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+++ b/doc-src/Classes/Thy/Classes.thy Wed Mar 04 11:05:29 2009 +0100
@@ -0,0 +1,634 @@
+theory Classes
+imports Main Setup
+begin
+
+section {* Introduction *}
+
+text {*
+ Type classes were introduces by Wadler and Blott \cite{wadler89how}
+ into the Haskell language, to allow for a reasonable implementation
+ of overloading\footnote{throughout this tutorial, we are referring
+ to classical Haskell 1.0 type classes, not considering
+ later additions in expressiveness}.
+ As a canonical example, a polymorphic equality function
+ @{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} which is overloaded on different
+ types for @{text "\<alpha>"}, which is achieved by splitting introduction
+ of the @{text eq} function from its overloaded definitions by means
+ of @{text class} and @{text instance} declarations:
+ \footnote{syntax here is a kind of isabellized Haskell}
+
+ \begin{quote}
+
+ \noindent@{text "class eq where"} \\
+ \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
+
+ \medskip\noindent@{text "instance nat \<Colon> eq where"} \\
+ \hspace*{2ex}@{text "eq 0 0 = True"} \\
+ \hspace*{2ex}@{text "eq 0 _ = False"} \\
+ \hspace*{2ex}@{text "eq _ 0 = False"} \\
+ \hspace*{2ex}@{text "eq (Suc n) (Suc m) = eq n m"}
+
+ \medskip\noindent@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
+ \hspace*{2ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
+
+ \medskip\noindent@{text "class ord extends eq where"} \\
+ \hspace*{2ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
+ \hspace*{2ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
+
+ \end{quote}
+
+ \noindent Type variables are annotated with (finitely many) classes;
+ these annotations are assertions that a particular polymorphic type
+ provides definitions for overloaded functions.
+
+ Indeed, type classes not only allow for simple overloading
+ but form a generic calculus, an instance of order-sorted
+ algebra \cite{Nipkow-Prehofer:1993,nipkow-sorts93,Wenzel:1997:TPHOL}.
+
+ From a software engeneering point of view, type classes
+ roughly correspond to interfaces in object-oriented languages like Java;
+ so, it is naturally desirable that type classes do not only
+ provide functions (class parameters) but also state specifications
+ implementations must obey. For example, the @{text "class eq"}
+ above could be given the following specification, demanding that
+ @{text "class eq"} is an equivalence relation obeying reflexivity,
+ symmetry and transitivity:
+
+ \begin{quote}
+
+ \noindent@{text "class eq where"} \\
+ \hspace*{2ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
+ @{text "satisfying"} \\
+ \hspace*{2ex}@{text "refl: eq x x"} \\
+ \hspace*{2ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
+ \hspace*{2ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
+
+ \end{quote}
+
+ \noindent From a theoretic point of view, type classes are lightweight
+ modules; Haskell type classes may be emulated by
+ SML functors \cite{classes_modules}.
+ Isabelle/Isar offers a discipline of type classes which brings
+ all those aspects together:
+
+ \begin{enumerate}
+ \item specifying abstract parameters together with
+ corresponding specifications,
+ \item instantiating those abstract parameters by a particular
+ type
+ \item in connection with a ``less ad-hoc'' approach to overloading,
+ \item with a direct link to the Isabelle module system
+ (aka locales \cite{kammueller-locales}).
+ \end{enumerate}
+
+ \noindent Isar type classes also directly support code generation
+ in a Haskell like fashion.
+
+ This tutorial demonstrates common elements of structured specifications
+ and abstract reasoning with type classes by the algebraic hierarchy of
+ semigroups, monoids and groups. Our background theory is that of
+ Isabelle/HOL \cite{isa-tutorial}, for which some
+ familiarity is assumed.
+
+ Here we merely present the look-and-feel for end users.
+ Internally, those are mapped to more primitive Isabelle concepts.
+ See \cite{Haftmann-Wenzel:2006:classes} for more detail.
+*}
+
+section {* A simple algebra example \label{sec:example} *}
+
+subsection {* Class definition *}
+
+text {*
+ Depending on an arbitrary type @{text "\<alpha>"}, class @{text
+ "semigroup"} introduces a binary operator @{text "(\<otimes>)"} that is
+ assumed to be associative:
+*}
+
+class %quote semigroup =
+ fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" (infixl "\<otimes>" 70)
+ assumes assoc: "(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+
+text {*
+ \noindent This @{command class} specification consists of two
+ parts: the \qn{operational} part names the class parameter
+ (@{element "fixes"}), the \qn{logical} part specifies properties on them
+ (@{element "assumes"}). The local @{element "fixes"} and
+ @{element "assumes"} are lifted to the theory toplevel,
+ yielding the global
+ parameter @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
+ global theorem @{fact "semigroup.assoc:"}~@{prop [source] "\<And>x y
+ z \<Colon> \<alpha>\<Colon>semigroup. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"}.
