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(* $Id$ *)
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theory Spec
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imports Main
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begin
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chapter {* Theory specifications *}
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section {* Defining theories \label{sec:begin-thy} *}
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text {*
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\begin{matharray}{rcl}
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@{command_def "theory"} & : & \isartrans{toplevel}{theory} \\
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@{command_def (global) "end"} & : & \isartrans{theory}{toplevel} \\
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\end{matharray}
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Isabelle/Isar theories are defined via theory file, which contain
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both specifications and proofs; occasionally definitional mechanisms
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also require some explicit proof. The theory body may be
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sub-structered by means of \emph{local theory} target mechanisms,
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notably @{command "locale"} and @{command "class"}.
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The first ``real'' command of any theory has to be @{command
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"theory"}, which starts a new theory based on the merge of existing
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ones. Just preceding the @{command "theory"} keyword, there may be
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an optional @{command "header"} declaration, which is relevant to
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document preparation only; it acts very much like a special
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pre-theory markup command (cf.\ \secref{sec:markup}). The @{command
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(global) "end"} command concludes a theory development; it has to be
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the very last command of any theory file loaded in batch-mode.
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\begin{rail}
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'theory' name 'imports' (name +) uses? 'begin'
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;
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uses: 'uses' ((name | parname) +);
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\end{rail}
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\begin{descr}
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\item [@{command "theory"}~@{text "A \<IMPORTS> B\<^sub>1 \<dots>
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B\<^sub>n \<BEGIN>"}] starts a new theory @{text A} based on the
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merge of existing theories @{text "B\<^sub>1 \<dots> B\<^sub>n"}.
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Due to inclusion of several ancestors, the overall theory structure
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emerging in an Isabelle session forms a directed acyclic graph
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(DAG). Isabelle's theory loader ensures that the sources
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contributing to the development graph are always up-to-date.
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Changed files are automatically reloaded when processing theory
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headers.
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The optional @{keyword_def "uses"} specification declares additional
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dependencies on extra files (usually ML sources). Files will be
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loaded immediately (as ML), unless the name is put in parentheses,
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which merely documents the dependency to be resolved later in the
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text (typically via explicit @{command_ref "use"} in the body text,
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see \secref{sec:ML}).
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\item [@{command (global) "end"}] concludes the current theory
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definition.
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\end{descr}
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*}
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section {* Local theory targets \label{sec:target} *}
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text {*
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A local theory target is a context managed separately within the
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enclosing theory. Contexts may introduce parameters (fixed
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variables) and assumptions (hypotheses). Definitions and theorems
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depending on the context may be added incrementally later on. Named
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contexts refer to locales (cf.\ \secref{sec:locale}) or type classes
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(cf.\ \secref{sec:class}); the name ``@{text "-"}'' signifies the
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global theory context.
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\begin{matharray}{rcll}
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@{command_def "context"} & : & \isartrans{theory}{local{\dsh}theory} \\
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@{command_def (local) "end"} & : & \isartrans{local{\dsh}theory}{theory} \\
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\end{matharray}
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\indexouternonterm{target}
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\begin{rail}
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'context' name 'begin'
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;
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target: '(' 'in' name ')'
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;
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\end{rail}
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\begin{descr}
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\item [@{command "context"}~@{text "c \<BEGIN>"}] recommences an
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existing locale or class context @{text c}. Note that locale and
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class definitions allow to include the @{keyword "begin"} keyword as
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well, in order to continue the local theory immediately after the
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initial specification.
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\item [@{command (local) "end"}] concludes the current local theory
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and continues the enclosing global theory. Note that a global
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@{command (global) "end"} has a different meaning: it concludes the
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theory itself (\secref{sec:begin-thy}).
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\item [@{text "(\<IN> c)"}] given after any local theory command
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specifies an immediate target, e.g.\ ``@{command
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"definition"}~@{text "(\<IN> c) \<dots>"}'' or ``@{command
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"theorem"}~@{text "(\<IN> c) \<dots>"}''. This works both in a local or
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global theory context; the current target context will be suspended
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for this command only. Note that ``@{text "(\<IN> -)"}'' will
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always produce a global result independently of the current target
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context.
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\end{descr}
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The exact meaning of results produced within a local theory context
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depends on the underlying target infrastructure (locale, type class
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etc.). The general idea is as follows, considering a context named
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@{text c} with parameter @{text x} and assumption @{text "A[x]"}.
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Definitions are exported by introducing a global version with
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additional arguments; a syntactic abbreviation links the long form
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with the abstract version of the target context. For example,
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@{text "a \<equiv> t[x]"} becomes @{text "c.a ?x \<equiv> t[?x]"} at the theory
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level (for arbitrary @{text "?x"}), together with a local
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abbreviation @{text "c \<equiv> c.a x"} in the target context (for the
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fixed parameter @{text x}).
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Theorems are exported by discharging the assumptions and
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generalizing the parameters of the context. For example, @{text "a:
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B[x]"} becomes @{text "c.a: A[?x] \<Longrightarrow> B[?x]"}, again for arbitrary
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@{text "?x"}.
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*}
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section {* Basic specification elements *}
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text {*
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\begin{matharray}{rcll}
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@{command_def "axiomatization"} & : & \isarkeep{local{\dsh}theory} & (axiomatic!)\\
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@{command_def "definition"} & : & \isarkeep{local{\dsh}theory} \\
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@{attribute_def "defn"} & : & \isaratt \\
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@{command_def "abbreviation"} & : & \isarkeep{local{\dsh}theory} \\
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@{command_def "print_abbrevs"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
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@{command_def "notation"} & : & \isarkeep{local{\dsh}theory} \\
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@{command_def "no_notation"} & : & \isarkeep{local{\dsh}theory} \\
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\end{matharray}
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These specification mechanisms provide a slightly more abstract view
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than the underlying primitives of @{command "consts"}, @{command
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"defs"} (see \secref{sec:consts}), and @{command "axioms"} (see
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\secref{sec:axms-thms}). In particular, type-inference is commonly
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available, and result names need not be given.
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\begin{rail}
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'axiomatization' target? fixes? ('where' specs)?
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;
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'definition' target? (decl 'where')? thmdecl? prop
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;
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'abbreviation' target? mode? (decl 'where')? prop
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;
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('notation' | 'no\_notation') target? mode? (nameref structmixfix + 'and')
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;
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fixes: ((name ('::' type)? mixfix? | vars) + 'and')
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;
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specs: (thmdecl? props + 'and')
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;
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decl: name ('::' type)? mixfix?
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;
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\end{rail}
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\begin{descr}
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\item [@{command "axiomatization"}~@{text "c\<^sub>1 \<dots> c\<^sub>m
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\<WHERE> \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}] introduces several constants
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simultaneously and states axiomatic properties for these. The
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constants are marked as being specified once and for all, which
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prevents additional specifications being issued later on.
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Note that axiomatic specifications are only appropriate when
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declaring a new logical system. Normal applications should only use
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definitional mechanisms!
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\item [@{command "definition"}~@{text "c \<WHERE> eq"}] produces an
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internal definition @{text "c \<equiv> t"} according to the specification
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given as @{text eq}, which is then turned into a proven fact. The
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given proposition may deviate from internal meta-level equality
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according to the rewrite rules declared as @{attribute defn} by the
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object-logic. This usually covers object-level equality @{text "x =
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y"} and equivalence @{text "A \<leftrightarrow> B"}. End-users normally need not
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change the @{attribute defn} setup.
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Definitions may be presented with explicit arguments on the LHS, as
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well as additional conditions, e.g.\ @{text "f x y = t"} instead of
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@{text "f \<equiv> \<lambda>x y. t"} and @{text "y \<noteq> 0 \<Longrightarrow> g x y = u"} instead of an
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unrestricted @{text "g \<equiv> \<lambda>x y. u"}.
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\item [@{command "abbreviation"}~@{text "c \<WHERE> eq"}] introduces
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a syntactic constant which is associated with a certain term
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according to the meta-level equality @{text eq}.
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Abbreviations participate in the usual type-inference process, but
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are expanded before the logic ever sees them. Pretty printing of
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terms involves higher-order rewriting with rules stemming from
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reverted abbreviations. This needs some care to avoid overlapping
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or looping syntactic replacements!
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The optional @{text mode} specification restricts output to a
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particular print mode; using ``@{text input}'' here achieves the
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effect of one-way abbreviations. The mode may also include an
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``@{keyword "output"}'' qualifier that affects the concrete syntax
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declared for abbreviations, cf.\ @{command "syntax"} in
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\secref{sec:syn-trans}.
