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theory Syntax |
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imports Base |
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begin |
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chapter {* Concrete syntax and type-checking *} |
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text {* Pure @{text "\<lambda>"}-calculus as introduced in \chref{ch:logic} is |
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an adequate foundation for logical languages --- in the tradition of |
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\emph{higher-order abstract syntax} --- but end-users require |
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additional means for reading and printing of terms and types. This |
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important add-on outside the logical core is called \emph{inner |
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syntax} in Isabelle jargon, as opposed to the \emph{outer syntax} of |
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the theory and proof language \cite{isabelle-isar-ref}. |
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For example, according to \cite{church40} quantifiers are represented as |
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higher-order constants @{text "All :: ('a \<Rightarrow> bool) \<Rightarrow> bool"} such that @{text |
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"All (\<lambda>x::'a. B x)"} faithfully represents the idea that is displayed in |
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Isabelle as @{text "\<forall>x::'a. B x"} via @{keyword "binder"} notation. |
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Moreover, type-inference in the style of Hindley-Milner \cite{hindleymilner} |
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(and extensions) enables users to write @{text "\<forall>x. B x"} concisely, when |
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the type @{text "'a"} is already clear from the |
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context.\footnote{Type-inference taken to the extreme can easily confuse |
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users. Beginners often stumble over unexpectedly general types inferred by |
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the system.} |
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\medskip The main inner syntax operations are \emph{read} for |
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parsing together with type-checking, and \emph{pretty} for formatted |
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output. See also \secref{sec:read-print}. |
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Furthermore, the input and output syntax layers are sub-divided into |
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separate phases for \emph{concrete syntax} versus \emph{abstract |
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syntax}, see also \secref{sec:parse-unparse} and |
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\secref{sec:term-check}, respectively. This results in the |
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following decomposition of the main operations: |
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\begin{itemize} |
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\item @{text "read = parse; check"} |
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\item @{text "pretty = uncheck; unparse"} |
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\end{itemize} |
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For example, some specification package might thus intercept syntax |
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processing at a well-defined stage after @{text "parse"}, to a augment the |
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resulting pre-term before full type-reconstruction is performed by @{text |
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"check"}. Note that the formal status of bound variables, versus free |
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variables, versus constants must not be changed between these phases. |
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\medskip In general, @{text check} and @{text uncheck} operate |
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simultaneously on a list of terms. This is particular important for |
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type-checking, to reconstruct types for several terms of the same context |
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and scope. In contrast, @{text parse} and @{text unparse} operate separately |
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on single terms. |
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There are analogous operations to read and print types, with the same |
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sub-division into phases. |
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*} |
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section {* Reading and pretty printing \label{sec:read-print} *} |
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text {* |
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Read and print operations are roughly dual to each other, such that for the |
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user @{text "s' = pretty (read s)"} looks similar to the original source |
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text @{text "s"}, but the details depend on many side-conditions. There are |
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also explicit options to control the removal of type information in the |
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output. The default configuration routinely looses information, so @{text |
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"t' = read (pretty t)"} might fail, or produce a differently typed term, or |
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a completely different term in the face of syntactic overloading. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML Syntax.read_typs: "Proof.context -> string list -> typ list"} \\ |
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@{index_ML Syntax.read_terms: "Proof.context -> string list -> term list"} \\ |
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@{index_ML Syntax.read_props: "Proof.context -> string list -> term list"} \\[0.5ex] |
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@{index_ML Syntax.read_typ: "Proof.context -> string -> typ"} \\ |
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@{index_ML Syntax.read_term: "Proof.context -> string -> term"} \\ |
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@{index_ML Syntax.read_prop: "Proof.context -> string -> term"} \\[0.5ex] |
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@{index_ML Syntax.pretty_typ: "Proof.context -> typ -> Pretty.T"} \\ |
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@{index_ML Syntax.pretty_term: "Proof.context -> term -> Pretty.T"} \\ |
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@{index_ML Syntax.string_of_typ: "Proof.context -> typ -> string"} \\ |
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@{index_ML Syntax.string_of_term: "Proof.context -> term -> string"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML Syntax.read_typs}~@{text "ctxt strs"} parses and checks a |
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simultaneous list of source strings as types of the logic. |
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\item @{ML Syntax.read_terms}~@{text "ctxt strs"} parses and checks a |
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simultaneous list of source strings as terms of the logic. |
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Type-reconstruction puts all parsed terms into the same scope: types of |