+*}
+
+
+subsection {* Class instantiation \label{sec:class_inst} *}
+
+text {*
+ The concrete type @{typ int} is made a @{class semigroup}
+ instance by providing a suitable definition for the class parameter
+ @{text "(\<otimes>)"} and a proof for the specification of @{fact assoc}.
+ This is accomplished by the @{command instantiation} target:
+*}
+
+instantiation %quote int :: semigroup
+begin
+
+definition %quote
+ mult_int_def: "i \<otimes> j = i + (j\<Colon>int)"
+
+instance %quote proof
+ fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
+ then show "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"
+ unfolding mult_int_def .
+qed
+
+end %quote
+
+text {*
+ \noindent @{command instantiation} allows to define class parameters
+ at a particular instance using common specification tools (here,
+ @{command definition}). The concluding @{command instance}
+ opens a proof that the given parameters actually conform
+ to the class specification. Note that the first proof step
+ is the @{method default} method,
+ which for such instance proofs maps to the @{method intro_classes} method.
+ This boils down an instance judgement to the relevant primitive
+ proof goals and should conveniently always be the first method applied
+ in an instantiation proof.
+
+ From now on, the type-checker will consider @{typ int}
+ as a @{class semigroup} automatically, i.e.\ any general results
+ are immediately available on concrete instances.
+
+ \medskip Another instance of @{class semigroup} are the natural numbers:
+*}
+
+instantiation %quote nat :: semigroup
+begin
+
+primrec %quote mult_nat where
+ "(0\<Colon>nat) \<otimes> n = n"
+ | "Suc m \<otimes> n = Suc (m \<otimes> n)"
+
+instance %quote proof
+ fix m n q :: nat
+ show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)"
+ by (induct m) auto
+qed
+
+end %quote
+
+text {*
+ \noindent Note the occurence of the name @{text mult_nat}
+ in the primrec declaration; by default, the local name of
+ a class operation @{text f} to instantiate on type constructor
+ @{text \<kappa>} are mangled as @{text f_\<kappa>}. In case of uncertainty,
+ these names may be inspected using the @{command "print_context"} command
+ or the corresponding ProofGeneral button.
+*}
+
+subsection {* Lifting and parametric types *}
+
+text {*
+ Overloaded definitions giving on class instantiation
+ may include recursion over the syntactic structure of types.
+ As a canonical example, we model product semigroups
+ using our simple algebra:
+*}
+
+instantiation %quote * :: (semigroup, semigroup) semigroup
+begin
+
+definition %quote
+ mult_prod_def: "p\<^isub>1 \<otimes> p\<^isub>2 = (fst p\<^isub>1 \<otimes> fst p\<^isub>2, snd p\<^isub>1 \<otimes> snd p\<^isub>2)"
+
+instance %quote proof
+ fix p\<^isub>1 p\<^isub>2 p\<^isub>3 :: "\<alpha>\<Colon>semigroup \<times> \<beta>\<Colon>semigroup"
+ show "p\<^isub>1 \<otimes> p\<^isub>2 \<otimes> p\<^isub>3 = p\<^isub>1 \<otimes> (p\<^isub>2 \<otimes> p\<^isub>3)"
+ unfolding mult_prod_def by (simp add: assoc)
+qed
+
+end %quote
+
+text {*
+ \noindent Associativity from product semigroups is
+ established using
+ the definition of @{text "(\<otimes>)"} on products and the hypothetical
+ associativity of the type components; these hypotheses
+ are facts due to the @{class semigroup} constraints imposed
+ on the type components by the @{command instance} proposition.
+ Indeed, this pattern often occurs with parametric types
+ and type classes.
