(* $Id$ *)
(*<*)
theory Classes
imports Main Pretty_Int
begin
ML {*
CodeTarget.code_width := 74;
*}
syntax
"_alpha" :: "type" ("\<alpha>")
"_alpha_ofsort" :: "sort \<Rightarrow> type" ("\<alpha>()\<Colon>_" [0] 1000)
"_beta" :: "type" ("\<beta>")
"_beta_ofsort" :: "sort \<Rightarrow> type" ("\<beta>()\<Colon>_" [0] 1000)
"_gamma" :: "type" ("\<gamma>")
"_gamma_ofsort" :: "sort \<Rightarrow> type" ("\<gamma>()\<Colon>_" [0] 1000)
"_alpha_f" :: "type" ("\<alpha>\<^sub>f")
"_alpha_f_ofsort" :: "sort \<Rightarrow> type" ("\<alpha>\<^sub>f()\<Colon>_" [0] 1000)
"_beta_f" :: "type" ("\<beta>\<^sub>f")
"_beta_f_ofsort" :: "sort \<Rightarrow> type" ("\<beta>\<^sub>f()\<Colon>_" [0] 1000)
"_gamma_f" :: "type" ("\<gamma>\<^sub>f")
"_gamma_ofsort_f" :: "sort \<Rightarrow> type" ("\<gamma>\<^sub>f()\<Colon>_" [0] 1000)
parse_ast_translation {*
let
fun alpha_ast_tr [] = Syntax.Variable "'a"
| alpha_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
fun alpha_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'a", ast]
| alpha_ofsort_ast_tr asts = raise Syntax.AST ("alpha_ast_tr", asts);
fun beta_ast_tr [] = Syntax.Variable "'b"
| beta_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
fun beta_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'b", ast]
| beta_ofsort_ast_tr asts = raise Syntax.AST ("beta_ast_tr", asts);
fun gamma_ast_tr [] = Syntax.Variable "'c"
| gamma_ast_tr asts = raise Syntax.AST ("gamma_ast_tr", asts);
fun gamma_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'c", ast]
| gamma_ofsort_ast_tr asts = raise Syntax.AST ("gamma_ast_tr", asts);
fun alpha_f_ast_tr [] = Syntax.Variable "'a_f"
| alpha_f_ast_tr asts = raise Syntax.AST ("alpha_f_ast_tr", asts);
fun alpha_f_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'a_f", ast]
| alpha_f_ofsort_ast_tr asts = raise Syntax.AST ("alpha_f_ast_tr", asts);
fun beta_f_ast_tr [] = Syntax.Variable "'b_f"
| beta_f_ast_tr asts = raise Syntax.AST ("beta_f_ast_tr", asts);
fun beta_f_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'b_f", ast]
| beta_f_ofsort_ast_tr asts = raise Syntax.AST ("beta_f_ast_tr", asts);
fun gamma_f_ast_tr [] = Syntax.Variable "'c_f"
| gamma_f_ast_tr asts = raise Syntax.AST ("gamma_f_ast_tr", asts);
fun gamma_f_ofsort_ast_tr [ast] =
Syntax.Appl [Syntax.Constant "_ofsort", Syntax.Variable "'c_f", ast]
| gamma_f_ofsort_ast_tr asts = raise Syntax.AST ("gamma_f_ast_tr", asts);
in [
("_alpha", alpha_ast_tr), ("_alpha_ofsort", alpha_ofsort_ast_tr),
("_beta", beta_ast_tr), ("_beta_ofsort", beta_ofsort_ast_tr),
("_gamma", gamma_ast_tr), ("_gamma_ofsort", gamma_ofsort_ast_tr),
("_alpha_f", alpha_f_ast_tr), ("_alpha_f_ofsort", alpha_f_ofsort_ast_tr),
("_beta_f", beta_f_ast_tr), ("_beta_f_ofsort", beta_f_ofsort_ast_tr),
("_gamma_f", gamma_f_ast_tr), ("_gamma_f_ofsort", gamma_f_ofsort_ast_tr)
] end
*}
(*>*)
chapter {* Haskell-style classes with Isabelle/Isar *}
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:
\medskip\noindent\hspace*{2ex}@{text "class eq where"}\footnote{syntax here is a kind of isabellized Haskell} \\
\hspace*{4ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
\medskip\noindent\hspace*{2ex}@{text "instance nat \<Colon> eq where"} \\
\hspace*{4ex}@{text "eq 0 0 = True"} \\
