%
\begin{isabellebody}%
\def\isabellecontext{Spec}%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ Spec\isanewline
\isakeyword{imports}\ Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
%
\isamarkupchapter{Theory 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{{\isachardoublequote}toplevel\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{global}{command}{end}\hypertarget{command.global.end}{\hyperlink{command.global.end}{\mbox{\isa{\isacommand{end}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ toplevel{\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{rail}
'theory' name 'imports' (name +) uses? 'begin'
;
uses: 'uses' ((name | parname) +);
\end{rail}
\begin{description}
\item \hyperlink{command.theory}{\mbox{\isa{\isacommand{theory}}}}~\isa{{\isachardoublequote}A\ {\isasymIMPORTS}\ B\isactrlsub {\isadigit{1}}\ {\isasymdots}\ B\isactrlsub n\ {\isasymBEGIN}{\isachardoublequote}}
starts a new theory \isa{A} based on the merge of existing
theories \isa{{\isachardoublequote}B\isactrlsub {\isadigit{1}}\ {\isasymdots}\ B\isactrlsub n{\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's theory loader ensures that the
sources contributing to the development graph are always up-to-date:
changed files are automatically reloaded whenever a theory header
specification is processed.
The optional \indexdef{}{keyword}{uses}\hypertarget{keyword.uses}{\hyperlink{keyword.uses}{\mbox{\isa{\isakeyword{uses}}}}} specification declares additional
dependencies on extra files (usually 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 local theory targets involve a local
\hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}}, which is clear from the nesting.
\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{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' signifies the
global theory context.
\begin{matharray}{rcll}
\indexdef{}{command}{context}\hypertarget{command.context}{\hyperlink{command.context}{\mbox{\isa{\isacommand{context}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{local}{command}{end}\hypertarget{command.local.end}{\hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\end{matharray}
\indexouternonterm{target}
\begin{rail}
'context' name 'begin'
;
target: '(' 'in' name ')'
;
\end{rail}
\begin{description}
\item \hyperlink{command.context}{\mbox{\isa{\isacommand{context}}}}~\isa{{\isachardoublequote}c\ {\isasymBEGIN}{\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{{\isachardoublequote}{\isacharparenleft}{\isachardoublequote}}\indexdef{}{keyword}{in}\hypertarget{keyword.in}{\hyperlink{keyword.in}{\mbox{\isa{\isakeyword{in}}}}}~\isa{{\isachardoublequote}c{\isacharparenright}{\isachardoublequote}} given after any
local theory command specifies an immediate target, e.g.\
``\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIN}\ c{\isacharparenright}\ {\isasymdots}{\isachardoublequote}}'' or ``\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIN}\ c{\isacharparenright}\ {\isasymdots}{\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{{\isachardoublequote}{\isacharparenleft}{\isasymIN}\ {\isacharminus}{\isacharparenright}{\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{{\isachardoublequote}A{\isacharbrackleft}x{\isacharbrackright}{\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{{\isachardoublequote}a\ {\isasymequiv}\ t{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} becomes \isa{{\isachardoublequote}c{\isachardot}a\ {\isacharquery}x\ {\isasymequiv}\ t{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}{\isachardoublequote}} at the theory
level (for arbitrary \isa{{\isachardoublequote}{\isacharquery}x{\isachardoublequote}}), together with a local
abbreviation \isa{{\isachardoublequote}c\ {\isasymequiv}\ c{\isachardot}a\ x{\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{{\isachardoublequote}a{\isacharcolon}\ B{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} becomes \isa{{\isachardoublequote}c{\isachardot}a{\isacharcolon}\ A{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}\ {\isasymLongrightarrow}\ B{\isacharbrackleft}{\isacharquery}x{\isacharbrackright}{\isachardoublequote}}, again for arbitrary
\isa{{\isachardoublequote}{\isacharquery}x{\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{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} & (axiomatic!)\\
\indexdef{}{command}{definition}\hypertarget{command.definition}{\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\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{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{print\_abbrevs}\hypertarget{command.print-abbrevs}{\hyperlink{command.print-abbrevs}{\mbox{\isa{\isacommand{print{\isacharunderscore}abbrevs}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}\ {\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{rail}
'axiomatization' target? fixes? ('where' specs)?
;
'definition' target? (decl 'where')? thmdecl? prop
;
'abbreviation' target? mode? (decl 'where')? prop
;
fixes: ((name ('::' type)? mixfix? | vars) + 'and')
;
specs: (thmdecl? props + 'and')
;
decl: name ('::' type)? mixfix?
;
\end{rail}
\begin{description}
\item \hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}~\isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub m\ {\isasymWHERE}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\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{{\isachardoublequote}c\ {\isasymWHERE}\ eq{\isachardoublequote}} produces an
internal definition \isa{{\isachardoublequote}c\ {\isasymequiv}\ t{\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{{\isachardoublequote}x\ {\isacharequal}\ y{\isachardoublequote}} and equivalence \isa{{\isachardoublequote}A\ {\isasymleftrightarrow}\ B{\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{{\isachardoublequote}f\ x\ y\ {\isacharequal}\ t{\isachardoublequote}} instead of
\isa{{\isachardoublequote}f\ {\isasymequiv}\ {\isasymlambda}x\ y{\isachardot}\ t{\isachardoublequote}} and \isa{{\isachardoublequote}y\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ g\ x\ y\ {\isacharequal}\ u{\isachardoublequote}} instead of an
unrestricted \isa{{\isachardoublequote}g\ {\isasymequiv}\ {\isasymlambda}x\ y{\isachardot}\ u{\isachardoublequote}}.
\item \hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}~\isa{{\isachardoublequote}c\ {\isasymWHERE}\ eq{\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{\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{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{declare}\hypertarget{command.declare}{\hyperlink{command.declare}{\mbox{\isa{\isacommand{declare}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'declaration' target? text
;
'declare' target? (thmrefs + 'and')
;
\end{rail}
\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.
