(* Title: HOL/ex/Locales.thy
ID: $Id$
Author: Markus Wenzel, LMU München
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
header {* Using locales in Isabelle/Isar *}
theory Locales = Main:
text_raw {*
\newcommand{\isasyminv}{\isasyminverse}
\newcommand{\isasymIN}{\isatext{\isakeyword{in}}}
\newcommand{\isasymUSES}{\isatext{\isakeyword{uses}}}
*}
subsection {* Overview *}
text {*
Locales provide a mechanism for encapsulating local contexts. The
original version due to Florian Kammüller \cite{Kamm-et-al:1999}
refers directly to the raw meta-logic of Isabelle. Semantically, a
locale is just a (curried) predicate of the pure meta-logic
\cite{Paulson:1989}.
The present version for Isabelle/Isar builds on top of the rich
infrastructure of proof contexts
\cite{Wenzel:1999,Wenzel:2001:isar-ref,Wenzel:2001:Thesis},
achieving a tight integration with Isar proof texts, and a slightly
more abstract view of the underlying concepts. An Isar proof
context encapsulates certain language elements that correspond to
@{text \<And>} (universal quantification), @{text \<Longrightarrow>} (implication), and
@{text "\<equiv>"} (definitions) of the pure Isabelle framework. Moreover,
there are extra-logical concepts like term abbreviations or local
theorem attributes (declarations of simplification rules etc.) that
are indispensable in realistic applications.
Locales support concrete syntax, providing a localized version of
the existing concept of mixfix annotations of Isabelle
\cite{Paulson:1994:Isabelle}. Furthermore, there is a separate
concept of ``implicit structures'' that admits to refer to
particular locale parameters in a casual manner (essentially derived
from the basic idea of ``anti-quotations'' or generalized de-Bruijn
indexes demonstrated in \cite[\S13--14]{Wenzel:2001:Isar-examples}).
Implicit structures work particular well together with extensible
records in HOL \cite{Naraschewski-Wenzel:1998} (the
``object-oriented'' features discussed there are not required here).
Thus we shall essentially use record types to represent polymorphic
signatures of mathematical structures, while locales describe their
logical properties as a predicate. Due to type inference of
simply-typed records we are able to give succinct specifications,
without caring about signature morphisms ourselves. Further
eye-candy is provided by the independent concept of ``indexed
concrete syntax'' for record selectors.
Operations for building up locale contexts from existing definitions
cover common operations of \emph{merge} (disjunctive union in
canonical order) and \emph{rename} (of term parameters; types are
always inferred automatically).
\medskip Note that the following further concepts are still
\emph{absent}:
\begin{itemize}
\item Specific language elements for \emph{instantiation} of
locales.
Currently users may simulate this to some extend by having primitive
Isabelle/Isar operations (@{text of} for substitution and @{text OF}
for composition, \cite{Wenzel:2001:isar-ref}) act on the
automatically exported results stemming from different contexts.
\item Interpretation of locales, analogous to ``functors'' in
abstract algebra.
In principle one could directly work with functions over structures
(extensible records), and predicates being derived from locale
definitions.
\end{itemize}
\medskip Subsequently, we demonstrate some readily available
concepts of Isabelle/Isar locales by some simple examples of
abstract algebraic reasoning.
*}
subsection {* Local contexts as mathematical structures *}
text {*
The following definitions of @{text group} and @{text abelian_group}
merely encapsulate local parameters (with private syntax) and
assumptions; local definitions may be given as well, but are not
shown here.
*}
locale group_context =
fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<cdot>" 70)
and inv :: "'a \<Rightarrow> 'a" ("(_\<inv>)" [1000] 999)
and one :: 'a ("\<one>")
assumes assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
and left_inv: "x\<inv> \<cdot> x = \<one>"
and left_one: "\<one> \<cdot> x = x"
locale abelian_group_context = group_context +
assumes commute: "x \<cdot> y = y \<cdot> x"
text {*
\medskip We may now prove theorems within a local context, just by
including a directive ``@{text "(\<IN> name)"}'' in the goal
specification. The final result will be stored within the named
locale; a second copy is exported to the enclosing theory context.