\hspace*{4ex}@{text "eq 0 _ = False"} \\
\hspace*{4ex}@{text "eq _ 0 = False"} \\
\hspace*{4ex}@{text "eq (Suc n) (Suc m) = eq n m"}
\medskip\noindent\hspace*{2ex}@{text "instance (\<alpha>\<Colon>eq, \<beta>\<Colon>eq) pair \<Colon> eq where"} \\
\hspace*{4ex}@{text "eq (x1, y1) (x2, y2) = eq x1 x2 \<and> eq y1 y2"}
\medskip\noindent\hspace*{2ex}@{text "class ord extends eq where"} \\
\hspace*{4ex}@{text "less_eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
\hspace*{4ex}@{text "less \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"}
\medskip\noindent Type variables are annotated with (finitly 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 enigineering point of view, type classes
correspond to interfaces in object-oriented languages like Java;
so, it is naturally desirable that type classes do not only
provide functions (class operations) 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:
\medskip\noindent\hspace*{2ex}@{text "class eq where"} \\
\hspace*{4ex}@{text "eq \<Colon> \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool"} \\
\hspace*{2ex}@{text "satisfying"} \\
\hspace*{4ex}@{text "refl: eq x x"} \\
\hspace*{4ex}@{text "sym: eq x y \<longleftrightarrow> eq x y"} \\
\hspace*{4ex}@{text "trans: eq x y \<and> eq y z \<longrightarrow> eq x z"}
\medskip\noindent From a theoretic point of view, type classes are leightweight
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 operations togehter with
corresponding specifications,
\item instantating those abstract operations 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 operation @{text "\<circ>"} that is
assumed to be associative:
*}
class semigroup = type +
fixes mult :: "\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" (infixl "\<^loc>\<otimes>" 70)
assumes assoc: "(x \<^loc>\<otimes> y) \<^loc>\<otimes> z = x \<^loc>\<otimes> (y \<^loc>\<otimes> z)"
text {*
\noindent This @{text "\<CLASS>"} specification consists of two
parts: the \qn{operational} part names the class operation (@{text
"\<FIXES>"}), the \qn{logical} part specifies properties on them
(@{text "\<ASSUMES>"}). The local @{text "\<FIXES>"} and @{text
"\<ASSUMES>"} are lifted to the theory toplevel, yielding the global
operation @{term [source] "mult \<Colon> \<alpha>\<Colon>semigroup \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>"} and the
global theorem @{text "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 @{text "int"} is made a @{text "semigroup"}
instance by providing a suitable definition for the class operation
@{text "mult"} and a proof for the specification of @{text "assoc"}.
*}
instance int :: semigroup
mult_int_def: "i \<otimes> j \<equiv> i + j"
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
text {*
\noindent From now on, the type-checker will consider @{text "int"}
as a @{text "semigroup"} automatically, i.e.\ any general results
are immediately available on concrete instances.
Note that the first proof step is the @{text default} method,
which for instantiation proofs maps to the @{text intro_classes} method.
This boils down an instantiation judgement to the relevant primitive
proof goals and should conveniently always be the first method applied
in an instantiation proof.