\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 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 specifications%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{locale}\hypertarget{command.locale}{\hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{print\_locale}\hypertarget{command.print-locale}{\hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isacharunderscore}locale}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{command}{print\_locales}\hypertarget{command.print-locales}{\hyperlink{command.print-locales}{\mbox{\isa{\isacommand{print{\isacharunderscore}locales}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{method}{intro\_locales}\hypertarget{method.intro-locales}{\hyperlink{method.intro-locales}{\mbox{\isa{intro{\isacharunderscore}locales}}}} & : & \isa{method} \\
\indexdef{}{method}{unfold\_locales}\hypertarget{method.unfold-locales}{\hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isacharunderscore}locales}}}} & : & \isa{method} \\
\end{matharray}
\indexouternonterm{contextexpr}\indexouternonterm{contextelem}
\indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}
\indexisarelem{defines}\indexisarelem{notes}
\begin{rail}
'locale' name ('=' localeexpr)? 'begin'?
;
'print\_locale' '!'? localeexpr
;
localeexpr: ((contextexpr '+' (contextelem+)) | contextexpr | (contextelem+))
;
contextexpr: nameref | '(' contextexpr ')' |
(contextexpr (name mixfix? +)) | (contextexpr + '+')
;
contextelem: fixes | constrains | assumes | defines | notes
;
fixes: 'fixes' ((name ('::' type)? structmixfix? | vars) + 'and')
;
constrains: 'constrains' (name '::' type + 'and')
;
assumes: 'assumes' (thmdecl? props + 'and')
;
defines: 'defines' (thmdecl? prop proppat? + 'and')
;
notes: 'notes' (thmdef? thmrefs + 'and')
;
\end{rail}
\begin{description}
\item \hyperlink{command.locale}{\mbox{\isa{\isacommand{locale}}}}~\isa{{\isachardoublequote}loc\ {\isacharequal}\ import\ {\isacharplus}\ body{\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 context expression,
consisting of references to existing locales, renamed contexts, or
merged contexts. Renaming uses positional notation: \isa{{\isachardoublequote}c\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n{\isachardoublequote}} means that (a prefix of) the fixed
parameters of context \isa{c} are named \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub n{\isachardoublequote}}; a ``\isa{{\isacharunderscore}}'' (underscore) means to skip that
position. Renaming by default deletes concrete syntax, but new
syntax may by specified with a mixfix annotation. An exeption of
this rule is the special syntax declared with ``\isa{{\isachardoublequote}{\isacharparenleft}{\isasymSTRUCTURE}{\isacharparenright}{\isachardoublequote}}'' (see below), which is neither deleted nor can it
be changed. Merging proceeds from left-to-right, suppressing any
duplicates stemming from different paths through the import
hierarchy.
The \isa{body} consists of basic context elements, further context
expressions may be included as well.
\begin{description}
\item \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}}~\isa{{\isachardoublequote}x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\ {\isacharparenleft}mx{\isacharparenright}{\isachardoublequote}} declares a local
parameter of type \isa{{\isasymtau}} and mixfix annotation \isa{mx} (both
are optional). The special syntax declaration ``\isa{{\isachardoublequote}{\isacharparenleft}{\isasymSTRUCTURE}{\isacharparenright}{\isachardoublequote}}'' means that \isa{x} may be referenced
implicitly in this context.
\item \hyperlink{element.constrains}{\mbox{\isa{\isakeyword{constrains}}}}~\isa{{\isachardoublequote}x\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}{\isachardoublequote}} introduces a type
constraint \isa{{\isasymtau}} on the local parameter \isa{x}.
\item \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\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{{\isachardoublequote}a{\isacharcolon}\ x\ {\isasymequiv}\ t{\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{{\isachardoublequote}f\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub n\ {\isasymequiv}\ t{\isachardoublequote}}''.
\item \hyperlink{element.notes}{\mbox{\isa{\isakeyword{notes}}}}~\isa{{\isachardoublequote}a\ {\isacharequal}\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\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{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\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 By default, locale specifications are ``closed up'' by
turning the given text into a predicate definition \isa{loc{\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{{\isasymAnd}} by \isa{{\isasymforall}}, and
\isa{{\isachardoublequote}{\isasymLongrightarrow}{\isachardoublequote}} by \isa{{\isachardoublequote}{\isasymlongrightarrow}{\isachardoublequote}} in HOL; see also
\secref{sec:object-logic}). Separate introduction rules \isa{loc{\isacharunderscore}axioms{\isachardot}intro} and \isa{loc{\isachardot}intro} are provided as well.
\item \hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isacharunderscore}locale}}}}~\isa{{\isachardoublequote}import\ {\isacharplus}\ body{\isachardoublequote}} prints the
specified locale expression in a flattened form. The notable
special case \hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isacharunderscore}locale}}}}~\isa{loc} just prints the
contents of the named locale, but keep in mind that type-inference
will normalize type variables according to the usual alphabetical
order. The command omits \hyperlink{element.notes}{\mbox{\isa{\isakeyword{notes}}}} elements by default.
Use \hyperlink{command.print-locale}{\mbox{\isa{\isacommand{print{\isacharunderscore}locale}}}}\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}} to get them included.
\item \hyperlink{command.print-locales}{\mbox{\isa{\isacommand{print{\isacharunderscore}locales}}}} prints the names of all locales
of the current theory.
\item \hyperlink{method.intro-locales}{\mbox{\isa{intro{\isacharunderscore}locales}}} and \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isacharunderscore}locales}}}
repeatedly expand all introduction rules of locale predicates of the
theory. While \hyperlink{method.intro-locales}{\mbox{\isa{intro{\isacharunderscore}locales}}} only applies the \isa{loc{\isachardot}intro} introduction rules and therefore does not decend to
assumptions, \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\isacharunderscore}locales}}} is more aggressive and applies
\isa{loc{\isacharunderscore}axioms{\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{Interpretation of locales%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
Locale expressions (more precisely, \emph{context 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 theories and locales (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{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
\indexdef{}{command}{interpret}\hypertarget{command.interpret}{\hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ proof{\isacharparenleft}chain\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
\end{matharray}
\indexouternonterm{interp}
\begin{rail}
'interpretation' (interp | name ('<' | subseteq) contextexpr)
;
'interpret' interp
;
instantiation: ('[' (inst+) ']')?
