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README
Overview
The parameterizedutils module contains a collection of typeclasses and datatypes for working with parameterized types, that is types that have a type argument. One example would be a algebraic data type for expressions, that use a type parameter to describe the type of the expression.
This packaged provides collections classes for these parameterized types.
Parameterized Types Motivation
Parameterized types are types with a single type parameter. One use of the type parameter is to embed the type system of an AST into Haskell, in order to have the Haskell compiler provide static guarantees of correctness. The notion of parameterized types in this library is similar to that of the singletons library, but in some ways more flexible but less automated.
A Simple Example
As an example of a parameterized type, consider the following:
{# LANGUAGE DataKinds #}
{# LANGUAGE GADTs #}
data EmbeddedType = EInt  EBool
data Expr (tp :: EmbeddedType) where
IntLit :: Int > Expr 'EInt
BoolLit :: Bool > Expr 'EBool
Add :: Expr 'EInt > Expr 'EInt > Expr 'EInt
Lt :: Expr 'EInt > Expr 'EInt > Expr 'EBool
The Expr
type is a parameterized type, as it has a single type parameter. The
GADT uses the type parameter to embed a simple type system into the language.
The datakind EmbeddedType
is used as a type index. GADT use comes with some
potential challenges, depending on use case. Creating collections of values of
this Expr
type can be slightly tricky due to the type parameter.
Attempting to define the value [IntLit 5, BoolLit False]
results in a type
error because the two terms in the list have different types: Expr 'EInt
and
Expr 'EBool
, respectively.
One option is to existentially quantify away the type parameter. There is a
helper type, Some
, defined in Data.Parameterized.Some that does just this:
[Some (IntLit 5), Some (BoolLit False)] :: [Some Expr]
. Because Expr
is
defined as a GADT, pattern matching on constructors allows us to recover the
type parameter.
Another option is to use a container designed to accommodate parameterized
types, such as List
defined in Data.Parameterized.List. This would look
something like (IntLit 5 :< BoolLit False :< Nil) :: List Expr '[EInt, EBool]
.
Note that the typelevel list reflects the types of the terms, allowing for some
powerful indexing and traversal patterns.
An Extended Example
In the previous example, it is possible to recover the type parameters after they have been existentially quantified away by pattern matching. In a more complicated example, that is not always possible:
{# LANGUAGE DataKinds #}
{# LANGUAGE GADTs #}
data EmbeddedType = EInt  EBool
data Expr (tp :: EmbeddedType) where
IntLit :: Int > Expr 'EInt
BoolLit :: Bool > Expr 'EBool
Add :: Expr 'EInt > Expr 'EInt > Expr 'EInt
Lt :: Expr 'EInt > Expr 'EInt > Expr 'EBool
IsEq :: Expr tp > Expr tp > Expr 'EBool
In this case, pattern matching on the IsEq
constructor does not recover the
types of the operands. IsEq
is polymorphic, so the parameters could either be
of type EBool
or EInt
, though we do learn that the types of the subterms
must at least be the same. We could pattern match on those subterms
individually, but doing so might introduce an unpredictable amount of recursion
and significantly complicate the code. One way to solve this issue is to
introduce runtime type representatives to allow us to more easily recover types.
{# LANGUAGE DataKinds #}
{# LANGUAGE GADTs #}
data EmbeddedType = EInt  EBool
data Repr tp where
IntRepr :: Repr 'EInt
BoolRepr :: Repr 'EBool
data Expr (tp :: EmbeddedType) where
IntLit :: Int > Expr 'EInt
BoolLit :: Bool > Expr 'EBool
Add :: Expr 'EInt > Expr 'EInt > Expr 'EInt
Lt :: Expr 'EInt > Expr 'EInt > Expr 'EBool
IsEq :: Repr tp > Expr tp > Expr tp > Expr 'EBool
The new type, Repr
, is a singleton type that establishes a connection between
a runtime value and a type. When we pattern match on IsEq
, we can simply
inspect (i.e., pattern match on) the contained Repr
value to determine the
types of the subterms:
withBoolExprs :: Expr tp > a > ([Expr 'EBool] > a) > a
withBoolExprs e def k =
case e of
BoolLit {} > k [e]
Lt {} > k [e]
IsEq rep e1 e2
 Just Refl < testEquality rep BoolRepr >
 Because we used a GADT pattern match, we know that tp ~ EBool
k [e, e1, e2]
 otherwise > def
_ > def
Package Structure
This package provides three main types of functionality:
 Typeclasses mirroring core Haskell classes, but adapted to parameterized types
 Data structures suitable for holding values of parameterized types
 Utilities for working with parameterized types, including tools for proving properties at the type level (dependentlytyped programming in Haskell)
Typeclasses
 Data.Parameterized.Classes
This module contains a number of basic classes lifted to parameterized types,
including EqF
, OrdF
, ShowF
, and HashableF
. It also reexports
a few types from base that are useful for working with parameterized types,
including TestEquality
.
