(*:maxLineLen=78:*)
theory Spec
imports Base Main
begin
chapter \<open>Specifications\<close>
text \<open>
The Isabelle/Isar theory format integrates specifications and proofs, with
support for interactive development by continuous document editing. There is
a separate 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
\<^theory_text>\<open>theorem\<close> or \<^theory_text>\<open>lemma\<close> at the theory level, and left again with the final
conclusion (e.g.\ via \<^theory_text>\<open>qed\<close>).
\<close>
section \<open>Defining theories \label{sec:begin-thy}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "theory"} & : & \<open>toplevel \<rightarrow> theory\<close> \\
@{command_def (global) "end"} & : & \<open>theory \<rightarrow> toplevel\<close> \\
@{command_def "thy_deps"}\<open>\<^sup>*\<close> & : & \<open>theory \<rightarrow>\<close> \\
\end{matharray}
Isabelle/Isar theories are defined via theory files, which consist of an
outermost sequence of definition--statement--proof elements. Some
definitions are self-sufficient (e.g.\ \<^theory_text>\<open>fun\<close> in Isabelle/HOL), with
foundational proofs performed internally. Other definitions require an
explicit proof as justification (e.g.\ \<^theory_text>\<open>function\<close> and \<^theory_text>\<open>termination\<close> in
Isabelle/HOL). Plain statements like \<^theory_text>\<open>theorem\<close> or \<^theory_text>\<open>lemma\<close> are merely a
special case of that, defining a theorem from a given proposition and its
proof.
The theory body may be sub-structured by means of \<^emph>\<open>local theory targets\<close>,
such as \<^theory_text>\<open>locale\<close> and \<^theory_text>\<open>class\<close>. It is also possible to use \<^theory_text>\<open>context begin \<dots>
end\<close> blocks to delimited a local theory context: a \<^emph>\<open>named context\<close> to
augment a locale or class specification, or an \<^emph>\<open>unnamed context\<close> to refer
to local parameters and assumptions that are discharged later. See
\secref{sec:target} for more details.
\<^medskip>
A theory is commenced by the \<^theory_text>\<open>theory\<close> command, which indicates imports of
previous theories, according to an acyclic foundational order. Before the
initial \<^theory_text>\<open>theory\<close> command, there may be optional document header material
(like \<^theory_text>\<open>section\<close> or \<^theory_text>\<open>text\<close>, see \secref{sec:markup}). The document header
is outside of the formal theory context, though.
A theory is concluded by a final @{command (global) "end"} command, one that
does not belong to a local theory target. No further commands may follow
such a global @{command (global) "end"}.
@{rail \<open>
@@{command theory} @{syntax name} imports? keywords? \<newline> @'begin'
;
imports: @'imports' (@{syntax name} +)
;
keywords: @'keywords' (keyword_decls + @'and')
;
keyword_decls: (@{syntax string} +)
('::' @{syntax name} @{syntax tags})? ('==' @{syntax name})?
;
@@{command thy_deps} (thy_bounds thy_bounds?)?
;
thy_bounds: @{syntax name} | '(' (@{syntax name} + @'|') ')'
\<close>}
\<^descr> \<^theory_text>\<open>theory A imports B\<^sub>1 \<dots> B\<^sub>n begin\<close> starts a new theory \<open>A\<close> based on the
merge of existing theories \<open>B\<^sub>1 \<dots> B\<^sub>n\<close>. Due to the possibility to import
more than one ancestor, the resulting theory structure of an Isabelle
session forms a directed acyclic graph (DAG). Isabelle takes care that
sources contributing to the development graph are always up-to-date: changed
files are automatically rechecked whenever a theory header specification is
processed.
Empty imports are only allowed in the bootstrap process of the special
theory @{theory Pure}, which is the start of any other formal development
based on Isabelle. Regular user theories usually refer to some more complex
entry point, such as theory @{theory Main} in Isabelle/HOL.
The optional @{keyword_def "keywords"} specification declares outer syntax
(\chref{ch:outer-syntax}) that is introduced in this theory later on (rare
in end-user applications). Both minor keywords and major keywords of the
Isar command language need to be specified, in order to make parsing of
proof documents work properly. Command keywords need to be classified
according to their structural role in the formal text. Examples may be seen
in Isabelle/HOL sources itself, such as @{keyword "keywords"}~\<^verbatim>\<open>"typedef"\<close>
\<open>:: thy_goal\<close> or @{keyword "keywords"}~\<^verbatim>\<open>"datatype"\<close> \<open>:: thy_decl\<close> for
theory-level declarations with and without proof, respectively. Additional
@{syntax tags} provide defaults for document preparation
(\secref{sec:tags}).
It is possible to specify an alternative completion via \<^verbatim>\<open>==\<close>~\<open>text\<close>, while
the default is the corresponding keyword name.
\<^descr> @{command (global) "end"} concludes the current theory definition. Note
that some other commands, e.g.\ local theory targets \<^theory_text>\<open>locale\<close> or \<^theory_text>\<open>class\<close>
may involve a \<^theory_text>\<open>begin\<close> that needs to be matched by @{command (local) "end"},
according to the usual rules for nested blocks.
\<^descr> \<^theory_text>\<open>thy_deps\<close> visualizes the theory hierarchy as a directed acyclic graph.
By default, all imported theories are shown, taking the base session as a
starting point. Alternatively, it is possibly to restrict the full theory
graph by giving bounds, analogously to @{command_ref class_deps}.
\<close>
section \<open>Local theory targets \label{sec:target}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "context"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
@{command_def (local) "end"} & : & \<open>local_theory \<rightarrow> theory\<close> \\
@{keyword_def "private"} \\
@{keyword_def "qualified"} \\
\end{matharray}
A local theory target is a specification context that is 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.
\<^emph>\<open>Named contexts\<close> refer to locales (cf.\ \secref{sec:locale}) or type
classes (cf.\ \secref{sec:class}); the name ``\<open>-\<close>'' signifies the global
theory context.
\<^emph>\<open>Unnamed contexts\<close> may introduce additional parameters and assumptions, and
results produced in the context are generalized accordingly. Such auxiliary
contexts may be nested within other targets, like \<^theory_text>\<open>locale\<close>, \<^theory_text>\<open>class\<close>,
\<^theory_text>\<open>instantiation\<close>, \<^theory_text>\<open>overloading\<close>.
@{rail \<open>
@@{command context} @{syntax nameref} @'begin'
;
@@{command context} @{syntax_ref "includes"}? (@{syntax context_elem} * ) @'begin'
;
@{syntax_def target}: '(' @'in' @{syntax nameref} ')'
\<close>}
\<^descr> \<^theory_text>\<open>context c begin\<close> opens a named context, by recommencing an existing
locale or class \<open>c\<close>. Note that locale and class definitions allow to include
the \<^theory_text>\<open>begin\<close> keyword as well, in order to continue the local theory
immediately after the initial specification.
\<^descr> \<^theory_text>\<open>context bundles elements begin\<close> opens an unnamed context, by extending
the enclosing global or local theory target by the given declaration bundles
(\secref{sec:bundle}) and context elements (\<^theory_text>\<open>fixes\<close>, \<^theory_text>\<open>assumes\<close> etc.). This
means any results stemming from definitions and proofs in the extended
context will be exported into the enclosing target by lifting over extra
parameters and premises.