;
interp: (name ':')? \\ (contextexpr instantiation |
name instantiation 'where' (thmdecl? prop + 'and'))
;
\end{rail}
\begin{description}
\item \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}~\isa{{\isachardoublequote}expr\ insts\ {\isasymWHERE}\ eqns{\isachardoublequote}}
The first form of \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}} interprets \isa{expr} in the theory. The instantiation is given as a list of terms
\isa{insts} and is positional. All parameters must receive an
instantiation term --- with the exception of defined parameters.
These are, if omitted, derived from the defining equation and other
instantiations. Use ``\isa{{\isacharunderscore}}'' to omit an instantiation term.
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 in facts during
post-processing, may be given after the keyword \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}}.
This is useful for interpreting concepts introduced through
definition specification elements. The equations must be proved.
Note that if equations are present, the context expression is
restricted to a locale name.
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{\isacharunderscore}locales}}}
or \hyperlink{method.unfold-locales}{\mbox{\isa{unfold{\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.
The context expression may be preceded by a name, which takes effect
in the post-processing of facts. It is used to prefix fact names,
for example to avoid accidental hiding of other facts.
Adding facts to locales has the effect of adding interpreted facts
to the theory for all active interpretations also. That is,
interpretations dynamically participate in any facts added to
locales.
\item \hyperlink{command.interpretation}{\mbox{\isa{\isacommand{interpretation}}}}~\isa{{\isachardoublequote}name\ {\isasymsubseteq}\ expr{\isachardoublequote}}
This form of the command interprets \isa{expr} in the locale
\isa{name}. It requires a proof that the specification of \isa{name} implies the specification of \isa{expr}. As in the
localized version of the theorem command, the proof is in the
context of \isa{name}. After the proof obligation has been
dischared, 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}.
Like facts of renamed context elements, facts obtained by
interpretation may be accessed by prefixing with the parameter
renaming (where the parameters are separated by ``\isa{{\isacharunderscore}}'').
Unlike interpretation in theories, instantiation is confined to the
renaming of parameters, which may be specified as part of the
context expression \isa{expr}. Using defined parameters in \isa{name} one may achieve an effect similar to instantiation, though.
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 by interpretation. This enables
circular interpretations.
If interpretations of \isa{name} exist in the current theory, the
command adds interpretations for \isa{expr} as well, with the same
prefix and attributes, although only for fragments of \isa{expr}
that are not interpreted in the theory already.
\item \hyperlink{command.interpret}{\mbox{\isa{\isacommand{interpret}}}}~\isa{{\isachardoublequote}expr\ insts\ {\isasymWHERE}\ eqns{\isachardoublequote}}
interprets \isa{expr} in the proof context and is otherwise
similar to interpretation in theories.
\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. Since the behavior of such
automated reasoning tools is \emph{not} stable under
interpretation morphisms, manual declarations might have to be
issued.
\end{warn}
\begin{warn}
An interpretation in a theory 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. A warning is issued, since it is likely that these could
have been generalized in the first place. 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{{\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
tutorial.
\begin{matharray}{rcl}
\indexdef{}{command}{class}\hypertarget{command.class}{\hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{instantiation}\hypertarget{command.instantiation}{\hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{instance}\hypertarget{command.instance}{\hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{subclass}\hypertarget{command.subclass}{\hyperlink{command.subclass}{\mbox{\isa{\isacommand{subclass}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{print\_classes}\hypertarget{command.print-classes}{\hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isacharunderscore}classes}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{command}{class\_deps}\hypertarget{command.class-deps}{\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{method}{intro\_classes}\hypertarget{method.intro-classes}{\hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}}} & : & \isa{method} \\
\end{matharray}
\begin{rail}
'class' name '=' ((superclassexpr '+' (contextelem+)) | superclassexpr | (contextelem+)) \\
'begin'?
;
'instantiation' (nameref + 'and') '::' arity 'begin'
;
'instance'
;
'subclass' target? nameref
;
'print\_classes'
;
'class\_deps'
;
superclassexpr: nameref | (nameref '+' superclassexpr)
;
\end{rail}
\begin{description}
\item \hyperlink{command.class}{\mbox{\isa{\isacommand{class}}}}~\isa{{\isachardoublequote}c\ {\isacharequal}\ superclasses\ {\isacharplus}\ body{\isachardoublequote}} defines
a new class \isa{c}, inheriting from \isa{superclasses}. This
introduces a locale \isa{c} with import of all locales \isa{superclasses}.
Any \hyperlink{element.fixes}{\mbox{\isa{\isakeyword{fixes}}}} in \isa{body} are lifted to the global
theory level (\emph{class operations} \isa{{\isachardoublequote}f\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ f\isactrlsub n{\isachardoublequote}} of class \isa{c}), mapping the local type parameter
\isa{{\isasymalpha}} to a schematic type variable \isa{{\isachardoublequote}{\isacharquery}{\isasymalpha}\ {\isacharcolon}{\isacharcolon}\ c{\isachardoublequote}}.
Likewise, \hyperlink{element.assumes}{\mbox{\isa{\isakeyword{assumes}}}} in \isa{body} are also lifted,
mapping each local parameter \isa{{\isachardoublequote}f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}{\isachardoublequote}} to its
corresponding global constant \isa{{\isachardoublequote}f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}{\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isacharcolon}{\isacharcolon}\ c{\isacharbrackright}{\isachardoublequote}}. The
corresponding introduction rule is provided as \isa{c{\isacharunderscore}class{\isacharunderscore}axioms{\isachardot}intro}. This rule should be rarely needed directly
--- the \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} method takes care of the details of
class membership proofs.