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free variables ultimately need to coincide. |
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If particular type-constraints are required for some of the arguments, the |
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read operations needs to be split into its parse and check phases. Then it |
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is possible to use @{ML Type.constraint} on the intermediate pre-terms |
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(\secref{sec:term-check}). |
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\item @{ML Syntax.read_props}~@{text "ctxt strs"} parses and checks a |
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simultaneous list of source strings as terms of the logic, with an implicit |
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type-constraint for each argument to enforce type @{typ prop}; this also |
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affects the inner syntax for parsing. The remaining type-reconstruction |
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works as for @{ML Syntax.read_terms}. |
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\item @{ML Syntax.read_typ}, @{ML Syntax.read_term}, @{ML Syntax.read_prop} |
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are like the simultaneous versions, but operate on a single argument only. |
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This convenient shorthand is adequate in situations where a single item in |
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its own scope is processed. Do not use @{ML "map o Syntax.read_term"} where |
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@{ML Syntax.read_terms} is actually intended! |
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\item @{ML Syntax.pretty_typ}~@{text "ctxt T"} and @{ML |
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Syntax.pretty_term}~@{text "ctxt t"} uncheck and pretty-print the given type |
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or term, respectively. Although the uncheck phase acts on a simultaneous |
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list as well, this is rarely used in practice, so only the singleton case is |
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provided as combined pretty operation. There is no distinction of term vs.\ |
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proposition. |
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\item @{ML Syntax.string_of_typ} and @{ML Syntax.string_of_term} are |
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convenient compositions of @{ML Syntax.pretty_typ} and @{ML |
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Syntax.pretty_term} with @{ML Pretty.string_of} for output. The result may |
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be concatenated with other strings, as long as there is no further |
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formatting and line-breaking involved. |
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\end{description} |
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@{ML Syntax.read_term}, @{ML Syntax.read_prop}, and @{ML |
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Syntax.string_of_term} are the most important operations in practice. |
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\medskip Note that the string values that are passed in and out are |
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annotated by the system, to carry further markup that is relevant for the |
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Prover IDE \cite{isabelle-jedit}. User code should neither compose its own |
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input strings, nor try to analyze the output strings. Conceptually this is |
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an abstract datatype, encoded as concrete string for historical reasons. |
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The standard way to provide the required position markup for input works via |
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the outer syntax parser wrapper @{ML Parse.inner_syntax}, which is already |
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part of @{ML Parse.typ}, @{ML Parse.term}, @{ML Parse.prop}. So a string |
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obtained from one of the latter may be directly passed to the corresponding |
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read operation: this yields PIDE markup of the input and precise positions |
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for warning and error messages. |
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*} |
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section {* Parsing and unparsing \label{sec:parse-unparse} *} |
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text {* |
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Parsing and unparsing converts between actual source text and a certain |
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\emph{pre-term} format, where all bindings and scopes are already resolved |
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faithfully. Thus the names of free variables or constants are determined in |
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the sense of the logical context, but type information might be still |
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missing. Pre-terms support an explicit language of \emph{type constraints} |
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that may be augmented by user code to guide the later \emph{check} phase. |
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Actual parsing is based on traditional lexical analysis and Earley parsing |
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for arbitrary context-free grammars. The user can specify the grammar |
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declaratively via mixfix annotations. Moreover, there are \emph{syntax |
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translations} that can be augmented by the user, either declaratively via |
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@{command translations} or programmatically via @{command |
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parse_translation}, @{command print_translation} \cite{isabelle-isar-ref}. |
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The final scope-resolution is performed by the system, according to name |
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spaces for types, term variables and constants determined by the context. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML Syntax.parse_typ: "Proof.context -> string -> typ"} \\ |
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@{index_ML Syntax.parse_term: "Proof.context -> string -> term"} \\ |
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@{index_ML Syntax.parse_prop: "Proof.context -> string -> term"} \\[0.5ex] |
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@{index_ML Syntax.unparse_typ: "Proof.context -> typ -> Pretty.T"} \\ |
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@{index_ML Syntax.unparse_term: "Proof.context -> term -> Pretty.T"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML Syntax.parse_typ}~@{text "ctxt str"} parses a source string as |
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pre-type that is ready to be used with subsequent check operations. |
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\item @{ML Syntax.parse_term}~@{text "ctxt str"} parses a source string as |
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pre-term that is ready to be used with subsequent check operations. |
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\item @{ML Syntax.parse_prop}~@{text "ctxt str"} parses a source string as |
