Theory Classes
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 ‹eq :: α ⇒ α ⇒ bool› which is overloaded on
different types for ‹α›, which is achieved by splitting
introduction of the ‹eq› function from its overloaded
definitions by means of ‹class› and ‹instance›
declarations: \footnote{syntax here is a kind of isabellized
Haskell}
\begin{quote}
⇤ ‹class eq where› \\
\hspace*{2ex}‹eq :: α ⇒ α ⇒ bool›
┉⇤ ‹instance nat :: eq where› \\
\hspace*{2ex}‹eq 0 0 = True› \\
\hspace*{2ex}‹eq 0 _ = False› \\
\hspace*{2ex}‹eq _ 0 = False› \\
\hspace*{2ex}‹eq (Suc n) (Suc m) = eq n m›
┉⇤ ‹instance (α::eq, β::eq) pair :: eq where› \\
\hspace*{2ex}‹eq (x1, y1) (x2, y2) = eq x1 x2 ∧ eq y1 y2›
┉⇤ ‹class ord extends eq where› \\
\hspace*{2ex}‹less_eq :: α ⇒ α ⇒ bool› \\
\hspace*{2ex}‹less :: α ⇒ α ⇒ bool›
\end{quote}
⇤ 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" and "Nipkow-Prehofer:1993" and "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 ‹class eq›
above could be given the following specification, demanding that
‹class eq› is an equivalence relation obeying reflexivity,
symmetry and transitivity:
\begin{quote}
⇤ ‹class eq where› \\
\hspace*{2ex}‹eq :: α ⇒ α ⇒ bool› \\
‹satisfying› \\
\hspace*{2ex}‹refl: eq x x› \\
\hspace*{2ex}‹sym: eq x y ⟷ eq x y› \\
\hspace*{2ex}‹trans: eq x y ∧ eq y z ⟶ eq x z›
\end{quote}
⇤ 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}
⇤ 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 ‹α›, class ‹semigroup› introduces a binary operator ‹(⊗)› that is
assumed to be associative:
›
class %quote semigroup =
fixes mult :: "α ⇒ α ⇒ α" (infixl "⊗" 70)
assumes assoc: "(x ⊗ y) ⊗ z = x ⊗ (y ⊗ z)"
text ‹
⇤ 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 :: α::semigroup ⇒ α ⇒ α"} and the
global theorem @{fact "semigroup.assoc:"}~@{prop [source] "⋀x y z ::
α::semigroup. (x ⊗ y) ⊗ z = x ⊗ (y ⊗ 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 ‹(⊗)› 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 ⊗ j = i + (j::int)"
instance %quote proof
fix i j k :: int have "(i + j) + k = i + (j + k)" by simp
then show "(i ⊗ j) ⊗ k = i ⊗ (j ⊗ k)"
unfolding mult_int_def .
qed
end %quote
text ‹
⇤ @{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
standard} 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.
┉ Another instance of \<^class>‹semigroup› yields the natural
numbers:
›
instantiation %quote nat :: semigroup
begin
primrec %quote mult_nat where
"(0::nat) ⊗ n = n"
| "Suc m ⊗ n = Suc (m ⊗ n)"
instance %quote proof
fix m n q :: nat
show "m ⊗ n ⊗ q = m ⊗ (n ⊗ q)"
by (induct m) auto
qed
end %quote
text ‹
⇤ Note the occurrence of the name ‹mult_nat› in the
primrec declaration; by default, the local name of a class operation
‹f› to be instantiated on type constructor ‹κ› is
mangled as ‹f_κ›. In case of uncertainty, these names may be
inspected using the @{command "print_context"} command.
›
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⇩1 ⊗ p⇩2 = (fst p⇩1 ⊗ fst p⇩2, snd p⇩1 ⊗ snd p⇩2)"
instance %quote proof
fix p⇩1 p⇩2 p⇩3 :: "α::semigroup × β::semigroup"
show "p⇩1 ⊗ p⇩2 ⊗ p⇩3 = p⇩1 ⊗ (p⇩2 ⊗ p⇩3)"
unfolding mult_prod_def by (simp add: assoc)
qed
end %quote
text ‹
⇤ Associativity of product semigroups is established using
the definition of ‹(⊗)› 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 ‹monoidl› (a semigroup with a left-hand
neutral) by extending \<^class>‹semigroup› with one additional
parameter ‹neutral› together with its characteristic property:
›
class %quote monoidl = semigroup +
fixes neutral :: "α" ("𝟭")
assumes neutl: "𝟭 ⊗ x = x"
text ‹
⇤ 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: "𝟭 = (0::nat)"
definition %quote
neutral_int_def: "𝟭 = (0::int)"
instance %quote proof
fix n :: nat
show "𝟭 ⊗ n = n"
unfolding neutral_nat_def by simp
next
fix k :: int
show "𝟭 ⊗ 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: "𝟭 = (𝟭, 𝟭)"
instance %quote proof
fix p :: "α::monoidl × β::monoidl"
show "𝟭 ⊗ p = p"
unfolding neutral_prod_def mult_prod_def by (simp add: neutl)
qed
end %quote
text ‹
⇤ 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 ⊗ 𝟭 = x"
instantiation %quote nat and int :: monoid
begin
instance %quote proof
fix n :: nat
show "n ⊗ 𝟭 = n"
unfolding neutral_nat_def by (induct n) simp_all
next
fix k :: int
show "k ⊗ 𝟭 = 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 :: "α::monoid × β::monoid"
show "p ⊗ 𝟭 = p"
unfolding neutral_prod_def mult_prod_def by (simp add: neutr)
qed
end %quote
text ‹
⇤ To finish our small algebra example, we add a ‹group› class with a corresponding instance:
›
class %quote group = monoidl +
fixes inverse :: "α ⇒ α" ("(_÷)" [1000] 999)
assumes invl: "x÷ ⊗ x = 𝟭"
instantiation %quote int :: group
begin
definition %quote
inverse_int_def: "i÷ = - (i::int)"
instance %quote proof
