author | wenzelm |
Sun, 31 Jan 2010 21:40:44 +0100 | |
changeset 34925 | 38a44d813a3c |
parent 34924 | 520727474bbe |
child 34926 | 19294b07e445 |
permissions | -rw-r--r-- |
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theory Prelim |
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imports Base |
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begin |
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chapter {* Preliminaries *} |
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section {* Contexts \label{sec:context} *} |
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text {* |
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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 @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<phi>"}, which means that a |
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proposition @{text "\<phi>"} is derivable from hypotheses @{text "\<Gamma>"} |
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within the theory @{text "\<Theta>"}. There are logical reasons for |
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keeping @{text "\<Theta>"} and @{text "\<Gamma>"} 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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@{text "\<Gamma> \<turnstile> \<phi>"} 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.\ @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> |
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\<phi>"} implies @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta>\<^sub>' \<phi>"} for contexts @{text "\<Theta>' |
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\<supseteq> \<Theta>"} and @{text "\<Gamma>' \<supseteq> \<Gamma>"}. |
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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.\ @{text "\<Gamma>' \<turnstile>\<^sub>\<Theta> \<phi>"} |
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implies @{text "\<Gamma> \<turnstile>\<^sub>\<Theta> \<Delta> \<Longrightarrow> \<phi>"} where @{text "\<Gamma>' \<supseteq> \<Gamma>"} and |
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@{text "\<Delta> = \<Gamma>' - \<Gamma>"}. Note that @{text "\<Theta>"} remains unchanged here, |
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only the @{text "\<Gamma>"} 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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@{text "\<Theta>"} and @{text "\<Gamma>"} 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 @{text |
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"\<Theta>"} and @{text "\<Gamma>"} 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 @{text "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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*} |
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subsection {* Theory context \label{sec:context-theory} *} |
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text {* A \emph{theory} is a data container with explicit name and |
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unique 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. To this end, the system maintains a set of |
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symbolic ``identification stamps'' within each theory. |
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In order to avoid the full-scale overhead of explicit sub-theory |
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identification of arbitrary intermediate stages, a theory is |
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switched into @{text "draft"} mode under certain circumstances. A |
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draft theory acts like a linear type, where updates invalidate |
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earlier versions. An invalidated draft is called \emph{stale}. |
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The @{text "checkpoint"} operation produces a safe stepping stone |
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that will survive the next update without becoming stale: both the |
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old and the new theory remain valid and are related by the |
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sub-theory relation. Checkpointing essentially recovers purely |
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functional theory values, at the expense of some extra internal |
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bookkeeping. |
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The @{text "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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The @{text "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 (according to the nominal sub-theory relation). |
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The @{text "begin"} operation starts a new theory by importing |
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several parent theories and entering a special mode of nameless |
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incremental updates, until the final @{text "end"} operation is |
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performed. |
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\medskip The example in \figref{fig:ex-theory} below shows a theory |
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graph derived from @{text "Pure"}, with theory @{text "Length"} |
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importing @{text "Nat"} and @{text "List"}. The body of @{text |
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"Length"} consists of a sequence of updates, working mostly on |
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drafts internally, while transaction boundaries of Isar top-level |
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commands (\secref{sec:isar-toplevel}) are guaranteed to be safe |
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checkpoints. |
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\begin{figure}[htb] |
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\begin{center} |
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\begin{tabular}{rcccl} |
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& & @{text "Pure"} \\ |
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& & @{text "\<down>"} \\ |
