support for 'restricted' modifier: only qualified accesses outside the local scope;
theory Spec
imports Base Main "~~/src/Tools/Permanent_Interpretation"
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 @{command theorem} or @{command lemma} at the theory
level, and left again with the final conclusion (e.g.\ via @{command
qed}).
\<close>
section \<open>Defining theories \label{sec:begin-thy}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "theory"} & : & @{text "toplevel \<rightarrow> theory"} \\
@{command_def (global) "end"} & : & @{text "theory \<rightarrow> toplevel"} \\
\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.\ @{command fun} in Isabelle/HOL),
with foundational proofs performed internally. Other definitions require
an explicit proof as justification (e.g.\ @{command function} and
@{command termination} in Isabelle/HOL). Plain statements like @{command
theorem} or @{command lemma} 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{local theory
targets}, such as @{command "locale"} and @{command "class"}. It is also
possible to use @{command context}~@{keyword "begin"}~\dots~@{command end}
blocks to delimited a local theory context: a \emph{named context} to
augment a locale or class specification, or an \emph{unnamed context} 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 @{command "theory"} command, which
indicates imports of previous theories, according to an acyclic
foundational order. Before the initial @{command "theory"} command, there
may be optional document header material (like @{command section} or
@{command text}, 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})?
\<close>}
\begin{description}
\item @{command "theory"}~@{text "A \<IMPORTS> B\<^sub>1 \<dots> B\<^sub>n \<BEGIN>"}
starts a new theory @{text A} based on the merge of existing
theories @{text "B\<^sub>1 \<dots> B\<^sub>n"}. 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>}
@{text ":: thy_goal"} or @{keyword "keywords"}~@{verbatim
\<open>"datatype"\<close>} @{text ":: thy_decl"} 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
"=="}~@{text "text"}, while the default is the corresponding keyword name.
\item @{command (global) "end"} concludes the current theory
definition. Note that some other commands, e.g.\ local theory
targets @{command locale} or @{command class} may involve a
@{keyword "begin"} that needs to be matched by @{command (local)
"end"}, according to the usual rules for nested blocks.
\end{description}
\<close>
section \<open>Local theory targets \label{sec:target}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "context"} & : & @{text "theory \<rightarrow> local_theory"} \\
@{command_def (local) "end"} & : & @{text "local_theory \<rightarrow> theory"} \\
@{keyword_def "private"} \\
@{keyword_def "restricted"} \\
\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{Named contexts} refer to locales (cf.\ \secref{sec:locale}) or
type classes (cf.\ \secref{sec:class}); the name ``@{text "-"}''
signifies the global theory context.
\emph{Unnamed contexts} 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 @{command "locale"}, @{command "class"}, @{command
"instantiation"}, @{command "overloading"}.
@{rail \<open>
@@{command context} @{syntax nameref} @'begin'
;
@@{command context} @{syntax_ref "includes"}? (@{syntax context_elem} * ) @'begin'
;
@{syntax_def target}: '(' @'in' @{syntax nameref} ')'
\<close>}
\begin{description}
\item @{command "context"}~@{text "c \<BEGIN>"} opens a named
context, by recommencing an existing locale or class @{text c}.
Note that locale and class definitions allow to include the
@{keyword "begin"} keyword as well, in order to continue the local
theory immediately after the initial specification.
\item @{command "context"}~@{text "bundles elements \<BEGIN>"} opens
an unnamed context, by extending the enclosing global or local
theory target by the given declaration bundles (\secref{sec:bundle})
and context elements (@{text "\<FIXES>"}, @{text "\<ASSUMES>"}
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.
\item @{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}).
\item @{keyword "private"} or @{keyword "restricted"} may be given as
modifiers before any local theory command. This limits name space accesses
to the local scope, as determined by the enclosing @{command
"context"}~@{keyword "begin"}~\dots~@{command "end"} block. Outside its
scope, a @{keyword "private"} name is inaccessible, and a @{keyword
"restricted"} name is only accessible with additional qualification.
Neither a global @{command theory} nor a @{command locale} target provides
a local scope by itself: an extra unnamed context is required to use
@{keyword "private"} or @{keyword "restricted"} here.
\item @{text "("}@{keyword_def "in"}~@{text "c)"} given after any local
theory command specifies an immediate target, e.g.\ ``@{command
"definition"}~@{text "(\<IN> c)"}'' or ``@{command "theorem"}~@{text
"(\<IN> c)"}''. This works both in a local or global theory context; the
current target context will be suspended for this command only. Note that
``@{text "(\<IN> -)"}'' will always produce a global result independently
of the current target context.