\<^descr> @{command (local) "end"} concludes the current local theory, according to
the nesting of contexts. Note that a global @{command (global) "end"} has a
different meaning: it concludes the theory itself (\secref{sec:begin-thy}).
\<^descr> \<^theory_text>\<open>private\<close> or \<^theory_text>\<open>qualified\<close> may be given as modifiers before any local
theory command. This restricts name space accesses to the local scope, as
determined by the enclosing \<^theory_text>\<open>context begin \<dots> end\<close> block. Outside its scope,
a \<^theory_text>\<open>private\<close> name is inaccessible, and a \<^theory_text>\<open>qualified\<close> name is only
accessible with some qualification.
Neither a global \<^theory_text>\<open>theory\<close> nor a \<^theory_text>\<open>locale\<close> target provides a local scope by
itself: an extra unnamed context is required to use \<^theory_text>\<open>private\<close> or
\<^theory_text>\<open>qualified\<close> here.
\<^descr> \<open>(\<close>@{keyword_def "in"}~\<open>c)\<close> given after any local theory command specifies
an immediate target, e.g.\ ``\<^theory_text>\<open>definition (in c)\<close>'' or
``\<^theory_text>\<open>theorem (in c)\<close>''. This works both in a local or global theory context;
the current target context will be suspended for this command only. Note
that ``\<^theory_text>\<open>(in -)\<close>'' will always produce a global result independently of the
current target context.
Any specification element that operates on \<open>local_theory\<close> according to this
manual implicitly allows the above target syntax \<^theory_text>\<open>(in c)\<close>, but individual
syntax diagrams omit that aspect for clarity.
\<^medskip>
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 \<open>c\<close> with parameter
\<open>x\<close> and assumption \<open>A[x]\<close>.
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, \<open>a \<equiv> t[x]\<close> becomes \<open>c.a ?x \<equiv>
t[?x]\<close> at the theory level (for arbitrary \<open>?x\<close>), together with a local
abbreviation \<open>c \<equiv> c.a x\<close> in the target context (for the fixed parameter
\<open>x\<close>).
Theorems are exported by discharging the assumptions and generalizing the
parameters of the context. For example, \<open>a: B[x]\<close> becomes \<open>c.a: A[?x] \<Longrightarrow>
B[?x]\<close>, again for arbitrary \<open>?x\<close>.
\<close>
section \<open>Bundled declarations \label{sec:bundle}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "bundle"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "print_bundles"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "include"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
@{command_def "including"} & : & \<open>proof(prove) \<rightarrow> proof(prove)\<close> \\
@{keyword_def "includes"} & : & syntax \\
\end{matharray}
The outer syntax of fact expressions (\secref{sec:syn-att}) involves
theorems and attributes, which are evaluated in the context and applied to
it. Attributes may declare theorems to the context, as in \<open>this_rule [intro]
that_rule [elim]\<close> for example. Configuration options (\secref{sec:config})
are special declaration attributes that operate on the context without a
theorem, as in \<open>[[show_types = false]]\<close> for example.
Expressions of this form may be defined as \<^emph>\<open>bundled declarations\<close> in the
context, and included in other situations later on. Including declaration
bundles augments a local context casually without logical dependencies,
which is in contrast to locales and locale interpretation
(\secref{sec:locale}).
@{rail \<open>
@@{command bundle} @{syntax name} '=' @{syntax thmrefs} @{syntax for_fixes}
;
@@{command print_bundles} ('!'?)
;
(@@{command include} | @@{command including}) (@{syntax nameref}+)
;
@{syntax_def "includes"}: @'includes' (@{syntax nameref}+)
\<close>}
\<^descr> \<^theory_text>\<open>bundle b = decls\<close> defines a bundle of declarations in the current
context. The RHS is similar to the one of the \<^theory_text>\<open>declare\<close> command. Bundles
defined in local theory targets are subject to transformations via
morphisms, when moved into different application contexts; this works
analogously to any other local theory specification.
\<^descr> \<^theory_text>\<open>print_bundles\<close> prints the named bundles that are available in the
current context; the ``\<open>!\<close>'' option indicates extra verbosity.
\<^descr> \<^theory_text>\<open>include b\<^sub>1 \<dots> b\<^sub>n\<close> includes the declarations from the given bundles into
the current proof body context. This is analogous to \<^theory_text>\<open>note\<close>
(\secref{sec:proof-facts}) with the expanded bundles.
\<^descr> \<^theory_text>\<open>including\<close> is similar to \<^theory_text>\<open>include\<close>, but works in proof refinement
(backward mode). This is analogous to \<^theory_text>\<open>using\<close> (\secref{sec:proof-facts})
with the expanded bundles.
\<^descr> \<^theory_text>\<open>includes b\<^sub>1 \<dots> b\<^sub>n\<close> is similar to \<^theory_text>\<open>include\<close>, but works in situations
where a specification context is constructed, notably for \<^theory_text>\<open>context\<close> and
long statements of \<^theory_text>\<open>theorem\<close> etc.
Here is an artificial example of bundling various configuration options:
\<close>
(*<*)experiment begin(*>*)
bundle trace = [[simp_trace, linarith_trace, metis_trace, smt_trace]]
lemma "x = x"
including trace by metis
(*<*)end(*>*)
section \<open>Term definitions \label{sec:term-definitions}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "definition"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{attribute_def "defn"} & : & \<open>attribute\<close> \\
@{command_def "print_defn_rules"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "abbreviation"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "print_abbrevs"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
\end{matharray}
Term definitions may either happen within the logic (as equational axioms of
a certain form (see also \secref{sec:overloading}), or outside of it as
rewrite system on abstract syntax. The second form is called
``abbreviation''.
@{rail \<open>
@@{command definition} (decl @'where')? @{syntax thmdecl}? @{syntax prop}
;
@@{command abbreviation} @{syntax mode}? \<newline>
(decl @'where')? @{syntax prop}
;
decl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?
;
@@{command print_abbrevs} ('!'?)
\<close>}
\<^descr> \<^theory_text>\<open>definition c where eq\<close> produces an internal definition \<open>c \<equiv> t\<close> according
to the specification given as \<open>eq\<close>, 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 @{attribute defn} by the
object-logic. This usually covers object-level equality \<open>x = y\<close> and
equivalence \<open>A \<longleftrightarrow> B\<close>. End-users normally need not change the @{attribute
defn} setup.
Definitions may be presented with explicit arguments on the LHS, as well as
additional conditions, e.g.\ \<open>f x y = t\<close> instead of \<open>f \<equiv> \<lambda>x y. t\<close> and \<open>y \<noteq> 0
\<Longrightarrow> g x y = u\<close> instead of an unrestricted \<open>g \<equiv> \<lambda>x y. u\<close>.
\<^descr> \<^theory_text>\<open>print_defn_rules\<close> prints the definitional rewrite rules declared via
@{attribute defn} in the current context.
\<^descr> \<^theory_text>\<open>abbreviation c where eq\<close> introduces a syntactic constant which is
associated with a certain term according to the meta-level equality \<open>eq\<close>.
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 \<open>mode\<close> specification restricts output to a particular print
mode; using ``\<open>input\<close>'' here achieves the effect of one-way abbreviations.