fix i :: int
have "-i + i = 0" by simp
then show "i÷ ⊗ i = 𝟭"
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 :: "α ⇒ α"
assumes idem: "f (f x) = f x"
text ‹
⇤ essentially introduces the locale
› setup %invisible ‹Sign.add_path "foo"›
locale %quote idem =
fixes f :: "α ⇒ α"
assumes idem: "f (f x) = f x"
text ‹⇤ together with corresponding constant(s):›
consts %quote f :: "α ⇒ α"
text ‹
⇤ The connection to the type system is done by means of a
primitive type class ‹idem›, together with a corresponding
interpretation:
›
interpretation %quote idem_class:
idem "f :: (α::idem) ⇒ α"
sorry
text ‹
⇤ This gives you the full power of the Isabelle module system;
conclusions in locale ‹idem› are implicitly propagated
to class ‹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 ‹left_cancel› lemma for groups, which
states that the function ‹(x ⊗)› is injective:
›
lemma %quote (in group) left_cancel: "x ⊗ y = x ⊗ z ⟷ y = z"
proof
assume "x ⊗ y = x ⊗ z"
then have "x÷ ⊗ (x ⊗ y) = x÷ ⊗ (x ⊗ z)" by simp
then have "(x÷ ⊗ x) ⊗ y = (x÷ ⊗ x) ⊗ z" using assoc by simp
then show "y = z" using neutl and invl by simp
next
assume "y = z"
then show "x ⊗ y = x ⊗ z" by simp
qed
text ‹
⇤ 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] "⋀x y z ::
α::group. x ⊗ y = x ⊗ z ⟷ y = z"}. Since type ‹int› has been
made an instance of ‹group› before, we may refer to that
fact as well: @{prop [source] "⋀x y z :: int. x ⊗ y = x ⊗ z ⟷ y =
z"}.
›
subsection ‹Derived definitions›
text ‹
Isabelle locales are targets which support local definitions:
›
primrec %quote (in monoid) pow_nat :: "nat ⇒ α ⇒ α" where
"pow_nat 0 x = 𝟭"
| "pow_nat (Suc n) x = x ⊗ pow_nat n x"
text ‹
⇤ If the locale ‹group› is also a class, this local
definition is propagated onto a global definition of @{term [source]
"pow_nat :: nat ⇒ α::monoid ⇒ α::monoid"} with corresponding theorems
@{thm pow_nat.simps [no_vars]}.
⇤ 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, ‹list›s also form a monoid with ‹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 ‹
⇤ 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 ⇒ α list ⇒ α list" where
"replicate 0 _ = []"
| "replicate (Suc n) xs = xs @ replicate n xs"
interpretation %quote list_monoid: monoid append "[]" rewrites
"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 ‹
⇤ This pattern is also helpful to reuse abstract
specifications on the \emph{same} type. For example, think of a
class ‹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 ‹preorder›, it is still possible to utilise the same abstract
specification again using @{command interpretation}.
›
subsection ‹Additional subclass relations›
text ‹
Any ‹group› is also a ‹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÷ ⊗ x = 𝟭" by simp
with assoc [symmetric] neutl invl have "x÷ ⊗ (x ⊗ 𝟭) = x÷ ⊗ x" by simp
with left_cancel show "x ⊗ 𝟭 = x" by simp
qed
text ‹
The logical proof is carried out on the locale level. Afterwards it
is propagated to the type system, making ‹group› an instance
of ‹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){‹semigroup›}}
\put(20,40){\makebox(0,0){‹monoidl›}}
\put(00,20){\makebox(0,0){‹monoid›}}
\put(40,00){\makebox(0,0){‹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){‹semigroup›}}
\put(20,40){\makebox(0,0){‹monoidl›}}
\put(00,20){\makebox(0,0){‹monoid›}}
\put(40,00){\makebox(0,0){‹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
‹group ⊆ monoid›; transitive edges are left out.}
\label{fig:subclass}
\end{center}
\end{figure}
For illustration, a derived definition in ‹group› using ‹pow_nat›
›
definition %quote (in group) pow_int :: "int ⇒ α ⇒ α" where
"pow_int k x = (if k >= 0
then pow_nat (nat k) x
else (pow_nat (nat (- k)) x)÷)"
text ‹
⇤ yields the global definition of @{term [source] "pow_int ::
int ⇒ α::group ⇒ α::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 ⊗ y"
term %quote "(x::nat) ⊗ y"
end %quote
term %quote "x ⊗ y"
text ‹
⇤ Here in example 1, the term refers to the local class
operation ‹mult [α]›, whereas in example 2 the type
constraint enforces the global class operation ‹mult [nat]›.
In the global context in example 3, the reference is to the
polymorphic global class operation ‹mult [?α :: 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 ‹
⇤ This maps to Haskell as follows:
›
text %quote ‹
@{code_stmts example (Haskell)}
›
text ‹
⇤ The code in SML has explicit dictionary passing:
›
text %quote ‹
@{code_stmts example (SML)}
›
text ‹
⇤ In Scala, implicits are used as dictionaries:
›
text %quote ‹
@{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. An optional first sort argument
constrains the set of classes to all subclasses of this sort,
an optional second sort argument to all superclasses of this sort.
\end{description}
›
end