\end{description}
Any specification element that operates on @{text local_theory} according
to this manual implicitly allows the above target syntax @{text
"("}@{keyword "in"}~@{text "c)"}, 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
@{text c} with parameter @{text x} and assumption @{text "A[x]"}.
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,
@{text "a \<equiv> t[x]"} becomes @{text "c.a ?x \<equiv> t[?x]"} at the theory
level (for arbitrary @{text "?x"}), together with a local
abbreviation @{text "c \<equiv> c.a x"} in the target context (for the
fixed parameter @{text x}).
Theorems are exported by discharging the assumptions and
generalizing the parameters of the context. For example, @{text "a:
B[x]"} becomes @{text "c.a: A[?x] \<Longrightarrow> B[?x]"}, again for arbitrary
@{text "?x"}.
\<close>
section \<open>Bundled declarations \label{sec:bundle}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "bundle"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "print_bundles"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\
@{command_def "include"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\
@{command_def "including"} & : & @{text "proof(prove) \<rightarrow> proof(prove)"} \\
@{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 @{text "this_rule [intro] that_rule [elim]"} for example.
Configuration options (\secref{sec:config}) are special declaration
attributes that operate on the context without a theorem, as in
@{text "[[show_types = false]]"} for example.
Expressions of this form may be defined as \emph{bundled
declarations} 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>}
\begin{description}
\item @{command bundle}~@{text "b = decls"} defines a bundle of
declarations in the current context. The RHS is similar to the one
of the @{command declare} 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.
\item @{command print_bundles} prints the named bundles that are available
in the current context; the ``@{text "!"}'' option indicates extra
verbosity.
\item @{command include}~@{text "b\<^sub>1 \<dots> b\<^sub>n"} includes the declarations
from the given bundles into the current proof body context. This is
analogous to @{command "note"} (\secref{sec:proof-facts}) with the
expanded bundles.
\item @{command including} is similar to @{command include}, but
works in proof refinement (backward mode). This is analogous to
@{command "using"} (\secref{sec:proof-facts}) with the expanded
bundles.
\item @{keyword "includes"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"} is similar to
@{command include}, but works in situations where a specification
context is constructed, notably for @{command context} and long
statements of @{command theorem} etc.
\end{description}
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"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{attribute_def "defn"} & : & @{text attribute} \\
@{command_def "print_defn_rules"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\
@{command_def "abbreviation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "print_abbrevs"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\
\end{matharray}
Term definitions may either happen within the logic (as equational axioms
of a certain form, see also see \secref{sec:consts}), 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>}
\begin{description}
\item @{command "definition"}~@{text "c \<WHERE> eq"} produces an
internal definition @{text "c \<equiv> t"} according to the specification
given as @{text 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 @{attribute defn} by the
object-logic. This usually covers object-level equality @{text "x =
y"} and equivalence @{text "A \<leftrightarrow> B"}. 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.\ @{text "f x y = t"} instead of
@{text "f \<equiv> \<lambda>x y. t"} and @{text "y \<noteq> 0 \<Longrightarrow> g x y = u"} instead of an
unrestricted @{text "g \<equiv> \<lambda>x y. u"}.
\item @{command "print_defn_rules"} prints the definitional rewrite rules
declared via @{attribute defn} in the current context.
\item @{command "abbreviation"}~@{text "c \<WHERE> eq"} introduces a
syntactic constant which is associated with a certain term according
to the meta-level equality @{text 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 @{text mode} specification restricts output to a
particular print mode; using ``@{text input}'' here achieves the
effect of one-way abbreviations. The mode may also include an
``@{keyword "output"}'' qualifier that affects the concrete syntax
declared for abbreviations, cf.\ @{command "syntax"} in
\secref{sec:syn-trans}.
\item @{command "print_abbrevs"} prints all constant abbreviations of the
current context; the ``@{text "!"}'' option indicates extra verbosity.
\end{description}
\<close>
section \<open>Axiomatizations \label{sec:axiomatizations}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "axiomatization"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\
\end{matharray}
@{rail \<open>
@@{command axiomatization} @{syntax "fixes"}? (@'where' specs)?
;
specs: (@{syntax thmdecl}? @{syntax props} + @'and')
\<close>}
\begin{description}
\item @{command "axiomatization"}~@{text "c\<^sub>1 \<dots> c\<^sub>m \<WHERE> \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}
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.