The mode may also include an ``\<^theory_text>\<open>output\<close>'' qualifier that affects the
concrete syntax declared for abbreviations, cf.\ \<^theory_text>\<open>syntax\<close> in
\secref{sec:syn-trans}.
\<^descr> \<^theory_text>\<open>print_abbrevs\<close> prints all constant abbreviations of the current context;
the ``\<open>!\<close>'' option indicates extra verbosity.
\<close>
section \<open>Axiomatizations \label{sec:axiomatizations}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "axiomatization"} & : & \<open>theory \<rightarrow> theory\<close> & (axiomatic!) \\
\end{matharray}
@{rail \<open>
@@{command axiomatization} @{syntax "fixes"}? (@'where' specs)?
;
specs: (@{syntax thmdecl}? @{syntax props} + @'and')
\<close>}
\<^descr> \<^theory_text>\<open>axiomatization c\<^sub>1 \<dots> c\<^sub>m where \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close> 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 for the same constants later on, but it is always possible do
emit axiomatizations without referring to particular constants. Note that
lack of precise dependency tracking of axiomatizations may disrupt the
well-formedness of an otherwise definitional theory.
Axiomatization is restricted to a global theory context: support for local
theory targets \secref{sec:target} would introduce an extra dimension of
uncertainty what the written specifications really are, and make it
infeasible to argue why they are correct.
Axiomatic specifications are required when declaring a new logical system
within Isabelle/Pure, but in an application environment like Isabelle/HOL
the user normally stays within definitional mechanisms provided by the logic
and its libraries.
\<close>
section \<open>Generic declarations\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "declaration"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "syntax_declaration"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "declare"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
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.
@{rail \<open>
(@@{command declaration} | @@{command syntax_declaration})
('(' @'pervasive' ')')? \<newline> @{syntax text}
;
@@{command declare} (@{syntax thmrefs} + @'and')
\<close>}
\<^descr> \<^theory_text>\<open>declaration d\<close> adds the declaration function \<open>d\<close> of ML type @{ML_type
declaration}, to the current local theory under construction. In later
application contexts, the function is transformed according to the morphisms
being involved in the interpretation hierarchy.
If the \<^theory_text>\<open>(pervasive)\<close> option is given, the corresponding declaration is
applied to all possible contexts involved, including the global background
theory.
\<^descr> \<^theory_text>\<open>syntax_declaration\<close> is similar to \<^theory_text>\<open>declaration\<close>, but is meant to affect
only ``syntactic'' tools by convention (such as notation and type-checking
information).
\<^descr> \<^theory_text>\<open>declare thms\<close> declares theorems to the current local theory context. No
theorem binding is involved here, unlike \<^theory_text>\<open>lemmas\<close> (cf.\
\secref{sec:theorems}), so \<^theory_text>\<open>declare\<close> only has the effect of applying
attributes as included in the theorem specification.
\<close>
section \<open>Locales \label{sec:locale}\<close>
text \<open>
A locale is a functor that maps parameters (including implicit type
parameters) and a specification to a list of declarations. The syntax of
locales is modeled after the Isar proof context commands (cf.\
\secref{sec:proof-context}).
Locale hierarchies are supported by maintaining a graph of dependencies
between locale instances in the global theory. Dependencies may be
introduced through import (where a locale is defined as sublocale of the
imported instances) or by proving that an existing locale is a sublocale of
one or several locale instances.
A locale may be opened with the purpose of appending to its list of
declarations (cf.\ \secref{sec:target}). When opening a locale declarations
from all dependencies are collected and are presented as a local theory. In
this process, which is called \<^emph>\<open>roundup\<close>, redundant locale instances are
omitted. A locale instance is redundant if it is subsumed by an instance
encountered earlier. A more detailed description of this process is
available elsewhere @{cite Ballarin2014}.
\<close>
subsection \<open>Locale expressions \label{sec:locale-expr}\<close>
text \<open>
A \<^emph>\<open>locale expression\<close> denotes a context composed of instances of existing
locales. The context consists of the declaration elements from the locale
instances. Redundant locale instances are omitted according to roundup.
@{rail \<open>
@{syntax_def locale_expr}: (instance + '+') @{syntax for_fixes}
;
instance: (qualifier ':')? @{syntax nameref} (pos_insts | named_insts)
;
qualifier: @{syntax name} ('?')?
;
pos_insts: ('_' | @{syntax term})*
;
named_insts: @'where' (@{syntax name} '=' @{syntax term} + @'and')
\<close>}
A locale instance consists of a reference to a locale and either positional
or named parameter instantiations. Identical instantiations (that is, those
that instantiate a parameter by itself) may be omitted. The notation ``\<open>_\<close>''
enables to omit the instantiation for a parameter inside a positional
instantiation.
Terms in instantiations are from the context the locale expressions is
declared in. Local names may be added to this context with the optional
\<^theory_text>\<open>for\<close> clause. This is useful for shadowing names bound in outer contexts,
and for declaring syntax. In addition, syntax declarations from one instance
are effective when parsing subsequent instances of the same expression.
Instances have an optional qualifier which applies to names in declarations.
Names include local definitions and theorem names. If present, the qualifier
itself is either mandatory (default) or non-mandatory (when followed by
``\<^verbatim>\<open>?\<close>''). Non-mandatory means that the qualifier may be omitted on input.
Qualifiers only affect name spaces; they play no role in determining whether
one locale instance subsumes another.
\<close>
subsection \<open>Locale declarations\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "locale"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
@{command_def "experiment"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
@{command_def "print_locale"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "print_locales"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "locale_deps"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{method_def intro_locales} & : & \<open>method\<close> \\
@{method_def unfold_locales} & : & \<open>method\<close> \\
\end{matharray}
\indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}
\indexisarelem{defines}\indexisarelem{notes}
@{rail \<open>
@@{command locale} @{syntax name} ('=' @{syntax locale})? @'begin'?
;
@@{command experiment} (@{syntax context_elem}*) @'begin'
;
@@{command print_locale} '!'? @{syntax nameref}
;
@@{command print_locales} ('!'?)
;
@{syntax_def locale}: @{syntax context_elem}+ |
@{syntax locale_expr} ('+' (@{syntax context_elem}+))?
;
@{syntax_def context_elem}:
@'fixes' @{syntax "fixes"} |
@'constrains' (@{syntax name} '::' @{syntax type} + @'and') |
@'assumes' (@{syntax props} + @'and') |
@'defines' (@{syntax thmdecl}? @{syntax prop} @{syntax prop_pat}? + @'and') |
@'notes' (@{syntax thmdef}? @{syntax thmrefs} + @'and')
\<close>}
\<^descr> \<^theory_text>\<open>locale loc = import + body\<close> defines a new locale \<open>loc\<close> as a context
consisting of a certain view of existing locales (\<open>import\<close>) plus some
additional elements (\<open>body\<close>). Both \<open>import\<close> and \<open>body\<close> are optional; the
degenerate form \<^theory_text>\<open>locale loc\<close> 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 \<open>import\<close> consists of a locale expression; see \secref{sec:proof-context}
above. Its \<^theory_text>\<open>for\<close> clause defines the parameters of \<open>import\<close>. These are
parameters of the defined locale. Locale parameters whose instantiation is
omitted automatically extend the (possibly empty) \<^theory_text>\<open>for\<close> clause: they are
inserted at its beginning. This means that these parameters may be referred
to from within the expression and also in the subsequent context elements
and provides a notational convenience for the inheritance of parameters in
locale declarations.
