(* Title: HOL/ex/Locales.thy ID: $Id: Locales.thy,v 1.19 2005/09/14 20:08:08 wenzelm Exp $ Author: Markus Wenzel, LMU München *) header {* Using locales in Isabelle/Isar -- outdated version! *} theory Locales imports Main begin text_raw {* \newcommand{\isasyminv}{\isasyminverse} \newcommand{\isasymIN}{\isatext{\isakeyword{in}}} \newcommand{\isasymINCLUDES}{\isatext{\isakeyword{includes}}} *} 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 Isabelle's meta-logic \cite{Paulson:1989}, which is minimal higher-order logic with connectives @{text "!!"} (universal quantification), @{text "==>"} (implication), and @{text "≡"} (equality). From this perspective, a locale is essentially a meta-level predicate, together with some infrastructure to manage the abstracted parameters (@{text "!!"}), assumptions (@{text "==>"}), and definitions for (@{text "≡"}) in a reasonable way during the proof process. This simple predicate view also provides a solid semantical basis for our specification concepts to be developed later. \medskip The present version of locales for Isabelle/Isar builds on top of the rich infrastructure of proof contexts \cite{Wenzel:1999,Wenzel:2001:isar-ref,Wenzel:2001:Thesis}, which in turn is based on the same meta-logic. Thus we achieve a tight integration with Isar proof texts, and a slightly more abstract view of the underlying logical concepts. An Isar proof context encapsulates certain language elements that correspond to @{text "!!/==>/≡"} at the level of structure proof texts. Moreover, there are extra-logical concepts like term abbreviations or local theorem attributes (declarations of simplification rules etc.) that are useful in applications (e.g.\ consider standard simplification rules declared in a group context). Locales also support concrete syntax, i.e.\ 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 (basically a simplified version of the idea of ``anti-quotations'', or generalized de-Bruijn indexes as demonstrated elsewhere \cite[\S13--14]{Wenzel:2001:Isar-examples}). Implicit structures work particular well together with extensible records in HOL \cite{Naraschewski-Wenzel:1998} (without the ``object-oriented'' features discussed there as well). Thus we achieve a specification technique where record type schemes represent polymorphic signatures of mathematical structures, and actual locales describe the corresponding logical properties. Semantically speaking, such abstract mathematical structures are just predicates over record types. Due to type inference of simply-typed records (which subsumes structural subtyping) we arrive at succinct specification texts --- ``signature morphisms'' degenerate to implicit type-instantiations. Additional eye-candy is provided by the separate concept of ``indexed concrete syntax'' used for record selectors, so we get close to informal mathematical notation. \medskip Operations for building up locale contexts from existing ones include \emph{merge} (disjoint union) and \emph{rename} (of term parameters only, types are inferred automatically). Here we draw from existing traditions of algebraic specification languages. A structured specification corresponds to a directed acyclic graph of potentially renamed nodes (due to distributivity renames may pushed inside of merges). The result is a ``flattened'' list of primitive context elements in canonical order (corresponding to left-to-right reading of merges, while suppressing duplicates). \medskip The present version of Isabelle/Isar locales still lacks some important specification concepts. \begin{itemize} \item Separate 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 (think of ``views'', ``functors'' etc.). 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_context} and @{text abelian_group_context} merely encapsulate local parameters (with private syntax) and assumptions; local definitions of derived concepts could be given, too, but are unused below. *} locale group_context = fixes prod :: "'a => 'a => 'a" (infixl "·" 70) and inv :: "'a => 'a" ("(_\<inv>)" [1000] 999) and one :: 'a ("\<one>") assumes assoc: "(x · y) · z = x · (y · z)" and left_inv: "x\<inv> · x = \<one>" and left_one: "\<one> · x = x" locale abelian_group_context = group_context + assumes commute: "x · y = y · 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, still holding the context; a second copy is exported to the enclosing theory context (with qualified name). *} theorem (in group_context) right_inv: "x · x\<inv> = \<one>" proof - have "x · x\<inv> = \<one> · (x · x\<inv>)" by (simp only: left_one) also have "… = \<one> · x · x\<inv>" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · x\<inv> · x · x\<inv>" by (simp only: left_inv) also have "… = (x\<inv>)\<inv> · (x\<inv> · x) · x\<inv>" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · \<one> · x\<inv>" by (simp only: left_inv) also have "… = (x\<inv>)\<inv> · (\<one> · x\<inv>)" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · x\<inv>" by (simp only: left_one) also have "… = \<one>" by (simp only: left_inv) finally show ?thesis . qed theorem (in group_context) right_one: "x · \<one> = x" proof - have "x · \<one> = x · (x\<inv> · x)" by (simp only: left_inv) also have "… = x · x\<inv> · x" by (simp only: assoc) also have "… = \<one> · x" by (simp only: right_inv) also have "… = x" by (simp only: left_one) finally show ?thesis . qed text {* Facts like @{text right_one} are available @{text group_context} as stated above. The exported version looses the additional infrastructure of Isar proof contexts (syntax etc.) retaining only the pure logical content: @{thm [source] group_context.right_one} becomes @{thm group_context.right_one} (in Isabelle outermost @{text !!