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& & @{text "FOL"} \\ |
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& $\swarrow$ & & $\searrow$ & \\ |
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@{text "Nat"} & & & & @{text "List"} \\ |
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& $\searrow$ & & $\swarrow$ \\ |
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& & @{text "Length"} \\ |
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& & \multicolumn{3}{l}{~~@{keyword "imports"}} \\ |
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& & \multicolumn{3}{l}{~~@{keyword "begin"}} \\ |
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& & $\vdots$~~ \\ |
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& & @{text "\<bullet>"}~~ \\ |
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& & $\vdots$~~ \\ |
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& & @{text "\<bullet>"}~~ \\ |
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& & $\vdots$~~ \\ |
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& & \multicolumn{3}{l}{~~@{command "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. Dynamic updating stops after |
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an explicit @{text "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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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML_type theory} \\ |
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@{index_ML Theory.subthy: "theory * theory -> bool"} \\ |
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@{index_ML Theory.checkpoint: "theory -> theory"} \\ |
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@{index_ML Theory.copy: "theory -> theory"} \\ |
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@{index_ML Theory.merge: "theory * theory -> theory"} \\ |
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@{index_ML Theory.begin_theory: "string -> theory list -> theory"} \\ |
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\end{mldecls} |
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\begin{mldecls} |
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@{index_ML_type theory_ref} \\ |
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@{index_ML Theory.deref: "theory_ref -> theory"} \\ |
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@{index_ML Theory.check_thy: "theory -> theory_ref"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML_type theory} represents theory contexts. This is |
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essentially a linear type, with explicit runtime checking! Most |
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internal theory operations destroy the original version, which then |
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becomes ``stale''. |
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\item @{ML "Theory.subthy"}~@{text "(thy\<^sub>1, thy\<^sub>2)"} compares theories |
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according to the intrinsic graph structure of the construction. |
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This sub-theory relation is a nominal approximation of inclusion |
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(@{text "\<subseteq>"}) of the corresponding content (according to the |
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semantics of the ML modules that implement the data). |
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\item @{ML "Theory.checkpoint"}~@{text "thy"} produces a safe |
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stepping stone in the linear development of @{text "thy"}. This |
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changes the old theory, but the next update will result in two |
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related, valid theories. |
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\item @{ML "Theory.copy"}~@{text "thy"} produces a variant of @{text |
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"thy"} with the same data. The copy is not related to the original, |
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but the original is unchanged. |
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\item @{ML "Theory.merge"}~@{text "(thy\<^sub>1, thy\<^sub>2)"} absorbs one theory |
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into the other, without changing @{text "thy\<^sub>1"} or @{text "thy\<^sub>2"}. |
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This version of ad-hoc theory merge fails for unrelated theories! |
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\item @{ML "Theory.begin_theory"}~@{text "name parents"} constructs |
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a new theory based on the given parents. This {\ML} function is |
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normally not invoked directly. |
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\item @{ML_type 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 @{ML "Theory.deref"}~@{text "thy_ref"} turns a @{ML_type |
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"theory_ref"} into an @{ML_type "theory"} value. As the referenced |
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theory evolves monotonically over time, later invocations of @{ML |
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"Theory.deref"} may refer to a larger context. |
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\item @{ML "Theory.check_thy"}~@{text "thy"} produces a @{ML_type |
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"theory_ref"} from a valid @{ML_type "theory"} value. |
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\end{description} |
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*} |
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text %mlex {* |
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The following artificial example demonstrates theory |
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data: we maintain a set of terms that are supposed to be wellformed |
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wrt.\ the enclosing theory. The public interface is as follows: |
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*} |
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ML {* |
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signature WELLFORMED_TERMS = |