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\item [@{command "print_abbrevs"}] prints all constant abbreviations
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of the current context.
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\item [@{command "notation"}~@{text "c (mx)"}] associates mixfix
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syntax with an existing constant or fixed variable. This is a
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robust interface to the underlying @{command "syntax"} primitive
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(\secref{sec:syn-trans}). Type declaration and internal syntactic
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representation of the given entity is retrieved from the context.
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\item [@{command "no_notation"}] is similar to @{command
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"notation"}, but removes the specified syntax annotation from the
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present context.
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\end{descr}
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All of these specifications support local theory targets (cf.\
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\secref{sec:target}).
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*}
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section {* Generic declarations *}
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text {*
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Arbitrary operations on the background context may be wrapped-up as
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generic declaration elements. Since the underlying concept of local
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theories may be subject to later re-interpretation, there is an
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additional dependency on a morphism that tells the difference of the
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original declaration context wrt.\ the application context
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encountered later on. A fact declaration is an important special
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case: it consists of a theorem which is applied to the context by
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means of an attribute.
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\begin{matharray}{rcl}
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@{command_def "declaration"} & : & \isarkeep{local{\dsh}theory} \\
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@{command_def "declare"} & : & \isarkeep{local{\dsh}theory} \\
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\end{matharray}
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\begin{rail}
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'declaration' target? text
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;
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'declare' target? (thmrefs + 'and')
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;
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\end{rail}
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\begin{descr}
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\item [@{command "declaration"}~@{text d}] adds the declaration
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function @{text d} of ML type @{ML_type declaration}, to the current
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local theory under construction. In later application contexts, the
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function is transformed according to the morphisms being involved in
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the interpretation hierarchy.
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\item [@{command "declare"}~@{text thms}] declares theorems to the
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current local theory context. No theorem binding is involved here,
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unlike @{command "theorems"} or @{command "lemmas"} (cf.\
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\secref{sec:axms-thms}), so @{command "declare"} only has the effect
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of applying attributes as included in the theorem specification.
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\end{descr}
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*}
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section {* Locales \label{sec:locale} *}
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text {*
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Locales are named local contexts, consisting of a list of
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declaration elements that are modeled after the Isar proof context
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commands (cf.\ \secref{sec:proof-context}).
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*}
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subsection {* Locale specifications *}
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text {*
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\begin{matharray}{rcl}
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@{command_def "locale"} & : & \isartrans{theory}{local{\dsh}theory} \\
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@{command_def "print_locale"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
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@{command_def "print_locales"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
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@{method_def intro_locales} & : & \isarmeth \\
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@{method_def unfold_locales} & : & \isarmeth \\
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\end{matharray}
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\indexouternonterm{contextexpr}\indexouternonterm{contextelem}
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\indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}
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\indexisarelem{defines}\indexisarelem{notes}\indexisarelem{includes}
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\begin{rail}
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'locale' ('(open)')? name ('=' localeexpr)? 'begin'?
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;
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'print\_locale' '!'? localeexpr
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;
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localeexpr: ((contextexpr '+' (contextelem+)) | contextexpr | (contextelem+))
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;
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contextexpr: nameref | '(' contextexpr ')' |
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(contextexpr (name mixfix? +)) | (contextexpr + '+')
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;
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contextelem: fixes | constrains | assumes | defines | notes
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;
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fixes: 'fixes' ((name ('::' type)? structmixfix? | vars) + 'and')
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;
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constrains: 'constrains' (name '::' type + 'and')
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;
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assumes: 'assumes' (thmdecl? props + 'and')
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;
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defines: 'defines' (thmdecl? prop proppat? + 'and')
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;
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notes: 'notes' (thmdef? thmrefs + 'and')
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;
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includes: 'includes' contextexpr
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;
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\end{rail}
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\begin{descr}
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\item [@{command "locale"}~@{text "loc = import + body"}] defines a
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new locale @{text loc} as a context consisting of a certain view of
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existing locales (@{text import}) plus some additional elements
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(@{text body}). Both @{text import} and @{text body} are optional;
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the degenerate form @{command "locale"}~@{text loc} defines an empty
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locale, which may still be useful to collect declarations of facts
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later on. Type-inference on locale expressions automatically takes
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care of the most general typing that the combined context elements
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may acquire.
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The @{text import} consists of a structured context expression,
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consisting of references to existing locales, renamed contexts, or
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merged contexts. Renaming uses positional notation: @{text "c
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x\<^sub>1 \<dots> x\<^sub>n"} means that (a prefix of) the fixed
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parameters of context @{text c} are named @{text "x\<^sub>1, \<dots>,
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x\<^sub>n"}; a ``@{text _}'' (underscore) means to skip that
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position. Renaming by default deletes concrete syntax, but new
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syntax may by specified with a mixfix annotation. An exeption of
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this rule is the special syntax declared with ``@{text
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"(\<STRUCTURE>)"}'' (see below), which is neither deleted nor can it
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be changed. Merging proceeds from left-to-right, suppressing any
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duplicates stemming from different paths through the import
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hierarchy.
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The @{text body} consists of basic context elements, further context
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expressions may be included as well.
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\begin{descr}
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\item [@{element "fixes"}~@{text "x :: \<tau> (mx)"}] declares a local
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parameter of type @{text \<tau>} and mixfix annotation @{text mx} (both
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are optional). The special syntax declaration ``@{text
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"(\<STRUCTURE>)"}'' means that @{text x} may be referenced
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implicitly in this context.
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\item [@{element "constrains"}~@{text "x :: \<tau>"}] introduces a type
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constraint @{text \<tau>} on the local parameter @{text x}.
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\item [@{element "assumes"}~@{text "a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}]
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introduces local premises, similar to @{command "assume"} within a
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proof (cf.\ \secref{sec:proof-context}).
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\item [@{element "defines"}~@{text "a: x \<equiv> t"}] defines a previously
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declared parameter. This is similar to @{command "def"} within a
|
|
373 |
proof (cf.\ \secref{sec:proof-context}), but @{element "defines"}
|
|
374 |
takes an equational proposition instead of variable-term pair. The
|
|
375 |
left-hand side of the equation may have additional arguments, e.g.\
|
|
376 |
``@{element "defines"}~@{text "f x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"}''.
|
|
377 |
|
|
378 |
\item [@{element "notes"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}]
|
|
379 |
reconsiders facts within a local context. Most notably, this may
|
|
380 |
include arbitrary declarations in any attribute specifications
|
|
381 |
included here, e.g.\ a local @{attribute simp} rule.
|
|
382 |
|
|
383 |
\item [@{element "includes"}~@{text c}] copies the specified context
|
|
384 |
in a statically scoped manner. Only available in the long goal
|
|
385 |
format of \secref{sec:goals}.
|
|
386 |
|
|
387 |
In contrast, the initial @{text import} specification of a locale
|
|
388 |
expression maintains a dynamic relation to the locales being
|
|
389 |
referenced (benefiting from any later fact declarations in the
|
|
390 |
obvious manner).
|
|
391 |
|
|
392 |
\end{descr}
|
|
393 |
|
|
394 |
Note that ``@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}'' patterns given
|
|
395 |
in the syntax of @{element "assumes"} and @{element "defines"} above
|
|
396 |
are illegal in locale definitions. In the long goal format of
|
|
397 |
\secref{sec:goals}, term bindings may be included as expected,
|
|
398 |
though.
|
|
399 |
|
|
400 |
\medskip By default, locale specifications are ``closed up'' by
|
|
401 |
turning the given text into a predicate definition @{text
|
|
402 |
loc_axioms} and deriving the original assumptions as local lemmas
|
|
403 |
(modulo local definitions). The predicate statement covers only the
|
|
404 |
newly specified assumptions, omitting the content of included locale
|
|
405 |
expressions. The full cumulative view is only provided on export,
|
|
406 |
involving another predicate @{text loc} that refers to the complete
|
|
407 |
specification text.
|
|
408 |
|
|
409 |
In any case, the predicate arguments are those locale parameters
|
|
410 |
that actually occur in the respective piece of text. Also note that
|
|
411 |
these predicates operate at the meta-level in theory, but the locale
|
|
412 |
packages attempts to internalize statements according to the
|
|
413 |
object-logic setup (e.g.\ replacing @{text \<And>} by @{text \<forall>}, and
|
|
414 |
@{text "\<Longrightarrow>"} by @{text "\<longrightarrow>"} in HOL; see also
|
|
415 |
\secref{sec:object-logic}). Separate introduction rules @{text
|
|
416 |
loc_axioms.intro} and @{text loc.intro} are provided as well.
