--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Locales/Locales/document/Locales.tex Fri Aug 19 22:50:20 2005 +0200
@@ -0,0 +1,1320 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Locales}%
+%
+\isadelimtheory
+\isanewline
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+\isamarkuptrue%
+%
+\isamarkupsection{Overview%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The present text is based on~\cite{Ballarin2004a}. It was updated
+ for for Isabelle2005, but does not cover locale interpretation.
+
+ Locales are an extension of the Isabelle proof assistant. They
+ provide support for modular reasoning. Locales were initially
+ developed by Kamm\"uller~\cite{Kammuller2000} to support reasoning
+ in abstract algebra, but are applied also in other domains --- for
+ example, bytecode verification~\cite{Klein2003}.
+
+ Kamm\"uller's original design, implemented in Isabelle99, provides, in
+ addition to
+ means for declaring locales, a set of ML functions that were used
+ along with ML tactics in a proof. In the meantime, the input format
+ for proof in Isabelle has changed and users write proof
+ scripts in ML only rarely if at all. Two new proof styles are
+ available, and can
+ be used interchangeably: linear proof scripts that closely resemble ML
+ tactics, and the structured Isar proof language by
+ Wenzel~\cite{Wenzel2002a}. Subsequently, Wenzel re-implemented
+ locales for
+ the new proof format. The implementation, available with
+ Isabelle2003, constitutes a complete re-design and exploits that
+ both Isar and locales are based on the notion of context,
+ and thus locales are seen as a natural extension of Isar.
+ Nevertheless, locales can also be used with proof scripts:
+ their use does not require a deep understanding of the structured
+ Isar proof style.
+
+ At the same time, Wenzel considerably extended locales. The most
+ important addition are locale expressions, which allow to combine
+ locales more freely. Previously only
+ linear inheritance was possible. Now locales support multiple
+ inheritance through a normalisation algorithm. New are also
+ structures, which provide special syntax for locale parameters that
+ represent algebraic structures.
+
+ Unfortunately, Wenzel provided only an implementation but hardly any
+ documentation. Besides providing documentation, the present paper
+ is a high-level description of locales, and in particular locale
+ expressions. It is meant as a first step towards the semantics of
+ locales, and also as a base for comparing locales with module concepts
+ in other provers. It also constitutes the base for future
+ extensions of locales in Isabelle.
+ The description was derived mainly by experimenting
+ with locales and partially also by inspecting the code.
+
+ The main contribution of the author of the present paper is the
+ abstract description of Wenzel's version of locales, and in
+ particular of the normalisation algorithm for locale expressions (see
+ Section~\ref{sec-normal-forms}). Contributions to the
+ implementation are confined to bug fixes and to provisions that
+ enable the use of locales with linear proof scripts.
+
+ Concepts are introduced along with examples, so that the text can be
+ used as tutorial. It is assumed that the reader is somewhat
+ familiar with Isabelle proof scripts. Examples have been phrased as
+ structured
+ Isar proofs. However, in order to understand the key concepts,
+ including locales expressions and their normalisation, detailed
+ knowledge of Isabelle is not necessary.
+
+\nocite{Nipkow2003,Wenzel2002b,Wenzel2003}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Locales: Beyond Proof Contexts%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In tactic-based provers the application of a sequence of proof
+ tactics leads to a proof state. This state is usually hard to
+ predict from looking at the tactic script, unless one replays the
+ proof step-by-step. The structured proof language Isar is
+ different. It is additionally based on \emph{proof contexts},
+ which are directly visible in Isar scripts, and since tactic
+ sequences tend to be short, this commonly leads to clearer proof
+ scripts.
+
+ Goals are stated with the \textbf{theorem}
+ command. This is followed by a proof. When discharging a goal
+ requires an elaborate argument
+ (rather than the application of a single tactic) a new context
+ may be entered (\textbf{proof}). Inside the context, variables may
+ be fixed (\textbf{fix}), assumptions made (\textbf{assume}) and
+ intermediate goals stated (\textbf{have}) and proved. The
+ assumptions must be dischargeable by premises of the surrounding
+ goal, and once this goal has been proved (\textbf{show}) the proof context
+ can be closed (\textbf{qed}). Contexts inherit from surrounding
+ contexts, but it is not possible
+ to export from them (with exception of the proved goal);
+ they ``disappear'' after the closing \textbf{qed}.
+ Facts may have attributes --- for example, identifying them as
+ default to the simplifier or classical reasoner.
+
+ Locales extend proof contexts in various ways:
+ \begin{itemize}
+ \item
+ Locales are usually \emph{named}. This makes them persistent.
+ \item
+ Fixed variables may have \emph{syntax}.
+ \item
+ It is possible to \emph{add} and \emph{export} facts.
+ \item
+ Locales can be combined and modified with \emph{locale
+ expressions}.
+ \end{itemize}
+ The Locales facility extends the Isar language: it provides new ways
+ of stating and managing facts, but it does not modify the language
+ for proofs. Its purpose is to support writing modular proofs.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Simple Locales%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{Syntax and Terminology%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The grammar of Isar is extended by commands for locales as shown in
+ Figure~\ref{fig-grammar}.
+ A key concept, introduced by Wenzel, is that
+ locales are (internally) lists
+ of \emph{context elements}. There are five kinds, identified
+ by the keywords \textbf{fixes}, \textbf{constrains},
+ \textbf{assumes}, \textbf{defines} and \textbf{notes}.
+
+ \begin{figure}
+ \hrule
+ \vspace{2ex}
+ \begin{small}
+ \begin{tabular}{l>$c<$l}
+ \textit{attr-name} & ::=
+ & \textit{name} $|$ \textit{attribute} $|$
+ \textit{name} \textit{attribute} \\
+
+ \textit{locale-expr} & ::=
+ & \textit{locale-expr1} ( ``\textbf{+}'' \textit{locale-expr1} )$^*$ \\
+ \textit{locale-expr1} & ::=
+ & ( \textit{qualified-name} $|$
+ ``\textbf{(}'' \textit{locale-expr} ``\textbf{)}'' ) \\
+ & & ( \textit{name} [ \textit{mixfix} ] $|$ ``\textbf{\_}'' )$^*$ \\
+
+ \textit{fixes} & ::=
+ & \textit{name} [ ``\textbf{::}'' \textit{type} ]
+ [ ``\textbf{(}'' \textbf{structure} ``\textbf{)}'' $|$
+ \textit{mixfix} ] \\
+ \textit{constrains} & ::=
+ & \textit{name} ``\textbf{::}'' \textit{type} \\
+ \textit{assumes} & ::=
+ & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+ \textit{defines} & ::=
+ & [ \textit{attr-name} ``\textbf{:}'' ] \textit{proposition} \\
+ \textit{notes} & ::=
+ & [ \textit{attr-name} ``\textbf{=}'' ]
+ ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
+
+ \textit{element} & ::=
+ & \textbf{fixes} \textit{fixes} ( \textbf{and} \textit{fixes} )$^*$ \\
+ & |
+ & \textbf{constrains} \textit{constrains}
+ ( \textbf{and} \textit{constrains} )$^*$ \\
+ & |
+ & \textbf{assumes} \textit{assumes} ( \textbf{and} \textit{assumes} )$^*$ \\
+ & |
+ & \textbf{defines} \textit{defines} ( \textbf{and} \textit{defines} )$^*$ \\
+ & |
+ & \textbf{notes} \textit{notes} ( \textbf{and} \textit{notes} )$^*$ \\
+ \textit{element1} & ::=
+ & \textit{element} \\
+ & | & \textbf{includes} \textit{locale-expr} \\
+
+ \textit{locale} & ::=
+ & \textit{element}$^+$ \\
+ & | & \textit{locale-expr} [ ``\textbf{+}'' \textit{element}$^+$ ] \\
+
+ \textit{in-target} & ::=
+ & ``\textbf{(}'' \textbf{in} \textit{qualified-name} ``\textbf{)}'' \\
+
+ \textit{theorem} & ::= & ( \textbf{theorem} $|$ \textbf{lemma} $|$
+ \textbf{corollary} ) [ \textit{in-target} ] [ \textit{attr-name} ] \\
+
+ \textit{theory-level} & ::= & \ldots \\
+ & | & \textbf{locale} \textit{name} [ ``\textbf{=}''
+ \textit{locale} ] \\
+ % note: legacy "locale (open)" omitted.