The related module Data.Parameterized.TH.GADT provides Template Haskell functions to automatically implement instances of some of these classes.
 Data.Parameterized.ClassesC
This module defines classes like Data.Parameterized.Classes, except that the class methods accept an additional parameter for comparing subterms.
 Data.Parameterized.TraversableFC
This module generalizes Functor
, Foldable
, and Traversable
to
parameterized types. In these operations, type parameters must be preserved.
 Data.Parameterized.TraversableF
This module is like Data.Parameterized.TraversableFC, but intended for types
that have a single parametric type parameter, rather than two. The most
common use of these functions and classes is with the MapF
type described
below.
Data Structures
This package provides data structures that are either lifted to hold parameterized types or otherwise type indexed. The following modules implement data structures:
 Data.Parameterized.Context (
Assignment (f :: k > Type) (ctx :: Ctx k)
)
Assignment
is a sequence type that holds values of parameterized types. It
is essentially a snoc list (i.e., a list that is extended on the right instead
of the left). The Ctx
(Context) type is a typelevel snoc list. In the
default implementation, indexing is O(log(n)) time and total.
There are technically two implementations of Assignment
: a safe
implementation based on a snoc list in pure Haskell and the default
implementation based on a balanced binary tree that uses unsafeCoerce
to
manipulate type indexes for efficiency. The safe implementation is a proof
that the API presented is safe, while the unsafe implementation is efficient
enough to use in practice.
 Data.Parameterized.List (
List (f :: k > Type) [k]
)
List
is the plain Haskell list lifted to hold values of parameterized
types. Moreover, it uses the data kind lifted list syntax instead of the
Ctx
type. Indexing into List
is total but O(n).
 Data.Parameterized.Map (
MapF (key :: k > Type) (value :: k > Type)
)
MapF
an associative map from keys to values where both keys and values are
parameterized types. The lookup operation is O(log(n)), and recovers the type
parameter of the value during lookup.
 Data.Parameterized.HashTable (
HashTable s (key :: k > Type) (value :: k > Type)
)
HashTable
is an associative container like MapF
, except is mutable in
ST
(or IO
via stToIO
) due to the s
type parameter.
 Data.Parameterized.Vector (
Vector (n :: Nat) (a :: Type)
)
This module implements a lengthindexed vector. Unlike the other data structures in parameterizedutils, the type parameter only describes the length of the vector as a typelevel natural; the elements in the vector do not have type indexes.
Additionally:
 Data.Parameterized.Pair (
data Pair a b = forall tp . Pair (a tp) (b tp)
)
This module provides an existentiallyquantified pair where both types in the pair are indexed by the same existentially quantified parameter. Pattern matching on the constructor recovers the equality. This type is primarily used in Data.Parameterized.Map, but is sometimes separately useful.
Note that there is another useful notion of type parameterized pair, which is
provided by Data.Functor.Product in base: data Product a b tp = Pair (a tp)
(b tp)
. The difference is that the type parameter of Product
is made
manifest in the type, and thus is not quantified away.
Utilities
 Data.Parameterized.NatRepr
This module provides runtime representative values for natural numbers lifted to the type level, as well as some utilities for proving properties over typelevel naturals.
 Data.Parameterized.Peano
This module provides an implementation of typelevel Peano numbers, as well as runtime representative values for them. It also provides some utilities for proving properties over Peano numbers.
 Data.Parameterized.SymbolRepr
This module provides runtime representative values for strings lifted to the type level (symbols).
 Data.Parameterized.BoolRepr
This module provides runtime representative values for booleans lifted to the type level.
 Data.Parameterized.Some
The Some
type is a wrapper that existentially quantifies away the type
parameter of a parameterized value. This can be used on any value with a
parameterized type, but is most useful when an operation exists to recover the
type parameter later (either via pattern matching over a GADT or by consulting
a runtime type representative value).
 Data.Parameterized.Nonce
Nonce
is a parameterized type backed by a Word64
. Its TestEquality
instance uses unsafeCoerce
to allow the type parameter to be recovered.
Similarly to a cryptographic nonce, the Nonce
type is safe as long as no
nonce value is reused.