\item \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}~\isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlsub n{\isacharparenright}s\ {\isasymBEGIN}{\isachardoublequote}} opens a theory target (cf.\ \secref{sec:target}) which
allows to specify class operations \isa{{\isachardoublequote}f\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ f\isactrlsub n{\isachardoublequote}} corresponding
to sort \isa{s} at the particular type instance \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ s\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ s\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}}. A plain \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} command in the
target body poses a goal stating these type arities. The target is
concluded by an \indexref{local}{command}{end}\hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}} command.
Note that a list of simultaneous type constructors may be given;
this corresponds nicely to mutual recursive type definitions, e.g.\
in Isabelle/HOL.
\item \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} in an instantiation target body sets
up a goal stating the type arities claimed at the opening \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}}. The proof would usually proceed by \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}}, and then establish the characteristic theorems of
the type classes involved. After finishing the proof, the
background theory will be augmented by the proven type arities.
\item \hyperlink{command.subclass}{\mbox{\isa{\isacommand{subclass}}}}~\isa{c} in a class context for class
\isa{d} sets up a goal stating that class \isa{c} is logically
contained in class \isa{d}. After finishing the proof, class
\isa{d} is proven to be subclass \isa{c} and the locale \isa{c} is interpreted into \isa{d} simultaneously.
\item \hyperlink{command.print-classes}{\mbox{\isa{\isacommand{print{\isacharunderscore}classes}}}} prints all classes in the current
theory.
\item \hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}} visualizes all classes and their
subclass relations as a Hasse diagram.
\item \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} repeatedly expands all class
introduction rules of this theory. Note that this method usually
needs not be named explicitly, as it is already included in the
default proof step (e.g.\ of \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}). In particular,
instantiation of trivial (syntactic) classes may be performed by a
single ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}'' proof step.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{The class target%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
%FIXME check
A named context may refer to a locale (cf.\ \secref{sec:target}).
If this locale is also a class \isa{c}, apart from the common
locale target behaviour the following happens.
\begin{itemize}
\item Local constant declarations \isa{{\isachardoublequote}g{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}{\isachardoublequote}} referring to the
local type parameter \isa{{\isasymalpha}} and local parameters \isa{{\isachardoublequote}f{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}{\isachardoublequote}}
are accompanied by theory-level constants \isa{{\isachardoublequote}g{\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isacharcolon}{\isacharcolon}\ c{\isacharbrackright}{\isachardoublequote}}
referring to theory-level class operations \isa{{\isachardoublequote}f{\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isacharcolon}{\isacharcolon}\ c{\isacharbrackright}{\isachardoublequote}}.
\item Local theorem bindings are lifted as are assumptions.
\item Local syntax refers to local operations \isa{{\isachardoublequote}g{\isacharbrackleft}{\isasymalpha}{\isacharbrackright}{\isachardoublequote}} and
global operations \isa{{\isachardoublequote}g{\isacharbrackleft}{\isacharquery}{\isasymalpha}\ {\isacharcolon}{\isacharcolon}\ c{\isacharbrackright}{\isachardoublequote}} uniformly. Type inference
resolves ambiguities. In rare cases, manual type annotations are
needed.
\end{itemize}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Old-style axiomatic type classes \label{sec:axclass}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{axclass}\hypertarget{command.axclass}{\hyperlink{command.axclass}{\mbox{\isa{\isacommand{axclass}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{instance}\hypertarget{command.instance}{\hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
\end{matharray}
Axiomatic type classes are Isabelle/Pure's primitive
\emph{definitional} interface to type classes. For practical
applications, you should consider using classes
(cf.~\secref{sec:classes}) which provide high level interface.
\begin{rail}
'axclass' classdecl (axmdecl prop +)
;
'instance' (nameref ('<' | subseteq) nameref | nameref '::' arity)
;
\end{rail}
\begin{description}
\item \hyperlink{command.axclass}{\mbox{\isa{\isacommand{axclass}}}}~\isa{{\isachardoublequote}c\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ axms{\isachardoublequote}} defines an
axiomatic type class as the intersection of existing classes, with
additional axioms holding. Class axioms may not contain more than
one type variable. The class axioms (with implicit sort constraints
added) are bound to the given names. Furthermore a class
introduction rule is generated (being bound as \isa{c{\isacharunderscore}class{\isachardot}intro}); this rule is employed by method \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}} to support instantiation proofs of this class.
The ``class axioms'' (which are derived from the internal class
definition) are stored as theorems according to the given name
specifications; the name space prefix \isa{{\isachardoublequote}c{\isacharunderscore}class{\isachardoublequote}} is added
here. The full collection of these facts is also stored as \isa{c{\isacharunderscore}class{\isachardot}axioms}.
\item \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}~\isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}{\isachardoublequote}} and \hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}~\isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlsub n{\isacharparenright}s{\isachardoublequote}} setup a goal stating a class
relation or type arity. The proof would usually proceed by \hyperlink{method.intro-classes}{\mbox{\isa{intro{\isacharunderscore}classes}}}, and then establish the characteristic theorems of
the type classes involved. After finishing the proof, the theory
will be augmented by a type signature declaration corresponding to
the resulting theorem.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Unrestricted overloading%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
Isabelle/Pure's definitional schemes support certain forms of
overloading (see \secref{sec:consts}). At most occassions
overloading will be used in a Haskell-like fashion together with
type classes by means of \hyperlink{command.instantiation}{\mbox{\isa{\isacommand{instantiation}}}} (see
\secref{sec:class}). Sometimes low-level overloading is desirable.