+ & | & ( \textbf{theorems} $|$ \textbf{lemmas} ) \\
+ & & [ \textit{in-target} ] [ \textit{attr-name} ``\textbf{=}'' ]
+ ( \textit{qualified-name} [ \textit{attribute} ] )$^+$ \\
+ & | & \textbf{declare} [ \textit{in-target} ] ( \textit{qualified-name}
+ [ \textit{attribute} ] )$^+$ \\
+ & | & \textit{theorem} \textit{proposition} \textit{proof} \\
+ & | & \textit{theorem} \textit{element1}$^*$
+ \textbf{shows} \textit{proposition} \textit{proof} \\
+ & | & \textbf{print\_locale} \textit{locale} \\
+ & | & \textbf{print\_locales}
+ \end{tabular}
+ \end{small}
+ \vspace{2ex}
+ \hrule
+ \caption{Locales extend the grammar of Isar.}
+ \label{fig-grammar}
+ \end{figure}
+
+ At the theory level --- that is, at the outer syntactic level of an
+ Isabelle input file --- \textbf{locale} declares a named
+ locale. Other kinds of locales,
+ locale expressions and unnamed locales, will be introduced later. When
+ declaring a named locale, it is possible to \emph{import} another
+ named locale, or indeed several ones by importing a locale
+ expression. The second part of the declaration, also optional,
+ consists of a number of context element declarations.
+
+ A number of Isar commands have an additional, optional \emph{target}
+ argument, which always refers to a named locale. These commands
+ are \textbf{theorem} (together with \textbf{lemma} and
+ \textbf{corollary}), \textbf{theorems} (and
+ \textbf{lemmas}), and \textbf{declare}. The effect of specifying a target is
+ that these commands focus on the specified locale, not the
+ surrounding theory. Commands that are used to
+ prove new theorems will add them not to the theory, but to the
+ locale. Similarly, \textbf{declare} modifies attributes of theorems
+ that belong to the specified target. Additionally, for
+ \textbf{theorem} (and related commands), theorems stored in the target
+ can be used in the associated proof scripts.
+
+ The Locales package provides a \emph{long goals format} for
+ propositions stated with \textbf{theorem} (and friends). While
+ normally a goal is just a formula, a long goal is a list of context
+ elements, followed by the keyword \textbf{shows}, followed by the
+ formula. Roughly speaking, the context elements are
+ (additional) premises. For an example, see
+ Section~\ref{sec-includes}. The list of context elements in a long goal
+ is also called \emph{unnamed locale}.
+
+ Finally, there are two commands to inspect locales when working in
+ interactive mode: \textbf{print\_locales} prints the names of all
+ targets
+ visible in the current theory, \textbf{print\_locale} outputs the
+ elements of a named locale or locale expression.
+
+ The following presentation will use notation of
+ Isabelle's meta logic, hence a few sentences to explain this.
+ The logical
+ primitives are universal quantification (\isa{{\isasymAnd}}), entailment
+ (\isa{{\isasymLongrightarrow}}) and equality (\isa{{\isasymequiv}}). Variables (not bound
+ variables) are sometimes preceded by a question mark. The logic is
+ typed. Type variables are denoted by \isa{{\isacharprime}a}, \isa{{\isacharprime}b}
+ etc., and \isa{{\isasymRightarrow}} is the function type. Double brackets \isa{{\isasymlbrakk}} and \isa{{\isasymrbrakk}} are used to abbreviate nested entailment.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Parameters, Assumptions and Facts%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+From a logical point of view a \emph{context} is a formula schema of
+ the form
+\[
+ \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ C\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}C\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
+\]
+ The variables $\isa{x\isactrlsub {\isadigit{1}}}, \ldots, \isa{x\isactrlsub n}$ are
+ called \emph{parameters}, the premises $\isa{C\isactrlsub {\isadigit{1}}}, \ldots,
+ \isa{C\isactrlsub n}$ \emph{assumptions}. A formula \isa{F}
+ holds in this context if
+\begin{equation}
+\label{eq-fact-in-context}
+ \isa{{\isasymAnd}x\isactrlsub {\isadigit{1}}{\isasymdots}x\isactrlsub n{\isachardot}\ {\isasymlbrakk}\ C\isactrlsub {\isadigit{1}}{\isacharsemicolon}\ {\isasymdots}\ {\isacharsemicolon}C\isactrlsub m\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ F}
+\end{equation}
+ is valid. The formula is called a \emph{fact} of the context.
+
+ A locale allows fixing the parameters \isa{x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub n} and making the assumptions \isa{C\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ C\isactrlsub m}. This implicitly builds the context in
+ which the formula \isa{F} can be established.
+ Parameters of a locale correspond to the context element
+ \textbf{fixes}, and assumptions may be declared with
+ \textbf{assumes}. Using these context elements one can define
+ the specification of semigroups.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ semi\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymcdot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z\ {\isacharequal}\ x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The parameter \isa{prod} has a
+ syntax annotation enabling the infix ``\isa{{\isasymcdot}}'' in the
+ assumption of associativity. Parameters may have arbitrary mixfix
+ syntax, like constants. In the example, the type of \isa{prod} is
+ specified explicitly. This is not necessary. If no type is
+ specified, a most general type is inferred simultaneously for all
+ parameters, taking into account all assumptions (and type
+ specifications of parameters, if present).%
+\footnote{Type inference also takes into account type constraints,
+ definitions and import, as introduced later.}
+
+ Free variables in assumptions are implicitly universally quantified,
+ unless they are parameters. Hence the context defined by the locale
+ \isa{semi} is
+\[
+ \isa{{\isasymAnd}prod{\isachardot}\ {\isasymlbrakk}\ {\isasymAnd}x\ y\ z{\isachardot}\ prod\ {\isacharparenleft}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ prod\ x\ {\isacharparenleft}prod\ y\ z{\isacharparenright}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
+\]
+ The locale can be extended to commutative semigroups.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ comm{\isacharunderscore}semi\ {\isacharequal}\ semi\ {\isacharplus}\isanewline
+\ \ \isakeyword{assumes}\ comm{\isacharcolon}\ {\isachardoublequote}x\ {\isasymcdot}\ y\ {\isacharequal}\ y\ {\isasymcdot}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This locale \emph{imports} all elements of \isa{semi}. The
+ latter locale is called the import of \isa{comm{\isacharunderscore}semi}. The
+ definition adds commutativity, hence its context is
+\begin{align*%
+}
+ \isa{{\isasymAnd}prod{\isachardot}\ {\isasymlbrakk}} &
+ \isa{{\isasymAnd}x\ y\ z{\isachardot}\ prod\ {\isacharparenleft}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ prod\ x\ {\isacharparenleft}prod\ y\ z{\isacharparenright}{\isacharsemicolon}} \\
+ & \isa{{\isasymAnd}x\ y{\isachardot}\ prod\ x\ y\ {\isacharequal}\ prod\ y\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymdots}}
+\end{align*%
+}
+ One may now derive facts --- for example, left-commutativity --- in
+ the context of \isa{comm{\isacharunderscore}semi} by specifying this locale as
+ target, and by referring to the names of the assumptions \isa{assoc} and \isa{comm} in the proof.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ {\isacharparenleft}\isakeyword{in}\ comm{\isacharunderscore}semi{\isacharparenright}\ lcomm{\isacharcolon}\isanewline
+\ \ {\isachardoublequote}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharminus}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymcdot}\ x{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ comm{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{finally}\ \isamarkupfalse%
+\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
+\isacommand{{\isachardot}}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In this equational Isar proof, ``\isa{{\isasymdots}}'' refers to the
+ right hand side of the preceding equation.