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sig |
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val get: theory -> term list |
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val add: term -> theory -> theory |
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end; |
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*} |
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text {* \noindent The implementation uses private theory data |
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internally, and only exposes an operation that involves explicit |
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argument checking wrt.\ the given theory. *} |
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ML {* |
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structure Wellformed_Terms: WELLFORMED_TERMS = |
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struct |
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structure Terms = Theory_Data |
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( |
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type T = term OrdList.T; |
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val empty = []; |
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val extend = I; |
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fun merge (ts1, ts2) = |
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OrdList.union TermOrd.fast_term_ord ts1 ts2; |
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) |
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val get = Terms.get; |
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fun add raw_t thy = |
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let val t = Sign.cert_term thy raw_t |
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in Terms.map (OrdList.insert TermOrd.fast_term_ord t) thy end; |
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end; |
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*} |
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text {* We use @{ML_type "term OrdList.T"} for reasonably efficient |
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representation of a set of terms: all operations are linear in the |
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number of stored elements. Here we assume that our users do not |
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care about the declaration order, since that data structure forces |
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its own arrangement of elements. |
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Observe how the @{verbatim merge} operation joins the data slots of |
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the two constituents: @{ML OrdList.union} prevents duplication of |
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common data from different branches, thus avoiding the danger of |
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exponential blowup. (Plain list append etc.\ must never be used for |
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theory data merges.) |
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\medskip Our intended invariant is achieved as follows: |
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\begin{enumerate} |
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\item @{ML Wellformed_Terms.add} only admits terms that have passed |
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the @{ML Sign.cert_term} check of the given theory at that point. |
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\item Wellformedness in the sense of @{ML Sign.cert_term} is |
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monotonic wrt.\ the sub-theory relation. So our data can move |
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upwards in the hierarchy (via extension or merges), and maintain |
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wellformedness without further checks. |
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\end{enumerate} |
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Note that all basic operations of the inference kernel (which |
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includes @{ML Sign.cert_term}) observe this monotonicity principle, |
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but other user-space tools don't. For example, fully-featured |
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type-inference via @{ML Syntax.check_term} (cf.\ |
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\secref{sec:term-check}) is not necessarily monotonic wrt.\ the |
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background theory, since constraints of term constants can be |
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strengthened by later declarations, for example. |
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In most cases, user-space context data does not have to take such |
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invariants too seriously. The situation is different in the |
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implementation of the inference kernel itself, which uses the very |
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same data mechanisms for types, constants, axioms etc. |
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*} |
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subsection {* Proof context \label{sec:context-proof} *} |
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text {* A proof context is a container for pure data with a |
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back-reference to the theory it belongs to. The @{text "init"} |
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operation creates a proof context from a given theory. |
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Modifications to draft theories are propagated to the proof context |
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as usual, but there is also an explicit @{text "transfer"} operation |
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to force resynchronization with more substantial updates to the |
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underlying theory. |
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Entities derived in a proof context need to record logical |