The \<open>body\<close> consists of context elements.
\<^descr> @{element "fixes"}~\<open>x :: \<tau> (mx)\<close> declares a local parameter of type \<open>\<tau>\<close>
and mixfix annotation \<open>mx\<close> (both are optional). The special syntax
declaration ``\<open>(\<close>@{keyword_ref "structure"}\<open>)\<close>'' means that \<open>x\<close> may be
referenced implicitly in this context.
\<^descr> @{element "constrains"}~\<open>x :: \<tau>\<close> introduces a type constraint \<open>\<tau>\<close> on the
local parameter \<open>x\<close>. This element is deprecated. The type constraint
should be introduced in the \<^theory_text>\<open>for\<close> clause or the relevant @{element
"fixes"} element.
\<^descr> @{element "assumes"}~\<open>a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close> introduces local premises, similar
to \<^theory_text>\<open>assume\<close> within a proof (cf.\ \secref{sec:proof-context}).
\<^descr> @{element "defines"}~\<open>a: x \<equiv> t\<close> defines a previously declared parameter.
This is similar to \<^theory_text>\<open>def\<close> within a proof (cf.\
\secref{sec:proof-context}), but @{element "defines"} takes an equational
proposition instead of variable-term pair. The left-hand side of the
equation may have additional arguments, e.g.\ ``@{element "defines"}~\<open>f
x\<^sub>1 \<dots> x\<^sub>n \<equiv> t\<close>''.
\<^descr> @{element "notes"}~\<open>a = b\<^sub>1 \<dots> b\<^sub>n\<close> reconsiders facts within a local
context. Most notably, this may include arbitrary declarations in any
attribute specifications included here, e.g.\ a local @{attribute simp}
rule.
Both @{element "assumes"} and @{element "defines"} elements contribute to
the locale specification. When defining an operation derived from the
parameters, \<^theory_text>\<open>definition\<close> (\secref{sec:term-definitions}) is usually more
appropriate.
Note that ``\<^theory_text>\<open>(is p\<^sub>1 \<dots> p\<^sub>n)\<close>'' patterns given in the syntax of @{element
"assumes"} and @{element "defines"} above are illegal in locale definitions.
In the long goal format of \secref{sec:goals}, term bindings may be included
as expected, though.
\<^medskip>
Locale specifications are ``closed up'' by turning the given text into a
predicate definition \<open>loc_axioms\<close> 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 \<open>loc\<close> 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 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
\<open>\<And>\<close> by \<open>\<forall>\<close>, and \<open>\<Longrightarrow>\<close> by \<open>\<longrightarrow>\<close> in HOL; see also \secref{sec:object-logic}).
Separate introduction rules \<open>loc_axioms.intro\<close> and \<open>loc.intro\<close> are provided
as well.
\<^descr> \<^theory_text>\<open>experiment exprs begin\<close> opens an anonymous locale context with private
naming policy. Specifications in its body are inaccessible from outside.
This is useful to perform experiments, without polluting the name space.
\<^descr> \<^theory_text>\<open>print_locale locale\<close> prints the contents of the named locale. The
command omits @{element "notes"} elements by default. Use \<^theory_text>\<open>print_locale!\<close>
to have them included.
\<^descr> \<^theory_text>\<open>print_locales\<close> prints the names of all locales of the current theory;
the ``\<open>!\<close>'' option indicates extra verbosity.
\<^descr> \<^theory_text>\<open>locale_deps\<close> visualizes all locales and their relations as a Hasse
diagram. This includes locales defined as type classes (\secref{sec:class}).
See also \<^theory_text>\<open>print_dependencies\<close> below.
\<^descr> @{method intro_locales} and @{method unfold_locales} repeatedly expand all
introduction rules of locale predicates of the theory. While @{method
intro_locales} only applies the \<open>loc.intro\<close> introduction rules and therefore
does not descend to assumptions, @{method unfold_locales} is more aggressive
and applies \<open>loc_axioms.intro\<close> 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.
\<close>
subsection \<open>Locale interpretation\<close>
text \<open>
\begin{matharray}{rcl}
@{command "interpretation"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
@{command_def "interpret"} & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
@{command_def "global_interpretation"} & : & \<open>theory | local_theory \<rightarrow> proof(prove)\<close> \\
@{command_def "sublocale"} & : & \<open>theory | local_theory \<rightarrow> proof(prove)\<close> \\
@{command_def "print_dependencies"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "print_interps"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
\end{matharray}
Locales may be instantiated, and the resulting instantiated declarations
added to the current context. This requires proof (of the instantiated
specification) and is called \<^emph>\<open>locale interpretation\<close>. Interpretation is
possible within arbitrary local theories (\<^theory_text>\<open>interpretation\<close>), within proof
bodies (\<^theory_text>\<open>interpret\<close>), into global theories (\<^theory_text>\<open>global_interpretation\<close>) and
into locales (\<^theory_text>\<open>sublocale\<close>).
@{rail \<open>
@@{command interpretation} @{syntax locale_expr} equations?
;
@@{command interpret} @{syntax locale_expr} equations?
;
@@{command global_interpretation} @{syntax locale_expr} \<newline>
definitions? equations?
;
@@{command sublocale} (@{syntax nameref} ('<' | '\<subseteq>'))? @{syntax locale_expr} \<newline>
definitions? equations?
;
@@{command print_dependencies} '!'? @{syntax locale_expr}
;
@@{command print_interps} @{syntax nameref}
;
definitions: @'defines' (@{syntax thmdecl}? @{syntax name} \<newline>
@{syntax mixfix}? @'=' @{syntax term} + @'and');
equations: @'rewrites' (@{syntax thmdecl}? @{syntax prop} + @'and')
\<close>}
The core of each interpretation command is a locale expression \<open>expr\<close>; the
command generates proof obligations for the instantiated specifications.
Once these are discharged by the user, instantiated declarations (in
particular, facts) are added to the context in a post-processing phase, in a
manner specific to each command.
Interpretation commands are aware of interpretations that are already
active: post-processing is achieved through a variant of roundup that takes
interpretations of the current global or local theory into account. In order
to simplify the proof obligations according to existing interpretations use
methods @{method intro_locales} or @{method unfold_locales}.
Given equations \<open>eqns\<close> amend the morphism through which \<open>expr\<close> is
interpreted, adding rewrite rules. This is particularly useful for
interpreting concepts introduced through definitions. The equations must be
proved the user.
Given definitions \<open>defs\<close> produce corresponding definitions in the local
theory's underlying target \<^emph>\<open>and\<close> amend the morphism with the equations
stemming from the symmetric of those definitions. Hence these need not be
proved explicitly the user. Such rewrite definitions are a even more useful
device for interpreting concepts introduced through definitions, but they
are only supported for interpretation commands operating in a local theory
whose implementing target actually supports this. Note that despite
the suggestive \<^theory_text>\<open>and\<close> connective, \<open>defs\<close>
are processed sequentially without mutual recursion.