+ After the proof is finished, the fact \isa{lcomm} is added to
+ the locale \isa{comm{\isacharunderscore}semi}. This is done by adding a
+ \textbf{notes} element to the internal representation of the locale,
+ as explained the next section.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Locale Predicates and the Internal Representation of
+ Locales%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec-locale-predicates}
+ In mathematical texts, often arbitrary but fixed objects with
+ certain properties are considered --- for instance, an arbitrary but
+ fixed group $G$ --- with the purpose of establishing facts valid for
+ any group. These facts are subsequently used on other objects that
+ also have these properties.
+
+ Locales permit the same style of reasoning. Exporting a fact $F$
+ generalises the fixed parameters and leads to a (valid) formula of the
+ form of equation~(\ref{eq-fact-in-context}). If a locale has many
+ assumptions
+ (possibly accumulated through a number of imports) this formula can
+ become large and cumbersome. Therefore, Wenzel introduced
+ predicates that abbreviate the assumptions of locales. These
+ predicates are not confined to the locale but are visible in the
+ surrounding theory.
+
+ The definition of the locale \isa{semi} generates the \emph{locale
+ predicate} \isa{semi} over the type of the parameter \isa{prod},
+ hence the predicate's type is \isa{{\isacharparenleft}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ bool}. Its
+ definition is
+\begin{equation}
+ \tag*{\isa{semi{\isacharunderscore}def}:} \isa{semi\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}}.
+\end{equation}
+ In the case where the locale has no import, the generated
+ predicate abbreviates all assumptions and is over the parameters
+ that occur in these assumptions.
+
+ The situation is more complicated when a locale extends
+ another locale, as is the case for \isa{comm{\isacharunderscore}semi}. Two
+ predicates are defined. The predicate
+ \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms} corresponds to the new assumptions and is
+ called \emph{delta predicate}, the locale
+ predicate \isa{comm{\isacharunderscore}semi} captures the content of all the locale,
+ including the import.
+ If a locale has neither assumptions nor import, no predicate is
+ defined. If a locale has import but no assumptions, only the locale
+ predicate is defined.%
+\end{isamarkuptext}%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The Locales package generates a number of theorems for locale and
+ delta predicates. All predicates have a definition and an
+ introduction rule. Locale predicates that are defined in terms of
+ other predicates (which is the case if and only if the locale has
+ import) also have a number of elimination rules (called
+ \emph{axioms}). All generated theorems for the predicates of the
+ locales \isa{semi} and \isa{comm{\isacharunderscore}semi} are shown in
+ Figures~\ref{fig-theorems-semi} and~\ref{fig-theorems-comm-semi},
+ respectively.
+ \begin{figure}
+ \hrule
+ \vspace{2ex}
+ Theorems generated for the predicate \isa{semi}.
+ \begin{gather}
+ \tag*{\isa{semi{\isacharunderscore}def}:} \isa{semi\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}} \\
+ \tag*{\isa{semi{\isachardot}intro}:} \isa{{\isacharparenleft}{\isasymAnd}x\ y\ z{\isachardot}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ x\ y{\isacharparenright}\ z\ {\isacharequal}\ {\isacharquery}prod\ x\ {\isacharparenleft}{\isacharquery}prod\ y\ z{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ semi\ {\isacharquery}prod}
+ \end{gather}
+ \hrule
+ \caption{Theorems for the locale predicate \isa{semi}.}
+ \label{fig-theorems-semi}
+ \end{figure}
+
+ \begin{figure}[h]
+ \hrule
+ \vspace{2ex}
+ Theorems generated for the predicate \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms}.
+ \begin{gather}
+ \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms{\isacharunderscore}def}:} \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod\ {\isasymequiv}\ {\isasymforall}x\ y{\isachardot}\ {\isacharquery}prod\ x\ y\ {\isacharequal}\ {\isacharquery}prod\ y\ x} \\
+ \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms{\isachardot}intro}:} \isa{{\isacharparenleft}{\isasymAnd}x\ y{\isachardot}\ {\isacharquery}prod\ x\ y\ {\isacharequal}\ {\isacharquery}prod\ y\ x{\isacharparenright}\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod}
+ \end{gather}
+ Theorems generated for the predicate \isa{comm{\isacharunderscore}semi}.
+ \begin{gather}
+ \tag*{\isa{comm{\isacharunderscore}semi{\isacharunderscore}def}:} \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymequiv}\ semi\ {\isacharquery}prod\ {\isasymand}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod} \\
+ \tag*{\isa{comm{\isacharunderscore}semi{\isachardot}intro}:} \isa{{\isasymlbrakk}semi\ {\isacharquery}prod{\isacharsemicolon}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod{\isasymrbrakk}\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi\ {\isacharquery}prod} \\
+ \tag*{\isa{comm{\isacharunderscore}semi{\isachardot}axioms}:} \mbox{} \\
+ \notag \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ semi\ {\isacharquery}prod} \\
+ \notag \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ comm{\isacharunderscore}semi{\isacharunderscore}axioms\ {\isacharquery}prod}
+ \end{gather}
+ \hrule
+ \caption{Theorems for the predicates \isa{comm{\isacharunderscore}semi{\isacharunderscore}axioms} and
+ \isa{comm{\isacharunderscore}semi}.}
+ \label{fig-theorems-comm-semi}
+ \end{figure}
+ Note that the theorems generated by a locale
+ definition may be inspected immediately after the definition in the
+ Proof General interface \cite{Aspinall2000} of Isabelle through
+ the menu item ``Isabelle/Isar$>$Show me $\ldots>$Theorems''.