\end{description}
\<close>
section \<open>Generic declarations\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "syntax_declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "declare"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
\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>}
\begin{description}
\item @{command "declaration"}~@{text d} adds the declaration
function @{text d} 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 @{text "(pervasive)"} option is given, the corresponding
declaration is applied to all possible contexts involved, including
the global background theory.
\item @{command "syntax_declaration"} is similar to @{command
"declaration"}, but is meant to affect only ``syntactic'' tools by
convention (such as notation and type-checking information).
\item @{command "declare"}~@{text thms} declares theorems to the
current local theory context. No theorem binding is involved here,
unlike @{command "theorems"} or @{command "lemmas"} (cf.\
\secref{sec:theorems}), so @{command "declare"} only has the effect
of applying attributes as included in the theorem specification.
\end{description}
\<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{roundup},
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{locale expression} 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 `@{text "_"}' 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 @{keyword "for"} 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 optional
(``\texttt{?}''), which means that it may be omitted on input of the
qualified name, or mandatory (``\texttt{!}''). If neither
``\texttt{?}'' nor ``\texttt{!}'' are present, the command's default
is used. For @{command "interpretation"} and @{command "interpret"}
the default is ``mandatory'', for @{command "locale"} and @{command
"sublocale"} the default is ``optional''. Qualifiers 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"} & : & @{text "theory \<rightarrow> local_theory"} \\
@{command_def "experiment"} & : & @{text "theory \<rightarrow> local_theory"} \\
@{command_def "print_locale"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "print_locales"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "locale_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{method_def intro_locales} & : & @{text method} \\
@{method_def unfold_locales} & : & @{text method} \\
\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>}
\begin{description}
\item @{command "locale"}~@{text "loc = import + body"} defines a
new locale @{text loc} as a context consisting of a certain view of
existing locales (@{text import}) plus some additional elements
(@{text body}). Both @{text import} and @{text body} are optional;
the degenerate form @{command "locale"}~@{text 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 @{text import} consists of a locale expression; see
\secref{sec:proof-context} above. Its @{keyword "for"} clause defines
the parameters of @{text import}. These are parameters of
the defined locale. Locale parameters whose instantiation is
omitted automatically extend the (possibly empty) @{keyword "for"}
clause: they are inserted at its beginning. This means that these
parameters may be referred to from within the expression and also in
the subsequent context elements and provides a notational
convenience for the inheritance of parameters in locale
declarations.
The @{text body} consists of context elements.
\begin{description}
\item @{element "fixes"}~@{text "x :: \<tau> (mx)"} declares a local
parameter of type @{text \<tau>} and mixfix annotation @{text mx} (both
are optional). The special syntax declaration ``@{text
"("}@{keyword_ref "structure"}@{text ")"}'' means that @{text x} may
be referenced implicitly in this context.
\item @{element "constrains"}~@{text "x :: \<tau>"} introduces a type
constraint @{text \<tau>} on the local parameter @{text x}. This
element is deprecated. The type constraint should be introduced in
the @{keyword "for"} clause or the relevant @{element "fixes"} element.
\item @{element "assumes"}~@{text "a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}
introduces local premises, similar to @{command "assume"} within a
proof (cf.\ \secref{sec:proof-context}).
\item @{element "defines"}~@{text "a: x \<equiv> t"} defines a previously
declared parameter. This is similar to @{command "def"} 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"}~@{text "f x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"}''.
\item @{element "notes"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}
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.
\end{description}
Both @{element "assumes"} and @{element "defines"} elements
contribute to the locale specification. When defining an operation
derived from the parameters, @{command "definition"}
(\secref{sec:term-definitions}) is usually more appropriate.
Note that ``@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}'' 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 @{text
loc_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 @{text 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 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 @{text \<And>} by @{text \<forall>}, and
@{text "\<Longrightarrow>"} by @{text "\<longrightarrow>"} in HOL; see also
\secref{sec:object-logic}). Separate introduction rules @{text
loc_axioms.intro} and @{text loc.intro} are provided as well.
\item @{command experiment}~@{text exprs}~@{keyword "begin"} 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.
\item @{command "print_locale"}~@{text "locale"} prints the
contents of the named locale. The command omits @{element "notes"}
elements by default. Use @{command "print_locale"}@{text "!"} to
have them included.
\item @{command "print_locales"} prints the names of all locales of the
current theory; the ``@{text "!"}'' option indicates extra verbosity.
\item @{command "locale_deps"} visualizes all locales and their
relations as a Hasse diagram. This includes locales defined as type
classes (\secref{sec:class}). See also @{command
"print_dependencies"} below.
