%

\begin{isabellebody}%

\def\isabellecontext{Spec}%

%

\isadelimtheory

%

\endisadelimtheory

%

\isatagtheory

\isacommand{theory}\isamarkupfalse%

\ Spec\isanewline

\isakeyword{imports}\ Base\ Main\isanewline

\isakeyword{begin}%

\endisatagtheory

{\isafoldtheory}%

%

\isadelimtheory

%

\endisadelimtheory

%

\isamarkupchapter{Specifications%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

The Isabelle/Isar theory format integrates specifications and

  proofs, supporting interactive development with unlimited undo

  operation.  There is an integrated document preparation system (see

  \chref{ch:document-prep}), for typesetting formal developments

  together with informal text.  The resulting hyper-linked PDF

  documents can be used both for WWW presentation and printed copies.

  The Isar proof language (see \chref{ch:proofs}) is embedded into the

  theory language as a proper sub-language.  Proof mode is entered by

  stating some \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}} or \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}} at the theory

  level, and left again with the final conclusion (e.g.\ via \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}).  Some theory specification mechanisms also require a proof,

  such as \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} in HOL, which demands non-emptiness of

  the representing sets.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Defining theories \label{sec:begin-thy}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{command}{theory}\hypertarget{command.theory}{\hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}toplevel\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{global}{command}{end}\hypertarget{command.global.end}{\hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ toplevel{\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  Isabelle/Isar theories are defined via theory files, which may

  contain both specifications and proofs; occasionally definitional

  mechanisms also require some explicit proof.  The theory body may be

  sub-structured by means of \emph{local theory targets}, such as

  \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}} and \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}}.

  The first proper command of a theory is \hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}}, which

  indicates imports of previous theories and optional dependencies on

  other source files (usually in ML).  Just preceding the initial

  \hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}} command there may be an optional \hyperlink{command.header}{\mbox{\isa{\isacommand{header}}}} declaration, which is only relevant to document

  preparation: see also the other section markup commands in

  \secref{sec:markup}.

  A theory is concluded by a final \hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}} command,

  one that does not belong to a local theory target.  No further

  commands may follow such a global \hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}},

  although some user-interfaces might pretend that trailing input is

  admissible.

  \begin{railoutput}

\rail@begin{4}{}

\rail@term{\hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nont{\isa{imports}}[]

\rail@cr{2}

\rail@bar

\rail@nextbar{3}

\rail@nont{\isa{keywords}}[]

\rail@endbar

\rail@bar

\rail@nextbar{3}

\rail@nont{\isa{uses}}[]

\rail@endbar

\rail@term{\isa{\isakeyword{begin}}}[]

\rail@end

\rail@begin{2}{\isa{imports}}

\rail@term{\isa{\isakeyword{imports}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@end

\rail@begin{3}{\isa{keywords}}

\rail@term{\isa{\isakeyword{keywords}}}[]

\rail@plus

\rail@plus

\rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nont{\hyperlink{syntax.tags}{\mbox{\isa{tags}}}}[]

\rail@endbar

\rail@nextplus{2}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\rail@begin{3}{\isa{uses}}

\rail@term{\isa{\isakeyword{uses}}}[]

\rail@plus

\rail@bar

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.parname}{\mbox{\isa{parname}}}}[]

\rail@endbar

\rail@nextplus{2}

\rail@endplus

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C494D504F5254533E}{\isasymIMPORTS}}\ B\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ B\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C424547494E3E}{\isasymBEGIN}}{\isaliteral{22}{\isachardoublequote}}}

  starts a new theory \isa{A} based on the merge of existing

  theories \isa{{\isaliteral{22}{\isachardoublequote}}B\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ B\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}.  Due to the possibility to import more

  than one ancestor, the resulting theory structure of an Isabelle

  session forms a directed acyclic graph (DAG).  Isabelle takes care

  that sources contributing to the development graph are always

  up-to-date: changed files are automatically rechecked whenever a

  theory header specification is processed.