+
+ Locale and delta predicates are used also in the internal
+ representation of locales as lists of context elements. While all
+ \textbf{fixes} in a
+ declaration generate internal \textbf{fixes}, all assumptions
+ of one locale declaration contribute to one internal \textbf{assumes}
+ element. The internal representation of \isa{semi} is
+\[
+\begin{array}{ll}
+ \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ (\textbf{infixl} \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} 70) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
+\end{array}
+\]
+ and the internal representation of \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isachardoublequote}} is
+\begin{equation}
+\label{eq-comm-semi}
+\begin{array}{ll}
+ \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
+ \textbf{notes} & \isa{lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
+\end{array}
+\end{equation}
+ The \textbf{notes} elements store facts of the
+ locales. The facts \isa{assoc} and \isa{comm} were added
+ during the declaration of the locales. They stem from assumptions,
+ which are trivially facts. The fact \isa{lcomm} was
+ added later, after finishing the proof in the respective
+ \textbf{theorem} command above.
+
+ By using \textbf{notes} in a declaration, facts can be added
+ to a locale directly. Of course, these must be theorems.
+ Typical use of this feature includes adding theorems that are not
+ usually used as a default rewrite rules by the simplifier to the
+ simpset of the locale by a \textbf{notes} element with the attribute
+ \isa{{\isacharbrackleft}simp{\isacharbrackright}}. This way it is also possible to add specialised
+ versions of
+ theorems to a locale by instantiating locale parameters for unknowns
+ or locale assumptions for premises.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Definitions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Definitions were available in Kamm\"uller's version of Locales, and
+ they are in Wenzel's.
+ The context element \textbf{defines} adds a definition of the form
+ \isa{p\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n\ {\isasymequiv}\ t} as an assumption, where \isa{p} is a
+ parameter of the locale (possibly an imported parameter), and \isa{t} a term that may contain the \isa{x\isactrlsub i}. The parameter may
+ neither occur in a previous \textbf{assumes} or \textbf{defines}
+ element, nor on the right hand side of the definition. Hence
+ recursion is not allowed.
+ The parameter may, however, occur in subsequent \textbf{assumes} and
+ on the right hand side of subsequent \textbf{defines}. We call
+ \isa{p} \emph{defined parameter}.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ semi{\isadigit{2}}\ {\isacharequal}\ semi\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ rprod\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymodot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{defines}\ rprod{\isacharunderscore}def{\isacharcolon}\ {\isachardoublequote}rprod\ x\ y\ {\isasymequiv}\ y\ {\isasymcdot}\ x\ {\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This locale extends \isa{semi} by a second binary operation \isa{{\isachardoublequote}{\isasymodot}{\isachardoublequote}} that is like \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} but with
+ reversed arguments. The
+ definition of the locale generates the predicate \isa{semi{\isadigit{2}}},
+ which is equivalent to \isa{semi}, but no \isa{semi{\isadigit{2}}{\isacharunderscore}axioms}.
+ The difference between \textbf{assumes} and \textbf{defines} lies in
+ the way parameters are treated on export.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Export%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+A fact is exported out
+ of a locale by generalising over the parameters and adding
+ assumptions as premises. For brevity of the exported theorems,
+ locale predicates are used. Exported facts are referenced by
+ writing qualified names consisting of the locale name and the fact name ---
+ for example,
+\begin{equation}
+ \tag*{\isa{semi{\isachardot}assoc}:} \isa{semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}x\ {\isacharquery}y{\isacharparenright}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}prod\ {\isacharquery}x\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}}.
+\end{equation}
+ Defined parameters receive special treatment. Instead of adding a
+ premise for the definition, the definition is unfolded in the
+ exported theorem. In order to illustrate this we prove that the
+ reverse operation \isa{{\isachardoublequote}{\isasymodot}{\isachardoublequote}} defined in the locale
+ \isa{semi{\isadigit{2}}} is also associative.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ {\isacharparenleft}\isakeyword{in}\ semi{\isadigit{2}}{\isacharparenright}\ r{\isacharunderscore}assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymodot}\ y{\isacharparenright}\ {\isasymodot}\ z\ {\isacharequal}\ x\ {\isasymodot}\ {\isacharparenleft}y\ {\isasymodot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ only{\isacharcolon}\ rprod{\isacharunderscore}def\ assoc{\isacharparenright}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The exported fact is
+\begin{equation}
+ \tag*{\isa{semi{\isadigit{2}}{\isachardot}r{\isacharunderscore}assoc}:} \isa{semi{\isadigit{2}}\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharquery}z\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}x{\isacharparenright}\ {\isacharequal}\ {\isacharquery}prod\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}z\ {\isacharquery}y{\isacharparenright}\ {\isacharquery}x}.
+\end{equation}
+ The defined parameter is not present but is replaced by its
+ definition. Note that the definition itself is not exported, hence
+ there is no \isa{semi{\isadigit{2}}{\isachardot}rprod{\isacharunderscore}def}.%
+\footnote{The definition could alternatively be exported using a
+ let-construct if there was one in Isabelle's meta-logic. Let is
+ usually defined in object-logics.}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Locale Expressions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locale expressions provide a simple language for combining
+ locales. They are an effective means of building complex
+ specifications from simple ones. Locale expressions are the main
+ innovation of the version of Locales discussed here. Locale
+ expressions are also reason for introducing locale predicates.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Rename and Merge%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The grammar of locale expressions is part of the grammar in
+ Figure~\ref{fig-grammar}. Locale names are locale
+ expressions, and
+ further expressions are obtained by \emph{rename} and \emph{merge}.
+\begin{description}
+\item[Rename.]
+ The locale expression $e\: q_1 \ldots q_n$ denotes
+ the locale of $e$ where pa\-ra\-me\-ters, in the order in
+ which they are fixed, are renamed to $q_1$ to $q_n$.
+ The expression is only well-formed if $n$ does not
+ exceed the number of parameters of $e$. Underscores denote
+ parameters that are not renamed.
+ Renaming by default removes mixfix syntax, but new syntax may be
+ specified.
+\item[Merge.]
+ The locale expression $e_1 + e_2$ denotes
+ the locale obtained by merging the locales of $e_1$
+ and $e_2$. This locale contains the context elements
+ of $e_1$, followed by the context elements of $e_2$.
+
+ In actual fact, the semantics of the merge operation
+ is more complicated. If $e_1$ and $e_2$ are expressions containing
+ the same name, followed by
+ identical parameter lists, then the merge of both will contain
+ the elements of those locales only once. Details are explained in
+ Section~\ref{sec-normal-forms} below.
+
+ The merge operation is associative but not commutative. The latter
+ is because parameters of $e_1$ appear
+ before parameters of $e_2$ in the composite expression.
+\end{description}
+
+ Rename can be used if a different parameter name seems more
+ appropriate --- for example, when moving from groups to rings, a
+ parameter \isa{G} representing the group could be changed to
+ \isa{R}. Besides of this stylistic use, renaming is important in
+ combination with merge. Both operations are used in the following
+ specification of semigroup homomorphisms.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ semi{\isacharunderscore}hom\ {\isacharequal}\ comm{\isacharunderscore}semi\ sum\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymoplus}{\isachardoublequote}\ {\isadigit{6}}{\isadigit{5}}{\isacharparenright}\ {\isacharplus}\ comm{\isacharunderscore}semi\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ hom\isanewline
+\ \ \isakeyword{assumes}\ hom{\isacharcolon}\ {\isachardoublequote}hom\ {\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymcdot}\ hom\ y{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This locale defines a context with three parameters \isa{sum},
+ \isa{prod} and \isa{hom}. The first two parameters have
+ mixfix syntax. They are associative operations,
+ the first of type \isa{{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a}, the second of
+ type \isa{{\isacharbrackleft}{\isacharprime}b{\isacharcomma}\ {\isacharprime}b{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}b}.