\<^descr> \<^theory_text>\<open>interpretation expr rewrites eqns\<close> interprets \<open>expr\<close> into a local theory
such that its lifetime is limited to the current context block (e.g. a
locale or unnamed context). At the closing @{command end} of the block the
interpretation and its declarations disappear. Hence facts based on
interpretation can be established without creating permanent links to the
interpreted locale instances, as would be the case with @{command
sublocale}.
When used on the level of a global theory, there is no end of a current
context block, hence \<^theory_text>\<open>interpretation\<close> behaves identically to
\<^theory_text>\<open>global_interpretation\<close> then.
\<^descr> \<^theory_text>\<open>interpret expr rewrites eqns\<close> interprets \<open>expr\<close> into a proof context:
the interpretation and its declarations disappear when closing the current
proof block. Note that for \<^theory_text>\<open>interpret\<close> the \<open>eqns\<close> should be explicitly
universally quantified.
\<^descr> \<^theory_text>\<open>global_interpretation defines defs rewrites eqns\<close> interprets \<open>expr\<close>
into a global theory.
When adding declarations to locales, interpreted versions of these
declarations are added to the global theory for all interpretations in the
global theory as well. That is, interpretations into global theories
dynamically participate in any declarations added to locales.
Free variables in the interpreted expression are allowed. They are turned
into schematic variables in the generated declarations. In order to use a
free variable whose name is already bound in the context --- for example,
because a constant of that name exists --- add it to the \<^theory_text>\<open>for\<close> clause.
\<^descr> \<^theory_text>\<open>sublocale name \<subseteq> defines defs expr rewrites eqns\<close> interprets \<open>expr\<close>
into the locale \<open>name\<close>. A proof that the specification of \<open>name\<close> implies the
specification of \<open>expr\<close> is required. As in the localized version of the
theorem command, the proof is in the context of \<open>name\<close>. After the proof
obligation has been discharged, the locale hierarchy is changed as if \<open>name\<close>
imported \<open>expr\<close> (hence the name \<^theory_text>\<open>sublocale\<close>). When the context of \<open>name\<close> is
subsequently entered, traversing the locale hierarchy will involve the
locale instances of \<open>expr\<close>, and their declarations will be added to the
context. This makes \<^theory_text>\<open>sublocale\<close> dynamic: extensions of a locale that is
instantiated in \<open>expr\<close> may take place after the \<^theory_text>\<open>sublocale\<close> declaration and
still become available in the context. Circular \<^theory_text>\<open>sublocale\<close> declarations
are allowed as long as they do not lead to infinite chains.
If interpretations of \<open>name\<close> exist in the current global theory, the command
adds interpretations for \<open>expr\<close> as well, with the same qualifier, although
only for fragments of \<open>expr\<close> that are not interpreted in the theory already.
Using equations \<open>eqns\<close> or rewrite definitions \<open>defs\<close> can help break infinite
chains induced by circular \<^theory_text>\<open>sublocale\<close> declarations.
In a named context block the \<^theory_text>\<open>sublocale\<close> command may also be used, but the
locale argument must be omitted. The command then refers to the locale (or
class) target of the context block.
\<^descr> \<^theory_text>\<open>print_dependencies expr\<close> is useful for understanding the effect of an
interpretation of \<open>expr\<close> in the current context. It lists all locale
instances for which interpretations would be added to the current context.
Variant \<^theory_text>\<open>print_dependencies!\<close> does not generalize parameters and assumes an
empty context --- that is, it prints all locale instances that would be
considered for interpretation. The latter is useful for understanding the
dependencies of a locale expression.
\<^descr> \<^theory_text>\<open>print_interps locale\<close> lists all interpretations of \<open>locale\<close> in the
current theory or proof context, including those due to a combination of an
\<^theory_text>\<open>interpretation\<close> or \<^theory_text>\<open>interpret\<close> and one or several \<^theory_text>\<open>sublocale\<close>
declarations.
\begin{warn}
If a global theory inherits declarations (body elements) for a locale from
one parent and an interpretation of that locale from another parent, the
interpretation will not be applied to the declarations.
\end{warn}
\begin{warn}
Since attributes are applied to interpreted theorems, interpretation may
modify the context of common proof tools, e.g.\ the Simplifier or
Classical Reasoner. As the behaviour of such tools is \<^emph>\<open>not\<close> stable under
interpretation morphisms, manual declarations might have to be added to
the target context of the interpretation to revert such declarations.
\end{warn}
\begin{warn}
An interpretation in a local theory or proof context may subsume previous
interpretations. This happens if the same specification fragment is
interpreted twice and the instantiation of the second interpretation is
more general than the interpretation of the first. The locale package does
not attempt to remove subsumed interpretations.
\end{warn}
\begin{warn}
Due to a technical limitation, the specific context of a interpretation
given by a \<^theory_text>\<open>for\<close> clause can get lost between a \<^theory_text>\<open>defines\<close> and
\<^theory_text>\<open>rewrites\<close> clause and must then be recovered manually using
explicit sort constraints and quantified term variables.
\end{warn}
\begin{warn}
While \<^theory_text>\<open>interpretation (in c) \<dots>\<close> is admissible, it is not useful since
its result is discarded immediately.
\end{warn}
\<close>
section \<open>Classes \label{sec:class}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "class"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
@{command_def "instantiation"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
@{command_def "instance"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command "instance"} & : & \<open>theory \<rightarrow> proof(prove)\<close> \\
@{command_def "subclass"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "print_classes"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def "class_deps"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{method_def intro_classes} & : & \<open>method\<close> \\
\end{matharray}
A class is a particular locale with \<^emph>\<open>exactly one\<close> type variable \<open>\<alpha>\<close>. 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.
@{rail \<open>
@@{command class} class_spec @'begin'?
;
class_spec: @{syntax name} '='
((@{syntax nameref} '+' (@{syntax context_elem}+)) |
@{syntax nameref} | (@{syntax context_elem}+))
;
@@{command instantiation} (@{syntax nameref} + @'and') '::' @{syntax arity} @'begin'
;
@@{command instance} (() | (@{syntax nameref} + @'and') '::' @{syntax arity} |
@{syntax nameref} ('<' | '\<subseteq>') @{syntax nameref} )
;
@@{command subclass} @{syntax nameref}
;
@@{command class_deps} (class_bounds class_bounds?)?
;
class_bounds: @{syntax sort} | '(' (@{syntax sort} + @'|') ')'
\<close>}
\<^descr> \<^theory_text>\<open>class c = superclasses + body\<close> defines a new class \<open>c\<close>, inheriting from
\<open>superclasses\<close>. This introduces a locale \<open>c\<close> with import of all locales
\<open>superclasses\<close>.
Any @{element "fixes"} in \<open>body\<close> are lifted to the global theory level
(\<^emph>\<open>class operations\<close> \<open>f\<^sub>1, \<dots>, f\<^sub>n\<close> of class \<open>c\<close>), mapping the local type
parameter \<open>\<alpha>\<close> to a schematic type variable \<open>?\<alpha> :: c\<close>.
Likewise, @{element "assumes"} in \<open>body\<close> are also lifted, mapping each local
parameter \<open>f :: \<tau>[\<alpha>]\<close> to its corresponding global constant \<open>f :: \<tau>[?\<alpha> ::
c]\<close>. The corresponding introduction rule is provided as
\<open>c_class_axioms.intro\<close>. This rule should be rarely needed directly --- the
@{method intro_classes} method takes care of the details of class membership
proofs.
