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Quelle Local_Theory.thySprache: Isabelle |
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(*:maxLineLen=78:*) theory Local_Theory imports Base begin chapter ‹Local theory specifications \label{ch:local-theory}› text ‹ java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "m \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 🪙‹target context›. The target is usually derived from the background theory by adding local ‹🪙› and ‹🪙› 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: ‹🪙› for terms and ‹🪙› 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. section ‹Definitional elements› text ‹ There are separate elements ‹🪙 c ≡ t› for terms, and ‹🪙 b = thm› for theorems. Types are treated implicitly, according to Hindley-Milner discipline (cf.\ \secref{sec:variables}). These definitional primitives essentially act like ‹let›-bindings within a local context that may already contain earlier ‹let›-bindings and some initial ‹λ›-bindings. Thus we gain 🪙‹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 & ‹λ›-binding & ‹let›-binding \\ \hline types & fixed ‹α› & arbitrary ‹β› \\ terms & ‹🪙 x :: τ› & ‹🪙 c ≡ t› \\ theorems & ‹🪙 a: A› & ‹🪙 b = 🪙B \hline \end{tabular} \end{center} A user package merely needs to produce suitable ‹🪙› and ‹🪙› elements according to the application. For example, a package for inductive definitions might first ‹🪙› a certain predicate as some fixed-point construction, then ‹🪙› a proven result about monotonicity of the functor involved here, and then produce further derived concepts via additional ‹🪙› and ‹🪙› elements. The cumulative sequence of ‹🪙› and ‹🪙› produced at package runtime is managed by the local theory infrastructure by means of an 🪙‹auxiliary context›. Thus the system holds up the impression of working within a fully abstract situation with hypothetical entities: ‹🪙 c ≡ t› always results in a literal fact ‹🪙c ≡ t🪙›, where ‹c› is a fixed variable ‹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\fram \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 ‹🪙› and ‹🪙› elements, transformed by the particular target policy (see cite‹‹\S4--5› in "Haftmann-Wenzel:2009"› for details). › text %mlref ‹ \begin{mldecls} @{define_ML_type local_theory = Proof.context} \\ @{define_ML Named_Target.init: "Bundle.name list -> string -> theory -> local_th @{define_ML Local_Theory.define: "(binding * mixfix) * (Attrib.binding * term) - local_theory -> (term * (string * thm)) * local_theory"} \\ @{define_ML Local_Theory.note: "Attrib.binding * thm list -> local_theory -> (string * thm list) * local_theory"} \\ \end{mldecls} 🪙 Type 🪙‹local_theory› represents local theories. Although this is merely an alias for 🪙‹Proof.context›, it is semantically a subtype of the same: a 🪙‹local_theory› holds target information as special context data. Subtyping means that any value ‹lthy:›~🪙‹local_theory› can be also used with operations on expecting a regular ‹ctxt:›~🪙‹Proof.context 🪙 🪙‹Named_Target.init›~‹includes name thy› initializes a local theory derived from the given background theory. An empty name refers to a 🪙‹global theory› 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. 🪙 🪙‹Local_Theory.define›~‹((b, mx), (a, rhs)) lthy› defines a local entity according to the specification that is given relatively to the current ‹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 ‹b_def›. Any given attributes are applied to that same fact --- immediately in the auxiliary context 🪙‹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. 🪙 🪙‹Local_Theory.note›~‹(a, ths) lthy› is analogous to 🪙‹Local_Theory.define› slightly more general variant 🪙‹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. section ‹Morphisms and declarations \label{sec:morphisms}› text ‹ %FIXME See also cite‹"Chaieb-Wenzel:2007"›. › end
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2026-06-09