|
|
417 |
|
|
418 |
The @{text "(open)"} option of a locale specification prevents both
|
|
419 |
the current @{text loc_axioms} and cumulative @{text loc} predicate
|
|
420 |
constructions. Predicates are also omitted for empty specification
|
|
421 |
texts.
|
|
422 |
|
|
423 |
\item [@{command "print_locale"}~@{text "import + body"}] prints the
|
|
424 |
specified locale expression in a flattened form. The notable
|
|
425 |
special case @{command "print_locale"}~@{text loc} just prints the
|
|
426 |
contents of the named locale, but keep in mind that type-inference
|
|
427 |
will normalize type variables according to the usual alphabetical
|
|
428 |
order. The command omits @{element "notes"} elements by default.
|
|
429 |
Use @{command "print_locale"}@{text "!"} to get them included.
|
|
430 |
|
|
431 |
\item [@{command "print_locales"}] prints the names of all locales
|
|
432 |
of the current theory.
|
|
433 |
|
|
434 |
\item [@{method intro_locales} and @{method unfold_locales}]
|
|
435 |
repeatedly expand all introduction rules of locale predicates of the
|
|
436 |
theory. While @{method intro_locales} only applies the @{text
|
|
437 |
loc.intro} introduction rules and therefore does not decend to
|
|
438 |
assumptions, @{method unfold_locales} is more aggressive and applies
|
|
439 |
@{text loc_axioms.intro} as well. Both methods are aware of locale
|
|
440 |
specifications entailed by the context, both from target and
|
|
441 |
@{element "includes"} statements, and from interpretations (see
|
|
442 |
below). New goals that are entailed by the current context are
|
|
443 |
discharged automatically.
|
|
444 |
|
|
445 |
\end{descr}
|
|
446 |
*}
|
|
447 |
|
|
448 |
|
|
449 |
subsection {* Interpretation of locales *}
|
|
450 |
|
|
451 |
text {*
|
|
452 |
Locale expressions (more precisely, \emph{context expressions}) may
|
|
453 |
be instantiated, and the instantiated facts added to the current
|
|
454 |
context. This requires a proof of the instantiated specification
|
|
455 |
and is called \emph{locale interpretation}. Interpretation is
|
|
456 |
possible in theories and locales (command @{command
|
|
457 |
"interpretation"}) and also within a proof body (command @{command
|
|
458 |
"interpret"}).
|
|
459 |
|
|
460 |
\begin{matharray}{rcl}
|
|
461 |
@{command_def "interpretation"} & : & \isartrans{theory}{proof(prove)} \\
|
|
462 |
@{command_def "interpret"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
|
|
463 |
@{command_def "print_interps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
|
|
464 |
\end{matharray}
|
|
465 |
|
|
466 |
\indexouternonterm{interp}
|
|
467 |
\begin{rail}
|
|
468 |
'interpretation' (interp | name ('<' | subseteq) contextexpr)
|
|
469 |
;
|
|
470 |
'interpret' interp
|
|
471 |
;
|
|
472 |
'print\_interps' '!'? name
|
|
473 |
;
|
|
474 |
instantiation: ('[' (inst+) ']')?
|
|
475 |
;
|
|
476 |
interp: thmdecl? \\ (contextexpr instantiation |
|
|
477 |
name instantiation 'where' (thmdecl? prop + 'and'))
|
|
478 |
;
|
|
479 |
\end{rail}
|
|
480 |
|
|
481 |
\begin{descr}
|
|
482 |
|
|
483 |
\item [@{command "interpretation"}~@{text "expr insts \<WHERE> eqns"}]
|
|
484 |
|
|
485 |
The first form of @{command "interpretation"} interprets @{text
|
|
486 |
expr} in the theory. The instantiation is given as a list of terms
|
|
487 |
@{text insts} and is positional. All parameters must receive an
|
|
488 |
instantiation term --- with the exception of defined parameters.
|
|
489 |
These are, if omitted, derived from the defining equation and other
|
|
490 |
instantiations. Use ``@{text _}'' to omit an instantiation term.
|
|
491 |
|
|
492 |
The command generates proof obligations for the instantiated
|
|
493 |
specifications (assumes and defines elements). Once these are
|
|
494 |
discharged by the user, instantiated facts are added to the theory
|
|
495 |
in a post-processing phase.
|
|
496 |
|
|
497 |
Additional equations, which are unfolded in facts during
|
|
498 |
post-processing, may be given after the keyword @{keyword "where"}.
|
|
499 |
This is useful for interpreting concepts introduced through
|
|
500 |
definition specification elements. The equations must be proved.
|
|
501 |
Note that if equations are present, the context expression is
|
|
502 |
restricted to a locale name.
|
|
503 |
|
|
504 |
The command is aware of interpretations already active in the
|
|
505 |
theory. No proof obligations are generated for those, neither is
|
|
506 |
post-processing applied to their facts. This avoids duplication of
|
|
507 |
interpreted facts, in particular. Note that, in the case of a
|
|
508 |
locale with import, parts of the interpretation may already be
|
|
509 |
active. The command will only generate proof obligations and
|
|
510 |
process facts for new parts.
|
|
511 |
|
|
512 |
The context expression may be preceded by a name and/or attributes.
|
|
513 |
These take effect in the post-processing of facts. The name is used
|
|
514 |
to prefix fact names, for example to avoid accidental hiding of
|
|
515 |
other facts. Attributes are applied after attributes of the
|
|
516 |
interpreted facts.
|
|
517 |
|
|
518 |
Adding facts to locales has the effect of adding interpreted facts
|
|
519 |
to the theory for all active interpretations also. That is,
|
|
520 |
interpretations dynamically participate in any facts added to
|
|
521 |
locales.
|
|
522 |
|
|
523 |
\item [@{command "interpretation"}~@{text "name \<subseteq> expr"}]
|
|
524 |
|
|
525 |
This form of the command interprets @{text expr} in the locale
|
|
526 |
@{text name}. It requires a proof that the specification of @{text
|
|
527 |
name} implies the specification of @{text expr}. As in the
|
|
528 |
localized version of the theorem command, the proof is in the
|
|
529 |
context of @{text name}. After the proof obligation has been
|
|
530 |
dischared, the facts of @{text expr} become part of locale @{text
|
|
531 |
name} as \emph{derived} context elements and are available when the
|
|
532 |
context @{text name} is subsequently entered. Note that, like
|
|
533 |
import, this is dynamic: facts added to a locale part of @{text
|
|
534 |
expr} after interpretation become also available in @{text name}.
|
|
535 |
Like facts of renamed context elements, facts obtained by
|
|
536 |
interpretation may be accessed by prefixing with the parameter
|
|
537 |
renaming (where the parameters are separated by ``@{text _}'').
|
|
538 |
|
|
539 |
Unlike interpretation in theories, instantiation is confined to the
|
|
540 |
renaming of parameters, which may be specified as part of the
|
|
541 |
context expression @{text expr}. Using defined parameters in @{text
|
|
542 |
name} one may achieve an effect similar to instantiation, though.
|
|
543 |
|
|
544 |
Only specification fragments of @{text expr} that are not already
|
|
545 |
part of @{text name} (be it imported, derived or a derived fragment
|
|
546 |
of the import) are considered by interpretation. This enables
|
|
547 |
circular interpretations.
|
|
548 |
|
|
549 |
If interpretations of @{text name} exist in the current theory, the
|
|
550 |
command adds interpretations for @{text expr} as well, with the same
|
|
551 |
prefix and attributes, although only for fragments of @{text expr}
|
|
552 |
that are not interpreted in the theory already.
|
|
553 |
|
|
554 |
\item [@{command "interpret"}~@{text "expr insts \<WHERE> eqns"}]
|
|
555 |
interprets @{text expr} in the proof context and is otherwise
|
|
556 |
similar to interpretation in theories.
|
|
557 |
|
|
558 |
\item [@{command "print_interps"}~@{text loc}] prints the
|
|
559 |
interpretations of a particular locale @{text loc} that are active
|
|
560 |
in the current context, either theory or proof context. The
|
|
561 |
exclamation point argument triggers printing of \emph{witness}
|
|
562 |
theorems justifying interpretations. These are normally omitted
|
|
563 |
from the output.
|
|
564 |
|
|
565 |
\end{descr}
|
|
566 |
|
|
567 |
\begin{warn}
|
|
568 |
Since attributes are applied to interpreted theorems,
|
|
569 |
interpretation may modify the context of common proof tools, e.g.\
|
|
570 |
the Simplifier or Classical Reasoner. Since the behavior of such
|
|
571 |
automated reasoning tools is \emph{not} stable under
|
|
572 |
interpretation morphisms, manual declarations might have to be
|
|
573 |
issued.