  The optional \indexdef{}{keyword}{keywords}\hypertarget{keyword.keywords}{\hyperlink{keyword.keywords}{\mbox{\isa{\isakeyword{keywords}}}}} specification declares outer

  syntax (\chref{ch:outer-syntax}) that is introduced in this theory

  later on (rare in end-user applications).  Both minor keywords and

  major keywords of the Isar command language need to be specified, in

  order to make parsing of proof documents work properly.  Command

  keywords need to be classified according to their structural role in

  the formal text.  Examples may be seen in Isabelle/HOL sources

  itself, such as \hyperlink{keyword.keywords}{\mbox{\isa{\isakeyword{keywords}}}}~\verb|"typedef"|

  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ thy{\isaliteral{5F}{\isacharunderscore}}goal{\isaliteral{22}{\isachardoublequote}}} or \hyperlink{keyword.keywords}{\mbox{\isa{\isakeyword{keywords}}}}~\verb|"datatype"| \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ thy{\isaliteral{5F}{\isacharunderscore}}decl{\isaliteral{22}{\isachardoublequote}}} for theory-level declarations

  with and without proof, respectively.  Additional \hyperlink{syntax.tags}{\mbox{\isa{tags}}}

  provide defaults for document preparation (\secref{sec:tags}).

  The optional \indexdef{}{keyword}{uses}\hypertarget{keyword.uses}{\hyperlink{keyword.uses}{\mbox{\isa{\isakeyword{uses}}}}} specification declares additional

  dependencies on external files (notably ML sources).  Files will be

  loaded immediately (as ML), unless the name is parenthesized.  The

  latter case records a dependency that needs to be resolved later in

  the text, usually via explicit \indexref{}{command}{use}\hyperlink{command.use}{\mbox{\isa{\isacommand{use}}}} for ML files;

  other file formats require specific load commands defined by the

  corresponding tools or packages.

  \item \hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}} concludes the current theory

  definition.  Note that some other commands, e.g.\ local theory

  targets \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}} or \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}} may involve a

  \hyperlink{keyword.begin}{\mbox{\isa{\isakeyword{begin}}}} that needs to be matched by \hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}}, according to the usual rules for nested blocks.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Local theory targets \label{sec:target}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

A local theory target is a context managed separately within the

  enclosing theory.  Contexts may introduce parameters (fixed

  variables) and assumptions (hypotheses).  Definitions and theorems

  depending on the context may be added incrementally later on.  Named

  contexts refer to locales (cf.\ \secref{sec:locale}) or type classes

  (cf.\ \secref{sec:class}); the name ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' signifies the

  global theory context.

  \begin{matharray}{rcll}

    \indexdef{}{command}{context}\hypertarget{command.context}{\hyperlink{command.context}{\mbox{\isa{\isacommand{context}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{local}{command}{end}\hypertarget{command.local.end}{\hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{1}{}

\rail@term{\hyperlink{command.context}{\mbox{\isa{\isacommand{context}}}}}[]

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@term{\isa{\isakeyword{begin}}}[]

\rail@end

\rail@begin{1}{\indexdef{}{syntax}{target}\hypertarget{syntax.target}{\hyperlink{syntax.target}{\mbox{\isa{target}}}}}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@term{\isa{\isakeyword{in}}}[]

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.context}{\mbox{\isa{\isacommand{context}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{5C3C424547494E3E}{\isasymBEGIN}}{\isaliteral{22}{\isachardoublequote}}} recommences an

  existing locale or class context \isa{c}.  Note that locale and

  class definitions allow to include the \hyperlink{keyword.begin}{\mbox{\isa{\isakeyword{begin}}}} keyword as

  well, in order to continue the local theory immediately after the

  initial specification.

  \item \hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}} concludes the current local theory

  and continues the enclosing global theory.  Note that a global

  \hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}} has a different meaning: it concludes the

  theory itself (\secref{sec:begin-thy}).

  \item \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequote}}}\indexdef{}{keyword}{in}\hypertarget{keyword.in}{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} given after any

  local theory command specifies an immediate target, e.g.\

  ``\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C494E3E}{\isasymIN}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' or ``\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C494E3E}{\isasymIN}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}''.  This works both in a local or

  global theory context; the current target context will be suspended

  for this command only.  Note that ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C494E3E}{\isasymIN}}\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' will

  always produce a global result independently of the current target

  context.

  \end{description}

  The exact meaning of results produced within a local theory context

  depends on the underlying target infrastructure (locale, type class

  etc.).  The general idea is as follows, considering a context named

  \isa{c} with parameter \isa{x} and assumption \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}.

  Definitions are exported by introducing a global version with

  additional arguments; a syntactic abbreviation links the long form

  with the abstract version of the target context.  For example,

  \isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} becomes \isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{2E}{\isachardot}}a\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{3F}{\isacharquery}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} at the theory

  level (for arbitrary \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}x{\isaliteral{22}{\isachardoublequote}}}), together with a local

  abbreviation \isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ c{\isaliteral{2E}{\isachardot}}a\ x{\isaliteral{22}{\isachardoublequote}}} in the target context (for the

  fixed parameter \isa{x}).