*}
theorem (in group_context)
right_inv: "x \<cdot> x\<inv> = \<one>"
proof -
have "x \<cdot> x\<inv> = \<one> \<cdot> (x \<cdot> x\<inv>)" by (simp only: left_one)
also have "\<dots> = \<one> \<cdot> x \<cdot> x\<inv>" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> x\<inv> \<cdot> x \<cdot> x\<inv>" by (simp only: left_inv)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> (x\<inv> \<cdot> x) \<cdot> x\<inv>" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> \<one> \<cdot> x\<inv>" by (simp only: left_inv)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> (\<one> \<cdot> x\<inv>)" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> x\<inv>" by (simp only: left_one)
also have "\<dots> = \<one>" by (simp only: left_inv)
finally show ?thesis .
qed
theorem (in group_context)
right_one: "x \<cdot> \<one> = x"
proof -
have "x \<cdot> \<one> = x \<cdot> (x\<inv> \<cdot> x)" by (simp only: left_inv)
also have "\<dots> = x \<cdot> x\<inv> \<cdot> x" by (simp only: assoc)
also have "\<dots> = \<one> \<cdot> x" by (simp only: right_inv)
also have "\<dots> = x" by (simp only: left_one)
finally show ?thesis .
qed
text {*
\medskip Apart from the named context we may also refer to further
context elements (parameters, assumptions, etc.) in a casual manner;
these are discharged when the proof is finished.
*}
theorem (in group_context)
assumes eq: "e \<cdot> x = x"
shows one_equality: "\<one> = e"
proof -
have "\<one> = x \<cdot> x\<inv>" by (simp only: right_inv)
also have "\<dots> = (e \<cdot> x) \<cdot> x\<inv>" by (simp only: eq)
also have "\<dots> = e \<cdot> (x \<cdot> x\<inv>)" by (simp only: assoc)
also have "\<dots> = e \<cdot> \<one>" by (simp only: right_inv)
also have "\<dots> = e" by (simp only: right_one)
finally show ?thesis .
qed
theorem (in group_context)
assumes eq: "x' \<cdot> x = \<one>"
shows inv_equality: "x\<inv> = x'"
proof -
have "x\<inv> = \<one> \<cdot> x\<inv>" by (simp only: left_one)
also have "\<dots> = (x' \<cdot> x) \<cdot> x\<inv>" by (simp only: eq)
also have "\<dots> = x' \<cdot> (x \<cdot> x\<inv>)" by (simp only: assoc)
also have "\<dots> = x' \<cdot> \<one>" by (simp only: right_inv)
also have "\<dots> = x'" by (simp only: right_one)
finally show ?thesis .
qed
theorem (in group_context)
inv_prod: "(x \<cdot> y)\<inv> = y\<inv> \<cdot> x\<inv>"
proof (rule inv_equality)
show "(y\<inv> \<cdot> x\<inv>) \<cdot> (x \<cdot> y) = \<one>"
proof -
have "(y\<inv> \<cdot> x\<inv>) \<cdot> (x \<cdot> y) = (y\<inv> \<cdot> (x\<inv> \<cdot> x)) \<cdot> y" by (simp only: assoc)
also have "\<dots> = (y\<inv> \<cdot> \<one>) \<cdot> y" by (simp only: left_inv)
also have "\<dots> = y\<inv> \<cdot> y" by (simp only: right_one)
also have "\<dots> = \<one>" by (simp only: left_inv)
finally show ?thesis .
qed
qed
text {*
\medskip Established results are automatically propagated through
the hierarchy of locales. Below we establish a trivial fact in
commutative groups, while referring both to theorems of @{text
group} and the additional assumption of @{text abelian_group}.
*}
theorem (in abelian_group_context)
inv_prod': "(x \<cdot> y)\<inv> = x\<inv> \<cdot> y\<inv>"
proof -
have "(x \<cdot> y)\<inv> = y\<inv> \<cdot> x\<inv>" by (rule inv_prod)
also have "\<dots> = x\<inv> \<cdot> y\<inv>" by (rule commute)
finally show ?thesis .
qed
text {*
We see that the initial import of @{text group} within the
definition of @{text abelian_group} is actually evaluated
dynamically. Thus any results in @{text group} are made available
to the derived context of @{text abelian_group} as well. Note that
the alternative context element @{text \<USES>} would import
existing locales in a static fashion, without participating in
further facts emerging later on.
\medskip Some more properties of inversion in general group theory
follow.
*}
theorem (in group_context)
inv_inv: "(x\<inv>)\<inv> = x"
proof (rule inv_equality)
show "x \<cdot> x\<inv> = \<one>" by (simp only: right_inv)
qed
theorem (in group_context)
assumes eq: "x\<inv> = y\<inv>"
shows inv_inject: "x = y"
proof -
have "x = x \<cdot> \<one>" by (simp only: right_one)
also have "\<dots> = x \<cdot> (y\<inv> \<cdot> y)" by (simp only: left_inv)
also have "\<dots> = x \<cdot> (x\<inv> \<cdot> y)" by (simp only: eq)
also have "\<dots> = (x \<cdot> x\<inv>) \<cdot> y" by (simp only: assoc)
also have "\<dots> = \<one> \<cdot> y" by (simp only: right_inv)
also have "\<dots> = y" by (simp only: left_one)
finally show ?thesis .
qed
text {*
\bigskip We see that this representation of structures as local
contexts is rather light-weight and convenient to use for abstract
reasoning. Here the ``components'' (the group operations) have been
exhibited directly as context parameters; logically this corresponds
to a curried predicate definition. Occasionally, this
``externalized'' version of the informal idea of classes of tuple
structures may cause some inconveniences, especially in
meta-theoretical studies (involving functors from groups to groups,
for example).