\item @{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 @{text
loc.intro} introduction rules and therefore does not descend to
assumptions, @{method unfold_locales} is more aggressive and applies
@{text loc_axioms.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}
\<close>
subsection \<open>Locale interpretation\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "interpretation"} & : & @{text "theory | local_theory \<rightarrow> proof(prove)"} \\
@{command_def "interpret"} & : & @{text "proof(state) | proof(chain) \<rightarrow> proof(prove)"} \\
@{command_def "sublocale"} & : & @{text "theory | local_theory \<rightarrow> proof(prove)"} \\
@{command_def "print_dependencies"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "print_interps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
\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{locale
interpretation}. Interpretation is possible in locales (@{command
"sublocale"}), global and local theories (@{command
"interpretation"}) and also within proof bodies (@{command
"interpret"}).
@{rail \<open>
@@{command interpretation} @{syntax locale_expr} equations?
;
@@{command interpret} @{syntax locale_expr} equations?
;
@@{command sublocale} (@{syntax nameref} ('<' | '\<subseteq>'))? @{syntax locale_expr} \<newline>
equations?
;
@@{command print_dependencies} '!'? @{syntax locale_expr}
;
@@{command print_interps} @{syntax nameref}
;
equations: @'where' (@{syntax thmdecl}? @{syntax prop} + @'and')
\<close>}
\begin{description}
\item @{command "interpretation"}~@{text "expr \<WHERE> eqns"}
interprets @{text expr} in a global or local theory. 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 theory in a post-processing
phase.
The command is aware of interpretations that are already active.
Post-processing is achieved through a variant of roundup that takes
the 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}.
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 in global
theories dynamically participate in any declarations added to
locales.
In contrast, the lifetime of an interpretation in a local theory is
limited to the current context block. At the closing @{command end}
of the block the interpretation and its declarations disappear.
This enables establishing facts based on interpretations without
creating permanent links to the interpreted locale instances, as
would be the case with @{command sublocale}.
While @{command "interpretation"}~@{text "(\<IN> c)
\<dots>"} is technically possible, it is not useful since its result is
discarded immediately.
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 @{keyword "for"} clause.
The equations @{text eqns} yield \emph{rewrite morphisms}, which are
unfolded during post-processing and are useful for interpreting
concepts introduced through definitions. The equations must be
proved.
\item @{command "interpret"}~@{text "expr \<WHERE> eqns"} interprets
@{text expr} in the proof context and is otherwise similar to
interpretation in local theories. Note that for @{command
"interpret"} the @{text eqns} should be
explicitly universally quantified.
\item @{command "sublocale"}~@{text "name \<subseteq> expr \<WHERE>
eqns"}
interprets @{text expr} in the locale @{text name}. A proof that
the specification of @{text name} implies the specification of
@{text expr} is required. As in the localized version of the
theorem command, the proof is in the context of @{text name}. After
the proof obligation has been discharged, the locale hierarchy is
changed as if @{text name} imported @{text expr} (hence the name
@{command "sublocale"}). When the context of @{text name} is
subsequently entered, traversing the locale hierarchy will involve
the locale instances of @{text expr}, and their declarations will be
added to the context. This makes @{command "sublocale"}
dynamic: extensions of a locale that is instantiated in @{text expr}
may take place after the @{command "sublocale"} declaration and
still become available in the context. Circular @{command
"sublocale"} declarations are allowed as long as they do not lead to
infinite chains.
If interpretations of @{text name} exist in the current global
theory, the command adds interpretations for @{text expr} as well,
with the same qualifier, although only for fragments of @{text expr}
that are not interpreted in the theory already.
The equations @{text eqns} amend the morphism through
which @{text expr} is interpreted. This enables mapping definitions
from the interpreted locales to entities of @{text name} and can
help break infinite chains induced by circular @{command
"sublocale"} declarations.
In a named context block the @{command sublocale} 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.
\item @{command "print_dependencies"}~@{text "expr"} is useful for
understanding the effect of an interpretation of @{text "expr"} in
the current context. It lists all locale instances for which
interpretations would be added to the current context. Variant
@{command "print_dependencies"}@{text "!"} 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.
\item @{command "print_interps"}~@{text "locale"} lists all
interpretations of @{text "locale"} in the current theory or proof
context, including those due to a combination of an @{command
"interpretation"} or @{command "interpret"} and one or several
@{command "sublocale"} declarations.