  Theorems are exported by discharging the assumptions and

  generalizing the parameters of the context.  For example, \isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} becomes \isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{2E}{\isachardot}}a{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{3F}{\isacharquery}}x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{3F}{\isacharquery}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}, again for arbitrary

  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}x{\isaliteral{22}{\isachardoublequote}}}.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Basic specification elements%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcll}

    \indexdef{}{command}{axiomatization}\hypertarget{command.axiomatization}{\hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} & (axiomatic!) \\

    \indexdef{}{command}{definition}\hypertarget{command.definition}{\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{attribute}{defn}\hypertarget{attribute.defn}{\hyperlink{attribute.defn}{\mbox{\isa{defn}}}} & : & \isa{attribute} \\

    \indexdef{}{command}{abbreviation}\hypertarget{command.abbreviation}{\hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{print\_abbrevs}\hypertarget{command.print-abbrevs}{\hyperlink{command.print-abbrevs}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}abbrevs}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ {\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  These specification mechanisms provide a slightly more abstract view

  than the underlying primitives of \hyperlink{command.consts}{\mbox{\isa{\isacommand{consts}}}}, \hyperlink{command.defs}{\mbox{\isa{\isacommand{defs}}}} (see \secref{sec:consts}), and \hyperlink{command.axioms}{\mbox{\isa{\isacommand{axioms}}}} (see

  \secref{sec:axms-thms}).  In particular, type-inference is commonly

  available, and result names need not be given.

  \begin{railoutput}

\rail@begin{2}{}

\rail@term{\hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{\isakeyword{where}}}[]

\rail@nont{\isa{specs}}[]

\rail@endbar

\rail@end

\rail@begin{5}{}

\rail@term{\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]

\rail@endbar

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@nont{\isa{decl}}[]

\rail@term{\isa{\isakeyword{where}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{4}

\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]

\rail@end

\rail@begin{5}{}

\rail@term{\hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.mode}{\mbox{\isa{mode}}}}[]

\rail@endbar

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@nont{\isa{decl}}[]

\rail@term{\isa{\isakeyword{where}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]

\rail@end

\rail@begin{4}{\indexdef{}{syntax}{fixes}\hypertarget{syntax.fixes}{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}}

\rail@plus

\rail@bar

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]

\rail@endbar

\rail@nextbar{2}

\rail@nont{\hyperlink{syntax.vars}{\mbox{\isa{vars}}}}[]

\rail@endbar

\rail@nextplus{3}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\rail@begin{3}{\isa{specs}}

\rail@plus

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[]

\rail@nextplus{2}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\rail@begin{2}{\isa{decl}}

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}

  introduces several constants simultaneously and states axiomatic

  properties for these.  The constants are marked as being specified

  once and for all, which prevents additional specifications being

  issued later on.

  Note that axiomatic specifications are only appropriate when

  declaring a new logical system; axiomatic specifications are

  restricted to global theory contexts.  Normal applications should

  only use definitional mechanisms!

  \item \hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ eq{\isaliteral{22}{\isachardoublequote}}} produces an

  internal definition \isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} according to the specification

  given as \isa{eq}, which is then turned into a proven fact.  The

  given proposition may deviate from internal meta-level equality

  according to the rewrite rules declared as \hyperlink{attribute.defn}{\mbox{\isa{defn}}} by the

  object-logic.  This usually covers object-level equality \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequote}}} and equivalence \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C65667472696768746172726F773E}{\isasymleftrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}.  End-users normally need not

  change the \hyperlink{attribute.defn}{\mbox{\isa{defn}}} setup.

  Definitions may be presented with explicit arguments on the LHS, as

  well as additional conditions, e.g.\ \isa{{\isaliteral{22}{\isachardoublequote}}f\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{22}{\isachardoublequote}}} instead of

  \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x\ y{\isaliteral{2E}{\isachardot}}\ t{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}y\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ g\ x\ y\ {\isaliteral{3D}{\isacharequal}}\ u{\isaliteral{22}{\isachardoublequote}}} instead of an

  unrestricted \isa{{\isaliteral{22}{\isachardoublequote}}g\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x\ y{\isaliteral{2E}{\isachardot}}\ u{\isaliteral{22}{\isachardoublequote}}}.

  \item \hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ eq{\isaliteral{22}{\isachardoublequote}}} introduces a

  syntactic constant which is associated with a certain term according

  to the meta-level equality \isa{eq}.