The \hyperlink{command.overloading}{\mbox{\isa{\isacommand{overloading}}}} target provides a convenient view for
end-users.
\begin{matharray}{rcl}
\indexdef{}{command}{overloading}\hypertarget{command.overloading}{\hyperlink{command.overloading}{\mbox{\isa{\isacommand{overloading}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'overloading' \\
( string ( '==' | equiv ) term ( '(' 'unchecked' ')' )? + ) 'begin'
\end{rail}
\begin{description}
\item \hyperlink{command.overloading}{\mbox{\isa{\isacommand{overloading}}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymequiv}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\isactrlsub {\isadigit{1}}\ {\isasymAND}\ {\isasymdots}\ x\isactrlsub n\ {\isasymequiv}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}\isactrlsub n\ {\isasymBEGIN}{\isachardoublequote}}
opens a theory target (cf.\ \secref{sec:target}) which allows to
specify constants with overloaded definitions. These are identified
by an explicitly given mapping from variable names \isa{{\isachardoublequote}x\isactrlsub i{\isachardoublequote}} to
constants \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} at particular type instances. The
definitions themselves are established using common specification
tools, using the names \isa{{\isachardoublequote}x\isactrlsub i{\isachardoublequote}} as reference to the
corresponding constants. The target is concluded by \hyperlink{command.local.end}{\mbox{\isa{\isacommand{end}}}}.
A \isa{{\isachardoublequote}{\isacharparenleft}unchecked{\isacharparenright}{\isachardoublequote}} option disables global dependency checks for
the corresponding definition, which is occasionally useful for
exotic overloading. It is at the discretion of the user to avoid
malformed theory specifications!
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Incorporating ML code \label{sec:ML}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{use}\hypertarget{command.use}{\hyperlink{command.use}{\mbox{\isa{\isacommand{use}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{ML}\hypertarget{command.ML}{\hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{ML\_prf}\hypertarget{command.ML-prf}{\hyperlink{command.ML-prf}{\mbox{\isa{\isacommand{ML{\isacharunderscore}prf}}}}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}\ proof{\isachardoublequote}} \\
\indexdef{}{command}{ML\_val}\hypertarget{command.ML-val}{\hyperlink{command.ML-val}{\mbox{\isa{\isacommand{ML{\isacharunderscore}val}}}}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{command}{ML\_command}\hypertarget{command.ML-command}{\hyperlink{command.ML-command}{\mbox{\isa{\isacommand{ML{\isacharunderscore}command}}}}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
\indexdef{}{command}{setup}\hypertarget{command.setup}{\hyperlink{command.setup}{\mbox{\isa{\isacommand{setup}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\end{matharray}
\begin{mldecls}
\indexml{bind\_thms}\verb|bind_thms: string * thm list -> unit| \\
\indexml{bind\_thm}\verb|bind_thm: string * thm -> unit| \\
\end{mldecls}
\begin{rail}
'use' name
;
('ML' | 'ML\_prf' | 'ML\_val' | 'ML\_command' | 'setup') text
;
\end{rail}
\begin{description}
\item \hyperlink{command.use}{\mbox{\isa{\isacommand{use}}}}~\isa{{\isachardoublequote}file{\isachardoublequote}} reads and executes ML
commands from \isa{{\isachardoublequote}file{\isachardoublequote}}. The current theory context is passed
down to the ML toplevel and may be modified, using \verb|Context.>>| or derived ML commands. The file name is checked with
the \indexref{}{keyword}{uses}\hyperlink{keyword.uses}{\mbox{\isa{\isakeyword{uses}}}} dependency declaration given in the theory
header (see also \secref{sec:begin-thy}).
Top-level ML bindings are stored within the (global or local) theory
context.
\item \hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}}~\isa{{\isachardoublequote}text{\isachardoublequote}} is similar to \hyperlink{command.use}{\mbox{\isa{\isacommand{use}}}},
but executes ML commands directly from the given \isa{{\isachardoublequote}text{\isachardoublequote}}.
Top-level ML bindings are stored within the (global or local) theory
context.
\item \hyperlink{command.ML-prf}{\mbox{\isa{\isacommand{ML{\isacharunderscore}prf}}}} is analogous to \hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}} but works
within a proof context.
Top-level ML bindings are stored within the proof context in a
purely sequential fashion, disregarding the nested proof structure.
ML bindings introduced by \hyperlink{command.ML-prf}{\mbox{\isa{\isacommand{ML{\isacharunderscore}prf}}}} are discarded at the
end of the proof.
\item \hyperlink{command.ML-val}{\mbox{\isa{\isacommand{ML{\isacharunderscore}val}}}} and \hyperlink{command.ML-command}{\mbox{\isa{\isacommand{ML{\isacharunderscore}command}}}} are diagnostic
versions of \hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}}, which means that the context may not be
updated. \hyperlink{command.ML-val}{\mbox{\isa{\isacommand{ML{\isacharunderscore}val}}}} echos the bindings produced at the ML
toplevel, but \hyperlink{command.ML-command}{\mbox{\isa{\isacommand{ML{\isacharunderscore}command}}}} is silent.
\item \hyperlink{command.setup}{\mbox{\isa{\isacommand{setup}}}}~\isa{{\isachardoublequote}text{\isachardoublequote}} changes the current theory
context by applying \isa{{\isachardoublequote}text{\isachardoublequote}}, which refers to an ML expression
of type \verb|theory -> theory|. This enables
to initialize any object-logic specific tools and packages written
in ML, for example.
\item \verb|bind_thms|~\isa{{\isachardoublequote}{\isacharparenleft}name{\isacharcomma}\ thms{\isacharparenright}{\isachardoublequote}} stores a list of
theorems produced in ML both in the theory context and the ML
toplevel, associating it with the provided name. Theorems are put
into a global ``standard'' format before being stored.