\end{description}
\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 behavior of such
tools is \emph{not} stable under interpretation morphisms, manual
declarations might have to be added to the target context of the
interpretation to revert such declarations.
\end{warn}
\begin{warn}
An interpretation in a 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}
\<close>
subsubsection \<open>Permanent locale interpretation\<close>
text \<open>
Permanent locale interpretation is a library extension on top
of @{command "interpretation"} and @{command "sublocale"}. It is
available by importing theory @{file "~~/src/Tools/Permanent_Interpretation.thy"}
and provides
\begin{enumerate}
\item a unified view on arbitrary suitable local theories as interpretation target;
\item rewrite morphisms by means of \emph{rewrite definitions}.
\end{enumerate}
\begin{matharray}{rcl}
@{command_def "permanent_interpretation"} & : & @{text "local_theory \<rightarrow> proof(prove)"}
\end{matharray}
@{rail \<open>
@@{command permanent_interpretation} @{syntax locale_expr} \<newline>
definitions? equations?
;
definitions: @'defining' (@{syntax thmdecl}? @{syntax name} \<newline>
@{syntax mixfix}? @'=' @{syntax term} + @'and');
equations: @'where' (@{syntax thmdecl}? @{syntax prop} + @'and')
\<close>}
\begin{description}
\item @{command "permanent_interpretation"}~@{text "expr \<DEFINING> defs \<WHERE> eqns"}
interprets @{text expr} in the current local theory. The command
generates proof obligations for the instantiated specifications.
Instantiated declarations (in particular, facts) are added to the
local theory's underlying target in a post-processing phase. When
adding declarations to locales, interpreted versions of these
declarations are added to any target for all interpretations
in that target as well. That is, permanent interpretations
dynamically participate in any declarations added to
locales.
The local theory's underlying target must explicitly support
permanent interpretations. Prominent examples are global theories
(where @{command "permanent_interpretation"} is technically
corresponding to @{command "interpretation"}) and locales
(where @{command "permanent_interpretation"} is technically
corresponding to @{command "sublocale"}). Unnamed contexts
(see \secref{sec:target}) are not admissible since they are
non-permanent due to their very nature.
In additions to \emph{rewrite morphisms} specified by @{text eqns},
also \emph{rewrite definitions} may be specified. Semantically, a
rewrite definition
\begin{itemize}
\item produces a corresponding definition in
the local theory's underlying target \emph{and}
\item augments the rewrite morphism with the equation
stemming from the symmetric of the corresponding definition.
\end{itemize}
This is technically different to to a naive combination
of a conventional definition and an explicit rewrite equation:
\begin{itemize}
\item Definitions are parsed in the syntactic interpretation
context, just like equations.
\item The proof needs not consider the equations stemming from
definitions -- they are proved implicitly by construction.
\end{itemize}
Rewrite definitions yield a pattern for introducing new explicit
operations for existing terms after interpretation.
\end{description}
\<close>
section \<open>Classes \label{sec:class}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "class"} & : & @{text "theory \<rightarrow> local_theory"} \\
@{command_def "instantiation"} & : & @{text "theory \<rightarrow> local_theory"} \\
@{command_def "instance"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command "instance"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
@{command_def "subclass"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "print_classes"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "class_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{method_def intro_classes} & : & @{text method} \\
\end{matharray}
A class is a particular locale with \emph{exactly one} type variable
@{text \<alpha>}. 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} ( ( @{syntax sort} | ( '(' ( @{syntax sort} + @'|' ) ')' ) ) \<newline>
( @{syntax sort} | ( '(' ( @{syntax sort} + @'|' ) ')' ) )? )?
\<close>}
\begin{description}
\item @{command "class"}~@{text "c = superclasses + body"} defines
a new class @{text c}, inheriting from @{text superclasses}. This
introduces a locale @{text c} with import of all locales @{text
superclasses}.
Any @{element "fixes"} in @{text body} are lifted to the global
theory level (\emph{class operations} @{text "f\<^sub>1, \<dots>,
f\<^sub>n"} of class @{text c}), mapping the local type parameter
@{text \<alpha>} to a schematic type variable @{text "?\<alpha> :: c"}.
Likewise, @{element "assumes"} in @{text body} are also lifted,
mapping each local parameter @{text "f :: \<tau>[\<alpha>]"} to its
corresponding global constant @{text "f :: \<tau>[?\<alpha> :: c]"}. The
corresponding introduction rule is provided as @{text
c_class_axioms.intro}. This rule should be rarely needed directly
--- the @{method intro_classes} method takes care of the details of
class membership proofs.