+*}
+
+
+subsection {* Subclassing *}
+
+text {*
+ We define a subclass @{text monoidl} (a semigroup with a left-hand neutral)
+ by extending @{class semigroup}
+ with one additional parameter @{text neutral} together
+ with its property:
+*}
+
+class %quote monoidl = semigroup +
+ fixes neutral :: "\<alpha>" ("\<one>")
+ assumes neutl: "\<one> \<otimes> x = x"
+
+text {*
+ \noindent Again, we prove some instances, by
+ providing suitable parameter definitions and proofs for the
+ additional specifications. Observe that instantiations
+ for types with the same arity may be simultaneous:
+*}
+
+instantiation %quote nat and int :: monoidl
+begin
+
+definition %quote
+ neutral_nat_def: "\<one> = (0\<Colon>nat)"
+
+definition %quote
+ neutral_int_def: "\<one> = (0\<Colon>int)"
+
+instance %quote proof
+ fix n :: nat
+ show "\<one> \<otimes> n = n"
+ unfolding neutral_nat_def by simp
+next
+ fix k :: int
+ show "\<one> \<otimes> k = k"
+ unfolding neutral_int_def mult_int_def by simp
+qed
+
+end %quote
+
+instantiation %quote * :: (monoidl, monoidl) monoidl
+begin
+
+definition %quote
+ neutral_prod_def: "\<one> = (\<one>, \<one>)"
+
+instance %quote proof
+ fix p :: "\<alpha>\<Colon>monoidl \<times> \<beta>\<Colon>monoidl"
+ show "\<one> \<otimes> p = p"
+ unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
+qed
+
+end %quote
+
+text {*
+ \noindent Fully-fledged monoids are modelled by another subclass
+ which does not add new parameters but tightens the specification:
+*}
+
+class %quote monoid = monoidl +
+ assumes neutr: "x \<otimes> \<one> = x"
+
+instantiation %quote nat and int :: monoid
+begin
+
+instance %quote proof
+ fix n :: nat
+ show "n \<otimes> \<one> = n"
+ unfolding neutral_nat_def by (induct n) simp_all
+next
+ fix k :: int
+ show "k \<otimes> \<one> = k"
+ unfolding neutral_int_def mult_int_def by simp
+qed
+
+end %quote
+
+instantiation %quote * :: (monoid, monoid) monoid
+begin
+
+instance %quote proof
+ fix p :: "\<alpha>\<Colon>monoid \<times> \<beta>\<Colon>monoid"
+ show "p \<otimes> \<one> = p"
+ unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
+qed
+
+end %quote
+
+text {*
+ \noindent To finish our small algebra example, we add a @{text group} class
+ with a corresponding instance:
+*}
+
+class %quote group = monoidl +
+ fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>" ("(_\<div>)" [1000] 999)
+ assumes invl: "x\<div> \<otimes> x = \<one>"
+
+instantiation %quote int :: group
+begin
+
+definition %quote
+ inverse_int_def: "i\<div> = - (i\<Colon>int)"
+
+instance %quote proof
+ fix i :: int
+ have "-i + i = 0" by simp
+ then show "i\<div> \<otimes> i = \<one>"
+ unfolding mult_int_def neutral_int_def inverse_int_def .
+qed
+
+end %quote
+
+
+section {* Type classes as locales *}
+
+subsection {* A look behind the scene *}
+
+text {*
+ The example above gives an impression how Isar type classes work
+ in practice. As stated in the introduction, classes also provide
+ a link to Isar's locale system. Indeed, the logical core of a class
+ is nothing else than a locale:
+*}
+
+class %quote idem =
+ fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
+ assumes idem: "f (f x) = f x"
+
+text {*
+ \noindent essentially introduces the locale
+*} (*<*)setup %invisible {* Sign.add_path "foo" *}
+(*>*)
+locale %quote idem =
+ fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
+ assumes idem: "f (f x) = f x"
+
+text {* \noindent together with corresponding constant(s): *}
+
+consts %quote f :: "\<alpha> \<Rightarrow> \<alpha>"
+
+text {*
+ \noindent The connection to the type system is done by means
+ of a primitive axclass
+*} (*<*)setup %invisible {* Sign.add_path "foo" *}
+(*>*)
+axclass %quote idem < type
+ idem: "f (f x) = f x" (*<*)setup %invisible {* Sign.parent_path *}(*>*)
+
+text {* \noindent together with a corresponding interpretation: *}
+
+interpretation %quote idem_class:
+ idem "f \<Colon> (\<alpha>\<Colon>idem) \<Rightarrow> \<alpha>"
+proof qed (rule idem)
+
+text {*
+ \noindent This gives you at hand the full power of the Isabelle module system;
+ conclusions in locale @{text idem} are implicitly propagated
+ to class @{text idem}.