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requirements explicitly, since there is no separate context |
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identification or symbolic inclusion as for theories. For example, |
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hypotheses used in primitive derivations (cf.\ \secref{sec:thms}) |
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are recorded separately within the sequent @{text "\<Gamma> \<turnstile> \<phi>"}, just to |
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make double sure. Results could still leak into an alien proof |
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context due to programming errors, but Isabelle/Isar includes some |
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extra validity checks in critical positions, notably at the end of a |
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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 an Isar proof state models |
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block-structured reasoning explicitly, using a stack of proof |
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contexts internally. For various technical reasons, the background |
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theory of an Isar proof state must not be changed while the proof is |
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still under construction! |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML_type Proof.context} \\ |
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@{index_ML ProofContext.init: "theory -> Proof.context"} \\ |
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@{index_ML ProofContext.theory_of: "Proof.context -> theory"} \\ |
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@{index_ML ProofContext.transfer: "theory -> Proof.context -> Proof.context"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML_type 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 @{ML ProofContext.init}~@{text "thy"} produces a proof context |
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derived from @{text "thy"}, initializing all data. |
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\item @{ML ProofContext.theory_of}~@{text "ctxt"} selects the |
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background theory from @{text "ctxt"}, dereferencing its internal |
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@{ML_type theory_ref}. |
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\item @{ML ProofContext.transfer}~@{text "thy ctxt"} promotes the |
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background theory of @{text "ctxt"} to the super theory @{text |
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"thy"}. |
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\end{description} |
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*} |
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subsection {* Generic contexts \label{sec:generic-context} *} |
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text {* |
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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 @{text "theory_of"} and @{text |
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"proof_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 @{text "init"} operation, which |
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incurs a small runtime overhead. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML_type Context.generic} \\ |
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@{index_ML Context.theory_of: "Context.generic -> theory"} \\ |
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@{index_ML Context.proof_of: "Context.generic -> Proof.context"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML_type Context.generic} is the direct sum of @{ML_type |
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"theory"} and @{ML_type "Proof.context"}, with the datatype |
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constructors @{ML "Context.Theory"} and @{ML "Context.Proof"}. |
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\item @{ML Context.theory_of}~@{text "context"} always produces a |
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theory from the generic @{text "context"}, using @{ML |
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"ProofContext.theory_of"} as required. |
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\item @{ML Context.proof_of}~@{text "context"} always produces a |
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proof context from the generic @{text "context"}, using @{ML |
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"ProofContext.init"} as required (note that this re-initializes the |
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context data with each invocation). |
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\end{description} |
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*} |
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subsection {* Context data \label{sec:context-data} *} |
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|
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a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
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33174
diff
changeset
|
396 |
text {* The main purpose of theory and proof contexts is to manage |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
397 |
arbitrary (pure) data. New data types can be declared incrementally |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
398 |
at compile time. There are separate declaration mechanisms for any |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
399 |
of the three kinds of contexts: theory, proof, generic. |
20449 | 400 |
|
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
401 |
\paragraph{Theory data} declarations need to implement the following |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
402 |
SML signature: |
20449 | 403 |
|
404 |
\medskip |
|
405 |
\begin{tabular}{ll} |
|
22869 | 406 |
@{text "\<type> T"} & representing type \\ |
407 |
@{text "\<val> empty: T"} & empty default value \\ |
|
408 |
@{text "\<val> extend: T \<rightarrow> T"} & re-initialize on import \\ |
|
409 |
@{text "\<val> merge: T \<times> T \<rightarrow> T"} & join on import \\ |
|
