purescript-typelevel
Type level natural numbers, booleans and sized vectors.
This package is a direct port of the Haskell type-level package by Acosta, Kiselyov, Jeltsch et al, with the addition of the Vec datatype. The original BSD 3-clause licence is preserved for this derivative work.
Type Level Numbers
This package provides a way to represent type level natural numbers and booleans, with a useful set of operations for both.
A natural number is expressed at the type level using uninhabited types with a D prefixed to the number: D0, D1, D2 etc. You can use the :* type operator to combine digits: eg. D1 :* D3 :* D3 :* D7 makes the number 1337. Aliases are provided for two digit numbers: D10, D11 etc. up to D99. Values of each type are also provided, with the lower case prefix d, so: d0, d1, d2 etc. up to d99.
Two type classes are provided acting as sets: Nat is the set of natural numbers (zero to infinity), and Pos is the set of positive numbers (1 to infinity). Nat defines the function toInt, which converts type level numbers to the value level. To move integer values to the type level, use the reifyInt function (which will throw a runtime exception for values which are not natural numbers).
You can express arithmetic operators as type class constraints on type level numbers. For instance, to express that a type c is the sum of types a and b, you would use the type signature ∀ a b c. (Nat a, Nat b, Add a b c) ⇒ a → b → c. To express that a type a must be less than another type b, you would use the type signature ∀ a b. (Nat a, Nat b, Lt a b) ⇒ a → b.
Most of these constraints are fully relational, so that (even though a Sub constraint exists) you could make c the result of subtracting b from a by saying ∀ a b c. (Add c b a) ⇒ a → b → c.
Note that while most of the arithmetic operations are reasonably performant, division (DivMod and friends) is not: division with a three digit divisor takes more than five minutes to type check on my machine.
Sized Vectors
The sized vector datatype (Nat s) ⇒ Vec s a is implemented as a newtype wrapper around an Array, but has a known size at the type level. Thus, the type checker is able to catch things like index out of bounds errors instead of forcing you to deal with Maybe values. Of course, this comes with its own set of limitations: operations where the size of a resulting vector can't be determined from the input types are impossible, including classics like bind and filter. However, you do at least have Functor, Applicative and Foldable implementations available, in addition to the full set of equivalents to Array operations allowed by Vec's constraints.
You can construct Vecs in only two ways: through consing 1 +> 2 +> 3 +> empty or through factory functions singleton 1 and replicate d5 "this string repeated 5 times". Note the d5 in the last example: Vec operations always take type level numbers where Array operations would take integer values.
LICENCE
Copyright (c) 2016 Bodil Stokke
Copyright (c) 2008 Alfonso Acosta, Oleg Kiselyov, Wolfgang Jeltsch and SAM Group at the School of Information and Communication Technology, (Royal Institute of Technology, Stockholm, Sweden)
All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
- Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
- Neither the name of The ForSyDe Team nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS TEAM BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