\item @{command "instantiation"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s
\<BEGIN>"} opens a target (cf.\ \secref{sec:target}) which
allows to specify class operations @{text "f\<^sub>1, \<dots>, f\<^sub>n"} corresponding
to sort @{text s} at the particular type instance @{text "(\<alpha>\<^sub>1 :: s\<^sub>1,
\<dots>, \<alpha>\<^sub>n :: s\<^sub>n) t"}. A plain @{command "instance"} 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.
\item @{command "instance"} in an instantiation target body sets
up a goal stating the type arities claimed at the opening @{command
"instantiation"}. 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, @{command "instance"}~@{text "t :: (s\<^sub>1, \<dots>,
s\<^sub>n)s"} provides a convenient way to instantiate a type class with no
need to specify operations: one can continue with the
instantiation proof immediately.
\item @{command "subclass"}~@{text c} in a class context for class
@{text d} sets up a goal stating that class @{text c} is logically
contained in class @{text d}. After finishing the proof, class
@{text d} is proven to be subclass @{text c} and the locale @{text
c} is interpreted into @{text d} simultaneously.
A weakened form of this is available through a further variant of
@{command instance}: @{command instance}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} opens
a proof that class @{text "c\<^sub>2"} implies @{text "c\<^sub>1"} 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.
\item @{command "print_classes"} prints all classes in the current
theory.
\item @{command "class_deps"} visualizes all classes and their
subclass relations as a Hasse diagram. An optional first argument
constrains the set of classes to all subclasses of at least one given
sort, an optional second rgument to all superclasses of at least one given
sort.
\item @{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 @{command "proof"}). In particular,
instantiation of trivial (syntactic) classes may be performed by a
single ``@{command ".."}'' proof step.
\end{description}
\<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 @{text c}, apart from the common
locale target behaviour the following happens.
\begin{itemize}
\item Local constant declarations @{text "g[\<alpha>]"} referring to the
local type parameter @{text \<alpha>} and local parameters @{text "f[\<alpha>]"}
are accompanied by theory-level constants @{text "g[?\<alpha> :: c]"}
referring to theory-level class operations @{text "f[?\<alpha> :: c]"}.
\item Local theorem bindings are lifted as are assumptions.
\item Local syntax refers to local operations @{text "g[\<alpha>]"} and
global operations @{text "g[?\<alpha> :: c]"} uniformly. Type inference
resolves ambiguities. In rare cases, manual type annotations are
needed.
\end{itemize}
\<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{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 @{text "t :: (\<^vec>s)c"} is
completed such that @{text "t :: (\<^vec>s)c"} and @{text "c \<subseteq> c'"}
implies @{text "t :: (\<^vec>s)c'"} for all such declarations.
Treating sorts as finite sets of classes (meaning the intersection),
the class relation @{text "c\<^sub>1 \<subseteq> c\<^sub>2"} is extended to sorts as
follows:
\[
@{text "s\<^sub>1 \<subseteq> s\<^sub>2 \<equiv> \<forall>c\<^sub>2 \<in> s\<^sub>2. \<exists>c\<^sub>1 \<in> s\<^sub>1. c\<^sub>1 \<subseteq> c\<^sub>2"}
\]
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:
\[
@{text "t :: (\<^vec>s\<^sub>1)c\<^sub>1 \<Longrightarrow> t :: (\<^vec>s\<^sub>2)c\<^sub>2 \<Longrightarrow> c\<^sub>1 \<subseteq> c\<^sub>2 \<Longrightarrow> \<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2"}
\]
\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"}.
\<close>
section \<open>Unrestricted overloading\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "overloading"} & : & @{text "theory \<rightarrow> local_theory"} \\
\end{matharray}
Isabelle/Pure's definitional schemes support certain forms of
overloading (see \secref{sec:consts}). Overloading means that a
constant being declared as @{text "c :: \<alpha> decl"} may be
defined separately on type instances
@{text "c :: (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t decl"}
for each type constructor @{text t}. At most occasions
overloading will be used in a Haskell-like fashion together with
type classes by means of @{command "instantiation"} (see
\secref{sec:class}). Sometimes low-level overloading is desirable.
The @{command "overloading"} target provides a convenient view for
end-users.
@{rail \<open>
@@{command overloading} ( spec + ) @'begin'
;
spec: @{syntax name} ( '==' | '\<equiv>' ) @{syntax term} ( '(' @'unchecked' ')' )?
