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src/Doc/Classes/Classes.thy

author | wenzelm |

Wed, 27 Mar 2013 16:38:25 +0100 | |

changeset 51553 | 63327f679cff |

parent 51143 | 0a2371e7ced3 |

child 53015 | a1119cf551e8 |

permissions | -rw-r--r-- |

more ambitious Goal.skip_proofs: covers Goal.prove forms as well, and do not insist in quick_and_dirty (for the sake of Isabelle/jEdit);

theory Classes imports Main Setup begin section {* Introduction *} text {* Type classes were introduced 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-sorts93,Nipkow-Prehofer:1993,Wenzel:1997:TPHOL}. From a software engineering 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 theoretical 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: locales \cite{kammueller-locales}. \end{enumerate} \noindent Isar type classes also directly support code generation in a Haskell like fashion. Internally, they are mapped to more primitive Isabelle concepts \cite{Haftmann-Wenzel:2006:classes}. 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. *} 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} defines 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 reduces an instance judgement to the relevant primitive proof goals; typically it is 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} yields 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 be instantiated on type constructor @{text \<kappa>} is 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 given at a 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 prod :: (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 of product semigroups is established using the definition of @{text "(\<otimes>)"} on products and the hypothetical associativity of the type components; these hypotheses are legitimate 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 characteristic 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 prod :: (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 prod :: (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 scenes *} 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 other 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 type class *} (*<*)setup %invisible {* Sign.add_path "foo" *} (*>*) classes %quote idem < type (*<*)axiomatization where idem: "f (f (x::\<alpha>\<Colon>idem)) = 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 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 are targets which support local definitions: *} 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, such as Krauss's recursive function package \cite{krauss2006}. *} subsection {* A functor analogy *} text {* We introduced Isar classes by analogy to type classes in 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 that have a canonical interpretation as type classes. There is also the possibility of other interpretations. For example, @{text list}s also 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 can simply make 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 text {* \noindent This pattern is also helpful to reuse abstract specifications on the \emph{same} type. For example, think of a class @{text preorder}; for type @{typ nat}, there are at least two possible instances: the natural order or the order induced by the divides relation. But only one of these instances can be used for @{command instantiation}; using the locale behind the class @{text preorder}, it is still possible to utilise the same abstract specification again using @{command interpretation}. *} 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 (\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} using @{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 convenience, class context syntax allows references 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. If the target language (e.g.~SML) lacks type classes, then they are implemented by an 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 follows: *} text %quotetypewriter {* @{code_stmts example (Haskell)} *} text {* \noindent The code in SML has explicit dictionary passing: *} text %quotetypewriter {* @{code_stmts example (SML)} *} text {* \noindent In Scala, implicts are used as dictionaries: *} text %quotetypewriter {* @{code_stmts example (Scala)} *} 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