+
+ How are facts that are imported via a locale expression identified?
+ Facts are always introduced in a named locale (either in the
+ locale's declaration, or by using the locale as target in
+ \textbf{theorem}), and their names are
+ qualified by the parameter names of this locale.
+ Hence the full name of associativity in \isa{semi} is
+ \isa{prod{\isachardot}assoc}. Renaming parameters of a target also renames
+ the qualifier of facts. Hence, associativity of \isa{sum} is
+ \isa{sum{\isachardot}assoc}. Several parameters are separated by
+ underscores in qualifiers. For example, the full name of the fact
+ \isa{hom} in the locale \isa{semi{\isacharunderscore}hom} is \isa{sum{\isacharunderscore}prod{\isacharunderscore}hom{\isachardot}hom}.
+
+ The following example is quite artificial, it illustrates the use of
+ facts, though.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ {\isacharparenleft}\isakeyword{in}\ semi{\isacharunderscore}hom{\isacharparenright}\ {\isachardoublequote}hom\ x\ {\isasymcdot}\ {\isacharparenleft}hom\ y\ {\isasymcdot}\ hom\ z{\isacharparenright}\ {\isacharequal}\ hom\ {\isacharparenleft}x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharminus}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}hom\ x\ {\isasymcdot}\ {\isacharparenleft}hom\ y\ {\isasymcdot}\ hom\ z{\isacharparenright}\ {\isacharequal}\ hom\ y\ {\isasymcdot}\ {\isacharparenleft}hom\ x\ {\isasymcdot}\ hom\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\ \ \ \ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ prod{\isachardot}lcomm{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ hom\ {\isacharparenleft}y\ {\isasymoplus}\ {\isacharparenleft}x\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ hom{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ hom\ {\isacharparenleft}x\ {\isasymoplus}\ {\isacharparenleft}y\ {\isasymoplus}\ z{\isacharparenright}{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ sum{\isachardot}lcomm{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{finally}\ \isamarkupfalse%
+\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
+\isacommand{{\isachardot}}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Importing via a locale expression imports all facts of
+ the imported locales, hence both \isa{sum{\isachardot}lcomm} and \isa{prod{\isachardot}lcomm} are
+ available in \isa{hom{\isacharunderscore}semi}. The import is dynamic --- that is,
+ whenever facts are added to a locale, they automatically
+ become available in subsequent \textbf{theorem} commands that use
+ the locale as a target, or a locale importing the locale.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Normal Forms%
+}
+%
+\label{sec-normal-forms}
+\newcommand{\I}{\mathcal{I}}
+\newcommand{\F}{\mathcal{F}}
+\newcommand{\N}{\mathcal{N}}
+\newcommand{\C}{\mathcal{C}}
+\newcommand{\App}{\mathbin{\overline{@}}}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locale expressions are interpreted in a two-step process. First, an
+ expression is normalised, then it is converted to a list of context
+ elements.
+
+ Normal forms are based on \textbf{locale} declarations. These
+ consist of an import section followed by a list of context
+ elements. Let $\I(l)$ denote the locale expression imported by
+ locale $l$. If $l$ has no import then $\I(l) = \varepsilon$.
+ Likewise, let $\F(l)$ denote the list of context elements, also
+ called the \emph{context fragment} of $l$. Note that $\F(l)$
+ contains only those context elements that are stated in the
+ declaration of $l$, not imported ones.
+
+\paragraph{Example 1.} Consider the locales \isa{semi} and \isa{comm{\isacharunderscore}semi}. We have $\I(\isa{semi}) = \varepsilon$ and
+ $\I(\isa{comm{\isacharunderscore}semi}) = \isa{semi}$, and the context fragments
+ are
+\begin{align*%
+}
+ \F(\isa{semi}) & = \left[
+\begin{array}{ll}
+ \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
+\end{array} \right], \\
+ \F(\isa{comm{\isacharunderscore}semi}) & = \left[
+\begin{array}{ll}
+ \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
+ \textbf{notes} & \isa{lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
+\end{array} \right].
+\end{align*%
+}
+ Let $\pi_0(\F(l))$ denote the list of parameters defined in the
+ \textbf{fixes} elements of $\F(l)$ in the order of their occurrence.
+ The list of parameters of a locale expression $\pi(e)$ is defined as
+ follows:
+\begin{align*%
+}
+ \pi(l) & = \pi(\I(l)) \App \pi_0(\F(l)) \text{, for named locale $l$.} \\
+ \pi(e\: q_1 \ldots q_n) & = \text{$[q_1, \ldots, q_n, p_{n+1}, \ldots,
+ p_{m}]$, where $\pi(e) = [p_1, \ldots, p_m]$.} \\
+ \pi(e_1 + e_2) & = \pi(e_1) \App \pi(e_2)
+\end{align*%
+}
+ The operation $\App$ concatenates two lists but omits elements from
+ the second list that are also present in the first list.
+ It is not possible to rename more parameters than there
+ are present in an expression --- that is, $n \le m$ --- otherwise
+ the renaming is illegal. If $q_i
+ = \_$ then the $i$th entry of the resulting list is $p_i$.
+
+ In the normalisation phase, imports of named locales are unfolded, and
+ renames and merges are recursively propagated to the imported locale
+ expressions. The result is a list of locale names, each with a full
+ list of parameters, where locale names occurring with the same parameter
+ list twice are removed. Let $\N$ denote normalisation. It is defined
+ by these equations:
+\begin{align*%
+}
+ \N(l) & = \N(\I(l)) \App [l\:\pi(l)] \text{, for named locale $l$.} \\
+ \N(e\: q_1 \ldots q_n) & = \N(e)\:[q_1 \ldots q_n/\pi(e)] \\
+ \N(e_1 + e_2) & = \N(e_1) \App \N(e_2)
+\end{align*%
+}
+ Normalisation yields a list of \emph{identifiers}. An identifier
+ consists of a locale name and a (possibly empty) list of parameters.
+
+ In the second phase, the list of identifiers $\N(e)$ is converted to
+ a list of context elements $\C(e)$ by converting each identifier to
+ a list of context elements, and flattening the obtained list.
+ Conversion of the identifier $l\:q_1 \ldots q_n$ yields the list of
+ context elements $\F(l)$, but with the following modifications:
+\begin{itemize}
+\item
+ Rename the parameter in the $i$th \textbf{fixes} element of $\F(l)$
+ to $q_i$, $i = 1, \ldots, n$. If the parameter name is actually
+ changed then delete the syntax annotation. Renaming a parameter may
+ also change its type.
+\item
+ Perform the same renamings on all occurrences of parameters (fixed
+ variables) in \textbf{assumes}, \textbf{defines} and \textbf{notes}
+ elements.
+\item
+ Qualify names of facts by $q_1\_\ldots\_q_n$.
+\end{itemize}
+ The locale expression is \emph{well-formed} if it contains no
+ illegal renamings and the following conditions on $\C(e)$ hold,
+ otherwise the expression is rejected:
+\begin{itemize}
+\item Parameters in \textbf{fixes} are distinct;
+\item Free variables in \textbf{assumes} and
+ \textbf{defines} occur in preceding \textbf{fixes};%
+\footnote{This restriction is relaxed for contexts obtained with
+ \textbf{includes}, see Section~\ref{sec-includes}.}
+\item Parameters defined in \textbf{defines} must neither occur in
+ preceding \textbf{assumes} nor \textbf{defines}.