} quantification is replaced by schematic variables). \medskip Apart from a named locale we may also refer to further context elements (parameters, assumptions, etc.) in an ad-hoc fashion, just for this particular statement. In the result (local or global), any additional elements are discharged as usual. *} theorem (in group_context) assumes eq: "e · x = x" shows one_equality: "\<one> = e" proof - have "\<one> = x · x\<inv>" by (simp only: right_inv) also have "… = (e · x) · x\<inv>" by (simp only: eq) also have "… = e · (x · x\<inv>)" by (simp only: assoc) also have "… = e · \<one>" by (simp only: right_inv) also have "… = e" by (simp only: right_one) finally show ?thesis . qed theorem (in group_context) assumes eq: "x' · x = \<one>" shows inv_equality: "x\<inv> = x'" proof - have "x\<inv> = \<one> · x\<inv>" by (simp only: left_one) also have "… = (x' · x) · x\<inv>" by (simp only: eq) also have "… = x' · (x · x\<inv>)" by (simp only: assoc) also have "… = x' · \<one>" by (simp only: right_inv) also have "… = x'" by (simp only: right_one) finally show ?thesis . qed theorem (in group_context) inv_prod: "(x · y)\<inv> = y\<inv> · x\<inv>" proof (rule inv_equality) show "(y\<inv> · x\<inv>) · (x · y) = \<one>" proof - have "(y\<inv> · x\<inv>) · (x · y) = (y\<inv> · (x\<inv> · x)) · y" by (simp only: assoc) also have "… = (y\<inv> · \<one>) · y" by (simp only: left_inv) also have "… = y\<inv> · y" by (simp only: right_one) also have "… = \<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 · y)\<inv> = x\<inv> · y\<inv>" proof - have "(x · y)\<inv> = y\<inv> · x\<inv>" by (rule inv_prod) also have "… = x\<inv> · 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 \<INCLUDES>} 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 · 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 · \<one>" by (simp only: right_one) also have "… = x · (y\<inv> · y)" by (simp only: left_inv) also have "… = x · (x\<inv> · y)" by (simp only: eq) also have "… = (x · x\<inv>) · y" by (simp only: assoc) also have "… = \<one> · y" by (simp only: right_inv) also have "… = 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: @{thm [display] group_context_def [no_vars]} The corresponding introduction rule is as follows: @{thm [display, mode = no_brackets] group_context.intro [no_vars]} 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 => 'a => 'a" (infixl "·\<index>" 70) record 'a group = "'a semigroup" + inv :: "'a => 'a" ("(_\<inv>\<index>)" [1000] 999) one :: 'a ("\<one>\<index>") text {* The mixfix annotations above include a special ``structure index indicator'' @{text \<index>} that makes grammar 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 · 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 (structure) assumes assoc: "(x · y) · z = x · (y · z)" locale group = semigroup G + assumes left_inv: "x\<inv> · x = \<one>" and left_one: "\<one> · x = x" declare semigroup.intro [intro?] group.intro [intro?] group_axioms.intro [intro?] 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 · x\<inv> = \<one>" proof - have "x · x\<inv> = \<one> · (x · x\<inv>)" by (simp only: left_one) also have "… = \<one> · x · x\<inv>" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · x\<inv> · x · x\<inv>" by (simp only: left_inv) also have "… = (x\<inv>)\<inv> · (x\<inv> · x) · x\<inv>" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · \<one> · x\<inv>" by (simp only: left_inv) also have "… = (x\<inv>)\<inv> · (\<one> · x\<inv>)" by (simp only: assoc) also have "… = (x\<inv>)\<inv> · x\<inv>" by (simp only: left_one) also have "… = \<one>" by (simp only: left_inv) finally show ?thesis . qed theorem (in group) right_one: "x · \<one> = x" proof - have "x · \<one> = x · (x\<inv> · x)" by (simp only: left_inv) also have "… = x · x\<inv> · x" by (simp only: assoc) also have "… = \<one> · x" by (simp only: right_inv) also have "… = 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 includes group G includes group H shows "x · y · \<one> = prod G (prod G x y) (one G)" and "x ·2 y ·2 \<one>2 = prod H (prod H x y) (one H)" and "x · \<one>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. *} subsection {* Simple meta-theory of structures *} text {* The packaging of the logical specification as