  Abbreviations participate in the usual type-inference process, but

  are expanded before the logic ever sees them.  Pretty printing of

  terms involves higher-order rewriting with rules stemming from

  reverted abbreviations.  This needs some care to avoid overlapping

  or looping syntactic replacements!

  The optional \isa{mode} specification restricts output to a

  particular print mode; using ``\isa{input}'' here achieves the

  effect of one-way abbreviations.  The mode may also include an

  ``\hyperlink{keyword.output}{\mbox{\isa{\isakeyword{output}}}}'' qualifier that affects the concrete syntax

  declared for abbreviations, cf.\ \hyperlink{command.syntax}{\mbox{\isa{\isacommand{syntax}}}} in

  \secref{sec:syn-trans}.

  \item \hyperlink{command.print-abbrevs}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}abbrevs}}}} prints all constant abbreviations

  of the current context.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Generic declarations%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Arbitrary operations on the background context may be wrapped-up as

  generic declaration elements.  Since the underlying concept of local

  theories may be subject to later re-interpretation, there is an

  additional dependency on a morphism that tells the difference of the

  original declaration context wrt.\ the application context

  encountered later on.  A fact declaration is an important special

  case: it consists of a theorem which is applied to the context by

  means of an attribute.

  \begin{matharray}{rcl}

    \indexdef{}{command}{declaration}\hypertarget{command.declaration}{\hyperlink{command.declaration}{\mbox{\isa{\isacommand{declaration}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{syntax\_declaration}\hypertarget{command.syntax-declaration}{\hyperlink{command.syntax-declaration}{\mbox{\isa{\isacommand{syntax{\isaliteral{5F}{\isacharunderscore}}declaration}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{declare}\hypertarget{command.declare}{\hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{5}{}

\rail@bar

\rail@term{\hyperlink{command.declaration}{\mbox{\isa{\isacommand{declaration}}}}}[]

\rail@nextbar{1}

\rail@term{\hyperlink{command.syntax-declaration}{\mbox{\isa{\isacommand{syntax{\isaliteral{5F}{\isacharunderscore}}declaration}}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]

\rail@term{\isa{\isakeyword{pervasive}}}[]

\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]

\rail@endbar

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[]

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]

\rail@endbar

\rail@plus

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@nextplus{1}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.declaration}{\mbox{\isa{\isacommand{declaration}}}}~\isa{d} adds the declaration

  function \isa{d} of ML type \verb|declaration|, to the current

  local theory under construction.  In later application contexts, the

  function is transformed according to the morphisms being involved in

  the interpretation hierarchy.

  If the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}pervasive{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} option is given, the corresponding

  declaration is applied to all possible contexts involved, including

  the global background theory.

  \item \hyperlink{command.syntax-declaration}{\mbox{\isa{\isacommand{syntax{\isaliteral{5F}{\isacharunderscore}}declaration}}}} is similar to \hyperlink{command.declaration}{\mbox{\isa{\isacommand{declaration}}}}, but is meant to affect only ``syntactic'' tools by

  convention (such as notation and type-checking information).

  \item \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}}~\isa{thms} declares theorems to the

  current local theory context.  No theorem binding is involved here,

  unlike \hyperlink{command.theorems}{\mbox{\isa{\isacommand{theorems}}}} or \hyperlink{command.lemmas}{\mbox{\isa{\isacommand{lemmas}}}} (cf.\

  \secref{sec:axms-thms}), so \hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}} only has the effect

  of applying attributes as included in the theorem specification.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Locales \label{sec:locale}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Locales are parametric named local contexts, consisting of a list of

  declaration elements that are modeled after the Isar proof context

  commands (cf.\ \secref{sec:proof-context}).%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Locale expressions \label{sec:locale-expr}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

A \emph{locale expression} denotes a structured context composed of

  instances of existing locales.  The context consists of a list of

  instances of declaration elements from the locales.  Two locale

  instances are equal if they are of the same locale and the

  parameters are instantiated with equivalent terms.  Declaration

  elements from equal instances are never repeated, thus avoiding

  duplicate declarations.

  \begin{railoutput}

\rail@begin{3}{\indexdef{}{syntax}{locale\_expr}\hypertarget{syntax.locale-expr}{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}}

\rail@plus

\rail@nont{\isa{instance}}[]

\rail@nextplus{1}

\rail@cterm{\isa{{\isaliteral{2B}{\isacharplus}}}}[]

\rail@endplus

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{\isakeyword{for}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]

\rail@nextplus{2}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@endbar

\rail@end

\rail@begin{2}{\isa{instance}}

\rail@bar

\rail@nextbar{1}

\rail@nont{\isa{qualifier}}[]