+\end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Examples%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\paragraph{Example 2.}
+ We obtain the context fragment $\C(\isa{comm{\isacharunderscore}semi})$ of the
+ locale \isa{comm{\isacharunderscore}semi}. First, the parameters are computed.
+\begin{align*%
+}
+ \pi(\isa{semi}) & = [\isa{prod}] \\
+ \pi(\isa{comm{\isacharunderscore}semi}) & = \pi(\isa{semi}) \App []
+ = [\isa{prod}]
+\end{align*%
+}
+ Next, the normal form of the locale expression
+ \isa{comm{\isacharunderscore}semi} is obtained.
+\begin{align*%
+}
+ \N(\isa{semi}) & = [\isa{semi} \isa{prod}] \\
+ \N(\isa{comm{\isacharunderscore}semi}) & = \N(\isa{semi}) \App
+ [\isa{comm{\isacharunderscore}semi\ prod}]
+ = [\isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}]
+\end{align*%
+}
+ Converting this to a list of context elements leads to the
+ list~(\ref{eq-comm-semi}) shown in
+ Section~\ref{sec-locale-predicates}, but with fact names qualified
+ by \isa{prod} --- for example, \isa{prod{\isachardot}assoc}.
+ Qualification was omitted to keep the presentation simple.
+ Isabelle's scoping rules identify the most recent fact with
+ qualified name $x.a$ when a fact with name $a$ is requested.
+
+\paragraph{Example 3.}
+ The locale expression \isa{comm{\isacharunderscore}semi\ sum} involves
+ renaming. Computing parameters yields $\pi(\isa{comm{\isacharunderscore}semi\ sum})
+ = [\isa{sum}]$, the normal form is
+\begin{align*%
+}
+ \N(\isa{comm{\isacharunderscore}semi\ sum}) & =
+ \N(\isa{comm{\isacharunderscore}semi})[\isa{sum}/\isa{prod}] =
+ [\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum}]
+\end{align*%
+}
+ and the list of context elements
+\[
+\begin{array}{ll}
+ \textbf{fixes} & \isa{sum} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymoplus}{\isachardoublequote}}~65) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ sum{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y{\isacharparenright}\ {\isasymoplus}\ {\isacharquery}z\ {\isacharequal}\ sum\ {\isacharquery}x\ {\isacharparenleft}sum\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharquery}x{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}}
+\end{array}
+\]
+
+\paragraph{Example 4.}
+ The context defined by the locale \isa{semi{\isacharunderscore}hom} involves
+ merging two copies of \isa{comm{\isacharunderscore}semi}. We obtain the parameter
+ list and normal form:
+\begin{align*%
+}
+ \pi(\isa{semi{\isacharunderscore}hom}) & = \pi(\isa{comm{\isacharunderscore}semi\ sum} +
+ \isa{comm{\isacharunderscore}semi}) \App [\isa{hom}] \\
+ & = (\pi(\isa{comm{\isacharunderscore}semi\ sum}) \App \pi(\isa{comm{\isacharunderscore}semi}))
+ \App [\isa{hom}] \\
+ & = ([\isa{sum}] \App [\isa{prod}]) \App [\isa{hom}]
+ = [\isa{sum}, \isa{prod}, \isa{hom}] \\
+ \N(\isa{semi{\isacharunderscore}hom}) & =
+ \N(\isa{comm{\isacharunderscore}semi\ sum} + \isa{comm{\isacharunderscore}semi}) \App \\
+ & \phantom{==}
+ [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
+ & = (\N(\isa{comm{\isacharunderscore}semi\ sum}) \App \N(\isa{comm{\isacharunderscore}semi})) \App \\
+ & \phantom{==}
+ [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
+ & = ([\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum}] \App %\\
+% & \phantom{==}
+ [\isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}]) \App \\
+ & \phantom{==}
+ [\isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}] \\
+ & = [\isa{semi\ sum}, \isa{comm{\isacharunderscore}semi\ sum},
+ \isa{semi\ prod}, \isa{comm{\isacharunderscore}semi\ prod}, \\
+ & \phantom{==}
+ \isa{semi{\isacharunderscore}hom\ sum\ prod\ hom}].
+\end{align*%
+}
+ Hence $\C(\isa{semi{\isacharunderscore}hom})$, shown below, is again well-formed.
+\[
+\begin{array}{ll}
+ \textbf{fixes} & \isa{sum} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymoplus}{\isachardoublequote}}~65) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ sum{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y{\isacharparenright}\ {\isasymoplus}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharquery}x{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}y\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymoplus}\ {\isacharparenleft}{\isacharquery}x\ {\isasymoplus}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{fixes} & \isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}b{\isacharcomma}\ {\isacharprime}b{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70) \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{prod{\isachardot}assoc}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z\ {\isacharequal}\ {\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{assumes} & \isa{{\isachardoublequote}comm{\isacharunderscore}semi{\isacharunderscore}axioms\ prod{\isachardoublequote}} \\
+ \textbf{notes} & \isa{prod{\isachardot}comm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharquery}x{\isachardoublequote}} \\
+ \textbf{notes} & \isa{prod{\isachardot}lcomm}: \isa{{\isachardoublequote}{\isacharquery}x\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}y\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}y\ {\isasymcdot}\ {\isacharparenleft}{\isacharquery}x\ {\isasymcdot}\ {\isacharquery}z{\isacharparenright}{\isachardoublequote}} \\
+ \textbf{fixes} & \isa{hom} :: \isa{{\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b{\isachardoublequote}} \\
+ \textbf{assumes} & \isa{{\isachardoublequote}semi{\isacharunderscore}hom{\isacharunderscore}axioms\ sum{\isachardoublequote}} \\
+ \textbf{notes} & \isa{sum{\isacharunderscore}prod{\isacharunderscore}hom{\isachardot}hom}:
+ \isa{hom\ {\isacharparenleft}x\ {\isasymoplus}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymcdot}\ hom\ y}
+\end{array}
+\]
+
+\paragraph{Example 5.}
+ In this example, a locale expression leading to a list of context
+ elements that is not well-defined is encountered, and it is illustrated
+ how normalisation deals with multiple inheritance.
+ Consider the specification of monads (in the algebraic sense)
+ and monoids.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ monad\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymcdot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{and}\ one\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ {\isacharparenleft}{\isachardoublequote}{\isasymone}{\isachardoublequote}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymcdot}\ x\ {\isacharequal}\ x{\isachardoublequote}\ \isakeyword{and}\ r{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequote}x\ {\isasymcdot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Monoids are both semigroups and monads and one would want to
+ specify them as locale expression \isa{semi\ {\isacharplus}\ monad}.