20449 | 410 |
\end{tabular} |
411 |
\medskip |
|
412 |
||
22869 | 413 |
\noindent The @{text "empty"} value acts as initial default for |
414 |
\emph{any} theory that does not declare actual data content; @{text |
|
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
415 |
"extend"} is acts like a unitary version of @{text "merge"}. |
20449 | 416 |
|
34921 | 417 |
Implementing @{text "merge"} can be tricky. The general idea is |
418 |
that @{text "merge (data\<^sub>1, data\<^sub>2)"} inserts those parts of @{text |
|
419 |
"data\<^sub>2"} into @{text "data\<^sub>1"} that are not yet present, while |
|
420 |
keeping the general order of things. The @{ML Library.merge} |
|
421 |
function on plain lists may serve as canonical template. |
|
422 |
||
423 |
Particularly note that shared parts of the data must not be |
|
424 |
duplicated by naive concatenation, or a theory graph that is like a |
|
425 |
chain of diamonds would cause an exponential blowup! |
|
426 |
||
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
427 |
\paragraph{Proof context data} declarations need to implement the |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
428 |
following SML signature: |
20449 | 429 |
|
430 |
\medskip |
|
431 |
\begin{tabular}{ll} |
|
22869 | 432 |
@{text "\<type> T"} & representing type \\ |
433 |
@{text "\<val> init: theory \<rightarrow> T"} & produce initial value \\ |
|
20449 | 434 |
\end{tabular} |
435 |
\medskip |
|
436 |
||
437 |
\noindent The @{text "init"} operation is supposed to produce a pure |
|
34921 | 438 |
value from the given background theory and should be somehow |
439 |
``immediate''. Whenever a proof context is initialized, which |
|
440 |
happens frequently, the the system invokes the @{text "init"} |
|
441 |
operation of \emph{all} theory data slots ever declared. |
|
20449 | 442 |
|
20451 | 443 |
\paragraph{Generic data} provides a hybrid interface for both theory |
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
444 |
and proof data. The @{text "init"} operation for proof contexts is |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
445 |
predefined to select the current data value from the background |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
446 |
theory. |
20449 | 447 |
|
34921 | 448 |
\bigskip Any of these data declaration over type @{text "T"} result |
449 |
in an ML structure with the following signature: |
|
20449 | 450 |
|
451 |
\medskip |
|
452 |
\begin{tabular}{ll} |
|
453 |
@{text "get: context \<rightarrow> T"} \\ |
|
454 |
@{text "put: T \<rightarrow> context \<rightarrow> context"} \\ |
|
455 |
@{text "map: (T \<rightarrow> T) \<rightarrow> context \<rightarrow> context"} \\ |
|
456 |
\end{tabular} |
|
457 |
\medskip |
|
458 |
||
34921 | 459 |
\noindent These other operations provide exclusive access for the |
460 |
particular kind of context (theory, proof, or generic context). |
|
461 |
This interface fully observes the ML discipline for types and |
|
462 |
scopes: there is no other way to access the corresponding data slot |
|
463 |
of a context. By keeping these operations private, an Isabelle/ML |
|
464 |
module may maintain abstract values authentically. |
|
20447 | 465 |
*} |
466 |
||
20450 | 467 |
text %mlref {* |
468 |
\begin{mldecls} |
|
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
469 |
@{index_ML_functor Theory_Data} \\ |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
470 |
@{index_ML_functor Proof_Data} \\ |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
471 |
@{index_ML_functor Generic_Data} \\ |
20450 | 472 |
\end{mldecls} |
473 |
||
474 |
\begin{description} |
|
475 |
||
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
476 |
\item @{ML_functor Theory_Data}@{text "(spec)"} declares data for |
20450 | 477 |
type @{ML_type theory} according to the specification provided as |
20451 | 478 |
argument structure. The resulting structure provides data init and |
479 |
access operations as described above. |
|
20450 | 480 |
|
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
481 |
\item @{ML_functor Proof_Data}@{text "(spec)"} is analogous to |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
482 |
@{ML_functor Theory_Data} for type @{ML_type Proof.context}. |
20450 | 483 |
|
33524
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
484 |
\item @{ML_functor Generic_Data}@{text "(spec)"} is analogous to |
a08e6c1cbc04
updated functor Theory_Data, Proof_Data, Generic_Data;
wenzelm
parents:
33174
diff
changeset
|
485 |
@{ML_functor Theory_Data} for type @{ML_type Context.generic}. |
20450 | 486 |
|
487 |
\end{description} |
|
488 |
*} |
|
489 |
||
20447 | 490 |
|
26872 | 491 |
section {* Names \label{sec:names} *} |
20451 | 492 |
|
34925 | 493 |
text {* In principle, a name is just a string, but there are various |
494 |
conventions for representing additional structure. For example, |
|
495 |
``@{text "Foo.bar.baz"}'' is considered as a qualified name |
|
496 |
consisting of three basic name components. The individual |
|
497 |
constituents of a name may have further substructure, e.g.\ the |
|
498 |
string ``\verb,\,\verb,<alpha>,'' encodes as a single symbol. |
|
20451 | 499 |
*} |
20437 | 500 |
|
501 |
||
502 |
subsection {* Strings of symbols *} |
|
503 |
||
34925 | 504 |
text {* A \emph{symbol} constitutes the smallest textual unit in |
505 |
Isabelle --- raw ML characters are normally not encountered at all! |
|
506 |
Isabelle strings consist of a sequence of symbols, represented as a |
|
507 |
packed string or an exploded list of strings. Each symbol is in |
|
508 |
itself a small string, which has either one of the following forms: |
|
20437 | 509 |
|
20451 | 510 |
\begin{enumerate} |
20437 | 511 |
|
34925 | 512 |
\item a single ASCII character ``@{text "c"}'' or raw byte in the |
513 |
range of 128\dots 255, for example ``\verb,a,'', |
|
20437 | 514 |
|
20488 | 515 |
\item a regular symbol ``\verb,\,\verb,<,@{text "ident"}\verb,>,'', |
20476 | 516 |
for example ``\verb,\,\verb,<alpha>,'', |
20437 | 517 |
|
20488 | 518 |
\item a control symbol ``\verb,\,\verb,<^,@{text "ident"}\verb,>,'', |
20476 | 519 |
for example ``\verb,\,\verb,<^bold>,'', |
20437 | 520 |
|
20488 | 521 |
\item a raw symbol ``\verb,\,\verb,<^raw:,@{text text}\verb,>,'' |
34925 | 522 |
where @{text text} consists of printable characters excluding |
20476 | 523 |
``\verb,.,'' and ``\verb,>,'', for example |
524 |