Another minor drawback of the naive approach above is that concrete
syntax will get lost on any kind of operation on the locale itself
(such as renaming, copying, or instantiation). Whenever the
particular terminology of local parameters is affected the
associated syntax would have to be changed as well, which is hard to
achieve formally.
*}
subsection {* Explicit structures referenced implicitly *}
text {*
We introduce the same hierarchy of basic group structures as above,
this time using extensible record types for the signature part,
together with concrete syntax for selector functions.
*}
record 'a semigroup =
prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<cdot>\<index>" 70)
record 'a group = "'a semigroup" +
inv :: "'a \<Rightarrow> 'a" ("(_\<inv>\<index>)" [1000] 999)
one :: 'a ("\<one>\<index>")
text {*
The mixfix annotations above include a special ``structure index
indicator'' @{text \<index>} that makes gammer productions dependent on
certain parameters that have been declared as ``structure'' in a
locale context later on. Thus we achieve casual notation as
encountered in informal mathematics, e.g.\ @{text "x \<cdot> y"} for
@{text "prod G x y"}.
\medskip The following locale definitions introduce operate on a
single parameter declared as ``\isakeyword{structure}''. Type
inference takes care to fill in the appropriate record type schemes
internally.
*}
locale semigroup =
fixes S :: "'a group" (structure)
assumes assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
locale group = semigroup G +
assumes left_inv: "x\<inv> \<cdot> x = \<one>"
and left_one: "\<one> \<cdot> x = x"
text {*
Note that we prefer to call the @{text group} record structure
@{text G} rather than @{text S} inherited from @{text semigroup}.
This does not affect our concrete syntax, which is only dependent on
the \emph{positional} arrangements of currently active structures
(actually only one above), rather than names. In fact, these
parameter names rarely occur in the term language at all (due to the
``indexed syntax'' facility of Isabelle). On the other hand, names
of locale facts will get qualified accordingly, e.g. @{text S.assoc}
versus @{text G.assoc}.
\medskip We may now proceed to prove results within @{text group}
just as before for @{text group}. The subsequent proof texts are
exactly the same as despite the more advanced internal arrangement.
*}
theorem (in group)
right_inv: "x \<cdot> x\<inv> = \<one>"
proof -
have "x \<cdot> x\<inv> = \<one> \<cdot> (x \<cdot> x\<inv>)" by (simp only: left_one)
also have "\<dots> = \<one> \<cdot> x \<cdot> x\<inv>" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> x\<inv> \<cdot> x \<cdot> x\<inv>" by (simp only: left_inv)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> (x\<inv> \<cdot> x) \<cdot> x\<inv>" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> \<one> \<cdot> x\<inv>" by (simp only: left_inv)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> (\<one> \<cdot> x\<inv>)" by (simp only: assoc)
also have "\<dots> = (x\<inv>)\<inv> \<cdot> x\<inv>" by (simp only: left_one)
also have "\<dots> = \<one>" by (simp only: left_inv)
finally show ?thesis .
qed
theorem (in group)
right_one: "x \<cdot> \<one> = x"
proof -
have "x \<cdot> \<one> = x \<cdot> (x\<inv> \<cdot> x)" by (simp only: left_inv)
also have "\<dots> = x \<cdot> x\<inv> \<cdot> x" by (simp only: assoc)
also have "\<dots> = \<one> \<cdot> x" by (simp only: right_inv)
also have "\<dots> = x" by (simp only: left_one)
finally show ?thesis .
qed
text {*
\medskip Several implicit structures may be active at the same time.
The concrete syntax facility for locales actually maintains indexed
structures that may be references implicitly --- via mixfix
annotations that have been decorated by an ``index argument''
(@{text \<index>}).
The following synthetic example demonstrates how to refer to several
structures of type @{text group} succinctly. We work with two
versions of the @{text group} locale above.
*}
lemma
uses group G + group H
shows "x \<cdot> y \<cdot> \<one> = prod G (prod G x y) (one G)" and
"x \<cdot>\<^sub>2 y \<cdot>\<^sub>2 \<one>\<^sub>2 = prod H (prod H x y) (one H)" and
"x \<cdot> \<one>\<^sub>2 = prod G x (one H)" by (rule refl)+
text {*
Note that the trivial statements above need to be given as a
simultaneous goal in order to have type-inference make the implicit
typing of structures @{text G} and @{text H} agree.
*}
end