+ Unfortunately, this expression is not well-formed. Its normal form
+\begin{align*%
+}
+ \N(\isa{monad}) & = [\isa{monad\ prod}] \\
+ \N(\isa{semi}+\isa{monad}) & =
+ \N(\isa{semi}) \App \N(\isa{monad})
+ = [\isa{semi\ prod}, \isa{monad\ prod}]
+\end{align*%
+}
+ leads to a list containing the context element
+\[
+ \textbf{fixes}~\isa{prod} :: \isa{{\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}}
+ ~(\textbf{infixl}~\isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}}~70)
+\]
+ twice and thus violating the first criterion of well-formedness. To
+ avoid this problem, one can
+ introduce a new locale \isa{magma} with the sole purpose of fixing the
+ parameter and defining its syntax. The specifications of semigroup
+ and monad are changed so that they import \isa{magma}.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ magma\ {\isacharequal}\ \isakeyword{fixes}\ prod\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymcdot}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
+\isanewline
+\isamarkupfalse%
+\isacommand{locale}\ semi{\isacharprime}\ {\isacharequal}\ magma\ {\isacharplus}\ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z\ {\isacharequal}\ x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+\isamarkupfalse%
+\isacommand{locale}\ monad{\isacharprime}\ {\isacharequal}\ magma\ {\isacharplus}\ \isakeyword{fixes}\ one\ {\isacharparenleft}{\isachardoublequote}{\isasymone}{\isachardoublequote}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequote}{\isasymone}\ {\isasymcdot}\ x\ {\isacharequal}\ x{\isachardoublequote}\ \isakeyword{and}\ r{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequote}x\ {\isasymcdot}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Normalisation now yields
+\begin{align*%
+}
+ \N(\isa{semi{\isacharprime}\ {\isacharplus}\ monad{\isacharprime}}) & =
+ \N(\isa{semi{\isacharprime}}) \App \N(\isa{monad{\isacharprime}}) \\
+ & = (\N(\isa{magma}) \App [\isa{semi{\isacharprime}\ prod}]) \App
+ (\N(\isa{magma}) \App [\isa{monad{\isacharprime}\ prod}]) \\
+ & = [\isa{magma\ prod}, \isa{semi{\isacharprime}\ prod}] \App
+ [\isa{magma\ prod}, \isa{monad{\isacharprime}\ prod}]) \\
+ & = [\isa{magma\ prod}, \isa{semi{\isacharprime}\ prod},
+ \isa{monad{\isacharprime}\ prod}]
+\end{align*%
+}
+ where the second occurrence of \isa{magma\ prod} is eliminated.
+ The reader is encouraged to check, using the \textbf{print\_locale}
+ command, that the list of context elements generated from this is
+ indeed well-formed.
+
+ It follows that importing
+ parameters is more flexible than fixing them using a context element.
+ The Locale package provides the predefined locale \isa{var} that
+ can be used to import parameters if no
+ particular mixfix syntax is required.
+ Its definition is
+\begin{center}
+ \textbf{locale} \isa{var} = \textbf{fixes} \isa{x{\isacharunderscore}}
+\end{center}
+ The use of the internal variable \isa{x{\isacharunderscore}}
+ enforces that the parameter is renamed before being used, because
+ internal variables may not occur in the input syntax. Using
+ \isa{var}, the locale \isa{magma} could have been replaced by
+ the locale expression
+\begin{center}
+ \isa{var} \isa{prod} (\textbf{infixl} \isa{{\isachardoublequote}{\isasymcdot}{\isachardoublequote}} 70)
+\end{center}
+ in the above locale declarations.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Includes%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec-includes}
+ The context element \textbf{includes} takes a locale expression $e$
+ as argument. It can only occur in long goals, where it
+ adds, like a target, locale context to the proof context. Unlike
+ with targets, the proved theorem is not stored
+ in the locale. Instead, it is exported immediately.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{theorem}\ lcomm{\isadigit{2}}{\isacharcolon}\isanewline
+\ \ \isakeyword{includes}\ comm{\isacharunderscore}semi\ \isakeyword{shows}\ {\isachardoublequote}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{proof}\ {\isacharminus}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}x\ {\isasymcdot}\ {\isacharparenleft}y\ {\isasymcdot}\ z{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcdot}\ y{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}y\ {\isasymcdot}\ x{\isacharparenright}\ {\isasymcdot}\ z{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ comm{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{also}\ \isamarkupfalse%
+\isacommand{have}\ {\isachardoublequote}{\isasymdots}\ {\isacharequal}\ y\ {\isasymcdot}\ {\isacharparenleft}x\ {\isasymcdot}\ z{\isacharparenright}{\isachardoublequote}\ \isamarkupfalse%
+\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ assoc{\isacharparenright}\isanewline
+\ \ \isamarkupfalse%
+\isacommand{finally}\ \isamarkupfalse%
+\isacommand{show}\ {\isacharquery}thesis\ \isamarkupfalse%
+\isacommand{{\isachardot}}\isanewline
+\isamarkupfalse%
+\isacommand{qed}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This proof is identical to the proof of \isa{lcomm}. The use of
+ \textbf{includes} provides the same context and facts as when using
+ \isa{comm{\isacharunderscore}semi} as target. On the other hand, \isa{lcomm{\isadigit{2}}} is not added as a fact to the locale \isa{comm{\isacharunderscore}semi}, but
+ is directly visible in the theory. The theorem is
+\[
+ \isa{comm{\isacharunderscore}semi\ {\isacharquery}prod\ {\isasymLongrightarrow}\ {\isacharquery}prod\ {\isacharquery}x\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}y\ {\isacharquery}z{\isacharparenright}\ {\isacharequal}\ {\isacharquery}prod\ {\isacharquery}y\ {\isacharparenleft}{\isacharquery}prod\ {\isacharquery}x\ {\isacharquery}z{\isacharparenright}}.
+\]
+ Note that it is possible to
+ combine a target and (several) \textbf{includes} in a goal statement, thus
+ using contexts of several locales but storing the theorem in only
+ one of them.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Structures%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\label{sec-structures}
+ The specifications of semigroups and monoids that served as examples
+ in previous sections modelled each operation of an algebraic
+ structure as a single parameter. This is rather inconvenient for
+ structures with many operations, and also unnatural. In accordance
+ to mathematical texts, one would rather fix two groups instead of
+ two sets of operations.
+
+ The approach taken in Isabelle is to encode algebraic structures
+ with suitable types (in Isabelle/HOL usually records). An issue to
+ be addressed by
+ locales is syntax for algebraic structures. This is the purpose of
+ the \textbf{(structure)} annotation in \textbf{fixes}, introduced by
+ Wenzel. We illustrate this, independently of record types, with a
+ different formalisation of semigroups.
+
+ Let \isa{{\isacharprime}a\ semi{\isacharunderscore}type} be a not further specified type that
+ represents semigroups over the carrier type \isa{{\isacharprime}a}. Let \isa{s{\isacharunderscore}op} be an operation that maps an object of \isa{{\isacharprime}a\ semi{\isacharunderscore}type} to
+ a binary operation.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{typedecl}\ {\isacharprime}a\ semi{\isacharunderscore}type\isanewline
+\isamarkupfalse%
+\isacommand{consts}\ s{\isacharunderscore}op\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharbrackleft}{\isacharprime}a\ semi{\isacharunderscore}type{\isacharcomma}\ {\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymstar}{\isasymindex}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Although \isa{s{\isacharunderscore}op} is a ternary operation, it is declared
+ infix. The syntax annotation contains the token \isa{{\isasymindex}}
+ (\verb.\<index>.), which refers to the first argument. This syntax is only
+ effective in the context of a locale, and only if the first argument
+ is a
+ \emph{structural} parameter --- that is, a parameter with annotation
+ \textbf{(structure)}. The token has the effect of subscripting the
+ parameter --- by bracketing it between \verb.\<^bsub>. and \verb.\<^esub>..