|
|
574 |
\end{warn}
|
|
575 |
|
|
576 |
\begin{warn}
|
|
577 |
An interpretation in a theory may subsume previous
|
|
578 |
interpretations. This happens if the same specification fragment
|
|
579 |
is interpreted twice and the instantiation of the second
|
|
580 |
interpretation is more general than the interpretation of the
|
|
581 |
first. A warning is issued, since it is likely that these could
|
|
582 |
have been generalized in the first place. The locale package does
|
|
583 |
not attempt to remove subsumed interpretations.
|
|
584 |
\end{warn}
|
|
585 |
*}
|
|
586 |
|
|
587 |
|
|
588 |
section {* Classes \label{sec:class} *}
|
|
589 |
|
|
590 |
text {*
|
|
591 |
A class is a particular locale with \emph{exactly one} type variable
|
|
592 |
@{text \<alpha>}. Beyond the underlying locale, a corresponding type class
|
|
593 |
is established which is interpreted logically as axiomatic type
|
|
594 |
class \cite{Wenzel:1997:TPHOL} whose logical content are the
|
|
595 |
assumptions of the locale. Thus, classes provide the full
|
|
596 |
generality of locales combined with the commodity of type classes
|
|
597 |
(notably type-inference). See \cite{isabelle-classes} for a short
|
|
598 |
tutorial.
|
|
599 |
|
|
600 |
\begin{matharray}{rcl}
|
|
601 |
@{command_def "class"} & : & \isartrans{theory}{local{\dsh}theory} \\
|
|
602 |
@{command_def "instantiation"} & : & \isartrans{theory}{local{\dsh}theory} \\
|
|
603 |
@{command_def "instance"} & : & \isartrans{local{\dsh}theory}{local{\dsh}theory} \\
|
|
604 |
@{command_def "subclass"} & : & \isartrans{local{\dsh}theory}{local{\dsh}theory} \\
|
|
605 |
@{command_def "print_classes"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
|
|
606 |
@{method_def intro_classes} & : & \isarmeth \\
|
|
607 |
\end{matharray}
|
|
608 |
|
|
609 |
\begin{rail}
|
|
610 |
'class' name '=' ((superclassexpr '+' (contextelem+)) | superclassexpr | (contextelem+)) \\
|
|
611 |
'begin'?
|
|
612 |
;
|
|
613 |
'instantiation' (nameref + 'and') '::' arity 'begin'
|
|
614 |
;
|
|
615 |
'instance'
|
|
616 |
;
|
|
617 |
'subclass' target? nameref
|
|
618 |
;
|
|
619 |
'print\_classes'
|
|
620 |
;
|
|
621 |
|
|
622 |
superclassexpr: nameref | (nameref '+' superclassexpr)
|
|
623 |
;
|
|
624 |
\end{rail}
|
|
625 |
|
|
626 |
\begin{descr}
|
|
627 |
|
|
628 |
\item [@{command "class"}~@{text "c = superclasses + body"}] defines
|
|
629 |
a new class @{text c}, inheriting from @{text superclasses}. This
|
|
630 |
introduces a locale @{text c} with import of all locales @{text
|
|
631 |
superclasses}.
|
|
632 |
|
|
633 |
Any @{element "fixes"} in @{text body} are lifted to the global
|
|
634 |
theory level (\emph{class operations} @{text "f\<^sub>1, \<dots>,
|
|
635 |
f\<^sub>n"} of class @{text c}), mapping the local type parameter
|
|
636 |
@{text \<alpha>} to a schematic type variable @{text "?\<alpha> :: c"}.
|
|
637 |
|
|
638 |
Likewise, @{element "assumes"} in @{text body} are also lifted,
|
|
639 |
mapping each local parameter @{text "f :: \<tau>[\<alpha>]"} to its
|
|
640 |
corresponding global constant @{text "f :: \<tau>[?\<alpha> :: c]"}. The
|
|
641 |
corresponding introduction rule is provided as @{text
|
|
642 |
c_class_axioms.intro}. This rule should be rarely needed directly
|
|
643 |
--- the @{method intro_classes} method takes care of the details of
|
|
644 |
class membership proofs.
|
|
645 |
|
|
646 |
\item [@{command "instantiation"}~@{text "t :: (s\<^sub>1, \<dots>,
|
|
647 |
s\<^sub>n) s \<BEGIN>"}] opens a theory target (cf.\
|
|
648 |
\secref{sec:target}) which allows to specify class operations @{text
|
|
649 |
"f\<^sub>1, \<dots>, f\<^sub>n"} corresponding to sort @{text s} at the
|
|
650 |
particular type instance @{text "(\<alpha>\<^sub>1 :: s\<^sub>1, \<dots>,
|
|
651 |
\<alpha>\<^sub>n :: s\<^sub>n) t"}. A plain @{command "instance"} command
|
|
652 |
in the target body poses a goal stating these type arities. The
|
|
653 |
target is concluded by an @{command_ref (local) "end"} command.
|
|
654 |
|
|
655 |
Note that a list of simultaneous type constructors may be given;
|
|
656 |
this corresponds nicely to mutual recursive type definitions, e.g.\
|
|
657 |
in Isabelle/HOL.
|
|
658 |
|
|
659 |
\item [@{command "instance"}] in an instantiation target body sets
|
|
660 |
up a goal stating the type arities claimed at the opening @{command
|
|
661 |
"instantiation"}. The proof would usually proceed by @{method
|
|
662 |
intro_classes}, and then establish the characteristic theorems of
|
|
663 |
the type classes involved. After finishing the proof, the
|
|
664 |
background theory will be augmented by the proven type arities.
|
|
665 |
|
|
666 |
\item [@{command "subclass"}~@{text c}] in a class context for class
|
|
667 |
@{text d} sets up a goal stating that class @{text c} is logically
|
|
668 |
contained in class @{text d}. After finishing the proof, class
|
|
669 |
@{text d} is proven to be subclass @{text c} and the locale @{text
|
|
670 |
c} is interpreted into @{text d} simultaneously.
|
|
671 |
|
|
672 |
\item [@{command "print_classes"}] prints all classes in the current
|
|
673 |
theory.
|
|
674 |
|
|
675 |
\item [@{method intro_classes}] repeatedly expands all class
|
|
676 |
introduction rules of this theory. Note that this method usually
|
|
677 |
needs not be named explicitly, as it is already included in the
|
|
678 |
default proof step (e.g.\ of @{command "proof"}). In particular,
|
|
679 |
instantiation of trivial (syntactic) classes may be performed by a
|
|
680 |
single ``@{command ".."}'' proof step.
|
26870
|
681 |
|
|
682 |
\end{descr}
|
|
683 |
*}
|
|
684 |
|
27040
|
685 |
|
|
686 |
subsection {* The class target *}
|
|
687 |
|
|
688 |
text {*
|
|
689 |
%FIXME check
|
|
690 |
|
|
691 |
A named context may refer to a locale (cf.\ \secref{sec:target}).
|
|
692 |
If this locale is also a class @{text c}, apart from the common
|
|
693 |
locale target behaviour the following happens.
|
|
694 |
|
|
695 |
\begin{itemize}
|
|
696 |
|
|
697 |
\item Local constant declarations @{text "g[\<alpha>]"} referring to the
|
|
698 |
local type parameter @{text \<alpha>} and local parameters @{text "f[\<alpha>]"}
|
|
699 |
are accompanied by theory-level constants @{text "g[?\<alpha> :: c]"}
|
|
700 |
referring to theory-level class operations @{text "f[?\<alpha> :: c]"}.
|
|
701 |
|
|
702 |
\item Local theorem bindings are lifted as are assumptions.
|
|
703 |
|
|
704 |
\item Local syntax refers to local operations @{text "g[\<alpha>]"} and
|
|
705 |
global operations @{text "g[?\<alpha> :: c]"} uniformly. Type inference
|
|
706 |
resolves ambiguities. In rare cases, manual type annotations are
|
|
707 |
needed.
|
|
708 |
|
|
709 |
\end{itemize}
|
|
710 |
*}
|
|
711 |
|
|
712 |
|
|
713 |
section {* Axiomatic type classes \label{sec:axclass} *}
|
|
714 |
|
|
715 |
text {*
|
|
716 |
\begin{warn}
|
|
717 |
This describes the old interface to axiomatic type-classes in
|
|
718 |
Isabelle. See \secref{sec:class} for a more recent higher-level
|
|
719 |
view on the same ideas.