\item \verb|bind_thm| is similar to \verb|bind_thms| but refers to a
singleton theorem.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Primitive specification elements%
}
\isamarkuptrue%
%
\isamarkupsubsection{Type classes and sorts \label{sec:classes}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcll}
\indexdef{}{command}{classes}\hypertarget{command.classes}{\hyperlink{command.classes}{\mbox{\isa{\isacommand{classes}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{classrel}\hypertarget{command.classrel}{\hyperlink{command.classrel}{\mbox{\isa{\isacommand{classrel}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} & (axiomatic!) \\
\indexdef{}{command}{defaultsort}\hypertarget{command.defaultsort}{\hyperlink{command.defaultsort}{\mbox{\isa{\isacommand{defaultsort}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{class\_deps}\hypertarget{command.class-deps}{\hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'classes' (classdecl +)
;
'classrel' (nameref ('<' | subseteq) nameref + 'and')
;
'defaultsort' sort
;
\end{rail}
\begin{description}
\item \hyperlink{command.classes}{\mbox{\isa{\isacommand{classes}}}}~\isa{{\isachardoublequote}c\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n{\isachardoublequote}} declares class
\isa{c} to be a subclass of existing classes \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n{\isachardoublequote}}.
Isabelle implicitly maintains the transitive closure of the class
hierarchy. Cyclic class structures are not permitted.
\item \hyperlink{command.classrel}{\mbox{\isa{\isacommand{classrel}}}}~\isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}{\isachardoublequote}} states subclass
relations between existing classes \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}{\isachardoublequote}} and \isa{{\isachardoublequote}c\isactrlsub {\isadigit{2}}{\isachardoublequote}}.
This is done axiomatically! The \indexref{}{command}{instance}\hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}} command
(see \secref{sec:axclass}) provides a way to introduce proven class
relations.
\item \hyperlink{command.defaultsort}{\mbox{\isa{\isacommand{defaultsort}}}}~\isa{s} makes sort \isa{s} the
new default sort for any type variable that is given explicitly in
the text, but lacks a sort constraint (wrt.\ the current context).
Type variables generated by type inference are not affected.
Usually the default sort is only changed when defining a new
object-logic. For example, the default sort in Isabelle/HOL is
\isa{type}, the class of all HOL types. %FIXME sort antiq?
When merging theories, the default sorts of the parents are
logically intersected, i.e.\ the representations as lists of classes
are joined.
\item \hyperlink{command.class-deps}{\mbox{\isa{\isacommand{class{\isacharunderscore}deps}}}} visualizes the subclass relation,
using Isabelle's graph browser tool (see also \cite{isabelle-sys}).
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Types and type abbreviations \label{sec:types-pure}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcll}
\indexdef{}{command}{types}\hypertarget{command.types}{\hyperlink{command.types}{\mbox{\isa{\isacommand{types}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{typedecl}\hypertarget{command.typedecl}{\hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{arities}\hypertarget{command.arities}{\hyperlink{command.arities}{\mbox{\isa{\isacommand{arities}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} & (axiomatic!) \\
\end{matharray}
\begin{rail}
'types' (typespec '=' type infix? +)
;
'typedecl' typespec infix?
;
'arities' (nameref '::' arity +)
;
\end{rail}
\begin{description}
\item \hyperlink{command.types}{\mbox{\isa{\isacommand{types}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}{\isachardoublequote}} introduces a
\emph{type synonym} \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}} for the existing type
\isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}}. Unlike actual type definitions, as are available in
Isabelle/HOL for example, type synonyms are merely syntactic
abbreviations without any logical significance. Internally, type
synonyms are fully expanded.
\item \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}} declares a new
type constructor \isa{t}. If the object-logic defines a base sort
\isa{s}, then the constructor is declared to operate on that, via
the axiomatic specification \hyperlink{command.arities}{\mbox{\isa{\isacommand{arities}}}}~\isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s{\isacharparenright}s{\isachardoublequote}}.
\item \hyperlink{command.arities}{\mbox{\isa{\isacommand{arities}}}}~\isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}s\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ s\isactrlsub n{\isacharparenright}s{\isachardoublequote}} augments
Isabelle's order-sorted signature of types by new type constructor
arities. This is done axiomatically! The \indexref{}{command}{instance}\hyperlink{command.instance}{\mbox{\isa{\isacommand{instance}}}}
command (see \secref{sec:axclass}) provides a way to introduce
proven type arities.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Co-regularity of type classes and arities%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
The class relation together with the collection of
type-constructor arities must obey the principle of
\emph{co-regularity} as defined below.
\medskip For the subsequent formulation of co-regularity we assume
that the class relation is closed by transitivity and reflexivity.
Moreover the collection of arities \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c{\isachardoublequote}} is
completed such that \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c{\isachardoublequote}} and \isa{{\isachardoublequote}c\ {\isasymsubseteq}\ c{\isacharprime}{\isachardoublequote}}
implies \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s{\isacharparenright}c{\isacharprime}{\isachardoublequote}} for all such declarations.
Treating sorts as finite sets of classes (meaning the intersection),
the class relation \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}{\isachardoublequote}} is extended to sorts as
follows:
\[
\isa{{\isachardoublequote}s\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ s\isactrlsub {\isadigit{2}}\ {\isasymequiv}\ {\isasymforall}c\isactrlsub {\isadigit{2}}\ {\isasymin}\ s\isactrlsub {\isadigit{2}}{\isachardot}\ {\isasymexists}c\isactrlsub {\isadigit{1}}\ {\isasymin}\ s\isactrlsub {\isadigit{1}}{\isachardot}\ c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}{\isachardoublequote}}
\]
This relation on sorts is further extended to tuples of sorts (of
the same length) in the component-wise way.
\smallskip Co-regularity of the class relation together with the
arities relation means:
\[
\isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlsub {\isadigit{1}}{\isacharparenright}c\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}\isactrlvec s\isactrlsub {\isadigit{2}}{\isacharparenright}c\isactrlsub {\isadigit{2}}\ {\isasymLongrightarrow}\ c\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ c\isactrlsub {\isadigit{2}}\ {\isasymLongrightarrow}\ \isactrlvec s\isactrlsub {\isadigit{1}}\ {\isasymsubseteq}\ \isactrlvec s\isactrlsub {\isadigit{2}}{\isachardoublequote}}
\]
\noindent for all such arities. In other words, whenever the result
classes of some type-constructor arities are related, then the
argument sorts need to be related in the same way.