\medskip Another instance of @{text "semigroup"} are the natural numbers:
*}
instance nat :: semigroup
mult_nat_def: "m \<otimes> n \<equiv> m + n"
proof
fix m n q :: nat
show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)" unfolding mult_nat_def by simp
qed
text {*
\noindent Also @{text "list"}s form a semigroup with @{const "op @"} as
operation:
*}
instance list :: (type) semigroup
mult_list_def: "xs \<otimes> ys \<equiv> xs @ ys"
proof
fix xs ys zs :: "\<alpha> list"
show "xs \<otimes> ys \<otimes> zs = xs \<otimes> (ys \<otimes> zs)"
proof -
from mult_list_def have "\<And>xs ys\<Colon>\<alpha> list. xs \<otimes> ys \<equiv> xs @ ys" .
thus ?thesis by simp
qed
qed
subsection {* Subclasses *}
text {*
We define a subclass @{text "monoidl"} (a semigroup with a left-hand neutral)
by extending @{text "semigroup"}
with one additional operation @{text "neutral"} together
with its property:
*}
class monoidl = semigroup +
fixes neutral :: "\<alpha>" ("\<^loc>\<one>")
assumes neutl: "\<^loc>\<one> \<^loc>\<otimes> x = x"
text {*
\noindent Again, we make some instances, by
providing suitable operation definitions and proofs for the
additional specifications.
*}
instance nat :: monoidl
neutral_nat_def: "\<one> \<equiv> 0"
proof
fix n :: nat
show "\<one> \<otimes> n = n" unfolding neutral_nat_def mult_nat_def by simp
qed
instance int :: monoidl
neutral_int_def: "\<one> \<equiv> 0"
proof
fix k :: int
show "\<one> \<otimes> k = k" unfolding neutral_int_def mult_int_def by simp
qed
instance list :: (type) monoidl
neutral_list_def: "\<one> \<equiv> []"
proof
fix xs :: "\<alpha> list"
show "\<one> \<otimes> xs = xs"
proof -
from mult_list_def have "\<And>xs ys\<Colon>\<alpha> list. xs \<otimes> ys \<equiv> xs @ ys" .
moreover from mult_list_def neutral_list_def have "\<one> \<equiv> []\<Colon>\<alpha> list" by simp
ultimately show ?thesis by simp
qed
qed
text {*
\noindent Fully-fledged monoids are modelled by another subclass
which does not add new operations but tightens the specification:
*}
class monoid = monoidl +
assumes neutr: "x \<^loc>\<otimes> \<^loc>\<one> = x"
text {*
\noindent Instantiations may also be given simultaneously for different
type constructors:
*}
instance nat :: monoid and int :: monoid and list :: (type) monoid
proof
fix n :: nat
show "n \<otimes> \<one> = n" unfolding neutral_nat_def mult_nat_def by simp
next
fix k :: int
show "k \<otimes> \<one> = k" unfolding neutral_int_def mult_int_def by simp
next
fix xs :: "\<alpha> list"
show "xs \<otimes> \<one> = xs"
proof -
from mult_list_def have "\<And>xs ys\<Colon>\<alpha> list. xs \<otimes> ys \<equiv> xs @ ys" .
moreover from mult_list_def neutral_list_def have "\<one> \<equiv> []\<Colon>\<alpha> list" by simp
ultimately show ?thesis by simp
qed
qed
text {*
\noindent To finish our small algebra example, we add a @{text "group"} class
with a corresponding instance:
*}
class group = monoidl +
fixes inverse :: "\<alpha> \<Rightarrow> \<alpha>" ("(_\<^loc>\<div>)" [1000] 999)
assumes invl: "x\<^loc>\<div> \<^loc>\<otimes> x = \<^loc>\<one>"
instance int :: group
inverse_int_def: "i\<div> \<equiv> - i"
proof
fix i :: int
have "-i + i = 0" by simp
then show "i\<div> \<otimes> i = \<one>"
unfolding mult_int_def and neutral_int_def and inverse_int_def .
qed
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 idem = type +
fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
assumes idem: "f (f x) = f x"
text {*
\noindent essentially introduces the locale
*}
(*<*) setup {* Sign.add_path "foo" *} (*>*)
locale idem =
fixes f :: "\<alpha> \<Rightarrow> \<alpha>"
assumes idem: "f (f x) = f x"
text {* \noindent together with corresponding constant(s): *}
consts f :: "\<alpha> \<Rightarrow> \<alpha>"
text {*
\noindent The connection to the type system is done by means
of a primitive axclass
*}
axclass idem < type
idem: "f (f x) = f x"
text {* \noindent together with a corresponding interpretation: *}
interpretation idem_class:
idem ["f \<Colon> ('a\<Colon>idem) \<Rightarrow> \<alpha>"]
by unfold_locales (rule idem)
(*<*) setup {* Sign.parent_path *} (*>*)
text {*
This give you at hand the full power of the Isabelle module system;
conclusions in locale @{text idem} are implicitly propagated
to class @{text idem}.