|
|
720 |
\end{warn}
|
|
721 |
|
|
722 |
\begin{matharray}{rcl}
|
|
723 |
@{command_def "axclass"} & : & \isartrans{theory}{theory} \\
|
|
724 |
@{command_def "instance"} & : & \isartrans{theory}{proof(prove)} \\
|
|
725 |
\end{matharray}
|
|
726 |
|
|
727 |
Axiomatic type classes are Isabelle/Pure's primitive
|
|
728 |
\emph{definitional} interface to type classes. For practical
|
|
729 |
applications, you should consider using classes
|
|
730 |
(cf.~\secref{sec:classes}) which provide high level interface.
|
|
731 |
|
|
732 |
\begin{rail}
|
|
733 |
'axclass' classdecl (axmdecl prop +)
|
|
734 |
;
|
|
735 |
'instance' (nameref ('<' | subseteq) nameref | nameref '::' arity)
|
|
736 |
;
|
|
737 |
\end{rail}
|
|
738 |
|
|
739 |
\begin{descr}
|
|
740 |
|
|
741 |
\item [@{command "axclass"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n
|
|
742 |
axms"}] defines an axiomatic type class as the intersection of
|
|
743 |
existing classes, with additional axioms holding. Class axioms may
|
|
744 |
not contain more than one type variable. The class axioms (with
|
|
745 |
implicit sort constraints added) are bound to the given names.
|
|
746 |
Furthermore a class introduction rule is generated (being bound as
|
|
747 |
@{text c_class.intro}); this rule is employed by method @{method
|
|
748 |
intro_classes} to support instantiation proofs of this class.
|
|
749 |
|
|
750 |
The ``class axioms'' are stored as theorems according to the given
|
|
751 |
name specifications, adding @{text "c_class"} as name space prefix;
|
|
752 |
the same facts are also stored collectively as @{text
|
|
753 |
c_class.axioms}.
|
|
754 |
|
|
755 |
\item [@{command "instance"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} and
|
|
756 |
@{command "instance"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n) s"}]
|
|
757 |
setup a goal stating a class relation or type arity. The proof
|
|
758 |
would usually proceed by @{method intro_classes}, and then establish
|
|
759 |
the characteristic theorems of the type classes involved. After
|
|
760 |
finishing the proof, the theory will be augmented by a type
|
|
761 |
signature declaration corresponding to the resulting theorem.
|
|
762 |
|
|
763 |
\end{descr}
|
|
764 |
*}
|
|
765 |
|
|
766 |
|
|
767 |
section {* Unrestricted overloading *}
|
|
768 |
|
|
769 |
text {*
|
|
770 |
Isabelle/Pure's definitional schemes support certain forms of
|
|
771 |
overloading (see \secref{sec:consts}). At most occassions
|
|
772 |
overloading will be used in a Haskell-like fashion together with
|
|
773 |
type classes by means of @{command "instantiation"} (see
|
|
774 |
\secref{sec:class}). Sometimes low-level overloading is desirable.
|
|
775 |
The @{command "overloading"} target provides a convenient view for
|
|
776 |
end-users.
|
|
777 |
|
|
778 |
\begin{matharray}{rcl}
|
|
779 |
@{command_def "overloading"} & : & \isartrans{theory}{local{\dsh}theory} \\
|
|
780 |
\end{matharray}
|
|
781 |
|
|
782 |
\begin{rail}
|
|
783 |
'overloading' \\
|
|
784 |
( string ( '==' | equiv ) term ( '(' 'unchecked' ')' )? + ) 'begin'
|
|
785 |
\end{rail}
|
|
786 |
|
|
787 |
\begin{descr}
|
|
788 |
|
|
789 |
\item [@{command "overloading"}~@{text "x\<^sub>1 \<equiv> c\<^sub>1 ::
|
|
790 |
\<tau>\<^sub>1 \<AND> \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n \<BEGIN>"}]
|
|
791 |
opens a theory target (cf.\ \secref{sec:target}) which allows to
|
|
792 |
specify constants with overloaded definitions. These are identified
|
|
793 |
by an explicitly given mapping from variable names @{text
|
|
794 |
"x\<^sub>i"} to constants @{text "c\<^sub>i"} at particular type
|
|
795 |
instances. The definitions themselves are established using common
|
|
796 |
specification tools, using the names @{text "x\<^sub>i"} as
|
|
797 |
reference to the corresponding constants. The target is concluded
|
|
798 |
by @{command (local) "end"}.
|
|
799 |
|
|
800 |
A @{text "(unchecked)"} option disables global dependency checks for
|
|
801 |
the corresponding definition, which is occasionally useful for
|
|
802 |
exotic overloading. It is at the discretion of the user to avoid
|
|
803 |
malformed theory specifications!
|
|
804 |
|
|
805 |
\end{descr}
|
|
806 |
*}
|
|
807 |
|
|
808 |
|
|
809 |
section {* Incorporating ML code \label{sec:ML} *}
|
|
810 |
|
|
811 |
text {*
|
|
812 |
\begin{matharray}{rcl}
|
|
813 |
@{command_def "use"} & : & \isarkeep{theory~|~local{\dsh}theory} \\
|
|
814 |
@{command_def "ML"} & : & \isarkeep{theory~|~local{\dsh}theory} \\
|
|
815 |
@{command_def "ML_val"} & : & \isartrans{\cdot}{\cdot} \\
|
|
816 |
@{command_def "ML_command"} & : & \isartrans{\cdot}{\cdot} \\
|
|
817 |
@{command_def "setup"} & : & \isartrans{theory}{theory} \\
|
|
818 |
@{command_def "method_setup"} & : & \isartrans{theory}{theory} \\
|
|
819 |
\end{matharray}
|
|
820 |
|
|
821 |
\begin{rail}
|
|
822 |
'use' name
|
|
823 |
;
|
|
824 |
('ML' | 'ML\_val' | 'ML\_command' | 'setup') text
|
|
825 |
;
|
|
826 |
'method\_setup' name '=' text text
|
|
827 |
;
|
|
828 |
\end{rail}
|
|
829 |
|
|
830 |
\begin{descr}
|
|
831 |
|
|
832 |
\item [@{command "use"}~@{text "file"}] reads and executes ML
|
|
833 |
commands from @{text "file"}. The current theory context is passed
|
|
834 |
down to the ML toplevel and may be modified, using @{ML
|
|
835 |
"Context.>>"} or derived ML commands. The file name is checked with
|
|
836 |
the @{keyword_ref "uses"} dependency declaration given in the theory
|
|
837 |
header (see also \secref{sec:begin-thy}).
|
|
838 |
|
|
839 |
\item [@{command "ML"}~@{text "text"}] is similar to @{command
|
|
840 |
"use"}, but executes ML commands directly from the given @{text
|
|
841 |
"text"}.
|
|
842 |
|
|
843 |
\item [@{command "ML_val"} and @{command "ML_command"}] are
|
|
844 |
diagnostic versions of @{command "ML"}, which means that the context
|
|
845 |
may not be updated. @{command "ML_val"} echos the bindings produced
|
|
846 |
at the ML toplevel, but @{command "ML_command"} is silent.
|
|
847 |
|
|
848 |
\item [@{command "setup"}~@{text "text"}] changes the current theory
|
|
849 |
context by applying @{text "text"}, which refers to an ML expression
|
|
850 |
of type @{ML_type "theory -> theory"}. This enables to initialize
|
|
851 |
any object-logic specific tools and packages written in ML, for
|
|
852 |
example.
|
|
853 |
|
|
854 |
\item [@{command "method_setup"}~@{text "name = text description"}]
|
|
855 |
defines a proof method in the current theory. The given @{text
|
|
856 |
"text"} has to be an ML expression of type @{ML_type "Args.src ->
|
|
857 |
Proof.context -> Proof.method"}. Parsing concrete method syntax
|
|
858 |
from @{ML_type Args.src} input can be quite tedious in general. The
|
|
859 |
following simple examples are for methods without any explicit
|
|
860 |
arguments, or a list of theorems, respectively.
|
|
861 |
|
|
862 |
%FIXME proper antiquotations
|
|
863 |
{\footnotesize
|
|
864 |
\begin{verbatim}
|
|
865 |
Method.no_args (Method.METHOD (fn facts => foobar_tac))
|
|
866 |
Method.thms_args (fn thms => Method.METHOD (fn facts => foobar_tac))
|
|
867 |
Method.ctxt_args (fn ctxt => Method.METHOD (fn facts => foobar_tac))
|
|
868 |
Method.thms_ctxt_args (fn thms => fn ctxt =>
|
|
869 |
Method.METHOD (fn facts => foobar_tac))
|
|
870 |
\end{verbatim}
|
|
871 |
}
|
|
872 |
|
|
873 |
Note that mere tactic emulations may ignore the @{text facts}
|
|
874 |
parameter above. Proper proof methods would do something
|
|
875 |
appropriate with the list of current facts, though. Single-rule
|
|
876 |
methods usually do strict forward-chaining (e.g.\ by using @{ML
|
|
877 |
Drule.multi_resolves}), while automatic ones just insert the facts
|
|
878 |
using @{ML Method.insert_tac} before applying the main tactic.