``\verb,\,\verb,<^raw:$\sum_{i = 1}^n$>,'', |
|
20437 | 525 |
|
20488 | 526 |
\item a numbered raw control symbol ``\verb,\,\verb,<^raw,@{text |
20476 | 527 |
n}\verb,>, where @{text n} consists of digits, for example |
20451 | 528 |
``\verb,\,\verb,<^raw42>,''. |
20437 | 529 |
|
20451 | 530 |
\end{enumerate} |
20437 | 531 |
|
20476 | 532 |
\noindent The @{text "ident"} syntax for symbol names is @{text |
533 |
"letter (letter | digit)\<^sup>*"}, where @{text "letter = |
|
534 |
A..Za..z"} and @{text "digit = 0..9"}. There are infinitely many |
|
535 |
regular symbols and control symbols, but a fixed collection of |
|
536 |
standard symbols is treated specifically. For example, |
|
20488 | 537 |
``\verb,\,\verb,<alpha>,'' is classified as a letter, which means it |
538 |
may occur within regular Isabelle identifiers. |
|
20437 | 539 |
|
20488 | 540 |
Since the character set underlying Isabelle symbols is 7-bit ASCII |
34925 | 541 |
and 8-bit characters are passed through transparently, Isabelle can |
542 |
also process Unicode/UCS data in UTF-8 encoding.\footnote{When |
|
543 |
counting precise source positions internally, bytes in the range of |
|
544 |
128\dots 191 are ignored. In UTF-8 encoding, this interval covers |
|
545 |
the additional trailer bytes, so Isabelle happens to count Unicode |
|
546 |
characters here, not bytes in memory. In ISO-Latin encoding, the |
|
547 |
ignored range merely includes some extra punctuation characters that |
|
548 |
even have replacements within the standard collection of Isabelle |
|
549 |
symbols; the accented letters range is counted properly.} Unicode |
|
550 |
provides its own collection of mathematical symbols, but within the |
|
551 |
core Isabelle/ML world there is no link to the standard collection |
|
552 |
of Isabelle regular symbols. |
|
20476 | 553 |
|
554 |
\medskip Output of Isabelle symbols depends on the print mode |
|
29758 | 555 |
(\secref{print-mode}). For example, the standard {\LaTeX} setup of |
556 |
the Isabelle document preparation system would present |
|
20451 | 557 |
``\verb,\,\verb,<alpha>,'' as @{text "\<alpha>"}, and |
558 |
``\verb,\,\verb,<^bold>,\verb,\,\verb,<alpha>,'' as @{text |
|
34925 | 559 |
"\<^bold>\<alpha>"}. On-screen rendering usually works by mapping a finite |
560 |
subset of Isabelle symbols to suitable Unicode characters. |
|
20451 | 561 |
*} |
20437 | 562 |
|
563 |
text %mlref {* |
|
564 |
\begin{mldecls} |
|
34921 | 565 |
@{index_ML_type "Symbol.symbol": string} \\ |
20437 | 566 |
@{index_ML Symbol.explode: "string -> Symbol.symbol list"} \\ |
567 |
@{index_ML Symbol.is_letter: "Symbol.symbol -> bool"} \\ |
|
568 |
@{index_ML Symbol.is_digit: "Symbol.symbol -> bool"} \\ |
|
569 |
@{index_ML Symbol.is_quasi: "Symbol.symbol -> bool"} \\ |
|
20547 | 570 |
@{index_ML Symbol.is_blank: "Symbol.symbol -> bool"} \\ |
571 |
\end{mldecls} |
|
572 |
\begin{mldecls} |
|
20437 | 573 |
@{index_ML_type "Symbol.sym"} \\ |
574 |
@{index_ML Symbol.decode: "Symbol.symbol -> Symbol.sym"} \\ |
|
575 |
\end{mldecls} |
|
576 |
||
577 |
\begin{description} |
|
578 |
||
20488 | 579 |
\item @{ML_type "Symbol.symbol"} represents individual Isabelle |
34921 | 580 |
symbols. |
20437 | 581 |
|
20476 | 582 |
\item @{ML "Symbol.explode"}~@{text "str"} produces a symbol list |
20488 | 583 |
from the packed form. This function supercedes @{ML |
20476 | 584 |
"String.explode"} for virtually all purposes of manipulating text in |
34925 | 585 |
Isabelle!\footnote{The runtime overhead for exploded strings is |
586 |
mainly that of the list structure: individual symbols that happen to |
|
587 |
be a singleton string --- which is the most common case --- do not |
|
588 |
require extra memory in Poly/ML.} |
|
20437 | 589 |
|
590 |
\item @{ML "Symbol.is_letter"}, @{ML "Symbol.is_digit"}, @{ML |
|
20476 | 591 |
"Symbol.is_quasi"}, @{ML "Symbol.is_blank"} classify standard |
592 |
symbols according to fixed syntactic conventions of Isabelle, cf.\ |
|
593 |
\cite{isabelle-isar-ref}. |
|
20437 | 594 |
|
595 |
\item @{ML_type "Symbol.sym"} is a concrete datatype that represents |
|
20488 | 596 |
the different kinds of symbols explicitly, with constructors @{ML |
597 |
"Symbol.Char"}, @{ML "Symbol.Sym"}, @{ML "Symbol.Ctrl"}, @{ML |
|
20451 | 598 |
"Symbol.Raw"}. |
20437 | 599 |
|
600 |
\item @{ML "Symbol.decode"} converts the string representation of a |
|
20451 | 601 |
symbol into the datatype version. |
20437 | 602 |
|
603 |
\end{description} |
|
34925 | 604 |
|
605 |
\paragraph{Historical note.} In the original SML90 standard the |
|
606 |
primitive ML type @{ML_type char} did not exists, and the basic @{ML |
|
607 |
"explode: string -> string list"} operation would produce a list of |
|
608 |
singleton strings as in Isabelle/ML today. When SML97 came out, |
|
609 |
Isabelle ignored its slightly anachronistic 8-bit characters, but |
|
610 |
the idea of exploding a string into a list of small strings was |
|
611 |
extended to ``symbols'' as explained above. Thus Isabelle sources |
|
612 |
can refer to an infinite store of user-defined symbols, without |
|
613 |
having to worry about the multitude of Unicode encodings. |
|
20437 | 614 |
*} |
615 |
||
616 |
||
20476 | 617 |
subsection {* Basic names \label{sec:basic-names} *} |
618 |
||
619 |
text {* |
|
620 |
A \emph{basic name} essentially consists of a single Isabelle |
|
621 |
identifier. There are conventions to mark separate classes of basic |
|
29761 | 622 |
names, by attaching a suffix of underscores: one underscore means |
623 |
\emph{internal name}, two underscores means \emph{Skolem name}, |
|
624 |
three underscores means \emph{internal Skolem name}. |
|
20476 | 625 |
|
626 |
For example, the basic name @{text "foo"} has the internal version |
|
627 |
@{text "foo_"}, with Skolem versions @{text "foo__"} and @{text |
|
628 |
"foo___"}, respectively. |
|
629 |
||
20488 | 630 |
These special versions provide copies of the basic name space, apart |
631 |
from anything that normally appears in the user text. For example, |
|
632 |
system generated variables in Isar proof contexts are usually marked |
|
633 |
as internal, which prevents mysterious name references like @{text |
|
634 |
"xaa"} to appear in the text. |
|
20476 | 635 |
|
20488 | 636 |
\medskip Manipulating binding scopes often requires on-the-fly |
637 |
renamings. A \emph{name context} contains a collection of already |
|
638 |
used names. The @{text "declare"} operation adds names to the |