\medskip Co-regularity is a very fundamental property of the
order-sorted algebra of types. For example, it entails principle
types and most general unifiers, e.g.\ see \cite{nipkow-prehofer}.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsubsection{Constants and definitions \label{sec:consts}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
Definitions essentially express abbreviations within the logic. The
simplest form of a definition is \isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\ {\isasymequiv}\ t{\isachardoublequote}}, where \isa{c} is a newly declared constant. Isabelle also allows derived forms
where the arguments of \isa{c} appear on the left, abbreviating a
prefix of \isa{{\isasymlambda}}-abstractions, e.g.\ \isa{{\isachardoublequote}c\ {\isasymequiv}\ {\isasymlambda}x\ y{\isachardot}\ t{\isachardoublequote}} may be
written more conveniently as \isa{{\isachardoublequote}c\ x\ y\ {\isasymequiv}\ t{\isachardoublequote}}. Moreover,
definitions may be weakened by adding arbitrary pre-conditions:
\isa{{\isachardoublequote}A\ {\isasymLongrightarrow}\ c\ x\ y\ {\isasymequiv}\ t{\isachardoublequote}}.
\medskip The built-in well-formedness conditions for definitional
specifications are:
\begin{itemize}
\item Arguments (on the left-hand side) must be distinct variables.
\item All variables on the right-hand side must also appear on the
left-hand side.
\item All type variables on the right-hand side must also appear on
the left-hand side; this prohibits \isa{{\isachardoublequote}{\isadigit{0}}\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymequiv}\ length\ {\isacharparenleft}{\isacharbrackleft}{\isacharbrackright}\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ list{\isacharparenright}{\isachardoublequote}} for example.
\item The definition must not be recursive. Most object-logics
provide definitional principles that can be used to express
recursion safely.
\end{itemize}
Overloading means that a constant being declared as \isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ decl{\isachardoublequote}} may be defined separately on type instances \isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isasymbeta}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymbeta}\isactrlsub n{\isacharparenright}\ t\ decl{\isachardoublequote}} for each type constructor \isa{t}. The right-hand side may mention overloaded constants
recursively at type instances corresponding to the immediate
argument types \isa{{\isachardoublequote}{\isasymbeta}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymbeta}\isactrlsub n{\isachardoublequote}}. Incomplete
specification patterns impose global constraints on all occurrences,
e.g.\ \isa{{\isachardoublequote}d\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymtimes}\ {\isasymalpha}{\isachardoublequote}} on the left-hand side means that all
corresponding occurrences on some right-hand side need to be an
instance of this, general \isa{{\isachardoublequote}d\ {\isacharcolon}{\isacharcolon}\ {\isasymalpha}\ {\isasymtimes}\ {\isasymbeta}{\isachardoublequote}} will be disallowed.
\begin{matharray}{rcl}
\indexdef{}{command}{consts}\hypertarget{command.consts}{\hyperlink{command.consts}{\mbox{\isa{\isacommand{consts}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{defs}\hypertarget{command.defs}{\hyperlink{command.defs}{\mbox{\isa{\isacommand{defs}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{constdefs}\hypertarget{command.constdefs}{\hyperlink{command.constdefs}{\mbox{\isa{\isacommand{constdefs}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'consts' ((name '::' type mixfix?) +)
;
'defs' ('(' 'unchecked'? 'overloaded'? ')')? \\ (axmdecl prop +)
;
\end{rail}
\begin{rail}
'constdefs' structs? (constdecl? constdef +)
;
structs: '(' 'structure' (vars + 'and') ')'
;
constdecl: ((name '::' type mixfix | name '::' type | name mixfix) 'where'?) | name 'where'
;
constdef: thmdecl? prop
;
\end{rail}
\begin{description}
\item \hyperlink{command.consts}{\mbox{\isa{\isacommand{consts}}}}~\isa{{\isachardoublequote}c\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}{\isachardoublequote}} declares constant \isa{c} to have any instance of type scheme \isa{{\isasymsigma}}. The optional
mixfix annotations may attach concrete syntax to the constants
declared.
\item \hyperlink{command.defs}{\mbox{\isa{\isacommand{defs}}}}~\isa{{\isachardoublequote}name{\isacharcolon}\ eqn{\isachardoublequote}} introduces \isa{eqn}
as a definitional axiom for some existing constant.
The \isa{{\isachardoublequote}{\isacharparenleft}unchecked{\isacharparenright}{\isachardoublequote}} option disables global dependency checks
for this definition, which is occasionally useful for exotic
overloading. It is at the discretion of the user to avoid malformed
theory specifications!
The \isa{{\isachardoublequote}{\isacharparenleft}overloaded{\isacharparenright}{\isachardoublequote}} option declares definitions to be
potentially overloaded. Unless this option is given, a warning
message would be issued for any definitional equation with a more
special type than that of the corresponding constant declaration.
\item \hyperlink{command.constdefs}{\mbox{\isa{\isacommand{constdefs}}}} combines constant declarations and
definitions, with type-inference taking care of the most general
typing of the given specification (the optional type constraint may
refer to type-inference dummies ``\isa{{\isacharunderscore}}'' as usual). The
resulting type declaration needs to agree with that of the
specification; overloading is \emph{not} supported here!
The constant name may be omitted altogether, if neither type nor
syntax declarations are given. The canonical name of the
definitional axiom for constant \isa{c} will be \isa{c{\isacharunderscore}def},
unless specified otherwise. Also note that the given list of
specifications is processed in a strictly sequential manner, with
type-checking being performed independently.
An optional initial context of \isa{{\isachardoublequote}{\isacharparenleft}structure{\isacharparenright}{\isachardoublequote}} declarations
admits use of indexed syntax, using the special symbol \verb|\<index>| (printed as ``\isa{{\isachardoublequote}{\isasymindex}{\isachardoublequote}}''). The latter concept is
particularly useful with locales (see also \secref{sec:locale}).