+*} (*<*)setup %invisible {* Sign.parent_path *}
+(*>*)
+subsection {* Abstract reasoning *}
+
+text {*
+ Isabelle locales enable reasoning at a general level, while results
+ are implicitly transferred to all instances. For example, we can
+ now establish the @{text "left_cancel"} lemma for groups, which
+ states that the function @{text "(x \<otimes>)"} is injective:
+*}
+
+lemma %quote (in group) left_cancel: "x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"
+proof
+ assume "x \<otimes> y = x \<otimes> z"
+ then have "x\<div> \<otimes> (x \<otimes> y) = x\<div> \<otimes> (x \<otimes> z)" by simp
+ then have "(x\<div> \<otimes> x) \<otimes> y = (x\<div> \<otimes> x) \<otimes> z" using assoc by simp
+ then show "y = z" using neutl and invl by simp
+next
+ assume "y = z"
+ then show "x \<otimes> y = x \<otimes> z" by simp
+qed
+
+text {*
+ \noindent Here the \qt{@{keyword "in"} @{class group}} target specification
+ indicates that the result is recorded within that context for later
+ use. This local theorem is also lifted to the global one @{fact
+ "group.left_cancel:"} @{prop [source] "\<And>x y z \<Colon> \<alpha>\<Colon>group. x \<otimes> y = x \<otimes>
+ z \<longleftrightarrow> y = z"}. Since type @{text "int"} has been made an instance of
+ @{text "group"} before, we may refer to that fact as well: @{prop
+ [source] "\<And>x y z \<Colon> int. x \<otimes> y = x \<otimes> z \<longleftrightarrow> y = z"}.
+*}
+
+
+subsection {* Derived definitions *}
+
+text {*
+ Isabelle locales support a concept of local definitions
+ in locales:
+*}
+
+primrec %quote (in monoid) pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
+ "pow_nat 0 x = \<one>"
+ | "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
+
+text {*
+ \noindent If the locale @{text group} is also a class, this local
+ definition is propagated onto a global definition of
+ @{term [source] "pow_nat \<Colon> nat \<Rightarrow> \<alpha>\<Colon>monoid \<Rightarrow> \<alpha>\<Colon>monoid"}
+ with corresponding theorems
+
+ @{thm pow_nat.simps [no_vars]}.
+
+ \noindent As you can see from this example, for local
+ definitions you may use any specification tool
+ which works together with locales (e.g. \cite{krauss2006}).
+*}
+
+
+subsection {* A functor analogy *}
+
+text {*
+ We introduced Isar classes by analogy to type classes
+ functional programming; if we reconsider this in the
+ context of what has been said about type classes and locales,
+ we can drive this analogy further by stating that type
+ classes essentially correspond to functors which have
+ a canonical interpretation as type classes.
+ Anyway, there is also the possibility of other interpretations.
+ For example, also @{text list}s form a monoid with
+ @{text append} and @{term "[]"} as operations, but it
+ seems inappropriate to apply to lists
+ the same operations as for genuinely algebraic types.
+ In such a case, we simply can do a particular interpretation
+ of monoids for lists:
+*}
+
+interpretation %quote list_monoid!: monoid append "[]"
+ proof qed auto
+
+text {*
+ \noindent This enables us to apply facts on monoids
+ to lists, e.g. @{thm list_monoid.neutl [no_vars]}.
+
+ When using this interpretation pattern, it may also
+ be appropriate to map derived definitions accordingly:
+*}
+
+primrec %quote replicate :: "nat \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list" where
+ "replicate 0 _ = []"
+ | "replicate (Suc n) xs = xs @ replicate n xs"
+
+interpretation %quote list_monoid!: monoid append "[]" where
+ "monoid.pow_nat append [] = replicate"
+proof -
+ interpret monoid append "[]" ..
+ show "monoid.pow_nat append [] = replicate"
+ proof
+ fix n
+ show "monoid.pow_nat append [] n = replicate n"
+ by (induct n) auto
+ qed
+qed intro_locales
+
+
+subsection {* Additional subclass relations *}
+
+text {*
+ Any @{text "group"} is also a @{text "monoid"}; this
+ can be made explicit by claiming an additional
+ subclass relation,
+ together with a proof of the logical difference:
+*}
+
+subclass %quote (in group) monoid
+proof
+ fix x
+ from invl have "x\<div> \<otimes> x = \<one>" by simp
+ with assoc [symmetric] neutl invl have "x\<div> \<otimes> (x \<otimes> \<one>) = x\<div> \<otimes> x" by simp
+ with left_cancel show "x \<otimes> \<one> = x" by simp
+qed
+
+text {*
+ The logical proof is carried out on the locale level.