|
639 |
context. |
|
20476 | 640 |
|
20488 | 641 |
The @{text "invents"} operation derives a number of fresh names from |
642 |
a given starting point. For example, the first three names derived |
|
643 |
from @{text "a"} are @{text "a"}, @{text "b"}, @{text "c"}. |
|
20476 | 644 |
|
645 |
The @{text "variants"} operation produces fresh names by |
|
20488 | 646 |
incrementing tentative names as base-26 numbers (with digits @{text |
647 |
"a..z"}) until all clashes are resolved. For example, name @{text |
|
648 |
"foo"} results in variants @{text "fooa"}, @{text "foob"}, @{text |
|
649 |
"fooc"}, \dots, @{text "fooaa"}, @{text "fooab"} etc.; each renaming |
|
650 |
step picks the next unused variant from this sequence. |
|
20476 | 651 |
*} |
652 |
||
653 |
text %mlref {* |
|
654 |
\begin{mldecls} |
|
655 |
@{index_ML Name.internal: "string -> string"} \\ |
|
20547 | 656 |
@{index_ML Name.skolem: "string -> string"} \\ |
657 |
\end{mldecls} |
|
658 |
\begin{mldecls} |
|
20476 | 659 |
@{index_ML_type Name.context} \\ |
660 |
@{index_ML Name.context: Name.context} \\ |
|
661 |
@{index_ML Name.declare: "string -> Name.context -> Name.context"} \\ |
|
662 |
@{index_ML Name.invents: "Name.context -> string -> int -> string list"} \\ |
|
663 |
@{index_ML Name.variants: "string list -> Name.context -> string list * Name.context"} \\ |
|
664 |
\end{mldecls} |
|
665 |
||
666 |
\begin{description} |
|
667 |
||
668 |
\item @{ML Name.internal}~@{text "name"} produces an internal name |
|
669 |
by adding one underscore. |
|
670 |
||
671 |
\item @{ML Name.skolem}~@{text "name"} produces a Skolem name by |
|
672 |
adding two underscores. |
|
673 |
||
674 |
\item @{ML_type Name.context} represents the context of already used |
|
675 |
names; the initial value is @{ML "Name.context"}. |
|
676 |
||
20488 | 677 |
\item @{ML Name.declare}~@{text "name"} enters a used name into the |
678 |
context. |
|
20437 | 679 |
|
20488 | 680 |
\item @{ML Name.invents}~@{text "context name n"} produces @{text |
681 |
"n"} fresh names derived from @{text "name"}. |
|
682 |
||
683 |
\item @{ML Name.variants}~@{text "names context"} produces fresh |
|
29761 | 684 |
variants of @{text "names"}; the result is entered into the context. |
20476 | 685 |
|
686 |
\end{description} |
|
687 |
*} |
|
688 |
||
689 |
||
690 |
subsection {* Indexed names *} |
|
691 |
||
692 |
text {* |
|
693 |
An \emph{indexed name} (or @{text "indexname"}) is a pair of a basic |
|
20488 | 694 |
name and a natural number. This representation allows efficient |
695 |
renaming by incrementing the second component only. The canonical |
|
696 |
way to rename two collections of indexnames apart from each other is |
|
697 |
this: determine the maximum index @{text "maxidx"} of the first |
|
698 |
collection, then increment all indexes of the second collection by |
|
699 |
@{text "maxidx + 1"}; the maximum index of an empty collection is |
|
700 |
@{text "-1"}. |
|
20476 | 701 |
|
20488 | 702 |
Occasionally, basic names and indexed names are injected into the |
703 |
same pair type: the (improper) indexname @{text "(x, -1)"} is used |
|
704 |
to encode basic names. |
|
705 |
||
706 |
\medskip Isabelle syntax observes the following rules for |
|
707 |
representing an indexname @{text "(x, i)"} as a packed string: |
|
20476 | 708 |
|
709 |
\begin{itemize} |
|
710 |
||
20479 | 711 |
\item @{text "?x"} if @{text "x"} does not end with a digit and @{text "i = 0"}, |
20476 | 712 |
|
713 |
\item @{text "?xi"} if @{text "x"} does not end with a digit, |
|
714 |
||
20488 | 715 |
\item @{text "?x.i"} otherwise. |
20476 | 716 |
|
717 |
\end{itemize} |
|
20470 | 718 |
|
20488 | 719 |
Indexnames may acquire large index numbers over time. Results are |
720 |
normalized towards @{text "0"} at certain checkpoints, notably at |
|
721 |
the end of a proof. This works by producing variants of the |
|
722 |
corresponding basic name components. For example, the collection |
|
723 |
@{text "?x1, ?x7, ?x42"} becomes @{text "?x, ?xa, ?xb"}. |
|
20476 | 724 |
*} |
725 |
||
726 |
text %mlref {* |
|
727 |
\begin{mldecls} |
|
728 |
@{index_ML_type indexname} \\ |
|
729 |
\end{mldecls} |
|
730 |
||
731 |
\begin{description} |
|
732 |
||
733 |
\item @{ML_type indexname} represents indexed names. This is an |
|
734 |
abbreviation for @{ML_type "string * int"}. The second component is |
|
735 |
usually non-negative, except for situations where @{text "(x, -1)"} |
|
20488 | 736 |
is used to embed basic names into this type. |
20476 | 737 |
|
738 |
\end{description} |
|
739 |
*} |
|
740 |
||
741 |
||
742 |
subsection {* Qualified names and name spaces *} |
|
743 |
||
744 |
text {* |
|
745 |
A \emph{qualified name} consists of a non-empty sequence of basic |
|
20488 | 746 |
name components. The packed representation uses a dot as separator, |
747 |
as in ``@{text "A.b.c"}''. The last component is called \emph{base} |
|
748 |
name, the remaining prefix \emph{qualifier} (which may be empty). |
|
749 |
The idea of qualified names is to encode nested structures by |
|
750 |
recording the access paths as qualifiers. For example, an item |
|
751 |
named ``@{text "A.b.c"}'' may be understood as a local entity @{text |
|
752 |
"c"}, within a local structure @{text "b"}, within a global |
|
753 |
structure @{text "A"}. Typically, name space hierarchies consist of |
|
754 |
1--2 levels of qualification, but this need not be always so. |
|
20437 | 755 |
|
20476 | 756 |
The empty name is commonly used as an indication of unnamed |
20488 | 757 |
entities, whenever this makes any sense. The basic operations on |
758 |
qualified names are smart enough to pass through such improper names |
|
20476 | 759 |
unchanged. |
760 |
||
761 |
\medskip A @{text "naming"} policy tells how to turn a name |
|
762 |
specification into a fully qualified internal name (by the @{text |
|
20488 | 763 |
"full"} operation), and how fully qualified names may be accessed |
764 |
externally. For example, the default naming policy is to prefix an |
|
765 |
implicit path: @{text "full x"} produces @{text "path.x"}, and the |
|
766 |
standard accesses for @{text "path.x"} include both @{text "x"} and |
|
767 |
@{text "path.x"}. Normally, the naming is implicit in the theory or |
|
768 |
proof context; there are separate versions of the corresponding. |
|
20437 | 769 |
|
20476 | 770 |
\medskip A @{text "name space"} manages a collection of fully |