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@bar

\rail@nont{\isa{pos{\isaliteral{5F}{\isacharunderscore}}insts}}[]

\rail@nextbar{1}

\rail@nont{\isa{named{\isaliteral{5F}{\isacharunderscore}}insts}}[]

\rail@endbar

\rail@end

\rail@begin{3}{\isa{qualifier}}

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@bar

\rail@term{\isa{{\isaliteral{3F}{\isacharquery}}}}[]

\rail@nextbar{2}

\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[]

\rail@endbar

\rail@endbar

\rail@end

\rail@begin{3}{\isa{pos{\isaliteral{5F}{\isacharunderscore}}insts}}

\rail@plus

\rail@nextplus{1}

\rail@bar

\rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[]

\rail@nextbar{2}

\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]

\rail@endbar

\rail@endplus

\rail@end

\rail@begin{2}{\isa{named{\isaliteral{5F}{\isacharunderscore}}insts}}

\rail@term{\isa{\isakeyword{where}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]

\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]

\rail@nextplus{1}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\end{railoutput}

  A locale instance consists of a reference to a locale and either

  positional or named parameter instantiations.  Identical

  instantiations (that is, those that instante a parameter by itself)

  may be omitted.  The notation `\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequote}}}' enables to omit the

  instantiation for a parameter inside a positional instantiation.

  Terms in instantiations are from the context the locale expressions

  is declared in.  Local names may be added to this context with the

  optional for clause.  In addition, syntax declarations from one

  instance are effective when parsing subsequent instances of the same

  expression.

  Instances have an optional qualifier which applies to names in

  declarations.  Names include local definitions and theorem names.

  If present, the qualifier itself is either optional

  (``\texttt{?}''), which means that it may be omitted on input of the

  qualified name, or mandatory (``\texttt{!}'').  If neither

  ``\texttt{?}'' nor ``\texttt{!}'' are present, the command's default

  is used.  For \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}} and \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}

  the default is ``mandatory'', for \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}} and \hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}} the default is ``optional''.%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Locale declarations%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

\begin{matharray}{rcl}

    \indexdef{}{command}{locale}\hypertarget{command.locale}{\hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{print\_locale}\hypertarget{command.print-locale}{\hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locale}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{print\_locales}\hypertarget{command.print-locales}{\hyperlink{command.print-locales}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locales}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{method}{intro\_locales}\hypertarget{method.intro-locales}{\hyperlink{method.intro-locales}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}locales}}}} & : & \isa{method} \\

    \indexdef{}{method}{unfold\_locales}\hypertarget{method.unfold-locales}{\hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}}}} & : & \isa{method} \\

  \end{matharray}

  \indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}

  \indexisarelem{defines}\indexisarelem{notes}

  \begin{railoutput}

\rail@begin{2}{}

\rail@term{\hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}}[]

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]

\rail@nont{\hyperlink{syntax.locale}{\mbox{\isa{locale}}}}[]

\rail@endbar

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{\isakeyword{begin}}}[]

\rail@endbar

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locale}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@end

\rail@begin{5}{\indexdef{}{syntax}{locale}\hypertarget{syntax.locale}{\hyperlink{syntax.locale}{\mbox{\isa{locale}}}}}

\rail@bar

\rail@plus

\rail@nont{\hyperlink{syntax.context-elem}{\mbox{\isa{context{\isaliteral{5F}{\isacharunderscore}}elem}}}}[]

\rail@nextplus{1}

\rail@endplus

\rail@nextbar{2}

\rail@nont{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}[]

\rail@bar

\rail@nextbar{3}

\rail@term{\isa{{\isaliteral{2B}{\isacharplus}}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.context-elem}{\mbox{\isa{context{\isaliteral{5F}{\isacharunderscore}}elem}}}}[]

\rail@nextplus{4}

\rail@endplus

\rail@endbar

\rail@endbar

\rail@end

\rail@begin{12}{\indexdef{}{syntax}{context\_elem}\hypertarget{syntax.context-elem}{\hyperlink{syntax.context-elem}{\mbox{\isa{context{\isaliteral{5F}{\isacharunderscore}}elem}}}}}

\rail@bar

\rail@term{\isa{\isakeyword{fixes}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]

\rail@nextplus{1}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@nextbar{2}

\rail@term{\isa{\isakeyword{constrains}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]

\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]

\rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]

\rail@nextplus{3}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@nextbar{4}

\rail@term{\isa{\isakeyword{assumes}}}[]

\rail@plus

\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[]

\rail@nextplus{5}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@nextbar{6}