|
|
879 |
|
|
880 |
\end{descr}
|
|
881 |
*}
|
|
882 |
|
|
883 |
|
|
884 |
section {* Primitive specification elements *}
|
|
885 |
|
|
886 |
subsection {* Type classes and sorts \label{sec:classes} *}
|
|
887 |
|
|
888 |
text {*
|
|
889 |
\begin{matharray}{rcll}
|
|
890 |
@{command_def "classes"} & : & \isartrans{theory}{theory} \\
|
|
891 |
@{command_def "classrel"} & : & \isartrans{theory}{theory} & (axiomatic!) \\
|
|
892 |
@{command_def "defaultsort"} & : & \isartrans{theory}{theory} \\
|
|
893 |
@{command_def "class_deps"} & : & \isarkeep{theory~|~proof} \\
|
|
894 |
\end{matharray}
|
|
895 |
|
|
896 |
\begin{rail}
|
|
897 |
'classes' (classdecl +)
|
|
898 |
;
|
|
899 |
'classrel' (nameref ('<' | subseteq) nameref + 'and')
|
|
900 |
;
|
|
901 |
'defaultsort' sort
|
|
902 |
;
|
|
903 |
\end{rail}
|
|
904 |
|
|
905 |
\begin{descr}
|
|
906 |
|
|
907 |
\item [@{command "classes"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n"}]
|
|
908 |
declares class @{text c} to be a subclass of existing classes @{text
|
|
909 |
"c\<^sub>1, \<dots>, c\<^sub>n"}. Cyclic class structures are not permitted.
|
|
910 |
|
|
911 |
\item [@{command "classrel"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"}] states
|
|
912 |
subclass relations between existing classes @{text "c\<^sub>1"} and
|
|
913 |
@{text "c\<^sub>2"}. This is done axiomatically! The @{command_ref
|
|
914 |
"instance"} command (see \secref{sec:axclass}) provides a way to
|
|
915 |
introduce proven class relations.
|
|
916 |
|
|
917 |
\item [@{command "defaultsort"}~@{text s}] makes sort @{text s} the
|
|
918 |
new default sort for any type variables given without sort
|
|
919 |
constraints. Usually, the default sort would be only changed when
|
|
920 |
defining a new object-logic.
|
|
921 |
|
|
922 |
\item [@{command "class_deps"}] visualizes the subclass relation,
|
|
923 |
using Isabelle's graph browser tool (see also \cite{isabelle-sys}).
|
|
924 |
|
|
925 |
\end{descr}
|
|
926 |
*}
|
|
927 |
|
|
928 |
|
|
929 |
subsection {* Types and type abbreviations \label{sec:types-pure} *}
|
|
930 |
|
|
931 |
text {*
|
|
932 |
\begin{matharray}{rcll}
|
|
933 |
@{command_def "types"} & : & \isartrans{theory}{theory} \\
|
|
934 |
@{command_def "typedecl"} & : & \isartrans{theory}{theory} \\
|
|
935 |
@{command_def "nonterminals"} & : & \isartrans{theory}{theory} \\
|
|
936 |
@{command_def "arities"} & : & \isartrans{theory}{theory} & (axiomatic!) \\
|
|
937 |
\end{matharray}
|
|
938 |
|
|
939 |
\begin{rail}
|
|
940 |
'types' (typespec '=' type infix? +)
|
|
941 |
;
|
|
942 |
'typedecl' typespec infix?
|
|
943 |
;
|
|
944 |
'nonterminals' (name +)
|
|
945 |
;
|
|
946 |
'arities' (nameref '::' arity +)
|
|
947 |
;
|
|
948 |
\end{rail}
|
|
949 |
|
|
950 |
\begin{descr}
|
|
951 |
|
|
952 |
\item [@{command "types"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = \<tau>"}]
|
|
953 |
introduces \emph{type synonym} @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"}
|
|
954 |
for existing type @{text "\<tau>"}. Unlike actual type definitions, as
|
|
955 |
are available in Isabelle/HOL for example, type synonyms are just
|
|
956 |
purely syntactic abbreviations without any logical significance.
|
|
957 |
Internally, type synonyms are fully expanded.
|
|
958 |
|
|
959 |
\item [@{command "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"}]
|
|
960 |
declares a new type constructor @{text t}, intended as an actual
|
|
961 |
logical type (of the object-logic, if available).
|
|
962 |
|
|
963 |
\item [@{command "nonterminals"}~@{text c}] declares type
|
|
964 |
constructors @{text c} (without arguments) to act as purely
|
|
965 |
syntactic types, i.e.\ nonterminal symbols of Isabelle's inner
|
|
966 |
syntax of terms or types.
|
|
967 |
|
|
968 |
\item [@{command "arities"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)
|
|
969 |
s"}] augments Isabelle's order-sorted signature of types by new type
|
|
970 |
constructor arities. This is done axiomatically! The @{command_ref
|
|
971 |
"instance"} command (see \S\ref{sec:axclass}) provides a way to
|
|
972 |
introduce proven type arities.
|
|
973 |
|
|
974 |
\end{descr}
|
|
975 |
*}
|
|
976 |
|
|
977 |
|
|
978 |
subsection {* Constants and definitions \label{sec:consts} *}
|
|
979 |
|
|
980 |
text {*
|
|
981 |
Definitions essentially express abbreviations within the logic. The
|
|
982 |
simplest form of a definition is @{text "c :: \<sigma> \<equiv> t"}, where @{text
|
|
983 |
c} is a newly declared constant. Isabelle also allows derived forms
|
|
984 |
where the arguments of @{text c} appear on the left, abbreviating a
|
|
985 |
prefix of @{text \<lambda>}-abstractions, e.g.\ @{text "c \<equiv> \<lambda>x y. t"} may be
|
|
986 |
written more conveniently as @{text "c x y \<equiv> t"}. Moreover,
|
|
987 |
definitions may be weakened by adding arbitrary pre-conditions:
|
|
988 |
@{text "A \<Longrightarrow> c x y \<equiv> t"}.
|
|
989 |
|
|
990 |
\medskip The built-in well-formedness conditions for definitional
|
|
991 |
specifications are:
|
|
992 |
|
|
993 |
\begin{itemize}
|
|
994 |
|
|
995 |
\item Arguments (on the left-hand side) must be distinct variables.
|
|
996 |
|
|
997 |
\item All variables on the right-hand side must also appear on the
|
|
998 |
left-hand side.
|
|
999 |
|
|
1000 |
\item All type variables on the right-hand side must also appear on
|
|
1001 |
the left-hand side; this prohibits @{text "0 :: nat \<equiv> length ([] ::
|
|
1002 |
\<alpha> list)"} for example.
|
|
1003 |
|
|
1004 |
\item The definition must not be recursive. Most object-logics
|
|
1005 |
provide definitional principles that can be used to express
|
|
1006 |
recursion safely.
|
|
1007 |
|
|
1008 |
\end{itemize}
|
|
1009 |
|
|
1010 |
Overloading means that a constant being declared as @{text "c :: \<alpha>
|
|
1011 |
decl"} may be defined separately on type instances @{text "c ::
|
|
1012 |
(\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t decl"} for each type constructor @{text
|
|
1013 |
t}. The right-hand side may mention overloaded constants
|
|
1014 |
recursively at type instances corresponding to the immediate
|
|
1015 |
argument types @{text "\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n"}. Incomplete
|
|
1016 |
specification patterns impose global constraints on all occurrences,
|
|
1017 |
e.g.\ @{text "d :: \<alpha> \<times> \<alpha>"} on the left-hand side means that all
|
|
1018 |
corresponding occurrences on some right-hand side need to be an
|
|
1019 |
instance of this, general @{text "d :: \<alpha> \<times> \<beta>"} will be disallowed.