771 |
internalized names, together with a mapping between external names |
|
772 |
and internal names (in both directions). The corresponding @{text |
|
773 |
"intern"} and @{text "extern"} operations are mostly used for |
|
774 |
parsing and printing only! The @{text "declare"} operation augments |
|
20488 | 775 |
a name space according to the accesses determined by the naming |
776 |
policy. |
|
20476 | 777 |
|
20488 | 778 |
\medskip As a general principle, there is a separate name space for |
779 |
each kind of formal entity, e.g.\ logical constant, type |
|
780 |
constructor, type class, theorem. It is usually clear from the |
|
781 |
occurrence in concrete syntax (or from the scope) which kind of |
|
782 |
entity a name refers to. For example, the very same name @{text |
|
783 |
"c"} may be used uniformly for a constant, type constructor, and |
|
784 |
type class. |
|
20476 | 785 |
|
20479 | 786 |
There are common schemes to name theorems systematically, according |
20488 | 787 |
to the name of the main logical entity involved, e.g.\ @{text |
788 |
"c.intro"} for a canonical theorem related to constant @{text "c"}. |
|
789 |
This technique of mapping names from one space into another requires |
|
790 |
some care in order to avoid conflicts. In particular, theorem names |
|
791 |
derived from a type constructor or type class are better suffixed in |
|
792 |
addition to the usual qualification, e.g.\ @{text "c_type.intro"} |
|
793 |
and @{text "c_class.intro"} for theorems related to type @{text "c"} |
|
794 |
and class @{text "c"}, respectively. |
|
20437 | 795 |
*} |
796 |
||
20476 | 797 |
text %mlref {* |
798 |
\begin{mldecls} |
|
30365 | 799 |
@{index_ML Long_Name.base_name: "string -> string"} \\ |
800 |
@{index_ML Long_Name.qualifier: "string -> string"} \\ |
|
801 |
@{index_ML Long_Name.append: "string -> string -> string"} \\ |
|
802 |
@{index_ML Long_Name.implode: "string list -> string"} \\ |
|
803 |
@{index_ML Long_Name.explode: "string -> string list"} \\ |
|
20547 | 804 |
\end{mldecls} |
805 |
\begin{mldecls} |
|
33174 | 806 |
@{index_ML_type Name_Space.naming} \\ |
807 |
@{index_ML Name_Space.default_naming: Name_Space.naming} \\ |
|
808 |
@{index_ML Name_Space.add_path: "string -> Name_Space.naming -> Name_Space.naming"} \\ |
|
809 |
@{index_ML Name_Space.full_name: "Name_Space.naming -> binding -> string"} \\ |
|
20547 | 810 |
\end{mldecls} |
811 |
\begin{mldecls} |
|
33174 | 812 |
@{index_ML_type Name_Space.T} \\ |
813 |
@{index_ML Name_Space.empty: "string -> Name_Space.T"} \\ |
|
814 |
@{index_ML Name_Space.merge: "Name_Space.T * Name_Space.T -> Name_Space.T"} \\ |
|
815 |
@{index_ML Name_Space.declare: "bool -> Name_Space.naming -> binding -> Name_Space.T -> |
|
816 |
string * Name_Space.T"} \\ |
|
817 |
@{index_ML Name_Space.intern: "Name_Space.T -> string -> string"} \\ |
|
818 |
@{index_ML Name_Space.extern: "Name_Space.T -> string -> string"} \\ |
|
20476 | 819 |
\end{mldecls} |
20437 | 820 |
|
20476 | 821 |
\begin{description} |
822 |
||
30365 | 823 |
\item @{ML Long_Name.base_name}~@{text "name"} returns the base name of a |
20476 | 824 |
qualified name. |
825 |
||
30365 | 826 |
\item @{ML Long_Name.qualifier}~@{text "name"} returns the qualifier |
20476 | 827 |
of a qualified name. |
20437 | 828 |
|
30365 | 829 |
\item @{ML Long_Name.append}~@{text "name\<^isub>1 name\<^isub>2"} |
20476 | 830 |
appends two qualified names. |
20437 | 831 |
|
30365 | 832 |
\item @{ML Long_Name.implode}~@{text "names"} and @{ML |
833 |
Long_Name.explode}~@{text "name"} convert between the packed string |
|
20488 | 834 |
representation and the explicit list form of qualified names. |
20476 | 835 |
|
33174 | 836 |
\item @{ML_type Name_Space.naming} represents the abstract concept of |
20476 | 837 |
a naming policy. |
20437 | 838 |
|
33174 | 839 |
\item @{ML Name_Space.default_naming} is the default naming policy. |
20476 | 840 |
In a theory context, this is usually augmented by a path prefix |
841 |
consisting of the theory name. |
|
842 |
||
33174 | 843 |
\item @{ML Name_Space.add_path}~@{text "path naming"} augments the |
20488 | 844 |
naming policy by extending its path component. |
20437 | 845 |
|
33174 | 846 |
\item @{ML Name_Space.full_name}~@{text "naming binding"} turns a |
30281
9ad15d8ed311
renamed NameSpace.base to NameSpace.base_name (in accordance with "full_name");
wenzelm
parents:
30272
diff
changeset
|
847 |
name binding (usually a basic name) into the fully qualified |
29008 | 848 |
internal name, according to the given naming policy. |
20476 | 849 |
|
33174 | 850 |
\item @{ML_type Name_Space.T} represents name spaces. |
20476 | 851 |
|
33174 | 852 |
\item @{ML Name_Space.empty}~@{text "kind"} and @{ML Name_Space.merge}~@{text |
20488 | 853 |
"(space\<^isub>1, space\<^isub>2)"} are the canonical operations for |
854 |
maintaining name spaces according to theory data management |
|
33174 | 855 |
(\secref{sec:context-data}); @{text "kind"} is a formal comment |
856 |
to characterize the purpose of a name space. |
|
20437 | 857 |
|
33174 | 858 |
\item @{ML Name_Space.declare}~@{text "strict naming bindings |
859 |
space"} enters a name binding as fully qualified internal name into |
|
860 |
the name space, with external accesses determined by the naming |
|
861 |
policy. |
|
20476 | 862 |
|
33174 | 863 |
\item @{ML Name_Space.intern}~@{text "space name"} internalizes a |
20476 | 864 |
(partially qualified) external name. |
20437 | 865 |
|
20488 | 866 |
This operation is mostly for parsing! Note that fully qualified |
20476 | 867 |
names stemming from declarations are produced via @{ML |
33174 | 868 |
"Name_Space.full_name"} and @{ML "Name_Space.declare"} |
29008 | 869 |
(or their derivatives for @{ML_type theory} and |
20488 | 870 |
@{ML_type Proof.context}). |
20437 | 871 |
|
33174 | 872 |
\item @{ML Name_Space.extern}~@{text "space name"} externalizes a |
20476 | 873 |
(fully qualified) internal name. |
874 |
||
30281
9ad15d8ed311
renamed NameSpace.base to NameSpace.base_name (in accordance with "full_name");
wenzelm
parents:
30272
diff
changeset
|
875 |
This operation is mostly for printing! User code should not rely on |
9ad15d8ed311
renamed NameSpace.base to NameSpace.base_name (in accordance with "full_name");
wenzelm
parents:
30272
diff
changeset
|
876 |
the precise result too much. |
20476 | 877 |
|
878 |
\end{description} |
|
879 |
*} |
|
30272 | 880 |
|
18537 | 881 |
end |