\rail@term{\isa{\isakeyword{defines}}}[]

\rail@plus

\rail@bar

\rail@nextbar{7}

\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]

\rail@bar

\rail@nextbar{7}

\rail@nont{\hyperlink{syntax.prop-pat}{\mbox{\isa{prop{\isaliteral{5F}{\isacharunderscore}}pat}}}}[]

\rail@endbar

\rail@nextplus{8}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@nextbar{9}

\rail@term{\isa{\isakeyword{notes}}}[]

\rail@plus

\rail@bar

\rail@nextbar{10}

\rail@nont{\hyperlink{syntax.thmdef}{\mbox{\isa{thmdef}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]

\rail@nextplus{11}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@endbar

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}loc\ {\isaliteral{3D}{\isacharequal}}\ import\ {\isaliteral{2B}{\isacharplus}}\ body{\isaliteral{22}{\isachardoublequote}}} defines a

  new locale \isa{loc} as a context consisting of a certain view of

  existing locales (\isa{import}) plus some additional elements

  (\isa{body}).  Both \isa{import} and \isa{body} are optional;

  the degenerate form \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}~\isa{loc} defines an empty

  locale, which may still be useful to collect declarations of facts

  later on.  Type-inference on locale expressions automatically takes

  care of the most general typing that the combined context elements

  may acquire.

  The \isa{import} consists of a structured locale expression; see

  \secref{sec:proof-context} above.  Its for clause defines the local

  parameters of the \isa{import}.  In addition, locale parameters

  whose instantance is omitted automatically extend the (possibly

  empty) for clause: they are inserted at its beginning.  This means

  that these parameters may be referred to from within the expression

  and also in the subsequent context elements and provides a

  notational convenience for the inheritance of parameters in locale

  declarations.

  The \isa{body} consists of context elements.

  \begin{description}

  \item \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{28}{\isacharparenleft}}mx{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} declares a local

  parameter of type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} and mixfix annotation \isa{mx} (both

  are optional).  The special syntax declaration ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C5354525543545552453E}{\isasymSTRUCTURE}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' means that \isa{x} may be referenced

  implicitly in this context.

  \item \hyperlink{element.constrains}{\mbox{\isa{\isakeyword{constrains}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}{\isaliteral{22}{\isachardoublequote}}} introduces a type

  constraint \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} on the local parameter \isa{x}.  This

  element is deprecated.  The type constraint should be introduced in

  the for clause or the relevant \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} element.

  \item \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}

  introduces local premises, similar to \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} within a

  proof (cf.\ \secref{sec:proof-context}).

  \item \hyperlink{element.defines}{\mbox{\isa{\isakeyword{defines}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} defines a previously

  declared parameter.  This is similar to \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} within a

  proof (cf.\ \secref{sec:proof-context}), but \hyperlink{element.defines}{\mbox{\isa{\isakeyword{defines}}}}

  takes an equational proposition instead of variable-term pair.  The

  left-hand side of the equation may have additional arguments, e.g.\

  ``\hyperlink{element.defines}{\mbox{\isa{\isakeyword{defines}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}f\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}}''.

  \item \hyperlink{element.notes}{\mbox{\isa{\isakeyword{notes}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{3D}{\isacharequal}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}

  reconsiders facts within a local context.  Most notably, this may

  include arbitrary declarations in any attribute specifications

  included here, e.g.\ a local \hyperlink{attribute.simp}{\mbox{\isa{simp}}} rule.

  The initial \isa{import} specification of a locale expression

  maintains a dynamic relation to the locales being referenced

  (benefiting from any later fact declarations in the obvious manner).

  \end{description}

  Note that ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C49533E}{\isasymIS}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' patterns given

  in the syntax of \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} and \hyperlink{element.defines}{\mbox{\isa{\isakeyword{defines}}}} above

  are illegal in locale definitions.  In the long goal format of

  \secref{sec:goals}, term bindings may be included as expected,

  though.

  \medskip Locale specifications are ``closed up'' by

  turning the given text into a predicate definition \isa{loc{\isaliteral{5F}{\isacharunderscore}}axioms} and deriving the original assumptions as local lemmas

  (modulo local definitions).  The predicate statement covers only the

  newly specified assumptions, omitting the content of included locale

  expressions.  The full cumulative view is only provided on export,

  involving another predicate \isa{loc} that refers to the complete

  specification text.