\<^descr> \<^theory_text>\<open>instantiation t :: (s\<^sub>1, \<dots>, s\<^sub>n)s begin\<close> opens a target (cf.\
\secref{sec:target}) which allows to specify class operations \<open>f\<^sub>1, \<dots>, f\<^sub>n\<close>
corresponding to sort \<open>s\<close> at the particular type instance \<open>(\<alpha>\<^sub>1 :: s\<^sub>1, \<dots>,
\<alpha>\<^sub>n :: s\<^sub>n) t\<close>. A plain \<^theory_text>\<open>instance\<close> command in the target body poses a goal
stating these type arities. The target is concluded by an @{command_ref
(local) "end"} command.
Note that a list of simultaneous type constructors may be given; this
corresponds nicely to mutually recursive type definitions, e.g.\ in
Isabelle/HOL.
\<^descr> \<^theory_text>\<open>instance\<close> in an instantiation target body sets up a goal stating the
type arities claimed at the opening \<^theory_text>\<open>instantiation\<close>. The proof would
usually proceed by @{method intro_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.
On the theory level, \<^theory_text>\<open>instance t :: (s\<^sub>1, \<dots>, s\<^sub>n)s\<close> provides a convenient
way to instantiate a type class with no need to specify operations: one can
continue with the instantiation proof immediately.
\<^descr> \<^theory_text>\<open>subclass c\<close> in a class context for class \<open>d\<close> sets up a goal stating that
class \<open>c\<close> is logically contained in class \<open>d\<close>. After finishing the proof,
class \<open>d\<close> is proven to be subclass \<open>c\<close> and the locale \<open>c\<close> is interpreted
into \<open>d\<close> simultaneously.
A weakened form of this is available through a further variant of @{command
instance}: \<^theory_text>\<open>instance c\<^sub>1 \<subseteq> c\<^sub>2\<close> opens a proof that class \<open>c\<^sub>2\<close> implies
\<open>c\<^sub>1\<close> without reference to the underlying locales; this is useful if the
properties to prove the logical connection are not sufficient on the locale
level but on the theory level.
\<^descr> \<^theory_text>\<open>print_classes\<close> prints all classes in the current theory.
\<^descr> \<^theory_text>\<open>class_deps\<close> visualizes classes and their subclass relations as a
directed acyclic graph. By default, all classes from the current theory
context are show. This may be restricted by optional bounds as follows:
\<^theory_text>\<open>class_deps upper\<close> or \<^theory_text>\<open>class_deps upper lower\<close>. A class is visualized, iff
it is a subclass of some sort from \<open>upper\<close> and a superclass of some sort
from \<open>lower\<close>.
\<^descr> @{method intro_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 \<^theory_text>\<open>proof\<close>). In
particular, instantiation of trivial (syntactic) classes may be performed by
a single ``\<^theory_text>\<open>..\<close>'' proof step.
\<close>
subsection \<open>The class target\<close>
text \<open>
%FIXME check
A named context may refer to a locale (cf.\ \secref{sec:target}). If this
locale is also a class \<open>c\<close>, apart from the common locale target behaviour
the following happens.
\<^item> Local constant declarations \<open>g[\<alpha>]\<close> referring to the local type parameter
\<open>\<alpha>\<close> and local parameters \<open>f[\<alpha>]\<close> are accompanied by theory-level constants
\<open>g[?\<alpha> :: c]\<close> referring to theory-level class operations \<open>f[?\<alpha> :: c]\<close>.
\<^item> Local theorem bindings are lifted as are assumptions.
\<^item> Local syntax refers to local operations \<open>g[\<alpha>]\<close> and global operations
\<open>g[?\<alpha> :: c]\<close> uniformly. Type inference resolves ambiguities. In rare
cases, manual type annotations are needed.
\<close>
subsection \<open>Co-regularity of type classes and arities\<close>
text \<open>
The class relation together with the collection of type-constructor arities
must obey the principle of \<^emph>\<open>co-regularity\<close> 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 \<open>t :: (\<^vec>s)c\<close> is completed such that \<open>t :: (\<^vec>s)c\<close> and
\<open>c \<subseteq> c'\<close> implies \<open>t :: (\<^vec>s)c'\<close> for all such declarations.
Treating sorts as finite sets of classes (meaning the intersection), the
class relation \<open>c\<^sub>1 \<subseteq> c\<^sub>2\<close> is extended to sorts as follows:
\[
\<open>s\<^sub>1 \<subseteq> s\<^sub>2 \<equiv> \<forall>c\<^sub>2 \<in> s\<^sub>2. \<exists>c\<^sub>1 \<in> s\<^sub>1. c\<^sub>1 \<subseteq> c\<^sub>2\<close>
\]
This relation on sorts is further extended to tuples of sorts (of the same
length) in the component-wise way.
\<^medskip>
Co-regularity of the class relation together with the arities relation
means:
\[
\<open>t :: (\<^vec>s\<^sub>1)c\<^sub>1 \<Longrightarrow> t :: (\<^vec>s\<^sub>2)c\<^sub>2 \<Longrightarrow> c\<^sub>1 \<subseteq> c\<^sub>2 \<Longrightarrow> \<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2\<close>
\]
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"}.
\<close>
section \<open>Overloaded constant definitions \label{sec:overloading}\<close>
text \<open>
Definitions essentially express abbreviations within the logic. The simplest
form of a definition is \<open>c :: \<sigma> \<equiv> t\<close>, where \<open>c\<close> is a new constant and \<open>t\<close> is
a closed term that does not mention \<open>c\<close>. Moreover, so-called \<^emph>\<open>hidden
polymorphism\<close> is excluded: all type variables in \<open>t\<close> need to occur in its
type \<open>\<sigma>\<close>.
\<^emph>\<open>Overloading\<close> means that a constant being declared as \<open>c :: \<alpha> decl\<close> may be
defined separately on type instances \<open>c :: (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n)\<kappa> decl\<close> for each
type constructor \<open>\<kappa>\<close>. At most occasions overloading will be used in a
Haskell-like fashion together with type classes by means of \<^theory_text>\<open>instantiation\<close>
(see \secref{sec:class}). Sometimes low-level overloading is desirable; this
is supported by \<^theory_text>\<open>consts\<close> and \<^theory_text>\<open>overloading\<close> explained below.
The right-hand side of overloaded definitions may mention overloaded
constants recursively at type instances corresponding to the immediate
argument types \<open>\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n\<close>. Incomplete specification patterns impose
global constraints on all occurrences. E.g.\ \<open>d :: \<alpha> \<times> \<alpha>\<close> on the left-hand
side means that all corresponding occurrences on some right-hand side need
to be an instance of this, and general \<open>d :: \<alpha> \<times> \<beta>\<close> will be disallowed. Full
details are given by Kun\v{c}ar @{cite "Kuncar:2015"}.