\<close>}
\begin{description}
\item @{command "overloading"}~@{text "x\<^sub>1 \<equiv> c\<^sub>1 :: \<tau>\<^sub>1 \<AND> \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n \<BEGIN>"}
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 @{text "x\<^sub>i"} to
constants @{text "c\<^sub>i"} at particular type instances. The
definitions themselves are established using common specification
tools, using the names @{text "x\<^sub>i"} as reference to the
corresponding constants. The target is concluded by @{command
(local) "end"}.
A @{text "(unchecked)"} option disables global dependency checks for
the corresponding definition, which is occasionally useful for
exotic overloading (see \secref{sec:consts} for a precise description).
It is at the discretion of the user to avoid
malformed theory specifications!
\end{description}
\<close>
section \<open>Incorporating ML code \label{sec:ML}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "SML_file"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "ML_file"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "ML"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "ML_prf"} & : & @{text "proof \<rightarrow> proof"} \\
@{command_def "ML_val"} & : & @{text "any \<rightarrow>"} \\
@{command_def "ML_command"} & : & @{text "any \<rightarrow>"} \\
@{command_def "setup"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "local_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "attribute_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
\end{matharray}
\begin{tabular}{rcll}
@{attribute_def ML_print_depth} & : & @{text attribute} & default 10 \\
@{attribute_def ML_source_trace} & : & @{text attribute} & default @{text false} \\
@{attribute_def ML_exception_trace} & : & @{text attribute} & default @{text false} \\
\end{tabular}
@{rail \<open>
(@@{command SML_file} | @@{command ML_file}) @{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>}
\begin{description}
\item @{command "SML_file"}~@{text "name"} reads and evaluates the
given Standard ML file. Top-level SML bindings are stored within
the theory context; the initial environment is restricted to the
Standard ML implementation of Poly/ML, without the many add-ons of
Isabelle/ML. Multiple @{command "SML_file"} commands may be used to
build larger Standard ML projects, independently of the regular
Isabelle/ML environment.
\item @{command "ML_file"}~@{text "name"} 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.
\item @{command "ML"}~@{text "text"} is similar to @{command
"ML_file"}, but evaluates directly the given @{text "text"}.
Top-level ML bindings are stored within the (global or local) theory
context.
\item @{command "ML_prf"} is analogous to @{command "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 @{command
"ML_prf"} are discarded at the end of the proof.
\item @{command "ML_val"} and @{command "ML_command"} are diagnostic
versions of @{command "ML"}, which means that the context may not be
updated. @{command "ML_val"} echos the bindings produced at the ML
toplevel, but @{command "ML_command"} is silent.
\item @{command "setup"}~@{text "text"} changes the current theory
context by applying @{text "text"}, 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.
\item @{command "local_setup"} is similar to @{command "setup"} 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.
\item @{command "attribute_setup"}~@{text "name = text description"}
defines an attribute in the current context. The given @{text
"text"} 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:
\end{description}
\<close>
(*<*)experiment begin(*>*)
attribute_setup my_rule =
\<open>Attrib.thms >> (fn ths =>
Thm.rule_attribute
(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>
\begin{description}
\item @{attribute ML_print_depth} controls the printing depth of the ML
toplevel pretty printer; the precise effect depends on the ML compiler and
run-time system. Typically the limit should be less than 10. Bigger values
such as 100--1000 are occasionally useful for debugging.
\item @{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.
\item @{attribute ML_exception_trace} indicates whether the ML run-time
system should print a detailed stack trace on exceptions. The result is
dependent on the particular ML compiler version. Note that after Poly/ML
5.3 some optimizations in the run-time systems may hinder exception
traces.
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.
\end{description}
\<close>
section \<open>Primitive specification elements\<close>
subsection \<open>Sorts\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "default_sort"} & : & @{text "local_theory \<rightarrow> local_theory"}
\end{matharray}
@{rail \<open>
@@{command default_sort} @{syntax sort}
\<close>}
\begin{description}
\item @{command "default_sort"}~@{text s} makes sort @{text 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
@{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.
\end{description}
\<close>
subsection \<open>Types \label{sec:types-pure}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "type_synonym"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "typedecl"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
\end{matharray}
@{rail \<open>
@@{command type_synonym} (@{syntax typespec} '=' @{syntax type} @{syntax mixfix}?)
;
@@{command typedecl} @{syntax typespec} @{syntax mixfix}?
