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 * :: (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 * :: (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 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 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 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, 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 %quote {*@{code_stmts example (Haskell)}*}
text {*
\noindent The code in SML has 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