  In any case, the predicate arguments are those locale parameters

  that actually occur in the respective piece of text.  Also note that

  these predicates operate at the meta-level in theory, but the locale

  packages attempts to internalize statements according to the

  object-logic setup (e.g.\ replacing \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}} by \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}}, and

  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} by \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} in HOL; see also

  \secref{sec:object-logic}).  Separate introduction rules \isa{loc{\isaliteral{5F}{\isacharunderscore}}axioms{\isaliteral{2E}{\isachardot}}intro} and \isa{loc{\isaliteral{2E}{\isachardot}}intro} are provided as well.

  \item \hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locale}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}locale{\isaliteral{22}{\isachardoublequote}}} prints the

  contents of the named locale.  The command omits \hyperlink{element.notes}{\mbox{\isa{\isakeyword{notes}}}}

  elements by default.  Use \hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locale}}}}\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}} to

  have them included.

  \item \hyperlink{command.print-locales}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}locales}}}} prints the names of all locales

  of the current theory.

  \item \hyperlink{method.intro-locales}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}locales}}} and \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}}}

  repeatedly expand all introduction rules of locale predicates of the

  theory.  While \hyperlink{method.intro-locales}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}locales}}} only applies the \isa{loc{\isaliteral{2E}{\isachardot}}intro} introduction rules and therefore does not decend to

  assumptions, \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}}} is more aggressive and applies

  \isa{loc{\isaliteral{5F}{\isacharunderscore}}axioms{\isaliteral{2E}{\isachardot}}intro} as well.  Both methods are aware of locale

  specifications entailed by the context, both from target statements,

  and from interpretations (see below).  New goals that are entailed

  by the current context are discharged automatically.

  \end{description}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsubsection{Locale interpretations%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

Locale expressions may be instantiated, and the instantiated facts

  added to the current context.  This requires a proof of the

  instantiated specification and is called \emph{locale

  interpretation}.  Interpretation is possible in locales \hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}}, theories (command \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}) and

  also within a proof body (command \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}).

  \begin{matharray}{rcl}

    \indexdef{}{command}{interpretation}\hypertarget{command.interpretation}{\hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{interpret}\hypertarget{command.interpret}{\hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{sublocale}\hypertarget{command.sublocale}{\hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{print\_dependencies}\hypertarget{command.print-dependencies}{\hyperlink{command.print-dependencies}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}dependencies}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

    \indexdef{}{command}{print\_interps}\hypertarget{command.print-interps}{\hyperlink{command.print-interps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}interps}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\

  \end{matharray}

  \begin{railoutput}

\rail@begin{2}{}

\rail@term{\hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}}[]

\rail@nont{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\isa{equations}}[]

\rail@endbar

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}}[]

\rail@nont{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@nont{\isa{equations}}[]

\rail@endbar

\rail@end

\rail@begin{5}{}

\rail@term{\hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}}}[]

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@bar

\rail@term{\isa{{\isaliteral{3C}{\isacharless}}}}[]

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}[]

\rail@cr{3}

\rail@bar

\rail@nextbar{4}

\rail@nont{\isa{equations}}[]

\rail@endbar

\rail@end

\rail@begin{2}{}

\rail@term{\hyperlink{command.print-dependencies}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}dependencies}}}}}[]

\rail@bar

\rail@nextbar{1}

\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.locale-expr}{\mbox{\isa{locale{\isaliteral{5F}{\isacharunderscore}}expr}}}}[]

\rail@end

\rail@begin{1}{}

\rail@term{\hyperlink{command.print-interps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}interps}}}}}[]

\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]

\rail@end

\rail@begin{3}{\isa{equations}}

\rail@term{\isa{\isakeyword{where}}}[]

\rail@plus

\rail@bar

\rail@nextbar{1}

\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]

\rail@endbar

\rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]

\rail@nextplus{2}

\rail@cterm{\isa{\isakeyword{and}}}[]

\rail@endplus

\rail@end

\end{railoutput}

  \begin{description}

  \item \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}expr\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ eqns{\isaliteral{22}{\isachardoublequote}}}

  interprets \isa{expr} in the theory.  The command generates proof

  obligations for the instantiated specifications (assumes and defines

  elements).  Once these are discharged by the user, instantiated

  facts are added to the theory in a post-processing phase.

  Additional equations, which are unfolded during

  post-processing, may be given after the keyword \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}}.

  This is useful for interpreting concepts introduced through

  definitions.  The equations must be proved.