|
|
1020 |
|
|
1021 |
\begin{matharray}{rcl}
|
|
1022 |
@{command_def "consts"} & : & \isartrans{theory}{theory} \\
|
|
1023 |
@{command_def "defs"} & : & \isartrans{theory}{theory} \\
|
|
1024 |
@{command_def "constdefs"} & : & \isartrans{theory}{theory} \\
|
|
1025 |
\end{matharray}
|
|
1026 |
|
|
1027 |
\begin{rail}
|
|
1028 |
'consts' ((name '::' type mixfix?) +)
|
|
1029 |
;
|
|
1030 |
'defs' ('(' 'unchecked'? 'overloaded'? ')')? \\ (axmdecl prop +)
|
|
1031 |
;
|
|
1032 |
\end{rail}
|
|
1033 |
|
|
1034 |
\begin{rail}
|
|
1035 |
'constdefs' structs? (constdecl? constdef +)
|
|
1036 |
;
|
|
1037 |
|
|
1038 |
structs: '(' 'structure' (vars + 'and') ')'
|
|
1039 |
;
|
|
1040 |
constdecl: ((name '::' type mixfix | name '::' type | name mixfix) 'where'?) | name 'where'
|
|
1041 |
;
|
|
1042 |
constdef: thmdecl? prop
|
|
1043 |
;
|
|
1044 |
\end{rail}
|
|
1045 |
|
|
1046 |
\begin{descr}
|
|
1047 |
|
|
1048 |
\item [@{command "consts"}~@{text "c :: \<sigma>"}] declares constant
|
|
1049 |
@{text c} to have any instance of type scheme @{text \<sigma>}. The
|
|
1050 |
optional mixfix annotations may attach concrete syntax to the
|
|
1051 |
constants declared.
|
|
1052 |
|
|
1053 |
\item [@{command "defs"}~@{text "name: eqn"}] introduces @{text eqn}
|
|
1054 |
as a definitional axiom for some existing constant.
|
|
1055 |
|
|
1056 |
The @{text "(unchecked)"} option disables global dependency checks
|
|
1057 |
for this definition, which is occasionally useful for exotic
|
|
1058 |
overloading. It is at the discretion of the user to avoid malformed
|
|
1059 |
theory specifications!
|
|
1060 |
|
|
1061 |
The @{text "(overloaded)"} option declares definitions to be
|
|
1062 |
potentially overloaded. Unless this option is given, a warning
|
|
1063 |
message would be issued for any definitional equation with a more
|
|
1064 |
special type than that of the corresponding constant declaration.
|
|
1065 |
|
|
1066 |
\item [@{command "constdefs"}] provides a streamlined combination of
|
|
1067 |
constants declarations and definitions: type-inference takes care of
|
|
1068 |
the most general typing of the given specification (the optional
|
|
1069 |
type constraint may refer to type-inference dummies ``@{text
|
|
1070 |
_}'' as usual). The resulting type declaration needs to agree with
|
|
1071 |
that of the specification; overloading is \emph{not} supported here!
|
|
1072 |
|
|
1073 |
The constant name may be omitted altogether, if neither type nor
|
|
1074 |
syntax declarations are given. The canonical name of the
|
|
1075 |
definitional axiom for constant @{text c} will be @{text c_def},
|
|
1076 |
unless specified otherwise. Also note that the given list of
|
|
1077 |
specifications is processed in a strictly sequential manner, with
|
|
1078 |
type-checking being performed independently.
|
|
1079 |
|
|
1080 |
An optional initial context of @{text "(structure)"} declarations
|
|
1081 |
admits use of indexed syntax, using the special symbol @{verbatim
|
|
1082 |
"\<index>"} (printed as ``@{text "\<index>"}''). The latter concept is
|
|
1083 |
particularly useful with locales (see also \S\ref{sec:locale}).
|
|
1084 |
|
|
1085 |
\end{descr}
|
|
1086 |
*}
|
|
1087 |
|
|
1088 |
|
|
1089 |
section {* Axioms and theorems \label{sec:axms-thms} *}
|
|
1090 |
|
|
1091 |
text {*
|
|
1092 |
\begin{matharray}{rcll}
|
|
1093 |
@{command_def "axioms"} & : & \isartrans{theory}{theory} & (axiomatic!) \\
|
|
1094 |
@{command_def "lemmas"} & : & \isarkeep{local{\dsh}theory} \\
|
27046
|
1095 |
@{command_def "theorems"} & : & \isarkeep{local{\dsh}theory} \\
|
27040
|
1096 |
\end{matharray}
|
|
1097 |
|
|
1098 |
\begin{rail}
|
|
1099 |
'axioms' (axmdecl prop +)
|
|
1100 |
;
|
|
1101 |
('lemmas' | 'theorems') target? (thmdef? thmrefs + 'and')
|
|
1102 |
;
|
|
1103 |
\end{rail}
|
|
1104 |
|
|
1105 |
\begin{descr}
|
|
1106 |
|
|
1107 |
\item [@{command "axioms"}~@{text "a: \<phi>"}] introduces arbitrary
|
|
1108 |
statements as axioms of the meta-logic. In fact, axioms are
|
|
1109 |
``axiomatic theorems'', and may be referred later just as any other
|
|
1110 |
theorem.
|
|
1111 |
|
|
1112 |
Axioms are usually only introduced when declaring new logical
|
|
1113 |
systems. Everyday work is typically done the hard way, with proper
|
|
1114 |
definitions and proven theorems.
|
|
1115 |
|
|
1116 |
\item [@{command "lemmas"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}]
|
|
1117 |
retrieves and stores existing facts in the theory context, or the
|
|
1118 |
specified target context (see also \secref{sec:target}). Typical
|
|
1119 |
applications would also involve attributes, to declare Simplifier
|
|
1120 |
rules, for example.
|
|
1121 |
|
|
1122 |
\item [@{command "theorems"}] is essentially the same as @{command
|
|
1123 |
"lemmas"}, but marks the result as a different kind of facts.
|
|
1124 |
|
|
1125 |
\end{descr}
|
|
1126 |
*}
|
|
1127 |
|
|
1128 |
|
|
1129 |
section {* Oracles *}
|
|
1130 |
|
|
1131 |
text {*
|
|
1132 |
\begin{matharray}{rcl}
|
|
1133 |
@{command_def "oracle"} & : & \isartrans{theory}{theory} \\
|
|
1134 |
\end{matharray}
|
|
1135 |
|
|
1136 |
The oracle interface promotes a given ML function @{ML_text
|
|
1137 |
"theory -> T -> term"} to @{ML_text "theory -> T -> thm"}, for some
|
|
1138 |
type @{ML_text T} given by the user. This acts like an infinitary
|
|
1139 |
specification of axioms -- there is no internal check of the
|
|
1140 |
correctness of the results! The inference kernel records oracle
|
|
1141 |
invocations within the internal derivation object of theorems, and
|
|
1142 |
the pretty printer attaches ``@{text "[!]"}'' to indicate results
|
|
1143 |
that are not fully checked by Isabelle inferences.
|
|
1144 |
|
|
1145 |
\begin{rail}
|
|
1146 |
'oracle' name '(' type ')' '=' text
|
|
1147 |
;
|
|
1148 |
\end{rail}
|
|
1149 |
|
|
1150 |
\begin{descr}
|
|
1151 |
|
|
1152 |
\item [@{command "oracle"}~@{text "name (type) = text"}] turns the
|
|
1153 |
given ML expression @{text "text"} of type
|
|
1154 |
@{ML_text "theory ->"}~@{text "type"}~@{ML_text "-> term"} into an
|
|
1155 |
ML function of type
|
|
1156 |
@{ML_text "theory ->"}~@{text "type"}~@{ML_text "-> thm"}, which is
|
|
1157 |
bound to the global identifier @{ML_text name}.
|
|
1158 |
|
|
1159 |
\end{descr}
|
|
1160 |
*}
|
|
1161 |
|
|
1162 |
|
|
1163 |
section {* Name spaces *}
|
|
1164 |
|
|
1165 |
text {*
|
|
1166 |
\begin{matharray}{rcl}
|
|
1167 |
@{command_def "global"} & : & \isartrans{theory}{theory} \\
|
|
1168 |
@{command_def "local"} & : & \isartrans{theory}{theory} \\
|
|
1169 |
@{command_def "hide"} & : & \isartrans{theory}{theory} \\
|
|
1170 |
\end{matharray}
|
|
1171 |
|
|
1172 |
\begin{rail}
|
|
1173 |
'hide' ('(open)')? name (nameref + )
|
|
1174 |
;
|
|
1175 |
\end{rail}
|
|
1176 |
|
|
1177 |
Isabelle organizes any kind of name declarations (of types,
|
|
1178 |
constants, theorems etc.) by separate hierarchically structured name
|
|
1179 |
spaces. Normally the user does not have to control the behavior of
|
|
1180 |
name spaces by hand, yet the following commands provide some way to
|
|
1181 |
do so.