+ Afterwards it is propagated
+ to the type system, making @{text group} an instance of
+ @{text monoid} by adding an additional edge
+ to the graph of subclass relations
+ (cf.\ \figref{fig:subclass}).
+
+ \begin{figure}[htbp]
+ \begin{center}
+ \small
+ \unitlength 0.6mm
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){@{text semigroup}}}
+ \put(20,40){\makebox(0,0){@{text monoidl}}}
+ \put(00,20){\makebox(0,0){@{text monoid}}}
+ \put(40,00){\makebox(0,0){@{text group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(25,35){\vector(1,-3){10}}
+ \end{picture}
+ \hspace{8em}
+ \begin{picture}(40,60)(0,0)
+ \put(20,60){\makebox(0,0){@{text semigroup}}}
+ \put(20,40){\makebox(0,0){@{text monoidl}}}
+ \put(00,20){\makebox(0,0){@{text monoid}}}
+ \put(40,00){\makebox(0,0){@{text group}}}
+ \put(20,55){\vector(0,-1){10}}
+ \put(15,35){\vector(-1,-1){10}}
+ \put(05,15){\vector(3,-1){30}}
+ \end{picture}
+ \caption{Subclass relationship of monoids and groups:
+ before and after establishing the relationship
+ @{text "group \<subseteq> monoid"}; transitive edges are left out.}
+ \label{fig:subclass}
+ \end{center}
+ \end{figure}
+
+ For illustration, a derived definition
+ in @{text group} which uses @{text pow_nat}:
+*}
+
+definition %quote (in group) pow_int :: "int \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
+ "pow_int k x = (if k >= 0
+ then pow_nat (nat k) x
+ else (pow_nat (nat (- k)) x)\<div>)"
+
+text {*
+ \noindent yields the global definition of
+ @{term [source] "pow_int \<Colon> int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group"}
+ with the corresponding theorem @{thm pow_int_def [no_vars]}.
+*}
+
+subsection {* A note on syntax *}
+
+text {*
+ As a commodity, class context syntax allows to refer
+ to local class operations and their global counterparts
+ uniformly; type inference resolves ambiguities. For example:
+*}
+
+context %quote semigroup
+begin
+
+term %quote "x \<otimes> y" -- {* example 1 *}
+term %quote "(x\<Colon>nat) \<otimes> y" -- {* example 2 *}
+
+end %quote
+
+term %quote "x \<otimes> y" -- {* example 3 *}
+
+text {*
+ \noindent Here in example 1, the term refers to the local class operation
+ @{text "mult [\<alpha>]"}, whereas in example 2 the type constraint
+ enforces the global class operation @{text "mult [nat]"}.
+ In the global context in example 3, the reference is
+ to the polymorphic global class operation @{text "mult [?\<alpha> \<Colon> semigroup]"}.
+*}
+
+section {* Further issues *}
+
+subsection {* Type classes and code generation *}
+
+text {*
+ Turning back to the first motivation for type classes,
+ namely overloading, it is obvious that overloading
+ stemming from @{command class} statements and
+ @{command instantiation}
+ targets naturally maps to Haskell type classes.
+ The code generator framework \cite{isabelle-codegen}
+ takes this into account. Concerning target languages
+ lacking type classes (e.g.~SML), type classes
+ are implemented by explicit dictionary construction.
+ As example, let's go back to the power function:
+*}
+
+definition %quote example :: int where
+ "example = pow_int 10 (-2)"
+
+text {*
+ \noindent This maps to Haskell as:
+*}
+
+text %quote {*@{code_stmts example (Haskell)}*}
+
+text {*
+ \noindent The whole code in SML with explicit dictionary passing:
+*}
+
+text %quote {*@{code_stmts example (SML)}*}
+
+subsection {* Inspecting the type class universe *}
+
+text {*
+ To facilitate orientation in complex subclass structures,
+ two diagnostics commands are provided:
+
+ \begin{description}
+
+ \item[@{command "print_classes"}] print a list of all classes
+ together with associated operations etc.
+
+ \item[@{command "class_deps"}] visualizes the subclass relation
+ between all classes as a Hasse diagram.
+
+ \end{description}
+*}
+
+end