+ Additionally, the subscript of the first structural parameter may be
+ omitted, as in this specification of semigroups with structures:%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ comm{\isacharunderscore}semi{\isacharprime}\ {\isacharequal}\isanewline
+\ \ \isakeyword{fixes}\ G\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ semi{\isacharunderscore}type{\isachardoublequote}\ {\isacharparenleft}\isakeyword{structure}{\isacharparenright}\isanewline
+\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymstar}\ y{\isacharparenright}\ {\isasymstar}\ z\ {\isacharequal}\ x\ {\isasymstar}\ {\isacharparenleft}y\ {\isasymstar}\ z{\isacharparenright}{\isachardoublequote}\ \isakeyword{and}\ comm{\isacharcolon}\ {\isachardoublequote}x\ {\isasymstar}\ y\ {\isacharequal}\ y\ {\isasymstar}\ x{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Here \isa{x\ {\isasymstar}\ y} is equivalent to \isa{x\ {\isasymstar}\isactrlbsub G\isactrlesub \ y} and
+ abbreviates \isa{s{\isacharunderscore}op\ G\ x\ y}. A specification of homomorphisms
+ requires a second structural parameter.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ semi{\isacharprime}{\isacharunderscore}hom\ {\isacharequal}\ comm{\isacharunderscore}semi{\isacharprime}\ {\isacharplus}\ comm{\isacharunderscore}semi{\isacharprime}\ H\ {\isacharplus}\isanewline
+\ \ \isakeyword{fixes}\ hom\isanewline
+\ \ \isakeyword{assumes}\ hom{\isacharcolon}\ {\isachardoublequote}hom\ {\isacharparenleft}x\ {\isasymstar}\ y{\isacharparenright}\ {\isacharequal}\ hom\ x\ {\isasymstar}\isactrlbsub H\isactrlesub \ hom\ y{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The parameter \isa{H} is defined in the second \textbf{fixes}
+ element of $\C(\isa{semi{\isacharprime}{\isacharunderscore}comm})$. Hence \isa{{\isasymstar}\isactrlbsub H\isactrlesub }
+ abbreviates \isa{s{\isacharunderscore}op\ H\ x\ y}. The same construction can be done
+ with records instead of an \textit{ad-hoc} type.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{record}\ {\isacharprime}a\ semi\ {\isacharequal}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharbrackleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}a{\isacharbrackright}\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequote}{\isasymbullet}{\isasymindex}{\isachardoublequote}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+This declares the types \isa{{\isacharprime}a\ semi} and \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ semi{\isacharunderscore}scheme}. The latter is an extensible record, where the second
+ type argument is the type of the extension field. For details on
+ records, see \cite{NipkowEtAl2002} Chapter~8.3.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{locale}\ semi{\isacharunderscore}w{\isacharunderscore}records\ {\isacharequal}\ struct\ G\ {\isacharplus}\isanewline
+\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}x\ {\isasymbullet}\ y{\isacharparenright}\ {\isasymbullet}\ z\ {\isacharequal}\ x\ {\isasymbullet}\ {\isacharparenleft}y\ {\isasymbullet}\ z{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The type \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}\ {\isacharprime}b{\isacharparenright}\ semi{\isacharunderscore}scheme} is inferred for the parameter \isa{G}. Using subtyping on records, the specification can be extended
+ to groups easily.%
+\end{isamarkuptext}%
+\isamarkupfalse%
+\isacommand{record}\ {\isacharprime}a\ group\ {\isacharequal}\ {\isachardoublequote}{\isacharprime}a\ semi{\isachardoublequote}\ {\isacharplus}\isanewline
+\ \ one\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}{\isachardoublequote}l{\isasymindex}{\isachardoublequote}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}\isanewline
+\ \ inv\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequote}\ {\isacharparenleft}{\isachardoublequote}inv{\isasymindex}\ {\isacharunderscore}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{8}}{\isadigit{1}}{\isacharbrackright}\ {\isadigit{8}}{\isadigit{0}}{\isacharparenright}\isanewline
+\isamarkupfalse%
+\isacommand{locale}\ group{\isacharunderscore}w{\isacharunderscore}records\ {\isacharequal}\ semi{\isacharunderscore}w{\isacharunderscore}records\ {\isacharplus}\isanewline
+\ \ \isakeyword{assumes}\ l{\isacharunderscore}one{\isacharcolon}\ {\isachardoublequote}l\ {\isasymbullet}\ x\ {\isacharequal}\ x{\isachardoublequote}\ \isakeyword{and}\ l{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequote}inv\ x\ {\isasymbullet}\ x\ {\isacharequal}\ l{\isachardoublequote}\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Finally, the predefined locale
+\begin{center}
+ \textbf{locale} \textit{struct} = \textbf{fixes} \isa{S{\isacharunderscore}}
+ \textbf{(structure)}.
+\end{center}
+ is analogous to \isa{var}.
+ More examples on the use of structures, including groups, rings and
+ polynomials can be found in the Isabelle distribution in the
+ session HOL-Algebra.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Conclusions and Outlook%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Locales provide a simple means of modular reasoning. They enable to
+ abbreviate frequently occurring context statements and maintain facts
+ valid in these contexts. Importantly, using structures, they allow syntax to be
+ effective only in certain contexts, and thus to mimic common
+ practice in mathematics, where notation is chosen very flexibly.
+ This is also known as literate formalisation \cite{Bailey1998}.
+ Locale expressions allow to duplicate and merge
+ specifications. This is a necessity, for example, when reasoning about
+ homomorphisms. Normalisation makes it possible to deal with
+ diamond-shaped inheritance structures, and generally with directed
+ acyclic graphs. The combination of locales with record
+ types in higher-order logic provides an effective means for
+ specifying algebraic structures: locale import and record subtyping
+ provide independent hierarchies for specifications and structure
+ elements. Rich examples for this can be found in
+ the Isabelle distribution in the session HOL-Algebra.
+
+ The primary reason for writing this report was to provide a better
+ understanding of locales in Isar. Wenzel provided hardly any
+ documentation, with the exception of \cite{Wenzel2002b}. The
+ present report should make it easier for users of Isabelle to take
+ advantage of locales.
+
+ The report is also a base for future extensions. These include
+ improved syntax for structures. Identifying them by numbers seems
+ unnatural and can be confusing if more than two structures are
+ involved --- for example, when reasoning about universal
+ properties --- and numbering them by order of occurrence seems
+ arbitrary. Another desirable feature is \emph{instantiation}. One
+ may, in the course of a theory development, construct objects that
+ fulfil the specification of a locale. These objects are possibly
+ defined in the context of another locale. Instantiation should make it
+ simple to specialise abstract facts for the object under
+ consideration and to use the specified facts.
+
+ A detailed comparison of locales with module systems in type theory
+ has not been undertaken yet, but could be beneficial. For example,
+ a module system for Coq has recently been presented by Chrzaszcz
+ \cite{Chrzaszcz2003,Chrzaszcz2004}. While the
+ latter usually constitute extensions of the calculus, locales are
+ a rather thin layer that does not change Isabelle's meta logic.
+ Locales mainly manage specifications and facts. Functors, like
+ the constructor for polynomial rings, remain objects of the
+ logic.
+
+ \textbf{Acknowledgements.} Lawrence C.\ Paulson and Norbert
+ Schirmer made useful comments on a draft of this paper.%
+\end{isamarkuptext}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End: