Overview

The parameterized-utils 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 type-level 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 sub-terms must at least be the same. We could pattern match on those sub-terms 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 run-time 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 run-time 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 sub-terms:

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 (dependently-typed 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 re-exports 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 sub-terms.

• 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 type-level 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 length-indexed vector. Unlike the other data structures in parameterized-utils, the type parameter only describes the length of the vector as a type-level 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 existentially-quantified 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 run-time representative values for natural numbers lifted to the type level, as well as some utilities for proving properties over type-level naturals.

• Data.Parameterized.Peano

  This module provides an implementation of type-level Peano numbers, as well as run-time representative values for them. It also provides some utilities for proving properties over Peano numbers.

• Data.Parameterized.Fin

  Fin n is a finite type with n (terminating/non-bottom) inhabitants. It can be used to index into a Vector n or other size-indexed datatype.

• Data.Parameterized.SymbolRepr

  This module provides run-time representative values for strings lifted to the type level (symbols).

• Data.Parameterized.BoolRepr

  This module provides run-time 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 run-time 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.