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pre-term that is ready to be used with subsequent check operations. The |
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inner syntax category is @{typ prop} and a suitable type-constraint is |
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included to ensure that this information is observed in subsequent type |
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reconstruction. |
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\item @{ML Syntax.unparse_typ}~@{text "ctxt T"} unparses a type after |
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uncheck operations, to turn it into a pretty tree. |
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\item @{ML Syntax.unparse_term}~@{text "ctxt T"} unparses a term after |
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uncheck operations, to turn it into a pretty tree. There is no distinction |
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for propositions here. |
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\end{description} |
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These operations always operate on a single item; use the combinator @{ML |
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map} to apply them to a list. |
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*} |
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section {* Checking and unchecking \label{sec:term-check} *} |
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text {* These operations define the transition from pre-terms and |
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fully-annotated terms in the sense of the logical core |
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(\chref{ch:logic}). |
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The \emph{check} phase is meant to subsume a variety of mechanisms |
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in the manner of ``type-inference'' or ``type-reconstruction'' or |
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``type-improvement'', not just type-checking in the narrow sense. |
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The \emph{uncheck} phase is roughly dual, it prunes type-information |
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before pretty printing. |
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A typical add-on for the check/uncheck syntax layer is the @{command |
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abbreviation} mechanism \cite{isabelle-isar-ref}. Here the user specifies |
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syntactic definitions that are managed by the system as polymorphic @{text |
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"let"} bindings. These are expanded during the @{text "check"} phase, and |
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contracted during the @{text "uncheck"} phase, without affecting the |
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type-assignment of the given terms. |
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\medskip The precise meaning of type checking depends on the context --- |
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additional check/uncheck modules might be defined in user space. |
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For example, the @{command class} command defines a context where |
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@{text "check"} treats certain type instances of overloaded |
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constants according to the ``dictionary construction'' of its |
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logical foundation. This involves ``type improvement'' |
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(specialization of slightly too general types) and replacement by |
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certain locale parameters. See also \cite{Haftmann-Wenzel:2009}. |
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*} |
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text %mlref {* |
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\begin{mldecls} |
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@{index_ML Syntax.check_typs: "Proof.context -> typ list -> typ list"} \\ |
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@{index_ML Syntax.check_terms: "Proof.context -> term list -> term list"} \\ |
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@{index_ML Syntax.check_props: "Proof.context -> term list -> term list"} \\[0.5ex] |
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@{index_ML Syntax.uncheck_typs: "Proof.context -> typ list -> typ list"} \\ |
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@{index_ML Syntax.uncheck_terms: "Proof.context -> term list -> term list"} \\ |
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\end{mldecls} |
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\begin{description} |
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\item @{ML Syntax.check_typs}~@{text "ctxt Ts"} checks a simultaneous list |
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of pre-types as types of the logic. Typically, this involves normalization |
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of type synonyms. |
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\item @{ML Syntax.check_terms}~@{text "ctxt ts"} checks a simultaneous list |
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of pre-terms as terms of the logic. Typically, this involves type-inference |
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and normalization term abbreviations. The types within the given terms are |
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treated in the same way as for @{ML Syntax.check_typs}. |
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Applications sometimes need to check several types and terms together. The |
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standard approach uses @{ML Logic.mk_type} to embed the language of types |
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into that of terms; all arguments are appended into one list of terms that |
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is checked; afterwards the type arguments are recovered with @{ML |
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Logic.dest_type}. |
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\item @{ML Syntax.check_props}~@{text "ctxt ts"} checks a simultaneous list |
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of pre-terms as terms of the logic, such that all terms are constrained by |
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type @{typ prop}. The remaining check operation works as @{ML |
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Syntax.check_terms} above. |
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\item @{ML Syntax.uncheck_typs}~@{text "ctxt Ts"} unchecks a simultaneous |
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list of types of the logic, in preparation of pretty printing. |
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\item @{ML Syntax.uncheck_terms}~@{text "ctxt ts"} unchecks a simultaneous |
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list of terms of the logic, in preparation of pretty printing. There is no |
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distinction for propositions here. |
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\end{description} |
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These operations always operate simultaneously on a list; use the combinator |
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@{ML singleton} to apply them to a single item. |
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*} |
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end |