\<close>}
\begin{description}
\item @{command "type_synonym"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = \<tau>"} introduces a
\emph{type synonym} @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} for the existing type @{text
"\<tau>"}. Unlike the semantic type definitions in Isabelle/HOL, type synonyms
are merely syntactic abbreviations without any logical significance.
Internally, type synonyms are fully expanded.
\item @{command "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} declares a new
type constructor @{text t}. If the object-logic defines a base sort
@{text s}, then the constructor is declared to operate on that, via
the axiomatic type-class instance @{text "t :: (s, \<dots>, s)s"}.
\end{description}
\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>
subsection \<open>Constants and definitions \label{sec:consts}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def "consts"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "defs"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
Definitions essentially express abbreviations within the logic. The
simplest form of a definition is @{text "c :: \<sigma> \<equiv> t"}, where @{text
c} is a newly declared constant. Isabelle also allows derived forms
where the arguments of @{text c} appear on the left, abbreviating a
prefix of @{text \<lambda>}-abstractions, e.g.\ @{text "c \<equiv> \<lambda>x y. t"} may be
written more conveniently as @{text "c x y \<equiv> t"}. Moreover,
definitions may be weakened by adding arbitrary pre-conditions:
@{text "A \<Longrightarrow> c x y \<equiv> t"}.
\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 @{text "0 :: nat \<equiv> length ([] ::
\<alpha> list)"} 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}
The right-hand side of overloaded definitions may mention overloaded constants
recursively at type instances corresponding to the immediate
argument types @{text "\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n"}. Incomplete
specification patterns impose global constraints on all occurrences,
e.g.\ @{text "d :: \<alpha> \<times> \<alpha>"} on the left-hand side means that all
corresponding occurrences on some right-hand side need to be an
instance of this, general @{text "d :: \<alpha> \<times> \<beta>"} will be disallowed.
@{rail \<open>
@@{command consts} ((@{syntax name} '::' @{syntax type} @{syntax mixfix}?) +)
;
@@{command defs} opt? (@{syntax axmdecl} @{syntax prop} +)
;
opt: '(' @'unchecked'? @'overloaded'? ')'
\<close>}
\begin{description}
\item @{command "consts"}~@{text "c :: \<sigma>"} declares constant @{text
c} to have any instance of type scheme @{text \<sigma>}. The optional
mixfix annotations may attach concrete syntax to the constants
declared.
\item @{command "defs"}~@{text "name: eqn"} introduces @{text eqn}
as a definitional axiom for some existing constant.
The @{text "(unchecked)"} 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 @{text "(overloaded)"} 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.
\end{description}
\<close>
section \<open>Naming existing theorems \label{sec:theorems}\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "lemmas"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "theorems"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "named_theorems"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
\end{matharray}
@{rail \<open>
(@@{command lemmas} | @@{command theorems}) (@{syntax thmdef}? @{syntax thmrefs} + @'and')
@{syntax for_fixes}
;
@@{command named_theorems} (@{syntax name} @{syntax text}? + @'and')
\<close>}
\begin{description}
\item @{command "lemmas"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}~@{keyword_def
"for"}~@{text "x\<^sub>1 \<dots> x\<^sub>m"} 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 @{text "0"} uniformly.
\item @{command "theorems"} is the same as @{command "lemmas"}, but
marks the result as a different kind of facts.
\item @{command "named_theorems"}~@{text "name description"} declares a
dynamic fact within the context. The same @{text name} is used to define
an attribute with the usual @{text add}/@{text del} syntax (e.g.\ see
\secref{sec:simp-rules}) to maintain the content incrementally, in
canonical declaration order of the text structure.
\end{description}
\<close>
section \<open>Oracles\<close>
text \<open>
\begin{matharray}{rcll}
@{command_def "oracle"} & : & @{text "theory \<rightarrow> theory"} & (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>}
\begin{description}
\item @{command "oracle"}~@{text "name = text"} turns the given ML
expression @{text "text"} 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.
\end{description}
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"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "hide_type"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "hide_const"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "hide_fact"} & : & @{text "theory \<rightarrow> theory"} \\
\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 behavior of
name spaces by hand, yet the following commands provide some way to
do so.
\begin{description}
\item @{command "hide_class"}~@{text names} fully removes class
declarations from a given name space; with the @{text "(open)"}
option, only the 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
``@{text "??"}'' prefixed to the full internal name.
\item @{command "hide_type"}, @{command "hide_const"}, and @{command
"hide_fact"} are similar to @{command "hide_class"}, but hide types,
constants, and facts, respectively.
\end{description}
\<close>
end