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+++ b/doc-src/IsarImplementation/Thy/Local_Theory.thy Wed Mar 04 11:05:29 2009 +0100
@@ -0,0 +1,168 @@
+theory Local_Theory
+imports Base
+begin
+
+chapter {* Local theory specifications *}
+
+text {*
+ 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 @{text "\<FIX>"} and @{text "\<ASSUME>"} 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: @{text "\<DEFINE>"} for terms and
+ @{text "\<NOTE>"} 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.
+*}
+
+
+section {* Definitional elements *}
+
+text {*
+ There are separate elements @{text "\<DEFINE> c \<equiv> t"} for terms, and
+ @{text "\<NOTE> b = thm"} for theorems. Types are treated
+ implicitly, according to Hindley-Milner discipline (cf.\
+ \secref{sec:variables}). These definitional primitives essentially
+ act like @{text "let"}-bindings within a local context that may
+ already contain earlier @{text "let"}-bindings and some initial
+ @{text "\<lambda>"}-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
+ & @{text "\<lambda>"}-binding & @{text "let"}-binding \\
+ \hline
+ types & fixed @{text "\<alpha>"} & arbitrary @{text "\<beta>"} \\
+ terms & @{text "\<FIX> x :: \<tau>"} & @{text "\<DEFINE> c \<equiv> t"} \\
+ theorems & @{text "\<ASSUME> a: A"} & @{text "\<NOTE> b = \<^BG>B\<^EN>"} \\
+ \hline
+ \end{tabular}
+ \end{center}
+
+ A user package merely needs to produce suitable @{text "\<DEFINE>"}
+ and @{text "\<NOTE>"} elements according to the application. For
+ example, a package for inductive definitions might first @{text
+ "\<DEFINE>"} a certain predicate as some fixed-point construction,
+ then @{text "\<NOTE>"} a proven result about monotonicity of the
+ functor involved here, and then produce further derived concepts via
+ additional @{text "\<DEFINE>"} and @{text "\<NOTE>"} elements.
+
+ The cumulative sequence of @{text "\<DEFINE>"} and @{text "\<NOTE>"}
+ 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: @{text "\<DEFINE> c \<equiv> t"}
+ always results in a literal fact @{text "\<^BG>c \<equiv> t\<^EN>"}, where
+ @{text "c"} is a fixed variable @{text "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
+ @{text "\<DEFINE>"} and @{text "\<NOTE>"} elements, transformed by
+ the particular target policy (see
+ \cite[\S4--5]{Haftmann-Wenzel:2009} for details).
+*}
+
+text %mlref {*
+ \begin{mldecls}
+ @{index_ML_type local_theory: Proof.context} \\
+ @{index_ML TheoryTarget.init: "string option -> theory -> local_theory"} \\[1ex]
+ @{index_ML LocalTheory.define: "string ->
+ (binding * mixfix) * (Attrib.binding * term) -> local_theory ->
+ (term * (string * thm)) * local_theory"} \\
+ @{index_ML LocalTheory.note: "string ->
+ Attrib.binding * thm list -> local_theory ->
+ (string * thm list) * local_theory"} \\
+ \end{mldecls}
+
+ \begin{description}
+
+ \item @{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 @{text "lthy:"}~@{ML_type local_theory} can be also used
+ with operations on expecting a regular @{text "ctxt:"}~@{ML_type
+ Proof.context}.
+
+ \item @{ML TheoryTarget.init}~@{text "NONE thy"} initializes a
+ trivial local theory from the given background theory.
+ Alternatively, @{text "SOME name"} may be given to initialize 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.
+
+ \item @{ML LocalTheory.define}~@{text "kind ((b, mx), (a, rhs))
+ lthy"} defines a local entity according to the specification that is
+ given relatively to the current @{text "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 @{text "b_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 @{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.
+
+ The @{text kind} determines the theorem kind tag of the resulting
+ fact. Typical examples are @{ML Thm.definitionK}, @{ML
+ Thm.theoremK}, or @{ML Thm.internalK}.
+
+ \item @{ML LocalTheory.note}~@{text "kind (a, ths) lthy"} is
+ analogous to @{ML LocalTheory.define}, but defines facts instead of
+ terms. There is also a slightly more general variant @{ML
+ LocalTheory.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.
+
+ \end{description}
+*}
+
+
+section {* Morphisms and declarations *}
+
+text FIXME
+
+end