quad auxiliary context \quad\framebox{\quad target context \quad\framebox{\quad background theory\quad}}}
\end{center}
\noindent When a definitional package is finished, the auxiliary
context is reset to the target context. The target now holds
definitions for terms and theorems that stem from the hypothetical
\isa{{\isasymDEFINE}} and \isa{{\isasymNOTE}} elements, transformed by
the particular target policy (see
\cite[\S4--5]{Haftmann-Wenzel:2009} for details).%
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\begin{mldecls}
\indexdef{}{ML type}{local\_theory}\verb|type local_theory = Proof.context| \\
\indexdef{}{ML}{Theory\_Target.init}\verb|Theory_Target.init: string option -> theory -> local_theory| \\[1ex]
\indexdef{}{ML}{Local\_Theory.define}\verb|Local_Theory.define: string ->|\isasep\isanewline%
\verb| (binding * mixfix) * (Attrib.binding * term) -> local_theory ->|\isasep\isanewline%
\verb| (term * (string * thm)) * local_theory| \\
\indexdef{}{ML}{Local\_Theory.note}\verb|Local_Theory.note: Attrib.binding * thm list ->|\isasep\isanewline%
\verb| local_theory -> (string * thm list) * local_theory| \\
\end{mldecls}
\begin{description}
\item \verb|local_theory| represents local theories. Although
this is merely an alias for \verb|Proof.context|, it is
semantically a subtype of the same: a \verb|local_theory| holds
target information as special context data. Subtyping means that
any value \isa{lthy{\isacharcolon}}~\verb|local_theory| can be also used
with operations on expecting a regular \isa{ctxt{\isacharcolon}}~\verb|Proof.context|.
\item \verb|Theory_Target.init|~\isa{NONE\ thy} initializes a
trivial local theory from the given background theory.
Alternatively, \isa{SOME\ name} may be given to initialize a
\hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}} or \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} context (a fully-qualified
internal name is expected here). This is useful for experimentation
--- normally the Isar toplevel already takes care to initialize the
local theory context.
\item \verb|Local_Theory.define|~\isa{kind\ {\isacharparenleft}{\isacharparenleft}b{\isacharcomma}\ mx{\isacharparenright}{\isacharcomma}\ {\isacharparenleft}a{\isacharcomma}\ rhs{\isacharparenright}{\isacharparenright}\ lthy} defines a local entity according to the specification that is
given relatively to the current \isa{lthy} context. In
particular the term of the RHS may refer to earlier local entities
from the auxiliary context, or hypothetical parameters from the
target context. The result is the newly defined term (which is
always a fixed variable with exactly the same name as specified for
the LHS), together with an equational theorem that states the
definition as a hypothetical fact.
Unless an explicit name binding is given for the RHS, the resulting
fact will be called \isa{b{\isacharunderscore}def}. Any given attributes are
applied to that same fact --- immediately in the auxiliary context
\emph{and} in any transformed versions stemming from target-specific
policies or any later interpretations of results from the target
context (think of \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}} and \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}},
for example). This means that attributes should be usually plain
declarations such as \hyperlink{attribute.simp}{\mbox{\isa{simp}}}, while non-trivial rules like
\hyperlink{attribute.simplified}{\mbox{\isa{simplified}}} are better avoided.
The \isa{kind} determines the theorem kind tag of the resulting
fact. Typical examples are \verb|Thm.definitionK| or \verb|Thm.theoremK|.
\item \verb|Local_Theory.note|~\isa{{\isacharparenleft}a{\isacharcomma}\ ths{\isacharparenright}\ lthy} is
analogous to \verb|Local_Theory.define|, but defines facts instead of
terms. There is also a slightly more general variant \verb|Local_Theory.notes| that defines several facts (with attribute
expressions) simultaneously.
This is essentially the internal version of the \hyperlink{command.lemmas}{\mbox{\isa{\isacommand{lemmas}}}}
command, or \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}} if an empty name binding is given.
\end{description}%
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\isamarkupsection{Morphisms and declarations%
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FIXME%
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