(*:maxLineLen=78:*)
theory Local_Theory
imports Base
begin
chapter \<open>Local theory specifications \label{ch:local-theory}\<close>
text \<open>
A \<^emph>\<open>local theory\<close> 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>\<open>target
context\<close>.
The target is usually derived from the background theory by adding local
\<open>\<FIX>\<close> and \<open>\<ASSUME>\<close> 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:
\<open>\<DEFINE>\<close> for terms and \<open>\<NOTE>\<close> 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 standard way
of implementing definitional packages in Isabelle is via the local theory
interface.
\<close>
section \<open>Definitional elements\<close>
text \<open>
There are separate elements \<open>\<DEFINE> c \<equiv> t\<close> for terms, and \<open>\<NOTE> b =
thm\<close> for theorems. Types are treated implicitly, according to Hindley-Milner
discipline (cf.\ \secref{sec:variables}). These definitional primitives
essentially act like \<open>let\<close>-bindings within a local context that may already
contain earlier \<open>let\<close>-bindings and some initial \<open>\<lambda>\<close>-bindings. Thus we gain
\<^emph>\<open>dependent definitions\<close> 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
& \<open>\<lambda>\<close>-binding & \<open>let\<close>-binding \\
\hline
types & fixed \<open>\<alpha>\<close> & arbitrary \<open>\<beta>\<close> \\
terms & \<open>\<FIX> x :: \<tau>\<close> & \<open>\<DEFINE> c \<equiv> t\<close> \\
theorems & \<open>\<ASSUME> a: A\<close> & \<open>\<NOTE> b = \<^BG>B\<^EN>\<close> \\
\hline
\end{tabular}
\end{center}
A user package merely needs to produce suitable \<open>\<DEFINE>\<close> and \<open>\<NOTE>\<close>
elements according to the application. For example, a package for inductive
definitions might first \<open>\<DEFINE>\<close> a certain predicate as some fixed-point
construction, then \<open>\<NOTE>\<close> a proven result about monotonicity of the
functor involved here, and then produce further derived concepts via
additional \<open>\<DEFINE>\<close> and \<open>\<NOTE>\<close> elements.
The cumulative sequence of \<open>\<DEFINE>\<close> and \<open>\<NOTE>\<close> produced at package
runtime is managed by the local theory infrastructure by means of an
\<^emph>\<open>auxiliary context\<close>. Thus the system holds up the impression of working
within a fully abstract situation with hypothetical entities: \<open>\<DEFINE> c \<equiv>
t\<close> always results in a literal fact \<open>\<^BG>c \<equiv> t\<^EN>\<close>, where \<open>c\<close> is a
fixed variable \<open>c\<close>. 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}
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 \<open>\<DEFINE>\<close> and \<open>\<NOTE>\<close> elements,
transformed by the particular target policy (see @{cite \<open>\S4--5\<close>
"Haftmann-Wenzel:2009"} for details).
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML_type local_theory: Proof.context} \\
@{index_ML Named_Target.init: "(local_theory -> local_theory) -> string ->
theory -> local_theory"} \\[1ex]
@{index_ML Local_Theory.define: "(binding * mixfix) * (Attrib.binding * term) ->
local_theory -> (term * (string * thm)) * local_theory"} \\
@{index_ML Local_Theory.note: "Attrib.binding * thm list ->
local_theory -> (string * thm list) * local_theory"} \\
\end{mldecls}
\<^descr> Type @{ML_type local_theory} represents local theories. Although this is
merely an alias for @{ML_type Proof.context}, it is semantically a subtype
of the same: a @{ML_type local_theory} holds target information as special
context data. Subtyping means that any value \<open>lthy:\<close>~@{ML_type local_theory}
can be also used with operations on expecting a regular \<open>ctxt:\<close>~@{ML_type
Proof.context}.
\<^descr> @{ML Named_Target.init}~\<open>before_exit name thy\<close> initializes a local theory
derived from the given background theory. An empty name refers to a \<^emph>\<open>global
theory\<close> context, and a non-empty name refers to a @{command locale} or
@{command 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.
\<^descr> @{ML Local_Theory.define}~\<open>((b, mx), (a, rhs)) lthy\<close> defines a local
entity according to the specification that is given relatively to the
current \<open>lthy\<close> 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 \<open>b_def\<close>. Any given attributes are applied to that same fact
--- immediately in the auxiliary context \<^emph>\<open>and\<close> in any transformed versions
stemming from target-specific policies or any later interpretations of
results from the target context (think of @{command locale} and @{command
interpretation}, for example). This means that attributes should be usually
plain declarations such as @{attribute simp}, while non-trivial rules like
@{attribute simplified} are better avoided.
\<^descr> @{ML Local_Theory.note}~\<open>(a, ths) lthy\<close> is analogous to @{ML
Local_Theory.define}, but defines facts instead of terms. There is also a
slightly more general variant @{ML Local_Theory.notes} that defines several
facts (with attribute expressions) simultaneously.
This is essentially the internal version of the @{command lemmas} command,
or @{command declare} if an empty name binding is given.
\<close>
section \<open>Morphisms and declarations \label{sec:morphisms}\<close>
text \<open>
%FIXME
See also @{cite "Chaieb-Wenzel:2007"}.
\<close>
end