  The command is aware of interpretations already active in the

  theory, but does not simplify the goal automatically.  In order to

  simplify the proof obligations use methods \hyperlink{method.intro-locales}{\mbox{\isa{intro{\isaliteral{5F}{\isacharunderscore}}locales}}}

  or \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isaliteral{5F}{\isacharunderscore}}locales}}}.  Post-processing is not applied to

  facts of interpretations that are already active.  This avoids

  duplication of interpreted facts, in particular.  Note that, in the

  case of a locale with import, parts of the interpretation may

  already be active.  The command will only process facts for new

  parts.

  Adding facts to locales has the effect of adding interpreted facts

  to the theory for all interpretations as well.  That is,

  interpretations dynamically participate in any facts added to

  locales.  Note that if a theory inherits additional facts for a

  locale through one parent and an interpretation of that locale

  through another parent, the additional facts will not be

  interpreted.

  \item \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}expr\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ eqns{\isaliteral{22}{\isachardoublequote}}} interprets

  \isa{expr} in the proof context and is otherwise similar to

  interpretation in theories.  Note that rewrite rules given to

  \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}} after the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} keyword should be

  explicitly universally quantified.

  \item \hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ expr\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ eqns{\isaliteral{22}{\isachardoublequote}}}

  interprets \isa{expr} in the locale \isa{name}.  A proof that

  the specification of \isa{name} implies the specification of

  \isa{expr} is required.  As in the localized version of the

  theorem command, the proof is in the context of \isa{name}.  After

  the proof obligation has been discharged, the facts of \isa{expr}

  become part of locale \isa{name} as \emph{derived} context

  elements and are available when the context \isa{name} is

  subsequently entered.  Note that, like import, this is dynamic:

  facts added to a locale part of \isa{expr} after interpretation

  become also available in \isa{name}.

  Only specification fragments of \isa{expr} that are not already

  part of \isa{name} (be it imported, derived or a derived fragment

  of the import) are considered in this process.  This enables

  circular interpretations provided that no infinite chains are

  generated in the locale hierarchy.

  If interpretations of \isa{name} exist in the current theory, the

  command adds interpretations for \isa{expr} as well, with the same

  qualifier, although only for fragments of \isa{expr} that are not

  interpreted in the theory already.

  Equations given after \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} amend the morphism through

  which \isa{expr} is interpreted.  This enables to map definitions

  from the interpreted locales to entities of \isa{name}.  This

  feature is experimental.

  \item \hyperlink{command.print-dependencies}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}dependencies}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}expr{\isaliteral{22}{\isachardoublequote}}} is useful for

  understanding the effect of an interpretation of \isa{{\isaliteral{22}{\isachardoublequote}}expr{\isaliteral{22}{\isachardoublequote}}}.  It

  lists all locale instances for which interpretations would be added

  to the current context.  Variant \hyperlink{command.print-dependencies}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}dependencies}}}}\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}} prints all locale instances that

  would be considered for interpretation, and would be interpreted in

  an empty context (that is, without interpretations).

  \item \hyperlink{command.print-interps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}interps}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}locale{\isaliteral{22}{\isachardoublequote}}} lists all

  interpretations of \isa{{\isaliteral{22}{\isachardoublequote}}locale{\isaliteral{22}{\isachardoublequote}}} in the current theory or proof

  context, including those due to a combination of a \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}} or \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}} and one or several

  \hyperlink{command.sublocale}{\mbox{\isa{\isacommand{sublocale}}}} declarations.

  \end{description}

  \begin{warn}

    Since attributes are applied to interpreted theorems,

    interpretation may modify the context of common proof tools, e.g.\

    the Simplifier or Classical Reasoner.  As the behavior of such

    tools is \emph{not} stable under interpretation morphisms, manual

    declarations might have to be added to the target context of the

    interpretation to revert such declarations.

  \end{warn}

  \begin{warn}

    An interpretation in a theory or proof context may subsume previous

    interpretations.  This happens if the same specification fragment

    is interpreted twice and the instantiation of the second

    interpretation is more general than the interpretation of the

    first.  The locale package does not attempt to remove subsumed

    interpretations.

  \end{warn}%

\end{isamarkuptext}%

\isamarkuptrue%

%

\isamarkupsection{Classes \label{sec:class}%

}

\isamarkuptrue%

%

\begin{isamarkuptext}%

A class is a particular locale with \emph{exactly one} type variable

  \isa{{\isaliteral{5C3C616C7068613E}{\isasymalpha}}}.  Beyond the underlying locale, a corresponding type class

  is established which is interpreted logically as axiomatic type

  class \cite{Wenzel:1997:TPHOL} whose logical content are the

  assumptions of the locale.  Thus, classes provide the full

  generality of locales combined with the commodity of type classes

  (notably type-inference).  See \cite{isabelle-classes} for a short