|
|
1182 |
|
|
1183 |
\begin{descr}
|
|
1184 |
|
|
1185 |
\item [@{command "global"} and @{command "local"}] change the
|
|
1186 |
current name declaration mode. Initially, theories start in
|
|
1187 |
@{command "local"} mode, causing all names to be automatically
|
|
1188 |
qualified by the theory name. Changing this to @{command "global"}
|
|
1189 |
causes all names to be declared without the theory prefix, until
|
|
1190 |
@{command "local"} is declared again.
|
|
1191 |
|
|
1192 |
Note that global names are prone to get hidden accidently later,
|
|
1193 |
when qualified names of the same base name are introduced.
|
|
1194 |
|
|
1195 |
\item [@{command "hide"}~@{text "space names"}] fully removes
|
|
1196 |
declarations from a given name space (which may be @{text "class"},
|
|
1197 |
@{text "type"}, @{text "const"}, or @{text "fact"}); with the @{text
|
|
1198 |
"(open)"} option, only the base name is hidden. Global
|
|
1199 |
(unqualified) names may never be hidden.
|
|
1200 |
|
|
1201 |
Note that hiding name space accesses has no impact on logical
|
|
1202 |
declarations -- they remain valid internally. Entities that are no
|
|
1203 |
longer accessible to the user are printed with the special qualifier
|
|
1204 |
``@{text "??"}'' prefixed to the full internal name.
|
|
1205 |
|
|
1206 |
\end{descr}
|
|
1207 |
*}
|
|
1208 |
|
|
1209 |
|
|
1210 |
section {* Syntax and translations \label{sec:syn-trans} *}
|
|
1211 |
|
|
1212 |
text {*
|
|
1213 |
\begin{matharray}{rcl}
|
|
1214 |
@{command_def "syntax"} & : & \isartrans{theory}{theory} \\
|
|
1215 |
@{command_def "no_syntax"} & : & \isartrans{theory}{theory} \\
|
|
1216 |
@{command_def "translations"} & : & \isartrans{theory}{theory} \\
|
|
1217 |
@{command_def "no_translations"} & : & \isartrans{theory}{theory} \\
|
|
1218 |
\end{matharray}
|
|
1219 |
|
|
1220 |
\begin{rail}
|
|
1221 |
('syntax' | 'no\_syntax') mode? (constdecl +)
|
|
1222 |
;
|
|
1223 |
('translations' | 'no\_translations') (transpat ('==' | '=>' | '<=' | rightleftharpoons | rightharpoonup | leftharpoondown) transpat +)
|
|
1224 |
;
|
|
1225 |
|
|
1226 |
mode: ('(' ( name | 'output' | name 'output' ) ')')
|
|
1227 |
;
|
|
1228 |
transpat: ('(' nameref ')')? string
|
|
1229 |
;
|
|
1230 |
\end{rail}
|
|
1231 |
|
|
1232 |
\begin{descr}
|
|
1233 |
|
|
1234 |
\item [@{command "syntax"}~@{text "(mode) decls"}] is similar to
|
|
1235 |
@{command "consts"}~@{text decls}, except that the actual logical
|
|
1236 |
signature extension is omitted. Thus the context free grammar of
|
|
1237 |
Isabelle's inner syntax may be augmented in arbitrary ways,
|
|
1238 |
independently of the logic. The @{text mode} argument refers to the
|
|
1239 |
print mode that the grammar rules belong; unless the @{keyword_ref
|
|
1240 |
"output"} indicator is given, all productions are added both to the
|
|
1241 |
input and output grammar.
|
|
1242 |
|
|
1243 |
\item [@{command "no_syntax"}~@{text "(mode) decls"}] removes
|
|
1244 |
grammar declarations (and translations) resulting from @{text
|
|
1245 |
decls}, which are interpreted in the same manner as for @{command
|
|
1246 |
"syntax"} above.
|
|
1247 |
|
|
1248 |
\item [@{command "translations"}~@{text rules}] specifies syntactic
|
|
1249 |
translation rules (i.e.\ macros): parse~/ print rules (@{text "\<rightleftharpoons>"}),
|
|
1250 |
parse rules (@{text "\<rightharpoonup>"}), or print rules (@{text "\<leftharpoondown>"}).
|
|
1251 |
Translation patterns may be prefixed by the syntactic category to be
|
|
1252 |
used for parsing; the default is @{text logic}.
|
|
1253 |
|
|
1254 |
\item [@{command "no_translations"}~@{text rules}] removes syntactic
|
|
1255 |
translation rules, which are interpreted in the same manner as for
|
|
1256 |
@{command "translations"} above.
|
|
1257 |
|
|
1258 |
\end{descr}
|
|
1259 |
*}
|
|
1260 |
|
|
1261 |
|
|
1262 |
section {* Syntax translation functions *}
|
|
1263 |
|
|
1264 |
text {*
|
|
1265 |
\begin{matharray}{rcl}
|
|
1266 |
@{command_def "parse_ast_translation"} & : & \isartrans{theory}{theory} \\
|
|
1267 |
@{command_def "parse_translation"} & : & \isartrans{theory}{theory} \\
|
|
1268 |
@{command_def "print_translation"} & : & \isartrans{theory}{theory} \\
|
|
1269 |
@{command_def "typed_print_translation"} & : & \isartrans{theory}{theory} \\
|
|
1270 |
@{command_def "print_ast_translation"} & : & \isartrans{theory}{theory} \\
|
|
1271 |
@{command_def "token_translation"} & : & \isartrans{theory}{theory} \\
|
|
1272 |
\end{matharray}
|
|
1273 |
|
|
1274 |
\begin{rail}
|
|
1275 |
( 'parse\_ast\_translation' | 'parse\_translation' | 'print\_translation' |
|
|
1276 |
'typed\_print\_translation' | 'print\_ast\_translation' ) ('(advanced)')? text
|
|
1277 |
;
|
|
1278 |
|
|
1279 |
'token\_translation' text
|
|
1280 |
;
|
|
1281 |
\end{rail}
|
|
1282 |
|
|
1283 |
Syntax translation functions written in ML admit almost arbitrary
|
|
1284 |
manipulations of Isabelle's inner syntax. Any of the above commands
|
|
1285 |
have a single \railqtok{text} argument that refers to an ML
|
|
1286 |
expression of appropriate type, which are as follows by default:
|
|
1287 |
|
|
1288 |
%FIXME proper antiquotations
|
|
1289 |
\begin{ttbox}
|
|
1290 |
val parse_ast_translation : (string * (ast list -> ast)) list
|
|
1291 |
val parse_translation : (string * (term list -> term)) list
|
|
1292 |
val print_translation : (string * (term list -> term)) list
|
|
1293 |
val typed_print_translation :
|
|
1294 |
(string * (bool -> typ -> term list -> term)) list
|
|
1295 |
val print_ast_translation : (string * (ast list -> ast)) list
|
|
1296 |
val token_translation :
|
|
1297 |
(string * string * (string -> string * real)) list
|
|
1298 |
\end{ttbox}
|
|
1299 |
|
|
1300 |
If the @{text "(advanced)"} option is given, the corresponding
|
|
1301 |
translation functions may depend on the current theory or proof
|
|
1302 |
context. This allows to implement advanced syntax mechanisms, as
|
|
1303 |
translations functions may refer to specific theory declarations or
|
|
1304 |
auxiliary proof data.
|
|
1305 |
|
|
1306 |
See also \cite[\S8]{isabelle-ref} for more information on the
|
|
1307 |
general concept of syntax transformations in Isabelle.
|
|
1308 |
|
|
1309 |
%FIXME proper antiquotations
|
|
1310 |
\begin{ttbox}
|
|
1311 |
val parse_ast_translation:
|
27046
|
1312 |
(string * (Proof.context -> ast list -> ast)) list
|
27040
|
1313 |
val parse_translation:
|
27046
|
1314 |
(string * (Proof.context -> term list -> term)) list
|
27040
|
1315 |
val print_translation:
|
27046
|
1316 |
(string * (Proof.context -> term list -> term)) list
|
27040
|
1317 |
val typed_print_translation:
|
27046
|
1318 |
(string * (Proof.context -> bool -> typ -> term list -> term)) list
|
27040
|
1319 |
val print_ast_translation:
|
27046
|
1320 |
(string * (Proof.context -> ast list -> ast)) list
|
27040
|
1321 |
\end{ttbox}
|
|
1322 |
*}
|
|
1323 |
|
26869
|
1324 |
end
|