*}
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 \<circ>)"} is injective:
*}
lemma (in group) left_cancel: "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z \<longleftrightarrow> y = z"
proof
assume "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z"
then have "x\<^loc>\<div> \<^loc>\<otimes> (x \<^loc>\<otimes> y) = x\<^loc>\<div> \<^loc>\<otimes> (x \<^loc>\<otimes> z)" by simp
then have "(x\<^loc>\<div> \<^loc>\<otimes> x) \<^loc>\<otimes> y = (x\<^loc>\<div> \<^loc>\<otimes> x) \<^loc>\<otimes> z" using assoc by simp
then show "y = z" using neutl and invl by simp
next
assume "y = z"
then show "x \<^loc>\<otimes> y = x \<^loc>\<otimes> z" by simp
qed
text {*
\noindent Here the \qt{@{text "\<IN> 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 @{text
"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:
*}
fun (in monoid)
pow_nat :: "nat \<Rightarrow> \<alpha> \<Rightarrow> \<alpha>" where
"pow_nat 0 x = \<^loc>\<one>"
| "pow_nat (Suc n) x = x \<^loc>\<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 {* 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:
*}
instance advanced group < monoid
proof unfold_locales
fix x
from invl have "x\<^loc>\<div> \<^loc>\<otimes> x = \<^loc>\<one>" by simp
with assoc [symmetric] neutl invl have "x\<^loc>\<div> \<^loc>\<otimes> (x \<^loc>\<otimes> \<^loc>\<one>) = x\<^loc>\<div> \<^loc>\<otimes> x" by simp
with left_cancel show "x \<^loc>\<otimes> \<^loc>\<one> = x" by simp
qed
text {*
The logical proof is carried out on the locale level
and thus conveniently is opened using the @{text unfold_locales}
method which only leaves the logical differences still
open to proof to the user. After the proof it is propagated
to the type system, making @{text group} an instance of
@{text monoid}. For illustration, a derived definition
in @{text group} which uses @{text of_nat}:
*}
definition (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)\<^loc>\<div>)"
text {*
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]}.
*} *)
section {* Further issues *}
subsection {* Code generation *}
text {*
Turning back to the first motivation for type classes,
namely overloading, it is obvious that overloading
stemming from @{text "\<CLASS>"} and @{text "\<INSTANCE>"}
statements 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.
For example, lets go back to the power function:
*}
fun
pow_nat :: "nat \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group" where
"pow_nat 0 x = \<one>"
| "pow_nat (Suc n) x = x \<otimes> pow_nat n x"
definition
pow_int :: "int \<Rightarrow> \<alpha>\<Colon>group \<Rightarrow> \<alpha>\<Colon>group" where
"pow_int k x = (if k >= 0
then pow_nat (nat k) x
else (pow_nat (nat (- k)) x)\<div>)"
definition
example :: int where
"example = pow_int 10 (-2)"
text {*
\noindent This maps to Haskell as:
*}
export_code example in Haskell module_name Classes file "code_examples/"
(* NOTE: you may use Haskell only once in this document, otherwise
you have to work in distinct subdirectories *)
text {*
\lsthaskell{Thy/code_examples/Classes.hs}
\noindent The whole code in SML with explicit dictionary passing:
*}
export_code example (*<*)in SML module_name Classes(*>*)in SML module_name Classes file "code_examples/classes.ML"
text {*
\lstsml{Thy/code_examples/classes.ML}
*}
(* subsection {* Different syntax for same specifications *}
text {*
subsection {* Syntactic classes *}
*} *)
end