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Axioms and theorems \label{sec:axms-thms}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcll}
\indexdef{}{command}{axioms}\hypertarget{command.axioms}{\hyperlink{command.axioms}{\mbox{\isa{\isacommand{axioms}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} & (axiomatic!) \\
\indexdef{}{command}{lemmas}\hypertarget{command.lemmas}{\hyperlink{command.lemmas}{\mbox{\isa{\isacommand{lemmas}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\indexdef{}{command}{theorems}\hypertarget{command.theorems}{\hyperlink{command.theorems}{\mbox{\isa{\isacommand{theorems}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'axioms' (axmdecl prop +)
;
('lemmas' | 'theorems') target? (thmdef? thmrefs + 'and')
;
\end{rail}
\begin{description}
\item \hyperlink{command.axioms}{\mbox{\isa{\isacommand{axioms}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} introduces arbitrary
statements as axioms of the meta-logic. In fact, axioms are
``axiomatic theorems'', and may be referred later just as any other
theorem.
Axioms are usually only introduced when declaring new logical
systems. Everyday work is typically done the hard way, with proper
definitions and proven theorems.
\item \hyperlink{command.lemmas}{\mbox{\isa{\isacommand{lemmas}}}}~\isa{{\isachardoublequote}a\ {\isacharequal}\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} retrieves and stores
existing facts in the theory context, or the specified target
context (see also \secref{sec:target}). Typical applications would
also involve attributes, to declare Simplifier rules, for example.
\item \hyperlink{command.theorems}{\mbox{\isa{\isacommand{theorems}}}} is essentially the same as \hyperlink{command.lemmas}{\mbox{\isa{\isacommand{lemmas}}}}, but marks the result as a different kind of facts.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Oracles%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
Oracles allow Isabelle to take advantage of external reasoners
such as arithmetic decision procedures, model checkers, fast
tautology checkers or computer algebra systems. Invoked as an
oracle, an external reasoner can create arbitrary Isabelle theorems.
It is the responsibility of the user to ensure that the external
reasoner is as trustworthy as the application requires. Another
typical source of errors is the linkup between Isabelle and the
external tool, not just its concrete implementation, but also the
required translation between two different logical environments.
Isabelle merely guarantees well-formedness of the propositions being
asserted, and records within the internal derivation object how
presumed theorems depend on unproven suppositions.
\begin{matharray}{rcl}
\indexdef{}{command}{oracle}\hypertarget{command.oracle}{\hyperlink{command.oracle}{\mbox{\isa{\isacommand{oracle}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'oracle' name '=' text
;
\end{rail}
\begin{description}
\item \hyperlink{command.oracle}{\mbox{\isa{\isacommand{oracle}}}}~\isa{{\isachardoublequote}name\ {\isacharequal}\ text{\isachardoublequote}} turns the given ML
expression \isa{{\isachardoublequote}text{\isachardoublequote}} of type \verb|'a -> cterm| into an
ML function of type \verb|'a -> thm|, which is bound to the
global identifier \verb|name|. This acts like an infinitary
specification of axioms! Invoking the oracle only works within the
scope of the resulting theory.
\end{description}
See \hyperlink{file.~~/src/FOL/ex/Iff-Oracle.thy}{\mbox{\isa{\isatt{{\isachartilde}{\isachartilde}{\isacharslash}src{\isacharslash}FOL{\isacharslash}ex{\isacharslash}Iff{\isacharunderscore}Oracle{\isachardot}thy}}}} for a worked example of
defining a new primitive rule as oracle, and turning it into a proof
method.%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isamarkupsection{Name spaces%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{}{command}{global}\hypertarget{command.global}{\hyperlink{command.global}{\mbox{\isa{\isacommand{global}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{local}\hypertarget{command.local}{\hyperlink{command.local}{\mbox{\isa{\isacommand{local}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\indexdef{}{command}{hide}\hypertarget{command.hide}{\hyperlink{command.hide}{\mbox{\isa{\isacommand{hide}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
\end{matharray}
\begin{rail}
'hide' ('(open)')? name (nameref + )
;
\end{rail}
Isabelle organizes any kind of name declarations (of types,
constants, theorems etc.) by separate hierarchically structured name
spaces. Normally the user does not have to control the behavior of
name spaces by hand, yet the following commands provide some way to
do so.
\begin{description}
\item \hyperlink{command.global}{\mbox{\isa{\isacommand{global}}}} and \hyperlink{command.local}{\mbox{\isa{\isacommand{local}}}} change the current
name declaration mode. Initially, theories start in \hyperlink{command.local}{\mbox{\isa{\isacommand{local}}}} mode, causing all names to be automatically qualified by
the theory name. Changing this to \hyperlink{command.global}{\mbox{\isa{\isacommand{global}}}} causes all
names to be declared without the theory prefix, until \hyperlink{command.local}{\mbox{\isa{\isacommand{local}}}} is declared again.
Note that global names are prone to get hidden accidently later,
when qualified names of the same base name are introduced.
\item \hyperlink{command.hide}{\mbox{\isa{\isacommand{hide}}}}~\isa{{\isachardoublequote}space\ names{\isachardoublequote}} fully removes
declarations from a given name space (which may be \isa{{\isachardoublequote}class{\isachardoublequote}},
\isa{{\isachardoublequote}type{\isachardoublequote}}, \isa{{\isachardoublequote}const{\isachardoublequote}}, or \isa{{\isachardoublequote}fact{\isachardoublequote}}); with the \isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}} option, only the base name is hidden. Global
(unqualified) names may never be hidden.
Note that hiding name space accesses has no impact on logical
declarations --- they remain valid internally. Entities that are no
longer accessible to the user are printed with the special qualifier
``\isa{{\isachardoublequote}{\isacharquery}{\isacharquery}{\isachardoublequote}}'' prefixed to the full internal name.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
%
\isadelimtheory
%
\endisadelimtheory
%
\isatagtheory
\isacommand{end}\isamarkupfalse%
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
%
\endisadelimtheory
\isanewline
\end{isabellebody}%
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