a predicate and the syntactic structure as a record type provides a reasonable starting point for simple meta-theoretic studies of mathematical structures. This includes operations on structures (also known as ``functors''), and statements about such constructions. For example, the direct product of semigroups works as follows. *} constdefs semigroup_product :: "'a semigroup => 'b semigroup => ('a × 'b) semigroup" "semigroup_product S T ≡ (|prod = λp q. (prod S (fst p) (fst q), prod T (snd p) (snd q))|))," lemma semigroup_product [intro]: assumes S: "semigroup S" and T: "semigroup T" shows "semigroup (semigroup_product S T)" proof fix p q r :: "'a × 'b" have "prod S (prod S (fst p) (fst q)) (fst r) = prod S (fst p) (prod S (fst q) (fst r))" by (rule semigroup.assoc [OF S]) moreover have "prod T (prod T (snd p) (snd q)) (snd r) = prod T (snd p) (prod T (snd q) (snd r))" by (rule semigroup.assoc [OF T]) ultimately show "prod (semigroup_product S T) (prod (semigroup_product S T) p q) r = prod (semigroup_product S T) p (prod (semigroup_product S T) q r)" by (simp add: semigroup_product_def) qed text {* The above proof is fairly easy, but obscured by the lack of concrete syntax. In fact, we didn't make use of the infrastructure of locales, apart from the raw predicate definition of @{text semigroup}. The alternative version below uses local context expressions to achieve a succinct proof body. The resulting statement is exactly the same as before, even though its specification is a bit more complex. *} lemma includes semigroup S + semigroup T fixes U (structure) defines "U ≡ semigroup_product S T" shows "semigroup U" proof fix p q r :: "'a × 'b" have "(fst p ·1 fst q) ·1 fst r = fst p ·1 (fst q ·1 fst r)" by (rule S.assoc) moreover have "(snd p ·2 snd q) ·2 snd r = snd p ·2 (snd q ·2 snd r)" by (rule T.assoc) ultimately show "(p ·3 q) ·3 r = p ·3 (q ·3 r)" by (simp add: U_def semigroup_product_def semigroup.defs) qed text {* Direct products of group structures may be defined in a similar manner, taking two further operations into account. Subsequently, we use high-level record operations to convert between different signature types explicitly; see also \cite[\S8.3]{isabelle-hol-book}. *} constdefs group_product :: "'a group => 'b group => ('a × 'b) group" "group_product G H ≡ semigroup.extend (semigroup_product (semigroup.truncate G) (semigroup.truncate H)) (group.fields (λp. (inv G (fst p), inv H (snd p))) (one G, one H))" lemma group_product_aux: includes group G + group H fixes I (structure) defines "I ≡ group_product G H" shows "group I" proof show "semigroup I" proof - let ?G' = "semigroup.truncate G" and ?H' = "semigroup.truncate H" have "prod (semigroup_product ?G' ?H') = prod I" by (simp add: I_def group_product_def group.defs semigroup_product_def semigroup.defs) moreover have "semigroup ?G'" and "semigroup ?H'" using prems by (simp_all add: semigroup_def semigroup.defs) then have "semigroup (semigroup_product ?G' ?H')" .. ultimately show ?thesis by (simp add: I_def semigroup_def) qed show "group_axioms I" proof fix p :: "'a × 'b" have "(fst p)\<inv>1 ·1 fst p = \<one>1" by (rule G.left_inv) moreover have "(snd p)\<inv>2 ·2 snd p = \<one>2" by (rule H.left_inv) ultimately show "p\<inv>3 ·3 p = \<one>3" by (simp add: I_def group_product_def group.defs semigroup_product_def semigroup.defs) have "\<one>1 ·1 fst p = fst p" by (rule G.left_one) moreover have "\<one>2 ·2 snd p = snd p" by (rule H.left_one) ultimately show "\<one>3 ·3 p = p" by (simp add: I_def group_product_def group.defs semigroup_product_def semigroup.defs) qed qed theorem group_product: "group G ==> group H ==> group (group_product G H)" by (rule group_product_aux) (assumption | rule group.axioms)+ end
theorem right_inv:
group_context prod inv one ==> prod x (inv x) = one
theorem right_one:
group_context prod inv one ==> prod x one = x
theorem one_equality:
[| group_context prod inv one; prod e x = x |] ==> one = e
theorem inv_equality:
[| group_context prod inv one; prod x' x = one |] ==> inv x = x'
theorem inv_prod:
group_context prod inv one ==> inv (prod x y) = prod (inv y) (inv x)
theorem inv_prod':
abelian_group_context prod inv one ==> inv (prod x y) = prod (inv x) (inv y)
theorem inv_inv:
group_context prod inv one ==> inv (inv x) = x
theorem inv_inject:
[| group_context prod inv one; inv x = inv y |] ==> x = y
theorem right_inv:
group G ==> x ·G group.inv G x = \<one>G
theorem right_one:
group G ==> x ·G \<one>G = x
lemma
[| group G; group H |] ==> x ·G y ·G \<one>G = x ·G y ·G \<one>G
and
[| group G; group H |] ==> x ·H y ·H \<one>H = x ·H y ·H \<one>H
and
[| group G; group H |] ==> x ·G \<one>H = x ·G \<one>H
lemma semigroup_product:
[| semigroup S; semigroup T |] ==> semigroup (semigroup_product S T)
lemma
[| semigroup S; semigroup T |] ==> semigroup (semigroup_product S T)
lemma group_product_aux:
[| group G; group H |] ==> group (group_product G H)
theorem group_product:
[| group G; group H |] ==> group (group_product G H)