\<^medskip>
The \<^theory_text>\<open>consts\<close> command and the \<^theory_text>\<open>overloading\<close> target provide a convenient
interface for end-users. Regular specification elements such as @{command
definition}, @{command inductive}, @{command function} may be used in the
body. It is also possible to use \<^theory_text>\<open>consts c :: \<sigma>\<close> with later \<^theory_text>\<open>overloading c
\<equiv> c :: \<sigma>\<close> to keep the declaration and definition of a constant separate.
\begin{matharray}{rcl}
@{command_def "consts"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def "overloading"} & : & \<open>theory \<rightarrow> local_theory\<close> \\
\end{matharray}
@{rail \<open>
@@{command consts} ((@{syntax name} '::' @{syntax type} @{syntax mixfix}?) +)
;
@@{command overloading} ( spec + ) @'begin'
;
spec: @{syntax name} ( '\<equiv>' | '==' ) @{syntax term} ( '(' @'unchecked' ')' )?
\<close>}
\<^descr> \<^theory_text>\<open>consts c :: \<sigma>\<close> declares constant \<open>c\<close> to have any instance of type scheme
\<open>\<sigma>\<close>. The optional mixfix annotations may attach concrete syntax to the
constants declared.
\<^descr> \<^theory_text>\<open>overloading x\<^sub>1 \<equiv> c\<^sub>1 :: \<tau>\<^sub>1 \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n begin \<dots> end\<close> defines
a theory target (cf.\ \secref{sec:target}) which allows to specify already
declared constants via definitions in the body. These are identified by an
explicitly given mapping from variable names \<open>x\<^sub>i\<close> to constants \<open>c\<^sub>i\<close> at
particular type instances. The definitions themselves are established using
common specification tools, using the names \<open>x\<^sub>i\<close> as reference to the
corresponding constants.
Option \<^theory_text>\<open>(unchecked)\<close> disables global dependency checks for the
corresponding definition, which is occasionally useful for exotic
overloading; this is a form of axiomatic specification. It is at the
discretion of the user to avoid malformed theory specifications!
\<close>
subsubsection \<open>Example\<close>
consts Length :: "'a \<Rightarrow> nat"
overloading
Length\<^sub>0 \<equiv> "Length :: unit \<Rightarrow> nat"
Length\<^sub>1 \<equiv> "Length :: 'a \<times> unit \<Rightarrow> nat"
Length\<^sub>2 \<equiv> "Length :: 'a \<times> 'b \<times> unit \<Rightarrow> nat"
Length\<^sub>3 \<equiv> "Length :: 'a \<times> 'b \<times> 'c \<times> unit \<Rightarrow> nat"
begin
fun Length\<^sub>0 :: "unit \<Rightarrow> nat" where "Length\<^sub>0 () = 0"
fun Length\<^sub>1 :: "'a \<times> unit \<Rightarrow> nat" where "Length\<^sub>1 (a, ()) = 1"
fun Length\<^sub>2 :: "'a \<times> 'b \<times> unit \<Rightarrow> nat" where "Length\<^sub>2 (a, b, ()) = 2"
fun Length\<^sub>3 :: "'a \<times> 'b \<times> 'c \<times> unit \<Rightarrow> nat" where "Length\<^sub>3 (a, b, c, ()) = 3"
end
lemma "Length (a, b, c, ()) = 3" by simp
lemma "Length ((a, b), (c, d), ()) = 2" by simp
lemma "Length ((a, b, c, d, e), ()) = 1" by simp
section \<open>Incorporating ML code \label{sec:ML}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "SML_file"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "SML_file_debug"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "SML_file_no_debug"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "ML_file"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "ML_file_debug"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "ML_file_no_debug"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "ML"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "ML_prf"} & : & \<open>proof \<rightarrow> proof\<close> \\
@{command_def "ML_val"} & : & \<open>any \<rightarrow>\<close> \\
@{command_def "ML_command"} & : & \<open>any \<rightarrow>\<close> \\
@{command_def "setup"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def "local_setup"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "attribute_setup"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
\begin{tabular}{rcll}
@{attribute_def ML_print_depth} & : & \<open>attribute\<close> & default 10 \\
@{attribute_def ML_source_trace} & : & \<open>attribute\<close> & default \<open>false\<close> \\
@{attribute_def ML_debugger} & : & \<open>attribute\<close> & default \<open>false\<close> \\
@{attribute_def ML_exception_trace} & : & \<open>attribute\<close> & default \<open>false\<close> \\
@{attribute_def ML_exception_debugger} & : & \<open>attribute\<close> & default \<open>false\<close> \\
\end{tabular}
@{rail \<open>
(@@{command SML_file} |
@@{command SML_file_debug} |
@@{command SML_file_no_debug} |
@@{command ML_file} |
@@{command ML_file_debug} |
@@{command ML_file_no_debug}) @{syntax name} ';'?
;
(@@{command ML} | @@{command ML_prf} | @@{command ML_val} |
@@{command ML_command} | @@{command setup} | @@{command local_setup}) @{syntax text}
;
@@{command attribute_setup} @{syntax name} '=' @{syntax text} @{syntax text}?
\<close>}
\<^descr> \<^theory_text>\<open>SML_file name\<close> reads and evaluates the given Standard ML file. Top-level
SML bindings are stored within the (global or local) theory context; the
initial environment is restricted to the Standard ML implementation of
Poly/ML, without the many add-ons of Isabelle/ML. Multiple \<^theory_text>\<open>SML_file\<close>
commands may be used to build larger Standard ML projects, independently of
the regular Isabelle/ML environment.
\<^descr> \<^theory_text>\<open>ML_file name\<close> reads and evaluates the given ML file. The current theory
context is passed down to the ML toplevel and may be modified, using @{ML
"Context.>>"} or derived ML commands. Top-level ML bindings are stored
within the (global or local) theory context.
\<^descr> \<^theory_text>\<open>SML_file_debug\<close>, \<^theory_text>\<open>SML_file_no_debug\<close>, \<^theory_text>\<open>ML_file_debug\<close>, and
\<^theory_text>\<open>ML_file_no_debug\<close> change the @{attribute ML_debugger} option locally while
the given file is compiled.
\<^descr> \<^theory_text>\<open>ML text\<close> is similar to \<^theory_text>\<open>ML_file\<close>, but evaluates directly the given
\<open>text\<close>. Top-level ML bindings are stored within the (global or local) theory
context.
\<^descr> \<^theory_text>\<open>ML_prf\<close> is analogous to \<^theory_text>\<open>ML\<close> 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 \<^theory_text>\<open>ML_prf\<close> are discarded at the end of the proof.
\<^descr> \<^theory_text>\<open>ML_val\<close> and \<^theory_text>\<open>ML_command\<close> are diagnostic versions of \<^theory_text>\<open>ML\<close>, which means
that the context may not be updated. \<^theory_text>\<open>ML_val\<close> echos the bindings produced
at the ML toplevel, but \<^theory_text>\<open>ML_command\<close> is silent.
\<^descr> \<^theory_text>\<open>setup "text"\<close> changes the current theory context by applying \<open>text\<close>,
which refers to an ML expression of type @{ML_type "theory -> theory"}. This
enables to initialize any object-logic specific tools and packages written
in ML, for example.
\<^descr> \<^theory_text>\<open>local_setup\<close> is similar to \<^theory_text>\<open>setup\<close> for a local theory context, and an
ML expression of type @{ML_type "local_theory -> local_theory"}. This allows
to invoke local theory specification packages without going through concrete
outer syntax, for example.
\<^descr> \<^theory_text>\<open>attribute_setup name = "text" description\<close> defines an attribute in the
current context. The given \<open>text\<close> has to be an ML expression of type
@{ML_type "attribute context_parser"}, cf.\ basic parsers defined in
structure @{ML_structure Args} and @{ML_structure Attrib}.
In principle, attributes can operate both on a given theorem and the
implicit context, although in practice only one is modified and the other
serves as parameter. Here are examples for these two cases:
\<close>
(*<*)experiment begin(*>*)
attribute_setup my_rule =
\<open>Attrib.thms >> (fn ths =>
Thm.rule_attribute ths
(fn context: Context.generic => fn th: thm =>
let val th' = th OF ths
in th' end))\<close>
attribute_setup my_declaration =
\<open>Attrib.thms >> (fn ths =>
Thm.declaration_attribute
(fn th: thm => fn context: Context.generic =>
let val context' = context
in context' end))\<close>
(*<*)end(*>*)
text \<open>
\<^descr> @{attribute ML_print_depth} controls the printing depth of the ML toplevel
pretty printer. Typically the limit should be less than 10. Bigger values
such as 100--1000 are occasionally useful for debugging.
\<^descr> @{attribute ML_source_trace} indicates whether the source text that is
given to the ML compiler should be output: it shows the raw Standard ML
after expansion of Isabelle/ML antiquotations.
\<^descr> @{attribute ML_debugger} controls compilation of sources with or without
debugging information. The global system option @{system_option_ref
ML_debugger} does the same when building a session image. It is also
possible use commands like \<^theory_text>\<open>ML_file_debug\<close> etc. The ML debugger is
explained further in @{cite "isabelle-jedit"}.
\<^descr> @{attribute ML_exception_trace} indicates whether the ML run-time system
should print a detailed stack trace on exceptions. The result is dependent
on various ML compiler optimizations. The boundary for the exception trace
is the current Isar command transactions: it is occasionally better to
insert the combinator @{ML Runtime.exn_trace} into ML code for debugging
@{cite "isabelle-implementation"}, closer to the point where it actually
happens.
\<^descr> @{attribute ML_exception_debugger} controls detailed exception trace via
the Poly/ML debugger, at the cost of extra compile-time and run-time
overhead. Relevant ML modules need to be compiled beforehand with debugging
enabled, see @{attribute ML_debugger} above.
\<close>
section \<open>Primitive specification elements\<close>
subsection \<open>Sorts\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "default_sort"} & : & \<open>local_theory \<rightarrow> local_theory\<close>
\end{matharray}
@{rail \<open>
@@{command default_sort} @{syntax sort}
\<close>}
\<^descr> \<^theory_text>\<open>default_sort s\<close> makes sort \<open>s\<close> 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 @{class type}, the class of
all HOL types.
When merging theories, the default sorts of the parents are logically
intersected, i.e.\ the representations as lists of classes are joined.
\<close>
subsection \<open>Types \label{sec:types-pure}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "type_synonym"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "typedecl"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
@{rail \<open>
@@{command type_synonym} (@{syntax typespec} '=' @{syntax type} @{syntax mixfix}?)
;
@@{command typedecl} @{syntax typespec} @{syntax mixfix}?
\<close>}
\<^descr> \<^theory_text>\<open>type_synonym (\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = \<tau>\<close> introduces a \<^emph>\<open>type synonym\<close> \<open>(\<alpha>\<^sub>1, \<dots>,
\<alpha>\<^sub>n) t\<close> for the existing type \<open>\<tau>\<close>. Unlike the semantic type definitions in
Isabelle/HOL, type synonyms are merely syntactic abbreviations without any
logical significance. Internally, type synonyms are fully expanded.
\<^descr> \<^theory_text>\<open>typedecl (\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t\<close> declares a new type constructor \<open>t\<close>. If the
object-logic defines a base sort \<open>s\<close>, then the constructor is declared to
operate on that, via the axiomatic type-class instance \<open>t :: (s, \<dots>, s)s\<close>.
\begin{warn}
If you introduce a new type axiomatically, i.e.\ via @{command_ref
typedecl} and @{command_ref axiomatization}
(\secref{sec:axiomatizations}), the minimum requirement is that it has a
non-empty model, to avoid immediate collapse of the logical environment.
Moreover, one needs to demonstrate that the interpretation of such
free-form axiomatizations can coexist with other axiomatization schemes
for types, notably @{command_def typedef} in Isabelle/HOL
(\secref{sec:hol-typedef}), or any other extension that people might have
introduced elsewhere.
\end{warn}
\<close>
section \<open>Naming existing theorems \label{sec:theorems}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "lemmas"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "named_theorems"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
@{rail \<open>
@@{command lemmas} (@{syntax thmdef}? @{syntax thmrefs} + @'and')
@{syntax for_fixes}
;
@@{command named_theorems} (@{syntax name} @{syntax text}? + @'and')
\<close>}
\<^descr> \<^theory_text>\<open>lemmas a = b\<^sub>1 \<dots> b\<^sub>n\<close>~@{keyword_def "for"}~\<open>x\<^sub>1 \<dots> x\<^sub>m\<close> evaluates given
facts (with attributes) in the current context, which may be augmented by
local variables. Results are standardized before being stored, i.e.\
schematic variables are renamed to enforce index \<open>0\<close> uniformly.
\<^descr> \<^theory_text>\<open>named_theorems name description\<close> declares a dynamic fact within the
context. The same \<open>name\<close> is used to define an attribute with the usual
\<open>add\<close>/\<open>del\<close> syntax (e.g.\ see \secref{sec:simp-rules}) to maintain the
content incrementally, in canonical declaration order of the text structure.
\<close>
section \<open>Oracles\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "oracle"} & : & \<open>theory \<rightarrow> theory\<close> & (axiomatic!) \\
\end{matharray}
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.
@{rail \<open>
@@{command oracle} @{syntax name} '=' @{syntax text}
\<close>}
\<^descr> \<^theory_text>\<open>oracle name = "text"\<close> turns the given ML expression \<open>text\<close> of type
@{ML_text "'a -> cterm"} into an ML function of type @{ML_text "'a -> thm"},
which is bound to the global identifier @{ML_text name}. This acts like an
infinitary specification of axioms! Invoking the oracle only works within
the scope of the resulting theory.
See @{file "~~/src/HOL/ex/Iff_Oracle.thy"} for a worked example of defining
a new primitive rule as oracle, and turning it into a proof method.
\<close>
section \<open>Name spaces\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "hide_class"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def "hide_type"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def "hide_const"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def "hide_fact"} & : & \<open>theory \<rightarrow> theory\<close> \\
\end{matharray}
@{rail \<open>
( @{command hide_class} | @{command hide_type} |
@{command hide_const} | @{command hide_fact} ) ('(' @'open' ')')? (@{syntax nameref} + )
\<close>}
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 behaviour of name spaces by hand, yet
the following commands provide some way to do so.
\<^descr> \<^theory_text>\<open>hide_class names\<close> fully removes class declarations from a given name
space; with the \<open>(open)\<close> option, only the unqualified base name is 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 ``\<open>??\<close>'' prefixed to the
full internal name.
\<^descr> \<^theory_text>\<open>hide_type\<close>, \<^theory_text>\<open>hide_const\<close>, and \<^theory_text>\<open>hide_fact\<close> are similar to
\<^theory_text>\<open>hide_class\<close>, but hide types, constants, and facts, respectively.
\<close>
end